updated minictx v1.5

Updated imported module and premises metadata.
Updated commit and time

PFR-declarations/PFR.ApproxHomPFR.jsonl

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+{"name":"approx_hom_pfr","declaration":"/-- Let $G, G'$ be finite abelian $2$-groups.\nLet $f : G \\to G'$ be a function, and suppose that there are at least\n$|G|^2 / K$ pairs $(x,y) \\in G^2$ such that $$ f(x+y) = f(x) + f(y).$$\nThen there exists a homomorphism $\\phi : G \\to G'$ and a constant $c \\in G'$ such that\n$f(x) = \\phi(x)+c$ for at least $|G| / (2 ^ {172} * K ^ {146})$ values of $x \\in G$. -/\ntheorem approx_hom_pfr {G : Type u_1} {G' : Type u_2} [AddCommGroup G] [Fintype G] [AddCommGroup G'] [Fintype G'] [ElementaryAddCommGroup G 2] [ElementaryAddCommGroup G' 2] (f : G → G') (K : ℝ) (hK : K > 0) (hf : ↑(Nat.card ↑{x | f (x.1 + x.2) = f x.1 + f x.2}) ≥ ↑(Nat.card G) ^ 2 / K) : ∃ φ c, ↑(Nat.card ↑{x | f x = φ x + c}) ≥ ↑(Nat.card G) / (2 ^ 172 * K ^ 146)"}

PFR-declarations/PFR.Endgame.jsonl

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+{"name":"hV","declaration":"theorem hV {G : Type u_1} [AddCommGroup G] [hG : MeasurableSpace G] [MeasurableAdd₂ G] {Ω : Type u_4} [mΩ : MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (X₁ : Ω → G) (X₂ : Ω → G) (X₁' : Ω → G) (X₂' : Ω → G) (h₁ : ProbabilityTheory.IdentDistrib X₁ X₁' MeasureTheory.volume MeasureTheory.volume) (h₂ : ProbabilityTheory.IdentDistrib X₂ X₂' MeasureTheory.volume MeasureTheory.volume) (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![X₁, X₂, X₁', X₂'] MeasureTheory.volume) : H[X₁' + X₂] = H[X₁ + X₂']"}
+{"name":"independenceCondition5","declaration":"theorem independenceCondition5 {G : Type u_1} [AddCommGroup G] [hG : MeasurableSpace G] [MeasurableAdd₂ G] {Ω : Type u_4} [mΩ : MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {X₁ : Ω → G} {X₂ : Ω → G} {X₁' : Ω → G} {X₂' : Ω → G} (hX₁ : Measurable X₁) (hX₂ : Measurable X₂) (hX₁' : Measurable X₁') (hX₂' : Measurable X₂') (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![X₁, X₂, X₁', X₂'] MeasureTheory.volume) : ProbabilityTheory.iIndepFun (fun x => hG) ![X₁, X₁', X₂ + X₂'] MeasureTheory.volume"}
+{"name":"independenceCondition3","declaration":"theorem independenceCondition3 {G : Type u_1} [AddCommGroup G] [hG : MeasurableSpace G] [MeasurableAdd₂ G] {Ω : Type u_4} [mΩ : MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {X₁ : Ω → G} {X₂ : Ω → G} {X₁' : Ω → G} {X₂' : Ω → G} (hX₁ : Measurable X₁) (hX₂ : Measurable X₂) (hX₁' : Measurable X₁') (hX₂' : Measurable X₂') (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![X₁, X₂, X₁', X₂'] MeasureTheory.volume) : ProbabilityTheory.iIndepFun (fun x => hG) ![X₁', X₂, X₁ + X₂'] MeasureTheory.volume"}
+{"name":"independenceCondition4","declaration":"theorem independenceCondition4 {G : Type u_1} [AddCommGroup G] [hG : MeasurableSpace G] [MeasurableAdd₂ G] {Ω : Type u_4} [mΩ : MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {X₁ : Ω → G} {X₂ : Ω → G} {X₁' : Ω → G} {X₂' : Ω → G} (hX₁ : Measurable X₁) (hX₂ : Measurable X₂) (hX₁' : Measurable X₁') (hX₂' : Measurable X₂') (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![X₁, X₂, X₁', X₂'] MeasureTheory.volume) : ProbabilityTheory.iIndepFun (fun x => hG) ![X₂, X₁', X₁ + X₂'] MeasureTheory.volume"}
+{"name":"sum_dist_diff_le","declaration":"/-- $$ \\sum_{i=1}^2 \\sum_{A\\in\\{U,V,W\\}} \\big(d[X^0_i;A|S] - d[X^0_i;X_i]\\big)$$\nis less than or equal to\n$$ \\leq (6 - 3\\eta) k + 3(2 \\eta k - I_1).$$\n-/\ntheorem sum_dist_diff_le {G : Type u_1} [AddCommGroup G] [Fintype G] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [elem : ElementaryAddCommGroup G 2] [MeasurableAdd₂ G] {Ω₀₁ : Type u_2} {Ω₀₂ : Type u_3} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (p : refPackage Ω₀₁ Ω₀₂ G) {Ω : Type u_4} [mΩ : MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (X₁ : Ω → G) (X₂ : Ω → G) (X₁' : Ω → G) (X₂' : Ω → G) (hX₁ : Measurable X₁) (hX₂ : Measurable X₂) (hX₁' : Measurable X₁') (hX₂' : Measurable X₂') (h₁ : ProbabilityTheory.IdentDistrib X₁ X₁' MeasureTheory.volume MeasureTheory.volume) (h₂ : ProbabilityTheory.IdentDistrib X₂ X₂' MeasureTheory.volume MeasureTheory.volume) (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![X₁, X₂, X₁', X₂'] MeasureTheory.volume) (h_min : tau_minimizes p X₁ X₂) : d[p.X₀₁ # X₁ + X₂ | X₁ + X₂ + X₁' + X₂'] - d[p.X₀₁ # X₁] + (d[p.X₀₂ # X₁ + X₂ | X₁ + X₂ + X₁' + X₂'] - d[p.X₀₂ # X₂]) +\n      (d[p.X₀₁ # X₁' + X₂ | X₁ + X₂ + X₁' + X₂'] - d[p.X₀₁ # X₁] +\n        (d[p.X₀₂ # X₁' + X₂ | X₁ + X₂ + X₁' + X₂'] - d[p.X₀₂ # X₂])) +\n    (d[p.X₀₁ # X₁' + X₁ | X₁ + X₂ + X₁' + X₂'] - d[p.X₀₁ # X₁] +\n      (d[p.X₀₂ # X₁' + X₁ | X₁ + X₂ + X₁' + X₂'] - d[p.X₀₂ # X₂])) ≤\n  (6 - 3 * p.η) * d[X₁ # X₂] + 3 * (2 * p.η * d[X₁ # X₂] - I[X₁ + X₂ : X₁' + X₂|X₁ + X₂ + X₁' + X₂'])"}
+{"name":"independenceCondition2","declaration":"theorem independenceCondition2 {G : Type u_1} [AddCommGroup G] [hG : MeasurableSpace G] [MeasurableAdd₂ G] {Ω : Type u_4} [mΩ : MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {X₁ : Ω → G} {X₂ : Ω → G} {X₁' : Ω → G} {X₂' : Ω → G} (hX₁ : Measurable X₁) (hX₂ : Measurable X₂) (hX₁' : Measurable X₁') (hX₂' : Measurable X₂') (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![X₁, X₂, X₁', X₂'] MeasureTheory.volume) : ProbabilityTheory.iIndepFun (fun x => hG) ![X₂, X₁, X₁' + X₂'] MeasureTheory.volume"}
+{"name":"construct_good'","declaration":"theorem construct_good' {G : Type u_1} [AddCommGroup G] [Fintype G] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [elem : ElementaryAddCommGroup G 2] [MeasurableAdd₂ G] {Ω₀₁ : Type u_2} {Ω₀₂ : Type u_3} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (p : refPackage Ω₀₁ Ω₀₂ G) {Ω : Type u_4} [mΩ : MeasureTheory.MeasureSpace Ω] (X₁ : Ω → G) (X₂ : Ω → G) (h_min : tau_minimizes p X₁ X₂) {Ω' : Type u_5} [MeasureTheory.MeasureSpace Ω'] {T₁ : Ω' → G} {T₂ : Ω' → G} {T₃ : Ω' → G} (hT : T₁ + T₂ + T₃ = 0) (hT₁ : Measurable T₁) (hT₂ : Measurable T₂) (hT₃ : Measurable T₃) (μ : MeasureTheory.Measure Ω') [MeasureTheory.IsProbabilityMeasure μ] : d[X₁ # X₂] ≤\n  I[T₁ : T₂ ; μ] + I[T₂ : T₃ ; μ] + I[T₃ : T₁ ; μ] +\n    p.η / 3 *\n      (I[T₁ : T₂ ; μ] + I[T₂ : T₃ ; μ] + I[T₃ : T₁ ; μ] +\n            (d[p.X₀₁ ; MeasureTheory.volume # T₁ ; μ] - d[p.X₀₁ # X₁] +\n              (d[p.X₀₂ ; MeasureTheory.volume # T₁ ; μ] - d[p.X₀₂ # X₂])) +\n          (d[p.X₀₁ ; MeasureTheory.volume # T₂ ; μ] - d[p.X₀₁ # X₁] +\n            (d[p.X₀₂ ; MeasureTheory.volume # T₂ ; μ] - d[p.X₀₂ # X₂])) +\n        (d[p.X₀₁ ; MeasureTheory.volume # T₃ ; μ] - d[p.X₀₁ # X₁] +\n          (d[p.X₀₂ ; MeasureTheory.volume # T₃ ; μ] - d[p.X₀₂ # X₂])))"}
+{"name":"I₃_eq","declaration":"/-- The quantity $I_3 = I[V:W|S]$ is equal to $I_2$. -/\ntheorem I₃_eq {G : Type u_1} [AddCommGroup G] [Fintype G] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [MeasurableAdd₂ G] {Ω : Type u_4} [mΩ : MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (X₁ : Ω → G) (X₂ : Ω → G) (X₁' : Ω → G) (X₂' : Ω → G) (hX₁ : Measurable X₁) (hX₂ : Measurable X₂) (hX₁' : Measurable X₁') (hX₂' : Measurable X₂') (h₁ : ProbabilityTheory.IdentDistrib X₁ X₁' MeasureTheory.volume MeasureTheory.volume) (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![X₁, X₂, X₁', X₂'] MeasureTheory.volume) : I[X₁' + X₂ : X₁' + X₁|X₁ + X₂ + X₁' + X₂'] = I[X₁ + X₂ : X₁' + X₁|X₁ + X₂ + X₁' + X₂']"}
+{"name":"cond_c_eq_integral","declaration":"theorem cond_c_eq_integral {G : Type u_1} [AddCommGroup G] [Fintype G] [hG : MeasurableSpace G] [MeasurableSingletonClass G] {Ω₀₁ : Type u_2} {Ω₀₂ : Type u_3} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] (p : refPackage Ω₀₁ Ω₀₂ G) {Ω : Type u_4} [mΩ : MeasureTheory.MeasureSpace Ω] (X₁ : Ω → G) (X₂ : Ω → G) {Ω' : Type u_5} [MeasureTheory.MeasureSpace Ω'] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {Y : Ω' → G} {Z : Ω' → G} (hY : Measurable Y) (hZ : Measurable Z) : d[p.X₀₁ # Y | Z] - d[p.X₀₁ # X₁] + (d[p.X₀₂ # Y | Z] - d[p.X₀₂ # X₂]) =\n  ∫ (x : G),\n    (fun z =>\n        d[p.X₀₁ ; MeasureTheory.volume # Y ; ProbabilityTheory.cond MeasureTheory.volume (Z ⁻¹' {z})] - d[p.X₀₁ # X₁] +\n          (d[p.X₀₂ ; MeasureTheory.volume # Y ; ProbabilityTheory.cond MeasureTheory.volume (Z ⁻¹' {z})] -\n            d[p.X₀₂ # X₂]))\n      x ∂MeasureTheory.Measure.map Z MeasureTheory.volume"}
+{"name":"independenceCondition1","declaration":"theorem independenceCondition1 {G : Type u_1} [AddCommGroup G] [hG : MeasurableSpace G] [MeasurableAdd₂ G] {Ω : Type u_4} [mΩ : MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {X₁ : Ω → G} {X₂ : Ω → G} {X₁' : Ω → G} {X₂' : Ω → G} (hX₁ : Measurable X₁) (hX₂ : Measurable X₂) (hX₁' : Measurable X₁') (hX₂' : Measurable X₂') (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![X₁, X₂, X₁', X₂'] MeasureTheory.volume) : ProbabilityTheory.iIndepFun (fun x => hG) ![X₁, X₂, X₁' + X₂'] MeasureTheory.volume"}
+{"name":"construct_good","declaration":"/-- If $T_1, T_2, T_3$ are $G$-valued random variables with $T_1+T_2+T_3=0$ holds identically and\n-\n$$ \\delta := \\sum_{1 \\leq i < j \\leq 3} I[T_i;T_j]$$\n\nThen there exist random variables $T'_1, T'_2$ such that\n\n$$ d[T'_1;T'_2] + \\eta (d[X_1^0;T'_1] - d[X_1^0;X _1]) + \\eta(d[X_2^0;T'_2] - d[X_2^0;X_2])$$\n\nis at most\n\n$$\\delta + \\frac{\\eta}{3} \\biggl( \\delta + \\sum_{i=1}^2 \\sum_{j = 1}^3\n    (d[X^0_i;T_j] - d[X^0_i; X_i]) \\biggr).$$\n-/\ntheorem construct_good {G : Type u_1} [AddCommGroup G] [Fintype G] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [elem : ElementaryAddCommGroup G 2] [MeasurableAdd₂ G] {Ω₀₁ : Type u_2} {Ω₀₂ : Type u_3} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (p : refPackage Ω₀₁ Ω₀₂ G) {Ω : Type u_4} [mΩ : MeasureTheory.MeasureSpace Ω] (X₁ : Ω → G) (X₂ : Ω → G) (h_min : tau_minimizes p X₁ X₂) {Ω' : Type u_5} [MeasureTheory.MeasureSpace Ω'] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {T₁ : Ω' → G} {T₂ : Ω' → G} {T₃ : Ω' → G} (hT : T₁ + T₂ + T₃ = 0) (hT₁ : Measurable T₁) (hT₂ : Measurable T₂) (hT₃ : Measurable T₃) : d[X₁ # X₂] ≤\n  I[T₁ : T₂] + I[T₂ : T₃] + I[T₃ : T₁] +\n    p.η / 3 *\n      (I[T₁ : T₂] + I[T₂ : T₃] + I[T₃ : T₁] + (d[p.X₀₁ # T₁] - d[p.X₀₁ # X₁] + (d[p.X₀₂ # T₁] - d[p.X₀₂ # X₂])) +\n          (d[p.X₀₁ # T₂] - d[p.X₀₁ # X₁] + (d[p.X₀₂ # T₂] - d[p.X₀₂ # X₂])) +\n        (d[p.X₀₁ # T₃] - d[p.X₀₁ # X₁] + (d[p.X₀₂ # T₃] - d[p.X₀₂ # X₂])))"}
+{"name":"independenceCondition6","declaration":"theorem independenceCondition6 {G : Type u_1} [AddCommGroup G] [hG : MeasurableSpace G] [MeasurableAdd₂ G] {Ω : Type u_4} [mΩ : MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {X₁ : Ω → G} {X₂ : Ω → G} {X₁' : Ω → G} {X₂' : Ω → G} (hX₁ : Measurable X₁) (hX₂ : Measurable X₂) (hX₁' : Measurable X₁') (hX₂' : Measurable X₂') (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![X₁, X₂, X₁', X₂'] MeasureTheory.volume) : ProbabilityTheory.iIndepFun (fun x => hG) ![X₂, X₂', X₁' + X₁] MeasureTheory.volume"}
+{"name":"sum_condMutual_le","declaration":"/-- $$ I(U : V | S) + I(V : W | S) + I(W : U | S) $$\nis less than or equal to\n$$ 6 \\eta k - \\frac{1 - 5 \\eta}{1-\\eta} (2 \\eta k - I_1).$$\n-/\ntheorem sum_condMutual_le {G : Type u_1} [AddCommGroup G] [Fintype G] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [elem : ElementaryAddCommGroup G 2] [MeasurableAdd₂ G] {Ω₀₁ : Type u_2} {Ω₀₂ : Type u_3} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (p : refPackage Ω₀₁ Ω₀₂ G) {Ω : Type u_4} [mΩ : MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (X₁ : Ω → G) (X₂ : Ω → G) (X₁' : Ω → G) (X₂' : Ω → G) (hX₁ : Measurable X₁) (hX₂ : Measurable X₂) (hX₁' : Measurable X₁') (hX₂' : Measurable X₂') (h₁ : ProbabilityTheory.IdentDistrib X₁ X₁' MeasureTheory.volume MeasureTheory.volume) (h₂ : ProbabilityTheory.IdentDistrib X₂ X₂' MeasureTheory.volume MeasureTheory.volume) (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![X₁, X₂, X₁', X₂'] MeasureTheory.volume) (h_min : tau_minimizes p X₁ X₂) : I[X₁ + X₂ : X₁' + X₂|X₁ + X₂ + X₁' + X₂'] + I[X₁' + X₂ : X₁' + X₁|X₁ + X₂ + X₁' + X₂'] +\n    I[X₁' + X₁ : X₁ + X₂|X₁ + X₂ + X₁' + X₂'] ≤\n  6 * p.η * d[X₁ # X₂] - (1 - 5 * p.η) / (1 - p.η) * (2 * p.η * d[X₁ # X₂] - I[X₁ + X₂ : X₁' + X₂|X₁ + X₂ + X₁' + X₂'])"}
+{"name":"cond_construct_good","declaration":"theorem cond_construct_good {G : Type u_1} [AddCommGroup G] [Fintype G] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [elem : ElementaryAddCommGroup G 2] [MeasurableAdd₂ G] {Ω₀₁ : Type u_2} {Ω₀₂ : Type u_3} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (p : refPackage Ω₀₁ Ω₀₂ G) {Ω : Type u_4} [mΩ : MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (X₁ : Ω → G) (X₂ : Ω → G) (hX₁ : Measurable X₁) (hX₂ : Measurable X₂) (h_min : tau_minimizes p X₁ X₂) {Ω' : Type u_5} [MeasureTheory.MeasureSpace Ω'] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {T₁ : Ω' → G} {T₂ : Ω' → G} {T₃ : Ω' → G} (hT : T₁ + T₂ + T₃ = 0) (hT₁ : Measurable T₁) (hT₂ : Measurable T₂) (hT₃ : Measurable T₃) {R : Ω' → G} (hR : Measurable R) : d[X₁ # X₂] ≤\n  I[T₁ : T₂|R] + I[T₂ : T₃|R] + I[T₃ : T₁|R] +\n    p.η / 3 *\n      (I[T₁ : T₂|R] + I[T₂ : T₃|R] + I[T₃ : T₁|R] +\n            (d[p.X₀₁ # T₁ | R] - d[p.X₀₁ # X₁] + (d[p.X₀₂ # T₁ | R] - d[p.X₀₂ # X₂])) +\n          (d[p.X₀₁ # T₂ | R] - d[p.X₀₁ # X₁] + (d[p.X₀₂ # T₂ | R] - d[p.X₀₂ # X₂])) +\n        (d[p.X₀₁ # T₃ | R] - d[p.X₀₁ # X₁] + (d[p.X₀₂ # T₃ | R] - d[p.X₀₂ # X₂])))"}
+{"name":"tau_strictly_decreases_aux","declaration":"/-- If $d[X_1;X_2] > 0$ then there are $G$-valued random variables $X'_1, X'_2$ such that\nPhrased in the contrapositive form for convenience of proof. -/\ntheorem tau_strictly_decreases_aux {G : Type u_1} [AddCommGroup G] [Fintype G] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [elem : ElementaryAddCommGroup G 2] [MeasurableAdd₂ G] {Ω₀₁ : Type u_2} {Ω₀₂ : Type u_3} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (p : refPackage Ω₀₁ Ω₀₂ G) {Ω : Type u_4} [mΩ : MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (X₁ : Ω → G) (X₂ : Ω → G) (X₁' : Ω → G) (X₂' : Ω → G) (hX₁ : Measurable X₁) (hX₂ : Measurable X₂) (hX₁' : Measurable X₁') (hX₂' : Measurable X₂') (h₁ : ProbabilityTheory.IdentDistrib X₁ X₁' MeasureTheory.volume MeasureTheory.volume) (h₂ : ProbabilityTheory.IdentDistrib X₂ X₂' MeasureTheory.volume MeasureTheory.volume) (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![X₁, X₂, X₁', X₂'] MeasureTheory.volume) (h_min : tau_minimizes p X₁ X₂) (hpη : p.η = 1 / 9) : d[X₁ # X₂] = 0"}
+{"name":"sum_uvw_eq_zero","declaration":"/-- $U+V+W=0$. -/\ntheorem sum_uvw_eq_zero {G : Type u_1} [AddCommGroup G] [elem : ElementaryAddCommGroup G 2] {Ω : Type u_4} (X₁ : Ω → G) (X₂ : Ω → G) (X₁' : Ω → G) : X₁ + X₂ + (X₁' + X₂) + (X₁' + X₁) = 0"}
+{"name":"delta'_eq_integral","declaration":"theorem delta'_eq_integral {G : Type u_1} [Fintype G] [hG : MeasurableSpace G] [MeasurableSingletonClass G] {Ω' : Type u_5} [MeasureTheory.MeasureSpace Ω'] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {T₁ : Ω' → G} {T₂ : Ω' → G} {T₃ : Ω' → G} {R : Ω' → G} : I[T₁ : T₂|R] + I[T₂ : T₃|R] + I[T₃ : T₁|R] =\n  ∫ (x : G),\n    (fun r =>\n        I[T₁ : T₂ ; ProbabilityTheory.cond MeasureTheory.volume (R ⁻¹' {r})] +\n            I[T₂ : T₃ ; ProbabilityTheory.cond MeasureTheory.volume (R ⁻¹' {r})] +\n          I[T₃ : T₁ ; ProbabilityTheory.cond MeasureTheory.volume (R ⁻¹' {r})])\n      x ∂MeasureTheory.Measure.map R MeasureTheory.volume"}
+{"name":"construct_good_prelim","declaration":"/-- If $T_1, T_2, T_3$ are $G$-valued random variables with $T_1+T_2+T_3=0$ holds identically and\n$$ \\delta := \\sum_{1 \\leq i < j \\leq 3} I[T_i;T_j]$$\nThen there exist random variables $T'_1, T'_2$ such that\n$$ d[T'_1;T'_2] + \\eta (d[X_1^0;T'_1] - d[X_1^0;X_1]) + \\eta(d[X_2^0;T'_2] - d[X_2^0;X_2]) $$\nis at most\n$$ \\delta + \\eta ( d[X^0_1;T_1]-d[X^0_1;X_1]) + \\eta (d[X^0_2;T_2]-d[X^0_2;X_2]) $$\n$$ + \\tfrac12 \\eta I[T_1: T_3] + \\tfrac12 \\eta I[T_2: T_3].$$\n-/\ntheorem construct_good_prelim {G : Type u_1} [AddCommGroup G] [Fintype G] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [elem : ElementaryAddCommGroup G 2] [MeasurableAdd₂ G] {Ω₀₁ : Type u_2} {Ω₀₂ : Type u_3} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (p : refPackage Ω₀₁ Ω₀₂ G) {Ω : Type u_4} [mΩ : MeasureTheory.MeasureSpace Ω] (X₁ : Ω → G) (X₂ : Ω → G) (h_min : tau_minimizes p X₁ X₂) {Ω' : Type u_5} [MeasureTheory.MeasureSpace Ω'] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {T₁ : Ω' → G} {T₂ : Ω' → G} {T₃ : Ω' → G} (hT : T₁ + T₂ + T₃ = 0) (hT₁ : Measurable T₁) (hT₂ : Measurable T₂) (hT₃ : Measurable T₃) : d[X₁ # X₂] ≤\n  I[T₁ : T₂] + I[T₂ : T₃] + I[T₃ : T₁] + p.η * (d[p.X₀₁ # T₁] - d[p.X₀₁ # X₁] + (d[p.X₀₂ # T₂] - d[p.X₀₂ # X₂])) +\n    p.η * (I[T₁ : T₃] + I[T₂ : T₃]) / 2"}
+{"name":"hU","declaration":"theorem hU {G : Type u_1} [AddCommGroup G] [hG : MeasurableSpace G] [MeasurableAdd₂ G] {Ω : Type u_4} [mΩ : MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (X₁ : Ω → G) (X₂ : Ω → G) (X₁' : Ω → G) (X₂' : Ω → G) (h₁ : ProbabilityTheory.IdentDistrib X₁ X₁' MeasureTheory.volume MeasureTheory.volume) (h₂ : ProbabilityTheory.IdentDistrib X₂ X₂' MeasureTheory.volume MeasureTheory.volume) (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![X₁, X₂, X₁', X₂'] MeasureTheory.volume) : H[X₁ + X₂] = H[X₁' + X₂']"}

PFR-declarations/PFR.EntropyPFR.jsonl

@@ -0,0 +1,3 @@
+{"name":"entropic_PFR_conjecture","declaration":"/-- `entropic_PFR_conjecture`: For two $G$-valued random variables $X^0_1, X^0_2$, there is some\nsubgroup $H \\leq G$ such that $d[X^0_1;U_H] + d[X^0_2;U_H] \\le 11 d[X^0_1;X^0_2]$. -/\ntheorem entropic_PFR_conjecture {Ω₀₁ : Type u_1} {Ω₀₂ : Type u_2} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {G : Type uG} [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] [MeasurableSpace G] [MeasurableSingletonClass G] (p : refPackage Ω₀₁ Ω₀₂ G) (hpη : p.η = 1 / 9) : ∃ H Ω mΩ U,\n  MeasureTheory.IsProbabilityMeasure MeasureTheory.volume ∧\n    Measurable U ∧\n      ProbabilityTheory.IsUniform (↑H) U MeasureTheory.volume ∧ d[p.X₀₁ # U] + d[p.X₀₂ # U] ≤ 11 * d[p.X₀₁ # p.X₀₂]"}
+{"name":"tau_strictly_decreases","declaration":"/-- If $d[X_1;X_2] > 0$ then there are $G$-valued random variables $X'_1, X'_2$ such that $\\tau[X'_1;X'_2] < \\tau[X_1;X_2]$.\nPhrased in the contrapositive form for convenience of proof. -/\ntheorem tau_strictly_decreases {Ω₀₁ : Type u_1} {Ω₀₂ : Type u_2} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] {Ω : Type u_3} [mΩ : MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {G : Type uG} [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] [MeasurableSpace G] [MeasurableSingletonClass G] (p : refPackage Ω₀₁ Ω₀₂ G) {X₁ : Ω → G} {X₂ : Ω → G} (hX₁ : Measurable X₁) (hX₂ : Measurable X₂) (h_min : tau_minimizes p X₁ X₂) (hpη : p.η = 1 / 9) : d[X₁ # X₂] = 0"}
+{"name":"entropic_PFR_conjecture'","declaration":"theorem entropic_PFR_conjecture' {Ω₀₁ : Type u_1} {Ω₀₂ : Type u_2} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {G : Type uG} [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] [MeasurableSpace G] [MeasurableSingletonClass G] (p : refPackage Ω₀₁ Ω₀₂ G) (hpη : p.η = 1 / 9) : ∃ H Ω mΩ U,\n  ProbabilityTheory.IsUniform (↑H) U MeasureTheory.volume ∧\n    d[p.X₀₁ # U] ≤ 6 * d[p.X₀₁ # p.X₀₂] ∧ d[p.X₀₂ # U] ≤ 6 * d[p.X₀₁ # p.X₀₂]"}

PFR-declarations/PFR.Fibring.jsonl

@@ -0,0 +1,7 @@
+{"name":"sum_of_rdist_eq","declaration":"/-- Let $Y_1,Y_2,Y_3$ and $Y_4$ be independent $G$-valued random variables.\n  Then\n$$d[Y_1-Y_3; Y_2-Y_4] + d[Y_1|Y_1-Y_3; Y_2|Y_2-Y_4] $$\n$$ + I[Y_1-Y_2 : Y_2 - Y_4 | Y_1-Y_2-Y_3+Y_4] = d[Y_1; Y_2] + d[Y_3; Y_4].$$\n-/\ntheorem sum_of_rdist_eq {G : Type u_1} [AddCommGroup G] [Fintype G] [hG : MeasurableSpace G] [MeasurableSingletonClass G] {Ω : Type u_2} [mΩ : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] (Y : Fin 4 → Ω → G) (h_indep : ProbabilityTheory.iIndepFun (fun x => hG) Y μ) (h_meas : ∀ (i : Fin 4), Measurable (Y i)) : d[Y 0 ; μ # Y 1 ; μ] + d[Y 2 ; μ # Y 3 ; μ] =\n  d[Y 0 - Y 2 ; μ # Y 1 - Y 3 ; μ] + d[Y 0 | Y 0 - Y 2 ; μ # Y 1 | Y 1 - Y 3 ; μ] +\n    I[Y 0 - Y 1 : Y 1 - Y 3|Y 0 - Y 1 - Y 2 + Y 3;μ]"}
+{"name":"rdist_of_hom_le","declaration":"/-- \\[d[X;Y]\\geq d[\\pi(X);\\pi(Y)].\\] -/\ntheorem rdist_of_hom_le {H : Type u_1} [AddCommGroup H] [Countable H] [hH : MeasurableSpace H] [MeasurableSingletonClass H] {H' : Type u_2} [AddCommGroup H'] [Countable H'] [hH' : MeasurableSpace H'] [MeasurableSingletonClass H'] (π : H →+ H') {Ω : Type u_3} {Ω' : Type u_4} [mΩ : MeasurableSpace Ω] [mΩ' : MeasurableSpace Ω'] {μ : MeasureTheory.Measure Ω} {μ' : MeasureTheory.Measure Ω'} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure μ'] {Z_1 : Ω → H} {Z_2 : Ω' → H} (h1 : Measurable Z_1) (h2 : Measurable Z_2) [FiniteRange Z_1] [FiniteRange Z_2] : d[⇑π ∘ Z_1 ; μ # ⇑π ∘ Z_2 ; μ'] ≤ d[Z_1 ; μ # Z_2 ; μ']"}
+{"name":"rdist_le_sum_fibre","declaration":"theorem rdist_le_sum_fibre {H : Type u_1} [AddCommGroup H] [Countable H] [hH : MeasurableSpace H] [MeasurableSingletonClass H] {H' : Type u_2} [AddCommGroup H'] [Countable H'] [hH' : MeasurableSpace H'] [MeasurableSingletonClass H'] (π : H →+ H') {Ω : Type u_3} {Ω' : Type u_4} [mΩ : MeasurableSpace Ω] [mΩ' : MeasurableSpace Ω'] {μ : MeasureTheory.Measure Ω} {μ' : MeasureTheory.Measure Ω'} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure μ'] {Z_1 : Ω → H} {Z_2 : Ω' → H} (h1 : Measurable Z_1) (h2 : Measurable Z_2) [FiniteRange Z_1] [FiniteRange Z_2] : d[⇑π ∘ Z_1 ; μ # ⇑π ∘ Z_2 ; μ'] + d[Z_1 | ⇑π ∘ Z_1 ; μ # Z_2 | ⇑π ∘ Z_2 ; μ'] ≤ d[Z_1 ; μ # Z_2 ; μ']"}
+{"name":"sum_of_rdist_eq_step_condMutualInfo","declaration":"/-- The conditional mutual information step of `sum_of_rdist_eq` -/\ntheorem sum_of_rdist_eq_step_condMutualInfo {G : Type u_1} [AddCommGroup G] [Fintype G] [hG : MeasurableSpace G] [MeasurableSingletonClass G] {Ω : Type u_2} [mΩ : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {Y : Fin 4 → Ω → G} (h_meas : ∀ (i : Fin 4), Measurable (Y i)) : I[⟨Y 0 - Y 1, Y 2 - Y 3⟩ : ⟨Y 0 - Y 2, Y 1 - Y 3⟩|Y 0 - Y 1 - (Y 2 - Y 3);μ] =\n  I[Y 0 - Y 1 : Y 1 - Y 3|Y 0 - Y 1 - Y 2 + Y 3;μ]"}
+{"name":"rdist_of_indep_eq_sum_fibre","declaration":"/-- If $Z_1, Z_2$ are independent, then $d[Z_1; Z_2]$ is equal to\n$$ d[\\pi(Z_1);\\pi(Z_2)] + d[Z_1|\\pi(Z_1); Z_2 |\\pi(Z_2)]$$\nplus\n$$I( Z_1 - Z_2 : (\\pi(Z_1), \\pi(Z_2)) | \\pi(Z_1 - Z_2) ).$$\n-/\ntheorem rdist_of_indep_eq_sum_fibre {H : Type u_1} [AddCommGroup H] [Countable H] [hH : MeasurableSpace H] [MeasurableSingletonClass H] {H' : Type u_2} [AddCommGroup H'] [Countable H'] [hH' : MeasurableSpace H'] [MeasurableSingletonClass H'] (π : H →+ H') {Ω : Type u_3} [mΩ : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {Z_1 : Ω → H} {Z_2 : Ω → H} (h : ProbabilityTheory.IndepFun Z_1 Z_2 μ) (h1 : Measurable Z_1) (h2 : Measurable Z_2) [FiniteRange Z_1] [FiniteRange Z_2] : d[Z_1 ; μ # Z_2 ; μ] =\n  d[⇑π ∘ Z_1 ; μ # ⇑π ∘ Z_2 ; μ] + d[Z_1 | ⇑π ∘ Z_1 ; μ # Z_2 | ⇑π ∘ Z_2 ; μ] +\n    I[Z_1 - Z_2 : ⟨⇑π ∘ Z_1, ⇑π ∘ Z_2⟩|⇑π ∘ (Z_1 - Z_2);μ]"}
+{"name":"sum_of_rdist_eq_step_condRuzsaDist","declaration":"/-- The conditional Ruzsa Distance step of `sum_of_rdist_eq` -/\ntheorem sum_of_rdist_eq_step_condRuzsaDist {G : Type u_1} [AddCommGroup G] [Fintype G] [hG : MeasurableSpace G] [MeasurableSingletonClass G] {Ω : Type u_2} [mΩ : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {Y : Fin 4 → Ω → G} (h_indep : ProbabilityTheory.iIndepFun (fun x => hG) Y μ) (h_meas : ∀ (i : Fin 4), Measurable (Y i)) : d[⟨Y 0, Y 2⟩ | Y 0 - Y 2 ; μ # ⟨Y 1, Y 3⟩ | Y 1 - Y 3 ; μ] = d[Y 0 | Y 0 - Y 2 ; μ # Y 1 | Y 1 - Y 3 ; μ]"}
+{"name":"sum_of_rdist_eq_char_2","declaration":"/-- Let $Y_1,Y_2,Y_3$ and $Y_4$ be independent $G$-valued random variables.\n  Then\n$$d[Y_1+Y_3; Y_2+Y_4] + d[Y_1|Y_1+Y_3; Y_2|Y_2+Y_4] $$\n$$ + I[Y_1+Y_2 : Y_2 + Y_4 | Y_1+Y_2+Y_3+Y_4] = d[Y_1; Y_2] + d[Y_3; Y_4].$$\n-/\ntheorem sum_of_rdist_eq_char_2 {G : Type u_1} [AddCommGroup G] [Fintype G] [hG : MeasurableSpace G] [MeasurableSingletonClass G] {Ω : Type u_2} [mΩ : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] [ElementaryAddCommGroup G 2] (Y : Fin 4 → Ω → G) (h_indep : ProbabilityTheory.iIndepFun (fun x => hG) Y μ) (h_meas : ∀ (i : Fin 4), Measurable (Y i)) : d[Y 0 ; μ # Y 1 ; μ] + d[Y 2 ; μ # Y 3 ; μ] =\n  d[Y 0 + Y 2 ; μ # Y 1 + Y 3 ; μ] + d[Y 0 | Y 0 + Y 2 ; μ # Y 1 | Y 1 + Y 3 ; μ] +\n    I[Y 0 + Y 1 : Y 1 + Y 3|Y 0 + Y 1 + Y 2 + Y 3;μ]"}

PFR-declarations/PFR.FirstEstimate.jsonl

@@ -0,0 +1,9 @@
+{"name":"first_estimate","declaration":"/-- We have $I_1 \\leq 2 \\eta k$ -/\ntheorem first_estimate {G : Type u_1} [addgroup : AddCommGroup G] [Fintype G] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [elem : ElementaryAddCommGroup G 2] [MeasurableAdd₂ G] {Ω₀₁ : Type u_2} {Ω₀₂ : Type u_3} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (p : refPackage Ω₀₁ Ω₀₂ G) {Ω : Type u_4} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (X₁ : Ω → G) (X₂ : Ω → G) (X₁' : Ω → G) (X₂' : Ω → G) (hX₁ : Measurable X₁) (hX₂ : Measurable X₂) (hX₁' : Measurable X₁') (hX₂' : Measurable X₂') (h₁ : ProbabilityTheory.IdentDistrib X₁ X₁' MeasureTheory.volume MeasureTheory.volume) (h₂ : ProbabilityTheory.IdentDistrib X₂ X₂' MeasureTheory.volume MeasureTheory.volume) (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![X₁, X₂, X₂', X₁'] MeasureTheory.volume) (h_min : tau_minimizes p X₁ X₂) : I[X₁ + X₂ : X₁' + X₂|X₁ + X₂ + X₁' + X₂'] ≤ 2 * p.η * d[X₁ # X₂]"}
+{"name":"diff_rdist_le_1","declaration":"/-- $$d[X^0_1; X_1+\\tilde X_2] - d[X^0_1; X_1] \\leq \\tfrac{1}{2} k + \\tfrac{1}{4} \\bbH[X_2] - \\tfrac{1}{4} \\bbH[X_1].$$ -/\ntheorem diff_rdist_le_1 {G : Type u_1} [addgroup : AddCommGroup G] [Fintype G] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [elem : ElementaryAddCommGroup G 2] [MeasurableAdd₂ G] {Ω₀₁ : Type u_2} {Ω₀₂ : Type u_3} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (p : refPackage Ω₀₁ Ω₀₂ G) {Ω : Type u_4} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (X₁ : Ω → G) (X₂ : Ω → G) (X₁' : Ω → G) (X₂' : Ω → G) (hX₁ : Measurable X₁) (hX₂' : Measurable X₂') (h₂ : ProbabilityTheory.IdentDistrib X₂ X₂' MeasureTheory.volume MeasureTheory.volume) (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![X₁, X₂, X₂', X₁'] MeasureTheory.volume) : d[p.X₀₁ # X₁ + X₂'] - d[p.X₀₁ # X₁] ≤ d[X₁ # X₂] / 2 + H[X₂] / 4 - H[X₁] / 4"}
+{"name":"condRuzsaDist_of_sums_ge","declaration":"/-- The distance $d[X_1|X_1+\\tilde X_2; X_2|X_2+\\tilde X_1]$ is at least\n$$ k - \\eta (d[X^0_1; X_1 | X_1 + \\tilde X_2] - d[X^0_1; X_1]) - \\eta(d[X^0_2; X_2 | X_2 + \\tilde X_1] - d[X^0_2; X_2]).$$\n-/\ntheorem condRuzsaDist_of_sums_ge {G : Type u_1} [addgroup : AddCommGroup G] [Fintype G] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [MeasurableAdd₂ G] {Ω₀₁ : Type u_2} {Ω₀₂ : Type u_3} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] (p : refPackage Ω₀₁ Ω₀₂ G) {Ω : Type u_4} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (X₁ : Ω → G) (X₂ : Ω → G) (X₁' : Ω → G) (X₂' : Ω → G) (hX₁ : Measurable X₁) (hX₂ : Measurable X₂) (hX₁' : Measurable X₁') (hX₂' : Measurable X₂') (h_min : tau_minimizes p X₁ X₂) : d[X₁ | X₁ + X₂' # X₂ | X₂ + X₁'] ≥\n  d[X₁ # X₂] - p.η * (d[p.X₀₁ # X₁ | X₁ + X₂'] - d[p.X₀₁ # X₁]) - p.η * (d[p.X₀₂ # X₂ | X₂ + X₁'] - d[p.X₀₂ # X₂])"}
+{"name":"diff_rdist_le_2","declaration":"/-- $$ d[X^0_2;X_2+\\tilde X_1] - d[X^0_2; X_2] \\leq \\tfrac{1}{2} k + \\tfrac{1}{4} \\mathbb{H}[X_1] - \\tfrac{1}{4} \\mathbb{H}[X_2].$$ -/\ntheorem diff_rdist_le_2 {G : Type u_1} [addgroup : AddCommGroup G] [Fintype G] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [elem : ElementaryAddCommGroup G 2] [MeasurableAdd₂ G] {Ω₀₁ : Type u_2} {Ω₀₂ : Type u_3} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (p : refPackage Ω₀₁ Ω₀₂ G) {Ω : Type u_4} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (X₁ : Ω → G) (X₂ : Ω → G) (X₁' : Ω → G) (X₂' : Ω → G) (hX₂ : Measurable X₂) (hX₁' : Measurable X₁') (h₁ : ProbabilityTheory.IdentDistrib X₁ X₁' MeasureTheory.volume MeasureTheory.volume) (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![X₁, X₂, X₂', X₁'] MeasureTheory.volume) : d[p.X₀₂ # X₂ + X₁'] - d[p.X₀₂ # X₂] ≤ d[X₁ # X₂] / 2 + H[X₁] / 4 - H[X₂] / 4"}
+{"name":"diff_rdist_le_3","declaration":"/-- $$ d[X_1^0;X_1|X_1+\\tilde X_2] - d[X_1^0;X_1] \\leq\n\\tfrac{1}{2} k + \\tfrac{1}{4} \\mathbb{H}[X_1] - \\tfrac{1}{4} \\mathbb{H}[X_2].$$ -/\ntheorem diff_rdist_le_3 {G : Type u_1} [addgroup : AddCommGroup G] [Fintype G] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [elem : ElementaryAddCommGroup G 2] [MeasurableAdd₂ G] {Ω₀₁ : Type u_2} {Ω₀₂ : Type u_3} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (p : refPackage Ω₀₁ Ω₀₂ G) {Ω : Type u_4} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (X₁ : Ω → G) (X₂ : Ω → G) (X₁' : Ω → G) (X₂' : Ω → G) (hX₁ : Measurable X₁) (hX₂' : Measurable X₂') (h₂ : ProbabilityTheory.IdentDistrib X₂ X₂' MeasureTheory.volume MeasureTheory.volume) (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![X₁, X₂, X₂', X₁'] MeasureTheory.volume) : d[p.X₀₁ # X₁ | X₁ + X₂'] - d[p.X₀₁ # X₁] ≤ d[X₁ # X₂] / 2 + H[X₁] / 4 - H[X₂] / 4"}
+{"name":"rdist_of_sums_ge","declaration":"/-- The distance $d[X_1+\\tilde X_2; X_2+\\tilde X_1]$ is at least\n$$ k - \\eta (d[X^0_1; X_1+\\tilde X_2] - d[X^0_1; X_1]) - \\eta (d[X^0_2; X_2+\\tilde X_1] - d[X^0_2; X_2]).$$ -/\ntheorem rdist_of_sums_ge {G : Type u_1} [addgroup : AddCommGroup G] [hG : MeasurableSpace G] [MeasurableAdd₂ G] {Ω₀₁ : Type u_2} {Ω₀₂ : Type u_3} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] (p : refPackage Ω₀₁ Ω₀₂ G) {Ω : Type u_4} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (X₁ : Ω → G) (X₂ : Ω → G) (X₁' : Ω → G) (X₂' : Ω → G) (hX₁ : Measurable X₁) (hX₂ : Measurable X₂) (hX₁' : Measurable X₁') (hX₂' : Measurable X₂') (h_min : tau_minimizes p X₁ X₂) : d[X₁ + X₂' # X₂ + X₁'] ≥\n  d[X₁ # X₂] - p.η * (d[p.X₀₁ # X₁ + X₂'] - d[p.X₀₁ # X₁]) - p.η * (d[p.X₀₂ # X₂ + X₁'] - d[p.X₀₂ # X₂])"}
+{"name":"rdist_add_rdist_add_condMutual_eq","declaration":"/-- The sum of\n$$ d[X_1+\\tilde X_2;X_2+\\tilde X_1] + d[X_1|X_1+\\tilde X_2; X_2|X_2+\\tilde X_1] $$\nand\n$$ I[X_1+ X_2 : \\tilde X_1 + X_2 \\,|\\, X_1 + X_2 + \\tilde X_1 + \\tilde X_2] $$\nis equal to $2k$. -/\ntheorem rdist_add_rdist_add_condMutual_eq {G : Type u_1} [addgroup : AddCommGroup G] [Fintype G] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [elem : ElementaryAddCommGroup G 2] {Ω : Type u_4} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (X₁ : Ω → G) (X₂ : Ω → G) (X₁' : Ω → G) (X₂' : Ω → G) (hX₁ : Measurable X₁) (hX₂ : Measurable X₂) (hX₁' : Measurable X₁') (hX₂' : Measurable X₂') (h₁ : ProbabilityTheory.IdentDistrib X₁ X₁' MeasureTheory.volume MeasureTheory.volume) (h₂ : ProbabilityTheory.IdentDistrib X₂ X₂' MeasureTheory.volume MeasureTheory.volume) (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![X₁, X₂, X₂', X₁'] MeasureTheory.volume) : d[X₁ + X₂' # X₂ + X₁'] + d[X₁ | X₁ + X₂' # X₂ | X₂ + X₁'] + I[X₁ + X₂ : X₁' + X₂|X₁ + X₂ + X₁' + X₂'] = 2 * d[X₁ # X₂]"}
+{"name":"ent_ofsum_le","declaration":"/-- $$\\mathbb{H}[X_1+X_2+\\tilde X_1+\\tilde X_2] \\le \\tfrac{1}{2} \\mathbb{H}[X_1]+\\tfrac{1}{2} \\mathbb{H}[X_2] + (2 + \\eta) k - I_1.$$\n-/\ntheorem ent_ofsum_le {G : Type u_1} [addgroup : AddCommGroup G] [Fintype G] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [elem : ElementaryAddCommGroup G 2] [MeasurableAdd₂ G] {Ω₀₁ : Type u_2} {Ω₀₂ : Type u_3} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (p : refPackage Ω₀₁ Ω₀₂ G) {Ω : Type u_4} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (X₁ : Ω → G) (X₂ : Ω → G) (X₁' : Ω → G) (X₂' : Ω → G) (hX₁ : Measurable X₁) (hX₂ : Measurable X₂) (hX₁' : Measurable X₁') (hX₂' : Measurable X₂') (h₁ : ProbabilityTheory.IdentDistrib X₁ X₁' MeasureTheory.volume MeasureTheory.volume) (h₂ : ProbabilityTheory.IdentDistrib X₂ X₂' MeasureTheory.volume MeasureTheory.volume) (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![X₁, X₂, X₂', X₁'] MeasureTheory.volume) (h_min : tau_minimizes p X₁ X₂) : H[X₁ + X₂ + X₁' + X₂'] ≤ H[X₁] / 2 + H[X₂] / 2 + (2 + p.η) * d[X₁ # X₂] - I[X₁ + X₂ : X₁' + X₂|X₁ + X₂ + X₁' + X₂']"}
+{"name":"diff_rdist_le_4","declaration":"/-- $$ d[X_2^0; X_2|X_2+\\tilde X_1] - d[X_2^0; X_2] \\leq\n\\tfrac{1}{2}k + \\tfrac{1}{4} \\mathbb{H}[X_2] - \\tfrac{1}{4} \\mathbb{H}[X_1].$$ -/\ntheorem diff_rdist_le_4 {G : Type u_1} [addgroup : AddCommGroup G] [Fintype G] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [elem : ElementaryAddCommGroup G 2] [MeasurableAdd₂ G] {Ω₀₁ : Type u_2} {Ω₀₂ : Type u_3} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (p : refPackage Ω₀₁ Ω₀₂ G) {Ω : Type u_4} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (X₁ : Ω → G) (X₂ : Ω → G) (X₁' : Ω → G) (X₂' : Ω → G) (hX₂ : Measurable X₂) (hX₁' : Measurable X₁') (h₁ : ProbabilityTheory.IdentDistrib X₁ X₁' MeasureTheory.volume MeasureTheory.volume) (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![X₁, X₂, X₂', X₁'] MeasureTheory.volume) : d[p.X₀₂ # X₂ | X₂ + X₁'] - d[p.X₀₂ # X₂] ≤ d[X₁ # X₂] / 2 + H[X₂] / 4 - H[X₁] / 4"}

PFR-declarations/PFR.ForMathlib.CompactProb.jsonl

@@ -0,0 +1,9 @@
+{"name":"continuous_pmf_apply","declaration":"theorem continuous_pmf_apply {X : Type u_1} [MeasurableSpace X] [TopologicalSpace X] [DiscreteTopology X] [BorelSpace X] (i : X) : Continuous fun μ => (fun s => (↑↑↑μ s).toNNReal) {i}"}
+{"name":"probabilityMeasureEquivStdSimplex_symm_coe_apply","declaration":"theorem probabilityMeasureEquivStdSimplex_symm_coe_apply {X : Type u_1} [MeasurableSpace X] [Fintype X] [MeasurableSingletonClass X] (p : ↑(stdSimplex ℝ X)) : ↑(probabilityMeasureEquivStdSimplex.symm p) =\n  Finset.sum Finset.univ fun i => ENNReal.ofReal (↑p i) • MeasureTheory.Measure.dirac i"}
+{"name":"tendsto_lintegral_of_forall_of_finite","declaration":"theorem tendsto_lintegral_of_forall_of_finite {X : Type u_1} [MeasurableSpace X] [TopologicalSpace X] [DiscreteTopology X] [BorelSpace X] [Finite X] {ι : Type u_2} {L : Filter ι} (μs : ι → MeasureTheory.Measure X) (μ : MeasureTheory.Measure X) (f : BoundedContinuousFunction X NNReal) (h : ∀ (x : X), Filter.Tendsto (fun i => ↑↑(μs i) {x}) L (nhds (↑↑μ {x}))) : Filter.Tendsto (fun i => ∫⁻ (x : X), ↑(f x) ∂μs i) L (nhds (∫⁻ (x : X), ↑(f x) ∂μ))"}
+{"name":"continuous_pmf_apply'","declaration":"theorem continuous_pmf_apply' {X : Type u_1} [MeasurableSpace X] [TopologicalSpace X] [DiscreteTopology X] [BorelSpace X] (i : X) : Continuous fun μ => (↑μ).real {i}"}
+{"name":"probabilityMeasureEquivStdSimplex_coe_apply","declaration":"theorem probabilityMeasureEquivStdSimplex_coe_apply {X : Type u_1} [MeasurableSpace X] [Fintype X] [MeasurableSingletonClass X] (μ : MeasureTheory.ProbabilityMeasure X) (i : X) : ↑(probabilityMeasureEquivStdSimplex μ) i = ↑((fun s => (↑↑↑μ s).toNNReal) {i})"}
+{"name":"probabilityMeasureEquivStdSimplex","declaration":"/-- The canonical bijection between the set of probability measures on a fintype and the set of\nnonnegative functions on the points adding up to one. -/\ndef probabilityMeasureEquivStdSimplex {X : Type u_1} [MeasurableSpace X] [Fintype X] [MeasurableSingletonClass X] : MeasureTheory.ProbabilityMeasure X ≃ ↑(stdSimplex ℝ X)"}
+{"name":"probabilityMeasureHomeoStdSimplex","declaration":"/-- The canonical homeomorphism between the space of probability measures on a finite space and the\nstandard simplex. -/\ndef probabilityMeasureHomeoStdSimplex {X : Type u_1} [MeasurableSpace X] [Fintype X] [TopologicalSpace X] [DiscreteTopology X] [BorelSpace X] : MeasureTheory.ProbabilityMeasure X ≃ₜ ↑(stdSimplex ℝ X)"}
+{"name":"instCompactSpaceProbabilityMeasureInstTopologicalSpaceProbabilityMeasure","declaration":"/-- This is still true when `X` is a metrizable compact space, but the proof requires Riesz\nrepresentation theorem.\nTODO: remove once the general version is proved. -/\ninstance instCompactSpaceProbabilityMeasureInstTopologicalSpaceProbabilityMeasure {X : Type u_1} [MeasurableSpace X] [TopologicalSpace X] [OpensMeasurableSpace X] [Finite X] [DiscreteTopology X] [BorelSpace X] : CompactSpace (MeasureTheory.ProbabilityMeasure X)"}
+{"name":"instSecondCountableTopologyProbabilityMeasureInstTopologicalSpaceProbabilityMeasure","declaration":"instance instSecondCountableTopologyProbabilityMeasureInstTopologicalSpaceProbabilityMeasure {X : Type u_1} [MeasurableSpace X] [TopologicalSpace X] [OpensMeasurableSpace X] [Finite X] [DiscreteTopology X] [BorelSpace X] : SecondCountableTopology (MeasureTheory.ProbabilityMeasure X)"}

PFR-declarations/PFR.ForMathlib.Elementary.jsonl

@@ -0,0 +1,25 @@
+{"name":"ElementaryAddCommGroup.mk","declaration":"ctor ElementaryAddCommGroup.mk {G : Type u_1} [AddCommGroup G] {p : outParam ℕ} (orderOf_of_ne : ∀ {x : G}, x ≠ 0 → addOrderOf x = p) : ElementaryAddCommGroup G p"}
+{"name":"ElementaryAddCommGroup.finite_closure","declaration":"theorem ElementaryAddCommGroup.finite_closure {G : Type u_1} [AddCommGroup G] {n : ℕ} [ElementaryAddCommGroup G (n + 1)] {A : Set G} (h : Set.Finite A) : Set.Finite ↑(AddSubgroup.closure A)"}
+{"name":"ElementaryAddCommGroup.char_smul_eq_zero'","declaration":"theorem ElementaryAddCommGroup.char_smul_eq_zero' {p : ℕ} {Γ : Type u_1} [AddCommGroup Γ] [ElementaryAddCommGroup Γ p] (x : Γ) (k : ℤ) : (k * ↑p) • x = 0"}
+{"name":"ElementaryAddCommGroup.two_le_char_of_ne_zero","declaration":"theorem ElementaryAddCommGroup.two_le_char_of_ne_zero {p : ℕ} {Γ : Type u_1} [NeZero p] [AddCommGroup Γ] [ElementaryAddCommGroup Γ p] {x : Γ} (x_ne_zero : x ≠ 0) : 2 ≤ p"}
+{"name":"ElementaryAddCommGroup.quotient_group","declaration":"theorem ElementaryAddCommGroup.quotient_group {G : Type u_1} [AddCommGroup G] {p : ℕ} (hp : Nat.Prime p) {H : AddSubgroup G} (hH : ∀ (x : G), p • x ∈ H) : ElementaryAddCommGroup (G ⧸ H) p"}
+{"name":"ElementaryAddCommGroup.of_torsion","declaration":"theorem ElementaryAddCommGroup.of_torsion {G : Type u_1} [AddCommGroup G] {p : ℕ} (hp : Nat.Prime p) (h : ∀ (x : G), p • x = 0) : ElementaryAddCommGroup G p"}
+{"name":"ElementaryAddCommGroup.char_ne_one_of_ne_zero","declaration":"theorem ElementaryAddCommGroup.char_ne_one_of_ne_zero {p : ℕ} {Γ : Type u_1} [AddCommGroup Γ] [ElementaryAddCommGroup Γ p] {x : Γ} (x_ne_zero : x ≠ 0) : p ≠ 1"}
+{"name":"ElementaryAddCommGroup.sum_add_sum_eq_sum","declaration":"theorem ElementaryAddCommGroup.sum_add_sum_eq_sum {G : Type u_1} [AddCommGroup G] [elem : ElementaryAddCommGroup G 2] (x : G) (y : G) (z : G) : x + y + (y + z) = x + z"}
+{"name":"ElementaryAddCommGroup.torsion","declaration":"theorem ElementaryAddCommGroup.torsion {G : Type u_1} [AddCommGroup G] (p : ℕ) [elem : ElementaryAddCommGroup G p] (x : G) : p • x = 0"}
+{"name":"ElementaryAddCommGroup.sub_eq_add","declaration":"theorem ElementaryAddCommGroup.sub_eq_add {G : Type u_1} [AddCommGroup G] [elem : ElementaryAddCommGroup G 2] (x : G) (y : G) : x - y = x + y"}
+{"name":"ElementaryAddCommGroup.ofModule","declaration":"/-- A vector space over Z/p is an elementary abelian p-group. -/\ntheorem ElementaryAddCommGroup.ofModule {G : Type u_1} {p : ℕ} [AddCommGroup G] [Module (ZMod p) G] [Fact (Nat.Prime p)] : ElementaryAddCommGroup G p"}
+{"name":"ElementaryAddCommGroup.char_smul_eq_zero","declaration":"theorem ElementaryAddCommGroup.char_smul_eq_zero {p : ℕ} {Γ : Type u_1} [AddCommGroup Γ] [ElementaryAddCommGroup Γ p] (x : Γ) : p • x = 0"}
+{"name":"ElementaryAddCommGroup.sum_add_sum_add_sum_eq_zero","declaration":"theorem ElementaryAddCommGroup.sum_add_sum_add_sum_eq_zero {G : Type u_1} [AddCommGroup G] [elem : ElementaryAddCommGroup G 2] (x : G) (y : G) (z : G) : x + y + (y + z) + (z + x) = 0"}
+{"name":"ElementaryAddCommGroup.exists_subgroup_subset_card_le","declaration":"/-- In an elementary abelian $p$-group, every finite subgroup $H$ contains a further subgroup of\ncardinality between $k$ and $pk$, if $k \\leq |H|$.-/\ntheorem ElementaryAddCommGroup.exists_subgroup_subset_card_le {G : Type u_1} {p : ℕ} (hp : Nat.Prime p) [AddCommGroup G] [h : ElementaryAddCommGroup G p] {k : ℕ} (H : AddSubgroup G) (hk : k ≤ Nat.card ↥H) (h'k : k ≠ 0) : ∃ H', Nat.card ↥H' ≤ k ∧ k < p * Nat.card ↥H' ∧ H' ≤ H"}
+{"name":"ElementaryAddCommGroup.neg_eq_self","declaration":"theorem ElementaryAddCommGroup.neg_eq_self {G : Type u_1} [AddCommGroup G] [elem : ElementaryAddCommGroup G 2] (x : G) : -x = x"}
+{"name":"ElementaryAddCommGroup.Int.mod_eq","declaration":"theorem ElementaryAddCommGroup.Int.mod_eq (n : ℤ) (m : ℤ) : n % m = n - n / m * m"}
+{"name":"ElementaryAddCommGroup.orderOf_of_ne","declaration":"def ElementaryAddCommGroup.orderOf_of_ne {G : Type u_1} [AddCommGroup G] {p : outParam ℕ} [self : ElementaryAddCommGroup G p] {x : G} (hx : x ≠ 0) : addOrderOf x = p"}
+{"name":"ElementaryAddCommGroup.module","declaration":"instance ElementaryAddCommGroup.module {G : Type u_1} {n : ℕ} [AddCommGroup G] [ElementaryAddCommGroup G n] : Module (ZMod n) G"}
+{"name":"ElementaryAddCommGroup.instElementaryAddCommGroupForAllAddCommGroup","declaration":"instance ElementaryAddCommGroup.instElementaryAddCommGroupForAllAddCommGroup (Ω : Type u_1) (Γ : Type u_2) (p : ℕ) [NeZero p] [AddCommGroup Γ] [ElementaryAddCommGroup Γ p] : ElementaryAddCommGroup (Ω → Γ) p"}
+{"name":"ElementaryAddCommGroup.subgroup","declaration":"theorem ElementaryAddCommGroup.subgroup {G : Type u_1} [AddCommGroup G] {n : ℕ} [ElementaryAddCommGroup G n] (H : AddSubgroup G) : ElementaryAddCommGroup (↥H) n"}
+{"name":"ElementaryAddCommGroup.mem_periodicPts","declaration":"theorem ElementaryAddCommGroup.mem_periodicPts {p : ℕ} {Γ : Type u_1} [NeZero p] [AddCommGroup Γ] [ElementaryAddCommGroup Γ p] {x : Γ} (y : Γ) : y ∈ Function.periodicPts fun z => x + z"}
+{"name":"ElementaryAddCommGroup.add_self","declaration":"theorem ElementaryAddCommGroup.add_self {G : Type u_1} [AddCommGroup G] [elem : ElementaryAddCommGroup G 2] (x : G) : x + x = 0"}
+{"name":"ElementaryAddCommGroup.exists_finsupp","declaration":"theorem ElementaryAddCommGroup.exists_finsupp {G : Type u_1} [AddCommGroup G] {n : ℕ} [ElementaryAddCommGroup G (n + 1)] {A : Set G} {x : G} (hx : x ∈ Submodule.span ℤ A) : ∃ μ, (Finsupp.sum μ fun a r => ZMod.cast r • ↑a) = x"}
+{"name":"ElementaryAddCommGroup","declaration":"/-- An elementary `p`-group, i.e., a commutative additive group in which every nonzero element has\norder exactly `p`. -/\nclass ElementaryAddCommGroup (G : Type u_1) [AddCommGroup G] (p : outParam ℕ) : Prop"}
+{"name":"ElementaryAddCommGroup.instElementaryAddCommGroupOfNatNatInstOfNatNat","declaration":"instance ElementaryAddCommGroup.instElementaryAddCommGroupOfNatNatInstOfNatNat {G : Type u_1} [AddCommGroup G] [Module (ZMod 2) G] : ElementaryAddCommGroup G 2"}

PFR-declarations/PFR.ForMathlib.Entropy.Basic.jsonl

@@ -0,0 +1,111 @@
+{"name":"ProbabilityTheory.entropy_const","declaration":"/-- The entropy of any constant is zero. -/\ntheorem ProbabilityTheory.entropy_const {Ω : Type uΩ} {S : Type uS} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] [MeasurableSingletonClass S] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] (c : S) : H[fun x => c ; μ] = 0"}
+{"name":"ProbabilityTheory.condEntropy_le_entropy","declaration":"/-- $$ H[X|Y] ≤ H[X] $$ -/\ntheorem ProbabilityTheory.condEntropy_le_entropy {Ω : Type uΩ} {S : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [Countable S] [Countable T] [Nonempty S] [Nonempty T] [MeasurableSpace S] [MeasurableSpace T] {X : Ω → S} {Y : Ω → T} (μ : MeasureTheory.Measure Ω) [MeasurableSingletonClass S] [MeasurableSingletonClass T] (hX : Measurable X) (hY : Measurable Y) [MeasureTheory.IsProbabilityMeasure μ] [FiniteRange X] [FiniteRange Y] : H[X | Y ; μ] ≤ H[X ; μ]"}
+{"name":"ProbabilityTheory.condMutualInfo_eq_kernel_mutualInfo","declaration":"/-- The conditional mutual information agrees with the information of the conditional kernel.\n-/\ntheorem ProbabilityTheory.condMutualInfo_eq_kernel_mutualInfo {Ω : Type uΩ} {S : Type uS} {T : Type uT} {U : Type uU} [mΩ : MeasurableSpace Ω] [Countable S] [Countable T] [Countable U] [Nonempty S] [Nonempty T] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U] [MeasurableSingletonClass S] [MeasurableSingletonClass T] [MeasurableSingletonClass U] {X : Ω → S} {Y : Ω → T} {Z : Ω → U} (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] [FiniteRange Z] : I[X : Y|Z;μ] = Ik[ProbabilityTheory.condDistrib (⟨X, Y⟩) Z μ , MeasureTheory.Measure.map Z μ]"}
+{"name":"ProbabilityTheory.condMutualInfo_eq_integral_mutualInfo","declaration":"theorem ProbabilityTheory.condMutualInfo_eq_integral_mutualInfo {Ω : Type uΩ} {S : Type uS} {T : Type uT} {U : Type uU} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U] {X : Ω → S} {Y : Ω → T} {Z : Ω → U} {μ : MeasureTheory.Measure Ω} : I[X : Y|Z;μ] = ∫ (x : U), (fun z => I[X : Y ; ProbabilityTheory.cond μ (Z ⁻¹' {z})]) x ∂MeasureTheory.Measure.map Z μ"}
+{"name":"ProbabilityTheory.«termI[_:_|_]»","declaration":"/-- The conditional mutual information $I[X : Y| Z]$ is the mutual information of $X| Z=z$ and\n$Y| Z=z$, integrated over $z$. -/\ndef ProbabilityTheory.«termI[_:_|_]»  : Lean.ParserDescr"}
+{"name":"ProbabilityTheory.mutualInfo","declaration":"/-- The mutual information $I[X : Y]$ of two random variables is defined to be $H[X] + H[Y] - H[X ; Y]$. -/\ndef ProbabilityTheory.mutualInfo {Ω : Type uΩ} {S : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] [MeasurableSpace T] (X : Ω → S) (Y : Ω → T) (μ : autoParam (MeasureTheory.Measure Ω) _auto✝) : ℝ"}
+{"name":"ProbabilityTheory.entropy_pair_eq_add","declaration":"/-- $H[X, Y] = H[X] + H[Y]$ if and only if $X, Y$ are independent. -/\ntheorem ProbabilityTheory.entropy_pair_eq_add {Ω : Type uΩ} {S : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSingletonClass S] [MeasurableSingletonClass T] {X : Ω → S} {Y : Ω → T} (hX : Measurable X) (hY : Measurable Y) {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] [FiniteRange X] [FiniteRange Y] : H[⟨X, Y⟩ ; μ] = H[X ; μ] + H[Y ; μ] ↔ ProbabilityTheory.IndepFun X Y μ"}
+{"name":"ProbabilityTheory.«termI[_:_|_]».delab","declaration":"/-- Pretty printer defined by `notation3` command. -/\ndef ProbabilityTheory.«termI[_:_|_]».delab  : Lean.PrettyPrinter.Delaborator.Delab"}
+{"name":"ProbabilityTheory.condEntropy_le_log_card","declaration":"/-- Conditional entropy is at most the logarithm of the cardinality of the range. -/\ntheorem ProbabilityTheory.condEntropy_le_log_card {Ω : Type uΩ} {S : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSingletonClass S] [Fintype S] (X : Ω → S) (Y : Ω → T) (hY : Measurable Y) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] : H[X | Y ; μ] ≤ Real.log ↑(Fintype.card S)"}
+{"name":"ProbabilityTheory.condMutualInfo_comm","declaration":"/-- $I[X : Y | Z] = I[Y : X | Z]$. -/\ntheorem ProbabilityTheory.condMutualInfo_comm {Ω : Type uΩ} {S : Type uS} {T : Type uT} {U : Type uU} [mΩ : MeasurableSpace Ω] [Countable S] [Countable T] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U] [MeasurableSingletonClass S] [MeasurableSingletonClass T] {X : Ω → S} {Y : Ω → T} (hX : Measurable X) (hY : Measurable Y) (Z : Ω → U) (μ : MeasureTheory.Measure Ω) : I[X : Y|Z;μ] = I[Y : X|Z;μ]"}
+{"name":"ProbabilityTheory.mutualInfo_eq_entropy_sub_condEntropy'","declaration":"/-- $I[X : Y] = H[Y] - H[Y | X]$. -/\ntheorem ProbabilityTheory.mutualInfo_eq_entropy_sub_condEntropy' {Ω : Type uΩ} {S : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [Countable S] [Countable T] [Nonempty S] [Nonempty T] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSingletonClass S] [MeasurableSingletonClass T] {X : Ω → S} {Y : Ω → T} (hX : Measurable X) (hY : Measurable Y) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] [FiniteRange X] [FiniteRange Y] : I[X : Y ; μ] = H[Y ; μ] - H[Y | X ; μ]"}
+{"name":"ProbabilityTheory.entropy_add_entropy_sub_mutualInfo","declaration":"theorem ProbabilityTheory.entropy_add_entropy_sub_mutualInfo {Ω : Type uΩ} {S : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] [MeasurableSpace T] (X : Ω → S) (Y : Ω → T) (μ : MeasureTheory.Measure Ω) : H[X ; μ] + H[Y ; μ] - I[X : Y ; μ] = H[⟨X, Y⟩ ; μ]"}
+{"name":"ProbabilityTheory.entropy_le_log_card_of_mem","declaration":"/-- Entropy is at most the logarithm of the cardinality of a set in which X almost surely takes values in. -/\ntheorem ProbabilityTheory.entropy_le_log_card_of_mem {Ω : Type uΩ} {S : Type uS} [mΩ : MeasurableSpace Ω] [Countable S] [MeasurableSpace S] [MeasurableSingletonClass S] {A : Finset S} {μ : MeasureTheory.Measure Ω} {X : Ω → S} (hX : Measurable X) (h : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ A) : H[X ; μ] ≤ Real.log ↑(Nat.card { x // x ∈ A })"}
+{"name":"ProbabilityTheory.entropy_of_comp_eq_of_comp","declaration":"/-- A Schroder-Bernstein type theorem for entropy : if two random variables are functions of each\nother, then they have the same entropy. Can be used as a substitute for\n`entropy_comp_of_injective` if one doesn't want to establish the injectivity. -/\ntheorem ProbabilityTheory.entropy_of_comp_eq_of_comp {Ω : Type uΩ} {S : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [Countable S] [Countable T] [Nonempty S] [Nonempty T] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSingletonClass S] [MeasurableSingletonClass T] {X : Ω → S} {Y : Ω → T} (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (hX : Measurable X) (hY : Measurable Y) (f : S → T) (g : T → S) (h1 : Y = f ∘ X) (h2 : X = g ∘ Y) [FiniteRange X] [FiniteRange Y] : H[X ; μ] = H[Y ; μ]"}
+{"name":"ProbabilityTheory.«termH[_;_]».delab","declaration":"/-- Pretty printer defined by `notation3` command. -/\ndef ProbabilityTheory.«termH[_;_]».delab  : Lean.PrettyPrinter.Delaborator.Delab"}
+{"name":"ProbabilityTheory.entropy_eq_sum_finset'","declaration":"theorem ProbabilityTheory.entropy_eq_sum_finset' {Ω : Type uΩ} {S : Type uS} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] {X : Ω → S} (hX : Measurable X) {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {A : Finset S} (hA : ↑↑(MeasureTheory.Measure.map X μ) (↑A)ᶜ = 0) : H[X ; μ] = Finset.sum A fun x => Real.negMulLog ((MeasureTheory.Measure.map X μ).real {x})"}
+{"name":"ProbabilityTheory.condEntropy_eq_sum_prod","declaration":"/-- Same as previous lemma, but with a sum over a product space rather than a double sum. -/\ntheorem ProbabilityTheory.condEntropy_eq_sum_prod {Ω : Type uΩ} {S : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSingletonClass S] {X : Ω → S} [MeasurableSingletonClass T] (hX : Measurable X) {Y : Ω → T} (hY : Measurable Y) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] [FiniteRange X] [FiniteRange Y] : H[X | Y ; μ] =\n  Finset.sum (FiniteRange.toFinset X ×ˢ FiniteRange.toFinset Y) fun p =>\n    (↑↑(MeasureTheory.Measure.map Y μ) {p.2}).toReal *\n      Real.negMulLog (↑↑(MeasureTheory.Measure.map X (ProbabilityTheory.cond μ (Y ⁻¹' {p.2}))) {p.1}).toReal"}
+{"name":"ProbabilityTheory.entropy_eq_sum","declaration":"/-- $H[X] = \\sum_s P[X=s] \\log \\frac{1}{P[X=s]}$. -/\ntheorem ProbabilityTheory.entropy_eq_sum {Ω : Type uΩ} {S : Type uS} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] {X : Ω → S} (hX : Measurable X) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] : H[X ; μ] = ∑' (x : S), Real.negMulLog (↑↑(MeasureTheory.Measure.map X μ) {x}).toReal"}
+{"name":"ProbabilityTheory.mutualInfo_const","declaration":"/-- The mutual information with a constant is always zero. -/\ntheorem ProbabilityTheory.mutualInfo_const {Ω : Type uΩ} {S : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSingletonClass S] [MeasurableSingletonClass T] {X : Ω → S} (hX : Measurable X) (c : T) {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] [FiniteRange X] : I[X : fun x => c ; μ] = 0"}
+{"name":"ProbabilityTheory.condEntropy_eq_sum","declaration":"/-- $H[X|Y] = \\sum_y P[Y=y] H[X|Y=y]$.-/\ntheorem ProbabilityTheory.condEntropy_eq_sum {Ω : Type uΩ} {S : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSingletonClass T] (X : Ω → S) (Y : Ω → T) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] (hY : Measurable Y) [FiniteRange Y] : H[X | Y ; μ] =\n  Finset.sum (FiniteRange.toFinset Y) fun y => (↑↑(MeasureTheory.Measure.map Y μ) {y}).toReal * H[X | Y ← y ; μ]"}
+{"name":"ProbabilityTheory.const_of_nonpos_entropy","declaration":"/-- If $X$ is an $S$-valued random variable of non-positive entropy, then $X$ is almost surely constant. -/\ntheorem ProbabilityTheory.const_of_nonpos_entropy {S : Type uS} [Nonempty S] [MeasurableSpace S] [MeasurableSingletonClass S] {Ω : Type u_1} [MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {X : Ω → S} (hX : Measurable X) [FiniteRange X] (hent : H[X ; μ] ≤ 0) : ∃ s, μ.real (X ⁻¹' {s}) = 1"}
+{"name":"ProbabilityTheory.IdentDistrib.mutualInfo_eq","declaration":"/-- Substituting variables for ones with the same distributions doesn't change the mutual information. -/\ntheorem ProbabilityTheory.IdentDistrib.mutualInfo_eq {Ω : Type uΩ} {S : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] [MeasurableSpace T] {X : Ω → S} {Y : Ω → T} {μ : MeasureTheory.Measure Ω} {Ω' : Type u_1} [MeasurableSpace Ω'] {μ' : MeasureTheory.Measure Ω'} {X' : Ω' → S} {Y' : Ω' → T} (hXY : ProbabilityTheory.IdentDistrib (⟨X, Y⟩) (⟨X', Y'⟩) μ μ') : I[X : Y ; μ] = I[X' : Y' ; μ']"}
+{"name":"ProbabilityTheory.«termH[_|_;_]»","declaration":"/-- Conditional entropy of a random variable w.r.t. another.\nThis is the expectation under the law of `Y` of the entropy of the law of `X` conditioned on the\nevent `Y = y`. -/\ndef ProbabilityTheory.«termH[_|_;_]»  : Lean.ParserDescr"}
+{"name":"ProbabilityTheory.condEntropy_comm","declaration":"/-- $H[X, Y| Z] = H[Y, X| Z]$. -/\ntheorem ProbabilityTheory.condEntropy_comm {Ω : Type uΩ} {S : Type uS} {T : Type uT} {U : Type uU} [mΩ : MeasurableSpace Ω] [Countable S] [Countable T] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U] [MeasurableSingletonClass S] [MeasurableSingletonClass T] {X : Ω → S} {Y : Ω → T} {Z : Ω → U} (hX : Measurable X) (hY : Measurable Y) (μ : MeasureTheory.Measure Ω) : H[⟨X, Y⟩ | Z ; μ] = H[⟨Y, X⟩ | Z ; μ]"}
+{"name":"ProbabilityTheory.entropy_nonneg","declaration":"/-- Entropy is always non-negative. -/\ntheorem ProbabilityTheory.entropy_nonneg {Ω : Type uΩ} {S : Type uS} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] (X : Ω → S) (μ : MeasureTheory.Measure Ω) : 0 ≤ H[X ; μ]"}
+{"name":"ProbabilityTheory.«termI[_:_|_;_]».delab","declaration":"/-- Pretty printer defined by `notation3` command. -/\ndef ProbabilityTheory.«termI[_:_|_;_]».delab  : Lean.PrettyPrinter.Delaborator.Delab"}
+{"name":"ProbabilityTheory.entropy_eq_sum'","declaration":"theorem ProbabilityTheory.entropy_eq_sum' {Ω : Type uΩ} {S : Type uS} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] {X : Ω → S} (hX : Measurable X) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] : H[X ; μ] = ∑' (x : S), Real.negMulLog ((MeasureTheory.Measure.map X μ).real {x})"}
+{"name":"ProbabilityTheory.condEntropy_of_injective","declaration":"/-- If $X : \\Omega \\to S$, $Y : \\Omega \\to T$ are random variables, and $f : T \\times S → U$ is\ninjective for each fixed $t \\in T$, then $H[f(Y, X)|Y] = H[X|Y]$.\nThus for instance $H[X-Y|Y] = H[X|Y]$. -/\ntheorem ProbabilityTheory.condEntropy_of_injective {Ω : Type uΩ} {S : Type uS} {T : Type uT} {U : Type uU} [mΩ : MeasurableSpace Ω] [Countable S] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U] [MeasurableSingletonClass S] [MeasurableSingletonClass T] {X : Ω → S} {Y : Ω → T} [MeasurableSingletonClass U] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] (hX : Measurable X) (hY : Measurable Y) (f : T → S → U) (hf : ∀ (t : T), Function.Injective (f t)) [FiniteRange Y] : H[fun ω => f (Y ω) (X ω) | Y ; μ] = H[X | Y ; μ]"}
+{"name":"ProbabilityTheory.condMutualInfo_eq_sum'","declaration":"/-- A variant of `condMutualInfo_eq_sum` when `Z` has finite codomain. -/\ntheorem ProbabilityTheory.condMutualInfo_eq_sum' {Ω : Type uΩ} {S : Type uS} {T : Type uT} {U : Type uU} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U] [MeasurableSingletonClass U] {X : Ω → S} {Y : Ω → T} {Z : Ω → U} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] (hZ : Measurable Z) [Fintype U] : I[X : Y|Z;μ] =\n  Finset.sum Finset.univ fun z => (↑↑μ (Z ⁻¹' {z})).toReal * I[X : Y ; ProbabilityTheory.cond μ (Z ⁻¹' {z})]"}
+{"name":"ProbabilityTheory.entropy_sub_mutualInfo_eq_condEntropy'","declaration":"/-- $H[Y] - I[X : Y] = H[Y | X]$. -/\ntheorem ProbabilityTheory.entropy_sub_mutualInfo_eq_condEntropy' {Ω : Type uΩ} {S : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [Countable S] [Countable T] [Nonempty S] [Nonempty T] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSingletonClass S] [MeasurableSingletonClass T] {X : Ω → S} {Y : Ω → T} (hX : Measurable X) (hY : Measurable Y) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] [FiniteRange X] [FiniteRange Y] : H[Y ; μ] - I[X : Y ; μ] = H[Y | X ; μ]"}
+{"name":"ProbabilityTheory.«termI[_:_;_]»","declaration":"/-- The mutual information $I[X : Y]$ of two random variables is defined to be $H[X] + H[Y] - H[X ; Y]$. -/\ndef ProbabilityTheory.«termI[_:_;_]»  : Lean.ParserDescr"}
+{"name":"ProbabilityTheory.IsUniform.entropy_eq'","declaration":"/-- Variant of `IsUniform.entropy_eq` where `H` is a finite `Set` rather than `Finset`. -/\ntheorem ProbabilityTheory.IsUniform.entropy_eq' {Ω : Type uΩ} {S : Type uS} [mΩ : MeasurableSpace Ω] [Countable S] [MeasurableSpace S] [MeasurableSingletonClass S] {H : Set S} [Finite ↑H] {X : Ω → S} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] (hX : ProbabilityTheory.IsUniform H X μ) (hX' : Measurable X) : H[X ; μ] = Real.log ↑(Nat.card ↑H)"}
+{"name":"ProbabilityTheory.«termH[_|_←_]»","declaration":"/-- Entropy of a random variable with values in a finite measurable space. -/\ndef ProbabilityTheory.«termH[_|_←_]»  : Lean.ParserDescr"}
+{"name":"ProbabilityTheory.«termH[_;_]»","declaration":"/-- Entropy of a random variable with values in a finite measurable space. -/\ndef ProbabilityTheory.«termH[_;_]»  : Lean.ParserDescr"}
+{"name":"ProbabilityTheory.entropy_congr","declaration":"/-- Two variables that agree almost everywhere, have the same entropy. -/\ntheorem ProbabilityTheory.entropy_congr {Ω : Type uΩ} {S : Type uS} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] {μ : MeasureTheory.Measure Ω} {X : Ω → S} {X' : Ω → S} (h : X =ᶠ[MeasureTheory.Measure.ae μ] X') : H[X ; μ] = H[X' ; μ]"}
+{"name":"ProbabilityTheory.condEntropy_zero_measure","declaration":"/-- Any random variable on a zero measure space has zero conditional entropy. -/\ntheorem ProbabilityTheory.condEntropy_zero_measure {Ω : Type uΩ} {S : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] [MeasurableSpace T] (X : Ω → S) (Y : Ω → T) : H[X | Y ; 0] = 0"}
+{"name":"ProbabilityTheory.IdentDistrib.entropy_eq","declaration":"/-- Two variables that have the same distribution, have the same entropy. -/\ntheorem ProbabilityTheory.IdentDistrib.entropy_eq {Ω : Type uΩ} {S : Type uS} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] {X : Ω → S} {μ : MeasureTheory.Measure Ω} {Ω' : Type u_1} [MeasurableSpace Ω'] {μ' : MeasureTheory.Measure Ω'} {X' : Ω' → S} (h : ProbabilityTheory.IdentDistrib X X' μ μ') : H[X ; μ] = H[X' ; μ']"}
+{"name":"ProbabilityTheory.condEntropy","declaration":"/-- Conditional entropy of a random variable w.r.t. another.\nThis is the expectation under the law of `Y` of the entropy of the law of `X` conditioned on the\nevent `Y = y`. -/\ndef ProbabilityTheory.condEntropy {Ω : Type uΩ} {S : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] [MeasurableSpace T] (X : Ω → S) (Y : Ω → T) (μ : autoParam (MeasureTheory.Measure Ω) _auto✝) : ℝ"}
+{"name":"ProbabilityTheory.entropy_eq_sum_finiteRange","declaration":"theorem ProbabilityTheory.entropy_eq_sum_finiteRange {Ω : Type uΩ} {S : Type uS} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] [MeasurableSingletonClass S] {X : Ω → S} (hX : Measurable X) {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] [FiniteRange X] : H[X ; μ] = Finset.sum (FiniteRange.toFinset X) fun x => Real.negMulLog (↑↑(MeasureTheory.Measure.map X μ) {x}).toReal"}
+{"name":"ProbabilityTheory.chain_rule''","declaration":"/-- Another form of the chain rule : $H[X|Y] = H[X, Y] - H[Y]$. -/\ntheorem ProbabilityTheory.chain_rule'' {Ω : Type uΩ} {S : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [Countable S] [Countable T] [Nonempty S] [Nonempty T] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSingletonClass S] [MeasurableSingletonClass T] {X : Ω → S} {Y : Ω → T} (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (hX : Measurable X) (hY : Measurable Y) [FiniteRange X] [FiniteRange Y] : H[X | Y ; μ] = H[⟨X, Y⟩ ; μ] - H[Y ; μ]"}
+{"name":"ProbabilityTheory.IndepFun.condEntropy_eq_entropy","declaration":"theorem ProbabilityTheory.IndepFun.condEntropy_eq_entropy {Ω : Type uΩ} {S : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [Countable S] [Countable T] [Nonempty S] [Nonempty T] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSingletonClass S] [MeasurableSingletonClass T] {X : Ω → S} {Y : Ω → T} {μ : MeasureTheory.Measure Ω} (h : ProbabilityTheory.IndepFun X Y μ) (hX : Measurable X) (hY : Measurable Y) [MeasureTheory.IsProbabilityMeasure μ] [FiniteRange X] [FiniteRange Y] : H[X | Y ; μ] = H[X ; μ]"}
+{"name":"ProbabilityTheory.«termH[_|_]».delab","declaration":"/-- Pretty printer defined by `notation3` command. -/\ndef ProbabilityTheory.«termH[_|_]».delab  : Lean.PrettyPrinter.Delaborator.Delab"}
+{"name":"ProbabilityTheory.«termH[_|_←_;_]»","declaration":"/-- Entropy of a random variable with values in a finite measurable space. -/\ndef ProbabilityTheory.«termH[_|_←_;_]»  : Lean.ParserDescr"}
+{"name":"ProbabilityTheory.IsUniform.entropy_eq","declaration":"/-- If $X$ is uniformly distributed on $H$, then $H[X] = \\log |H|$.\n-/\ntheorem ProbabilityTheory.IsUniform.entropy_eq {Ω : Type uΩ} {S : Type uS} [mΩ : MeasurableSpace Ω] [Countable S] [MeasurableSpace S] [MeasurableSingletonClass S] (H : Finset S) (X : Ω → S) {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] (hX : ProbabilityTheory.IsUniform (↑H) X μ) (hX' : Measurable X) : H[X ; μ] = Real.log ↑(Nat.card { x // x ∈ H })"}
+{"name":"ProbabilityTheory.«termI[_:_]».delab","declaration":"/-- Pretty printer defined by `notation3` command. -/\ndef ProbabilityTheory.«termI[_:_]».delab  : Lean.PrettyPrinter.Delaborator.Delab"}
+{"name":"ProbabilityTheory.cond_chain_rule","declaration":"/-- $$ H[X, Y | Z] = H[Y | Z] + H[X|Y, Z].$$ -/\ntheorem ProbabilityTheory.cond_chain_rule {Ω : Type uΩ} {S : Type uS} {T : Type uT} {U : Type uU} [mΩ : MeasurableSpace Ω] [Countable S] [Countable T] [Countable U] [Nonempty S] [Nonempty T] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U] [MeasurableSingletonClass S] [MeasurableSingletonClass T] [MeasurableSingletonClass U] {X : Ω → S} {Y : Ω → T} {Z : Ω → U} (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) [FiniteRange X] [FiniteRange Y] [FiniteRange Z] : H[⟨X, Y⟩ | Z ; μ] = H[Y | Z ; μ] + H[X | ⟨Y, Z⟩ ; μ]"}
+{"name":"ProbabilityTheory.mutualInfo_def","declaration":"theorem ProbabilityTheory.mutualInfo_def {Ω : Type uΩ} {S : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] [MeasurableSpace T] (X : Ω → S) (Y : Ω → T) (μ : MeasureTheory.Measure Ω) : I[X : Y ; μ] = H[X ; μ] + H[Y ; μ] - H[⟨X, Y⟩ ; μ]"}
+{"name":"ProbabilityTheory.IndepFun.mutualInfo_eq_zero","declaration":"/-- **Alias** of the reverse direction of `ProbabilityTheory.mutualInfo_eq_zero`.\n\n---\n\n$I[X : Y] = 0$ iff $X, Y$ are independent. -/\ntheorem ProbabilityTheory.IndepFun.mutualInfo_eq_zero {Ω : Type uΩ} {S : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSingletonClass S] [MeasurableSingletonClass T] {X : Ω → S} {Y : Ω → T} (hX : Measurable X) (hY : Measurable Y) {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] [FiniteRange X] [FiniteRange Y] : ProbabilityTheory.IndepFun X Y μ → I[X : Y ; μ] = 0"}
+{"name":"ProbabilityTheory.entropy_eq_sum_finiteRange'","declaration":"theorem ProbabilityTheory.entropy_eq_sum_finiteRange' {Ω : Type uΩ} {S : Type uS} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] [MeasurableSingletonClass S] {X : Ω → S} (hX : Measurable X) {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] [FiniteRange X] : H[X ; μ] = Finset.sum (FiniteRange.toFinset X) fun x => Real.negMulLog ((MeasureTheory.Measure.map X μ).real {x})"}
+{"name":"ProbabilityTheory.entropy_comp_le","declaration":"/-- Data-processing inequality for the entropy :\n$$ H[f(X)] \\leq H[X].$$\nTo upgrade this to equality, see `entropy_of_comp_eq_of_comp` or `entropy_comp_of_injective`. -/\ntheorem ProbabilityTheory.entropy_comp_le {Ω : Type uΩ} {S : Type uS} {U : Type uU} [mΩ : MeasurableSpace Ω] [Countable S] [Countable U] [Nonempty S] [Nonempty U] [MeasurableSpace S] [MeasurableSpace U] [MeasurableSingletonClass S] [MeasurableSingletonClass U] {X : Ω → S} (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (hX : Measurable X) (f : S → U) [FiniteRange X] : H[f ∘ X ; μ] ≤ H[X ; μ]"}
+{"name":"ProbabilityTheory.«termH[_|_;_]».delab","declaration":"/-- Pretty printer defined by `notation3` command. -/\ndef ProbabilityTheory.«termH[_|_;_]».delab  : Lean.PrettyPrinter.Delaborator.Delab"}
+{"name":"ProbabilityTheory.map_prod_comap_swap","declaration":"/-- The law of $(X, Z)$ is the image of the law of $(Z, X)$.-/\ntheorem ProbabilityTheory.map_prod_comap_swap {Ω : Type uΩ} {S : Type uS} {U : Type uU} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] [MeasurableSpace U] {Z : Ω → U} {X : Ω → S} (hX : Measurable X) (hZ : Measurable Z) (μ : MeasureTheory.Measure Ω) : MeasureTheory.Measure.comap Prod.swap (MeasureTheory.Measure.map (fun ω => (X ω, Z ω)) μ) =\n  MeasureTheory.Measure.map (fun ω => (Z ω, X ω)) μ"}
+{"name":"ProbabilityTheory.entropy_eq_kernel_entropy","declaration":"/-- Entropy of a random variable is also the kernel entropy of the distribution over a Dirac mass. -/\ntheorem ProbabilityTheory.entropy_eq_kernel_entropy {Ω : Type uΩ} {S : Type uS} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] (X : Ω → S) (μ : MeasureTheory.Measure Ω) : H[X ; μ] = Hk[ProbabilityTheory.kernel.const Unit (MeasureTheory.Measure.map X μ) , MeasureTheory.Measure.dirac ()]"}
+{"name":"ProbabilityTheory.IndepFun.entropy_pair_eq_add","declaration":"/-- If $X, Y$ are independent, then $H[X, Y] = H[X] + H[Y]$. -/\ntheorem ProbabilityTheory.IndepFun.entropy_pair_eq_add {Ω : Type uΩ} {S : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSingletonClass S] [MeasurableSingletonClass T] {X : Ω → S} {Y : Ω → T} (hX : Measurable X) (hY : Measurable Y) {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] [FiniteRange X] [FiniteRange Y] : ProbabilityTheory.IndepFun X Y μ → H[⟨X, Y⟩ ; μ] = H[X ; μ] + H[Y ; μ]"}
+{"name":"ProbabilityTheory.condMutualInfo_nonneg","declaration":"/-- Conditional information is non-nonegative. -/\ntheorem ProbabilityTheory.condMutualInfo_nonneg {Ω : Type uΩ} {S : Type uS} {T : Type uT} {U : Type uU} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U] [MeasurableSingletonClass S] [MeasurableSingletonClass T] {X : Ω → S} {Y : Ω → T} (hX : Measurable X) (hY : Measurable Y) (Z : Ω → U) (μ : MeasureTheory.Measure Ω) [FiniteRange X] [FiniteRange Y] : 0 ≤ I[X : Y|Z;μ]"}
+{"name":"ProbabilityTheory.condMutualInfo_eq'","declaration":"/-- $$ I[X : Y| Z] = H[X| Z] - H[X|Y, Z].$$ -/\ntheorem ProbabilityTheory.condMutualInfo_eq' {Ω : Type uΩ} {S : Type uS} {T : Type uT} {U : Type uU} [mΩ : MeasurableSpace Ω] [Countable S] [Countable T] [Countable U] [Nonempty S] [Nonempty T] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U] [MeasurableSingletonClass S] [MeasurableSingletonClass T] [MeasurableSingletonClass U] {X : Ω → S} {Y : Ω → T} {Z : Ω → U} (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] [FiniteRange X] [FiniteRange Y] [FiniteRange Z] : I[X : Y|Z;μ] = H[X | Z ; μ] - H[X | ⟨Y, Z⟩ ; μ]"}
+{"name":"ProbabilityTheory.«termI[_:_|_;_]»","declaration":"/-- The conditional mutual information $I[X : Y| Z]$ is the mutual information of $X| Z=z$ and\n$Y| Z=z$, integrated over $z$. -/\ndef ProbabilityTheory.«termI[_:_|_;_]»  : Lean.ParserDescr"}
+{"name":"ProbabilityTheory.«termH[_|_←_;_]».delab","declaration":"/-- Pretty printer defined by `notation3` command. -/\ndef ProbabilityTheory.«termH[_|_←_;_]».delab  : Lean.PrettyPrinter.Delaborator.Delab"}
+{"name":"ProbabilityTheory.condEntropy_eq_kernel_entropy","declaration":"/-- Conditional entropy of a random variable is equal to the entropy of its conditional kernel. -/\ntheorem ProbabilityTheory.condEntropy_eq_kernel_entropy {Ω : Type uΩ} {S : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [Countable S] [Countable T] [Nonempty S] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSingletonClass S] [MeasurableSingletonClass T] {X : Ω → S} {Y : Ω → T} (hX : Measurable X) (hY : Measurable Y) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] [FiniteRange Y] : H[X | Y ; μ] = Hk[ProbabilityTheory.condDistrib X Y μ , MeasureTheory.Measure.map Y μ]"}
+{"name":"ProbabilityTheory.chain_rule","declaration":"/-- Another form of the chain rule : $H[X, Y] = H[Y] + H[X|Y]$. -/\ntheorem ProbabilityTheory.chain_rule {Ω : Type uΩ} {S : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [Countable S] [Countable T] [Nonempty S] [Nonempty T] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSingletonClass S] [MeasurableSingletonClass T] {X : Ω → S} {Y : Ω → T} (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (hX : Measurable X) (hY : Measurable Y) [FiniteRange X] [FiniteRange Y] : H[⟨X, Y⟩ ; μ] = H[Y ; μ] + H[X | Y ; μ]"}
+{"name":"ProbabilityTheory.condEntropy_eq_sum_sum","declaration":"/-- $H[X|Y] = \\sum_y \\sum_x P[Y=y] P[X=x|Y=y] log \\frac{1}{P[X=x|Y=y]}$.-/\ntheorem ProbabilityTheory.condEntropy_eq_sum_sum {Ω : Type uΩ} {S : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSingletonClass S] {X : Ω → S} [MeasurableSingletonClass T] (hX : Measurable X) {Y : Ω → T} (hY : Measurable Y) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] [FiniteRange X] [FiniteRange Y] : H[X | Y ; μ] =\n  Finset.sum (FiniteRange.toFinset Y) fun y =>\n    Finset.sum (FiniteRange.toFinset X) fun x =>\n      (↑↑(MeasureTheory.Measure.map Y μ) {y}).toReal *\n        Real.negMulLog (↑↑(MeasureTheory.Measure.map X (ProbabilityTheory.cond μ (Y ⁻¹' {y}))) {x}).toReal"}
+{"name":"ProbabilityTheory.condMutualInfo_def","declaration":"theorem ProbabilityTheory.condMutualInfo_def {Ω : Type uΩ} {S : Type uS} {T : Type uT} {U : Type uU} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U] (X : Ω → S) (Y : Ω → T) (Z : Ω → U) (μ : MeasureTheory.Measure Ω) : I[X : Y|Z;μ] =\n  ∫ (x : U), (fun z => H[X | Z ← z ; μ] + H[Y | Z ← z ; μ] - H[⟨X, Y⟩ | Z ← z ; μ]) x ∂MeasureTheory.Measure.map Z μ"}
+{"name":"ProbabilityTheory.prob_ge_exp_neg_entropy'","declaration":"/-- If $X$ is an $S$-valued random variable, then there exists $s \\in S$ such that\n$P[X=s] \\geq \\exp(-H[X])$. -/\ntheorem ProbabilityTheory.prob_ge_exp_neg_entropy' {S : Type uS} [Nonempty S] [MeasurableSpace S] [MeasurableSingletonClass S] {Ω : Type u_1} [MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] (X : Ω → S) (hX : Measurable X) [FiniteRange X] : ∃ s, Real.exp (-H[X ; μ]) ≤ μ.real (X ⁻¹' {s})"}
+{"name":"ProbabilityTheory.condMutualInfo_of_inj_map","declaration":"/-- If $f(Z, X)$ is injective for each fixed $Z$, then $I[f(Z, X) : Y| Z] = I[X : Y| Z]$.-/\ntheorem ProbabilityTheory.condMutualInfo_of_inj_map {Ω : Type uΩ} {S : Type uS} {T : Type uT} {U : Type uU} [mΩ : MeasurableSpace Ω] [Countable S] [Countable T] [Countable U] [Nonempty S] [Nonempty T] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U] [MeasurableSingletonClass S] [MeasurableSingletonClass T] [MeasurableSingletonClass U] {X : Ω → S} {Y : Ω → T} {Z : Ω → U} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) {V : Type u_1} [Nonempty V] [MeasurableSpace V] [MeasurableSingletonClass V] [Countable V] (f : U → S → V) (hf : ∀ (t : U), Function.Injective (f t)) [FiniteRange Z] : I[fun ω => f (Z ω) (X ω) : Y|Z;μ] = I[X : Y|Z;μ]"}
+{"name":"ProbabilityTheory.mutualInfo_eq_zero","declaration":"/-- $I[X : Y] = 0$ iff $X, Y$ are independent. -/\ntheorem ProbabilityTheory.mutualInfo_eq_zero {Ω : Type uΩ} {S : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSingletonClass S] [MeasurableSingletonClass T] {X : Ω → S} {Y : Ω → T} (hX : Measurable X) (hY : Measurable Y) {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] [FiniteRange X] [FiniteRange Y] : I[X : Y ; μ] = 0 ↔ ProbabilityTheory.IndepFun X Y μ"}
+{"name":"ProbabilityTheory.condMutualInfo","declaration":"/-- The conditional mutual information $I[X : Y| Z]$ is the mutual information of $X| Z=z$ and\n$Y| Z=z$, integrated over $z$. -/\ndef ProbabilityTheory.condMutualInfo {Ω : Type uΩ} {S : Type uS} {T : Type uT} {U : Type uU} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U] (X : Ω → S) (Y : Ω → T) (Z : Ω → U) (μ : autoParam (MeasureTheory.Measure Ω) _auto✝) : ℝ"}
+{"name":"ProbabilityTheory.entropy_submodular","declaration":"/-- $H[X | Y, Z] \\leq H[X | Z]$ -/\ntheorem ProbabilityTheory.entropy_submodular {Ω : Type uΩ} {S : Type uS} {T : Type uT} {U : Type uU} [mΩ : MeasurableSpace Ω] [Countable S] [Countable T] [Countable U] [Nonempty S] [Nonempty T] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U] [MeasurableSingletonClass U] {X : Ω → S} {Y : Ω → T} {Z : Ω → U} (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] [MeasurableSingletonClass S] [MeasurableSingletonClass T] (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) [FiniteRange X] [FiniteRange Y] [FiniteRange Z] : H[X | ⟨Y, Z⟩ ; μ] ≤ H[X | Z ; μ]"}
+{"name":"ProbabilityTheory.entropy_comp_of_injective","declaration":"/-- If $X$, $Y$ are $S$-valued and $T$-valued random variables, and $Y = f(X)$ for\nsome injection $f : S \\to T$, then $H[Y] = H[X]$. One can also use `entropy_of_comp_eq_of_comp` as an alternative if verifying injectivity is fiddly. For the upper bound only, see `entropy_comp_le`. -/\ntheorem ProbabilityTheory.entropy_comp_of_injective {Ω : Type uΩ} {S : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [Countable S] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSingletonClass S] [MeasurableSingletonClass T] {X : Ω → S} (μ : MeasureTheory.Measure Ω) (hX : Measurable X) (f : S → T) (hf : Function.Injective f) : H[f ∘ X ; μ] = H[X ; μ]"}
+{"name":"ProbabilityTheory.entropy_prod_comp","declaration":"/-- $H[X, f(X)] = H[X]$.-/\ntheorem ProbabilityTheory.entropy_prod_comp {Ω : Type uΩ} {S : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [Countable S] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSingletonClass S] [MeasurableSingletonClass T] {X : Ω → S} (hX : Measurable X) (μ : MeasureTheory.Measure Ω) (f : S → T) : H[⟨X, f ∘ X⟩ ; μ] = H[X ; μ]"}
+{"name":"ProbabilityTheory.entropy_sub_mutualInfo_eq_condEntropy","declaration":"/-- $H[X] - I[X : Y] = H[X | Y]$. -/\ntheorem ProbabilityTheory.entropy_sub_mutualInfo_eq_condEntropy {Ω : Type uΩ} {S : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [Countable S] [Countable T] [Nonempty S] [Nonempty T] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSingletonClass S] [MeasurableSingletonClass T] {X : Ω → S} {Y : Ω → T} (hX : Measurable X) (hY : Measurable Y) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] [FiniteRange X] [FiniteRange Y] : H[X ; μ] - I[X : Y ; μ] = H[X | Y ; μ]"}
+{"name":"ProbabilityTheory.«termH[_]»","declaration":"/-- Entropy of a random variable with values in a finite measurable space. -/\ndef ProbabilityTheory.«termH[_]»  : Lean.ParserDescr"}
+{"name":"ProbabilityTheory.IdentDistrib.condEntropy_eq","declaration":"/-- Two pairs of variables that have the same joint distribution, have the same\nconditional entropy. -/\ntheorem ProbabilityTheory.IdentDistrib.condEntropy_eq {Ω : Type uΩ} {S : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [Countable S] [Countable T] [Nonempty S] [Nonempty T] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSingletonClass S] [MeasurableSingletonClass T] {μ : MeasureTheory.Measure Ω} {Ω' : Type u_1} [MeasurableSpace Ω'] {X : Ω → S} {Y : Ω → T} {μ' : MeasureTheory.Measure Ω'} {X' : Ω' → S} {Y' : Ω' → T} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure μ'] (hX : Measurable X) (hY : Measurable Y) (hX' : Measurable X') (hY' : Measurable Y') (h : ProbabilityTheory.IdentDistrib (⟨X, Y⟩) (⟨X', Y'⟩) μ μ') [FiniteRange X] [FiniteRange Y] [FiniteRange X'] [FiniteRange Y'] : H[X | Y ; μ] = H[X' | Y' ; μ']"}
+{"name":"ProbabilityTheory.«termH[_|_]»","declaration":"/-- Conditional entropy of a random variable w.r.t. another.\nThis is the expectation under the law of `Y` of the entropy of the law of `X` conditioned on the\nevent `Y = y`. -/\ndef ProbabilityTheory.«termH[_|_]»  : Lean.ParserDescr"}
+{"name":"ProbabilityTheory.entropy_eq_log_card","declaration":"/-- If $X$ is $S$-valued random variable, then $H[X] = \\log |S|$ if and only if $X$ is uniformly\ndistributed. -/\ntheorem ProbabilityTheory.entropy_eq_log_card {Ω : Type uΩ} {S : Type uS} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] [MeasurableSingletonClass S] {X : Ω → S} [Fintype S] (hX : Measurable X) (μ : MeasureTheory.Measure Ω) [hμ : NeZero μ] [MeasureTheory.IsFiniteMeasure μ] : H[X ; μ] = Real.log ↑(Fintype.card S) ↔\n  ∀ (s : S), ↑↑(MeasureTheory.Measure.map X μ) {s} = ↑↑μ Set.univ / ↑(Fintype.card S)"}
+{"name":"ProbabilityTheory.cond_chain_rule'","declaration":"/-- If $X : \\Omega \\to S$, $Y : \\Omega \\to T$,$Z : \\Omega \\to U$ are random variables, then\n$$H[X, Y | Z] = H[X | Z] + H[Y|X, Z]$$. -/\ntheorem ProbabilityTheory.cond_chain_rule' {Ω : Type uΩ} {S : Type uS} {T : Type uT} {U : Type uU} [mΩ : MeasurableSpace Ω] [Countable S] [Countable T] [Countable U] [Nonempty S] [Nonempty T] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U] [MeasurableSingletonClass S] [MeasurableSingletonClass T] [MeasurableSingletonClass U] {X : Ω → S} {Y : Ω → T} {Z : Ω → U} (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) [FiniteRange X] [FiniteRange Y] [FiniteRange Z] : H[⟨X, Y⟩ | Z ; μ] = H[X | Z ; μ] + H[Y | ⟨X, Z⟩ ; μ]"}
+{"name":"ProbabilityTheory.entropy_def","declaration":"/-- Entropy of a random variable agrees with entropy of its distribution. -/\ntheorem ProbabilityTheory.entropy_def {Ω : Type uΩ} {S : Type uS} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] (X : Ω → S) (μ : MeasureTheory.Measure Ω) : H[X ; μ] = Hm[MeasureTheory.Measure.map X μ]"}
+{"name":"ProbabilityTheory.chain_rule'","declaration":"/-- One form of the chain rule : $H[X, Y] = H[X] + H[Y|X]. -/\ntheorem ProbabilityTheory.chain_rule' {Ω : Type uΩ} {S : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [Countable S] [Countable T] [Nonempty S] [Nonempty T] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSingletonClass S] [MeasurableSingletonClass T] {X : Ω → S} {Y : Ω → T} (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (hX : Measurable X) (hY : Measurable Y) [FiniteRange X] [FiniteRange Y] : H[⟨X, Y⟩ ; μ] = H[X ; μ] + H[Y | X ; μ]"}
+{"name":"ProbabilityTheory.condEntropy_eq_sum_fintype","declaration":"/-- $H[X|Y] = \\sum_y P[Y=y] H[X|Y=y]$.-/\ntheorem ProbabilityTheory.condEntropy_eq_sum_fintype {Ω : Type uΩ} {S : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSingletonClass T] (X : Ω → S) (Y : Ω → T) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] (hY : Measurable Y) [Fintype T] : H[X | Y ; μ] = Finset.sum Finset.univ fun y => (↑↑μ (Y ⁻¹' {y})).toReal * H[X | Y ← y ; μ]"}
+{"name":"ProbabilityTheory.condEntropy_eq_zero","declaration":"theorem ProbabilityTheory.condEntropy_eq_zero {Ω : Type uΩ} {S : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSingletonClass T] {X : Ω → S} {Y : Ω → T} (hY : Measurable Y) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] (t : T) (ht : (↑↑(MeasureTheory.Measure.map Y μ) {t}).toReal = 0) : H[X | Y ← t ; μ] = 0"}
+{"name":"ProbabilityTheory.entropy_triple_add_entropy_le","declaration":"/-- The submodularity inequality:\n$$ H[X, Y, Z] + H[Z] \\leq H[X, Z] + H[Y, Z].$$ -/\ntheorem ProbabilityTheory.entropy_triple_add_entropy_le {Ω : Type uΩ} {S : Type uS} {T : Type uT} {U : Type uU} [mΩ : MeasurableSpace Ω] [Countable S] [Countable T] [Countable U] [Nonempty S] [Nonempty T] [Nonempty U] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U] [MeasurableSingletonClass U] {X : Ω → S} {Y : Ω → T} {Z : Ω → U} (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] [MeasurableSingletonClass S] [MeasurableSingletonClass T] (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) [FiniteRange X] [FiniteRange Y] [FiniteRange Z] : H[⟨X, ⟨Y, Z⟩⟩ ; μ] + H[Z ; μ] ≤ H[⟨X, Z⟩ ; μ] + H[⟨Y, Z⟩ ; μ]"}
+{"name":"ProbabilityTheory.condEntropy_prod_eq_of_indepFun","declaration":"theorem ProbabilityTheory.condEntropy_prod_eq_of_indepFun {Ω : Type uΩ} {S : Type uS} {T : Type uT} {U : Type uU} [mΩ : MeasurableSpace Ω] [Countable S] [Countable T] [Nonempty S] [Nonempty T] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U] [MeasurableSingletonClass S] [MeasurableSingletonClass T] [MeasurableSingletonClass U] {X : Ω → S} {Y : Ω → T} {Z : Ω → U} {μ : MeasureTheory.Measure Ω} [Fintype T] [Fintype U] [MeasureTheory.IsProbabilityMeasure μ] (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) [FiniteRange X] (h : ProbabilityTheory.IndepFun (⟨X, Y⟩) Z μ) : H[X | ⟨Y, Z⟩ ; μ] = H[X | Y ; μ]"}
+{"name":"ProbabilityTheory.«termI[_:_;_]».delab","declaration":"/-- Pretty printer defined by `notation3` command. -/\ndef ProbabilityTheory.«termI[_:_;_]».delab  : Lean.PrettyPrinter.Delaborator.Delab"}
+{"name":"ProbabilityTheory.entropy_sub_condEntropy","declaration":"/-- $$ H[X] - H[X|Y] = I[X : Y] $$ -/\ntheorem ProbabilityTheory.entropy_sub_condEntropy {Ω : Type uΩ} {S : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [Countable S] [Countable T] [Nonempty S] [Nonempty T] [MeasurableSpace S] [MeasurableSpace T] {X : Ω → S} {Y : Ω → T} (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] [MeasurableSingletonClass S] [MeasurableSingletonClass T] (hX : Measurable X) (hY : Measurable Y) [FiniteRange X] [FiniteRange Y] : H[X ; μ] - H[X | Y ; μ] = I[X : Y ; μ]"}
+{"name":"ProbabilityTheory.entropy_zero_measure","declaration":"/-- Any variable on a zero measure space has zero entropy. -/\ntheorem ProbabilityTheory.entropy_zero_measure {Ω : Type uΩ} {S : Type uS} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] (X : Ω → S) : H[X ; 0] = 0"}
+{"name":"ProbabilityTheory.condMutualInfo_eq_zero","declaration":"/-- $I[X : Y| Z]=0$ iff $X, Y$ are conditionally independent over $Z$. -/\ntheorem ProbabilityTheory.condMutualInfo_eq_zero {Ω : Type uΩ} {S : Type uS} {T : Type uT} {U : Type uU} [mΩ : MeasurableSpace Ω] [Countable U] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U] [MeasurableSingletonClass U] {X : Ω → S} {Y : Ω → T} {Z : Ω → U} [MeasurableSingletonClass S] [MeasurableSingletonClass T] {μ : MeasureTheory.Measure Ω} (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) [MeasureTheory.IsProbabilityMeasure μ] [FiniteRange X] [FiniteRange Y] [FiniteRange Z] : I[X : Y|Z;μ] = 0 ↔ ProbabilityTheory.CondIndepFun X Y Z μ"}
+{"name":"ProbabilityTheory.entropy_le_log_card","declaration":"/-- Entropy is at most the logarithm of the cardinality of the range. -/\ntheorem ProbabilityTheory.entropy_le_log_card {Ω : Type uΩ} {S : Type uS} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] [MeasurableSingletonClass S] [Fintype S] (X : Ω → S) (μ : MeasureTheory.Measure Ω) : H[X ; μ] ≤ Real.log ↑(Fintype.card S)"}
+{"name":"ProbabilityTheory.condEntropy_comp_ge","declaration":"/-- Data-processing inequality for the conditional entropy:\n$$ H[Y|f(X)] \\geq H[Y|X]$$\nTo upgrade this to equality, see `condEntropy_of_injective'` -/\ntheorem ProbabilityTheory.condEntropy_comp_ge {Ω : Type uΩ} {S : Type uS} {T : Type uT} {U : Type uU} [mΩ : MeasurableSpace Ω] [Countable S] [Countable T] [Countable U] [Nonempty S] [Nonempty T] [Nonempty U] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U] [MeasurableSingletonClass U] {X : Ω → S} {Y : Ω → T} [MeasurableSingletonClass S] [MeasurableSingletonClass T] [FiniteRange X] [FiniteRange Y] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (hX : Measurable X) (hY : Measurable Y) (f : S → U) : H[Y | f ∘ X ; μ] ≥ H[Y | X ; μ]"}
+{"name":"ProbabilityTheory.mutualInfo_comm","declaration":"/-- $I[X : Y] = I[Y : X]$. -/\ntheorem ProbabilityTheory.mutualInfo_comm {Ω : Type uΩ} {S : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [Countable S] [Countable T] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSingletonClass S] [MeasurableSingletonClass T] {X : Ω → S} {Y : Ω → T} (hX : Measurable X) (hY : Measurable Y) (μ : MeasureTheory.Measure Ω) : I[X : Y ; μ] = I[Y : X ; μ]"}
+{"name":"ProbabilityTheory.mutualInfo_nonneg","declaration":"/-- Mutual information is non-negative. -/\ntheorem ProbabilityTheory.mutualInfo_nonneg {Ω : Type uΩ} {S : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSingletonClass S] [MeasurableSingletonClass T] {X : Ω → S} {Y : Ω → T} (hX : Measurable X) (hY : Measurable Y) (μ : MeasureTheory.Measure Ω) [FiniteRange X] [FiniteRange Y] : 0 ≤ I[X : Y ; μ]"}
+{"name":"ProbabilityTheory.ent_of_cond_indep","declaration":"/-- If $X, Y$ are conditionally independent over $Z$, then $H[X, Y, Z] = H[X, Z] + H[Y, Z] - H[Z]$. -/\ntheorem ProbabilityTheory.ent_of_cond_indep {Ω : Type uΩ} {S : Type uS} {T : Type uT} {U : Type uU} [mΩ : MeasurableSpace Ω] [Countable S] [Countable T] [Countable U] [Nonempty S] [Nonempty T] [Nonempty U] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U] [MeasurableSingletonClass U] {X : Ω → S} {Y : Ω → T} {Z : Ω → U} [MeasurableSingletonClass S] [MeasurableSingletonClass T] {μ : MeasureTheory.Measure Ω} (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) (h : ProbabilityTheory.CondIndepFun X Y Z μ) [MeasureTheory.IsProbabilityMeasure μ] [FiniteRange X] [FiniteRange Y] [FiniteRange Z] : H[⟨X, ⟨Y, Z⟩⟩ ; μ] = H[⟨X, Z⟩ ; μ] + H[⟨Y, Z⟩ ; μ] - H[Z ; μ]"}
+{"name":"ProbabilityTheory.condEntropy_nonneg","declaration":"/-- Conditional entropy is non-negative. -/\ntheorem ProbabilityTheory.condEntropy_nonneg {Ω : Type uΩ} {S : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] [MeasurableSpace T] (X : Ω → S) (Y : Ω → T) (μ : MeasureTheory.Measure Ω) : 0 ≤ H[X | Y ; μ]"}
+{"name":"ProbabilityTheory.condEntropy_comp_self","declaration":"/-- $H[X|f(X)] = H[X] - H[f(X)]$. -/\ntheorem ProbabilityTheory.condEntropy_comp_self {Ω : Type uΩ} {S : Type uS} {U : Type uU} [mΩ : MeasurableSpace Ω] [Countable S] [Countable U] [Nonempty S] [Nonempty U] [MeasurableSpace S] [MeasurableSpace U] [MeasurableSingletonClass S] [MeasurableSingletonClass U] {X : Ω → S} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] (hX : Measurable X) {f : S → U} (hf : Measurable f) [FiniteRange X] : H[X | f ∘ X ; μ] = H[X ; μ] - H[f ∘ X ; μ]"}
+{"name":"ProbabilityTheory.condEntropy_of_injective'","declaration":"/-- If $X : \\Omega \\to S$ and $Y : \\Omega \\to T$ are random variables, and $f : T \\to U$ is an\ninjection then $H[X|f(Y)] = H[X|Y]$.\n -/\ntheorem ProbabilityTheory.condEntropy_of_injective' {Ω : Type uΩ} {S : Type uS} {T : Type uT} {U : Type uU} [mΩ : MeasurableSpace Ω] [Countable S] [Countable T] [Countable U] [Nonempty S] [Nonempty T] [Nonempty U] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U] [MeasurableSingletonClass T] [MeasurableSingletonClass U] {X : Ω → S} {Y : Ω → T} [MeasurableSingletonClass S] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (hX : Measurable X) (hY : Measurable Y) (f : T → U) (hf : Function.Injective f) (hfY : Measurable (f ∘ Y)) [FiniteRange X] [FiniteRange Y] : H[X | f ∘ Y ; μ] = H[X | Y ; μ]"}
+{"name":"ProbabilityTheory.prob_ge_exp_neg_entropy","declaration":"/-- If $X$ is an $S$-valued random variable, then there exists $s \\in S$ such that\n$P[X=s] \\geq \\exp(-H[X])$.  TODO: remove the probability measure hypothesis, which is unncessary here. -/\ntheorem ProbabilityTheory.prob_ge_exp_neg_entropy {Ω : Type uΩ} {S : Type uS} [mΩ : MeasurableSpace Ω] [Nonempty S] [MeasurableSpace S] [MeasurableSingletonClass S] (X : Ω → S) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (hX : Measurable X) [hX' : FiniteRange X] : ∃ s, ↑↑(MeasureTheory.Measure.map X μ) {s} ≥ ↑↑μ Set.univ * ↑(Real.toNNReal (Real.exp (-H[X ; μ])))"}
+{"name":"ProbabilityTheory.mutualInfo_eq_entropy_sub_condEntropy","declaration":"/-- $I[X : Y] = H[X] - H[X|Y]$. -/\ntheorem ProbabilityTheory.mutualInfo_eq_entropy_sub_condEntropy {Ω : Type uΩ} {S : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [Countable S] [Countable T] [Nonempty S] [Nonempty T] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSingletonClass S] [MeasurableSingletonClass T] {X : Ω → S} {Y : Ω → T} (hX : Measurable X) (hY : Measurable Y) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] [FiniteRange X] [FiniteRange Y] : I[X : Y ; μ] = H[X ; μ] - H[X | Y ; μ]"}
+{"name":"ProbabilityTheory.condMutualInfo_eq_sum","declaration":"theorem ProbabilityTheory.condMutualInfo_eq_sum {Ω : Type uΩ} {S : Type uS} {T : Type uT} {U : Type uU} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U] [MeasurableSingletonClass U] {X : Ω → S} {Y : Ω → T} {Z : Ω → U} {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsFiniteMeasure μ] (hZ : Measurable Z) [FiniteRange Z] : I[X : Y|Z;μ] =\n  Finset.sum (FiniteRange.toFinset Z) fun z =>\n    (↑↑μ (Z ⁻¹' {z})).toReal * I[X : Y ; ProbabilityTheory.cond μ (Z ⁻¹' {z})]"}
+{"name":"ProbabilityTheory.entropy_assoc","declaration":"/-- $H[(X, Y), Z] = H[X, (Y, Z)]$. -/\ntheorem ProbabilityTheory.entropy_assoc {Ω : Type uΩ} {S : Type uS} {T : Type uT} {U : Type uU} [mΩ : MeasurableSpace Ω] [Countable S] [Countable T] [Countable U] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U] [MeasurableSingletonClass S] [MeasurableSingletonClass T] [MeasurableSingletonClass U] {X : Ω → S} {Y : Ω → T} {Z : Ω → U} (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) (μ : MeasureTheory.Measure Ω) : H[⟨X, ⟨Y, Z⟩⟩ ; μ] = H[⟨⟨X, Y⟩, Z⟩ ; μ]"}
+{"name":"ProbabilityTheory.entropy_eq_sum_finset","declaration":"theorem ProbabilityTheory.entropy_eq_sum_finset {Ω : Type uΩ} {S : Type uS} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] {X : Ω → S} (hX : Measurable X) {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {A : Finset S} (hA : ↑↑(MeasureTheory.Measure.map X μ) (↑A)ᶜ = 0) : H[X ; μ] = Finset.sum A fun x => Real.negMulLog (↑↑(MeasureTheory.Measure.map X μ) {x}).toReal"}
+{"name":"ProbabilityTheory.entropy","declaration":"/-- Entropy of a random variable with values in a finite measurable space. -/\ndef ProbabilityTheory.entropy {Ω : Type uΩ} {S : Type uS} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] (X : Ω → S) (μ : autoParam (MeasureTheory.Measure Ω) _auto✝) : ℝ"}
+{"name":"ProbabilityTheory.entropy_comm","declaration":"/-- $H[X, Y] = H[Y, X]$. -/\ntheorem ProbabilityTheory.entropy_comm {Ω : Type uΩ} {S : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [Countable S] [Countable T] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSingletonClass S] [MeasurableSingletonClass T] {X : Ω → S} {Y : Ω → T} (hX : Measurable X) (hY : Measurable Y) (μ : MeasureTheory.Measure Ω) : H[⟨X, Y⟩ ; μ] = H[⟨Y, X⟩ ; μ]"}
+{"name":"ProbabilityTheory.entropy_pair_le_add","declaration":"/-- Subadditivity of entropy. -/\ntheorem ProbabilityTheory.entropy_pair_le_add {Ω : Type uΩ} {S : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSingletonClass S] [MeasurableSingletonClass T] {X : Ω → S} {Y : Ω → T} (hX : Measurable X) (hY : Measurable Y) (μ : MeasureTheory.Measure Ω) [FiniteRange X] [FiniteRange Y] : H[⟨X, Y⟩ ; μ] ≤ H[X ; μ] + H[Y ; μ]"}
+{"name":"ProbabilityTheory.entropy_cond_eq_sum","declaration":"/-- $H[X|Y=y] = \\sum_s P[X=s|Y=y] \\log \\frac{1}{P[X=s|Y=y]}$. -/\ntheorem ProbabilityTheory.entropy_cond_eq_sum {Ω : Type uΩ} {S : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] {X : Ω → S} {Y : Ω → T} (hX : Measurable X) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (y : T) : H[X | Y ← y ; μ] =\n  ∑' (x : S), Real.negMulLog (↑↑(MeasureTheory.Measure.map X (ProbabilityTheory.cond μ (Y ⁻¹' {y}))) {x}).toReal"}
+{"name":"ProbabilityTheory.condEntropy_comp_of_injective","declaration":"/-- A weaker version of the above lemma in which f is independent of Y. -/\ntheorem ProbabilityTheory.condEntropy_comp_of_injective {Ω : Type uΩ} {S : Type uS} {T : Type uT} {U : Type uU} [mΩ : MeasurableSpace Ω] [Countable S] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U] {X : Ω → S} {Y : Ω → T} [MeasurableSingletonClass S] [MeasurableSingletonClass U] (μ : MeasureTheory.Measure Ω) (hX : Measurable X) (f : S → U) (hf : Function.Injective f) : H[f ∘ X | Y ; μ] = H[X | Y ; μ]"}
+{"name":"ProbabilityTheory.condEntropy_def","declaration":"theorem ProbabilityTheory.condEntropy_def {Ω : Type uΩ} {S : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] [MeasurableSpace T] (X : Ω → S) (Y : Ω → T) (μ : MeasureTheory.Measure Ω) : H[X | Y ; μ] = ∫ (x : T), (fun y => H[X | Y ← y ; μ]) x ∂MeasureTheory.Measure.map Y μ"}
+{"name":"ProbabilityTheory.«termI[_:_]»","declaration":"/-- The mutual information $I[X : Y]$ of two random variables is defined to be $H[X] + H[Y] - H[X ; Y]$. -/\ndef ProbabilityTheory.«termI[_:_]»  : Lean.ParserDescr"}
+{"name":"ProbabilityTheory.condEntropy_prod_eq_sum","declaration":"theorem ProbabilityTheory.condEntropy_prod_eq_sum {Ω : Type uΩ} {S : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSingletonClass T] {T' : Type u_1} {X : Ω → S} {Y : Ω → T} {Z : Ω → T'} [MeasurableSpace T'] [MeasurableSingletonClass T'] (μ : MeasureTheory.Measure Ω) (hY : Measurable Y) (hZ : Measurable Z) [MeasureTheory.IsFiniteMeasure μ] [Fintype T] [Fintype T'] : H[X | ⟨Y, Z⟩ ; μ] =\n  Finset.sum Finset.univ fun z => (↑↑μ (Z ⁻¹' {z})).toReal * H[X | Y ; ProbabilityTheory.cond μ (Z ⁻¹' {z})]"}
+{"name":"ProbabilityTheory.«termH[_|_←_]».delab","declaration":"/-- Pretty printer defined by `notation3` command. -/\ndef ProbabilityTheory.«termH[_|_←_]».delab  : Lean.PrettyPrinter.Delaborator.Delab"}
+{"name":"ProbabilityTheory.«termH[_]».delab","declaration":"/-- Pretty printer defined by `notation3` command. -/\ndef ProbabilityTheory.«termH[_]».delab  : Lean.PrettyPrinter.Delaborator.Delab"}
+{"name":"ProbabilityTheory.entropy_cond_eq_sum_finiteRange","declaration":"theorem ProbabilityTheory.entropy_cond_eq_sum_finiteRange {Ω : Type uΩ} {S : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] [MeasurableSingletonClass S] {X : Ω → S} {Y : Ω → T} (hX : Measurable X) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (y : T) [FiniteRange X] : H[X | Y ← y ; μ] =\n  Finset.sum (FiniteRange.toFinset X) fun x =>\n    Real.negMulLog (↑↑(MeasureTheory.Measure.map X (ProbabilityTheory.cond μ (Y ⁻¹' {y}))) {x}).toReal"}
+{"name":"ProbabilityTheory.condMutualInfo_eq","declaration":"/-- $$ I[X : Y| Z] = H[X| Z] + H[Y| Z] - H[X, Y| Z].$$ -/\ntheorem ProbabilityTheory.condMutualInfo_eq {Ω : Type uΩ} {S : Type uS} {T : Type uT} {U : Type uU} [mΩ : MeasurableSpace Ω] [Countable S] [Countable T] [Countable U] [Nonempty S] [Nonempty T] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U] [MeasurableSingletonClass S] [MeasurableSingletonClass T] [MeasurableSingletonClass U] {X : Ω → S} {Y : Ω → T} {Z : Ω → U} (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] [FiniteRange Z] : I[X : Y|Z;μ] = H[X | Z ; μ] + H[Y | Z ; μ] - H[⟨X, Y⟩ | Z ; μ]"}
+{"name":"ProbabilityTheory.condEntropy_two_eq_kernel_entropy","declaration":"theorem ProbabilityTheory.condEntropy_two_eq_kernel_entropy {Ω : Type uΩ} {S : Type uS} {T : Type uT} {U : Type uU} [mΩ : MeasurableSpace Ω] [Countable S] [Countable T] [Countable U] [Nonempty S] [Nonempty T] [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U] [MeasurableSingletonClass S] [MeasurableSingletonClass T] [MeasurableSingletonClass U] {Z : Ω → U} {X : Ω → S} {Y : Ω → T} (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] [FiniteRange Y] [FiniteRange Z] : H[X | ⟨Y, Z⟩ ; μ] =\n  Hk[ProbabilityTheory.kernel.condKernel (ProbabilityTheory.condDistrib (fun a => (Y a, X a)) Z μ) ,\n    MeasureTheory.Measure.compProd (MeasureTheory.Measure.map Z μ)\n      (ProbabilityTheory.kernel.fst (ProbabilityTheory.condDistrib (fun a => (Y a, X a)) Z μ))]"}

PFR-declarations/PFR.ForMathlib.Entropy.Group.jsonl

@@ -0,0 +1,56 @@
+{"name":"ProbabilityTheory.max_condEntropy_sub_condMutualInfo_le_condEntropy_add","declaration":"/-- $$\\max(H[X | Z], H[Y | Z]) - I[X : Y | Z] \\leq H[X + Y | Z]$$-/\ntheorem ProbabilityTheory.max_condEntropy_sub_condMutualInfo_le_condEntropy_add {Ω : Type uΩ} {G : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [Countable G] [Countable T] [Nonempty G] [Nonempty T] [MeasurableSpace G] [MeasurableSpace T] [MeasurableSingletonClass G] [MeasurableSingletonClass T] [AddGroup G] {X : Ω → G} {μ : MeasureTheory.Measure Ω} [FiniteRange X] [MeasureTheory.IsProbabilityMeasure μ] {Y : Ω → G} [FiniteRange Y] {Z : Ω → T} [FiniteRange Z] (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) : max H[X | Z ; μ] H[Y | Z ; μ] - I[X : Y|Z;μ] ≤ H[X + Y | Z ; μ]"}
+{"name":"ProbabilityTheory.entropy_div_left","declaration":"/-- $H[Y / X, Y] = H[X, Y]$ -/\ntheorem ProbabilityTheory.entropy_div_left {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] {X : Ω → G} {Y : Ω → G} (hX : Measurable X) (hY : Measurable Y) (μ : MeasureTheory.Measure Ω) : H[⟨Y / X, Y⟩ ; μ] = H[⟨X, Y⟩ ; μ]"}
+{"name":"ProbabilityTheory.entropy_neg","declaration":"/-- If $X$ is $G$-valued, then $H[-X]=H[X]$.-/\ntheorem ProbabilityTheory.entropy_neg {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] {X : Ω → G} {μ : MeasureTheory.Measure Ω} (hX : Measurable X) : H[-X ; μ] = H[X ; μ]"}
+{"name":"ProbabilityTheory.max_condEntropy_sub_condMutualInfo_le_condEntropy_sub","declaration":"/-- $$\\max(H[X | Z], H[Y | Z]) - I[X : Y | Z] \\leq H[X - Y | Z]$$-/\ntheorem ProbabilityTheory.max_condEntropy_sub_condMutualInfo_le_condEntropy_sub {Ω : Type uΩ} {G : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [Countable G] [Countable T] [Nonempty G] [Nonempty T] [MeasurableSpace G] [MeasurableSpace T] [MeasurableSingletonClass G] [MeasurableSingletonClass T] [AddGroup G] {X : Ω → G} {μ : MeasureTheory.Measure Ω} [FiniteRange X] {Y : Ω → G} [FiniteRange Y] {Z : Ω → T} (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) [MeasureTheory.IsProbabilityMeasure μ] [FiniteRange Z] : max H[X | Z ; μ] H[Y | Z ; μ] - I[X : Y|Z;μ] ≤ H[X - Y | Z ; μ]"}
+{"name":"ProbabilityTheory.entropy_sub_left","declaration":"/-- $H[Y - X, Y] = H[X, Y]$-/\ntheorem ProbabilityTheory.entropy_sub_left {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] {X : Ω → G} {Y : Ω → G} (hX : Measurable X) (hY : Measurable Y) (μ : MeasureTheory.Measure Ω) : H[⟨Y - X, Y⟩ ; μ] = H[⟨X, Y⟩ ; μ]"}
+{"name":"ProbabilityTheory.entropy_sub_mutualInfo_le_entropy_add'","declaration":"/-- $$H[Y] - I[X : Y] \\leq H[X + Y]$$-/\ntheorem ProbabilityTheory.entropy_sub_mutualInfo_le_entropy_add' {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [Nonempty G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] {X : Ω → G} {μ : MeasureTheory.Measure Ω} [FiniteRange X] [MeasureTheory.IsProbabilityMeasure μ] {Y : Ω → G} [FiniteRange Y] (hX : Measurable X) (hY : Measurable Y) : H[Y ; μ] - I[X : Y ; μ] ≤ H[X + Y ; μ]"}
+{"name":"ProbabilityTheory.entropy_sub_mutualInfo_le_entropy_mul","declaration":"/-- $$H[X] - I[X : Y] \\leq H[X * Y]$$ -/\ntheorem ProbabilityTheory.entropy_sub_mutualInfo_le_entropy_mul {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [Nonempty G] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] {X : Ω → G} {μ : MeasureTheory.Measure Ω} [FiniteRange X] [MeasureTheory.IsProbabilityMeasure μ] {Y : Ω → G} [FiniteRange Y] (hX : Measurable X) (hY : Measurable Y) : H[X ; μ] - I[X : Y ; μ] ≤ H[X * Y ; μ]"}
+{"name":"ProbabilityTheory.entropy_sub_mutualInfo_le_entropy_add","declaration":"/-- $$H[X] - I[X : Y] \\leq H[X + Y]$$-/\ntheorem ProbabilityTheory.entropy_sub_mutualInfo_le_entropy_add {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [Nonempty G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] {X : Ω → G} {μ : MeasureTheory.Measure Ω} [FiniteRange X] [MeasureTheory.IsProbabilityMeasure μ] {Y : Ω → G} [FiniteRange Y] (hX : Measurable X) (hY : Measurable Y) : H[X ; μ] - I[X : Y ; μ] ≤ H[X + Y ; μ]"}
+{"name":"ProbabilityTheory.condEntropy_sub_left","declaration":"/-- $$H[Y - X | Y] = H[X | Y]$$-/\ntheorem ProbabilityTheory.condEntropy_sub_left {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] {X : Ω → G} {Y : Ω → G} {μ : MeasureTheory.Measure Ω} [FiniteRange Y] [MeasureTheory.IsFiniteMeasure μ] (hX : Measurable X) (hY : Measurable Y) : H[Y - X | Y ; μ] = H[X | Y ; μ]"}
+{"name":"ProbabilityTheory.condEntropy_add_right","declaration":"/-- $$H[X + Y | Y] = H[X | Y]$$-/\ntheorem ProbabilityTheory.condEntropy_add_right {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] {X : Ω → G} {Y : Ω → G} {μ : MeasureTheory.Measure Ω} [FiniteRange Y] [MeasureTheory.IsFiniteMeasure μ] (hX : Measurable X) (hY : Measurable Y) : H[X + Y | Y ; μ] = H[X | Y ; μ]"}
+{"name":"ProbabilityTheory.max_entropy_sub_mutualInfo_le_entropy_mul","declaration":"/-- $$\\max(H[X], H[Y]) - I[X : Y] \\leq H[X * Y]$$ -/\ntheorem ProbabilityTheory.max_entropy_sub_mutualInfo_le_entropy_mul {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [Nonempty G] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] {X : Ω → G} {μ : MeasureTheory.Measure Ω} [FiniteRange X] [MeasureTheory.IsProbabilityMeasure μ] {Y : Ω → G} [FiniteRange Y] (hX : Measurable X) (hY : Measurable Y) : max H[X ; μ] H[Y ; μ] - I[X : Y ; μ] ≤ H[X * Y ; μ]"}
+{"name":"ProbabilityTheory.max_entropy_sub_mutualInfo_le_entropy_div","declaration":"/-- $$\\max(H[X], H[Y]) - I[X : Y] \\leq H[X / Y]$$ -/\ntheorem ProbabilityTheory.max_entropy_sub_mutualInfo_le_entropy_div {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [Nonempty G] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] {X : Ω → G} {μ : MeasureTheory.Measure Ω} [FiniteRange X] [MeasureTheory.IsProbabilityMeasure μ] {Y : Ω → G} [FiniteRange Y] (hX : Measurable X) (hY : Measurable Y) : max H[X ; μ] H[Y ; μ] - I[X : Y ; μ] ≤ H[X / Y ; μ]"}
+{"name":"ProbabilityTheory.entropy_mul_left","declaration":"/-- $H[Y * X, Y] = H[X, Y]$ -/\ntheorem ProbabilityTheory.entropy_mul_left {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] {X : Ω → G} {Y : Ω → G} (hX : Measurable X) (hY : Measurable Y) (μ : MeasureTheory.Measure Ω) : H[⟨Y * X, Y⟩ ; μ] = H[⟨X, Y⟩ ; μ]"}
+{"name":"ProbabilityTheory.entropy_neg_left","declaration":"/-- $H[-X, Y] = H[X, Y]$-/\ntheorem ProbabilityTheory.entropy_neg_left {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] {X : Ω → G} {Y : Ω → G} (hX : Measurable X) (hY : Measurable Y) (μ : MeasureTheory.Measure Ω) : H[⟨-X, Y⟩ ; μ] = H[⟨X, Y⟩ ; μ]"}
+{"name":"ProbabilityTheory.condEntropy_add_left","declaration":"/-- $$H[Y + X | Y] = H[X | Y]$$-/\ntheorem ProbabilityTheory.condEntropy_add_left {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] {X : Ω → G} {Y : Ω → G} {μ : MeasureTheory.Measure Ω} [FiniteRange Y] [MeasureTheory.IsFiniteMeasure μ] (hX : Measurable X) (hY : Measurable Y) : H[Y + X | Y ; μ] = H[X | Y ; μ]"}
+{"name":"ProbabilityTheory.entropy_sub_mutualInfo_le_entropy_sub'","declaration":"/-- $$H[Y] - I[X : Y] \\leq H[X - Y]$$-/\ntheorem ProbabilityTheory.entropy_sub_mutualInfo_le_entropy_sub' {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [Nonempty G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] {X : Ω → G} {μ : MeasureTheory.Measure Ω} [FiniteRange X] [MeasureTheory.IsProbabilityMeasure μ] {Y : Ω → G} [FiniteRange Y] (hX : Measurable X) (hY : Measurable Y) : H[Y ; μ] - I[X : Y ; μ] ≤ H[X - Y ; μ]"}
+{"name":"ProbabilityTheory.condEntropy_mul_right","declaration":"/-- $$H[X * Y | Y] = H[X | Y]$$ -/\ntheorem ProbabilityTheory.condEntropy_mul_right {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] {X : Ω → G} {Y : Ω → G} {μ : MeasureTheory.Measure Ω} [FiniteRange Y] [MeasureTheory.IsFiniteMeasure μ] (hX : Measurable X) (hY : Measurable Y) : H[X * Y | Y ; μ] = H[X | Y ; μ]"}
+{"name":"ProbabilityTheory.entropy_sub_right'","declaration":"/-- $H[X, Y - X] = H[X, Y]$-/\ntheorem ProbabilityTheory.entropy_sub_right' {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] {X : Ω → G} {Y : Ω → G} (hX : Measurable X) (hY : Measurable Y) (μ : MeasureTheory.Measure Ω) : H[⟨X, Y - X⟩ ; μ] = H[⟨X, Y⟩ ; μ]"}
+{"name":"ProbabilityTheory.entropy_div_left'","declaration":"/-- $H[X / Y, Y] = H[X, Y]$ -/\ntheorem ProbabilityTheory.entropy_div_left' {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] {X : Ω → G} {Y : Ω → G} (hX : Measurable X) (hY : Measurable Y) (μ : MeasureTheory.Measure Ω) : H[⟨X / Y, Y⟩ ; μ] = H[⟨X, Y⟩ ; μ]"}
+{"name":"ProbabilityTheory.max_entropy_sub_mutualInfo_le_entropy_add","declaration":"/-- $$\\max(H[X], H[Y]) - I[X : Y] \\leq H[X + Y]$$-/\ntheorem ProbabilityTheory.max_entropy_sub_mutualInfo_le_entropy_add {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [Nonempty G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] {X : Ω → G} {μ : MeasureTheory.Measure Ω} [FiniteRange X] [MeasureTheory.IsProbabilityMeasure μ] {Y : Ω → G} [FiniteRange Y] (hX : Measurable X) (hY : Measurable Y) : max H[X ; μ] H[Y ; μ] - I[X : Y ; μ] ≤ H[X + Y ; μ]"}
+{"name":"ProbabilityTheory.condEntropy_mul_left","declaration":"/-- $$H[Y * X | Y] = H[X | Y]$$ -/\ntheorem ProbabilityTheory.condEntropy_mul_left {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] {X : Ω → G} {Y : Ω → G} {μ : MeasureTheory.Measure Ω} [FiniteRange Y] [MeasureTheory.IsFiniteMeasure μ] (hX : Measurable X) (hY : Measurable Y) : H[Y * X | Y ; μ] = H[X | Y ; μ]"}
+{"name":"ProbabilityTheory.max_entropy_le_entropy_mul","declaration":"/-- If $X, Y$ are independent, then $$\\max(H[X], H[Y]) \\leq H[X * Y]$$. -/\ntheorem ProbabilityTheory.max_entropy_le_entropy_mul {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [Nonempty G] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] {X : Ω → G} {μ : MeasureTheory.Measure Ω} [FiniteRange X] [MeasureTheory.IsProbabilityMeasure μ] {Y : Ω → G} [FiniteRange Y] (hX : Measurable X) (hY : Measurable Y) (h : ProbabilityTheory.IndepFun X Y μ) : max H[X ; μ] H[Y ; μ] ≤ H[X * Y ; μ]"}
+{"name":"ProbabilityTheory.max_entropy_le_entropy_add","declaration":"/-- If $X, Y$ are independent, then $$\\max(H[X], H[Y]) \\leq H[X + Y]$$-/\ntheorem ProbabilityTheory.max_entropy_le_entropy_add {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [Nonempty G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] {X : Ω → G} {μ : MeasureTheory.Measure Ω} [FiniteRange X] [MeasureTheory.IsProbabilityMeasure μ] {Y : Ω → G} [FiniteRange Y] (hX : Measurable X) (hY : Measurable Y) (h : ProbabilityTheory.IndepFun X Y μ) : max H[X ; μ] H[Y ; μ] ≤ H[X + Y ; μ]"}
+{"name":"ProbabilityTheory.entropy_add_const","declaration":"theorem ProbabilityTheory.entropy_add_const {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] {X : Ω → G} {μ : MeasureTheory.Measure Ω} (hX : Measurable X) (c : G) : H[X + fun x => c ; μ] = H[X ; μ]"}
+{"name":"ProbabilityTheory.entropy_sub_mutualInfo_le_entropy_div'","declaration":"/-- $$H[Y] - I[X : Y] \\leq H[X / Y]$$ -/\ntheorem ProbabilityTheory.entropy_sub_mutualInfo_le_entropy_div' {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [Nonempty G] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] {X : Ω → G} {μ : MeasureTheory.Measure Ω} [FiniteRange X] [MeasureTheory.IsProbabilityMeasure μ] {Y : Ω → G} [FiniteRange Y] (hX : Measurable X) (hY : Measurable Y) : H[Y ; μ] - I[X : Y ; μ] ≤ H[X / Y ; μ]"}
+{"name":"ProbabilityTheory.entropy_sub_mutualInfo_le_entropy_div","declaration":"/-- $$H[X] - I[X : Y] \\leq H[X / Y]$$ -/\ntheorem ProbabilityTheory.entropy_sub_mutualInfo_le_entropy_div {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [Nonempty G] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] {X : Ω → G} {μ : MeasureTheory.Measure Ω} [FiniteRange X] [MeasureTheory.IsProbabilityMeasure μ] {Y : Ω → G} [FiniteRange Y] (hX : Measurable X) (hY : Measurable Y) : H[X ; μ] - I[X : Y ; μ] ≤ H[X / Y ; μ]"}
+{"name":"ProbabilityTheory.mutualInfo_add_right","declaration":"/-- $I[X : X + Y] = H[X + Y] - H[Y]$ iff $X, Y$ are independent.-/\ntheorem ProbabilityTheory.mutualInfo_add_right {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] {X : Ω → G} {Y : Ω → G} [FiniteRange X] [FiniteRange Y] (hX : Measurable X) (hY : Measurable Y) {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] (h : ProbabilityTheory.IndepFun X Y μ) : I[X : X + Y ; μ] = H[X + Y ; μ] - H[Y ; μ]"}
+{"name":"ProbabilityTheory.entropy_mul_right","declaration":"/-- $H[X, X * Y] = H[X, Y]$ -/\ntheorem ProbabilityTheory.entropy_mul_right {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] {X : Ω → G} {Y : Ω → G} (hX : Measurable X) (hY : Measurable Y) (μ : MeasureTheory.Measure Ω) : H[⟨X, X * Y⟩ ; μ] = H[⟨X, Y⟩ ; μ]"}
+{"name":"ProbabilityTheory.entropy_add_left","declaration":"/-- $H[Y + X, Y] = H[X, Y]$-/\ntheorem ProbabilityTheory.entropy_add_left {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] {X : Ω → G} {Y : Ω → G} (hX : Measurable X) (hY : Measurable Y) (μ : MeasureTheory.Measure Ω) : H[⟨Y + X, Y⟩ ; μ] = H[⟨X, Y⟩ ; μ]"}
+{"name":"ProbabilityTheory.mutualInfo_mul_right","declaration":"/-- $I[X : X * Y] = H[X * Y] - H[Y]$ iff $X, Y$ are independent. -/\ntheorem ProbabilityTheory.mutualInfo_mul_right {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] {X : Ω → G} {Y : Ω → G} [FiniteRange X] [FiniteRange Y] (hX : Measurable X) (hY : Measurable Y) {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] (h : ProbabilityTheory.IndepFun X Y μ) : I[X : X * Y ; μ] = H[X * Y ; μ] - H[Y ; μ]"}
+{"name":"ProbabilityTheory.condEntropy_sub_right","declaration":"/-- $$H[X - Y | Y] = H[X | Y]$$-/\ntheorem ProbabilityTheory.condEntropy_sub_right {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] {X : Ω → G} {Y : Ω → G} {μ : MeasureTheory.Measure Ω} [FiniteRange Y] [MeasureTheory.IsFiniteMeasure μ] (hX : Measurable X) (hY : Measurable Y) : H[X - Y | Y ; μ] = H[X | Y ; μ]"}
+{"name":"ProbabilityTheory.max_entropy_le_entropy_sub","declaration":"/-- If $X, Y$ are independent, then $$\\max(H[X], H[Y]) \\leq H[X - Y]$$.-/\ntheorem ProbabilityTheory.max_entropy_le_entropy_sub {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [Nonempty G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] {X : Ω → G} {μ : MeasureTheory.Measure Ω} [FiniteRange X] [MeasureTheory.IsProbabilityMeasure μ] {Y : Ω → G} [FiniteRange Y] (hX : Measurable X) (hY : Measurable Y) (h : ProbabilityTheory.IndepFun X Y μ) : max H[X ; μ] H[Y ; μ] ≤ H[X - Y ; μ]"}
+{"name":"ProbabilityTheory.entropy_sub_comm","declaration":"/-- $$H[X - Y] = H[Y - X]$$-/\ntheorem ProbabilityTheory.entropy_sub_comm {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] {X : Ω → G} {μ : MeasureTheory.Measure Ω} {Y : Ω → G} (hX : Measurable X) (hY : Measurable Y) : H[X - Y ; μ] = H[Y - X ; μ]"}
+{"name":"ProbabilityTheory.max_condEntropy_sub_condMutualInfo_le_condEntropy_div","declaration":"/-- $$\\max(H[X | Z], H[Y | Z]) - I[X : Y | Z] \\leq H[X / Y | Z]$$ -/\ntheorem ProbabilityTheory.max_condEntropy_sub_condMutualInfo_le_condEntropy_div {Ω : Type uΩ} {G : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [Countable G] [Countable T] [Nonempty G] [Nonempty T] [MeasurableSpace G] [MeasurableSpace T] [MeasurableSingletonClass G] [MeasurableSingletonClass T] [Group G] {X : Ω → G} {μ : MeasureTheory.Measure Ω} [FiniteRange X] {Y : Ω → G} [FiniteRange Y] {Z : Ω → T} (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) [MeasureTheory.IsProbabilityMeasure μ] [FiniteRange Z] : max H[X | Z ; μ] H[Y | Z ; μ] - I[X : Y|Z;μ] ≤ H[X / Y | Z ; μ]"}
+{"name":"ProbabilityTheory.entropy_add_right","declaration":"/-- $H[X, X + Y] = H[X, Y]$-/\ntheorem ProbabilityTheory.entropy_add_right {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] {X : Ω → G} {Y : Ω → G} (hX : Measurable X) (hY : Measurable Y) (μ : MeasureTheory.Measure Ω) : H[⟨X, X + Y⟩ ; μ] = H[⟨X, Y⟩ ; μ]"}
+{"name":"ProbabilityTheory.entropy_mul_const","declaration":"theorem ProbabilityTheory.entropy_mul_const {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] {X : Ω → G} {μ : MeasureTheory.Measure Ω} (hX : Measurable X) (c : G) : H[X * fun x => c ; μ] = H[X ; μ]"}
+{"name":"ProbabilityTheory.max_condEntropy_sub_condMutualInfo_le_condEntropy_mul","declaration":"/-- $$\\max(H[X | Z], H[Y | Z]) - I[X : Y | Z] \\leq H[X * Y | Z]$$ -/\ntheorem ProbabilityTheory.max_condEntropy_sub_condMutualInfo_le_condEntropy_mul {Ω : Type uΩ} {G : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] [Countable G] [Countable T] [Nonempty G] [Nonempty T] [MeasurableSpace G] [MeasurableSpace T] [MeasurableSingletonClass G] [MeasurableSingletonClass T] [Group G] {X : Ω → G} {μ : MeasureTheory.Measure Ω} [FiniteRange X] [MeasureTheory.IsProbabilityMeasure μ] {Y : Ω → G} [FiniteRange Y] {Z : Ω → T} [FiniteRange Z] (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) : max H[X | Z ; μ] H[Y | Z ; μ] - I[X : Y|Z;μ] ≤ H[X * Y | Z ; μ]"}
+{"name":"ProbabilityTheory.max_entropy_sub_mutualInfo_le_entropy_sub","declaration":"/-- $$\\max(H[X], H[Y]) - I[X : Y] \\leq H[X - Y]$$-/\ntheorem ProbabilityTheory.max_entropy_sub_mutualInfo_le_entropy_sub {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [Nonempty G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] {X : Ω → G} {μ : MeasureTheory.Measure Ω} [FiniteRange X] [MeasureTheory.IsProbabilityMeasure μ] {Y : Ω → G} [FiniteRange Y] (hX : Measurable X) (hY : Measurable Y) : max H[X ; μ] H[Y ; μ] - I[X : Y ; μ] ≤ H[X - Y ; μ]"}
+{"name":"ProbabilityTheory.entropy_div_right","declaration":"/-- $H[X, X / Y] = H[X, Y]$ -/\ntheorem ProbabilityTheory.entropy_div_right {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] {X : Ω → G} {Y : Ω → G} (hX : Measurable X) (hY : Measurable Y) (μ : MeasureTheory.Measure Ω) : H[⟨X, X / Y⟩ ; μ] = H[⟨X, Y⟩ ; μ]"}
+{"name":"ProbabilityTheory.entropy_inv_left","declaration":"/-- $H[X⁻¹, Y] = H[X, Y]$ -/\ntheorem ProbabilityTheory.entropy_inv_left {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] {X : Ω → G} {Y : Ω → G} (hX : Measurable X) (hY : Measurable Y) (μ : MeasureTheory.Measure Ω) : H[⟨X⁻¹, Y⟩ ; μ] = H[⟨X, Y⟩ ; μ]"}
+{"name":"ProbabilityTheory.entropy_div_comm","declaration":"/-- $$H[X / Y] = H[Y / X]$$ -/\ntheorem ProbabilityTheory.entropy_div_comm {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] {X : Ω → G} {μ : MeasureTheory.Measure Ω} {Y : Ω → G} (hX : Measurable X) (hY : Measurable Y) : H[X / Y ; μ] = H[Y / X ; μ]"}
+{"name":"ProbabilityTheory.entropy_neg_right","declaration":"/-- $H[X, -Y] = H[X, Y]$-/\ntheorem ProbabilityTheory.entropy_neg_right {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] {X : Ω → G} {Y : Ω → G} (hX : Measurable X) (hY : Measurable Y) (μ : MeasureTheory.Measure Ω) : H[⟨X, -Y⟩ ; μ] = H[⟨X, Y⟩ ; μ]"}
+{"name":"ProbabilityTheory.entropy_add_right'","declaration":"/-- $H[X, Y + X] = H[X, Y]$-/\ntheorem ProbabilityTheory.entropy_add_right' {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] {X : Ω → G} {Y : Ω → G} (hX : Measurable X) (hY : Measurable Y) (μ : MeasureTheory.Measure Ω) : H[⟨X, Y + X⟩ ; μ] = H[⟨X, Y⟩ ; μ]"}
+{"name":"ProbabilityTheory.entropy_mul_left'","declaration":"/-- $H[X * Y, Y] = H[X, Y]$ -/\ntheorem ProbabilityTheory.entropy_mul_left' {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] {X : Ω → G} {Y : Ω → G} (hX : Measurable X) (hY : Measurable Y) (μ : MeasureTheory.Measure Ω) : H[⟨X * Y, Y⟩ ; μ] = H[⟨X, Y⟩ ; μ]"}
+{"name":"ProbabilityTheory.max_entropy_le_entropy_div","declaration":"/-- If $X, Y$ are independent, then $$\\max(H[X], H[Y]) \\leq H[X / Y]$$. -/\ntheorem ProbabilityTheory.max_entropy_le_entropy_div {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [Nonempty G] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] {X : Ω → G} {μ : MeasureTheory.Measure Ω} [FiniteRange X] [MeasureTheory.IsProbabilityMeasure μ] {Y : Ω → G} [FiniteRange Y] (hX : Measurable X) (hY : Measurable Y) (h : ProbabilityTheory.IndepFun X Y μ) : max H[X ; μ] H[Y ; μ] ≤ H[X / Y ; μ]"}
+{"name":"ProbabilityTheory.condEntropy_div_left","declaration":"/-- $$H[Y / X | Y] = H[X | Y]$$ -/\ntheorem ProbabilityTheory.condEntropy_div_left {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] {X : Ω → G} {Y : Ω → G} {μ : MeasureTheory.Measure Ω} [FiniteRange Y] [MeasureTheory.IsFiniteMeasure μ] (hX : Measurable X) (hY : Measurable Y) : H[Y / X | Y ; μ] = H[X | Y ; μ]"}
+{"name":"ProbabilityTheory.entropy_add_left'","declaration":"/-- $H[X + Y, Y] = H[X, Y]$-/\ntheorem ProbabilityTheory.entropy_add_left' {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] {X : Ω → G} {Y : Ω → G} (hX : Measurable X) (hY : Measurable Y) (μ : MeasureTheory.Measure Ω) : H[⟨X + Y, Y⟩ ; μ] = H[⟨X, Y⟩ ; μ]"}
+{"name":"ProbabilityTheory.entropy_inv_right","declaration":"/-- $H[X, Y⁻¹] = H[X, Y]$ -/\ntheorem ProbabilityTheory.entropy_inv_right {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] {X : Ω → G} {Y : Ω → G} (hX : Measurable X) (hY : Measurable Y) (μ : MeasureTheory.Measure Ω) : H[⟨X, Y⁻¹⟩ ; μ] = H[⟨X, Y⟩ ; μ]"}
+{"name":"ProbabilityTheory.entropy_sub_right","declaration":"/-- $H[X, X - Y] = H[X, Y]$-/\ntheorem ProbabilityTheory.entropy_sub_right {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] {X : Ω → G} {Y : Ω → G} (hX : Measurable X) (hY : Measurable Y) (μ : MeasureTheory.Measure Ω) : H[⟨X, X - Y⟩ ; μ] = H[⟨X, Y⟩ ; μ]"}
+{"name":"ProbabilityTheory.entropy_sub_mutualInfo_le_entropy_sub","declaration":"/-- $$H[X] - I[X : Y] \\leq H[X - Y]$$-/\ntheorem ProbabilityTheory.entropy_sub_mutualInfo_le_entropy_sub {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [Nonempty G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] {X : Ω → G} {μ : MeasureTheory.Measure Ω} [FiniteRange X] [MeasureTheory.IsProbabilityMeasure μ] {Y : Ω → G} [FiniteRange Y] (hX : Measurable X) (hY : Measurable Y) : H[X ; μ] - I[X : Y ; μ] ≤ H[X - Y ; μ]"}
+{"name":"ProbabilityTheory.entropy_div_right'","declaration":"/-- $H[X, Y / X] = H[X, Y]$ -/\ntheorem ProbabilityTheory.entropy_div_right' {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] {X : Ω → G} {Y : Ω → G} (hX : Measurable X) (hY : Measurable Y) (μ : MeasureTheory.Measure Ω) : H[⟨X, Y / X⟩ ; μ] = H[⟨X, Y⟩ ; μ]"}
+{"name":"ProbabilityTheory.entropy_inv","declaration":"/-- If $X$ is $G$-valued, then $H[X⁻¹]=H[X]$. -/\ntheorem ProbabilityTheory.entropy_inv {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] {X : Ω → G} {μ : MeasureTheory.Measure Ω} (hX : Measurable X) : H[X⁻¹ ; μ] = H[X ; μ]"}
+{"name":"ProbabilityTheory.entropy_mul_right'","declaration":"/-- $H[X, Y * X] = H[X, Y]$ -/\ntheorem ProbabilityTheory.entropy_mul_right' {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] {X : Ω → G} {Y : Ω → G} (hX : Measurable X) (hY : Measurable Y) (μ : MeasureTheory.Measure Ω) : H[⟨X, Y * X⟩ ; μ] = H[⟨X, Y⟩ ; μ]"}
+{"name":"ProbabilityTheory.entropy_sub_left'","declaration":"/-- $H[X - Y, Y] = H[X, Y]$-/\ntheorem ProbabilityTheory.entropy_sub_left' {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] {X : Ω → G} {Y : Ω → G} (hX : Measurable X) (hY : Measurable Y) (μ : MeasureTheory.Measure Ω) : H[⟨X - Y, Y⟩ ; μ] = H[⟨X, Y⟩ ; μ]"}
+{"name":"ProbabilityTheory.condEntropy_div_right","declaration":"/-- $$H[X / Y | Y] = H[X | Y]$$ -/\ntheorem ProbabilityTheory.condEntropy_div_right {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] {X : Ω → G} {Y : Ω → G} {μ : MeasureTheory.Measure Ω} [FiniteRange Y] [MeasureTheory.IsFiniteMeasure μ] (hX : Measurable X) (hY : Measurable Y) : H[X / Y | Y ; μ] = H[X | Y ; μ]"}
+{"name":"ProbabilityTheory.entropy_sub_mutualInfo_le_entropy_mul'","declaration":"/-- $$H[Y] - I[X : Y] \\leq H[X * Y]$$ -/\ntheorem ProbabilityTheory.entropy_sub_mutualInfo_le_entropy_mul' {Ω : Type uΩ} {G : Type uS} [mΩ : MeasurableSpace Ω] [Countable G] [Nonempty G] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] {X : Ω → G} {μ : MeasureTheory.Measure Ω} [FiniteRange X] [MeasureTheory.IsProbabilityMeasure μ] {Y : Ω → G} [FiniteRange Y] (hX : Measurable X) (hY : Measurable Y) : H[Y ; μ] - I[X : Y ; μ] ≤ H[X * Y ; μ]"}

PFR-declarations/PFR.ForMathlib.Entropy.Kernel.Basic.jsonl

@@ -0,0 +1,39 @@
+{"name":"ProbabilityTheory.kernel.entropy_const","declaration":"theorem ProbabilityTheory.kernel.entropy_const {S : Type u_2} {T : Type u_3} [MeasurableSpace S] [MeasurableSpace T] (ν : MeasureTheory.Measure S) (μ : MeasureTheory.Measure T) : Hk[ProbabilityTheory.kernel.const T ν , μ] = (↑↑μ Set.univ).toReal * Hm[ν]"}
+{"name":"ProbabilityTheory.kernel.entropy_compProd","declaration":"theorem ProbabilityTheory.kernel.entropy_compProd {S : Type u_2} {T : Type u_3} {U : Type u_4} [Countable S] [Nonempty S] [MeasurableSpace S] [MeasurableSingletonClass S] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] [Nonempty U] [MeasurableSpace U] [MeasurableSingletonClass U] {μ : MeasureTheory.Measure T} [MeasureTheory.IsFiniteMeasure μ] {κ : ↥(ProbabilityTheory.kernel T S)} [ProbabilityTheory.IsMarkovKernel κ] {η : ↥(ProbabilityTheory.kernel (T × S) U)} [ProbabilityTheory.IsMarkovKernel η] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) (hη : ProbabilityTheory.kernel.AEFiniteKernelSupport η (MeasureTheory.Measure.compProd μ κ)) : Hk[ProbabilityTheory.kernel.compProd κ η , μ] = Hk[κ , μ] + Hk[η , MeasureTheory.Measure.compProd μ κ]"}
+{"name":"ProbabilityTheory.kernel.entropy_prodMkRight","declaration":"theorem ProbabilityTheory.kernel.entropy_prodMkRight {S : Type u_2} {T : Type u_3} {U : Type u_4} [Countable S] [Nonempty S] [MeasurableSpace S] [MeasurableSingletonClass S] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] [MeasurableSpace U] {κ : ↥(ProbabilityTheory.kernel T S)} {η : ↥(ProbabilityTheory.kernel T U)} [ProbabilityTheory.IsMarkovKernel κ] {μ : MeasureTheory.Measure T} [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : Hk[ProbabilityTheory.kernel.prodMkRight S η , MeasureTheory.Measure.compProd μ κ] = Hk[η , μ]"}
+{"name":"ProbabilityTheory.kernel.entropy_snd_le","declaration":"theorem ProbabilityTheory.kernel.entropy_snd_le {S : Type u_2} {T : Type u_3} {U : Type u_4} [Countable S] [Nonempty S] [MeasurableSpace S] [MeasurableSingletonClass S] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] [Countable U] [Nonempty U] [MeasurableSpace U] [MeasurableSingletonClass U] {κ : ↥(ProbabilityTheory.kernel T (S × U))} [ProbabilityTheory.IsMarkovKernel κ] {μ : MeasureTheory.Measure T} [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : Hk[ProbabilityTheory.kernel.snd κ , μ] ≤ Hk[κ , μ]"}
+{"name":"ProbabilityTheory.kernel.aefiniteKernelSupport_condDistrib","declaration":"theorem ProbabilityTheory.kernel.aefiniteKernelSupport_condDistrib {Ω : Type u_1} {S : Type u_2} {T : Type u_3} [mΩ : MeasurableSpace Ω] [Countable S] [Nonempty S] [MeasurableSpace S] [MeasurableSingletonClass S] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] (X : Ω → S) (Y : Ω → T) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] (hX : Measurable X) (hY : Measurable Y) [FiniteRange X] [FiniteRange Y] : ProbabilityTheory.kernel.AEFiniteKernelSupport (ProbabilityTheory.condDistrib X Y μ) (MeasureTheory.Measure.map Y μ)"}
+{"name":"ProbabilityTheory.kernel.entropy_nonneg","declaration":"theorem ProbabilityTheory.kernel.entropy_nonneg {S : Type u_2} {T : Type u_3} [MeasurableSpace S] [MeasurableSpace T] (κ : ↥(ProbabilityTheory.kernel T S)) (μ : MeasureTheory.Measure T) : 0 ≤ Hk[κ , μ]"}
+{"name":"ProbabilityTheory.kernel.entropy_prodMkLeft_unit","declaration":"theorem ProbabilityTheory.kernel.entropy_prodMkLeft_unit {S : Type u_2} {T : Type u_3} [MeasurableSpace S] [MeasurableSpace T] [MeasurableSingletonClass T] (κ : ↥(ProbabilityTheory.kernel T S)) {μ : MeasureTheory.Measure T} [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] : Hk[ProbabilityTheory.kernel.prodMkLeft Unit κ , MeasureTheory.Measure.map (Prod.mk ()) μ] = Hk[κ , μ]"}
+{"name":"ProbabilityTheory.kernel.chain_rule'","declaration":"theorem ProbabilityTheory.kernel.chain_rule' {S : Type u_2} {T : Type u_3} {U : Type u_4} [Countable S] [Nonempty S] [MeasurableSpace S] [MeasurableSingletonClass S] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] [Countable U] [Nonempty U] [MeasurableSpace U] [MeasurableSingletonClass U] {κ : ↥(ProbabilityTheory.kernel T (S × U))} [ProbabilityTheory.IsMarkovKernel κ] {μ : MeasureTheory.Measure T} [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : Hk[κ , μ] =\n  Hk[ProbabilityTheory.kernel.snd κ , μ] +\n    Hk[ProbabilityTheory.kernel.condKernel (ProbabilityTheory.kernel.swapRight κ) ,\n      MeasureTheory.Measure.compProd μ (ProbabilityTheory.kernel.snd κ)]"}
+{"name":"ProbabilityTheory.kernel.finiteSupport_of_compProd'","declaration":"/-- Composing a finitely supported measure with a finitely supported kernel gives a finitely supported kernel. -/\ntheorem ProbabilityTheory.kernel.finiteSupport_of_compProd' {S : Type u_2} {T : Type u_3} [Countable S] [MeasurableSpace S] [MeasurableSingletonClass S] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] {μ : MeasureTheory.Measure T} [MeasureTheory.IsFiniteMeasure μ] {κ : ↥(ProbabilityTheory.kernel T S)} [ProbabilityTheory.IsMarkovKernel κ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.FiniteKernelSupport κ) : ProbabilityTheory.FiniteSupport (MeasureTheory.Measure.compProd μ κ)"}
+{"name":"ProbabilityTheory.kernel.entropy_comap","declaration":"theorem ProbabilityTheory.kernel.entropy_comap {S : Type u_2} {T : Type u_3} [MeasurableSpace S] [MeasurableSpace T] [MeasurableSingletonClass T] {T' : Type u_5} [MeasurableSpace T'] [MeasurableSingletonClass T'] (κ : ↥(ProbabilityTheory.kernel T S)) (μ : MeasureTheory.Measure T) (f : T' → T) (hf : MeasurableEmbedding f) (hf_range : Set.range f =ᶠ[MeasureTheory.Measure.ae μ] Set.univ) [MeasureTheory.IsFiniteMeasure μ] [MeasureTheory.IsFiniteMeasure (MeasureTheory.Measure.comap f μ)] (hfμ : ProbabilityTheory.FiniteSupport (MeasureTheory.Measure.comap f μ)) : Hk[ProbabilityTheory.kernel.comap κ f ⋯ , MeasureTheory.Measure.comap f μ] = Hk[κ , μ]"}
+{"name":"ProbabilityTheory.kernel.«termHk[_,_]».delab","declaration":"/-- Pretty printer defined by `notation3` command. -/\ndef ProbabilityTheory.kernel.«termHk[_,_]».delab  : Lean.PrettyPrinter.Delaborator.Delab"}
+{"name":"ProbabilityTheory.kernel.entropy_comap_swap","declaration":"theorem ProbabilityTheory.kernel.entropy_comap_swap {S : Type u_2} {T : Type u_3} [MeasurableSpace S] [MeasurableSpace T] [MeasurableSingletonClass T] {T' : Type u_5} [MeasurableSpace T'] [MeasurableSingletonClass T'] [Nonempty T'] (κ : ↥(ProbabilityTheory.kernel (T' × T) S)) {μ : MeasureTheory.Measure (T' × T)} [MeasureTheory.IsFiniteMeasure μ] [ProbabilityTheory.FiniteSupport μ] : Hk[ProbabilityTheory.kernel.comap κ Prod.swap ⋯ , MeasureTheory.Measure.comap Prod.swap μ] = Hk[κ , μ]"}
+{"name":"ProbabilityTheory.kernel.chain_rule","declaration":"theorem ProbabilityTheory.kernel.chain_rule {S : Type u_2} {T : Type u_3} {U : Type u_4} [Countable S] [Nonempty S] [MeasurableSpace S] [MeasurableSingletonClass S] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] [Countable U] [MeasurableSpace U] [MeasurableSingletonClass U] {κ : ↥(ProbabilityTheory.kernel T (S × U))} [ProbabilityTheory.IsMarkovKernel κ] [hU : Nonempty U] {μ : MeasureTheory.Measure T} [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : Hk[κ , μ] =\n  Hk[ProbabilityTheory.kernel.fst κ , μ] +\n    Hk[ProbabilityTheory.kernel.condKernel κ , MeasureTheory.Measure.compProd μ (ProbabilityTheory.kernel.fst κ)]"}
+{"name":"ProbabilityTheory.kernel.finiteKernelSupport_of_const","declaration":"/-- Constant kernels with finite support, have finite kernel support. -/\ntheorem ProbabilityTheory.kernel.finiteKernelSupport_of_const {S : Type u_2} {T : Type u_3} [MeasurableSpace S] [MeasurableSpace T] (ν : MeasureTheory.Measure S) [ProbabilityTheory.FiniteSupport ν] : ProbabilityTheory.kernel.FiniteKernelSupport (ProbabilityTheory.kernel.const T ν)"}
+{"name":"ProbabilityTheory.kernel.entropy_map_of_injective","declaration":"theorem ProbabilityTheory.kernel.entropy_map_of_injective {S : Type u_2} {T : Type u_3} {U : Type u_4} [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U] [MeasurableSingletonClass U] (κ : ↥(ProbabilityTheory.kernel T S)) (μ : MeasureTheory.Measure T) {f : S → U} (hf : Function.Injective f) (hmes : Measurable f) : Hk[ProbabilityTheory.kernel.map κ f hmes , μ] = Hk[κ , μ]"}
+{"name":"ProbabilityTheory.kernel.entropy_prod","declaration":"theorem ProbabilityTheory.kernel.entropy_prod {S : Type u_2} {T : Type u_3} {U : Type u_4} [Countable S] [Nonempty S] [MeasurableSpace S] [MeasurableSingletonClass S] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] [Countable U] [Nonempty U] [MeasurableSpace U] [MeasurableSingletonClass U] {κ : ↥(ProbabilityTheory.kernel T S)} {η : ↥(ProbabilityTheory.kernel T U)} [ProbabilityTheory.IsMarkovKernel κ] [ProbabilityTheory.IsMarkovKernel η] {μ : MeasureTheory.Measure T} [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) (hη : ProbabilityTheory.kernel.AEFiniteKernelSupport η μ) : Hk[ProbabilityTheory.kernel.prod κ η , μ] = Hk[κ , μ] + Hk[η , μ]"}
+{"name":"ProbabilityTheory.kernel.entropy_map_le","declaration":"/-- Data-processing inequality for the kernel entropy. -/\ntheorem ProbabilityTheory.kernel.entropy_map_le {S : Type u_2} {T : Type u_3} {U : Type u_4} [Countable S] [Nonempty S] [MeasurableSpace S] [MeasurableSingletonClass S] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] [Countable U] [Nonempty U] [MeasurableSpace U] [MeasurableSingletonClass U] {κ : ↥(ProbabilityTheory.kernel T S)} [ProbabilityTheory.IsMarkovKernel κ] {μ : MeasureTheory.Measure T} [MeasureTheory.IsProbabilityMeasure μ] (f : S → U) [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : Hk[ProbabilityTheory.kernel.map κ f ⋯ , μ] ≤ Hk[κ , μ]"}
+{"name":"ProbabilityTheory.kernel.entropy_deterministic","declaration":"theorem ProbabilityTheory.kernel.entropy_deterministic {S : Type u_2} {T : Type u_3} [MeasurableSpace S] [MeasurableSingletonClass S] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] (f : T → S) (μ : MeasureTheory.Measure T) : Hk[ProbabilityTheory.kernel.deterministic f ⋯ , μ] = 0"}
+{"name":"ProbabilityTheory.kernel.entropy_fst_le","declaration":"theorem ProbabilityTheory.kernel.entropy_fst_le {S : Type u_2} {T : Type u_3} {U : Type u_4} [Countable S] [Nonempty S] [MeasurableSpace S] [MeasurableSingletonClass S] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] [Countable U] [Nonempty U] [MeasurableSpace U] [MeasurableSingletonClass U] (κ : ↥(ProbabilityTheory.kernel T (S × U))) [ProbabilityTheory.IsMarkovKernel κ] (μ : MeasureTheory.Measure T) [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : Hk[ProbabilityTheory.kernel.fst κ , μ] ≤ Hk[κ , μ]"}
+{"name":"ProbabilityTheory.kernel.entropy_zero_measure","declaration":"theorem ProbabilityTheory.kernel.entropy_zero_measure {S : Type u_2} {T : Type u_3} [MeasurableSpace S] [MeasurableSpace T] (κ : ↥(ProbabilityTheory.kernel T S)) : Hk[κ , 0] = 0"}
+{"name":"ProbabilityTheory.kernel.entropy_swapRight","declaration":"theorem ProbabilityTheory.kernel.entropy_swapRight {S : Type u_2} {T : Type u_3} {U : Type u_4} [MeasurableSpace S] [MeasurableSingletonClass S] [MeasurableSpace T] [MeasurableSpace U] [MeasurableSingletonClass U] (κ : ↥(ProbabilityTheory.kernel T (S × U))) (μ : MeasureTheory.Measure T) : Hk[ProbabilityTheory.kernel.swapRight κ , μ] = Hk[κ , μ]"}
+{"name":"ProbabilityTheory.kernel.prodMKLeft_unit_equiv","declaration":"/-- Measurable equivalence with the product with the one-point space `Unit`.-/\ndef ProbabilityTheory.kernel.prodMKLeft_unit_equiv (T : Type u_5) [MeasurableSpace T] : Unit × T ≃ᵐ T"}
+{"name":"ProbabilityTheory.kernel.entropy_prodMkLeft","declaration":"theorem ProbabilityTheory.kernel.entropy_prodMkLeft {S : Type u_2} {T : Type u_3} {U : Type u_4} [Countable S] [MeasurableSpace S] [MeasurableSingletonClass S] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] [MeasurableSpace U] {η : ↥(ProbabilityTheory.kernel T U)} {ν : MeasureTheory.Measure S} [MeasureTheory.IsProbabilityMeasure ν] {μ : MeasureTheory.Measure T} [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] [ProbabilityTheory.FiniteSupport ν] : Hk[ProbabilityTheory.kernel.prodMkLeft S η , MeasureTheory.Measure.prod ν μ] = Hk[η , μ]"}
+{"name":"ProbabilityTheory.kernel.entropy_of_map_eq_of_map","declaration":"theorem ProbabilityTheory.kernel.entropy_of_map_eq_of_map {S : Type u_2} {T : Type u_3} {U : Type u_4} [Countable S] [Nonempty S] [MeasurableSpace S] [MeasurableSingletonClass S] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] [Countable U] [Nonempty U] [MeasurableSpace U] [MeasurableSingletonClass U] {κ : ↥(ProbabilityTheory.kernel T S)} {η : ↥(ProbabilityTheory.kernel T U)} [ProbabilityTheory.IsMarkovKernel κ] [ProbabilityTheory.IsMarkovKernel η] {μ : MeasureTheory.Measure T} [MeasureTheory.IsProbabilityMeasure μ] (f : S → U) (g : U → S) (h1 : η = ProbabilityTheory.kernel.map κ f ⋯) (h2 : κ = ProbabilityTheory.kernel.map η g ⋯) [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) (hη : ProbabilityTheory.kernel.AEFiniteKernelSupport η μ) : Hk[κ , μ] = Hk[η , μ]"}
+{"name":"ProbabilityTheory.kernel.entropy_zero_kernel","declaration":"theorem ProbabilityTheory.kernel.entropy_zero_kernel {S : Type u_2} {T : Type u_3} [MeasurableSpace S] [MeasurableSpace T] (μ : MeasureTheory.Measure T) : Hk[0 , μ] = 0"}
+{"name":"ProbabilityTheory.kernel.entropy_le_log_card","declaration":"theorem ProbabilityTheory.kernel.entropy_le_log_card {S : Type u_2} {T : Type u_3} [MeasurableSpace S] [MeasurableSingletonClass S] [MeasurableSpace T] (κ : ↥(ProbabilityTheory.kernel T S)) (μ : MeasureTheory.Measure T) [Fintype S] [MeasureTheory.IsProbabilityMeasure μ] : Hk[κ , μ] ≤ Real.log ↑(Fintype.card S)"}
+{"name":"ProbabilityTheory.kernel.FiniteSupport.comap_equiv","declaration":"theorem ProbabilityTheory.kernel.FiniteSupport.comap_equiv {T : Type u_3} [MeasurableSpace T] [MeasurableSingletonClass T] {T' : Type u_5} [MeasurableSpace T'] [MeasurableSingletonClass T'] {μ : MeasureTheory.Measure T} (f : T' ≃ᵐ T) [ProbabilityTheory.FiniteSupport μ] : ProbabilityTheory.FiniteSupport (MeasureTheory.Measure.comap (⇑f) μ)"}
+{"name":"ProbabilityTheory.kernel.entropy","declaration":"/-- Entropy of a kernel with respect to a measure. -/\ndef ProbabilityTheory.kernel.entropy {S : Type u_2} {T : Type u_3} [MeasurableSpace S] [MeasurableSpace T] (κ : ↥(ProbabilityTheory.kernel T S)) (μ : MeasureTheory.Measure T) : ℝ"}
+{"name":"ProbabilityTheory.kernel.«termHk[_,_]»","declaration":"/-- Entropy of a kernel with respect to a measure. -/\ndef ProbabilityTheory.kernel.«termHk[_,_]»  : Lean.ParserDescr"}
+{"name":"ProbabilityTheory.kernel.entropy_compProd_aux","declaration":"theorem ProbabilityTheory.kernel.entropy_compProd_aux {S : Type u_2} {T : Type u_3} {U : Type u_4} [MeasurableSpace S] [MeasurableSingletonClass S] [MeasurableSpace T] [MeasurableSingletonClass T] [MeasurableSpace U] [MeasurableSingletonClass U] {μ : MeasureTheory.Measure T} [MeasureTheory.IsFiniteMeasure μ] {κ : ↥(ProbabilityTheory.kernel T S)} [ProbabilityTheory.IsMarkovKernel κ] {η : ↥(ProbabilityTheory.kernel (T × S) U)} [ProbabilityTheory.IsMarkovKernel η] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.FiniteKernelSupport κ) (hη : ProbabilityTheory.kernel.FiniteKernelSupport η) : Hk[ProbabilityTheory.kernel.compProd κ η , μ] =\n  Hk[κ , μ] + ∫ (x : T), (fun t => Hk[ProbabilityTheory.kernel.comap η (Prod.mk t) ⋯ , κ t]) x ∂μ"}
+{"name":"ProbabilityTheory.kernel.entropy_compProd'","declaration":"theorem ProbabilityTheory.kernel.entropy_compProd' {S : Type u_2} {T : Type u_3} {U : Type u_4} [Countable S] [MeasurableSpace S] [MeasurableSingletonClass S] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] [MeasurableSpace U] [MeasurableSingletonClass U] {μ : MeasureTheory.Measure T} [MeasureTheory.IsFiniteMeasure μ] {κ : ↥(ProbabilityTheory.kernel T S)} [ProbabilityTheory.IsMarkovKernel κ] {η : ↥(ProbabilityTheory.kernel (T × S) U)} [ProbabilityTheory.IsMarkovKernel η] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.FiniteKernelSupport κ) (hη : ProbabilityTheory.kernel.FiniteKernelSupport η) : Hk[ProbabilityTheory.kernel.compProd κ η , μ] = Hk[κ , μ] + Hk[η , MeasureTheory.Measure.compProd μ κ]"}
+{"name":"ProbabilityTheory.kernel.finiteSupport_of_compProd","declaration":"theorem ProbabilityTheory.kernel.finiteSupport_of_compProd {S : Type u_2} {T : Type u_3} [Countable S] [Nonempty S] [MeasurableSpace S] [MeasurableSingletonClass S] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] {μ : MeasureTheory.Measure T} [MeasureTheory.IsFiniteMeasure μ] {κ : ↥(ProbabilityTheory.kernel T S)} [ProbabilityTheory.IsMarkovKernel κ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : ProbabilityTheory.FiniteSupport (MeasureTheory.Measure.compProd μ κ)"}
+{"name":"ProbabilityTheory.kernel.entropy_prodMkRight'","declaration":"theorem ProbabilityTheory.kernel.entropy_prodMkRight' {S : Type u_2} {T : Type u_3} {U : Type u_4} [Countable S] [Nonempty S] [MeasurableSpace S] [MeasurableSingletonClass S] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] [MeasurableSpace U] {η : ↥(ProbabilityTheory.kernel T U)} {μ : MeasureTheory.Measure T} [MeasureTheory.IsProbabilityMeasure μ] {ν : MeasureTheory.Measure S} [MeasureTheory.IsProbabilityMeasure ν] [ProbabilityTheory.FiniteSupport μ] [ProbabilityTheory.FiniteSupport ν] : Hk[ProbabilityTheory.kernel.prodMkRight S η , MeasureTheory.Measure.prod μ ν] = Hk[η , μ]"}
+{"name":"ProbabilityTheory.kernel.entropy_comap_equiv","declaration":"theorem ProbabilityTheory.kernel.entropy_comap_equiv {S : Type u_2} {T : Type u_3} [MeasurableSpace S] [MeasurableSpace T] [MeasurableSingletonClass T] {T' : Type u_5} [MeasurableSpace T'] [MeasurableSingletonClass T'] (κ : ↥(ProbabilityTheory.kernel T S)) {μ : MeasureTheory.Measure T} (f : T' ≃ᵐ T) [MeasureTheory.IsFiniteMeasure μ] [ProbabilityTheory.FiniteSupport μ] : Hk[ProbabilityTheory.kernel.comap κ ⇑f ⋯ , MeasureTheory.Measure.comap (⇑f) μ] = Hk[κ , μ]"}
+{"name":"ProbabilityTheory.kernel.entropy_congr","declaration":"theorem ProbabilityTheory.kernel.entropy_congr {S : Type u_2} {T : Type u_3} [MeasurableSpace S] [MeasurableSpace T] {μ : MeasureTheory.Measure T} {κ : ↥(ProbabilityTheory.kernel T S)} {η : ↥(ProbabilityTheory.kernel T S)} (h : ⇑κ =ᶠ[MeasureTheory.Measure.ae μ] ⇑η) : Hk[κ , μ] = Hk[η , μ]"}
+{"name":"ProbabilityTheory.kernel.entropy_map_swap","declaration":"theorem ProbabilityTheory.kernel.entropy_map_swap {S : Type u_2} {T : Type u_3} {U : Type u_4} [MeasurableSpace S] [MeasurableSingletonClass S] [MeasurableSpace T] [MeasurableSpace U] [MeasurableSingletonClass U] (κ : ↥(ProbabilityTheory.kernel T (S × U))) (μ : MeasureTheory.Measure T) : Hk[ProbabilityTheory.kernel.map κ Prod.swap ⋯ , μ] = Hk[κ , μ]"}
+{"name":"ProbabilityTheory.kernel.entropy_compProd_deterministic","declaration":"theorem ProbabilityTheory.kernel.entropy_compProd_deterministic {S : Type u_2} {T : Type u_3} {U : Type u_4} [Countable S] [Nonempty S] [MeasurableSpace S] [MeasurableSingletonClass S] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] [Nonempty U] [MeasurableSpace U] [MeasurableSingletonClass U] (κ : ↥(ProbabilityTheory.kernel T S)) [ProbabilityTheory.IsMarkovKernel κ] (μ : MeasureTheory.Measure T) [MeasureTheory.IsFiniteMeasure μ] (f : T × S → U) [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : Hk[ProbabilityTheory.kernel.compProd κ (ProbabilityTheory.kernel.deterministic f ⋯) , μ] = Hk[κ , μ]"}
+{"name":"ProbabilityTheory.kernel.entropy_snd_compProd_deterministic_of_injective","declaration":"theorem ProbabilityTheory.kernel.entropy_snd_compProd_deterministic_of_injective {S : Type u_2} {T : Type u_3} {U : Type u_4} [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U] [MeasurableSingletonClass U] (κ : ↥(ProbabilityTheory.kernel T S)) [ProbabilityTheory.IsMarkovKernel κ] (μ : MeasureTheory.Measure T) {f : T × S → U} (hf : ∀ (t : T), Function.Injective fun x => f (t, x)) (hmes : Measurable f) : Hk[ProbabilityTheory.kernel.snd (ProbabilityTheory.kernel.compProd κ (ProbabilityTheory.kernel.deterministic f hmes)) ,\n    μ] =\n  Hk[κ , μ]"}
+{"name":"ProbabilityTheory.kernel.entropy_eq_integral_sum","declaration":"theorem ProbabilityTheory.kernel.entropy_eq_integral_sum {S : Type u_2} {T : Type u_3} [MeasurableSpace S] [MeasurableSpace T] (κ : ↥(ProbabilityTheory.kernel T S)) [ProbabilityTheory.IsMarkovKernel κ] (μ : MeasureTheory.Measure T) : Hk[κ , μ] = ∫ (x : T), (fun y => ∑' (x : S), Real.negMulLog (↑↑(κ y) {x}).toReal) x ∂μ"}

PFR-declarations/PFR.ForMathlib.Entropy.Kernel.Group.jsonl

@@ -0,0 +1,30 @@
+{"name":"ProbabilityTheory.kernel.entropy_snd_sub_mutualInfo_le_entropy_map_add","declaration":"theorem ProbabilityTheory.kernel.entropy_snd_sub_mutualInfo_le_entropy_map_add {G : Type u_5} {T : Type u_6} [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] [MeasurableAdd₂ G] [Countable G] (κ : ↥(ProbabilityTheory.kernel T (G × G))) [ProbabilityTheory.IsMarkovKernel κ] (μ : MeasureTheory.Measure T) [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : Hk[ProbabilityTheory.kernel.snd κ , μ] - Ik[κ , μ] ≤ Hk[ProbabilityTheory.kernel.map κ (fun p => p.1 + p.2) ⋯ , μ]"}
+{"name":"ProbabilityTheory.kernel.entropy_fst_sub_mutualInfo_le_entropy_map_div","declaration":"theorem ProbabilityTheory.kernel.entropy_fst_sub_mutualInfo_le_entropy_map_div {G : Type u_5} {T : Type u_6} [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] [MeasurableDiv₂ G] [Countable G] (κ : ↥(ProbabilityTheory.kernel T (G × G))) [ProbabilityTheory.IsMarkovKernel κ] (μ : MeasureTheory.Measure T) [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : Hk[ProbabilityTheory.kernel.fst κ , μ] - Ik[κ , μ] ≤ Hk[ProbabilityTheory.kernel.map κ (fun p => p.1 / p.2) ⋯ , μ]"}
+{"name":"ProbabilityTheory.kernel.entropy_div_comm","declaration":"theorem ProbabilityTheory.kernel.entropy_div_comm {G : Type u_5} {T : Type u_6} [MeasurableSpace T] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] [MeasurableDiv₂ G] [Countable G] (κ : ↥(ProbabilityTheory.kernel T (G × G))) (μ : MeasureTheory.Measure T) : Hk[ProbabilityTheory.kernel.map κ (fun p => p.1 / p.2) ⋯ , μ] =\n  Hk[ProbabilityTheory.kernel.map κ (fun p => p.2 / p.1) ⋯ , μ]"}
+{"name":"ProbabilityTheory.kernel.max_entropy_sub_mutualInfo_le_entropy_div","declaration":"theorem ProbabilityTheory.kernel.max_entropy_sub_mutualInfo_le_entropy_div {G : Type u_5} {T : Type u_6} [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] [MeasurableDiv₂ G] [Countable G] (κ : ↥(ProbabilityTheory.kernel T (G × G))) [ProbabilityTheory.IsMarkovKernel κ] (μ : MeasureTheory.Measure T) [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : max (Hk[ProbabilityTheory.kernel.fst κ , μ]) (Hk[ProbabilityTheory.kernel.snd κ , μ]) - Ik[κ , μ] ≤\n  Hk[ProbabilityTheory.kernel.map κ (fun p => p.1 / p.2) ⋯ , μ]"}
+{"name":"measureEntropy_inv","declaration":"theorem measureEntropy_inv {G : Type u_5} [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] [Countable G] (μ : MeasureTheory.Measure G) : Hm[MeasureTheory.Measure.map (fun x => x⁻¹) μ] = Hm[μ]"}
+{"name":"ProbabilityTheory.kernel.max_entropy_le_entropy_div_prod","declaration":"theorem ProbabilityTheory.kernel.max_entropy_le_entropy_div_prod {G : Type u_5} {T : Type u_6} [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] [MeasurableDiv₂ G] [Countable G] (κ : ↥(ProbabilityTheory.kernel T G)) [ProbabilityTheory.IsMarkovKernel κ] (η : ↥(ProbabilityTheory.kernel T G)) [ProbabilityTheory.IsMarkovKernel η] (μ : MeasureTheory.Measure T) [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) (hη : ProbabilityTheory.kernel.AEFiniteKernelSupport η μ) : max (Hk[κ , μ]) (Hk[η , μ]) ≤\n  Hk[ProbabilityTheory.kernel.map (ProbabilityTheory.kernel.prod κ η) (fun p => p.1 / p.2) ⋯ , μ]"}
+{"name":"ProbabilityTheory.kernel.max_entropy_le_entropy_add_sum","declaration":"theorem ProbabilityTheory.kernel.max_entropy_le_entropy_add_sum {G : Type u_5} {T : Type u_6} [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] [MeasurableAdd₂ G] [Countable G] (κ : ↥(ProbabilityTheory.kernel T G)) [ProbabilityTheory.IsMarkovKernel κ] (η : ↥(ProbabilityTheory.kernel T G)) [ProbabilityTheory.IsMarkovKernel η] (μ : MeasureTheory.Measure T) [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) (hη : ProbabilityTheory.kernel.AEFiniteKernelSupport η μ) : max (Hk[κ , μ]) (Hk[η , μ]) ≤\n  Hk[ProbabilityTheory.kernel.map (ProbabilityTheory.kernel.prod κ η) (fun p => p.1 + p.2) ⋯ , μ]"}
+{"name":"ProbabilityTheory.kernel.entropy_snd_sub_mutualInfo_le_entropy_map_sub","declaration":"theorem ProbabilityTheory.kernel.entropy_snd_sub_mutualInfo_le_entropy_map_sub {G : Type u_5} {T : Type u_6} [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] [MeasurableSub₂ G] [Countable G] (κ : ↥(ProbabilityTheory.kernel T (G × G))) [ProbabilityTheory.IsMarkovKernel κ] (μ : MeasureTheory.Measure T) [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : Hk[ProbabilityTheory.kernel.snd κ , μ] - Ik[κ , μ] ≤ Hk[ProbabilityTheory.kernel.map κ (fun p => p.1 - p.2) ⋯ , μ]"}
+{"name":"ProbabilityTheory.kernel.entropy_fst_sub_mutualInfo_le_entropy_map_mul","declaration":"theorem ProbabilityTheory.kernel.entropy_fst_sub_mutualInfo_le_entropy_map_mul {G : Type u_5} {T : Type u_6} [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] [MeasurableMul₂ G] [Countable G] (κ : ↥(ProbabilityTheory.kernel T (G × G))) [ProbabilityTheory.IsMarkovKernel κ] (μ : MeasureTheory.Measure T) [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : Hk[ProbabilityTheory.kernel.fst κ , μ] - Ik[κ , μ] ≤ Hk[ProbabilityTheory.kernel.map κ (fun p => p.1 * p.2) ⋯ , μ]"}
+{"name":"measureEntropy_sub_comm","declaration":"theorem measureEntropy_sub_comm {G : Type u_5} [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] [MeasurableSub₂ G] [Countable G] (μ : MeasureTheory.Measure (G × G)) : Hm[MeasureTheory.Measure.map (fun p => p.1 - p.2) μ] = Hm[MeasureTheory.Measure.map (fun p => p.2 - p.1) μ]"}
+{"name":"ProbabilityTheory.kernel.entropy_inv","declaration":"theorem ProbabilityTheory.kernel.entropy_inv {G : Type u_5} {T : Type u_6} [MeasurableSpace T] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] [Countable G] (κ : ↥(ProbabilityTheory.kernel T G)) (μ : MeasureTheory.Measure T) : Hk[ProbabilityTheory.kernel.map κ (fun x => x⁻¹) ⋯ , μ] = Hk[κ , μ]"}
+{"name":"ProbabilityTheory.kernel.entropy_fst_sub_mutualInfo_le_entropy_map_sub","declaration":"theorem ProbabilityTheory.kernel.entropy_fst_sub_mutualInfo_le_entropy_map_sub {G : Type u_5} {T : Type u_6} [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] [MeasurableSub₂ G] [Countable G] (κ : ↥(ProbabilityTheory.kernel T (G × G))) [ProbabilityTheory.IsMarkovKernel κ] (μ : MeasureTheory.Measure T) [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : Hk[ProbabilityTheory.kernel.fst κ , μ] - Ik[κ , μ] ≤ Hk[ProbabilityTheory.kernel.map κ (fun p => p.1 - p.2) ⋯ , μ]"}
+{"name":"ProbabilityTheory.kernel.entropy_snd_sub_mutualInfo_le_entropy_map_mul","declaration":"theorem ProbabilityTheory.kernel.entropy_snd_sub_mutualInfo_le_entropy_map_mul {G : Type u_5} {T : Type u_6} [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] [MeasurableMul₂ G] [Countable G] (κ : ↥(ProbabilityTheory.kernel T (G × G))) [ProbabilityTheory.IsMarkovKernel κ] (μ : MeasureTheory.Measure T) [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : Hk[ProbabilityTheory.kernel.snd κ , μ] - Ik[κ , μ] ≤ Hk[ProbabilityTheory.kernel.map κ (fun p => p.1 * p.2) ⋯ , μ]"}
+{"name":"measureEntropy_neg","declaration":"theorem measureEntropy_neg {G : Type u_5} [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] [Countable G] (μ : MeasureTheory.Measure G) : Hm[MeasureTheory.Measure.map (fun x => -x) μ] = Hm[μ]"}
+{"name":"ProbabilityTheory.kernel.max_entropy_le_entropy_sub_prod","declaration":"theorem ProbabilityTheory.kernel.max_entropy_le_entropy_sub_prod {G : Type u_5} {T : Type u_6} [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] [MeasurableSub₂ G] [Countable G] (κ : ↥(ProbabilityTheory.kernel T G)) [ProbabilityTheory.IsMarkovKernel κ] (η : ↥(ProbabilityTheory.kernel T G)) [ProbabilityTheory.IsMarkovKernel η] (μ : MeasureTheory.Measure T) [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) (hη : ProbabilityTheory.kernel.AEFiniteKernelSupport η μ) : max (Hk[κ , μ]) (Hk[η , μ]) ≤\n  Hk[ProbabilityTheory.kernel.map (ProbabilityTheory.kernel.prod κ η) (fun p => p.1 - p.2) ⋯ , μ]"}
+{"name":"ProbabilityTheory.kernel.max_entropy_sub_mutualInfo_le_entropy_mul","declaration":"theorem ProbabilityTheory.kernel.max_entropy_sub_mutualInfo_le_entropy_mul {G : Type u_5} {T : Type u_6} [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] [MeasurableMul₂ G] [Countable G] (κ : ↥(ProbabilityTheory.kernel T (G × G))) [ProbabilityTheory.IsMarkovKernel κ] (μ : MeasureTheory.Measure T) [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : max (Hk[ProbabilityTheory.kernel.fst κ , μ]) (Hk[ProbabilityTheory.kernel.snd κ , μ]) - Ik[κ , μ] ≤\n  Hk[ProbabilityTheory.kernel.map κ (fun p => p.1 * p.2) ⋯ , μ]"}
+{"name":"ProbabilityTheory.kernel.max_entropy_sub_mutualInfo_le_entropy_mul'","declaration":"theorem ProbabilityTheory.kernel.max_entropy_sub_mutualInfo_le_entropy_mul' {G : Type u_5} {T : Type u_6} [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] [MeasurableMul₂ G] [Countable G] (κ : ↥(ProbabilityTheory.kernel T (G × G))) [ProbabilityTheory.IsMarkovKernel κ] (μ : MeasureTheory.Measure T) [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : max (Hk[ProbabilityTheory.kernel.fst κ , μ]) (Hk[ProbabilityTheory.kernel.snd κ , μ]) - Ik[κ , μ] ≤\n  Hk[ProbabilityTheory.kernel.map κ (fun p => p.2 * p.1) ⋯ , μ]"}
+{"name":"ProbabilityTheory.kernel.entropy_neg","declaration":"theorem ProbabilityTheory.kernel.entropy_neg {G : Type u_5} {T : Type u_6} [MeasurableSpace T] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] [Countable G] (κ : ↥(ProbabilityTheory.kernel T G)) (μ : MeasureTheory.Measure T) : Hk[ProbabilityTheory.kernel.map κ (fun x => -x) ⋯ , μ] = Hk[κ , μ]"}
+{"name":"ProbabilityTheory.kernel.max_entropy_sub_mutualInfo_le_entropy_sub","declaration":"theorem ProbabilityTheory.kernel.max_entropy_sub_mutualInfo_le_entropy_sub {G : Type u_5} {T : Type u_6} [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] [MeasurableSub₂ G] [Countable G] (κ : ↥(ProbabilityTheory.kernel T (G × G))) [ProbabilityTheory.IsMarkovKernel κ] (μ : MeasureTheory.Measure T) [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : max (Hk[ProbabilityTheory.kernel.fst κ , μ]) (Hk[ProbabilityTheory.kernel.snd κ , μ]) - Ik[κ , μ] ≤\n  Hk[ProbabilityTheory.kernel.map κ (fun p => p.1 - p.2) ⋯ , μ]"}
+{"name":"ProbabilityTheory.kernel.entropy_fst_sub_mutualInfo_le_entropy_map_add","declaration":"theorem ProbabilityTheory.kernel.entropy_fst_sub_mutualInfo_le_entropy_map_add {G : Type u_5} {T : Type u_6} [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] [MeasurableAdd₂ G] [Countable G] (κ : ↥(ProbabilityTheory.kernel T (G × G))) [ProbabilityTheory.IsMarkovKernel κ] (μ : MeasureTheory.Measure T) [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : Hk[ProbabilityTheory.kernel.fst κ , μ] - Ik[κ , μ] ≤ Hk[ProbabilityTheory.kernel.map κ (fun p => p.1 + p.2) ⋯ , μ]"}
+{"name":"ProbabilityTheory.kernel.max_entropy_le_entropy_mul_prod","declaration":"theorem ProbabilityTheory.kernel.max_entropy_le_entropy_mul_prod {G : Type u_5} {T : Type u_6} [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] [MeasurableMul₂ G] [Countable G] (κ : ↥(ProbabilityTheory.kernel T G)) [ProbabilityTheory.IsMarkovKernel κ] (η : ↥(ProbabilityTheory.kernel T G)) [ProbabilityTheory.IsMarkovKernel η] (μ : MeasureTheory.Measure T) [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) (hη : ProbabilityTheory.kernel.AEFiniteKernelSupport η μ) : max (Hk[κ , μ]) (Hk[η , μ]) ≤\n  Hk[ProbabilityTheory.kernel.map (ProbabilityTheory.kernel.prod κ η) (fun p => p.1 * p.2) ⋯ , μ]"}
+{"name":"ProbabilityTheory.kernel.entropy_fst_sub_mutualInfo_le_entropy_map_add'","declaration":"theorem ProbabilityTheory.kernel.entropy_fst_sub_mutualInfo_le_entropy_map_add' {G : Type u_5} {T : Type u_6} [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] [MeasurableAdd₂ G] [Countable G] (κ : ↥(ProbabilityTheory.kernel T (G × G))) [ProbabilityTheory.IsMarkovKernel κ] (μ : MeasureTheory.Measure T) [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : Hk[ProbabilityTheory.kernel.fst κ , μ] - Ik[κ , μ] ≤ Hk[ProbabilityTheory.kernel.map κ (fun p => p.2 + p.1) ⋯ , μ]"}
+{"name":"ProbabilityTheory.kernel.entropy_fst_sub_mutualInfo_le_entropy_map_mul'","declaration":"theorem ProbabilityTheory.kernel.entropy_fst_sub_mutualInfo_le_entropy_map_mul' {G : Type u_5} {T : Type u_6} [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] [MeasurableMul₂ G] [Countable G] (κ : ↥(ProbabilityTheory.kernel T (G × G))) [ProbabilityTheory.IsMarkovKernel κ] (μ : MeasureTheory.Measure T) [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : Hk[ProbabilityTheory.kernel.fst κ , μ] - Ik[κ , μ] ≤ Hk[ProbabilityTheory.kernel.map κ (fun p => p.2 * p.1) ⋯ , μ]"}
+{"name":"ProbabilityTheory.kernel.entropy_snd_sub_mutualInfo_le_entropy_map_add'","declaration":"theorem ProbabilityTheory.kernel.entropy_snd_sub_mutualInfo_le_entropy_map_add' {G : Type u_5} {T : Type u_6} [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] [Countable G] (κ : ↥(ProbabilityTheory.kernel T (G × G))) [ProbabilityTheory.IsMarkovKernel κ] (μ : MeasureTheory.Measure T) [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : Hk[ProbabilityTheory.kernel.snd κ , μ] - Ik[κ , μ] ≤ Hk[ProbabilityTheory.kernel.map κ (fun p => p.2 + p.1) ⋯ , μ]"}
+{"name":"ProbabilityTheory.kernel.entropy_sub_comm","declaration":"theorem ProbabilityTheory.kernel.entropy_sub_comm {G : Type u_5} {T : Type u_6} [MeasurableSpace T] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] [MeasurableSub₂ G] [Countable G] (κ : ↥(ProbabilityTheory.kernel T (G × G))) (μ : MeasureTheory.Measure T) : Hk[ProbabilityTheory.kernel.map κ (fun p => p.1 - p.2) ⋯ , μ] =\n  Hk[ProbabilityTheory.kernel.map κ (fun p => p.2 - p.1) ⋯ , μ]"}
+{"name":"ProbabilityTheory.kernel.max_entropy_sub_mutualInfo_le_entropy_add","declaration":"theorem ProbabilityTheory.kernel.max_entropy_sub_mutualInfo_le_entropy_add {G : Type u_5} {T : Type u_6} [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] [MeasurableAdd₂ G] [Countable G] (κ : ↥(ProbabilityTheory.kernel T (G × G))) [ProbabilityTheory.IsMarkovKernel κ] (μ : MeasureTheory.Measure T) [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : max (Hk[ProbabilityTheory.kernel.fst κ , μ]) (Hk[ProbabilityTheory.kernel.snd κ , μ]) - Ik[κ , μ] ≤\n  Hk[ProbabilityTheory.kernel.map κ (fun p => p.1 + p.2) ⋯ , μ]"}
+{"name":"ProbabilityTheory.kernel.max_entropy_sub_mutualInfo_le_entropy_add'","declaration":"theorem ProbabilityTheory.kernel.max_entropy_sub_mutualInfo_le_entropy_add' {G : Type u_5} {T : Type u_6} [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T] [MeasurableSpace G] [MeasurableSingletonClass G] [AddGroup G] [MeasurableAdd₂ G] [Countable G] (κ : ↥(ProbabilityTheory.kernel T (G × G))) [ProbabilityTheory.IsMarkovKernel κ] (μ : MeasureTheory.Measure T) [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : max (Hk[ProbabilityTheory.kernel.fst κ , μ]) (Hk[ProbabilityTheory.kernel.snd κ , μ]) - Ik[κ , μ] ≤\n  Hk[ProbabilityTheory.kernel.map κ (fun p => p.2 + p.1) ⋯ , μ]"}
+{"name":"measureEntropy_div_comm","declaration":"theorem measureEntropy_div_comm {G : Type u_5} [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] [MeasurableDiv₂ G] [Countable G] (μ : MeasureTheory.Measure (G × G)) : Hm[MeasureTheory.Measure.map (fun p => p.1 / p.2) μ] = Hm[MeasureTheory.Measure.map (fun p => p.2 / p.1) μ]"}
+{"name":"ProbabilityTheory.kernel.entropy_snd_sub_mutualInfo_le_entropy_map_mul'","declaration":"theorem ProbabilityTheory.kernel.entropy_snd_sub_mutualInfo_le_entropy_map_mul' {G : Type u_5} {T : Type u_6} [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] [Countable G] (κ : ↥(ProbabilityTheory.kernel T (G × G))) [ProbabilityTheory.IsMarkovKernel κ] (μ : MeasureTheory.Measure T) [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : Hk[ProbabilityTheory.kernel.snd κ , μ] - Ik[κ , μ] ≤ Hk[ProbabilityTheory.kernel.map κ (fun p => p.2 * p.1) ⋯ , μ]"}
+{"name":"ProbabilityTheory.kernel.entropy_snd_sub_mutualInfo_le_entropy_map_div","declaration":"theorem ProbabilityTheory.kernel.entropy_snd_sub_mutualInfo_le_entropy_map_div {G : Type u_5} {T : Type u_6} [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T] [MeasurableSpace G] [MeasurableSingletonClass G] [Group G] [MeasurableDiv₂ G] [Countable G] (κ : ↥(ProbabilityTheory.kernel T (G × G))) [ProbabilityTheory.IsMarkovKernel κ] (μ : MeasureTheory.Measure T) [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : Hk[ProbabilityTheory.kernel.snd κ , μ] - Ik[κ , μ] ≤ Hk[ProbabilityTheory.kernel.map κ (fun p => p.1 / p.2) ⋯ , μ]"}

PFR-declarations/PFR.ForMathlib.Entropy.Kernel.MutualInfo.jsonl

@@ -0,0 +1,27 @@
+{"name":"ProbabilityTheory.kernel.compProd_assoc","declaration":"theorem ProbabilityTheory.kernel.compProd_assoc {V : Type u_5} {S : Type u_2} {T : Type u_3} {U : Type u_4} [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U] [MeasurableSpace V] (ξ : ↥(ProbabilityTheory.kernel T S)) [ProbabilityTheory.IsMarkovKernel ξ] (κ : ↥(ProbabilityTheory.kernel (T × S) U)) [ProbabilityTheory.IsMarkovKernel κ] (η : ↥(ProbabilityTheory.kernel (T × S × U) V)) [ProbabilityTheory.IsMarkovKernel η] : ProbabilityTheory.kernel.map (ProbabilityTheory.kernel.compProd (ProbabilityTheory.kernel.compProd ξ κ) η)\n    ⇑MeasurableEquiv.prodAssoc ⋯ =\n  ProbabilityTheory.kernel.compProd ξ\n    (ProbabilityTheory.kernel.compProd κ (ProbabilityTheory.kernel.comap η ⇑MeasurableEquiv.prodAssoc ⋯))"}
+{"name":"ProbabilityTheory.kernel.entropy_triple_add_entropy_le'","declaration":"/-- The submodularity inequality:\n$$ H[X,Y,Z] + H[X] \\leq H[X,Z] + H[X,Y].$$ -/\ntheorem ProbabilityTheory.kernel.entropy_triple_add_entropy_le' {V : Type u_5} {S : Type u_2} {T : Type u_3} {U : Type u_4} [Nonempty S] [Countable S] [MeasurableSpace S] [MeasurableSingletonClass S] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] [Nonempty U] [Countable U] [MeasurableSpace U] [MeasurableSingletonClass U] [Nonempty V] [Countable V] [MeasurableSpace V] [MeasurableSingletonClass V] {κ : ↥(ProbabilityTheory.kernel T (S × U × V))} [ProbabilityTheory.IsMarkovKernel κ] {μ : MeasureTheory.Measure T} [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : Hk[κ , μ] + Hk[ProbabilityTheory.kernel.fst κ , μ] ≤\n  Hk[ProbabilityTheory.kernel.deleteMiddle κ , μ] + Hk[ProbabilityTheory.kernel.deleteRight κ , μ]"}
+{"name":"ProbabilityTheory.kernel.Measure.compProd_compProd''","declaration":"theorem ProbabilityTheory.kernel.Measure.compProd_compProd'' {S : Type u_2} {T : Type u_3} {U : Type u_4} [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U] (μ : MeasureTheory.Measure T) [MeasureTheory.IsProbabilityMeasure μ] (ξ : ↥(ProbabilityTheory.kernel T S)) [ProbabilityTheory.IsMarkovKernel ξ] (κ : ↥(ProbabilityTheory.kernel (T × S) U)) [ProbabilityTheory.IsMarkovKernel κ] : MeasureTheory.Measure.compProd (MeasureTheory.Measure.compProd μ ξ) κ =\n  MeasureTheory.Measure.comap (⇑MeasurableEquiv.prodAssoc)\n    (MeasureTheory.Measure.compProd μ (ProbabilityTheory.kernel.compProd ξ κ))"}
+{"name":"ProbabilityTheory.kernel.entropy_condKernel_compProd_triple","declaration":"theorem ProbabilityTheory.kernel.entropy_condKernel_compProd_triple {V : Type u_5} {S : Type u_2} {T : Type u_3} {U : Type u_4} [Countable S] [MeasurableSpace S] [MeasurableSingletonClass S] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] [Countable U] [MeasurableSpace U] [MeasurableSingletonClass U] [Nonempty V] [Countable V] [MeasurableSpace V] [MeasurableSingletonClass V] (ξ : ↥(ProbabilityTheory.kernel T S)) [ProbabilityTheory.IsMarkovKernel ξ] (κ : ↥(ProbabilityTheory.kernel (T × S) U)) [ProbabilityTheory.IsMarkovKernel κ] (η : ↥(ProbabilityTheory.kernel (T × S × U) V)) [ProbabilityTheory.IsMarkovKernel η] (μ : MeasureTheory.Measure T) [MeasureTheory.IsProbabilityMeasure μ] : Hk[ProbabilityTheory.kernel.condKernel (ProbabilityTheory.kernel.compProd (ProbabilityTheory.kernel.compProd ξ κ) η) ,\n    MeasureTheory.Measure.compProd μ (ProbabilityTheory.kernel.compProd ξ κ)] =\n  Hk[η , MeasureTheory.Measure.compProd μ (ProbabilityTheory.kernel.compProd ξ κ)]"}
+{"name":"ProbabilityTheory.kernel.«termIk[_,_]»","declaration":"/-- Mutual information of a kernel into a product space with respect to a measure. -/\ndef ProbabilityTheory.kernel.«termIk[_,_]»  : Lean.ParserDescr"}
+{"name":"ProbabilityTheory.kernel.mutualInfo_nonneg","declaration":"theorem ProbabilityTheory.kernel.mutualInfo_nonneg {S : Type u_2} {T : Type u_3} {U : Type u_4} [Nonempty S] [MeasurableSpace S] [MeasurableSingletonClass S] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] [Nonempty U] [MeasurableSpace U] [MeasurableSingletonClass U] {κ : ↥(ProbabilityTheory.kernel T (S × U))} {μ : MeasureTheory.Measure T} [MeasureTheory.IsFiniteMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : 0 ≤ Ik[κ , μ]"}
+{"name":"ProbabilityTheory.kernel.mutualInfo_eq_fst_sub","declaration":"theorem ProbabilityTheory.kernel.mutualInfo_eq_fst_sub {S : Type u_2} {T : Type u_3} {U : Type u_4} [Nonempty S] [Countable S] [MeasurableSpace S] [MeasurableSingletonClass S] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] [Nonempty U] [Countable U] [MeasurableSpace U] [MeasurableSingletonClass U] {κ : ↥(ProbabilityTheory.kernel T (S × U))} [ProbabilityTheory.IsMarkovKernel κ] {μ : MeasureTheory.Measure T} [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : Ik[κ , μ] =\n  Hk[ProbabilityTheory.kernel.fst κ , μ] -\n    Hk[ProbabilityTheory.kernel.condKernel (ProbabilityTheory.kernel.swapRight κ) ,\n      MeasureTheory.Measure.compProd μ (ProbabilityTheory.kernel.snd κ)]"}
+{"name":"ProbabilityTheory.kernel.entropy_reverse","declaration":"theorem ProbabilityTheory.kernel.entropy_reverse {V : Type u_5} {S : Type u_2} {T : Type u_3} {U : Type u_4} [Nonempty S] [Countable S] [MeasurableSpace S] [MeasurableSingletonClass S] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] [Nonempty U] [Countable U] [MeasurableSpace U] [MeasurableSingletonClass U] [Nonempty V] [Countable V] [MeasurableSpace V] [MeasurableSingletonClass V] {κ : ↥(ProbabilityTheory.kernel T (S × U × V))} [ProbabilityTheory.IsMarkovKernel κ] {μ : MeasureTheory.Measure T} [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : Hk[ProbabilityTheory.kernel.reverse κ , μ] = Hk[κ , μ]"}
+{"name":"ProbabilityTheory.kernel.mutualInfo_nonneg'","declaration":"theorem ProbabilityTheory.kernel.mutualInfo_nonneg' {S : Type u_2} {T : Type u_3} {U : Type u_4} [MeasurableSpace S] [MeasurableSingletonClass S] [MeasurableSpace T] [MeasurableSingletonClass T] [MeasurableSpace U] [MeasurableSingletonClass U] {κ : ↥(ProbabilityTheory.kernel T (S × U))} {μ : MeasureTheory.Measure T} [MeasureTheory.IsFiniteMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.FiniteKernelSupport κ) : 0 ≤ Ik[κ , μ]"}
+{"name":"ProbabilityTheory.kernel.entropy_compProd_triple_add_entropy_le","declaration":"theorem ProbabilityTheory.kernel.entropy_compProd_triple_add_entropy_le {V : Type u_5} {S : Type u_2} {T : Type u_3} {U : Type u_4} [Nonempty S] [Countable S] [MeasurableSpace S] [MeasurableSingletonClass S] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] [Nonempty U] [Countable U] [MeasurableSpace U] [MeasurableSingletonClass U] [Nonempty V] [Countable V] [MeasurableSpace V] [MeasurableSingletonClass V] {ξ : ↥(ProbabilityTheory.kernel T S)} [ProbabilityTheory.IsMarkovKernel ξ] {κ : ↥(ProbabilityTheory.kernel (T × S) U)} [ProbabilityTheory.IsMarkovKernel κ] {η : ↥(ProbabilityTheory.kernel (T × S × U) V)} [ProbabilityTheory.IsMarkovKernel η] {μ : MeasureTheory.Measure T} [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ (MeasureTheory.Measure.compProd μ ξ)) (hη : ProbabilityTheory.kernel.AEFiniteKernelSupport η\n  (MeasureTheory.Measure.compProd μ (ProbabilityTheory.kernel.compProd ξ κ))) (hξ : ProbabilityTheory.kernel.AEFiniteKernelSupport ξ μ) : Hk[ProbabilityTheory.kernel.compProd (ProbabilityTheory.kernel.compProd ξ κ) η , μ] + Hk[ξ , μ] ≤\n  Hk[ProbabilityTheory.kernel.compProd ξ\n        (ProbabilityTheory.kernel.snd\n          (ProbabilityTheory.kernel.compProd κ (ProbabilityTheory.kernel.comap η ⇑MeasurableEquiv.prodAssoc ⋯))) ,\n      μ] +\n    Hk[ProbabilityTheory.kernel.compProd ξ κ , μ]"}
+{"name":"ProbabilityTheory.kernel.mutualInfo_prod","declaration":"theorem ProbabilityTheory.kernel.mutualInfo_prod {S : Type u_2} {T : Type u_3} {U : Type u_4} [Nonempty S] [Countable S] [MeasurableSpace S] [MeasurableSingletonClass S] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] [Nonempty U] [Countable U] [MeasurableSpace U] [MeasurableSingletonClass U] {κ : ↥(ProbabilityTheory.kernel T S)} {η : ↥(ProbabilityTheory.kernel T U)} [ProbabilityTheory.IsMarkovKernel κ] [ProbabilityTheory.IsMarkovKernel η] (μ : MeasureTheory.Measure T) [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) (hη : ProbabilityTheory.kernel.AEFiniteKernelSupport η μ) : Ik[ProbabilityTheory.kernel.prod κ η , μ] = 0"}
+{"name":"ProbabilityTheory.kernel.mutualInfo_swapRight","declaration":"theorem ProbabilityTheory.kernel.mutualInfo_swapRight {S : Type u_2} {T : Type u_3} {U : Type u_4} [MeasurableSpace S] [MeasurableSingletonClass S] [MeasurableSpace T] [MeasurableSpace U] [MeasurableSingletonClass U] (κ : ↥(ProbabilityTheory.kernel T (S × U))) (μ : MeasureTheory.Measure T) : Ik[ProbabilityTheory.kernel.swapRight κ , μ] = Ik[κ , μ]"}
+{"name":"ProbabilityTheory.kernel.entropy_condKernel_le_entropy_snd","declaration":"theorem ProbabilityTheory.kernel.entropy_condKernel_le_entropy_snd {S : Type u_2} {T : Type u_3} {U : Type u_4} [Nonempty S] [Countable S] [MeasurableSpace S] [MeasurableSingletonClass S] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] [Nonempty U] [Countable U] [MeasurableSpace U] [MeasurableSingletonClass U] {κ : ↥(ProbabilityTheory.kernel T (S × U))} [ProbabilityTheory.IsMarkovKernel κ] {μ : MeasureTheory.Measure T} [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : Hk[ProbabilityTheory.kernel.condKernel κ , MeasureTheory.Measure.compProd μ (ProbabilityTheory.kernel.fst κ)] ≤\n  Hk[ProbabilityTheory.kernel.snd κ , μ]"}
+{"name":"ProbabilityTheory.kernel.mutualInfo_zero_measure","declaration":"theorem ProbabilityTheory.kernel.mutualInfo_zero_measure {S : Type u_2} {T : Type u_3} {U : Type u_4} [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U] (κ : ↥(ProbabilityTheory.kernel T (S × U))) : Ik[κ , 0] = 0"}
+{"name":"ProbabilityTheory.kernel.mutualInfo_def","declaration":"theorem ProbabilityTheory.kernel.mutualInfo_def {S : Type u_2} {T : Type u_3} {U : Type u_4} [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U] (κ : ↥(ProbabilityTheory.kernel T (S × U))) (μ : MeasureTheory.Measure T) : Ik[κ , μ] = Hk[ProbabilityTheory.kernel.fst κ , μ] + Hk[ProbabilityTheory.kernel.snd κ , μ] - Hk[κ , μ]"}
+{"name":"ProbabilityTheory.kernel.entropy_snd_sub_mutualInfo_le_entropy_map_of_injective","declaration":"theorem ProbabilityTheory.kernel.entropy_snd_sub_mutualInfo_le_entropy_map_of_injective {S : Type u_2} {T : Type u_3} {U : Type u_4} [Nonempty S] [Countable S] [MeasurableSpace S] [MeasurableSingletonClass S] [Nonempty T] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] [Nonempty U] [Countable U] [MeasurableSpace U] [MeasurableSingletonClass U] {V : Type u_5} [Nonempty V] [Countable V] [MeasurableSpace V] [MeasurableSingletonClass V] (κ : ↥(ProbabilityTheory.kernel T (S × U))) [ProbabilityTheory.IsMarkovKernel κ] (μ : MeasureTheory.Measure T) [MeasureTheory.IsProbabilityMeasure μ] (f : S × U → V) (hfi : ∀ (x : S), Function.Injective fun y => f (x, y)) [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : Hk[ProbabilityTheory.kernel.snd κ , μ] - Ik[κ , μ] ≤ Hk[ProbabilityTheory.kernel.map κ f ⋯ , μ]"}
+{"name":"ProbabilityTheory.kernel.Measure.compProd_compProd'","declaration":"theorem ProbabilityTheory.kernel.Measure.compProd_compProd' {S : Type u_2} {T : Type u_3} {U : Type u_4} [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U] (μ : MeasureTheory.Measure T) [MeasureTheory.IsProbabilityMeasure μ] (ξ : ↥(ProbabilityTheory.kernel T S)) [ProbabilityTheory.IsMarkovKernel ξ] (κ : ↥(ProbabilityTheory.kernel (T × S) U)) [ProbabilityTheory.IsMarkovKernel κ] : MeasureTheory.Measure.compProd μ (ProbabilityTheory.kernel.compProd ξ κ) =\n  MeasureTheory.Measure.comap (⇑MeasurableEquiv.prodAssoc.symm)\n    (MeasureTheory.Measure.compProd (MeasureTheory.Measure.compProd μ ξ) κ)"}
+{"name":"ProbabilityTheory.kernel.mutualInfo_congr","declaration":"theorem ProbabilityTheory.kernel.mutualInfo_congr {S : Type u_2} {T : Type u_3} {U : Type u_4} [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U] {κ : ↥(ProbabilityTheory.kernel T (S × U))} {η : ↥(ProbabilityTheory.kernel T (S × U))} {μ : MeasureTheory.Measure T} (h : ⇑κ =ᶠ[MeasureTheory.Measure.ae μ] ⇑η) : Ik[κ , μ] = Ik[η , μ]"}
+{"name":"ProbabilityTheory.kernel.entropy_triple_add_entropy_le","declaration":"/-- The submodularity inequality:\n$$ H[X,Y,Z] + H[Z] \\leq H[X,Z] + H[Y,Z].$$ -/\ntheorem ProbabilityTheory.kernel.entropy_triple_add_entropy_le {V : Type u_5} {S : Type u_2} {T : Type u_3} {U : Type u_4} [Nonempty S] [Countable S] [MeasurableSpace S] [MeasurableSingletonClass S] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] [Nonempty U] [Countable U] [MeasurableSpace U] [MeasurableSingletonClass U] [Nonempty V] [Countable V] [MeasurableSpace V] [MeasurableSingletonClass V] (κ : ↥(ProbabilityTheory.kernel T (S × U × V))) [ProbabilityTheory.IsMarkovKernel κ] (μ : MeasureTheory.Measure T) [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : Hk[κ , μ] + Hk[ProbabilityTheory.kernel.snd (ProbabilityTheory.kernel.snd κ) , μ] ≤\n  Hk[ProbabilityTheory.kernel.deleteMiddle κ , μ] + Hk[ProbabilityTheory.kernel.snd κ , μ]"}
+{"name":"ProbabilityTheory.kernel.«termIk[_,_]».delab","declaration":"/-- Pretty printer defined by `notation3` command. -/\ndef ProbabilityTheory.kernel.«termIk[_,_]».delab  : Lean.PrettyPrinter.Delaborator.Delab"}
+{"name":"ProbabilityTheory.kernel.mutualInfo_compProd","declaration":"theorem ProbabilityTheory.kernel.mutualInfo_compProd {S : Type u_2} {T : Type u_3} {U : Type u_4} [Nonempty S] [Countable S] [MeasurableSpace S] [MeasurableSingletonClass S] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] [Nonempty U] [MeasurableSpace U] [MeasurableSingletonClass U] {κ : ↥(ProbabilityTheory.kernel T S)} [ProbabilityTheory.IsMarkovKernel κ] {η : ↥(ProbabilityTheory.kernel (T × S) U)} [ProbabilityTheory.IsMarkovKernel η] {μ : MeasureTheory.Measure T} [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) (hη : ProbabilityTheory.kernel.AEFiniteKernelSupport η (MeasureTheory.Measure.compProd μ κ)) : Ik[ProbabilityTheory.kernel.compProd κ η , μ] =\n  Hk[κ , μ] + Hk[ProbabilityTheory.kernel.snd (ProbabilityTheory.kernel.compProd κ η) , μ] -\n    Hk[ProbabilityTheory.kernel.compProd κ η , μ]"}
+{"name":"ProbabilityTheory.kernel.mutualInfo_eq_snd_sub","declaration":"theorem ProbabilityTheory.kernel.mutualInfo_eq_snd_sub {S : Type u_2} {T : Type u_3} {U : Type u_4} [Nonempty S] [Countable S] [MeasurableSpace S] [MeasurableSingletonClass S] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] [Nonempty U] [Countable U] [MeasurableSpace U] [MeasurableSingletonClass U] {κ : ↥(ProbabilityTheory.kernel T (S × U))} [ProbabilityTheory.IsMarkovKernel κ] {μ : MeasureTheory.Measure T} [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : Ik[κ , μ] =\n  Hk[ProbabilityTheory.kernel.snd κ , μ] -\n    Hk[ProbabilityTheory.kernel.condKernel κ , MeasureTheory.Measure.compProd μ (ProbabilityTheory.kernel.fst κ)]"}
+{"name":"ProbabilityTheory.kernel.Measure.compProd_compProd","declaration":"theorem ProbabilityTheory.kernel.Measure.compProd_compProd {S : Type u_2} {T : Type u_3} {U : Type u_4} [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U] (μ : MeasureTheory.Measure T) [MeasureTheory.IsProbabilityMeasure μ] (ξ : ↥(ProbabilityTheory.kernel T S)) [ProbabilityTheory.IsMarkovKernel ξ] (κ : ↥(ProbabilityTheory.kernel (T × S) U)) [ProbabilityTheory.IsMarkovKernel κ] : MeasureTheory.Measure.compProd μ (ProbabilityTheory.kernel.compProd ξ κ) =\n  MeasureTheory.Measure.map (⇑MeasurableEquiv.prodAssoc)\n    (MeasureTheory.Measure.compProd (MeasureTheory.Measure.compProd μ ξ) κ)"}
+{"name":"ProbabilityTheory.kernel.entropy_condKernel_le_entropy_fst","declaration":"theorem ProbabilityTheory.kernel.entropy_condKernel_le_entropy_fst {S : Type u_2} {T : Type u_3} {U : Type u_4} [Nonempty S] [Countable S] [MeasurableSpace S] [MeasurableSingletonClass S] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] [Nonempty U] [Countable U] [MeasurableSpace U] [MeasurableSingletonClass U] (κ : ↥(ProbabilityTheory.kernel T (S × U))) [ProbabilityTheory.IsMarkovKernel κ] (μ : MeasureTheory.Measure T) [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) : Hk[ProbabilityTheory.kernel.condKernel (ProbabilityTheory.kernel.swapRight κ) ,\n    MeasureTheory.Measure.compProd μ (ProbabilityTheory.kernel.snd κ)] ≤\n  Hk[ProbabilityTheory.kernel.fst κ , μ]"}
+{"name":"ProbabilityTheory.kernel.mutualInfo_zero_kernel","declaration":"theorem ProbabilityTheory.kernel.mutualInfo_zero_kernel {S : Type u_2} {T : Type u_3} {U : Type u_4} [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U] (μ : MeasureTheory.Measure T) : Ik[0 , μ] = 0"}
+{"name":"ProbabilityTheory.kernel.mutualInfo","declaration":"/-- Mutual information of a kernel into a product space with respect to a measure. -/\ndef ProbabilityTheory.kernel.mutualInfo {S : Type u_2} {T : Type u_3} {U : Type u_4} [MeasurableSpace S] [MeasurableSpace T] [MeasurableSpace U] (κ : ↥(ProbabilityTheory.kernel T (S × U))) (μ : MeasureTheory.Measure T) : ℝ"}
+{"name":"ProbabilityTheory.kernel.entropy_submodular_compProd","declaration":"theorem ProbabilityTheory.kernel.entropy_submodular_compProd {V : Type u_5} {S : Type u_2} {T : Type u_3} {U : Type u_4} [Nonempty S] [Countable S] [MeasurableSpace S] [MeasurableSingletonClass S] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] [Nonempty U] [Countable U] [MeasurableSpace U] [MeasurableSingletonClass U] [Nonempty V] [Countable V] [MeasurableSpace V] [MeasurableSingletonClass V] {ξ : ↥(ProbabilityTheory.kernel T S)} [ProbabilityTheory.IsMarkovKernel ξ] {κ : ↥(ProbabilityTheory.kernel (T × S) U)} [ProbabilityTheory.IsMarkovKernel κ] {η : ↥(ProbabilityTheory.kernel (T × S × U) V)} [ProbabilityTheory.IsMarkovKernel η] {μ : MeasureTheory.Measure T} [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ (MeasureTheory.Measure.compProd μ ξ)) (hη : ProbabilityTheory.kernel.AEFiniteKernelSupport η\n  (MeasureTheory.Measure.compProd μ (ProbabilityTheory.kernel.compProd ξ κ))) (hξ : ProbabilityTheory.kernel.AEFiniteKernelSupport ξ μ) : Hk[η , MeasureTheory.Measure.compProd μ (ProbabilityTheory.kernel.compProd ξ κ)] ≤\n  Hk[ProbabilityTheory.kernel.snd\n      (ProbabilityTheory.kernel.compProd κ (ProbabilityTheory.kernel.comap η ⇑MeasurableEquiv.prodAssoc ⋯)) ,\n    MeasureTheory.Measure.compProd μ ξ]"}

PFR-declarations/PFR.ForMathlib.Entropy.Kernel.RuzsaDist.jsonl

@@ -0,0 +1,20 @@
+{"name":"ProbabilityTheory.kernel.rdist_zero_kernel_left","declaration":"theorem ProbabilityTheory.kernel.rdist_zero_kernel_left {T : Type u_1} {T' : Type u_2} {G : Type u_4} [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T] [Countable T'] [MeasurableSpace T'] [MeasurableSingletonClass T'] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [MeasurableSub₂ G] {η : ↥(ProbabilityTheory.kernel T' G)} [ProbabilityTheory.IsFiniteKernel η] {μ : MeasureTheory.Measure T} {ν : MeasureTheory.Measure T'} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν] [ProbabilityTheory.FiniteSupport μ] [ProbabilityTheory.FiniteSupport ν] : dk[0 ; μ # η ; ν] = -Hk[η , ν] / 2"}
+{"name":"ProbabilityTheory.kernel.rdist","declaration":"/-- The Rusza distance between two kernels taking values in the same space, defined as the average\nRusza distance between the image measures. -/\ndef ProbabilityTheory.kernel.rdist {T : Type u_1} {T' : Type u_2} {G : Type u_4} [MeasurableSpace T] [MeasurableSpace T'] [MeasurableSpace G] [AddCommGroup G] (κ : ↥(ProbabilityTheory.kernel T G)) (η : ↥(ProbabilityTheory.kernel T' G)) (μ : MeasureTheory.Measure T) (ν : MeasureTheory.Measure T') : ℝ"}
+{"name":"ProbabilityTheory.kernel.rdist_zero_left","declaration":"theorem ProbabilityTheory.kernel.rdist_zero_left {T : Type u_1} {T' : Type u_2} {G : Type u_4} [MeasurableSpace T] [MeasurableSpace T'] [MeasurableSpace G] [AddCommGroup G] (κ : ↥(ProbabilityTheory.kernel T G)) (η : ↥(ProbabilityTheory.kernel T' G)) (ν' : MeasureTheory.Measure T') : dk[κ ; 0 # η ; ν'] = 0"}
+{"name":"ProbabilityTheory.kernel.rdist_symm","declaration":"theorem ProbabilityTheory.kernel.rdist_symm {T : Type u_1} {T' : Type u_2} {G : Type u_4} [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T] [Countable T'] [MeasurableSpace T'] [MeasurableSingletonClass T'] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [MeasurableSub₂ G] {κ : ↥(ProbabilityTheory.kernel T G)} {η : ↥(ProbabilityTheory.kernel T' G)} [ProbabilityTheory.IsFiniteKernel κ] [ProbabilityTheory.IsFiniteKernel η] {μ : MeasureTheory.Measure T} {ν : MeasureTheory.Measure T'} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν] [ProbabilityTheory.FiniteSupport μ] [ProbabilityTheory.FiniteSupport ν] : dk[κ ; μ # η ; ν] = dk[η ; ν # κ ; μ]"}
+{"name":"ProbabilityTheory.kernel.rdist_dirac_zero_left","declaration":"theorem ProbabilityTheory.kernel.rdist_dirac_zero_left {T : Type u_1} {T' : Type u_2} {G : Type u_4} [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T] [Countable T'] [MeasurableSpace T'] [MeasurableSingletonClass T'] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [MeasurableSub₂ G] {η : ↥(ProbabilityTheory.kernel T' G)} [ProbabilityTheory.IsFiniteKernel η] {μ : MeasureTheory.Measure T} {ν : MeasureTheory.Measure T'} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν] [ProbabilityTheory.FiniteSupport μ] [ProbabilityTheory.FiniteSupport ν] : dk[ProbabilityTheory.kernel.const T (MeasureTheory.Measure.dirac 0) ; μ # η ; ν] = Hk[η , ν] / 2"}
+{"name":"ProbabilityTheory.kernel.abs_sub_entropy_le_rdist","declaration":"theorem ProbabilityTheory.kernel.abs_sub_entropy_le_rdist {T : Type u_1} {T' : Type u_2} {G : Type u_4} [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T] [Countable T'] [Nonempty T'] [MeasurableSpace T'] [MeasurableSingletonClass T'] [Countable G] [Nonempty G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [MeasurableSub₂ G] {κ : ↥(ProbabilityTheory.kernel T G)} {η : ↥(ProbabilityTheory.kernel T' G)} [ProbabilityTheory.IsMarkovKernel κ] [ProbabilityTheory.IsMarkovKernel η] {μ : MeasureTheory.Measure T} {ν : MeasureTheory.Measure T'} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν] [ProbabilityTheory.FiniteSupport μ] [ProbabilityTheory.FiniteSupport ν] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) (hη : ProbabilityTheory.kernel.AEFiniteKernelSupport η ν) : |Hk[κ , μ] - Hk[η , ν]| ≤ 2 * dk[κ ; μ # η ; ν]"}
+{"name":"ProbabilityTheory.kernel.rdist_zero_kernel_right","declaration":"theorem ProbabilityTheory.kernel.rdist_zero_kernel_right {T : Type u_1} {T' : Type u_2} {G : Type u_4} [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] [Countable T'] [MeasurableSpace T'] [MeasurableSingletonClass T'] [MeasurableSpace G] [AddCommGroup G] [MeasurableSub₂ G] {κ : ↥(ProbabilityTheory.kernel T G)} [ProbabilityTheory.IsFiniteKernel κ] {μ : MeasureTheory.Measure T} {ν : MeasureTheory.Measure T'} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν] [ProbabilityTheory.FiniteSupport μ] [ProbabilityTheory.FiniteSupport ν] : dk[κ ; μ # 0 ; ν] = -Hk[κ , μ] / 2"}
+{"name":"ProbabilityTheory.kernel.rdist_zero_right","declaration":"theorem ProbabilityTheory.kernel.rdist_zero_right {T : Type u_1} {T' : Type u_2} {G : Type u_4} [MeasurableSpace T] [MeasurableSpace T'] [MeasurableSpace G] [AddCommGroup G] (κ : ↥(ProbabilityTheory.kernel T G)) (η : ↥(ProbabilityTheory.kernel T' G)) (μ : MeasureTheory.Measure T) : dk[κ ; μ # η ; 0] = 0"}
+{"name":"ProbabilityTheory.kernel.rdist_nonneg","declaration":"theorem ProbabilityTheory.kernel.rdist_nonneg {T : Type u_1} {T' : Type u_2} {G : Type u_4} [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T] [Countable T'] [Nonempty T'] [MeasurableSpace T'] [MeasurableSingletonClass T'] [Countable G] [Nonempty G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [MeasurableSub₂ G] {κ : ↥(ProbabilityTheory.kernel T G)} {η : ↥(ProbabilityTheory.kernel T' G)} [ProbabilityTheory.IsMarkovKernel κ] [ProbabilityTheory.IsMarkovKernel η] {μ : MeasureTheory.Measure T} {ν : MeasureTheory.Measure T'} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν] [ProbabilityTheory.FiniteSupport μ] [ProbabilityTheory.FiniteSupport ν] (hκ : ProbabilityTheory.kernel.AEFiniteKernelSupport κ μ) (hη : ProbabilityTheory.kernel.AEFiniteKernelSupport η ν) : 0 ≤ dk[κ ; μ # η ; ν]"}
+{"name":"ProbabilityTheory.kernel.ent_of_diff_le","declaration":"/-- The **improved entropic Ruzsa triangle inequality**. -/\ntheorem ProbabilityTheory.kernel.ent_of_diff_le {T : Type u_1} {G : Type u_4} [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] [Countable G] [Nonempty G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [MeasurableSub₂ G] (κ : ↥(ProbabilityTheory.kernel T (G × G))) (η : ↥(ProbabilityTheory.kernel T G)) [ProbabilityTheory.IsMarkovKernel κ] [ProbabilityTheory.IsMarkovKernel η] (μ : MeasureTheory.Measure T) [MeasureTheory.IsProbabilityMeasure μ] [ProbabilityTheory.FiniteSupport μ] (hκ : ProbabilityTheory.kernel.FiniteKernelSupport κ) (hη : ProbabilityTheory.kernel.FiniteKernelSupport η) : Hk[ProbabilityTheory.kernel.map κ (fun p => p.1 - p.2) ⋯ , μ] ≤\n  Hk[ProbabilityTheory.kernel.map (ProbabilityTheory.kernel.prod (ProbabilityTheory.kernel.fst κ) η)\n          (fun p => p.1 - p.2) ⋯ ,\n        μ] +\n      Hk[ProbabilityTheory.kernel.map (ProbabilityTheory.kernel.prod η (ProbabilityTheory.kernel.snd κ))\n          (fun p => p.1 - p.2) ⋯ ,\n        μ] -\n    Hk[η , μ]"}
+{"name":"ProbabilityTheory.kernel.«termDk[_;_#_;_]»","declaration":"/-- The Rusza distance between two kernels taking values in the same space, defined as the average\nRusza distance between the image measures. -/\ndef ProbabilityTheory.kernel.«termDk[_;_#_;_]»  : Lean.ParserDescr"}
+{"name":"ProbabilityTheory.kernel.«termDk[_;_#_;_]».delab","declaration":"/-- Pretty printer defined by `notation3` command. -/\ndef ProbabilityTheory.kernel.«termDk[_;_#_;_]».delab  : Lean.PrettyPrinter.Delaborator.Delab"}
+{"name":"ProbabilityTheory.kernel.ruzsa_triangle_aux","declaration":"theorem ProbabilityTheory.kernel.ruzsa_triangle_aux {T : Type u_1} {G : Type u_4} [MeasurableSpace T] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] (κ : ↥(ProbabilityTheory.kernel T (G × G))) (η : ↥(ProbabilityTheory.kernel T G)) [ProbabilityTheory.IsMarkovKernel κ] [ProbabilityTheory.IsMarkovKernel η] : ProbabilityTheory.kernel.map (ProbabilityTheory.kernel.prod κ η) (fun p => p.2 - p.1.2) ⋯ =\n  ProbabilityTheory.kernel.map (ProbabilityTheory.kernel.prod η (ProbabilityTheory.kernel.snd κ)) (fun p => p.1 - p.2) ⋯"}
+{"name":"ProbabilityTheory.kernel.rdist_eq","declaration":"theorem ProbabilityTheory.kernel.rdist_eq {T : Type u_1} {T' : Type u_2} {G : Type u_4} [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] [Countable T'] [MeasurableSpace T'] [MeasurableSingletonClass T'] [MeasurableSpace G] [AddCommGroup G] {κ : ↥(ProbabilityTheory.kernel T G)} {η : ↥(ProbabilityTheory.kernel T' G)} {μ : MeasureTheory.Measure T} {ν : MeasureTheory.Measure T'} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν] [ProbabilityTheory.FiniteSupport μ] [ProbabilityTheory.FiniteSupport ν] : dk[κ ; μ # η ; ν] =\n  ∫ (x : T × T'),\n        (fun p => Hm[MeasureTheory.Measure.map (fun x => x.1 - x.2) (MeasureTheory.Measure.prod (κ p.1) (η p.2))])\n          x ∂MeasureTheory.Measure.prod μ ν -\n      Hk[κ , μ] / 2 -\n    Hk[η , ν] / 2"}
+{"name":"ProbabilityTheory.kernel.rdist_eq'","declaration":"theorem ProbabilityTheory.kernel.rdist_eq' {T : Type u_1} {T' : Type u_2} {G : Type u_4} [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] [Countable T'] [MeasurableSpace T'] [MeasurableSingletonClass T'] [MeasurableSpace G] [AddCommGroup G] [MeasurableSub₂ G] {κ : ↥(ProbabilityTheory.kernel T G)} {η : ↥(ProbabilityTheory.kernel T' G)} [ProbabilityTheory.IsFiniteKernel κ] [ProbabilityTheory.IsFiniteKernel η] {μ : MeasureTheory.Measure T} {ν : MeasureTheory.Measure T'} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν] [ProbabilityTheory.FiniteSupport μ] [ProbabilityTheory.FiniteSupport ν] : dk[κ ; μ # η ; ν] =\n  Hk[ProbabilityTheory.kernel.map\n          (ProbabilityTheory.kernel.prod (ProbabilityTheory.kernel.prodMkRight T' κ)\n            (ProbabilityTheory.kernel.prodMkLeft T η))\n          (fun x => x.1 - x.2) ⋯ ,\n        MeasureTheory.Measure.prod μ ν] -\n      Hk[κ , μ] / 2 -\n    Hk[η , ν] / 2"}
+{"name":"ProbabilityTheory.kernel.rdist_dirac_zero_right","declaration":"theorem ProbabilityTheory.kernel.rdist_dirac_zero_right {T : Type u_1} {T' : Type u_2} {G : Type u_4} [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] [Countable T'] [Nonempty T'] [MeasurableSpace T'] [MeasurableSingletonClass T'] [MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [MeasurableSub₂ G] {κ : ↥(ProbabilityTheory.kernel T G)} [ProbabilityTheory.IsFiniteKernel κ] {μ : MeasureTheory.Measure T} {ν : MeasureTheory.Measure T'} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν] [ProbabilityTheory.FiniteSupport μ] [ProbabilityTheory.FiniteSupport ν] : dk[κ ; μ # ProbabilityTheory.kernel.const T' (MeasureTheory.Measure.dirac 0) ; ν] = Hk[κ , μ] / 2"}
+{"name":"ProbabilityTheory.kernel.rdistm","declaration":"/-- The Rusza distance between two measures, defined as `H[X - Y] - H[X]/2 - H[Y]/2` where `X`\nand `Y` are independent variables distributed according to the two measures. -/\ndef ProbabilityTheory.kernel.rdistm {G : Type u_4} [MeasurableSpace G] [AddCommGroup G] (μ : MeasureTheory.Measure G) (ν : MeasureTheory.Measure G) : ℝ"}
+{"name":"ProbabilityTheory.kernel.rdist_triangle_aux1","declaration":"theorem ProbabilityTheory.kernel.rdist_triangle_aux1 {T : Type u_1} {T' : Type u_2} {T'' : Type u_3} {G : Type u_4} [MeasurableSpace T] [MeasurableSingletonClass T] [MeasurableSpace T'] [MeasurableSingletonClass T'] [MeasurableSpace T''] [MeasurableSingletonClass T''] [MeasurableSpace G] [AddCommGroup G] [MeasurableSub₂ G] (κ : ↥(ProbabilityTheory.kernel T G)) (η : ↥(ProbabilityTheory.kernel T' G)) [ProbabilityTheory.IsMarkovKernel κ] [ProbabilityTheory.IsMarkovKernel η] (μ : MeasureTheory.Measure T) (μ' : MeasureTheory.Measure T') (μ'' : MeasureTheory.Measure T'') [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure μ'] [MeasureTheory.IsProbabilityMeasure μ''] [ProbabilityTheory.FiniteSupport μ] [ProbabilityTheory.FiniteSupport μ'] [ProbabilityTheory.FiniteSupport μ''] : Hk[ProbabilityTheory.kernel.map\n      (ProbabilityTheory.kernel.prod\n        (ProbabilityTheory.kernel.prodMkRight T' (ProbabilityTheory.kernel.prodMkRight T'' κ))\n        (ProbabilityTheory.kernel.prodMkLeft (T × T'') η))\n      (fun p => p.1 - p.2) ⋯ ,\n    MeasureTheory.Measure.prod (MeasureTheory.Measure.prod μ μ'') μ'] =\n  Hk[ProbabilityTheory.kernel.map\n      (ProbabilityTheory.kernel.prod (ProbabilityTheory.kernel.prodMkRight T' κ)\n        (ProbabilityTheory.kernel.prodMkLeft T η))\n      (fun x => x.1 - x.2) ⋯ ,\n    MeasureTheory.Measure.prod μ μ']"}
+{"name":"ProbabilityTheory.kernel.rdist_triangle_aux2","declaration":"theorem ProbabilityTheory.kernel.rdist_triangle_aux2 {T : Type u_1} {T' : Type u_2} {T'' : Type u_3} {G : Type u_4} [MeasurableSpace T] [MeasurableSingletonClass T] [MeasurableSpace T'] [MeasurableSingletonClass T'] [MeasurableSpace T''] [MeasurableSingletonClass T''] [MeasurableSpace G] [AddCommGroup G] [MeasurableSub₂ G] (η : ↥(ProbabilityTheory.kernel T' G)) (ξ : ↥(ProbabilityTheory.kernel T'' G)) [ProbabilityTheory.IsMarkovKernel η] [ProbabilityTheory.IsMarkovKernel ξ] (μ : MeasureTheory.Measure T) (μ' : MeasureTheory.Measure T') (μ'' : MeasureTheory.Measure T'') [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure μ'] [MeasureTheory.IsProbabilityMeasure μ''] [ProbabilityTheory.FiniteSupport μ] [ProbabilityTheory.FiniteSupport μ'] [ProbabilityTheory.FiniteSupport μ''] : Hk[ProbabilityTheory.kernel.map\n      (ProbabilityTheory.kernel.prod (ProbabilityTheory.kernel.prodMkLeft (T × T'') η)\n        (ProbabilityTheory.kernel.prodMkRight T' (ProbabilityTheory.kernel.prodMkLeft T ξ)))\n      (fun p => p.1 - p.2) ⋯ ,\n    MeasureTheory.Measure.prod (MeasureTheory.Measure.prod μ μ'') μ'] =\n  Hk[ProbabilityTheory.kernel.map\n      (ProbabilityTheory.kernel.prod (ProbabilityTheory.kernel.prodMkRight T'' η)\n        (ProbabilityTheory.kernel.prodMkLeft T' ξ))\n      (fun x => x.1 - x.2) ⋯ ,\n    MeasureTheory.Measure.prod μ' μ'']"}
+{"name":"ProbabilityTheory.kernel.rdist_triangle","declaration":"theorem ProbabilityTheory.kernel.rdist_triangle {T : Type u_1} {T' : Type u_2} {T'' : Type u_3} {G : Type u_4} [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] [Countable T'] [Nonempty T'] [MeasurableSpace T'] [MeasurableSingletonClass T'] [Countable T''] [MeasurableSpace T''] [MeasurableSingletonClass T''] [Countable G] [Nonempty G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [MeasurableSub₂ G] (κ : ↥(ProbabilityTheory.kernel T G)) (η : ↥(ProbabilityTheory.kernel T' G)) (ξ : ↥(ProbabilityTheory.kernel T'' G)) [ProbabilityTheory.IsMarkovKernel κ] [ProbabilityTheory.IsMarkovKernel η] [ProbabilityTheory.IsMarkovKernel ξ] (μ : MeasureTheory.Measure T) (μ' : MeasureTheory.Measure T') (μ'' : MeasureTheory.Measure T'') [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure μ'] [MeasureTheory.IsProbabilityMeasure μ''] [ProbabilityTheory.FiniteSupport μ] [ProbabilityTheory.FiniteSupport μ'] [ProbabilityTheory.FiniteSupport μ''] (hκ : ProbabilityTheory.kernel.FiniteKernelSupport κ) (hη : ProbabilityTheory.kernel.FiniteKernelSupport η) (hξ : ProbabilityTheory.kernel.FiniteKernelSupport ξ) : dk[κ ; μ # ξ ; μ''] ≤ dk[κ ; μ # η ; μ'] + dk[η ; μ' # ξ ; μ'']"}

PFR-declarations/PFR.ForMathlib.Entropy.Measure.jsonl

@@ -0,0 +1,55 @@
+{"name":"ProbabilityTheory.measureEntropy_nonneg","declaration":"theorem ProbabilityTheory.measureEntropy_nonneg {S : Type u_2} [MeasurableSpace S] (μ : MeasureTheory.Measure S) : 0 ≤ Hm[μ]"}
+{"name":"ProbabilityTheory.measureEntropy_of_isProbabilityMeasure_finite","declaration":"theorem ProbabilityTheory.measureEntropy_of_isProbabilityMeasure_finite {S : Type u_2} [MeasurableSpace S] {μ : MeasureTheory.Measure S} {A : Finset S} (hA : ↑↑μ (↑A)ᶜ = 0) [MeasureTheory.IsProbabilityMeasure μ] : Hm[μ] = Finset.sum A fun s => Real.negMulLog (↑↑μ {s}).toReal"}
+{"name":"ProbabilityTheory.finiteSupport_zero","declaration":"instance ProbabilityTheory.finiteSupport_zero {S : Type u_2} [MeasurableSpace S] : ProbabilityTheory.FiniteSupport 0"}
+{"name":"ProbabilityTheory.«termHm[_]»","declaration":"/-- Entropy of a measure on a finite measurable space.\n\nWe normalize the measure by `(μ Set.univ)⁻¹` to extend the entropy definition to finite measures.\nWhat we really want to do is deal with `μ=0` or `IsProbabilityMeasure μ`, but we don't have\na typeclass for that (we could create one though).\nThe added complexity in the expression is not an issue because if `μ` is a probability measure,\na call to `simp` will simplify `(μ Set.univ)⁻¹ • μ` to `μ`. -/\ndef ProbabilityTheory.«termHm[_]»  : Lean.ParserDescr"}
+{"name":"ProbabilityTheory.finiteSupport_of_prod","declaration":"instance ProbabilityTheory.finiteSupport_of_prod {S : Type u_2} {T : Type u_3} [MeasurableSpace S] [MeasurableSpace T] {μ : MeasureTheory.Measure S} [ProbabilityTheory.FiniteSupport μ] {ν : MeasureTheory.Measure T} [MeasureTheory.SigmaFinite ν] [ProbabilityTheory.FiniteSupport ν] : ProbabilityTheory.FiniteSupport (MeasureTheory.Measure.prod μ ν)"}
+{"name":"ProbabilityTheory.measureEntropy","declaration":"/-- Entropy of a measure on a finite measurable space.\n\nWe normalize the measure by `(μ Set.univ)⁻¹` to extend the entropy definition to finite measures.\nWhat we really want to do is deal with `μ=0` or `IsProbabilityMeasure μ`, but we don't have\na typeclass for that (we could create one though).\nThe added complexity in the expression is not an issue because if `μ` is a probability measure,\na call to `simp` will simplify `(μ Set.univ)⁻¹ • μ` to `μ`. -/\ndef ProbabilityTheory.measureEntropy {S : Type u_2} [MeasurableSpace S] (μ : autoParam (MeasureTheory.Measure S) _auto✝) : ℝ"}
+{"name":"ProbabilityTheory.integrable_of_finiteSupport","declaration":"/-- The countability hypothesis can probably be dropped here. Proof is unwieldy and can probably\nbe golfed. -/\ntheorem ProbabilityTheory.integrable_of_finiteSupport {S : Type u_2} [MeasurableSpace S] [MeasurableSingletonClass S] (μ : MeasureTheory.Measure S) [ProbabilityTheory.FiniteSupport μ] {β : Type u_5} [NormedAddCommGroup β] [MeasureTheory.IsFiniteMeasure μ] [Countable S] {f : S → β} : MeasureTheory.Integrable f μ"}
+{"name":"ProbabilityTheory.measureEntropy_zero","declaration":"theorem ProbabilityTheory.measureEntropy_zero {S : Type u_2} [MeasurableSpace S] : Hm[0] = 0"}
+{"name":"ProbabilityTheory.measureEntropy_prod","declaration":"/-- An ambitious goal would be to replace FiniteSupport with finite entropy. -/\ntheorem ProbabilityTheory.measureEntropy_prod {S : Type u_2} {T : Type u_3} [MeasurableSpace S] [MeasurableSingletonClass S] [MeasurableSpace T] [MeasurableSingletonClass T] {μ : MeasureTheory.Measure S} {ν : MeasureTheory.Measure T} [ProbabilityTheory.FiniteSupport μ] [ProbabilityTheory.FiniteSupport ν] [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν] : Hm[MeasureTheory.Measure.prod μ ν] = Hm[μ] + Hm[ν]"}
+{"name":"ProbabilityTheory.FiniteSupport","declaration":"/-- A measure has finite support if there exsists a finite set whose complement has zero measure. -/\nclass ProbabilityTheory.FiniteSupport {S : Type u_2} [MeasurableSpace S] (μ : autoParam (MeasureTheory.Measure S) _auto✝) : Prop"}
+{"name":"ProbabilityTheory.«termIm[_]»","declaration":"/-- The mutual information between the marginals of a measure on a product space. -/\ndef ProbabilityTheory.«termIm[_]»  : Lean.ParserDescr"}
+{"name":"ProbabilityTheory.finiteSupport_of_mul","declaration":"instance ProbabilityTheory.finiteSupport_of_mul {S : Type u_2} [MeasurableSpace S] {μ : MeasureTheory.Measure S} [ProbabilityTheory.FiniteSupport μ] (c : ENNReal) : ProbabilityTheory.FiniteSupport (c • μ)"}
+{"name":"ProbabilityTheory.measureMutualInfo_swap","declaration":"theorem ProbabilityTheory.measureMutualInfo_swap {S : Type u_2} {T : Type u_3} [MeasurableSpace S] [MeasurableSingletonClass S] [MeasurableSpace T] [MeasurableSingletonClass T] (μ : MeasureTheory.Measure (S × T)) : Im[MeasureTheory.Measure.map Prod.swap μ] = Im[μ]"}
+{"name":"ProbabilityTheory.full_measure_of_finiteRange","declaration":"/-- duplicate of `FiniteRange.null_of_compl` -/\ntheorem ProbabilityTheory.full_measure_of_finiteRange {Ω : Type u_1} {S : Type u_2} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] [MeasurableSingletonClass S] {μ : MeasureTheory.Measure Ω} {X : Ω → S} (hX : Measurable X) [hX' : FiniteRange X] : ↑↑(MeasureTheory.Measure.map X μ) (↑(FiniteRange.toFinset X))ᶜ = 0"}
+{"name":"ProbabilityTheory.finiteSupport_of_finiteRange","declaration":"instance ProbabilityTheory.finiteSupport_of_finiteRange {Ω : Type u_1} {S : Type u_2} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] [MeasurableSingletonClass S] {μ : MeasureTheory.Measure Ω} {X : Ω → S} [hX' : FiniteRange X] : ProbabilityTheory.FiniteSupport (MeasureTheory.Measure.map X μ)"}
+{"name":"ProbabilityTheory.FiniteSupport.finite","declaration":"def ProbabilityTheory.FiniteSupport.finite {S : Type u_2} [MeasurableSpace S] {μ : autoParam (MeasureTheory.Measure S) _auto✝} [self : ProbabilityTheory.FiniteSupport μ] : ∃ A, ↑↑μ (↑A)ᶜ = 0"}
+{"name":"ProbabilityTheory.measureEntropy_univ_smul","declaration":"theorem ProbabilityTheory.measureEntropy_univ_smul {S : Type u_2} [MeasurableSpace S] {μ : MeasureTheory.Measure S} : Hm[(↑↑μ Set.univ)⁻¹ • μ] = Hm[μ]"}
+{"name":"ProbabilityTheory.measureMutualInfo_univ_smul","declaration":"theorem ProbabilityTheory.measureMutualInfo_univ_smul {S : Type u_2} {U : Type u_4} [MeasurableSpace S] [MeasurableSpace U] (μ : MeasureTheory.Measure (S × U)) : Im[(↑↑μ Set.univ)⁻¹ • μ] = Im[μ]"}
+{"name":"ProbabilityTheory.FiniteEntropy","declaration":"/-- TODO: replace FiniteSupport hypotheses in these files with FiniteEntropy hypotheses. -/\ndef ProbabilityTheory.FiniteEntropy {S : Type u_2} [MeasurableSpace S] (μ : autoParam (MeasureTheory.Measure S) _auto✝) : Prop"}
+{"name":"ProbabilityTheory.measureEntropy_of_isProbabilityMeasure","declaration":"theorem ProbabilityTheory.measureEntropy_of_isProbabilityMeasure {S : Type u_2} [MeasurableSpace S] (μ : MeasureTheory.Measure S) [MeasureTheory.IsProbabilityMeasure μ] : Hm[μ] = ∑' (s : S), Real.negMulLog (↑↑μ {s}).toReal"}
+{"name":"ProbabilityTheory.measureMutualInfo_eq_zero_iff","declaration":"theorem ProbabilityTheory.measureMutualInfo_eq_zero_iff {S : Type u_2} {U : Type u_4} [MeasurableSpace S] [MeasurableSingletonClass S] [MeasurableSpace U] [MeasurableSingletonClass U] {μ : MeasureTheory.Measure (S × U)} [ProbabilityTheory.FiniteSupport μ] [MeasureTheory.IsProbabilityMeasure μ] : Im[μ] = 0 ↔\n  ∀ (p : S × U),\n    μ.real {p} = (MeasureTheory.Measure.map Prod.fst μ).real {p.1} * (MeasureTheory.Measure.map Prod.snd μ).real {p.2}"}
+{"name":"ProbabilityTheory.measureEntropy_le_log_card_of_mem","declaration":"theorem ProbabilityTheory.measureEntropy_le_log_card_of_mem {S : Type u_2} [MeasurableSpace S] [MeasurableSingletonClass S] {A : Finset S} (μ : MeasureTheory.Measure S) (hμA : ↑↑μ (↑A)ᶜ = 0) : Hm[μ] ≤ Real.log ↑(Nat.card { x // x ∈ A })"}
+{"name":"ProbabilityTheory.measureMutualInfo_prod","declaration":"theorem ProbabilityTheory.measureMutualInfo_prod {S : Type u_2} {T : Type u_3} [MeasurableSpace S] [MeasurableSingletonClass S] [MeasurableSpace T] [MeasurableSingletonClass T] {μ : MeasureTheory.Measure S} {ν : MeasureTheory.Measure T} [ProbabilityTheory.FiniteSupport μ] [ProbabilityTheory.FiniteSupport ν] [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν] : Im[MeasureTheory.Measure.prod μ ν] = 0"}
+{"name":"MeasureTheory.Measure.support","declaration":"def MeasureTheory.Measure.support {S : Type u_2} [MeasurableSpace S] (μ : MeasureTheory.Measure S) [hμ : ProbabilityTheory.FiniteSupport μ] : Finset S"}
+{"name":"ProbabilityTheory.finiteSupport_of_dirac","declaration":"instance ProbabilityTheory.finiteSupport_of_dirac {S : Type u_2} [MeasurableSpace S] [MeasurableSingletonClass S] (x : S) : ProbabilityTheory.FiniteSupport (MeasureTheory.Measure.dirac x)"}
+{"name":"ProbabilityTheory.measureEntropy_of_isProbabilityMeasure_finite'","declaration":"theorem ProbabilityTheory.measureEntropy_of_isProbabilityMeasure_finite' {S : Type u_2} [MeasurableSpace S] {μ : MeasureTheory.Measure S} {A : Finset S} (hA : ↑↑μ (↑A)ᶜ = 0) [MeasureTheory.IsProbabilityMeasure μ] : Hm[μ] = Finset.sum A fun s => Real.negMulLog (μ.real {s})"}
+{"name":"ProbabilityTheory.measureEntropy_eq_card_iff_measure_eq_aux","declaration":"theorem ProbabilityTheory.measureEntropy_eq_card_iff_measure_eq_aux {S : Type u_2} [MeasurableSpace S] [MeasurableSingletonClass S] (μ : MeasureTheory.Measure S) [Fintype S] [MeasureTheory.IsProbabilityMeasure μ] : Hm[μ] = Real.log ↑(Fintype.card S) ↔ ∀ (s : S), ↑↑μ {s} = ↑(↑(Fintype.card S))⁻¹"}
+{"name":"ProbabilityTheory.FiniteSupport.mk","declaration":"ctor ProbabilityTheory.FiniteSupport.mk {S : Type u_2} [MeasurableSpace S] {μ : autoParam (MeasureTheory.Measure S) _auto✝} (finite : ∃ A, ↑↑μ (↑A)ᶜ = 0) : ProbabilityTheory.FiniteSupport μ"}
+{"name":"ProbabilityTheory.measureEntropy_le_card_aux","declaration":"theorem ProbabilityTheory.measureEntropy_le_card_aux {S : Type u_2} [MeasurableSpace S] [MeasurableSingletonClass S] {μ : MeasureTheory.Measure S} [MeasureTheory.IsProbabilityMeasure μ] (A : Finset S) (hμ : ↑��μ (↑A)ᶜ = 0) : Hm[μ] ≤ Real.log ↑A.card"}
+{"name":"ProbabilityTheory.measureEntropy_of_not_isFiniteMeasure","declaration":"theorem ProbabilityTheory.measureEntropy_of_not_isFiniteMeasure {S : Type u_2} [MeasurableSpace S] {μ : MeasureTheory.Measure S} (h : ¬MeasureTheory.IsFiniteMeasure μ) : Hm[μ] = 0"}
+{"name":"ProbabilityTheory.Measure.ext_iff_measureReal_singleton_finiteSupport","declaration":"theorem ProbabilityTheory.Measure.ext_iff_measureReal_singleton_finiteSupport {S : Type u_2} [MeasurableSpace S] [MeasurableSingletonClass S] {μ1 : MeasureTheory.Measure S} {μ2 : MeasureTheory.Measure S} [ProbabilityTheory.FiniteSupport μ1] [ProbabilityTheory.FiniteSupport μ2] [MeasureTheory.IsFiniteMeasure μ1] [MeasureTheory.IsFiniteMeasure μ2] : μ1 = μ2 ↔ ∀ (x : S), μ1.real {x} = μ2.real {x}"}
+{"name":"ProbabilityTheory.finiteSupport_of_fintype","declaration":"instance ProbabilityTheory.finiteSupport_of_fintype {S : Type u_2} [MeasurableSpace S] {μ : MeasureTheory.Measure S} [Fintype S] : ProbabilityTheory.FiniteSupport μ"}
+{"name":"ProbabilityTheory.measureMutualInfo_def","declaration":"theorem ProbabilityTheory.measureMutualInfo_def {S : Type u_2} {T : Type u_3} [MeasurableSpace S] [MeasurableSpace T] (μ : MeasureTheory.Measure (S × T)) : Im[μ] = Hm[MeasureTheory.Measure.map Prod.fst μ] + Hm[MeasureTheory.Measure.map Prod.snd μ] - Hm[μ]"}
+{"name":"ProbabilityTheory.measureMutualInfo","declaration":"/-- The mutual information between the marginals of a measure on a product space. -/\ndef ProbabilityTheory.measureMutualInfo {S : Type u_2} {T : Type u_3} [MeasurableSpace S] [MeasurableSpace T] (μ : autoParam (MeasureTheory.Measure (S × T)) _auto✝) : ℝ"}
+{"name":"ProbabilityTheory.measureEntropy_def","declaration":"theorem ProbabilityTheory.measureEntropy_def {S : Type u_2} [MeasurableSpace S] (μ : MeasureTheory.Measure S) : Hm[μ] = ∑' (s : S), Real.negMulLog (↑↑((↑↑μ Set.univ)⁻¹ • μ) {s}).toReal"}
+{"name":"ProbabilityTheory.measure_compl_support","declaration":"theorem ProbabilityTheory.measure_compl_support {S : Type u_2} [MeasurableSpace S] (μ : MeasureTheory.Measure S) [hμ : ProbabilityTheory.FiniteSupport μ] : ↑↑μ (↑(MeasureTheory.Measure.support μ))ᶜ = 0"}
+{"name":"ProbabilityTheory.measureEntropy_le_log_card","declaration":"theorem ProbabilityTheory.measureEntropy_le_log_card {S : Type u_2} [MeasurableSpace S] [MeasurableSingletonClass S] [Fintype S] (μ : MeasureTheory.Measure S) : Hm[μ] ≤ Real.log ↑(Fintype.card S)"}
+{"name":"ProbabilityTheory.Measure.ext_iff_singleton_finiteSupport","declaration":"/-- This generalizes Measure.ext_iff_singleton in MeasureReal -/\ntheorem ProbabilityTheory.Measure.ext_iff_singleton_finiteSupport {S : Type u_2} [MeasurableSpace S] [MeasurableSingletonClass S] {μ1 : MeasureTheory.Measure S} {μ2 : MeasureTheory.Measure S} [ProbabilityTheory.FiniteSupport μ1] [ProbabilityTheory.FiniteSupport μ2] : μ1 = μ2 ↔ ∀ (x : S), ↑↑μ1 {x} = ↑↑μ2 {x}"}
+{"name":"ProbabilityTheory.measureEntropy_eq_card_iff_measureReal_eq_aux","declaration":"theorem ProbabilityTheory.measureEntropy_eq_card_iff_measureReal_eq_aux {S : Type u_2} [MeasurableSpace S] [MeasurableSingletonClass S] [Fintype S] (μ : MeasureTheory.Measure S) [MeasureTheory.IsProbabilityMeasure μ] : Hm[μ] = Real.log ↑(Fintype.card S) ↔ ∀ (s : S), μ.real {s} = (↑(Fintype.card S))⁻¹"}
+{"name":"ProbabilityTheory.measureMutualInfo_of_not_isFiniteMeasure","declaration":"theorem ProbabilityTheory.measureMutualInfo_of_not_isFiniteMeasure {S : Type u_2} {U : Type u_4} [MeasurableSpace S] [MeasurableSpace U] {μ : MeasureTheory.Measure (S × U)} (h : ¬MeasureTheory.IsFiniteMeasure μ) : Im[μ] = 0"}
+{"name":"ProbabilityTheory.measureEntropy_def_finite'","declaration":"theorem ProbabilityTheory.measureEntropy_def_finite' {S : Type u_2} [MeasurableSpace S] {μ : MeasureTheory.Measure S} {A : Finset S} (hA : ↑↑μ (↑A)ᶜ = 0) : Hm[μ] = Finset.sum A fun s => Real.negMulLog (((μ.real Set.univ)⁻¹ • μ.real) {s})"}
+{"name":"ProbabilityTheory.measureEntropy_map_of_injective","declaration":"theorem ProbabilityTheory.measureEntropy_map_of_injective {S : Type u_2} {T : Type u_3} [MeasurableSpace S] [MeasurableSpace T] [MeasurableSingletonClass T] (μ : MeasureTheory.Measure S) (f : S → T) (hf_m : Measurable f) (hf : Function.Injective f) : Hm[MeasureTheory.Measure.map f μ] = Hm[μ]"}
+{"name":"ProbabilityTheory.measureEntropy_comap_equiv","declaration":"theorem ProbabilityTheory.measureEntropy_comap_equiv {S : Type u_2} {T : Type u_3} [MeasurableSpace S] [MeasurableSpace T] [MeasurableSingletonClass T] (μ : MeasureTheory.Measure S) (f : T ≃ᵐ S) : Hm[MeasureTheory.Measure.comap (⇑f) μ] = Hm[μ]"}
+{"name":"ProbabilityTheory.measureEntropy_comap","declaration":"theorem ProbabilityTheory.measureEntropy_comap {S : Type u_2} {T : Type u_3} [MeasurableSpace S] [MeasurableSpace T] [MeasurableSingletonClass T] (μ : MeasureTheory.Measure S) (f : T → S) (hf : MeasurableEmbedding f) (hf_range : Set.range f =ᶠ[MeasureTheory.Measure.ae μ] Set.univ) : Hm[MeasureTheory.Measure.comap f μ] = Hm[μ]"}
+{"name":"ProbabilityTheory.measureEntropy_dirac","declaration":"theorem ProbabilityTheory.measureEntropy_dirac {S : Type u_2} [MeasurableSpace S] [MeasurableSingletonClass S] (x : S) : Hm[MeasureTheory.Measure.dirac x] = 0"}
+{"name":"ProbabilityTheory.measureMutualInfo_zero_measure","declaration":"theorem ProbabilityTheory.measureMutualInfo_zero_measure {S : Type u_2} {T : Type u_3} [MeasurableSpace S] [MeasurableSpace T] : Im[0] = 0"}
+{"name":"ProbabilityTheory.measureMutualInfo_nonneg_aux","declaration":"/-- An ambitious goal would be to replace FiniteSupport with finite entropy.  Proof is long and slow; needs to be optimized -/\ntheorem ProbabilityTheory.measureMutualInfo_nonneg_aux {S : Type u_2} {U : Type u_4} [MeasurableSpace S] [MeasurableSingletonClass S] [MeasurableSpace U] [MeasurableSingletonClass U] {μ : MeasureTheory.Measure (S × U)} [ProbabilityTheory.FiniteSupport μ] [MeasureTheory.IsProbabilityMeasure μ] : 0 ≤ Im[μ] ∧\n  (Im[μ] = 0 ↔\n    ∀ (p : S × U),\n      μ.real {p} =\n        (MeasureTheory.Measure.map Prod.fst μ).real {p.1} * (MeasureTheory.Measure.map Prod.snd μ).real {p.2})"}
+{"name":"ProbabilityTheory.measureEntropy_def'","declaration":"theorem ProbabilityTheory.measureEntropy_def' {S : Type u_2} [MeasurableSpace S] (μ : MeasureTheory.Measure S) : Hm[μ] = ∑' (s : S), Real.negMulLog (((μ.real Set.univ)⁻¹ • μ.real) {s})"}
+{"name":"ProbabilityTheory.measureEntropy_def_finite","declaration":"theorem ProbabilityTheory.measureEntropy_def_finite {S : Type u_2} [MeasurableSpace S] {μ : MeasureTheory.Measure S} {A : Finset S} (hA : ↑↑μ (↑A)ᶜ = 0) : Hm[μ] = Finset.sum A fun s => Real.negMulLog (↑↑((↑↑μ Set.univ)⁻¹ • μ) {s}).toReal"}
+{"name":"ProbabilityTheory.measureEntropy_of_isProbabilityMeasure'","declaration":"theorem ProbabilityTheory.measureEntropy_of_isProbabilityMeasure' {S : Type u_2} [MeasurableSpace S] (μ : MeasureTheory.Measure S) [MeasureTheory.IsProbabilityMeasure μ] : Hm[μ] = ∑' (s : S), Real.negMulLog (μ.real {s})"}
+{"name":"ProbabilityTheory.measureEntropy_eq_card_iff_measureReal_eq","declaration":"theorem ProbabilityTheory.measureEntropy_eq_card_iff_measureReal_eq {S : Type u_2} [MeasurableSpace S] {μ : MeasureTheory.Measure S} [MeasurableSingletonClass S] [Fintype S] [MeasureTheory.IsFiniteMeasure μ] [NeZero μ] : Hm[μ] = Real.log ↑(Fintype.card S) ↔ ∀ (s : S), μ.real {s} = μ.real Set.univ / ↑(Fintype.card S)"}
+{"name":"ProbabilityTheory.integral_congr_finiteSupport","declaration":"theorem ProbabilityTheory.integral_congr_finiteSupport {Ω : Type u_1} [mΩ : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} {G : Type u_5} [MeasurableSingletonClass Ω] [NormedAddCommGroup G] [NormedSpace ℝ G] [CompleteSpace G] {f : Ω → G} {g : Ω → G} [ProbabilityTheory.FiniteSupport μ] [MeasureTheory.IsFiniteMeasure μ] (hfg : ∀ (x : Ω), ↑↑μ {x} ≠ 0 → f x = g x) : ∫ (x : Ω), f x ∂μ = ∫ (x : Ω), g x ∂μ"}
+{"name":"ProbabilityTheory.measureMutualInfo_nonneg","declaration":"theorem ProbabilityTheory.measureMutualInfo_nonneg {S : Type u_2} {U : Type u_4} [MeasurableSpace S] [MeasurableSingletonClass S] [MeasurableSpace U] [MeasurableSingletonClass U] {μ : MeasureTheory.Measure (S × U)} [ProbabilityTheory.FiniteSupport μ] : 0 ≤ Im[μ]"}
+{"name":"ProbabilityTheory.finiteSupport_of_comp","declaration":"theorem ProbabilityTheory.finiteSupport_of_comp {Ω : Type u_1} {S : Type u_2} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] [MeasurableSingletonClass S] {μ : MeasureTheory.Measure Ω} [ProbabilityTheory.FiniteSupport μ] {X : Ω → S} (hX : Measurable X) : ProbabilityTheory.FiniteSupport (MeasureTheory.Measure.map X μ)"}
+{"name":"ProbabilityTheory.measureEntropy_eq_card_iff_measure_eq","declaration":"theorem ProbabilityTheory.measureEntropy_eq_card_iff_measure_eq {S : Type u_2} [MeasurableSpace S] {μ : MeasureTheory.Measure S} [MeasurableSingletonClass S] [Fintype S] [MeasureTheory.IsFiniteMeasure μ] [NeZero μ] : Hm[μ] = Real.log ↑(Fintype.card S) ↔ ∀ (s : S), ↑↑μ {s} = ↑↑μ Set.univ / ↑(Fintype.card S)"}

PFR-declarations/PFR.ForMathlib.Entropy.RuzsaDist.jsonl

@@ -0,0 +1,75 @@
+{"name":"ent_of_diff_le","declaration":"/-- The **improved entropic Ruzsa triangle inequality**. -/\ntheorem ent_of_diff_le {Ω : Type u_1} {G : Type u_5} [mΩ : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [MeasurableSub₂ G] [Countable G] (X : Ω → G) (Y : Ω → G) (Z : Ω → G) (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) (h : ProbabilityTheory.IndepFun (⟨X, Y⟩) Z μ) [MeasureTheory.IsProbabilityMeasure μ] [FiniteRange X] [FiniteRange Y] [FiniteRange Z] : H[X - Y ; μ] ≤ H[X - Z ; μ] + H[Z - Y ; μ] - H[Z ; μ]"}
+{"name":"condRuzsaDist'_eq_sum","declaration":"/-- Explicit formula for conditional Ruzsa distance $d[X ; Y|W]$. -/\ntheorem condRuzsaDist'_eq_sum {Ω : Type u_1} {Ω' : Type u_2} {G : Type u_5} {T : Type u_6} [mΩ : MeasurableSpace Ω] [mΩ' : MeasurableSpace Ω'] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [Countable G] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] {X : Ω → G} {Y : Ω' → G} {W : Ω' → T} (hY : Measurable Y) (hW : Measurable W) (μ : MeasureTheory.Measure Ω) (μ' : MeasureTheory.Measure Ω') [MeasureTheory.IsFiniteMeasure μ'] [FiniteRange W] : d[X ; μ # Y | W ; μ'] =\n  Finset.sum (FiniteRange.toFinset W) fun w =>\n    (↑↑μ' (W ⁻¹' {w})).toReal * d[X ; μ # Y ; ProbabilityTheory.cond μ' (W ⁻¹' {w})]"}
+{"name":"«termD[_|_#_|_]».delab","declaration":"/-- Pretty printer defined by `notation3` command. -/\ndef «termD[_|_#_|_]».delab  : Lean.PrettyPrinter.Delaborator.Delab"}
+{"name":"condRuzsaDist'_of_copy","declaration":"theorem condRuzsaDist'_of_copy {Ω : Type u_1} {Ω' : Type u_2} {Ω'' : Type u_3} {Ω''' : Type u_4} {G : Type u_5} {T : Type u_6} [mΩ : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [mΩ' : MeasurableSpace Ω'] {μ' : MeasureTheory.Measure Ω'} [mΩ'' : MeasurableSpace Ω''] {μ'' : MeasureTheory.Measure Ω''} [mΩ''' : MeasurableSpace Ω'''] {μ''' : MeasureTheory.Measure Ω'''} [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [Countable G] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] (X : Ω → G) {Y : Ω' → G} (hY : Measurable Y) {W : Ω' → T} (hW : Measurable W) (X' : Ω'' → G) {Y' : Ω''' → G} (hY' : Measurable Y') {W' : Ω''' → T} (hW' : Measurable W') [MeasureTheory.IsFiniteMeasure μ'] [MeasureTheory.IsFiniteMeasure μ'''] (h1 : ProbabilityTheory.IdentDistrib X X' μ μ'') (h2 : ProbabilityTheory.IdentDistrib (⟨Y, W⟩) (⟨Y', W'⟩) μ' μ''') [FiniteRange W] [FiniteRange W'] : d[X ; μ # Y | W ; μ'] = d[X' ; μ'' # Y' | W' ; μ''']"}
+{"name":"condRuzsaDist_def","declaration":"theorem condRuzsaDist_def {S : Type u_7} {Ω : Type u_1} {Ω' : Type u_2} {G : Type u_5} {T : Type u_6} [mΩ : MeasurableSpace Ω] [mΩ' : MeasurableSpace Ω'] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [Countable G] [MeasurableSpace S] [MeasurableSpace T] (X : Ω → G) (Z : Ω → S) (Y : Ω' → G) (W : Ω' → T) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] (μ' : MeasureTheory.Measure Ω') [MeasureTheory.IsFiniteMeasure μ'] : d[X | Z ; μ # Y | W ; μ'] =\n  dk[ProbabilityTheory.condDistrib X Z μ ; MeasureTheory.Measure.map Z μ # ProbabilityTheory.condDistrib Y W μ' ;\n    MeasureTheory.Measure.map W μ']"}
+{"name":"continuous_rdist_restrict_probabilityMeasure","declaration":"/-- Ruzsa distance depends continuously on the measure. -/\ntheorem continuous_rdist_restrict_probabilityMeasure {G : Type u_5} [hG : MeasurableSpace G] [AddCommGroup G] [Countable G] [Fintype G] [TopologicalSpace G] [DiscreteTopology G] [BorelSpace G] : Continuous fun μ => d[id ; ↑μ.1 # id ; ↑μ.2]"}
+{"name":"«termD[_;_#_|_;_]».delab","declaration":"/-- Pretty printer defined by `notation3` command. -/\ndef «termD[_;_#_|_;_]».delab  : Lean.PrettyPrinter.Delaborator.Delab"}
+{"name":"condRuzsaDist_le'_prod","declaration":"theorem condRuzsaDist_le'_prod {Ω : Type u_1} {Ω' : Type u_2} {G : Type u_5} {T : Type u_6} [mΩ : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) [mΩ' : MeasurableSpace Ω'] (μ' : MeasureTheory.Measure Ω') [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [MeasurableSub₂ G] [Countable G] [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T] {X : Ω → G} {Y : Ω' → G} {W : Ω' → T} {Z : Ω' → T} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure μ'] (hX : Measurable X) (hY : Measurable Y) (hW : Measurable W) (hZ : Measurable Z) [FiniteRange X] [FiniteRange Y] [FiniteRange W] [FiniteRange Z] : d[X ; μ # Y | ⟨W, Z⟩ ; μ'] ≤ d[X ; μ # Y | Z ; μ'] + I[Y : W|Z;μ'] / 2"}
+{"name":"rdist_add_const'","declaration":"/-- A variant of `rdist_add_const` where one adds constants to both variables. -/\ntheorem rdist_add_const' {Ω : Type u_1} {Ω' : Type u_2} {G : Type u_5} [mΩ : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [mΩ' : MeasurableSpace Ω'] {μ' : MeasureTheory.Measure Ω'} [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [MeasurableSub₂ G] [MeasurableAdd₂ G] [Countable G] {X : Ω → G} {Y : Ω' → G} [FiniteRange X] [FiniteRange Y] [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure μ'] (c : G) (c' : G) (hX : Measurable X) (hY : Measurable Y) : d[X + fun x => c ; μ # Y + fun x => c' ; μ'] = d[X ; μ # Y ; μ']"}
+{"name":"condRuzsaDist_symm","declaration":"/-- $$ d[X|Z; Y|W] = d[Y|W; X|Z]$$-/\ntheorem condRuzsaDist_symm {S : Type u_7} {Ω : Type u_1} {Ω' : Type u_2} {G : Type u_5} {T : Type u_6} [mΩ : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [mΩ' : MeasurableSpace Ω'] {μ' : MeasureTheory.Measure Ω'} [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [MeasurableSub₂ G] [Countable G] [Countable S] [Nonempty S] [MeasurableSpace S] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass S] [MeasurableSingletonClass T] {X : Ω → G} {Z : Ω → S} {Y : Ω' → G} {W : Ω' → T} (hZ : Measurable Z) (hW : Measurable W) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure μ'] [FiniteRange Z] [FiniteRange W] : d[X | Z ; μ # Y | W ; μ'] = d[Y | W ; μ' # X | Z ; μ]"}
+{"name":"entropy_sub_entropy_eq_condRuzsaDist_add","declaration":"theorem entropy_sub_entropy_eq_condRuzsaDist_add {Ω : Type u_1} {Ω' : Type u_2} {G : Type u_5} [mΩ : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) [mΩ' : MeasurableSpace Ω'] {μ' : MeasureTheory.Measure Ω'} [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [MeasurableSub₂ G] [MeasurableAdd₂ G] [Countable G] [ElementaryAddCommGroup G 2] [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure μ'] {X : Ω → G} {Y : Ω' → G} {Z : Ω' → G} (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) (h : ProbabilityTheory.IndepFun Y Z μ') [FiniteRange X] [FiniteRange Z] [FiniteRange Y] : H[Y + Z ; μ'] - H[Y ; μ'] = d[Y ; μ' # Z ; μ'] + H[Z ; μ'] / 2 - H[Y ; μ'] / 2"}
+{"name":"condRuzsaDist_diff_le''","declaration":"theorem condRuzsaDist_diff_le'' {Ω : Type u_1} {Ω' : Type u_2} {G : Type u_5} [mΩ : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) [mΩ' : MeasurableSpace Ω'] {μ' : MeasureTheory.Measure Ω'} [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [MeasurableSub₂ G] [MeasurableAdd₂ G] [Countable G] [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure μ'] {X : Ω → G} {Y : Ω' → G} {Z : Ω' → G} (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) (h : ProbabilityTheory.IndepFun Y Z μ') [FiniteRange X] [FiniteRange Z] [FiniteRange Y] : d[X ; μ # Y | Y + Z ; μ'] - d[X ; μ # Y ; μ'] ≤ (H[Y + Z ; μ'] - H[Z ; μ']) / 2"}
+{"name":"condRuszaDist_zero_left","declaration":"theorem condRuszaDist_zero_left {S : Type u_7} {Ω : Type u_1} {Ω' : Type u_2} {G : Type u_5} {T : Type u_6} [mΩ : MeasurableSpace Ω] [mΩ' : MeasurableSpace Ω'] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [Countable G] [MeasurableSpace S] [MeasurableSpace T] (X : Ω → G) (Z : Ω → S) (Y : Ω' → G) (W : Ω' → T) (μ' : MeasureTheory.Measure Ω') [MeasureTheory.IsFiniteMeasure μ'] : d[X | Z ; 0 # Y | W ; μ'] = 0"}
+{"name":"ent_of_proj_le","declaration":"/-- If $G$ is an additive group and $X$ is a $G$-valued random variable and\n$H\\leq G$ is a finite subgroup then, with $\\pi:G\\to G/H$ the natural homomorphism we have\n(where $U_H$ is uniform on $H$) $\\mathbb{H}(\\pi(X))\\leq 2d[X;U_H].$ -/\ntheorem ent_of_proj_le {Ω : Type u_1} {Ω' : Type u_2} {G : Type u_5} [mΩ : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [mΩ' : MeasurableSpace Ω'] {μ' : MeasureTheory.Measure Ω'} [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [MeasurableSub₂ G] [Countable G] {X : Ω → G} {UH : Ω' → G} [FiniteRange X] [FiniteRange UH] [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure μ'] (hX : Measurable X) (hU : Measurable UH) {H : AddSubgroup G} [Finite ↥H] (hunif : ProbabilityTheory.IsUniform (↑H) UH μ') : H[⇑(QuotientAddGroup.mk' H) ∘ X ; μ] ≤ 2 * d[X ; μ # UH ; μ']"}
+{"name":"ProbabilityTheory.IndepFun.rdist_eq","declaration":"/-- If $X, Y$ are independent $G$-random variables then\n$$ d[X ; Y] := H[X - Y] - H[X]/2 - H[Y]/2$$-/\ntheorem ProbabilityTheory.IndepFun.rdist_eq {Ω : Type u_1} {G : Type u_5} [mΩ : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [hG : MeasurableSpace G] [AddCommGroup G] [MeasurableSub₂ G] {X : Ω → G} [MeasureTheory.IsFiniteMeasure μ] {Y : Ω → G} (h : ProbabilityTheory.IndepFun X Y μ) (hX : Measurable X) (hY : Measurable Y) : d[X ; μ # Y ; μ] = H[X - Y ; μ] - H[X ; μ] / 2 - H[Y ; μ] / 2"}
+{"name":"kaimanovich_vershik'","declaration":"/-- A version of the **Kaimanovich-Vershik inequality** with some variables negated. -/\ntheorem kaimanovich_vershik' {Ω : Type u_1} {G : Type u_5} [mΩ : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [MeasurableAdd₂ G] [Countable G] {X : Ω → G} {Y : Ω → G} {Z : Ω → G} (h : ProbabilityTheory.iIndepFun (fun x => hG) ![X, Y, Z] μ) (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) [MeasureTheory.IsProbabilityMeasure μ] [FiniteRange X] [FiniteRange Z] [FiniteRange Y] : H[X - (Y + Z) ; μ] - H[X - Y ; μ] ≤ H[Y + Z ; μ] - H[Y ; μ]"}
+{"name":"comparison_of_ruzsa_distances","declaration":"theorem comparison_of_ruzsa_distances {Ω : Type u_1} {Ω' : Type u_2} {G : Type u_5} [mΩ : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) [mΩ' : MeasurableSpace Ω'] {μ' : MeasureTheory.Measure Ω'} [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [MeasurableSub₂ G] [MeasurableAdd₂ G] [Countable G] [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure μ'] {X : Ω → G} {Y : Ω' → G} {Z : Ω' → G} (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) (h : ProbabilityTheory.IndepFun Y Z μ') [FiniteRange X] [FiniteRange Z] [FiniteRange Y] : d[X ; μ # Y + Z ; μ'] - d[X ; μ # Y ; μ'] ≤ (H[Y + Z ; μ'] - H[Y ; μ']) / 2 ∧\n  (ElementaryAddCommGroup G 2 → H[Y + Z ; μ'] - H[Y ; μ'] = d[Y ; μ' # Z ; μ'] + H[Z ; μ'] / 2 - H[Y ; μ'] / 2)"}
+{"name":"«termD[_|_;_#_|_;_]».delab","declaration":"/-- Pretty printer defined by `notation3` command. -/\ndef «termD[_|_;_#_|_;_]».delab  : Lean.PrettyPrinter.Delaborator.Delab"}
+{"name":"ent_bsg","declaration":"/-- The **entropic Balog-Szemerédi-Gowers inequality**. Let $A, B$ be $G$-valued random variables on\n$\\Omega$, and set $Z := A+B$. Then\n$$\\sum_{z} P[Z=z] d[(A | Z = z) ; (B | Z = z)] \\leq 3 I[A :B] + 2 H[Z] - H[A] - H[B]. $$\nTODO: remove the hypothesis of `Fintype G` from here and from `condIndep_copies'` -/\ntheorem ent_bsg {Ω : Type u_1} {G : Type u_5} [mΩ : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [MeasurableSub₂ G] [MeasurableAdd₂ G] [Countable G] [MeasureTheory.IsProbabilityMeasure μ] {A : Ω → G} {B : Ω → G} (hA : Measurable A) (hB : Measurable B) [Fintype G] : ∫ (x : G),\n    (fun z => d[A ; ProbabilityTheory.cond μ ((A + B) ⁻¹' {z}) # B ; ProbabilityTheory.cond μ ((A + B) ⁻¹' {z})])\n      x ∂MeasureTheory.Measure.map (A + B) μ ≤\n  3 * I[A : B ; μ] + 2 * H[A + B ; μ] - H[A ; μ] - H[B ; μ]"}
+{"name":"rdist_zero_eq_half_ent","declaration":"/-- $$ d[X ; 0] = H[X] / 2 $$ -/\ntheorem rdist_zero_eq_half_ent {Ω : Type u_1} {Ω' : Type u_2} {G : Type u_5} [mΩ : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [mΩ' : MeasurableSpace Ω'] {μ' : MeasureTheory.Measure Ω'} [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [Countable G] {X : Ω → G} [MeasureTheory.IsFiniteMeasure μ] [MeasureTheory.IsProbabilityMeasure μ'] : d[X ; μ # fun x => 0 ; μ'] = H[X ; μ] / 2"}
+{"name":"condRuzsaDist_of_const","declaration":"/-- Conditioning by a constant does not affect Ruzsa distance. -/\ntheorem condRuzsaDist_of_const {S : Type u_7} {Ω : Type u_1} {Ω' : Type u_2} {G : Type u_5} {T : Type u_6} [mΩ : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [mΩ' : MeasurableSpace Ω'] {μ' : MeasureTheory.Measure Ω'} [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [Countable G] [Countable S] [MeasurableSpace S] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass S] [MeasurableSingletonClass T] {X : Ω → G} (hX : Measurable X) (Y : Ω' → G) (W : Ω' → T) (c : S) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure μ'] [FiniteRange W] : d[X | fun x => c ; μ # Y | W ; μ'] = d[X ; μ # Y | W ; μ']"}
+{"name":"«termD[_;_#_;_]»","declaration":"/-- The Ruzsa distance `rdist X Y` or $d[X ;Y]$ between two random variables is defined as\n$H[X'- Y'] - H[X']/2 - H[Y']/2$, where $X', Y'$ are independent copies of $X, Y$. -/\ndef «termD[_;_#_;_]»  : Lean.ParserDescr"}
+{"name":"condRuzsaDist'_of_inj_map'","declaration":"theorem condRuzsaDist'_of_inj_map' {Ω : Type u_1} {Ω'' : Type u_3} {G : Type u_5} [mΩ : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [mΩ'' : MeasurableSpace Ω''] {μ'' : MeasureTheory.Measure Ω''} [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [MeasurableSub₂ G] [MeasurableAdd₂ G] [Countable G] [elem : ElementaryAddCommGroup G 2] [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure μ''] {A : Ω'' → G} {B : Ω → G} {C : Ω → G} (hA : Measurable A) (hB : Measurable B) (hC : Measurable C) [FiniteRange A] [FiniteRange B] [FiniteRange C] : d[A ; μ'' # B | B + C ; μ] = d[A ; μ'' # C | B + C ; μ]"}
+{"name":"rdist_add_const","declaration":"/-- Adding a constant to a random variable does not change the Rusza distance. -/\ntheorem rdist_add_const {Ω : Type u_1} {Ω' : Type u_2} {G : Type u_5} [mΩ : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [mΩ' : MeasurableSpace Ω'] {μ' : MeasureTheory.Measure Ω'} [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [MeasurableSub₂ G] [Countable G] {X : Ω → G} {Y : Ω' → G} [FiniteRange X] [FiniteRange Y] {c : G} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure μ'] (hX : Measurable X) (hY : Measurable Y) : d[X ; μ # Y + fun x => c ; μ'] = d[X ; μ # Y ; μ']"}
+{"name":"diff_ent_le_rdist","declaration":"/-- $$|H[X] - H[Y]| \\leq 2 d[X ; Y]$$ -/\ntheorem diff_ent_le_rdist {Ω : Type u_1} {Ω' : Type u_2} {G : Type u_5} [mΩ : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [mΩ' : MeasurableSpace Ω'] {μ' : MeasureTheory.Measure Ω'} [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [MeasurableSub₂ G] [Countable G] {X : Ω → G} {Y : Ω' → G} [FiniteRange X] [FiniteRange Y] [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure μ'] (hX : Measurable X) (hY : Measurable Y) : |H[X ; μ] - H[Y ; μ']| ≤ 2 * d[X ; μ # Y ; μ']"}
+{"name":"«termD[_#_|_]»","declaration":"/-- The conditional Ruzsa distance `d[X ; Y|W]`. -/\ndef «termD[_#_|_]»  : Lean.ParserDescr"}
+{"name":"condRuzsaDist_eq_sum","declaration":"/-- Explicit formula for conditional Ruzsa distance $d[X|Z; Y|W]$. -/\ntheorem condRuzsaDist_eq_sum {S : Type u_7} {Ω : Type u_1} {Ω' : Type u_2} {G : Type u_5} {T : Type u_6} [mΩ : MeasurableSpace Ω] [mΩ' : MeasurableSpace Ω'] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [Countable G] [Countable S] [MeasurableSpace S] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass S] [MeasurableSingletonClass T] {X : Ω → G} {Z : Ω → S} {Y : Ω' → G} {W : Ω' → T} (hX : Measurable X) (hZ : Measurable Z) (hY : Measurable Y) (hW : Measurable W) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] (μ' : MeasureTheory.Measure Ω') [MeasureTheory.IsFiniteMeasure μ'] [FiniteRange Z] [FiniteRange W] : d[X | Z ; μ # Y | W ; μ'] =\n  Finset.sum (FiniteRange.toFinset Z) fun z =>\n    Finset.sum (FiniteRange.toFinset W) fun w =>\n      (↑↑μ (Z ⁻¹' {z})).toReal * (↑↑μ' (W ⁻¹' {w})).toReal *\n        d[X ; ProbabilityTheory.cond μ (Z ⁻¹' {z}) # Y ; ProbabilityTheory.cond μ' (W ⁻¹' {w})]"}
+{"name":"condRuzsaDist'_of_indep","declaration":"/-- Formula for conditional Ruzsa distance for independent sets of variables. -/\ntheorem condRuzsaDist'_of_indep {Ω : Type u_1} {G : Type u_5} {T : Type u_6} [mΩ : MeasurableSpace Ω] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [MeasurableSub₂ G] [Countable G] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] {X : Ω → G} {Y : Ω → G} {W : Ω → T} (hX : Measurable X) (hY : Measurable Y) (hW : Measurable W) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (h : ProbabilityTheory.IndepFun X (⟨Y, W⟩) μ) [FiniteRange W] : d[X ; μ # Y | W ; μ] = H[X - Y | W ; μ] - H[X ; μ] / 2 - H[Y | W ; μ] / 2"}
+{"name":"condRuzsaDist'_eq_integral","declaration":"/-- Explicit formula for conditional Ruzsa distance $d[X ; Y|W]$, in integral form. -/\ntheorem condRuzsaDist'_eq_integral {Ω : Type u_1} {Ω' : Type u_2} {G : Type u_5} {T : Type u_6} [mΩ : MeasurableSpace Ω] [mΩ' : MeasurableSpace Ω'] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [Countable G] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] (X : Ω → G) {Y : Ω' → G} {W : Ω' → T} (hY : Measurable Y) (hW : Measurable W) (μ : MeasureTheory.Measure Ω) (μ' : MeasureTheory.Measure Ω') [MeasureTheory.IsFiniteMeasure μ'] [FiniteRange W] : d[X ; μ # Y | W ; μ'] =\n  ∫ (x : T), (fun w => d[X ; μ # Y ; ProbabilityTheory.cond μ' (W ⁻¹' {w})]) x ∂MeasureTheory.Measure.map W μ'"}
+{"name":"rdist_nonneg","declaration":"/-- $$ d[X ; Y] \\geq 0$$ -/\ntheorem rdist_nonneg {Ω : Type u_1} {Ω' : Type u_2} {G : Type u_5} [mΩ : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [mΩ' : MeasurableSpace Ω'] {μ' : MeasureTheory.Measure Ω'} [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [MeasurableSub₂ G] [Countable G] {X : Ω → G} {Y : Ω' → G} [FiniteRange X] [FiniteRange Y] [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure μ'] (hX : Measurable X) (hY : Measurable Y) : 0 ≤ d[X ; μ # Y ; μ']"}
+{"name":"ProbabilityTheory.IdentDistrib.rdist_eq","declaration":"/-- If $X', Y'$ are copies of $X, Y$ respectively then $d[X' ; Y']=d[X ; Y]$. -/\ntheorem ProbabilityTheory.IdentDistrib.rdist_eq {Ω : Type u_1} {Ω' : Type u_2} {Ω'' : Type u_3} {Ω''' : Type u_4} {G : Type u_5} [mΩ : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [mΩ' : MeasurableSpace Ω'] {μ' : MeasureTheory.Measure Ω'} [mΩ'' : MeasurableSpace Ω''] {μ'' : MeasureTheory.Measure Ω''} [mΩ''' : MeasurableSpace Ω'''] {μ''' : MeasureTheory.Measure Ω'''} [hG : MeasurableSpace G] [AddCommGroup G] {X : Ω → G} {Y : Ω' → G} {X' : Ω'' → G} {Y' : Ω''' → G} (hX : ProbabilityTheory.IdentDistrib X X' μ μ'') (hY : ProbabilityTheory.IdentDistrib Y Y' μ' μ''') : d[X ; μ # Y ; μ'] = d[X' ; μ'' # Y' ; μ''']"}
+{"name":"condRuzsaDist_comp_right","declaration":"theorem condRuzsaDist_comp_right {Ω : Type u_1} {Ω' : Type u_2} {G : Type u_5} {T : Type u_6} [mΩ : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) [mΩ' : MeasurableSpace Ω'] (μ' : MeasureTheory.Measure Ω') [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [Countable G] [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T] {T' : Type u_7} [Fintype T] [Fintype T'] [MeasurableSpace T'] [MeasurableSingletonClass T'] [MeasureTheory.IsFiniteMeasure μ'] (X : Ω → G) (Y : Ω' → G) (W : Ω' → T) (e : T → T') (hY : Measurable Y) (hW : Measurable W) (he : Measurable e) (h'e : Function.Injective e) : d[X ; μ # Y | e ∘ W ; μ'] = d[X ; μ # Y | W ; μ']"}
+{"name":"diff_ent_le_rdist'","declaration":"/-- $$H[X - Y] - H[X] \\leq 2d[X ; Y]$$ -/\ntheorem diff_ent_le_rdist' {Ω : Type u_1} {G : Type u_5} [mΩ : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [MeasurableSub₂ G] [Countable G] {X : Ω → G} [FiniteRange X] [MeasureTheory.IsProbabilityMeasure μ] {Y : Ω → G} (hX : Measurable X) (hY : Measurable Y) (h : ProbabilityTheory.IndepFun X Y μ) [FiniteRange Y] : H[X - Y ; μ] - H[X ; μ] ≤ 2 * d[X ; μ # Y ; μ]"}
+{"name":"«termD[_|_#_|_]»","declaration":"/-- The conditional Ruzsa distance `d[X|Z ; Y|W]`. -/\ndef «termD[_|_#_|_]»  : Lean.ParserDescr"}
+{"name":"condRuzsaDist_le'","declaration":"theorem condRuzsaDist_le' {Ω : Type u_1} {Ω' : Type u_2} {G : Type u_5} {T : Type u_6} [mΩ : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) [mΩ' : MeasurableSpace Ω'] (μ' : MeasureTheory.Measure Ω') [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [MeasurableSub₂ G] [Countable G] [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass T] {X : Ω → G} {Y : Ω' → G} {W : Ω' → T} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure μ'] (hX : Measurable X) (hY : Measurable Y) (hW : Measurable W) [FiniteRange X] [FiniteRange Y] [FiniteRange W] : d[X ; μ # Y | W ; μ'] ≤ d[X ; μ # Y ; μ'] + I[Y : W ; μ'] / 2"}
+{"name":"condRuszaDist_zero_right","declaration":"theorem condRuszaDist_zero_right {S : Type u_7} {Ω : Type u_1} {Ω' : Type u_2} {G : Type u_5} {T : Type u_6} [mΩ : MeasurableSpace Ω] [mΩ' : MeasurableSpace Ω'] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [Countable G] [MeasurableSpace S] [MeasurableSpace T] (X : Ω → G) (Z : Ω → S) (Y : Ω' → G) (W : Ω' → T) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] : d[X | Z ; μ # Y | W ; 0] = 0"}
+{"name":"condRuzsaDist_eq_sum'","declaration":"/-- Explicit formula for conditional Ruzsa distance $d[X|Z; Y|W]$ in a fintype. -/\ntheorem condRuzsaDist_eq_sum' {S : Type u_7} {Ω : Type u_1} {Ω' : Type u_2} {G : Type u_5} {T : Type u_6} [mΩ : MeasurableSpace Ω] [mΩ' : MeasurableSpace Ω'] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [Countable G] [Countable S] [MeasurableSpace S] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass S] [MeasurableSingletonClass T] {X : Ω → G} {Z : Ω → S} {Y : Ω' → G} {W : Ω' → T} (hX : Measurable X) (hZ : Measurable Z) (hY : Measurable Y) (hW : Measurable W) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] (μ' : MeasureTheory.Measure Ω') [MeasureTheory.IsFiniteMeasure μ'] [Fintype S] [Fintype T] : d[X | Z ; μ # Y | W ; μ'] =\n  Finset.sum Finset.univ fun z =>\n    Finset.sum Finset.univ fun w =>\n      (↑↑μ (Z ⁻¹' {z})).toReal * (↑↑μ' (W ⁻¹' {w})).toReal *\n        d[X ; ProbabilityTheory.cond μ (Z ⁻¹' {z}) # Y ; ProbabilityTheory.cond μ' (W ⁻¹' {w})]"}
+{"name":"continuous_entropy_restrict_probabilityMeasure","declaration":"theorem continuous_entropy_restrict_probabilityMeasure {G : Type u_5} [hG : MeasurableSpace G] [Fintype G] [TopologicalSpace G] [DiscreteTopology G] [BorelSpace G] : Continuous fun μ => H[id ; ↑μ]"}
+{"name":"condRuzsaDist'_of_inj_map","declaration":"theorem condRuzsaDist'_of_inj_map {Ω : Type u_1} {G : Type u_5} [mΩ : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [MeasurableSub₂ G] [MeasurableAdd₂ G] [Countable G] [MeasureTheory.IsProbabilityMeasure μ] [elem : ElementaryAddCommGroup G 2] {X : Ω → G} {B : Ω → G} {C : Ω → G} (hX : Measurable X) (hB : Measurable B) (hC : Measurable C) (h_indep : ProbabilityTheory.IndepFun X (⟨B, C⟩) μ) [FiniteRange X] [FiniteRange B] [FiniteRange C] : d[X ; μ # B | B + C ; μ] = d[X ; μ # C | B + C ; μ]"}
+{"name":"rdist_def","declaration":"/-- Explicit formula for the Ruzsa distance. -/\ntheorem rdist_def {Ω : Type u_1} {Ω' : Type u_2} {G : Type u_5} [mΩ : MeasurableSpace Ω] [mΩ' : MeasurableSpace Ω'] [hG : MeasurableSpace G] [AddCommGroup G] (X : Ω → G) (Y : Ω' → G) (μ : MeasureTheory.Measure Ω) (μ' : MeasureTheory.Measure Ω') : d[X ; μ # Y ; μ'] =\n  H[fun x => x.1 - x.2 ; MeasureTheory.Measure.prod (MeasureTheory.Measure.map X μ) (MeasureTheory.Measure.map Y μ')] -\n      H[X ; μ] / 2 -\n    H[Y ; μ'] / 2"}
+{"name":"condRuzsaDist'","declaration":"/-- The conditional Ruzsa distance `d[X ; Y|W]`. -/\ndef condRuzsaDist' {Ω : Type u_1} {Ω' : Type u_2} {G : Type u_5} {T : Type u_6} [mΩ : MeasurableSpace Ω] [mΩ' : MeasurableSpace Ω'] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [Countable G] [MeasurableSpace T] (X : Ω → G) (Y : Ω' → G) (W : Ω' → T) (μ : autoParam (MeasureTheory.Measure Ω) _auto✝) (μ' : autoParam (MeasureTheory.Measure Ω') _auto✝) [MeasureTheory.IsFiniteMeasure μ'] : ℝ"}
+{"name":"condRuzsaDist_diff_ofsum_le","declaration":"theorem condRuzsaDist_diff_ofsum_le {Ω : Type u_1} {Ω' : Type u_2} {G : Type u_5} [mΩ : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) [mΩ' : MeasurableSpace Ω'] {μ' : MeasureTheory.Measure Ω'} [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [MeasurableSub₂ G] [MeasurableAdd₂ G] [Countable G] [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure μ'] {X : Ω → G} {Y : Ω' → G} {Z : Ω' → G} {Z' : Ω' → G} (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) (hZ' : Measurable Z') (h : ProbabilityTheory.iIndepFun (fun x => hG) ![Y, Z, Z'] μ') [FiniteRange X] [FiniteRange Z] [FiniteRange Y] [FiniteRange Z'] : d[X ; μ # Y + Z | Y + Z + Z' ; μ'] - d[X ; μ # Y ; μ'] ≤\n  (H[Y + Z + Z' ; μ'] + H[Y + Z ; μ'] - H[Y ; μ'] - H[Z' ; μ']) / 2"}
+{"name":"rdist_symm","declaration":"/-- $$ d[X ; Y] = d[Y ; X]$$ -/\ntheorem rdist_symm {Ω : Type u_1} {Ω' : Type u_2} {G : Type u_5} [mΩ : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [mΩ' : MeasurableSpace Ω'] {μ' : MeasureTheory.Measure Ω'} [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [MeasurableSub₂ G] [Countable G] {X : Ω → G} {Y : Ω' → G} [MeasureTheory.IsFiniteMeasure μ] [MeasureTheory.IsFiniteMeasure μ'] : d[X ; μ # Y ; μ'] = d[Y ; μ' # X ; μ]"}
+{"name":"rdist_of_inj","declaration":"/-- Applying an injective homomorphism does not affect Ruzsa distance. -/\ntheorem rdist_of_inj {Ω : Type u_1} {Ω' : Type u_2} {G : Type u_5} [mΩ : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [mΩ' : MeasurableSpace Ω'] {μ' : MeasureTheory.Measure Ω'} [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [Countable G] {X : Ω → G} {Y : Ω' → G} {H : Type u_7} [hH : MeasurableSpace H] [MeasurableSingletonClass H] [AddCommGroup H] [MeasurableSub₂ H] [MeasurableAdd₂ H] [Countable H] (hX : Measurable X) (hY : Measurable Y) (φ : G →+ H) (hφ : Function.Injective ⇑φ) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure μ'] : d[⇑φ ∘ X ; μ # ⇑φ ∘ Y ; μ'] = d[X ; μ # Y ; μ']"}
+{"name":"condRuzsaDist'_eq_sum'","declaration":"/-- Alternate formula for conditional Ruzsa distance $d[X ; Y|W]$ when T is a Fintype. -/\ntheorem condRuzsaDist'_eq_sum' {Ω : Type u_1} {Ω' : Type u_2} {G : Type u_5} {T : Type u_6} [mΩ : MeasurableSpace Ω] [mΩ' : MeasurableSpace Ω'] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [Countable G] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] {X : Ω → G} {Y : Ω' → G} {W : Ω' → T} (hY : Measurable Y) (hW : Measurable W) (μ : MeasureTheory.Measure Ω) (μ' : MeasureTheory.Measure Ω') [MeasureTheory.IsFiniteMeasure μ'] [Fintype T] : d[X ; μ # Y | W ; μ'] =\n  Finset.sum Finset.univ fun w => (↑↑μ' (W ⁻¹' {w})).toReal * d[X ; μ # Y ; ProbabilityTheory.cond μ' (W ⁻¹' {w})]"}
+{"name":"condRuzsaDist_diff_le'","declaration":"theorem condRuzsaDist_diff_le' {Ω : Type u_1} {Ω' : Type u_2} {G : Type u_5} [mΩ : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) [mΩ' : MeasurableSpace Ω'] {μ' : MeasureTheory.Measure Ω'} [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [MeasurableSub₂ G] [MeasurableAdd₂ G] [Countable G] [ElementaryAddCommGroup G 2] [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure μ'] {X : Ω → G} {Y : Ω' → G} {Z : Ω' → G} (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) (h : ProbabilityTheory.IndepFun Y Z μ') [FiniteRange X] [FiniteRange Z] [FiniteRange Y] : d[X ; μ # Y + Z ; μ'] - d[X ; μ # Y ; μ'] ≤ d[Y ; μ' # Z ; μ'] / 2 + H[Z ; μ'] / 4 - H[Y ; μ'] / 4"}
+{"name":"condRuzsaDist_of_indep","declaration":"/-- If $(X,Z)$ and $(Y,W)$ are independent, then\n$$ d[X | Z ; Y | W] = H[X'- Y'|Z', W'] - H[X'|Z']/2 - H[Y'|W']/2$$\n-/\ntheorem condRuzsaDist_of_indep {S : Type u_7} {Ω : Type u_1} {G : Type u_5} {T : Type u_6} [mΩ : MeasurableSpace Ω] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [MeasurableSub₂ G] [Countable G] [Countable S] [MeasurableSpace S] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass S] [MeasurableSingletonClass T] {X : Ω → G} {Z : Ω → S} {Y : Ω → G} {W : Ω → T} (hX : Measurable X) (hZ : Measurable Z) (hY : Measurable Y) (hW : Measurable W) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (h : ProbabilityTheory.IndepFun (⟨X, Z⟩) (⟨Y, W⟩) μ) [FiniteRange Z] [FiniteRange W] : d[X | Z ; μ # Y | W ; μ] = H[X - Y | ⟨Z, W⟩ ; μ] - H[X | Z ; μ] / 2 - H[Y | W ; μ] / 2"}
+{"name":"kaimanovich_vershik","declaration":"/-- The **Kaimanovich-Vershik inequality**. $$H[X + Y + Z] - H[X + Y] \\leq H[Y+ Z] - H[Y]$$ -/\ntheorem kaimanovich_vershik {Ω : Type u_1} {G : Type u_5} [mΩ : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [MeasurableAdd₂ G] [Countable G] {X : Ω → G} {Y : Ω → G} {Z : Ω → G} (h : ProbabilityTheory.iIndepFun (fun x => hG) ![X, Y, Z] μ) (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) [MeasureTheory.IsProbabilityMeasure μ] [FiniteRange X] [FiniteRange Z] [FiniteRange Y] : H[X + Y + Z ; μ] - H[X + Y ; μ] ≤ H[Y + Z ; μ] - H[Y ; μ]"}
+{"name":"condRuzsaDist_diff_le","declaration":"/-- Let $X, Y, Z$ be random variables taking values in some abelian group, and with $Y, Z$\nindependent. Then we have\n$$d[X ; Y + Z] -d[X ; Y] \\leq \\tfrac{1}{2} (H[Y+ Z] - H[Y])$$\n$$= \\tfrac{1}{2} d[Y ; Z] + \\tfrac{1}{4} H[Z] - \\tfrac{1}{4} H[Y]$$\nand\n$$d[X ; Y|Y+ Z] - d[X ; Y] \\leq \\tfrac{1}{2} \\bigl(H[Y+ Z] - H[Z]\\bigr)$$\n$$= \\tfrac{1}{2} d[Y ; Z] + \\tfrac{1}{4} H[Y] - \\tfrac{1}{4} H[Z]$$\n-/\ntheorem condRuzsaDist_diff_le {Ω : Type u_1} {Ω' : Type u_2} {G : Type u_5} [mΩ : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) [mΩ' : MeasurableSpace Ω'] {μ' : MeasureTheory.Measure Ω'} [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [MeasurableSub₂ G] [MeasurableAdd₂ G] [Countable G] [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure μ'] {X : Ω → G} {Y : Ω' → G} {Z : Ω' → G} (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) (h : ProbabilityTheory.IndepFun Y Z μ') [FiniteRange X] [FiniteRange Z] [FiniteRange Y] : d[X ; μ # Y + Z ; μ'] - d[X ; μ # Y ; μ'] ≤ (H[Y + Z ; μ'] - H[Y ; μ']) / 2"}
+{"name":"«termD[_#_|_]».delab","declaration":"/-- Pretty printer defined by `notation3` command. -/\ndef «termD[_#_|_]».delab  : Lean.PrettyPrinter.Delaborator.Delab"}
+{"name":"condRuzsaDist'_zero_right","declaration":"theorem condRuzsaDist'_zero_right {Ω : Type u_1} {Ω' : Type u_2} {G : Type u_5} {T : Type u_6} [mΩ : MeasurableSpace Ω] [mΩ' : MeasurableSpace Ω'] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [Countable G] [MeasurableSpace T] (X : Ω → G) (Y : Ω' → G) (W : Ω' → T) (μ : MeasureTheory.Measure Ω) : d[X ; μ # Y | W ; 0] = 0"}
+{"name":"continuous_measureEntropy_probabilityMeasure","declaration":"/-- Entropy depends continuously on the measure. -/\ntheorem continuous_measureEntropy_probabilityMeasure {Ω : Type u_7} [Fintype Ω] [TopologicalSpace Ω] [DiscreteTopology Ω] [MeasurableSpace Ω] [OpensMeasurableSpace Ω] : Continuous fun μ => Hm[↑μ]"}
+{"name":"«termD[_|_;_#_|_;_]»","declaration":"/-- The conditional Ruzsa distance `d[X|Z ; Y|W]`. -/\ndef «termD[_|_;_#_|_;_]»  : Lean.ParserDescr"}
+{"name":"condRuzsaDist","declaration":"/-- The conditional Ruzsa distance `d[X|Z ; Y|W]`. -/\ndef condRuzsaDist {S : Type u_7} {Ω : Type u_1} {Ω' : Type u_2} {G : Type u_5} {T : Type u_6} [mΩ : MeasurableSpace Ω] [mΩ' : MeasurableSpace Ω'] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [Countable G] [MeasurableSpace S] [MeasurableSpace T] (X : Ω → G) (Z : Ω → S) (Y : Ω' → G) (W : Ω' → T) (μ : autoParam (MeasureTheory.Measure Ω) _auto✝) [MeasureTheory.IsFiniteMeasure μ] (μ' : autoParam (MeasureTheory.Measure Ω') _auto✝) [MeasureTheory.IsFiniteMeasure μ'] : ℝ"}
+{"name":"condRuzsaDist'_prod_eq_sum'","declaration":"/-- Version of `condRuzsaDist'_prod_eq_sum` when `W` has finite codomain. -/\ntheorem condRuzsaDist'_prod_eq_sum' {Ω : Type u_1} {Ω' : Type u_2} {G : Type u_5} {T : Type u_6} [mΩ : MeasurableSpace Ω] [mΩ' : MeasurableSpace Ω'] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [Countable G] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] {X : Ω → G} {Y : Ω' → G} {W : Ω' → T} {W' : Ω' → T} (μ : MeasureTheory.Measure Ω) (μ' : MeasureTheory.Measure Ω') (hY : Measurable Y) (hW' : Measurable W') (hW : Measurable W) [MeasureTheory.IsFiniteMeasure μ'] [Fintype T] : d[X ; μ # Y | ⟨W', W⟩ ; μ'] =\n  Finset.sum Finset.univ fun w => (↑↑μ' (W ⁻¹' {w})).toReal * d[X ; μ # Y | W' ; ProbabilityTheory.cond μ' (W ⁻¹' {w})]"}
+{"name":"rdist_eq_rdistm","declaration":"/-- Ruzsa distance of random variables equals Ruzsa distance of the kernels. -/\ntheorem rdist_eq_rdistm {Ω : Type u_1} {Ω' : Type u_2} {G : Type u_5} [mΩ : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [mΩ' : MeasurableSpace Ω'] {μ' : MeasureTheory.Measure Ω'} [hG : MeasurableSpace G] [AddCommGroup G] {X : Ω → G} {Y : Ω' → G} : d[X ; μ # Y ; μ'] = ProbabilityTheory.kernel.rdistm (MeasureTheory.Measure.map X μ) (MeasureTheory.Measure.map Y μ')"}
+{"name":"rdist_triangle","declaration":"/-- The **entropic Ruzsa triangle inequality** -/\ntheorem rdist_triangle {Ω : Type u_1} {Ω' : Type u_2} {Ω'' : Type u_3} {G : Type u_5} [mΩ : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [mΩ' : MeasurableSpace Ω'] {μ' : MeasureTheory.Measure Ω'} [mΩ'' : MeasurableSpace Ω''] {μ'' : MeasureTheory.Measure Ω''} [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [MeasurableSub₂ G] [Countable G] {X : Ω → G} {Y : Ω' → G} {Z : Ω'' → G} (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) [hμ : MeasureTheory.IsProbabilityMeasure μ] [hμ' : MeasureTheory.IsProbabilityMeasure μ'] [hμ'' : MeasureTheory.IsProbabilityMeasure μ''] [FiniteRange X] [FiniteRange Y] [FiniteRange Z] : d[X ; μ # Z ; μ''] ≤ d[X ; μ # Y ; μ'] + d[Y ; μ' # Z ; μ'']"}
+{"name":"«termD[_;_#_;_]».delab","declaration":"/-- Pretty printer defined by `notation3` command. -/\ndef «termD[_;_#_;_]».delab  : Lean.PrettyPrinter.Delaborator.Delab"}
+{"name":"condRuzsaDist_nonneg","declaration":"theorem condRuzsaDist_nonneg {S : Type u_7} {Ω : Type u_1} {Ω' : Type u_2} {G : Type u_5} {T : Type u_6} [mΩ : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [mΩ' : MeasurableSpace Ω'] {μ' : MeasureTheory.Measure Ω'} [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [MeasurableSub₂ G] [Countable G] [Countable S] [Nonempty S] [MeasurableSpace S] [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass S] [MeasurableSingletonClass T] {X : Ω → G} (hX : Measurable X) [FiniteRange X] {Z : Ω → S} (hZ : Measurable Z) [FiniteRange Z] {Y : Ω' → G} (hY : Measurable Y) [FiniteRange Y] {W : Ω' → T} (hW : Measurable W) [FiniteRange W] [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure μ'] : 0 ≤ d[X | Z ; μ # Y | W ; μ']"}
+{"name":"condRuszaDist_prod_eq_of_indepFun","declaration":"theorem condRuszaDist_prod_eq_of_indepFun {Ω : Type u_1} {Ω' : Type u_2} {G : Type u_5} {T : Type u_6} [mΩ : MeasurableSpace Ω] [mΩ' : MeasurableSpace Ω'] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [Countable G] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] {μ : MeasureTheory.Measure Ω} {μ' : MeasureTheory.Measure Ω'} {X : Ω → G} {Y : Ω' → G} {W : Ω' → T} {W' : Ω' → T} (hX : Measurable X) (hY : Measurable Y) (hW : Measurable W) (hW' : Measurable W') (h : ProbabilityTheory.IndepFun (⟨Y, W⟩) W' μ') [MeasureTheory.IsProbabilityMeasure μ'] [Fintype T] : d[X ; μ # Y | ⟨W, W'⟩ ; μ'] = d[X ; μ # Y | W ; μ']"}
+{"name":"«termD[_#_]»","declaration":"/-- The Ruzsa distance `rdist X Y` or $d[X ;Y]$ between two random variables is defined as\n$H[X'- Y'] - H[X']/2 - H[Y']/2$, where $X', Y'$ are independent copies of $X, Y$. -/\ndef «termD[_#_]»  : Lean.ParserDescr"}
+{"name":"condRuzsaDist'_prod_eq_sum","declaration":"theorem condRuzsaDist'_prod_eq_sum {Ω : Type u_1} {Ω' : Type u_2} {G : Type u_5} {T : Type u_6} [mΩ : MeasurableSpace Ω] [mΩ' : MeasurableSpace Ω'] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [Countable G] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass T] {X : Ω → G} {Y : Ω' → G} {W : Ω' → T} {W' : Ω' → T} (μ : MeasureTheory.Measure Ω) (μ' : MeasureTheory.Measure Ω') (hY : Measurable Y) (hW' : Measurable W') (hW : Measurable W) [MeasureTheory.IsFiniteMeasure μ'] [FiniteRange W] [FiniteRange W'] : d[X ; μ # Y | ⟨W', W⟩ ; μ'] =\n  Finset.sum (FiniteRange.toFinset W) fun w =>\n    (↑↑μ' (W ⁻¹' {w})).toReal * d[X ; μ # Y | W' ; ProbabilityTheory.cond μ' (W ⁻¹' {w})]"}
+{"name":"condRuzsaDist'_def","declaration":"/-- Conditional Ruzsa distance equals Ruzsa distance of associated kernels. -/\ntheorem condRuzsaDist'_def {Ω : Type u_1} {Ω' : Type u_2} {G : Type u_5} {T : Type u_6} [mΩ : MeasurableSpace Ω] [mΩ' : MeasurableSpace Ω'] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [Countable G] [MeasurableSpace T] (X : Ω → G) (Y : Ω' → G) (W : Ω' → T) (μ : MeasureTheory.Measure Ω) (μ' : MeasureTheory.Measure Ω') [MeasureTheory.IsFiniteMeasure μ'] : d[X ; μ # Y | W ; μ'] =\n  dk[ProbabilityTheory.kernel.const Unit (MeasureTheory.Measure.map X μ) ; MeasureTheory.Measure.dirac () #\n    ProbabilityTheory.condDistrib Y W μ' ; MeasureTheory.Measure.map W μ']"}
+{"name":"rdist_le_avg_ent","declaration":"/-- $d[X;Y] ≤ H[X]/2 + H[Y]/2$. -/\ntheorem rdist_le_avg_ent {Ω : Type u_1} {Ω' : Type u_2} {G : Type u_5} [mΩ : MeasurableSpace Ω] [mΩ' : MeasurableSpace Ω'] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [MeasurableSub₂ G] [Countable G] {X : Ω → G} {Y : Ω' → G} [FiniteRange X] [FiniteRange Y] (hX : Measurable X) (hY : Measurable Y) (μ : autoParam (MeasureTheory.Measure Ω) _auto✝) (μ' : autoParam (MeasureTheory.Measure Ω') _auto✝) [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure μ'] : d[X ; μ # Y ; μ'] ≤ (H[X ; μ] + H[Y ; μ']) / 2"}
+{"name":"condRuzsaDist_diff_le'''","declaration":"theorem condRuzsaDist_diff_le''' {Ω : Type u_1} {Ω' : Type u_2} {G : Type u_5} [mΩ : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) [mΩ' : MeasurableSpace Ω'] {μ' : MeasureTheory.Measure Ω'} [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [MeasurableSub₂ G] [MeasurableAdd₂ G] [Countable G] [ElementaryAddCommGroup G 2] [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure μ'] {X : Ω → G} {Y : Ω' → G} {Z : Ω' → G} (hX : Measurable X) (hY : Measurable Y) (hZ : Measurable Z) (h : ProbabilityTheory.IndepFun Y Z μ') [FiniteRange X] [FiniteRange Z] [FiniteRange Y] : d[X ; μ # Y | Y + Z ; μ'] - d[X ; μ # Y ; μ'] ≤ d[Y ; μ' # Z ; μ'] / 2 + H[Y ; μ'] / 4 - H[Z ; μ'] / 4"}
+{"name":"condRuzsaDist_le","declaration":"/-- Suppose that $(X, Z)$ and $(Y, W)$ are random variables, where $X, Y$ take values in an abelian\ngroup. Then $$d[X | Z ; Y | W] \\leq d[X ; Y] + \\tfrac{1}{2} I[X : Z] + \\tfrac{1}{2} I[Y : W]$$ -/\ntheorem condRuzsaDist_le {S : Type u_7} {Ω : Type u_1} {Ω' : Type u_2} {G : Type u_5} {T : Type u_6} [mΩ : MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) [mΩ' : MeasurableSpace Ω'] (μ' : MeasureTheory.Measure Ω') [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [MeasurableSub₂ G] [Countable G] [Countable S] [MeasurableSpace S] [Countable T] [Nonempty T] [MeasurableSpace T] [MeasurableSingletonClass S] [MeasurableSingletonClass T] {X : Ω → G} {Z : Ω → S} {Y : Ω' → G} {W : Ω' → T} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure μ'] [Nonempty S] (hX : Measurable X) (hZ : Measurable Z) (hY : Measurable Y) (hW : Measurable W) [FiniteRange X] [FiniteRange Z] [FiniteRange Y] [FiniteRange W] : d[X | Z ; μ # Y | W ; μ'] ≤ d[X ; μ # Y ; μ'] + I[X : Z ; μ] / 2 + I[Y : W ; μ'] / 2"}
+{"name":"continuous_rdist_restrict_probabilityMeasure₁'","declaration":"/-- Ruzsa distance depends continuously on the second measure. -/\ntheorem continuous_rdist_restrict_probabilityMeasure₁' {Ω : Type u_1} {G : Type u_5} [mΩ : MeasurableSpace Ω] [hG : MeasurableSpace G] [AddCommGroup G] [Countable G] [Fintype G] [TopologicalSpace G] [DiscreteTopology G] [BorelSpace G] (X : Ω → G) (P : autoParam (MeasureTheory.Measure Ω) _auto✝) [MeasureTheory.IsProbabilityMeasure P] (X_mble : Measurable X) : Continuous fun μ => d[X ; P # id ; ↑μ]"}
+{"name":"«termD[_;_#_|_;_]»","declaration":"/-- The conditional Ruzsa distance `d[X ; Y|W]`. -/\ndef «termD[_;_#_|_;_]»  : Lean.ParserDescr"}
+{"name":"condRuzsaDist_of_inj_map","declaration":"theorem condRuzsaDist_of_inj_map {Ω : Type u_1} {G : Type u_5} [mΩ : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [MeasurableSub₂ G] [Countable G] {G' : Type u_7} [Countable G'] [AddCommGroup G'] [MeasurableSpace G'] [MeasurableSingletonClass G'] [MeasureTheory.IsProbabilityMeasure μ] (Y : Fin 4 → Ω → G) (h_indep : ProbabilityTheory.IndepFun (⟨Y 0, Y 2⟩) (⟨Y 1, Y 3⟩) μ) (h_meas : ∀ (i : Fin 4), Measurable (Y i)) (π : G × G →+ G') (hπ : ∀ (h : G), Function.Injective fun g => π (g, h)) [FiniteRange (Y 2)] [FiniteRange (Y 3)] : d[⇑π ∘ ⟨Y 0, Y 2⟩ | Y 2 ; μ # ⇑π ∘ ⟨Y 1, Y 3⟩ | Y 3 ; μ] = d[Y 0 | Y 2 ; μ # Y 1 | Y 3 ; μ]"}
+{"name":"condRuzsaDist_of_copy","declaration":"/-- The conditional Ruzsa distance is unchanged if the sets of random variables are replaced with\ncopies. -/\ntheorem condRuzsaDist_of_copy {S : Type u_7} {Ω : Type u_1} {Ω' : Type u_2} {Ω'' : Type u_3} {Ω''' : Type u_4} {G : Type u_5} {T : Type u_6} [mΩ : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [mΩ' : MeasurableSpace Ω'] {μ' : MeasureTheory.Measure Ω'} [mΩ'' : MeasurableSpace Ω''] {μ'' : MeasureTheory.Measure Ω''} [mΩ''' : MeasurableSpace Ω'''] {μ''' : MeasureTheory.Measure Ω'''} [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [Countable G] [Countable S] [MeasurableSpace S] [Countable T] [MeasurableSpace T] [MeasurableSingletonClass S] [MeasurableSingletonClass T] {X : Ω → G} (hX : Measurable X) {Z : Ω → S} (hZ : Measurable Z) {Y : Ω' → G} (hY : Measurable Y) {W : Ω' → T} (hW : Measurable W) {X' : Ω'' → G} (hX' : Measurable X') {Z' : Ω'' → S} (hZ' : Measurable Z') {Y' : Ω''' → G} (hY' : Measurable Y') {W' : Ω''' → T} (hW' : Measurable W') [MeasureTheory.IsFiniteMeasure μ] [MeasureTheory.IsFiniteMeasure μ'] [MeasureTheory.IsFiniteMeasure μ''] [MeasureTheory.IsFiniteMeasure μ'''] (h1 : ProbabilityTheory.IdentDistrib (⟨X, Z⟩) (⟨X', Z'⟩) μ μ'') (h2 : ProbabilityTheory.IdentDistrib (⟨Y, W⟩) (⟨Y', W'⟩) μ' μ''') [FiniteRange Z] [FiniteRange W] [FiniteRange Z'] [FiniteRange W'] : d[X | Z ; μ # Y | W ; μ'] = d[X' | Z' ; μ'' # Y' | W' ; μ''']"}
+{"name":"rdist_eq_rdist_id_map","declaration":"/-- Ruzsa distance between random variables equals Ruzsa distance between their distributions.-/\ntheorem rdist_eq_rdist_id_map {Ω : Type u_1} {Ω' : Type u_2} {G : Type u_5} [mΩ : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [mΩ' : MeasurableSpace Ω'] {μ' : MeasureTheory.Measure Ω'} [hG : MeasurableSpace G] [AddCommGroup G] {X : Ω → G} {Y : Ω' → G} : d[X ; μ # Y ; μ'] = d[id ; MeasureTheory.Measure.map X μ # id ; MeasureTheory.Measure.map Y μ']"}
+{"name":"continuous_rdist_restrict_probabilityMeasure₁","declaration":"theorem continuous_rdist_restrict_probabilityMeasure₁ {Ω : Type u_1} {G : Type u_5} [mΩ : MeasurableSpace Ω] [hG : MeasurableSpace G] [AddCommGroup G] [Countable G] [Fintype G] [TopologicalSpace G] [DiscreteTopology G] [BorelSpace G] (X : Ω → G) (P : autoParam (MeasureTheory.Measure Ω) _auto✝) [MeasureTheory.IsProbabilityMeasure P] (X_mble : Measurable X) : Continuous fun μ => d[id ; MeasureTheory.Measure.map X P # id ; ↑μ]"}
+{"name":"«termD[_#_]».delab","declaration":"/-- Pretty printer defined by `notation3` command. -/\ndef «termD[_#_]».delab  : Lean.PrettyPrinter.Delaborator.Delab"}
+{"name":"rdist","declaration":"/-- The Ruzsa distance `rdist X Y` or $d[X ;Y]$ between two random variables is defined as\n$H[X'- Y'] - H[X']/2 - H[Y']/2$, where $X', Y'$ are independent copies of $X, Y$. -/\ndef rdist {Ω : Type u_1} {Ω' : Type u_2} {G : Type u_5} [mΩ : MeasurableSpace Ω] [mΩ' : MeasurableSpace Ω'] [hG : MeasurableSpace G] [AddCommGroup G] (X : Ω → G) (Y : Ω' → G) (μ : autoParam (MeasureTheory.Measure Ω) _auto✝) (μ' : autoParam (MeasureTheory.Measure Ω') _auto✝) : ℝ"}
+{"name":"diff_ent_le_rdist''","declaration":"/-- $$H[X - Y] - H[Y] \\leq 2d[X ; Y]$$ -/\ntheorem diff_ent_le_rdist'' {Ω : Type u_1} {G : Type u_5} [mΩ : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [hG : MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] [MeasurableSub₂ G] [Countable G] {X : Ω → G} [FiniteRange X] [MeasureTheory.IsProbabilityMeasure μ] {Y : Ω → G} (hX : Measurable X) (hY : Measurable Y) (h : ProbabilityTheory.IndepFun X Y μ) [FiniteRange Y] : H[X - Y ; μ] - H[Y ; μ] ≤ 2 * d[X ; μ # Y ; μ]"}

PFR-declarations/PFR.ForMathlib.Entropy.RuzsaSetDist.jsonl

@@ -0,0 +1,18 @@
+{"name":"MeasureTheory.Measure.discreteUniform_apply","declaration":"/-- The usual formula for the discrete uniform measure applied to an arbitrary set. -/\ntheorem MeasureTheory.Measure.discreteUniform_apply {S : Type u_1} [MeasurableSpace S] (H : Set S) [MeasurableSingletonClass S] [Finite ↑H] (A : Set S) : ↑↑(MeasureTheory.Measure.discreteUniform H) A = ↑(Nat.card ↑(A ∩ H)) / ↑(Nat.card ↑H)"}
+{"name":"ProbabilityTheory.rdist_set_eq_rdist","declaration":"/-- Relating Ruzsa distance between sets to Ruzsa distance between random variables -/\ntheorem ProbabilityTheory.rdist_set_eq_rdist {G : Type u_1} [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] {A : Set G} {B : Set G} [Finite ↑A] [Finite ↑B] [Nonempty ↑A] [Nonempty ↑B] {Ω : Type u_2} {Ω' : Type u_3} [mΩ : MeasureTheory.MeasureSpace Ω] [mΩ' : MeasureTheory.MeasureSpace Ω'] {μ : MeasureTheory.Measure Ω} {μ' : MeasureTheory.Measure Ω'} (hμ : MeasureTheory.IsProbabilityMeasure μ) (hμ' : MeasureTheory.IsProbabilityMeasure μ') {UA : Ω → G} {UB : Ω' → G} (hUA : ProbabilityTheory.IsUniform A UA μ) (hUB : ProbabilityTheory.IsUniform B UB μ') (hUA_mes : Measurable UA) (hUB_mes : Measurable UB) : dᵤ[A # B] = d[UA ; μ # UB ; μ']"}
+{"name":"ProbabilityTheory.rdist_set_nonneg","declaration":"/-- Ruzsa distance between sets is nonnegative. -/\ntheorem ProbabilityTheory.rdist_set_nonneg {G : Type u_1} [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] (A : Set G) (B : Set G) [Finite ↑A] [Finite ↑B] [Nonempty ↑A] [Nonempty ↑B] : 0 ≤ dᵤ[A # B]"}
+{"name":"MeasureTheory.Measure.map_discreteUniform_of_inj","declaration":"/-- injective map of discrete uniform is discrete uniform -/\ntheorem MeasureTheory.Measure.map_discreteUniform_of_inj {S : Type u_1} [MeasurableSpace S] (H : Set S) [MeasurableSingletonClass S] [Finite ↑H] {T : Type u_2} [MeasurableSpace T] [MeasurableSingletonClass T] {f : S → T} (hmes : Measurable f) (hf : Function.Injective f) : MeasureTheory.Measure.map f (MeasureTheory.Measure.discreteUniform H) = MeasureTheory.Measure.discreteUniform (f '' H)"}
+{"name":"ProbabilityTheory.rdist_set_le","declaration":"/-- Ruzsa distance between sets is controlled by the doubling constant. -/\ntheorem ProbabilityTheory.rdist_set_le {G : Type u_1} [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] (A : Set G) (B : Set G) [Finite ↑A] [Finite ↑B] (hA : Set.Nonempty A) (hB : Set.Nonempty B) : dᵤ[A # B] ≤ Real.log ↑(Nat.card ↑(A - B)) - Real.log ↑(Nat.card ↑A) / 2 - Real.log ↑(Nat.card ↑B) / 2"}
+{"name":"MeasureTheory.Measure.isUniform_iff_uniform_dist","declaration":"/-- A random variable is uniform iff its distribution is. -/\ntheorem MeasureTheory.Measure.isUniform_iff_uniform_dist {S : Type u_1} [MeasurableSpace S] (H : Set S) [MeasurableSingletonClass S] [Finite ↑H] [Nonempty ↑H] {Ω : Type u_2} [mΩ : MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [Countable S] (hμ : MeasureTheory.IsProbabilityMeasure μ) {U : Ω → S} (hU : Measurable U) : ProbabilityTheory.IsUniform H U μ ↔ MeasureTheory.Measure.map U μ = MeasureTheory.Measure.discreteUniform H"}
+{"name":"ProbabilityTheory.«termDᵤ[_#_]»","declaration":"def ProbabilityTheory.«termDᵤ[_#_]»  : Lean.ParserDescr"}
+{"name":"ProbabilityTheory.«termDᵤ[_#_]».delab","declaration":"/-- Pretty printer defined by `notation3` command. -/\ndef ProbabilityTheory.«termDᵤ[_#_]».delab  : Lean.PrettyPrinter.Delaborator.Delab"}
+{"name":"ProbabilityTheory.rdist_set_add_const","declaration":"/-- Ruzsa distance between sets is translation invariant. -/\ntheorem ProbabilityTheory.rdist_set_add_const {G : Type u_1} [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] (A : Set G) (B : Set G) [Finite ↑A] [Finite ↑B] [Nonempty ↑A] [Nonempty ↑B] (c : G) (c' : G) : dᵤ[A + {c} # B + {c'}] = dᵤ[A # B]"}
+{"name":"ProbabilityTheory.rdist_set_triangle","declaration":"/-- Ruzsa distance between sets obeys the triangle inequality. -/\ntheorem ProbabilityTheory.rdist_set_triangle {G : Type u_1} [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] (A : Set G) (B : Set G) (C : Set G) [Finite ↑A] [Finite ↑B] [Finite ↑C] [Nonempty ↑A] [Nonempty ↑B] [Nonempty ↑C] : dᵤ[A # C] ≤ dᵤ[A # B] + dᵤ[B # C]"}
+{"name":"MeasureTheory.Measure.discreteUniform_apply'","declaration":"/-- Variant of `discreteUniform_apply' using real-valued measures. -/\ntheorem MeasureTheory.Measure.discreteUniform_apply' {S : Type u_1} [MeasurableSpace S] (H : Set S) [MeasurableSingletonClass S] [Finite ↑H] (A : Set S) : (MeasureTheory.Measure.discreteUniform H).real A = ↑(Nat.card ↑(A ∩ H)) / ↑(Nat.card ↑H)"}
+{"name":"MeasureTheory.Measure.discreteUniform.isProbabilityMeasure","declaration":"instance MeasureTheory.Measure.discreteUniform.isProbabilityMeasure {S : Type u_1} [MeasurableSpace S] (H : Set S) [MeasurableSingletonClass S] [Finite ↑H] [Nonempty ↑H] : MeasureTheory.IsProbabilityMeasure (MeasureTheory.Measure.discreteUniform H)"}
+{"name":"MeasureTheory.Measure.discreteUniform","declaration":"/-- In practice one would also impose the conditions `MeasurableSingletonClass S`, `Finite H` and `Nonempty H` before attempting to use this definition. -/\ndef MeasureTheory.Measure.discreteUniform {S : Type u_1} [MeasurableSpace S] (H : Set S) : MeasureTheory.Measure S"}
+{"name":"ProbabilityTheory.rdist_set_of_inj","declaration":"/-- Ruzsa distance between sets is preserved by injective homomorphisms. -/\ntheorem ProbabilityTheory.rdist_set_of_inj {G : Type u_1} [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] (A : Set G) (B : Set G) [Finite ↑A] [Finite ↑B] [Nonempty ↑A] [Nonempty ↑B] {H : Type u_2} [hH : MeasurableSpace H] [MeasurableSingletonClass H] [AddCommGroup H] [Countable H] {φ : G →+ H} (hφ : Function.Injective ⇑φ) : dᵤ[⇑φ '' A # ⇑φ '' B] = dᵤ[A # B]"}
+{"name":"ProbabilityTheory.rdist_set_symm","declaration":"/-- Ruzsa distance between sets is symmetric. -/\ntheorem ProbabilityTheory.rdist_set_symm {G : Type u_1} [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G] [AddCommGroup G] (A : Set G) (B : Set G) [Finite ↑A] [Finite ↑B] [Nonempty ↑A] [Nonempty ↑B] : dᵤ[A # B] = dᵤ[B # A]"}
+{"name":"MeasureTheory.Measure.discreteUniform_of_infinite","declaration":"/-- The uniform distribution on an infinite set vanishes by definition. -/\ntheorem MeasureTheory.Measure.discreteUniform_of_infinite {S : Type u_1} [MeasurableSpace S] (H : Set S) (h : Set.Infinite H) : MeasureTheory.Measure.discreteUniform H = 0"}
+{"name":"ProbabilityTheory.entropy_of_discreteUniform","declaration":"/-- The entropy of a uniform measure is the log of the cardinality of its support. -/\ntheorem ProbabilityTheory.entropy_of_discreteUniform {S : Type u_1} [MeasurableSpace S] (H : Set S) [MeasurableSingletonClass S] [Finite ↑H] [Nonempty ↑H] : Hm[MeasureTheory.Measure.discreteUniform H] = Real.log ↑(Nat.card ↑H)"}
+{"name":"ProbabilityTheory.rdist_set","declaration":"/-- The Ruzsa distance between two subsets `A`, `B` of a group `G` is defined to be the Ruzsa distance between their uniform probability distributions.  Is only intended for use when `A`, `B` are finite and non-empty. -/\ndef ProbabilityTheory.rdist_set {G : Type u_1} [MeasurableSpace G] [AddCommGroup G] (A : Set G) (B : Set G) : ℝ"}

PFR-declarations/PFR.ForMathlib.FiniteMeasureComponent.jsonl

@@ -0,0 +1,2 @@
+{"name":"continuous_finiteMeasure_apply_of_isClopen","declaration":"/-- The measure of any connected component depends continuously on the `FiniteMeasure`. -/\ntheorem continuous_finiteMeasure_apply_of_isClopen {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] {s : Set α} (s_clopen : IsClopen s) : Continuous fun μ => (↑μ).real s"}
+{"name":"continuous_probabilityMeasure_apply_of_isClopen","declaration":"/-- The probability of any connected component depends continuously on the `ProbabilityMeasure`. -/\ntheorem continuous_probabilityMeasure_apply_of_isClopen {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] {s : Set α} (s_clopen : IsClopen s) : Continuous fun μ => (↑μ).real s"}

PFR-declarations/PFR.ForMathlib.FiniteMeasureProd.jsonl

@@ -0,0 +1,25 @@
+{"name":"MeasureTheory.ProbabilityMeasure.map_prod_map","declaration":"theorem MeasureTheory.ProbabilityMeasure.map_prod_map {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.ProbabilityMeasure α) (ν : MeasureTheory.ProbabilityMeasure β) {α' : Type u_3} [MeasurableSpace α'] {β' : Type u_4} [MeasurableSpace β'] {f : α → α'} {g : β → β'} (f_mble : Measurable f) (g_mble : Measurable g) : MeasureTheory.ProbabilityMeasure.prod (MeasureTheory.ProbabilityMeasure.map μ ⋯)\n    (MeasureTheory.ProbabilityMeasure.map ν ⋯) =\n  MeasureTheory.ProbabilityMeasure.map (MeasureTheory.ProbabilityMeasure.prod μ ν) ⋯"}
+{"name":"MeasureTheory.FiniteMeasure.prod","declaration":"/-- The binary product of finite measures. -/\ndef MeasureTheory.FiniteMeasure.prod {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.FiniteMeasure α) (ν : MeasureTheory.FiniteMeasure β) : MeasureTheory.FiniteMeasure (α × β)"}
+{"name":"MeasureTheory.ProbabilityMeasure.prod_prod","declaration":"theorem MeasureTheory.ProbabilityMeasure.prod_prod {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.ProbabilityMeasure α) (ν : MeasureTheory.ProbabilityMeasure β) (s : Set α) (t : Set β) : (fun s => (↑↑↑(MeasureTheory.ProbabilityMeasure.prod μ ν) s).toNNReal) (s ×ˢ t) =\n  (fun s => (↑↑↑μ s).toNNReal) s * (fun s => (↑↑↑ν s).toNNReal) t"}
+{"name":"MeasureTheory.ProbabilityMeasure.prod_swap","declaration":"theorem MeasureTheory.ProbabilityMeasure.prod_swap {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.ProbabilityMeasure α) (ν : MeasureTheory.ProbabilityMeasure β) : MeasureTheory.ProbabilityMeasure.map (MeasureTheory.ProbabilityMeasure.prod μ ν) ⋯ =\n  MeasureTheory.ProbabilityMeasure.prod ν μ"}
+{"name":"MeasureTheory.FiniteMeasure.measure_ae_null_of_prod_null","declaration":"theorem MeasureTheory.FiniteMeasure.measure_ae_null_of_prod_null {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.FiniteMeasure α) (ν : MeasureTheory.FiniteMeasure β) {s : Set (α × β)} (h : (fun s => (↑↑↑(MeasureTheory.FiniteMeasure.prod μ ν) s).toNNReal) s = 0) : (fun x => (fun s => (↑↑↑ν s).toNNReal) (Prod.mk x ⁻¹' s)) =ᶠ[MeasureTheory.Measure.ae ↑μ] 0"}
+{"name":"MeasureTheory.ProbabilityMeasure.toMeasure_prod","declaration":"theorem MeasureTheory.ProbabilityMeasure.toMeasure_prod {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.ProbabilityMeasure α) (ν : MeasureTheory.ProbabilityMeasure β) : ↑(MeasureTheory.ProbabilityMeasure.prod μ ν) = MeasureTheory.Measure.prod ↑μ ↑ν"}
+{"name":"MeasureTheory.FiniteMeasure.mass_prod","declaration":"theorem MeasureTheory.FiniteMeasure.mass_prod {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.FiniteMeasure α) (ν : MeasureTheory.FiniteMeasure β) : MeasureTheory.FiniteMeasure.mass (MeasureTheory.FiniteMeasure.prod μ ν) =\n  MeasureTheory.FiniteMeasure.mass μ * MeasureTheory.FiniteMeasure.mass ν"}
+{"name":"MeasureTheory.FiniteMeasure.prod_swap","declaration":"theorem MeasureTheory.FiniteMeasure.prod_swap {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.FiniteMeasure α) (ν : MeasureTheory.FiniteMeasure β) : MeasureTheory.FiniteMeasure.map (MeasureTheory.FiniteMeasure.prod μ ν) Prod.swap = MeasureTheory.FiniteMeasure.prod ν μ"}
+{"name":"MeasureTheory.FiniteMeasure.map_fst_prod","declaration":"theorem MeasureTheory.FiniteMeasure.map_fst_prod {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.FiniteMeasure α) (ν : MeasureTheory.FiniteMeasure β) : MeasureTheory.FiniteMeasure.map (MeasureTheory.FiniteMeasure.prod μ ν) Prod.fst =\n  (fun s => (↑↑↑ν s).toNNReal) Set.univ • μ"}
+{"name":"MeasureTheory.ProbabilityMeasure.map_fst_prod","declaration":"theorem MeasureTheory.ProbabilityMeasure.map_fst_prod {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.ProbabilityMeasure α) (ν : MeasureTheory.ProbabilityMeasure β) : MeasureTheory.ProbabilityMeasure.map (MeasureTheory.ProbabilityMeasure.prod μ ν) ⋯ = μ"}
+{"name":"MeasureTheory.ProbabilityMeasure.prod_apply_null","declaration":"theorem MeasureTheory.ProbabilityMeasure.prod_apply_null {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.ProbabilityMeasure α) (ν : MeasureTheory.ProbabilityMeasure β) {s : Set (α × β)} (hs : MeasurableSet s) : (fun s => (↑↑↑(MeasureTheory.ProbabilityMeasure.prod μ ν) s).toNNReal) s = 0 ↔\n  (fun x => (fun s => (↑↑↑ν s).toNNReal) (Prod.mk x ⁻¹' s)) =ᶠ[MeasureTheory.Measure.ae ↑μ] 0"}
+{"name":"MeasureTheory.FiniteMeasure.prod_apply","declaration":"theorem MeasureTheory.FiniteMeasure.prod_apply {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.FiniteMeasure α) (ν : MeasureTheory.FiniteMeasure β) (s : Set (α × β)) (s_mble : MeasurableSet s) : (fun s => (↑↑↑(MeasureTheory.FiniteMeasure.prod μ ν) s).toNNReal) s = (∫⁻ (x : α), ↑↑↑ν (Prod.mk x ⁻¹' s) ∂↑μ).toNNReal"}
+{"name":"MeasureTheory.FiniteMeasure.prod_apply_symm","declaration":"theorem MeasureTheory.FiniteMeasure.prod_apply_symm {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.FiniteMeasure α) (ν : MeasureTheory.FiniteMeasure β) (s : Set (α × β)) (s_mble : MeasurableSet s) : (fun s => (↑↑↑(MeasureTheory.FiniteMeasure.prod μ ν) s).toNNReal) s =\n  (∫⁻ (y : β), ↑↑↑μ ((fun x => (x, y)) ⁻¹' s) ∂↑ν).toNNReal"}
+{"name":"MeasureTheory.FiniteMeasure.prod_prod","declaration":"theorem MeasureTheory.FiniteMeasure.prod_prod {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.FiniteMeasure α) (ν : MeasureTheory.FiniteMeasure β) (s : Set α) (t : Set β) : (fun s => (↑↑↑(MeasureTheory.FiniteMeasure.prod μ ν) s).toNNReal) (s ×ˢ t) =\n  (fun s => (↑↑↑μ s).toNNReal) s * (fun s => (↑↑↑ν s).toNNReal) t"}
+{"name":"MeasureTheory.FiniteMeasure.prod_apply_null","declaration":"theorem MeasureTheory.FiniteMeasure.prod_apply_null {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.FiniteMeasure α) (ν : MeasureTheory.FiniteMeasure β) {s : Set (α × β)} (hs : MeasurableSet s) : (fun s => (↑↑↑(MeasureTheory.FiniteMeasure.prod μ ν) s).toNNReal) s = 0 ↔\n  (fun x => (fun s => (↑↑↑ν s).toNNReal) (Prod.mk x ⁻¹' s)) =ᶠ[MeasureTheory.Measure.ae ↑μ] 0"}
+{"name":"MeasureTheory.ProbabilityMeasure.map_snd_prod","declaration":"theorem MeasureTheory.ProbabilityMeasure.map_snd_prod {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.ProbabilityMeasure α) (ν : MeasureTheory.ProbabilityMeasure β) : MeasureTheory.ProbabilityMeasure.map (MeasureTheory.ProbabilityMeasure.prod μ ν) ⋯ = ν"}
+{"name":"MeasureTheory.ProbabilityMeasure.prod","declaration":"/-- The binary product of probability measures. -/\ndef MeasureTheory.ProbabilityMeasure.prod {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.ProbabilityMeasure α) (ν : MeasureTheory.ProbabilityMeasure β) : MeasureTheory.ProbabilityMeasure (α × β)"}
+{"name":"MeasureTheory.ProbabilityMeasure.prod_apply_symm","declaration":"theorem MeasureTheory.ProbabilityMeasure.prod_apply_symm {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.ProbabilityMeasure α) (ν : MeasureTheory.ProbabilityMeasure β) (s : Set (α × β)) (s_mble : MeasurableSet s) : (fun s => (↑↑↑(MeasureTheory.ProbabilityMeasure.prod μ ν) s).toNNReal) s =\n  (∫⁻ (y : β), ↑↑↑μ ((fun x => (x, y)) ⁻¹' s) ∂↑ν).toNNReal"}
+{"name":"MeasureTheory.FiniteMeasure.map_snd_prod","declaration":"theorem MeasureTheory.FiniteMeasure.map_snd_prod {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.FiniteMeasure α) (ν : MeasureTheory.FiniteMeasure β) : MeasureTheory.FiniteMeasure.map (MeasureTheory.FiniteMeasure.prod μ ν) Prod.snd =\n  (fun s => (↑↑↑μ s).toNNReal) Set.univ • ν"}
+{"name":"MeasureTheory.ProbabilityMeasure.measure_ae_null_of_prod_null","declaration":"theorem MeasureTheory.ProbabilityMeasure.measure_ae_null_of_prod_null {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.ProbabilityMeasure α) (ν : MeasureTheory.ProbabilityMeasure β) {s : Set (α × β)} (h : (fun s => (↑↑↑(MeasureTheory.ProbabilityMeasure.prod μ ν) s).toNNReal) s = 0) : (fun x => (fun s => (↑↑↑ν s).toNNReal) (Prod.mk x ⁻¹' s)) =ᶠ[MeasureTheory.Measure.ae ↑μ] 0"}
+{"name":"MeasureTheory.ProbabilityMeasure.prod_apply","declaration":"theorem MeasureTheory.ProbabilityMeasure.prod_apply {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.ProbabilityMeasure α) (ν : MeasureTheory.ProbabilityMeasure β) (s : Set (α × β)) (s_mble : MeasurableSet s) : (fun s => (↑↑↑(MeasureTheory.ProbabilityMeasure.prod μ ν) s).toNNReal) s =\n  (∫⁻ (x : α), ↑↑↑ν (Prod.mk x ⁻¹' s) ∂↑μ).toNNReal"}
+{"name":"MeasureTheory.FiniteMeasure.map_prod_map","declaration":"theorem MeasureTheory.FiniteMeasure.map_prod_map {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.FiniteMeasure α) (ν : MeasureTheory.FiniteMeasure β) {α' : Type u_3} [MeasurableSpace α'] {β' : Type u_4} [MeasurableSpace β'] {f : α → α'} {g : β → β'} (f_mble : Measurable f) (g_mble : Measurable g) : MeasureTheory.FiniteMeasure.prod (MeasureTheory.FiniteMeasure.map μ f) (MeasureTheory.FiniteMeasure.map ν g) =\n  MeasureTheory.FiniteMeasure.map (MeasureTheory.FiniteMeasure.prod μ ν) (Prod.map f g)"}
+{"name":"MeasureTheory.FiniteMeasure.zero_prod","declaration":"theorem MeasureTheory.FiniteMeasure.zero_prod {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (ν : MeasureTheory.FiniteMeasure β) : MeasureTheory.FiniteMeasure.prod 0 ν = 0"}
+{"name":"MeasureTheory.FiniteMeasure.toMeasure_prod","declaration":"theorem MeasureTheory.FiniteMeasure.toMeasure_prod {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.FiniteMeasure α) (ν : MeasureTheory.FiniteMeasure β) : ↑(MeasureTheory.FiniteMeasure.prod μ ν) = MeasureTheory.Measure.prod ↑μ ↑ν"}
+{"name":"MeasureTheory.FiniteMeasure.prod_zero","declaration":"theorem MeasureTheory.FiniteMeasure.prod_zero {α : Type u_1} [MeasurableSpace α] {β : Type u_2} [MeasurableSpace β] (μ : MeasureTheory.FiniteMeasure α) : MeasureTheory.FiniteMeasure.prod μ 0 = 0"}

PFR-declarations/PFR.ForMathlib.FiniteRange.jsonl

@@ -0,0 +1,24 @@
+{"name":"FiniteRange.pow","declaration":"/-- A function of finite range raised to a constant power, has finite range.  -/\ninstance FiniteRange.pow {Ω : Type u_1} {G : Type u_2} (X : Ω → G) [Group G] [hX : FiniteRange X] (c : ℤ) : FiniteRange (X ^ c)"}
+{"name":"FiniteRange.finite","declaration":"def FiniteRange.finite {Ω : Type u_1} {G : Type u_2} {X : Ω → G} [self : FiniteRange X] : Set.Finite (Set.range X)"}
+{"name":"FiniteRange.mk","declaration":"ctor FiniteRange.mk {Ω : Type u_1} {G : Type u_2} {X : Ω → G} (finite : Set.Finite (Set.range X)) : FiniteRange X"}
+{"name":"instFiniteRangeComp_1","declaration":"/-- If X has finite range, then X of any function has finite range.  -/\ninstance instFiniteRangeComp_1 {Ω : Type u_1} {Ω' : Type u_2} {G : Type u_3} (X : Ω → G) (f : Ω' → Ω) [hX : FiniteRange X] : FiniteRange (X ∘ f)"}
+{"name":"FiniteRange.toFinset","declaration":"/-- The range of a finite range map, as a finset. -/\ndef FiniteRange.toFinset {Ω : Type u_1} {G : Type u_2} (X : Ω → G) [hX : FiniteRange X] : Finset G"}
+{"name":"FiniteRange.mem","declaration":"theorem FiniteRange.mem {Ω : Type u_1} {G : Type u_2} (X : Ω → G) [FiniteRange X] (ω : Ω) : X ω ∈ FiniteRange.toFinset X"}
+{"name":"FiniteRange.sub","declaration":"/-- The difference of functions of finite range, has finite range.-/\ninstance FiniteRange.sub {Ω : Type u_1} {G : Type u_2} (X : Ω → G) (Y : Ω → G) [AddGroup G] [hX : FiniteRange X] [hY : FiniteRange Y] : FiniteRange (X - Y)"}
+{"name":"instFiniteRange_1","declaration":"/-- Constants have finite range -/\ninstance instFiniteRange_1 {Ω : Type u_1} {G : Type u_2} (c : G) : FiniteRange fun x => c"}
+{"name":"instFiniteRangeProdProd","declaration":"/-- If X, Y have finite range, then so does the pair ⟨X, Y⟩. -/\ninstance instFiniteRangeProdProd {Ω : Type u_1} {G : Type u_2} {H : Type u_3} (X : Ω → G) (Y : Ω → H) [hX : FiniteRange X] [hY : FiniteRange Y] : FiniteRange (⟨X, Y⟩)"}
+{"name":"FiniteRange.null_of_compl","declaration":"theorem FiniteRange.null_of_compl {Ω : Type u_1} {G : Type u_2} [MeasurableSpace Ω] [MeasurableSpace G] [MeasurableSingletonClass G] (μ : MeasureTheory.Measure Ω) (X : Ω → G) [FiniteRange X] : ↑↑(MeasureTheory.Measure.map X μ) (↑(FiniteRange.toFinset X))ᶜ = 0"}
+{"name":"FiniteRange.range","declaration":"theorem FiniteRange.range {Ω : Type u_1} {G : Type u_2} (X : Ω → G) [hX : FiniteRange X] : Set.range X = ↑(FiniteRange.toFinset X)"}
+{"name":"FiniteRange.nsmul","declaration":"/-- The multiple of a function of finite range by a constant, has finite range.-/\ninstance FiniteRange.nsmul {Ω : Type u_1} {G : Type u_2} (X : Ω → G) [AddGroup G] [hX : FiniteRange X] (c : ℤ) : FiniteRange (c • X)"}
+{"name":"FiniteRange.sum","declaration":"/-- The sum of functions of finite range, has finite range.-/\ninstance FiniteRange.sum {Ω : Type u_1} {G : Type u_2} (X : Ω → G) (Y : Ω → G) [AddGroup G] [hX : FiniteRange X] [hY : FiniteRange Y] : FiniteRange (X + Y)"}
+{"name":"FiniteRange.neg","declaration":"/-- The negation of a function of finite range, has finite range.-/\ninstance FiniteRange.neg {Ω : Type u_1} {G : Type u_2} (X : Ω → G) [AddGroup G] [hX : FiniteRange X] : FiniteRange (-X)"}
+{"name":"FiniteRange","declaration":"/-- The property of having a finite range. -/\nclass FiniteRange {Ω : Type u_1} {G : Type u_2} (X : Ω → G) : Prop"}
+{"name":"FiniteRange.fintype","declaration":"/-- fintype structure on the range of a finite range map. -/\ndef FiniteRange.fintype {Ω : Type u_1} {G : Type u_2} (X : Ω → G) [hX : FiniteRange X] : Fintype ↑(Set.range X)"}
+{"name":"instFiniteRangeComp","declaration":"/-- If X has finite range, then any function of X has finite range.  -/\ninstance instFiniteRangeComp {Ω : Type u_1} {G : Type u_2} {H : Type u_3} (X : Ω → G) (f : G → H) [hX : FiniteRange X] : FiniteRange (f ∘ X)"}
+{"name":"FiniteRange.mem_iff","declaration":"theorem FiniteRange.mem_iff {Ω : Type u_1} {G : Type u_2} (X : Ω → G) [FiniteRange X] (x : G) : x ∈ FiniteRange.toFinset X ↔ ∃ ω, X ω = x"}
+{"name":"instFiniteRange","declaration":"/-- If the codomain of X is finite, then X has finite range.  -/\ninstance instFiniteRange {Ω : Type u_1} {G : Type u_2} (X : Ω → G) [Fintype G] : FiniteRange X"}
+{"name":"finiteRange_of_finset","declaration":"/-- Functions ranging in a Finset have finite range -/\ntheorem finiteRange_of_finset {Ω : Type u_1} {G : Type u_2} (f : Ω → G) (A : Finset G) (h : ∀ (ω : Ω), f ω ∈ A) : FiniteRange f"}
+{"name":"FiniteRange.full","declaration":"theorem FiniteRange.full {Ω : Type u_1} {G : Type u_2} [MeasurableSpace Ω] [MeasurableSpace G] [MeasurableSingletonClass G] {X : Ω → G} (hX : Measurable X) [FiniteRange X] (μ : MeasureTheory.Measure Ω) : ↑↑(MeasureTheory.Measure.map X μ) ↑(FiniteRange.toFinset X) = ↑↑μ Set.univ"}
+{"name":"FiniteRange.div","declaration":"/-- The quotient of two functions with finite range, has finite range. -/\ninstance FiniteRange.div {Ω : Type u_1} {G : Type u_2} (X : Ω → G) (Y : Ω → G) [Group G] [hX : FiniteRange X] [hY : FiniteRange Y] : FiniteRange (X / Y)"}
+{"name":"FiniteRange.prod","declaration":"/-- The product of functions of finite range, has finite range.   -/\ninstance FiniteRange.prod {Ω : Type u_1} {G : Type u_2} (X : Ω → G) (Y : Ω → G) [Group G] [hX : FiniteRange X] [hY : FiniteRange Y] : FiniteRange (X * Y)"}
+{"name":"FiniteRange.inv","declaration":"/-- The inverse of a function of finite range, has finite range.-/\ninstance FiniteRange.inv {Ω : Type u_1} {G : Type u_2} (X : Ω → G) [Group G] [hX : FiniteRange X] : FiniteRange X⁻¹"}

PFR-declarations/PFR.ForMathlib.Graph.jsonl

@@ -0,0 +1,11 @@
+{"name":"Set.graph_comp","declaration":"theorem Set.graph_comp {A : Type u_3} {B : Type u_4} {C : Type u_5} {f : A → B} (g : B → C) : Set.graph (g ∘ f) = (fun p => (p.1, g p.2)) '' Set.graph f"}
+{"name":"Set.image_snd_graph","declaration":"theorem Set.image_snd_graph {G : Type u_1} {G' : Type u_2} {f : G → G'} : Prod.snd '' Set.graph f = f '' Set.univ"}
+{"name":"Set.graph_nonempty","declaration":"theorem Set.graph_nonempty {G : Type u_1} {G' : Type u_2} [Nonempty G] (f : G → G') : Set.Nonempty (Set.graph f)"}
+{"name":"Set.graph","declaration":"def Set.graph {G : Type u_1} {G' : Type u_2} (f : G → G') : Set (G × G')"}
+{"name":"Set.fst_injOn_graph","declaration":"theorem Set.fst_injOn_graph {G : Type u_1} {G' : Type u_2} (f : G → G') : Set.InjOn Prod.fst (Set.graph f)"}
+{"name":"Set.mem_graph","declaration":"theorem Set.mem_graph {G : Type u_1} {G' : Type u_2} {f : G → G'} (x : G × G') : x ∈ Set.graph f ↔ f x.1 = x.2"}
+{"name":"Set.image_fst_graph","declaration":"theorem Set.image_fst_graph {G : Type u_1} {G' : Type u_2} {f : G → G'} : Prod.fst '' Set.graph f = Set.univ"}
+{"name":"Set.graph_def","declaration":"theorem Set.graph_def {G : Type u_1} {G' : Type u_2} (f : G → G') : Set.graph f = {x | ∃ x_1, (x_1, f x_1) = x}"}
+{"name":"Set.card_graph","declaration":"theorem Set.card_graph {G : Type u_1} {G' : Type u_2} (f : G → G') : Nat.card ↑(Set.graph f) = Nat.card G"}
+{"name":"Set.graph_add","declaration":"theorem Set.graph_add {G : Type u_1} {G' : Type u_2} [AddGroup G] [AddCommGroup G'] {f : G →+ G'} {c : G × G'} : (fun x => c + x) '' Set.graph ⇑f = {x | ∃ g, (g, f g + (c.2 - f c.1)) = x}"}
+{"name":"Set.equiv_filter_graph","declaration":"theorem Set.equiv_filter_graph {G : Type u_3} {G' : Type u_4} [AddCommGroup G] [Fintype G] [AddCommGroup G'] [Fintype G'] [DecidableEq G] [DecidableEq G'] (f : G → G') : let A := Set.Finite.toFinset ⋯;\n{ x //\n    x ∈\n      Finset.filter\n        (fun x =>\n          match x with\n          | (a, a') => a + a' ∈ A)\n        (A ×ˢ A) } ≃\n  ↑{x | f (x.1 + x.2) = f x.1 + f x.2}"}

PFR-declarations/PFR.ForMathlib.MeasureReal.jsonl

@@ -0,0 +1,74 @@
+{"name":"sum_measure_preimage_singleton'","declaration":"/-- Variant of `sum_measure_preimage_singleton` using real numbers rather than extended nonnegative\nreals. -/\ntheorem sum_measure_preimage_singleton' {Ω : Type u_1} [MeasurableSpace Ω] (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] {T : Type u} [Fintype T] [MeasurableSpace T] [MeasurableSingletonClass T] {Y : Ω → T} (hY : Measurable Y) : (Finset.sum Finset.univ fun y => (↑↑μ (Y ⁻¹' {y})).toReal) = 1"}
+{"name":"MeasureTheory.measureReal_symmDiff_le","declaration":"theorem MeasureTheory.measureReal_symmDiff_le {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} (s t u : Set α),\n  autoParam (↑↑μ s ≠ ⊤) _auto✝ →\n    autoParam (↑↑μ t ≠ ⊤) _auto✝¹ → μ.real (symmDiff s u) ≤ μ.real (symmDiff s t) + μ.real (symmDiff t u)"}
+{"name":"Finset.sum_toReal_measure_singleton","declaration":"theorem Finset.sum_toReal_measure_singleton {S : Type u_1} {s : Finset S} : ∀ {x : MeasurableSpace S} [inst : MeasurableSingletonClass S] (μ : MeasureTheory.Measure S)\n  [inst : MeasureTheory.IsFiniteMeasure μ], (Finset.sum s fun x_1 => (↑↑μ {x_1}).toReal) = (↑↑μ ↑s).toReal"}
+{"name":"MeasureTheory.exists_nonempty_inter_of_measureReal_univ_lt_sum_measureReal","declaration":"/-- Pigeonhole principle for measure spaces: if `s` is a `Finset` and\n`∑ i in s, μ.real (t i) > μ.real univ`, then one of the intersections `t i ∩ t j` is not empty. -/\ntheorem MeasureTheory.exists_nonempty_inter_of_measureReal_univ_lt_sum_measureReal {α : Type u_1} {ι : Type u_3} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) [MeasureTheory.IsFiniteMeasure μ] {s : Finset ι} {t : ι → Set α} (h : ∀ i ∈ s, MeasurableSet (t i)) (H : μ.real Set.univ < Finset.sum s fun i => μ.real (t i)) : ∃ i ∈ s, ∃ j ∈ s, ∃ (_ : i ≠ j), Set.Nonempty (t i ∩ t j)"}
+{"name":"MeasureTheory.measureReal_le_measureReal_union_left","declaration":"theorem MeasureTheory.measureReal_le_measureReal_union_left {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α},\n  autoParam (↑↑μ t ≠ ⊤) _auto✝ → μ.real s ≤ μ.real (s ∪ t)"}
+{"name":"sum_toReal_measure_singleton","declaration":"theorem sum_toReal_measure_singleton {S : Type u_1} [Fintype S] : ∀ {x : MeasurableSpace S} [inst : MeasurableSingletonClass S] (μ : MeasureTheory.Measure S)\n  [inst : MeasureTheory.IsFiniteMeasure μ],\n  (Finset.sum Finset.univ fun x_1 => (↑↑μ {x_1}).toReal) = (↑↑μ Set.univ).toReal"}
+{"name":"MeasureTheory.measure_symmDiff_eq_top","declaration":"theorem MeasureTheory.measure_symmDiff_eq_top {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α}, ↑↑μ s ≠ ⊤ → ↑↑μ t = ⊤ → ↑↑μ (symmDiff s t) = ⊤"}
+{"name":"MeasureTheory.measureReal_mono","declaration":"theorem MeasureTheory.measureReal_mono {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s₁ s₂ : Set α},\n  s₁ ⊆ s₂ → autoParam (↑↑μ s₂ ≠ ⊤) _auto✝ → μ.real s₁ ≤ μ.real s₂"}
+{"name":"MeasureTheory.measureReal_symmDiff_eq","declaration":"theorem MeasureTheory.measureReal_symmDiff_eq {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α},\n  MeasurableSet s →\n    MeasurableSet t →\n      autoParam (↑↑μ s ≠ ⊤) _auto✝ →\n        autoParam (↑↑μ t ≠ ⊤) _auto✝¹ → μ.real (symmDiff s t) = μ.real (s \\ t) + μ.real (t \\ s)"}
+{"name":"MeasureTheory.measureReal_eq_measureReal_of_between_null_diff","declaration":"theorem MeasureTheory.measureReal_eq_measureReal_of_between_null_diff {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s₁ s₂ s₃ : Set α},\n  s₁ ⊆ s₂ →\n    s₂ ⊆ s₃ →\n      μ.real (s₃ \\ s₁) = 0 → autoParam (↑↑μ (s₃ \\ s₁) ≠ ⊤) _auto✝ → μ.real s₁ = μ.real s₂ ∧ μ.real s₂ = μ.real s₃"}
+{"name":"MeasureTheory.measureReal_empty","declaration":"theorem MeasureTheory.measureReal_empty {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α}, μ.real ∅ = 0"}
+{"name":"MeasureTheory.nonempty_inter_of_measureReal_lt_add'","declaration":"/-- If two sets `s` and `t` are included in a set `u` of finite measure,\nand `μ.real s + μ.real t > μ.real u`, then `s` intersects `t`.\nVersion assuming that `s` is measurable. -/\ntheorem MeasureTheory.nonempty_inter_of_measureReal_lt_add' {α : Type u_1} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) {s : Set α} {t : Set α} {u : Set α} (hs : MeasurableSet s) (h's : s ⊆ u) (h't : t ⊆ u) (h : μ.real u < μ.real s + μ.real t) (hu : autoParam (↑↑μ u ≠ ⊤) _auto✝) : Set.Nonempty (s ∩ t)"}
+{"name":"MeasureTheory.measureReal_union_null_iff","declaration":"theorem MeasureTheory.measureReal_union_null_iff {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s₁ s₂ : Set α},\n  autoParam (↑↑μ s₁ ≠ ⊤) _auto✝ →\n    autoParam (↑↑μ s₂ ≠ ⊤) _auto✝¹ → (μ.real (s₁ ∪ s₂) = 0 ↔ μ.real s₁ = 0 ∧ μ.real s₂ = 0)"}
+{"name":"MeasureTheory.measureReal_nonneg","declaration":"theorem MeasureTheory.measureReal_nonneg {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α}, 0 ≤ μ.real s"}
+{"name":"MeasureTheory.Measure.real","declaration":"/-- The real-valued version of a measure. Maps infinite measure sets to zero. Use as `μ.real s`. -/\ndef MeasureTheory.Measure.real {α : Type u_1} : {x : MeasurableSpace α} → MeasureTheory.Measure α → Set α → ℝ"}
+{"name":"MeasureTheory.measureReal_union_congr_of_subset","declaration":"theorem MeasureTheory.measureReal_union_congr_of_subset {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s₁ s₂ t₁ t₂ : Set α},\n  s₁ ⊆ s₂ →\n    μ.real s₂ ≤ μ.real s₁ →\n      t₁ ⊆ t₂ →\n        μ.real t₂ ≤ μ.real t₁ →\n          autoParam (↑↑μ s₂ ≠ ⊤) _auto✝ → autoParam (↑↑μ t₂ ≠ ⊤) _auto✝¹ → μ.real (s₁ ∪ t₁) = μ.real (s₂ ∪ t₂)"}
+{"name":"MeasureTheory.measureReal_zero","declaration":"theorem MeasureTheory.measureReal_zero {α : Type u_1} : ∀ {x : MeasurableSpace α} (s : Set α), 0.real s = 0"}
+{"name":"MeasureTheory.measureReal_union_le","declaration":"theorem MeasureTheory.measureReal_union_le {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} (s₁ s₂ : Set α), μ.real (s₁ ∪ s₂) ≤ μ.real s₁ + μ.real s₂"}
+{"name":"MeasureTheory.measureReal_union₀'","declaration":"theorem MeasureTheory.measureReal_union₀' {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α},\n  MeasureTheory.NullMeasurableSet s μ →\n    MeasureTheory.AEDisjoint μ s t →\n      autoParam (↑↑μ s ≠ ⊤) _auto✝ → autoParam (↑↑μ t ≠ ⊤) _auto✝¹ → μ.real (s ∪ t) = μ.real s + μ.real t"}
+{"name":"MeasureTheory.measureReal_biUnion_finset","declaration":"theorem MeasureTheory.measureReal_biUnion_finset {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {ι : Type u_3} {s : Finset ι} {f : ι → Set α},\n  Set.PairwiseDisjoint (↑s) f →\n    (∀ b ∈ s, MeasurableSet (f b)) →\n      autoParam (∀ b ∈ s, ↑↑μ (f b) ≠ ⊤) _auto✝ → μ.real (⋃ b ∈ s, f b) = Finset.sum s fun p => μ.real (f p)"}
+{"name":"MeasureTheory.IsProbabilityMeasure.measureReal_univ","declaration":"theorem MeasureTheory.IsProbabilityMeasure.measureReal_univ {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst : MeasureTheory.IsProbabilityMeasure μ],\n  μ.real Set.univ = 1"}
+{"name":"MeasureTheory.measureReal_inter_add_diff₀","declaration":"theorem MeasureTheory.measureReal_inter_add_diff₀ {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {t : Set α} (s : Set α),\n  MeasureTheory.NullMeasurableSet t μ → autoParam (↑↑μ s ≠ ⊤) _auto✝ → μ.real (s ∩ t) + μ.real (s \\ t) = μ.real s"}
+{"name":"measureReal_preimage_snd_singleton_eq_sum","declaration":"theorem measureReal_preimage_snd_singleton_eq_sum {S : Type u_1} {T : Type u_2} [Fintype S] : ∀ {x : MeasurableSpace S} [inst : MeasurableSingletonClass S] {x_1 : MeasurableSpace T}\n  [inst : MeasurableSingletonClass T] (μ : MeasureTheory.Measure (S × T)) [inst : MeasureTheory.IsFiniteMeasure μ]\n  (y : T), μ.real (Prod.snd ⁻¹' {y}) = Finset.sum Finset.univ fun x_2 => μ.real {(x_2, y)}"}
+{"name":"MeasureTheory.measureReal_add_measureReal_compl₀","declaration":"theorem MeasureTheory.measureReal_add_measureReal_compl₀ {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst : MeasureTheory.IsFiniteMeasure μ] {s : Set α},\n  MeasureTheory.NullMeasurableSet s μ → μ.real s + μ.real sᶜ = μ.real Set.univ"}
+{"name":"MeasureTheory.measureReal_diff","declaration":"theorem MeasureTheory.measureReal_diff {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s₁ s₂ : Set α},\n  s₂ ⊆ s₁ → MeasurableSet s₂ → autoParam (↑↑μ s₁ ≠ ⊤) _auto✝ → μ.real (s₁ \\ s₂) = μ.real s₁ - μ.real s₂"}
+{"name":"MeasureTheory.measureReal_iUnion_fintype","declaration":"theorem MeasureTheory.measureReal_iUnion_fintype {α : Type u_1} {β : Type u_2} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst : Fintype β] {f : β → Set α},\n  Pairwise (Disjoint on f) →\n    (∀ (i : β), MeasurableSet (f i)) →\n      autoParam (∀ (i : β), ↑↑μ (f i) ≠ ⊤) _auto✝ → μ.real (⋃ b, f b) = Finset.sum Finset.univ fun p => μ.real (f p)"}
+{"name":"MeasureTheory.measureReal_diff_null'","declaration":"theorem MeasureTheory.measureReal_diff_null' {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s₁ s₂ : Set α},\n  μ.real (s₁ ∩ s₂) = 0 → autoParam (↑↑μ s₁ ≠ ⊤) _auto✝ → μ.real (s₁ \\ s₂) = μ.real s₁"}
+{"name":"MeasureTheory.measureReal_smul_apply","declaration":"theorem MeasureTheory.measureReal_smul_apply {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} (c : ENNReal), (c • μ).real s = c.toReal • μ.real s"}
+{"name":"Finset.sum_measure_singleton","declaration":"theorem Finset.sum_measure_singleton {S : Type u_1} {s : Finset S} : ∀ {x : MeasurableSpace S} [inst : MeasurableSingletonClass S] (μ : MeasureTheory.Measure S),\n  (Finset.sum s fun x_1 => ↑↑μ {x_1}) = ↑↑μ ↑s"}
+{"name":"MeasureTheory.measureReal_union_add_inter₀","declaration":"theorem MeasureTheory.measureReal_union_add_inter₀ {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {t : Set α} (s : Set α),\n  MeasureTheory.NullMeasurableSet t μ →\n    autoParam (↑↑μ s ≠ ⊤) _auto✝ → autoParam (↑↑μ t ≠ ⊤) _auto✝¹ → μ.real (s ∪ t) + μ.real (s ∩ t) = μ.real s + μ.real t"}
+{"name":"MeasureTheory.nonempty_of_measureReal_ne_zero","declaration":"theorem MeasureTheory.nonempty_of_measureReal_ne_zero {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α}, μ.real s ≠ 0 → Set.Nonempty s"}
+{"name":"Mathlib.Meta.Positivity.evalMeasureReal","declaration":"/-- Extension for the `positivity` tactic: applications of `μ.real` are nonnegative. -/\ndef Mathlib.Meta.Positivity.evalMeasureReal  : Mathlib.Meta.Positivity.PositivityExt"}
+{"name":"MeasureTheory.measureReal_mono_null","declaration":"theorem MeasureTheory.measureReal_mono_null {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s₁ s₂ : Set α},\n  s₁ ⊆ s₂ → μ.real s₂ = 0 → autoParam (↑↑μ s₂ ≠ ⊤) _auto✝ → μ.real s₁ = 0"}
+{"name":"measureReal_preimage_fst_singleton_eq_sum","declaration":"theorem measureReal_preimage_fst_singleton_eq_sum {S : Type u_1} {T : Type u_2} : ∀ {x : MeasurableSpace S} [inst : MeasurableSingletonClass S] [inst : Fintype T] {x_1 : MeasurableSpace T}\n  [inst_1 : MeasurableSingletonClass T] (μ : MeasureTheory.Measure (S × T)) [inst_2 : MeasureTheory.IsFiniteMeasure μ]\n  (x_2 : S), μ.real (Prod.fst ⁻¹' {x_2}) = Finset.sum Finset.univ fun y => μ.real {(x_2, y)}"}
+{"name":"MeasureTheory.measureReal_eq_measureReal_smaller_of_between_null_diff","declaration":"theorem MeasureTheory.measureReal_eq_measureReal_smaller_of_between_null_diff {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s₁ s₂ s₃ : Set α},\n  s₁ ⊆ s₂ → s₂ ⊆ s₃ → μ.real (s₃ \\ s₁) = 0 → autoParam (↑↑μ (s₃ \\ s₁) ≠ ⊤) _auto✝ → μ.real s₁ = μ.real s₂"}
+{"name":"MeasureTheory.sum_measureReal_preimage_singleton","declaration":"/-- If `s` is a `Finset`, then the measure of its preimage can be found as the sum of measures\nof the fibers `f ⁻¹' {y}`. -/\ntheorem MeasureTheory.sum_measureReal_preimage_singleton {α : Type u_1} {β : Type u_2} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} (s : Finset β) {f : α → β},\n  (∀ y ∈ s, MeasurableSet (f ⁻¹' {y})) →\n    autoParam (∀ a ∈ s, ↑↑μ (f ⁻¹' {a}) ≠ ⊤) _auto✝ → (Finset.sum s fun b => μ.real (f ⁻¹' {b})) = μ.real (f ⁻¹' ↑s)"}
+{"name":"MeasureTheory.measureReal_union_null","declaration":"theorem MeasureTheory.measureReal_union_null {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s₁ s₂ : Set α},\n  μ.real s₁ = 0 → μ.real s₂ = 0 → μ.real (s₁ ∪ s₂) = 0"}
+{"name":"MeasureTheory.measureReal_union_add_inter₀'","declaration":"theorem MeasureTheory.measureReal_union_add_inter₀' {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α},\n  MeasureTheory.NullMeasurableSet s μ →\n    ∀ (t : Set α),\n      autoParam (↑↑μ s ≠ ⊤) _auto✝ →\n        autoParam (↑↑μ t ≠ ⊤) _auto✝¹ → μ.real (s ∪ t) + μ.real (s ∩ t) = μ.real s + μ.real t"}
+{"name":"MeasureTheory.measureReal_union₀","declaration":"theorem MeasureTheory.measureReal_union₀ {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α},\n  MeasureTheory.NullMeasurableSet t μ →\n    MeasureTheory.AEDisjoint μ s t →\n      autoParam (↑↑μ s ≠ ⊤) _auto✝ → autoParam (↑↑μ t ≠ ⊤) _auto✝¹ → μ.real (s ∪ t) = μ.real s + μ.real t"}
+{"name":"MeasureTheory.measureReal_univ_pos","declaration":"theorem MeasureTheory.measureReal_univ_pos {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst : MeasureTheory.IsFiniteMeasure μ] [inst : NeZero μ],\n  0 < μ.real Set.univ"}
+{"name":"sum_measure_singleton","declaration":"theorem sum_measure_singleton {S : Type u_1} [Fintype S] : ∀ {x : MeasurableSpace S} [inst : MeasurableSingletonClass S] (μ : MeasureTheory.Measure S),\n  (Finset.sum Finset.univ fun x_1 => ↑↑μ {x_1}) = ↑↑μ Set.univ"}
+{"name":"MeasureTheory.measureReal_iUnion_fintype_le","declaration":"theorem MeasureTheory.measureReal_iUnion_fintype_le {α : Type u_1} {β : Type u_2} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst : Fintype β] (f : β → Set α),\n  μ.real (⋃ b, f b) ≤ Finset.sum Finset.univ fun p => μ.real (f p)"}
+{"name":"MeasureTheory.measureReal_univ_ne_zero","declaration":"theorem MeasureTheory.measureReal_univ_ne_zero {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst : MeasureTheory.IsFiniteMeasure μ] [inst : NeZero μ],\n  μ.real Set.univ ≠ 0"}
+{"name":"MeasureTheory.measure_diff_eq_top","declaration":"theorem MeasureTheory.measure_diff_eq_top {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α}, ↑↑μ s = ⊤ → ↑↑μ t ≠ ⊤ → ↑↑μ (s \\ t) = ⊤"}
+{"name":"MeasureTheory.measureReal_congr","declaration":"/-- If two sets are equal modulo a set of measure zero, then `μ.real s = μ.real t`. -/\ntheorem MeasureTheory.measureReal_congr {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α},\n  s =ᶠ[MeasureTheory.Measure.ae μ] t → μ.real s = μ.real t"}
+{"name":"MeasureTheory.measure_ne_top_of_subset","declaration":"theorem MeasureTheory.measure_ne_top_of_subset {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α}, s ⊆ t → ↑↑μ t ≠ ⊤ → ↑↑μ s ≠ ⊤"}
+{"name":"MeasureTheory.measureReal_union","declaration":"theorem MeasureTheory.measureReal_union {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s₁ s₂ : Set α},\n  Disjoint s₁ s₂ →\n    MeasurableSet s₂ →\n      autoParam (↑↑μ s₁ ≠ ⊤) _auto✝ → autoParam (↑↑μ s₂ ≠ ⊤) _auto✝¹ → μ.real (s₁ ∪ s₂) = μ.real s₁ + μ.real s₂"}
+{"name":"MeasureTheory.measureReal_le_measureReal_union_right","declaration":"theorem MeasureTheory.measureReal_le_measureReal_union_right {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α},\n  autoParam (↑↑μ s ≠ ⊤) _auto✝ → μ.real t ≤ μ.real (s ∪ t)"}
+{"name":"MeasureTheory.map_measureReal_apply","declaration":"theorem MeasureTheory.map_measureReal_apply {α : Type u_1} {β : Type u_2} : ∀ {x : MeasurableSpace α} [inst : MeasurableSpace β] {μ : MeasureTheory.Measure α} {f : α → β},\n  Measurable f → ∀ {s : Set β}, MeasurableSet s → (MeasureTheory.Measure.map f μ).real s = μ.real (f ⁻¹' s)"}
+{"name":"MeasureTheory.measureReal_compl","declaration":"theorem MeasureTheory.measureReal_compl {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} [inst : MeasureTheory.IsFiniteMeasure μ],\n  MeasurableSet s → μ.real sᶜ = μ.real Set.univ - μ.real s"}
+{"name":"MeasureTheory.measureReal_diff'","declaration":"theorem MeasureTheory.measureReal_diff' {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {t : Set α} (s : Set α),\n  MeasurableSet t →\n    autoParam (↑↑μ s ≠ ⊤) _auto✝ → autoParam (↑↑μ t ≠ ⊤) _auto✝¹ → μ.real (s \\ t) = μ.real (s ∪ t) - μ.real t"}
+{"name":"MeasureTheory.measureReal_union_add_inter'","declaration":"theorem MeasureTheory.measureReal_union_add_inter' {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α},\n  MeasurableSet s →\n    ∀ (t : Set α),\n      autoParam (↑↑μ s ≠ ⊤) _auto✝ →\n        autoParam (↑↑μ t ≠ ⊤) _auto✝¹ → μ.real (s ∪ t) + μ.real (s ∩ t) = μ.real s + μ.real t"}
+{"name":"MeasureTheory.measureReal_union'","declaration":"theorem MeasureTheory.measureReal_union' {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s₁ s₂ : Set α},\n  Disjoint s₁ s₂ →\n    MeasurableSet s₁ →\n      autoParam (↑↑μ s₁ ≠ ⊤) _auto✝ → autoParam (↑↑μ s₂ ≠ ⊤) _auto✝¹ → μ.real (s₁ ∪ s₂) = μ.real s₁ + μ.real s₂"}
+{"name":"MeasureTheory.measureReal_def","declaration":"theorem MeasureTheory.measureReal_def {α : Type u_1} : ∀ {x : MeasurableSpace α} (μ : MeasureTheory.Measure α) (s : Set α), μ.real s = (↑↑μ s).toReal"}
+{"name":"MeasureTheory.measureReal_eq_measureReal_of_null_diff","declaration":"theorem MeasureTheory.measureReal_eq_measureReal_of_null_diff {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α},\n  s ⊆ t → μ.real (t \\ s) = 0 → autoParam (↑↑μ (t \\ s) ≠ ⊤) _auto✝ → μ.real s = μ.real t"}
+{"name":"MeasureTheory.measureReal_prod_prod","declaration":"theorem MeasureTheory.measureReal_prod_prod {α : Type u_1} {β : Type u_2} : ∀ {x : MeasurableSpace α} [inst : MeasurableSpace β] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β}\n  [inst_1 : MeasureTheory.SigmaFinite ν] (s : Set α) (t : Set β),\n  (MeasureTheory.Measure.prod μ ν).real (s ×ˢ t) = μ.real s * ν.real t"}
+{"name":"MeasureTheory.Measure.ext_iff_singleton","declaration":"/-- Generalized in Measure.ext_iff_singleton_finiteSupport at Entropy.Measure -/\ntheorem MeasureTheory.Measure.ext_iff_singleton {S : Type u_3} [Fintype S] [MeasurableSpace S] [MeasurableSingletonClass S] {μ1 : MeasureTheory.Measure S} {μ2 : MeasureTheory.Measure S} : μ1 = μ2 ↔ ∀ (x : S), ↑↑μ1 {x} = ↑↑μ2 {x}"}
+{"name":"MeasureTheory.measureReal_diff_null","declaration":"theorem MeasureTheory.measureReal_diff_null {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s₁ s₂ : Set α},\n  μ.real s₂ = 0 → autoParam (↑↑μ s₂ ≠ ⊤) _auto✝ → μ.real (s₁ \\ s₂) = μ.real s₁"}
+{"name":"MeasureTheory.measureReal_biUnion_finset_le","declaration":"theorem MeasureTheory.measureReal_biUnion_finset_le {α : Type u_1} {β : Type u_2} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} (s : Finset β) (f : β → Set α),\n  μ.real (⋃ b ∈ s, f b) ≤ Finset.sum s fun p => μ.real (f p)"}
+{"name":"MeasureTheory.measureReal_eq_measureReal_larger_of_between_null_diff","declaration":"theorem MeasureTheory.measureReal_eq_measureReal_larger_of_between_null_diff {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s₁ s₂ s₃ : Set α},\n  s₁ ⊆ s₂ → s₂ ⊆ s₃ → μ.real (s₃ \\ s₁) = 0 → autoParam (↑↑μ (s₃ \\ s₁) ≠ ⊤) _auto✝ → μ.real s₂ = μ.real s₃"}
+{"name":"MeasureTheory.le_measureReal_diff","declaration":"theorem MeasureTheory.le_measureReal_diff {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s₁ s₂ : Set α},\n  autoParam (↑↑μ s₂ ≠ ⊤) _auto✝ → μ.real s₁ - μ.real s₂ ≤ μ.real (s₁ \\ s₂)"}
+{"name":"MeasureTheory.measureReal_biUnion_finset₀","declaration":"theorem MeasureTheory.measureReal_biUnion_finset₀ {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {ι : Type u_3} {s : Finset ι} {f : ι → Set α},\n  Set.Pairwise (↑s) (MeasureTheory.AEDisjoint μ on f) →\n    (∀ b ∈ s, MeasureTheory.NullMeasurableSet (f b) μ) →\n      autoParam (∀ b ∈ s, ↑↑μ (f b) ≠ ⊤) _auto✝ → μ.real (⋃ b ∈ s, f b) = Finset.sum s fun p => μ.real (f p)"}
+{"name":"MeasureTheory.measureReal_add_diff","declaration":"theorem MeasureTheory.measureReal_add_diff {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α},\n  MeasurableSet s →\n    ∀ (t : Set α),\n      autoParam (↑↑μ s ≠ ⊤) _auto✝ → autoParam (↑↑μ t ≠ ⊤) _auto✝¹ → μ.real s + μ.real (t \\ s) = μ.real (s ∪ t)"}
+{"name":"MeasureTheory.Finset.sum_realMeasure_singleton","declaration":"/-- If `s` is a `Finset`, then the sums of the real measures of the singletons in the set is the\nreal measure of the set. -/\ntheorem MeasureTheory.Finset.sum_realMeasure_singleton {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst : MeasurableSingletonClass α]\n  [inst : MeasureTheory.IsFiniteMeasure μ] (s : Finset α), (Finset.sum s fun b => μ.real {b}) = μ.real ↑s"}
+{"name":"MeasureTheory.sum_measureReal_le_measureReal_univ","declaration":"theorem MeasureTheory.sum_measureReal_le_measureReal_univ {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {ι : Type u_3} [inst : MeasureTheory.IsFiniteMeasure μ]\n  {s : Finset ι} {t : ι → Set α},\n  (∀ i ∈ s, MeasurableSet (t i)) → Set.PairwiseDisjoint (↑s) t → (Finset.sum s fun i => μ.real (t i)) ≤ μ.real Set.univ"}
+{"name":"MeasureTheory.measureReal_eq_zero_iff","declaration":"theorem MeasureTheory.measureReal_eq_zero_iff {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α},\n  autoParam (↑↑μ s ≠ ⊤) _auto✝ → (μ.real s = 0 ↔ ↑↑μ s = 0)"}
+{"name":"MeasureTheory.measureReal_diff_le_iff_le_add","declaration":"theorem MeasureTheory.measureReal_diff_le_iff_le_add {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α},\n  MeasurableSet s → s ⊆ t → ∀ (ε : ℝ), autoParam (↑↑μ t ≠ ⊤) _auto✝ → (μ.real (t \\ s) ≤ ε ↔ μ.real t ≤ μ.real s + ε)"}
+{"name":"MeasureTheory.measureReal_add_measureReal_compl","declaration":"theorem MeasureTheory.measureReal_add_measureReal_compl {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s : Set α} [inst : MeasureTheory.IsFiniteMeasure μ],\n  MeasurableSet s → μ.real s + μ.real sᶜ = μ.real Set.univ"}
+{"name":"MeasureTheory.measureReal_union_add_inter","declaration":"theorem MeasureTheory.measureReal_union_add_inter {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {t : Set α} (s : Set α),\n  MeasurableSet t →\n    autoParam (↑↑μ s ≠ ⊤) _auto✝ → autoParam (↑↑μ t ≠ ⊤) _auto✝¹ → μ.real (s ∪ t) + μ.real (s ∩ t) = μ.real s + μ.real t"}
+{"name":"MeasureTheory.measureReal_diff_add_inter","declaration":"theorem MeasureTheory.measureReal_diff_add_inter {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {t : Set α} (s : Set α),\n  MeasurableSet t → autoParam (↑↑μ s ≠ ⊤) _auto✝ → μ.real (s \\ t) + μ.real (s ∩ t) = μ.real s"}
+{"name":"MeasureTheory.measureReal_inter_add_diff","declaration":"theorem MeasureTheory.measureReal_inter_add_diff {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {t : Set α} (s : Set α),\n  MeasurableSet t → autoParam (↑↑μ s ≠ ⊤) _auto✝ → μ.real (s ∩ t) + μ.real (s \\ t) = μ.real s"}
+{"name":"MeasureTheory.ext_iff_measureReal_singleton","declaration":"theorem MeasureTheory.ext_iff_measureReal_singleton {S : Type u_3} [Fintype S] [MeasurableSpace S] [MeasurableSingletonClass S] {μ1 : MeasureTheory.Measure S} {μ2 : MeasureTheory.Measure S} [MeasureTheory.IsFiniteMeasure μ1] [MeasureTheory.IsFiniteMeasure μ2] : μ1 = μ2 ↔ ∀ (x : S), μ1.real {x} = μ2.real {x}"}
+{"name":"MeasureTheory.measureReal_diff_lt_of_lt_add","declaration":"theorem MeasureTheory.measureReal_diff_lt_of_lt_add {α : Type u_1} : ∀ {x : MeasurableSpace α} {μ : MeasureTheory.Measure α} {s t : Set α},\n  MeasurableSet s → s ⊆ t → ∀ (ε : ℝ), μ.real t < μ.real s + ε → autoParam (↑↑μ t ≠ ⊤) _auto✝ → μ.real (t \\ s) < ε"}
+{"name":"MeasureTheory.nonempty_inter_of_measureReal_lt_add","declaration":"/-- If two sets `s` and `t` are included in a set `u` of finite measure,\nand `μ.real s + μ.real t > μ.real u`, then `s` intersects `t`.\nVersion assuming that `t` is measurable. -/\ntheorem MeasureTheory.nonempty_inter_of_measureReal_lt_add {α : Type u_1} {m : MeasurableSpace α} (μ : MeasureTheory.Measure α) {s : Set α} {t : Set α} {u : Set α} (ht : MeasurableSet t) (h's : s ⊆ u) (h't : t ⊆ u) (h : μ.real u < μ.real s + μ.real t) (hu : autoParam (↑↑μ u ≠ ⊤) _auto✝) : Set.Nonempty (s ∩ t)"}

PFR-declarations/PFR.ForMathlib.Pair.jsonl

@@ -0,0 +1,3 @@
+{"name":"prod","declaration":"/-- The pair of two random variables -/\ndef prod {Ω : Type u_1} {S : Type u_2} {T : Type u_3} (X : Ω → S) (Y : Ω → T) (ω : Ω) : S × T"}
+{"name":"«term⟨_,_⟩».delab","declaration":"/-- Pretty printer defined by `notation3` command. -/\ndef «term⟨_,_⟩».delab  : Lean.PrettyPrinter.Delaborator.Delab"}
+{"name":"«term⟨_,_⟩»","declaration":"/-- The pair of two random variables -/\ndef «term⟨_,_⟩»  : Lean.ParserDescr"}

PFR-declarations/PFR.ForMathlib.ProbabilityMeasureProdCont.jsonl

@@ -0,0 +1,4 @@
+{"name":"MeasureTheory.ProbabilityMeasure.continuous_prod_of_finite","declaration":"/-- The product of two probability measures on finite spaces depend continuously on the two\nprobability measures.\nTODO: In Mathlib, this should be done on all separable metrizable spaces. -/\ntheorem MeasureTheory.ProbabilityMeasure.continuous_prod_of_finite {α : Type u_1} {β : Type u_2} [Finite α] [TopologicalSpace α] [DiscreteTopology α] [MeasurableSpace α] [BorelSpace α] [Finite β] [TopologicalSpace β] [DiscreteTopology β] [MeasurableSpace β] [BorelSpace β] : Continuous fun x =>\n  match x with\n  | (μ, ν) => MeasureTheory.ProbabilityMeasure.prod μ ν"}
+{"name":"MeasureTheory.ProbabilityMeasure.tendsto_iff_forall_apply_tendsto","declaration":"/-- Probability measures on a finite space tend to a limit if and only if the probability masses\nof all points tend to the corresponding limits. -/\ntheorem MeasureTheory.ProbabilityMeasure.tendsto_iff_forall_apply_tendsto {ι : Type u_1} {α : Type u_2} {L : Filter ι} [Finite α] [TopologicalSpace α] [DiscreteTopology α] [MeasurableSpace α] [BorelSpace α] (μs : ι → MeasureTheory.ProbabilityMeasure α) (μ : MeasureTheory.ProbabilityMeasure α) : Filter.Tendsto μs L (nhds μ) ↔\n  ∀ (a : α), Filter.Tendsto (fun x => (fun s => (↑↑↑(μs x) s).toNNReal) {a}) L (nhds ((fun s => (↑↑↑μ s).toNNReal) {a}))"}
+{"name":"MeasureTheory.t1Space_probabilityMeasure_of_finite","declaration":"instance MeasureTheory.t1Space_probabilityMeasure_of_finite {α : Type u_1} [Finite α] [TopologicalSpace α] [DiscreteTopology α] [MeasurableSpace α] [BorelSpace α] : T1Space (MeasureTheory.ProbabilityMeasure α)"}
+{"name":"MeasureTheory.ProbabilityMeasure.tendsto_prod_of_tendsto_of_tendsto","declaration":"/-- If probability measures on two finite spaces tend to limits, then the products of them\non the product space tend to the product of the limits.\nTODO: In Mathlib, this should be done on all separable metrizable spaces. -/\ntheorem MeasureTheory.ProbabilityMeasure.tendsto_prod_of_tendsto_of_tendsto {ι : Type u_1} {L : Filter ι} {α : Type u_2} {β : Type u_3} [Finite α] [TopologicalSpace α] [DiscreteTopology α] [MeasurableSpace α] [BorelSpace α] [Finite β] [TopologicalSpace β] [DiscreteTopology β] [MeasurableSpace β] [BorelSpace β] (μs : ι → MeasureTheory.ProbabilityMeasure α) (μ : MeasureTheory.ProbabilityMeasure α) (μs_lim : Filter.Tendsto μs L (nhds μ)) (νs : ι → MeasureTheory.ProbabilityMeasure β) (ν : MeasureTheory.ProbabilityMeasure β) (νs_lim : Filter.Tendsto νs L (nhds ν)) : Filter.Tendsto (fun i => MeasureTheory.ProbabilityMeasure.prod (μs i) (νs i)) L\n  (nhds (MeasureTheory.ProbabilityMeasure.prod μ ν))"}

PFR-declarations/PFR.ForMathlib.Summable.jsonl

@@ -0,0 +1,2 @@
+{"name":"tsum_of_not_summable","declaration":"/-- Currently not needed. -/\ntheorem tsum_of_not_summable {S : Type u_1} {f : S → ℝ} (hf : ∀ (s : S), 0 ≤ f s) (hsum : ¬Summable f) : ∑' (s : S), ENNReal.ofReal (f s) = ⊤"}
+{"name":"tsum_eq_toReal_tsum_ofReal","declaration":"/-- Currently not needed. -/\ntheorem tsum_eq_toReal_tsum_ofReal {S : Type u_1} {f : S → ℝ} (hf : ∀ (s : S), 0 ≤ f s) : ∑' (s : S), f s = (∑' (s : S), ENNReal.ofReal (f s)).toReal"}

PFR-declarations/PFR.ForMathlib.Uniform.jsonl

@@ -0,0 +1,23 @@
+{"name":"ProbabilityTheory.exists_isUniform_measureSpace","declaration":"/-- Uniform distributions exist, version giving a measure space -/\ntheorem ProbabilityTheory.exists_isUniform_measureSpace {S : Type u} [MeasurableSpace S] [MeasurableSingletonClass S] (H : Finset S) (h : H.Nonempty) : ∃ Ω mΩ U,\n  MeasureTheory.IsProbabilityMeasure MeasureTheory.volume ∧\n    Measurable U ∧ ProbabilityTheory.IsUniform (↑H) U MeasureTheory.volume ∧ (∀ (ω : Ω), U ω ∈ H) ∧ FiniteRange U"}
+{"name":"ProbabilityTheory.IsUniform.mk","declaration":"ctor ProbabilityTheory.IsUniform.mk {Ω : Type uΩ} {S : Type uS} [mΩ : MeasurableSpace Ω] {H : Set S} {X : Ω → S} {μ : autoParam (MeasureTheory.Measure Ω) _auto✝} (eq_of_mem : ∀ (x y : S), x ∈ H → y ∈ H → ↑↑μ (X ⁻¹' {x}) = ↑↑μ (X ⁻¹' {y})) (measure_preimage_compl : ↑↑μ (X ⁻¹' Hᶜ) = 0) : ProbabilityTheory.IsUniform H X μ"}
+{"name":"ProbabilityTheory.IsUniform.measureReal_preimage","declaration":"/-- $\\mathbb{P}(U_H \\in H') = \\dfrac{|H' \\cap H|}{|H|}$ -/\ntheorem ProbabilityTheory.IsUniform.measureReal_preimage {Ω : Type uΩ} {S : Type uS} [mΩ : MeasurableSpace Ω] [Countable S] [MeasurableSpace S] [MeasurableSingletonClass S] {X : Ω → S} {μ : MeasureTheory.Measure Ω} {H : Finset S} (h : ProbabilityTheory.IsUniform (↑H) X μ) (hX : Measurable X) (H' : Set S) : μ.real (X ⁻¹' H') = μ.real Set.univ * ↑(Nat.card ↑(H' ∩ ↑H)) / ↑(Nat.card { x // x ∈ H })"}
+{"name":"ProbabilityTheory.exists_isUniform","declaration":"/-- Uniform distributions exist. -/\ntheorem ProbabilityTheory.exists_isUniform {S : Type uS} [MeasurableSpace S] [MeasurableSingletonClass S] (H : Finset S) (h : H.Nonempty) : ∃ Ω mΩ X μ,\n  MeasureTheory.IsProbabilityMeasure μ ∧\n    Measurable X ∧ ProbabilityTheory.IsUniform (↑H) X μ ∧ (∀ (ω : Ω), X ω ∈ H) ∧ FiniteRange X"}
+{"name":"ProbabilityTheory.IsUniform.of_identDistrib","declaration":"/-- A copy of a uniform random variable is also uniform.-/\ntheorem ProbabilityTheory.IsUniform.of_identDistrib {Ω : Type uΩ} {S : Type uS} [mΩ : MeasurableSpace Ω] [MeasurableSpace S] [MeasurableSingletonClass S] {X : Ω → S} {μ : MeasureTheory.Measure Ω} {H : Set S} {Ω' : Type u_1} [MeasurableSpace Ω'] (h : ProbabilityTheory.IsUniform H X μ) {X' : Ω' → S} {μ' : MeasureTheory.Measure Ω'} (h' : ProbabilityTheory.IdentDistrib X X' μ μ') (hH : MeasurableSet H) : ProbabilityTheory.IsUniform H X' μ'"}
+{"name":"ProbabilityTheory.IsUniform.nonempty_preimage_of_mem","declaration":"theorem ProbabilityTheory.IsUniform.nonempty_preimage_of_mem {Ω : Type uΩ} {S : Type uS} [mΩ : MeasurableSpace Ω] [Countable S] [MeasurableSpace S] [MeasurableSingletonClass S] {X : Ω → S} {μ : MeasureTheory.Measure Ω} [NeZero μ] {H : Finset S} (h : ProbabilityTheory.IsUniform (↑H) X μ) (hX : Measurable X) {s : S} (hs : s ∈ H) : Set.Nonempty (X ⁻¹' {s})"}
+{"name":"ProbabilityTheory.IsUniform.full_measure","declaration":"theorem ProbabilityTheory.IsUniform.full_measure {Ω : Type uΩ} {S : Type uS} [mΩ : MeasurableSpace Ω] [Countable S] [MeasurableSpace S] [MeasurableSingletonClass S] {X : Ω → S} {μ : MeasureTheory.Measure Ω} {H : Set S} (h : ProbabilityTheory.IsUniform H X μ) (hX : Measurable X) : ↑↑(MeasureTheory.Measure.map X μ) H = ↑↑μ Set.univ"}
+{"name":"ProbabilityTheory.IsUniform.comp","declaration":"/-- The image of a uniform random variable under an injective map is uniform on the image. -/\ntheorem ProbabilityTheory.IsUniform.comp {Ω : Type uΩ} {S : Type uS} {T : Type uT} [mΩ : MeasurableSpace Ω] {X : Ω → S} {μ : MeasureTheory.Measure Ω} [DecidableEq T] {H : Finset S} (h : ProbabilityTheory.IsUniform (↑H) X μ) {f : S → T} (hf : Function.Injective f) : ProbabilityTheory.IsUniform (↑(Finset.image f H)) (f ∘ X) μ"}
+{"name":"ProbabilityTheory.IsUniform.measureReal_preimage_of_mem","declaration":"/-- A \"unit test\" for the definition of uniform distribution. -/\ntheorem ProbabilityTheory.IsUniform.measureReal_preimage_of_mem {Ω : Type uΩ} {S : Type uS} [mΩ : MeasurableSpace Ω] [Countable S] [MeasurableSpace S] [MeasurableSingletonClass S] {X : Ω → S} {μ : MeasureTheory.Measure Ω} {H : Finset S} [MeasureTheory.IsProbabilityMeasure μ] (h : ProbabilityTheory.IsUniform (↑H) X μ) (hX : Measurable X) {s : S} (hs : s ∈ H) : μ.real (X ⁻¹' {s}) = 1 / ↑(Nat.card { x // x ∈ H })"}
+{"name":"ProbabilityTheory.IsUniform.measureReal_preimage_of_nmem","declaration":"/-- Another \"unit test\" for the definition of uniform distribution. -/\ntheorem ProbabilityTheory.IsUniform.measureReal_preimage_of_nmem {Ω : Type uΩ} {S : Type uS} [mΩ : MeasurableSpace Ω] {X : Ω → S} {μ : MeasureTheory.Measure Ω} {H : Set S} (h : ProbabilityTheory.IsUniform H X μ) {s : S} (hs : s ∉ H) : μ.real (X ⁻¹' {s}) = 0"}
+{"name":"ProbabilityTheory.exists_isUniform_measureSpace'","declaration":"/-- Uniform distributions exist, version with a Finite set rather than a Finset and giving a measure space -/\ntheorem ProbabilityTheory.exists_isUniform_measureSpace' {S : Type u} [MeasurableSpace S] [MeasurableSingletonClass S] (H : Set S) [Finite ↑H] [Nonempty ↑H] : ∃ Ω mΩ U,\n  MeasureTheory.IsProbabilityMeasure MeasureTheory.volume ∧\n    Measurable U ∧ ProbabilityTheory.IsUniform H U MeasureTheory.volume ∧ (∀ (ω : Ω), U ω ∈ H) ∧ FiniteRange U"}
+{"name":"ProbabilityTheory.IsUniform.restrict","declaration":"/-- If $X$ is uniform w.r.t. $\\mu$ on $H$, then $X$ is uniform w.r.t. $\\mu$ conditioned by\n$H'$ on $H' \\cap H$. -/\ntheorem ProbabilityTheory.IsUniform.restrict {Ω : Type uΩ} {S : Type uS} [mΩ : MeasurableSpace Ω] [Countable S] [MeasurableSpace S] [MeasurableSingletonClass S] {X : Ω → S} {μ : MeasureTheory.Measure Ω} {H : Set S} (h : ProbabilityTheory.IsUniform H X μ) (hX : Measurable X) (H' : Set S) : ProbabilityTheory.IsUniform (H' ∩ H) X (ProbabilityTheory.cond μ (X ⁻¹' H'))"}
+{"name":"ProbabilityTheory.IsUniform.ae_mem","declaration":"/-- A uniform random variable on H almost surely takes values in H. -/\ntheorem ProbabilityTheory.IsUniform.ae_mem {Ω : Type uΩ} {S : Type uS} [mΩ : MeasurableSpace Ω] {X : Ω → S} {μ : MeasureTheory.Measure Ω} {H : Set S} (h : ProbabilityTheory.IsUniform H X μ) : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ H"}
+{"name":"ProbabilityTheory.IsUniform.measure_preimage_of_mem","declaration":"/-- A \"unit test\" for the definition of uniform distribution. -/\ntheorem ProbabilityTheory.IsUniform.measure_preimage_of_mem {Ω : Type uΩ} {S : Type uS} [mΩ : MeasurableSpace Ω] [Countable S] [MeasurableSpace S] [MeasurableSingletonClass S] {X : Ω → S} {μ : MeasureTheory.Measure Ω} {H : Finset S} (h : ProbabilityTheory.IsUniform (↑H) X μ) (hX : Measurable X) {s : S} (hs : s ∈ H) : ↑↑μ (X ⁻¹' {s}) = ↑↑μ Set.univ / ↑(Nat.card { x // x ∈ H })"}
+{"name":"ProbabilityTheory.IsUniform.nonempty","declaration":"/-- Uniform random variables only exist for non-empty sets H. -/\ntheorem ProbabilityTheory.IsUniform.nonempty {Ω : Type uΩ} {S : Type uS} [mΩ : MeasurableSpace Ω] {X : Ω → S} {μ : MeasureTheory.Measure Ω} {H : Finset S} (h : ProbabilityTheory.IsUniform (↑H) X μ) [hμ : NeZero μ] : H.Nonempty"}
+{"name":"ProbabilityTheory.IsUniform.measure_preimage_of_nmem","declaration":"/-- Another \"unit test\" for the definition of uniform distribution. -/\ntheorem ProbabilityTheory.IsUniform.measure_preimage_of_nmem {Ω : Type uΩ} {S : Type uS} [mΩ : MeasurableSpace Ω] {X : Ω → S} {μ : MeasureTheory.Measure Ω} {H : Set S} (h : ProbabilityTheory.IsUniform H X μ) {s : S} (hs : s ∉ H) : ↑↑μ (X ⁻¹' {s}) = 0"}
+{"name":"ProbabilityTheory.IsUniform.measure_preimage_compl","declaration":"def ProbabilityTheory.IsUniform.measure_preimage_compl {Ω : Type uΩ} {S : Type uS} [mΩ : MeasurableSpace Ω] {H : Set S} {X : Ω → S} {μ : autoParam (MeasureTheory.Measure Ω) _auto✝} (self : ProbabilityTheory.IsUniform H X μ) : ↑↑μ (X ⁻¹' Hᶜ) = 0"}
+{"name":"ProbabilityTheory.IsUniform.eq_of_mem","declaration":"def ProbabilityTheory.IsUniform.eq_of_mem {Ω : Type uΩ} {S : Type uS} [mΩ : MeasurableSpace Ω] {H : Set S} {X : Ω → S} {μ : autoParam (MeasureTheory.Measure Ω) _auto✝} (self : ProbabilityTheory.IsUniform H X μ) (x : S) (y : S) : x ∈ H → y ∈ H → ↑↑μ (X ⁻¹' {x}) = ↑↑μ (X ⁻¹' {y})"}
+{"name":"ProbabilityTheory.IdentDistrib.of_isUniform","declaration":"theorem ProbabilityTheory.IdentDistrib.of_isUniform {Ω : Type uΩ} {S : Type uS} [mΩ : MeasurableSpace Ω] [Countable S] [MeasurableSpace S] [MeasurableSingletonClass S] {μ : MeasureTheory.Measure Ω} {H : Set S} {Ω' : Type u_1} [MeasurableSpace Ω'] {μ' : MeasureTheory.Measure Ω'} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure μ'] [Finite ↑H] {X : Ω → S} {X' : Ω' → S} (hX : Measurable X) (hX' : Measurable X') (hX_unif : ProbabilityTheory.IsUniform H X μ) (hX'_unif : ProbabilityTheory.IsUniform H X' μ') : ProbabilityTheory.IdentDistrib X X' μ μ'"}
+{"name":"ProbabilityTheory.IsUniform","declaration":"/-- The assertion that the law of $X$ is the uniform probability measure on a finite set $H$.\nWhile in applications $H$ will be non-empty finite set, $X$ measurable, and and $μ$ a probability\nmeasure, it could be technically convenient to have a definition that works even without these\nhypotheses.  (For instance, `isUniform` would be well-defined, but false, for infinite `H`) -/\nstructure ProbabilityTheory.IsUniform {Ω : Type uΩ} {S : Type uS} [mΩ : MeasurableSpace Ω] (H : Set S) (X : Ω → S) (μ : autoParam (MeasureTheory.Measure Ω) _auto✝) : Prop"}
+{"name":"ProbabilityTheory.IsUniform.measure_preimage_ne_zero","declaration":"/-- $\\mathbb{P}(U_H \\in H') \\neq 0$ if $H'$ intersects $H$ and the measure is non-zero. -/\ntheorem ProbabilityTheory.IsUniform.measure_preimage_ne_zero {Ω : Type uΩ} {S : Type uS} [mΩ : MeasurableSpace Ω] [Countable S] [MeasurableSpace S] [MeasurableSingletonClass S] {X : Ω → S} {μ : MeasureTheory.Measure Ω} {H : Finset S} [NeZero μ] (h : ProbabilityTheory.IsUniform (↑H) X μ) (hX : Measurable X) (H' : Set S) [Nonempty ↑(H' ∩ ↑H)] : ↑↑μ (X ⁻¹' H') ≠ 0"}
+{"name":"ProbabilityTheory.IsUniform.measureReal_preimage_of_mem'","declaration":"theorem ProbabilityTheory.IsUniform.measureReal_preimage_of_mem' {Ω : Type uΩ} {S : Type uS} [mΩ : MeasurableSpace Ω] [Countable S] [MeasurableSpace S] [MeasurableSingletonClass S] {X : Ω → S} {μ : MeasureTheory.Measure Ω} {H : Finset S} [MeasureTheory.IsProbabilityMeasure μ] (h : ProbabilityTheory.IsUniform (↑H) X μ) (hX : Measurable X) {s : S} (hs : s ∈ H) : (MeasureTheory.Measure.map X μ).real {s} = 1 / ↑(Nat.card { x // x ∈ H })"}
+{"name":"ProbabilityTheory.IsUniform.measure_preimage","declaration":"/-- $\\mathbb{P}(U_H \\in H') = \\dfrac{|H' \\cap H|}{|H|}$ -/\ntheorem ProbabilityTheory.IsUniform.measure_preimage {Ω : Type uΩ} {S : Type uS} [mΩ : MeasurableSpace Ω] [Countable S] [MeasurableSpace S] [MeasurableSingletonClass S] {X : Ω → S} {μ : MeasureTheory.Measure Ω} {H : Finset S} (h : ProbabilityTheory.IsUniform (↑H) X μ) (hX : Measurable X) (H' : Set S) : ↑↑μ (X ⁻¹' H') = ↑↑μ Set.univ * ↑(Nat.card ↑(H' ∩ ↑H)) / ↑(Nat.card { x // x ∈ H })"}

PFR-declarations/PFR.HomPFR.jsonl

@@ -0,0 +1,3 @@
+{"name":"hahn_banach","declaration":"/-- Let $H_0$ be a subgroup of $G$.  Then every homomorphism $\\phi: H_0 \\to G'$ can be extended to a\nhomomorphism $\\tilde \\phi: G \\to G'$. -/\ntheorem hahn_banach {G : Type u_1} {G' : Type u_2} [AddCommGroup G] [AddCommGroup G'] [ElementaryAddCommGroup G 2] [ElementaryAddCommGroup G' 2] (H₀ : AddSubgroup G) (φ : ↥H₀ →+ G') : ∃ φ', ∀ (x : ↥H₀), φ x = φ' ↑x"}
+{"name":"homomorphism_pfr","declaration":"/-- Let $f: G \\to G'$ be a function, and let $S$ denote the set\n$$ S := \\{ f(x+y)-f(x)-f(y): x,y \\in G \\}.$$\nThen there exists a homomorphism $\\phi: G \\to G'$ such that\n$$ |\\{f(x) - \\phi(x)\\}| \\leq |S|^{12}. $$ -/\ntheorem homomorphism_pfr {G : Type u_1} {G' : Type u_2} [AddCommGroup G] [Fintype G] [AddCommGroup G'] [Fintype G'] [ElementaryAddCommGroup G 2] [ElementaryAddCommGroup G' 2] (f : G → G') (S : Set G') (hS : ∀ (x y : G), f (x + y) - f x - f y ∈ S) : ∃ φ T, Nat.card ↑T ≤ Nat.card ↑S ^ 12 ∧ ∀ (x : G), f x - φ x ∈ T"}
+{"name":"goursat","declaration":"/-- Let $H$ be a subgroup of $G \\times G'$.  Then there exists a subgroup $H_0$ of $G$, a\nsubgroup $H_1$ of $G'$, and a homomorphism $\\phi: G \\to G'$ such that\n$$ H := \\{ (x, \\phi(x) + y): x \\in H_0, y \\in H_1 \\}.$$\nIn particular, $|H| = |H_0| |H_1|$. -/\ntheorem goursat {G : Type u_1} {G' : Type u_2} [AddCommGroup G] [AddCommGroup G'] [ElementaryAddCommGroup G 2] [ElementaryAddCommGroup G' 2] (H : AddSubgroup (G × G')) : ∃ H₀ H₁ φ, (∀ (x : G × G'), x ∈ H ↔ x.1 ∈ H₀ ∧ x.2 - φ x.1 ∈ H₁) ∧ Nat.card ↥H = Nat.card ↥H₀ * Nat.card ↥H₁"}

PFR-declarations/PFR.HundredPercent.jsonl

@@ -0,0 +1,7 @@
+{"name":"mem_symmGroup","declaration":"theorem mem_symmGroup {Ω : Type u_1} {G : Type u_2} [MeasureTheory.MeasureSpace Ω] [AddCommGroup G] [MeasurableSpace G] [MeasurableAdd₂ G] {X : Ω → G} (hX : Measurable X) {x : G} : x ∈ symmGroup X hX ↔ ProbabilityTheory.IdentDistrib X (fun ω => X ω + x) MeasureTheory.volume MeasureTheory.volume"}
+{"name":"exists_isUniform_of_rdist_eq_zero","declaration":"/-- If $d[X_1;X_2]=0$, then there exists a subgroup $H \\leq G$ such that\n$d[X_1;U_H] = d[X_2;U_H] = 0$. Follows from the preceding claim by the triangle inequality. -/\ntheorem exists_isUniform_of_rdist_eq_zero {Ω : Type u_1} {G : Type u_2} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] [AddCommGroup G] [Fintype G] [MeasurableSpace G] [MeasurableAdd₂ G] [MeasurableSub₂ G] {X : Ω → G} [MeasurableSingletonClass G] {Ω' : Type u_3} [MeasureTheory.MeasureSpace Ω'] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {X' : Ω' → G} (hX : Measurable X) (hX' : Measurable X') (hdist : d[X # X'] = 0) : ∃ H U, Measurable U ∧ ProbabilityTheory.IsUniform (↑H) U MeasureTheory.volume ∧ d[X # U] = 0 ∧ d[X' # U] = 0"}
+{"name":"sub_mem_symmGroup","declaration":"/-- If $d[X ;X]=0$, and $x,y \\in G$ are such that $P[X=x], P[X=y]>0$,\nthen $x-y \\in \\mathrm{Sym}[X]$. -/\ntheorem sub_mem_symmGroup {Ω : Type u_1} {G : Type u_2} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] [AddCommGroup G] [Fintype G] [MeasurableSpace G] [MeasurableAdd₂ G] [MeasurableSub₂ G] {X : Ω → G} [MeasurableSingletonClass G] (hX : Measurable X) (hdist : d[X # X] = 0) {x : G} {y : G} (hx : ↑↑MeasureTheory.volume (X ⁻¹' {x}) ≠ 0) (hy : ↑↑MeasureTheory.volume (X ⁻¹' {y}) ≠ 0) : x - y ∈ symmGroup X hX"}
+{"name":"exists_isUniform_of_rdist_self_eq_zero","declaration":"/-- If $d[X ;X]=0$, then there exists a subgroup $H \\leq G$ such that $d[X ;U_H] = 0$. -/\ntheorem exists_isUniform_of_rdist_self_eq_zero {Ω : Type u_1} {G : Type u_2} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] [AddCommGroup G] [Fintype G] [MeasurableSpace G] [MeasurableAdd₂ G] [MeasurableSub₂ G] {X : Ω → G} [MeasurableSingletonClass G] (hX : Measurable X) (hdist : d[X # X] = 0) : ∃ H U, Measurable U ∧ ProbabilityTheory.IsUniform (↑H) U MeasureTheory.volume ∧ d[X # U] = 0"}
+{"name":"symmGroup","declaration":"/-- The symmetry group Sym of $X$: the set of all $h ∈ G$ such that $X + h$ has an identical\ndistribution to $X$. -/\ndef symmGroup {Ω : Type u_1} {G : Type u_2} [MeasureTheory.MeasureSpace Ω] [AddCommGroup G] [MeasurableSpace G] [MeasurableAdd₂ G] (X : Ω → G) (hX : Measurable X) : AddSubgroup G"}
+{"name":"ProbabilityTheory.IdentDistrib.symmGroup_eq","declaration":"theorem ProbabilityTheory.IdentDistrib.symmGroup_eq {Ω : Type u_1} {G : Type u_2} [MeasureTheory.MeasureSpace Ω] [AddCommGroup G] [MeasurableSpace G] [MeasurableAdd₂ G] {X : Ω → G} {Ω' : Type u_3} [MeasureTheory.MeasureSpace Ω'] {X' : Ω' → G} (hX : Measurable X) (hX' : Measurable X') (h : ProbabilityTheory.IdentDistrib X X' MeasureTheory.volume MeasureTheory.volume) : symmGroup X hX = symmGroup X' hX'"}
+{"name":"isUniform_sub_const_of_rdist_eq_zero","declaration":"/-- If `d[X # X] = 0`, then `X - x₀` is the uniform distribution on the subgroup of `G`\nstabilizing the distribution of `X`, for any `x₀` of positive probability. -/\ntheorem isUniform_sub_const_of_rdist_eq_zero {Ω : Type u_1} {G : Type u_2} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] [AddCommGroup G] [Fintype G] [MeasurableSpace G] [MeasurableAdd₂ G] [MeasurableSub₂ G] {X : Ω → G} [MeasurableSingletonClass G] (hX : Measurable X) (hdist : d[X # X] = 0) {x₀ : G} (hx₀ : ↑↑MeasureTheory.volume (X ⁻¹' {x₀}) ≠ 0) : ProbabilityTheory.IsUniform (↑(symmGroup X hX)) (fun ω => X ω - x₀) MeasureTheory.volume"}

PFR-declarations/PFR.ImprovedPFR.jsonl

@@ -0,0 +1,20 @@
+{"name":"entropic_PFR_conjecture_improv","declaration":"/-- `entropic_PFR_conjecture_improv`: For two $G$-valued random variables $X^0_1, X^0_2$, there is some\nsubgroup $H \\leq G$ such that $d[X^0_1;U_H] + d[X^0_2;U_H] \\le 10 d[X^0_1;X^0_2]$. -/\ntheorem entropic_PFR_conjecture_improv {Ω₀₁ : Type u_1} {Ω₀₂ : Type u_2} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {G : Type uG} [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] [MeasurableSpace G] [MeasurableSingletonClass G] (p : refPackage Ω₀₁ Ω₀₂ G) (hpη : p.η = 1 / 8) : ∃ H Ω mΩ U,\n  MeasureTheory.IsProbabilityMeasure MeasureTheory.volume ∧\n    Measurable U ∧\n      ProbabilityTheory.IsUniform (↑H) U MeasureTheory.volume ∧ d[p.X₀₁ # U] + d[p.X₀₂ # U] ≤ 10 * d[p.X₀₁ # p.X₀₂]"}
+{"name":"gen_ineq_aux1","declaration":"theorem gen_ineq_aux1 {G : Type u_1} [AddCommGroup G] [Fintype G] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [ElementaryAddCommGroup G 2] [MeasurableAdd₂ G] {Ω : Type u_2} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {Ω₀ : Type u_3} [MeasureTheory.MeasureSpace Ω₀] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (Y : Ω₀ → G) (hY : Measurable Y) (Z₁ : Ω → G) (Z₂ : Ω → G) (Z₃ : Ω → G) (Z₄ : Ω → G) (hZ₁ : Measurable Z₁) (hZ₂ : Measurable Z₂) (hZ₃ : Measurable Z₃) (hZ₄ : Measurable Z₄) (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![Z₁, Z₂, Z₃, Z₄] MeasureTheory.volume) : d[Y # Z₁ + Z₂ | ⟨Z₁ + Z₃, Z₁ + Z₂ + Z₃ + Z₄⟩] ≤\n  d[Y # Z₁] + (d[Z₁ # Z₂] + d[Z₁ # Z₃] + d[Z₂ # Z₄] - d[Z₁ | Z₁ + Z₂ # Z₃ | Z₃ + Z₄]) / 2 +\n    (H[Z₁ + Z₂] - H[Z₃ + Z₄] + H[Z₂] - H[Z₁]) / 4"}
+{"name":"tau_strictly_decreases_aux'","declaration":"/-- Suppose $0 < \\eta < 1/8$.  Let $X_1, X_2$ be tau-minimizers.  Then $d[X_1;X_2] = 0$. The proof\nof this lemma uses copies `X₁', X₂'` already in the context. For a version that does not assume\nthese are given and constructs them instead, use `tau_strictly_decreases'`.\n-/\ntheorem tau_strictly_decreases_aux' {G : Type u_1} [AddCommGroup G] [Fintype G] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [ElementaryAddCommGroup G 2] [MeasurableAdd₂ G] {Ω₀₁ : Type u_2} {Ω₀₂ : Type u_3} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (p : refPackage Ω₀₁ Ω₀₂ G) {Ω : Type u_4} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {X₁ : Ω → G} {X₂ : Ω → G} {X₁' : Ω → G} {X₂' : Ω → G} (hX₁ : Measurable X₁) (hX₂ : Measurable X₂) (hX₁' : Measurable X₁') (hX₂' : Measurable X₂') (h₁ : ProbabilityTheory.IdentDistrib X₁ X₁' MeasureTheory.volume MeasureTheory.volume) (h₂ : ProbabilityTheory.IdentDistrib X₂ X₂' MeasureTheory.volume MeasureTheory.volume) (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![X₁, X₂, X₂', X₁'] MeasureTheory.volume) (h_min : tau_minimizes p X₁ X₂) (hp : 8 * p.η < 1) : d[X₁ # X₂] = 0"}
+{"name":"entropic_PFR_conjecture_improv'","declaration":"/-- `entropic_PFR_conjecture_improv'`: For two $G$-valued random variables $X^0_1, X^0_2$, there is\nsome subgroup $H \\leq G$ such that $d[X^0_1;U_H] + d[X^0_2;U_H] \\le 10 d[X^0_1;X^0_2]$., and\nd[X^0_1; U_H] and d[X^0_2; U_H] are at most 5/2 * d[X^0_1;X^0_2] -/\ntheorem entropic_PFR_conjecture_improv' {Ω₀₁ : Type u_1} {Ω₀₂ : Type u_2} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {G : Type uG} [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] [MeasurableSpace G] [MeasurableSingletonClass G] (p : refPackage Ω₀₁ Ω₀₂ G) (hpη : p.η = 1 / 8) : ∃ H Ω mΩ U,\n  MeasureTheory.IsProbabilityMeasure MeasureTheory.volume ∧\n    Measurable U ∧\n      ProbabilityTheory.IsUniform (↑H) U MeasureTheory.volume ∧\n        d[p.X₀₁ # U] + d[p.X₀₂ # U] ≤ 10 * d[p.X₀₁ # p.X₀₂] ∧\n          d[p.X₀₁ # U] ≤ 11 / 2 * d[p.X₀₁ # p.X₀₂] ∧ d[p.X₀₂ # U] ≤ 11 / 2 * d[p.X₀₁ # p.X₀₂]"}
+{"name":"dist_diff_bound_1","declaration":"theorem dist_diff_bound_1 {G : Type u_1} [AddCommGroup G] [Fintype G] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [ElementaryAddCommGroup G 2] [MeasurableAdd₂ G] {Ω₀₁ : Type u_2} {Ω₀₂ : Type u_3} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (p : refPackage Ω₀₁ Ω₀₂ G) {Ω : Type u_4} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {X₁ : Ω → G} {X₂ : Ω → G} {X₁' : Ω → G} {X₂' : Ω → G} (hX₁ : Measurable X₁) (hX₂ : Measurable X₂) (hX₁' : Measurable X₁') (hX₂' : Measurable X₂') (h₁ : ProbabilityTheory.IdentDistrib X₁ X₁' MeasureTheory.volume MeasureTheory.volume) (h₂ : ProbabilityTheory.IdentDistrib X₂ X₂' MeasureTheory.volume MeasureTheory.volume) (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![X₁, X₂, X₂', X₁'] MeasureTheory.volume) : d[p.X₀₁ # X₁ + X₂ | ⟨X₁' + X₂, X₁ + X₂ + X₁' + X₂'⟩] - d[p.X₀₁ # X₁] +\n            (d[p.X₀₁ # X₁ + X₂ | ⟨X₁' + X₁, X₁ + X₂ + X₁' + X₂'⟩] - d[p.X₀₁ # X₁]) +\n          (d[p.X₀₁ # X₁' + X₂ | ⟨X₁ + X₂, X₁ + X₂ + X₁' + X₂'⟩] - d[p.X₀₁ # X₁]) +\n        (d[p.X₀₁ # X₁' + X₂ | ⟨X₁' + X₁, X₁ + X₂ + X₁' + X₂'⟩] - d[p.X₀₁ # X₁]) +\n      (d[p.X₀₁ # X₁' + X₁ | ⟨X₁ + X₂, X₁ + X₂ + X₁' + X₂'⟩] - d[p.X₀₁ # X₁]) +\n    (d[p.X₀₁ # X₁' + X₁ | ⟨X₁' + X₂, X₁ + X₂ + X₁' + X₂'⟩] - d[p.X₀₁ # X₁]) ≤\n  (16 * d[X₁ # X₂] + 6 * d[X₁ # X₁] + 2 * d[X₂ # X₂]) / 4 + (H[X₁ + X₁'] - H[X₂ + X₂']) / 4 +\n    (H[X₂ | X₂ + X₂'] - H[X₁ | X₁ + X₁']) / 4"}
+{"name":"PFR_conjecture_improv'","declaration":"/-- Corollary of `PFR_conjecture_improv` in which the ambient group is not required to be finite\n(but) then $H$ and $c$ are finite. -/\ntheorem PFR_conjecture_improv' {G : Type u_3} [AddCommGroup G] [ElementaryAddCommGroup G 2] {A : Set G} {K : ℝ} (h₀A : Set.Nonempty A) (Afin : Set.Finite A) (hA : ↑(Nat.card ↑(A + A)) ≤ K * ↑(Nat.card ↑A)) : ∃ H c, Set.Finite c ∧ Set.Finite ↑H ∧ ↑(Nat.card ↑c) < 2 * K ^ 11 ∧ Nat.card ↥H ≤ Nat.card ↑A ∧ A ⊆ c + ↑H"}
+{"name":"PFR_conjecture_improv_aux","declaration":"/-- Auxiliary statement towards the polynomial Freiman-Ruzsa (PFR) conjecture: if $A$ is a subset of\nan elementary abelian 2-group of doubling constant at most $K$, then there exists a subgroup $H$\nsuch that $A$ can be covered by at most $K^6 |A|^{1/2} / |H|^{1/2}$ cosets of $H$, and $H$ has\nthe same cardinality as $A$ up to a multiplicative factor $K^10$. -/\ntheorem PFR_conjecture_improv_aux {G : Type u_1} [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] {A : Set G} {K : ℝ} (h₀A : Set.Nonempty A) (hA : ↑(Nat.card ↑(A + A)) ≤ K * ↑(Nat.card ↑A)) : ∃ H c,\n  ↑(Nat.card ↑c) ≤ K ^ 6 * ↑(Nat.card ↑A) ^ (1 / 2) * ↑(Nat.card ↑↑H) ^ (-1 / 2) ∧\n    ↑(Nat.card ↥H) ≤ K ^ 10 * ↑(Nat.card ↑A) ∧ ↑(Nat.card ↑A) ≤ K ^ 10 * ↑(Nat.card ↥H) ∧ A ⊆ c + ↑H"}
+{"name":"tau_strictly_decreases'","declaration":"theorem tau_strictly_decreases' {G : Type u_1} [AddCommGroup G] [Fintype G] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [ElementaryAddCommGroup G 2] [MeasurableAdd₂ G] {Ω₀₁ : Type u_2} {Ω₀₂ : Type u_3} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (p : refPackage Ω₀₁ Ω₀₂ G) {Ω : Type u_4} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {X₁ : Ω → G} {X₂ : Ω → G} (hX₁ : Measurable X₁) (hX₂ : Measurable X₂) (h_min : tau_minimizes p X₁ X₂) (hp : 8 * p.η < 1) : d[X₁ # X₂] = 0"}
+{"name":"gen_ineq_00","declaration":"/-- Let $Z_1, Z_2, Z_3, Z_4$ be independent $G$-valued random variables, and let $Y$ be another\n$G$-valued random variable.  Set $S := Z_1+Z_2+Z_3+Z_4$. Then\n$d[Y; Z_1+Z_2|Z_1 + Z_3, S] - d[Y; Z_1]$ is at most\n$$ \\tfrac{1}{4} (d[Z_1;Z_2] + 2d[Z_1;Z_3] + d[Z_2;Z_4])$$\n$$+ \\tfrac{1}{4}(d[Z_1|Z_1 + Z_3 ; Z_2|Z_2+Z_4] - d[Z_1|Z_1+Z_2 ; Z_3|Z_3+Z_4]])$$\n$$+ \\tfrac{1}{8} (\\bbH[Z_1+Z_2] - \\bbH[Z_3+Z_4] + \\bbH[Z_2] - \\bbH[Z_3]$$\n$$ + \\bbH[Z_2|Z_2+Z_4] - \\bbH[Z_1|Z_1+Z_3]).$$\n-/\ntheorem gen_ineq_00 {G : Type u_1} [AddCommGroup G] [Fintype G] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [ElementaryAddCommGroup G 2] [MeasurableAdd₂ G] {Ω : Type u_2} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {Ω₀ : Type u_3} [MeasureTheory.MeasureSpace Ω₀] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (Y : Ω₀ → G) (hY : Measurable Y) (Z₁ : Ω → G) (Z₂ : Ω → G) (Z₃ : Ω → G) (Z₄ : Ω → G) (hZ₁ : Measurable Z₁) (hZ₂ : Measurable Z₂) (hZ₃ : Measurable Z₃) (hZ₄ : Measurable Z₄) (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![Z₁, Z₂, Z₃, Z₄] MeasureTheory.volume) : d[Y # Z₁ + Z₂ | ⟨Z₁ + Z₃, Z₁ + Z₂ + Z₃ + Z₄⟩] - d[Y # Z₁] ≤\n  (d[Z₁ # Z₂] + 2 * d[Z₁ # Z₃] + d[Z₂ # Z₄]) / 4 +\n      (d[Z₁ | Z₁ + Z₃ # Z₂ | Z₂ + Z₄] - d[Z₁ | Z₁ + Z₂ # Z₃ | Z₃ + Z₄]) / 4 +\n    (H[Z₁ + Z₂] - H[Z₃ + Z₄] + H[Z₂] - H[Z₃] + H[Z₂ | Z₂ + Z₄] - H[Z₁ | Z₁ + Z₃]) / 8"}
+{"name":"tau_minimizer_exists_rdist_eq_zero","declaration":"/-- For `p.η ≤ 1/8`, there exist τ-minimizers `X₁, X₂` at zero Rusza distance. For `p.η < 1/8`,\nall minimizers are fine, by `tau_strictly_decreases'`. For `p.η = 1/8`, we use a limit of\nminimizers for `η < 1/8`, which exists by compactness. -/\ntheorem tau_minimizer_exists_rdist_eq_zero {Ω₀₁ : Type u_1} {Ω₀₂ : Type u_2} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {G : Type uG} [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] [MeasurableSpace G] [MeasurableSingletonClass G] (p : refPackage Ω₀₁ Ω₀₂ G) : ∃ Ω mΩ X₁ X₂,\n  Measurable X₁ ∧\n    Measurable X₂ ∧ MeasureTheory.IsProbabilityMeasure MeasureTheory.volume ∧ tau_minimizes p X₁ X₂ ∧ d[X₁ # X₂] = 0"}
+{"name":"construct_good_improved''","declaration":"/-- Rephrase `construct_good_improved'` with an explicit probability measure, as we will\napply it to (varying) conditional measures. -/\ntheorem construct_good_improved'' {G : Type u_1} [AddCommGroup G] [Fintype G] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [ElementaryAddCommGroup G 2] [MeasurableAdd₂ G] {Ω₀₁ : Type u_2} {Ω₀₂ : Type u_3} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] {p : refPackage Ω₀₁ Ω₀₂ G} {Ω : Type u_4} [MeasureTheory.MeasureSpace Ω] {X₁ : Ω → G} {X₂ : Ω → G} (h_min : tau_minimizes p X₁ X₂) {Ω' : Type u_6} [MeasurableSpace Ω'] (μ : MeasureTheory.Measure Ω') [MeasureTheory.IsProbabilityMeasure μ] {T₁ : Ω' → G} {T₂ : Ω' → G} {T₃ : Ω' → G} (hT : T₁ + T₂ + T₃ = 0) (hT₁ : Measurable T₁) (hT₂ : Measurable T₂) (hT₃ : Measurable T₃) : d[X₁ # X₂] ≤\n  I[T₁ : T₂ ; μ] + I[T₂ : T₃ ; μ] + I[T₃ : T₁ ; μ] +\n    p.η / 6 *\n      (d[p.X₀₁ ; MeasureTheory.volume # T₁ | T₂ ; μ] - d[p.X₀₁ # X₁] +\n                            (d[p.X₀₁ ; MeasureTheory.volume # T₁ | T₃ ; μ] - d[p.X₀₁ # X₁]) +\n                          (d[p.X₀₁ ; MeasureTheory.volume # T₂ | T₁ ; μ] - d[p.X₀₁ # X₁]) +\n                        (d[p.X₀₁ ; MeasureTheory.volume # T₂ | T₃ ; μ] - d[p.X₀₁ # X₁]) +\n                      (d[p.X₀₁ ; MeasureTheory.volume # T₃ | T₁ ; μ] - d[p.X₀₁ # X₁]) +\n                    (d[p.X₀₁ ; MeasureTheory.volume # T₃ | T₂ ; μ] - d[p.X₀₁ # X₁]) +\n                  (d[p.X₀₂ ; MeasureTheory.volume # T₁ | T₂ ; μ] - d[p.X₀₂ # X₂]) +\n                (d[p.X₀₂ ; MeasureTheory.volume # T₁ | T₃ ; μ] - d[p.X₀₂ # X₂]) +\n              (d[p.X₀₂ ; MeasureTheory.volume # T₂ | T₁ ; μ] - d[p.X₀₂ # X₂]) +\n            (d[p.X₀₂ ; MeasureTheory.volume # T₂ | T₃ ; μ] - d[p.X₀₂ # X₂]) +\n          (d[p.X₀₂ ; MeasureTheory.volume # T₃ | T₁ ; μ] - d[p.X₀₂ # X₂]) +\n        (d[p.X₀₂ ; MeasureTheory.volume # T₃ | T₂ ; μ] - d[p.X₀₂ # X₂]))"}
+{"name":"construct_good_prelim'","declaration":"/-- For any $T_1, T_2, T_3$ adding up to $0$, then $k$ is at most\n$$ \\delta + \\eta (d[X^0_1;T_1|T_3]-d[X^0_1;X_1]) + \\eta (d[X^0_2;T_2|T_3]-d[X^0_2;X_2])$$\nwhere $\\delta = I[T₁ : T₂ ; μ] + I[T₂ : T₃ ; μ] + I[T₃ : T₁ ; μ]$. -/\ntheorem construct_good_prelim' {G : Type u_1} [AddCommGroup G] [Fintype G] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [ElementaryAddCommGroup G 2] [MeasurableAdd₂ G] {Ω₀₁ : Type u_2} {Ω₀₂ : Type u_3} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] {p : refPackage Ω₀₁ Ω₀₂ G} {Ω : Type u_4} [MeasureTheory.MeasureSpace Ω] {X₁ : Ω → G} {X₂ : Ω → G} (h_min : tau_minimizes p X₁ X₂) {Ω' : Type u_5} [MeasureTheory.MeasureSpace Ω'] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {T₁ : Ω' → G} {T₂ : Ω' → G} {T₃ : Ω' → G} (hT : T₁ + T₂ + T₃ = 0) (hT₁ : Measurable T₁) (hT₂ : Measurable T₂) (hT₃ : Measurable T₃) : d[X₁ # X₂] ≤\n  I[T₁ : T₂] + I[T₂ : T₃] + I[T₃ : T₁] +\n    p.η * (d[p.X₀₁ # T₁ | T₃] - d[p.X₀₁ # X₁] + (d[p.X₀₂ # T₂ | T₃] - d[p.X₀₂ # X₂]))"}
+{"name":"gen_ineq_10","declaration":"/-- Other version of `gen_ineq_00`, in which we switch to the complement in the first term. -/\ntheorem gen_ineq_10 {G : Type u_1} [AddCommGroup G] [Fintype G] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [ElementaryAddCommGroup G 2] [MeasurableAdd₂ G] {Ω : Type u_2} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {Ω₀ : Type u_3} [MeasureTheory.MeasureSpace Ω₀] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (Y : Ω₀ → G) (hY : Measurable Y) (Z₁ : Ω → G) (Z₂ : Ω → G) (Z₃ : Ω → G) (Z₄ : Ω → G) (hZ₁ : Measurable Z₁) (hZ₂ : Measurable Z₂) (hZ₃ : Measurable Z₃) (hZ₄ : Measurable Z₄) (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![Z₁, Z₂, Z₃, Z₄] MeasureTheory.volume) : d[Y # Z₃ + Z₄ | ⟨Z₁ + Z₃, Z₁ + Z₂ + Z₃ + Z₄⟩] - d[Y # Z₁] ≤\n  (d[Z₁ # Z₂] + 2 * d[Z₁ # Z₃] + d[Z₂ # Z₄]) / 4 +\n      (d[Z₁ | Z₁ + Z₃ # Z₂ | Z₂ + Z₄] - d[Z₁ | Z₁ + Z₂ # Z₃ | Z₃ + Z₄]) / 4 +\n    (H[Z₁ + Z₂] - H[Z₃ + Z₄] + H[Z₂] - H[Z₃] + H[Z₂ | Z₂ + Z₄] - H[Z₁ | Z₁ + Z₃]) / 8"}
+{"name":"PFR_conjecture_improv","declaration":"/-- The polynomial Freiman-Ruzsa (PFR) conjecture: if $A$ is a subset of an elementary abelian\n2-group of doubling constant at most $K$, then $A$ can be covered by at most $2K^{11$} cosets of\na subgroup of cardinality at most $|A|$. -/\ntheorem PFR_conjecture_improv {G : Type u_1} [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] {A : Set G} {K : ℝ} (h₀A : Set.Nonempty A) (hA : ↑(Nat.card ↑(A + A)) ≤ K * ↑(Nat.card ↑A)) : ∃ H c, ↑(Nat.card ↑c) < 2 * K ^ 11 ∧ Nat.card ↥H ≤ Nat.card ↑A ∧ A ⊆ c + ↑H"}
+{"name":"averaged_construct_good","declaration":"/-- $k$ is at most\n$$ \\leq I(U : V \\, | \\, S) + I(V : W \\, | \\,S) + I(W : U \\, | \\, S) + \\frac{\\eta}{6}  \\sum_{i=1}^2 \\sum_{A,B \\in \\{U,V,W\\}: A \\neq B} (d[X^0_i;A|B,S] - d[X^0_i; X_i]).$$\n-/\ntheorem averaged_construct_good {G : Type u_1} [AddCommGroup G] [Fintype G] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [ElementaryAddCommGroup G 2] [MeasurableAdd₂ G] {Ω₀₁ : Type u_2} {Ω₀₂ : Type u_3} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] {p : refPackage Ω₀₁ Ω₀₂ G} {Ω : Type u_4} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {X₁ : Ω → G} {X₂ : Ω → G} {X₁' : Ω → G} {X₂' : Ω → G} (hX₁ : Measurable X₁) (hX₂ : Measurable X₂) (hX₁' : Measurable X₁') (hX₂' : Measurable X₂') (h_min : tau_minimizes p X₁ X₂) : d[X₁ # X₂] ≤\n  I[X₁ + X₂ : X₁' + X₂|X₁ + X₂ + X₁' + X₂'] + I[X₁' + X₂ : X₁' + X₁|X₁ + X₂ + X₁' + X₂'] +\n      I[X₁' + X₁ : X₁ + X₂|X₁ + X₂ + X₁' + X₂'] +\n    p.η / 6 *\n      (d[p.X₀₁ # X₁ + X₂ | ⟨X₁' + X₂, X₁ + X₂ + X₁' + X₂'⟩] - d[p.X₀₁ # X₁] +\n                  (d[p.X₀₁ # X₁ + X₂ | ⟨X₁' + X₁, X₁ + X₂ + X₁' + X₂'⟩] - d[p.X₀₁ # X₁]) +\n                (d[p.X₀₁ # X₁' + X₂ | ⟨X₁ + X₂, X₁ + X₂ + X₁' + X₂'⟩] - d[p.X₀₁ # X₁]) +\n              (d[p.X₀₁ # X₁' + X₂ | ⟨X₁' + X₁, X₁ + X₂ + X₁' + X₂'⟩] - d[p.X₀₁ # X₁]) +\n            (d[p.X₀₁ # X₁' + X₁ | ⟨X₁ + X₂, X₁ + X₂ + X₁' + X₂'⟩] - d[p.X₀₁ # X₁]) +\n          (d[p.X₀₁ # X₁' + X₁ | ⟨X₁' + X₂, X₁ + X₂ + X₁' + X₂'⟩] - d[p.X₀₁ # X₁]) +\n        (d[p.X₀₂ # X₁ + X₂ | ⟨X₁' + X₂, X₁ + X₂ + X₁' + X₂'⟩] - d[p.X₀₂ # X₂] +\n                  (d[p.X₀₂ # X₁ + X₂ | ⟨X₁' + X₁, X₁ + X₂ + X₁' + X₂'⟩] - d[p.X₀₂ # X₂]) +\n                (d[p.X₀₂ # X₁' + X₂ | ⟨X₁ + X₂, X₁ + X₂ + X₁' + X₂'⟩] - d[p.X₀₂ # X₂]) +\n              (d[p.X₀₂ # X₁' + X₂ | ⟨X₁' + X₁, X₁ + X₂ + X₁' + X₂'⟩] - d[p.X₀₂ # X₂]) +\n            (d[p.X₀₂ # X₁' + X₁ | ⟨X₁ + X₂, X₁ + X₂ + X₁' + X₂'⟩] - d[p.X₀₂ # X₂]) +\n          (d[p.X₀₂ # X₁' + X₁ | ⟨X₁' + X₂, X₁ + X₂ + X₁' + X₂'⟩] - d[p.X₀₂ # X₂])))"}
+{"name":"averaged_final","declaration":"theorem averaged_final {G : Type u_1} [AddCommGroup G] [Fintype G] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [ElementaryAddCommGroup G 2] [MeasurableAdd₂ G] {Ω₀₁ : Type u_2} {Ω₀₂ : Type u_3} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (p : refPackage Ω₀₁ Ω₀₂ G) {Ω : Type u_4} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {X₁ : Ω → G} {X₂ : Ω → G} {X₁' : Ω → G} {X₂' : Ω → G} (hX₁ : Measurable X₁) (hX₂ : Measurable X₂) (hX₁' : Measurable X₁') (hX₂' : Measurable X₂') (h₁ : ProbabilityTheory.IdentDistrib X₁ X₁' MeasureTheory.volume MeasureTheory.volume) (h₂ : ProbabilityTheory.IdentDistrib X₂ X₂' MeasureTheory.volume MeasureTheory.volume) (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![X₁, X₂, X₂', X₁'] MeasureTheory.volume) (h_min : tau_minimizes p X₁ X₂) : d[X₁ # X₂] ≤\n  6 * p.η * d[X₁ # X₂] -\n      (1 - 5 * p.η) / (1 - p.η) * (2 * p.η * d[X₁ # X₂] - I[X₁ + X₂ : X₁' + X₂|X₁ + X₂ + X₁' + X₂']) +\n    p.η / 6 * (8 * d[X₁ # X₂] + 2 * (d[X₁ # X₁] + d[X₂ # X₂]))"}
+{"name":"construct_good_improved'","declaration":"/-- In fact $k$ is at most\n$$ \\delta + \\frac{\\eta}{6}  \\sum_{i=1}^2 \\sum_{1 \\leq j,l \\leq 3; j \\neq l}\n    (d[X^0_i;T_j|T_l] - d[X^0_i; X_i]).$$\n-/\ntheorem construct_good_improved' {G : Type u_1} [AddCommGroup G] [Fintype G] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [ElementaryAddCommGroup G 2] [MeasurableAdd₂ G] {Ω₀₁ : Type u_2} {Ω₀₂ : Type u_3} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] {p : refPackage Ω₀₁ Ω₀₂ G} {Ω : Type u_4} [MeasureTheory.MeasureSpace Ω] {X₁ : Ω → G} {X₂ : Ω → G} (h_min : tau_minimizes p X₁ X₂) {Ω' : Type u_5} [MeasureTheory.MeasureSpace Ω'] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {T₁ : Ω' → G} {T₂ : Ω' → G} {T₃ : Ω' → G} (hT : T₁ + T₂ + T₃ = 0) (hT₁ : Measurable T₁) (hT₂ : Measurable T₂) (hT₃ : Measurable T₃) : d[X₁ # X₂] ≤\n  I[T₁ : T₂] + I[T₂ : T₃] + I[T₃ : T₁] +\n    p.η / 6 *\n      (d[p.X₀₁ # T₁ | T₂] - d[p.X₀₁ # X₁] + (d[p.X₀₁ # T₁ | T₃] - d[p.X₀₁ # X₁]) +\n                          (d[p.X₀₁ # T₂ | T₁] - d[p.X₀₁ # X₁]) +\n                        (d[p.X₀₁ # T₂ | T₃] - d[p.X₀₁ # X₁]) +\n                      (d[p.X₀₁ # T₃ | T₁] - d[p.X₀₁ # X₁]) +\n                    (d[p.X₀₁ # T₃ | T₂] - d[p.X₀₁ # X₁]) +\n                  (d[p.X₀₂ # T₁ | T₂] - d[p.X₀₂ # X₂]) +\n                (d[p.X₀₂ # T₁ | T₃] - d[p.X₀₂ # X₂]) +\n              (d[p.X₀₂ # T₂ | T₁] - d[p.X₀₂ # X₂]) +\n            (d[p.X₀₂ # T₂ | T₃] - d[p.X₀₂ # X₂]) +\n          (d[p.X₀₂ # T₃ | T₁] - d[p.X₀₂ # X₂]) +\n        (d[p.X₀₂ # T₃ | T₂] - d[p.X₀₂ # X₂]))"}
+{"name":"dist_diff_bound_2","declaration":"theorem dist_diff_bound_2 {G : Type u_1} [AddCommGroup G] [Fintype G] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [ElementaryAddCommGroup G 2] [MeasurableAdd₂ G] {Ω₀₁ : Type u_2} {Ω₀₂ : Type u_3} [MeasureTheory.MeasureSpace Ω₀₁] [MeasureTheory.MeasureSpace Ω₀₂] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (p : refPackage Ω₀₁ Ω₀₂ G) {Ω : Type u_4} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {X₁ : Ω → G} {X₂ : Ω → G} {X₁' : Ω → G} {X₂' : Ω → G} (hX₁ : Measurable X₁) (hX₂ : Measurable X₂) (hX₁' : Measurable X₁') (hX₂' : Measurable X₂') (h₁ : ProbabilityTheory.IdentDistrib X₁ X₁' MeasureTheory.volume MeasureTheory.volume) (h₂ : ProbabilityTheory.IdentDistrib X₂ X₂' MeasureTheory.volume MeasureTheory.volume) (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![X₁, X₂, X₂', X₁'] MeasureTheory.volume) : d[p.X₀₂ # X₁ + X₂ | ⟨X₁' + X₂, X₁ + X₂ + X₁' + X₂'⟩] - d[p.X₀₂ # X₂] +\n            (d[p.X₀₂ # X₁ + X₂ | ⟨X₁' + X₁, X₁ + X₂ + X₁' + X₂'⟩] - d[p.X₀₂ # X₂]) +\n          (d[p.X₀₂ # X₁' + X₂ | ⟨X₁ + X₂, X₁ + X₂ + X₁' + X₂'⟩] - d[p.X₀₂ # X₂]) +\n        (d[p.X₀₂ # X₁' + X₂ | ⟨X₁' + X₁, X₁ + X₂ + X₁' + X₂'⟩] - d[p.X₀₂ # X₂]) +\n      (d[p.X₀₂ # X₁' + X₁ | ⟨X₁ + X₂, X₁ + X₂ + X₁' + X₂'⟩] - d[p.X₀₂ # X₂]) +\n    (d[p.X₀₂ # X₁' + X₁ | ⟨X₁' + X₂, X₁ + X₂ + X₁' + X₂'⟩] - d[p.X₀₂ # X₂]) ≤\n  (16 * d[X₁ # X₂] + 6 * d[X₂ # X₂] + 2 * d[X₁ # X₁]) / 4 + (H[X₂ + X₂'] - H[X₁ + X₁']) / 4 +\n    (H[X₁ | X₁ + X₁'] - H[X₂ | X₂ + X₂']) / 4"}
+{"name":"gen_ineq_01","declaration":"/-- Other version of `gen_ineq_00`, in which we switch to the complement in the second term. -/\ntheorem gen_ineq_01 {G : Type u_1} [AddCommGroup G] [Fintype G] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [ElementaryAddCommGroup G 2] [MeasurableAdd₂ G] {Ω : Type u_2} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {Ω₀ : Type u_3} [MeasureTheory.MeasureSpace Ω₀] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (Y : Ω₀ → G) (hY : Measurable Y) (Z₁ : Ω → G) (Z₂ : Ω → G) (Z₃ : Ω → G) (Z₄ : Ω → G) (hZ₁ : Measurable Z₁) (hZ₂ : Measurable Z₂) (hZ₃ : Measurable Z₃) (hZ₄ : Measurable Z₄) (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![Z₁, Z₂, Z₃, Z₄] MeasureTheory.volume) : d[Y # Z₁ + Z₂ | ⟨Z₂ + Z₄, Z₁ + Z₂ + Z₃ + Z₄⟩] - d[Y # Z₁] ≤\n  (d[Z₁ # Z₂] + 2 * d[Z₁ # Z₃] + d[Z₂ # Z₄]) / 4 +\n      (d[Z₁ | Z₁ + Z₃ # Z₂ | Z₂ + Z₄] - d[Z₁ | Z₁ + Z₂ # Z₃ | Z₃ + Z₄]) / 4 +\n    (H[Z₁ + Z₂] - H[Z₃ + Z₄] + H[Z₂] - H[Z₃] + H[Z₂ | Z₂ + Z₄] - H[Z₁ | Z₁ + Z₃]) / 8"}
+{"name":"gen_ineq_aux2","declaration":"theorem gen_ineq_aux2 {G : Type u_1} [AddCommGroup G] [Fintype G] [hG : MeasurableSpace G] [MeasurableSingletonClass G] [ElementaryAddCommGroup G 2] [MeasurableAdd₂ G] {Ω : Type u_2} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {Ω₀ : Type u_3} [MeasureTheory.MeasureSpace Ω₀] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] (Y : Ω₀ → G) (hY : Measurable Y) (Z₁ : Ω → G) (Z₂ : Ω → G) (Z₃ : Ω → G) (Z₄ : Ω → G) (hZ₁ : Measurable Z₁) (hZ₂ : Measurable Z₂) (hZ₃ : Measurable Z₃) (hZ₄ : Measurable Z₄) (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![Z₁, Z₂, Z₃, Z₄] MeasureTheory.volume) : d[Y # Z₁ + Z₂ | ⟨Z₁ + Z₃, Z₁ + Z₂ + Z₃ + Z₄⟩] ≤\n  d[Y # Z₁] + (d[Z₁ # Z₃] + d[Z₁ | Z₁ + Z₃ # Z₂ | Z₂ + Z₄]) / 2 +\n    (H[Z₂ | Z₂ + Z₄] - H[Z₁ | Z₁ + Z₃] + H[Z₁] - H[Z₃]) / 4"}

PFR-declarations/PFR.Main.jsonl

@@ -0,0 +1,9 @@
+{"name":"PFR_conjecture_pos_aux","declaration":"/-- Record positivity results that are useful in the proof of PFR. -/\ntheorem PFR_conjecture_pos_aux {G : Type u_1} [AddCommGroup G] {A : Set G} [Finite ↑A] {K : ℝ} (h₀A : Set.Nonempty A) (hA : ↑(Nat.card ↑(A - A)) ≤ K * ↑(Nat.card ↑A)) : 0 < ↑(Nat.card ↑A) ∧ 0 < ↑(Nat.card ↑(A - A)) ∧ 0 < K"}
+{"name":"PFR_conjecture","declaration":"/-- The polynomial Freiman-Ruzsa (PFR) conjecture: if $A$ is a subset of an elementary abelian\n2-group of doubling constant at most $K$, then $A$ can be covered by at most $2K^{12}$ cosets of\na subgroup of cardinality at most $|A|$. -/\ntheorem PFR_conjecture {G : Type u_1} [AddCommGroup G] {A : Set G} [Finite ↑A] {K : ℝ} [Countable G] [ElementaryAddCommGroup G 2] [Fintype G] (h₀A : Set.Nonempty A) (hA : ↑(Nat.card ↑(A + A)) ≤ K * ↑(Nat.card ↑A)) : ∃ H c, ↑(Nat.card ↑c) < 2 * K ^ 12 ∧ Nat.card ↥H ≤ Nat.card ↑A ∧ A ⊆ c + ↑H"}
+{"name":"PFR_conjecture_pos_aux'","declaration":"theorem PFR_conjecture_pos_aux' {G : Type u_1} [AddCommGroup G] {A : Set G} [Finite ↑A] {K : ℝ} (h₀A : Set.Nonempty A) (hA : ↑(Nat.card ↑(A + A)) ≤ K * ↑(Nat.card ↑A)) : 0 < ↑(Nat.card ↑A) ∧ 0 < ↑(Nat.card ↑(A + A)) ∧ 0 < K"}
+{"name":"rdist_le_of_isUniform_of_card_add_le","declaration":"/-- A uniform distribution on a set with doubling constant `K` has self Rusza distance\nat most `log K`. -/\ntheorem rdist_le_of_isUniform_of_card_add_le {G : Type u_1} [AddCommGroup G] [MeasurableSpace G] [MeasurableSingletonClass G] {A : Set G} [Finite ↑A] {K : ℝ} [Countable G] (h₀A : Set.Nonempty A) (hA : ↑(Nat.card ↑(A - A)) ≤ K * ↑(Nat.card ↑A)) {Ω : Type u_2} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {U₀ : Ω → G} (U₀unif : ProbabilityTheory.IsUniform A U₀ MeasureTheory.volume) (U₀meas : Measurable U₀) : d[U₀ # U₀] ≤ Real.log K"}
+{"name":"ProbabilityTheory.IsUniform.measureReal_preimage_sub","declaration":"/-- Given two independent random variables `U` and `V` uniformly distributed respectively on `A`\nand `B`, then `U = V + x` with probability `# (A ∩ (B + x)) / #A ⬝ #B`. -/\ntheorem ProbabilityTheory.IsUniform.measureReal_preimage_sub {G : Type u_1} {Ω : Type u_2} [AddCommGroup G] [Fintype G] [MeasurableSpace G] [MeasurableSingletonClass G] {A : Finset G} {B : Finset G} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {U : Ω → G} {V : Ω → G} (Uunif : ProbabilityTheory.IsUniform (↑A) U MeasureTheory.volume) (Umeas : Measurable U) (Vunif : ProbabilityTheory.IsUniform (↑B) V MeasureTheory.volume) (Vmeas : Measurable V) (hindep : ProbabilityTheory.IndepFun U V MeasureTheory.volume) (x : G) : MeasureTheory.volume.real ((U - V) ⁻¹' {x}) =\n  ↑(Nat.card ↑(↑A ∩ (↑B + {x}))) / (↑(Nat.card { x // x ∈ A }) * ↑(Nat.card { x // x ∈ B }))"}
+{"name":"sumset_eq_sub","declaration":"theorem sumset_eq_sub {G : Type u_1} [AddCommGroup G] {A : Set G} [ElementaryAddCommGroup G 2] : A + A = A - A"}
+{"name":"ProbabilityTheory.IsUniform.measureReal_preimage_sub_zero","declaration":"/-- Given two independent random variables `U` and `V` uniformly distributed respectively on `A`\nand `B`, then `U = V` with probability `# (A ∩ B) / #A ⬝ #B`. -/\ntheorem ProbabilityTheory.IsUniform.measureReal_preimage_sub_zero {G : Type u_1} {Ω : Type u_2} [AddCommGroup G] [Fintype G] [MeasurableSpace G] [MeasurableSingletonClass G] {A : Finset G} {B : Finset G} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {U : Ω → G} {V : Ω → G} (Uunif : ProbabilityTheory.IsUniform (↑A) U MeasureTheory.volume) (Umeas : Measurable U) (Vunif : ProbabilityTheory.IsUniform (↑B) V MeasureTheory.volume) (Vmeas : Measurable V) (hindep : ProbabilityTheory.IndepFun U V MeasureTheory.volume) : MeasureTheory.volume.real ((U - V) ⁻¹' {0}) =\n  ↑(Nat.card ↑(↑A ∩ ↑B)) / (↑(Nat.card { x // x ∈ A }) * ↑(Nat.card { x // x ∈ B }))"}
+{"name":"PFR_conjecture_aux","declaration":"/-- Auxiliary statement towards the polynomial Freiman-Ruzsa (PFR) conjecture: if $A$ is a subset of\nan elementary abelian 2-group of doubling constant at most $K$, then there exists a subgroup $H$\nsuch that $A$ can be covered by at most $K^{13/2} |A|^{1/2} / |H|^{1/2}$ cosets of $H$, and $H$ has\nthe same cardinality as $A$ up to a multiplicative factor $K^11$. -/\ntheorem PFR_conjecture_aux {G : Type u_1} [AddCommGroup G] {A : Set G} [Finite ↑A] {K : ℝ} [Countable G] [ElementaryAddCommGroup G 2] [Fintype G] (h₀A : Set.Nonempty A) (hA : ↑(Nat.card ↑(A + A)) ≤ K * ↑(Nat.card ↑A)) : ∃ H c,\n  ↑(Nat.card ↑c) ≤ K ^ (13 / 2) * ↑(Nat.card ↑A) ^ (1 / 2) * ↑(Nat.card ↑↑H) ^ (-1 / 2) ∧\n    ↑(Nat.card ↥H) ≤ K ^ 11 * ↑(Nat.card ↑A) ∧ ↑(Nat.card ↑A) ≤ K ^ 11 * ↑(Nat.card ↥H) ∧ A ⊆ c + ↑H"}
+{"name":"PFR_conjecture'","declaration":"/-- Corollary of `PFR_conjecture` in which the ambient group is not required to be finite (but) then\n$H$ and $c$ are finite. -/\ntheorem PFR_conjecture' {G : Type u_2} [AddCommGroup G] [ElementaryAddCommGroup G 2] {A : Set G} {K : ℝ} (h₀A : Set.Nonempty A) (Afin : Set.Finite A) (hA : ↑(Nat.card ↑(A + A)) ≤ K * ↑(Nat.card ↑A)) : ∃ H c, Set.Finite c ∧ Set.Finite ↑H ∧ ↑(Nat.card ↑c) < 2 * K ^ 12 ∧ Nat.card ↥H ≤ Nat.card ↑A ∧ A ⊆ c + ↑H"}

PFR-declarations/PFR.Mathlib.Data.Set.Pointwise.SMul.jsonl

@@ -0,0 +1,6 @@
+{"name":"Set.singleton_add'","declaration":"theorem Set.singleton_add' {α : Type u_1} [Add α] (a : α) (s : Set α) : {a} + s = a +ᵥ s"}
+{"name":"Set.vadd_sub_vadd_comm","declaration":"theorem Set.vadd_sub_vadd_comm {α : Type u_1} [AddCommGroup α] (a : α) (s : Set α) (b : α) (t : Set α) : a +ᵥ s - (b +ᵥ t) = a - b +ᵥ (s - t)"}
+{"name":"Set.singleton_mul'","declaration":"theorem Set.singleton_mul' {α : Type u_1} [Mul α] (a : α) (s : Set α) : {a} * s = a • s"}
+{"name":"Set.smul_div_smul_comm","declaration":"theorem Set.smul_div_smul_comm {α : Type u_1} [CommGroup α] (a : α) (s : Set α) (b : α) (t : Set α) : a • s / b • t = (a / b) • (s / t)"}
+{"name":"Set.mul_singleton'","declaration":"theorem Set.mul_singleton' {α : Type u_1} [Mul α] (s : Set α) (a : α) : s * {a} = MulOpposite.op a • s"}
+{"name":"Set.add_singleton'","declaration":"theorem Set.add_singleton' {α : Type u_1} [Add α] (s : Set α) (a : α) : s + {a} = AddOpposite.op a +ᵥ s"}

PFR-declarations/PFR.Mathlib.GroupTheory.Subgroup.Pointwise.jsonl

@@ -0,0 +1,6 @@
+{"name":"AddSubgroupClass.coe_sub_coe","declaration":"theorem AddSubgroupClass.coe_sub_coe {S : Type u_1} {G : Type u_2} [SetLike S G] [SubtractionMonoid G] [AddSubgroupClass S G] (H : S) : ↑H - ↑H = ↑H"}
+{"name":"AddSubgroupClass.neg_coe","declaration":"theorem AddSubgroupClass.neg_coe {S : Type u_1} {G : Type u_2} [SetLike S G] [SubtractionMonoid G] [AddSubgroupClass S G] (H : S) : -↑H = ↑H"}
+{"name":"AddSubgroupClass.coe_add_coe","declaration":"theorem AddSubgroupClass.coe_add_coe {S : Type u_1} {G : Type u_2} [SetLike S G] [SubNegMonoid G] [AddSubgroupClass S G] (H : S) : ↑H + ↑H = ↑H"}
+{"name":"SubgroupClass.inv_coe","declaration":"theorem SubgroupClass.inv_coe {S : Type u_1} {G : Type u_2} [SetLike S G] [DivisionMonoid G] [SubgroupClass S G] (H : S) : (↑H)⁻¹ = ↑H"}
+{"name":"SubgroupClass.coe_div_coe","declaration":"theorem SubgroupClass.coe_div_coe {S : Type u_1} {G : Type u_2} [SetLike S G] [DivisionMonoid G] [SubgroupClass S G] (H : S) : ↑H / ↑H = ↑H"}
+{"name":"SubgroupClass.coe_mul_coe","declaration":"theorem SubgroupClass.coe_mul_coe {S : Type u_1} {G : Type u_2} [SetLike S G] [DivInvMonoid G] [SubgroupClass S G] (H : S) : ↑H * ↑H = ↑H"}

PFR-declarations/PFR.Mathlib.GroupTheory.Torsion.jsonl

@@ -0,0 +1 @@
+{"name":"AddMonoid.IsTorsionFree.noZeroNsmulDivisors","declaration":"/-- See note [reducible non-instances]. -/\ndef AddMonoid.IsTorsionFree.noZeroNsmulDivisors {M : Type u_1} [AddMonoid M] (hM : AddMonoid.IsTorsionFree M) : NoZeroSMulDivisors ℕ M"}

PFR-declarations/PFR.Mathlib.LinearAlgebra.Basis.VectorSpace.jsonl

@@ -0,0 +1 @@
+{"name":"Submodule.exists_equiv_fst_sndModFst","declaration":"/-- Given a submodule $E$ of $B \\times F$, there is an equivalence $f : E \\to B' \\times F'$\ngiven by the projections $E \\to B$ and $E \\to F$ \"modulo\" $φ : B \\to F$. -/\ntheorem Submodule.exists_equiv_fst_sndModFst {B : Type u_1} {F : Type u_2} {R : Type u_3} [DivisionRing R] [AddCommGroup B] [AddCommGroup F] [Module R B] [Module R F] (E : Submodule R (B × F)) : ∃ B' F' f φ,\n  (∀ (x : ↥E), ↑(f x).1 = (↑x).1 ∧ ↑(f x).2 = (↑x).2 - φ (↑x).1) ∧\n    ∀ (x₁ : ↥B') (x₂ : ↥F'), ↑((LinearEquiv.symm f) (x₁, x₂)) = (↑x₁, ↑x₂ + φ ↑x₁)"}

PFR-declarations/PFR.Mathlib.MeasureTheory.Constructions.Pi.jsonl

@@ -0,0 +1,5 @@
+{"name":"MeasureTheory.Measure.pi_pi_finset","declaration":"theorem MeasureTheory.Measure.pi_pi_finset {ι : Type u_1} {α : ι → Type u_2} [Fintype ι] [(i : ι) → MeasurableSpace (α i)] (μ : (i : ι) → MeasureTheory.Measure (α i)) [∀ (i : ι), MeasureTheory.IsProbabilityMeasure (μ i)] (t : Finset ι) (s : (i : ι) → Set (α i)) : ↑↑(MeasureTheory.Measure.pi μ) (Set.pi (↑t) s) = Finset.prod t fun i => ↑↑(μ i) (s i)"}
+{"name":"MeasureTheory.Measure.instIsProbabilityMeasureForAllPiPi","declaration":"instance MeasureTheory.Measure.instIsProbabilityMeasureForAllPiPi {ι : Type u_1} {α : ι → Type u_2} [Fintype ι] [(i : ι) → MeasurableSpace (α i)] (μ : (i : ι) → MeasureTheory.Measure (α i)) [∀ (i : ι), MeasureTheory.IsProbabilityMeasure (μ i)] : MeasureTheory.IsProbabilityMeasure (MeasureTheory.Measure.pi μ)"}
+{"name":"MeasureTheory.Measure.pi_eval_preimage","declaration":"theorem MeasureTheory.Measure.pi_eval_preimage {ι : Type u_1} {α : ι → Type u_2} [Fintype ι] [(i : ι) → MeasurableSpace (α i)] (μ : (i : ι) → MeasureTheory.Measure (α i)) [∀ (i : ι), MeasureTheory.IsProbabilityMeasure (μ i)] (i : ι) (s : Set (α i)) : ↑↑(MeasureTheory.Measure.pi μ) (Function.eval i ⁻¹' s) = ↑↑(μ i) s"}
+{"name":"MeasureTheory.Measure.map_eval_pi","declaration":"theorem MeasureTheory.Measure.map_eval_pi {ι : Type u_1} {α : ι → Type u_2} [Fintype ι] [(i : ι) → MeasurableSpace (α i)] (μ : (i : ι) → MeasureTheory.Measure (α i)) [∀ (i : ι), MeasureTheory.IsProbabilityMeasure (μ i)] (i : ι) : MeasureTheory.Measure.map (Function.eval i) (MeasureTheory.Measure.pi μ) = μ i"}
+{"name":"MeasureTheory.Measure.pi_pi_set","declaration":"theorem MeasureTheory.Measure.pi_pi_set {ι : Type u_1} {α : ι → Type u_2} [Fintype ι] [(i : ι) → MeasurableSpace (α i)] (μ : (i : ι) → MeasureTheory.Measure (α i)) [∀ (i : ι), MeasureTheory.IsProbabilityMeasure (μ i)] (t : Set ι) [DecidablePred fun x => x ∈ t] (s : (i : ι) → Set (α i)) : ↑↑(MeasureTheory.Measure.pi μ) (Set.pi t s) =\n  Finset.prod (Finset.filter (fun x => x ∈ t) Finset.univ) fun i => ↑↑(μ i) (s i)"}

PFR-declarations/PFR.Mathlib.MeasureTheory.Constructions.Prod.Basic.jsonl

@@ -0,0 +1,3 @@
+{"name":"MeasureTheory.Measure.map_prod_comap_swap","declaration":"/-- The law of $(X, Z)$ is the image of the law of $(Z,X)$.-/\ntheorem MeasureTheory.Measure.map_prod_comap_swap {Ω : Type u_1} {α : Type u_2} {γ : Type u_4} [MeasurableSpace Ω] [MeasurableSpace α] [MeasurableSpace γ] {X : Ω → α} {Z : Ω → γ} (hX : Measurable X) (hZ : Measurable Z) (μ : MeasureTheory.Measure Ω) : MeasureTheory.Measure.comap Prod.swap (MeasureTheory.Measure.map (fun ω => (X ω, Z ω)) μ) =\n  MeasureTheory.Measure.map (fun ω => (Z ω, X ω)) μ"}
+{"name":"MeasureTheory.Measure.prod_apply_singleton","declaration":"theorem MeasureTheory.Measure.prod_apply_singleton {α : Type u_5} {β : Type u_6} : ∀ {x : MeasurableSpace α} {x_1 : MeasurableSpace β} (μ : MeasureTheory.Measure α) (ν : MeasureTheory.Measure β)\n  [inst : MeasureTheory.SigmaFinite ν] (x_2 : α × β),\n  ↑↑(MeasureTheory.Measure.prod μ ν) {x_2} = ↑↑μ {x_2.1} * ↑↑ν {x_2.2}"}
+{"name":"MeasureTheory.Measure.prod_of_full_measure_finset","declaration":"theorem MeasureTheory.Measure.prod_of_full_measure_finset {α : Type u_2} {β : Type u_3} [MeasurableSpace α] [MeasurableSpace β] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} [MeasureTheory.SigmaFinite ν] {A : Finset α} {B : Finset β} (hA : ↑↑μ (↑A)ᶜ = 0) (hB : ↑↑ν (↑B)ᶜ = 0) : ↑↑(MeasureTheory.Measure.prod μ ν) (↑(A ×ˢ B))ᶜ = 0"}

PFR-declarations/PFR.Mathlib.MeasureTheory.Integral.Bochner.jsonl

@@ -0,0 +1 @@
+{"name":"MeasureTheory.integral_eq_sum","declaration":"theorem MeasureTheory.integral_eq_sum {α : Type u_1} {E : Type u_2} [MeasurableSpace α] [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] (μ : MeasureTheory.Measure α) [MeasureTheory.IsFiniteMeasure μ] [MeasurableSingletonClass α] [Fintype α] (f : α → E) : ∫ (x : α), f x ∂μ = Finset.sum Finset.univ fun x => (↑↑μ {x}).toReal • f x"}

PFR-declarations/PFR.Mathlib.MeasureTheory.Integral.Lebesgue.jsonl

@@ -0,0 +1,5 @@
+{"name":"MeasureTheory.lintegral_eq_sum'","declaration":"theorem MeasureTheory.lintegral_eq_sum' {α : Type u_1} [MeasurableSpace α] [MeasurableSingletonClass α] (μ : MeasureTheory.Measure α) {s : Finset α} (hA : ↑↑μ (↑s)ᶜ = 0) (f : α → ENNReal) : ∫⁻ (x : α), f x ∂μ = Finset.sum s fun x => f x * ↑↑μ {x}"}
+{"name":"MeasureTheory.lintegral_eq_single","declaration":"theorem MeasureTheory.lintegral_eq_single {α : Type u_1} [MeasurableSpace α] [MeasurableSingletonClass α] (μ : MeasureTheory.Measure α) (a : α) (f : α → ENNReal) (ha : ∀ (b : α), b ≠ a → f b = 0) : ∫⁻ (x : α), f x ∂μ = f a * ↑↑μ {a}"}
+{"name":"MeasureTheory.lintegral_eq_sum","declaration":"theorem MeasureTheory.lintegral_eq_sum {α : Type u_1} [MeasurableSpace α] [MeasurableSingletonClass α] (μ : MeasureTheory.Measure α) (f : α → ENNReal) [Fintype α] : ∫⁻ (x : α), f x ∂μ = Finset.sum Finset.univ fun x => ↑↑μ {x} * f x"}
+{"name":"MeasureTheory.lintegral_eq_zero_of_ae_zero","declaration":"theorem MeasureTheory.lintegral_eq_zero_of_ae_zero {α : Type u_1} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {f : α → ENNReal} {E : Set α} (hE : ↑↑μ Eᶜ = 0) (hf : ∀ x ∈ E, f x = 0) (hmes : MeasurableSet E) : ∫⁻ (x : α), f x ∂μ = 0"}
+{"name":"MeasureTheory.lintegral_eq_sum_countable","declaration":"theorem MeasureTheory.lintegral_eq_sum_countable {α : Type u_1} [MeasurableSpace α] [MeasurableSingletonClass α] (μ : MeasureTheory.Measure α) (f : α → ENNReal) [Countable α] : ∫⁻ (x : α), f x ∂μ = ∑' (x : α), ↑↑μ {x} * f x"}

PFR-declarations/PFR.Mathlib.MeasureTheory.Integral.SetIntegral.jsonl

@@ -0,0 +1,2 @@
+{"name":"MeasureTheory.setIntegral_eq_sum","declaration":"theorem MeasureTheory.setIntegral_eq_sum {α : Type u_1} {E : Type u_2} [MeasurableSpace α] [MeasurableSingletonClass α] [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] (μ : MeasureTheory.Measure α) [MeasureTheory.IsFiniteMeasure μ] (s : Finset α) (f : α → E) : ∫ (x : α) in ↑s, f x ∂μ = Finset.sum s fun x => (↑↑μ {x}).toReal • f x"}
+{"name":"MeasureTheory.integral_eq_sum'","declaration":"theorem MeasureTheory.integral_eq_sum' {α : Type u_1} {E : Type u_2} [MeasurableSpace α] [MeasurableSingletonClass α] [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] (μ : MeasureTheory.Measure α) [MeasureTheory.IsFiniteMeasure μ] {s : Finset α} (hs : ↑↑μ (↑s)ᶜ = 0) (f : α → E) : ∫ (x : α), f x ∂μ = Finset.sum s fun x => (↑↑μ {x}).toReal • f x"}

PFR-declarations/PFR.Mathlib.MeasureTheory.MeasurableSpace.Basic.jsonl

@@ -0,0 +1,2 @@
+{"name":"prodMKLeft_unit_equiv","declaration":"/-- Measurable equivalence with the product with the one-point space `Unit`.-/\ndef prodMKLeft_unit_equiv (α : Type u_4) [MeasurableSpace α] : Unit × α ≃ᵐ α"}
+{"name":"prodAssoc","declaration":"/-- Canonical bijection between `(α × β) × γ` and `α × β × γ`. -/\ndef prodAssoc {α : Type u_1} {β : Type u_2} {γ : Type u_3} : {x : MeasurableSpace α} → {x_1 : MeasurableSpace β} → {x_2 : MeasurableSpace γ} → (α × β) × γ ≃ᵐ α × β × γ"}

PFR-declarations/PFR.Mathlib.MeasureTheory.Measure.MeasureSpace.jsonl

@@ -0,0 +1 @@
+{"name":"MeasureTheory.full_measure_of_null_compl","declaration":"theorem MeasureTheory.full_measure_of_null_compl {α : Type u_1} [MeasurableSpace α] [MeasurableSingletonClass α] {μ : MeasureTheory.Measure α} {A : Finset α} (hA : ↑↑μ (↑A)ᶜ = 0) : ↑↑μ ↑A = ↑↑μ Set.univ"}

PFR-declarations/PFR.Mathlib.MeasureTheory.Measure.NullMeasurable.jsonl

@@ -0,0 +1,4 @@
+{"name":"MeasureTheory.measure_preimage_snd_singleton_eq_sum","declaration":"theorem MeasureTheory.measure_preimage_snd_singleton_eq_sum {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSingletonClass α] [MeasurableSpace β] [MeasurableSingletonClass β] [Fintype α] (μ : MeasureTheory.Measure (α × β)) (y : β) : ↑↑μ (Prod.snd ⁻¹' {y}) = Finset.sum Finset.univ fun x => ↑↑μ {(x, y)}"}
+{"name":"MeasureTheory.measure_preimage_snd_singleton_eq_sum_countable","declaration":"theorem MeasureTheory.measure_preimage_snd_singleton_eq_sum_countable {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSingletonClass α] [MeasurableSpace β] [MeasurableSingletonClass β] [Countable α] (μ : MeasureTheory.Measure (α × β)) (y : β) : ↑↑μ (Prod.snd ⁻¹' {y}) = ∑' (x : α), ↑↑μ {(x, y)}"}
+{"name":"MeasureTheory.measure_preimage_fst_singleton_eq_sum_countable","declaration":"theorem MeasureTheory.measure_preimage_fst_singleton_eq_sum_countable {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSingletonClass α] [MeasurableSpace β] [MeasurableSingletonClass β] [Countable β] (μ : MeasureTheory.Measure (α × β)) (x : α) : ↑↑μ (Prod.fst ⁻¹' {x}) = ∑' (y : β), ↑↑μ {(x, y)}"}
+{"name":"MeasureTheory.measure_preimage_fst_singleton_eq_sum","declaration":"theorem MeasureTheory.measure_preimage_fst_singleton_eq_sum {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSingletonClass α] [MeasurableSpace β] [MeasurableSingletonClass β] [Fintype β] (μ : MeasureTheory.Measure (α × β)) (x : α) : ↑↑μ (Prod.fst ⁻¹' {x}) = Finset.sum Finset.univ fun y => ↑↑μ {(x, y)}"}

PFR-declarations/PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure.jsonl

@@ -0,0 +1,6 @@
+{"name":"lintegral_indicatorBCF","declaration":"theorem lintegral_indicatorBCF {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] (μ : MeasureTheory.Measure α) {s : Set α} (s_clopen : IsClopen s) (s_mble : MeasurableSet s) : ∫⁻ (x : α), ENNReal.ofReal ((indicatorBCF s_clopen) x) ∂μ = ↑↑μ s"}
+{"name":"integral_indicatorBCF","declaration":"theorem integral_indicatorBCF {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] (μ : MeasureTheory.Measure α) {s : Set α} (s_clopen : IsClopen s) (s_mble : MeasurableSet s) : ∫ (x : α), (indicatorBCF s_clopen) x ∂μ = (↑↑μ s).toReal"}
+{"name":"continuous_integral_finiteMeasure","declaration":"theorem continuous_integral_finiteMeasure {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] (f : BoundedContinuousFunction α ℝ) : Continuous fun μ => ∫ (x : α), f x ∂↑μ"}
+{"name":"indicatorBCF_apply","declaration":"theorem indicatorBCF_apply {α : Type u_1} [TopologicalSpace α] {s : Set α} (s_clopen : IsClopen s) (x : α) : (indicatorBCF s_clopen) x = Set.indicator s (fun x => 1) x"}
+{"name":"indicatorBCF","declaration":"/-- The indicator function of a clopen set, as a bounded continuous function. -/\ndef indicatorBCF {α : Type u_1} [TopologicalSpace α] {s : Set α} (s_clopen : IsClopen s) : BoundedContinuousFunction α ℝ"}
+{"name":"continuous_integral_probabilityMeasure","declaration":"theorem continuous_integral_probabilityMeasure {α : Type u_1} [TopologicalSpace α] [MeasurableSpace α] [OpensMeasurableSpace α] (f : BoundedContinuousFunction α ℝ) : Continuous fun μ => ∫ (x : α), f x ∂↑μ"}

PFR-declarations/PFR.Mathlib.MeasureTheory.Measure.Typeclasses.jsonl

@@ -0,0 +1 @@
+{"name":"MeasureTheory.IsFiniteMeasure_comap_equiv","declaration":"instance MeasureTheory.IsFiniteMeasure_comap_equiv {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] {μ : MeasureTheory.Measure α} (f : β ≃ᵐ α) [MeasureTheory.IsFiniteMeasure μ] : MeasureTheory.IsFiniteMeasure (MeasureTheory.Measure.comap (⇑f) μ)"}

PFR-declarations/PFR.Mathlib.Probability.IdentDistrib.jsonl

@@ -0,0 +1,28 @@
+{"name":"ProbabilityTheory.identDistrib_map","declaration":"/-- A random variable is identically distributed to its pullbacks. -/\ntheorem ProbabilityTheory.identDistrib_map {Ω : Type u_5} {α : Type u_7} {β : Type u_9} {mΩ : MeasurableSpace Ω} [MeasurableSpace α] [MeasurableSpace β] {X : Ω → α} (hX : Measurable X) {f : α → β} (hf : Measurable f) (μ : MeasureTheory.Measure Ω) : ProbabilityTheory.IdentDistrib f (f ∘ X) (MeasureTheory.Measure.map X μ) μ"}
+{"name":"ProbabilityTheory.identDistrib_of_finiteRange","declaration":"/-- If `X` has identical distribution to `X₀`, and `X₀` has finite range, then `X` is almost everywhere equivalent to a random variable of finite range. -/\ntheorem ProbabilityTheory.identDistrib_of_finiteRange {Ω : Type u_11} {Ω₀ : Type u_12} {S : Type u_13} [MeasurableSpace Ω] [MeasurableSpace Ω₀] [MeasurableSpace S] [MeasurableSingletonClass S] [hS : Nonempty S] {μ : MeasureTheory.Measure Ω} {μ₀ : MeasureTheory.Measure Ω₀} {X₀ : Ω₀ → S} [FiniteRange X₀] {X : Ω → S} (hX : Measurable X) (hi : ProbabilityTheory.IdentDistrib X₀ X μ₀ μ) : ∃ X', Measurable X' ∧ FiniteRange X' ∧ X' =ᶠ[MeasureTheory.Measure.ae μ] X"}
+{"name":"ProbabilityTheory.IdentDistrib.snd_id","declaration":"/-- The second projection in a product space with measure `μ.prod ν` is distributed like `ν`. -/\ntheorem ProbabilityTheory.IdentDistrib.snd_id {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure ν] : ProbabilityTheory.IdentDistrib Prod.snd id (MeasureTheory.Measure.prod μ ν) ν"}
+{"name":"ProbabilityTheory.identDistrib_ulift_self","declaration":"theorem ProbabilityTheory.identDistrib_ulift_self {Ω : Type u_5} {α : Type u_7} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [MeasurableSpace α] {X : Ω → α} (hX : Measurable X) : ProbabilityTheory.IdentDistrib X (X ∘ ULift.down) μ (MeasureTheory.Measure.comap ULift.down μ)"}
+{"name":"ProbabilityTheory.independent_copies4_nondep","declaration":"/-- A version with exactly 4 random variables that have the same codomain.\nIt's unfortunately incredibly painful to prove this from the general case. -/\ntheorem ProbabilityTheory.independent_copies4_nondep {α : Type u} [mS : MeasurableSpace α] {Ω₁ : Type u_1} {Ω₂ : Type u_2} {Ω₃ : Type u_3} {Ω₄ : Type u_4} [mΩ₁ : MeasurableSpace Ω₁] [mΩ₂ : MeasurableSpace Ω₂] [mΩ₃ : MeasurableSpace Ω₃] [mΩ₄ : MeasurableSpace Ω₄] {X₁ : Ω₁ → α} {X₂ : Ω₂ → α} {X₃ : Ω₃ → α} {X₄ : Ω₄ → α} (hX₁ : Measurable X₁) (hX₂ : Measurable X₂) (hX₃ : Measurable X₃) (hX₄ : Measurable X₄) (μ₁ : MeasureTheory.Measure Ω₁) (μ₂ : MeasureTheory.Measure Ω₂) (μ₃ : MeasureTheory.Measure Ω₃) (μ₄ : MeasureTheory.Measure Ω₄) [hμ₁ : MeasureTheory.IsProbabilityMeasure μ₁] [hμ₂ : MeasureTheory.IsProbabilityMeasure μ₂] [hμ₃ : MeasureTheory.IsProbabilityMeasure μ₃] [hμ₄ : MeasureTheory.IsProbabilityMeasure μ₄] : ∃ A mA μA X₁' X₂' X₃' X₄',\n  MeasureTheory.IsProbabilityMeasure μA ∧\n    ProbabilityTheory.iIndepFun (fun x => mS) ![X₁', X₂', X₃', X₄'] μA ∧\n      Measurable X₁' ∧\n        Measurable X₂' ∧\n          Measurable X₃' ∧\n            Measurable X₄' ∧\n              ProbabilityTheory.IdentDistrib X₁' X₁ μA μ₁ ∧\n                ProbabilityTheory.IdentDistrib X₂' X₂ μA μ₂ ∧\n                  ProbabilityTheory.IdentDistrib X₃' X₃ μA μ₃ ∧ ProbabilityTheory.IdentDistrib X₄' X₄ μA μ₄"}
+{"name":"ProbabilityTheory.identDistrib_comp_snd","declaration":"/-- A random variable is identically distributed to its lift to a product space (in the second factor). -/\ntheorem ProbabilityTheory.identDistrib_comp_snd {Ω : Type u_5} {Ω' : Type u_6} {α : Type u_7} {mΩ : MeasurableSpace Ω} {mΩ' : MeasurableSpace Ω'} [MeasurableSpace α] {X : Ω → α} (hX : Measurable X) (μ : MeasureTheory.Measure Ω) (μ' : MeasureTheory.Measure Ω') [MeasureTheory.SigmaFinite μ] [MeasureTheory.IsProbabilityMeasure μ'] : ProbabilityTheory.IdentDistrib (X ∘ Prod.snd) X (MeasureTheory.Measure.prod μ' μ) μ"}
+{"name":"ProbabilityTheory.identDistrib_comp_left","declaration":"/-- A function is identically distributed to itself composed with a measurable embedding of conull\nrange. -/\ntheorem ProbabilityTheory.identDistrib_comp_left {α : Type u_1} {γ : Type u_3} {δ : Type u_4} [MeasurableSpace α] [MeasurableSpace γ] [MeasurableSpace δ] {μ : MeasureTheory.Measure α} {f : α → γ} {i : δ → α} (hi : MeasurableEmbedding i) (hi' : ∀ᵐ (a : α) ∂μ, a ∈ Set.range i) (hf : Measurable f) : ProbabilityTheory.IdentDistrib (f ∘ i) f (MeasureTheory.Measure.comap i μ) μ"}
+{"name":"ProbabilityTheory.independent_copies_finiteRange","declaration":"/-- A version of `independent_copies` that guarantees that the copies have `FiniteRange` if the original variables do. -/\ntheorem ProbabilityTheory.independent_copies_finiteRange {Ω : Type u_5} {Ω' : Type u_6} {α : Type u_7} {β : Type u_9} {mΩ : MeasurableSpace Ω} {mΩ' : MeasurableSpace Ω'} [MeasurableSpace α] [MeasurableSpace β] {X : Ω → α} {Y : Ω' → β} (hX : Measurable X) (hY : Measurable Y) [FiniteRange X] [FiniteRange Y] [MeasurableSingletonClass α] [Nonempty α] [MeasurableSingletonClass β] [Nonempty β] (μ : MeasureTheory.Measure Ω) (μ' : MeasureTheory.Measure Ω') [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure μ'] : ∃ ν X' Y',\n  MeasureTheory.IsProbabilityMeasure ν ∧\n    Measurable X' ∧\n      Measurable Y' ∧\n        ProbabilityTheory.IndepFun X' Y' ν ∧\n          ProbabilityTheory.IdentDistrib X' X ν μ ∧\n            ProbabilityTheory.IdentDistrib Y' Y ν μ' ∧ FiniteRange X' ∧ FiniteRange Y'"}
+{"name":"ProbabilityTheory.IdentDistrib.cond","declaration":"theorem ProbabilityTheory.IdentDistrib.cond {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {f : α → γ} {f' : α → γ} {g : β → γ} {g' : β → γ} {s : Set γ} (hs : MeasurableSet s) (hf' : Measurable f') (hg' : Measurable g') (hfg : ProbabilityTheory.IdentDistrib (fun a => (f a, f' a)) (fun b => (g b, g' b)) μ ν) : ProbabilityTheory.IdentDistrib f g (ProbabilityTheory.cond μ (f' ⁻¹' s)) (ProbabilityTheory.cond ν (g' ⁻¹' s))"}
+{"name":"ProbabilityTheory.independent_copies","declaration":"/-- For $X, Y$ random variables, one can find independent copies $X', Y'$ of $X, Y$. -/\ntheorem ProbabilityTheory.independent_copies {Ω : Type u_5} {Ω' : Type u_6} {α : Type u_7} {β : Type u_9} {mΩ : MeasurableSpace Ω} {mΩ' : MeasurableSpace Ω'} [MeasurableSpace α] [MeasurableSpace β] {X : Ω → α} {Y : Ω' → β} (hX : Measurable X) (hY : Measurable Y) (μ : MeasureTheory.Measure Ω) (μ' : MeasureTheory.Measure Ω') [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure μ'] : ∃ ν X' Y',\n  MeasureTheory.IsProbabilityMeasure ν ∧\n    Measurable X' ∧\n      Measurable Y' ∧\n        ProbabilityTheory.IndepFun X' Y' ν ∧\n          ProbabilityTheory.IdentDistrib X' X ν μ ∧ ProbabilityTheory.IdentDistrib Y' Y ν μ'"}
+{"name":"ProbabilityTheory.independent_copies3_nondep_finiteRange","declaration":"/-- A version of `independent_copies3_nondep` that guarantees that the copies have `FiniteRange` if the original variables do. -/\ntheorem ProbabilityTheory.independent_copies3_nondep_finiteRange {α : Type u} [mS : MeasurableSpace α] [MeasurableSingletonClass α] [Nonempty α] {Ω₁ : Type u_1} {Ω₂ : Type u_2} {Ω₃ : Type u_3} [MeasurableSpace Ω₁] [MeasurableSpace Ω₂] [MeasurableSpace Ω₃] {X₁ : Ω₁ → α} {X₂ : Ω₂ → α} {X₃ : Ω₃ → α} (hX₁ : Measurable X₁) (hX₂ : Measurable X₂) (hX₃ : Measurable X₃) [FiniteRange X₁] [FiniteRange X₂] [FiniteRange X₃] (μ₁ : MeasureTheory.Measure Ω₁) (μ₂ : MeasureTheory.Measure Ω₂) (μ₃ : MeasureTheory.Measure Ω₃) [hμ₁ : MeasureTheory.IsProbabilityMeasure μ₁] [hμ₂ : MeasureTheory.IsProbabilityMeasure μ₂] [hμ₃ : MeasureTheory.IsProbabilityMeasure μ₃] : ∃ A mA μA X₁' X₂' X₃',\n  MeasureTheory.IsProbabilityMeasure μA ∧\n    ProbabilityTheory.iIndepFun (fun x => mS) ![X₁', X₂', X₃'] μA ∧\n      Measurable X₁' ∧\n        Measurable X₂' ∧\n          Measurable X₃' ∧\n            ProbabilityTheory.IdentDistrib X₁' X₁ μA μ₁ ∧\n              ProbabilityTheory.IdentDistrib X₂' X₂ μA μ₂ ∧\n                ProbabilityTheory.IdentDistrib X₃' X₃ μA μ₃ ∧ FiniteRange X₁' ∧ FiniteRange X₂' ∧ FiniteRange X₃'"}
+{"name":"ProbabilityTheory.IdentDistrib.mul","declaration":"theorem ProbabilityTheory.IdentDistrib.mul {Ω : Type u_5} {Ω' : Type u_6} {β : Type u_9} {mΩ : MeasurableSpace Ω} {mΩ' : MeasurableSpace Ω'} {mβ : MeasurableSpace β} {μ : MeasureTheory.Measure Ω} {ν : MeasureTheory.Measure Ω'} {f : Ω → β} {g : Ω → β} {f' : Ω' → β} {g' : Ω' → β} [Mul β] [MeasurableMul₂ β] [MeasureTheory.IsFiniteMeasure μ] [MeasureTheory.IsFiniteMeasure ν] (hff' : ProbabilityTheory.IdentDistrib f f' μ ν) (hgg' : ProbabilityTheory.IdentDistrib g g' μ ν) (h : ProbabilityTheory.IndepFun f g μ) (h' : ProbabilityTheory.IndepFun f' g' ν) : ProbabilityTheory.IdentDistrib (f * g) (f' * g') μ ν"}
+{"name":"ProbabilityTheory.independent_copies'","declaration":"/-- Let $X_i : \\Omega_i \\to S_i$ be random variables for $i=1,\\dots,k$.\nThen there exist jointly independent random variables $X'_i : \\Omega' \\to S_i$ for $i=1,\\dots,k$\nsuch that each $X'_i$ is a copy of $X_i$. -/\ntheorem ProbabilityTheory.independent_copies' {I : Type u} [Fintype I] {α : I → Type u'} [mS : (i : I) → MeasurableSpace (α i)] {Ω : I → Type v} [mΩ : (i : I) → MeasurableSpace (Ω i)] (X : (i : I) → Ω i → α i) (hX : ��� (i : I), Measurable (X i)) (μ : (i : I) → MeasureTheory.Measure (Ω i)) [∀ (i : I), MeasureTheory.IsProbabilityMeasure (μ i)] : ∃ A mA μA X',\n  MeasureTheory.IsProbabilityMeasure μA ∧\n    ProbabilityTheory.iIndepFun mS X' μA ∧\n      ∀ (i : I), Measurable (X' i) ∧ ProbabilityTheory.IdentDistrib (X' i) (X i) μA (μ i)"}
+{"name":"ProbabilityTheory.identDistrib_id_left","declaration":"theorem ProbabilityTheory.identDistrib_id_left {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] {μ : MeasureTheory.Measure α} {X : α → β} (hX : AEMeasurable X μ) : ProbabilityTheory.IdentDistrib id X (MeasureTheory.Measure.map X μ) μ"}
+{"name":"ProbabilityTheory.IdentDistrib.add","declaration":"theorem ProbabilityTheory.IdentDistrib.add {Ω : Type u_5} {Ω' : Type u_6} {β : Type u_9} {mΩ : MeasurableSpace Ω} {mΩ' : MeasurableSpace Ω'} {mβ : MeasurableSpace β} {μ : MeasureTheory.Measure Ω} {ν : MeasureTheory.Measure Ω'} {f : Ω → β} {g : Ω → β} {f' : Ω' → β} {g' : Ω' → β} [Add β] [MeasurableAdd₂ β] [MeasureTheory.IsFiniteMeasure μ] [MeasureTheory.IsFiniteMeasure ν] (hff' : ProbabilityTheory.IdentDistrib f f' μ ν) (hgg' : ProbabilityTheory.IdentDistrib g g' μ ν) (h : ProbabilityTheory.IndepFun f g μ) (h' : ProbabilityTheory.IndepFun f' g' ν) : ProbabilityTheory.IdentDistrib (f + g) (f' + g') μ ν"}
+{"name":"ProbabilityTheory.identDistrib_of_sum","declaration":"/-- To show identical distribution of two random variables on a mixture of probability measures, it suffices to do so on each non-trivial component. -/\ntheorem ProbabilityTheory.identDistrib_of_sum {Ω : Type u_5} {Ω' : Type u_6} {α : Type u_7} {mΩ : MeasurableSpace Ω} {mΩ' : MeasurableSpace Ω'} [MeasurableSpace α] {T : Type u_11} {X : Ω → α} {Y : Ω' → α} [Fintype T] {μ : T → MeasureTheory.Measure Ω} {μ' : T → MeasureTheory.Measure Ω'} {w : T → ENNReal} (hX : Measurable X) (hY : Measurable Y) (h_ident : ∀ (y : T), w y ≠ 0 → ProbabilityTheory.IdentDistrib X Y (μ y) (μ' y)) : ProbabilityTheory.IdentDistrib X Y (Finset.sum Finset.univ fun y => w y • μ y)\n  (Finset.sum Finset.univ fun y => w y • μ' y)"}
+{"name":"ProbabilityTheory.identDistrib_id","declaration":"theorem ProbabilityTheory.identDistrib_id {α : Type u_1} [MeasurableSpace α] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure α} : ProbabilityTheory.IdentDistrib id id μ ν ↔ μ = ν"}
+{"name":"ProbabilityTheory.independent_copies3_nondep","declaration":"/-- A version with exactly 3 random variables that have the same codomain.\nIt's unfortunately incredibly painful to prove this from the general case. -/\ntheorem ProbabilityTheory.independent_copies3_nondep {α : Type u} [mS : MeasurableSpace α] {Ω₁ : Type u_1} {Ω₂ : Type u_2} {Ω₃ : Type u_3} [MeasurableSpace Ω₁] [MeasurableSpace Ω₂] [MeasurableSpace Ω₃] {X₁ : Ω₁ → α} {X₂ : Ω₂ → α} {X₃ : Ω₃ → α} (hX₁ : Measurable X₁) (hX₂ : Measurable X₂) (hX₃ : Measurable X₃) (μ₁ : MeasureTheory.Measure Ω₁) (μ₂ : MeasureTheory.Measure Ω₂) (μ₃ : MeasureTheory.Measure Ω₃) [hμ₁ : MeasureTheory.IsProbabilityMeasure μ₁] [hμ₂ : MeasureTheory.IsProbabilityMeasure μ₂] [hμ₃ : MeasureTheory.IsProbabilityMeasure μ₃] : ∃ A mA μA X₁' X₂' X₃',\n  MeasureTheory.IsProbabilityMeasure μA ∧\n    ProbabilityTheory.iIndepFun (fun x => mS) ![X₁', X₂', X₃'] μA ∧\n      Measurable X₁' ∧\n        Measurable X₂' ∧\n          Measurable X₃' ∧\n            ProbabilityTheory.IdentDistrib X₁' X₁ μA μ₁ ∧\n              ProbabilityTheory.IdentDistrib X₂' X₂ μA μ₂ ∧ ProbabilityTheory.IdentDistrib X₃' X₃ μA μ₃"}
+{"name":"ProbabilityTheory.independent_copies4_nondep_finiteRange","declaration":"/-- A version of `independent_copies4_nondep` that guarantees that the copies have `FiniteRange` if the original variables do. -/\ntheorem ProbabilityTheory.independent_copies4_nondep_finiteRange {α : Type u} [mS : MeasurableSpace α] [MeasurableSingletonClass α] [Nonempty α] {Ω₁ : Type u_1} {Ω₂ : Type u_2} {Ω₃ : Type u_3} {Ω₄ : Type u_4} [MeasurableSpace Ω₁] [MeasurableSpace Ω₂] [MeasurableSpace Ω₃] [MeasurableSpace Ω₄] {X₁ : Ω₁ → α} {X₂ : Ω₂ → α} {X₃ : Ω₃ → α} {X₄ : Ω₄ → α} (hX₁ : Measurable X₁) (hX₂ : Measurable X₂) (hX₃ : Measurable X₃) (hX₄ : Measurable X₄) [FiniteRange X₁] [FiniteRange X₂] [FiniteRange X₃] [FiniteRange X₄] (μ₁ : MeasureTheory.Measure Ω₁) (μ₂ : MeasureTheory.Measure Ω₂) (μ₃ : MeasureTheory.Measure Ω₃) (μ₄ : MeasureTheory.Measure Ω₄) [hμ₁ : MeasureTheory.IsProbabilityMeasure μ₁] [hμ₂ : MeasureTheory.IsProbabilityMeasure μ₂] [hμ₃ : MeasureTheory.IsProbabilityMeasure μ₃] [hμ₄ : MeasureTheory.IsProbabilityMeasure μ₄] : ∃ A mA μA X₁' X₂' X₃' X₄',\n  MeasureTheory.IsProbabilityMeasure μA ∧\n    ProbabilityTheory.iIndepFun (fun x => mS) ![X₁', X₂', X₃', X₄'] μA ∧\n      Measurable X₁' ∧\n        Measurable X₂' ∧\n          Measurable X₃' ∧\n            Measurable X₄' ∧\n              ProbabilityTheory.IdentDistrib X₁' X₁ μA μ₁ ∧\n                ProbabilityTheory.IdentDistrib X₂' X₂ μA μ₂ ∧\n                  ProbabilityTheory.IdentDistrib X₃' X₃ μA μ₃ ∧\n                    ProbabilityTheory.IdentDistrib X₄' X₄ μA μ₄ ∧\n                      FiniteRange X₁' ∧ FiniteRange X₂' ∧ FiniteRange X₃' ∧ FiniteRange X₄'"}
+{"name":"ProbabilityTheory.identDistrib_id_right","declaration":"theorem ProbabilityTheory.identDistrib_id_right {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] {μ : MeasureTheory.Measure α} {X : α → β} (hX : AEMeasurable X μ) : ProbabilityTheory.IdentDistrib X id μ (MeasureTheory.Measure.map X μ)"}
+{"name":"ProbabilityTheory.IdentDistrib.comp_left","declaration":"/-- Composing identically distributed functions with a measurable embedding of conull range\ngives identically distributed functions. -/\ntheorem ProbabilityTheory.IdentDistrib.comp_left {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] [MeasurableSpace δ] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {f : α → γ} {g : β → γ} {i : δ → α} (hi : MeasurableEmbedding i) (hi' : ∀ᵐ (a : α) ∂μ, a ∈ Set.range i) (hf : Measurable f) (hfg : ProbabilityTheory.IdentDistrib f g μ ν) : ProbabilityTheory.IdentDistrib (f ∘ i) g (MeasureTheory.Measure.comap i μ) ν"}
+{"name":"ProbabilityTheory.identDistrib_comp_fst","declaration":"/-- A random variable is identically distributed to its lift to a product space (in the first factor). -/\ntheorem ProbabilityTheory.identDistrib_comp_fst {Ω : Type u_5} {Ω' : Type u_6} {α : Type u_7} {mΩ : MeasurableSpace Ω} {mΩ' : MeasurableSpace Ω'} [MeasurableSpace α] {X : Ω → α} (hX : Measurable X) (μ : MeasureTheory.Measure Ω) (μ' : MeasureTheory.Measure Ω') [MeasureTheory.IsProbabilityMeasure μ'] : ProbabilityTheory.IdentDistrib (X ∘ Prod.fst) X (MeasureTheory.Measure.prod μ μ') μ"}
+{"name":"ProbabilityTheory.independent_copies_two","declaration":"/-- For $X, Y$ random variables, one can find independent copies $X', Y'$ of $X, Y$. Version\nformulated in spaces with a canonical measures. -/\ntheorem ProbabilityTheory.independent_copies_two {α : Type u_7} {β : Type u_9} [MeasurableSpace α] [MeasurableSpace β] {Ω : Type u} {Ω' : Type v} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.MeasureSpace Ω'] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {X : Ω → α} {Y : Ω' → β} (hX : Measurable X) (hY : Measurable Y) : ∃ Ω'' m'' X' Y',\n  MeasureTheory.IsProbabilityMeasure MeasureTheory.volume ∧\n    Measurable X' ∧\n      Measurable Y' ∧\n        ProbabilityTheory.IndepFun X' Y' MeasureTheory.volume ∧\n          ProbabilityTheory.IdentDistrib X' X MeasureTheory.volume MeasureTheory.volume ∧\n            ProbabilityTheory.IdentDistrib Y' Y MeasureTheory.volume MeasureTheory.volume"}
+{"name":"ProbabilityTheory.identDistrib_iff","declaration":"theorem ProbabilityTheory.identDistrib_iff {α : Type u_1} {β : Type u_2} {γ : Type u_3} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] (f : α → γ) (g : β → γ) (μ : autoParam (MeasureTheory.Measure α) _auto✝) (ν : autoParam (MeasureTheory.Measure β) _auto✝) : ProbabilityTheory.IdentDistrib f g μ ν ↔\n  AEMeasurable f μ ∧ AEMeasurable g ν ∧ MeasureTheory.Measure.map f μ = MeasureTheory.Measure.map g ν"}
+{"name":"ProbabilityTheory.IdentDistrib.comp_right","declaration":"/-- Composing identically distributed functions with a measurable embedding of conull range\ngives identically distributed functions. -/\ntheorem ProbabilityTheory.IdentDistrib.comp_right {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] [MeasurableSpace δ] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} {f : α → γ} {g : β → γ} {i : δ → β} (hi : MeasurableEmbedding i) (hi' : ∀ᵐ (a : β) ∂ν, a ∈ Set.range i) (hg : Measurable g) (hfg : ProbabilityTheory.IdentDistrib f g μ ν) : ProbabilityTheory.IdentDistrib f (g ∘ i) μ (MeasureTheory.Measure.comap i ν)"}
+{"name":"ProbabilityTheory.IdentDistrib.prod_mk","declaration":"theorem ProbabilityTheory.IdentDistrib.prod_mk {Ω : Type u_5} {Ω' : Type u_6} {β : Type u_9} {mΩ : MeasurableSpace Ω} {mΩ' : MeasurableSpace Ω'} {mβ : MeasurableSpace β} {μ : MeasureTheory.Measure Ω} {ν : MeasureTheory.Measure Ω'} {f : Ω → β} {g : Ω → β} {f' : Ω' → β} {g' : Ω' → β} [MeasureTheory.IsFiniteMeasure μ] [MeasureTheory.IsFiniteMeasure ν] (hff' : ProbabilityTheory.IdentDistrib f f' μ ν) (hgg' : ProbabilityTheory.IdentDistrib g g' μ ν) (h : ProbabilityTheory.IndepFun f g μ) (h' : ProbabilityTheory.IndepFun f' g' ν) : ProbabilityTheory.IdentDistrib (fun x => (f x, g x)) (fun x => (f' x, g' x)) μ ν"}
+{"name":"ProbabilityTheory.identDistrib_comp_right","declaration":"/-- A function is identically distributed to itself composed with a measurable embedding of conull\nrange. -/\ntheorem ProbabilityTheory.identDistrib_comp_right {α : Type u_1} {γ : Type u_3} {δ : Type u_4} [MeasurableSpace α] [MeasurableSpace γ] [MeasurableSpace δ] {μ : MeasureTheory.Measure α} {f : α → γ} {i : δ → α} (hi : MeasurableEmbedding i) (hi' : ∀ᵐ (a : α) ∂μ, a ∈ Set.range i) (hf : Measurable f) : ProbabilityTheory.IdentDistrib f (f ∘ i) μ (MeasureTheory.Measure.comap i μ)"}
+{"name":"ProbabilityTheory.IdentDistrib.fst_id","declaration":"/-- The first projection in a product space with measure `μ.prod ν` is distributed like `μ`. -/\ntheorem ProbabilityTheory.IdentDistrib.fst_id {α : Type u_1} {β : Type u_2} [MeasurableSpace α] [MeasurableSpace β] {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} [MeasureTheory.IsProbabilityMeasure ν] : ProbabilityTheory.IdentDistrib Prod.fst id (MeasureTheory.Measure.prod μ ν) μ"}

PFR-declarations/PFR.Mathlib.Probability.Independence.Basic.jsonl

@@ -0,0 +1,28 @@
+{"name":"ProbabilityTheory.iIndepFun_iff'","declaration":"theorem ProbabilityTheory.iIndepFun_iff' {Ω : Type u_1} {ι : Type u_2} [MeasurableSpace Ω] {β : ι → Type u_11} (m : (i : ι) → MeasurableSpace (β i)) (f : (i : ι) → Ω → β i) (μ : MeasureTheory.Measure Ω) : ProbabilityTheory.iIndepFun m f μ ↔\n  ∀ (s : Finset ι) ⦃f' : ι → Set Ω⦄,\n    (∀ (i : ι), MeasurableSet (f' i)) → ↑↑μ (⋂ i ∈ s, f' i) = Finset.prod s fun i => ↑↑μ (f' i)"}
+{"name":"ProbabilityTheory.EventuallyEq.finite_iInter","declaration":"/-- The new Mathlib tool `Finset.eventuallyEq_iInter` will supersede this result. -/\ntheorem ProbabilityTheory.EventuallyEq.finite_iInter {ι : Type u_14} {α : Type u_2} {l : Filter α} (s : Finset ι) {E : ι → Set α} {F : ι → Set α} (h : ∀ i ∈ s, E i =ᶠ[l] F i) : ⋂ i ∈ s, E i =ᶠ[l] ⋂ i ∈ s, F i"}
+{"name":"ProbabilityTheory.iIndepFun.pi","declaration":"/-- If a family of functions `(i, j) ↦ f i j` is independent, then the family of function tuples\n`i ↦ (f i j)ⱼ` is independent. -/\ntheorem ProbabilityTheory.iIndepFun.pi {Ω : Type u_2} [MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {ι : Type u_5} {κ : ι → Type u_6} [(i : ι) → Fintype (κ i)] {α : (i : ι) → κ i → Type u_7} {f : (i : ι) → (j : κ i) → Ω → α i j} [m : (i : ι) → (j : κ i) → MeasurableSpace (α i j)] (f_meas : ∀ (i : ι) (j : κ i), Measurable (f i j)) (hf : ProbabilityTheory.iIndepFun (fun ij => m ij.fst ij.snd) (fun ij => f ij.fst ij.snd) μ) : ProbabilityTheory.iIndepFun (fun i => MeasurableSpace.pi) (fun i ω j => f i j ω) μ"}
+{"name":"ProbabilityTheory.iIndepFun.neg","declaration":"theorem ProbabilityTheory.iIndepFun.neg {Ω : Type u_10} {ι : Type u_11} [MeasurableSpace Ω] {α : ι → Type u_13} [n : (i : ι) → MeasurableSpace (α i)] {f : (i : ι) → Ω → α i} {μ : MeasureTheory.Measure Ω} (i : ι) [Neg (α i)] [MeasurableNeg (α i)] [DecidableEq ι] (h : ProbabilityTheory.iIndepFun n f μ) : ProbabilityTheory.iIndepFun n (Function.update f i (-f i)) μ"}
+{"name":"ProbabilityTheory.IndepFun.comp_right","declaration":"/-- Composing independent functions with a measurable embedding of conull range gives independent\nfunctions. -/\ntheorem ProbabilityTheory.IndepFun.comp_right {α : Type u_11} {Ω : Type u_1} {β : Type u_6} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {Ω' : Type u_10} [MeasurableSpace Ω'] [MeasurableSpace α] [MeasurableSpace β] {f : Ω → α} {g : Ω → β} {i : Ω' → Ω} (hi : MeasurableEmbedding i) (hi' : ∀ᵐ (a : Ω) ∂μ, a ∈ Set.range i) (hf : Measurable f) (hg : Measurable g) (hfg : ProbabilityTheory.IndepFun f g μ) : ProbabilityTheory.IndepFun (f ∘ i) (g ∘ i) (MeasureTheory.Measure.comap i μ)"}
+{"name":"ProbabilityTheory.IndepFun.measureReal_inter_preimage_eq_mul","declaration":"theorem ProbabilityTheory.IndepFun.measureReal_inter_preimage_eq_mul {Ω : Type u_1} {β : Type u_10} {β' : Type u_11} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {f : Ω → β} {g : Ω → β'} {_mβ : MeasurableSpace β} {_mβ' : MeasurableSpace β'} (h : ProbabilityTheory.IndepFun f g μ) {s : Set β} {t : Set β'} (hs : MeasurableSet s) (ht : MeasurableSet t) : μ.real (f ⁻¹' s ∩ g ⁻¹' t) = μ.real (f ⁻¹' s) * μ.real (g ⁻¹' t)"}
+{"name":"Finset.prod_univ_prod","declaration":"theorem Finset.prod_univ_prod {ι : Type u_1} {κ : ι → Type u_3} [Fintype ι] [(i : ι) → Fintype (κ i)] {β : Type u_5} [CommMonoid β] (f : (i : ι) → κ i → β) : (Finset.prod Finset.univ fun ij => f ij.fst ij.snd) =\n  Finset.prod Finset.univ fun i => Finset.prod Finset.univ fun j => f i j"}
+{"name":"ProbabilityTheory.iIndepFun_reindex_iff","declaration":"theorem ProbabilityTheory.iIndepFun_reindex_iff {Ω : Type u_10} {ι : Type u_11} {ι' : Type u_12} [MeasurableSpace Ω] {α : ι → Type u_13} [n : (i : ι) → MeasurableSpace (α i)] {f : (i : ι) → Ω → α i} {μ : MeasureTheory.Measure Ω} (g : ι' ≃ ι) : ProbabilityTheory.iIndepFun ((fun {x} => n) ∘' ⇑g) ((fun {x} => f) ∘' ⇑g) μ ↔ ProbabilityTheory.iIndepFun n f μ"}
+{"name":"ProbabilityTheory.iIndepFun.reindex_of_injective","declaration":"theorem ProbabilityTheory.iIndepFun.reindex_of_injective {Ω : Type u_10} {ι : Type u_11} {ι' : Type u_12} [MeasurableSpace Ω] {α : ι → Type u_13} [n : (i : ι) → MeasurableSpace (α i)] {f : (i : ι) → Ω → α i} {μ : MeasureTheory.Measure Ω} (h : ProbabilityTheory.iIndepFun n f μ) (g : ι' → ι) (hg : Function.Injective g) : ProbabilityTheory.iIndepFun ((fun {x} => n) ∘' g) ((fun {x} => f) ∘' g) μ"}
+{"name":"ProbabilityTheory.iIndepFun.reindex_symm","declaration":"theorem ProbabilityTheory.iIndepFun.reindex_symm {Ω : Type u_10} {ι : Type u_11} {ι' : Type u_12} [MeasurableSpace Ω] {α : ι → Type u_13} [n : (i : ι) → MeasurableSpace (α i)] {f : (i : ι) → Ω → α i} {μ : MeasureTheory.Measure Ω} (g : ι' ≃ ι) (h : ProbabilityTheory.iIndepFun n f μ) : ProbabilityTheory.iIndepFun ((fun {x} => n) ∘' ⇑g) ((fun {x} => f) ∘' ⇑g) μ"}
+{"name":"ProbabilityTheory.indepFun_const","declaration":"/-- Random variables are always independent of constants. -/\ntheorem ProbabilityTheory.indepFun_const {α : Type u_11} {Ω : Type u_1} {β : Type u_6} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {f : Ω → β} [MeasurableSpace α] [MeasurableSpace β] [MeasureTheory.IsProbabilityMeasure μ] (c : α) : ProbabilityTheory.IndepFun f (fun x => c) μ"}
+{"name":"ProbabilityTheory.iIndepFun.pi'","declaration":"/-- If a family of functions `(i, j) ↦ f i j` is independent, then the family of function tuples\n`i ↦ (f i j)ⱼ` is independent. -/\ntheorem ProbabilityTheory.iIndepFun.pi' {Ω : Type u_2} [MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {ι : Type u_5} {κ : ι → Type u_6} [(i : ι) → Fintype (κ i)] {α : (i : ι) → κ i → Type u_7} [m : (i : ι) → (j : κ i) → MeasurableSpace (α i j)] {f : (ij : (i : ι) × κ i) → Ω → α ij.fst ij.snd} (f_meas : ∀ (i : (i : ι) × κ i), Measurable (f i)) (hf : ProbabilityTheory.iIndepFun (fun ij => m ij.fst ij.snd) f μ) : ProbabilityTheory.iIndepFun (fun _i => MeasurableSpace.pi) (fun i ω j => f { fst := i, snd := j } ω) μ"}
+{"name":"Finset.sum_univ_sum","declaration":"theorem Finset.sum_univ_sum {ι : Type u_1} {κ : ι → Type u_3} [Fintype ι] [(i : ι) → Fintype (κ i)] {β : Type u_5} [AddCommMonoid β] (f : (i : ι) → κ i → β) : (Finset.sum Finset.univ fun ij => f ij.fst ij.snd) =\n  Finset.sum Finset.univ fun i => Finset.sum Finset.univ fun j => f i j"}
+{"name":"Finset.sum_univ_sum'","declaration":"theorem Finset.sum_univ_sum' {ι : Type u_1} {κ : ι → Type u_3} [Fintype ι] [(i : ι) → Fintype (κ i)] {β : Type u_5} [AddCommMonoid β] (f : (i : ι) × κ i → β) : (Finset.sum Finset.univ fun ij => f ij) =\n  Finset.sum Finset.univ fun i => Finset.sum Finset.univ fun j => f { fst := i, snd := j }"}
+{"name":"ProbabilityTheory.IndepFun.symm'","declaration":"/-- in mathlib as of `4d385393cd569f08ac30425ef886a57bb10daaa5` (TODO: bump) -/\ntheorem ProbabilityTheory.IndepFun.symm' {γ : Type u_14} {β : Type u_15} {Ω : Type u_16} : ∀ {x : MeasurableSpace γ} {x_1 : MeasurableSpace β} {x_2 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {f : Ω → β}\n  {g : Ω → γ}, ProbabilityTheory.IndepFun f g μ → ProbabilityTheory.IndepFun g f μ"}
+{"name":"ProbabilityTheory.kernel.IndepFun.symm'","declaration":"/-- in mathlib as of `4d385393cd569f08ac30425ef886a57bb10daaa5` (TODO: bump) -/\ntheorem ProbabilityTheory.kernel.IndepFun.symm' {Ω : Type u_14} {α : Type u_15} {β : Type u_16} {γ : Type u_17} : ∀ {x : MeasurableSpace Ω} {x_1 : MeasurableSpace α} {x_2 : MeasurableSpace β} {x_3 : MeasurableSpace γ}\n  {κ : ↥(ProbabilityTheory.kernel α Ω)} {f : Ω → β} {g : Ω → γ} {μ : MeasureTheory.Measure α},\n  ProbabilityTheory.kernel.IndepFun f g κ μ → ProbabilityTheory.kernel.IndepFun g f κ μ"}
+{"name":"ProbabilityTheory.IndepFun.measure_inter_preimage_eq_mul","declaration":"theorem ProbabilityTheory.IndepFun.measure_inter_preimage_eq_mul {Ω : Type u_1} {β : Type u_10} {β' : Type u_11} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {f : Ω → β} {g : Ω → β'} {_mβ : MeasurableSpace β} {_mβ' : MeasurableSpace β'} (h : ProbabilityTheory.IndepFun f g μ) {s : Set β} {t : Set β'} (hs : MeasurableSet s) (ht : MeasurableSet t) : ↑↑μ (f ⁻¹' s ∩ g ⁻¹' t) = ↑↑μ (f ⁻¹' s) * ↑↑μ (g ⁻¹' t)"}
+{"name":"ProbabilityTheory.iIndepFun.ae_eq","declaration":"/-- TODO: a kernel version of this theorem-/\ntheorem ProbabilityTheory.iIndepFun.ae_eq {Ω : Type u_13} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {ι : Type u_14} {β : ι → Type u_15} {m : (i : ι) → MeasurableSpace (β i)} {f : (i : ι) → Ω → β i} {g : (i : ι) → Ω → β i} (hf_Indep : ProbabilityTheory.iIndepFun m f μ) (hfg : ∀ (i : ι), f i =ᶠ[MeasureTheory.Measure.ae μ] g i) : ProbabilityTheory.iIndepFun m g μ"}
+{"name":"Finset.prod_univ_prod'","declaration":"theorem Finset.prod_univ_prod' {ι : Type u_1} {κ : ι → Type u_3} [Fintype ι] [(i : ι) → Fintype (κ i)] {β : Type u_5} [CommMonoid β] (f : (i : ι) × κ i → β) : (Finset.prod Finset.univ fun ij => f ij) =\n  Finset.prod Finset.univ fun i => Finset.prod Finset.univ fun j => f { fst := i, snd := j }"}
+{"name":"ProbabilityTheory.measurable_sigmaCurry","declaration":"theorem ProbabilityTheory.measurable_sigmaCurry {ι : Type u_1} {κ : ι → Type u_3} {α : (i : ι) → κ i → Type u_4} [m : (i : ι) → (j : κ i) → MeasurableSpace (α i j)] : Measurable Sigma.curry"}
+{"name":"ProbabilityTheory.IndepFun.ae_eq'","declaration":"/-- in mathlib as of `4d385393cd569f08ac30425ef886a57bb10daaa5` (TODO: bump) -/\ntheorem ProbabilityTheory.IndepFun.ae_eq' {β : Type u_11} {β' : Type u_12} {Ω : Type u_13} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {mβ : MeasurableSpace β} {mβ' : MeasurableSpace β'} {f : Ω → β} {f' : Ω → β} {g : Ω → β'} {g' : Ω → β'} (hfg : ProbabilityTheory.IndepFun f g μ) (hf : f =ᶠ[MeasureTheory.Measure.ae μ] f') (hg : g =ᶠ[MeasureTheory.Measure.ae μ] g') : ProbabilityTheory.IndepFun f' g' μ"}
+{"name":"ProbabilityTheory.iIndepFun.prod","declaration":"theorem ProbabilityTheory.iIndepFun.prod {Ω : Type u_2} [MeasurableSpace Ω] {μ : MeasureTheory.Measure Ω} [MeasureTheory.IsProbabilityMeasure μ] {ι : Type u_8} {ι' : Type u_9} {α : ι → Type u_10} {n : (i : ι) → MeasurableSpace (α i)} {f : (i : ι) → Ω → α i} {hf : ∀ (i : ι), Measurable (f i)} {ST : ι' → Finset ι} (hS : Pairwise (Disjoint on ST)) (h : ProbabilityTheory.iIndepFun n f μ) : let β := fun k => (i : { x // x ∈ ST k }) → α ↑i;\nProbabilityTheory.iIndepFun (fun k => MeasurableSpace.pi) (fun k x i => f (↑i) x) μ"}
+{"name":"ProbabilityTheory.indepFun_iff_map_prod_eq_prod_map_map'","declaration":"theorem ProbabilityTheory.indepFun_iff_map_prod_eq_prod_map_map' {Ω : Type u_1} {β : Type u_6} {β' : Type u_7} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {mβ : MeasurableSpace β} {mβ' : MeasurableSpace β'} {f : Ω → β} {g : Ω → β'} [MeasureTheory.IsFiniteMeasure μ] (hf : AEMeasurable f μ) (hg : AEMeasurable g μ) : ProbabilityTheory.IndepFun f g μ ↔\n  MeasureTheory.Measure.map (fun ω => (f ω, g ω)) μ =\n    MeasureTheory.Measure.prod (MeasureTheory.Measure.map f μ) (MeasureTheory.Measure.map g μ)"}
+{"name":"ProbabilityTheory.iIndepFun.reindex","declaration":"theorem ProbabilityTheory.iIndepFun.reindex {Ω : Type u_10} {ι : Type u_11} {ι' : Type u_12} [MeasurableSpace Ω] {α : ι → Type u_13} [n : (i : ι) → MeasurableSpace (α i)] {f : (i : ι) → Ω → α i} {μ : MeasureTheory.Measure Ω} (g : ι' ≃ ι) (h : ProbabilityTheory.iIndepFun ((fun {x} => n) ∘' ⇑g) ((fun {x} => f) ∘' ⇑g) μ) : ProbabilityTheory.iIndepFun n f μ"}
+{"name":"ProbabilityTheory.iIndepFun.comp","declaration":"theorem ProbabilityTheory.iIndepFun.comp {Ω : Type u_10} {ι : Type u_11} [MeasurableSpace Ω] {α : ι → Type u_13} {β : ι → Type u_14} [n : (i : ι) → MeasurableSpace (α i)] [m : (i : ι) → MeasurableSpace (β i)] {f : (i : ι) → Ω → α i} {μ : MeasureTheory.Measure Ω} (h : ProbabilityTheory.iIndepFun n f μ) (g : (i : ι) → α i → β i) (hg : ∀ (i : ι), Measurable (g i)) : ProbabilityTheory.iIndepFun m (fun i => g i ∘ f i) μ"}
+{"name":"ProbabilityTheory.indepFun_fst_snd","declaration":"theorem ProbabilityTheory.indepFun_fst_snd {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {Ω' : Type u_10} [MeasurableSpace Ω'] {μ' : MeasureTheory.Measure Ω'} [MeasureTheory.IsProbabilityMeasure μ] [MeasureTheory.IsProbabilityMeasure μ'] : ProbabilityTheory.IndepFun Prod.fst Prod.snd (MeasureTheory.Measure.prod μ μ')"}
+{"name":"ProbabilityTheory.iIndepFun_iff_pi_map_eq_map","declaration":"theorem ProbabilityTheory.iIndepFun_iff_pi_map_eq_map {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {ι : Type u_11} {β : ι → Type u_12} [Fintype ι] (f : (x : ι) → Ω → β x) [m : (x : ι) → MeasurableSpace (β x)] [MeasureTheory.IsProbabilityMeasure μ] (hf : ∀ (x : ι), Measurable (f x)) : ProbabilityTheory.iIndepFun m f μ ↔\n  (MeasureTheory.Measure.pi fun i => MeasureTheory.Measure.map (f i) μ) = MeasureTheory.Measure.map (fun ω i => f i ω) μ"}
+{"name":"ProbabilityTheory.iIndepFun.inv","declaration":"theorem ProbabilityTheory.iIndepFun.inv {Ω : Type u_10} {ι : Type u_11} [MeasurableSpace Ω] {α : ι → Type u_13} [n : (i : ι) → MeasurableSpace (α i)] {f : (i : ι) → Ω → α i} {μ : MeasureTheory.Measure Ω} (i : ι) [Inv (α i)] [MeasurableInv (α i)] [DecidableEq ι] (h : ProbabilityTheory.iIndepFun n f μ) : ProbabilityTheory.iIndepFun n (Function.update f i (f i)⁻¹) μ"}

PFR-declarations/PFR.Mathlib.Probability.Independence.Conditional.jsonl

@@ -0,0 +1,9 @@
+{"name":"ProbabilityTheory.CondIndepFun.comp_right","declaration":"/-- Composing independent functions with a measurable embedding of conull range gives independent\nfunctions. -/\ntheorem ProbabilityTheory.CondIndepFun.comp_right {Ω : Type u_1} {Ω' : Type u_2} {α : Type u_3} {β : Type u_4} {γ : Type u_5} [MeasurableSpace Ω] [MeasurableSpace Ω'] [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] {μ : MeasureTheory.Measure Ω} {f : Ω → α} {g : Ω → β} {h : Ω → γ} [MeasurableSingletonClass γ] {i : Ω' → Ω} (hi : MeasurableEmbedding i) (hi' : ∀ᵐ (a : Ω) ∂μ, a ∈ Set.range i) (hf : Measurable f) (hg : Measurable g) (hh : Measurable h) (hfg : ProbabilityTheory.CondIndepFun f g h μ) : ProbabilityTheory.CondIndepFun (f ∘ i) (g ∘ i) (h ∘ i) (MeasureTheory.Measure.comap i μ)"}
+{"name":"ProbabilityTheory.IndepFun.identDistrib_cond","declaration":"/-- If `A` is independent from `B`, then conditioning on an event given by `B` does not change\nthe distribution of `A`. -/\ntheorem ProbabilityTheory.IndepFun.identDistrib_cond {Ω : Type u_1} {α : Type u_2} {β : Type u_3} : ∀ {x : MeasurableSpace Ω} {x_1 : MeasurableSpace α} {x_2 : MeasurableSpace β} {μ : MeasureTheory.Measure Ω} {A : Ω → α}\n  {B : Ω → β} [inst : MeasureTheory.IsProbabilityMeasure μ],\n  ProbabilityTheory.IndepFun A B μ →\n    ∀ {s : Set β},\n      MeasurableSet s →\n        Measurable A →\n          Measurable B → ↑↑μ (B ⁻¹' s) ≠ 0 → ProbabilityTheory.IdentDistrib A A μ (ProbabilityTheory.cond μ (B ⁻¹' s))"}
+{"name":"ProbabilityTheory.condIndep_copies","declaration":"/-- For $X, Y$ random variables, there exist conditionally independent trials $X_1, X_2, Y'$. -/\ntheorem ProbabilityTheory.condIndep_copies {Ω : Type u_1} {α : Type u} {β : Type u} [MeasurableSpace Ω] [MeasurableSpace α] [MeasurableSpace β] [MeasurableSingletonClass β] [Fintype β] (X : Ω → α) (Y : Ω → β) (hX : Measurable X) (hY : Measurable Y) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] : ∃ Ω' mΩ' X₁ X₂ Y' ν,\n  MeasureTheory.IsProbabilityMeasure ν ∧\n    Measurable X₁ ∧\n      Measurable X₂ ∧\n        Measurable Y' ∧\n          ProbabilityTheory.CondIndepFun X₁ X₂ Y' ν ∧\n            ProbabilityTheory.IdentDistrib (⟨X₁, Y'⟩) (⟨X, Y⟩) ν μ ∧\n              ProbabilityTheory.IdentDistrib (⟨X₂, Y'⟩) (⟨X, Y⟩) ν μ"}
+{"name":"ProbabilityTheory.IndepFun.cond_right","declaration":"/-- If `A` is independent of `B`, then they remain independent when conditioning on an event\nof the form `B ∈ t` of positive probability. -/\ntheorem ProbabilityTheory.IndepFun.cond_right {Ω : Type u_1} {α : Type u_2} {β : Type u_3} : ∀ {x : MeasurableSpace Ω} {x_1 : MeasurableSpace α} {x_2 : MeasurableSpace β} {μ : MeasureTheory.Measure Ω} {A : Ω → α}\n  {B : Ω → β},\n  ProbabilityTheory.IndepFun A B μ →\n    ∀ {t : Set β}, MeasurableSet t → Measurable B → ProbabilityTheory.IndepFun A B (ProbabilityTheory.cond μ (B ⁻¹' t))"}
+{"name":"ProbabilityTheory.IndepFun.cond","declaration":"/-- If `A` is independent of `B`, then they remain independent when conditioning on an event\nof the form `A ∈ s ∩ B ∈ t` of positive probability. -/\ntheorem ProbabilityTheory.IndepFun.cond {Ω : Type u_1} {α : Type u_2} {β : Type u_3} : ∀ {x : MeasurableSpace Ω} {x_1 : MeasurableSpace α} {x_2 : MeasurableSpace β} {μ : MeasureTheory.Measure Ω} {A : Ω → α}\n  {B : Ω → β},\n  ProbabilityTheory.IndepFun A B μ →\n    ∀ {s : Set α} {t : Set β},\n      MeasurableSet s →\n        MeasurableSet t →\n          Measurable A → Measurable B → ProbabilityTheory.IndepFun A B (ProbabilityTheory.cond μ (A ⁻¹' s ∩ B ⁻¹' t))"}
+{"name":"ProbabilityTheory.condIndep_copies'","declaration":"/-- For $X, Y$ random variables, there exist conditionally independent trials $X_1, X_2, Y'$. -/\ntheorem ProbabilityTheory.condIndep_copies' {Ω : Type u_1} {α : Type u} {β : Type u} [MeasurableSpace Ω] [MeasurableSpace α] [MeasurableSpace β] [MeasurableSingletonClass β] [Fintype β] (X : Ω → α) (Y : Ω → β) (hX : Measurable X) (hY : Measurable Y) (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsProbabilityMeasure μ] (p : α → β → Prop) (hp : Measurable (Function.uncurry p)) (hp' : ∀ᵐ (ω : Ω) ∂μ, p (X ω) (Y ω)) : ∃ Ω' mΩ' X₁ X₂ Y' ν,\n  MeasureTheory.IsProbabilityMeasure ν ∧\n    Measurable X₁ ∧\n      Measurable X₂ ∧\n        Measurable Y' ∧\n          ProbabilityTheory.CondIndepFun X₁ X₂ Y' ν ∧\n            ProbabilityTheory.IdentDistrib (⟨X₁, Y'⟩) (⟨X, Y⟩) ν μ ∧\n              ProbabilityTheory.IdentDistrib (⟨X₂, Y'⟩) (⟨X, Y⟩) ν μ ∧\n                (∀ (ω : Ω'), p (X₁ ω) (Y' ω)) ∧ ∀ (ω : Ω'), p (X₂ ω) (Y' ω)"}
+{"name":"ProbabilityTheory.condIndepFun_iff","declaration":"theorem ProbabilityTheory.condIndepFun_iff {Ω : Type u_1} {α : Type u_3} {β : Type u_4} {γ : Type u_5} [MeasurableSpace ��] [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] {μ : MeasureTheory.Measure Ω} {f : Ω → α} {g : Ω → β} {h : Ω → γ} : ProbabilityTheory.CondIndepFun f g h μ ↔\n  ∀ᵐ (z : γ) ∂MeasureTheory.Measure.map h μ, ProbabilityTheory.IndepFun f g (ProbabilityTheory.cond μ (h ⁻¹' {z}))"}
+{"name":"ProbabilityTheory.IndepFun.cond_left","declaration":"/-- If `A` is independent of `B`, then they remain independent when conditioning on an event\nof the form `A ∈ s` of positive probability. -/\ntheorem ProbabilityTheory.IndepFun.cond_left {Ω : Type u_1} {α : Type u_2} {β : Type u_3} : ∀ {x : MeasurableSpace Ω} {x_1 : MeasurableSpace α} {x_2 : MeasurableSpace β} {μ : MeasureTheory.Measure Ω} {A : Ω → α}\n  {B : Ω → β},\n  ProbabilityTheory.IndepFun A B μ →\n    ∀ {s : Set α}, MeasurableSet s → Measurable A → ProbabilityTheory.IndepFun A B (ProbabilityTheory.cond μ (A ⁻¹' s))"}
+{"name":"ProbabilityTheory.CondIndepFun","declaration":"/-- The assertion that `f` and `g` are conditionally independent relative to `h`. -/\ndef ProbabilityTheory.CondIndepFun {Ω : Type u_1} {α : Type u_3} {β : Type u_4} {γ : Type u_5} [MeasurableSpace Ω] [MeasurableSpace α] [MeasurableSpace β] [MeasurableSpace γ] (f : Ω → α) (g : Ω → β) (h : Ω → γ) (μ : autoParam (MeasureTheory.Measure Ω) _auto✝) : Prop"}

PFR-declarations/PFR.Mathlib.Probability.Independence.FourVariables.jsonl

@@ -0,0 +1,19 @@
+{"name":"ProbabilityTheory.iIndepFun.reindex_four_abdc","declaration":"theorem ProbabilityTheory.iIndepFun.reindex_four_abdc {Ω : Type u_1} [MeasureTheory.MeasureSpace Ω] {G : Type u_2} [hG : MeasurableSpace G] {Z₁ : Ω → G} {Z₂ : Ω → G} {Z₃ : Ω → G} {Z₄ : Ω → G} (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![Z₁, Z₂, Z₃, Z₄] MeasureTheory.volume) : ProbabilityTheory.iIndepFun (fun x => hG) ![Z₁, Z₂, Z₄, Z₃] MeasureTheory.volume"}
+{"name":"ProbabilityTheory.iIndepFun.reindex_four_badc","declaration":"theorem ProbabilityTheory.iIndepFun.reindex_four_badc {Ω : Type u_1} [MeasureTheory.MeasureSpace Ω] {G : Type u_2} [hG : MeasurableSpace G] {Z₁ : Ω → G} {Z₂ : Ω → G} {Z₃ : Ω → G} {Z₄ : Ω → G} (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![Z₁, Z₂, Z₃, Z₄] MeasureTheory.volume) : ProbabilityTheory.iIndepFun (fun x => hG) ![Z₂, Z₁, Z₄, Z₃] MeasureTheory.volume"}
+{"name":"ProbabilityTheory.iIndepFun.reindex_four_bdca","declaration":"theorem ProbabilityTheory.iIndepFun.reindex_four_bdca {Ω : Type u_1} [MeasureTheory.MeasureSpace Ω] {G : Type u_2} [hG : MeasurableSpace G] {Z₁ : Ω → G} {Z₂ : Ω → G} {Z₃ : Ω → G} {Z₄ : Ω → G} (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![Z₁, Z₂, Z₃, Z₄] MeasureTheory.volume) : ProbabilityTheory.iIndepFun (fun x => hG) ![Z₂, Z₄, Z₃, Z₁] MeasureTheory.volume"}
+{"name":"ProbabilityTheory.iIndepFun.reindex_four_adcb","declaration":"theorem ProbabilityTheory.iIndepFun.reindex_four_adcb {Ω : Type u_1} [MeasureTheory.MeasureSpace Ω] {G : Type u_2} [hG : MeasurableSpace G] {Z₁ : Ω → G} {Z₂ : Ω → G} {Z₃ : Ω → G} {Z₄ : Ω → G} (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![Z₁, Z₂, Z₃, Z₄] MeasureTheory.volume) : ProbabilityTheory.iIndepFun (fun x => hG) ![Z₁, Z₄, Z₃, Z₂] MeasureTheory.volume"}
+{"name":"ProbabilityTheory.iIndepFun.reindex_four_dabc","declaration":"theorem ProbabilityTheory.iIndepFun.reindex_four_dabc {Ω : Type u_1} [MeasureTheory.MeasureSpace Ω] {G : Type u_2} [hG : MeasurableSpace G] {Z₁ : Ω → G} {Z₂ : Ω → G} {Z₃ : Ω → G} {Z₄ : Ω → G} (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![Z₁, Z₂, Z₃, Z₄] MeasureTheory.volume) : ProbabilityTheory.iIndepFun (fun x => hG) ![Z₄, Z₁, Z₂, Z₃] MeasureTheory.volume"}
+{"name":"ProbabilityTheory.iIndepFun.reindex_four_bcad","declaration":"theorem ProbabilityTheory.iIndepFun.reindex_four_bcad {Ω : Type u_1} [MeasureTheory.MeasureSpace Ω] {G : Type u_2} [hG : MeasurableSpace G] {Z₁ : Ω → G} {Z₂ : Ω → G} {Z₃ : Ω → G} {Z₄ : Ω → G} (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![Z₁, Z₂, Z₃, Z₄] MeasureTheory.volume) : ProbabilityTheory.iIndepFun (fun x => hG) ![Z₂, Z₃, Z₁, Z₄] MeasureTheory.volume"}
+{"name":"ProbabilityTheory.iIndepFun.reindex_four_bcda","declaration":"theorem ProbabilityTheory.iIndepFun.reindex_four_bcda {Ω : Type u_1} [MeasureTheory.MeasureSpace Ω] {G : Type u_2} [hG : MeasurableSpace G] {Z₁ : Ω → G} {Z₂ : Ω → G} {Z₃ : Ω → G} {Z₄ : Ω → G} (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![Z₁, Z₂, Z₃, Z₄] MeasureTheory.volume) : ProbabilityTheory.iIndepFun (fun x => hG) ![Z₂, Z₃, Z₄, Z₁] MeasureTheory.volume"}
+{"name":"ProbabilityTheory.iIndepFun.reindex_four_dbca","declaration":"theorem ProbabilityTheory.iIndepFun.reindex_four_dbca {Ω : Type u_1} [MeasureTheory.MeasureSpace Ω] {G : Type u_2} [hG : MeasurableSpace G] {Z₁ : Ω → G} {Z₂ : Ω → G} {Z₃ : Ω → G} {Z₄ : Ω → G} (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![Z₁, Z₂, Z₃, Z₄] MeasureTheory.volume) : ProbabilityTheory.iIndepFun (fun x => hG) ![Z₄, Z₂, Z₃, Z₁] MeasureTheory.volume"}
+{"name":"ProbabilityTheory.iIndepFun.κ","declaration":"def ProbabilityTheory.iIndepFun.κ  : Fin 3 → Type"}
+{"name":"ProbabilityTheory.iIndepFun.reindex_four_acdb","declaration":"theorem ProbabilityTheory.iIndepFun.reindex_four_acdb {Ω : Type u_1} [MeasureTheory.MeasureSpace Ω] {G : Type u_2} [hG : MeasurableSpace G] {Z₁ : Ω → G} {Z₂ : Ω → G} {Z₃ : Ω → G} {Z₄ : Ω → G} (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![Z₁, Z₂, Z₃, Z₄] MeasureTheory.volume) : ProbabilityTheory.iIndepFun (fun x => hG) ![Z₁, Z₃, Z₄, Z₂] MeasureTheory.volume"}
+{"name":"ProbabilityTheory.iIndepFun.reindex_four_bacd","declaration":"theorem ProbabilityTheory.iIndepFun.reindex_four_bacd {Ω : Type u_1} [MeasureTheory.MeasureSpace Ω] {G : Type u_2} [hG : MeasurableSpace G] {Z₁ : Ω → G} {Z₂ : Ω → G} {Z₃ : Ω → G} {Z₄ : Ω → G} (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![Z₁, Z₂, Z₃, Z₄] MeasureTheory.volume) : ProbabilityTheory.iIndepFun (fun x => hG) ![Z₂, Z₁, Z₃, Z₄] MeasureTheory.volume"}
+{"name":"ProbabilityTheory.iIndepFun.reindex_four_acbd","declaration":"theorem ProbabilityTheory.iIndepFun.reindex_four_acbd {Ω : Type u_1} [MeasureTheory.MeasureSpace Ω] {G : Type u_2} [hG : MeasurableSpace G] {Z₁ : Ω → G} {Z₂ : Ω → G} {Z₃ : Ω → G} {Z₄ : Ω → G} (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![Z₁, Z₂, Z₃, Z₄] MeasureTheory.volume) : ProbabilityTheory.iIndepFun (fun x => hG) ![Z₁, Z₃, Z₂, Z₄] MeasureTheory.volume"}
+{"name":"ProbabilityTheory.iIndepFun.reindex_four_cadb","declaration":"theorem ProbabilityTheory.iIndepFun.reindex_four_cadb {Ω : Type u_1} [MeasureTheory.MeasureSpace Ω] {G : Type u_2} [hG : MeasurableSpace G] {Z₁ : Ω → G} {Z₂ : Ω → G} {Z₃ : Ω → G} {Z₄ : Ω → G} (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![Z₁, Z₂, Z₃, Z₄] MeasureTheory.volume) : ProbabilityTheory.iIndepFun (fun x => hG) ![Z₃, Z₁, Z₄, Z₂] MeasureTheory.volume"}
+{"name":"ProbabilityTheory.iIndepFun.reindex_four_cbad","declaration":"theorem ProbabilityTheory.iIndepFun.reindex_four_cbad {Ω : Type u_1} [MeasureTheory.MeasureSpace Ω] {G : Type u_2} [hG : MeasurableSpace G] {Z₁ : Ω → G} {Z₂ : Ω → G} {Z₃ : Ω → G} {Z₄ : Ω → G} (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![Z₁, Z₂, Z₃, Z₄] MeasureTheory.volume) : ProbabilityTheory.iIndepFun (fun x => hG) ![Z₃, Z₂, Z₁, Z₄] MeasureTheory.volume"}
+{"name":"ProbabilityTheory.iIndepFun.reindex_four_abcd","declaration":"theorem ProbabilityTheory.iIndepFun.reindex_four_abcd {Ω : Type u_1} [MeasureTheory.MeasureSpace Ω] {G : Type u_2} [hG : MeasurableSpace G] {Z₁ : Ω → G} {Z₂ : Ω → G} {Z₃ : Ω → G} {Z₄ : Ω → G} (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![Z₁, Z₂, Z₃, Z₄] MeasureTheory.volume) : ProbabilityTheory.iIndepFun (fun x => hG) ![Z₁, Z₂, Z₃, Z₄] MeasureTheory.volume"}
+{"name":"ProbabilityTheory.iIndepFun.reindex_four_bdac","declaration":"theorem ProbabilityTheory.iIndepFun.reindex_four_bdac {Ω : Type u_1} [MeasureTheory.MeasureSpace Ω] {G : Type u_2} [hG : MeasurableSpace G] {Z₁ : Ω → G} {Z₂ : Ω → G} {Z₃ : Ω → G} {Z₄ : Ω → G} (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![Z₁, Z₂, Z₃, Z₄] MeasureTheory.volume) : ProbabilityTheory.iIndepFun (fun x => hG) ![Z₂, Z₄, Z₁, Z₃] MeasureTheory.volume"}
+{"name":"ProbabilityTheory.iIndepFun.reindex_four_adbc","declaration":"theorem ProbabilityTheory.iIndepFun.reindex_four_adbc {Ω : Type u_1} [MeasureTheory.MeasureSpace Ω] {G : Type u_2} [hG : MeasurableSpace G] {Z₁ : Ω → G} {Z₂ : Ω → G} {Z₃ : Ω → G} {Z₄ : Ω → G} (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![Z₁, Z₂, Z₃, Z₄] MeasureTheory.volume) : ProbabilityTheory.iIndepFun (fun x => hG) ![Z₁, Z₄, Z₂, Z₃] MeasureTheory.volume"}
+{"name":"ProbabilityTheory.iIndepFun.apply_two_last","declaration":"/-- If `(Z₁, Z₂, Z₃, Z₄)` are independent, so are `(Z₁, Z₂, φ Z₃ Z₄)` for any measurable `φ`. -/\ntheorem ProbabilityTheory.iIndepFun.apply_two_last {Ω : Type u_1} [MeasureTheory.MeasureSpace Ω] [MeasureTheory.IsProbabilityMeasure MeasureTheory.volume] {G : Type u_2} [hG : MeasurableSpace G] {Z₁ : Ω → G} {Z₂ : Ω → G} {Z₃ : Ω → G} {Z₄ : Ω → G} (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![Z₁, Z₂, Z₃, Z₄] MeasureTheory.volume) (hZ₁ : Measurable Z₁) (hZ₂ : Measurable Z₂) (hZ₃ : Measurable Z₃) (hZ₄ : Measurable Z₄) {phi : G → G → G} (hphi : Measurable (Function.uncurry phi)) : ProbabilityTheory.iIndepFun (fun x => hG) ![Z₁, Z₂, fun ω => phi (Z₃ ω) (Z₄ ω)] MeasureTheory.volume"}
+{"name":"ProbabilityTheory.iIndepFun.reindex_four_dbac","declaration":"theorem ProbabilityTheory.iIndepFun.reindex_four_dbac {Ω : Type u_1} [MeasureTheory.MeasureSpace Ω] {G : Type u_2} [hG : MeasurableSpace G] {Z₁ : Ω → G} {Z₂ : Ω → G} {Z₃ : Ω → G} {Z₄ : Ω → G} (h_indep : ProbabilityTheory.iIndepFun (fun _i => hG) ![Z₁, Z₂, Z₃, Z₄] MeasureTheory.volume) : ProbabilityTheory.iIndepFun (fun x => hG) ![Z₄, Z₂, Z₁, Z₃] MeasureTheory.volume"}

PFR-declarations/PFR.Mathlib.Probability.Independence.Kernel.jsonl

@@ -0,0 +1 @@
+{"name":"ProbabilityTheory.kernel.IndepFun.ae_eq'","declaration":"/-- in mathlib as of `4d385393cd569f08ac30425ef886a57bb10daaa5` (TODO: bump) -/\ntheorem ProbabilityTheory.kernel.IndepFun.ae_eq' {α : Type u_6} {Ω : Type u_5} {β : Type u_1} {β' : Type u_2} {_mα : MeasurableSpace α} {_mΩ : MeasurableSpace Ω} {κ : ↥(ProbabilityTheory.kernel α Ω)} {μ : MeasureTheory.Measure α} {mβ : MeasurableSpace β} {mβ' : MeasurableSpace β'} {f : Ω → β} {f' : Ω → β} {g : Ω → β'} {g' : Ω → β'} (hfg : ProbabilityTheory.kernel.IndepFun f g κ μ) (hf : ∀ᵐ (a : α) ∂μ, f =ᶠ[MeasureTheory.Measure.ae (κ a)] f') (hg : ∀ᵐ (a : α) ∂μ, g =ᶠ[MeasureTheory.Measure.ae (κ a)] g') : ProbabilityTheory.kernel.IndepFun f' g' κ μ"}