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.CategoryTheory.Shift.Twist | {
"line": 57,
"column": 2
} | {
"line": 57,
"column": 13
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nA : Type w\ninst✝¹ : AddMonoid A\ninst✝ : HasShift C A\nt : TwistShiftData C A\na b c : A\nX : C\n⊢ (shiftFunctor C c).map ((↑(t.z a b)).app X) = (↑(t.z a b)).app ((shiftFunctor C c).obj X)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.BifunctorHomotopy | {
"line": 172,
"column": 10
} | {
"line": 172,
"column": 46
} | [
{
"pp": "case neg\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 : Comp... | rw [zero₁ _ _ _ _ _ _ h₅, comp_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Homology.BifunctorHomotopy | {
"line": 172,
"column": 10
} | {
"line": 172,
"column": 46
} | [
{
"pp": "case neg\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 : Comp... | rw [zero₁ _ _ _ _ _ _ h₅, comp_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.BifunctorHomotopy | {
"line": 172,
"column": 10
} | {
"line": 172,
"column": 46
} | [
{
"pp": "case neg\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 : Comp... | rw [zero₁ _ _ _ _ _ _ h₅, comp_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.BifunctorHomotopy | {
"line": 227,
"column": 22
} | {
"line": 227,
"column": 48
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.BifunctorHomotopy | {
"line": 228,
"column": 17
} | {
"line": 228,
"column": 43
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.CochainComplexOpposite | {
"line": 135,
"column": 34
} | {
"line": 135,
"column": 45
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nK L : CochainComplex C ℤ\nf g : K ⟶ L\nh : Homotopy ((opEquivalence C).functor.map f.op) ((opEquivalence C).functor.map g.op)\nn p q p' q' : ℤ\nhp : p = p'\nhq : q = q'\n⊢ ComplexShape.embeddingUpIntDownInt.f p' = ComplexShape.embeddi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.BifunctorAssociator | {
"line": 241,
"column": 2
} | {
"line": 242,
"column": 88
} | [
{
"pp": "C₁ : Type u_1\nC₂ : Type u_2\nC₁₂ : Type u_3\nC₃ : Type u_5\nC₄ : Type u_6\ninst✝¹⁹ : Category.{v_1, u_1} C₁\ninst✝¹⁸ : Category.{v_2, u_2} C₂\ninst✝¹⁷ : Category.{v_3, u_5} C₃\ninst✝¹⁶ : Category.{v_4, u_6} C₄\ninst✝¹⁵ : Category.{v_5, u_3} C₁₂\ninst✝¹⁴ : HasZeroMorphisms C₁\ninst✝¹³ : HasZeroMorphism... | dsimp [d₂]
rw [shape _ _ _ h, Functor.map_zero, Functor.map_zero, zero_app, zero_comp, smul_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.BifunctorAssociator | {
"line": 241,
"column": 2
} | {
"line": 242,
"column": 88
} | [
{
"pp": "C₁ : Type u_1\nC₂ : Type u_2\nC₁₂ : Type u_3\nC₃ : Type u_5\nC₄ : Type u_6\ninst✝¹⁹ : Category.{v_1, u_1} C₁\ninst✝¹⁸ : Category.{v_2, u_2} C₂\ninst✝¹⁷ : Category.{v_3, u_5} C₃\ninst✝¹⁶ : Category.{v_4, u_6} C₄\ninst✝¹⁵ : Category.{v_5, u_3} C₁₂\ninst✝¹⁴ : HasZeroMorphisms C₁\ninst✝¹³ : HasZeroMorphism... | dsimp [d₂]
rw [shape _ _ _ h, Functor.map_zero, Functor.map_zero, zero_app, zero_comp, smul_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.CommSq | {
"line": 93,
"column": 18
} | {
"line": 93,
"column": 29
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preadditive C\nX₁ X₂ X₃ X₄ : C\ninst✝ : HasBinaryBiproduct X₂ X₃\nf : X₁ ⟶ X₂\ng : X₁ ⟶ X₃\ninl : X₂ ⟶ X₄\ninr : X₃ ⟶ X₄\nsq : CommSq f g inl inr\nh : IsColimit sq.cokernelCofork\ns : PushoutCocone f g\n⊢ inl ≫ (fun s ↦ h.desc (CokernelCofork.ofπ (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.CommSq | {
"line": 96,
"column": 18
} | {
"line": 96,
"column": 29
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preadditive C\nX₁ X₂ X₃ X₄ : C\ninst✝ : HasBinaryBiproduct X₂ X₃\nf : X₁ ⟶ X₂\ng : X₁ ⟶ X₃\ninl : X₂ ⟶ X₄\ninr : X₃ ⟶ X₄\nsq : CommSq f g inl inr\nh : IsColimit sq.cokernelCofork\ns : PushoutCocone f g\n⊢ inr ≫ (fun s ↦ h.desc (CokernelCofork.ofπ (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.CommSq | {
"line": 117,
"column": 63
} | {
"line": 117,
"column": 74
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preadditive C\nX₁ X₂ X₃ X₄ : C\ninst✝ : HasBinaryBiproduct X₂ X₃\nf : X₁ ⟶ X₂\ng : X₁ ⟶ X₃\ninl : X₂ ⟶ X₄\ninr : X₃ ⟶ X₄\nh : IsPushout f g inl inr\nR✝ : C\nb : ⋯.shortComplex.X₃ ⟶ R✝\nhb : ⋯.shortComplex.g ≫ b = 0\n⊢ Cofork.π ⋯.cokernelCofork ≫ b ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.ConcreteCategory | {
"line": 113,
"column": 2
} | {
"line": 124,
"column": 91
} | [
{
"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 ... | refine hS.δ_apply' i j hij _ ((forget₂ C Ab).map (S.X₂.pOpcycles i) x₂) _ ?_ ?_
· rw [← ConcreteCategory.forget₂_comp_apply, ← ConcreteCategory.forget₂_comp_apply,
HomologicalComplex.p_opcyclesMap, Functor.map_comp, ConcreteCategory.comp_apply,
HomologicalComplex.homology_π_ι, ConcreteCategory.forget₂_com... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.ConcreteCategory | {
"line": 113,
"column": 2
} | {
"line": 124,
"column": 91
} | [
{
"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 ... | refine hS.δ_apply' i j hij _ ((forget₂ C Ab).map (S.X₂.pOpcycles i) x₂) _ ?_ ?_
· rw [← ConcreteCategory.forget₂_comp_apply, ← ConcreteCategory.forget₂_comp_apply,
HomologicalComplex.p_opcyclesMap, Functor.map_comp, ConcreteCategory.comp_apply,
HomologicalComplex.homology_π_ι, ConcreteCategory.forget₂_com... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.CommSq | {
"line": 173,
"column": 18
} | {
"line": 173,
"column": 29
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preadditive C\nX₁ X₂ X₃ X₄ : C\ninst✝ : HasBinaryBiproduct X₂ X₃\nfst : X₁ ⟶ X₂\nsnd : X₁ ⟶ X₃\nf : X₂ ⟶ X₄\ng : X₃ ⟶ X₄\nsq : CommSq fst snd f g\nh : IsLimit sq.kernelFork\ns : PullbackCone f g\n⊢ (fun s ↦ h.lift (KernelFork.ofι (biprod.lift s.fst... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.CommSq | {
"line": 175,
"column": 18
} | {
"line": 175,
"column": 29
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preadditive C\nX₁ X₂ X₃ X₄ : C\ninst✝ : HasBinaryBiproduct X₂ X₃\nfst : X₁ ⟶ X₂\nsnd : X₁ ⟶ X₃\nf : X₂ ⟶ X₄\ng : X₃ ⟶ X₄\nsq : CommSq fst snd f g\nh : IsLimit sq.kernelFork\ns : PullbackCone f g\n⊢ (fun s ↦ h.lift (KernelFork.ofι (biprod.lift s.fst... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.CommSq | {
"line": 195,
"column": 53
} | {
"line": 195,
"column": 64
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preadditive C\nX₁ X₂ X₃ X₄ : C\ninst✝ : HasBinaryBiproduct X₂ X₃\nfst : X₁ ⟶ X₂\nsnd : X₁ ⟶ X₃\nf : X₂ ⟶ X₄\ng : X₃ ⟶ X₄\nh : IsPullback fst snd f g\nP✝ : C\nb : P✝ ⟶ ⋯.shortComplex'.X₁\nhb : b ≫ ⋯.shortComplex'.f = 0\n⊢ b ≫ Fork.ι ⋯.kernelFork = 0... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.DerivedCategory.Ext.MapBijective | {
"line": 47,
"column": 12
} | {
"line": 47,
"column": 46
} | [
{
"pp": "case zero\nC : Type u\ninst✝¹² : Category.{v, u} C\ninst✝¹¹ : Abelian C\nD : Type u'\ninst✝¹⁰ : Category.{v', u'} D\ninst✝⁹ : Abelian D\nF : C ⥤ D\ninst✝⁸ : F.Additive\ninst✝⁷ : PreservesFiniteLimits F\ninst✝⁶ : PreservesFiniteColimits F\ninst✝⁵ : F.Full\ninst✝⁴ : F.Faithful\ninst✝³ : HasExt C\ninst✝² ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.DerivedCategory.Ext.MapBijective | {
"line": 69,
"column": 12
} | {
"line": 69,
"column": 46
} | [
{
"pp": "case zero\nC : Type u\ninst✝¹² : Category.{v, u} C\ninst✝¹¹ : Abelian C\nD : Type u'\ninst✝¹⁰ : Category.{v', u'} D\ninst✝⁹ : Abelian D\nF : C ⥤ D\ninst✝⁸ : F.Additive\ninst✝⁷ : PreservesFiniteLimits F\ninst✝⁶ : PreservesFiniteColimits F\ninst✝⁵ : F.Full\ninst✝⁴ : F.Faithful\ninst✝³ : HasExt C\ninst✝² ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.DerivedCategory.Ext.Map | {
"line": 215,
"column": 2
} | {
"line": 216,
"column": 75
} | [
{
"pp": "case e_f\nC : Type u\ninst✝⁸ : Category.{v, u} C\ninst✝⁷ : Abelian C\nD : Type u'\ninst✝⁶ : Category.{v', u'} D\ninst✝⁵ : Abelian D\nF : C ⥤ D\ninst✝⁴ : F.Additive\ninst✝³ : PreservesFiniteLimits F\ninst✝² : PreservesFiniteColimits F\ninst✝¹ : HasExt C\ninst✝ : HasExt D\nX Y : C\nf : X ⟶ Y\n⊢ ((F.mapCo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.DerivedCategory.TStructure | {
"line": 100,
"column": 6
} | {
"line": 100,
"column": 28
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : HasDerivedCategory C\nX : DerivedCategory C\nn : ℤ\nhX : ∀ i < n, IsZero ((homologyFunctor C i).obj X)\ni : ℤ\nhi : i < n\n⊢ IsZero (HomologicalComplex.homology (Q.objPreimage X) i)",
"usedConstants": [
"CategoryTheory.Abelia... | apply (hX i hi).of_iso | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Algebra.Homology.DerivedCategory.TStructure | {
"line": 117,
"column": 6
} | {
"line": 117,
"column": 28
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : HasDerivedCategory C\nX : DerivedCategory C\nn : ℤ\nhX : ∀ (i : ℤ), n < i → IsZero ((homologyFunctor C i).obj X)\ni : ℤ\nhi : n < i\n⊢ IsZero (HomologicalComplex.homology (Q.objPreimage X) i)",
"usedConstants": [
"CategoryThe... | apply (hX i hi).of_iso | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Algebra.Homology.DerivedCategory.Ext.TStructure | {
"line": 55,
"column": 22
} | {
"line": 55,
"column": 59
} | [
{
"pp": "C : Type u\ninst✝⁶ : Category.{v, u} C\ninst✝⁵ : Abelian C\ninst✝⁴ : HasExt C\nK L : CochainComplex C ℤ\na b : ℤ\ninst✝³ : K.IsGE a\ninst✝² : K.IsLE a\ninst✝¹ : L.IsGE b\ninst✝ : L.IsLE b\nthis : (C : Type u) → [inst : Category.{v, u} C] → [inst_1 : Abelian C] → HasDerivedCategory C :=\n HasDerivedCat... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.HomotopyCategory.HomComplexCohomology | {
"line": 162,
"column": 33
} | {
"line": 162,
"column": 48
} | [
{
"pp": "case refine_1\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nK L : CochainComplex C ℤ\nn : ℤ\nx y : Cocycle K L n\nh : toHom (mk x) = toHom (mk y)\n⊢ mk (x - y) = 0",
"usedConstants": [
"CochainComplex.HomComplex.coboundaries",
"AddGroup.toSubtractionMonoid",
"Eq.... | mk_eq_zero_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.BifunctorShift | {
"line": 235,
"column": 26
} | {
"line": 246,
"column": 44
} | [
{
"pp": "C₁ : Type u_1\nC₂ : Type u_2\nD : Type u_3\ninst✝⁸ : Category.{v_1, u_1} C₁\ninst✝⁷ : Category.{v_2, u_2} C₂\ninst✝⁶ : Category.{v_3, u_3} D\ninst✝⁵ : Preadditive C₁\ninst✝⁴ : Preadditive C₂\ninst✝³ : Preadditive D\nF : C₁ ⥤ C₂ ⥤ D\ninst✝² : F.Additive\ninst✝¹ : ∀ (X₁ : C₁), (F.obj X₁).Additive\ninst✝ ... | by
ext K₂ n
dsimp
ext p q h
dsimp at h
simp [CochainComplex.ι_mapBifunctorShift₂Iso_hom_f _ _ F (a + b) p q n h
(q + a + b) (n + a + b) (by lia) (by lia),
CochainComplex.ι_mapBifunctorShift₂Iso_hom_f_assoc _ _ F b p q n h _ _ rfl rfl,
CochainComplex.ι_mapBifunctorShift₂Iso_hom_f_... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.HomotopyCategory.HomComplexCohomology | {
"line": 199,
"column": 10
} | {
"line": 199,
"column": 38
} | [
{
"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\nhm : n + 1 = m\nhp : m + 1 = p\ns : Cofork ((Cocycle.isKernel K L m p hp).lift (KernelFork.ofι (HomologicalComplex.sc' (K.HomComplex L) n m p).f ⋯)) 0\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.HomotopyCategory.HomComplexCohomology | {
"line": 204,
"column": 8
} | {
"line": 204,
"column": 19
} | [
{
"pp": "case hf.h\nC : 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\nhm : n + 1 = m\nhp : m + 1 = p\ns : Cofork ((Cocycle.isKernel K L m p hp).lift (KernelFork.ofι (HomologicalComplex.sc' (K.HomComplex L) n m ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.BifunctorAssociator | {
"line": 609,
"column": 14
} | {
"line": 609,
"column": 73
} | [
{
"pp": "C₁ : Type u_1\nC₂ : Type u_2\nC₂₃ : Type u_4\nC₃ : Type u_5\nC₄ : Type u_6\ninst✝²² : Category.{v_1, u_1} C₁\ninst✝²¹ : Category.{v_2, u_2} C₂\ninst✝²⁰ : Category.{v_3, u_5} C₃\ninst✝¹⁹ : Category.{v_4, u_6} C₄\ninst✝¹⁸ : Category.{v_6, u_4} C₂₃\ninst✝¹⁷ : HasZeroMorphisms C₁\ninst✝¹⁶ : HasZeroMorphism... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.HomotopyCategory.HomComplexInduction | {
"line": 56,
"column": 2
} | {
"line": 63,
"column": 20
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nK L : CochainComplex C ℤ\nd : ℤ\nX : ℕ → Set (Cochain K L d)\nφ : (n : ℕ) → ↑(X n) → ↑(X (n + 1))\np₀ : ℤ\nhφ : ∀ (n : ℕ) (x : ↑(X n)), (↑(φ n x)).EqUpTo (↑x) (p₀ + ↑n)\nx₀ : ↑(X 0)\nn₁ n₂ : ℕ\nh : n₁ ≤ n₂\n⊢ (↑(sequence φ x₀ n₁)).EqUpTo (↑... | obtain ⟨k, rfl⟩ := Nat.exists_eq_add_of_le h
clear h
induction k generalizing n₁ with
| zero => intro _ _ _ _; simp
| succ k hk =>
intro p q hpq hp
rw [hk n₁ p q hpq hp, ← hφ (n₁ + k) (sequence φ x₀ (n₁ + k)) p q hpq (by lia)]
dsimp [sequence] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.HomotopyCategory.HomComplexInduction | {
"line": 56,
"column": 2
} | {
"line": 63,
"column": 20
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nK L : CochainComplex C ℤ\nd : ℤ\nX : ℕ → Set (Cochain K L d)\nφ : (n : ℕ) → ↑(X n) → ↑(X (n + 1))\np₀ : ℤ\nhφ : ∀ (n : ℕ) (x : ↑(X n)), (↑(φ n x)).EqUpTo (↑x) (p₀ + ↑n)\nx₀ : ↑(X 0)\nn₁ n₂ : ℕ\nh : n₁ ≤ n₂\n⊢ (↑(sequence φ x₀ n₁)).EqUpTo (↑... | obtain ⟨k, rfl⟩ := Nat.exists_eq_add_of_le h
clear h
induction k generalizing n₁ with
| zero => intro _ _ _ _; simp
| succ k hk =>
intro p q hpq hp
rw [hk n₁ p q hpq hp, ← hφ (n₁ + k) (sequence φ x₀ (n₁ + k)) p q hpq (by lia)]
dsimp [sequence] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Triangulated.Orthogonal | {
"line": 81,
"column": 6
} | {
"line": 81,
"column": 31
} | [
{
"pp": "C : Type u\ninst✝⁶ : Category.{v, u} C\nP : ObjectProperty C\ninst✝⁵ : HasZeroObject C\ninst✝⁴ : HasShift C ℤ\ninst✝³ : Preadditive C\ninst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝¹ : Pretriangulated C\ninst✝ : P.IsTriangulated\nY : C\nhY : P.rightOrthogonal Y\nX₁ X₂ : C\nf : X₁ ⟶ X₂\nx✝ : P.t... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Triangulated.Orthogonal | {
"line": 81,
"column": 51
} | {
"line": 81,
"column": 76
} | [
{
"pp": "C : Type u\ninst✝⁶ : Category.{v, u} C\nP : ObjectProperty C\ninst✝⁵ : HasZeroObject C\ninst✝⁴ : HasShift C ℤ\ninst✝³ : Preadditive C\ninst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝¹ : Pretriangulated C\ninst✝ : P.IsTriangulated\nY : C\nhY : P.rightOrthogonal Y\nX₁ X₂ : C\nf : X₁ ⟶ X₂\nx✝ : P.t... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Triangulated.Orthogonal | {
"line": 99,
"column": 8
} | {
"line": 99,
"column": 33
} | [
{
"pp": "C : Type u\ninst✝⁶ : Category.{v, u} C\nP : ObjectProperty C\ninst✝⁵ : HasZeroObject C\ninst✝⁴ : HasShift C ℤ\ninst✝³ : Preadditive C\ninst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝¹ : Pretriangulated C\ninst✝ : P.IsTriangulated\nX : C\nhX : P.leftOrthogonal X\nY₂ Y₃ : C\nh : Y₂ ⟶ Y₃\nY₁ : C\nf... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Triangulated.Orthogonal | {
"line": 117,
"column": 4
} | {
"line": 117,
"column": 65
} | [
{
"pp": "case refine_2\nC : Type u\ninst✝⁹ : Category.{v, u} C\nD : Type u'\ninst✝⁸ : Category.{v', u'} D\nP : ObjectProperty C\ninst✝⁷ : HasZeroObject C\ninst✝⁶ : HasShift C ℤ\ninst✝⁵ : Preadditive C\ninst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝³ : Pretriangulated C\ninst✝² : P.IsTriangulated\ninst✝¹... | obtain ⟨φ, hφ⟩ := Localization.exists_rightFraction L P.trW g | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.CategoryTheory.Triangulated.Orthogonal | {
"line": 117,
"column": 4
} | {
"line": 121,
"column": 29
} | [
{
"pp": "case refine_2\nC : Type u\ninst✝⁹ : Category.{v, u} C\nD : Type u'\ninst✝⁸ : Category.{v', u'} D\nP : ObjectProperty C\ninst✝⁷ : HasZeroObject C\ninst✝⁶ : HasShift C ℤ\ninst✝⁵ : Preadditive C\ninst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝³ : Pretriangulated C\ninst✝² : P.IsTriangulated\ninst✝¹... | obtain ⟨φ, hφ⟩ := Localization.exists_rightFraction L P.trW g
obtain ⟨α, hα⟩ := (hY _ φ.hs).2 φ.f
refine ⟨α, ?_⟩
rw [hφ, ← cancel_epi (L.map φ.s), MorphismProperty.RightFraction.map_s_comp_map,
← hα, Functor.map_comp] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Triangulated.Orthogonal | {
"line": 117,
"column": 4
} | {
"line": 121,
"column": 29
} | [
{
"pp": "case refine_2\nC : Type u\ninst✝⁹ : Category.{v, u} C\nD : Type u'\ninst✝⁸ : Category.{v', u'} D\nP : ObjectProperty C\ninst✝⁷ : HasZeroObject C\ninst✝⁶ : HasShift C ℤ\ninst✝⁵ : Preadditive C\ninst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝³ : Pretriangulated C\ninst✝² : P.IsTriangulated\ninst✝¹... | obtain ⟨φ, hφ⟩ := Localization.exists_rightFraction L P.trW g
obtain ⟨α, hα⟩ := (hY _ φ.hs).2 φ.f
refine ⟨α, ?_⟩
rw [hφ, ← cancel_epi (L.map φ.s), MorphismProperty.RightFraction.map_s_comp_map,
← hα, Functor.map_comp] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.Embedding.ExtendHomotopy | {
"line": 76,
"column": 10
} | {
"line": 76,
"column": 30
} | [
{
"pp": "case pos\nι : Type u_1\nι' : Type u_2\nc : ComplexShape ι\nc' : ComplexShape ι'\nC : Type u_3\ninst✝³ : Category.{v_1, u_3} C\ninst✝² : HasZeroObject C\ninst✝¹ : Preadditive C\nK L : HomologicalComplex C c\nf g : K ⟶ L\nh : Homotopy f g\ne : c.Embedding c'\ninst✝ : e.IsRelIff\ni : ι\n⊢ (extendMap f e).... | extendMap_f _ _ rfl, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.Embedding.ExtendHomotopy | {
"line": 76,
"column": 31
} | {
"line": 76,
"column": 51
} | [
{
"pp": "case pos\nι : Type u_1\nι' : Type u_2\nc : ComplexShape ι\nc' : ComplexShape ι'\nC : Type u_3\ninst✝³ : Category.{v_1, u_3} C\ninst✝² : HasZeroObject C\ninst✝¹ : Preadditive C\nK L : HomologicalComplex C c\nf g : K ⟶ L\nh : Homotopy f g\ne : c.Embedding c'\ninst✝ : e.IsRelIff\ni : ι\n⊢ (K.extendXIso e ... | extendMap_f _ _ rfl, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.Embedding.ExtendHomotopy | {
"line": 141,
"column": 18
} | {
"line": 141,
"column": 79
} | [
{
"pp": "ι : Type u_1\nι' : Type u_2\nc : ComplexShape ι\nc' : ComplexShape ι'\nC : Type u_3\ninst✝³ : Category.{v_1, u_3} C\ninst✝² : HasZeroObject C\ninst✝¹ : Preadditive C\nK L : HomologicalComplex C c\nf g : K ⟶ L\ne : c.Embedding c'\ninst✝ : e.IsRelIff\nh : Homotopy (extendMap f e) (extendMap g e)\ni j : ι... | by rw [h.zero _ _ (by rwa [e.rel_iff]), zero_comp, comp_zero] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.BifunctorAssociator | {
"line": 705,
"column": 8
} | {
"line": 706,
"column": 15
} | [
{
"pp": "case e_a.h\nC₁ : Type u_1\nC₂ : Type u_2\nC₂₃ : Type u_4\nC₃ : Type u_5\nC₄ : Type u_6\ninst✝²³ : Category.{v_1, u_1} C₁\ninst✝²² : Category.{v_2, u_2} C₂\ninst✝²¹ : Category.{v_3, u_5} C₃\ninst✝²⁰ : Category.{v_4, u_6} C₄\ninst✝¹⁹ : Category.{v_6, u_4} C₂₃\ninst✝¹⁸ : HasZeroMorphisms C₁\ninst✝¹⁷ : Has... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.BifunctorAssociator | {
"line": 710,
"column": 8
} | {
"line": 711,
"column": 15
} | [
{
"pp": "case e_a.h\nC₁ : Type u_1\nC₂ : Type u_2\nC₂₃ : Type u_4\nC₃ : Type u_5\nC₄ : Type u_6\ninst✝²³ : Category.{v_1, u_1} C₁\ninst✝²² : Category.{v_2, u_2} C₂\ninst✝²¹ : Category.{v_3, u_5} C₃\ninst✝²⁰ : Category.{v_4, u_6} C₄\ninst✝¹⁹ : Category.{v_6, u_4} C₂₃\ninst✝¹⁸ : HasZeroMorphisms C₁\ninst✝¹⁷ : Has... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.HomotopyCategory.KInjective | {
"line": 183,
"column": 4
} | {
"line": 183,
"column": 46
} | [
{
"pp": "case a\nC : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Abelian C\nK L : CochainComplex C ℤ\nn : ℤ\nz : Cocycle K L n\ninst✝ : L.IsKInjective\nhK : HomologicalComplex.Acyclic K\nm : ℤ\nhm : m + 1 = n\nφ : K ⟶ (CategoryTheory.shiftFunctor (CochainComplex C ℤ) n).obj L\nhφ : Cochain.ofHom φ = (↑z)... | Cochain.δ_rightUnshift _ _ _ _ 0 (by simp) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.DifferentialObject | {
"line": 234,
"column": 52
} | {
"line": 234,
"column": 92
} | [
{
"pp": "S : Type u_1\ninst✝⁵ : AddMonoidWithOne S\nC : Type (u + 1)\ninst✝⁴ : LargeCategory C\ninst✝³ : HasZeroMorphisms C\nFC : C → C → Type u_2\nCC : C → Type u_3\ninst✝² : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)\ninst✝¹ : ConcreteCategory C FC\ninst✝ : HasShift C S\nX✝ Y✝ : DifferentialObject S C\nf : Ho... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.Embedding.StupidTrunc | {
"line": 63,
"column": 6
} | {
"line": 63,
"column": 17
} | [
{
"pp": "case neg.hi'\nι : Type u_1\nι' : Type u_2\nc : ComplexShape ι\nc' : ComplexShape ι'\nC : Type u_3\ninst✝⁴ : Category.{v_1, u_3} C\ninst✝³ : HasZeroMorphisms C\ninst✝² : HasZeroObject C\nK L M : HomologicalComplex C c'\nφ : K ⟶ L\nφ' : L ⟶ M\ne : c.Embedding c'\ninst✝¹ : e.IsRelIff\nι'' : Type u_4\nc'' ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.Embedding.Connect | {
"line": 98,
"column": 39
} | {
"line": 98,
"column": 60
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasZeroMorphisms C\nK : ChainComplex C ℕ\nL : CochainComplex C ℕ\nh : ConnectData K L\nn✝ m✝ : ℤ\nn m : ℕ\nhnm : Int.ofNat n + 1 ≠ Int.ofNat m\n⊢ ¬(ComplexShape.up ℕ).Rel n m",
"usedConstants": [
"Nat.instOne",
"_private.Mathlib.Algebra.Ho... | by simp at hnm ⊢; lia | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.Embedding.Connect | {
"line": 100,
"column": 4
} | {
"line": 100,
"column": 32
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasZeroMorphisms C\nK : ChainComplex C ℕ\nL : CochainComplex C ℕ\nh : ConnectData K L\nn✝ m✝ : ℤ\nn m : ℕ\nhnm : Int.negSucc n + 1 ≠ Int.negSucc m\n⊢ h.d (Int.negSucc n) (Int.negSucc m) = 0",
"usedConstants": [
"Eq.mpr",
"Nat.instOne",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.Embedding.Connect | {
"line": 100,
"column": 46
} | {
"line": 100,
"column": 67
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasZeroMorphisms C\nK : ChainComplex C ℕ\nL : CochainComplex C ℕ\nh : ConnectData K L\nn✝ m✝ : ℤ\nn m : ℕ\nhnm : Int.negSucc n + 1 ≠ Int.negSucc m\n⊢ ¬(ComplexShape.down ℕ).Rel n m",
"usedConstants": [
"Nat.instOne",
"AddRightCancelSemigro... | by simp at hnm ⊢; lia | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.MorphismProperty.LiftingProperty | {
"line": 138,
"column": 4
} | {
"line": 138,
"column": 28
} | [
{
"pp": "case a\nC : Type u\ninst✝ : Category.{v, u} C\nT : MorphismProperty C\n⊢ T.rlp ≤ T.pushouts.rlp",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.MorphismProperty",
"CategoryTheory.MorphismProperty.pushouts",
"CategoryTheory.MorphismProperty.llp",
"CategoryTheory.Morphism... | rw [← le_llp_iff_le_rlp] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.MorphismProperty.LiftingProperty | {
"line": 156,
"column": 4
} | {
"line": 156,
"column": 28
} | [
{
"pp": "case a\nC : Type u\ninst✝ : Category.{v, u} C\nT : MorphismProperty C\n⊢ T.rlp ≤ (coproducts.{w, v, u} T).rlp",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.MorphismProperty",
"CategoryTheory.MorphismProperty.llp",
"CategoryTheory.MorphismProperty.instCompleteBooleanAlgebra"... | rw [← le_llp_iff_le_rlp] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.MorphismProperty.LiftingProperty | {
"line": 166,
"column": 4
} | {
"line": 166,
"column": 28
} | [
{
"pp": "case a\nC : Type u\ninst✝ : Category.{v, u} C\nT : MorphismProperty C\n⊢ T.rlp ≤ T.retracts.rlp",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.MorphismProperty",
"CategoryTheory.MorphismProperty.llp",
"CategoryTheory.MorphismProperty.instCompleteBooleanAlgebra",
"congr... | rw [← le_llp_iff_le_rlp] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Homology.ExactSequenceFour | {
"line": 47,
"column": 35
} | {
"line": 47,
"column": 46
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{?u.170, u_1} C\ninst✝ : HasZeroMorphisms C\nn : ℕ\nS : ComposableArrows C (n + 3)\nhS : S.IsComplex\nk : ℕ\nhk : k ≤ n\ncc : CokernelCofork (S.map' k (k + 1) ⋯ ⋯)\nkf : KernelFork (S.map' (k + 2) (k + 3) ⋯ ⋯)\nhcc : IsColimit cc\nhkf : IsLimit kf\n⊢ (S.map' k (k + 1) ⋯ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Generator.HomologicalComplex | {
"line": 55,
"column": 2
} | {
"line": 55,
"column": 41
} | [
{
"pp": "case h.a.a\nC : Type u\ninst✝³ : Category.{v, u} C\nι : Type w\nc : ComplexShape ι\ninst✝² : c.HasNoLoop\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasZeroObject C\nα : Type t\nX : α → C\nhX : (ObjectProperty.ofObj X).IsSeparating\nK L : HomologicalComplex C c\nf g : K ⟶ L\nh : ∀ (G : HomologicalComplex C c... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Generator.HomologicalComplex | {
"line": 66,
"column": 67
} | {
"line": 66,
"column": 78
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\nι : Type w\nc : ComplexShape ι\ninst✝³ : c.HasNoLoop\ninst✝² : HasCoproductsOfShape ι C\ninst✝¹ : Preadditive C\ninst✝ : HasZeroObject C\nX : C\nhX : IsSeparator X\nφ : ι → HomologicalComplex C c := fun i ↦ separatingFamily c (fun x ↦ X) (PUnit.unit, i)\n⊢ (Objec... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.GrothendieckAbelian | {
"line": 43,
"column": 6
} | {
"line": 43,
"column": 23
} | [
{
"pp": "case h\nC : Type u\ninst✝³ : Category.{v, u} C\nι : Type t\nc : ComplexShape ι\ninst✝² : HasZeroMorphisms C\ninst✝¹ : LocallySmall.{w, v, u} C\ninst✝ : Small.{w, t} ι\nK L : HomologicalComplex C c\nemb : (K ⟶ L) → (i : Shrink.{w, t} ι) → Shrink.{w, v} (K.X ((equivShrink ι).symm i) ⟶ L.X ((equivShrink ι... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.MorphismProperty.RetractArgument | {
"line": 61,
"column": 4
} | {
"line": 61,
"column": 15
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\nW₁ W₂ : MorphismProperty C\ninst✝¹ : W₁.HasFactorization W₂\ninst✝ : W₁.IsStableUnderRetracts\nh₁ : W₁ ≤ W₂.llp\nA B : C\ni : A ⟶ B\nhi : W₂.llp i\nh : W₁.MapFactorizationData W₂ i\nthis : HasLiftingProperty i h.p\n⊢ W₁ i",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.MorphismProperty.RetractArgument | {
"line": 69,
"column": 4
} | {
"line": 69,
"column": 15
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\nW₁ W₂ : MorphismProperty C\ninst✝¹ : W₁.HasFactorization W₂\ninst✝ : W₂.IsStableUnderRetracts\nh₂ : W₂ ≤ W₁.rlp\nX Y : C\np : X ⟶ Y\nhp : W₁.rlp p\nh : W₁.MapFactorizationData W₂ p\nthis : HasLiftingProperty h.i p\n⊢ W₂ p",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.HomotopyFiber | {
"line": 92,
"column": 26
} | {
"line": 92,
"column": 37
} | [
{
"pp": "C : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : Preadditive C\nα : Type u_2\nc : ComplexShape α\nK : HomologicalComplex C c\ninst✝² : DecidableRel c.Rel\ninst✝¹ : ∀ (i : α), HasBinaryBiproduct (K.X i) (K.X i)\ninst✝ : K.HasPathObject\ni : α\nh₁ : IsZero (K.X i)\nh₂ : ∀ (j : α), c.Rel j i → IsZer... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicTopology.ModelCategory.CategoryWithCofibrations | {
"line": 205,
"column": 71
} | {
"line": 206,
"column": 26
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nX✝ Y✝ : C\nf✝ : X✝ ⟶ Y✝\ninst✝¹ : CategoryWithFibrations C\nX Y : Cᵒᵖ\nf : X ⟶ Y\ninst✝ : Cofibration f\n⊢ Fibration f.unop",
"usedConstants": [
"Eq.mpr",
"Opposite",
"CategoryTheory.CategoryStruct.toQuiver",
"congrArg",
"Quiver.... | by
rwa [fibration_unop_iff] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicTopology.ModelCategory.Instances | {
"line": 267,
"column": 44
} | {
"line": 270,
"column": 48
} | [
{
"pp": "C : Type u\ninst✝⁷ : Category.{v, u} C\ninst✝⁶ : CategoryWithWeakEquivalences C\ninst✝⁵ : CategoryWithCofibrations C\ninst✝⁴ : CategoryWithFibrations C\nJ✝ : Type w\nJ : Type u_1\nX Y : J → C\nf : (i : J) → X i ⟶ Y i\ninst✝³ : HasCoproduct X\ninst✝² : HasCoproduct Y\nh : ∀ (i : J), Cofibration (f i)\ni... | by
rw [weakEquivalence_iff]
exact (MorphismProperty.colimMap (W := (trivialCofibrations C)) _
(fun ⟨i⟩ ↦ mem_trivialCofibrations (f i))).2 | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicTopology.ModelCategory.Instances | {
"line": 322,
"column": 2
} | {
"line": 322,
"column": 36
} | [
{
"pp": "C : Type u\ninst✝⁵ : Category.{v, u} C\ninst✝⁴ : CategoryWithWeakEquivalences C\ninst✝³ : CategoryWithCofibrations C\ninst✝² : CategoryWithFibrations C\nX Y : C\nf : X ⟶ Y\ninst✝¹ : (trivialCofibrations C).IsWeakFactorizationSystem (fibrations C)\ninst✝ : IsIso f\nthis : trivialCofibrations C f\n⊢ Cofi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicTopology.ModelCategory.Instances | {
"line": 328,
"column": 2
} | {
"line": 328,
"column": 34
} | [
{
"pp": "C : Type u\ninst✝⁵ : Category.{v, u} C\ninst✝⁴ : CategoryWithWeakEquivalences C\ninst✝³ : CategoryWithCofibrations C\ninst✝² : CategoryWithFibrations C\nX Y : C\nf : X ⟶ Y\ninst✝¹ : (cofibrations C).IsWeakFactorizationSystem (trivialFibrations C)\ninst✝ : IsIso f\nthis : trivialFibrations C f\n⊢ Fibrat... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicTopology.ModelCategory.Instances | {
"line": 369,
"column": 2
} | {
"line": 369,
"column": 36
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : CategoryWithWeakEquivalences C\ninst✝¹ : CategoryWithCofibrations C\ninst✝ : CategoryWithFibrations C\nX Y : C\nf : X ⟶ Y\nh : (cofibrations C).MapFactorizationData (trivialFibrations C) f\n⊢ Cofibration h.i",
"usedConstants": [
"Eq.mpr",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicTopology.ModelCategory.Instances | {
"line": 372,
"column": 2
} | {
"line": 372,
"column": 34
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : CategoryWithWeakEquivalences C\ninst✝¹ : CategoryWithCofibrations C\ninst✝ : CategoryWithFibrations C\nX Y : C\nf : X ⟶ Y\nh : (cofibrations C).MapFactorizationData (trivialFibrations C) f\n⊢ Fibration h.p",
"usedConstants": [
"Eq.mpr",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicTopology.ModelCategory.Instances | {
"line": 375,
"column": 2
} | {
"line": 375,
"column": 40
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : CategoryWithWeakEquivalences C\ninst✝¹ : CategoryWithCofibrations C\ninst✝ : CategoryWithFibrations C\nX Y : C\nf : X ⟶ Y\nh : (cofibrations C).MapFactorizationData (trivialFibrations C) f\n⊢ WeakEquivalence h.p",
"usedConstants": [
"Eq.mpr",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicTopology.ModelCategory.Instances | {
"line": 384,
"column": 2
} | {
"line": 384,
"column": 36
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : CategoryWithWeakEquivalences C\ninst✝¹ : CategoryWithCofibrations C\ninst✝ : CategoryWithFibrations C\nX Y : C\nf : X ⟶ Y\nh : (trivialCofibrations C).MapFactorizationData (fibrations C) f\n⊢ Cofibration h.i",
"usedConstants": [
"Eq.mpr",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicTopology.ModelCategory.Instances | {
"line": 387,
"column": 2
} | {
"line": 387,
"column": 40
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : CategoryWithWeakEquivalences C\ninst✝¹ : CategoryWithCofibrations C\ninst✝ : CategoryWithFibrations C\nX Y : C\nf : X ⟶ Y\nh : (trivialCofibrations C).MapFactorizationData (fibrations C) f\n⊢ WeakEquivalence h.i",
"usedConstants": [
"Eq.mpr",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicTopology.ModelCategory.Instances | {
"line": 390,
"column": 2
} | {
"line": 390,
"column": 34
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : CategoryWithWeakEquivalences C\ninst✝¹ : CategoryWithCofibrations C\ninst✝ : CategoryWithFibrations C\nX Y : C\nf : X ⟶ Y\nh : (trivialCofibrations C).MapFactorizationData (fibrations C) f\n⊢ Fibration h.p",
"usedConstants": [
"Eq.mpr",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicTopology.ModelCategory.Basic | {
"line": 120,
"column": 2
} | {
"line": 120,
"column": 40
} | [
{
"pp": "C : Type u\ninst✝⁸ : Category.{v, u} C\ninst✝⁷ : CategoryWithFibrations C\ninst✝⁶ : CategoryWithCofibrations C\ninst✝⁵ : CategoryWithWeakEquivalences C\ninst✝⁴ : (weakEquivalences C).HasTwoOutOfThreeProperty\ninst✝³ : (trivialCofibrations C).IsWeakFactorizationSystem (fibrations C)\ninst✝² : (cofibrati... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicTopology.ModelCategory.Basic | {
"line": 136,
"column": 6
} | {
"line": 136,
"column": 40
} | [
{
"pp": "C : Type u\ninst✝⁸ : Category.{v, u} C\ninst✝⁷ : CategoryWithFibrations C\ninst✝⁶ : CategoryWithCofibrations C\ninst✝⁵ : CategoryWithWeakEquivalences C\ninst✝⁴ : HasFiniteLimits C\ninst✝³ : HasFiniteColimits C\ninst✝² : (weakEquivalences C).HasTwoOutOfThreeProperty\ninst✝¹ : (cofibrations C).IsWeakFact... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicTopology.ModelCategory.Basic | {
"line": 138,
"column": 6
} | {
"line": 138,
"column": 44
} | [
{
"pp": "C : Type u\ninst✝⁸ : Category.{v, u} C\ninst✝⁷ : CategoryWithFibrations C\ninst✝⁶ : CategoryWithCofibrations C\ninst✝⁵ : CategoryWithWeakEquivalences C\ninst✝⁴ : HasFiniteLimits C\ninst✝³ : HasFiniteColimits C\ninst✝² : (weakEquivalences C).HasTwoOutOfThreeProperty\ninst✝¹ : (cofibrations C).IsWeakFact... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicTopology.ModelCategory.Basic | {
"line": 141,
"column": 32
} | {
"line": 141,
"column": 64
} | [
{
"pp": "C : Type u\ninst✝⁸ : Category.{v, u} C\ninst✝⁷ : CategoryWithFibrations C\ninst✝⁶ : CategoryWithCofibrations C\ninst✝⁵ : CategoryWithWeakEquivalences C\ninst✝⁴ : HasFiniteLimits C\ninst✝³ : HasFiniteColimits C\ninst✝² : (weakEquivalences C).HasTwoOutOfThreeProperty\ninst✝¹ : (cofibrations C).IsWeakFact... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.HomotopyCategory.HomComplexSingle | {
"line": 257,
"column": 15
} | {
"line": 257,
"column": 26
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Preadditive C\ninst✝ : HasZeroObject C\nX : C\nK : CochainComplex C ℤ\np n : ℤ\nα : Cocycle ((singleFunctor C p).obj X) K n\nq : ℤ\nh : p + n = q\nq' : ℤ\nhq' : q + 1 = q'\nf : X ⟶ K.X q\nhf : Cochain.fromSingleMk f h = ↑α\nhα : (Cochain.fromSingleMk (f ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicTopology.ModelCategory.Cylinder | {
"line": 248,
"column": 22
} | {
"line": 248,
"column": 33
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : ModelCategory C\nA : C\nP : Cylinder A\nh : (cofibrations C).MapFactorizationData (trivialFibrations C) (codiag A)\n⊢ Cofibration (ofFactorizationData h).i",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Limits.hasFiniteCoproducts_of_hasFin... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicTopology.ModelCategory.Cylinder | {
"line": 291,
"column": 6
} | {
"line": 291,
"column": 21
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : ModelCategory C\nA : C\nP✝ : Cylinder A\ninst✝² : IsCofibrant A\nP P' : Cylinder A\ninst✝¹ : P.IsGood\ninst✝ : P'.IsGood\nψ : P.I ⨿ A ⟶ (P.trans P').I := coprod.desc (pushout.inl P.i₁ P'.i₀) (P'.i₁ ≫ pushout.inr P.i₁ P'.i₀)\nfac : coprod.map P.i₁ (𝟙 A) ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.HomotopyCategory.HomComplexSingle | {
"line": 294,
"column": 4
} | {
"line": 294,
"column": 55
} | [
{
"pp": "case mp\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Preadditive C\ninst✝ : HasZeroObject C\nX : C\nK : CochainComplex C ℤ\np q : ℤ\nf : X ⟶ K.X q\nn : ℤ\nh : p + n = q\nq' : ℤ\nhq' : q + 1 = q'\nhf : f ≫ K.d q q' = 0\nq'' : ℤ\nhq'' : q'' + 1 = q\nα : Cochain ((singleFunctor C p).obj X) K (n - 1)\... | exact (Cochain.fromSingleEquiv h).symm.injective hα | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Homology.HomotopyCategory.HomComplexSingle | {
"line": 332,
"column": 4
} | {
"line": 332,
"column": 83
} | [
{
"pp": "case refine_1\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Preadditive C\ninst✝ : HasZeroObject C\nX : C\nK : CochainComplex C ℤ\nq n : ℤ\nα : Cocycle K ((singleFunctor C q).obj X) n\np : ℤ\nh : p + n = q\np' : ℤ\nhp' : p' + 1 = p\nf : K.X p ⟶ X\nhf : Cochain.toSingleMk f h = ↑α\nhα : Cochain.toSi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Localization.Quotient | {
"line": 64,
"column": 21
} | {
"line": 64,
"column": 37
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_2, u_2} D\nhomRel : HomRel C\nW : MorphismProperty C\nh : homRel.FactorsThroughLocalization W\nW' : MorphismProperty (CategoryTheory.Quotient homRel)\nhW : W = W'.inverseImage (Quotient.functor homRel)\nE : Type u_3\ninst✝... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.Factorizations.CM5a | {
"line": 193,
"column": 4
} | {
"line": 193,
"column": 68
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Abelian C\nK L : CochainComplex C ℤ\nf : K ⟶ L\ninst✝ : EnoughInjectives C\nn₀✝ n₁ : ℤ\nhn₁ : n₀✝ + 1 = n₁\nhf : ∀ i ≤ n₀✝, QuasiIsoAt f i\nn₀ : ℤ := n₁ - 1\nA : C\nx₁ : A ⟶ K.X n₁\nx✝ : x₁ ≫ K.d n₁ (n₁ + 1) = 0\ny₀ : A ⟶ L.X n₀\nh₁ : x₁ ≫ (i K n₁)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicTopology.ModelCategory.PathObject | {
"line": 254,
"column": 20
} | {
"line": 254,
"column": 31
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : ModelCategory C\nA : C\nP : PathObject A\nh : (trivialCofibrations C).MapFactorizationData (fibrations C) (diag A)\n⊢ Fibration (ofFactorizationData h).p",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.MorphismProperty.MapFactorizationData.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicTopology.ModelCategory.PathObject | {
"line": 295,
"column": 6
} | {
"line": 295,
"column": 21
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : ModelCategory C\nA : C\nP✝ : PathObject A\ninst✝² : IsFibrant A\nP P' : PathObject A\ninst✝¹ : P.IsGood\ninst✝ : P'.IsGood\nψ : (P.trans P').P ⟶ P.P ⨯ A := prod.lift (pullback.fst P.p₁ P'.p₀) (pullback.snd P.p₁ P'.p₀ ≫ P'.p₁)\nfac : ψ ≫ prod.map P.p₁ (𝟙... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.Factorizations.CM5a | {
"line": 331,
"column": 2
} | {
"line": 345,
"column": 40
} | [
{
"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\ninst✝¹ : EnoughInjectives C\nn : ℤ\ninst✝ : Mono f\n⊢ (homologyShortComplex f n).Exact",
"usedConstants": [
"CategoryTheory.Abelian.toPreadditive",
"Eq.mpr",
"CategoryTheory.Shor... | let T := ShortComplex.mk (homologyMap f n) (homologyMap (cokernel.π f) n)
(by rw [← homologyMap_comp, cokernel.condition, homologyMap_zero])
let φ : T ⟶ homologyShortComplex f n :=
{ τ₁ := 𝟙 _
τ₂ := 𝟙 _
τ₃ := homologyMap ((cokernel f).πTruncGE n ≫ p f n) n
comm₂₃ := by
dsimp
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.Factorizations.CM5a | {
"line": 331,
"column": 2
} | {
"line": 345,
"column": 40
} | [
{
"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\ninst✝¹ : EnoughInjectives C\nn : ℤ\ninst✝ : Mono f\n⊢ (homologyShortComplex f n).Exact",
"usedConstants": [
"CategoryTheory.Abelian.toPreadditive",
"Eq.mpr",
"CategoryTheory.Shor... | let T := ShortComplex.mk (homologyMap f n) (homologyMap (cokernel.π f) n)
(by rw [← homologyMap_comp, cokernel.condition, homologyMap_zero])
let φ : T ⟶ homologyShortComplex f n :=
{ τ₁ := 𝟙 _
τ₂ := 𝟙 _
τ₃ := homologyMap ((cokernel f).πTruncGE n ≫ p f n) n
comm₂₃ := by
dsimp
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.HomotopyCategory.Plus | {
"line": 44,
"column": 4
} | {
"line": 44,
"column": 15
} | [
{
"pp": "case refine_1\nC : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preadditive C\ninst✝ : HasBinaryBiproducts C\nK : CochainComplex C ℤ\nn : ℤ\nhn : K.IsStrictlyGE n\ni : ℤ\nhi : autoParam (i < n - 1) isStrictlyGE_iff._auto_1\n⊢ IsZero ((eval C (ComplexShape.up ℤ) i).obj K ⊞ (eval C (ComplexShape.up... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.Factorizations.CM5a | {
"line": 433,
"column": 6
} | {
"line": 438,
"column": 41
} | [
{
"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\nn : ℤ\ninst✝¹ : K.IsStrictlyGE (n + 1)\ninst✝ : L.IsStrictlyGE (n + 1)\ni : ℤ\nhi : i ≤ n + ↑0\n⊢ QuasiIsoAt { obj := { mid := L, ι := f, π := 𝟙 L, ι_π :... | dsimp
rw [quasiIsoAt_iff_isIso_homologyMap]
apply IsZero.isIso
all_goals
· rw [← exactAt_iff_isZero_homology]
exact exactAt_of_isGE _ (n + 1) i | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.Factorizations.CM5a | {
"line": 433,
"column": 6
} | {
"line": 438,
"column": 41
} | [
{
"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\nn : ℤ\ninst✝¹ : K.IsStrictlyGE (n + 1)\ninst✝ : L.IsStrictlyGE (n + 1)\ni : ℤ\nhi : i ≤ n + ↑0\n⊢ QuasiIsoAt { obj := { mid := L, ι := f, π := 𝟙 L, ι_π :... | dsimp
rw [quasiIsoAt_iff_isIso_homologyMap]
apply IsZero.isIso
all_goals
· rw [← exactAt_iff_isZero_homology]
exact exactAt_of_isGE _ (n + 1) i | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.HomotopyCategory.Plus | {
"line": 125,
"column": 6
} | {
"line": 128,
"column": 28
} | [
{
"pp": "case refine_1\nC : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : Preadditive C\nA : Type u_2\ninst✝³ : Category.{v_2, u_2} A\ninst✝² : Abelian A\ninst✝¹ : HasZeroObject C\ninst✝ : HasBinaryBiproducts C\nT : Triangle (HomotopyCategory C (ComplexShape.up ℤ))\nhT : T ∈ distinguishedTriangles\nh₁ : pl... | dsimp
simp only [plus_quotient_obj_iff]
exact ⟨min (n₁ - 1) n₂, CochainComplex.isStrictlyGE_mappingCone f n₁ n₂ _
(by simp) (by simp)⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.HomotopyCategory.Plus | {
"line": 125,
"column": 6
} | {
"line": 128,
"column": 28
} | [
{
"pp": "case refine_1\nC : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : Preadditive C\nA : Type u_2\ninst✝³ : Category.{v_2, u_2} A\ninst✝² : Abelian A\ninst✝¹ : HasZeroObject C\ninst✝ : HasBinaryBiproducts C\nT : Triangle (HomotopyCategory C (ComplexShape.up ℤ))\nhT : T ∈ distinguishedTriangles\nh₁ : pl... | dsimp
simp only [plus_quotient_obj_iff]
exact ⟨min (n₁ - 1) n₂, CochainComplex.isStrictlyGE_mappingCone f n₁ n₂ _
(by simp) (by simp)⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.HomotopyCategory.Plus | {
"line": 210,
"column": 6
} | {
"line": 211,
"column": 51
} | [
{
"pp": "C : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : Preadditive C\nA : Type u_2\ninst✝³ : Category.{v_2, u_2} A\ninst✝² : Abelian A\ninst✝¹ : HasZeroObject C\ninst✝ : HasBinaryBiproducts C\nx✝¹ x✝ : CochainComplex.Plus C\nf : x✝¹ ⟶ x✝\nhf : (homotopyEquivalences C (ComplexShape.up ℤ)).inverseImage (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Homology.Factorizations.CM5a | {
"line": 507,
"column": 33
} | {
"line": 507,
"column": 44
} | [
{
"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\nthis : ∀ {q₁ q₂ : ℕ} (hq : q₁ ≤ q₂), i ≤ n₀ + ↑q₁ → q₁ + 1 = q₂ → IsIso ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Idempotents.FunctorExtension | {
"line": 51,
"column": 38
} | {
"line": 51,
"column": 80
} | [
{
"pp": "C : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_2, u_2} D\ninst✝ : Category.{v_3, u_3} E\nF : C ⥤ Karoubi D\nP : Karoubi C\n⊢ (F.map P.p).f ≫ (F.map P.p).f = (F.map P.p).f",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Idempotents.FunctorExtension | {
"line": 52,
"column": 30
} | {
"line": 52,
"column": 72
} | [
{
"pp": "C : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_2, u_2} D\ninst✝ : Category.{v_3, u_3} E\nF : C ⥤ Karoubi D\nX✝ Y✝ : Karoubi C\nf : X✝ ⟶ Y✝\n⊢ { X := (F.obj X✝.X).X, p := (F.map X✝.p).f, idem := ⋯ }.p ≫\n (F.map f.f).f ≫ { X := (F.obj Y✝.X).X, p := (F... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Abelian.Projective.Resolution | {
"line": 184,
"column": 4
} | {
"line": 184,
"column": 25
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nX : C\nP Q : ProjectiveResolution X\n⊢ Homotopy (lift (𝟙 X ≫ 𝟙 X) P P) (𝟙 P.complex)",
"usedConstants": [
"CategoryTheory.Abelian.toPreadditive",
"Eq.mpr",
"ChainComplex",
"HomologicalComplex.instCategory",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Abelian.Projective.Resolution | {
"line": 186,
"column": 4
} | {
"line": 186,
"column": 25
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nX : C\nP Q : ProjectiveResolution X\n⊢ Homotopy (lift (𝟙 X ≫ 𝟙 X) Q Q) (𝟙 Q.complex)",
"usedConstants": [
"CategoryTheory.Abelian.toPreadditive",
"Eq.mpr",
"ChainComplex",
"HomologicalComplex.instCategory",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Idempotents.FunctorExtension | {
"line": 125,
"column": 18
} | {
"line": 125,
"column": 70
} | [
{
"pp": "C : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_2, u_2} D\ninst✝ : Category.{v_3, u_3} E\nG : Karoubi C ⥤ Karoubi D\nP : Karoubi C\n⊢ ((((whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C) ⋙ functorExtension₁ C D).obj G).obj P).p ≫\n (G.map P... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Idempotents.FunctorExtension | {
"line": 129,
"column": 14
} | {
"line": 130,
"column": 21
} | [
{
"pp": "C : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_2, u_2} D\ninst✝ : Category.{v_3, u_3} E\nG : Karoubi C ⥤ Karoubi D\nP Q : Karoubi C\nf : P ⟶ Q\n⊢ (((whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C) ⋙ functorExtension₁ C D).obj G).map f ≫\n ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Idempotents.FunctorExtension | {
"line": 135,
"column": 18
} | {
"line": 135,
"column": 70
} | [
{
"pp": "C : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_2, u_2} D\ninst✝ : Category.{v_3, u_3} E\nG : Karoubi C ⥤ Karoubi D\nP : Karoubi C\n⊢ (((𝟭 (Karoubi C ⥤ Karoubi D)).obj G).obj P).p ≫\n (G.map P.decompId_i).f ≫\n ((((whiskeringLeft C (Karoubi C)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Idempotents.FunctorExtension | {
"line": 139,
"column": 14
} | {
"line": 139,
"column": 56
} | [
{
"pp": "C : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_2, u_2} D\ninst✝ : Category.{v_3, u_3} E\nG : Karoubi C ⥤ Karoubi D\nP Q : Karoubi C\nf : P ⟶ Q\n⊢ ((𝟭 (Karoubi C ⥤ Karoubi D)).obj G).map f ≫ { f := (G.map Q.decompId_i).f, comm := ⋯ } =\n { f := (G.map ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Idempotents.FunctorExtension | {
"line": 142,
"column": 10
} | {
"line": 142,
"column": 62
} | [
{
"pp": "case w.h.h\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_2, u_2} D\ninst✝ : Category.{v_3, u_3} E\nG : Karoubi C ⥤ Karoubi D\nP : Karoubi C\n⊢ (({ app := fun P ↦ { f := (G.map P.decompId_p).f, comm := ⋯ }, naturality := ⋯ } ≫\n { app := fun P... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Idempotents.FunctorExtension | {
"line": 145,
"column": 10
} | {
"line": 145,
"column": 62
} | [
{
"pp": "case w.h.h\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_2, u_2} D\ninst✝ : Category.{v_3, u_3} E\nG : Karoubi C ⥤ Karoubi D\nP : Karoubi C\n⊢ (({ app := fun P ↦ { f := (G.map P.decompId_i).f, comm := ⋯ }, naturality := ⋯ } ≫\n { app := fun P... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Abelian.LeftDerived | {
"line": 153,
"column": 2
} | {
"line": 153,
"column": 85
} | [
{
"pp": "C : Type u\ninst✝⁶ : Category.{v, u} C\nD : Type u_1\ninst✝⁵ : Category.{v_1, u_1} D\ninst✝⁴ : Abelian C\ninst✝³ : HasProjectiveResolutions C\ninst✝² : Abelian D\nF : C ⥤ D\ninst✝¹ : F.Additive\nn : ℕ\nX : C\ninst✝ : Projective X\n⊢ IsZero ((F.leftDerived (n + 1)).obj X)",
"usedConstants": [
... | refine IsZero.of_iso ?_ ((ProjectiveResolution.self X).isoLeftDerivedObj F (n + 1)) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
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