module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 375
values | kind stringclasses 379
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.RingTheory.Ideal.Cotangent | {
"line": 177,
"column": 12
} | {
"line": 177,
"column": 14
} | {
"line": 177,
"column": 14
} | [
{
"pp": "case refine_1\nR : Type u\nS : Type v\nS' : Type w\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommSemiring S\ninst✝⁸ : Algebra S R\ninst✝⁷ : CommSemiring S'\ninst✝⁶ : Algebra S' R\ninst✝⁵ : Algebra S S'\ninst✝⁴ : IsScalarTower S S' R\nI : Ideal R\nA : Type u_1\nB : Type u_2\ninst✝³ : CommRing A\ninst✝² : CommRing... | [
"case refine_1\nR : Type u\nS : Type v\nS' : Type w\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommSemiring S\ninst✝⁸ : Algebra S R\ninst✝⁷ : CommSemiring S'\ninst✝⁶ : Algebra S' R\ninst✝⁵ : Algebra S S'\ninst✝⁴ : IsScalarTower S S' R\nI : Ideal R\nA : Type u_1\nB : Type u_2\ninst✝³ : CommRing A\ninst✝² : CommRing B\ninst✝¹ :... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.RingTheory.Ideal.Cotangent | {
"line": 177,
"column": 4
} | {
"line": 177,
"column": 54
} | {
"line": 178,
"column": 2
} | [
{
"pp": "case refine_1\nR : Type u\nS : Type v\nS' : Type w\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommSemiring S\ninst✝⁸ : Algebra S R\ninst✝⁷ : CommSemiring S'\ninst✝⁶ : Algebra S' R\ninst✝⁵ : Algebra S S'\ninst✝⁴ : IsScalarTower S S' R\nI : Ideal R\nA : Type u_1\nB : Type u_2\ninst✝³ : CommRing A\ninst✝² : CommRing... | [] | intro a ha; exact Ideal.pow_le_self two_ne_zero ha | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Ideal.Cotangent | {
"line": 177,
"column": 4
} | {
"line": 177,
"column": 54
} | {
"line": 178,
"column": 2
} | [
{
"pp": "case refine_1\nR : Type u\nS : Type v\nS' : Type w\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommSemiring S\ninst✝⁸ : Algebra S R\ninst✝⁷ : CommSemiring S'\ninst✝⁶ : Algebra S' R\ninst✝⁵ : Algebra S S'\ninst✝⁴ : IsScalarTower S S' R\nI : Ideal R\nA : Type u_1\nB : Type u_2\ninst✝³ : CommRing A\ninst✝² : CommRing... | [] | intro a ha; exact Ideal.pow_le_self two_ne_zero ha | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Ideal.Cotangent | {
"line": 424,
"column": 2
} | {
"line": 424,
"column": 58
} | {
"line": 425,
"column": 2
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝² : CommRing A\ninst✝¹ : CommRing B\ninst✝ : Algebra A B\nsurj : Function.Surjective ⇑(algebraMap A B)\nI : Ideal B\nJ : Ideal A\neq : comap (algebraMap A B) I = RingHom.ker (algebraMap A B) ⊔ J\n⊢ (J.mapCotangent I (Algebra.ofId A B) ⋯).ker =\n Submodule.map J.toCot... | [
"A : Type u_1\nB : Type u_2\ninst✝² : CommRing A\ninst✝¹ : CommRing B\ninst✝ : Algebra A B\nsurj : Function.Surjective ⇑(algebraMap A B)\nI : Ideal B\nJ : Ideal A\neq : comap (algebraMap A B) I = RingHom.ker (algebraMap A B) ⊔ J\neqmap : I = map (algebraMap A B) J\n⊢ (J.mapCotangent I (Algebra.ofId A B) ⋯).ker =\n ... | have eqmap := Ideal.eq_map_of_comap_eq_ker_sup _ surj eq | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.CategoryTheory.GradedObject | {
"line": 99,
"column": 77
} | {
"line": 101,
"column": 16
} | {
"line": 103,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nβ : Type u_1\nX Y : GradedObject β C\nf : X ⟶ Y\ninst✝ : IsIso f\ni : β\n⊢ IsIso (f i)",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"CategoryTheory.IsIso",
"CategoryTheory.GradedObject.eval",
"inferInstance",
"id",... | [] | by
change IsIso ((eval i).map f)
infer_instance | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.GradedObject | {
"line": 115,
"column": 31
} | {
"line": 117,
"column": 44
} | {
"line": 119,
"column": 0
} | [
{
"pp": "C : Type u_1\nJ : Type u_4\ninst✝ : Category.{v_1, u_1} C\nX Y : GradedObject J C\ne : X ≅ Y\nj : J\n⊢ e.hom j ≫ e.inv j = 𝟙 (X j)",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",... | [] | by
rw [← GradedObject.categoryOfGradedObjects_comp, e.hom_inv_id,
GradedObject.categoryOfGradedObjects_id] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.GradedObject | {
"line": 406,
"column": 2
} | {
"line": 406,
"column": 34
} | {
"line": 408,
"column": 0
} | [
{
"pp": "I : Type u_1\nJ : Type u_2\nC : Type u_4\ninst✝² : Category.{v_1, u_4} C\nX Y : GradedObject I C\nφ : X ⟶ Y\np : I → J\ninst✝¹ : X.HasMap p\ninst✝ : Y.HasMap p\ni : I\nj : J\nhij : p i = j\n⊢ X.ιMapObj p i j hij ≫ mapMap φ p j = φ i ≫ Y.ιMapObj p i j hij",
"ppTerm": "?m.45",
"assigned": true,
... | [] | simp only [mapMap, ι_descMapObj] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.GradedObject | {
"line": 406,
"column": 2
} | {
"line": 406,
"column": 34
} | {
"line": 408,
"column": 0
} | [
{
"pp": "I : Type u_1\nJ : Type u_2\nC : Type u_4\ninst✝² : Category.{v_1, u_4} C\nX Y : GradedObject I C\nφ : X ⟶ Y\np : I → J\ninst✝¹ : X.HasMap p\ninst✝ : Y.HasMap p\ni : I\nj : J\nhij : p i = j\n⊢ X.ιMapObj p i j hij ≫ mapMap φ p j = φ i ≫ Y.ιMapObj p i j hij",
"ppTerm": "?m.45",
"assigned": true,
... | [] | simp only [mapMap, ι_descMapObj] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.GradedObject | {
"line": 406,
"column": 2
} | {
"line": 406,
"column": 34
} | {
"line": 408,
"column": 0
} | [
{
"pp": "I : Type u_1\nJ : Type u_2\nC : Type u_4\ninst✝² : Category.{v_1, u_4} C\nX Y : GradedObject I C\nφ : X ⟶ Y\np : I → J\ninst✝¹ : X.HasMap p\ninst✝ : Y.HasMap p\ni : I\nj : J\nhij : p i = j\n⊢ X.ιMapObj p i j hij ≫ mapMap φ p j = φ i ≫ Y.ιMapObj p i j hij",
"ppTerm": "?m.45",
"assigned": true,
... | [] | simp only [mapMap, ι_descMapObj] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Monoidal.End | {
"line": 316,
"column": 4
} | {
"line": 317,
"column": 54
} | {
"line": 319,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\nM : Type u_1\ninst✝² : Category.{v_1, u_1} M\ninst✝¹ : MonoidalCategory M\nF : M ⥤ C ⥤ C\nm n : M\nh₁ : m ⊗ n ≅ 𝟙_ M\nh₂ : n ⊗ m ≅ 𝟙_ M\nH : h₁.hom ▷ m ≫ (λ_ m).hom = (α_ m n m).hom ≫ m ◁ h₂.hom ≫ (ρ_ m).hom\ninst✝ : F.Monoidal\nX : C\n⊢ (F.map (λ_ m).inv).app ... | [] | simp only [← NatTrans.comp_app, ← F.map_comp, ← H, inv_hom_whiskerRight_assoc,
Iso.inv_hom_id, Functor.map_id, NatTrans.id_app] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Homology.Single | {
"line": 74,
"column": 2
} | {
"line": 74,
"column": 16
} | {
"line": 75,
"column": 2
} | [
{
"pp": "V : Type u\ninst✝³ : Category.{v, u} V\ninst✝² : HasZeroMorphisms V\ninst✝¹ : HasZeroObject V\nι : Type u_1\ninst✝ : DecidableEq ι\nc : ComplexShape ι\nj : ι\nA : V\ni : ι\nhi : i ≠ j\n⊢ IsZero (if i = j then A else 0)",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr"... | [
"V : Type u\ninst✝³ : Category.{v, u} V\ninst✝² : HasZeroMorphisms V\ninst✝¹ : HasZeroObject V\nι : Type u_1\ninst✝ : DecidableEq ι\nc : ComplexShape ι\nj : ι\nA : V\ni : ι\nhi : i ≠ j\n⊢ IsZero 0"
] | rw [if_neg hi] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Shift.Basic | {
"line": 101,
"column": 6
} | {
"line": 101,
"column": 94
} | {
"line": 102,
"column": 4
} | [
{
"pp": "C : Type u\nA : Type u_1\ninst✝¹ : Category.{v, u} C\ninst✝ : AddMonoid A\nh : ShiftMkCore C A\nm₁ m₂ m₃ : A\nX : C\n⊢ (h.F m₃).map ((h.add m₁ m₂).inv.app X) ≫ (h.add (m₁ + m₂) m₃).inv.app X =\n (h.add m₂ m₃).inv.app ((h.F m₁).obj X) ≫ (h.add m₁ (m₂ + m₃)).inv.app X ≫ eqToHom ⋯",
"ppTerm": "?m.1... | [
"C : Type u\nA : Type u_1\ninst✝¹ : Category.{v, u} C\ninst✝ : AddMonoid A\nh : ShiftMkCore C A\nm₁ m₂ m₃ : A\nX : C\n⊢ ((h.F m₃).map ((h.add m₁ m₂).inv.app X) ≫ (h.add (m₁ + m₂) m₃).inv.app X) ≫\n (h.add (m₁ + m₂) m₃).hom.app X ≫ (h.F m₃).map ((h.add m₁ m₂).hom.app X) =\n ((h.add m₂ m₃).inv.app ((h.F m₁).o... | ← cancel_mono ((h.add (m₁ + m₂) m₃).hom.app X ≫ (h.F m₃).map ((h.add m₁ m₂).hom.app X)), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.Single | {
"line": 132,
"column": 47
} | {
"line": 132,
"column": 67
} | {
"line": 133,
"column": 6
} | [
{
"pp": "V : Type u\ninst✝³ : Category.{v, u} V\ninst✝² : HasZeroMorphisms V\ninst✝¹ : HasZeroObject V\nι : Type u_1\ninst✝ : DecidableEq ι\nc : ComplexShape ι\nj : ι\nA B : V\nf g : A ⟶ B\nw : (single V c j).map f = (single V c j).map g\n⊢ (singleObjXSelf c j A).hom ≫ f ≫ (singleObjXSelf c j B).inv =\n (sin... | [
"V : Type u\ninst✝³ : Category.{v, u} V\ninst✝² : HasZeroMorphisms V\ninst✝¹ : HasZeroObject V\nι : Type u_1\ninst✝ : DecidableEq ι\nc : ComplexShape ι\nj : ι\nA B : V\nf g : A ⟶ B\nw : (single V c j).map f = (single V c j).map g\n⊢ ((single V c j).map f).f j = (singleObjXSelf c j A).hom ≫ g ≫ (singleObjXSelf c j B... | ← single_map_f_self, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.Single | {
"line": 133,
"column": 6
} | {
"line": 133,
"column": 26
} | {
"line": 133,
"column": 27
} | [
{
"pp": "V : Type u\ninst✝³ : Category.{v, u} V\ninst✝² : HasZeroMorphisms V\ninst✝¹ : HasZeroObject V\nι : Type u_1\ninst✝ : DecidableEq ι\nc : ComplexShape ι\nj : ι\nA B : V\nf g : A ⟶ B\nw : (single V c j).map f = (single V c j).map g\n⊢ ((single V c j).map f).f j = (singleObjXSelf c j A).hom ≫ g ≫ (singleOb... | [
"V : Type u\ninst✝³ : Category.{v, u} V\ninst✝² : HasZeroMorphisms V\ninst✝¹ : HasZeroObject V\nι : Type u_1\ninst✝ : DecidableEq ι\nc : ComplexShape ι\nj : ι\nA B : V\nf g : A ⟶ B\nw : (single V c j).map f = (single V c j).map g\n⊢ ((single V c j).map f).f j = ((single V c j).map g).f j"
] | ← single_map_f_self, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Shift.Basic | {
"line": 117,
"column": 64
} | {
"line": 117,
"column": 83
} | {
"line": 118,
"column": 4
} | [
{
"pp": "C : Type u\nA : Type u_1\ninst✝¹ : Category.{v, u} C\ninst✝ : AddMonoid A\nh : ShiftMkCore C A\nn : A\nX : C\n⊢ 𝟙 ((h.F (n + 0)).obj X) = (h.add n 0).hom.app X ≫ h.zero.hom.app ((h.F n).obj X) ≫ eqToHom ⋯",
"ppTerm": "?m.92",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Cate... | [
"C : Type u\nA : Type u_1\ninst✝¹ : Category.{v, u} C\ninst✝ : AddMonoid A\nh : ShiftMkCore C A\nn : A\nX : C\n⊢ 𝟙 ((h.F (n + 0)).obj X) = (eqToHom ⋯ ≫ h.zero.inv.app ((h.F n).obj X)) ≫ h.zero.hom.app ((h.F n).obj X) ≫ eqToHom ⋯"
] | h.add_zero_hom_app, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Shift.Basic | {
"line": 198,
"column": 18
} | {
"line": 198,
"column": 20
} | {
"line": 198,
"column": 20
} | [
{
"pp": "C : Type u\nA : Type u_1\ninst✝² : Category.{v, u} C\ninst✝¹ : AddMonoid A\ninst✝ : HasShift C A\na : A\nha : a = 0\n⊢ shiftFunctor C a = shiftFunctor C 0",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Functor",
"congrArg",
"AddMo... | [
"C : Type u\nA : Type u_1\ninst✝² : Category.{v, u} C\ninst✝¹ : AddMonoid A\ninst✝ : HasShift C A\na : A\nha : a = 0\n⊢ shiftFunctor C 0 = shiftFunctor C 0"
] | ha | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Kaehler.Basic | {
"line": 424,
"column": 14
} | {
"line": 424,
"column": 16
} | {
"line": 425,
"column": 6
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : EssFiniteType R S\nx : S\nI : Ideal (S ⊗[R] S) := Ideal.span ↑(Finset.image (fun s ↦ 1 ⊗ₜ[R] s - s ⊗ₜ[R] 1) (EssFiniteType.finset R S))\na : S\n⊢ a ∈ EssFiniteType.finset R S →\n ((IsScalarTower.toAlgHom ... | [
"R : Type u\nS : Type v\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : EssFiniteType R S\nx : S\nI : Ideal (S ⊗[R] S) := Ideal.span ↑(Finset.image (fun s ↦ 1 ⊗ₜ[R] s - s ⊗ₜ[R] 1) (EssFiniteType.finset R S))\na : S\nha : a ∈ EssFiniteType.finset R S\n⊢ ((IsScalarTower.toAlgHom R (S ⊗[R] S) ... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex | {
"line": 288,
"column": 30
} | {
"line": 289,
"column": 79
} | {
"line": 291,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : HasZeroMorphisms C\nι : Type u_2\nc : ComplexShape ι\nK : HomologicalComplex C c\ni j : ι\ninst✝ : K.HasHomology i\nhij : ¬c.Rel i j\n⊢ K.fromOpcycles i j = 0",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | by
rw [← cancel_epi (K.pOpcycles i), p_fromOpcycles, comp_zero, K.shape _ _ hij] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex | {
"line": 310,
"column": 4
} | {
"line": 310,
"column": 24
} | {
"line": 312,
"column": 0
} | [
{
"pp": "case neg\nC : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : HasZeroMorphisms C\nι : Type u_2\nc : ComplexShape ι\nK : HomologicalComplex C c\ni : ι\ninst✝ : K.HasHomology i\nA : C\nk : K.X i ⟶ A\nj : ι\nhj : c.prev i = j\ni' : ι\nx : K.X i' ⟶ A\nhx : k = K.d i i' ≫ x\nh : ¬c.Rel i i'\nthis : K.des... | [] | rw [this, comp_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex | {
"line": 458,
"column": 60
} | {
"line": 459,
"column": 95
} | {
"line": 461,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : HasZeroMorphisms C\nι : Type u_2\nc : ComplexShape ι\nK L : HomologicalComplex C c\ni : ι\ninst✝¹ : K.HasHomology i\ninst✝ : L.HasHomology i\nA : C\nk : A ⟶ K.X i\nj : ι\nhj : c.next i = j\nhk : k ≫ K.d i j = 0\nφ : K ⟶ L\n⊢ K.liftCycles k j hj hk ... | [] | by
simp only [← cancel_mono (L.iCycles i), assoc, cyclesMap_i, liftCycles_i_assoc, liftCycles_i] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Kaehler.Basic | {
"line": 677,
"column": 2
} | {
"line": 688,
"column": 8
} | {
"line": 690,
"column": 0
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝¹¹ : CommRing R\ninst✝¹⁰ : CommRing S\ninst✝⁹ : Algebra R S\nA : Type u_2\nB : Type u_3\ninst✝⁸ : CommRing A\ninst✝⁷ : CommRing B\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra A B\ninst✝⁴ : Algebra S B\ninst✝³ : Algebra R B\ninst✝² : IsScalarTower R A B\ninst✝¹ : IsScalarTower R ... | [] | rw [← Submodule.map_comap_eq_of_surjective (linearCombination_surjective R A) (LinearMap.ker _)]
congr 1
ext x
simp only [Submodule.mem_comap, LinearMap.mem_ker, Finsupp.apply_linearCombination, ← kerTotal_eq,
Submodule.restrictScalars_mem]
simp only [linearCombination_apply, Function.comp_apply, LinearMap.... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Kaehler.Basic | {
"line": 677,
"column": 2
} | {
"line": 688,
"column": 8
} | {
"line": 690,
"column": 0
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝¹¹ : CommRing R\ninst✝¹⁰ : CommRing S\ninst✝⁹ : Algebra R S\nA : Type u_2\nB : Type u_3\ninst✝⁸ : CommRing A\ninst✝⁷ : CommRing B\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra A B\ninst✝⁴ : Algebra S B\ninst✝³ : Algebra R B\ninst✝² : IsScalarTower R A B\ninst✝¹ : IsScalarTower R ... | [] | rw [← Submodule.map_comap_eq_of_surjective (linearCombination_surjective R A) (LinearMap.ker _)]
congr 1
ext x
simp only [Submodule.mem_comap, LinearMap.mem_ker, Finsupp.apply_linearCombination, ← kerTotal_eq,
Submodule.restrictScalars_mem]
simp only [linearCombination_apply, Function.comp_apply, LinearMap.... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Kaehler.Basic | {
"line": 808,
"column": 8
} | {
"line": 808,
"column": 57
} | {
"line": 808,
"column": 57
} | [
{
"pp": "case mpr\nR : Type u\ninst✝⁶ : CommRing R\nA : Type u_2\nB : Type u_3\ninst✝⁵ : CommRing A\ninst✝⁴ : CommRing B\ninst✝³ : Algebra R A\ninst✝² : Algebra A B\ninst✝¹ : Algebra R B\ninst✝ : IsScalarTower R A B\nh : Function.Surjective ⇑(algebraMap A B)\nx : Ω[A⁄R]\nhx : x ∈ (map R R A B).ker\n⊢ (LinearMap... | [
"case mpr\nR : Type u\ninst✝⁶ : CommRing R\nA : Type u_2\nB : Type u_3\ninst✝⁵ : CommRing A\ninst✝⁴ : CommRing B\ninst✝³ : Algebra R A\ninst✝² : Algebra A B\ninst✝¹ : Algebra R B\ninst✝ : IsScalarTower R A B\nh : Function.Surjective ⇑(algebraMap A B)\nx : Ω[A⁄R]\nhx :\n x ∈\n Submodule.map (linearCombination A ... | KaehlerDifferential.ker_map_of_surjective R A B h | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.Homotopy | {
"line": 76,
"column": 21
} | {
"line": 78,
"column": 32
} | {
"line": 80,
"column": 0
} | [
{
"pp": "ι : Type u_1\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC D : HomologicalComplex V c\nf : (i j : ι) → C.X i ⟶ D.X j\ni : ι\nhi : ¬c.Rel (c.prev i) i\n⊢ (prevD i) f = 0",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"C... | [] | by
dsimp [prevD]
rw [shape _ _ _ hi, comp_zero] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.HomotopyCategory.HomComplex | {
"line": 145,
"column": 2
} | {
"line": 145,
"column": 11
} | {
"line": 146,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nF G : CochainComplex C ℤ\nψ : (p : ℤ) → F.X p ⟶ G.X p\np q q' : ℤ\nhpq✝ : p + 0 = q\nhpq : p = q\n⊢ (ofHoms ψ).v p q hpq✝ ≫ G.d q q' = ψ p ≫ G.d p q'",
"ppTerm": "?m.77",
"assigned": true,
"usedConstants": [
"CategoryTheor... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nF G : CochainComplex C ℤ\nψ : (p : ℤ) → F.X p ⟶ G.X p\np q' : ℤ\nhpq : p + 0 = p\n⊢ (ofHoms ψ).v p p hpq ≫ G.d p q' = ψ p ≫ G.d p q'"
] | subst hpq | Lean.Elab.Tactic.evalSubst | Lean.Parser.Tactic.subst |
Mathlib.Algebra.Homology.HomotopyCategory.HomComplex | {
"line": 152,
"column": 2
} | {
"line": 152,
"column": 11
} | {
"line": 153,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nF G : CochainComplex C ℤ\nψ : (p : ℤ) → F.X p ⟶ G.X p\np' p q : ℤ\nhpq✝ : p + 0 = q\nhpq : p = q\n⊢ F.d p' p ≫ (ofHoms ψ).v p q hpq✝ = F.d p' q ≫ ψ q",
"ppTerm": "?m.77",
"assigned": true,
"usedConstants": [
"CategoryTheor... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nF G : CochainComplex C ℤ\nψ : (p : ℤ) → F.X p ⟶ G.X p\np' p : ℤ\nhpq : p + 0 = p\n⊢ F.d p' p ≫ (ofHoms ψ).v p p hpq = F.d p' p ≫ ψ p"
] | subst hpq | Lean.Elab.Tactic.evalSubst | Lean.Parser.Tactic.subst |
Mathlib.Algebra.Homology.Homotopy | {
"line": 263,
"column": 2
} | {
"line": 263,
"column": 68
} | {
"line": 264,
"column": 2
} | [
{
"pp": "ι : Type u_1\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nhom : (i j : ι) → C.X i ⟶ D.X j\ng : D ⟶ E\nn : ι\n⊢ (nullHomotopicMap hom ≫ g).f n = (nullHomotopicMap fun i j ↦ hom i j ≫ g.f j).f n",
"ppTerm": "?m.81",
"assigned"... | [
"ι : Type u_1\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nhom : (i j : ι) → C.X i ⟶ D.X j\ng : D ⟶ E\nn : ι\n⊢ (C.dFrom n ≫ hom (c.next n) n + hom n (c.prev n) ≫ D.dTo n) ≫ g.f n =\n C.dFrom n ≫ hom (c.next n) n ≫ g.f n + (hom n (c.prev n) ≫... | dsimp [nullHomotopicMap, fromNext, toPrev, AddMonoidHom.mk'_apply] | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.Algebra.Homology.Homotopy | {
"line": 283,
"column": 2
} | {
"line": 283,
"column": 68
} | {
"line": 284,
"column": 2
} | [
{
"pp": "ι : Type u_1\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nf : C ⟶ D\nhom : (i j : ι) → D.X i ⟶ E.X j\nn : ι\n⊢ (f ≫ nullHomotopicMap hom).f n = (nullHomotopicMap fun i j ↦ f.f i ≫ hom i j).f n",
"ppTerm": "?m.81",
"assigned"... | [
"ι : Type u_1\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nf : C ⟶ D\nhom : (i j : ι) → D.X i ⟶ E.X j\nn : ι\n⊢ f.f n ≫ (D.dFrom n ≫ hom (c.next n) n + hom n (c.prev n) ≫ E.dTo n) =\n C.dFrom n ≫ f.f (c.next n) ≫ hom (c.next n) n + (f.f n ≫ h... | dsimp [nullHomotopicMap, fromNext, toPrev, AddMonoidHom.mk'_apply] | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.Algebra.Homology.HomotopyCategory.HomComplex | {
"line": 340,
"column": 64
} | {
"line": 342,
"column": 64
} | {
"line": 344,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nF G K : CochainComplex C ℤ\nn₁ n₂ n₁₂ : ℤ\nz₁ : Cochain F G n₁\nh : n₁ + n₂ = n₁₂\n⊢ z₁.comp 0 h = 0",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
... | [] | by
ext p q hpq
simp only [comp_v _ _ h p _ q rfl (by lia), zero_v, comp_zero] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.HomotopyCategory.MappingCone | {
"line": 118,
"column": 71
} | {
"line": 120,
"column": 71
} | {
"line": 122,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v, u_1} C\ninst✝¹ : Preadditive C\nF G : CochainComplex C ℤ\nφ : F ⟶ G\ninst✝ : HasHomotopyCofiber φ\n⊢ (inl φ).comp ↑(fst φ) ⋯ = Cochain.ofHom (𝟙 F)",
"ppTerm": "?m.55",
"assigned": true,
"usedConstants": [
"_private.Mathlib.Algebra.Homology.Homotopy... | [] | by
ext p
simp [Cochain.comp_v _ _ (neg_add_cancel 1) p (p - 1) p rfl (by lia)] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.Homotopy | {
"line": 449,
"column": 2
} | {
"line": 449,
"column": 25
} | {
"line": 450,
"column": 2
} | [
{
"pp": "V : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : Preadditive V\nP Q : ChainComplex V ℕ\nf : (i j : ℕ) → P.X i ⟶ Q.X j\n⊢ P.d 0 ((ComplexShape.down ℕ).next 0) ≫ f ((ComplexShape.down ℕ).next 0) 0 = 0",
"ppTerm": "?m.55",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instOne"... | [
"case a\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : Preadditive V\nP Q : ChainComplex V ℕ\nf : (i j : ℕ) → P.X i ⟶ Q.X j\n⊢ ¬(ComplexShape.down ℕ).Rel 0 ((ComplexShape.down ℕ).next 0)"
] | rw [P.shape, zero_comp] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Homology.Homotopy | {
"line": 543,
"column": 8
} | {
"line": 543,
"column": 63
} | {
"line": 544,
"column": 8
} | [
{
"pp": "case e_a.succ\nι : Type u_1\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nf g : C ⟶ D\nh k : D ⟶ E\ni : ι\nP Q : ChainComplex V ℕ\ne : P ⟶ Q\nzero : P.X 0 ⟶ Q.X 1\ncomm_zero : e.f 0 = zero ≫ Q.d 1 0\none : P.X 1 ⟶ Q.X 2\ncomm_one : e... | [
"case e_a.succ\nι : Type u_1\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nf g : C ⟶ D\nh k : D ⟶ E\ni : ι\nP Q : ChainComplex V ℕ\ne : P ⟶ Q\nzero : P.X 0 ⟶ Q.X 1\ncomm_zero : e.f 0 = zero ≫ Q.d 1 0\none : P.X 1 ⟶ Q.X 2\ncomm_one : e.f 1 = P.d 1... | rw [mkInductiveAux₃ e zero comm_zero one comm_one succ] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Homology.Homotopy | {
"line": 792,
"column": 4
} | {
"line": 795,
"column": 41
} | {
"line": 796,
"column": 2
} | [
{
"pp": "ι : Type u_1\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nf✝ g : C ⟶ D\nh k : D ⟶ E\ni : ι\nX✝ Y✝ Z✝ : HomologicalComplex V c\nf : X✝ ⟶ Y✝\nx✝ : Y✝ ⟶ Z✝\n⊢ homotopyEquivalences V c x✝ → homotopyEquivalences V c (f ≫ x✝) → homotopyEq... | [] | rintro ⟨g, rfl⟩ ⟨e, he⟩
refine (e.trans g.symm).homotopyEquivalences_hom.of_homotopy ?_
simp only [HomotopyEquiv.trans_hom, HomotopyEquiv.symm_hom, he, Category.assoc]
exact g.homotopyHomInvId.compLeftId f | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.Homotopy | {
"line": 792,
"column": 4
} | {
"line": 795,
"column": 41
} | {
"line": 796,
"column": 2
} | [
{
"pp": "ι : Type u_1\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nf✝ g : C ⟶ D\nh k : D ⟶ E\ni : ι\nX✝ Y✝ Z✝ : HomologicalComplex V c\nf : X✝ ⟶ Y✝\nx✝ : Y✝ ⟶ Z✝\n⊢ homotopyEquivalences V c x✝ → homotopyEquivalences V c (f ≫ x✝) → homotopyEq... | [] | rintro ⟨g, rfl⟩ ⟨e, he⟩
refine (e.trans g.symm).homotopyEquivalences_hom.of_homotopy ?_
simp only [HomotopyEquiv.trans_hom, HomotopyEquiv.symm_hom, he, Category.assoc]
exact g.homotopyHomInvId.compLeftId f | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Quotient | {
"line": 87,
"column": 2
} | {
"line": 89,
"column": 50
} | {
"line": 91,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\nr : HomRel C\ninst✝¹ : IsStableUnderPrecomp r\ninst✝ : IsStableUnderPostcomp r\nX Y : C\nf g : X ⟶ Y\n⊢ CompClosure r f g ↔ r f g",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Qu... | [] | refine ⟨?_, CompClosure.of⟩
rintro ⟨_, _, _, _, _, _, h⟩
exact HomRel.comp_left _ (HomRel.comp_right _ h) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Quotient | {
"line": 87,
"column": 2
} | {
"line": 89,
"column": 50
} | {
"line": 91,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\nr : HomRel C\ninst✝¹ : IsStableUnderPrecomp r\ninst✝ : IsStableUnderPostcomp r\nX Y : C\nf g : X ⟶ Y\n⊢ CompClosure r f g ↔ r f g",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Qu... | [] | refine ⟨?_, CompClosure.of⟩
rintro ⟨_, _, _, _, _, _, h⟩
exact HomRel.comp_left _ (HomRel.comp_right _ h) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.Homotopy | {
"line": 885,
"column": 11
} | {
"line": 891,
"column": 10
} | {
"line": 893,
"column": 0
} | [
{
"pp": "ι✝ : Type u_1\nV : Type u\ninst✝⁴ : Category.{v, u} V\ninst✝³ : Preadditive V\nc✝ : ComplexShape ι✝\nC✝ D E : HomologicalComplex V c✝\nf✝ g✝ : C✝ ⟶ D\nh k : D ⟶ E\ni✝ : ι✝\nC : Type u_2\ninst✝² : Category.{v_1, u_2} C\ninst✝¹ : Preadditive C\nι : Type ?u.42\nc : ComplexShape ι\ninst✝ : DecidableRel c.R... | [] | by
dsimp
split_ifs with h
· rw [ho.comm (c.next i)]
dsimp [dFrom, dTo, fromNext, toPrev]
rw [congr_arg (fun j => ho.hom (c.next i) j ≫ L.d j (c.next i)) (c.prev_eq' h)]
· abel | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.HomotopyCategory | {
"line": 131,
"column": 45
} | {
"line": 133,
"column": 6
} | {
"line": 135,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝² : Semiring R\nι : Type u_2\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC D : HomologicalComplex V c\nf : C ⟶ D\n⊢ Homotopy (Quot.out ((quotient V c).map f)) f",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Homolo... | [] | by
apply homotopyOfEq
simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Shift.Quotient | {
"line": 138,
"column": 8
} | {
"line": 138,
"column": 98
} | {
"line": 138,
"column": 98
} | [
{
"pp": "case e_a\nC : Type u\ninst✝⁶ : Category.{v, u} C\nD : Type u'\ninst✝⁵ : Category.{v', u'} D\nF : C ⥤ D\nr : HomRel C\nA : Type w\ninst✝⁴ : AddMonoid A\ninst✝³ : HasShift C A\ninst✝² : HasShift D A\ninst✝¹ : r.IsCompatibleWithShift A\ninst✝ : F.CommShift A\nhF : ∀ (x y : C) (f₁ f₂ : x ⟶ y), r f₁ f₂ → F.... | [
"case e_a\nC : Type u\ninst✝⁶ : Category.{v, u} C\nD : Type u'\ninst✝⁵ : Category.{v', u'} D\nF : C ⥤ D\nr : HomRel C\nA : Type w\ninst✝⁴ : AddMonoid A\ninst✝³ : HasShift C A\ninst✝² : HasShift D A\ninst✝¹ : r.IsCompatibleWithShift A\ninst✝ : F.CommShift A\nhF : ∀ (x y : C) (f₁ f₂ : x ⟶ y), r f₁ f₂ → F.map f₁ = F.m... | (LiftCommShift.iso F r hF b).hom.naturality_assoc (((functor r).commShiftIso a).hom.app X) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.HomotopyCategory.Shift | {
"line": 107,
"column": 6
} | {
"line": 110,
"column": 84
} | {
"line": 110,
"column": 84
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nD : Type u'\ninst✝¹ : Category.{v', u'} D\ninst✝ : Preadditive D\nn₁ n₂ n₁₂ : ℤ\nh : n₁ + n₂ = n₁₂\nK : CochainComplex C ℤ\nx✝² x✝¹ : ℤ\nx✝ : (ComplexShape.up ℤ).Rel x✝² x✝¹\n⊢ (K.shiftFunctorObjXIso n₁₂ x✝² (x✝² + n₂ + n₁) ⋯).hom ≫ ((shif... | [] | subst h
dsimp
simp only [add_comm n₁ n₂, Int.negOnePow_add, Linear.units_smul_comp,
Linear.comp_units_smul, d_comp_XIsoOfEq_hom, smul_smul, XIsoOfEq_hom_comp_d] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.HomotopyCategory.Shift | {
"line": 107,
"column": 6
} | {
"line": 110,
"column": 84
} | {
"line": 110,
"column": 84
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nD : Type u'\ninst✝¹ : Category.{v', u'} D\ninst✝ : Preadditive D\nn₁ n₂ n₁₂ : ℤ\nh : n₁ + n₂ = n₁₂\nK : CochainComplex C ℤ\nx✝² x✝¹ : ℤ\nx✝ : (ComplexShape.up ℤ).Rel x✝² x✝¹\n⊢ (K.shiftFunctorObjXIso n₁₂ x✝² (x✝² + n₂ + n₁) ⋯).hom ≫ ((shif... | [] | subst h
dsimp
simp only [add_comm n₁ n₂, Int.negOnePow_add, Linear.units_smul_comp,
Linear.comp_units_smul, d_comp_XIsoOfEq_hom, smul_smul, XIsoOfEq_hom_comp_d] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.HomotopyCategory.Shift | {
"line": 251,
"column": 40
} | {
"line": 251,
"column": 61
} | {
"line": 251,
"column": 61
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : Preadditive C\nD : Type u'\ninst✝² : Category.{v', u'} D\ninst✝¹ : Preadditive D\nF : C ⥤ D\ninst✝ : F.Additive\nn : ℤ\nX✝ Y✝ : HomologicalComplex C (ComplexShape.up ℤ)\nx✝ : X✝ ⟶ Y✝\ni✝ : ℤ\n⊢ F.map (x✝.f (i✝ + n)) ≫ 𝟙 (F.obj (Y✝.X (i✝ + n))) = 𝟙 (F.o... | [] | rw [id_comp, comp_id] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Homology.HomotopyCategory.HomComplexShift | {
"line": 506,
"column": 2
} | {
"line": 506,
"column": 23
} | {
"line": 508,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nK L : CochainComplex C ℤ\nn : ℤ\nγ : Cochain K L n\na n' : ℤ\nhn' : n' + a = n\np q : ℤ\nhpq : p + n = q\n⊢ (a * n + a * (a - 1) / 2).negOnePow • 𝟙 (K.X (p + a)) ≫ γ.v (p + a) (q + a) ⋯ ≫ 𝟙 (L.X (q + a)) =\n (a * n + a * (a - 1) / 2).n... | [] | rw [id_comp, comp_id] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Homology.HomotopyCategory.HomComplexShift | {
"line": 517,
"column": 2
} | {
"line": 517,
"column": 23
} | {
"line": 519,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nK L : CochainComplex C ℤ\nn : ℤ\nγ : Cochain K L n\na n' : ℤ\nhn' : n + a = n'\np q : ℤ\nhpq : p + n = q\n⊢ ((a * n' + a * (a - 1) / 2).negOnePow • 𝟙 (K.X (p + a)) ≫ γ.v (p + a) (q + a) ⋯) ≫ 𝟙 (L.X (q + a)) =\n (a * n' + a * (a - 1) / ... | [] | rw [id_comp, comp_id] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Homology.HomotopyCategory.Pretriangulated | {
"line": 170,
"column": 39
} | {
"line": 170,
"column": 60
} | {
"line": 170,
"column": 60
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : Category.{v_2, u_2} D\ninst✝³ : Preadditive C\ninst✝² : HasBinaryBiproducts C\ninst✝¹ : Preadditive D\ninst✝ : HasBinaryBiproducts D\nK L : CochainComplex C ℤ\nφ : K ⟶ L\nK₁ L₁ K₂ L₂ K₃ L₃ : CochainComplex C ℤ\nφ₁ : K₁ ⟶ L₁\nφ₂ : K₂ ⟶... | [] | rw [id_comp, comp_id] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Homology.HomotopyCategory.Pretriangulated | {
"line": 170,
"column": 39
} | {
"line": 170,
"column": 60
} | {
"line": 170,
"column": 60
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : Category.{v_2, u_2} D\ninst✝³ : Preadditive C\ninst✝² : HasBinaryBiproducts C\ninst✝¹ : Preadditive D\ninst✝ : HasBinaryBiproducts D\nK L : CochainComplex C ℤ\nφ : K ⟶ L\nK₁ L₁ K₂ L₂ K₃ L₃ : CochainComplex C ℤ\nφ₁ : K₁ ⟶ L₁\nφ₂ : K₂ ⟶... | [] | rw [id_comp, comp_id] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.HomotopyCategory.Pretriangulated | {
"line": 170,
"column": 39
} | {
"line": 170,
"column": 60
} | {
"line": 170,
"column": 60
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : Category.{v_2, u_2} D\ninst✝³ : Preadditive C\ninst✝² : HasBinaryBiproducts C\ninst✝¹ : Preadditive D\ninst✝ : HasBinaryBiproducts D\nK L : CochainComplex C ℤ\nφ : K ⟶ L\nK₁ L₁ K₂ L₂ K₃ L₃ : CochainComplex C ℤ\nφ₁ : K₁ ⟶ L₁\nφ₂ : K₂ ⟶... | [] | rw [id_comp, comp_id] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Triangulated.Pretriangulated | {
"line": 654,
"column": 18
} | {
"line": 654,
"column": 20
} | {
"line": 655,
"column": 6
} | [
{
"pp": "C : Type u\ninst✝⁸ : Category.{v, u} C\ninst✝⁷ : HasZeroObject C\ninst✝⁶ : HasShift C ℤ\ninst✝⁵ : Preadditive C\ninst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive\nhC : Pretriangulated C\nJ : Type u_1\nT : J → Triangle C\nhT : ∀ (j : J), T j ∈ distinguishedTriangles\ninst✝³ : HasProduct fun j ↦ (T j).obj... | [
"C : Type u\ninst✝⁸ : Category.{v, u} C\ninst✝⁷ : HasZeroObject C\ninst✝⁶ : HasShift C ℤ\ninst✝⁵ : Preadditive C\ninst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive\nhC : Pretriangulated C\nJ : Type u_1\nT : J → Triangle C\nhT : ∀ (j : J), T j ∈ distinguishedTriangles\ninst✝³ : HasProduct fun j ↦ (T j).obj₁\ninst✝² : ... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.CategoryTheory.Triangulated.Functor | {
"line": 50,
"column": 25
} | {
"line": 50,
"column": 58
} | {
"line": 51,
"column": 6
} | [
{
"pp": "C : Type u_1\nD : Type u_2\nE : Type u_3\ninst✝⁷ : Category.{v_1, u_1} C\ninst✝⁶ : Category.{v_2, u_2} D\ninst✝⁵ : Category.{v_3, u_3} E\ninst✝⁴ : HasShift C ℤ\ninst✝³ : HasShift D ℤ\ninst✝² : HasShift E ℤ\nF : C ⥤ D\ninst✝¹ : F.CommShift ℤ\nG : D ⥤ E\ninst✝ : G.CommShift ℤ\nX✝ Y✝ : Triangle C\nf : X✝ ... | [] | simp only [← F.map_comp, f.comm₂] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Homology.HomotopyCategory.DegreewiseSplit | {
"line": 100,
"column": 4
} | {
"line": 100,
"column": 13
} | {
"line": 101,
"column": 4
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v, u_1} C\ninst✝¹ : Preadditive C\nS : ShortComplex (CochainComplex C ℤ)\nσ : (n : ℤ) → (S.map (eval C (ComplexShape.up ℤ) n)).Splitting\ninst✝ : HasBinaryBiproducts C\np q : ℤ\nhpq : p + 1 = q\n⊢ ((↑(mappingCone.fst (homOfDegreewiseSplit S σ))).v p q hpq ≫ (σ q).s -\n ... | [
"C : Type u_1\ninst✝² : Category.{v, u_1} C\ninst✝¹ : Preadditive C\nS : ShortComplex (CochainComplex C ℤ)\nσ : (n : ℤ) → (S.map (eval C (ComplexShape.up ℤ) n)).Splitting\ninst✝ : HasBinaryBiproducts C\np : ℤ\n⊢ ((↑(mappingCone.fst (homOfDegreewiseSplit S σ))).v p (p + 1) ⋯ ≫ (σ (p + 1)).s -\n (mappingCone.s... | subst hpq | Lean.Elab.Tactic.evalSubst | Lean.Parser.Tactic.subst |
Mathlib.Algebra.Homology.HomotopyCategory.DegreewiseSplit | {
"line": 116,
"column": 4
} | {
"line": 116,
"column": 13
} | {
"line": 117,
"column": 4
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v, u_1} C\ninst✝¹ : Preadditive C\nS : ShortComplex (CochainComplex C ℤ)\nσ : (n : ℤ) → (S.map (eval C (ComplexShape.up ℤ) n)).Splitting\ninst✝ : HasBinaryBiproducts C\np q : ℤ\nhpq : p + 1 = q\n⊢ (S.g.f q ≫ (mappingCone.inl (homOfDegreewiseSplit S σ)).v q p ⋯ -\n ... | [
"C : Type u_1\ninst✝² : Category.{v, u_1} C\ninst✝¹ : Preadditive C\nS : ShortComplex (CochainComplex C ℤ)\nσ : (n : ℤ) → (S.map (eval C (ComplexShape.up ℤ) n)).Splitting\ninst✝ : HasBinaryBiproducts C\np : ℤ\n⊢ (S.g.f (p + 1) ≫ (mappingCone.inl (homOfDegreewiseSplit S σ)).v (p + 1) p ⋯ -\n (σ (p + 1)).r ≫ (... | subst hpq | Lean.Elab.Tactic.evalSubst | Lean.Parser.Tactic.subst |
Mathlib.CategoryTheory.PathCategory.Basic | {
"line": 121,
"column": 18
} | {
"line": 130,
"column": 29
} | {
"line": 132,
"column": 0
} | [
{
"pp": "V : Type u₁\ninst✝¹ : Quiver V\nC : Type ?u.6\ninst✝ : Category.{v_1, ?u.6} C\nφ : V ⥤q C\nX✝ Y✝ Z✝ : Paths V\nf : X✝ ⟶ Y✝\ng : Y✝ ⟶ Z✝\n⊢ Quiver.Path.rec (𝟙 (φ.obj X✝)) (fun {b c} x f ihp ↦ ihp ≫ φ.map f) (f ≫ g) =\n Quiver.Path.rec (𝟙 (φ.obj X✝)) (fun {b c} x f ihp ↦ ihp ≫ φ.map f) f ≫\n Qu... | [] | by
induction g with
| nil =>
rw [Category.comp_id]
rfl
| cons g' p ih =>
have : f ≫ Quiver.Path.cons g' p = (f ≫ g').cons p := by apply Quiver.Path.comp_cons
rw [this]
simp only at ih ⊢
rw [ih, Category.assoc] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.PathCategory.Basic | {
"line": 236,
"column": 62
} | {
"line": 239,
"column": 28
} | {
"line": 241,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nX Y Z : C\nf : Path X Y\ng : Path Y Z\n⊢ composePath (f.comp g) = composePath f ≫ composePath g",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"CategoryTheory.composePath",
"CategoryTheory.Category.assoc",
"CategoryTheor... | [] | by
induction g with
| nil => simp
| cons g e ih => simp [ih] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.PathCategory.Basic | {
"line": 301,
"column": 28
} | {
"line": 301,
"column": 78
} | {
"line": 301,
"column": 78
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nX✝ Y✝ : Quotient (pathsHomRel C)\nf : X✝.as ⟶ Y✝.as\n⊢ (𝟭 (Quotient (pathsHomRel C))).map (Quot.mk (HomRel.CompClosure (pathsHomRel C)) f) ≫\n (Quotient.casesOn (motive := fun t ↦\n Y✝ = t → ((𝟭 (Quotient (pathsHomRel C))).obj Y✝ ≅ (quotientPath... | [] | exact Quot.sound (HomRel.CompClosure.of (by simp)) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.PathCategory.Basic | {
"line": 301,
"column": 28
} | {
"line": 301,
"column": 78
} | {
"line": 301,
"column": 78
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nX✝ Y✝ : Quotient (pathsHomRel C)\nf : X✝.as ⟶ Y✝.as\n⊢ (𝟭 (Quotient (pathsHomRel C))).map (Quot.mk (HomRel.CompClosure (pathsHomRel C)) f) ≫\n (Quotient.casesOn (motive := fun t ↦\n Y✝ = t → ((𝟭 (Quotient (pathsHomRel C))).obj Y✝ ≅ (quotientPath... | [] | exact Quot.sound (HomRel.CompClosure.of (by simp)) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.PathCategory.Basic | {
"line": 301,
"column": 28
} | {
"line": 301,
"column": 78
} | {
"line": 301,
"column": 78
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nX✝ Y✝ : Quotient (pathsHomRel C)\nf : X✝.as ⟶ Y✝.as\n⊢ (𝟭 (Quotient (pathsHomRel C))).map (Quot.mk (HomRel.CompClosure (pathsHomRel C)) f) ≫\n (Quotient.casesOn (motive := fun t ↦\n Y✝ = t → ((𝟭 (Quotient (pathsHomRel C))).obj Y✝ ≅ (quotientPath... | [] | exact Quot.sound (HomRel.CompClosure.of (by simp)) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.HomologySequence | {
"line": 258,
"column": 4
} | {
"line": 259,
"column": 43
} | {
"line": 260,
"column": 2
} | [
{
"pp": "C : Type u_1\nι : Type u_2\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Abelian C\nc : ComplexShape ι\nS : ShortComplex (HomologicalComplex C c)\nhS✝ : S.ShortExact\ni✝ j✝ : ι\nhij✝ : c.Rel i✝ j✝\nhS : S.ShortExact\ni j : ι\nhij : c.Rel i j\n⊢ ((opcyclesFunctor C c i).mapShortComplex.obj S).Exact",
"pp... | [] | have := hS.epi_g
exact opcycles_right_exact S hS.exact i | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.HomologySequence | {
"line": 258,
"column": 4
} | {
"line": 259,
"column": 43
} | {
"line": 260,
"column": 2
} | [
{
"pp": "C : Type u_1\nι : Type u_2\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Abelian C\nc : ComplexShape ι\nS : ShortComplex (HomologicalComplex C c)\nhS✝ : S.ShortExact\ni✝ j✝ : ι\nhij✝ : c.Rel i✝ j✝\nhS : S.ShortExact\ni j : ι\nhij : c.Rel i j\n⊢ ((opcyclesFunctor C c i).mapShortComplex.obj S).Exact",
"pp... | [] | have := hS.epi_g
exact opcycles_right_exact S hS.exact i | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Category.Quiv | {
"line": 275,
"column": 26
} | {
"line": 275,
"column": 40
} | {
"line": 275,
"column": 41
} | [
{
"pp": "V : Type u\nC : Type u₁\ninst✝¹ : Quiver V\ninst✝ : Category.{v₁, u₁} C\nF : Paths V ⥤ C\n⊢ (Cat.freeMap (Paths.of V) ⋙ Cat.freeMap F.toPrefunctor) ⋙ pathComposition C = F",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Functor",
"Catego... | [
"V : Type u\nC : Type u₁\ninst✝¹ : Quiver V\ninst✝ : Category.{v₁, u₁} C\nF : Paths V ⥤ C\n⊢ Cat.freeMap (Paths.of V) ⋙ Cat.freeMap F.toPrefunctor ⋙ pathComposition C = F"
] | Functor.assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Localization.Construction | {
"line": 176,
"column": 4
} | {
"line": 177,
"column": 31
} | {
"line": 178,
"column": 4
} | [
{
"pp": "case refine_1\nC : Type uC\ninst✝¹ : Category.{uC', uC} C\nW : MorphismProperty C\nD : Type uD\ninst✝ : Category.{uD', uD} D\nG₁ G₂ : W.Localization ⥤ D\nh : W.Q ⋙ G₁ = W.Q ⋙ G₂\nh' : Quotient.functor (relations W) ⋙ G₁ = Quotient.functor (relations W) ⋙ G₂\n⊢ ∀ (X : W.Localization), G₁.obj X = G₂.obj ... | [
"case refine_2\nC : Type uC\ninst✝¹ : Category.{uC', uC} C\nW : MorphismProperty C\nD : Type uD\ninst✝ : Category.{uD', uD} D\nG₁ G₂ : W.Localization ⥤ D\nh : W.Q ⋙ G₁ = W.Q ⋙ G₂\nh' : Quotient.functor (relations W) ⋙ G₁ = Quotient.functor (relations W) ⋙ G₂\n⊢ ∀ (X Y : W.Localization) (f : X ⟶ Y), G₁.map f = eqToH... | · rintro ⟨⟨X⟩⟩
apply Functor.congr_obj h | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.CatCommSq | {
"line": 164,
"column": 6
} | {
"line": 164,
"column": 74
} | {
"line": 164,
"column": 74
} | [
{
"pp": "C₁ : Type u_1\nC₂ : Type u_2\nC₃ : Type u_3\nC₄ : Type u_4\ninst✝³ : Category.{v_1, u_1} C₁\ninst✝² : Category.{v_2, u_2} C₂\ninst✝¹ : Category.{v_3, u_3} C₃\ninst✝ : Category.{v_4, u_4} C₄\nT : C₁ ⥤ C₂\nL : C₁ ≌ C₃\nR : C₂ ≌ C₄\nB : C₃ ⥤ C₄\nh : CatCommSq T L.functor R.functor B\nX : C₁\n⊢ R.functor.m... | [
"C₁ : Type u_1\nC₂ : Type u_2\nC₃ : Type u_3\nC₄ : Type u_4\ninst✝³ : Category.{v_1, u_1} C₁\ninst✝² : Category.{v_2, u_2} C₂\ninst✝¹ : Category.{v_3, u_3} C₃\ninst✝ : Category.{v_4, u_4} C₄\nT : C₁ ⥤ C₂\nL : C₁ ≌ C₃\nR : C₂ ≌ C₄\nB : C₃ ⥤ C₄\nh : CatCommSq T L.functor R.functor B\nX : C₁\n⊢ (R.functor.map (T.map (... | ← cancel_mono (B.map (L.functor.map (NatTrans.app L.unitIso.hom X))) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Localization.Construction | {
"line": 314,
"column": 39
} | {
"line": 314,
"column": 50
} | {
"line": 314,
"column": 50
} | [
{
"pp": "C : Type uC\ninst✝¹ : Category.{uC', uC} C\nW : MorphismProperty C\nD : Type uD\ninst✝ : Category.{uD', uD} D\nG : C ⥤ D\nhG : W.IsInvertedBy G\nX✝ Y✝ : W.FunctorsInverting D\nτ : X✝ ⟶ Y✝\n⊢ W.Q ⋙ lift X✝.obj ⋯ = X✝.obj",
"ppTerm": "?m.97",
"assigned": true,
"usedConstants": [
"Eq.mpr... | [] | by rw [fac] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Localization.Construction | {
"line": 314,
"column": 71
} | {
"line": 314,
"column": 82
} | {
"line": 314,
"column": 82
} | [
{
"pp": "C : Type uC\ninst✝¹ : Category.{uC', uC} C\nW : MorphismProperty C\nD : Type uD\ninst✝ : Category.{uD', uD} D\nG : C ⥤ D\nhG : W.IsInvertedBy G\nX✝ Y✝ : W.FunctorsInverting D\nτ : X✝ ⟶ Y✝\n⊢ Y✝.obj = W.Q ⋙ lift Y✝.obj ⋯",
"ppTerm": "?m.106",
"assigned": true,
"usedConstants": [
"Eq.mp... | [] | by rw [fac] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Localization.Predicate | {
"line": 444,
"column": 2
} | {
"line": 444,
"column": 15
} | {
"line": 446,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\nD₁ : Type u_4\nD₂ : Type u_5\ninst✝³ : Category.{v_4, u_4} D₁\ninst✝² : Category.{v_5, u_5} D₂\nL₁ : C ⥤ D₁\nL₂ : C ⥤ D₂\nW' : MorphismProperty C\ninst✝¹ : L₁.IsLocalization W'\ninst✝ : L₂.IsLocalization W'\n⊢ { functor := (equivalenceFromModel L₁ W').inver... | [] | ext <;> aesop | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.CategoryTheory.Localization.CalculusOfFractions.Preadditive | {
"line": 111,
"column": 6
} | {
"line": 111,
"column": 46
} | {
"line": 111,
"column": 46
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : Category.{v_2, u_2} D\ninst✝² : Preadditive C\nL : C ⥤ D\nW : MorphismProperty C\ninst✝¹ : L.IsLocalization W\ninst✝ : W.HasLeftCalculusOfFractions\nX Y : C\nφ φ₀ : W.LeftFraction X Y\nhφ : φ₀.map L ⋯ = φ.map L ⋯\nhφ₀ : neg' W (φ₀.map... | [
"C : Type u_1\nD : Type u_2\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : Category.{v_2, u_2} D\ninst✝² : Preadditive C\nL : C ⥤ D\nW : MorphismProperty C\ninst✝¹ : L.IsLocalization W\ninst✝ : W.HasLeftCalculusOfFractions\nX Y : C\nφ φ₀ : W.LeftFraction X Y\nhφ : LeftFractionRel φ₀ φ\nhφ₀ : neg' W (φ₀.map L ⋯) = φ₀.neg... | MorphismProperty.LeftFraction.map_eq_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Localization.CalculusOfFractions.Preadditive | {
"line": 112,
"column": 2
} | {
"line": 112,
"column": 41
} | {
"line": 113,
"column": 2
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : Category.{v_2, u_2} D\ninst✝² : Preadditive C\nL : C ⥤ D\nW : MorphismProperty C\ninst✝¹ : L.IsLocalization W\ninst✝ : W.HasLeftCalculusOfFractions\nX Y : C\nφ φ₀ : W.LeftFraction X Y\nhφ : LeftFractionRel φ₀ φ\nhφ₀ : neg' W (φ₀.map L... | [
"C : Type u_1\nD : Type u_2\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : Category.{v_2, u_2} D\ninst✝² : Preadditive C\nL : C ⥤ D\nW : MorphismProperty C\ninst✝¹ : L.IsLocalization W\ninst✝ : W.HasLeftCalculusOfFractions\nX Y : C\nφ φ₀ : W.LeftFraction X Y\nhφ₀ : neg' W (φ₀.map L ⋯) = φ₀.neg.map L ⋯\nY' : C\nt₁ : φ₀.Y... | obtain ⟨Y', t₁, t₂, hst, hft, ht⟩ := hφ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.CategoryTheory.Localization.CalculusOfFractions | {
"line": 318,
"column": 4
} | {
"line": 318,
"column": 45
} | {
"line": 319,
"column": 2
} | [
{
"pp": "case refine_1\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\nW : MorphismProperty C\ninst✝ : W.HasLeftCalculusOfFractions\nX Y✝ Z : C\nz₁ : W.LeftFraction X Y✝\nz₂ : W.LeftFraction Y✝ Z\nz₃ z₃' : W.LeftFraction z₁.Y' z₂.Y'\nh₃ : z₂.f ≫ z₃.s = z₁.s ≫ z₃.f\nh₃' : z₂.f ≫ z₃'.s = z₁.s ≫ z₃'.f\nz₄ : W.LeftF... | [] | simp only [comp₀, assoc, reassoc_of% fac] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Localization.CalculusOfFractions | {
"line": 318,
"column": 4
} | {
"line": 318,
"column": 45
} | {
"line": 319,
"column": 2
} | [
{
"pp": "case refine_1\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\nW : MorphismProperty C\ninst✝ : W.HasLeftCalculusOfFractions\nX Y✝ Z : C\nz₁ : W.LeftFraction X Y✝\nz₂ : W.LeftFraction Y✝ Z\nz₃ z₃' : W.LeftFraction z₁.Y' z₂.Y'\nh₃ : z₂.f ≫ z₃.s = z₁.s ≫ z₃.f\nh₃' : z₂.f ≫ z₃'.s = z₁.s ≫ z₃'.f\nz₄ : W.LeftF... | [] | simp only [comp₀, assoc, reassoc_of% fac] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Localization.CalculusOfFractions | {
"line": 318,
"column": 4
} | {
"line": 318,
"column": 45
} | {
"line": 319,
"column": 2
} | [
{
"pp": "case refine_1\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\nW : MorphismProperty C\ninst✝ : W.HasLeftCalculusOfFractions\nX Y✝ Z : C\nz₁ : W.LeftFraction X Y✝\nz₂ : W.LeftFraction Y✝ Z\nz₃ z₃' : W.LeftFraction z₁.Y' z₂.Y'\nh₃ : z₂.f ≫ z₃.s = z₁.s ≫ z₃.f\nh₃' : z₂.f ≫ z₃'.s = z₁.s ≫ z₃'.f\nz₄ : W.LeftF... | [] | simp only [comp₀, assoc, reassoc_of% fac] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Localization.LocalizerMorphism | {
"line": 283,
"column": 4
} | {
"line": 283,
"column": 86
} | {
"line": 284,
"column": 4
} | [
{
"pp": "C₁ : Type u₁\nC₂ : Type u₂\nC₃ : Type u₃\nD₁ : Type u₄\nD₂ : Type u₅\ninst✝⁷ : Category.{v₁, u₁} C₁\ninst✝⁶ : Category.{v₂, u₂} C₂\ninst✝⁵ : Category.{v₃, u₃} C₃\ninst✝⁴ : Category.{v₄, u₄} D₁\ninst✝³ : Category.{v₅, u₅} D₂\nW₁ : MorphismProperty C₁\nW₂ : MorphismProperty C₂\nW₃ : MorphismProperty C₃\n... | [
"C₁ : Type u₁\nC₂ : Type u₂\nC₃ : Type u₃\nD₁ : Type u₄\nD₂ : Type u₅\ninst✝⁷ : Category.{v₁, u₁} C₁\ninst✝⁶ : Category.{v₂, u₂} C₂\ninst✝⁵ : Category.{v₃, u₃} C₃\ninst✝⁴ : Category.{v₄, u₄} D₁\ninst✝³ : Category.{v₅, u₅} D₂\nW₁ : MorphismProperty C₁\nW₂ : MorphismProperty C₂\nW₃ : MorphismProperty C₃\nΦ : Localize... | rw [Φ.nonempty_fullyFaithful_iff L₁ L₂ G W₁.Q W₂.Q (Φ.localizedFunctor W₁.Q W₂.Q)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Localization.CalculusOfFractions | {
"line": 407,
"column": 6
} | {
"line": 409,
"column": 82
} | {
"line": 410,
"column": 4
} | [
{
"pp": "case refine_2.refine_1.refine_3\nC : Type u_1\nD : Type u_2\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_2, u_2} D\nW : MorphismProperty C\ninst✝ : W.HasLeftCalculusOfFractions\nX Y Z✝ : C\nz₁✝ : Hom W X Y\nz₂✝ : Hom W Y Z✝\na₁ a₂ : W.LeftFraction X Y\nb : W.LeftFraction Y Z✝\nU : C\nt₁ : a₁.Y... | [] | · dsimp
simp only [assoc]
exact W.comp_mem _ _ b.hs (W.comp_mem _ _ z₁.hs (W.comp_mem _ _ w₁.hs hu)) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Localization.CalculusOfFractions | {
"line": 680,
"column": 2
} | {
"line": 682,
"column": 11
} | {
"line": 683,
"column": 2
} | [
{
"pp": "case mp\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\nW : MorphismProperty C\ninst✝ : W.HasLeftCalculusOfFractions\nX Y : C\nf g : W.LeftFraction X Y\n⊢ (Hom.mk f).map (Q W) ⋯ = (Hom.mk g).map (Q W) ⋯ → LeftFractionRel f g",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Eq.m... | [
"case mpr\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\nW : MorphismProperty C\ninst✝ : W.HasLeftCalculusOfFractions\nX Y : C\nf g : W.LeftFraction X Y\n⊢ LeftFractionRel f g → (Hom.mk f).map (Q W) ⋯ = (Hom.mk g).map (Q W) ⋯"
] | · intro h
rw [← homMk_eq_iff_leftFractionRel, homMk_eq, homMk_eq]
exact h | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Localization.Composition | {
"line": 49,
"column": 21
} | {
"line": 49,
"column": 35
} | {
"line": 49,
"column": 36
} | [
{
"pp": "C₁ : Type u₁\nC₂ : Type u₂\nC₃ : Type u₃\nE : Type u₄\ninst✝³ : Category.{v₁, u₁} C₁\ninst✝² : Category.{v₂, u₂} C₂\ninst✝¹ : Category.{v₃, u₃} C₃\ninst✝ : Category.{v₄, u₄} E\nL₁ : C₁ ⥤ C₂\nL₂ : C₂ ⥤ C₃\nW₁ : MorphismProperty C₁\nW₂ : MorphismProperty C₂\nh₁ : StrictUniversalPropertyFixedTarget L₁ W₁ ... | [
"C₁ : Type u₁\nC₂ : Type u₂\nC₃ : Type u₃\nE : Type u₄\ninst✝³ : Category.{v₁, u₁} C₁\ninst✝² : Category.{v₂, u₂} C₂\ninst✝¹ : Category.{v₃, u₃} C₃\ninst✝ : Category.{v₄, u₄} E\nL₁ : C₁ ⥤ C₂\nL₂ : C₂ ⥤ C₃\nW₁ : MorphismProperty C₁\nW₂ : MorphismProperty C₂\nh₁ : StrictUniversalPropertyFixedTarget L₁ W₁ E\nh₂ : Stri... | Functor.assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Localization.Composition | {
"line": 76,
"column": 2
} | {
"line": 82,
"column": 32
} | {
"line": 83,
"column": 2
} | [
{
"pp": "C₁ : Type u₁\nC₂ : Type u₂\nC₃ : Type u₃\ninst✝⁴ : Category.{v₁, u₁} C₁\ninst✝³ : Category.{v₂, u₂} C₂\ninst✝² : Category.{v₃, u₃} C₃\nL₁ : C₁ ⥤ C₂\nL₂ : C₂ ⥤ C₃\nW₁ : MorphismProperty C₁\nW₂ : MorphismProperty C₂\ninst✝¹ : L₁.IsLocalization W₁\ninst✝ : L₂.IsLocalization W₂\nW₃ : MorphismProperty C₁\nh... | [
"C₁ : Type u₁\nC₂ : Type u₂\nC₃ : Type u₃\ninst✝⁴ : Category.{v₁, u₁} C₁\ninst✝³ : Category.{v₂, u₂} C₂\ninst✝² : Category.{v₃, u₃} C₃\nL₁ : C₁ ⥤ C₂\nL₂ : C₂ ⥤ C₃\nW₁ : MorphismProperty C₁\nW₂ : MorphismProperty C₂\ninst✝¹ : L₁.IsLocalization W₁\ninst✝ : L₂.IsLocalization W₂\nW₃ : MorphismProperty C₁\nhW₃ : W₃.IsIn... | let Φ : LocalizerMorphism W₂ W₂' :=
{ functor := E₂.functor
map := by
have eq := W₂.isoClosure.inverseImage_map_eq_of_isEquivalence E₂.functor
rw [MorphismProperty.map_isoClosure] at eq
rw [eq]
apply W₂.le_isoClosure } | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Algebra.Homology.HomotopyCategory.ShiftSequence | {
"line": 176,
"column": 12
} | {
"line": 176,
"column": 45
} | {
"line": 176,
"column": 46
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preadditive C\ninst✝ : CategoryWithHomology C\nK : CochainComplex C ℤ\nA : C\nn i : ℤ\nf : A ⟶ ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) n).obj K).X i\nj : ℤ\nhj : i + 1 = j\nhi' : n + i = i + n\nhj' : i + n + 1 = j + n\nhf : n.negOnePow •... | [
"C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preadditive C\ninst✝ : CategoryWithHomology C\nK : CochainComplex C ℤ\nA : C\nn i : ℤ\nf : A ⟶ ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) n).obj K).X i\nj : ℤ\nhj : i + 1 = j\nhi' : n + i = i + n\nhj' : i + n + 1 = j + n\nhf : n.negOnePow • f ≫ K.d (i ... | ← Int.units_mul_self n.negOnePow, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Shift.SingleFunctors | {
"line": 169,
"column": 8
} | {
"line": 170,
"column": 79
} | {
"line": 170,
"column": 80
} | [
{
"pp": "C : Type u_1\nD : Type u_2\nE : Type u_3\nE' : Type u_4\ninst✝⁷ : Category.{v_1, u_1} C\ninst✝⁶ : Category.{v_2, u_2} D\ninst✝⁵ : Category.{v_3, u_3} E\ninst✝⁴ : Category.{v_4, u_4} E'\nA : Type u_5\ninst✝³ : AddMonoid A\ninst✝² : HasShift D A\ninst✝¹ : HasShift E A\ninst✝ : HasShift E' A\nF G H : Sing... | [] | rw [← cancel_mono (iso a).hom, assoc, assoc, Iso.inv_hom_id, comp_id, comm,
← whiskerRight_comp_assoc, Iso.inv_hom_id, whiskerRight_id', id_comp] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Shift.SingleFunctors | {
"line": 169,
"column": 8
} | {
"line": 170,
"column": 79
} | {
"line": 170,
"column": 80
} | [
{
"pp": "C : Type u_1\nD : Type u_2\nE : Type u_3\nE' : Type u_4\ninst✝⁷ : Category.{v_1, u_1} C\ninst✝⁶ : Category.{v_2, u_2} D\ninst✝⁵ : Category.{v_3, u_3} E\ninst✝⁴ : Category.{v_4, u_4} E'\nA : Type u_5\ninst✝³ : AddMonoid A\ninst✝² : HasShift D A\ninst✝¹ : HasShift E A\ninst✝ : HasShift E' A\nF G H : Sing... | [] | rw [← cancel_mono (iso a).hom, assoc, assoc, Iso.inv_hom_id, comp_id, comm,
← whiskerRight_comp_assoc, Iso.inv_hom_id, whiskerRight_id', id_comp] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Shift.SingleFunctors | {
"line": 169,
"column": 8
} | {
"line": 170,
"column": 79
} | {
"line": 170,
"column": 80
} | [
{
"pp": "C : Type u_1\nD : Type u_2\nE : Type u_3\nE' : Type u_4\ninst✝⁷ : Category.{v_1, u_1} C\ninst✝⁶ : Category.{v_2, u_2} D\ninst✝⁵ : Category.{v_3, u_3} E\ninst✝⁴ : Category.{v_4, u_4} E'\nA : Type u_5\ninst✝³ : AddMonoid A\ninst✝² : HasShift D A\ninst✝¹ : HasShift E A\ninst✝ : HasShift E' A\nF G H : Sing... | [] | rw [← cancel_mono (iso a).hom, assoc, assoc, Iso.inv_hom_id, comp_id, comm,
← whiskerRight_comp_assoc, Iso.inv_hom_id, whiskerRight_id', id_comp] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.HomotopyCategory.MappingCocone | {
"line": 227,
"column": 2
} | {
"line": 227,
"column": 11
} | {
"line": 228,
"column": 2
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preadditive C\nK L : CochainComplex C ℤ\nφ : K ⟶ L\ninst✝ : HasHomotopyCofiber φ\nM : CochainComplex C ℤ\nn m : ℤ\nα : Cochain M K n\nβ : Cochain M L m\nh : m + 1 = n\np₁ p₂ p₃ : ℤ\nh₁₂ : p₁ + n = p₂\nh₂₃ : p₂ + -1 = p₃\n⊢ (liftCochain φ α β h).v p... | [
"C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preadditive C\nK L : CochainComplex C ℤ\nφ : K ⟶ L\ninst✝ : HasHomotopyCofiber φ\nM : CochainComplex C ℤ\nn m : ℤ\nα : Cochain M K n\nβ : Cochain M L m\nh : m + 1 = n\np₁ p₂ : ℤ\nh₁₂ : p₁ + n = p₂\n⊢ (liftCochain φ α β h).v p₁ p₂ h₁₂ ≫ (snd φ).v p₂ (p₂ + -1) ⋯... | subst h₂₃ | Lean.Elab.Tactic.evalSubst | Lean.Parser.Tactic.subst |
Mathlib.Algebra.Homology.DerivedCategory.HomologySequence | {
"line": 245,
"column": 30
} | {
"line": 245,
"column": 69
} | {
"line": 245,
"column": 70
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : HasDerivedCategory C\nT : Triangle (CochainComplex C ℤ)\nhT : DerivedCategory.Q.mapTriangle.obj T ∈ distinguishedTriangles\nn : ℤ\n⊢ (DerivedCategory.homologyFunctor C n).map (DerivedCategory.Q.map T.mor₁ ≫ DerivedCategory.Q.map T.mor₂... | [
"C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : HasDerivedCategory C\nT : Triangle (CochainComplex C ℤ)\nhT : DerivedCategory.Q.mapTriangle.obj T ∈ distinguishedTriangles\nn : ℤ\n⊢ (DerivedCategory.homologyFunctor C n).map 0 ≫ (DerivedCategory.homologyFunctorFactors C n).hom.app T.obj₃ =\n (... | dsimp% comp_distTriang_mor_zero₁₂ _ hT, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.HomotopyCategory.Triangulated | {
"line": 123,
"column": 6
} | {
"line": 124,
"column": 48
} | {
"line": 125,
"column": 6
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v, u_1} C\ninst✝¹ : Preadditive C\ninst✝ : HasBinaryBiproducts C\nX₁ X₂ X₃ : CochainComplex C ℤ\nf : X₁ ⟶ X₂\ng : X₂ ⟶ X₃\nn : ℤ\n⊢ (Cochain.ofHom (inv f g ≫ hom f g)).v n n ⋯ =\n (-((snd (mappingConeCompTriangle f g).mor₁).comp\n ((↑(fst (f ≫ g))).com... | [
"C : Type u_1\ninst✝² : Category.{v, u_1} C\ninst✝¹ : Preadditive C\ninst✝ : HasBinaryBiproducts C\nX₁ X₂ X₃ : CochainComplex C ℤ\nf : X₁ ⟶ X₂\ng : X₂ ⟶ X₃\nn : ℤ\n⊢ ((inl f).v (n + 2) (n + 1) ⋯ ≫\n (inl (mappingConeCompTriangle f g).mor₁).v (n + 1) n ⋯ ≫ (Cochain.ofHom (inv f g ≫ hom f g)).v n n ⋯ =\n ... | rw [ext_from_iff _ (n + 1) n rfl, ext_from_iff _ (n + 1) n rfl,
ext_from_iff _ (n + 2) (n + 1) (by lia)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Homology.Embedding.Extend | {
"line": 107,
"column": 2
} | {
"line": 107,
"column": 23
} | {
"line": 108,
"column": 2
} | [
{
"pp": "ι : Type u_1\nc : ComplexShape ι\nC : Type u_3\ninst✝² : Category.{v_1, u_3} C\ninst✝¹ : HasZeroObject C\ninst✝ : HasZeroMorphisms C\nK L : HomologicalComplex C c\nφ : K ⟶ L\na : ι\n⊢ mapX φ (some a) = 𝟙 (K.X a) ≫ φ.f a ≫ 𝟙 (L.X a)",
"ppTerm": "?m.77",
"assigned": true,
"usedConstants": [... | [
"ι : Type u_1\nc : ComplexShape ι\nC : Type u_3\ninst✝² : Category.{v_1, u_3} C\ninst✝¹ : HasZeroObject C\ninst✝ : HasZeroMorphisms C\nK L : HomologicalComplex C c\nφ : K ⟶ L\na : ι\n⊢ mapX φ (some a) = φ.f a"
] | rw [id_comp, comp_id] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Homology.Embedding.Boundary | {
"line": 82,
"column": 26
} | {
"line": 82,
"column": 59
} | {
"line": 82,
"column": 59
} | [
{
"pp": "ι : Type u_1\nι' : Type u_2\nc : ComplexShape ι\nc' : ComplexShape ι'\ne : c.Embedding c'\ninst✝ : e.IsRelIff\ni j : ι\nhij : c.prev j = i\nhj : ¬e.BoundaryGE j\nhij' : c.Rel i j\n⊢ c'.Rel (e.f i) (e.f j)",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"id",
... | [] | simpa only [e.rel_iff] using hij' | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Algebra.Homology.Embedding.Boundary | {
"line": 82,
"column": 26
} | {
"line": 82,
"column": 59
} | {
"line": 82,
"column": 59
} | [
{
"pp": "ι : Type u_1\nι' : Type u_2\nc : ComplexShape ι\nc' : ComplexShape ι'\ne : c.Embedding c'\ninst✝ : e.IsRelIff\ni j : ι\nhij : c.prev j = i\nhj : ¬e.BoundaryGE j\nhij' : c.Rel i j\n⊢ c'.Rel (e.f i) (e.f j)",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"id",
... | [] | simpa only [e.rel_iff] using hij' | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.Embedding.Boundary | {
"line": 82,
"column": 26
} | {
"line": 82,
"column": 59
} | {
"line": 82,
"column": 59
} | [
{
"pp": "ι : Type u_1\nι' : Type u_2\nc : ComplexShape ι\nc' : ComplexShape ι'\ne : c.Embedding c'\ninst✝ : e.IsRelIff\ni j : ι\nhij : c.prev j = i\nhj : ¬e.BoundaryGE j\nhij' : c.Rel i j\n⊢ c'.Rel (e.f i) (e.f j)",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"id",
... | [] | simpa only [e.rel_iff] using hij' | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.Embedding.HomEquiv | {
"line": 88,
"column": 6
} | {
"line": 88,
"column": 26
} | {
"line": 89,
"column": 6
} | [
{
"pp": "case neg\nι : Type u_1\nι' : Type u_2\nc : ComplexShape ι\nc' : ComplexShape ι'\ne : c.Embedding c'\nC : Type u_3\ninst✝³ : Category.{v_1, u_3} C\ninst✝² : HasZeroMorphisms C\ninst✝¹ : HasZeroObject C\nK : HomologicalComplex C c'\nL : HomologicalComplex C c\ninst✝ : e.IsRelIff\nφ : K.restriction e ⟶ L\... | [
"case neg\nι : Type u_1\nι' : Type u_2\nc : ComplexShape ι\nc' : ComplexShape ι'\ne : c.Embedding c'\nC : Type u_3\ninst✝³ : Category.{v_1, u_3} C\ninst✝² : HasZeroMorphisms C\ninst✝¹ : HasZeroObject C\nK : HomologicalComplex C c'\nL : HomologicalComplex C c\ninst✝ : e.IsRelIff\nφ : K.restriction e ⟶ L\nhφ : e.HasL... | rw [this, comp_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Homology.Embedding.HomEquiv | {
"line": 97,
"column": 8
} | {
"line": 97,
"column": 28
} | {
"line": 98,
"column": 2
} | [
{
"pp": "case neg\nι : Type u_1\nι' : Type u_2\nc : ComplexShape ι\nc' : ComplexShape ι'\ne : c.Embedding c'\nC : Type u_3\ninst✝³ : Category.{v_1, u_3} C\ninst✝² : HasZeroMorphisms C\ninst✝¹ : HasZeroObject C\nK : HomologicalComplex C c'\nL : HomologicalComplex C c\ninst✝ : e.IsRelIff\nφ : K.restriction e ⟶ L\... | [] | rw [this, comp_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Homology.Embedding.RestrictionHomology | {
"line": 170,
"column": 2
} | {
"line": 170,
"column": 62
} | {
"line": 171,
"column": 2
} | [
{
"pp": "ι : 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\nK : HomologicalComplex C c'\ne : c.Embedding c'\ninst✝² : e.IsRelIff\ni j k : ι\nhi : c.prev j = i\nhk : c.next j = k\ni' j' k' : ι'\nhi' : e.f i = i'\nhj' :... | [
"ι : 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\nK : HomologicalComplex C c'\ne : c.Embedding c'\ninst✝² : e.IsRelIff\ni j k : ι\nhi : c.prev j = i\nhk : c.next j = k\ni' j' k' : ι'\nhi' : e.f i = i'\nhj' : e.f j = j'\... | refine ((ShortComplex.homology_π_ι_assoc _ _).trans ?_).symm | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Algebra.Homology.Embedding.ExtendHomology | {
"line": 442,
"column": 4
} | {
"line": 442,
"column": 14
} | {
"line": 443,
"column": 4
} | [
{
"pp": "case mpr\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 : HomologicalComplex C c\nφ : K ⟶ L\ne : c.Embedding c'\ninst✝¹ : ∀ (j : ι), K.HasHomology j\ninst✝ : ∀ (j : ι), L.Ha... | [
"case mpr\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 : HomologicalComplex C c\nφ : K ⟶ L\ne : c.Embedding c'\ninst✝¹ : ∀ (j : ι), K.HasHomology j\ninst✝ : ∀ (j : ι), L.HasHomology j\... | intro h j' | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Algebra.Homology.Embedding.ExtendHomology | {
"line": 446,
"column": 10
} | {
"line": 446,
"column": 32
} | {
"line": 446,
"column": 32
} | [
{
"pp": "case neg\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 : HomologicalComplex C c\nφ : K ⟶ L\ne : c.Embedding c'\ninst✝¹ : ∀ (j : ι), K.HasHomology j\ninst✝ : ∀ (j : ι), L.Ha... | [
"case neg\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 : HomologicalComplex C c\nφ : K ⟶ L\ne : c.Embedding c'\ninst✝¹ : ∀ (j : ι), K.HasHomology j\ninst✝ : ∀ (j : ι), L.HasHomology j\... | quasiIsoAt_iff_exactAt | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.Embedding.ExtendHomology | {
"line": 446,
"column": 6
} | {
"line": 446,
"column": 33
} | {
"line": 447,
"column": 6
} | [
{
"pp": "case neg\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 : HomologicalComplex C c\nφ : K ⟶ L\ne : c.Embedding c'\ninst✝¹ : ∀ (j : ι), K.HasHomology j\ninst✝ : ∀ (j : ι), L.Ha... | [
"case neg\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 : HomologicalComplex C c\nφ : K ⟶ L\ne : c.Embedding c'\ninst✝¹ : ∀ (j : ι), K.HasHomology j\ninst✝ : ∀ (j : ι), L.HasHomology j\... | rw [quasiIsoAt_iff_exactAt] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Homology.Embedding.TruncGEHomology | {
"line": 250,
"column": 10
} | {
"line": 250,
"column": 32
} | {
"line": 250,
"column": 32
} | [
{
"pp": "case neg\nι : Type u_1\nι' : Type u_2\nc : ComplexShape ι\nc' : ComplexShape ι'\nC : Type u_3\ninst✝⁵ : Category.{v_1, u_3} C\ninst✝⁴ : HasZeroMorphisms C\nK L : HomologicalComplex C c'\nφ : K ⟶ L\ne : c.Embedding c'\ninst✝³ : e.IsTruncGE\ninst✝² : ∀ (i' : ι'), K.HasHomology i'\ninst✝¹ : ∀ (i' : ι'), L... | [
"case neg\nι : Type u_1\nι' : Type u_2\nc : ComplexShape ι\nc' : ComplexShape ι'\nC : Type u_3\ninst✝⁵ : Category.{v_1, u_3} C\ninst✝⁴ : HasZeroMorphisms C\nK L : HomologicalComplex C c'\nφ : K ⟶ L\ne : c.Embedding c'\ninst✝³ : e.IsTruncGE\ninst✝² : ∀ (i' : ι'), K.HasHomology i'\ninst✝¹ : ∀ (i' : ι'), L.HasHomology... | quasiIsoAt_iff_exactAt | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.Embedding.TruncGEHomology | {
"line": 250,
"column": 6
} | {
"line": 250,
"column": 33
} | {
"line": 251,
"column": 6
} | [
{
"pp": "case neg\nι : Type u_1\nι' : Type u_2\nc : ComplexShape ι\nc' : ComplexShape ι'\nC : Type u_3\ninst✝⁵ : Category.{v_1, u_3} C\ninst✝⁴ : HasZeroMorphisms C\nK L : HomologicalComplex C c'\nφ : K ⟶ L\ne : c.Embedding c'\ninst✝³ : e.IsTruncGE\ninst✝² : ∀ (i' : ι'), K.HasHomology i'\ninst✝¹ : ∀ (i' : ι'), L... | [
"case neg\nι : Type u_1\nι' : Type u_2\nc : ComplexShape ι\nc' : ComplexShape ι'\nC : Type u_3\ninst✝⁵ : Category.{v_1, u_3} C\ninst✝⁴ : HasZeroMorphisms C\nK L : HomologicalComplex C c'\nφ : K ⟶ L\ne : c.Embedding c'\ninst✝³ : e.IsTruncGE\ninst✝² : ∀ (i' : ι'), K.HasHomology i'\ninst✝¹ : ∀ (i' : ι'), L.HasHomology... | rw [quasiIsoAt_iff_exactAt] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Homology.SingleHomology | {
"line": 121,
"column": 44
} | {
"line": 123,
"column": 65
} | {
"line": 125,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : HasZeroMorphisms C\ninst✝¹ : HasZeroObject C\nι : Type u_1\ninst✝ : DecidableEq ι\nc : ComplexShape ι\nj : ι\nA : C\n⊢ (singleObjHomologySelfIso c j A).hom ≫ (singleObjOpcyclesSelfIso c j A).hom = ((single C c j).obj A).homologyι j",
"ppTerm": "?m.50... | [] | by
rw [← cancel_epi (singleObjHomologySelfIso _ _ _).inv,
Iso.inv_hom_id_assoc, singleObjHomologySelfIso_inv_homologyι] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Localization.SmallHom | {
"line": 271,
"column": 2
} | {
"line": 297,
"column": 50
} | {
"line": 299,
"column": 0
} | [
{
"pp": "C₁ : Type u₁\ninst✝⁷ : Category.{v₁, u₁} C₁\nW₁ : MorphismProperty C₁\nC₂ : Type u₂\ninst✝⁶ : Category.{v₂, u₂} C₂\nW₂ : MorphismProperty C₂\nD₁ : Type u₃\ninst✝⁵ : Category.{v₃, u₃} D₁\nD₂ : Type u₄\ninst✝⁴ : Category.{v₄, u₄} D₂\nΦ : LocalizerMorphism W₁ W₂\nL₁ : C₁ ⥤ D₁\ninst✝³ : L₁.IsLocalization W... | [] | obtain ⟨g, rfl⟩ := (SmallHom.equiv W₁ W₁.Q).symm.surjective f
simp only [smallHomMap, Equiv.apply_symm_apply]
let G' := Φ.localizedFunctor W₁.Q W₂.Q
let β := CatCommSq.iso Φ.functor W₁.Q W₂.Q G'
let E₁ := (uniq W₁.Q L₁ W₁).functor
let α₁ : W₁.Q ⋙ E₁ ≅ L₁ := compUniqFunctor W₁.Q L₁ W₁
let E₂ := (uniq W₂.Q L₂... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Localization.SmallHom | {
"line": 271,
"column": 2
} | {
"line": 297,
"column": 50
} | {
"line": 299,
"column": 0
} | [
{
"pp": "C₁ : Type u₁\ninst✝⁷ : Category.{v₁, u₁} C₁\nW₁ : MorphismProperty C₁\nC₂ : Type u₂\ninst✝⁶ : Category.{v₂, u₂} C₂\nW₂ : MorphismProperty C₂\nD₁ : Type u₃\ninst✝⁵ : Category.{v₃, u₃} D₁\nD₂ : Type u₄\ninst✝⁴ : Category.{v₄, u₄} D₂\nΦ : LocalizerMorphism W₁ W₂\nL₁ : C₁ ⥤ D₁\ninst✝³ : L₁.IsLocalization W... | [] | obtain ⟨g, rfl⟩ := (SmallHom.equiv W₁ W₁.Q).symm.surjective f
simp only [smallHomMap, Equiv.apply_symm_apply]
let G' := Φ.localizedFunctor W₁.Q W₂.Q
let β := CatCommSq.iso Φ.functor W₁.Q W₂.Q G'
let E₁ := (uniq W₁.Q L₁ W₁).functor
let α₁ : W₁.Q ⋙ E₁ ≅ L₁ := compUniqFunctor W₁.Q L₁ W₁
let E₂ := (uniq W₂.Q L₂... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic | {
"line": 93,
"column": 2
} | {
"line": 94,
"column": 38
} | {
"line": 96,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\n⊢ HasExt C",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"CategoryTheory.Abelian",
"CategoryTheory.hasExt_of_hasDerivedCategory",
"HasDerivedCategory.standard",
"HasDerivedCategory",
"Categor... | [] | let := HasDerivedCategory.standard
exact hasExt_of_hasDerivedCategory _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic | {
"line": 93,
"column": 2
} | {
"line": 94,
"column": 38
} | {
"line": 96,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\n⊢ HasExt C",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"CategoryTheory.Abelian",
"CategoryTheory.hasExt_of_hasDerivedCategory",
"HasDerivedCategory.standard",
"HasDerivedCategory",
"Categor... | [] | let := HasDerivedCategory.standard
exact hasExt_of_hasDerivedCategory _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.Refinements | {
"line": 71,
"column": 2
} | {
"line": 72,
"column": 63
} | {
"line": 74,
"column": 0
} | [
{
"pp": "C : Type u_1\nι : Type u_2\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Abelian C\nc : ComplexShape ι\nK : HomologicalComplex C c\ni j : ι\nhi : c.prev j = i\nA : C\nz₂ z₂' : A ⟶ K.cycles j\n⊢ z₂ ≫ K.homologyπ j = z₂' ≫ K.homologyπ j ↔ ∃ A' π, ∃ (_ : Epi π), ∃ x₁, π ≫ z₂ = π ≫ z₂' + x₁ ≫ K.toCycles i j",
... | [] | subst hi
exact (K.sc j).comp_homologyπ_eq_iff_up_to_refinements z₂ z₂' | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.Refinements | {
"line": 71,
"column": 2
} | {
"line": 72,
"column": 63
} | {
"line": 74,
"column": 0
} | [
{
"pp": "C : Type u_1\nι : Type u_2\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Abelian C\nc : ComplexShape ι\nK : HomologicalComplex C c\ni j : ι\nhi : c.prev j = i\nA : C\nz₂ z₂' : A ⟶ K.cycles j\n⊢ z₂ ≫ K.homologyπ j = z₂' ≫ K.homologyπ j ↔ ∃ A' π, ∃ (_ : Epi π), ∃ x₁, π ≫ z₂ = π ≫ z₂' + x₁ ≫ K.toCycles i j",
... | [] | subst hi
exact (K.sc j).comp_homologyπ_eq_iff_up_to_refinements z₂ z₂' | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.