module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.Algebra.Group.Action.Equidecomp | {
"line": 125,
"column": 51
} | {
"line": 125,
"column": 83
} | [
{
"pp": "X : Type u_1\nG : Type u_2\ninst✝ : SMul G X\nf : Equidecomp X G\nA : Set X\nhA : A ⊆ f.source\n⊢ f.source ∩ A = A",
"usedConstants": [
"Eq.mpr",
"congrArg",
"id",
"Set.instInter",
"Inter.inter",
"Equidecomp.toPartialEquiv",
"Set.inter_eq_self_of_subset_rig... | inter_eq_self_of_subset_right hA | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Group.Action.Pointwise.Finset | {
"line": 189,
"column": 60
} | {
"line": 192,
"column": 31
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝² : DecidableEq β\ninst✝¹ : Group α\ninst✝ : MulAction α β\ns t : Finset β\na : α\n⊢ s ⊆ a • t ↔ a⁻¹ • s ⊆ t",
"usedConstants": [
"Eq.mpr",
"instHSMul",
"DivInvOneMonoid.toInvOneClass",
"_private.Mathlib.Algebra.Group.Action.Pointwise.Finset.... | by
simp_rw [← coe_subset]
push_cast
exact Set.subset_smul_set_iff | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Quaternion | {
"line": 994,
"column": 2
} | {
"line": 994,
"column": 48
} | [
{
"pp": "R : Type u_3\ninst✝ : CommRing R\na : ℍ[R]\n⊢ star a + a = ↑(2 * a.re)",
"usedConstants": [
"Quaternion.coe",
"Eq.mpr",
"NegZeroClass.toNeg",
"QuaternionAlgebra.imI",
"HMul.hMul",
"congrArg",
"CommSemiring.toSemiring",
"MulZeroClass.zero_mul",
"... | simpa using QuaternionAlgebra.star_add_self' a | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Algebra.Quaternion | {
"line": 994,
"column": 2
} | {
"line": 994,
"column": 48
} | [
{
"pp": "R : Type u_3\ninst✝ : CommRing R\na : ℍ[R]\n⊢ star a + a = ↑(2 * a.re)",
"usedConstants": [
"Quaternion.coe",
"Eq.mpr",
"NegZeroClass.toNeg",
"QuaternionAlgebra.imI",
"HMul.hMul",
"congrArg",
"CommSemiring.toSemiring",
"MulZeroClass.zero_mul",
"... | simpa using QuaternionAlgebra.star_add_self' a | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Quaternion | {
"line": 994,
"column": 2
} | {
"line": 994,
"column": 48
} | [
{
"pp": "R : Type u_3\ninst✝ : CommRing R\na : ℍ[R]\n⊢ star a + a = ↑(2 * a.re)",
"usedConstants": [
"Quaternion.coe",
"Eq.mpr",
"NegZeroClass.toNeg",
"QuaternionAlgebra.imI",
"HMul.hMul",
"congrArg",
"CommSemiring.toSemiring",
"MulZeroClass.zero_mul",
"... | simpa using QuaternionAlgebra.star_add_self' a | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Quaternion | {
"line": 1146,
"column": 59
} | {
"line": 1150,
"column": 95
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na : ℍ[R]\n⊢ a ^ 2 = -↑(normSq a) ↔ a.re = 0",
"usedConstants": [
"Quaternion.star_mul_self",
"Quaternion.coe",
"Iff.mpr",
"Eq.mpr",
"NegZeroClass.toNeg",
"False",
"HMu... | by
simp_rw [← star_eq_neg]
obtain rfl | hq0 := eq_or_ne a 0
· simp
· rw [← star_mul_self, ← mul_neg, ← neg_sq, sq, mul_left_inj' (neg_ne_zero.mpr hq0), eq_comm] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Quaternion | {
"line": 1221,
"column": 22
} | {
"line": 1221,
"column": 88
} | [
{
"pp": "R : Type u_1\ninst✝² : Field R\na✝ b✝ : ℍ[R]\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na b : ℍ[R]\nx✝¹ : ℚ\nx✝ : ℍ[R]\n⊢ x✝¹ • x✝ = ↑x✝¹ * x✝",
"usedConstants": [
"Quaternion.coe",
"Eq.mpr",
"NegZeroClass.toNeg",
"Semigroup.toMul",
"instHSMul",
"Qua... | rw [← coe_ratCast, coe_mul_eq_smul]; ext <;> exact Rat.smul_def .. | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Quaternion | {
"line": 1221,
"column": 22
} | {
"line": 1221,
"column": 88
} | [
{
"pp": "R : Type u_1\ninst✝² : Field R\na✝ b✝ : ℍ[R]\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\na b : ℍ[R]\nx✝¹ : ℚ\nx✝ : ℍ[R]\n⊢ x✝¹ • x✝ = ↑x✝¹ * x✝",
"usedConstants": [
"Quaternion.coe",
"Eq.mpr",
"NegZeroClass.toNeg",
"Semigroup.toMul",
"instHSMul",
"Qua... | rw [← coe_ratCast, coe_mul_eq_smul]; ext <;> exact Rat.smul_def .. | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Group.ForwardDiff | {
"line": 272,
"column": 67
} | {
"line": 272,
"column": 80
} | [
{
"pp": "R : Type u_3\ninst✝ : CommRing R\nP : R[X]\nx : R\ni : ℕ\nhi : i ∈ range P.natDegree\n⊢ 0 x = 0",
"usedConstants": [
"Eq.mpr",
"congrArg",
"CommSemiring.toSemiring",
"AddCommGroup.toAddCommMonoid",
"AddMonoid.toAddZeroClass",
"AddZeroClass.toAddZero",
"Pi.z... | Pi.zero_apply | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.GroupWithZero.Range | {
"line": 114,
"column": 2
} | {
"line": 114,
"column": 28
} | [
{
"pp": "A : Type u_1\nB : Type u_2\nF : Type u_3\ninst✝³ : FunLike F A B\nf : F\ninst✝² : MonoidWithZero A\ninst✝¹ : MonoidWithZero B\ninst✝ : MonoidWithZeroHomClass F A B\nb : Bˣ\nhb : ↑b ∈ range ⇑f\n⊢ b ∈ valueMonoid f",
"usedConstants": [
"MonoidWithZeroHom.mem_valueMonoid"
]
}
] | exact mem_valueMonoid _ hb | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Group.Irreducible.Indecomposable | {
"line": 137,
"column": 8
} | {
"line": 137,
"column": 11
} | [
{
"pp": "case refine_2\nι : Type u_1\nG : Type u_3\nS : Type u_4\ninst✝⁵ : CommGroup G\ninst✝⁴ : LinearOrder S\ninst✝³ : Finite ι\ninst✝² : InvolutiveInv ι\ninst✝¹ : CommGroup S\ninst✝ : IsOrderedMonoid S\nv : ι → G\nhv_inv : ∀ (i : ι), v i⁻¹ = (v i)⁻¹\nf : G →* S\nhf : ∀ (i : ι), f (v i) ≠ 1\nthis : univ = {i ... | h₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.GroupWithZero.Range | {
"line": 276,
"column": 35
} | {
"line": 276,
"column": 43
} | [
{
"pp": "A : Type u_1\nB : Type u_2\nF : Type u_3\ninst✝³ : FunLike F A B\nf : F\ninst✝² : MonoidWithZero A\ninst✝¹ : CommGroupWithZero B\ninst✝ : MonoidWithZeroHomClass F A B\nr₁ s₁ r₂ s₂ : A\nhr₁ : f r₁ ≠ 0\nhs₁ : f s₁ ≠ 0\nhr₂ : f r₂ ≠ 0\nhs₂ : f s₂ ≠ 0\n⊢ f (r₁ * r₂) ≠ 0",
"usedConstants": [
"Grou... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Algebra.GroupWithZero.Range | {
"line": 276,
"column": 35
} | {
"line": 276,
"column": 43
} | [
{
"pp": "A : Type u_1\nB : Type u_2\nF : Type u_3\ninst✝³ : FunLike F A B\nf : F\ninst✝² : MonoidWithZero A\ninst✝¹ : CommGroupWithZero B\ninst✝ : MonoidWithZeroHomClass F A B\nr₁ s₁ r₂ s₂ : A\nhr₁ : f r₁ ≠ 0\nhs₁ : f s₁ ≠ 0\nhr₂ : f r₂ ≠ 0\nhs₂ : f s₂ ≠ 0\n⊢ f (r₁ * r₂) ≠ 0",
"usedConstants": [
"Grou... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.GroupWithZero.Range | {
"line": 276,
"column": 35
} | {
"line": 276,
"column": 43
} | [
{
"pp": "A : Type u_1\nB : Type u_2\nF : Type u_3\ninst✝³ : FunLike F A B\nf : F\ninst✝² : MonoidWithZero A\ninst✝¹ : CommGroupWithZero B\ninst✝ : MonoidWithZeroHomClass F A B\nr₁ s₁ r₂ s₂ : A\nhr₁ : f r₁ ≠ 0\nhs₁ : f s₁ ≠ 0\nhr₂ : f r₂ ≠ 0\nhs₂ : f s₂ ≠ 0\n⊢ f (r₁ * r₂) ≠ 0",
"usedConstants": [
"Grou... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.GroupWithZero.Range | {
"line": 276,
"column": 49
} | {
"line": 276,
"column": 57
} | [
{
"pp": "A : Type u_1\nB : Type u_2\nF : Type u_3\ninst✝³ : FunLike F A B\nf : F\ninst✝² : MonoidWithZero A\ninst✝¹ : CommGroupWithZero B\ninst✝ : MonoidWithZeroHomClass F A B\nr₁ s₁ r₂ s₂ : A\nhr₁ : f r₁ ≠ 0\nhs₁ : f s₁ ≠ 0\nhr₂ : f r₂ ≠ 0\nhs₂ : f s₂ ≠ 0\n⊢ f (s₁ * s₂) ≠ 0",
"usedConstants": [
"Grou... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Algebra.GroupWithZero.Range | {
"line": 276,
"column": 49
} | {
"line": 276,
"column": 57
} | [
{
"pp": "A : Type u_1\nB : Type u_2\nF : Type u_3\ninst✝³ : FunLike F A B\nf : F\ninst✝² : MonoidWithZero A\ninst✝¹ : CommGroupWithZero B\ninst✝ : MonoidWithZeroHomClass F A B\nr₁ s₁ r₂ s₂ : A\nhr₁ : f r₁ ≠ 0\nhs₁ : f s₁ ≠ 0\nhr₂ : f r₂ ≠ 0\nhs₂ : f s₂ ≠ 0\n⊢ f (s₁ * s₂) ≠ 0",
"usedConstants": [
"Grou... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.GroupWithZero.Range | {
"line": 276,
"column": 49
} | {
"line": 276,
"column": 57
} | [
{
"pp": "A : Type u_1\nB : Type u_2\nF : Type u_3\ninst✝³ : FunLike F A B\nf : F\ninst✝² : MonoidWithZero A\ninst✝¹ : CommGroupWithZero B\ninst✝ : MonoidWithZeroHomClass F A B\nr₁ s₁ r₂ s₂ : A\nhr₁ : f r₁ ≠ 0\nhs₁ : f s₁ ≠ 0\nhr₂ : f r₂ ≠ 0\nhs₂ : f s₂ ≠ 0\n⊢ f (s₁ * s₂) ≠ 0",
"usedConstants": [
"Grou... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Group.Irreducible.Indecomposable | {
"line": 223,
"column": 31
} | {
"line": 223,
"column": 39
} | [
{
"pp": "ι✝ : Type u_1\nG✝ : Type u_3\nS✝ : Type u_4\ninst✝⁶ : CommGroup G✝\ninst✝⁵ : LinearOrder S✝\nι : Type u_1\nG : Type u_3\nS : Type u_4\ninst✝⁴ : CommGroup G\ninst✝³ : LinearOrder S\ninst✝² : InvolutiveInv ι\ninst✝¹ : CommGroup S\ninst✝ : IsOrderedMonoid S\nv : ι → G\nf : G →* S\ns : Set ι\nhf : ∀ i ∈ s,... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Algebra.Group.Irreducible.Indecomposable | {
"line": 223,
"column": 31
} | {
"line": 223,
"column": 39
} | [
{
"pp": "ι✝ : Type u_1\nG✝ : Type u_3\nS✝ : Type u_4\ninst✝⁶ : CommGroup G✝\ninst✝⁵ : LinearOrder S✝\nι : Type u_1\nG : Type u_3\nS : Type u_4\ninst✝⁴ : CommGroup G\ninst✝³ : LinearOrder S\ninst✝² : InvolutiveInv ι\ninst✝¹ : CommGroup S\ninst✝ : IsOrderedMonoid S\nv : ι → G\nf : G →* S\ns : Set ι\nhf : ∀ i ∈ s,... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Group.Irreducible.Indecomposable | {
"line": 223,
"column": 31
} | {
"line": 223,
"column": 39
} | [
{
"pp": "ι✝ : Type u_1\nG✝ : Type u_3\nS✝ : Type u_4\ninst✝⁶ : CommGroup G✝\ninst✝⁵ : LinearOrder S✝\nι : Type u_1\nG : Type u_3\nS : Type u_4\ninst✝⁴ : CommGroup G\ninst✝³ : LinearOrder S\ninst✝² : InvolutiveInv ι\ninst✝¹ : CommGroup S\ninst✝ : IsOrderedMonoid S\nv : ι → G\nf : G →* S\ns : Set ι\nhf : ∀ i ∈ s,... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.MooreComplex | {
"line": 107,
"column": 46
} | {
"line": 107,
"column": 69
} | [
{
"pp": "case succ\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Abelian C\nX : SimplicialObject C\nn : ℕ\n⊢ (Finset.univ.inf fun k ↦ kernelSubobject (X.δ k.succ)).factorThru\n ((Finset.univ.inf fun k ↦ kernelSubobject (X.δ k.succ)).arrow ≫ X.δ 0) ⋯ ≫\n (Finset.univ.inf fun k ↦ kernelSubobj... | factorThru_arrow_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Idempotents.Karoubi | {
"line": 121,
"column": 24
} | {
"line": 121,
"column": 36
} | [
{
"pp": "C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nX : C\n⊢ 𝟙 X ≫ 𝟙 X = 𝟙 X",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
"CategoryTheory.CategoryStruct.id",
"id",
"CategoryTheory.Category.comp_id",
"C... | rw [comp_id] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Idempotents.Karoubi | {
"line": 121,
"column": 24
} | {
"line": 121,
"column": 36
} | [
{
"pp": "C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nX : C\n⊢ 𝟙 X ≫ 𝟙 X = 𝟙 X",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
"CategoryTheory.CategoryStruct.id",
"id",
"CategoryTheory.Category.comp_id",
"C... | rw [comp_id] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Idempotents.Karoubi | {
"line": 121,
"column": 24
} | {
"line": 121,
"column": 36
} | [
{
"pp": "C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nX : C\n⊢ 𝟙 X ≫ 𝟙 X = 𝟙 X",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
"CategoryTheory.CategoryStruct.id",
"id",
"CategoryTheory.Category.comp_id",
"C... | rw [comp_id] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Idempotents.Karoubi | {
"line": 140,
"column": 23
} | {
"line": 140,
"column": 35
} | [
{
"pp": "C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nX : C\n⊢ 𝟙 X ≫ 𝟙 X = 𝟙 X",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
"CategoryTheory.CategoryStruct.id",
"id",
"CategoryTheory.Category.comp_id",
"C... | rw [comp_id] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Idempotents.Karoubi | {
"line": 140,
"column": 23
} | {
"line": 140,
"column": 35
} | [
{
"pp": "C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nX : C\n⊢ 𝟙 X ≫ 𝟙 X = 𝟙 X",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
"CategoryTheory.CategoryStruct.id",
"id",
"CategoryTheory.Category.comp_id",
"C... | rw [comp_id] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Idempotents.Karoubi | {
"line": 140,
"column": 23
} | {
"line": 140,
"column": 35
} | [
{
"pp": "C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nX : C\n⊢ 𝟙 X ≫ 𝟙 X = 𝟙 X",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
"CategoryTheory.CategoryStruct.id",
"id",
"CategoryTheory.Category.comp_id",
"C... | rw [comp_id] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Idempotents.FunctorCategories | {
"line": 72,
"column": 2
} | {
"line": 74,
"column": 59
} | [
{
"pp": "J : Type u_1\nC : Type u_2\ninst✝² : Category.{v_1, u_1} J\ninst✝¹ : Category.{v_2, u_2} C\nP Q : Karoubi (J ⥤ C)\nf : P ⟶ Q\nX : J\ninst✝ : IsIdempotentComplete C\nF : J ⥤ C\np : F ⟶ F\nhp : p ≫ p = p\nhC : ∀ (X : C) (p : X ⟶ X), p ≫ p = p → HasEqualizer (𝟙 X) p\nthis : ∀ (j : J), HasEqualizer (𝟙 (F... | let i : Y ⟶ F :=
{ app := fun j => equalizer.ι _ _
naturality := fun _ _ _ => by rw [equalizer.lift_ι] } | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.AlgebraicTopology.SimplexCategory.Basic | {
"line": 630,
"column": 87
} | {
"line": 632,
"column": 42
} | [
{
"pp": "x y : SimplexCategory\nf : x ⟶ y\ninst✝ : Mono f\n⊢ x.len ≤ y.len",
"usedConstants": [
"Fintype.card_le_of_injective",
"Eq.mpr",
"Fintype.card_fin",
"Preorder.toLT",
"Nat.instIsOrderedAddMonoid",
"AddLeftCancelSemigroup.toIsLeftCancelAdd",
"CategoryTheory.M... | by
simpa using Fintype.card_le_of_injective f.toOrderHom.toFun
(by dsimp; rwa [← mono_iff_injective]) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicTopology.SimplexCategory.Basic | {
"line": 733,
"column": 2
} | {
"line": 733,
"column": 34
} | [
{
"pp": "case h.a.h.h\nn : ℕ\nΔ' : SimplexCategory\nθ : ⦋n + 1⦌ ⟶ Δ'\ni : Fin (n + 1)\nhi : (Hom.toOrderHom θ) i.castSucc = (Hom.toOrderHom θ) i.succ\nx : Fin (⦋n + 1⦌.len + 1)\n⊢ (Hom.toOrderHom θ) x = (Hom.toOrderHom θ) ((Hom.toOrderHom (δ i.succ)) (i.predAbove x))",
"usedConstants": [
"Fin.succ",
... | by_cases h' : x ≤ Fin.castSucc i | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.AlgebraicTopology.SimplicialObject.Homotopy | {
"line": 153,
"column": 26
} | {
"line": 154,
"column": 71
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nX Y : SimplicialObject C\nf g : X ⟶ Y\nH : Homotopy f g\nX' : SimplicialObject C\np : X' ⟶ X\nn : ℕ\n⊢ (p.app (op ⦋n⦌) ≫ H.h 0) ≫ Y.δ 0 = (p ≫ g).app (op ⦋n⦌)",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Category.assoc",
"Opposite",
"... | by
simpa [-h_zero_comp_δ_zero] using p.app _ ≫= H.h_zero_comp_δ_zero n | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicTopology.SimplicialSet.Degenerate | {
"line": 222,
"column": 2
} | {
"line": 222,
"column": 53
} | [
{
"pp": "X : SSet\nn : ℕ\nx : X _⦋n⦌\nm₁ m₂ : ℕ\nf₁ : ⦋n⦌ ⟶ ⦋m₁⦌\ninst✝¹ : Epi f₁\ny₁ : ↑(X.nonDegenerate m₁)\nhy₁ : x = (ConcreteCategory.hom (X.map f₁.op)) ↑y₁\nf₂ : ⦋n⦌ ⟶ ⦋m₂⦌\ninst✝ : Epi f₂\ny₂ : ↑(X.nonDegenerate m₂)\nhy₂ : x = (ConcreteCategory.hom (X.map f₂.op)) ↑y₂\nhf₁ : SplitEpi f₁\nhf₂ : SplitEpi f₂... | exact le_antisymm (le hf₁ hy₁ hy₂) (le hf₂ hy₂ hy₁) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.AlgebraicTopology.SimplicialSet.Dimension | {
"line": 98,
"column": 4
} | {
"line": 99,
"column": 83
} | [
{
"pp": "case h\nX Y : SSet\nf : X ⟶ Y\ninst✝¹ : Mono f\nd : ℕ\ninst✝ : Y.HasDimensionLT d\nn : ℕ\nhn : d ≤ n\nx : X _⦋n⦌\n⊢ x ∈ X.degenerate n ↔ x ∈ ⊤",
"usedConstants": [
"SSet.Subcomplex.toSSet",
"Eq.mpr",
"SSet.Subcomplex.range",
"Opposite",
"congrArg",
"CategoryTheor... | rw [← degenerate_iff_of_isIso (Subcomplex.toRange f),
Subcomplex.mem_degenerate_iff, Y.degenerate_eq_univ_of_hasDimensionLT d n hn] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicTopology.SimplicialSet.Degenerate | {
"line": 269,
"column": 4
} | {
"line": 270,
"column": 47
} | [
{
"pp": "case mp\nX : SSet\nA : X.Subcomplex\nn : ℕ\nx : ↑(A.obj (op ⦋n⦌))\n⊢ (∃ m, ∃ (_ : m < n), ∃ f, ∃ (_ : Epi f), x ∈ Set.range ⇑(ConcreteCategory.hom (A.toSSet.map f.op))) →\n ∃ m, ∃ (_ : m < n), ∃ f, ∃ (_ : Epi f), ↑x ∈ Set.range ⇑(ConcreteCategory.hom (X.map f.op))",
"usedConstants": [
"SSe... | rintro ⟨m, hm, f, _, y, rfl⟩
exact ⟨m, hm, f, inferInstance, y.val, rfl⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.SimplicialSet.Degenerate | {
"line": 269,
"column": 4
} | {
"line": 270,
"column": 47
} | [
{
"pp": "case mp\nX : SSet\nA : X.Subcomplex\nn : ℕ\nx : ↑(A.obj (op ⦋n⦌))\n⊢ (∃ m, ∃ (_ : m < n), ∃ f, ∃ (_ : Epi f), x ∈ Set.range ⇑(ConcreteCategory.hom (A.toSSet.map f.op))) →\n ∃ m, ∃ (_ : m < n), ∃ f, ∃ (_ : Epi f), ↑x ∈ Set.range ⇑(ConcreteCategory.hom (X.map f.op))",
"usedConstants": [
"SSe... | rintro ⟨m, hm, f, _, y, rfl⟩
exact ⟨m, hm, f, inferInstance, y.val, rfl⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.SimplexCategory.Truncated | {
"line": 45,
"column": 4
} | {
"line": 45,
"column": 79
} | [
{
"pp": "n : ℕ\ninst✝ : NeZero n\nthis✝ : 0 < n\nΔ : SimplexCategory\nthis : Nonempty (CostructuredArrow (inclusion n) Δ)\nΔ₁ : SimplexCategory\nhΔ₁ : Δ₁.len ≤ n\nf : (inclusion n).obj { obj := Δ₁, property := hΔ₁ } ⟶ (Functor.fromPUnit Δ).obj { as := PUnit.unit }\nΔ₂ : SimplexCategory\nhΔ₂ : Δ₂.len ≤ n\nf' : (... | · apply Zigzag.of_hom <| CostructuredArrow.homMk <| Hom.tr <| ⦋0⦌.const _ 1 | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicTopology.SimplicialSet.NerveNondegenerate | {
"line": 63,
"column": 4
} | {
"line": 63,
"column": 49
} | [
{
"pp": "case succ\nX : Type u_1\ninst✝ : PartialOrder X\nn : ℕ\ns : nerve X _⦋n + 1⦌\n⊢ (∃ i, s ∈ Set.range ⇑(ConcreteCategory.hom (SimplicialObject.σ (nerve X) i))) ↔\n ¬∀ (i : Fin (Opposite.unop (Opposite.op ⦋n + 1⦌)).len), s.obj i.castSucc < s.obj i.succ",
"usedConstants": [
"Eq.mpr",
"Pr... | simp only [mem_range_nerve_σ_iff, not_forall] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Homology.AlternatingConst | {
"line": 173,
"column": 53
} | {
"line": 173,
"column": 61
} | [
{
"pp": "case zero\nC : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasZeroObject C\nX : C\nhn : Even 0\nh₀ : 0 ≠ 0\n⊢ ∀ (Z : C) (g : Z ⟶ (HomologicalComplex.sc (alternatingConst.obj X) 0).X₂),\n g ≫ (HomologicalComplex.sc (alternatingConst.obj X) 0).g = 0 → g = 0",
"us... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Algebra.Homology.AlternatingConst | {
"line": 173,
"column": 53
} | {
"line": 173,
"column": 61
} | [
{
"pp": "case succ\nC : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasZeroObject C\nX : C\nn✝ : ℕ\nhn : Even (n✝ + 1)\nh₀ : n✝ + 1 ≠ 0\n⊢ ∀ (Z : C) (g : Z ⟶ (HomologicalComplex.sc (alternatingConst.obj X) (n✝ + 1)).X₂),\n g ≫ (HomologicalComplex.sc (alternatingConst.obj X)... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Algebra.Homology.Augment | {
"line": 216,
"column": 46
} | {
"line": 216,
"column": 54
} | [
{
"pp": "case zero.zero\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : HasZeroMorphisms V\nC : CochainComplex V ℕ\nX : V\nf : X ⟶ C.X 0\nw : f ≫ C.d 0 1 = 0\ns : ¬(ComplexShape.up ℕ).Rel 0 0\n⊢ (match 0, 0 with\n | 0, 1 => f\n | i.succ, j.succ => C.d i j\n | x, x_1 => 0) =\n 0",
"usedConstants... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Algebra.Homology.Augment | {
"line": 216,
"column": 46
} | {
"line": 216,
"column": 54
} | [
{
"pp": "case zero.succ\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : HasZeroMorphisms V\nC : CochainComplex V ℕ\nX : V\nf : X ⟶ C.X 0\nw : f ≫ C.d 0 1 = 0\nn✝ : ℕ\ns : ¬(ComplexShape.up ℕ).Rel (n✝ + 1) 0\n⊢ (match n✝ + 1, 0 with\n | 0, 1 => f\n | i.succ, j.succ => C.d i j\n | x, x_1 => 0) =\n 0"... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Algebra.Homology.Augment | {
"line": 216,
"column": 46
} | {
"line": 216,
"column": 54
} | [
{
"pp": "case succ.zero.zero\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : HasZeroMorphisms V\nC : CochainComplex V ℕ\nX : V\nf : X ⟶ C.X 0\nw : f ≫ C.d 0 1 = 0\ns : ¬(ComplexShape.up ℕ).Rel 0 (0 + 1)\n⊢ (match 0, 0 + 1 with\n | 0, 1 => f\n | i.succ, j.succ => C.d i j\n | x, x_1 => 0) =\n 0",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Algebra.Homology.Augment | {
"line": 216,
"column": 46
} | {
"line": 216,
"column": 54
} | [
{
"pp": "case succ.zero.succ\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : HasZeroMorphisms V\nC : CochainComplex V ℕ\nX : V\nf : X ⟶ C.X 0\nw : f ≫ C.d 0 1 = 0\nn✝ : ℕ\ns : ¬(ComplexShape.up ℕ).Rel (n✝ + 1) (0 + 1)\n⊢ (match n✝ + 1, 0 + 1 with\n | 0, 1 => f\n | i.succ, j.succ => C.d i j\n | x, x_1 ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Algebra.Homology.Augment | {
"line": 216,
"column": 46
} | {
"line": 216,
"column": 54
} | [
{
"pp": "case succ.succ.zero\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : HasZeroMorphisms V\nC : CochainComplex V ℕ\nX : V\nf : X ⟶ C.X 0\nw : f ≫ C.d 0 1 = 0\nj : ℕ\ns : ¬(ComplexShape.up ℕ).Rel 0 (j + 1 + 1)\n⊢ (match 0, j + 1 + 1 with\n | 0, 1 => f\n | i.succ, j.succ => C.d i j\n | x, x_1 => 0)... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Algebra.Homology.Augment | {
"line": 216,
"column": 46
} | {
"line": 216,
"column": 54
} | [
{
"pp": "case succ.succ.succ\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : HasZeroMorphisms V\nC : CochainComplex V ℕ\nX : V\nf : X ⟶ C.X 0\nw : f ≫ C.d 0 1 = 0\nj n✝ : ℕ\ns : ¬(ComplexShape.up ℕ).Rel (n✝ + 1) (j + 1 + 1)\n⊢ (match n✝ + 1, j + 1 + 1 with\n | 0, 1 => f\n | i.succ, j.succ => C.d i j\n ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Algebra.Homology.ComplexShapeSigns | {
"line": 174,
"column": 4
} | {
"line": 174,
"column": 52
} | [
{
"pp": "I₁ : Type u_1\nI₂ : Type u_2\nI₃ : Type u_3\nI₁₂ : Type u_4\nI₂₃ : Type u_5\nJ : Type u_6\nc₁ : ComplexShape I₁\nc₂ : ComplexShape I₂\nc₃ : ComplexShape I₃\nc₁₂ : ComplexShape I₁₂\nc₂₃ : ComplexShape I₂₃\nc✝ : ComplexShape J\ninst✝² : TotalComplexShape c₁ c₂ c₁₂\nI : Type u_7\ninst✝¹ : AddMonoid I\nc :... | rw [pow_add, pow_one, mul_neg, mul_one, neg_neg] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Homology.TotalComplex | {
"line": 220,
"column": 10
} | {
"line": 220,
"column": 32
} | [
{
"pp": "case neg\nC : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : Preadditive C\nI₁ : Type u_2\nI₂ : Type u_3\nI₁₂ : Type u_4\nc₁ : ComplexShape I₁\nc₂ : ComplexShape I₂\nK : HomologicalComplex₂ C c₁ c₂\nc₁₂ : ComplexShape I₁₂\ninst✝² : TotalComplexShape c₁ c₂ c₁₂\ninst✝¹ : DecidableEq I₁₂\ninst✝ : K.Ha... | K.D₂_shape c₁₂ _ _ h₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicTopology.ExtraDegeneracy | {
"line": 339,
"column": 6
} | {
"line": 339,
"column": 29
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\nf : Arrow C\ninst✝ : ∀ (n : ℕ), HasWidePullback f.right (fun x ↦ f.left) fun x ↦ f.hom\nS : SplitEpi f.hom\nn : ℕ\ni : Fin ((unop (op ⦋n + 1⦌)).len + 1)\n⊢ (fun i ↦ Fin.cases ((WidePullback.base fun x ↦ f.hom) ≫ S.section_) (WidePullback.π fun x ↦ f.hom) i)... | cases i using Fin.cases | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | Lean.Parser.Tactic.cases |
Mathlib.Algebra.Homology.TotalComplex | {
"line": 252,
"column": 33
} | {
"line": 252,
"column": 55
} | [
{
"pp": "case neg\nC : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : Preadditive C\nI₁ : Type u_2\nI₂ : Type u_3\nI₁₂ : Type u_4\nc₁ : ComplexShape I₁\nc₂ : ComplexShape I₂\nK : HomologicalComplex₂ C c₁ c₂\nc₁₂ : ComplexShape I₁₂\ninst✝² : TotalComplexShape c₁ c₂ c₁₂\ninst✝¹ : DecidableEq I₁₂\ninst✝ : K.Ha... | K.D₂_shape c₁₂ _ _ h₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.TotalComplex | {
"line": 277,
"column": 41
} | {
"line": 277,
"column": 44
} | [
{
"pp": "C : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : Preadditive C\nI₁ : Type u_2\nI₂ : Type u_3\nI₁₂ : Type u_4\nc₁ : ComplexShape I₁\nc₂ : ComplexShape I₂\nK L M : HomologicalComplex₂ C c₁ c₂\nφ : K ⟶ L\ne : K ≅ L\nψ : L ⟶ M\nc₁₂ : ComplexShape I₁₂\ninst✝² : TotalComplexShape c₁ c₂ c₁₂\ninst✝¹ : De... | h₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.BifunctorHomotopy | {
"line": 126,
"column": 2
} | {
"line": 126,
"column": 13
} | [
{
"pp": "C₁ : Type u_1\nC₂ : Type u_2\nD : Type u_3\nI₁ : Type u_4\nI₂ : Type u_5\nJ : Type u_6\ninst✝¹¹ : Category.{v_1, u_1} C₁\ninst✝¹⁰ : Category.{v_2, u_2} C₂\ninst✝⁹ : Category.{v_3, u_3} D\ninst✝⁸ : Preadditive C₁\ninst✝⁷ : Preadditive C₂\ninst✝⁶ : Preadditive D\nc₁ : ComplexShape I₁\nc₂ : ComplexShape I... | ext i₁ i₂ h | _private.Lean.Elab.Tactic.Ext.0.Lean.Elab.Tactic.Ext.evalExt | Lean.Elab.Tactic.Ext.ext |
Mathlib.Algebra.Homology.BifunctorHomotopy | {
"line": 127,
"column": 2
} | {
"line": 134,
"column": 51
} | [
{
"pp": "case h\nC₁ : Type u_1\nC₂ : Type u_2\nD : Type u_3\nI₁ : Type u_4\nI₂ : Type u_5\nJ : Type u_6\ninst✝¹¹ : Category.{v_1, u_1} C₁\ninst✝¹⁰ : Category.{v_2, u_2} C₂\ninst✝⁹ : Category.{v_3, u_3} D\ninst✝⁸ : Preadditive C₁\ninst✝⁷ : Preadditive C₂\ninst✝⁶ : Preadditive D\nc₁ : ComplexShape I₁\nc₂ : Comple... | simp? [HomologicalComplex₂.total_d, h₁.comm i₁, dFrom, fromNext, toPrev, dTo] says
simp only [ι_mapBifunctorMap, h₁.comm i₁, dNext_eq_dFrom_fromNext, dFrom, fromNext,
AddMonoidHom.mk'_apply, prevD_eq_toPrev_dTo, toPrev, dTo, Functor.map_add,
Functor.map_comp, NatTrans.app_add, NatTrans.comp_app,
P... | Mathlib.Tactic.Says._aux_Mathlib_Tactic_Says___elabRules_Mathlib_Tactic_Says_says_1 | Mathlib.Tactic.Says.says |
Mathlib.Algebra.Homology.TotalComplexShift | {
"line": 246,
"column": 16
} | {
"line": 251,
"column": 14
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preadditive C\nK L : HomologicalComplex₂ C (up ℤ) (up ℤ)\nf : K ⟶ L\nx y : ℤ\ninst✝ : K.HasTotal (up ℤ)\nn n' : ℤ\nh : n + y = n'\n⊢ ((K.totalDesc fun p q hpq ↦\n (p * y).negOnePow •\n (XXIsoOfEq C (up ℤ) (up ℤ) K ⋯ ⋯).inv ≫ ((shift... | by
ext
dsimp
simp only [ι_totalDesc_assoc, Linear.units_smul_comp, Category.assoc, ι_totalDesc,
Linear.comp_units_smul, XXIsoOfEq_inv_ιTotal, smul_smul, Int.units_mul_self, one_smul,
comp_id] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.ConcreteCategory | {
"line": 120,
"column": 6
} | {
"line": 121,
"column": 80
} | [
{
"pp": "C : Type u\ninst✝⁶ : Category.{v, u} C\nFC : C → C → Type u_1\nCC : C → Type v\ninst✝⁵ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)\ninst✝⁴ : ConcreteCategory C FC\ninst✝³ : HasForget₂ C Ab\ninst✝² : Abelian C\ninst✝¹ : (forget₂ C Ab).Additive\ninst✝ : (forget₂ C Ab).PreservesHomology\nι : Type u_2\nc ... | rw [← ConcreteCategory.forget₂_comp_apply, HomologicalComplex.cyclesMap_i,
ConcreteCategory.forget₂_comp_apply, HomologicalComplex.i_cyclesMk, hx₁] | Lean.Parser.Tactic.Conv._aux_Init_Conv___macroRules_Lean_Parser_Tactic_Conv_convRw___1 | Lean.Parser.Tactic.Conv.convRw__ |
Mathlib.Algebra.Homology.ConcreteCategory | {
"line": 120,
"column": 6
} | {
"line": 121,
"column": 80
} | [
{
"pp": "C : Type u\ninst✝⁶ : Category.{v, u} C\nFC : C → C → Type u_1\nCC : C → Type v\ninst✝⁵ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)\ninst✝⁴ : ConcreteCategory C FC\ninst✝³ : HasForget₂ C Ab\ninst✝² : Abelian C\ninst✝¹ : (forget₂ C Ab).Additive\ninst✝ : (forget₂ C Ab).PreservesHomology\nι : Type u_2\nc ... | rw [← ConcreteCategory.forget₂_comp_apply, HomologicalComplex.cyclesMap_i,
ConcreteCategory.forget₂_comp_apply, HomologicalComplex.i_cyclesMk, hx₁] | Lean.Elab.Tactic.Conv.evalConvSeq1Indented | Lean.Parser.Tactic.Conv.convSeq1Indented |
Mathlib.Algebra.Homology.ConcreteCategory | {
"line": 120,
"column": 6
} | {
"line": 121,
"column": 80
} | [
{
"pp": "C : Type u\ninst✝⁶ : Category.{v, u} C\nFC : C → C → Type u_1\nCC : C → Type v\ninst✝⁵ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)\ninst✝⁴ : ConcreteCategory C FC\ninst✝³ : HasForget₂ C Ab\ninst✝² : Abelian C\ninst✝¹ : (forget₂ C Ab).Additive\ninst✝ : (forget₂ C Ab).PreservesHomology\nι : Type u_2\nc ... | rw [← ConcreteCategory.forget₂_comp_apply, HomologicalComplex.cyclesMap_i,
ConcreteCategory.forget₂_comp_apply, HomologicalComplex.i_cyclesMk, hx₁] | Lean.Elab.Tactic.Conv.evalConvSeq | Lean.Parser.Tactic.Conv.convSeq |
Mathlib.Algebra.Homology.HomotopyCategory.HomComplexCohomology | {
"line": 53,
"column": 4
} | {
"line": 53,
"column": 27
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : Linear R C\nK L : CochainComplex C ℤ\nn m p : ℤ\n⊢ ∀ {x : Cocycle K L n},\n x ∈ {α | ∃ m, ∃ (_ : m + 1 = n), ∃ β, δ m n β = ↑α} → -x ∈ {α | ∃ m, ∃ (_ : m + 1 = n), ∃ β, δ m n β = ↑α}",
"usedCon... | rintro α ⟨m, hm, β, hβ⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Algebra.Homology.Double | {
"line": 95,
"column": 4
} | {
"line": 95,
"column": 40
} | [
{
"pp": "case pos\nC : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasZeroObject C\nX₀ X₁ : C\nf : X₀ ⟶ X₁\nι : Type u_2\nc : ComplexShape ι\ni₀ i₁ : ι\nhi₀₁ : c.Rel i₀ i₁\nK : HomologicalComplex C c\nφ φ' : double f hi₀₁ ⟶ K\nh₀ : φ.f i₀ = φ'.f i₀\nh₁ : φ.f i₁ = φ'.f i₁\nk : ... | obtain rfl | rfl := h <;> assumption | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Algebra.Homology.Double | {
"line": 95,
"column": 4
} | {
"line": 95,
"column": 40
} | [
{
"pp": "case pos\nC : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasZeroObject C\nX₀ X₁ : C\nf : X₀ ⟶ X₁\nι : Type u_2\nc : ComplexShape ι\ni₀ i₁ : ι\nhi₀₁ : c.Rel i₀ i₁\nK : HomologicalComplex C c\nφ φ' : double f hi₀₁ ⟶ K\nh₀ : φ.f i₀ = φ'.f i₀\nh₁ : φ.f i₁ = φ'.f i₁\nk : ... | obtain rfl | rfl := h <;> assumption | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.Double | {
"line": 95,
"column": 4
} | {
"line": 95,
"column": 40
} | [
{
"pp": "case pos\nC : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasZeroObject C\nX₀ X₁ : C\nf : X₀ ⟶ X₁\nι : Type u_2\nc : ComplexShape ι\ni₀ i₁ : ι\nhi₀₁ : c.Rel i₀ i₁\nK : HomologicalComplex C c\nφ φ' : double f hi₀₁ ⟶ K\nh₀ : φ.f i₀ = φ'.f i₀\nh₁ : φ.f i₁ = φ'.f i₁\nk : ... | obtain rfl | rfl := h <;> assumption | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.Double | {
"line": 105,
"column": 4
} | {
"line": 105,
"column": 40
} | [
{
"pp": "case pos\nC : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasZeroObject C\nX₀ X₁ : C\nf : X₀ ⟶ X₁\nι : Type u_2\nc : ComplexShape ι\ni₀ i₁ : ι\nhi₀₁ : c.Rel i₀ i₁\nK : HomologicalComplex C c\nφ φ' : K ⟶ double f hi₀₁\nh₀ : φ.f i₀ = φ'.f i₀\nh₁ : φ.f i₁ = φ'.f i₁\nk : ... | obtain rfl | rfl := h <;> assumption | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Algebra.Homology.Double | {
"line": 105,
"column": 4
} | {
"line": 105,
"column": 40
} | [
{
"pp": "case pos\nC : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasZeroObject C\nX₀ X₁ : C\nf : X₀ ⟶ X₁\nι : Type u_2\nc : ComplexShape ι\ni₀ i₁ : ι\nhi₀₁ : c.Rel i₀ i₁\nK : HomologicalComplex C c\nφ φ' : K ⟶ double f hi₀₁\nh₀ : φ.f i₀ = φ'.f i₀\nh₁ : φ.f i₁ = φ'.f i₁\nk : ... | obtain rfl | rfl := h <;> assumption | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.Double | {
"line": 105,
"column": 4
} | {
"line": 105,
"column": 40
} | [
{
"pp": "case pos\nC : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasZeroObject C\nX₀ X₁ : C\nf : X₀ ⟶ X₁\nι : Type u_2\nc : ComplexShape ι\ni₀ i₁ : ι\nhi₀₁ : c.Rel i₀ i₁\nK : HomologicalComplex C c\nφ φ' : K ⟶ double f hi₀₁\nh₀ : φ.f i₀ = φ'.f i₀\nh₁ : φ.f i₁ = φ'.f i₁\nk : ... | obtain rfl | rfl := h <;> assumption | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.MorphismProperty.LiftingProperty | {
"line": 130,
"column": 2
} | {
"line": 132,
"column": 41
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nT : MorphismProperty C\n⊢ T.pushouts ≤ T.rlp.llp",
"usedConstants": [
"CategoryTheory.MorphismProperty",
"CategoryTheory.MorphismProperty.pushouts",
"CategoryTheory.MorphismProperty.llp",
"CategoryTheory.CategoryStruct.toQuiver",
... | intro A B i hi
exact (T.rlp.llp.isStableUnderCobaseChange_iff_pushouts_le).1 inferInstance i
(pushouts_monotone T.le_llp_rlp _ hi) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.MorphismProperty.LiftingProperty | {
"line": 130,
"column": 2
} | {
"line": 132,
"column": 41
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nT : MorphismProperty C\n⊢ T.pushouts ≤ T.rlp.llp",
"usedConstants": [
"CategoryTheory.MorphismProperty",
"CategoryTheory.MorphismProperty.pushouts",
"CategoryTheory.MorphismProperty.llp",
"CategoryTheory.CategoryStruct.toQuiver",
... | intro A B i hi
exact (T.rlp.llp.isStableUnderCobaseChange_iff_pushouts_le).1 inferInstance i
(pushouts_monotone T.le_llp_rlp _ hi) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.GrothendieckAbelian | {
"line": 54,
"column": 66
} | {
"line": 59,
"column": 82
} | [
{
"pp": "C : Type u\ninst✝⁵ : Category.{v, u} C\nι : Type t\nc : ComplexShape ι\ninst✝⁴ : HasZeroMorphisms C\nJ : Type w\ninst✝³ : Category.{w', w} J\ninst✝² : HasFiniteLimits C\ninst✝¹ : HasColimitsOfShape J C\ninst✝ : HasExactColimitsOfShape J C\nK : Type\nx✝¹ : SmallCategory K\nx✝ : FinCategory K\nF : K ⥤ J ... | by
let e := preservesColimitNatIso (J := J) (eval C c i)
exact (IsLimit.postcomposeHomEquiv (Functor.isoWhiskerLeft F e) _).1
(IsLimit.ofIsoLimit
(isLimitOfPreserves ((Functor.whiskeringRight J _ _).obj (eval C c i) ⋙ colim) hc)
(Cone.ext (e.symm.app _) (fun k ↦ (NatIso.naturalit... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.HomotopyCategory.HomComplexSingle | {
"line": 83,
"column": 4
} | {
"line": 83,
"column": 24
} | [
{
"pp": "case h\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Preadditive C\ninst✝ : HasZeroObject C\nX : C\nK : CochainComplex C ℤ\np q n : ℤ\nh : p + n = q\nα : Cochain ((singleFunctor C p).obj X) K n\np' q' : ℤ\nhpq' : p' + n = q'\n⊢ ((fun f ↦ fromSingleMk f h)\n ((fun α ↦ (HomologicalComplex.si... | by_cases hp : p' = p | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.Algebra.Homology.Factorizations.CM5a | {
"line": 370,
"column": 2
} | {
"line": 370,
"column": 28
} | [
{
"pp": "C : Type u_1\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Abelian C\nK L : CochainComplex C ℤ\nf : K ⟶ L\nn : ℤ\nhf : ∀ i < n, QuasiIsoAt f i\ninst✝¹ : Mono f\ninst✝ : Mono (homologyMap f n)\n⊢ QuasiIso ((cokernel f).πTruncGE n)",
"usedConstants": [
"CategoryTheory.Abelian.toPreadditive",
... | rw [quasiIso_πTruncGE_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Homology.Factorizations.CM5a | {
"line": 503,
"column": 4
} | {
"line": 503,
"column": 13
} | [
{
"pp": "case inr\nC : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : Abelian C\nK L : CochainComplex C ℤ\nf : K ⟶ L\ninst✝³ : EnoughInjectives C\ninst✝² : Mono f\nn₀ : ℤ\ninst✝¹ : K.IsStrictlyGE (n₀ + 1)\ninst✝ : L.IsStrictlyGE (n₀ + 1)\nq₁ q₂ : ℕ\nhq : q₁ ≤ q₂\ni : ℤ\nhi : i ≤ n₀ + ↑q₁\nthis : ∀ {q₁ q₂ : ... | clear hq' | Lean.Elab.Tactic.evalClear | Lean.Parser.Tactic.clear |
Mathlib.Algebra.Homology.Factorizations.CM5a | {
"line": 618,
"column": 4
} | {
"line": 618,
"column": 60
} | [
{
"pp": "C : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : Abelian C\nK L : CochainComplex C ℤ\nf : K ⟶ L\ninst✝³ : EnoughInjectives C\ninst✝² : Mono f\nn₀ : ℤ\ninst✝¹ : K.IsStrictlyGE (n₀ + 1)\ninst✝ : L.IsStrictlyGE (n₀ + 1)\ni : ℤ\nq : ℕ\nhq : i + 1 ≤ n₀ + ↑q\nthis : QuasiIsoAt (midπ f n₀ q) i\n⊢ QuasiI... | rw [← quasiIsoAt_iff_comp_right _ (midπ f n₀ q), ι_midπ] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Abelian.LeftDerived | {
"line": 103,
"column": 6
} | {
"line": 103,
"column": 78
} | [
{
"pp": "C : Type u\ninst✝⁵ : Category.{v, u} C\nD : Type u_1\ninst✝⁴ : Category.{v_1, u_1} D\ninst✝³ : Abelian C\ninst✝² : HasProjectiveResolutions C\ninst✝¹ : Abelian D\nX Y : C\nf : X ⟶ Y\nP : ProjectiveResolution X\nQ : ProjectiveResolution Y\nφ : P.complex ⟶ Q.complex\ncomm : φ.f 0 ≫ Q.π.f 0 = P.π.f 0 ≫ f\... | isoLeftDerivedToHomotopyCategoryObj_inv_naturality_assoc f P Q φ comm F, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Abelian.LeftDerived | {
"line": 325,
"column": 53
} | {
"line": 325,
"column": 56
} | [
{
"pp": "C : Type u\ninst✝⁵ : Category.{v, u} C\nD : Type u_1\ninst✝⁴ : Category.{v_1, u_1} D\ninst✝³ : Abelian C\ninst✝² : HasProjectiveResolutions C\ninst✝¹ : Abelian D\nX : C\nP : ProjectiveResolution X\nF : C ⥤ D\ninst✝ : F.Additive\nh₂ :\n (P.isoLeftDerivedToHomotopyCategoryObj F).inv =\n (F.mapHomolog... | h₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.ModelCategory.Lifting | {
"line": 103,
"column": 4
} | {
"line": 113,
"column": 31
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Abelian C\nA B X Y : CochainComplex C ℤ\nt : A ⟶ X\ni : A ⟶ B\np : X ⟶ Y\nb : B ⟶ Y\nsq : CommSq t i p b\nhsq : (n : ℤ) → ⋯.LiftStruct\nQ : CochainComplex C ℤ\nπ : B ⟶ Q\nhπ : i ≫ π = 0\nhQ : IsColimit (CokernelCofork.ofπ π hπ)\nK : CochainComplex C... | have : Epi π := Cofork.IsColimit.epi hQ
have : Mono ι := Fork.IsLimit.mono hK
ext n _ rfl
have this := Cochain.congr_v ((cocycle₁' sq hsq).δ_eq_zero 2) n _ rfl
rw [Cochain.zero_v, δ_v _ _ (by simp) _ _ _ _ (n + 1) _ (by lia) rfl,
Int.negOnePow_even 2 ⟨1, by simp⟩, one_smul] at this ⊢
rwa [← ca... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.ModelCategory.Lifting | {
"line": 103,
"column": 4
} | {
"line": 113,
"column": 31
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Abelian C\nA B X Y : CochainComplex C ℤ\nt : A ⟶ X\ni : A ⟶ B\np : X ⟶ Y\nb : B ⟶ Y\nsq : CommSq t i p b\nhsq : (n : ℤ) → ⋯.LiftStruct\nQ : CochainComplex C ℤ\nπ : B ⟶ Q\nhπ : i ≫ π = 0\nhQ : IsColimit (CokernelCofork.ofπ π hπ)\nK : CochainComplex C... | have : Epi π := Cofork.IsColimit.epi hQ
have : Mono ι := Fork.IsLimit.mono hK
ext n _ rfl
have this := Cochain.congr_v ((cocycle₁' sq hsq).δ_eq_zero 2) n _ rfl
rw [Cochain.zero_v, δ_v _ _ (by simp) _ _ _ _ (n + 1) _ (by lia) rfl,
Int.negOnePow_even 2 ⟨1, by simp⟩, one_smul] at this ⊢
rwa [← ca... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.GradedObject.Single | {
"line": 76,
"column": 90
} | {
"line": 77,
"column": 44
} | [
{
"pp": "J : Type u_1\nC : Type u_2\ninst✝² : Category.{v_1, u_2} C\ninst✝¹ : HasInitial C\ninst✝ : DecidableEq J\nj : J\nX Y : C\nf : X ⟶ Y\n⊢ (single j).map f j ≫ (singleObjApplyIso j Y).hom = (singleObjApplyIso j X).hom ≫ f",
"usedConstants": [
"CategoryTheory.GradedObject.single_map_singleObjApply... | by
apply single_map_singleObjApplyIsoOfEq_hom | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.ModelCategory.Lifting | {
"line": 164,
"column": 8
} | {
"line": 167,
"column": 47
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Abelian C\nA B X Y : CochainComplex C ℤ\nt : A ⟶ X\ni : A ⟶ B\np : X ⟶ Y\nb : B ⟶ Y\nsq : CommSq t i p b\nhsq : (n : ℤ) → ⋯.LiftStruct\nQ : CochainComplex C ℤ\nπ : B ⟶ Q\nhπ : i ≫ π = 0\nhQ : IsColimit (CokernelCofork.ofπ π hπ)\nK : CochainComplex C... | ext n
have : ι.f n ≫ p.f n = 0 := by
simp [← HomologicalComplex.comp_f, hι]
simpa [l, this] using (hsq n).fac_right | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.ModelCategory.Lifting | {
"line": 164,
"column": 8
} | {
"line": 167,
"column": 47
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Abelian C\nA B X Y : CochainComplex C ℤ\nt : A ⟶ X\ni : A ⟶ B\np : X ⟶ Y\nb : B ⟶ Y\nsq : CommSq t i p b\nhsq : (n : ℤ) → ⋯.LiftStruct\nQ : CochainComplex C ℤ\nπ : B ⟶ Q\nhπ : i ≫ π = 0\nhQ : IsColimit (CokernelCofork.ofπ π hπ)\nK : CochainComplex C... | ext n
have : ι.f n ≫ p.f n = 0 := by
simp [← HomologicalComplex.comp_f, hι]
simpa [l, this] using (hsq n).fac_right | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.GradedObject.Unitor | {
"line": 131,
"column": 2
} | {
"line": 132,
"column": 25
} | [
{
"pp": "case h\nC : Type u_1\nD : Type u_2\nI : Type u_3\nJ : Type u_4\ninst✝⁷ : Category.{v_1, u_1} C\ninst✝⁶ : Category.{v_2, u_2} D\ninst✝⁵ : Zero I\ninst✝⁴ : DecidableEq I\ninst✝³ : HasInitial C\nF : C ⥤ D ⥤ D\nX : C\ne : F.obj X ≅ 𝟭 D\ninst✝² : ∀ (Y : D), PreservesColimit (Functor.empty C) (F.flip.obj Y)... | rw [mapBifunctorLeftUnitor_inv_apply, mapBifunctorLeftUnitor_inv_apply, assoc, assoc,
ι_mapBifunctorMapMap] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Homology.Monoidal | {
"line": 163,
"column": 10
} | {
"line": 163,
"column": 52
} | [
{
"pp": "case neg\nC : Type u_1\ninst✝⁹ : Category.{v_1, u_1} C\ninst✝⁸ : MonoidalCategory C\ninst✝⁷ : Preadditive C\ninst✝⁶ : HasZeroObject C\ninst✝⁵ : (curriedTensor C).Additive\ninst✝⁴ : ∀ (X₁ : C), ((curriedTensor C).obj X₁).Additive\nI : Type u_2\ninst✝³ : AddMonoid I\nc : ComplexShape I\ninst✝² : c.Tensor... | mapBifunctor.d₂_eq_zero' _ _ _ _ _ h₁ _ h₂ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.Monoidal | {
"line": 299,
"column": 4
} | {
"line": 299,
"column": 16
} | [
{
"pp": "C : Type u_1\ninst✝¹⁴ : Category.{v_1, u_1} C\ninst✝¹³ : MonoidalCategory C\ninst✝¹² : Preadditive C\ninst✝¹¹ : HasZeroObject C\ninst✝¹⁰ : (curriedTensor C).Additive\ninst✝⁹ : ∀ (X₁ : C), ((curriedTensor C).obj X₁).Additive\nI : Type u_2\ninst✝⁸ : AddMonoid I\nc : ComplexShape I\ninst✝⁷ : c.TensorSigns... | rw [comp_id] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Homology.Monoidal | {
"line": 304,
"column": 4
} | {
"line": 304,
"column": 16
} | [
{
"pp": "C : Type u_1\ninst✝¹⁴ : Category.{v_1, u_1} C\ninst✝¹³ : MonoidalCategory C\ninst✝¹² : Preadditive C\ninst✝¹¹ : HasZeroObject C\ninst✝¹⁰ : (curriedTensor C).Additive\ninst✝⁹ : ∀ (X₁ : C), ((curriedTensor C).obj X₁).Additive\nI : Type u_2\ninst✝⁸ : AddMonoid I\nc : ComplexShape I\ninst✝⁷ : c.TensorSigns... | rw [comp_id] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Homology.Monoidal | {
"line": 309,
"column": 4
} | {
"line": 309,
"column": 16
} | [
{
"pp": "C : Type u_1\ninst✝¹⁴ : Category.{v_1, u_1} C\ninst✝¹³ : MonoidalCategory C\ninst✝¹² : Preadditive C\ninst✝¹¹ : HasZeroObject C\ninst✝¹⁰ : (curriedTensor C).Additive\ninst✝⁹ : ∀ (X₁ : C), ((curriedTensor C).obj X₁).Additive\nI : Type u_2\ninst✝⁸ : AddMonoid I\nc : ComplexShape I\ninst✝⁷ : c.TensorSigns... | rw [comp_id] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Homology.SpectralObject.HasSpectralSequence | {
"line": 343,
"column": 6
} | {
"line": 343,
"column": 14
} | [
{
"pp": "case h\nC : Type u_1\nι : Type u_2\nκ : Type u_3\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Abelian C\ninst✝¹ : Preorder ι\nc : ℤ → ComplexShape κ\nr₀ : ℤ\nX : SpectralObject C ι\ndata : SpectralSequenceDataCore ι c r₀\ninst✝ : X.HasSpectralSequence data\nl : ℕ\nE : SpectralObject C (Fin (l + 1))\nn : ℤ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Algebra.Homology.SpectralObject.SpectralSequence | {
"line": 143,
"column": 50
} | {
"line": 143,
"column": 53
} | [
{
"pp": "C : Type u_1\nι : Type u_2\nκ : Type u_3\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Abelian C\ninst✝ : Preorder ι\nX : SpectralObject C ι\nc : ℤ → ComplexShape κ\nr₀ : ℤ\ndata : SpectralSequenceDataCore ι c r₀\nr : ℤ\nhr : r₀ ≤ r\npq pq' : κ\nhpq : (c r).Rel pq pq'\ni₀ i₁ i₂ i₃ i₄ i₅ : ι\nf₁ : i₀ ⟶ i₁\n... | h₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.SpectralObject.Page | {
"line": 907,
"column": 4
} | {
"line": 907,
"column": 29
} | [
{
"pp": "C : Type u_1\nι : Type u_2\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_2, u_2} ι\ninst✝ : Abelian C\nX : SpectralObject C ι\ni₀ i₁ i₂ i₃ : ι\nf₁ : i₀ ⟶ i₁\nf₂ : i₁ ⟶ i₂\nf₃ : i₂ ⟶ i₃\nf₁₂ : i₀ ⟶ i₂\nh₁₂ : f₁ ≫ f₂ = f₁₂\ni₀' i₁' i₂' i₃' : ι\nf₁' : i₀' ⟶ i₁'\nf₂' : i₁' ⟶ i₂'\nf₃' : i₂' ⟶ i₃'\nf... | X.p_opcyclesMap_assoc .., | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.SpectralObject.SpectralSequence | {
"line": 334,
"column": 18
} | {
"line": 334,
"column": 45
} | [
{
"pp": "C : Type u_1\nι : Type u_2\nκ : Type u_3\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Abelian C\ninst✝ : Preorder ι\nX : SpectralObject C ι\nc : ℤ → ComplexShape κ\nr₀ : ℤ\ndata : SpectralSequenceDataCore ι c r₀\nr r' : ℤ\nhrr' : r + 1 = r'\nhr : r₀ ≤ r\npq pq' : κ\ni₀ i₁ i₂ i₃ i₃' : ι\nhi₀ : i₀ = data.i₀... | rw [hi₂, data.hc₀₂ r _ _ h] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Homology.SpectralObject.SpectralSequence | {
"line": 334,
"column": 18
} | {
"line": 334,
"column": 45
} | [
{
"pp": "C : Type u_1\nι : Type u_2\nκ : Type u_3\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Abelian C\ninst✝ : Preorder ι\nX : SpectralObject C ι\nc : ℤ → ComplexShape κ\nr₀ : ℤ\ndata : SpectralSequenceDataCore ι c r₀\nr r' : ℤ\nhrr' : r + 1 = r'\nhr : r₀ ≤ r\npq pq' : κ\ni₀ i₁ i₂ i₃ i₃' : ι\nhi₀ : i₀ = data.i₀... | rw [hi₂, data.hc₀₂ r _ _ h] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.SpectralObject.SpectralSequence | {
"line": 334,
"column": 18
} | {
"line": 334,
"column": 45
} | [
{
"pp": "C : Type u_1\nι : Type u_2\nκ : Type u_3\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Abelian C\ninst✝ : Preorder ι\nX : SpectralObject C ι\nc : ℤ → ComplexShape κ\nr₀ : ℤ\ndata : SpectralSequenceDataCore ι c r₀\nr r' : ℤ\nhrr' : r + 1 = r'\nhr : r₀ ≤ r\npq pq' : κ\ni₀ i₁ i₂ i₃ i₃' : ι\nhi₀ : i₀ = data.i₀... | rw [hi₂, data.hc₀₂ r _ _ h] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.SpectralObject.SpectralSequence | {
"line": 391,
"column": 12
} | {
"line": 391,
"column": 39
} | [
{
"pp": "C : Type u_1\nι : Type u_2\nκ : Type u_3\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Abelian C\ninst✝¹ : Preorder ι\nX : SpectralObject C ι\nc : ℤ → ComplexShape κ\nr₀ : ℤ\ndata : SpectralSequenceDataCore ι c r₀\nr r' : ℤ\nhrr' : r + 1 = r'\nhr : r₀ ≤ r\npq pq' : κ\nhpq : (c r).prev pq' = pq\ni₀ i₁ i₂ i₃... | rw [hi₂, data.hc₀₂ r _ _ h] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Homology.SpectralObject.SpectralSequence | {
"line": 391,
"column": 12
} | {
"line": 391,
"column": 39
} | [
{
"pp": "C : Type u_1\nι : Type u_2\nκ : Type u_3\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Abelian C\ninst✝¹ : Preorder ι\nX : SpectralObject C ι\nc : ℤ → ComplexShape κ\nr₀ : ℤ\ndata : SpectralSequenceDataCore ι c r₀\nr r' : ℤ\nhrr' : r + 1 = r'\nhr : r₀ ≤ r\npq pq' : κ\nhpq : (c r).prev pq' = pq\ni₀ i₁ i₂ i₃... | rw [hi₂, data.hc₀₂ r _ _ h] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.SpectralObject.SpectralSequence | {
"line": 391,
"column": 12
} | {
"line": 391,
"column": 39
} | [
{
"pp": "C : Type u_1\nι : Type u_2\nκ : Type u_3\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Abelian C\ninst✝¹ : Preorder ι\nX : SpectralObject C ι\nc : ℤ → ComplexShape κ\nr₀ : ℤ\ndata : SpectralSequenceDataCore ι c r₀\nr r' : ℤ\nhrr' : r + 1 = r'\nhr : r₀ ≤ r\npq pq' : κ\nhpq : (c r).prev pq' = pq\ni₀ i₁ i₂ i₃... | rw [hi₂, data.hc₀₂ r _ _ h] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.SpectralObject.SpectralSequence | {
"line": 508,
"column": 20
} | {
"line": 508,
"column": 23
} | [
{
"pp": "C : Type u_1\nι : Type u_2\nκ : Type u_3\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Abelian C\ninst✝¹ : Preorder ι\nX : SpectralObject C ι\nc : ℤ → ComplexShape κ\nr₀ : ℤ\ndata : SpectralSequenceDataCore ι c r₀\ninst✝ : X.HasSpectralSequence data\nr : ℤ\nhr : r₀ ≤ r\npq pq' : κ\nhpq : (c r).Rel pq pq'\n... | h₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Artinian.Module | {
"line": 132,
"column": 91
} | {
"line": 132,
"column": 99
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : Nontrivial R\nh : IsArtinian R M\ns : Set M\nhs : LinearIndependent R Subtype.val\ni : ↑s\nx✝ : R ∙ ↑i = ⊥\n⊢ ↑i = 0",
"usedConstants": [
"Submodule.span_eq_bot._simp_1",
"Submodule",... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.Artinian.Module | {
"line": 132,
"column": 91
} | {
"line": 132,
"column": 99
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : Nontrivial R\nh : IsArtinian R M\ns : Set M\nhs : LinearIndependent R Subtype.val\ni : ↑s\nx✝ : R ∙ ↑i = ⊥\n⊢ ↑i = 0",
"usedConstants": [
"Submodule.span_eq_bot._simp_1",
"Submodule",... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Artinian.Module | {
"line": 132,
"column": 91
} | {
"line": 132,
"column": 99
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : Nontrivial R\nh : IsArtinian R M\ns : Set M\nhs : LinearIndependent R Subtype.val\ni : ↑s\nx✝ : R ∙ ↑i = ⊥\n⊢ ↑i = 0",
"usedConstants": [
"Submodule.span_eq_bot._simp_1",
"Submodule",... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Artinian.Module | {
"line": 187,
"column": 4
} | {
"line": 188,
"column": 37
} | [
{
"pp": "case h.succ\nR : Type u_1\nM : Type u_2\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : IsArtinian R M\nf : ℕ → Submodule R M\nh : ∀ (n : ℕ), Disjoint ((partialSups (⇑OrderDual.toDual ∘ f)) n) (OrderDual.toDual (f (n + 1)))\nn : ℕ\nw : ∀ (m : ℕ), n ≤ m → OrderDual.toDual f ... | · apply w
exact Nat.succ_le_succ_iff.mp p | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Artinian.Module | {
"line": 331,
"column": 7
} | {
"line": 331,
"column": 10
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁴ : Ring R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : IsArtinian R M\ninst✝ : IsNoetherian R M\nf : M →ₗ[R] M\nk : ℕ\nhk : ∀ b ≥ k, IsCompl (f ^ b).ker (f ^ b).range ∧ ⨅ m, (f ^ m).range = (f ^ b).range ∧ ⨆ m, (f ^ m).ker = (f ^ b).ker\nh₁ : IsCompl (f ^ k)... | h₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Artinian.Module | {
"line": 358,
"column": 11
} | {
"line": 358,
"column": 26
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : IsArtinian R M\nr : R\nx : M\nn : ℕ\nhn : ∀ (m : ℕ), n ≤ m → (r ^ n • LinearMap.id).range = (r ^ m • LinearMap.id).range\n⊢ ∃ n y, r ^ n.succ • y = r ^ n • x",
"usedConstants": [
"_priv... | SetLike.ext_iff | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.Artinian.Module | {
"line": 503,
"column": 51
} | {
"line": 504,
"column": 90
} | [
{
"pp": "R : Type u_1\ninst✝¹ : Ring R\ninst✝ : IsArtinianRing R\n⊢ IsUnit.submonoid R = nonZeroDivisorsRight R",
"usedConstants": [
"Eq.mpr",
"nonZeroDivisorsRight",
"isRightRegular_iff_mem_nonZeroDivisorsRight",
"Monoid.toMulOneClass",
"congrArg",
"Membership.mem",
... | by
ext; rw [← isRightRegular_iff_mem_nonZeroDivisorsRight]; exact isUnit_iff_isRightRegular | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Lie.Subalgebra | {
"line": 358,
"column": 6
} | {
"line": 358,
"column": 64
} | [
{
"pp": "R : Type u\nL : Type v\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\nL₂ : Type w\ninst✝¹ : LieRing L₂\ninst✝ : LieAlgebra R L₂\nf : L →ₗ⁅R⁆ L₂\nK K' : LieSubalgebra R L\nK₂ : LieSubalgebra R L₂\nx' : L\nhx' : x' ∈ ↑K.toSubmodule\ny' : L\nhy' : y' ∈ ↑K.toSubmodule\n⊢ ⁅↑f x', ↑f y'⁆ ... | simpa using ⟨⁅x', y'⁆, K.lie_mem hx' hy', f.map_lie x' y'⟩ | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
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