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tasksource 14

[ { "state_after": "no goals", "state_before": "p : Prop\ninst✝ : Decidable p\n⊢ decide p = true ↔ p", "tactic": "simp" } ]

[ 522, 87 ]

[ 522, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝ : DivisionCommMonoid α\na b : α\nn : ℕ\n⊢ (a / b) ^ n = a ^ n / b ^ n", "tactic": "simp only [div_eq_mul_inv, mul_pow, inv_pow]" } ]

[ 393, 47 ]

[ 392, 1 ]

[]

[ 721, 8 ]

[ 715, 1 ]

[ { "state_after": "ι : Type ?u.46031\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.46040\nπ : ι → Type ?u.46045\nr : α → α → Prop\ninst✝ : Preorder α\nf : ℤ → α\nhf : Antitone f\nn a : ℤ\nh1 : f (n + 1) < f a\nh2 : f a < f n\n⊢ False", "state_before": "ι : Type ?u.46031\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.46040\nπ : ι → Type ?u.46045\nr : α → α → Prop\ninst✝ : Preorder α\nf : ℤ → α\nhf : Antitone f\nn : ℤ\nx : α\nh1 : f (n + 1) < x\nh2 : x < f n\na : ℤ\n⊢ f a ≠ x", "tactic": "rintro rfl" }, { "state_after": "no goals", "state_before": "ι : Type ?u.46031\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.46040\nπ : ι → Type ?u.46045\nr : α → α → Prop\ninst✝ : Preorder α\nf : ℤ → α\nhf : Antitone f\nn a : ℤ\nh1 : f (n + 1) < f a\nh2 : f a < f n\n⊢ False", "tactic": "exact (hf.reflect_lt h2).not_le (Int.le_of_lt_add_one <| hf.reflect_lt h1)" } ]

[ 1128, 77 ]

[ 1125, 1 ]

[]

[ 420, 48 ]

[ 418, 1 ]

[ { "state_after": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝² : PseudoEMetricSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : PseudoEMetricSpace γ\nf : α × β → γ\nK : ℝ≥0\nha : ∀ (a : α), ContinuousOn (fun y => f (a, y)) univ\nhb : ∀ (b : β), LipschitzOnWith K (fun x => f (x, b)) univ\n⊢ ContinuousOn f (univ ×ˢ univ)", "state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝² : PseudoEMetricSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : PseudoEMetricSpace γ\nf : α × β → γ\nK : ℝ≥0\nha : ∀ (a : α), Continuous fun y => f (a, y)\nhb : ∀ (b : β), LipschitzWith K fun x => f (x, b)\n⊢ Continuous f", "tactic": "simp only [continuous_iff_continuousOn_univ, ← univ_prod_univ, ← lipschitz_on_univ] at *" }, { "state_after": "no goals", "state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝² : PseudoEMetricSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : PseudoEMetricSpace γ\nf : α × β → γ\nK : ℝ≥0\nha : ∀ (a : α), ContinuousOn (fun y => f (a, y)) univ\nhb : ∀ (b : β), LipschitzOnWith K (fun x => f (x, b)) univ\n⊢ ContinuousOn f (univ ×ˢ univ)", "tactic": "exact continuousOn_prod_of_continuousOn_lipschitz_on f K (fun a _ => ha a) fun b _ => hb b" } ]

[ 618, 93 ]

[ 614, 1 ]

[ { "state_after": "case left\nn : Type ?u.214920\np : Type ?u.214923\nR : Type u₂\n𝕜 : Type u_1\ninst✝³ : Field 𝕜\ninst✝² : DecidableEq n\ninst✝¹ : DecidableEq p\ninst✝ : CommRing R\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\nhM : M (inr ()) (inr ()) ≠ 0\n⊢ toBlocks₁₂ (List.prod (listTransvecCol M) ⬝ M ⬝ List.prod (listTransvecRow M)) = 0\n\ncase right\nn : Type ?u.214920\np : Type ?u.214923\nR : Type u₂\n𝕜 : Type u_1\ninst✝³ : Field 𝕜\ninst✝² : DecidableEq n\ninst✝¹ : DecidableEq p\ninst✝ : CommRing R\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\nhM : M (inr ()) (inr ()) ≠ 0\n⊢ toBlocks₂₁ (List.prod (listTransvecCol M) ⬝ M ⬝ List.prod (listTransvecRow M)) = 0", "state_before": "n : Type ?u.214920\np : Type ?u.214923\nR : Type u₂\n𝕜 : Type u_1\ninst✝³ : Field 𝕜\ninst✝² : DecidableEq n\ninst✝¹ : DecidableEq p\ninst✝ : CommRing R\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\nhM : M (inr ()) (inr ()) ≠ 0\n⊢ IsTwoBlockDiagonal (List.prod (listTransvecCol M) ⬝ M ⬝ List.prod (listTransvecRow M))", "tactic": "constructor" }, { "state_after": "case left.a.h\nn : Type ?u.214920\np : Type ?u.214923\nR : Type u₂\n𝕜 : Type u_1\ninst✝³ : Field 𝕜\ninst✝² : DecidableEq n\ninst✝¹ : DecidableEq p\ninst✝ : CommRing R\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\nhM : M (inr ()) (inr ()) ≠ 0\ni : Fin r\nj : Unit\n⊢ toBlocks₁₂ (List.prod (listTransvecCol M) ⬝ M ⬝ List.prod (listTransvecRow M)) i j = OfNat.ofNat 0 i j", "state_before": "case left\nn : Type ?u.214920\np : Type ?u.214923\nR : Type u₂\n𝕜 : Type u_1\ninst✝³ : Field 𝕜\ninst✝² : DecidableEq n\ninst✝¹ : DecidableEq p\ninst✝ : CommRing R\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\nhM : M (inr ()) (inr ()) ≠ 0\n⊢ toBlocks₁₂ (List.prod (listTransvecCol M) ⬝ M ⬝ List.prod (listTransvecRow M)) = 0", "tactic": "ext (i j)" }, { "state_after": "case left.a.h\nn : Type ?u.214920\np : Type ?u.214923\nR : Type u₂\n𝕜 : Type u_1\ninst✝³ : Field 𝕜\ninst✝² : DecidableEq n\ninst✝¹ : DecidableEq p\ninst✝ : CommRing R\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\nhM : M (inr ()) (inr ()) ≠ 0\ni : Fin r\nj : Unit\nthis : j = ()\n⊢ toBlocks₁₂ (List.prod (listTransvecCol M) ⬝ M ⬝ List.prod (listTransvecRow M)) i j = OfNat.ofNat 0 i j", "state_before": "case left.a.h\nn : Type ?u.214920\np : Type ?u.214923\nR : Type u₂\n𝕜 : Type u_1\ninst✝³ : Field 𝕜\ninst✝² : DecidableEq n\ninst✝¹ : DecidableEq p\ninst✝ : CommRing R\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\nhM : M (inr ()) (inr ()) ≠ 0\ni : Fin r\nj : Unit\n⊢ toBlocks₁₂ (List.prod (listTransvecCol M) ⬝ M ⬝ List.prod (listTransvecRow M)) i j = OfNat.ofNat 0 i j", "tactic": "have : j = unit := by simp only [eq_iff_true_of_subsingleton]" }, { "state_after": "no goals", "state_before": "case left.a.h\nn : Type ?u.214920\np : Type ?u.214923\nR : Type u₂\n𝕜 : Type u_1\ninst✝³ : Field 𝕜\ninst✝² : DecidableEq n\ninst✝¹ : DecidableEq p\ninst✝ : CommRing R\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\nhM : M (inr ()) (inr ()) ≠ 0\ni : Fin r\nj : Unit\nthis : j = ()\n⊢ toBlocks₁₂ (List.prod (listTransvecCol M) ⬝ M ⬝ List.prod (listTransvecRow M)) i j = OfNat.ofNat 0 i j", "tactic": "simp [toBlocks₁₂, this, listTransvecCol_mul_mul_listTransvecRow_last_row M hM]" }, { "state_after": "no goals", "state_before": "n : Type ?u.214920\np : Type ?u.214923\nR : Type u₂\n𝕜 : Type u_1\ninst✝³ : Field 𝕜\ninst✝² : DecidableEq n\ninst✝¹ : DecidableEq p\ninst✝ : CommRing R\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\nhM : M (inr ()) (inr ()) ≠ 0\ni : Fin r\nj : Unit\n⊢ j = ()", "tactic": "simp only [eq_iff_true_of_subsingleton]" }, { "state_after": "case right.a.h\nn : Type ?u.214920\np : Type ?u.214923\nR : Type u₂\n𝕜 : Type u_1\ninst✝³ : Field 𝕜\ninst✝² : DecidableEq n\ninst✝¹ : DecidableEq p\ninst✝ : CommRing R\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\nhM : M (inr ()) (inr ()) ≠ 0\ni : Unit\nj : Fin r\n⊢ toBlocks₂₁ (List.prod (listTransvecCol M) ⬝ M ⬝ List.prod (listTransvecRow M)) i j = OfNat.ofNat 0 i j", "state_before": "case right\nn : Type ?u.214920\np : Type ?u.214923\nR : Type u₂\n𝕜 : Type u_1\ninst✝³ : Field 𝕜\ninst✝² : DecidableEq n\ninst✝¹ : DecidableEq p\ninst✝ : CommRing R\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\nhM : M (inr ()) (inr ()) ≠ 0\n⊢ toBlocks₂₁ (List.prod (listTransvecCol M) ⬝ M ⬝ List.prod (listTransvecRow M)) = 0", "tactic": "ext (i j)" }, { "state_after": "case right.a.h\nn : Type ?u.214920\np : Type ?u.214923\nR : Type u₂\n𝕜 : Type u_1\ninst✝³ : Field 𝕜\ninst✝² : DecidableEq n\ninst✝¹ : DecidableEq p\ninst✝ : CommRing R\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\nhM : M (inr ()) (inr ()) ≠ 0\ni : Unit\nj : Fin r\nthis : i = ()\n⊢ toBlocks₂₁ (List.prod (listTransvecCol M) ⬝ M ⬝ List.prod (listTransvecRow M)) i j = OfNat.ofNat 0 i j", "state_before": "case right.a.h\nn : Type ?u.214920\np : Type ?u.214923\nR : Type u₂\n𝕜 : Type u_1\ninst✝³ : Field 𝕜\ninst✝² : DecidableEq n\ninst✝¹ : DecidableEq p\ninst✝ : CommRing R\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\nhM : M (inr ()) (inr ()) ≠ 0\ni : Unit\nj : Fin r\n⊢ toBlocks₂₁ (List.prod (listTransvecCol M) ⬝ M ⬝ List.prod (listTransvecRow M)) i j = OfNat.ofNat 0 i j", "tactic": "have : i = unit := by simp only [eq_iff_true_of_subsingleton]" }, { "state_after": "no goals", "state_before": "case right.a.h\nn : Type ?u.214920\np : Type ?u.214923\nR : Type u₂\n𝕜 : Type u_1\ninst✝³ : Field 𝕜\ninst✝² : DecidableEq n\ninst✝¹ : DecidableEq p\ninst✝ : CommRing R\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\nhM : M (inr ()) (inr ()) ≠ 0\ni : Unit\nj : Fin r\nthis : i = ()\n⊢ toBlocks₂₁ (List.prod (listTransvecCol M) ⬝ M ⬝ List.prod (listTransvecRow M)) i j = OfNat.ofNat 0 i j", "tactic": "simp [toBlocks₂₁, this, listTransvecCol_mul_mul_listTransvecRow_last_col M hM]" }, { "state_after": "no goals", "state_before": "n : Type ?u.214920\np : Type ?u.214923\nR : Type u₂\n𝕜 : Type u_1\ninst✝³ : Field 𝕜\ninst✝² : DecidableEq n\ninst✝¹ : DecidableEq p\ninst✝ : CommRing R\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\nhM : M (inr ()) (inr ()) ≠ 0\ni : Unit\nj : Fin r\n⊢ i = ()", "tactic": "simp only [eq_iff_true_of_subsingleton]" } ]

[ 533, 83 ]

[ 524, 1 ]

[ { "state_after": "no goals", "state_before": "F : Type ?u.268989\nα : Type u_1\nβ : Type ?u.268995\nγ : Type ?u.268998\nδ : Type ?u.269001\nε : Type ?u.269004\ninst✝ : Monoid α\nf g : Filter α\ns : Set α\na : α\nm n✝ n : ℕ\nx✝ : n + 2 ≠ 0\n⊢ ⊤ ^ (n + 2) = ⊤", "tactic": "rw [pow_succ, top_pow n.succ_ne_zero, top_mul_top]" } ]

[ 718, 76 ]

[ 715, 1 ]

[ { "state_after": "case h\na : ZFSet\np : ZFSet → Prop\nx✝ : ZFSet\n⊢ x✝ ∈ toSet (ZFSet.sep p a) ↔ x✝ ∈ {x | x ∈ toSet a ∧ p x}", "state_before": "a : ZFSet\np : ZFSet → Prop\n⊢ toSet (ZFSet.sep p a) = {x | x ∈ toSet a ∧ p x}", "tactic": "ext" }, { "state_after": "no goals", "state_before": "case h\na : ZFSet\np : ZFSet → Prop\nx✝ : ZFSet\n⊢ x✝ ∈ toSet (ZFSet.sep p a) ↔ x✝ ∈ {x | x ∈ toSet a ∧ p x}", "tactic": "simp" } ]

[ 992, 7 ]

[ 989, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u\nα' : Type w\nβ : Type v\nβ' : Type x\nγ : Type u_1\nδ : Type ?u.13935\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq (α ⊕ β)\nf : α → γ\ng : β → γ\ni : α\nx : γ\n⊢ x = Sum.elim (update f i x) g (inl i)", "tactic": "simp" }, { "state_after": "no goals", "state_before": "α : Type u\nα' : Type w\nβ : Type v\nβ' : Type x\nγ : Type u_1\nδ : Type ?u.13935\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq (α ⊕ β)\nf : α → γ\ng : β → γ\ni : α\nx : γ\n⊢ ∀ (x_1 : α ⊕ β), x_1 ≠ inl i → Sum.elim f g x_1 = Sum.elim (update f i x) g x_1", "tactic": "simp (config := { contextual := true })" } ]

[ 279, 72 ]

[ 277, 1 ]

[ { "state_after": "no goals", "state_before": "R : Type u_1\nV : Type u_2\nV' : Type ?u.215069\nP : Type u_3\nP' : Type ?u.215075\ninst✝⁷ : OrderedRing R\ninst✝⁶ : AddCommGroup V\ninst✝⁵ : Module R V\ninst✝⁴ : AddTorsor V P\ninst✝³ : AddCommGroup V'\ninst✝² : Module R V'\ninst✝¹ : AddTorsor V' P'\ninst✝ : NoZeroSMulDivisors R V\nx y z : P\nh : Wbtw R x y z\n⊢ Wbtw R y x z ↔ x = y", "tactic": "rw [← wbtw_swap_left_iff R z, and_iff_right h]" } ]

[ 392, 78 ]

[ 391, 1 ]

[ { "state_after": "α : Type u_1\nE : Type ?u.81134\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t✝ : Set α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\np : ℝ≥0∞\nf✝ f : α → ℝ\nt : Set α\nhf : IntegrableOn f t\nhf_zero : ∀ (s : Set α), MeasurableSet s → ↑↑μ (s ∩ t) < ⊤ → 0 ≤ ∫ (x : α) in s ∩ t, f x ∂μ\ns : Set α\nhs : MeasurableSet s\nh's : ↑↑(Measure.restrict μ t) s < ⊤\n⊢ 0 ≤ ∫ (x : α) in s, f x ∂Measure.restrict μ t", "state_before": "α : Type u_1\nE : Type ?u.81134\nm m0 : MeasurableSpace α\nμ : Measure α\ns t✝ : Set α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\np : ℝ≥0∞\nf✝ f : α → ℝ\nt : Set α\nhf : IntegrableOn f t\nhf_zero : ∀ (s : Set α), MeasurableSet s → ↑↑μ (s ∩ t) < ⊤ → 0 ≤ ∫ (x : α) in s ∩ t, f x ∂μ\n⊢ 0 ≤ᵐ[Measure.restrict μ t] f", "tactic": "refine' ae_nonneg_of_forall_set_integral_nonneg hf fun s hs h's => _" }, { "state_after": "α : Type u_1\nE : Type ?u.81134\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t✝ : Set α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\np : ℝ≥0∞\nf✝ f : α → ℝ\nt : Set α\nhf : IntegrableOn f t\nhf_zero : ∀ (s : Set α), MeasurableSet s → ↑↑μ (s ∩ t) < ⊤ → 0 ≤ ∫ (x : α) in s ∩ t, f x ∂μ\ns : Set α\nhs : MeasurableSet s\nh's : ↑↑(Measure.restrict μ t) s < ⊤\n⊢ 0 ≤ ∫ (x : α) in s ∩ t, f x ∂μ", "state_before": "α : Type u_1\nE : Type ?u.81134\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t✝ : Set α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\np : ℝ≥0∞\nf✝ f : α → ℝ\nt : Set α\nhf : IntegrableOn f t\nhf_zero : ∀ (s : Set α), MeasurableSet s → ↑↑μ (s ∩ t) < ⊤ → 0 ≤ ∫ (x : α) in s ∩ t, f x ∂μ\ns : Set α\nhs : MeasurableSet s\nh's : ↑↑(Measure.restrict μ t) s < ⊤\n⊢ 0 ≤ ∫ (x : α) in s, f x ∂Measure.restrict μ t", "tactic": "simp_rw [Measure.restrict_restrict hs]" }, { "state_after": "α : Type u_1\nE : Type ?u.81134\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t✝ : Set α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\np : ℝ≥0∞\nf✝ f : α → ℝ\nt : Set α\nhf : IntegrableOn f t\nhf_zero : ∀ (s : Set α), MeasurableSet s → ↑↑μ (s ∩ t) < ⊤ → 0 ≤ ∫ (x : α) in s ∩ t, f x ∂μ\ns : Set α\nhs : MeasurableSet s\nh's : ↑↑(Measure.restrict μ t) s < ⊤\n⊢ ↑↑μ (s ∩ t) < ⊤", "state_before": "α : Type u_1\nE : Type ?u.81134\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t✝ : Set α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\np : ℝ≥0∞\nf✝ f : α → ℝ\nt : Set α\nhf : IntegrableOn f t\nhf_zero : ∀ (s : Set α), MeasurableSet s → ↑↑μ (s ∩ t) < ⊤ → 0 ≤ ∫ (x : α) in s ∩ t, f x ∂μ\ns : Set α\nhs : MeasurableSet s\nh's : ↑↑(Measure.restrict μ t) s < ⊤\n⊢ 0 ≤ ∫ (x : α) in s ∩ t, f x ∂μ", "tactic": "apply hf_zero s hs" }, { "state_after": "no goals", "state_before": "α : Type u_1\nE : Type ?u.81134\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t✝ : Set α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\np : ℝ≥0∞\nf✝ f : α → ℝ\nt : Set α\nhf : IntegrableOn f t\nhf_zero : ∀ (s : Set α), MeasurableSet s → ↑↑μ (s ∩ t) < ⊤ → 0 ≤ ∫ (x : α) in s ∩ t, f x ∂μ\ns : Set α\nhs : MeasurableSet s\nh's : ↑↑(Measure.restrict μ t) s < ⊤\n⊢ ↑↑μ (s ∩ t) < ⊤", "tactic": "rwa [Measure.restrict_apply hs] at h's" } ]

[ 313, 41 ]

[ 306, 1 ]

[ { "state_after": "case intro.intro\nd : ℕ\na : ℤ√↑d\nx y : ℕ\nh : a ≤ { re := ↑x, im := ↑y }\n⊢ ∃ n, a ≤ ↑n", "state_before": "d : ℕ\na : ℤ√↑d\n⊢ ∃ n, a ≤ ↑n", "tactic": "obtain ⟨x, y, (h : a ≤ ⟨x, y⟩)⟩ : ∃ x y : ℕ, Nonneg (⟨x, y⟩ + -a) :=\n match -a with\n | ⟨Int.ofNat x, Int.ofNat y⟩ => ⟨0, 0, by trivial⟩\n | ⟨Int.ofNat x, -[y+1]⟩ => ⟨0, y + 1, by simp [add_def, Int.negSucc_coe, add_assoc]; trivial⟩\n | ⟨-[x+1], Int.ofNat y⟩ => ⟨x + 1, 0, by simp [Int.negSucc_coe, add_assoc]; trivial⟩\n | ⟨-[x+1], -[y+1]⟩ => ⟨x + 1, y + 1, by simp [Int.negSucc_coe, add_assoc]; trivial⟩" }, { "state_after": "case intro.intro\nd : ℕ\na : ℤ√↑d\nx y : ℕ\nh : a ≤ { re := ↑x, im := ↑y }\n⊢ { re := ↑x, im := ↑y } ≤ ↑(x + d * y)", "state_before": "case intro.intro\nd : ℕ\na : ℤ√↑d\nx y : ℕ\nh : a ≤ { re := ↑x, im := ↑y }\n⊢ ∃ n, a ≤ ↑n", "tactic": "refine' ⟨x + d * y, h.trans _⟩" }, { "state_after": "case intro.intro\nd : ℕ\na : ℤ√↑d\nx y : ℕ\nh : a ≤ { re := ↑x, im := ↑y }\n⊢ Nonneg { re := ↑x + ↑d * ↑y - ↑x, im := 0 - ↑y }", "state_before": "case intro.intro\nd : ℕ\na : ℤ√↑d\nx y : ℕ\nh : a ≤ { re := ↑x, im := ↑y }\n⊢ { re := ↑x, im := ↑y } ≤ ↑(x + d * y)", "tactic": "change Nonneg ⟨↑x + d * y - ↑x, 0 - ↑y⟩" }, { "state_after": "case intro.intro.zero\nd : ℕ\na : ℤ√↑d\nx : ℕ\nh : a ≤ { re := ↑x, im := ↑Nat.zero }\n⊢ Nonneg { re := ↑x + ↑d * ↑Nat.zero - ↑x, im := 0 - ↑Nat.zero }\n\ncase intro.intro.succ\nd : ℕ\na : ℤ√↑d\nx y : ℕ\nh : a ≤ { re := ↑x, im := ↑(Nat.succ y) }\n⊢ Nonneg { re := ↑x + ↑d * ↑(Nat.succ y) - ↑x, im := 0 - ↑(Nat.succ y) }", "state_before": "case intro.intro\nd : ℕ\na : ℤ√↑d\nx y : ℕ\nh : a ≤ { re := ↑x, im := ↑y }\n⊢ Nonneg { re := ↑x + ↑d * ↑y - ↑x, im := 0 - ↑y }", "tactic": "cases' y with y" }, { "state_after": "case intro.intro.succ\nd : ℕ\na : ℤ√↑d\nx y : ℕ\nh✝ : a ≤ { re := ↑x, im := ↑(Nat.succ y) }\nh : ∀ (y : ℕ), SqLe y d (d * y) 1\n⊢ Nonneg { re := ↑x + ↑d * ↑(Nat.succ y) - ↑x, im := 0 - ↑(Nat.succ y) }", "state_before": "case intro.intro.succ\nd : ℕ\na : ℤ√↑d\nx y : ℕ\nh : a ≤ { re := ↑x, im := ↑(Nat.succ y) }\n⊢ Nonneg { re := ↑x + ↑d * ↑(Nat.succ y) - ↑x, im := 0 - ↑(Nat.succ y) }", "tactic": "have h : ∀ y, SqLe y d (d * y) 1 := fun y => by\n simpa [SqLe, mul_comm, mul_left_comm] using Nat.mul_le_mul_right (y * y) (Nat.le_mul_self d)" }, { "state_after": "case intro.intro.succ\nd : ℕ\na : ℤ√↑d\nx y : ℕ\nh✝ : a ≤ { re := ↑x, im := ↑(Nat.succ y) }\nh : ∀ (y : ℕ), SqLe y d (d * y) 1\n⊢ Nonneg { re := ↑d * ↑(Nat.succ y), im := 0 - ↑(Nat.succ y) }", "state_before": "case intro.intro.succ\nd : ℕ\na : ℤ√↑d\nx y : ℕ\nh✝ : a ≤ { re := ↑x, im := ↑(Nat.succ y) }\nh : ∀ (y : ℕ), SqLe y d (d * y) 1\n⊢ Nonneg { re := ↑x + ↑d * ↑(Nat.succ y) - ↑x, im := 0 - ↑(Nat.succ y) }", "tactic": "rw [show (x : ℤ) + d * Nat.succ y - x = d * Nat.succ y by simp]" }, { "state_after": "no goals", "state_before": "case intro.intro.succ\nd : ℕ\na : ℤ√↑d\nx y : ℕ\nh✝ : a ≤ { re := ↑x, im := ↑(Nat.succ y) }\nh : ∀ (y : ℕ), SqLe y d (d * y) 1\n⊢ Nonneg { re := ↑d * ↑(Nat.succ y), im := 0 - ↑(Nat.succ y) }", "tactic": "exact h (y + 1)" }, { "state_after": "no goals", "state_before": "d : ℕ\na : ℤ√↑d\nx y : ℕ\n⊢ Nonneg ({ re := ↑0, im := ↑0 } + { re := ofNat x, im := ofNat y })", "tactic": "trivial" }, { "state_after": "d : ℕ\na : ℤ√↑d\nx y : ℕ\n⊢ Nonneg { re := ↑x, im := 0 }", "state_before": "d : ℕ\na : ℤ√↑d\nx y : ℕ\n⊢ Nonneg ({ re := ↑0, im := ↑(y + 1) } + { re := ofNat x, im := -[y+1] })", "tactic": "simp [add_def, Int.negSucc_coe, add_assoc]" }, { "state_after": "no goals", "state_before": "d : ℕ\na : ℤ√↑d\nx y : ℕ\n⊢ Nonneg { re := ↑x, im := 0 }", "tactic": "trivial" }, { "state_after": "d : ℕ\na : ℤ√↑d\nx y : ℕ\n⊢ Nonneg { re := 0, im := ↑y }", "state_before": "d : ℕ\na : ℤ√↑d\nx y : ℕ\n⊢ Nonneg ({ re := ↑(x + 1), im := ↑0 } + { re := -[x+1], im := ofNat y })", "tactic": "simp [Int.negSucc_coe, add_assoc]" }, { "state_after": "no goals", "state_before": "d : ℕ\na : ℤ√↑d\nx y : ℕ\n⊢ Nonneg { re := 0, im := ↑y }", "tactic": "trivial" }, { "state_after": "d : ℕ\na : ℤ√↑d\nx y : ℕ\n⊢ Nonneg { re := 0, im := 0 }", "state_before": "d : ℕ\na : ℤ√↑d\nx y : ℕ\n⊢ Nonneg ({ re := ↑(x + 1), im := ↑(y + 1) } + { re := -[x+1], im := -[y+1] })", "tactic": "simp [Int.negSucc_coe, add_assoc]" }, { "state_after": "no goals", "state_before": "d : ℕ\na : ℤ√↑d\nx y : ℕ\n⊢ Nonneg { re := 0, im := 0 }", "tactic": "trivial" }, { "state_after": "case intro.intro.zero\nd : ℕ\na : ℤ√↑d\nx : ℕ\nh : a ≤ { re := ↑x, im := ↑Nat.zero }\n⊢ Nonneg { re := 0, im := 0 }", "state_before": "case intro.intro.zero\nd : ℕ\na : ℤ√↑d\nx : ℕ\nh : a ≤ { re := ↑x, im := ↑Nat.zero }\n⊢ Nonneg { re := ↑x + ↑d * ↑Nat.zero - ↑x, im := 0 - ↑Nat.zero }", "tactic": "simp" }, { "state_after": "no goals", "state_before": "case intro.intro.zero\nd : ℕ\na : ℤ√↑d\nx : ℕ\nh : a ≤ { re := ↑x, im := ↑Nat.zero }\n⊢ Nonneg { re := 0, im := 0 }", "tactic": "trivial" }, { "state_after": "no goals", "state_before": "d : ℕ\na : ℤ√↑d\nx y✝ : ℕ\nh : a ≤ { re := ↑x, im := ↑(Nat.succ y✝) }\ny : ℕ\n⊢ SqLe y d (d * y) 1", "tactic": "simpa [SqLe, mul_comm, mul_left_comm] using Nat.mul_le_mul_right (y * y) (Nat.le_mul_self d)" }, { "state_after": "no goals", "state_before": "d : ℕ\na : ℤ√↑d\nx y : ℕ\nh✝ : a ≤ { re := ↑x, im := ↑(Nat.succ y) }\nh : ∀ (y : ℕ), SqLe y d (d * y) 1\n⊢ ↑x + ↑d * ↑(Nat.succ y) - ↑x = ↑d * ↑(Nat.succ y)", "tactic": "simp" } ]

[ 782, 18 ]

[ 767, 1 ]

[]

[ 385, 6 ]

[ 384, 1 ]

[]

[ 140, 6 ]

[ 138, 1 ]

[]

[ 4083, 6 ]

[ 4081, 1 ]

[ { "state_after": "case intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.301280\nP : Type u_3\nP' : Type ?u.301286\ninst✝⁶ : LinearOrderedField R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y z : P\nhyz : WOppSide s y z\nhy : ¬y ∈ s\np₁ : P\nhp₁ : p₁ ∈ s\np₂ : P\nhp₂ : p₂ ∈ s\nhxy : SameRay R (x -ᵥ p₁) (p₂ -ᵥ y)\n⊢ WSameSide s x z", "state_before": "R : Type u_1\nV : Type u_2\nV' : Type ?u.301280\nP : Type u_3\nP' : Type ?u.301286\ninst✝⁶ : LinearOrderedField R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y z : P\nhxy : WOppSide s x y\nhyz : WOppSide s y z\nhy : ¬y ∈ s\n⊢ WSameSide s x z", "tactic": "rcases hxy with ⟨p₁, hp₁, p₂, hp₂, hxy⟩" }, { "state_after": "case intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.301280\nP : Type u_3\nP' : Type ?u.301286\ninst✝⁶ : LinearOrderedField R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y z : P\nhy : ¬y ∈ s\np₁ : P\nhp₁ : p₁ ∈ s\np₂ : P\nhyz : ∃ p₂_1, p₂_1 ∈ s ∧ SameRay R (y -ᵥ p₂) (p₂_1 -ᵥ z)\nhp₂ : p₂ ∈ s\nhxy : SameRay R (x -ᵥ p₁) (p₂ -ᵥ y)\n⊢ WSameSide s x z", "state_before": "case intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.301280\nP : Type u_3\nP' : Type ?u.301286\ninst✝⁶ : LinearOrderedField R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y z : P\nhyz : WOppSide s y z\nhy : ¬y ∈ s\np₁ : P\nhp₁ : p₁ ∈ s\np₂ : P\nhp₂ : p₂ ∈ s\nhxy : SameRay R (x -ᵥ p₁) (p₂ -ᵥ y)\n⊢ WSameSide s x z", "tactic": "rw [wOppSide_iff_exists_left hp₂, or_iff_right hy] at hyz" }, { "state_after": "case intro.intro.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.301280\nP : Type u_3\nP' : Type ?u.301286\ninst✝⁶ : LinearOrderedField R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y z : P\nhy : ¬y ∈ s\np₁ : P\nhp₁ : p₁ ∈ s\np₂ : P\nhp₂ : p₂ ∈ s\nhxy : SameRay R (x -ᵥ p₁) (p₂ -ᵥ y)\np₃ : P\nhp₃ : p₃ ∈ s\nhyz : SameRay R (y -ᵥ p₂) (p₃ -ᵥ z)\n⊢ WSameSide s x z", "state_before": "case intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.301280\nP : Type u_3\nP' : Type ?u.301286\ninst✝⁶ : LinearOrderedField R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y z : P\nhy : ¬y ∈ s\np₁ : P\nhp₁ : p₁ ∈ s\np₂ : P\nhyz : ∃ p₂_1, p₂_1 ∈ s ∧ SameRay R (y -ᵥ p₂) (p₂_1 -ᵥ z)\nhp₂ : p₂ ∈ s\nhxy : SameRay R (x -ᵥ p₁) (p₂ -ᵥ y)\n⊢ WSameSide s x z", "tactic": "rcases hyz with ⟨p₃, hp₃, hyz⟩" }, { "state_after": "case intro.intro.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.301280\nP : Type u_3\nP' : Type ?u.301286\ninst✝⁶ : LinearOrderedField R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y z : P\nhy : ¬y ∈ s\np₁ : P\nhp₁ : p₁ ∈ s\np₂ : P\nhp₂ : p₂ ∈ s\nhxy : SameRay R (x -ᵥ p₁) (p₂ -ᵥ y)\np₃ : P\nhp₃ : p₃ ∈ s\nhyz : SameRay R (p₂ -ᵥ y) (z -ᵥ p₃)\n⊢ WSameSide s x z", "state_before": "case intro.intro.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.301280\nP : Type u_3\nP' : Type ?u.301286\ninst✝⁶ : LinearOrderedField R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y z : P\nhy : ¬y ∈ s\np₁ : P\nhp₁ : p₁ ∈ s\np₂ : P\nhp₂ : p₂ ∈ s\nhxy : SameRay R (x -ᵥ p₁) (p₂ -ᵥ y)\np₃ : P\nhp₃ : p₃ ∈ s\nhyz : SameRay R (y -ᵥ p₂) (p₃ -ᵥ z)\n⊢ WSameSide s x z", "tactic": "rw [← sameRay_neg_iff, neg_vsub_eq_vsub_rev, neg_vsub_eq_vsub_rev] at hyz" }, { "state_after": "case intro.intro.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.301280\nP : Type u_3\nP' : Type ?u.301286\ninst✝⁶ : LinearOrderedField R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y z : P\nhy : ¬y ∈ s\np₁ : P\nhp₁ : p₁ ∈ s\np₂ : P\nhp₂ : p₂ ∈ s\nhxy : SameRay R (x -ᵥ p₁) (p₂ -ᵥ y)\np₃ : P\nhp₃ : p₃ ∈ s\nhyz : SameRay R (p₂ -ᵥ y) (z -ᵥ p₃)\n⊢ p₂ -ᵥ y = 0 → x -ᵥ p₁ = 0 ∨ z -ᵥ p₃ = 0", "state_before": "case intro.intro.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.301280\nP : Type u_3\nP' : Type ?u.301286\ninst✝⁶ : LinearOrderedField R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y z : P\nhy : ¬y ∈ s\np₁ : P\nhp₁ : p₁ ∈ s\np₂ : P\nhp₂ : p₂ ∈ s\nhxy : SameRay R (x -ᵥ p₁) (p₂ -ᵥ y)\np₃ : P\nhp₃ : p₃ ∈ s\nhyz : SameRay R (p₂ -ᵥ y) (z -ᵥ p₃)\n⊢ WSameSide s x z", "tactic": "refine' ⟨p₁, hp₁, p₃, hp₃, hxy.trans hyz _⟩" }, { "state_after": "case intro.intro.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.301280\nP : Type u_3\nP' : Type ?u.301286\ninst✝⁶ : LinearOrderedField R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y z : P\nhy : ¬y ∈ s\np₁ : P\nhp₁ : p₁ ∈ s\np₂ : P\nhp₂ : p₂ ∈ s\nhxy : SameRay R (x -ᵥ p₁) (p₂ -ᵥ y)\np₃ : P\nhp₃ : p₃ ∈ s\nhyz : SameRay R (p₂ -ᵥ y) (z -ᵥ p₃)\nh : p₂ -ᵥ y = 0\n⊢ False", "state_before": "case intro.intro.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.301280\nP : Type u_3\nP' : Type ?u.301286\ninst✝⁶ : LinearOrderedField R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y z : P\nhy : ¬y ∈ s\np₁ : P\nhp₁ : p₁ ∈ s\np₂ : P\nhp₂ : p₂ ∈ s\nhxy : SameRay R (x -ᵥ p₁) (p₂ -ᵥ y)\np₃ : P\nhp₃ : p₃ ∈ s\nhyz : SameRay R (p₂ -ᵥ y) (z -ᵥ p₃)\n⊢ p₂ -ᵥ y = 0 → x -ᵥ p₁ = 0 ∨ z -ᵥ p₃ = 0", "tactic": "refine' fun h => False.elim _" }, { "state_after": "case intro.intro.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.301280\nP : Type u_3\nP' : Type ?u.301286\ninst✝⁶ : LinearOrderedField R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y z : P\nhy : ¬y ∈ s\np₁ : P\nhp₁ : p₁ ∈ s\np₂ : P\nhp₂ : p₂ ∈ s\nhxy : SameRay R (x -ᵥ p₁) (p₂ -ᵥ y)\np₃ : P\nhp₃ : p₃ ∈ s\nhyz : SameRay R (p₂ -ᵥ y) (z -ᵥ p₃)\nh : p₂ = y\n⊢ False", "state_before": "case intro.intro.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.301280\nP : Type u_3\nP' : Type ?u.301286\ninst✝⁶ : LinearOrderedField R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y z : P\nhy : ¬y ∈ s\np₁ : P\nhp₁ : p₁ ∈ s\np₂ : P\nhp₂ : p₂ ∈ s\nhxy : SameRay R (x -ᵥ p₁) (p₂ -ᵥ y)\np₃ : P\nhp₃ : p₃ ∈ s\nhyz : SameRay R (p₂ -ᵥ y) (z -ᵥ p₃)\nh : p₂ -ᵥ y = 0\n⊢ False", "tactic": "rw [vsub_eq_zero_iff_eq] at h" }, { "state_after": "no goals", "state_before": "case intro.intro.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.301280\nP : Type u_3\nP' : Type ?u.301286\ninst✝⁶ : LinearOrderedField R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y z : P\nhy : ¬y ∈ s\np₁ : P\nhp₁ : p₁ ∈ s\np₂ : P\nhp₂ : p₂ ∈ s\nhxy : SameRay R (x -ᵥ p₁) (p₂ -ᵥ y)\np₃ : P\nhp₃ : p₃ ∈ s\nhyz : SameRay R (p₂ -ᵥ y) (z -ᵥ p₃)\nh : p₂ = y\n⊢ False", "tactic": "exact hy (h ▸ hp₂)" } ]

[ 596, 21 ]

[ 587, 1 ]

[ { "state_after": "K : Type u_1\ninst✝¹ : LinearOrderedField K\ninst✝ : FloorRing K\nv : K\nn✝ : ℕ\na : ℤ\nn : ℕ\n⊢ Stream'.Seq.get? (of ↑a).s n = none", "state_before": "K : Type u_1\ninst✝¹ : LinearOrderedField K\ninst✝ : FloorRing K\nv : K\nn : ℕ\na : ℤ\n⊢ ∀ (n : ℕ), Stream'.Seq.get? (of ↑a).s n = none", "tactic": "intro n" }, { "state_after": "case zero\nK : Type u_1\ninst✝¹ : LinearOrderedField K\ninst✝ : FloorRing K\nv : K\nn : ℕ\na : ℤ\n⊢ Stream'.Seq.get? (of ↑a).s Nat.zero = none\n\ncase succ\nK : Type u_1\ninst✝¹ : LinearOrderedField K\ninst✝ : FloorRing K\nv : K\nn✝ : ℕ\na : ℤ\nn : ℕ\nih : Stream'.Seq.get? (of ↑a).s n = none\n⊢ Stream'.Seq.get? (of ↑a).s (Nat.succ n) = none", "state_before": "K : Type u_1\ninst✝¹ : LinearOrderedField K\ninst✝ : FloorRing K\nv : K\nn✝ : ℕ\na : ℤ\nn : ℕ\n⊢ Stream'.Seq.get? (of ↑a).s n = none", "tactic": "induction' n with n ih" }, { "state_after": "no goals", "state_before": "case zero\nK : Type u_1\ninst✝¹ : LinearOrderedField K\ninst✝ : FloorRing K\nv : K\nn : ℕ\na : ℤ\n⊢ Stream'.Seq.get? (of ↑a).s Nat.zero = none", "tactic": "rw [of_s_head_aux, stream_succ_of_int, Option.bind]" }, { "state_after": "no goals", "state_before": "case succ\nK : Type u_1\ninst✝¹ : LinearOrderedField K\ninst✝ : FloorRing K\nv : K\nn✝ : ℕ\na : ℤ\nn : ℕ\nih : Stream'.Seq.get? (of ↑a).s n = none\n⊢ Stream'.Seq.get? (of ↑a).s (Nat.succ n) = none", "tactic": "exact (of (a : K)).s.prop ih" } ]

[ 293, 69 ]

[ 287, 1 ]

[]

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[ 322, 1 ]

[]

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[]

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[]

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[ { "state_after": "no goals", "state_before": "B : Type u\ninst✝ : Bicategory B\na b c d e : B\nf g : a ⟶ b\nη θ : f ⟶ g\n⊢ 𝟙 a ◁ η = 𝟙 a ◁ θ ↔ η = θ", "tactic": "simp" } ]

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[ 417, 1 ]

[ { "state_after": "no goals", "state_before": "⊢ (!false) = true", "tactic": "decide" } ]

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[ { "state_after": "no goals", "state_before": "𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type w\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL L₁ L₂ : Filter 𝕜\nhs : ¬x ∈ s\n⊢ HasDerivWithinAt f f' s x ↔ Tendsto (slope f x) (𝓝[s] x) (𝓝 f')", "tactic": "rw [hasDerivWithinAt_iff_tendsto_slope, diff_singleton_eq_self hs]" } ]

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[]

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[ { "state_after": "no goals", "state_before": "ι : Type ?u.130781\nV✝ : Type u\ninst✝²⁰ : Category V✝\ninst✝¹⁹ : HasZeroMorphisms V✝\nA B C : V✝\nf : A ⟶ B\ninst✝¹⁸ : HasImage f\ng : B ⟶ C\ninst✝¹⁷ : HasKernel g\nw : f ≫ g = 0\nA' B' C' : V✝\nf' : A' ⟶ B'\ninst✝¹⁶ : HasImage f'\ng' : B' ⟶ C'\ninst✝¹⁵ : HasKernel g'\nw' : f' ≫ g' = 0\nα : Arrow.mk f ⟶ Arrow.mk f'\ninst✝¹⁴ : HasImageMap α\nβ : Arrow.mk g ⟶ Arrow.mk g'\nA₁✝ B₁✝ C₁✝ : V✝\nf₁✝ : A₁✝ ⟶ B₁✝\ninst✝¹³ : HasImage f₁✝\ng₁✝ : B₁✝ ⟶ C₁✝\ninst✝¹² : HasKernel g₁✝\nw₁ : f₁✝ ≫ g₁✝ = 0\nA₂✝ B₂✝ C₂✝ : V✝\nf₂✝ : A₂✝ ⟶ B₂✝\ninst✝¹¹ : HasImage f₂✝\ng₂✝ : B₂✝ ⟶ C₂✝\ninst✝¹⁰ : HasKernel g₂✝\nw₂ : f₂✝ ≫ g₂✝ = 0\nA₃✝ B₃✝ C₃✝ : V✝\nf₃✝ : A₃✝ ⟶ B₃✝\ninst✝⁹ : HasImage f₃✝\ng₃✝ : B₃✝ ⟶ C₃✝\ninst✝⁸ : HasKernel g₃✝\nw₃ : f₃✝ ≫ g₃✝ = 0\nα₁✝ : Arrow.mk f₁✝ ⟶ Arrow.mk f₂✝\ninst✝⁷ : HasImageMap α₁✝\nβ₁✝ : Arrow.mk g₁✝ ⟶ Arrow.mk g₂✝\nα₂✝ : Arrow.mk f₂✝ ⟶ Arrow.mk f₃✝\ninst✝⁶ : HasImageMap α₂✝\nβ₂✝ : Arrow.mk g₂✝ ⟶ Arrow.mk g₃✝\ninst✝⁵ : HasCokernel (imageToKernel f g w)\ninst✝⁴ : HasCokernel (imageToKernel f' g' w')\ninst✝³ : HasCokernel (imageToKernel f₁✝ g₁✝ w₁)\ninst✝² : HasCokernel (imageToKernel f₂✝ g₂✝ w₂)\ninst✝¹ : HasCokernel (imageToKernel f₃✝ g₃✝ w₃)\nV : Type u_1\ninst✝ : Category V\nA₁ B₁ C₁ A₂ B₂ C₂ A₃ B₃ C₃ : V\nf₁ : A₁ ⟶ B₁\ng₁ : B₁ ⟶ C₁\nf₂ : A₂ ⟶ B₂\ng₂ : B₂ ⟶ C₂\nf₃ : A₃ ⟶ B₃\ng₃ : B₃ ⟶ C₃\nα₁ : Arrow.mk f₁ ⟶ Arrow.mk f₂\nβ₁ : Arrow.mk g₁ ⟶ Arrow.mk g₂\nα₂ : Arrow.mk f₂ ⟶ Arrow.mk f₃\nβ₂ : Arrow.mk g₂ ⟶ Arrow.mk g₃\np₁ : α₁.right = β₁.left\np₂ : α₂.right = β₂.left\n⊢ (α₁ ≫ α₂).right = (β₁ ≫ β₂).left", "tactic": "simp only [Arrow.comp_left, Arrow.comp_right, p₁, p₂]" } ]

[ 312, 56 ]

[ 307, 1 ]

[ { "state_after": "no goals", "state_before": "ι : Sort ?u.19787\nα : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nx y : α\ns t : Set α\nΦ : α → β\nhΦ : Isometry Φ\n⊢ infEdist (Φ x) (Φ '' t) = infEdist x t", "tactic": "simp only [infEdist, iInf_image, hΦ.edist_eq]" } ]

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[ { "state_after": "no goals", "state_before": "R : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP : Type ?u.702381\ninst✝¹ : CommSemiring P\ninst✝ : IsLocalization M S\ns : { x // x ∈ M }\n⊢ mk' S 0 s = 0", "tactic": "rw [eq_comm, IsLocalization.eq_mk'_iff_mul_eq, zero_mul, map_zero]" } ]

[ 353, 69 ]

[ 352, 1 ]

[]

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[ { "state_after": "no goals", "state_before": "𝓕 : Type ?u.454837\n𝕜 : Type ?u.454840\nα : Type ?u.454843\nι : Type ?u.454846\nκ : Type ?u.454849\nE : Type u_1\nF : Type ?u.454855\nG : Type ?u.454858\ninst✝² : SeminormedGroup E\ninst✝¹ : SeminormedGroup F\ninst✝ : SeminormedGroup G\ns : Set E\na a₁ a₂ b b₁ b₂ : E\nr r₁ r₂ : ℝ\nx : E\n⊢ Tendsto (fun a => ‖a / x‖) (𝓝 x) (𝓝 0)", "tactic": "simpa [dist_eq_norm_div] using\n tendsto_id.dist (tendsto_const_nhds : Tendsto (fun _a => (x : E)) (𝓝 x) _)" } ]

[ 1075, 79 ]

[ 1073, 1 ]

[ { "state_after": "case mk\nV : Type u\ninst✝¹ : Quiver V\nW : Type u₂\ninst✝ : Quiver W\nG : Prefunctor V W\nF_obj : V → W\nmap✝ : {X Y : V} → (X ⟶ Y) → (F_obj X ⟶ F_obj Y)\nh_obj : ∀ (X : V), { obj := F_obj, map := map✝ }.obj X = G.obj X\nh_map :\n ∀ (X Y : V) (f : X ⟶ Y),\n { obj := F_obj, map := map✝ }.map f =\n Eq.recOn (_ : G.obj Y = { obj := F_obj, map := map✝ }.obj Y)\n (Eq.recOn (_ : G.obj X = { obj := F_obj, map := map✝ }.obj X) (G.map f))\n⊢ { obj := F_obj, map := map✝ } = G", "state_before": "V : Type u\ninst✝¹ : Quiver V\nW : Type u₂\ninst✝ : Quiver W\nF G : Prefunctor V W\nh_obj : ∀ (X : V), F.obj X = G.obj X\nh_map : ∀ (X Y : V) (f : X ⟶ Y), F.map f = Eq.recOn (_ : G.obj Y = F.obj Y) (Eq.recOn (_ : G.obj X = F.obj X) (G.map f))\n⊢ F = G", "tactic": "cases' F with F_obj _" }, { "state_after": "case mk.mk\nV : Type u\ninst✝¹ : Quiver V\nW : Type u₂\ninst✝ : Quiver W\nF_obj : V → W\nmap✝¹ : {X Y : V} → (X ⟶ Y) → (F_obj X ⟶ F_obj Y)\nG_obj : V → W\nmap✝ : {X Y : V} → (X ⟶ Y) → (G_obj X ⟶ G_obj Y)\nh_obj : ∀ (X : V), { obj := F_obj, map := map✝¹ }.obj X = { obj := G_obj, map := map✝ }.obj X\nh_map :\n ∀ (X Y : V) (f : X ⟶ Y),\n { obj := F_obj, map := map✝¹ }.map f =\n Eq.recOn (_ : { obj := G_obj, map := map✝ }.obj Y = { obj := F_obj, map := map✝¹ }.obj Y)\n (Eq.recOn (_ : { obj := G_obj, map := map✝ }.obj X = { obj := F_obj, map := map✝¹ }.obj X)\n ({ obj := G_obj, map := map✝ }.map f))\n⊢ { obj := F_obj, map := map✝¹ } = { obj := G_obj, map := map✝ }", "state_before": "case mk\nV : Type u\ninst✝¹ : Quiver V\nW : Type u₂\ninst✝ : Quiver W\nG : Prefunctor V W\nF_obj : V → W\nmap✝ : {X Y : V} → (X ⟶ Y) → (F_obj X ⟶ F_obj Y)\nh_obj : ∀ (X : V), { obj := F_obj, map := map✝ }.obj X = G.obj X\nh_map :\n ∀ (X Y : V) (f : X ⟶ Y),\n { obj := F_obj, map := map✝ }.map f =\n Eq.recOn (_ : G.obj Y = { obj := F_obj, map := map✝ }.obj Y)\n (Eq.recOn (_ : G.obj X = { obj := F_obj, map := map✝ }.obj X) (G.map f))\n⊢ { obj := F_obj, map := map✝ } = G", "tactic": "cases' G with G_obj _" }, { "state_after": "case mk.mk\nV : Type u\ninst✝¹ : Quiver V\nW : Type u₂\ninst✝ : Quiver W\nF_obj : V → W\nmap✝¹ map✝ : {X Y : V} → (X ⟶ Y) → (F_obj X ⟶ F_obj Y)\nh_obj : ∀ (X : V), { obj := F_obj, map := map✝¹ }.obj X = { obj := F_obj, map := map✝ }.obj X\nh_map :\n ∀ (X Y : V) (f : X ⟶ Y),\n { obj := F_obj, map := map✝¹ }.map f =\n Eq.recOn (_ : { obj := F_obj, map := map✝ }.obj Y = { obj := F_obj, map := map✝¹ }.obj Y)\n (Eq.recOn (_ : { obj := F_obj, map := map✝ }.obj X = { obj := F_obj, map := map✝¹ }.obj X)\n ({ obj := F_obj, map := map✝ }.map f))\n⊢ { obj := F_obj, map := map✝¹ } = { obj := F_obj, map := map✝ }", "state_before": "case mk.mk\nV : Type u\ninst✝¹ : Quiver V\nW : Type u₂\ninst✝ : Quiver W\nF_obj : V → W\nmap✝¹ : {X Y : V} → (X ⟶ Y) → (F_obj X ⟶ F_obj Y)\nG_obj : V → W\nmap✝ : {X Y : V} → (X ⟶ Y) → (G_obj X ⟶ G_obj Y)\nh_obj : ∀ (X : V), { obj := F_obj, map := map✝¹ }.obj X = { obj := G_obj, map := map✝ }.obj X\nh_map :\n ∀ (X Y : V) (f : X ⟶ Y),\n { obj := F_obj, map := map✝¹ }.map f =\n Eq.recOn (_ : { obj := G_obj, map := map✝ }.obj Y = { obj := F_obj, map := map✝¹ }.obj Y)\n (Eq.recOn (_ : { obj := G_obj, map := map✝ }.obj X = { obj := F_obj, map := map✝¹ }.obj X)\n ({ obj := G_obj, map := map✝ }.map f))\n⊢ { obj := F_obj, map := map✝¹ } = { obj := G_obj, map := map✝ }", "tactic": "obtain rfl : F_obj = G_obj := by\n ext X\n apply h_obj" }, { "state_after": "case mk.mk.e_map\nV : Type u\ninst✝¹ : Quiver V\nW : Type u₂\ninst✝ : Quiver W\nF_obj : V → W\nmap✝¹ map✝ : {X Y : V} → (X ⟶ Y) → (F_obj X ⟶ F_obj Y)\nh_obj : ∀ (X : V), { obj := F_obj, map := map✝¹ }.obj X = { obj := F_obj, map := map✝ }.obj X\nh_map :\n ∀ (X Y : V) (f : X ⟶ Y),\n { obj := F_obj, map := map✝¹ }.map f =\n Eq.recOn (_ : { obj := F_obj, map := map✝ }.obj Y = { obj := F_obj, map := map✝¹ }.obj Y)\n (Eq.recOn (_ : { obj := F_obj, map := map✝ }.obj X = { obj := F_obj, map := map✝¹ }.obj X)\n ({ obj := F_obj, map := map✝ }.map f))\n⊢ map✝¹ = map✝", "state_before": "case mk.mk\nV : Type u\ninst✝¹ : Quiver V\nW : Type u₂\ninst✝ : Quiver W\nF_obj : V → W\nmap✝¹ map✝ : {X Y : V} → (X ⟶ Y) → (F_obj X ⟶ F_obj Y)\nh_obj : ∀ (X : V), { obj := F_obj, map := map✝¹ }.obj X = { obj := F_obj, map := map✝ }.obj X\nh_map :\n ∀ (X Y : V) (f : X ⟶ Y),\n { obj := F_obj, map := map✝¹ }.map f =\n Eq.recOn (_ : { obj := F_obj, map := map✝ }.obj Y = { obj := F_obj, map := map✝¹ }.obj Y)\n (Eq.recOn (_ : { obj := F_obj, map := map✝ }.obj X = { obj := F_obj, map := map✝¹ }.obj X)\n ({ obj := F_obj, map := map✝ }.map f))\n⊢ { obj := F_obj, map := map✝¹ } = { obj := F_obj, map := map✝ }", "tactic": "congr" }, { "state_after": "case mk.mk.e_map.h.h.h\nV : Type u\ninst✝¹ : Quiver V\nW : Type u₂\ninst✝ : Quiver W\nF_obj : V → W\nmap✝¹ map✝ : {X Y : V} → (X ⟶ Y) → (F_obj X ⟶ F_obj Y)\nh_obj : ∀ (X : V), { obj := F_obj, map := map✝¹ }.obj X = { obj := F_obj, map := map✝ }.obj X\nh_map :\n ∀ (X Y : V) (f : X ⟶ Y),\n { obj := F_obj, map := map✝¹ }.map f =\n Eq.recOn (_ : { obj := F_obj, map := map✝ }.obj Y = { obj := F_obj, map := map✝¹ }.obj Y)\n (Eq.recOn (_ : { obj := F_obj, map := map✝ }.obj X = { obj := F_obj, map := map✝¹ }.obj X)\n ({ obj := F_obj, map := map✝ }.map f))\nX Y : V\nf : X ⟶ Y\n⊢ map✝¹ f = map✝ f", "state_before": "case mk.mk.e_map\nV : Type u\ninst✝¹ : Quiver V\nW : Type u₂\ninst✝ : Quiver W\nF_obj : V → W\nmap✝¹ map✝ : {X Y : V} → (X ⟶ Y) → (F_obj X ⟶ F_obj Y)\nh_obj : ∀ (X : V), { obj := F_obj, map := map✝¹ }.obj X = { obj := F_obj, map := map✝ }.obj X\nh_map :\n ∀ (X Y : V) (f : X ⟶ Y),\n { obj := F_obj, map := map✝¹ }.map f =\n Eq.recOn (_ : { obj := F_obj, map := map✝ }.obj Y = { obj := F_obj, map := map✝¹ }.obj Y)\n (Eq.recOn (_ : { obj := F_obj, map := map✝ }.obj X = { obj := F_obj, map := map✝¹ }.obj X)\n ({ obj := F_obj, map := map✝ }.map f))\n⊢ map✝¹ = map✝", "tactic": "funext X Y f" }, { "state_after": "no goals", "state_before": "case mk.mk.e_map.h.h.h\nV : Type u\ninst✝¹ : Quiver V\nW : Type u₂\ninst✝ : Quiver W\nF_obj : V → W\nmap✝¹ map✝ : {X Y : V} → (X ⟶ Y) → (F_obj X ⟶ F_obj Y)\nh_obj : ∀ (X : V), { obj := F_obj, map := map✝¹ }.obj X = { obj := F_obj, map := map✝ }.obj X\nh_map :\n ∀ (X Y : V) (f : X ⟶ Y),\n { obj := F_obj, map := map✝¹ }.map f =\n Eq.recOn (_ : { obj := F_obj, map := map✝ }.obj Y = { obj := F_obj, map := map✝¹ }.obj Y)\n (Eq.recOn (_ : { obj := F_obj, map := map✝ }.obj X = { obj := F_obj, map := map✝¹ }.obj X)\n ({ obj := F_obj, map := map✝ }.map f))\nX Y : V\nf : X ⟶ Y\n⊢ map✝¹ f = map✝ f", "tactic": "simpa using h_map X Y f" }, { "state_after": "case h\nV : Type u\ninst✝¹ : Quiver V\nW : Type u₂\ninst✝ : Quiver W\nF_obj : V → W\nmap✝¹ : {X Y : V} → (X ⟶ Y) → (F_obj X ⟶ F_obj Y)\nG_obj : V → W\nmap✝ : {X Y : V} → (X ⟶ Y) → (G_obj X ⟶ G_obj Y)\nh_obj : ∀ (X : V), { obj := F_obj, map := map✝¹ }.obj X = { obj := G_obj, map := map✝ }.obj X\nh_map :\n ∀ (X Y : V) (f : X ⟶ Y),\n { obj := F_obj, map := map✝¹ }.map f =\n Eq.recOn (_ : { obj := G_obj, map := map✝ }.obj Y = { obj := F_obj, map := map✝¹ }.obj Y)\n (Eq.recOn (_ : { obj := G_obj, map := map✝ }.obj X = { obj := F_obj, map := map✝¹ }.obj X)\n ({ obj := G_obj, map := map✝ }.map f))\nX : V\n⊢ F_obj X = G_obj X", "state_before": "V : Type u\ninst✝¹ : Quiver V\nW : Type u₂\ninst✝ : Quiver W\nF_obj : V → W\nmap✝¹ : {X Y : V} → (X ⟶ Y) → (F_obj X ⟶ F_obj Y)\nG_obj : V → W\nmap✝ : {X Y : V} → (X ⟶ Y) → (G_obj X ⟶ G_obj Y)\nh_obj : ∀ (X : V), { obj := F_obj, map := map✝¹ }.obj X = { obj := G_obj, map := map✝ }.obj X\nh_map :\n ∀ (X Y : V) (f : X ⟶ Y),\n { obj := F_obj, map := map✝¹ }.map f =\n Eq.recOn (_ : { obj := G_obj, map := map✝ }.obj Y = { obj := F_obj, map := map✝¹ }.obj Y)\n (Eq.recOn (_ : { obj := G_obj, map := map✝ }.obj X = { obj := F_obj, map := map✝¹ }.obj X)\n ({ obj := G_obj, map := map✝ }.map f))\n⊢ F_obj = G_obj", "tactic": "ext X" }, { "state_after": "no goals", "state_before": "case h\nV : Type u\ninst✝¹ : Quiver V\nW : Type u₂\ninst✝ : Quiver W\nF_obj : V → W\nmap✝¹ : {X Y : V} → (X ⟶ Y) → (F_obj X ⟶ F_obj Y)\nG_obj : V → W\nmap✝ : {X Y : V} → (X ⟶ Y) → (G_obj X ⟶ G_obj Y)\nh_obj : ∀ (X : V), { obj := F_obj, map := map✝¹ }.obj X = { obj := G_obj, map := map✝ }.obj X\nh_map :\n ∀ (X Y : V) (f : X ⟶ Y),\n { obj := F_obj, map := map✝¹ }.map f =\n Eq.recOn (_ : { obj := G_obj, map := map✝ }.obj Y = { obj := F_obj, map := map✝¹ }.obj Y)\n (Eq.recOn (_ : { obj := G_obj, map := map✝ }.obj X = { obj := F_obj, map := map✝¹ }.obj X)\n ({ obj := G_obj, map := map✝ }.map f))\nX : V\n⊢ F_obj X = G_obj X", "tactic": "apply h_obj" } ]

[ 92, 26 ]

[ 81, 1 ]

[ { "state_after": "α : Type u_1\na : Array α\nn : Nat\nd : α\n⊢ (if h : n < size a then a[n] else d) = Option.getD (if h : n < size a then some a[n] else none) d", "state_before": "α : Type u_1\na : Array α\nn : Nat\nd : α\n⊢ getD a n d = Option.getD (get? a n) d", "tactic": "simp [get?, getD]" }, { "state_after": "no goals", "state_before": "α : Type u_1\na : Array α\nn : Nat\nd : α\n⊢ (if h : n < size a then a[n] else d) = Option.getD (if h : n < size a then some a[n] else none) d", "tactic": "split <;> simp" } ]

[ 63, 36 ]

[ 62, 9 ]

[]

[ 276, 61 ]

[ 276, 1 ]

[ { "state_after": "J K : Type v\ninst✝¹⁰ : SmallCategory J\ninst✝⁹ : SmallCategory K\nF✝ : J × K ⥤ Type v\ninst✝⁸ : IsFiltered K\ninst✝⁷ : FinCategory J\nC : Type u\ninst✝⁶ : Category C\ninst✝⁵ : ConcreteCategory C\ninst✝⁴ : HasLimitsOfShape J C\ninst✝³ : HasColimitsOfShape K C\ninst✝² : ReflectsLimitsOfShape J (forget C)\ninst✝¹ : PreservesColimitsOfShape K (forget C)\ninst✝ : PreservesLimitsOfShape J (forget C)\nF : J ⥤ K ⥤ C\na : K\nb : J\n⊢ colimit.ι (limit F) a ≫\n ((IsLimit.conePointUniqueUpToIso (isLimitOfPreserves colim (limit.isLimit F)) (limit.isLimit (F ⋙ colim))).hom ≫\n (HasLimit.isoOfNatIso (colimitFlipIsoCompColim F).symm).hom) ≫\n limit.π (colimit (Functor.flip F)) b =\n (limit.π F b).app a ≫ (colimit.ι (Functor.flip F) a).app b", "state_before": "J K : Type v\ninst✝¹⁰ : SmallCategory J\ninst✝⁹ : SmallCategory K\nF✝ : J × K ⥤ Type v\ninst✝⁸ : IsFiltered K\ninst✝⁷ : FinCategory J\nC : Type u\ninst✝⁶ : Category C\ninst✝⁵ : ConcreteCategory C\ninst✝⁴ : HasLimitsOfShape J C\ninst✝³ : HasColimitsOfShape K C\ninst✝² : ReflectsLimitsOfShape J (forget C)\ninst✝¹ : PreservesColimitsOfShape K (forget C)\ninst✝ : PreservesLimitsOfShape J (forget C)\nF : J ⥤ K ⥤ C\na : K\nb : J\n⊢ colimit.ι (limit F) a ≫ (colimitLimitIso F).hom ≫ limit.π (colimit (Functor.flip F)) b =\n (limit.π F b).app a ≫ (colimit.ι (Functor.flip F) a).app b", "tactic": "dsimp [colimitLimitIso]" }, { "state_after": "J K : Type v\ninst✝¹⁰ : SmallCategory J\ninst✝⁹ : SmallCategory K\nF✝ : J × K ⥤ Type v\ninst✝⁸ : IsFiltered K\ninst✝⁷ : FinCategory J\nC : Type u\ninst✝⁶ : Category C\ninst✝⁵ : ConcreteCategory C\ninst✝⁴ : HasLimitsOfShape J C\ninst✝³ : HasColimitsOfShape K C\ninst✝² : ReflectsLimitsOfShape J (forget C)\ninst✝¹ : PreservesColimitsOfShape K (forget C)\ninst✝ : PreservesLimitsOfShape J (forget C)\nF : J ⥤ K ⥤ C\na : K\nb : J\n⊢ (limit.π F b).app a ≫\n colimit.ι (F.obj b) a ≫\n (HasColimit.isoOfNatIso (flipCompEvaluation F b)).inv ≫\n (colimitObjIsoColimitCompEvaluation (Functor.flip F) b).inv =\n (limit.π F b).app a ≫ (colimit.ι (Functor.flip F) a).app b", "state_before": "J K : Type v\ninst✝¹⁰ : SmallCategory J\ninst✝⁹ : SmallCategory K\nF✝ : J × K ⥤ Type v\ninst✝⁸ : IsFiltered K\ninst✝⁷ : FinCategory J\nC : Type u\ninst✝⁶ : Category C\ninst✝⁵ : ConcreteCategory C\ninst✝⁴ : HasLimitsOfShape J C\ninst✝³ : HasColimitsOfShape K C\ninst✝² : ReflectsLimitsOfShape J (forget C)\ninst✝¹ : PreservesColimitsOfShape K (forget C)\ninst✝ : PreservesLimitsOfShape J (forget C)\nF : J ⥤ K ⥤ C\na : K\nb : J\n⊢ colimit.ι (limit F) a ≫\n ((IsLimit.conePointUniqueUpToIso (isLimitOfPreserves colim (limit.isLimit F)) (limit.isLimit (F ⋙ colim))).hom ≫\n (HasLimit.isoOfNatIso (colimitFlipIsoCompColim F).symm).hom) ≫\n limit.π (colimit (Functor.flip F)) b =\n (limit.π F b).app a ≫ (colimit.ι (Functor.flip F) a).app b", "tactic": "simp only [Functor.mapCone_π_app, Iso.symm_hom,\n Limits.limit.conePointUniqueUpToIso_hom_comp_assoc, Limits.limit.cone_π,\n Limits.colimit.ι_map_assoc, Limits.colimitFlipIsoCompColim_inv_app, assoc,\n Limits.HasLimit.isoOfNatIso_hom_π]" }, { "state_after": "case e_a\nJ K : Type v\ninst✝¹⁰ : SmallCategory J\ninst✝⁹ : SmallCategory K\nF✝ : J × K ⥤ Type v\ninst✝⁸ : IsFiltered K\ninst✝⁷ : FinCategory J\nC : Type u\ninst✝⁶ : Category C\ninst✝⁵ : ConcreteCategory C\ninst✝⁴ : HasLimitsOfShape J C\ninst✝³ : HasColimitsOfShape K C\ninst✝² : ReflectsLimitsOfShape J (forget C)\ninst✝¹ : PreservesColimitsOfShape K (forget C)\ninst✝ : PreservesLimitsOfShape J (forget C)\nF : J ⥤ K ⥤ C\na : K\nb : J\n⊢ colimit.ι (F.obj b) a ≫\n (HasColimit.isoOfNatIso (flipCompEvaluation F b)).inv ≫\n (colimitObjIsoColimitCompEvaluation (Functor.flip F) b).inv =\n (colimit.ι (Functor.flip F) a).app b", "state_before": "J K : Type v\ninst✝¹⁰ : SmallCategory J\ninst✝⁹ : SmallCategory K\nF✝ : J × K ⥤ Type v\ninst✝⁸ : IsFiltered K\ninst✝⁷ : FinCategory J\nC : Type u\ninst✝⁶ : Category C\ninst✝⁵ : ConcreteCategory C\ninst✝⁴ : HasLimitsOfShape J C\ninst✝³ : HasColimitsOfShape K C\ninst✝² : ReflectsLimitsOfShape J (forget C)\ninst✝¹ : PreservesColimitsOfShape K (forget C)\ninst✝ : PreservesLimitsOfShape J (forget C)\nF : J ⥤ K ⥤ C\na : K\nb : J\n⊢ (limit.π F b).app a ≫\n colimit.ι (F.obj b) a ≫\n (HasColimit.isoOfNatIso (flipCompEvaluation F b)).inv ≫\n (colimitObjIsoColimitCompEvaluation (Functor.flip F) b).inv =\n (limit.π F b).app a ≫ (colimit.ι (Functor.flip F) a).app b", "tactic": "congr 1" }, { "state_after": "case e_a\nJ K : Type v\ninst✝¹⁰ : SmallCategory J\ninst✝⁹ : SmallCategory K\nF✝ : J × K ⥤ Type v\ninst✝⁸ : IsFiltered K\ninst✝⁷ : FinCategory J\nC : Type u\ninst✝⁶ : Category C\ninst✝⁵ : ConcreteCategory C\ninst✝⁴ : HasLimitsOfShape J C\ninst✝³ : HasColimitsOfShape K C\ninst✝² : ReflectsLimitsOfShape J (forget C)\ninst✝¹ : PreservesColimitsOfShape K (forget C)\ninst✝ : PreservesLimitsOfShape J (forget C)\nF : J ⥤ K ⥤ C\na : K\nb : J\n⊢ colimit.ι (F.obj b) a = (flipCompEvaluation F b).hom.app a ≫ colimit.ι (F.obj b) a", "state_before": "case e_a\nJ K : Type v\ninst✝¹⁰ : SmallCategory J\ninst✝⁹ : SmallCategory K\nF✝ : J × K ⥤ Type v\ninst✝⁸ : IsFiltered K\ninst✝⁷ : FinCategory J\nC : Type u\ninst✝⁶ : Category C\ninst✝⁵ : ConcreteCategory C\ninst✝⁴ : HasLimitsOfShape J C\ninst✝³ : HasColimitsOfShape K C\ninst✝² : ReflectsLimitsOfShape J (forget C)\ninst✝¹ : PreservesColimitsOfShape K (forget C)\ninst✝ : PreservesLimitsOfShape J (forget C)\nF : J ⥤ K ⥤ C\na : K\nb : J\n⊢ colimit.ι (F.obj b) a ≫\n (HasColimit.isoOfNatIso (flipCompEvaluation F b)).inv ≫\n (colimitObjIsoColimitCompEvaluation (Functor.flip F) b).inv =\n (colimit.ι (Functor.flip F) a).app b", "tactic": "simp only [← Category.assoc, Iso.comp_inv_eq,\n Limits.colimitObjIsoColimitCompEvaluation_ι_app_hom,\n Limits.HasColimit.isoOfNatIso_ι_hom, NatIso.ofComponents_hom_app]" }, { "state_after": "case e_a\nJ K : Type v\ninst✝¹⁰ : SmallCategory J\ninst✝⁹ : SmallCategory K\nF✝ : J × K ⥤ Type v\ninst✝⁸ : IsFiltered K\ninst✝⁷ : FinCategory J\nC : Type u\ninst✝⁶ : Category C\ninst✝⁵ : ConcreteCategory C\ninst✝⁴ : HasLimitsOfShape J C\ninst✝³ : HasColimitsOfShape K C\ninst✝² : ReflectsLimitsOfShape J (forget C)\ninst✝¹ : PreservesColimitsOfShape K (forget C)\ninst✝ : PreservesLimitsOfShape J (forget C)\nF : J ⥤ K ⥤ C\na : K\nb : J\n⊢ colimit.ι (F.obj b) a = 𝟙 ((F.obj b).obj a) ≫ colimit.ι (F.obj b) a", "state_before": "case e_a\nJ K : Type v\ninst✝¹⁰ : SmallCategory J\ninst✝⁹ : SmallCategory K\nF✝ : J × K ⥤ Type v\ninst✝⁸ : IsFiltered K\ninst✝⁷ : FinCategory J\nC : Type u\ninst✝⁶ : Category C\ninst✝⁵ : ConcreteCategory C\ninst✝⁴ : HasLimitsOfShape J C\ninst✝³ : HasColimitsOfShape K C\ninst✝² : ReflectsLimitsOfShape J (forget C)\ninst✝¹ : PreservesColimitsOfShape K (forget C)\ninst✝ : PreservesLimitsOfShape J (forget C)\nF : J ⥤ K ⥤ C\na : K\nb : J\n⊢ colimit.ι (F.obj b) a = (flipCompEvaluation F b).hom.app a ≫ colimit.ι (F.obj b) a", "tactic": "dsimp" }, { "state_after": "no goals", "state_before": "case e_a\nJ K : Type v\ninst✝¹⁰ : SmallCategory J\ninst✝⁹ : SmallCategory K\nF✝ : J × K ⥤ Type v\ninst✝⁸ : IsFiltered K\ninst✝⁷ : FinCategory J\nC : Type u\ninst✝⁶ : Category C\ninst✝⁵ : ConcreteCategory C\ninst✝⁴ : HasLimitsOfShape J C\ninst✝³ : HasColimitsOfShape K C\ninst✝² : ReflectsLimitsOfShape J (forget C)\ninst✝¹ : PreservesColimitsOfShape K (forget C)\ninst✝ : PreservesLimitsOfShape J (forget C)\nF : J ⥤ K ⥤ C\na : K\nb : J\n⊢ colimit.ι (F.obj b) a = 𝟙 ((F.obj b).obj a) ≫ colimit.ι (F.obj b) a", "tactic": "simp" } ]

[ 399, 7 ]

[ 386, 1 ]

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[ 1387, 47 ]

[ 1384, 1 ]

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[ 109, 1 ]

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[ 97, 1 ]

[ { "state_after": "no goals", "state_before": "C : Type u\ninst✝⁴ : Category C\nX Y Z✝ : C\ninst✝³ : HasZeroMorphisms C\nf✝ : X ⟶ Y\ninst✝² : HasKernel f✝\nf : X ⟶ Y\ninst✝¹ : HasKernel f\nZ : C\nh : Y ⟶ Z\ninst✝ : HasKernel (f ≫ h)\n⊢ arrow (kernelSubobject f) ≫ f ≫ h = 0", "tactic": "simp" } ]

[ 230, 35 ]

[ 228, 1 ]

[ { "state_after": "L : Language\nL' : Language\nM : Type w\nN : Type ?u.49784\nP : Type ?u.49787\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nγ : Type ?u.49815\nn k : ℕ\nφ : BoundedFormula L α k\n⊢ ∀ {m n : ℕ} (km : k ≤ m) (mn : m ≤ n), castLE mn (castLE km φ) = castLE (_ : k ≤ n) φ", "state_before": "L : Language\nL' : Language\nM : Type w\nN : Type ?u.49784\nP : Type ?u.49787\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nγ : Type ?u.49815\nn✝ k m n : ℕ\nkm : k ≤ m\nmn : m ≤ n\nφ : BoundedFormula L α k\n⊢ castLE mn (castLE km φ) = castLE (_ : k ≤ n) φ", "tactic": "revert m n" }, { "state_after": "case falsum\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.49784\nP : Type ?u.49787\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nγ : Type ?u.49815\nn✝¹ k n✝ m n : ℕ\nkm : n✝ ≤ m\nmn : m ≤ n\n⊢ castLE mn (castLE km falsum) = castLE (_ : n✝ ≤ n) falsum\n\ncase equal\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.49784\nP : Type ?u.49787\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nγ : Type ?u.49815\nn✝¹ k n✝ : ℕ\nt₁✝ t₂✝ : Term L (α ⊕ Fin n✝)\nm n : ℕ\nkm : n✝ ≤ m\nmn : m ≤ n\n⊢ castLE mn (castLE km (equal t₁✝ t₂✝)) = castLE (_ : n✝ ≤ n) (equal t₁✝ t₂✝)\n\ncase rel\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.49784\nP : Type ?u.49787\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nγ : Type ?u.49815\nn✝¹ k n✝ l✝ : ℕ\nR✝ : Relations L l✝\nts✝ : Fin l✝ → Term L (α ⊕ Fin n✝)\nm n : ℕ\nkm : n✝ ≤ m\nmn : m ≤ n\n⊢ castLE mn (castLE km (rel R✝ ts✝)) = castLE (_ : n✝ ≤ n) (rel R✝ ts✝)\n\ncase imp\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.49784\nP : Type ?u.49787\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nγ : Type ?u.49815\nn✝¹ k n✝ : ℕ\nf₁✝ f₂✝ : BoundedFormula L α n✝\nih1 : ∀ {m n : ℕ} (km : n✝ ≤ m) (mn : m ≤ n), castLE mn (castLE km f₁✝) = castLE (_ : n✝ ≤ n) f₁✝\nih2 : ∀ {m n : ℕ} (km : n✝ ≤ m) (mn : m ≤ n), castLE mn (castLE km f₂✝) = castLE (_ : n✝ ≤ n) f₂✝\nm n : ℕ\nkm : n✝ ≤ m\nmn : m ≤ n\n⊢ castLE mn (castLE km (imp f₁✝ f₂✝)) = castLE (_ : n✝ ≤ n) (imp f₁✝ f₂✝)\n\ncase all\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.49784\nP : Type ?u.49787\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nγ : Type ?u.49815\nn✝¹ k n✝ : ℕ\nf✝ : BoundedFormula L α (n✝ + 1)\nih3 : ∀ {m n : ℕ} (km : n✝ + 1 ≤ m) (mn : m ≤ n), castLE mn (castLE km f✝) = castLE (_ : n✝ + 1 ≤ n) f✝\nm n : ℕ\nkm : n✝ ≤ m\nmn : m ≤ n\n⊢ castLE mn (castLE km (all f✝)) = castLE (_ : n✝ ≤ n) (all f✝)", "state_before": "L : Language\nL' : Language\nM : Type w\nN : Type ?u.49784\nP : Type ?u.49787\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nγ : Type ?u.49815\nn k : ℕ\nφ : BoundedFormula L α k\n⊢ ∀ {m n : ℕ} (km : k ≤ m) (mn : m ≤ n), castLE mn (castLE km φ) = castLE (_ : k ≤ n) φ", "tactic": "induction' φ with _ _ _ _ _ _ _ _ _ _ _ ih1 ih2 _ _ ih3 <;> intro m n km mn" }, { "state_after": "no goals", "state_before": "case falsum\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.49784\nP : Type ?u.49787\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nγ : Type ?u.49815\nn✝¹ k n✝ m n : ℕ\nkm : n✝ ≤ m\nmn : m ≤ n\n⊢ castLE mn (castLE km falsum) = castLE (_ : n✝ ≤ n) falsum", "tactic": "rfl" }, { "state_after": "no goals", "state_before": "case equal\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.49784\nP : Type ?u.49787\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nγ : Type ?u.49815\nn✝¹ k n✝ : ℕ\nt₁✝ t₂✝ : Term L (α ⊕ Fin n✝)\nm n : ℕ\nkm : n✝ ≤ m\nmn : m ≤ n\n⊢ castLE mn (castLE km (equal t₁✝ t₂✝)) = castLE (_ : n✝ ≤ n) (equal t₁✝ t₂✝)", "tactic": "simp" }, { "state_after": "case rel\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.49784\nP : Type ?u.49787\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nγ : Type ?u.49815\nn✝¹ k n✝ l✝ : ℕ\nR✝ : Relations L l✝\nts✝ : Fin l✝ → Term L (α ⊕ Fin n✝)\nm n : ℕ\nkm : n✝ ≤ m\nmn : m ≤ n\n⊢ rel R✝ (Term.relabel (Sum.map id ↑(Fin.castLE mn)) ∘ Term.relabel (Sum.map id ↑(Fin.castLE km)) ∘ ts✝) =\n rel R✝ (Term.relabel (Sum.map id ↑(Fin.castLE (_ : n✝ ≤ n))) ∘ ts✝)", "state_before": "case rel\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.49784\nP : Type ?u.49787\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nγ : Type ?u.49815\nn✝¹ k n✝ l✝ : ℕ\nR✝ : Relations L l✝\nts✝ : Fin l✝ → Term L (α ⊕ Fin n✝)\nm n : ℕ\nkm : n✝ ≤ m\nmn : m ≤ n\n⊢ castLE mn (castLE km (rel R✝ ts✝)) = castLE (_ : n✝ ≤ n) (rel R✝ ts✝)", "tactic": "simp only [castLE, eq_self_iff_true, heq_iff_eq, true_and_iff]" }, { "state_after": "case rel\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.49784\nP : Type ?u.49787\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nγ : Type ?u.49815\nn✝¹ k n✝ l✝ : ℕ\nR✝ : Relations L l✝\nts✝ : Fin l✝ → Term L (α ⊕ Fin n✝)\nm n : ℕ\nkm : n✝ ≤ m\nmn : m ≤ n\n⊢ rel R✝ (Term.relabel (Sum.map id ↑(Fin.castLE mn) ∘ Sum.map id ↑(Fin.castLE km)) ∘ ts✝) =\n rel R✝ (Term.relabel (Sum.map id ↑(Fin.castLE (_ : n✝ ≤ n))) ∘ ts✝)", "state_before": "case rel\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.49784\nP : Type ?u.49787\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nγ : Type ?u.49815\nn✝¹ k n✝ l✝ : ℕ\nR✝ : Relations L l✝\nts✝ : Fin l✝ → Term L (α ⊕ Fin n✝)\nm n : ℕ\nkm : n✝ ≤ m\nmn : m ≤ n\n⊢ rel R✝ (Term.relabel (Sum.map id ↑(Fin.castLE mn)) ∘ Term.relabel (Sum.map id ↑(Fin.castLE km)) ∘ ts✝) =\n rel R✝ (Term.relabel (Sum.map id ↑(Fin.castLE (_ : n✝ ≤ n))) ∘ ts✝)", "tactic": "rw [← Function.comp.assoc, Term.relabel_comp_relabel]" }, { "state_after": "no goals", "state_before": "case rel\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.49784\nP : Type ?u.49787\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nγ : Type ?u.49815\nn✝¹ k n✝ l✝ : ℕ\nR✝ : Relations L l✝\nts✝ : Fin l✝ → Term L (α ⊕ Fin n✝)\nm n : ℕ\nkm : n✝ ≤ m\nmn : m ≤ n\n⊢ rel R✝ (Term.relabel (Sum.map id ↑(Fin.castLE mn) ∘ Sum.map id ↑(Fin.castLE km)) ∘ ts✝) =\n rel R✝ (Term.relabel (Sum.map id ↑(Fin.castLE (_ : n✝ ≤ n))) ∘ ts✝)", "tactic": "simp" }, { "state_after": "no goals", "state_before": "case imp\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.49784\nP : Type ?u.49787\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nγ : Type ?u.49815\nn✝¹ k n✝ : ℕ\nf₁✝ f₂✝ : BoundedFormula L α n✝\nih1 : ∀ {m n : ℕ} (km : n✝ ≤ m) (mn : m ≤ n), castLE mn (castLE km f₁✝) = castLE (_ : n✝ ≤ n) f₁✝\nih2 : ∀ {m n : ℕ} (km : n✝ ≤ m) (mn : m ≤ n), castLE mn (castLE km f₂✝) = castLE (_ : n✝ ≤ n) f₂✝\nm n : ℕ\nkm : n✝ ≤ m\nmn : m ≤ n\n⊢ castLE mn (castLE km (imp f₁✝ f₂✝)) = castLE (_ : n✝ ≤ n) (imp f₁✝ f₂✝)", "tactic": "simp [ih1, ih2]" }, { "state_after": "no goals", "state_before": "case all\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.49784\nP : Type ?u.49787\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nγ : Type ?u.49815\nn✝¹ k n✝ : ℕ\nf✝ : BoundedFormula L α (n✝ + 1)\nih3 : ∀ {m n : ℕ} (km : n✝ + 1 ≤ m) (mn : m ≤ n), castLE mn (castLE km f✝) = castLE (_ : n✝ + 1 ≤ n) f✝\nm n : ℕ\nkm : n✝ ≤ m\nmn : m ≤ n\n⊢ castLE mn (castLE km (all f✝)) = castLE (_ : n✝ ≤ n) (all f✝)", "tactic": "simp only [castLE, ih3]" } ]

[ 470, 28 ]

[ 460, 1 ]

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[ 809, 94 ]

[ 807, 1 ]

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[ 174, 49 ]

[ 172, 1 ]

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[ 120, 11 ]

[ 119, 1 ]

[ { "state_after": "no goals", "state_before": "J : Type v\ninst✝ : SmallCategory J\nF : J ⥤ TypeMax\nx : colimit F\n⊢ ∃ j y, colimit.ι F j y = x", "tactic": "exact jointly_surjective.{v, u} F (colimit.isColimit F) x" } ]

[ 377, 60 ]

[ 375, 1 ]

[]

[ 288, 41 ]

[ 287, 1 ]

[ { "state_after": "no goals", "state_before": "θ : ℂ\n⊢ sin θ ≠ 0 ↔ ∀ (k : ℤ), θ ≠ ↑k * ↑π", "tactic": "rw [← not_exists, not_iff_not, sin_eq_zero_iff]" } ]

[ 63, 50 ]

[ 62, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type ?u.2661\nγ : Type ?u.2664\ninst✝ : BooleanRing α\na b : α\n⊢ -a = -a + 0", "tactic": "rw [add_zero]" }, { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type ?u.2661\nγ : Type ?u.2664\ninst✝ : BooleanRing α\na b : α\n⊢ -a + 0 = -a + -a + a", "tactic": "rw [← neg_add_self, add_assoc]" }, { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type ?u.2661\nγ : Type ?u.2664\ninst✝ : BooleanRing α\na b : α\n⊢ -a + -a + a = a", "tactic": "rw [add_self, zero_add]" } ]

[ 85, 40 ]

[ 81, 1 ]

[ { "state_after": "case inl\n𝕜 : Type ?u.62428\ninst✝⁴ : NormedField 𝕜\nE : Type ?u.62434\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_1\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nx : F\ns : Set F\nhx : x ∈ s\nh₀ : infDist x (sᶜ) = 0\n⊢ closedBall x (infDist x (sᶜ)) ⊆ closure s\n\ncase inr\n𝕜 : Type ?u.62428\ninst✝⁴ : NormedField 𝕜\nE : Type ?u.62434\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_1\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nx : F\ns : Set F\nhx : x ∈ s\nh₀ : infDist x (sᶜ) ≠ 0\n⊢ closedBall x (infDist x (sᶜ)) ⊆ closure s", "state_before": "𝕜 : Type ?u.62428\ninst✝⁴ : NormedField 𝕜\nE : Type ?u.62434\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_1\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nx : F\ns : Set F\nhx : x ∈ s\n⊢ closedBall x (infDist x (sᶜ)) ⊆ closure s", "tactic": "cases' eq_or_ne (infDist x (sᶜ)) 0 with h₀ h₀" }, { "state_after": "case inl\n𝕜 : Type ?u.62428\ninst✝⁴ : NormedField 𝕜\nE : Type ?u.62434\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_1\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nx : F\ns : Set F\nhx : x ∈ s\nh₀ : infDist x (sᶜ) = 0\n⊢ closure {x} ⊆ closure s", "state_before": "case inl\n𝕜 : Type ?u.62428\ninst✝⁴ : NormedField 𝕜\nE : Type ?u.62434\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_1\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nx : F\ns : Set F\nhx : x ∈ s\nh₀ : infDist x (sᶜ) = 0\n⊢ closedBall x (infDist x (sᶜ)) ⊆ closure s", "tactic": "rw [h₀, closedBall_zero']" }, { "state_after": "no goals", "state_before": "case inl\n𝕜 : Type ?u.62428\ninst✝⁴ : NormedField 𝕜\nE : Type ?u.62434\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_1\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nx : F\ns : Set F\nhx : x ∈ s\nh₀ : infDist x (sᶜ) = 0\n⊢ closure {x} ⊆ closure s", "tactic": "exact closure_mono (singleton_subset_iff.2 hx)" }, { "state_after": "case inr\n𝕜 : Type ?u.62428\ninst✝⁴ : NormedField 𝕜\nE : Type ?u.62434\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_1\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nx : F\ns : Set F\nhx : x ∈ s\nh₀ : infDist x (sᶜ) ≠ 0\n⊢ closure (ball x (infDist x (sᶜ))) ⊆ closure s", "state_before": "case inr\n𝕜 : Type ?u.62428\ninst✝⁴ : NormedField 𝕜\nE : Type ?u.62434\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_1\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nx : F\ns : Set F\nhx : x ∈ s\nh₀ : infDist x (sᶜ) ≠ 0\n⊢ closedBall x (infDist x (sᶜ)) ⊆ closure s", "tactic": "rw [← closure_ball x h₀]" }, { "state_after": "no goals", "state_before": "case inr\n𝕜 : Type ?u.62428\ninst✝⁴ : NormedField 𝕜\nE : Type ?u.62434\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_1\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nx : F\ns : Set F\nhx : x ∈ s\nh₀ : infDist x (sᶜ) ≠ 0\n⊢ closure (ball x (infDist x (sᶜ))) ⊆ closure s", "tactic": "exact closure_mono ball_infDist_compl_subset" } ]

[ 118, 49 ]

[ 112, 1 ]

[ { "state_after": "α : Type u_1\nβ : Type u_3\nγ : Type u_2\nδ : Type ?u.465503\nr : α → β → Prop\np : γ → δ → Prop\ns : Multiset α\nt : Multiset γ\nf : γ → β\n⊢ Rel (flip fun a b => flip r (f a) b) s t ↔ Rel (fun a b => r a (f b)) s t", "state_before": "α : Type u_1\nβ : Type u_3\nγ : Type u_2\nδ : Type ?u.465503\nr : α → β → Prop\np : γ → δ → Prop\ns : Multiset α\nt : Multiset γ\nf : γ → β\n⊢ Rel r s (map f t) ↔ Rel (fun a b => r a (f b)) s t", "tactic": "rw [← rel_flip, rel_map_left, ← rel_flip]" }, { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type u_3\nγ : Type u_2\nδ : Type ?u.465503\nr : α → β → Prop\np : γ → δ → Prop\ns : Multiset α\nt : Multiset γ\nf : γ → β\n⊢ Rel (flip fun a b => flip r (f a) b) s t ↔ Rel (fun a b => r a (f b)) s t", "tactic": "rfl" } ]

[ 2772, 49 ]

[ 2770, 1 ]

[]

[ 688, 6 ]

[ 686, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.984360\nδ : Type ?u.984363\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝² : MeasurableSpace δ\ninst✝¹ : NormedAddCommGroup β\ninst✝ : NormedAddCommGroup γ\nf : α → β\nhf : AEStronglyMeasurable f μ\n⊢ (Integrable fun a => ‖f a‖) ↔ Integrable f", "tactic": "simp_rw [Integrable, and_iff_right hf, and_iff_right hf.norm, hasFiniteIntegral_norm_iff]" } ]

[ 768, 92 ]

[ 766, 1 ]

[]

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[ 1131, 1 ]

[]

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[ 464, 1 ]

[ { "state_after": "α : Type u\nβ : Type v\nγ : Type w\ns : WSeq α\n⊢ Computation.IsBisimulation fun c1 c2 =>\n ∃ l s,\n c1 =\n Computation.corec\n (fun x =>\n match x with\n | (n, s) =>\n match Seq.destruct s with\n | none => Sum.inl n\n | some (none, s') => Sum.inr (n, s')\n | some (some val, s') => Sum.inr (n + 1, s'))\n (List.length l, s) ∧\n c2 =\n Computation.map List.length\n (Computation.corec\n (fun x =>\n match x with\n | (l, s) =>\n match Seq.destruct s with\n | none => Sum.inl (List.reverse l)\n | some (none, s') => Sum.inr (l, s')\n | some (some a, s') => Sum.inr (a :: l, s'))\n (l, s))", "state_before": "α : Type u\nβ : Type v\nγ : Type w\ns : WSeq α\n⊢ length s = Computation.map List.length (toList s)", "tactic": "refine'\n Computation.eq_of_bisim\n (fun c1 c2 =>\n ∃ (l : List α) (s : WSeq α),\n c1 = Computation.corec (fun ⟨n, s⟩ =>\n match Seq.destruct s with\n | none => Sum.inl n\n | some (none, s') => Sum.inr (n, s')\n | some (some _, s') => Sum.inr (n + 1, s')) (l.length, s) ∧\n c2 = Computation.map List.length (Computation.corec (fun ⟨l, s⟩ =>\n match Seq.destruct s with\n | none => Sum.inl l.reverse\n | some (none, s') => Sum.inr (l, s')\n | some (some a, s') => Sum.inr (a::l, s')) (l, s)))\n _ ⟨[], s, rfl, rfl⟩" }, { "state_after": "α : Type u\nβ : Type v\nγ : Type w\ns : WSeq α\ns1 s2 : Computation ℕ\nh :\n ∃ l s,\n s1 =\n Computation.corec\n (fun x =>\n match x with\n | (n, s) =>\n match Seq.destruct s with\n | none => Sum.inl n\n | some (none, s') => Sum.inr (n, s')\n | some (some val, s') => Sum.inr (n + 1, s'))\n (List.length l, s) ∧\n s2 =\n Computation.map List.length\n (Computation.corec\n (fun x =>\n match x with\n | (l, s) =>\n match Seq.destruct s with\n | none => Sum.inl (List.reverse l)\n | some (none, s') => Sum.inr (l, s')\n | some (some a, s') => Sum.inr (a :: l, s'))\n (l, s))\n⊢ Computation.BisimO\n (fun c1 c2 =>\n ∃ l s,\n c1 =\n Computation.corec\n (fun x =>\n match x with\n | (n, s) =>\n match Seq.destruct s with\n | none => Sum.inl n\n | some (none, s') => Sum.inr (n, s')\n | some (some val, s') => Sum.inr (n + 1, s'))\n (List.length l, s) ∧\n c2 =\n Computation.map List.length\n (Computation.corec\n (fun x =>\n match x with\n | (l, s) =>\n match Seq.destruct s with\n | none => Sum.inl (List.reverse l)\n | some (none, s') => Sum.inr (l, s')\n | some (some a, s') => Sum.inr (a :: l, s'))\n (l, s)))\n (Computation.destruct s1) (Computation.destruct s2)", "state_before": "α : Type u\nβ : Type v\nγ : Type w\ns : WSeq α\n⊢ Computation.IsBisimulation fun c1 c2 =>\n ∃ l s,\n c1 =\n Computation.corec\n (fun x =>\n match x with\n | (n, s) =>\n match Seq.destruct s with\n | none => Sum.inl n\n | some (none, s') => Sum.inr (n, s')\n | some (some val, s') => Sum.inr (n + 1, s'))\n (List.length l, s) ∧\n c2 =\n Computation.map List.length\n (Computation.corec\n (fun x =>\n match x with\n | (l, s) =>\n match Seq.destruct s with\n | none => Sum.inl (List.reverse l)\n | some (none, s') => Sum.inr (l, s')\n | some (some a, s') => Sum.inr (a :: l, s'))\n (l, s))", "tactic": "intro s1 s2 h" }, { "state_after": "case intro.intro\nα : Type u\nβ : Type v\nγ : Type w\ns✝ : WSeq α\ns1 s2 : Computation ℕ\nl : List α\ns : WSeq α\nh :\n s1 =\n Computation.corec\n (fun x =>\n match x with\n | (n, s) =>\n match Seq.destruct s with\n | none => Sum.inl n\n | some (none, s') => Sum.inr (n, s')\n | some (some val, s') => Sum.inr (n + 1, s'))\n (List.length l, s) ∧\n s2 =\n Computation.map List.length\n (Computation.corec\n (fun x =>\n match x with\n | (l, s) =>\n match Seq.destruct s with\n | none => Sum.inl (List.reverse l)\n | some (none, s') => Sum.inr (l, s')\n | some (some a, s') => Sum.inr (a :: l, s'))\n (l, s))\n⊢ Computation.BisimO\n (fun c1 c2 =>\n ∃ l s,\n c1 =\n Computation.corec\n (fun x =>\n match x with\n | (n, s) =>\n match Seq.destruct s with\n | none => Sum.inl n\n | some (none, s') => Sum.inr (n, s')\n | some (some val, s') => Sum.inr (n + 1, s'))\n (List.length l, s) ∧\n c2 =\n Computation.map List.length\n (Computation.corec\n (fun x =>\n match x with\n | (l, s) =>\n match Seq.destruct s with\n | none => Sum.inl (List.reverse l)\n | some (none, s') => Sum.inr (l, s')\n | some (some a, s') => Sum.inr (a :: l, s'))\n (l, s)))\n (Computation.destruct s1) (Computation.destruct s2)", "state_before": "α : Type u\nβ : Type v\nγ : Type w\ns : WSeq α\ns1 s2 : Computation ℕ\nh :\n ∃ l s,\n s1 =\n Computation.corec\n (fun x =>\n match x with\n | (n, s) =>\n match Seq.destruct s with\n | none => Sum.inl n\n | some (none, s') => Sum.inr (n, s')\n | some (some val, s') => Sum.inr (n + 1, s'))\n (List.length l, s) ∧\n s2 =\n Computation.map List.length\n (Computation.corec\n (fun x =>\n match x with\n | (l, s) =>\n match Seq.destruct s with\n | none => Sum.inl (List.reverse l)\n | some (none, s') => Sum.inr (l, s')\n | some (some a, s') => Sum.inr (a :: l, s'))\n (l, s))\n⊢ Computation.BisimO\n (fun c1 c2 =>\n ∃ l s,\n c1 =\n Computation.corec\n (fun x =>\n match x with\n | (n, s) =>\n match Seq.destruct s with\n | none => Sum.inl n\n | some (none, s') => Sum.inr (n, s')\n | some (some val, s') => Sum.inr (n + 1, s'))\n (List.length l, s) ∧\n c2 =\n Computation.map List.length\n (Computation.corec\n (fun x =>\n match x with\n | (l, s) =>\n match Seq.destruct s with\n | none => Sum.inl (List.reverse l)\n | some (none, s') => Sum.inr (l, s')\n | some (some a, s') => Sum.inr (a :: l, s'))\n (l, s)))\n (Computation.destruct s1) (Computation.destruct s2)", "tactic": "rcases h with ⟨l, s, h⟩" }, { "state_after": "case intro.intro\nα : Type u\nβ : Type v\nγ : Type w\ns✝ : WSeq α\ns1 s2 : Computation ℕ\nl : List α\ns : WSeq α\nh :\n s1 =\n Computation.corec\n (fun x =>\n match x with\n | (n, s) =>\n match Seq.destruct s with\n | none => Sum.inl n\n | some (none, s') => Sum.inr (n, s')\n | some (some val, s') => Sum.inr (n + 1, s'))\n (List.length l, s) ∧\n s2 =\n Computation.map List.length\n (Computation.corec\n (fun x =>\n match x with\n | (l, s) =>\n match Seq.destruct s with\n | none => Sum.inl (List.reverse l)\n | some (none, s') => Sum.inr (l, s')\n | some (some a, s') => Sum.inr (a :: l, s'))\n (l, s))\n⊢ Computation.BisimO\n (fun c1 c2 =>\n ∃ l s,\n c1 =\n Computation.corec\n (fun x =>\n match x with\n | (n, s) =>\n match Seq.destruct s with\n | none => Sum.inl n\n | some (none, s') => Sum.inr (n, s')\n | some (some val, s') => Sum.inr (n + 1, s'))\n (List.length l, s) ∧\n c2 =\n Computation.map List.length\n (Computation.corec\n (fun x =>\n match x with\n | (l, s) =>\n match Seq.destruct s with\n | none => Sum.inl (List.reverse l)\n | some (none, s') => Sum.inr (l, s')\n | some (some a, s') => Sum.inr (a :: l, s'))\n (l, s)))\n (Computation.destruct\n (Computation.corec\n (fun x =>\n match x with\n | (n, s) =>\n match Seq.destruct s with\n | none => Sum.inl n\n | some (none, s') => Sum.inr (n, s')\n | some (some val, s') => Sum.inr (n + 1, s'))\n (List.length l, s)))\n (Computation.destruct\n (Computation.map List.length\n (Computation.corec\n (fun x =>\n match x with\n | (l, s) =>\n match Seq.destruct s with\n | none => Sum.inl (List.reverse l)\n | some (none, s') => Sum.inr (l, s')\n | some (some a, s') => Sum.inr (a :: l, s'))\n (l, s))))", "state_before": "case intro.intro\nα : Type u\nβ : Type v\nγ : Type w\ns✝ : WSeq α\ns1 s2 : Computation ℕ\nl : List α\ns : WSeq α\nh :\n s1 =\n Computation.corec\n (fun x =>\n match x with\n | (n, s) =>\n match Seq.destruct s with\n | none => Sum.inl n\n | some (none, s') => Sum.inr (n, s')\n | some (some val, s') => Sum.inr (n + 1, s'))\n (List.length l, s) ∧\n s2 =\n Computation.map List.length\n (Computation.corec\n (fun x =>\n match x with\n | (l, s) =>\n match Seq.destruct s with\n | none => Sum.inl (List.reverse l)\n | some (none, s') => Sum.inr (l, s')\n | some (some a, s') => Sum.inr (a :: l, s'))\n (l, s))\n⊢ Computation.BisimO\n (fun c1 c2 =>\n ∃ l s,\n c1 =\n Computation.corec\n (fun x =>\n match x with\n | (n, s) =>\n match Seq.destruct s with\n | none => Sum.inl n\n | some (none, s') => Sum.inr (n, s')\n | some (some val, s') => Sum.inr (n + 1, s'))\n (List.length l, s) ∧\n c2 =\n Computation.map List.length\n (Computation.corec\n (fun x =>\n match x with\n | (l, s) =>\n match Seq.destruct s with\n | none => Sum.inl (List.reverse l)\n | some (none, s') => Sum.inr (l, s')\n | some (some a, s') => Sum.inr (a :: l, s'))\n (l, s)))\n (Computation.destruct s1) (Computation.destruct s2)", "tactic": "rw [h.left, h.right]" }, { "state_after": "case intro.intro.h2\nα : Type u\nβ : Type v\nγ : Type w\ns✝¹ : WSeq α\ns1 s2 : Computation ℕ\nl : List α\ns✝ : WSeq α\nh✝ :\n s1 =\n Computation.corec\n (fun x =>\n match x with\n | (n, s) =>\n match Seq.destruct s with\n | none => Sum.inl n\n | some (none, s') => Sum.inr (n, s')\n | some (some val, s') => Sum.inr (n + 1, s'))\n (List.length l, s✝) ∧\n s2 =\n Computation.map List.length\n (Computation.corec\n (fun x =>\n match x with\n | (l, s) =>\n match Seq.destruct s with\n | none => Sum.inl (List.reverse l)\n | some (none, s') => Sum.inr (l, s')\n | some (some a, s') => Sum.inr (a :: l, s'))\n (l, s✝))\na : α\ns : WSeq α\nh :\n s1 =\n Computation.corec\n (fun x =>\n match x with\n | (n, s) =>\n match Seq.destruct s with\n | none => Sum.inl n\n | some (none, s') => Sum.inr (n, s')\n | some (some val, s') => Sum.inr (n + 1, s'))\n (List.length l, cons a s) ∧\n s2 =\n Computation.map List.length\n (Computation.corec\n (fun x =>\n match x with\n | (l, s) =>\n match Seq.destruct s with\n | none => Sum.inl (List.reverse l)\n | some (none, s') => Sum.inr (l, s')\n | some (some a, s') => Sum.inr (a :: l, s'))\n (l, cons a s))\n⊢ ∃ l_1 s_1,\n Computation.corec\n (fun x =>\n match Seq.destruct x.snd with\n | none => Sum.inl x.fst\n | some (none, s') => Sum.inr (x.fst, s')\n | some (some val, s') => Sum.inr (x.fst + 1, s'))\n (List.length l + 1, s) =\n Computation.corec\n (fun x =>\n match Seq.destruct x.snd with\n | none => Sum.inl x.fst\n | some (none, s') => Sum.inr (x.fst, s')\n | some (some val, s') => Sum.inr (x.fst + 1, s'))\n (List.length l_1, s_1) ∧\n Computation.map List.length\n (Computation.corec\n (fun x =>\n match Seq.destruct x.snd with\n | none => Sum.inl (List.reverse x.fst)\n | some (none, s') => Sum.inr (x.fst, s')\n | some (some a, s') => Sum.inr (a :: x.fst, s'))\n (a :: l, s)) =\n Computation.map List.length\n (Computation.corec\n (fun x =>\n match Seq.destruct x.snd with\n | none => Sum.inl (List.reverse x.fst)\n | some (none, s') => Sum.inr (x.fst, s')\n | some (some a, s') => Sum.inr (a :: x.fst, s'))\n (l_1, s_1))\n\ncase intro.intro.h3\nα : Type u\nβ : Type v\nγ : Type w\ns✝¹ : WSeq α\ns1 s2 : Computation ℕ\nl : List α\ns✝ : WSeq α\nh✝ :\n s1 =\n Computation.corec\n (fun x =>\n match x with\n | (n, s) =>\n match Seq.destruct s with\n | none => Sum.inl n\n | some (none, s') => Sum.inr (n, s')\n | some (some val, s') => Sum.inr (n + 1, s'))\n (List.length l, s✝) ∧\n s2 =\n Computation.map List.length\n (Computation.corec\n (fun x =>\n match x with\n | (l, s) =>\n match Seq.destruct s with\n | none => Sum.inl (List.reverse l)\n | some (none, s') => Sum.inr (l, s')\n | some (some a, s') => Sum.inr (a :: l, s'))\n (l, s✝))\ns : WSeq α\nh :\n s1 =\n Computation.corec\n (fun x =>\n match x with\n | (n, s) =>\n match Seq.destruct s with\n | none => Sum.inl n\n | some (none, s') => Sum.inr (n, s')\n | some (some val, s') => Sum.inr (n + 1, s'))\n (List.length l, think s) ∧\n s2 =\n Computation.map List.length\n (Computation.corec\n (fun x =>\n match x with\n | (l, s) =>\n match Seq.destruct s with\n | none => Sum.inl (List.reverse l)\n | some (none, s') => Sum.inr (l, s')\n | some (some a, s') => Sum.inr (a :: l, s'))\n (l, think s))\n⊢ ∃ l_1 s_1,\n Computation.corec\n (fun x =>\n match Seq.destruct x.snd with\n | none => Sum.inl x.fst\n | some (none, s') => Sum.inr (x.fst, s')\n | some (some val, s') => Sum.inr (x.fst + 1, s'))\n (List.length l, s) =\n Computation.corec\n (fun x =>\n match Seq.destruct x.snd with\n | none => Sum.inl x.fst\n | some (none, s') => Sum.inr (x.fst, s')\n | some (some val, s') => Sum.inr (x.fst + 1, s'))\n (List.length l_1, s_1) ∧\n Computation.map List.length\n (Computation.corec\n (fun x =>\n match Seq.destruct x.snd with\n | none => Sum.inl (List.reverse x.fst)\n | some (none, s') => Sum.inr (x.fst, s')\n | some (some a, s') => Sum.inr (a :: x.fst, s'))\n (l, s)) =\n Computation.map List.length\n (Computation.corec\n (fun x =>\n match Seq.destruct x.snd with\n | none => Sum.inl (List.reverse x.fst)\n | some (none, s') => Sum.inr (x.fst, s')\n | some (some a, s') => Sum.inr (a :: x.fst, s'))\n (l_1, s_1))", "state_before": "case intro.intro\nα : Type u\nβ : Type v\nγ : Type w\ns✝ : WSeq α\ns1 s2 : Computation ℕ\nl : List α\ns : WSeq α\nh :\n s1 =\n Computation.corec\n (fun x =>\n match x with\n | (n, s) =>\n match Seq.destruct s with\n | none => Sum.inl n\n | some (none, s') => Sum.inr (n, s')\n | some (some val, s') => Sum.inr (n + 1, s'))\n (List.length l, s) ∧\n s2 =\n Computation.map List.length\n (Computation.corec\n (fun x =>\n match x with\n | (l, s) =>\n match Seq.destruct s with\n | none => Sum.inl (List.reverse l)\n | some (none, s') => Sum.inr (l, s')\n | some (some a, s') => Sum.inr (a :: l, s'))\n (l, s))\n⊢ Computation.BisimO\n (fun c1 c2 =>\n ∃ l s,\n c1 =\n Computation.corec\n (fun x =>\n match x with\n | (n, s) =>\n match Seq.destruct s with\n | none => Sum.inl n\n | some (none, s') => Sum.inr (n, s')\n | some (some val, s') => Sum.inr (n + 1, s'))\n (List.length l, s) ∧\n c2 =\n Computation.map List.length\n (Computation.corec\n (fun x =>\n match x with\n | (l, s) =>\n match Seq.destruct s with\n | none => Sum.inl (List.reverse l)\n | some (none, s') => Sum.inr (l, s')\n | some (some a, s') => Sum.inr (a :: l, s'))\n (l, s)))\n (Computation.destruct\n (Computation.corec\n (fun x =>\n match x with\n | (n, s) =>\n match Seq.destruct s with\n | none => Sum.inl n\n | some (none, s') => Sum.inr (n, s')\n | some (some val, s') => Sum.inr (n + 1, s'))\n (List.length l, s)))\n (Computation.destruct\n (Computation.map List.length\n (Computation.corec\n (fun x =>\n match x with\n | (l, s) =>\n match Seq.destruct s with\n | none => Sum.inl (List.reverse l)\n | some (none, s') => Sum.inr (l, s')\n | some (some a, s') => Sum.inr (a :: l, s'))\n (l, s))))", "tactic": "induction' s using WSeq.recOn with a s s <;> simp [toList, nil, cons, think, length]" }, { "state_after": "no goals", "state_before": "case intro.intro.h2\nα : Type u\nβ : Type v\nγ : Type w\ns✝¹ : WSeq α\ns1 s2 : Computation ℕ\nl : List α\ns✝ : WSeq α\nh✝ :\n s1 =\n Computation.corec\n (fun x =>\n match x with\n | (n, s) =>\n match Seq.destruct s with\n | none => Sum.inl n\n | some (none, s') => Sum.inr (n, s')\n | some (some val, s') => Sum.inr (n + 1, s'))\n (List.length l, s✝) ∧\n s2 =\n Computation.map List.length\n (Computation.corec\n (fun x =>\n match x with\n | (l, s) =>\n match Seq.destruct s with\n | none => Sum.inl (List.reverse l)\n | some (none, s') => Sum.inr (l, s')\n | some (some a, s') => Sum.inr (a :: l, s'))\n (l, s✝))\na : α\ns : WSeq α\nh :\n s1 =\n Computation.corec\n (fun x =>\n match x with\n | (n, s) =>\n match Seq.destruct s with\n | none => Sum.inl n\n | some (none, s') => Sum.inr (n, s')\n | some (some val, s') => Sum.inr (n + 1, s'))\n (List.length l, cons a s) ∧\n s2 =\n Computation.map List.length\n (Computation.corec\n (fun x =>\n match x with\n | (l, s) =>\n match Seq.destruct s with\n | none => Sum.inl (List.reverse l)\n | some (none, s') => Sum.inr (l, s')\n | some (some a, s') => Sum.inr (a :: l, s'))\n (l, cons a s))\n⊢ ∃ l_1 s_1,\n Computation.corec\n (fun x =>\n match Seq.destruct x.snd with\n | none => Sum.inl x.fst\n | some (none, s') => Sum.inr (x.fst, s')\n | some (some val, s') => Sum.inr (x.fst + 1, s'))\n (List.length l + 1, s) =\n Computation.corec\n (fun x =>\n match Seq.destruct x.snd with\n | none => Sum.inl x.fst\n | some (none, s') => Sum.inr (x.fst, s')\n | some (some val, s') => Sum.inr (x.fst + 1, s'))\n (List.length l_1, s_1) ∧\n Computation.map List.length\n (Computation.corec\n (fun x =>\n match Seq.destruct x.snd with\n | none => Sum.inl (List.reverse x.fst)\n | some (none, s') => Sum.inr (x.fst, s')\n | some (some a, s') => Sum.inr (a :: x.fst, s'))\n (a :: l, s)) =\n Computation.map List.length\n (Computation.corec\n (fun x =>\n match Seq.destruct x.snd with\n | none => Sum.inl (List.reverse x.fst)\n | some (none, s') => Sum.inr (x.fst, s')\n | some (some a, s') => Sum.inr (a :: x.fst, s'))\n (l_1, s_1))", "tactic": "refine' ⟨a::l, s, _, _⟩ <;> simp" }, { "state_after": "no goals", "state_before": "case intro.intro.h3\nα : Type u\nβ : Type v\nγ : Type w\ns✝¹ : WSeq α\ns1 s2 : Computation ℕ\nl : List α\ns✝ : WSeq α\nh✝ :\n s1 =\n Computation.corec\n (fun x =>\n match x with\n | (n, s) =>\n match Seq.destruct s with\n | none => Sum.inl n\n | some (none, s') => Sum.inr (n, s')\n | some (some val, s') => Sum.inr (n + 1, s'))\n (List.length l, s✝) ∧\n s2 =\n Computation.map List.length\n (Computation.corec\n (fun x =>\n match x with\n | (l, s) =>\n match Seq.destruct s with\n | none => Sum.inl (List.reverse l)\n | some (none, s') => Sum.inr (l, s')\n | some (some a, s') => Sum.inr (a :: l, s'))\n (l, s✝))\ns : WSeq α\nh :\n s1 =\n Computation.corec\n (fun x =>\n match x with\n | (n, s) =>\n match Seq.destruct s with\n | none => Sum.inl n\n | some (none, s') => Sum.inr (n, s')\n | some (some val, s') => Sum.inr (n + 1, s'))\n (List.length l, think s) ∧\n s2 =\n Computation.map List.length\n (Computation.corec\n (fun x =>\n match x with\n | (l, s) =>\n match Seq.destruct s with\n | none => Sum.inl (List.reverse l)\n | some (none, s') => Sum.inr (l, s')\n | some (some a, s') => Sum.inr (a :: l, s'))\n (l, think s))\n⊢ ∃ l_1 s_1,\n Computation.corec\n (fun x =>\n match Seq.destruct x.snd with\n | none => Sum.inl x.fst\n | some (none, s') => Sum.inr (x.fst, s')\n | some (some val, s') => Sum.inr (x.fst + 1, s'))\n (List.length l, s) =\n Computation.corec\n (fun x =>\n match Seq.destruct x.snd with\n | none => Sum.inl x.fst\n | some (none, s') => Sum.inr (x.fst, s')\n | some (some val, s') => Sum.inr (x.fst + 1, s'))\n (List.length l_1, s_1) ∧\n Computation.map List.length\n (Computation.corec\n (fun x =>\n match Seq.destruct x.snd with\n | none => Sum.inl (List.reverse x.fst)\n | some (none, s') => Sum.inr (x.fst, s')\n | some (some a, s') => Sum.inr (a :: x.fst, s'))\n (l, s)) =\n Computation.map List.length\n (Computation.corec\n (fun x =>\n match Seq.destruct x.snd with\n | none => Sum.inl (List.reverse x.fst)\n | some (none, s') => Sum.inr (x.fst, s')\n | some (some a, s') => Sum.inr (a :: x.fst, s'))\n (l_1, s_1))", "tactic": "refine' ⟨l, s, _, _⟩ <;> simp" } ]

[ 1253, 34 ]

[ 1234, 1 ]

[]

[ 313, 15 ]

[ 312, 1 ]

[ { "state_after": "no goals", "state_before": "F : Type ?u.18184\nι : Type ?u.18187\nα : Type u_1\nβ : Type ?u.18193\ninst✝¹ : OrderedRing α\ninst✝ : Nontrivial α\nn : ℤ\n⊢ ↑n ≤ 0 ↔ n ≤ 0", "tactic": "rw [← cast_zero, cast_le]" } ]

[ 146, 28 ]

[ 145, 1 ]

[ { "state_after": "no goals", "state_before": "ctx : Context\ncs : Certificate\nh : PolyCnstr.denote ctx (combine cs) → False\n⊢ denote ctx cs", "tactic": "match cs with\n| [] => simp [combine, PolyCnstr.denote, Poly.denote_eq] at h\n| (k, c)::cs =>\n simp [denote, combine] at *\n intro h'\n apply of_combineHyps (h := h)\n simp [h']" }, { "state_after": "no goals", "state_before": "ctx : Context\ncs : Certificate\nh : PolyCnstr.denote ctx (combine []) → False\n⊢ denote ctx []", "tactic": "simp [combine, PolyCnstr.denote, Poly.denote_eq] at h" }, { "state_after": "ctx : Context\ncs✝ : Certificate\nk : Nat\nc : ExprCnstr\ncs : List (Nat × ExprCnstr)\nh : PolyCnstr.denote ctx (combineHyps (PolyCnstr.mul (k + 1) (ExprCnstr.toNormPoly c)) cs) → False\n⊢ ExprCnstr.denote ctx c → denote ctx cs", "state_before": "ctx : Context\ncs✝ : Certificate\nk : Nat\nc : ExprCnstr\ncs : List (Nat × ExprCnstr)\nh : PolyCnstr.denote ctx (combine ((k, c) :: cs)) → False\n⊢ denote ctx ((k, c) :: cs)", "tactic": "simp [denote, combine] at *" }, { "state_after": "ctx : Context\ncs✝ : Certificate\nk : Nat\nc : ExprCnstr\ncs : List (Nat × ExprCnstr)\nh : PolyCnstr.denote ctx (combineHyps (PolyCnstr.mul (k + 1) (ExprCnstr.toNormPoly c)) cs) → False\nh' : ExprCnstr.denote ctx c\n⊢ denote ctx cs", "state_before": "ctx : Context\ncs✝ : Certificate\nk : Nat\nc : ExprCnstr\ncs : List (Nat × ExprCnstr)\nh : PolyCnstr.denote ctx (combineHyps (PolyCnstr.mul (k + 1) (ExprCnstr.toNormPoly c)) cs) → False\n⊢ ExprCnstr.denote ctx c → denote ctx cs", "tactic": "intro h'" }, { "state_after": "ctx : Context\ncs✝ : Certificate\nk : Nat\nc : ExprCnstr\ncs : List (Nat × ExprCnstr)\nh : PolyCnstr.denote ctx (combineHyps (PolyCnstr.mul (k + 1) (ExprCnstr.toNormPoly c)) cs) → False\nh' : ExprCnstr.denote ctx c\n⊢ PolyCnstr.denote ctx (PolyCnstr.mul (k + 1) (ExprCnstr.toNormPoly c))", "state_before": "ctx : Context\ncs✝ : Certificate\nk : Nat\nc : ExprCnstr\ncs : List (Nat × ExprCnstr)\nh : PolyCnstr.denote ctx (combineHyps (PolyCnstr.mul (k + 1) (ExprCnstr.toNormPoly c)) cs) → False\nh' : ExprCnstr.denote ctx c\n⊢ denote ctx cs", "tactic": "apply of_combineHyps (h := h)" }, { "state_after": "no goals", "state_before": "ctx : Context\ncs✝ : Certificate\nk : Nat\nc : ExprCnstr\ncs : List (Nat × ExprCnstr)\nh : PolyCnstr.denote ctx (combineHyps (PolyCnstr.mul (k + 1) (ExprCnstr.toNormPoly c)) cs) → False\nh' : ExprCnstr.denote ctx c\n⊢ PolyCnstr.denote ctx (PolyCnstr.mul (k + 1) (ExprCnstr.toNormPoly c))", "tactic": "simp [h']" } ]

[ 689, 14 ]

[ 682, 1 ]

[]

[ 1022, 70 ]

[ 1020, 1 ]

[]

[ 123, 54 ]

[ 122, 1 ]

[]

[ 695, 6 ]

[ 693, 1 ]

[ { "state_after": "case H\nV : Type u_1\nW : Type u_2\nV₁ : Type ?u.501676\nV₂ : Type ?u.501679\nV₃ : Type ?u.501682\ninst✝⁷ : SeminormedAddCommGroup V\ninst✝⁶ : SeminormedAddCommGroup W\ninst✝⁵ : SeminormedAddCommGroup V₁\ninst✝⁴ : SeminormedAddCommGroup V₂\ninst✝³ : SeminormedAddCommGroup V₃\nf : NormedAddGroupHom V W\nW₁ : Type ?u.501714\nW₂ : Type ?u.501717\nW₃ : Type ?u.501720\ninst✝² : SeminormedAddCommGroup W₁\ninst✝¹ : SeminormedAddCommGroup W₂\ninst✝ : SeminormedAddCommGroup W₃\ng : NormedAddGroupHom V W\nf₁ g₁ : NormedAddGroupHom V₁ W₁\nf₂ g₂ : NormedAddGroupHom V₂ W₂\nf₃ g₃ : NormedAddGroupHom V₃ W₃\nx : { x // x ∈ equalizer f g }\n⊢ ↑(NormedAddGroupHom.comp f (ι f g)) x = ↑(NormedAddGroupHom.comp g (ι f g)) x", "state_before": "V : Type u_1\nW : Type u_2\nV₁ : Type ?u.501676\nV₂ : Type ?u.501679\nV₃ : Type ?u.501682\ninst✝⁷ : SeminormedAddCommGroup V\ninst✝⁶ : SeminormedAddCommGroup W\ninst✝⁵ : SeminormedAddCommGroup V₁\ninst✝⁴ : SeminormedAddCommGroup V₂\ninst✝³ : SeminormedAddCommGroup V₃\nf : NormedAddGroupHom V W\nW₁ : Type ?u.501714\nW₂ : Type ?u.501717\nW₃ : Type ?u.501720\ninst✝² : SeminormedAddCommGroup W₁\ninst✝¹ : SeminormedAddCommGroup W₂\ninst✝ : SeminormedAddCommGroup W₃\ng : NormedAddGroupHom V W\nf₁ g₁ : NormedAddGroupHom V₁ W₁\nf₂ g₂ : NormedAddGroupHom V₂ W₂\nf₃ g₃ : NormedAddGroupHom V₃ W₃\n⊢ NormedAddGroupHom.comp f (ι f g) = NormedAddGroupHom.comp g (ι f g)", "tactic": "ext x" }, { "state_after": "case H\nV : Type u_1\nW : Type u_2\nV₁ : Type ?u.501676\nV₂ : Type ?u.501679\nV₃ : Type ?u.501682\ninst✝⁷ : SeminormedAddCommGroup V\ninst✝⁶ : SeminormedAddCommGroup W\ninst✝⁵ : SeminormedAddCommGroup V₁\ninst✝⁴ : SeminormedAddCommGroup V₂\ninst✝³ : SeminormedAddCommGroup V₃\nf : NormedAddGroupHom V W\nW₁ : Type ?u.501714\nW₂ : Type ?u.501717\nW₃ : Type ?u.501720\ninst✝² : SeminormedAddCommGroup W₁\ninst✝¹ : SeminormedAddCommGroup W₂\ninst✝ : SeminormedAddCommGroup W₃\ng : NormedAddGroupHom V W\nf₁ g₁ : NormedAddGroupHom V₁ W₁\nf₂ g₂ : NormedAddGroupHom V₂ W₂\nf₃ g₃ : NormedAddGroupHom V₃ W₃\nx : { x // x ∈ equalizer f g }\n⊢ ↑(f - g) (↑(ι f g) x) = 0", "state_before": "case H\nV : Type u_1\nW : Type u_2\nV₁ : Type ?u.501676\nV₂ : Type ?u.501679\nV₃ : Type ?u.501682\ninst✝⁷ : SeminormedAddCommGroup V\ninst✝⁶ : SeminormedAddCommGroup W\ninst✝⁵ : SeminormedAddCommGroup V₁\ninst✝⁴ : SeminormedAddCommGroup V₂\ninst✝³ : SeminormedAddCommGroup V₃\nf : NormedAddGroupHom V W\nW₁ : Type ?u.501714\nW₂ : Type ?u.501717\nW₃ : Type ?u.501720\ninst✝² : SeminormedAddCommGroup W₁\ninst✝¹ : SeminormedAddCommGroup W₂\ninst✝ : SeminormedAddCommGroup W₃\ng : NormedAddGroupHom V W\nf₁ g₁ : NormedAddGroupHom V₁ W₁\nf₂ g₂ : NormedAddGroupHom V₂ W₂\nf₃ g₃ : NormedAddGroupHom V₃ W₃\nx : { x // x ∈ equalizer f g }\n⊢ ↑(NormedAddGroupHom.comp f (ι f g)) x = ↑(NormedAddGroupHom.comp g (ι f g)) x", "tactic": "rw [comp_apply, comp_apply, ← sub_eq_zero, ← NormedAddGroupHom.sub_apply]" }, { "state_after": "no goals", "state_before": "case H\nV : Type u_1\nW : Type u_2\nV₁ : Type ?u.501676\nV₂ : Type ?u.501679\nV₃ : Type ?u.501682\ninst✝⁷ : SeminormedAddCommGroup V\ninst✝⁶ : SeminormedAddCommGroup W\ninst✝⁵ : SeminormedAddCommGroup V₁\ninst✝⁴ : SeminormedAddCommGroup V₂\ninst✝³ : SeminormedAddCommGroup V₃\nf : NormedAddGroupHom V W\nW₁ : Type ?u.501714\nW₂ : Type ?u.501717\nW₃ : Type ?u.501720\ninst✝² : SeminormedAddCommGroup W₁\ninst✝¹ : SeminormedAddCommGroup W₂\ninst✝ : SeminormedAddCommGroup W₃\ng : NormedAddGroupHom V W\nf₁ g₁ : NormedAddGroupHom V₁ W₁\nf₂ g₂ : NormedAddGroupHom V₂ W₂\nf₃ g₃ : NormedAddGroupHom V₃ W₃\nx : { x // x ∈ equalizer f g }\n⊢ ↑(f - g) (↑(ι f g) x) = 0", "tactic": "exact x.2" } ]

[ 899, 12 ]

[ 896, 1 ]

[ { "state_after": "α : Type u_2\nι : Type u_1\nI : Set ι\nhI : Set.Nonempty I\nm : ι → OuterMeasure α\ns : Set α\nthis : Nonempty ↑I\n⊢ ↑(⨅ (i : ι) (_ : i ∈ I), m i) s =\n ⨅ (t : ℕ → Set α) (_ : s ⊆ iUnion t), ∑' (n : ℕ), ⨅ (i : ι) (_ : i ∈ I), ↑(m i) (t n)", "state_before": "α : Type u_2\nι : Type u_1\nI : Set ι\nhI : Set.Nonempty I\nm : ι → OuterMeasure α\ns : Set α\n⊢ ↑(⨅ (i : ι) (_ : i ∈ I), m i) s =\n ⨅ (t : ℕ → Set α) (_ : s ⊆ iUnion t), ∑' (n : ℕ), ⨅ (i : ι) (_ : i ∈ I), ↑(m i) (t n)", "tactic": "haveI := hI.to_subtype" }, { "state_after": "no goals", "state_before": "α : Type u_2\nι : Type u_1\nI : Set ι\nhI : Set.Nonempty I\nm : ι → OuterMeasure α\ns : Set α\nthis : Nonempty ↑I\n⊢ ↑(⨅ (i : ι) (_ : i ∈ I), m i) s =\n ⨅ (t : ℕ → Set α) (_ : s ⊆ iUnion t), ∑' (n : ℕ), ⨅ (i : ι) (_ : i ∈ I), ↑(m i) (t n)", "tactic": "simp only [← iInf_subtype'', iInf_apply]" } ]

[ 1211, 43 ]

[ 1208, 1 ]

[]

[ 77, 27 ]

[ 75, 1 ]

[]

[ 346, 95 ]

[ 346, 9 ]

[]

[ 699, 21 ]

[ 698, 1 ]

[]

[ 257, 9 ]

[ 256, 1 ]

[ { "state_after": "C : Type u\ninst✝² : Category C\nX✝ Y✝ W X Y Z : C\ninst✝¹ : HasBinaryCoproduct W Y\ninst✝ : HasBinaryCoproduct X Z\ng : W ⟶ X\ng' : Y ⟶ Z\n⊢ map g g' ≫ desc inl inr = map g g'", "state_before": "C : Type u\ninst✝² : Category C\nX✝ Y✝ W X Y Z : C\ninst✝¹ : HasBinaryCoproduct W Y\ninst✝ : HasBinaryCoproduct X Z\ng : W ⟶ X\ng' : Y ⟶ Z\n⊢ desc (g ≫ inl) (g' ≫ inr) = map g g'", "tactic": "rw [← coprod.map_desc]" }, { "state_after": "no goals", "state_before": "C : Type u\ninst✝² : Category C\nX✝ Y✝ W X Y Z : C\ninst✝¹ : HasBinaryCoproduct W Y\ninst✝ : HasBinaryCoproduct X Z\ng : W ⟶ X\ng' : Y ⟶ Z\n⊢ map g g' ≫ desc inl inr = map g g'", "tactic": "simp" } ]

[ 875, 31 ]

[ 872, 1 ]

[ { "state_after": "ι : Type ?u.4550285\nα : Type u_2\nβ✝ : Type ?u.4550291\nγ : Type ?u.4550294\nδ : Type ?u.4550297\nR : Type ?u.4550300\nm0 : MeasurableSpace α\ninst✝⁴ : MeasurableSpace β✝\ninst✝³ : MeasurableSpace γ\ninst✝² : MeasurableSpace δ\nf g✝ : α → β✝\nμ ν : MeasureTheory.Measure α\nβ : Type u_1\nmβ : MeasurableSpace β\ninst✝¹ : LinearOrder α\ninst✝ : IsCountablyGenerated atTop\nx : α\ng : α → β\ng_meas : ∀ (t : α), t > x → AEMeasurable g\nthis : Nonempty α\n⊢ AEMeasurable g", "state_before": "ι : Type ?u.4550285\nα : Type u_2\nβ✝ : Type ?u.4550291\nγ : Type ?u.4550294\nδ : Type ?u.4550297\nR : Type ?u.4550300\nm0 : MeasurableSpace α\ninst✝⁴ : MeasurableSpace β✝\ninst✝³ : MeasurableSpace γ\ninst✝² : MeasurableSpace δ\nf g✝ : α → β✝\nμ ν : MeasureTheory.Measure α\nβ : Type u_1\nmβ : MeasurableSpace β\ninst✝¹ : LinearOrder α\ninst✝ : IsCountablyGenerated atTop\nx : α\ng : α → β\ng_meas : ∀ (t : α), t > x → AEMeasurable g\n⊢ AEMeasurable g", "tactic": "haveI : Nonempty α := ⟨x⟩" }, { "state_after": "case intro\nι : Type ?u.4550285\nα : Type u_2\nβ✝ : Type ?u.4550291\nγ : Type ?u.4550294\nδ : Type ?u.4550297\nR : Type ?u.4550300\nm0 : MeasurableSpace α\ninst✝⁴ : MeasurableSpace β✝\ninst✝³ : MeasurableSpace γ\ninst✝² : MeasurableSpace δ\nf g✝ : α → β✝\nμ ν : MeasureTheory.Measure α\nβ : Type u_1\nmβ : MeasurableSpace β\ninst✝¹ : LinearOrder α\ninst✝ : IsCountablyGenerated atTop\nx : α\ng : α → β\ng_meas : ∀ (t : α), t > x → AEMeasurable g\nthis : Nonempty α\nu : ℕ → α\nhu_tendsto : Tendsto u atTop atTop\n⊢ AEMeasurable g", "state_before": "ι : Type ?u.4550285\nα : Type u_2\nβ✝ : Type ?u.4550291\nγ : Type ?u.4550294\nδ : Type ?u.4550297\nR : Type ?u.4550300\nm0 : MeasurableSpace α\ninst✝⁴ : MeasurableSpace β✝\ninst✝³ : MeasurableSpace γ\ninst✝² : MeasurableSpace δ\nf g✝ : α → β✝\nμ ν : MeasureTheory.Measure α\nβ : Type u_1\nmβ : MeasurableSpace β\ninst✝¹ : LinearOrder α\ninst✝ : IsCountablyGenerated atTop\nx : α\ng : α → β\ng_meas : ∀ (t : α), t > x → AEMeasurable g\nthis : Nonempty α\n⊢ AEMeasurable g", "tactic": "obtain ⟨u, hu_tendsto⟩ := exists_seq_tendsto (atTop : Filter α)" }, { "state_after": "case intro\nι : Type ?u.4550285\nα : Type u_2\nβ✝ : Type ?u.4550291\nγ : Type ?u.4550294\nδ : Type ?u.4550297\nR : Type ?u.4550300\nm0 : MeasurableSpace α\ninst✝⁴ : MeasurableSpace β✝\ninst✝³ : MeasurableSpace γ\ninst✝² : MeasurableSpace δ\nf g✝ : α → β✝\nμ ν : MeasureTheory.Measure α\nβ : Type u_1\nmβ : MeasurableSpace β\ninst✝¹ : LinearOrder α\ninst✝ : IsCountablyGenerated atTop\nx : α\ng : α → β\ng_meas : ∀ (t : α), t > x → AEMeasurable g\nthis : Nonempty α\nu : ℕ → α\nhu_tendsto : Tendsto u atTop atTop\nIoi_eq_iUnion : Ioi x = ⋃ (n : ℕ), Ioc x (u n)\n⊢ AEMeasurable g", "state_before": "case intro\nι : Type ?u.4550285\nα : Type u_2\nβ✝ : Type ?u.4550291\nγ : Type ?u.4550294\nδ : Type ?u.4550297\nR : Type ?u.4550300\nm0 : MeasurableSpace α\ninst✝⁴ : MeasurableSpace β✝\ninst✝³ : MeasurableSpace γ\ninst✝² : MeasurableSpace δ\nf g✝ : α → β✝\nμ ν : MeasureTheory.Measure α\nβ : Type u_1\nmβ : MeasurableSpace β\ninst✝¹ : LinearOrder α\ninst✝ : IsCountablyGenerated atTop\nx : α\ng : α → β\ng_meas : ∀ (t : α), t > x → AEMeasurable g\nthis : Nonempty α\nu : ℕ → α\nhu_tendsto : Tendsto u atTop atTop\n⊢ AEMeasurable g", "tactic": "have Ioi_eq_iUnion : Ioi x = ⋃ n : ℕ, Ioc x (u n) := by\n rw [iUnion_Ioc_eq_Ioi_self_iff.mpr _]\n exact fun y _ => (hu_tendsto.eventually (eventually_ge_atTop y)).exists" }, { "state_after": "case intro\nι : Type ?u.4550285\nα : Type u_2\nβ✝ : Type ?u.4550291\nγ : Type ?u.4550294\nδ : Type ?u.4550297\nR : Type ?u.4550300\nm0 : MeasurableSpace α\ninst✝⁴ : MeasurableSpace β✝\ninst✝³ : MeasurableSpace γ\ninst✝² : MeasurableSpace δ\nf g✝ : α → β✝\nμ ν : MeasureTheory.Measure α\nβ : Type u_1\nmβ : MeasurableSpace β\ninst✝¹ : LinearOrder α\ninst✝ : IsCountablyGenerated atTop\nx : α\ng : α → β\ng_meas : ∀ (t : α), t > x → AEMeasurable g\nthis : Nonempty α\nu : ℕ → α\nhu_tendsto : Tendsto u atTop atTop\nIoi_eq_iUnion : Ioi x = ⋃ (n : ℕ), Ioc x (u n)\n⊢ ∀ (i : ℕ), AEMeasurable g", "state_before": "case intro\nι : Type ?u.4550285\nα : Type u_2\nβ✝ : Type ?u.4550291\nγ : Type ?u.4550294\nδ : Type ?u.4550297\nR : Type ?u.4550300\nm0 : MeasurableSpace α\ninst✝⁴ : MeasurableSpace β✝\ninst✝³ : MeasurableSpace γ\ninst✝² : MeasurableSpace δ\nf g✝ : α → β✝\nμ ν : MeasureTheory.Measure α\nβ : Type u_1\nmβ : MeasurableSpace β\ninst✝¹ : LinearOrder α\ninst✝ : IsCountablyGenerated atTop\nx : α\ng : α → β\ng_meas : ∀ (t : α), t > x → AEMeasurable g\nthis : Nonempty α\nu : ℕ → α\nhu_tendsto : Tendsto u atTop atTop\nIoi_eq_iUnion : Ioi x = ⋃ (n : ℕ), Ioc x (u n)\n⊢ AEMeasurable g", "tactic": "rw [Ioi_eq_iUnion, aemeasurable_iUnion_iff]" }, { "state_after": "case intro\nι : Type ?u.4550285\nα : Type u_2\nβ✝ : Type ?u.4550291\nγ : Type ?u.4550294\nδ : Type ?u.4550297\nR : Type ?u.4550300\nm0 : MeasurableSpace α\ninst✝⁴ : MeasurableSpace β✝\ninst✝³ : MeasurableSpace γ\ninst✝² : MeasurableSpace δ\nf g✝ : α → β✝\nμ ν : MeasureTheory.Measure α\nβ : Type u_1\nmβ : MeasurableSpace β\ninst✝¹ : LinearOrder α\ninst✝ : IsCountablyGenerated atTop\nx : α\ng : α → β\ng_meas : ∀ (t : α), t > x → AEMeasurable g\nthis : Nonempty α\nu : ℕ → α\nhu_tendsto : Tendsto u atTop atTop\nIoi_eq_iUnion : Ioi x = ⋃ (n : ℕ), Ioc x (u n)\nn : ℕ\n⊢ AEMeasurable g", "state_before": "case intro\nι : Type ?u.4550285\nα : Type u_2\nβ✝ : Type ?u.4550291\nγ : Type ?u.4550294\nδ : Type ?u.4550297\nR : Type ?u.4550300\nm0 : MeasurableSpace α\ninst✝⁴ : MeasurableSpace β✝\ninst✝³ : MeasurableSpace γ\ninst✝² : MeasurableSpace δ\nf g✝ : α → β✝\nμ ν : MeasureTheory.Measure α\nβ : Type u_1\nmβ : MeasurableSpace β\ninst✝¹ : LinearOrder α\ninst✝ : IsCountablyGenerated atTop\nx : α\ng : α → β\ng_meas : ∀ (t : α), t > x → AEMeasurable g\nthis : Nonempty α\nu : ℕ → α\nhu_tendsto : Tendsto u atTop atTop\nIoi_eq_iUnion : Ioi x = ⋃ (n : ℕ), Ioc x (u n)\n⊢ ∀ (i : ℕ), AEMeasurable g", "tactic": "intro n" }, { "state_after": "case intro.inl\nι : Type ?u.4550285\nα : Type u_2\nβ✝ : Type ?u.4550291\nγ : Type ?u.4550294\nδ : Type ?u.4550297\nR : Type ?u.4550300\nm0 : MeasurableSpace α\ninst✝⁴ : MeasurableSpace β✝\ninst✝³ : MeasurableSpace γ\ninst✝² : MeasurableSpace δ\nf g✝ : α → β✝\nμ ν : MeasureTheory.Measure α\nβ : Type u_1\nmβ : MeasurableSpace β\ninst✝¹ : LinearOrder α\ninst✝ : IsCountablyGenerated atTop\nx : α\ng : α → β\ng_meas : ∀ (t : α), t > x → AEMeasurable g\nthis : Nonempty α\nu : ℕ → α\nhu_tendsto : Tendsto u atTop atTop\nIoi_eq_iUnion : Ioi x = ⋃ (n : ℕ), Ioc x (u n)\nn : ℕ\nh : x < u n\n⊢ AEMeasurable g\n\ncase intro.inr\nι : Type ?u.4550285\nα : Type u_2\nβ✝ : Type ?u.4550291\nγ : Type ?u.4550294\nδ : Type ?u.4550297\nR : Type ?u.4550300\nm0 : MeasurableSpace α\ninst✝⁴ : MeasurableSpace β✝\ninst✝³ : MeasurableSpace γ\ninst✝² : MeasurableSpace δ\nf g✝ : α → β✝\nμ ν : MeasureTheory.Measure α\nβ : Type u_1\nmβ : MeasurableSpace β\ninst✝¹ : LinearOrder α\ninst✝ : IsCountablyGenerated atTop\nx : α\ng : α → β\ng_meas : ∀ (t : α), t > x → AEMeasurable g\nthis : Nonempty α\nu : ℕ → α\nhu_tendsto : Tendsto u atTop atTop\nIoi_eq_iUnion : Ioi x = ⋃ (n : ℕ), Ioc x (u n)\nn : ℕ\nh : u n ≤ x\n⊢ AEMeasurable g", "state_before": "case intro\nι : Type ?u.4550285\nα : Type u_2\nβ✝ : Type ?u.4550291\nγ : Type ?u.4550294\nδ : Type ?u.4550297\nR : Type ?u.4550300\nm0 : MeasurableSpace α\ninst✝⁴ : MeasurableSpace β✝\ninst✝³ : MeasurableSpace γ\ninst✝² : MeasurableSpace δ\nf g✝ : α → β✝\nμ ν : MeasureTheory.Measure α\nβ : Type u_1\nmβ : MeasurableSpace β\ninst✝¹ : LinearOrder α\ninst✝ : IsCountablyGenerated atTop\nx : α\ng : α → β\ng_meas : ∀ (t : α), t > x → AEMeasurable g\nthis : Nonempty α\nu : ℕ → α\nhu_tendsto : Tendsto u atTop atTop\nIoi_eq_iUnion : Ioi x = ⋃ (n : ℕ), Ioc x (u n)\nn : ℕ\n⊢ AEMeasurable g", "tactic": "cases' lt_or_le x (u n) with h h" }, { "state_after": "ι : Type ?u.4550285\nα : Type u_2\nβ✝ : Type ?u.4550291\nγ : Type ?u.4550294\nδ : Type ?u.4550297\nR : Type ?u.4550300\nm0 : MeasurableSpace α\ninst✝⁴ : MeasurableSpace β✝\ninst✝³ : MeasurableSpace γ\ninst✝² : MeasurableSpace δ\nf g✝ : α → β✝\nμ ν : MeasureTheory.Measure α\nβ : Type u_1\nmβ : MeasurableSpace β\ninst✝¹ : LinearOrder α\ninst✝ : IsCountablyGenerated atTop\nx : α\ng : α → β\ng_meas : ∀ (t : α), t > x → AEMeasurable g\nthis : Nonempty α\nu : ℕ → α\nhu_tendsto : Tendsto u atTop atTop\n⊢ ∀ (x_1 : α), x < x_1 → ∃ i, x_1 ≤ u i", "state_before": "ι : Type ?u.4550285\nα : Type u_2\nβ✝ : Type ?u.4550291\nγ : Type ?u.4550294\nδ : Type ?u.4550297\nR : Type ?u.4550300\nm0 : MeasurableSpace α\ninst✝⁴ : MeasurableSpace β✝\ninst✝³ : MeasurableSpace γ\ninst✝² : MeasurableSpace δ\nf g✝ : α → β✝\nμ ν : MeasureTheory.Measure α\nβ : Type u_1\nmβ : MeasurableSpace β\ninst✝¹ : LinearOrder α\ninst✝ : IsCountablyGenerated atTop\nx : α\ng : α → β\ng_meas : ∀ (t : α), t > x → AEMeasurable g\nthis : Nonempty α\nu : ℕ → α\nhu_tendsto : Tendsto u atTop atTop\n⊢ Ioi x = ⋃ (n : ℕ), Ioc x (u n)", "tactic": "rw [iUnion_Ioc_eq_Ioi_self_iff.mpr _]" }, { "state_after": "no goals", "state_before": "ι : Type ?u.4550285\nα : Type u_2\nβ✝ : Type ?u.4550291\nγ : Type ?u.4550294\nδ : Type ?u.4550297\nR : Type ?u.4550300\nm0 : MeasurableSpace α\ninst✝⁴ : MeasurableSpace β✝\ninst✝³ : MeasurableSpace γ\ninst✝² : MeasurableSpace δ\nf g✝ : α → β✝\nμ ν : MeasureTheory.Measure α\nβ : Type u_1\nmβ : MeasurableSpace β\ninst✝¹ : LinearOrder α\ninst✝ : IsCountablyGenerated atTop\nx : α\ng : α → β\ng_meas : ∀ (t : α), t > x → AEMeasurable g\nthis : Nonempty α\nu : ℕ → α\nhu_tendsto : Tendsto u atTop atTop\n⊢ ∀ (x_1 : α), x < x_1 → ∃ i, x_1 ≤ u i", "tactic": "exact fun y _ => (hu_tendsto.eventually (eventually_ge_atTop y)).exists" }, { "state_after": "no goals", "state_before": "case intro.inl\nι : Type ?u.4550285\nα : Type u_2\nβ✝ : Type ?u.4550291\nγ : Type ?u.4550294\nδ : Type ?u.4550297\nR : Type ?u.4550300\nm0 : MeasurableSpace α\ninst✝⁴ : MeasurableSpace β✝\ninst✝³ : MeasurableSpace γ\ninst✝² : MeasurableSpace δ\nf g✝ : α → β✝\nμ ν : MeasureTheory.Measure α\nβ : Type u_1\nmβ : MeasurableSpace β\ninst✝¹ : LinearOrder α\ninst✝ : IsCountablyGenerated atTop\nx : α\ng : α → β\ng_meas : ∀ (t : α), t > x → AEMeasurable g\nthis : Nonempty α\nu : ℕ → α\nhu_tendsto : Tendsto u atTop atTop\nIoi_eq_iUnion : Ioi x = ⋃ (n : ℕ), Ioc x (u n)\nn : ℕ\nh : x < u n\n⊢ AEMeasurable g", "tactic": "exact g_meas (u n) h" }, { "state_after": "case intro.inr\nι : Type ?u.4550285\nα : Type u_2\nβ✝ : Type ?u.4550291\nγ : Type ?u.4550294\nδ : Type ?u.4550297\nR : Type ?u.4550300\nm0 : MeasurableSpace α\ninst✝⁴ : MeasurableSpace β✝\ninst✝³ : MeasurableSpace γ\ninst✝² : MeasurableSpace δ\nf g✝ : α → β✝\nμ ν : MeasureTheory.Measure α\nβ : Type u_1\nmβ : MeasurableSpace β\ninst✝¹ : LinearOrder α\ninst✝ : IsCountablyGenerated atTop\nx : α\ng : α → β\ng_meas : ∀ (t : α), t > x → AEMeasurable g\nthis : Nonempty α\nu : ℕ → α\nhu_tendsto : Tendsto u atTop atTop\nIoi_eq_iUnion : Ioi x = ⋃ (n : ℕ), Ioc x (u n)\nn : ℕ\nh : u n ≤ x\n⊢ AEMeasurable g", "state_before": "case intro.inr\nι : Type ?u.4550285\nα : Type u_2\nβ✝ : Type ?u.4550291\nγ : Type ?u.4550294\nδ : Type ?u.4550297\nR : Type ?u.4550300\nm0 : MeasurableSpace α\ninst✝⁴ : MeasurableSpace β✝\ninst✝³ : MeasurableSpace γ\ninst✝² : MeasurableSpace δ\nf g✝ : α → β✝\nμ ν : MeasureTheory.Measure α\nβ : Type u_1\nmβ : MeasurableSpace β\ninst✝¹ : LinearOrder α\ninst✝ : IsCountablyGenerated atTop\nx : α\ng : α → β\ng_meas : ∀ (t : α), t > x → AEMeasurable g\nthis : Nonempty α\nu : ℕ → α\nhu_tendsto : Tendsto u atTop atTop\nIoi_eq_iUnion : Ioi x = ⋃ (n : ℕ), Ioc x (u n)\nn : ℕ\nh : u n ≤ x\n⊢ AEMeasurable g", "tactic": "rw [Ioc_eq_empty (not_lt.mpr h), Measure.restrict_empty]" }, { "state_after": "no goals", "state_before": "case intro.inr\nι : Type ?u.4550285\nα : Type u_2\nβ✝ : Type ?u.4550291\nγ : Type ?u.4550294\nδ : Type ?u.4550297\nR : Type ?u.4550300\nm0 : MeasurableSpace α\ninst✝⁴ : MeasurableSpace β✝\ninst✝³ : MeasurableSpace γ\ninst✝² : MeasurableSpace δ\nf g✝ : α → β✝\nμ ν : MeasureTheory.Measure α\nβ : Type u_1\nmβ : MeasurableSpace β\ninst✝¹ : LinearOrder α\ninst✝ : IsCountablyGenerated atTop\nx : α\ng : α → β\ng_meas : ∀ (t : α), t > x → AEMeasurable g\nthis : Nonempty α\nu : ℕ → α\nhu_tendsto : Tendsto u atTop atTop\nIoi_eq_iUnion : Ioi x = ⋃ (n : ℕ), Ioc x (u n)\nn : ℕ\nh : u n ≤ x\n⊢ AEMeasurable g", "tactic": "exact aemeasurable_zero_measure" } ]

[ 331, 36 ]

[ 317, 1 ]

[ { "state_after": "case h.none\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\nκ : Type ?u.661354\nα : Option ι → Type v\ninst✝¹ : DecidableEq ι\ninst✝ : (i : Option ι) → Zero (α i)\ni : ι\nx : α (some i)\n⊢ ↑(extendWith 0 (single i x)) none = ↑(single (some i) x) none\n\ncase h.some\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\nκ : Type ?u.661354\nα : Option ι → Type v\ninst✝¹ : DecidableEq ι\ninst✝ : (i : Option ι) → Zero (α i)\ni : ι\nx : α (some i)\nj : ι\n⊢ ↑(extendWith 0 (single i x)) (some j) = ↑(single (some i) x) (some j)", "state_before": "ι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\nκ : Type ?u.661354\nα : Option ι → Type v\ninst✝¹ : DecidableEq ι\ninst✝ : (i : Option ι) → Zero (α i)\ni : ι\nx : α (some i)\n⊢ extendWith 0 (single i x) = single (some i) x", "tactic": "ext (_ | j)" }, { "state_after": "no goals", "state_before": "case h.none\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\nκ : Type ?u.661354\nα : Option ι → Type v\ninst✝¹ : DecidableEq ι\ninst✝ : (i : Option ι) → Zero (α i)\ni : ι\nx : α (some i)\n⊢ ↑(extendWith 0 (single i x)) none = ↑(single (some i) x) none", "tactic": "rw [extendWith_none, single_eq_of_ne (Option.some_ne_none _)]" }, { "state_after": "case h.some\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\nκ : Type ?u.661354\nα : Option ι → Type v\ninst✝¹ : DecidableEq ι\ninst✝ : (i : Option ι) → Zero (α i)\ni : ι\nx : α (some i)\nj : ι\n⊢ ↑(single i x) j = ↑(single (some i) x) (some j)", "state_before": "case h.some\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\nκ : Type ?u.661354\nα : Option ι → Type v\ninst✝¹ : DecidableEq ι\ninst✝ : (i : Option ι) → Zero (α i)\ni : ι\nx : α (some i)\nj : ι\n⊢ ↑(extendWith 0 (single i x)) (some j) = ↑(single (some i) x) (some j)", "tactic": "rw [extendWith_some]" }, { "state_after": "case h.some.inl\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\nκ : Type ?u.661354\nα : Option ι → Type v\ninst✝¹ : DecidableEq ι\ninst✝ : (i : Option ι) → Zero (α i)\ni : ι\nx : α (some i)\n⊢ ↑(single i x) i = ↑(single (some i) x) (some i)\n\ncase h.some.inr\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\nκ : Type ?u.661354\nα : Option ι → Type v\ninst✝¹ : DecidableEq ι\ninst✝ : (i : Option ι) → Zero (α i)\ni : ι\nx : α (some i)\nj : ι\nhij : i ≠ j\n⊢ ↑(single i x) j = ↑(single (some i) x) (some j)", "state_before": "case h.some\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\nκ : Type ?u.661354\nα : Option ι → Type v\ninst✝¹ : DecidableEq ι\ninst✝ : (i : Option ι) → Zero (α i)\ni : ι\nx : α (some i)\nj : ι\n⊢ ↑(single i x) j = ↑(single (some i) x) (some j)", "tactic": "obtain rfl | hij := Decidable.eq_or_ne i j" }, { "state_after": "no goals", "state_before": "case h.some.inl\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\nκ : Type ?u.661354\nα : Option ι → Type v\ninst✝¹ : DecidableEq ι\ninst✝ : (i : Option ι) → Zero (α i)\ni : ι\nx : α (some i)\n⊢ ↑(single i x) i = ↑(single (some i) x) (some i)", "tactic": "rw [single_eq_same, single_eq_same]" }, { "state_after": "no goals", "state_before": "case h.some.inr\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\nκ : Type ?u.661354\nα : Option ι → Type v\ninst✝¹ : DecidableEq ι\ninst✝ : (i : Option ι) → Zero (α i)\ni : ι\nx : α (some i)\nj : ι\nhij : i ≠ j\n⊢ ↑(single i x) j = ↑(single (some i) x) (some j)", "tactic": "rw [single_eq_of_ne hij, single_eq_of_ne ((Option.some_injective _).ne hij)]" } ]

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[]

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[]

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[]

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[]

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[ { "state_after": "a b c : ℚ\nn d : ℤ\n⊢ n /. 1 / (d /. 1) = n /. d", "state_before": "a b c : ℚ\nn d : ℤ\n⊢ ↑n / ↑d = n /. d", "tactic": "repeat' rw [coe_int_eq_divInt]" }, { "state_after": "no goals", "state_before": "a b c : ℚ\nn d : ℤ\n⊢ n /. 1 / (d /. 1) = n /. d", "tactic": "exact divInt_div_divInt_cancel_left one_ne_zero n d" }, { "state_after": "a b c : ℚ\nn d : ℤ\n⊢ n /. 1 / (d /. 1) = n /. d", "state_before": "a b c : ℚ\nn d : ℤ\n⊢ n /. 1 / (d /. 1) = n /. d", "tactic": "rw [coe_int_eq_divInt]" } ]

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[ 503, 1 ]

[ { "state_after": "case a.h\nl : Type ?u.279030\nm : Type u_1\nn : Type u_2\no : Type ?u.279039\nm' : o → Type ?u.279044\nn' : o → Type ?u.279049\nR : Type ?u.279052\nS : Type ?u.279055\nα : Type v\nβ : Type w\nγ : Type ?u.279062\ninst✝² : NonUnitalNonAssocSemiring α\ninst✝¹ : Fintype m\ninst✝ : DecidableEq m\nM : Matrix m n α\na : α\ni✝ : m\nx✝ : n\n⊢ (a • M) i✝ x✝ = ((diagonal fun x => a) ⬝ M) i✝ x✝", "state_before": "l : Type ?u.279030\nm : Type u_1\nn : Type u_2\no : Type ?u.279039\nm' : o → Type ?u.279044\nn' : o → Type ?u.279049\nR : Type ?u.279052\nS : Type ?u.279055\nα : Type v\nβ : Type w\nγ : Type ?u.279062\ninst✝² : NonUnitalNonAssocSemiring α\ninst✝¹ : Fintype m\ninst✝ : DecidableEq m\nM : Matrix m n α\na : α\n⊢ a • M = (diagonal fun x => a) ⬝ M", "tactic": "ext" }, { "state_after": "no goals", "state_before": "case a.h\nl : Type ?u.279030\nm : Type u_1\nn : Type u_2\no : Type ?u.279039\nm' : o → Type ?u.279044\nn' : o → Type ?u.279049\nR : Type ?u.279052\nS : Type ?u.279055\nα : Type v\nβ : Type w\nγ : Type ?u.279062\ninst✝² : NonUnitalNonAssocSemiring α\ninst✝¹ : Fintype m\ninst✝ : DecidableEq m\nM : Matrix m n α\na : α\ni✝ : m\nx✝ : n\n⊢ (a • M) i✝ x✝ = ((diagonal fun x => a) ⬝ M) i✝ x✝", "tactic": "simp" } ]

[ 1051, 7 ]

[ 1048, 1 ]

[]

[ 1010, 6 ]

[ 1007, 1 ]

[ { "state_after": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nf : ℂ → E\nc : ℂ\nR : ℝ\nn : ℕ\nw : ℂ\n⊢ (w ^ n • (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), (z - c)⁻¹ ^ n • (z - c)⁻¹ • f z) =\n (2 * ↑π * I)⁻¹ • w ^ n • ∮ (z : ℂ) in C(c, R), (z - c)⁻¹ ^ n • (z - c)⁻¹ • f z", "state_before": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nf : ℂ → E\nc : ℂ\nR : ℝ\nn : ℕ\nw : ℂ\n⊢ (↑(cauchyPowerSeries f c R n) fun x => w) = (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), (w / (z - c)) ^ n • (z - c)⁻¹ • f z", "tactic": "simp only [cauchyPowerSeries, ContinuousMultilinearMap.mkPiField_apply, Fin.prod_const,\n div_eq_mul_inv, mul_pow, mul_smul, circleIntegral.integral_smul]" }, { "state_after": "no goals", "state_before": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nf : ℂ → E\nc : ℂ\nR : ℝ\nn : ℕ\nw : ℂ\n⊢ (w ^ n • (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, R), (z - c)⁻¹ ^ n • (z - c)⁻¹ • f z) =\n (2 * ↑π * I)⁻¹ • w ^ n • ∮ (z : ℂ) in C(c, R), (z - c)⁻¹ ^ n • (z - c)⁻¹ • f z", "tactic": "rw [← smul_comm (w ^ n)]" } ]

[ 538, 27 ]

[ 533, 1 ]

[]

[ 1389, 6 ]

[ 1385, 1 ]

[]

[ 289, 6 ]

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[]

[ 502, 95 ]

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[ { "state_after": "case inl\nb n m : ℕ\nh : n ≤ m\nhb : b ≤ 1\n⊢ clog b n ≤ clog b m\n\ncase inr\nb n m : ℕ\nh : n ≤ m\nhb : 1 < b\n⊢ clog b n ≤ clog b m", "state_before": "b n m : ℕ\nh : n ≤ m\n⊢ clog b n ≤ clog b m", "tactic": "cases' le_or_lt b 1 with hb hb" }, { "state_after": "case inl\nb n m : ℕ\nh : n ≤ m\nhb : b ≤ 1\n⊢ 0 ≤ clog b m", "state_before": "case inl\nb n m : ℕ\nh : n ≤ m\nhb : b ≤ 1\n⊢ clog b n ≤ clog b m", "tactic": "rw [clog_of_left_le_one hb]" }, { "state_after": "no goals", "state_before": "case inl\nb n m : ℕ\nh : n ≤ m\nhb : b ≤ 1\n⊢ 0 ≤ clog b m", "tactic": "exact zero_le _" }, { "state_after": "case inr\nb n m : ℕ\nh : n ≤ m\nhb : 1 < b\n⊢ n ≤ b ^ clog b m", "state_before": "case inr\nb n m : ℕ\nh : n ≤ m\nhb : 1 < b\n⊢ clog b n ≤ clog b m", "tactic": "rw [← le_pow_iff_clog_le hb]" }, { "state_after": "no goals", "state_before": "case inr\nb n m : ℕ\nh : n ≤ m\nhb : 1 < b\n⊢ n ≤ b ^ clog b m", "tactic": "exact h.trans (le_pow_clog hb _)" } ]

[ 339, 37 ]

[ 334, 1 ]

[]

[ 251, 29 ]

[ 249, 1 ]

[]

[ 496, 40 ]

[ 494, 1 ]

[]

[ 214, 22 ]

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[ { "state_after": "no goals", "state_before": "L : Language\nL' : Language\nM : Type w\nN : Type ?u.109819\nP : Type ?u.109822\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nγ : Type ?u.109850\nn l : ℕ\nφ✝ ψ : BoundedFormula L α l\nθ : BoundedFormula L α (Nat.succ l)\nv : α → M\nxs : Fin l → M\nφ : BoundedFormula L α (n + 1)\ncon : IsAtomic (all φ)\n⊢ False", "tactic": "cases con" } ]

[ 688, 12 ]

[ 687, 1 ]

[]

[ 344, 6 ]

[ 342, 1 ]

[]

[ 303, 80 ]

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[]

[ 322, 83 ]

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[]

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[ 245, 1 ]

[]

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[ { "state_after": "a b c : Int\nh : a < b + c\n⊢ a + -b < c", "state_before": "a b c : Int\nh : a < b + c\n⊢ -b + a < c", "tactic": "rw [Int.add_comm]" }, { "state_after": "no goals", "state_before": "a b c : Int\nh : a < b + c\n⊢ a + -b < c", "tactic": "exact Int.sub_left_lt_of_lt_add h" } ]

[ 1093, 36 ]

[ 1091, 11 ]

[]

[ 165, 19 ]

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[]

[ 93, 51 ]

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[]

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[]

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[ { "state_after": "no goals", "state_before": "α✝ : Type u_1\nx : Option α✝\n⊢ isSome x = true ↔ ∃ a, x = some a", "tactic": "cases x <;> simp [isSome]" } ]

[ 55, 87 ]

[ 55, 1 ]

[ { "state_after": "C : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX✝ : SimplicialObject C\nq n m : ℕ\nhnm : ComplexShape.Rel c m n\nX Y : SimplicialObject C\nf : X ⟶ Y\nh : n + 1 = m\n⊢ f.app [n].op ≫ hσ' q n m hnm = hσ' q n m hnm ≫ f.app [m].op", "state_before": "C : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX✝ : SimplicialObject C\nq n m : ℕ\nhnm : ComplexShape.Rel c m n\nX Y : SimplicialObject C\nf : X ⟶ Y\n⊢ f.app [n].op ≫ hσ' q n m hnm = hσ' q n m hnm ≫ f.app [m].op", "tactic": "have h : n + 1 = m := hnm" }, { "state_after": "C : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX✝ : SimplicialObject C\nq n : ℕ\nX Y : SimplicialObject C\nf : X ⟶ Y\nhnm : ComplexShape.Rel c (n + 1) n\n⊢ f.app [n].op ≫ hσ' q n (n + 1) hnm = hσ' q n (n + 1) hnm ≫ f.app [n + 1].op", "state_before": "C : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX✝ : SimplicialObject C\nq n m : ℕ\nhnm : ComplexShape.Rel c m n\nX Y : SimplicialObject C\nf : X ⟶ Y\nh : n + 1 = m\n⊢ f.app [n].op ≫ hσ' q n m hnm = hσ' q n m hnm ≫ f.app [m].op", "tactic": "subst h" }, { "state_after": "C : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX✝ : SimplicialObject C\nq n : ℕ\nX Y : SimplicialObject C\nf : X ⟶ Y\nhnm : ComplexShape.Rel c (n + 1) n\n⊢ f.app [n].op ≫ hσ q n = hσ q n ≫ f.app [n + 1].op", "state_before": "C : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX✝ : SimplicialObject C\nq n : ℕ\nX Y : SimplicialObject C\nf : X ⟶ Y\nhnm : ComplexShape.Rel c (n + 1) n\n⊢ f.app [n].op ≫ hσ' q n (n + 1) hnm = hσ' q n (n + 1) hnm ≫ f.app [n + 1].op", "tactic": "simp only [hσ', eqToHom_refl, comp_id]" }, { "state_after": "C : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX✝ : SimplicialObject C\nq n : ℕ\nX Y : SimplicialObject C\nf : X ⟶ Y\nhnm : ComplexShape.Rel c (n + 1) n\n⊢ (f.app [n].op ≫ if n < q then 0 else (-1) ^ (n - q) • σ Y { val := n - q, isLt := (_ : n - q < Nat.succ n) }) =\n (if n < q then 0 else (-1) ^ (n - q) • σ X { val := n - q, isLt := (_ : n - q < Nat.succ n) }) ≫ f.app [n + 1].op", "state_before": "C : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX✝ : SimplicialObject C\nq n : ℕ\nX Y : SimplicialObject C\nf : X ⟶ Y\nhnm : ComplexShape.Rel c (n + 1) n\n⊢ f.app [n].op ≫ hσ q n = hσ q n ≫ f.app [n + 1].op", "tactic": "unfold hσ" }, { "state_after": "case inl\nC : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX✝ : SimplicialObject C\nq n : ℕ\nX Y : SimplicialObject C\nf : X ⟶ Y\nhnm : ComplexShape.Rel c (n + 1) n\nh✝ : n < q\n⊢ f.app [n].op ≫ 0 = 0 ≫ f.app [n + 1].op\n\ncase inr\nC : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX✝ : SimplicialObject C\nq n : ℕ\nX Y : SimplicialObject C\nf : X ⟶ Y\nhnm : ComplexShape.Rel c (n + 1) n\nh✝ : ¬n < q\n⊢ f.app [n].op ≫ ((-1) ^ (n - q) • σ Y { val := n - q, isLt := (_ : n - q < Nat.succ n) }) =\n ((-1) ^ (n - q) • σ X { val := n - q, isLt := (_ : n - q < Nat.succ n) }) ≫ f.app [n + 1].op", "state_before": "C : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX✝ : SimplicialObject C\nq n : ℕ\nX Y : SimplicialObject C\nf : X ⟶ Y\nhnm : ComplexShape.Rel c (n + 1) n\n⊢ (f.app [n].op ≫ if n < q then 0 else (-1) ^ (n - q) • σ Y { val := n - q, isLt := (_ : n - q < Nat.succ n) }) =\n (if n < q then 0 else (-1) ^ (n - q) • σ X { val := n - q, isLt := (_ : n - q < Nat.succ n) }) ≫ f.app [n + 1].op", "tactic": "split_ifs" }, { "state_after": "no goals", "state_before": "case inl\nC : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX✝ : SimplicialObject C\nq n : ℕ\nX Y : SimplicialObject C\nf : X ⟶ Y\nhnm : ComplexShape.Rel c (n + 1) n\nh✝ : n < q\n⊢ f.app [n].op ≫ 0 = 0 ≫ f.app [n + 1].op", "tactic": "rw [zero_comp, comp_zero]" }, { "state_after": "case inr\nC : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX✝ : SimplicialObject C\nq n : ℕ\nX Y : SimplicialObject C\nf : X ⟶ Y\nhnm : ComplexShape.Rel c (n + 1) n\nh✝ : ¬n < q\n⊢ (-1) ^ (n - q) • f.app [n].op ≫ σ Y { val := n - q, isLt := (_ : n - q < Nat.succ n) } =\n (-1) ^ (n - q) • σ X { val := n - q, isLt := (_ : n - q < Nat.succ n) } ≫ f.app [n + 1].op", "state_before": "case inr\nC : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX✝ : SimplicialObject C\nq n : ℕ\nX Y : SimplicialObject C\nf : X ⟶ Y\nhnm : ComplexShape.Rel c (n + 1) n\nh✝ : ¬n < q\n⊢ f.app [n].op ≫ ((-1) ^ (n - q) • σ Y { val := n - q, isLt := (_ : n - q < Nat.succ n) }) =\n ((-1) ^ (n - q) • σ X { val := n - q, isLt := (_ : n - q < Nat.succ n) }) ≫ f.app [n + 1].op", "tactic": "simp only [zsmul_comp, comp_zsmul]" }, { "state_after": "case inr\nC : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX✝ : SimplicialObject C\nq n : ℕ\nX Y : SimplicialObject C\nf : X ⟶ Y\nhnm : ComplexShape.Rel c (n + 1) n\nh✝ : ¬n < q\n⊢ (-1) ^ (n - q) • f.app [n].op ≫ σ Y { val := n - q, isLt := (_ : n - q < Nat.succ n) } =\n (-1) ^ (n - q) • f.app [n].op ≫ Y.map (SimplexCategory.σ { val := n - q, isLt := (_ : n - q < Nat.succ n) }).op", "state_before": "case inr\nC : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX✝ : SimplicialObject C\nq n : ℕ\nX Y : SimplicialObject C\nf : X ⟶ Y\nhnm : ComplexShape.Rel c (n + 1) n\nh✝ : ¬n < q\n⊢ (-1) ^ (n - q) • f.app [n].op ≫ σ Y { val := n - q, isLt := (_ : n - q < Nat.succ n) } =\n (-1) ^ (n - q) • σ X { val := n - q, isLt := (_ : n - q < Nat.succ n) } ≫ f.app [n + 1].op", "tactic": "erw [f.naturality]" }, { "state_after": "no goals", "state_before": "case inr\nC : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX✝ : SimplicialObject C\nq n : ℕ\nX Y : SimplicialObject C\nf : X ⟶ Y\nhnm : ComplexShape.Rel c (n + 1) n\nh✝ : ¬n < q\n⊢ (-1) ^ (n - q) • f.app [n].op ≫ σ Y { val := n - q, isLt := (_ : n - q < Nat.succ n) } =\n (-1) ^ (n - q) • f.app [n].op ≫ Y.map (SimplexCategory.σ { val := n - q, isLt := (_ : n - q < Nat.succ n) }).op", "tactic": "rfl" } ]

[ 171, 8 ]

[ 161, 1 ]