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[ { "state_after": "𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.89112\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace 𝕜 F\nK : Submodule 𝕜 E\n⊢ ⊤ ≤ ⊤ᗮᗮ", "state_before": "𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.89112\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace 𝕜 F\nK : Submodule 𝕜 E\n⊢ ⊥ᗮ = ⊤", "tactic": "rw [← top_orthogonal_eq_bot, eq_top_iff]" }, { "state_after": "no goals", "state_before": "𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.89112\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace 𝕜 F\nK : Submodule 𝕜 E\n⊢ ⊤ ≤ ⊤ᗮᗮ", "tactic": "exact le_orthogonal_orthogonal ⊤" } ]

[ 203, 35 ]

[ 201, 1 ]

[]

[ 1078, 17 ]

[ 1077, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type ?u.85057\nβ : Type ?u.85060\nγ : Type ?u.85063\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\no : Ordinal\nx✝ : ∃ a, o = succ a\na : Ordinal\ne : o = succ a\n⊢ succ (pred o) = o", "tactic": "simp only [e, pred_succ]" } ]

[ 204, 68 ]

[ 203, 1 ]

[ { "state_after": "α✝ : Type ?u.33186\ninst✝¹ : DecidableEq α✝\ns✝ s'✝ : Cycle α✝\nα : Type u_1\ninst✝ : DecidableEq α\ns s' : Cycle α\nhs : Nodup s\nhs' : Nodup s'\n⊢ formPerm s hs = formPerm s' hs' ↔ s = s' ∨ length s ≤ 1 ∧ length s' ≤ 1", "state_before": "α✝ : Type ?u.33186\ninst✝¹ : DecidableEq α✝\ns✝ s'✝ : Cycle α✝\nα : Type u_1\ninst✝ : DecidableEq α\ns s' : Cycle α\nhs : Nodup s\nhs' : Nodup s'\n⊢ formPerm s hs = formPerm s' hs' ↔ s = s' ∨ Subsingleton s ∧ Subsingleton s'", "tactic": "rw [Cycle.length_subsingleton_iff, Cycle.length_subsingleton_iff]" }, { "state_after": "α✝ : Type ?u.33186\ninst✝¹ : DecidableEq α✝\ns s' : Cycle α✝\nα : Type u_1\ninst✝ : DecidableEq α\n⊢ ∀ {s s' : Cycle α} {hs : Nodup s} {hs' : Nodup s'},\n formPerm s hs = formPerm s' hs' ↔ s = s' ∨ length s ≤ 1 ∧ length s' ≤ 1", "state_before": "α✝ : Type ?u.33186\ninst✝¹ : DecidableEq α✝\ns✝ s'✝ : Cycle α✝\nα : Type u_1\ninst✝ : DecidableEq α\ns s' : Cycle α\nhs : Nodup s\nhs' : Nodup s'\n⊢ formPerm s hs = formPerm s' hs' ↔ s = s' ∨ length s ≤ 1 ∧ length s' ≤ 1", "tactic": "revert s s'" }, { "state_after": "α✝ : Type ?u.33186\ninst✝¹ : DecidableEq α✝\ns✝ s'✝ : Cycle α✝\nα : Type u_1\ninst✝ : DecidableEq α\ns s' : Cycle α\n⊢ ∀ {hs : Nodup s} {hs' : Nodup s'}, formPerm s hs = formPerm s' hs' ↔ s = s' ∨ length s ≤ 1 ∧ length s' ≤ 1", "state_before": "α✝ : Type ?u.33186\ninst✝¹ : DecidableEq α✝\ns s' : Cycle α✝\nα : Type u_1\ninst✝ : DecidableEq α\n⊢ ∀ {s s' : Cycle α} {hs : Nodup s} {hs' : Nodup s'},\n formPerm s hs = formPerm s' hs' ↔ s = s' ∨ length s ≤ 1 ∧ length s' ≤ 1", "tactic": "intro s s'" }, { "state_after": "α✝ : Type ?u.33186\ninst✝¹ : DecidableEq α✝\ns✝ s'✝ : Cycle α✝\nα : Type u_1\ninst✝ : DecidableEq α\ns s' : Cycle α\n⊢ ∀ (a₁ a₂ : List α) {hs : Nodup (Quotient.mk'' a₁)} {hs' : Nodup (Quotient.mk'' a₂)},\n formPerm (Quotient.mk'' a₁) hs = formPerm (Quotient.mk'' a₂) hs' ↔\n Quotient.mk'' a₁ = Quotient.mk'' a₂ ∨ length (Quotient.mk'' a₁) ≤ 1 ∧ length (Quotient.mk'' a₂) ≤ 1", "state_before": "α✝ : Type ?u.33186\ninst✝¹ : DecidableEq α✝\ns✝ s'✝ : Cycle α✝\nα : Type u_1\ninst✝ : DecidableEq α\ns s' : Cycle α\n⊢ ∀ {hs : Nodup s} {hs' : Nodup s'}, formPerm s hs = formPerm s' hs' ↔ s = s' ∨ length s ≤ 1 ∧ length s' ≤ 1", "tactic": "apply @Quotient.inductionOn₂' _ _ _ _ _ s s'" }, { "state_after": "α✝ : Type ?u.33186\ninst✝¹ : DecidableEq α✝\ns✝ s'✝ : Cycle α✝\nα : Type u_1\ninst✝ : DecidableEq α\ns s' : Cycle α\nl l' : List α\n⊢ ∀ {hs : Nodup (Quotient.mk'' l)} {hs' : Nodup (Quotient.mk'' l')},\n formPerm (Quotient.mk'' l) hs = formPerm (Quotient.mk'' l') hs' ↔\n Quotient.mk'' l = Quotient.mk'' l' ∨ length (Quotient.mk'' l) ≤ 1 ∧ length (Quotient.mk'' l') ≤ 1", "state_before": "α✝ : Type ?u.33186\ninst✝¹ : DecidableEq α✝\ns✝ s'✝ : Cycle α✝\nα : Type u_1\ninst✝ : DecidableEq α\ns s' : Cycle α\n⊢ ∀ (a₁ a₂ : List α) {hs : Nodup (Quotient.mk'' a₁)} {hs' : Nodup (Quotient.mk'' a₂)},\n formPerm (Quotient.mk'' a₁) hs = formPerm (Quotient.mk'' a₂) hs' ↔\n Quotient.mk'' a₁ = Quotient.mk'' a₂ ∨ length (Quotient.mk'' a₁) ≤ 1 ∧ length (Quotient.mk'' a₂) ≤ 1", "tactic": "intro l l'" }, { "state_after": "α✝ : Type ?u.33186\ninst✝¹ : DecidableEq α✝\ns✝ s'✝ : Cycle α✝\nα : Type u_1\ninst✝ : DecidableEq α\ns s' : Cycle α\nl l' : List α\n⊢ ∀ {hs : List.Nodup l} {hs' : List.Nodup l'},\n List.formPerm l = List.formPerm l' ↔ l ~r l' ∨ List.length l ≤ 1 ∧ List.length l' ≤ 1", "state_before": "α✝ : Type ?u.33186\ninst✝¹ : DecidableEq α✝\ns✝ s'✝ : Cycle α✝\nα : Type u_1\ninst✝ : DecidableEq α\ns s' : Cycle α\nl l' : List α\n⊢ ∀ {hs : Nodup (Quotient.mk'' l)} {hs' : Nodup (Quotient.mk'' l')},\n formPerm (Quotient.mk'' l) hs = formPerm (Quotient.mk'' l') hs' ↔\n Quotient.mk'' l = Quotient.mk'' l' ∨ length (Quotient.mk'' l) ≤ 1 ∧ length (Quotient.mk'' l') ≤ 1", "tactic": "simp_all" }, { "state_after": "α✝ : Type ?u.33186\ninst✝¹ : DecidableEq α✝\ns✝ s'✝ : Cycle α✝\nα : Type u_1\ninst✝ : DecidableEq α\ns s' : Cycle α\nl l' : List α\nhs : List.Nodup l\nhs' : List.Nodup l'\n⊢ List.formPerm l = List.formPerm l' ↔ l ~r l' ∨ List.length l ≤ 1 ∧ List.length l' ≤ 1", "state_before": "α✝ : Type ?u.33186\ninst✝¹ : DecidableEq α✝\ns✝ s'✝ : Cycle α✝\nα : Type u_1\ninst✝ : DecidableEq α\ns s' : Cycle α\nl l' : List α\n⊢ ∀ {hs : List.Nodup l} {hs' : List.Nodup l'},\n List.formPerm l = List.formPerm l' ↔ l ~r l' ∨ List.length l ≤ 1 ∧ List.length l' ≤ 1", "tactic": "intro hs hs'" }, { "state_after": "no goals", "state_before": "α✝ : Type ?u.33186\ninst✝¹ : DecidableEq α✝\ns✝ s'✝ : Cycle α✝\nα : Type u_1\ninst✝ : DecidableEq α\ns s' : Cycle α\nl l' : List α\nhs : List.Nodup l\nhs' : List.Nodup l'\n⊢ List.formPerm l = List.formPerm l' ↔ l ~r l' ∨ List.length l ≤ 1 ∧ List.length l' ≤ 1", "tactic": "constructor <;> intro h <;> simp_all only [formPerm_eq_formPerm_iff]" } ]

[ 206, 71 ]

[ 195, 8 ]

[ { "state_after": "no goals", "state_before": "a b : Nat\nh : a ≤ b\n⊢ max a b = b", "tactic": "simp [Nat.max_def, h, Nat.not_lt.2 h]" } ]

[ 211, 40 ]

[ 210, 11 ]

[]

[ 1510, 6 ]

[ 1508, 1 ]

[ { "state_after": "no goals", "state_before": "l r : List Char\nit : Iterator\nh : ValidFor l r it\n⊢ Iterator.hasPrev it = true ↔ l ≠ []", "tactic": "simp [Iterator.hasPrev, h.pos, Nat.pos_iff_ne_zero]" } ]

[ 573, 68 ]

[ 572, 1 ]

[]

[ 701, 61 ]

[ 700, 1 ]

[ { "state_after": "α : Type ?u.981446\nm n : ZNum\nh : ↑m < ↑n\n⊢ ↑m < ↑n ↔ Ordering.lt = Ordering.lt", "state_before": "α : Type ?u.981446\nm n : ZNum\nh : Ordering.casesOn Ordering.lt (↑m < ↑n) (m = n) (↑n < ↑m)\n⊢ ↑m < ↑n ↔ Ordering.lt = Ordering.lt", "tactic": "simp at h" }, { "state_after": "no goals", "state_before": "α : Type ?u.981446\nm n : ZNum\nh : ↑m < ↑n\n⊢ ↑m < ↑n ↔ Ordering.lt = Ordering.lt", "tactic": "simp [h]" }, { "state_after": "α : Type ?u.981446\nm n : ZNum\nh : m = n\n⊢ ↑m < ↑n ↔ Ordering.eq = Ordering.lt", "state_before": "α : Type ?u.981446\nm n : ZNum\nh : Ordering.casesOn Ordering.eq (↑m < ↑n) (m = n) (↑n < ↑m)\n⊢ ↑m < ↑n ↔ Ordering.eq = Ordering.lt", "tactic": "simp at h" }, { "state_after": "no goals", "state_before": "α : Type ?u.981446\nm n : ZNum\nh : m = n\n⊢ ↑m < ↑n ↔ Ordering.eq = Ordering.lt", "tactic": "simp [h, lt_irrefl]" }, { "state_after": "no goals", "state_before": "α : Type ?u.981446\nm n : ZNum\nh : Ordering.casesOn Ordering.gt (↑m < ↑n) (m = n) (↑n < ↑m)\n⊢ ↑m < ↑n ↔ Ordering.gt = Ordering.lt", "tactic": "simp [not_lt_of_gt h]" } ]

[ 1398, 49 ]

[ 1393, 1 ]

[ { "state_after": "case w\nJ : Type\ninst✝⁴ : Fintype J\nK : Type\ninst✝³ : Fintype K\nC : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasFiniteBiproducts C\nf : J → C\ng : K → C\nm : (j : J) → (k : K) → f j ⟶ g k\nj : J\nj✝ : K\n⊢ (ι f j ≫ matrix m) ≫ π (fun k => g k) j✝ = (lift fun k => m j k) ≫ π (fun k => g k) j✝", "state_before": "J : Type\ninst✝⁴ : Fintype J\nK : Type\ninst✝³ : Fintype K\nC : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasFiniteBiproducts C\nf : J → C\ng : K → C\nm : (j : J) → (k : K) → f j ⟶ g k\nj : J\n⊢ ι f j ≫ matrix m = lift fun k => m j k", "tactic": "ext" }, { "state_after": "no goals", "state_before": "case w\nJ : Type\ninst✝⁴ : Fintype J\nK : Type\ninst✝³ : Fintype K\nC : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasFiniteBiproducts C\nf : J → C\ng : K → C\nm : (j : J) → (k : K) → f j ⟶ g k\nj : J\nj✝ : K\n⊢ (ι f j ≫ matrix m) ≫ π (fun k => g k) j✝ = (lift fun k => m j k) ≫ π (fun k => g k) j✝", "tactic": "simp [biproduct.matrix]" } ]

[ 840, 26 ]

[ 837, 1 ]

[]

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[ 363, 1 ]

[ { "state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.42243\ninst✝¹ : UniformSpace α\ninst✝ : UniformSpace β\ns : Set α\nf : α → β\nhs : IsCompact s\nhf : ContinuousOn f s\n⊢ UniformContinuous (restrict s f)", "state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.42243\ninst✝¹ : UniformSpace α\ninst✝ : UniformSpace β\ns : Set α\nf : α → β\nhs : IsCompact s\nhf : ContinuousOn f s\n⊢ UniformContinuousOn f s", "tactic": "rw [uniformContinuousOn_iff_restrict]" }, { "state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.42243\ninst✝¹ : UniformSpace α\ninst✝ : UniformSpace β\ns : Set α\nf : α → β\nhs : CompactSpace ↑s\nhf : ContinuousOn f s\n⊢ UniformContinuous (restrict s f)", "state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.42243\ninst✝¹ : UniformSpace α\ninst✝ : UniformSpace β\ns : Set α\nf : α → β\nhs : IsCompact s\nhf : ContinuousOn f s\n⊢ UniformContinuous (restrict s f)", "tactic": "rw [isCompact_iff_compactSpace] at hs" }, { "state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.42243\ninst✝¹ : UniformSpace α\ninst✝ : UniformSpace β\ns : Set α\nf : α → β\nhs : CompactSpace ↑s\nhf : Continuous (restrict s f)\n⊢ UniformContinuous (restrict s f)", "state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.42243\ninst✝¹ : UniformSpace α\ninst✝ : UniformSpace β\ns : Set α\nf : α → β\nhs : CompactSpace ↑s\nhf : ContinuousOn f s\n⊢ UniformContinuous (restrict s f)", "tactic": "rw [continuousOn_iff_continuous_restrict] at hf" }, { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.42243\ninst✝¹ : UniformSpace α\ninst✝ : UniformSpace β\ns : Set α\nf : α → β\nhs : CompactSpace ↑s\nhf : Continuous (restrict s f)\n⊢ UniformContinuous (restrict s f)", "tactic": "exact CompactSpace.uniformContinuous_of_continuous hf" } ]

[ 185, 56 ]

[ 180, 1 ]

[ { "state_after": "no goals", "state_before": "K R : Type v\nV M : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : End R M\n⊢ eigenspace f 0 = LinearMap.ker f", "tactic": "simp [eigenspace]" } ]

[ 71, 97 ]

[ 71, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type ?u.219951\nβ : Type ?u.219954\nγ : Type ?u.219957\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\no : Ordinal\n⊢ card (type (?m.220055 o)) = card o", "tactic": "rw [Ordinal.type_lt]" } ]

[ 1324, 62 ]

[ 1323, 1 ]

[]

[ 248, 53 ]

[ 246, 1 ]

[]

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[ 952, 1 ]

[]

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[ 1658, 1 ]

[]

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[ 235, 1 ]

[ { "state_after": "no goals", "state_before": "⊢ ∀ (x : Bool), (x || !x) = true", "tactic": "decide" } ]

[ 256, 57 ]

[ 256, 1 ]

[ { "state_after": "α : Type u_1\nβ : Type ?u.481748\nγ : Type ?u.481751\nι : Type ?u.481754\nM : Type u_2\nM' : Type ?u.481760\nN : Type ?u.481763\nP : Type ?u.481766\nG : Type ?u.481769\nH : Type ?u.481772\nR : Type ?u.481775\nS : Type ?u.481778\ninst✝ : Zero M\nf : α →₀ M\ny : M\n⊢ (∃ a, a ∈ f.support ∧ ↑f a = y) ↔ y ≠ 0 ∧ ∃ x, ↑f x = y", "state_before": "α : Type u_1\nβ : Type ?u.481748\nγ : Type ?u.481751\nι : Type ?u.481754\nM : Type u_2\nM' : Type ?u.481760\nN : Type ?u.481763\nP : Type ?u.481766\nG : Type ?u.481769\nH : Type ?u.481772\nR : Type ?u.481775\nS : Type ?u.481778\ninst✝ : Zero M\nf : α →₀ M\ny : M\n⊢ y ∈ frange f ↔ y ≠ 0 ∧ ∃ x, ↑f x = y", "tactic": "rw [frange, @Finset.mem_image _ _ (Classical.decEq _) _ f.support]" }, { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type ?u.481748\nγ : Type ?u.481751\nι : Type ?u.481754\nM : Type u_2\nM' : Type ?u.481760\nN : Type ?u.481763\nP : Type ?u.481766\nG : Type ?u.481769\nH : Type ?u.481772\nR : Type ?u.481775\nS : Type ?u.481778\ninst✝ : Zero M\nf : α →₀ M\ny : M\n⊢ (∃ a, a ∈ f.support ∧ ↑f a = y) ↔ y ≠ 0 ∧ ∃ x, ↑f x = y", "tactic": "exact ⟨fun ⟨x, hx1, hx2⟩ => ⟨hx2 ▸ mem_support_iff.1 hx1, x, hx2⟩, fun ⟨hy, x, hx⟩ =>\n ⟨x, mem_support_iff.2 (hx.symm ▸ hy), hx⟩⟩" } ]

[ 991, 47 ]

[ 988, 1 ]

[]

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[ 231, 1 ]

[ { "state_after": "no goals", "state_before": "S : Type u_3\nR : Type u_1\nR₁ : Type ?u.429167\nM : Type u_2\ninst✝⁷ : Ring R\ninst✝⁶ : CommRing R₁\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : Module R₁ M\ninst✝² : CommSemiring S\ninst✝¹ : Algebra S R\ninst✝ : Invertible 2\nB₁ : BilinForm R M\nQ : QuadraticForm R M\nx : M\n⊢ bilin (↑(associatedHom S) Q) x x = ↑Q x", "tactic": "rw [associated_apply, map_add_self, ← three_add_one_eq_four, ← two_add_one_eq_three,\n add_mul, add_mul, one_mul, add_sub_cancel, add_sub_cancel, invOf_mul_self_assoc]" } ]

[ 810, 85 ]

[ 808, 1 ]

[ { "state_after": "case h.h\nα : Type uu\nβ : Type vv\nl₁✝ l₂ : List α\nγ : Type ?u.38852\nδ : Type ?u.38855\nr : α → β → Prop\np : γ → δ → Prop\nl₁ : List α\nl₃ : List β\n⊢ (Forall₂ r ∘r Perm) l₁ l₃ = (Perm ∘r Forall₂ r) l₁ l₃", "state_before": "α : Type uu\nβ : Type vv\nl₁ l₂ : List α\nγ : Type ?u.38852\nδ : Type ?u.38855\nr : α → β → Prop\np : γ → δ → Prop\n⊢ Forall₂ r ∘r Perm = Perm ∘r Forall₂ r", "tactic": "funext l₁ l₃" }, { "state_after": "case h.h.a\nα : Type uu\nβ : Type vv\nl₁✝ l₂ : List α\nγ : Type ?u.38852\nδ : Type ?u.38855\nr : α → β → Prop\np : γ → δ → Prop\nl₁ : List α\nl₃ : List β\n⊢ (Forall₂ r ∘r Perm) l₁ l₃ ↔ (Perm ∘r Forall₂ r) l₁ l₃", "state_before": "case h.h\nα : Type uu\nβ : Type vv\nl₁✝ l₂ : List α\nγ : Type ?u.38852\nδ : Type ?u.38855\nr : α → β → Prop\np : γ → δ → Prop\nl₁ : List α\nl₃ : List β\n⊢ (Forall₂ r ∘r Perm) l₁ l₃ = (Perm ∘r Forall₂ r) l₁ l₃", "tactic": "apply propext" }, { "state_after": "case h.h.a.mp\nα : Type uu\nβ : Type vv\nl₁✝ l₂ : List α\nγ : Type ?u.38852\nδ : Type ?u.38855\nr : α → β → Prop\np : γ → δ → Prop\nl₁ : List α\nl₃ : List β\n⊢ (Forall₂ r ∘r Perm) l₁ l₃ → (Perm ∘r Forall₂ r) l₁ l₃\n\ncase h.h.a.mpr\nα : Type uu\nβ : Type vv\nl₁✝ l₂ : List α\nγ : Type ?u.38852\nδ : Type ?u.38855\nr : α → β → Prop\np : γ → δ → Prop\nl₁ : List α\nl₃ : List β\n⊢ (Perm ∘r Forall₂ r) l₁ l₃ → (Forall₂ r ∘r Perm) l₁ l₃", "state_before": "case h.h.a\nα : Type uu\nβ : Type vv\nl₁✝ l₂ : List α\nγ : Type ?u.38852\nδ : Type ?u.38855\nr : α → β → Prop\np : γ → δ → Prop\nl₁ : List α\nl₃ : List β\n⊢ (Forall₂ r ∘r Perm) l₁ l₃ ↔ (Perm ∘r Forall₂ r) l₁ l₃", "tactic": "constructor" }, { "state_after": "case h.h.a.mp\nα : Type uu\nβ : Type vv\nl₁✝ l₂ : List α\nγ : Type ?u.38852\nδ : Type ?u.38855\nr : α → β → Prop\np : γ → δ → Prop\nl₁ : List α\nl₃ : List β\nh : (Forall₂ r ∘r Perm) l₁ l₃\n⊢ (Perm ∘r Forall₂ r) l₁ l₃", "state_before": "case h.h.a.mp\nα : Type uu\nβ : Type vv\nl₁✝ l₂ : List α\nγ : Type ?u.38852\nδ : Type ?u.38855\nr : α → β → Prop\np : γ → δ → Prop\nl₁ : List α\nl₃ : List β\n⊢ (Forall₂ r ∘r Perm) l₁ l₃ → (Perm ∘r Forall₂ r) l₁ l₃", "tactic": "intro h" }, { "state_after": "case h.h.a.mp.intro.intro\nα : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\nγ : Type ?u.38852\nδ : Type ?u.38855\nr : α → β → Prop\np : γ → δ → Prop\nl₁ : List α\nl₃ l₂ : List β\nh₁₂ : Forall₂ r l₁ l₂\nh₂₃ : l₂ ~ l₃\n⊢ (Perm ∘r Forall₂ r) l₁ l₃", "state_before": "case h.h.a.mp\nα : Type uu\nβ : Type vv\nl₁✝ l₂ : List α\nγ : Type ?u.38852\nδ : Type ?u.38855\nr : α → β → Prop\np : γ → δ → Prop\nl₁ : List α\nl₃ : List β\nh : (Forall₂ r ∘r Perm) l₁ l₃\n⊢ (Perm ∘r Forall₂ r) l₁ l₃", "tactic": "rcases h with ⟨l₂, h₁₂, h₂₃⟩" }, { "state_after": "case h.h.a.mp.intro.intro\nα : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\nγ : Type ?u.38852\nδ : Type ?u.38855\nr : α → β → Prop\np : γ → δ → Prop\nl₁ : List α\nl₃ l₂ : List β\nh₁₂ : Forall₂ r l₁ l₂\nh₂₃ : l₂ ~ l₃\nthis : Forall₂ (flip r) l₂ l₁\n⊢ (Perm ∘r Forall₂ r) l₁ l₃", "state_before": "case h.h.a.mp.intro.intro\nα : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\nγ : Type ?u.38852\nδ : Type ?u.38855\nr : α → β → Prop\np : γ → δ → Prop\nl₁ : List α\nl₃ l₂ : List β\nh₁₂ : Forall₂ r l₁ l₂\nh₂₃ : l₂ ~ l₃\n⊢ (Perm ∘r Forall₂ r) l₁ l₃", "tactic": "have : Forall₂ (flip r) l₂ l₁ := h₁₂.flip" }, { "state_after": "case h.h.a.mp.intro.intro.intro.intro\nα : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\nγ : Type ?u.38852\nδ : Type ?u.38855\nr : α → β → Prop\np : γ → δ → Prop\nl₁ : List α\nl₃ l₂ : List β\nh₁₂ : Forall₂ r l₁ l₂\nh₂₃ : l₂ ~ l₃\nthis : Forall₂ (flip r) l₂ l₁\nl' : List α\nh₁ : Forall₂ (flip r) l₃ l'\nh₂ : l' ~ l₁\n⊢ (Perm ∘r Forall₂ r) l₁ l₃", "state_before": "case h.h.a.mp.intro.intro\nα : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\nγ : Type ?u.38852\nδ : Type ?u.38855\nr : α → β → Prop\np : γ → δ → Prop\nl₁ : List α\nl₃ l₂ : List β\nh₁₂ : Forall₂ r l₁ l₂\nh₂₃ : l₂ ~ l₃\nthis : Forall₂ (flip r) l₂ l₁\n⊢ (Perm ∘r Forall₂ r) l₁ l₃", "tactic": "rcases perm_comp_forall₂ h₂₃.symm this with ⟨l', h₁, h₂⟩" }, { "state_after": "no goals", "state_before": "case h.h.a.mp.intro.intro.intro.intro\nα : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\nγ : Type ?u.38852\nδ : Type ?u.38855\nr : α → β → Prop\np : γ → δ → Prop\nl₁ : List α\nl₃ l₂ : List β\nh₁₂ : Forall₂ r l₁ l₂\nh₂₃ : l₂ ~ l₃\nthis : Forall₂ (flip r) l₂ l₁\nl' : List α\nh₁ : Forall₂ (flip r) l₃ l'\nh₂ : l' ~ l₁\n⊢ (Perm ∘r Forall₂ r) l₁ l₃", "tactic": "exact ⟨l', h₂.symm, h₁.flip⟩" }, { "state_after": "no goals", "state_before": "case h.h.a.mpr\nα : Type uu\nβ : Type vv\nl₁✝ l₂ : List α\nγ : Type ?u.38852\nδ : Type ?u.38855\nr : α → β → Prop\np : γ → δ → Prop\nl₁ : List α\nl₃ : List β\n⊢ (Perm ∘r Forall₂ r) l₁ l₃ → (Forall₂ r ∘r Perm) l₁ l₃", "tactic": "exact fun ⟨l₂, h₁₂, h₂₃⟩ => perm_comp_forall₂ h₁₂ h₂₃" } ]

[ 368, 58 ]

[ 360, 1 ]

[ { "state_after": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.2519860\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx y : E\n⊢ ↑re (inner x y) = (‖x‖ * ‖x‖ + 2 * ↑re (inner x y) + ‖y‖ * ‖y‖ - ‖x‖ * ‖x‖ - ‖y‖ * ‖y‖) / 2", "state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.2519860\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx y : E\n⊢ ↑re (inner x y) = (‖x + y‖ * ‖x + y‖ - ‖x‖ * ‖x‖ - ‖y‖ * ‖y‖) / 2", "tactic": "rw [@norm_add_mul_self 𝕜]" }, { "state_after": "no goals", "state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.2519860\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx y : E\n⊢ ↑re (inner x y) = (‖x‖ * ‖x‖ + 2 * ↑re (inner x y) + ‖y‖ * ‖y‖ - ‖x‖ * ‖x‖ - ‖y‖ * ‖y‖) / 2", "tactic": "ring" } ]

[ 1139, 7 ]

[ 1136, 1 ]

[]

[ 89, 28 ]

[ 88, 1 ]

[]

[ 180, 47 ]

[ 179, 1 ]

[ { "state_after": "α : Type u_1\nβ : Type ?u.981881\nγ : Type ?u.981884\nδ : Type ?u.981887\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\n⊢ (∫⁻ (x : α), g x ∂μ) ≤ (∫⁻ (x : α), g x - f x ∂μ) + ∫⁻ (x : α), f x ∂μ", "state_before": "α : Type u_1\nβ : Type ?u.981881\nγ : Type ?u.981884\nδ : Type ?u.981887\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\n⊢ ((∫⁻ (x : α), g x ∂μ) - ∫⁻ (x : α), f x ∂μ) ≤ ∫⁻ (x : α), g x - f x ∂μ", "tactic": "rw [tsub_le_iff_right]" }, { "state_after": "case pos\nα : Type u_1\nβ : Type ?u.981881\nγ : Type ?u.981884\nδ : Type ?u.981887\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhfi : (∫⁻ (x : α), f x ∂μ) = ⊤\n⊢ (∫⁻ (x : α), g x ∂μ) ≤ (∫⁻ (x : α), g x - f x ∂μ) + ∫⁻ (x : α), f x ∂μ\n\ncase neg\nα : Type u_1\nβ : Type ?u.981881\nγ : Type ?u.981884\nδ : Type ?u.981887\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhfi : ¬(∫⁻ (x : α), f x ∂μ) = ⊤\n⊢ (∫⁻ (x : α), g x ∂μ) ≤ (∫⁻ (x : α), g x - f x ∂μ) + ∫⁻ (x : α), f x ∂μ", "state_before": "α : Type u_1\nβ : Type ?u.981881\nγ : Type ?u.981884\nδ : Type ?u.981887\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\n⊢ (∫⁻ (x : α), g x ∂μ) ≤ (∫⁻ (x : α), g x - f x ∂μ) + ∫⁻ (x : α), f x ∂μ", "tactic": "by_cases hfi : (∫⁻ x, f x ∂μ) = ∞" }, { "state_after": "case pos\nα : Type u_1\nβ : Type ?u.981881\nγ : Type ?u.981884\nδ : Type ?u.981887\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhfi : (∫⁻ (x : α), f x ∂μ) = ⊤\n⊢ (∫⁻ (x : α), g x ∂μ) ≤ ⊤", "state_before": "case pos\nα : Type u_1\nβ : Type ?u.981881\nγ : Type ?u.981884\nδ : Type ?u.981887\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhfi : (∫⁻ (x : α), f x ∂μ) = ⊤\n⊢ (∫⁻ (x : α), g x ∂μ) ≤ (∫⁻ (x : α), g x - f x ∂μ) + ∫⁻ (x : α), f x ∂μ", "tactic": "rw [hfi, add_top]" }, { "state_after": "no goals", "state_before": "case pos\nα : Type u_1\nβ : Type ?u.981881\nγ : Type ?u.981884\nδ : Type ?u.981887\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhfi : (∫⁻ (x : α), f x ∂μ) = ⊤\n⊢ (∫⁻ (x : α), g x ∂μ) ≤ ⊤", "tactic": "exact le_top" }, { "state_after": "case neg\nα : Type u_1\nβ : Type ?u.981881\nγ : Type ?u.981884\nδ : Type ?u.981887\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhfi : ¬(∫⁻ (x : α), f x ∂μ) = ⊤\n⊢ (∫⁻ (x : α), g x ∂μ) ≤ ∫⁻ (a : α), g a - f a + f a ∂μ", "state_before": "case neg\nα : Type u_1\nβ : Type ?u.981881\nγ : Type ?u.981884\nδ : Type ?u.981887\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhfi : ¬(∫⁻ (x : α), f x ∂μ) = ⊤\n⊢ (∫⁻ (x : α), g x ∂μ) ≤ (∫⁻ (x : α), g x - f x ∂μ) + ∫⁻ (x : α), f x ∂μ", "tactic": "rw [← lintegral_add_right' _ hf]" }, { "state_after": "no goals", "state_before": "case neg\nα : Type u_1\nβ : Type ?u.981881\nγ : Type ?u.981884\nδ : Type ?u.981887\nm : MeasurableSpace α\nμ ν : Measure α\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhfi : ¬(∫⁻ (x : α), f x ∂μ) = ⊤\n⊢ (∫⁻ (x : α), g x ∂μ) ≤ ∫⁻ (a : α), g a - f a + f a ∂μ", "tactic": "exact lintegral_mono fun x => le_tsub_add" } ]

[ 929, 46 ]

[ 922, 1 ]

[]

[ 512, 26 ]

[ 511, 1 ]

[]

[ 812, 13 ]

[ 810, 1 ]

[ { "state_after": "no goals", "state_before": "p : Prop\nh : Decidable p\nq : Prop\n⊢ (if p then False else q) = (¬p ∧ q)", "tactic": "by_cases p <;> simp [h]" } ]

[ 40, 70 ]

[ 39, 1 ]

[ { "state_after": "p : ℕ\np_prime : Nat.Prime p\nm n : ℤ\nk l : ℕ\nhpm : ↑(p ^ k) ∣ m\nhpn : ↑(p ^ l) ∣ n\nhpmn : ↑(p ^ (k + l + 1)) ∣ m * n\nhpm' : p ^ k ∣ natAbs m\nhpn' : p ^ l ∣ natAbs n\n⊢ p ^ (k + l + 1) ∣ natAbs (m * n)", "state_before": "p : ℕ\np_prime : Nat.Prime p\nm n : ℤ\nk l : ℕ\nhpm : ↑(p ^ k) ∣ m\nhpn : ↑(p ^ l) ∣ n\nhpmn : ↑(p ^ (k + l + 1)) ∣ m * n\nhpm' : p ^ k ∣ natAbs m\nhpn' : p ^ l ∣ natAbs n\n⊢ p ^ (k + l + 1) ∣ natAbs m * natAbs n", "tactic": "rw [← Int.natAbs_mul]" }, { "state_after": "no goals", "state_before": "p : ℕ\np_prime : Nat.Prime p\nm n : ℤ\nk l : ℕ\nhpm : ↑(p ^ k) ∣ m\nhpn : ↑(p ^ l) ∣ n\nhpmn : ↑(p ^ (k + l + 1)) ∣ m * n\nhpm' : p ^ k ∣ natAbs m\nhpn' : p ^ l ∣ natAbs n\n⊢ p ^ (k + l + 1) ∣ natAbs (m * n)", "tactic": "apply Int.coe_nat_dvd.1 <| Int.dvd_natAbs.2 hpmn" }, { "state_after": "p : ℕ\np_prime : Nat.Prime p\nm n : ℤ\nk l : ℕ\nhpm : ↑(p ^ k) ∣ m\nhpn : ↑(p ^ l) ∣ n\nhpmn : ↑(p ^ (k + l + 1)) ∣ m * n\nhpm' : p ^ k ∣ natAbs m\nhpn' : p ^ l ∣ natAbs n\nhpmn' : p ^ (k + l + 1) ∣ natAbs m * natAbs n\nhsd : p ^ (k + 1) ∣ natAbs m ∨ p ^ (l + 1) ∣ natAbs n :=\n Nat.succ_dvd_or_succ_dvd_of_succ_sum_dvd_mul p_prime hpm' hpn' hpmn'\nhsd1 : p ^ (k + 1) ∣ natAbs m\n⊢ ↑(p ^ (k + 1)) ∣ ↑(natAbs m)", "state_before": "p : ℕ\np_prime : Nat.Prime p\nm n : ℤ\nk l : ℕ\nhpm : ↑(p ^ k) ∣ m\nhpn : ↑(p ^ l) ∣ n\nhpmn : ↑(p ^ (k + l + 1)) ∣ m * n\nhpm' : p ^ k ∣ natAbs m\nhpn' : p ^ l ∣ natAbs n\nhpmn' : p ^ (k + l + 1) ∣ natAbs m * natAbs n\nhsd : p ^ (k + 1) ∣ natAbs m ∨ p ^ (l + 1) ∣ natAbs n :=\n Nat.succ_dvd_or_succ_dvd_of_succ_sum_dvd_mul p_prime hpm' hpn' hpmn'\nhsd1 : p ^ (k + 1) ∣ natAbs m\n⊢ ↑(p ^ (k + 1)) ∣ m", "tactic": "apply Int.dvd_natAbs.1" }, { "state_after": "no goals", "state_before": "p : ℕ\np_prime : Nat.Prime p\nm n : ℤ\nk l : ℕ\nhpm : ↑(p ^ k) ∣ m\nhpn : ↑(p ^ l) ∣ n\nhpmn : ↑(p ^ (k + l + 1)) ∣ m * n\nhpm' : p ^ k ∣ natAbs m\nhpn' : p ^ l ∣ natAbs n\nhpmn' : p ^ (k + l + 1) ∣ natAbs m * natAbs n\nhsd : p ^ (k + 1) ∣ natAbs m ∨ p ^ (l + 1) ∣ natAbs n :=\n Nat.succ_dvd_or_succ_dvd_of_succ_sum_dvd_mul p_prime hpm' hpn' hpmn'\nhsd1 : p ^ (k + 1) ∣ natAbs m\n⊢ ↑(p ^ (k + 1)) ∣ ↑(natAbs m)", "tactic": "apply Int.coe_nat_dvd.2 hsd1" }, { "state_after": "p : ℕ\np_prime : Nat.Prime p\nm n : ℤ\nk l : ℕ\nhpm : ↑(p ^ k) ∣ m\nhpn : ↑(p ^ l) ∣ n\nhpmn : ↑(p ^ (k + l + 1)) ∣ m * n\nhpm' : p ^ k ∣ natAbs m\nhpn' : p ^ l ∣ natAbs n\nhpmn' : p ^ (k + l + 1) ∣ natAbs m * natAbs n\nhsd : p ^ (k + 1) ∣ natAbs m ∨ p ^ (l + 1) ∣ natAbs n :=\n Nat.succ_dvd_or_succ_dvd_of_succ_sum_dvd_mul p_prime hpm' hpn' hpmn'\nhsd2 : p ^ (l + 1) ∣ natAbs n\n⊢ ↑(p ^ (l + 1)) ∣ ↑(natAbs n)", "state_before": "p : ℕ\np_prime : Nat.Prime p\nm n : ℤ\nk l : ℕ\nhpm : ↑(p ^ k) ∣ m\nhpn : ↑(p ^ l) ∣ n\nhpmn : ↑(p ^ (k + l + 1)) ∣ m * n\nhpm' : p ^ k ∣ natAbs m\nhpn' : p ^ l ∣ natAbs n\nhpmn' : p ^ (k + l + 1) ∣ natAbs m * natAbs n\nhsd : p ^ (k + 1) ∣ natAbs m ∨ p ^ (l + 1) ∣ natAbs n :=\n Nat.succ_dvd_or_succ_dvd_of_succ_sum_dvd_mul p_prime hpm' hpn' hpmn'\nhsd2 : p ^ (l + 1) ∣ natAbs n\n⊢ ↑(p ^ (l + 1)) ∣ n", "tactic": "apply Int.dvd_natAbs.1" }, { "state_after": "no goals", "state_before": "p : ℕ\np_prime : Nat.Prime p\nm n : ℤ\nk l : ℕ\nhpm : ↑(p ^ k) ∣ m\nhpn : ↑(p ^ l) ∣ n\nhpmn : ↑(p ^ (k + l + 1)) ∣ m * n\nhpm' : p ^ k ∣ natAbs m\nhpn' : p ^ l ∣ natAbs n\nhpmn' : p ^ (k + l + 1) ∣ natAbs m * natAbs n\nhsd : p ^ (k + 1) ∣ natAbs m ∨ p ^ (l + 1) ∣ natAbs n :=\n Nat.succ_dvd_or_succ_dvd_of_succ_sum_dvd_mul p_prime hpm' hpn' hpmn'\nhsd2 : p ^ (l + 1) ∣ natAbs n\n⊢ ↑(p ^ (l + 1)) ∣ ↑(natAbs n)", "tactic": "apply Int.coe_nat_dvd.2 hsd2" } ]

[ 36, 81 ]

[ 27, 1 ]

[]

[ 100, 41 ]

[ 96, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝² : PseudoMetricSpace α\ninst✝¹ : PseudoMetricSpace β\ninst✝ : PseudoMetricSpace γ\nK : ℝ≥0\nf : α → β\nx y : α\nr : ℝ\nh : ∀ (x y : α), dist (f x) (f y) ≤ dist x y\n⊢ ∀ (x y : α), dist (f x) (f y) ≤ ↑1 * dist x y", "tactic": "simpa only [NNReal.coe_one, one_mul] using h" } ]

[ 320, 68 ]

[ 319, 11 ]

[]

[ 775, 72 ]

[ 773, 1 ]

[ { "state_after": "no goals", "state_before": "𝕜 : Type ?u.3694\nι : Type ?u.3697\nκ : Type ?u.3700\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrderedField 𝕜\nr : α → β → Prop\ninst✝ : (a : α) → DecidablePred (r a)\ns✝ s₁ s₂ : Finset α\nt✝ t₁ t₂ : Finset β\na : α\nb : β\nδ : 𝕜\ns : Finset α\nt : Finset β\n⊢ card (interedges r s t) + card (interedges (fun x y => ¬r x y) s t) = card s * card t", "tactic": "classical\nrw [← card_product, interedges, interedges, ← card_union_eq, filter_union_filter_neg_eq]\nexact disjoint_filter.2 fun _ _ ↦ Classical.not_not.2" }, { "state_after": "𝕜 : Type ?u.3694\nι : Type ?u.3697\nκ : Type ?u.3700\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrderedField 𝕜\nr : α → β → Prop\ninst✝ : (a : α) → DecidablePred (r a)\ns✝ s₁ s₂ : Finset α\nt✝ t₁ t₂ : Finset β\na : α\nb : β\nδ : 𝕜\ns : Finset α\nt : Finset β\n⊢ Disjoint (filter (fun e => r e.fst e.snd) (s ×ˢ t)) (filter (fun e => ¬r e.fst e.snd) (s ×ˢ t))", "state_before": "𝕜 : Type ?u.3694\nι : Type ?u.3697\nκ : Type ?u.3700\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrderedField 𝕜\nr : α → β → Prop\ninst✝ : (a : α) → DecidablePred (r a)\ns✝ s₁ s₂ : Finset α\nt✝ t₁ t₂ : Finset β\na : α\nb : β\nδ : 𝕜\ns : Finset α\nt : Finset β\n⊢ card (interedges r s t) + card (interedges (fun x y => ¬r x y) s t) = card s * card t", "tactic": "rw [← card_product, interedges, interedges, ← card_union_eq, filter_union_filter_neg_eq]" }, { "state_after": "no goals", "state_before": "𝕜 : Type ?u.3694\nι : Type ?u.3697\nκ : Type ?u.3700\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrderedField 𝕜\nr : α → β → Prop\ninst✝ : (a : α) → DecidablePred (r a)\ns✝ s₁ s₂ : Finset α\nt✝ t₁ t₂ : Finset β\na : α\nb : β\nδ : 𝕜\ns : Finset α\nt : Finset β\n⊢ Disjoint (filter (fun e => r e.fst e.snd) (s ×ˢ t)) (filter (fun e => ¬r e.fst e.snd) (s ×ˢ t))", "tactic": "exact disjoint_filter.2 fun _ _ ↦ Classical.not_not.2" } ]

[ 85, 56 ]

[ 81, 1 ]

[]

[ 945, 6 ]

[ 943, 1 ]

[ { "state_after": "case intro\nC : Type u\ninst✝⁹ : Category C\ninst✝⁸ : HasZeroObject C\ninst✝⁷ : HasShift C ℤ\ninst✝⁶ : Preadditive C\ninst✝⁵ : ∀ (n : ℤ), Functor.Additive (shiftFunctor C n)\nD : Type u₂\ninst✝⁴ : Category D\ninst✝³ : HasZeroObject D\ninst✝² : HasShift D ℤ\ninst✝¹ : Preadditive D\ninst✝ : ∀ (n : ℤ), Functor.Additive (shiftFunctor D n)\nhC : Pretriangulated C\nT : Triangle C\nH : T ∈ distinguishedTriangles\nc : (contractibleTriangle T.obj₁).obj₃ ⟶ T.obj₃\nhc :\n (contractibleTriangle T.obj₁).mor₂ ≫ c = T.mor₁ ≫ T.mor₂ ∧\n (contractibleTriangle T.obj₁).mor₃ ≫ (shiftFunctor C 1).map (𝟙 T.obj₁) = c ≫ T.mor₃\n⊢ T.mor₁ ≫ T.mor₂ = 0", "state_before": "C : Type u\ninst✝⁹ : Category C\ninst✝⁸ : HasZeroObject C\ninst✝⁷ : HasShift C ℤ\ninst✝⁶ : Preadditive C\ninst✝⁵ : ∀ (n : ℤ), Functor.Additive (shiftFunctor C n)\nD : Type u₂\ninst✝⁴ : Category D\ninst✝³ : HasZeroObject D\ninst✝² : HasShift D ℤ\ninst✝¹ : Preadditive D\ninst✝ : ∀ (n : ℤ), Functor.Additive (shiftFunctor D n)\nhC : Pretriangulated C\nT : Triangle C\nH : T ∈ distinguishedTriangles\n⊢ T.mor₁ ≫ T.mor₂ = 0", "tactic": "obtain ⟨c, hc⟩ :=\n complete_distinguished_triangle_morphism _ _ (contractible_distinguished T.obj₁) H (𝟙 T.obj₁)\n T.mor₁ rfl" }, { "state_after": "no goals", "state_before": "case intro\nC : Type u\ninst✝⁹ : Category C\ninst✝⁸ : HasZeroObject C\ninst✝⁷ : HasShift C ℤ\ninst✝⁶ : Preadditive C\ninst✝⁵ : ∀ (n : ℤ), Functor.Additive (shiftFunctor C n)\nD : Type u₂\ninst✝⁴ : Category D\ninst✝³ : HasZeroObject D\ninst✝² : HasShift D ℤ\ninst✝¹ : Preadditive D\ninst✝ : ∀ (n : ℤ), Functor.Additive (shiftFunctor D n)\nhC : Pretriangulated C\nT : Triangle C\nH : T ∈ distinguishedTriangles\nc : (contractibleTriangle T.obj₁).obj₃ ⟶ T.obj₃\nhc :\n (contractibleTriangle T.obj₁).mor₂ ≫ c = T.mor₁ ≫ T.mor₂ ∧\n (contractibleTriangle T.obj₁).mor₃ ≫ (shiftFunctor C 1).map (𝟙 T.obj₁) = c ≫ T.mor₃\n⊢ T.mor₁ ≫ T.mor₂ = 0", "tactic": "simpa only [contractibleTriangle_mor₂, zero_comp] using hc.left.symm" } ]

[ 133, 71 ]

[ 129, 1 ]

[]

[ 350, 33 ]

[ 349, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.202052\ninst✝¹ : UniformSpace α\ninst✝ : TopologicalSpace β\nf : β → α\ns : Set β\n⊢ ContinuousOn f s ↔ ∀ (b : β), b ∈ s → Tendsto (fun x => (f x, f b)) (𝓝[s] b) (𝓤 α)", "tactic": "simp [ContinuousOn, continuousWithinAt_iff'_left]" } ]

[ 1956, 52 ]

[ 1954, 1 ]

[ { "state_after": "R : Type u\nM : Type v\nF : Type ?u.157879\nG : Type ?u.157882\ninst✝⁴ : CommSemiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nI J : Ideal R\nN✝ P : Submodule R M\nS : Set R\nT : Set M\nM' : Type w\ninst✝¹ : AddCommMonoid M'\ninst✝ : Module R M'\nN : Submodule R M\nx : { x // x ∈ N }\n⊢ x ∈ I • ⊤ ↔ ↑(Submodule.subtype N) x ∈ I • N", "state_before": "R : Type u\nM : Type v\nF : Type ?u.157879\nG : Type ?u.157882\ninst✝⁴ : CommSemiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nI J : Ideal R\nN✝ P : Submodule R M\nS : Set R\nT : Set M\nM' : Type w\ninst✝¹ : AddCommMonoid M'\ninst✝ : Module R M'\nN : Submodule R M\nx : { x // x ∈ N }\n⊢ x ∈ I • ⊤ ↔ ↑x ∈ I • N", "tactic": "change _ ↔ N.subtype x ∈ I • N" }, { "state_after": "R : Type u\nM : Type v\nF : Type ?u.157879\nG : Type ?u.157882\ninst✝⁴ : CommSemiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nI J : Ideal R\nN✝ P : Submodule R M\nS : Set R\nT : Set M\nM' : Type w\ninst✝¹ : AddCommMonoid M'\ninst✝ : Module R M'\nN : Submodule R M\nx : { x // x ∈ N }\nthis : map (Submodule.subtype N) (I • ⊤) = I • N\n⊢ x ∈ I • ⊤ ↔ ↑(Submodule.subtype N) x ∈ I • N", "state_before": "R : Type u\nM : Type v\nF : Type ?u.157879\nG : Type ?u.157882\ninst✝⁴ : CommSemiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nI J : Ideal R\nN✝ P : Submodule R M\nS : Set R\nT : Set M\nM' : Type w\ninst✝¹ : AddCommMonoid M'\ninst✝ : Module R M'\nN : Submodule R M\nx : { x // x ∈ N }\n⊢ x ∈ I • ⊤ ↔ ↑(Submodule.subtype N) x ∈ I • N", "tactic": "have : Submodule.map N.subtype (I • ⊤) = I • N := by\n rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype]" }, { "state_after": "R : Type u\nM : Type v\nF : Type ?u.157879\nG : Type ?u.157882\ninst✝⁴ : CommSemiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nI J : Ideal R\nN✝ P : Submodule R M\nS : Set R\nT : Set M\nM' : Type w\ninst✝¹ : AddCommMonoid M'\ninst✝ : Module R M'\nN : Submodule R M\nx : { x // x ∈ N }\nthis : map (Submodule.subtype N) (I • ⊤) = I • N\n⊢ x ∈ I • ⊤ ↔ ↑(Submodule.subtype N) x ∈ map (Submodule.subtype N) (I • ⊤)", "state_before": "R : Type u\nM : Type v\nF : Type ?u.157879\nG : Type ?u.157882\ninst✝⁴ : CommSemiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nI J : Ideal R\nN✝ P : Submodule R M\nS : Set R\nT : Set M\nM' : Type w\ninst✝¹ : AddCommMonoid M'\ninst✝ : Module R M'\nN : Submodule R M\nx : { x // x ∈ N }\nthis : map (Submodule.subtype N) (I • ⊤) = I • N\n⊢ x ∈ I • ⊤ ↔ ↑(Submodule.subtype N) x ∈ I • N", "tactic": "rw [← this]" }, { "state_after": "no goals", "state_before": "R : Type u\nM : Type v\nF : Type ?u.157879\nG : Type ?u.157882\ninst✝⁴ : CommSemiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nI J : Ideal R\nN✝ P : Submodule R M\nS : Set R\nT : Set M\nM' : Type w\ninst✝¹ : AddCommMonoid M'\ninst✝ : Module R M'\nN : Submodule R M\nx : { x // x ∈ N }\nthis : map (Submodule.subtype N) (I • ⊤) = I • N\n⊢ x ∈ I • ⊤ ↔ ↑(Submodule.subtype N) x ∈ map (Submodule.subtype N) (I • ⊤)", "tactic": "exact (Function.Injective.mem_set_image N.injective_subtype).symm" }, { "state_after": "no goals", "state_before": "R : Type u\nM : Type v\nF : Type ?u.157879\nG : Type ?u.157882\ninst✝⁴ : CommSemiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nI J : Ideal R\nN✝ P : Submodule R M\nS : Set R\nT : Set M\nM' : Type w\ninst✝¹ : AddCommMonoid M'\ninst✝ : Module R M'\nN : Submodule R M\nx : { x // x ∈ N }\n⊢ map (Submodule.subtype N) (I • ⊤) = I • N", "tactic": "rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype]" } ]

[ 339, 68 ]

[ 333, 1 ]

[ { "state_after": "α : Type u_1\ninst✝ : PseudoEMetricSpace α\nf : ℕ → α\nd : ℕ → ℝ≥0\nhf : ∀ (n : ℕ), edist (f n) (f (Nat.succ n)) ≤ ↑(d n)\nhd : Summable d\nε : ℝ≥0\nεpos : 0 < ε\n⊢ ∃ N, ∀ (n : ℕ), N ≤ n → edist (f n) (f N) < ↑ε", "state_before": "α : Type u_1\ninst✝ : PseudoEMetricSpace α\nf : ℕ → α\nd : ℕ → ℝ≥0\nhf : ∀ (n : ℕ), edist (f n) (f (Nat.succ n)) ≤ ↑(d n)\nhd : Summable d\n⊢ CauchySeq f", "tactic": "refine EMetric.cauchySeq_iff_NNReal.2 fun ε εpos => ?_" }, { "state_after": "α : Type u_1\ninst✝ : PseudoEMetricSpace α\nf : ℕ → α\nd : ℕ → ℝ≥0\nhf : ∀ (n : ℕ), edist (f n) (f (Nat.succ n)) ≤ ↑(d n)\nε : ℝ≥0\nεpos : 0 < ε\nhd : CauchySeq fun n => ∑ x in range n, d x\n⊢ ∃ N, ∀ (n : ℕ), N ≤ n → edist (f n) (f N) < ↑ε", "state_before": "α : Type u_1\ninst✝ : PseudoEMetricSpace α\nf : ℕ → α\nd : ℕ → ℝ≥0\nhf : ∀ (n : ℕ), edist (f n) (f (Nat.succ n)) ≤ ↑(d n)\nhd : Summable d\nε : ℝ≥0\nεpos : 0 < ε\n⊢ ∃ N, ∀ (n : ℕ), N ≤ n → edist (f n) (f N) < ↑ε", "tactic": "replace hd : CauchySeq fun n : ℕ => ∑ x in range n, d x :=\n let ⟨_, H⟩ := hd\n H.tendsto_sum_nat.cauchySeq" }, { "state_after": "α : Type u_1\ninst✝ : PseudoEMetricSpace α\nf : ℕ → α\nd : ℕ → ℝ≥0\nhf : ∀ (n : ℕ), edist (f n) (f (Nat.succ n)) ≤ ↑(d n)\nε : ℝ≥0\nεpos : 0 < ε\nhd : CauchySeq fun n => ∑ x in range n, d x\nN : ℕ\nhN : ∀ (n : ℕ), n ≥ N → dist (∑ x in range n, d x) (∑ x in range N, d x) < ↑ε\nn : ℕ\nhn : N ≤ n\n⊢ edist (f n) (f N) < ↑ε", "state_before": "α : Type u_1\ninst✝ : PseudoEMetricSpace α\nf : ℕ → α\nd : ℕ → ℝ≥0\nhf : ∀ (n : ℕ), edist (f n) (f (Nat.succ n)) ≤ ↑(d n)\nε : ℝ≥0\nεpos : 0 < ε\nhd : CauchySeq fun n => ∑ x in range n, d x\n⊢ ∃ N, ∀ (n : ℕ), N ≤ n → edist (f n) (f N) < ↑ε", "tactic": "refine (Metric.cauchySeq_iff'.1 hd ε (NNReal.coe_pos.2 εpos)).imp fun N hN n hn => ?_" }, { "state_after": "α : Type u_1\ninst✝ : PseudoEMetricSpace α\nf : ℕ → α\nd : ℕ → ℝ≥0\nhf : ∀ (n : ℕ), edist (f n) (f (Nat.succ n)) ≤ ↑(d n)\nε : ℝ≥0\nεpos : 0 < ε\nhd : CauchySeq fun n => ∑ x in range n, d x\nN n : ℕ\nhn : N ≤ n\nhN : dist (∑ x in range n, d x) (∑ x in range N, d x) < ↑ε\n⊢ edist (f n) (f N) < ↑ε", "state_before": "α : Type u_1\ninst✝ : PseudoEMetricSpace α\nf : ℕ → α\nd : ℕ → ℝ≥0\nhf : ∀ (n : ℕ), edist (f n) (f (Nat.succ n)) ≤ ↑(d n)\nε : ℝ≥0\nεpos : 0 < ε\nhd : CauchySeq fun n => ∑ x in range n, d x\nN : ℕ\nhN : ∀ (n : ℕ), n ≥ N → dist (∑ x in range n, d x) (∑ x in range N, d x) < ↑ε\nn : ℕ\nhn : N ≤ n\n⊢ edist (f n) (f N) < ↑ε", "tactic": "specialize hN n hn" }, { "state_after": "α : Type u_1\ninst✝ : PseudoEMetricSpace α\nf : ℕ → α\nd : ℕ → ℝ≥0\nhf : ∀ (n : ℕ), edist (f n) (f (Nat.succ n)) ≤ ↑(d n)\nε : ℝ≥0\nεpos : 0 < ε\nhd : CauchySeq fun n => ∑ x in range n, d x\nN n : ℕ\nhn : N ≤ n\nhN : ∑ k in Ico N n, d k < ε ∧ ∑ x in range N, d x - (∑ k in range N, d k + ∑ k in Ico N n, d k) < ε\n⊢ edist (f n) (f N) < ↑ε", "state_before": "α : Type u_1\ninst✝ : PseudoEMetricSpace α\nf : ℕ → α\nd : ℕ → ℝ≥0\nhf : ∀ (n : ℕ), edist (f n) (f (Nat.succ n)) ≤ ↑(d n)\nε : ℝ≥0\nεpos : 0 < ε\nhd : CauchySeq fun n => ∑ x in range n, d x\nN n : ℕ\nhn : N ≤ n\nhN : dist (∑ x in range n, d x) (∑ x in range N, d x) < ↑ε\n⊢ edist (f n) (f N) < ↑ε", "tactic": "rw [dist_nndist, NNReal.nndist_eq, ← sum_range_add_sum_Ico _ hn, add_tsub_cancel_left,\n NNReal.coe_lt_coe, max_lt_iff] at hN" }, { "state_after": "α : Type u_1\ninst✝ : PseudoEMetricSpace α\nf : ℕ → α\nd : ℕ → ℝ≥0\nhf : ∀ (n : ℕ), edist (f n) (f (Nat.succ n)) ≤ ↑(d n)\nε : ℝ≥0\nεpos : 0 < ε\nhd : CauchySeq fun n => ∑ x in range n, d x\nN n : ℕ\nhn : N ≤ n\nhN : ∑ k in Ico N n, d k < ε ∧ ∑ x in range N, d x - (∑ k in range N, d k + ∑ k in Ico N n, d k) < ε\n⊢ edist (f N) (f n) < ↑ε", "state_before": "α : Type u_1\ninst✝ : PseudoEMetricSpace α\nf : ℕ → α\nd : ℕ → ℝ≥0\nhf : ∀ (n : ℕ), edist (f n) (f (Nat.succ n)) ≤ ↑(d n)\nε : ℝ≥0\nεpos : 0 < ε\nhd : CauchySeq fun n => ∑ x in range n, d x\nN n : ℕ\nhn : N ≤ n\nhN : ∑ k in Ico N n, d k < ε ∧ ∑ x in range N, d x - (∑ k in range N, d k + ∑ k in Ico N n, d k) < ε\n⊢ edist (f n) (f N) < ↑ε", "tactic": "rw [edist_comm]" }, { "state_after": "α : Type u_1\ninst✝ : PseudoEMetricSpace α\nf : ℕ → α\nd : ℕ → ℝ≥0\nhf : ∀ (n : ℕ), edist (f n) (f (Nat.succ n)) ≤ ↑(d n)\nε : ℝ≥0\nεpos : 0 < ε\nhd : CauchySeq fun n => ∑ x in range n, d x\nN n : ℕ\nhn : N ≤ n\nhN : ∑ k in Ico N n, d k < ε ∧ ∑ x in range N, d x - (∑ k in range N, d k + ∑ k in Ico N n, d k) < ε\n⊢ ∑ i in Ico N n, ↑(d i) < ↑ε", "state_before": "α : Type u_1\ninst✝ : PseudoEMetricSpace α\nf : ℕ → α\nd : ℕ → ℝ≥0\nhf : ∀ (n : ℕ), edist (f n) (f (Nat.succ n)) ≤ ↑(d n)\nε : ℝ≥0\nεpos : 0 < ε\nhd : CauchySeq fun n => ∑ x in range n, d x\nN n : ℕ\nhn : N ≤ n\nhN : ∑ k in Ico N n, d k < ε ∧ ∑ x in range N, d x - (∑ k in range N, d k + ∑ k in Ico N n, d k) < ε\n⊢ edist (f N) (f n) < ↑ε", "tactic": "refine lt_of_le_of_lt (edist_le_Ico_sum_of_edist_le hn fun _ _ => hf _) ?_" }, { "state_after": "no goals", "state_before": "α : Type u_1\ninst✝ : PseudoEMetricSpace α\nf : ℕ → α\nd : ℕ → ℝ≥0\nhf : ∀ (n : ℕ), edist (f n) (f (Nat.succ n)) ≤ ↑(d n)\nε : ℝ≥0\nεpos : 0 < ε\nhd : CauchySeq fun n => ∑ x in range n, d x\nN n : ℕ\nhn : N ≤ n\nhN : ∑ k in Ico N n, d k < ε ∧ ∑ x in range N, d x - (∑ k in range N, d k + ∑ k in Ico N n, d k) < ε\n⊢ ∑ i in Ico N n, ↑(d i) < ↑ε", "tactic": "exact_mod_cast hN.1" } ]

[ 43, 22 ]

[ 27, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type u_2\nl : AssocList α β\n⊢ isEmpty l = List.isEmpty (toList l)", "tactic": "cases l <;> simp [*, isEmpty, List.isEmpty]" } ]

[ 44, 46 ]

[ 43, 9 ]

[ { "state_after": "case h\nM : Type u_2\nN : Type u_1\nP : Type ?u.144954\ninst✝³ : MulOneClass M\ninst✝² : MulOneClass N\ninst✝¹ : MulOneClass P\nS : Submonoid M\nA : Type ?u.144975\ninst✝ : SetLike A M\nhA : SubmonoidClass A M\nS' : A\nF : Type ?u.144999\nmc : MonoidHomClass F M N\nx : N\n⊢ x ∈ mker (inr M N) ↔ x ∈ ⊥", "state_before": "M : Type u_2\nN : Type u_1\nP : Type ?u.144954\ninst✝³ : MulOneClass M\ninst✝² : MulOneClass N\ninst✝¹ : MulOneClass P\nS : Submonoid M\nA : Type ?u.144975\ninst✝ : SetLike A M\nhA : SubmonoidClass A M\nS' : A\nF : Type ?u.144999\nmc : MonoidHomClass F M N\n⊢ mker (inr M N) = ⊥", "tactic": "ext x" }, { "state_after": "no goals", "state_before": "case h\nM : Type u_2\nN : Type u_1\nP : Type ?u.144954\ninst✝³ : MulOneClass M\ninst✝² : MulOneClass N\ninst✝¹ : MulOneClass P\nS : Submonoid M\nA : Type ?u.144975\ninst✝ : SetLike A M\nhA : SubmonoidClass A M\nS' : A\nF : Type ?u.144999\nmc : MonoidHomClass F M N\nx : N\n⊢ x ∈ mker (inr M N) ↔ x ∈ ⊥", "tactic": "simp [mem_mker]" } ]

[ 1242, 18 ]

[ 1240, 1 ]

[]

[ 825, 43 ]

[ 823, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type ?u.110781\nβ : Type ?u.110784\nγ : Type ?u.110787\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nf : Ordinal → Ordinal\no : Ordinal\nH : IsNormal f\np : Set Ordinal\np0 : Set.Nonempty p\nb : Ordinal\nH₂ : ∀ (o : Ordinal), b ≤ o ↔ ∀ (a : Ordinal), a ∈ p → a ≤ o\nh : ∀ (a : Ordinal), a ∈ p → f a ≤ o\n⊢ f b ≤ o", "tactic": "induction b using limitRecOn with\n| H₁ =>\n cases' p0 with x px\n have := Ordinal.le_zero.1 ((H₂ _).1 (Ordinal.zero_le _) _ px)\n rw [this] at px\n exact h _ px\n| H₂ S _ =>\n rcases not_ball.1 (mt (H₂ S).2 <| (lt_succ S).not_le) with ⟨a, h₁, h₂⟩\n exact (H.le_iff.2 <| succ_le_of_lt <| not_le.1 h₂).trans (h _ h₁)\n| H₃ S L _ =>\n refine' (H.2 _ L _).2 fun a h' => _\n rcases not_ball.1 (mt (H₂ a).2 h'.not_le) with ⟨b, h₁, h₂⟩\n exact (H.le_iff.2 <| (not_le.1 h₂).le).trans (h _ h₁)" }, { "state_after": "case H₁.intro\nα : Type ?u.110781\nβ : Type ?u.110784\nγ : Type ?u.110787\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nf : Ordinal → Ordinal\no : Ordinal\nH : IsNormal f\np : Set Ordinal\nh : ∀ (a : Ordinal), a ∈ p → f a ≤ o\nH₂ : ∀ (o : Ordinal), 0 ≤ o ↔ ∀ (a : Ordinal), a ∈ p → a ≤ o\nx : Ordinal\npx : x ∈ p\n⊢ f 0 ≤ o", "state_before": "case H₁\nα : Type ?u.110781\nβ : Type ?u.110784\nγ : Type ?u.110787\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nf : Ordinal → Ordinal\no : Ordinal\nH : IsNormal f\np : Set Ordinal\np0 : Set.Nonempty p\nh : ∀ (a : Ordinal), a ∈ p → f a ≤ o\nH₂ : ∀ (o : Ordinal), 0 ≤ o ↔ ∀ (a : Ordinal), a ∈ p → a ≤ o\n⊢ f 0 ≤ o", "tactic": "cases' p0 with x px" }, { "state_after": "case H₁.intro\nα : Type ?u.110781\nβ : Type ?u.110784\nγ : Type ?u.110787\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nf : Ordinal → Ordinal\no : Ordinal\nH : IsNormal f\np : Set Ordinal\nh : ∀ (a : Ordinal), a ∈ p → f a ≤ o\nH₂ : ∀ (o : Ordinal), 0 ≤ o ↔ ∀ (a : Ordinal), a ∈ p → a ≤ o\nx : Ordinal\npx : x ∈ p\nthis : x = 0\n⊢ f 0 ≤ o", "state_before": "case H₁.intro\nα : Type ?u.110781\nβ : Type ?u.110784\nγ : Type ?u.110787\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nf : Ordinal → Ordinal\no : Ordinal\nH : IsNormal f\np : Set Ordinal\nh : ∀ (a : Ordinal), a ∈ p → f a ≤ o\nH₂ : ∀ (o : Ordinal), 0 ≤ o ↔ ∀ (a : Ordinal), a ∈ p → a ≤ o\nx : Ordinal\npx : x ∈ p\n⊢ f 0 ≤ o", "tactic": "have := Ordinal.le_zero.1 ((H₂ _).1 (Ordinal.zero_le _) _ px)" }, { "state_after": "case H₁.intro\nα : Type ?u.110781\nβ : Type ?u.110784\nγ : Type ?u.110787\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nf : Ordinal → Ordinal\no : Ordinal\nH : IsNormal f\np : Set Ordinal\nh : ∀ (a : Ordinal), a ∈ p → f a ≤ o\nH₂ : ∀ (o : Ordinal), 0 ≤ o ↔ ∀ (a : Ordinal), a ∈ p → a ≤ o\nx : Ordinal\npx : 0 ∈ p\nthis : x = 0\n⊢ f 0 ≤ o", "state_before": "case H₁.intro\nα : Type ?u.110781\nβ : Type ?u.110784\nγ : Type ?u.110787\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nf : Ordinal → Ordinal\no : Ordinal\nH : IsNormal f\np : Set Ordinal\nh : ∀ (a : Ordinal), a ∈ p → f a ≤ o\nH₂ : ∀ (o : Ordinal), 0 ≤ o ↔ ∀ (a : Ordinal), a ∈ p → a ≤ o\nx : Ordinal\npx : x ∈ p\nthis : x = 0\n⊢ f 0 ≤ o", "tactic": "rw [this] at px" }, { "state_after": "no goals", "state_before": "case H₁.intro\nα : Type ?u.110781\nβ : Type ?u.110784\nγ : Type ?u.110787\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nf : Ordinal → Ordinal\no : Ordinal\nH : IsNormal f\np : Set Ordinal\nh : ∀ (a : Ordinal), a ∈ p → f a ≤ o\nH₂ : ∀ (o : Ordinal), 0 ≤ o ↔ ∀ (a : Ordinal), a ∈ p → a ≤ o\nx : Ordinal\npx : 0 ∈ p\nthis : x = 0\n⊢ f 0 ≤ o", "tactic": "exact h _ px" }, { "state_after": "case H₂.intro.intro\nα : Type ?u.110781\nβ : Type ?u.110784\nγ : Type ?u.110787\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nf : Ordinal → Ordinal\no : Ordinal\nH : IsNormal f\np : Set Ordinal\np0 : Set.Nonempty p\nh : ∀ (a : Ordinal), a ∈ p → f a ≤ o\nS : Ordinal\na✝ : (∀ (o : Ordinal), S ≤ o ↔ ∀ (a : Ordinal), a ∈ p → a ≤ o) → f S ≤ o\nH₂ : ∀ (o : Ordinal), succ S ≤ o ↔ ∀ (a : Ordinal), a ∈ p → a ≤ o\na : Ordinal\nh₁ : a ∈ p\nh₂ : ¬a ≤ S\n⊢ f (succ S) ≤ o", "state_before": "case H₂\nα : Type ?u.110781\nβ : Type ?u.110784\nγ : Type ?u.110787\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nf : Ordinal → Ordinal\no : Ordinal\nH : IsNormal f\np : Set Ordinal\np0 : Set.Nonempty p\nh : ∀ (a : Ordinal), a ∈ p → f a ≤ o\nS : Ordinal\na✝ : (∀ (o : Ordinal), S ≤ o ↔ ∀ (a : Ordinal), a ∈ p → a ≤ o) → f S ≤ o\nH₂ : ∀ (o : Ordinal), succ S ≤ o ↔ ∀ (a : Ordinal), a ∈ p → a ≤ o\n⊢ f (succ S) ≤ o", "tactic": "rcases not_ball.1 (mt (H₂ S).2 <| (lt_succ S).not_le) with ⟨a, h₁, h₂⟩" }, { "state_after": "no goals", "state_before": "case H₂.intro.intro\nα : Type ?u.110781\nβ : Type ?u.110784\nγ : Type ?u.110787\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nf : Ordinal → Ordinal\no : Ordinal\nH : IsNormal f\np : Set Ordinal\np0 : Set.Nonempty p\nh : ∀ (a : Ordinal), a ∈ p → f a ≤ o\nS : Ordinal\na✝ : (∀ (o : Ordinal), S ≤ o ↔ ∀ (a : Ordinal), a ∈ p → a ≤ o) → f S ≤ o\nH₂ : ∀ (o : Ordinal), succ S ≤ o ↔ ∀ (a : Ordinal), a ∈ p → a ≤ o\na : Ordinal\nh₁ : a ∈ p\nh₂ : ¬a ≤ S\n⊢ f (succ S) ≤ o", "tactic": "exact (H.le_iff.2 <| succ_le_of_lt <| not_le.1 h₂).trans (h _ h₁)" }, { "state_after": "case H₃\nα : Type ?u.110781\nβ : Type ?u.110784\nγ : Type ?u.110787\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nf : Ordinal → Ordinal\no : Ordinal\nH : IsNormal f\np : Set Ordinal\np0 : Set.Nonempty p\nh : ∀ (a : Ordinal), a ∈ p → f a ≤ o\nS : Ordinal\nL : IsLimit S\na✝ : ∀ (o' : Ordinal), o' < S → (∀ (o : Ordinal), o' ≤ o ↔ ∀ (a : Ordinal), a ∈ p → a ≤ o) → f o' ≤ o\nH₂ : ∀ (o : Ordinal), S ≤ o ↔ ∀ (a : Ordinal), a ∈ p → a ≤ o\na : Ordinal\nh' : a < S\n⊢ f a ≤ o", "state_before": "case H₃\nα : Type ?u.110781\nβ : Type ?u.110784\nγ : Type ?u.110787\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nf : Ordinal → Ordinal\no : Ordinal\nH : IsNormal f\np : Set Ordinal\np0 : Set.Nonempty p\nh : ∀ (a : Ordinal), a ∈ p → f a ≤ o\nS : Ordinal\nL : IsLimit S\na✝ : ∀ (o' : Ordinal), o' < S → (∀ (o : Ordinal), o' ≤ o ↔ ∀ (a : Ordinal), a ∈ p → a ≤ o) → f o' ≤ o\nH₂ : ∀ (o : Ordinal), S ≤ o ↔ ∀ (a : Ordinal), a ∈ p → a ≤ o\n⊢ f S ≤ o", "tactic": "refine' (H.2 _ L _).2 fun a h' => _" }, { "state_after": "case H₃.intro.intro\nα : Type ?u.110781\nβ : Type ?u.110784\nγ : Type ?u.110787\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nf : Ordinal → Ordinal\no : Ordinal\nH : IsNormal f\np : Set Ordinal\np0 : Set.Nonempty p\nh : ∀ (a : Ordinal), a ∈ p → f a ≤ o\nS : Ordinal\nL : IsLimit S\na✝ : ∀ (o' : Ordinal), o' < S → (∀ (o : Ordinal), o' ≤ o ↔ ∀ (a : Ordinal), a ∈ p → a ≤ o) → f o' ≤ o\nH₂ : ∀ (o : Ordinal), S ≤ o ↔ ∀ (a : Ordinal), a ∈ p → a ≤ o\na : Ordinal\nh' : a < S\nb : Ordinal\nh₁ : b ∈ p\nh₂ : ¬b ≤ a\n⊢ f a ≤ o", "state_before": "case H₃\nα : Type ?u.110781\nβ : Type ?u.110784\nγ : Type ?u.110787\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nf : Ordinal → Ordinal\no : Ordinal\nH : IsNormal f\np : Set Ordinal\np0 : Set.Nonempty p\nh : ∀ (a : Ordinal), a ∈ p → f a ≤ o\nS : Ordinal\nL : IsLimit S\na✝ : ∀ (o' : Ordinal), o' < S → (∀ (o : Ordinal), o' ≤ o ↔ ∀ (a : Ordinal), a ∈ p → a ≤ o) → f o' ≤ o\nH₂ : ∀ (o : Ordinal), S ≤ o ↔ ∀ (a : Ordinal), a ∈ p → a ≤ o\na : Ordinal\nh' : a < S\n⊢ f a ≤ o", "tactic": "rcases not_ball.1 (mt (H₂ a).2 h'.not_le) with ⟨b, h₁, h₂⟩" }, { "state_after": "no goals", "state_before": "case H₃.intro.intro\nα : Type ?u.110781\nβ : Type ?u.110784\nγ : Type ?u.110787\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nf : Ordinal → Ordinal\no : Ordinal\nH : IsNormal f\np : Set Ordinal\np0 : Set.Nonempty p\nh : ∀ (a : Ordinal), a ∈ p → f a ≤ o\nS : Ordinal\nL : IsLimit S\na✝ : ∀ (o' : Ordinal), o' < S → (∀ (o : Ordinal), o' ≤ o ↔ ∀ (a : Ordinal), a ∈ p → a ≤ o) → f o' ≤ o\nH₂ : ∀ (o : Ordinal), S ≤ o ↔ ∀ (a : Ordinal), a ∈ p → a ≤ o\na : Ordinal\nh' : a < S\nb : Ordinal\nh₁ : b ∈ p\nh₂ : ¬b ≤ a\n⊢ f a ≤ o", "tactic": "exact (H.le_iff.2 <| (not_le.1 h₂).le).trans (h _ h₁)" } ]

[ 463, 61 ]

[ 447, 1 ]

[]

[ 245, 83 ]

[ 244, 1 ]

[]

[ 564, 19 ]

[ 562, 11 ]

[]

[ 532, 6 ]

[ 531, 1 ]

[]

[ 516, 43 ]

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[]

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[ 221, 1 ]

[]

[ 889, 37 ]

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[]

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[]

[ 108, 47 ]

[ 104, 1 ]

[ { "state_after": "case h.e'_5.h.e'_6\nα : Type u_1\nβ : Type ?u.1531711\nι : Type ?u.1531714\nE : Type u_2\nF : Type ?u.1531720\n𝕜 : Type ?u.1531723\ninst✝² : MeasurableSpace α\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedAddCommGroup F\np : ℝ≥0∞\nμ : Measure α\nf : { x // x ∈ simpleFunc E p μ }\n⊢ ↑↑f = AEEqFun.mk ↑(toSimpleFunc f) (_ : AEStronglyMeasurable (↑(toSimpleFunc f)) μ)", "state_before": "α : Type u_1\nβ : Type ?u.1531711\nι : Type ?u.1531714\nE : Type u_2\nF : Type ?u.1531720\n𝕜 : Type ?u.1531723\ninst✝² : MeasurableSpace α\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedAddCommGroup F\np : ℝ≥0∞\nμ : Measure α\nf : { x // x ∈ simpleFunc E p μ }\n⊢ ↑(toSimpleFunc f) =ᵐ[μ] ↑↑↑f", "tactic": "convert (AEEqFun.coeFn_mk (toSimpleFunc f)\n (toSimpleFunc f).aestronglyMeasurable).symm using 2" }, { "state_after": "no goals", "state_before": "case h.e'_5.h.e'_6\nα : Type u_1\nβ : Type ?u.1531711\nι : Type ?u.1531714\nE : Type u_2\nF : Type ?u.1531720\n𝕜 : Type ?u.1531723\ninst✝² : MeasurableSpace α\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedAddCommGroup F\np : ℝ≥0∞\nμ : Measure α\nf : { x // x ∈ simpleFunc E p μ }\n⊢ ↑↑f = AEEqFun.mk ↑(toSimpleFunc f) (_ : AEStronglyMeasurable (↑(toSimpleFunc f)) μ)", "tactic": "exact (Classical.choose_spec f.2).symm" } ]

[ 625, 43 ]

[ 621, 1 ]

[]

[ 108, 50 ]

[ 106, 1 ]

[ { "state_after": "F : Type ?u.55319\nα : Type u_3\nβ : Type u_1\nγ : Type u_2\nι : Type ?u.55331\nκ : Type ?u.55334\ninst✝¹ : SemilatticeSup α\ninst✝ : OrderBot α\ns✝ s₁ s₂ : Finset β\nf✝ g : β → α\na : α\ns : Finset β\nt : Finset γ\nf : β × γ → α\n⊢ (∀ (a : β) (b : γ), a ∈ s → b ∈ t → f (a, b) ≤ sup s fun i => sup t fun i' => f (i, i')) ∧\n ∀ (b : β), b ∈ s → ∀ (b_1 : γ), b_1 ∈ t → f (b, b_1) ≤ sup (s ×ˢ t) f", "state_before": "F : Type ?u.55319\nα : Type u_3\nβ : Type u_1\nγ : Type u_2\nι : Type ?u.55331\nκ : Type ?u.55334\ninst✝¹ : SemilatticeSup α\ninst✝ : OrderBot α\ns✝ s₁ s₂ : Finset β\nf✝ g : β → α\na : α\ns : Finset β\nt : Finset γ\nf : β × γ → α\n⊢ sup (s ×ˢ t) f = sup s fun i => sup t fun i' => f (i, i')", "tactic": "simp only [le_antisymm_iff, Finset.sup_le_iff, mem_product, and_imp, Prod.forall]" }, { "state_after": "case refine_1\nF : Type ?u.55319\nα : Type u_3\nβ : Type u_1\nγ : Type u_2\nι : Type ?u.55331\nκ : Type ?u.55334\ninst✝¹ : SemilatticeSup α\ninst✝ : OrderBot α\ns✝ s₁ s₂ : Finset β\nf✝ g : β → α\na : α\ns : Finset β\nt : Finset γ\nf : β × γ → α\nb : β\nc : γ\nhb : b ∈ s\nhc : c ∈ t\n⊢ f (b, c) ≤ sup s fun i => sup t fun i' => f (i, i')\n\ncase refine_2\nF : Type ?u.55319\nα : Type u_3\nβ : Type u_1\nγ : Type u_2\nι : Type ?u.55331\nκ : Type ?u.55334\ninst✝¹ : SemilatticeSup α\ninst✝ : OrderBot α\ns✝ s₁ s₂ : Finset β\nf✝ g : β → α\na : α\ns : Finset β\nt : Finset γ\nf : β × γ → α\nb : β\nhb : b ∈ s\nc : γ\nhc : c ∈ t\n⊢ f (b, c) ≤ sup (s ×ˢ t) f", "state_before": "F : Type ?u.55319\nα : Type u_3\nβ : Type u_1\nγ : Type u_2\nι : Type ?u.55331\nκ : Type ?u.55334\ninst✝¹ : SemilatticeSup α\ninst✝ : OrderBot α\ns✝ s₁ s₂ : Finset β\nf✝ g : β → α\na : α\ns : Finset β\nt : Finset γ\nf : β × γ → α\n⊢ (∀ (a : β) (b : γ), a ∈ s → b ∈ t → f (a, b) ≤ sup s fun i => sup t fun i' => f (i, i')) ∧\n ∀ (b : β), b ∈ s → ∀ (b_1 : γ), b_1 ∈ t → f (b, b_1) ≤ sup (s ×ˢ t) f", "tactic": "refine ⟨fun b c hb hc => ?_, fun b hb c hc => ?_⟩" }, { "state_after": "case refine_1\nF : Type ?u.55319\nα : Type u_3\nβ : Type u_1\nγ : Type u_2\nι : Type ?u.55331\nκ : Type ?u.55334\ninst✝¹ : SemilatticeSup α\ninst✝ : OrderBot α\ns✝ s₁ s₂ : Finset β\nf✝ g : β → α\na : α\ns : Finset β\nt : Finset γ\nf : β × γ → α\nb : β\nc : γ\nhb : b ∈ s\nhc : c ∈ t\n⊢ f (b, c) ≤ sup t fun i' => f (b, i')", "state_before": "case refine_1\nF : Type ?u.55319\nα : Type u_3\nβ : Type u_1\nγ : Type u_2\nι : Type ?u.55331\nκ : Type ?u.55334\ninst✝¹ : SemilatticeSup α\ninst✝ : OrderBot α\ns✝ s₁ s₂ : Finset β\nf✝ g : β → α\na : α\ns : Finset β\nt : Finset γ\nf : β × γ → α\nb : β\nc : γ\nhb : b ∈ s\nhc : c ∈ t\n⊢ f (b, c) ≤ sup s fun i => sup t fun i' => f (i, i')", "tactic": "refine (le_sup hb).trans' ?_" }, { "state_after": "no goals", "state_before": "case refine_1\nF : Type ?u.55319\nα : Type u_3\nβ : Type u_1\nγ : Type u_2\nι : Type ?u.55331\nκ : Type ?u.55334\ninst✝¹ : SemilatticeSup α\ninst✝ : OrderBot α\ns✝ s₁ s₂ : Finset β\nf✝ g : β → α\na : α\ns : Finset β\nt : Finset γ\nf : β × γ → α\nb : β\nc : γ\nhb : b ∈ s\nhc : c ∈ t\n⊢ f (b, c) ≤ sup t fun i' => f (b, i')", "tactic": "exact @le_sup _ _ _ _ _ (fun c => f (b, c)) c hc" }, { "state_after": "no goals", "state_before": "case refine_2\nF : Type ?u.55319\nα : Type u_3\nβ : Type u_1\nγ : Type u_2\nι : Type ?u.55331\nκ : Type ?u.55334\ninst✝¹ : SemilatticeSup α\ninst✝ : OrderBot α\ns✝ s₁ s₂ : Finset β\nf✝ g : β → α\na : α\ns : Finset β\nt : Finset γ\nf : β × γ → α\nb : β\nhb : b ∈ s\nc : γ\nhc : c ∈ t\n⊢ f (b, c) ≤ sup (s ×ˢ t) f", "tactic": "exact le_sup <| mem_product.2 ⟨hb, hc⟩" } ]

[ 178, 43 ]

[ 171, 1 ]

[ { "state_after": "P : Type u_1\ninst✝ : SemilatticeInf P\nx✝ y✝ : P\nI : Ideal P\nhI : IsPrime I\nx y : P\n⊢ ¬x ∈ I ∧ ¬y ∈ I → ¬x ⊓ y ∈ I", "state_before": "P : Type u_1\ninst✝ : SemilatticeInf P\nx✝ y✝ : P\nI : Ideal P\nhI : IsPrime I\nx y : P\n⊢ x ⊓ y ∈ I → x ∈ I ∨ y ∈ I", "tactic": "contrapose!" }, { "state_after": "P : Type u_1\ninst✝ : SemilatticeInf P\nx✝ y✝ : P\nI : Ideal P\nhI : IsPrime I\nx y : P\nF : PFilter P := IsPFilter.toPFilter (_ : IsPFilter (↑Iᶜ))\n⊢ ¬x ∈ I ∧ ¬y ∈ I → ¬x ⊓ y ∈ I", "state_before": "P : Type u_1\ninst✝ : SemilatticeInf P\nx✝ y✝ : P\nI : Ideal P\nhI : IsPrime I\nx y : P\n⊢ ¬x ∈ I ∧ ¬y ∈ I → ¬x ⊓ y ∈ I", "tactic": "let F := hI.compl_filter.toPFilter" }, { "state_after": "P : Type u_1\ninst✝ : SemilatticeInf P\nx✝ y✝ : P\nI : Ideal P\nhI : IsPrime I\nx y : P\nF : PFilter P := IsPFilter.toPFilter (_ : IsPFilter (↑Iᶜ))\n⊢ x ∈ F ∧ y ∈ F → x ⊓ y ∈ F", "state_before": "P : Type u_1\ninst✝ : SemilatticeInf P\nx✝ y✝ : P\nI : Ideal P\nhI : IsPrime I\nx y : P\nF : PFilter P := IsPFilter.toPFilter (_ : IsPFilter (↑Iᶜ))\n⊢ ¬x ∈ I ∧ ¬y ∈ I → ¬x ⊓ y ∈ I", "tactic": "show x ∈ F ∧ y ∈ F → x ⊓ y ∈ F" }, { "state_after": "no goals", "state_before": "P : Type u_1\ninst✝ : SemilatticeInf P\nx✝ y✝ : P\nI : Ideal P\nhI : IsPrime I\nx y : P\nF : PFilter P := IsPFilter.toPFilter (_ : IsPFilter (↑Iᶜ))\n⊢ x ∈ F ∧ y ∈ F → x ⊓ y ∈ F", "tactic": "exact fun h => inf_mem h.1 h.2" } ]

[ 130, 33 ]

[ 126, 1 ]

[ { "state_after": "α : Type u_1\nβ : Type ?u.8808\ns : Set (Setoid α)\n⊢ (EqvGen.Setoid fun x y => ∃ r, r ∈ s ∧ Rel r x y) = EqvGen.Setoid (⨆ (a : Setoid α) (_ : a ∈ s), Rel a)", "state_before": "α : Type u_1\nβ : Type ?u.8808\ns : Set (Setoid α)\n⊢ sSup s = EqvGen.Setoid (sSup (Rel '' s))", "tactic": "rw [sSup_eq_eqvGen, sSup_image]" }, { "state_after": "case e_r.h.h.a\nα : Type u_1\nβ : Type ?u.8808\ns : Set (Setoid α)\nx y : α\n⊢ (∃ r, r ∈ s ∧ Rel r x y) ↔ iSup (fun a => ⨆ (_ : a ∈ s), Rel a) x y", "state_before": "α : Type u_1\nβ : Type ?u.8808\ns : Set (Setoid α)\n⊢ (EqvGen.Setoid fun x y => ∃ r, r ∈ s ∧ Rel r x y) = EqvGen.Setoid (⨆ (a : Setoid α) (_ : a ∈ s), Rel a)", "tactic": "congr with (x y)" }, { "state_after": "no goals", "state_before": "case e_r.h.h.a\nα : Type u_1\nβ : Type ?u.8808\ns : Set (Setoid α)\nx y : α\n⊢ (∃ r, r ∈ s ∧ Rel r x y) ↔ iSup (fun a => ⨆ (_ : a ∈ s), Rel a) x y", "tactic": "simp only [iSup_apply, iSup_Prop_eq, exists_prop]" } ]

[ 244, 52 ]

[ 241, 1 ]

[]

[ 495, 6 ]

[ 493, 1 ]

[]

[ 2557, 38 ]

[ 2555, 1 ]

[ { "state_after": "C : Type u_1\ninst✝³ : Category C\nD : Type u_3\ninst✝² : Category D\nE : Type ?u.5750\ninst✝¹ : Category E\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology E\nA : Type ?u.5802\ninst✝ : Category A\nG : C ⥤ D\nH✝ H : CoverDense K G\nℱ : SheafOfTypes K\nX : D\ns t : ℱ.val.obj X.op\nh : ∀ ⦃Y : C⦄ (f : G.obj Y ⟶ X), ℱ.val.map f.op s = ℱ.val.map f.op t\n⊢ ∀ ⦃Y : D⦄ ⦃f : Y ⟶ X⦄, (Sieve.coverByImage G X).arrows f → ℱ.val.map f.op s = ℱ.val.map f.op t", "state_before": "C : Type u_1\ninst✝³ : Category C\nD : Type u_3\ninst✝² : Category D\nE : Type ?u.5750\ninst✝¹ : Category E\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology E\nA : Type ?u.5802\ninst✝ : Category A\nG : C ⥤ D\nH✝ H : CoverDense K G\nℱ : SheafOfTypes K\nX : D\ns t : ℱ.val.obj X.op\nh : ∀ ⦃Y : C⦄ (f : G.obj Y ⟶ X), ℱ.val.map f.op s = ℱ.val.map f.op t\n⊢ s = t", "tactic": "apply (ℱ.cond (Sieve.coverByImage G X) (H.is_cover X)).isSeparatedFor.ext" }, { "state_after": "case intro.mk.refl\nC : Type u_1\ninst✝³ : Category C\nD : Type u_3\ninst✝² : Category D\nE : Type ?u.5750\ninst✝¹ : Category E\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology E\nA : Type ?u.5802\ninst✝ : Category A\nG : C ⥤ D\nH✝ H : CoverDense K G\nℱ : SheafOfTypes K\nX : D\ns t : ℱ.val.obj X.op\nh : ∀ ⦃Y : C⦄ (f : G.obj Y ⟶ X), ℱ.val.map f.op s = ℱ.val.map f.op t\nY : D\nZ : C\nf₁ : Y ⟶ G.obj Z\nf₂ : G.obj Z ⟶ X\n⊢ ℱ.val.map (f₁ ≫ f₂).op s = ℱ.val.map (f₁ ≫ f₂).op t", "state_before": "C : Type u_1\ninst✝³ : Category C\nD : Type u_3\ninst✝² : Category D\nE : Type ?u.5750\ninst✝¹ : Category E\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology E\nA : Type ?u.5802\ninst✝ : Category A\nG : C ⥤ D\nH✝ H : CoverDense K G\nℱ : SheafOfTypes K\nX : D\ns t : ℱ.val.obj X.op\nh : ∀ ⦃Y : C⦄ (f : G.obj Y ⟶ X), ℱ.val.map f.op s = ℱ.val.map f.op t\n⊢ ∀ ⦃Y : D⦄ ⦃f : Y ⟶ X⦄, (Sieve.coverByImage G X).arrows f → ℱ.val.map f.op s = ℱ.val.map f.op t", "tactic": "rintro Y _ ⟨Z, f₁, f₂, ⟨rfl⟩⟩" }, { "state_after": "no goals", "state_before": "case intro.mk.refl\nC : Type u_1\ninst✝³ : Category C\nD : Type u_3\ninst✝² : Category D\nE : Type ?u.5750\ninst✝¹ : Category E\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nL : GrothendieckTopology E\nA : Type ?u.5802\ninst✝ : Category A\nG : C ⥤ D\nH✝ H : CoverDense K G\nℱ : SheafOfTypes K\nX : D\ns t : ℱ.val.obj X.op\nh : ∀ ⦃Y : C⦄ (f : G.obj Y ⟶ X), ℱ.val.map f.op s = ℱ.val.map f.op t\nY : D\nZ : C\nf₁ : Y ⟶ G.obj Z\nf₂ : G.obj Z ⟶ X\n⊢ ℱ.val.map (f₁ ≫ f₂).op s = ℱ.val.map (f₁ ≫ f₂).op t", "tactic": "simp [h f₂]" } ]

[ 116, 14 ]

[ 112, 1 ]

[]

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[ 2146, 1 ]

[ { "state_after": "no goals", "state_before": "o₁ o₂ : Ordinal\n⊢ aleph o₁ * aleph o₂ = aleph (max o₁ o₂)", "tactic": "rw [Cardinal.mul_eq_max (aleph0_le_aleph o₁) (aleph0_le_aleph o₂), max_aleph_eq]" } ]

[ 566, 83 ]

[ 565, 1 ]

[]

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[ { "state_after": "α : Type u_2\ninst✝³ : Group α\ns t : Subgroup α\nH : Type u_1\ninst✝² : Group H\nK : Subgroup H\ninst✝¹ : Fintype { x // x ∈ K }\nf : α →* H\ninst✝ : Fintype { x // x ∈ comap f K }\nhf : Injective ↑f\nthis : Fintype { x // x ∈ map f (comap f K) }\n⊢ card { x // x ∈ comap f K } ∣ card { x // x ∈ K }", "state_before": "α : Type u_2\ninst✝³ : Group α\ns t : Subgroup α\nH : Type u_1\ninst✝² : Group H\nK : Subgroup H\ninst✝¹ : Fintype { x // x ∈ K }\nf : α →* H\ninst✝ : Fintype { x // x ∈ comap f K }\nhf : Injective ↑f\n⊢ card { x // x ∈ comap f K } ∣ card { x // x ∈ K }", "tactic": "haveI : Fintype ((K.comap f).map f) :=\n Fintype.ofEquiv _ (equivMapOfInjective _ _ hf).toEquiv" }, { "state_after": "no goals", "state_before": "α : Type u_2\ninst✝³ : Group α\ns t : Subgroup α\nH : Type u_1\ninst✝² : Group H\nK : Subgroup H\ninst✝¹ : Fintype { x // x ∈ K }\nf : α →* H\ninst✝ : Fintype { x // x ∈ comap f K }\nhf : Injective ↑f\nthis : Fintype { x // x ∈ map f (comap f K) }\n⊢ card { x // x ∈ comap f K } ∣ card { x // x ∈ K }", "tactic": "calc\n Fintype.card (K.comap f) = Fintype.card ((K.comap f).map f) :=\n Fintype.card_congr (equivMapOfInjective _ _ hf).toEquiv\n _ ∣ Fintype.card K := card_dvd_of_le (map_comap_le _ _)" } ]

[ 832, 62 ]

[ 825, 1 ]

[ { "state_after": "α✝ : Type ?u.31102\nα : Type u\ninst✝ : DecidableEq α\nl : List α\nl' : List (Multiset α)\nH : ∀ ⦃x : Multiset α × Multiset α⦄, x ∈ revzip l' → x.fst + x.snd = ↑l\nthis : Forall₂ (fun p s => p = (s, ↑l - s)) (revzip l') (List.map Prod.fst (revzip l'))\n⊢ revzip l' = List.map (fun x => (x, ↑l - x)) l'", "state_before": "α✝ : Type ?u.31102\nα : Type u\ninst✝ : DecidableEq α\nl : List α\nl' : List (Multiset α)\nH : ∀ ⦃x : Multiset α × Multiset α⦄, x ∈ revzip l' → x.fst + x.snd = ↑l\n⊢ revzip l' = List.map (fun x => (x, ↑l - x)) l'", "tactic": "have :\n Forall₂ (fun (p : Multiset α × Multiset α) (s : Multiset α) => p = (s, ↑l - s)) (revzip l')\n ((revzip l').map Prod.fst) := by\n rw [forall₂_map_right_iff, forall₂_same]\n rintro ⟨s, t⟩ h\n dsimp\n rw [← H h, add_tsub_cancel_left]" }, { "state_after": "α✝ : Type ?u.31102\nα : Type u\ninst✝ : DecidableEq α\nl : List α\nl' : List (Multiset α)\nH : ∀ ⦃x : Multiset α × Multiset α⦄, x ∈ revzip l' → x.fst + x.snd = ↑l\nthis : Forall₂ (fun p s => p = (s, ↑l - s)) (revzip l') (List.map Prod.fst (revzip l'))\n⊢ Forall₂ (fun a c => a = (c, ↑l - c)) (revzip l') l'", "state_before": "α✝ : Type ?u.31102\nα : Type u\ninst✝ : DecidableEq α\nl : List α\nl' : List (Multiset α)\nH : ∀ ⦃x : Multiset α × Multiset α⦄, x ∈ revzip l' → x.fst + x.snd = ↑l\nthis : Forall₂ (fun p s => p = (s, ↑l - s)) (revzip l') (List.map Prod.fst (revzip l'))\n⊢ revzip l' = List.map (fun x => (x, ↑l - x)) l'", "tactic": "rw [← forall₂_eq_eq_eq, forall₂_map_right_iff]" }, { "state_after": "no goals", "state_before": "α✝ : Type ?u.31102\nα : Type u\ninst✝ : DecidableEq α\nl : List α\nl' : List (Multiset α)\nH : ∀ ⦃x : Multiset α × Multiset α⦄, x ∈ revzip l' → x.fst + x.snd = ↑l\nthis : Forall₂ (fun p s => p = (s, ↑l - s)) (revzip l') (List.map Prod.fst (revzip l'))\n⊢ Forall₂ (fun a c => a = (c, ↑l - c)) (revzip l') l'", "tactic": "simpa using this" }, { "state_after": "α✝ : Type ?u.31102\nα : Type u\ninst✝ : DecidableEq α\nl : List α\nl' : List (Multiset α)\nH : ∀ ⦃x : Multiset α × Multiset α⦄, x ∈ revzip l' → x.fst + x.snd = ↑l\n⊢ ∀ (x : Multiset α × Multiset α), x ∈ revzip l' → x = (x.fst, ↑l - x.fst)", "state_before": "α✝ : Type ?u.31102\nα : Type u\ninst✝ : DecidableEq α\nl : List α\nl' : List (Multiset α)\nH : ∀ ⦃x : Multiset α × Multiset α⦄, x ∈ revzip l' → x.fst + x.snd = ↑l\n⊢ Forall₂ (fun p s => p = (s, ↑l - s)) (revzip l') (List.map Prod.fst (revzip l'))", "tactic": "rw [forall₂_map_right_iff, forall₂_same]" }, { "state_after": "case mk\nα✝ : Type ?u.31102\nα : Type u\ninst✝ : DecidableEq α\nl : List α\nl' : List (Multiset α)\nH : ∀ ⦃x : Multiset α × Multiset α⦄, x ∈ revzip l' → x.fst + x.snd = ↑l\ns t : Multiset α\nh : (s, t) ∈ revzip l'\n⊢ (s, t) = ((s, t).fst, ↑l - (s, t).fst)", "state_before": "α✝ : Type ?u.31102\nα : Type u\ninst✝ : DecidableEq α\nl : List α\nl' : List (Multiset α)\nH : ∀ ⦃x : Multiset α × Multiset α⦄, x ∈ revzip l' → x.fst + x.snd = ↑l\n⊢ ∀ (x : Multiset α × Multiset α), x ∈ revzip l' → x = (x.fst, ↑l - x.fst)", "tactic": "rintro ⟨s, t⟩ h" }, { "state_after": "case mk\nα✝ : Type ?u.31102\nα : Type u\ninst✝ : DecidableEq α\nl : List α\nl' : List (Multiset α)\nH : ∀ ⦃x : Multiset α × Multiset α⦄, x ∈ revzip l' → x.fst + x.snd = ↑l\ns t : Multiset α\nh : (s, t) ∈ revzip l'\n⊢ (s, t) = (s, ↑l - s)", "state_before": "case mk\nα✝ : Type ?u.31102\nα : Type u\ninst✝ : DecidableEq α\nl : List α\nl' : List (Multiset α)\nH : ∀ ⦃x : Multiset α × Multiset α⦄, x ∈ revzip l' → x.fst + x.snd = ↑l\ns t : Multiset α\nh : (s, t) ∈ revzip l'\n⊢ (s, t) = ((s, t).fst, ↑l - (s, t).fst)", "tactic": "dsimp" }, { "state_after": "no goals", "state_before": "case mk\nα✝ : Type ?u.31102\nα : Type u\ninst✝ : DecidableEq α\nl : List α\nl' : List (Multiset α)\nH : ∀ ⦃x : Multiset α × Multiset α⦄, x ∈ revzip l' → x.fst + x.snd = ↑l\ns t : Multiset α\nh : (s, t) ∈ revzip l'\n⊢ (s, t) = (s, ↑l - s)", "tactic": "rw [← H h, add_tsub_cancel_left]" } ]

[ 154, 19 ]

[ 143, 1 ]

[]

[ 118, 44 ]

[ 117, 11 ]

[]

[ 435, 6 ]

[ 433, 1 ]

[]

[ 245, 21 ]

[ 244, 1 ]

[]

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[ 579, 1 ]

[]

[ 449, 97 ]

[ 448, 1 ]

[ { "state_after": "no goals", "state_before": "ι : Sort uι\nR : Type u_1\nM : Type u_2\nN : Type u_4\nP : Type u_3\ninst✝⁶ : CommSemiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid N\ninst✝³ : AddCommMonoid P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N →ₗ[R] P\nm : M\nn : N\np : Submodule R M\nq : Submodule R N\nhm : m ∈ p\nhn : n ∈ q\n⊢ ↑(↑f ↑{ val := m, property := hm }) n = ↑(↑f m) n", "tactic": "rfl" } ]

[ 56, 55 ]

[ 54, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.130759\nι : Type ?u.130762\ninst✝ : CompleteLattice α\nf g : Filter β\np q : β → Prop\nu v : β → α\nh : ∀ (x : β), p x → q x\na : α\nha : a ∈ {a | ∀ᶠ (x : β) in f, q x → u x ≤ a}\n⊢ ∀ (x : β), (q x → u x ≤ a) → p x → u x ≤ a", "tactic": "tauto" } ]

[ 858, 47 ]

[ 857, 1 ]

[]

[ 310, 39 ]

[ 308, 1 ]

[ { "state_after": "R : Type u_1\ninst✝ : Semiring R\nφ✝ φ ψ : PowerSeries R\n⊢ ∀ (i : ℕ), ↑i < min (order φ) (order ψ) → ↑(coeff R i) (φ + ψ) = 0", "state_before": "R : Type u_1\ninst✝ : Semiring R\nφ✝ φ ψ : PowerSeries R\n⊢ min (order φ) (order ψ) ≤ order (φ + ψ)", "tactic": "refine' le_order _ _ _" }, { "state_after": "no goals", "state_before": "R : Type u_1\ninst✝ : Semiring R\nφ✝ φ ψ : PowerSeries R\n⊢ ∀ (i : ℕ), ↑i < min (order φ) (order ψ) → ↑(coeff R i) (φ + ψ) = 0", "tactic": "simp (config := { contextual := true }) [coeff_of_lt_order]" } ]

[ 2343, 62 ]

[ 2341, 1 ]

[ { "state_after": "no goals", "state_before": "⊢ comap re (comap Real.exp atTop) = comap re atTop", "tactic": "rw [Real.comap_exp_atTop]" } ]

[ 430, 55 ]

[ 426, 1 ]

[]

[ 2930, 61 ]

[ 2929, 1 ]

[]

[ 866, 26 ]

[ 865, 1 ]

[]

[ 825, 53 ]

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[]

[ 396, 29 ]

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[]

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[]

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[ { "state_after": "case intro.intro\nR : Type u_2\nA : Type ?u.738556\nB : Type ?u.738559\nS : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S\nx y : S\nr : R\nhr : ↑f r * y = 1\np : R[X]\np_monic : Monic p\nhp : eval₂ f (x * y) p = 0\n⊢ IsIntegralElem f x", "state_before": "R : Type u_2\nA : Type ?u.738556\nB : Type ?u.738559\nS : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S\nx y : S\nr : R\nhr : ↑f r * y = 1\nhx : IsIntegralElem f (x * y)\n⊢ IsIntegralElem f x", "tactic": "obtain ⟨p, ⟨p_monic, hp⟩⟩ := hx" }, { "state_after": "case intro.intro\nR : Type u_2\nA : Type ?u.738556\nB : Type ?u.738559\nS : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S\nx y : S\nr : R\nhr : ↑f r * y = 1\np : R[X]\np_monic : Monic p\nhp : eval₂ f (x * y) p = 0\n⊢ eval₂ f x (scaleRoots p r) = 0", "state_before": "case intro.intro\nR : Type u_2\nA : Type ?u.738556\nB : Type ?u.738559\nS : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S\nx y : S\nr : R\nhr : ↑f r * y = 1\np : R[X]\np_monic : Monic p\nhp : eval₂ f (x * y) p = 0\n⊢ IsIntegralElem f x", "tactic": "refine' ⟨scaleRoots p r, ⟨(monic_scaleRoots_iff r).2 p_monic, _⟩⟩" }, { "state_after": "case h.e'_2.h.e'_6\nR : Type u_2\nA : Type ?u.738556\nB : Type ?u.738559\nS : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S\nx y : S\nr : R\nhr : ↑f r * y = 1\np : R[X]\np_monic : Monic p\nhp : eval₂ f (x * y) p = 0\n⊢ x = ↑f r * (x * y)", "state_before": "case intro.intro\nR : Type u_2\nA : Type ?u.738556\nB : Type ?u.738559\nS : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S\nx y : S\nr : R\nhr : ↑f r * y = 1\np : R[X]\np_monic : Monic p\nhp : eval₂ f (x * y) p = 0\n⊢ eval₂ f x (scaleRoots p r) = 0", "tactic": "convert scaleRoots_eval₂_eq_zero f hp" }, { "state_after": "no goals", "state_before": "case h.e'_2.h.e'_6\nR : Type u_2\nA : Type ?u.738556\nB : Type ?u.738559\nS : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : CommRing S\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : R →+* S\nx y : S\nr : R\nhr : ↑f r * y = 1\np : R[X]\np_monic : Monic p\nhp : eval₂ f (x * y) p = 0\n⊢ x = ↑f r * (x * y)", "tactic": "rw [mul_comm x y, ← mul_assoc, hr, one_mul]" } ]

[ 613, 46 ]

[ 608, 1 ]

[]

[ 123, 101 ]

[ 123, 1 ]

[ { "state_after": "no goals", "state_before": "R : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q✝ : R[X]\nι : Type ?u.692387\np q : R[X]\nhq : Monic q\nH : leadingCoeff p = 0\n⊢ leadingCoeff (p * q) = leadingCoeff p", "tactic": "rw [H, leadingCoeff_eq_zero.1 H, zero_mul, leadingCoeff_zero]" }, { "state_after": "R : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q✝ : R[X]\nι : Type ?u.692387\np q : R[X]\nhq : Monic q\nH : leadingCoeff p ≠ 0\n⊢ leadingCoeff p * leadingCoeff q ≠ 0", "state_before": "R : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q✝ : R[X]\nι : Type ?u.692387\np q : R[X]\nhq : Monic q\nH : leadingCoeff p ≠ 0\n⊢ leadingCoeff (p * q) = leadingCoeff p", "tactic": "rw [leadingCoeff_mul', hq.leadingCoeff, mul_one]" }, { "state_after": "no goals", "state_before": "R : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q✝ : R[X]\nι : Type ?u.692387\np q : R[X]\nhq : Monic q\nH : leadingCoeff p ≠ 0\n⊢ leadingCoeff p * leadingCoeff q ≠ 0", "tactic": "rwa [hq.leadingCoeff, mul_one]" } ]

[ 1016, 84 ]

[ 1010, 1 ]

[ { "state_after": "no goals", "state_before": "ι : Type ?u.25998\nm : Type u_2\nn : Type u_3\np : Type ?u.26007\nα : Type ?u.26010\nR : Type u_1\nS : Type ?u.26016\ninst✝⁴ : Fintype m\ninst✝³ : Fintype n\ninst✝² : Fintype p\ninst✝¹ : AddCommMonoid R\ninst✝ : CommSemigroup R\nA : Matrix m n R\nB : Matrix n m R\n⊢ trace (A ⬝ B) = trace (B ⬝ A)", "tactic": "rw [← trace_transpose, ← trace_transpose_mul, transpose_mul]" } ]

[ 152, 101 ]

[ 151, 1 ]

[ { "state_after": "case mk\nm : ℕ\nb : Bool\nxs : List Bool\nP : List.length xs = m\n⊢ Bitvec.toNat ({ val := xs, property := P }++ₜb ::ᵥ Vector.nil) =\n Bitvec.toNat { val := xs, property := P } * 2 + Bitvec.toNat (b ::ᵥ Vector.nil)", "state_before": "m : ℕ\nxs : Bitvec m\nb : Bool\n⊢ Bitvec.toNat (xs++ₜb ::ᵥ Vector.nil) = Bitvec.toNat xs * 2 + Bitvec.toNat (b ::ᵥ Vector.nil)", "tactic": "cases' xs with xs P" }, { "state_after": "case mk\nm : ℕ\nb : Bool\nxs : List Bool\nP : List.length xs = m\n⊢ bitsToNat (xs ++ [b]) = bitsToNat [b] + bitsToNat xs * 2", "state_before": "case mk\nm : ℕ\nb : Bool\nxs : List Bool\nP : List.length xs = m\n⊢ Bitvec.toNat ({ val := xs, property := P }++ₜb ::ᵥ Vector.nil) =\n Bitvec.toNat { val := xs, property := P } * 2 + Bitvec.toNat (b ::ᵥ Vector.nil)", "tactic": "simp [bitsToNat_toList]" }, { "state_after": "case mk\nm : ℕ\nb : Bool\nxs : List Bool\n⊢ bitsToNat (xs ++ [b]) = bitsToNat [b] + bitsToNat xs * 2", "state_before": "case mk\nm : ℕ\nb : Bool\nxs : List Bool\nP : List.length xs = m\n⊢ bitsToNat (xs ++ [b]) = bitsToNat [b] + bitsToNat xs * 2", "tactic": "clear P" }, { "state_after": "case mk\nm : ℕ\nb : Bool\nxs : List Bool\n⊢ List.foldl addLsb 0 (xs ++ [b]) = List.foldl addLsb 0 [b] + List.foldl addLsb 0 xs * 2", "state_before": "case mk\nm : ℕ\nb : Bool\nxs : List Bool\n⊢ bitsToNat (xs ++ [b]) = bitsToNat [b] + bitsToNat xs * 2", "tactic": "unfold bitsToNat" }, { "state_after": "case mk\nm : ℕ\nb : Bool\nxs : List Bool\n⊢ List.foldl addLsb 0 (xs ++ [b]) = addLsb 0 b + List.foldl addLsb 0 xs * 2", "state_before": "case mk\nm : ℕ\nb : Bool\nxs : List Bool\n⊢ List.foldl addLsb 0 (xs ++ [b]) = List.foldl addLsb 0 [b] + List.foldl addLsb 0 xs * 2", "tactic": "rw [List.foldl, List.foldl]" }, { "state_after": "case mk\nm : ℕ\nb : Bool\nxs : List Bool\nx : ℕ\nh : 0 = x\n⊢ List.foldl addLsb x (xs ++ [b]) = addLsb x b + List.foldl addLsb x xs * 2", "state_before": "case mk\nm : ℕ\nb : Bool\nxs : List Bool\n⊢ List.foldl addLsb 0 (xs ++ [b]) = addLsb 0 b + List.foldl addLsb 0 xs * 2", "tactic": "generalize h : 0 = x" }, { "state_after": "case mk\nm : ℕ\nb : Bool\nxs : List Bool\nx : ℕ\nh : 0 = x\n⊢ List.foldl addLsb x (xs ++ [b]) = addLsb 0 b + List.foldl addLsb x xs * 2", "state_before": "case mk\nm : ℕ\nb : Bool\nxs : List Bool\nx : ℕ\nh : 0 = x\n⊢ List.foldl addLsb x (xs ++ [b]) = addLsb x b + List.foldl addLsb x xs * 2", "tactic": "conv in addLsb x b =>\n rw [← h]" }, { "state_after": "case mk\nm : ℕ\nb : Bool\nxs : List Bool\nx : ℕ\n⊢ List.foldl addLsb x (xs ++ [b]) = addLsb 0 b + List.foldl addLsb x xs * 2", "state_before": "case mk\nm : ℕ\nb : Bool\nxs : List Bool\nx : ℕ\nh : 0 = x\n⊢ List.foldl addLsb x (xs ++ [b]) = addLsb 0 b + List.foldl addLsb x xs * 2", "tactic": "clear h" }, { "state_after": "case mk\nm : ℕ\nb : Bool\nxs : List Bool\nx : ℕ\n⊢ addLsb (List.foldl addLsb x xs) b = addLsb 0 b + List.foldl addLsb x xs * 2", "state_before": "case mk\nm : ℕ\nb : Bool\nxs : List Bool\nx : ℕ\n⊢ List.foldl addLsb x (xs ++ [b]) = addLsb 0 b + List.foldl addLsb x xs * 2", "tactic": "simp" }, { "state_after": "case mk.nil\nm : ℕ\nb : Bool\nx✝ x : ℕ\n⊢ addLsb (List.foldl addLsb x []) b = addLsb 0 b + List.foldl addLsb x [] * 2\n\ncase mk.cons\nm : ℕ\nb : Bool\nx✝¹ : ℕ\nx✝ : Bool\nxs : List Bool\nxs_ih : ∀ (x : ℕ), addLsb (List.foldl addLsb x xs) b = addLsb 0 b + List.foldl addLsb x xs * 2\nx : ℕ\n⊢ addLsb (List.foldl addLsb x (x✝ :: xs)) b = addLsb 0 b + List.foldl addLsb x (x✝ :: xs) * 2", "state_before": "case mk\nm : ℕ\nb : Bool\nxs : List Bool\nx : ℕ\n⊢ addLsb (List.foldl addLsb x xs) b = addLsb 0 b + List.foldl addLsb x xs * 2", "tactic": "induction' xs with x xs xs_ih generalizing x" }, { "state_after": "case mk.nil\nm : ℕ\nb : Bool\nx✝ x : ℕ\n⊢ addLsb x b = addLsb 0 b + x * 2", "state_before": "case mk.nil\nm : ℕ\nb : Bool\nx✝ x : ℕ\n⊢ addLsb (List.foldl addLsb x []) b = addLsb 0 b + List.foldl addLsb x [] * 2", "tactic": "simp" }, { "state_after": "case mk.nil\nm : ℕ\nb : Bool\nx✝ x : ℕ\n⊢ (x + x + bif b then 1 else 0) = (0 + 0 + bif b then 1 else 0) + x * 2", "state_before": "case mk.nil\nm : ℕ\nb : Bool\nx✝ x : ℕ\n⊢ addLsb x b = addLsb 0 b + x * 2", "tactic": "unfold addLsb" }, { "state_after": "no goals", "state_before": "case mk.nil\nm : ℕ\nb : Bool\nx✝ x : ℕ\n⊢ (x + x + bif b then 1 else 0) = (0 + 0 + bif b then 1 else 0) + x * 2", "tactic": "simp [Nat.mul_succ]" }, { "state_after": "case mk.cons\nm : ℕ\nb : Bool\nx✝¹ : ℕ\nx✝ : Bool\nxs : List Bool\nxs_ih : ∀ (x : ℕ), addLsb (List.foldl addLsb x xs) b = addLsb 0 b + List.foldl addLsb x xs * 2\nx : ℕ\n⊢ addLsb (List.foldl addLsb (addLsb x x✝) xs) b = addLsb 0 b + List.foldl addLsb (addLsb x x✝) xs * 2", "state_before": "case mk.cons\nm : ℕ\nb : Bool\nx✝¹ : ℕ\nx✝ : Bool\nxs : List Bool\nxs_ih : ∀ (x : ℕ), addLsb (List.foldl addLsb x xs) b = addLsb 0 b + List.foldl addLsb x xs * 2\nx : ℕ\n⊢ addLsb (List.foldl addLsb x (x✝ :: xs)) b = addLsb 0 b + List.foldl addLsb x (x✝ :: xs) * 2", "tactic": "simp" }, { "state_after": "no goals", "state_before": "case mk.cons\nm : ℕ\nb : Bool\nx✝¹ : ℕ\nx✝ : Bool\nxs : List Bool\nxs_ih : ∀ (x : ℕ), addLsb (List.foldl addLsb x xs) b = addLsb 0 b + List.foldl addLsb x xs * 2\nx : ℕ\n⊢ addLsb (List.foldl addLsb (addLsb x x✝) xs) b = addLsb 0 b + List.foldl addLsb (addLsb x x✝) xs * 2", "tactic": "apply xs_ih" } ]

[ 55, 16 ]

[ 36, 1 ]

[]

[ 204, 42 ]

[ 203, 1 ]

[ { "state_after": "no goals", "state_before": "𝕜 : Type u_3\nE : Type u_1\nM : Type u_2\nH : Type u_4\nE' : Type ?u.140773\nM' : Type ?u.140776\nH' : Type ?u.140779\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : TopologicalSpace H\ninst✝⁴ : TopologicalSpace M\nf f' : LocalHomeomorph M H\nI : ModelWithCorners 𝕜 E H\ninst✝³ : NormedAddCommGroup E'\ninst✝² : NormedSpace 𝕜 E'\ninst✝¹ : TopologicalSpace H'\ninst✝ : TopologicalSpace M'\nI' : ModelWithCorners 𝕜 E' H'\nx : M\ns t : Set M\n⊢ (extend f I).target ⊆ range ↑I", "tactic": "simp only [mfld_simps]" } ]

[ 847, 96 ]

[ 847, 1 ]

[ { "state_after": "α β γ : Type u\nF G : Type u → Type u\ninst✝³ : Applicative F\ninst✝² : Applicative G\ninst✝¹ : CommApplicative F\ninst✝ : CommApplicative G\ng : α → G β\nh : β → γ\ns : Finset α\n⊢ Functor.map h <$> Multiset.toFinset <$> Multiset.traverse g s.val =\n Multiset.toFinset <$> Multiset.traverse (Functor.map h ∘ g) s.val", "state_before": "α β γ : Type u\nF G : Type u → Type u\ninst✝³ : Applicative F\ninst✝² : Applicative G\ninst✝¹ : CommApplicative F\ninst✝ : CommApplicative G\ng : α → G β\nh : β → γ\ns : Finset α\n⊢ Functor.map h <$> traverse g s = traverse (Functor.map h ∘ g) s", "tactic": "unfold traverse" }, { "state_after": "α β γ : Type u\nF G : Type u → Type u\ninst✝³ : Applicative F\ninst✝² : Applicative G\ninst✝¹ : CommApplicative F\ninst✝ : CommApplicative G\ng : α → G β\nh : β → γ\ns : Finset α\n⊢ (Multiset.toFinset ∘ Functor.map h) <$> Multiset.traverse g s.val =\n Multiset.toFinset <$> Multiset.traverse (Functor.map h ∘ g) s.val", "state_before": "α β γ : Type u\nF G : Type u → Type u\ninst✝³ : Applicative F\ninst✝² : Applicative G\ninst✝¹ : CommApplicative F\ninst✝ : CommApplicative G\ng : α → G β\nh : β → γ\ns : Finset α\n⊢ Functor.map h <$> Multiset.toFinset <$> Multiset.traverse g s.val =\n Multiset.toFinset <$> Multiset.traverse (Functor.map h ∘ g) s.val", "tactic": "simp only [map_comp_coe, functor_norm]" }, { "state_after": "no goals", "state_before": "α β γ : Type u\nF G : Type u → Type u\ninst✝³ : Applicative F\ninst✝² : Applicative G\ninst✝¹ : CommApplicative F\ninst✝ : CommApplicative G\ng : α → G β\nh : β → γ\ns : Finset α\n⊢ (Multiset.toFinset ∘ Functor.map h) <$> Multiset.traverse g s.val =\n Multiset.toFinset <$> Multiset.traverse (Functor.map h ∘ g) s.val", "tactic": "rw [LawfulFunctor.comp_map, Multiset.map_traverse]" } ]

[ 220, 53 ]

[ 216, 1 ]

[]

[ 314, 53 ]

[ 313, 1 ]

[ { "state_after": "c : Char\nh : isValidCharNat (toNat c)\n⊢ ofNatAux (toNat c) h = c", "state_before": "c : Char\nh : isValidCharNat (toNat c)\n⊢ ofNat (toNat c) = c", "tactic": "rw [Char.ofNat, dif_pos h]" }, { "state_after": "no goals", "state_before": "c : Char\nh : isValidCharNat (toNat c)\n⊢ ofNatAux (toNat c) h = c", "tactic": "rfl" } ]

[ 46, 6 ]

[ 44, 1 ]

[]

[ 50, 96 ]

[ 48, 1 ]

[ { "state_after": "case h\nl : Type ?u.916992\nm : Type u_2\nn : Type u_1\no : Type ?u.917001\nm' : o → Type ?u.917006\nn' : o → Type ?u.917011\nR : Type ?u.917014\nS : Type ?u.917017\nα : Type v\nβ : Type w\nγ : Type ?u.917024\ninst✝¹ : NonUnitalCommSemiring α\ninst✝ : Fintype n\nA : Matrix m n α\nx : n → α\nx✝ : m\n⊢ vecMul x Aᵀ x✝ = mulVec A x x✝", "state_before": "l : Type ?u.916992\nm : Type u_2\nn : Type u_1\no : Type ?u.917001\nm' : o → Type ?u.917006\nn' : o → Type ?u.917011\nR : Type ?u.917014\nS : Type ?u.917017\nα : Type v\nβ : Type w\nγ : Type ?u.917024\ninst✝¹ : NonUnitalCommSemiring α\ninst✝ : Fintype n\nA : Matrix m n α\nx : n → α\n⊢ vecMul x Aᵀ = mulVec A x", "tactic": "ext" }, { "state_after": "no goals", "state_before": "case h\nl : Type ?u.916992\nm : Type u_2\nn : Type u_1\no : Type ?u.917001\nm' : o → Type ?u.917006\nn' : o → Type ?u.917011\nR : Type ?u.917014\nS : Type ?u.917017\nα : Type v\nβ : Type w\nγ : Type ?u.917024\ninst✝¹ : NonUnitalCommSemiring α\ninst✝ : Fintype n\nA : Matrix m n α\nx : n → α\nx✝ : m\n⊢ vecMul x Aᵀ x✝ = mulVec A x x✝", "tactic": "apply dotProduct_comm" } ]

[ 1937, 24 ]

[ 1935, 1 ]

[]

[ 500, 6 ]

[ 499, 1 ]

[ { "state_after": "no goals", "state_before": "α : Type u_1\nβ : Type ?u.97009\nγ : Type ?u.97012\na✝ b✝ c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\nf : Filter α\nm : α → ℝ≥0∞\na b : ℝ≥0∞\nhm : Tendsto m f (𝓝 b)\nhb : b ≠ 0 ∨ a ≠ ⊤\nthis : a = 0\n⊢ Tendsto (fun b => a * m b) f (𝓝 (a * b))", "tactic": "simp [this, tendsto_const_nhds]" } ]

[ 379, 61 ]

[ 376, 11 ]

[]

[ 555, 31 ]

[ 554, 1 ]

[]

[ 401, 81 ]

[ 399, 1 ]

[]

[ 950, 77 ]

[ 947, 1 ]

[ { "state_after": "no goals", "state_before": "A : Type u\ninst✝¹ : Category A\ninst✝ : Abelian A\nX Y Z : A\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\nX' Y' Z' : A\nf' : X' ⟶ Y'\ng' : Y' ⟶ Z'\nw' : f' ≫ g' = 0\nα : Arrow.mk f ⟶ Arrow.mk f'\nβ : Arrow.mk g ⟶ Arrow.mk g'\nh : α.right = β.left\n⊢ (kernel.ι g ≫ α.right ≫ cokernel.π f') ≫ cokernel.desc f' g' w' = 0", "tactic": "simp [h]" }, { "state_after": "case h\nA : Type u\ninst✝¹ : Category A\ninst✝ : Abelian A\nX Y Z : A\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\nX' Y' Z' : A\nf' : X' ⟶ Y'\ng' : Y' ⟶ Z'\nw' : f' ≫ g' = 0\nα : Arrow.mk f ⟶ Arrow.mk f'\nβ : Arrow.mk g ⟶ Arrow.mk g'\nh : α.right = β.left\n⊢ (kernel.lift g f w ≫\n lift f' g' w' (kernel.ι g ≫ α.right ≫ cokernel.π f')\n (_ : (kernel.ι g ≫ α.right ≫ cokernel.π f') ≫ cokernel.desc f' g' w' = 0)) ≫\n ι f' g' w' =\n 0 ≫ ι f' g' w'", "state_before": "A : Type u\ninst✝¹ : Category A\ninst✝ : Abelian A\nX Y Z : A\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\nX' Y' Z' : A\nf' : X' ⟶ Y'\ng' : Y' ⟶ Z'\nw' : f' ≫ g' = 0\nα : Arrow.mk f ⟶ Arrow.mk f'\nβ : Arrow.mk g ⟶ Arrow.mk g'\nh : α.right = β.left\n⊢ kernel.lift g f w ≫\n lift f' g' w' (kernel.ι g ≫ α.right ≫ cokernel.π f')\n (_ : (kernel.ι g ≫ α.right ≫ cokernel.π f') ≫ cokernel.desc f' g' w' = 0) =\n 0", "tactic": "ext" }, { "state_after": "case h\nA : Type u\ninst✝¹ : Category A\ninst✝ : Abelian A\nX Y Z : A\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\nX' Y' Z' : A\nf' : X' ⟶ Y'\ng' : Y' ⟶ Z'\nw' : f' ≫ g' = 0\nα : Arrow.mk f ⟶ Arrow.mk f'\nβ : Arrow.mk g ⟶ Arrow.mk g'\nh : α.right = β.left\n⊢ f ≫ α.right ≫ cokernel.π f' = 0", "state_before": "case h\nA : Type u\ninst✝¹ : Category A\ninst✝ : Abelian A\nX Y Z : A\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\nX' Y' Z' : A\nf' : X' ⟶ Y'\ng' : Y' ⟶ Z'\nw' : f' ≫ g' = 0\nα : Arrow.mk f ⟶ Arrow.mk f'\nβ : Arrow.mk g ⟶ Arrow.mk g'\nh : α.right = β.left\n⊢ (kernel.lift g f w ≫\n lift f' g' w' (kernel.ι g ≫ α.right ≫ cokernel.π f')\n (_ : (kernel.ι g ≫ α.right ≫ cokernel.π f') ≫ cokernel.desc f' g' w' = 0)) ≫\n ι f' g' w' =\n 0 ≫ ι f' g' w'", "tactic": "simp only [Category.assoc, zero_comp, lift_ι, kernel.lift_ι_assoc]" }, { "state_after": "case h\nA : Type u\ninst✝¹ : Category A\ninst✝ : Abelian A\nX Y Z : A\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\nX' Y' Z' : A\nf' : X' ⟶ Y'\ng' : Y' ⟶ Z'\nw' : f' ≫ g' = 0\nα : Arrow.mk f ⟶ Arrow.mk f'\nβ : Arrow.mk g ⟶ Arrow.mk g'\nh : α.right = β.left\n⊢ α.left ≫ (Arrow.mk f').hom ≫ cokernel.π f' = 0", "state_before": "case h\nA : Type u\ninst✝¹ : Category A\ninst✝ : Abelian A\nX Y Z : A\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\nX' Y' Z' : A\nf' : X' ⟶ Y'\ng' : Y' ⟶ Z'\nw' : f' ≫ g' = 0\nα : Arrow.mk f ⟶ Arrow.mk f'\nβ : Arrow.mk g ⟶ Arrow.mk g'\nh : α.right = β.left\n⊢ f ≫ α.right ≫ cokernel.π f' = 0", "tactic": "erw [← reassoc_of% α.w]" }, { "state_after": "no goals", "state_before": "case h\nA : Type u\ninst✝¹ : Category A\ninst✝ : Abelian A\nX Y Z : A\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\nX' Y' Z' : A\nf' : X' ⟶ Y'\ng' : Y' ⟶ Z'\nw' : f' ≫ g' = 0\nα : Arrow.mk f ⟶ Arrow.mk f'\nβ : Arrow.mk g ⟶ Arrow.mk g'\nh : α.right = β.left\n⊢ α.left ≫ (Arrow.mk f').hom ≫ cokernel.π f' = 0", "tactic": "simp" }, { "state_after": "A : Type u\ninst✝¹ : Category A\ninst✝ : Abelian A\nX Y Z : A\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\nX' Y' Z' : A\nf' : X' ⟶ Y'\ng' : Y' ⟶ Z'\nw' : f' ≫ g' = 0\nα : Arrow.mk f ⟶ Arrow.mk f'\nβ : Arrow.mk g ⟶ Arrow.mk g'\nh : α.right = β.left\n⊢ desc' f g w\n (lift f' g' w' (kernel.ι g ≫ β.left ≫ cokernel.π f')\n (_ : (kernel.ι g ≫ β.left ≫ cokernel.π f') ≫ cokernel.desc f' g' w' = 0))\n (_ :\n kernel.lift g f w ≫\n lift f' g' w' (kernel.ι g ≫ β.left ≫ cokernel.π f')\n (_ : (kernel.ι g ≫ β.left ≫ cokernel.π f') ≫ cokernel.desc f' g' w' = 0) =\n 0) =\n desc' f g w\n (lift f' g' w' (kernel.ι g ≫ α.right ≫ cokernel.π f')\n (_ : (kernel.ι g ≫ α.right ≫ cokernel.π f') ≫ cokernel.desc f' g' w' = 0))\n (_ :\n kernel.lift g f w ≫\n lift f' g' w' (kernel.ι g ≫ α.right ≫ cokernel.π f')\n (_ : (kernel.ι g ≫ α.right ≫ cokernel.π f') ≫ cokernel.desc f' g' w' = 0) =\n 0)", "state_before": "A : Type u\ninst✝¹ : Category A\ninst✝ : Abelian A\nX Y Z : A\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\nX' Y' Z' : A\nf' : X' ⟶ Y'\ng' : Y' ⟶ Z'\nw' : f' ≫ g' = 0\nα : Arrow.mk f ⟶ Arrow.mk f'\nβ : Arrow.mk g ⟶ Arrow.mk g'\nh : α.right = β.left\n⊢ map w w' α β h =\n desc' f g w\n (lift f' g' w' (kernel.ι g ≫ α.right ≫ cokernel.π f')\n (_ : (kernel.ι g ≫ α.right ≫ cokernel.π f') ≫ cokernel.desc f' g' w' = 0))\n (_ :\n kernel.lift g f w ≫\n lift f' g' w' (kernel.ι g ≫ α.right ≫ cokernel.π f')\n (_ : (kernel.ι g ≫ α.right ≫ cokernel.π f') ≫ cokernel.desc f' g' w' = 0) =\n 0)", "tactic": "rw [map_eq_desc'_lift_left]" }, { "state_after": "case h.h\nA : Type u\ninst✝¹ : Category A\ninst✝ : Abelian A\nX Y Z : A\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\nX' Y' Z' : A\nf' : X' ⟶ Y'\ng' : Y' ⟶ Z'\nw' : f' ≫ g' = 0\nα : Arrow.mk f ⟶ Arrow.mk f'\nβ : Arrow.mk g ⟶ Arrow.mk g'\nh : α.right = β.left\n⊢ (π' f g w ≫\n desc' f g w\n (lift f' g' w' (kernel.ι g ≫ β.left ≫ cokernel.π f')\n (_ : (kernel.ι g ≫ β.left ≫ cokernel.π f') ≫ cokernel.desc f' g' w' = 0))\n (_ :\n kernel.lift g f w ≫\n lift f' g' w' (kernel.ι g ≫ β.left ≫ cokernel.π f')\n (_ : (kernel.ι g ≫ β.left ≫ cokernel.π f') ≫ cokernel.desc f' g' w' = 0) =\n 0)) ≫\n ι f' g' w' =\n (π' f g w ≫\n desc' f g w\n (lift f' g' w' (kernel.ι g ≫ α.right ≫ cokernel.π f')\n (_ : (kernel.ι g ≫ α.right ≫ cokernel.π f') ≫ cokernel.desc f' g' w' = 0))\n (_ :\n kernel.lift g f w ≫\n lift f' g' w' (kernel.ι g ≫ α.right ≫ cokernel.π f')\n (_ : (kernel.ι g ≫ α.right ≫ cokernel.π f') ≫ cokernel.desc f' g' w' = 0) =\n 0)) ≫\n ι f' g' w'", "state_before": "A : Type u\ninst✝¹ : Category A\ninst✝ : Abelian A\nX Y Z : A\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\nX' Y' Z' : A\nf' : X' ⟶ Y'\ng' : Y' ⟶ Z'\nw' : f' ≫ g' = 0\nα : Arrow.mk f ⟶ Arrow.mk f'\nβ : Arrow.mk g ⟶ Arrow.mk g'\nh : α.right = β.left\n⊢ desc' f g w\n (lift f' g' w' (kernel.ι g ≫ β.left ≫ cokernel.π f')\n (_ : (kernel.ι g ≫ β.left ≫ cokernel.π f') ≫ cokernel.desc f' g' w' = 0))\n (_ :\n kernel.lift g f w ≫\n lift f' g' w' (kernel.ι g ≫ β.left ≫ cokernel.π f')\n (_ : (kernel.ι g ≫ β.left ≫ cokernel.π f') ≫ cokernel.desc f' g' w' = 0) =\n 0) =\n desc' f g w\n (lift f' g' w' (kernel.ι g ≫ α.right ≫ cokernel.π f')\n (_ : (kernel.ι g ≫ α.right ≫ cokernel.π f') ≫ cokernel.desc f' g' w' = 0))\n (_ :\n kernel.lift g f w ≫\n lift f' g' w' (kernel.ι g ≫ α.right ≫ cokernel.π f')\n (_ : (kernel.ι g ≫ α.right ≫ cokernel.π f') ≫ cokernel.desc f' g' w' = 0) =\n 0)", "tactic": "ext" }, { "state_after": "no goals", "state_before": "case h.h\nA : Type u\ninst✝¹ : Category A\ninst✝ : Abelian A\nX Y Z : A\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\nX' Y' Z' : A\nf' : X' ⟶ Y'\ng' : Y' ⟶ Z'\nw' : f' ≫ g' = 0\nα : Arrow.mk f ⟶ Arrow.mk f'\nβ : Arrow.mk g ⟶ Arrow.mk g'\nh : α.right = β.left\n⊢ (π' f g w ≫\n desc' f g w\n (lift f' g' w' (kernel.ι g ≫ β.left ≫ cokernel.π f')\n (_ : (kernel.ι g ≫ β.left ≫ cokernel.π f') ≫ cokernel.desc f' g' w' = 0))\n (_ :\n kernel.lift g f w ≫\n lift f' g' w' (kernel.ι g ≫ β.left ≫ cokernel.π f')\n (_ : (kernel.ι g ≫ β.left ≫ cokernel.π f') ≫ cokernel.desc f' g' w' = 0) =\n 0)) ≫\n ι f' g' w' =\n (π' f g w ≫\n desc' f g w\n (lift f' g' w' (kernel.ι g ≫ α.right ≫ cokernel.π f')\n (_ : (kernel.ι g ≫ α.right ≫ cokernel.π f') ≫ cokernel.desc f' g' w' = 0))\n (_ :\n kernel.lift g f w ≫\n lift f' g' w' (kernel.ι g ≫ α.right ≫ cokernel.π f')\n (_ : (kernel.ι g ≫ α.right ≫ cokernel.π f') ≫ cokernel.desc f' g' w' = 0) =\n 0)) ≫\n ι f' g' w'", "tactic": "simp [h]" } ]

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[ 279, 1 ]