mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.CategoryTheory.GuitartExact.VerticalComposition

Goal state before:
pp: case mp
C₁ : Type u_1
C₂ : Type u_2
C₃ : Type u_3
D₁ : Type u_4
D₂ : Type u_5
D₃ : Type u_6
inst✝⁵ : Category.{v_1, u_1} C₁
inst✝⁴ : Category.{v_2, u_2} C₂
inst✝³ : Category.{v_3, u_3} C₃
inst✝² : Category.{v_4, u_4} D₁
inst✝¹ : Category.{v_5, u_5} D₂
inst✝ : Category.{v_6, u_6} D₃
H₁ : C₁ ⥤ D₁
L₁ : C₁ ⥤ C₂
R₁ : D₁ ⥤ D₂
H₂ : C₂ ⥤ D₂
w : TwoSquare H₁ L₁ R₁ H₂
H₃ : C₃ ⥤ D₃
eL : C₂ ≌ C₃
eR : D₂ ≌ D₃
w' : H₂ ⋙ eR.functor ≅ eL.functor ⋙ H₃
hww' : (w ≫ᵥ w'.hom).GuitartExact
this✝¹ : CatCommSq H₂ eL.functor eR.functor H₃ := { iso := w' }
hw' : CatCommSq.iso H₂ eL.functor eR.functor H₃ = w'
this✝ : CatCommSq H₃ eL.inverse eR.inverse H₂ := (CatCommSq.vInvEquiv H₂ eL eR H₃) inferInstance
w'' : H₃ ⋙ eR.inverse ≅ eL.inverse ⋙ H₂ := CatCommSq.iso H₃ eL.inverse eR.inverse H₂
α : (L₁ ⋙ eL.functor) ⋙ eL.inverse ≅ L₁ :=
  L₁.associator eL.functor eL.inverse ≪≫ L₁.isoWhiskerLeft eL.unitIso.symm ≪≫ L₁.rightUnitor
β : (R₁ ⋙ eR.functor) ⋙ eR.inverse ≅ R₁ :=
  R₁.associator eR.functor eR.inverse ≪≫ R₁.isoWhiskerLeft eR.unitIso.symm ≪≫ R₁.rightUnitor
this : w = (w ≫ᵥ w'.hom).vComp' w''.hom α β
⊢ w.GuitartExact

Tactic:
rw [this]

Goal state after:
case mp
C₁ : Type u_1
C₂ : Type u_2
C₃ : Type u_3
D₁ : Type u_4
D₂ : Type u_5
D₃ : Type u_6
inst✝⁵ : Category.{v_1, u_1} C₁
inst✝⁴ : Category.{v_2, u_2} C₂
inst✝³ : Category.{v_3, u_3} C₃
inst✝² : Category.{v_4, u_4} D₁
inst✝¹ : Category.{v_5, u_5} D₂
inst✝ : Category.{v_6, u_6} D₃
H₁ : C₁ ⥤ D₁
L₁ : C₁ ⥤ C₂
R₁ : D₁ ⥤ D₂
H₂ : C₂ ⥤ D₂
w : TwoSquare H₁ L₁ R₁ H₂
H₃ : C₃ ⥤ D₃
eL : C₂ ≌ C₃
eR : D₂ ≌ D₃
w' : H₂ ⋙ eR.functor ≅ eL.functor ⋙ H₃
hww' : (w ≫ᵥ w'.hom).GuitartExact
this✝¹ : CatCommSq H₂ eL.functor eR.functor H₃ := { iso := w' }
hw' : CatCommSq.iso H₂ eL.functor eR.functor H₃ = w'
this✝ : CatCommSq H₃ eL.inverse eR.inverse H₂ := (CatCommSq.vInvEquiv H₂ eL eR H₃) inferInstance
w'' : H₃ ⋙ eR.inverse ≅ eL.inverse ⋙ H₂ := CatCommSq.iso H₃ eL.inverse eR.inverse H₂
α : (L₁ ⋙ eL.functor) ⋙ eL.inverse ≅ L₁ :=
  L₁.associator eL.functor eL.inverse ≪≫ L₁.isoWhiskerLeft eL.unitIso.symm ≪≫ L₁.rightUnitor
β : (R₁ ⋙ eR.functor) ⋙ eR.inverse ≅ R₁ :=
  R₁.associator eR.functor eR.inverse ≪≫ R₁.isoWhiskerLeft eR.unitIso.symm ≪≫ R₁.rightUnitor
this : w = (w ≫ᵥ w'.hom).vComp' w''.hom α β
⊢ ((w ≫ᵥ w'.hom).vComp' w''.hom α β).GuitartExact
Module: Mathlib.CategoryTheory.Localization.DerivabilityStructure.Basic

Goal state before:
pp: C₁ : Type u₁
C₂ : Type u₂
inst✝⁶ : Category.{v₁, u₁} C₁
inst✝⁵ : Category.{v₂, u₂} C₂
W₁ : MorphismProperty C₁
W₂ : MorphismProperty C₂
Φ : LocalizerMorphism W₁ W₂
D₁ : Type u_1
D₂ : Type u_2
inst✝⁴ : Category.{v_1, u_1} D₁
inst✝³ : Category.{v_2, u_2} D₂
L₁ : C₁ ⥤ D₁
L₂ : C₂ ⥤ D₂
inst✝² : L₁.IsLocalization W₁
inst✝¹ : L₂.IsLocalization W₂
F : D₁ ⥤ D₂
inst✝ : Φ.HasRightResolutions
e : Φ.functor ⋙ L₂ ≅ L₁ ⋙ F
this :
  Φ.IsRightDerivabilityStructure ↔
    TwoSquare.GuitartExact (CatCommSq.iso Φ.functor W₁.Q W₂.Q (Φ.localizedFunctor W₁.Q W₂.Q)).hom
⊢ Φ.IsRightDerivabilityStructure ↔ TwoSquare.GuitartExact e.hom

Tactic:
rw [this]

Goal state after:
C₁ : Type u₁
C₂ : Type u₂
inst✝⁶ : Category.{v₁, u₁} C₁
inst✝⁵ : Category.{v₂, u₂} C₂
W₁ : MorphismProperty C₁
W₂ : MorphismProperty C₂
Φ : LocalizerMorphism W₁ W₂
D₁ : Type u_1
D₂ : Type u_2
inst✝⁴ : Category.{v_1, u_1} D₁
inst✝³ : Category.{v_2, u_2} D₂
L₁ : C₁ ⥤ D₁
L₂ : C₂ ⥤ D₂
inst✝² : L₁.IsLocalization W₁
inst✝¹ : L₂.IsLocalization W₂
F : D₁ ⥤ D₂
inst✝ : Φ.HasRightResolutions
e : Φ.functor ⋙ L₂ ≅ L₁ ⋙ F
this :
  Φ.IsRightDerivabilityStructure ↔
    TwoSquare.GuitartExact (CatCommSq.iso Φ.functor W₁.Q W₂.Q (Φ.localizedFunctor W₁.Q W₂.Q)).hom
⊢ TwoSquare.GuitartExact (CatCommSq.iso Φ.functor W₁.Q W₂.Q (Φ.localizedFunctor W₁.Q W₂.Q)).hom ↔
    TwoSquare.GuitartExact e.hom
Module: Mathlib.CategoryTheory.Functor.Derived.RightDerived

Goal state before:
pp: C : Type u_1
D : Type u_3
H : Type u_2
inst✝³ : Category.{v_1, u_1} C
inst✝² : Category.{v_3, u_3} D
inst✝¹ : Category.{v_5, u_2} H
F : C ⥤ H
L : C ⥤ D
W : MorphismProperty C
inst✝ : L.IsLocalization W
⊢ F.HasRightDerivedFunctor W ↔ L.HasLeftKanExtension F

Tactic:
by
  have : HasRightDerivedFunctor F W ↔ HasLeftKanExtension W.Q F :=
    ⟨fun h => h.hasLeftKanExtension', fun h => ⟨h⟩⟩
  rw [this, hasLeftExtension_iff_postcomp₁ (Localization.compUniqFunctor W.Q L W) F]

Goal state after:
Module: Mathlib.CategoryTheory.Localization.DerivabilityStructure.PointwiseRightDerived

Goal state before:
pp: case mp
C₁ : Type u₁
C₂ : Type u₂
H : Type u₃
inst✝³ : Category.{v₁, u₁} C₁
inst✝² : Category.{v₂, u₂} C₂
inst✝¹ : Category.{v₃, u₃} H
W₁ : MorphismProperty C₁
W₂ : MorphismProperty C₂
Φ : LocalizerMorphism W₁ W₂
F : C₂ ⥤ H
inst✝ : Φ.IsRightDerivabilityStructure
hF : F.HasPointwiseRightDerivedFunctor W₂
⊢ (Φ.functor ⋙ F).HasPointwiseRightDerivedFunctor W₁

Tactic:
X₁

Goal state after:
case mp
C₁ : Type u₁
C₂ : Type u₂
H : Type u₃
inst✝³ : Category.{v₁, u₁} C₁
inst✝² : Category.{v₂, u₂} C₂
inst✝¹ : Category.{v₃, u₃} H
W₁ : MorphismProperty C₁
W₂ : MorphismProperty C₂
Φ : LocalizerMorphism W₁ W₂
F : C₂ ⥤ H
inst✝ : Φ.IsRightDerivabilityStructure
hF : F.HasPointwiseRightDerivedFunctor W₂
X₁ : C₁
⊢ (Φ.functor ⋙ F).HasPointwiseRightDerivedFunctorAt W₁ X₁
Module: Mathlib.CategoryTheory.Functor.Derived.PointwiseRightDerived

Goal state before:
pp: C : Type u₁
D : Type u₂
H : Type u₃
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : Category.{v₂, u₂} D
inst✝² : Category.{v₃, u₃} H
L : C ⥤ D
W : MorphismProperty C
inst✝¹ : L.IsLocalization W
F : C ⥤ H
RF : D ⥤ H
α : F ⟶ L ⋙ RF
hF : W.IsInvertedBy F
inst✝ : RF.IsRightDerivedFunctor α W
this :
  α =
    (Localization.fac F hF L).inv ≫
      L.whiskerLeft ((Localization.lift F hF L).rightDerivedUnique RF (Localization.fac F hF L).inv α W).hom
⊢ IsIso α

Tactic:
rw [this]

Goal state after:
C : Type u₁
D : Type u₂
H : Type u₃
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : Category.{v₂, u₂} D
inst✝² : Category.{v₃, u₃} H
L : C ⥤ D
W : MorphismProperty C
inst✝¹ : L.IsLocalization W
F : C ⥤ H
RF : D ⥤ H
α : F ⟶ L ⋙ RF
hF : W.IsInvertedBy F
inst✝ : RF.IsRightDerivedFunctor α W
this :
  α =
    (Localization.fac F hF L).inv ≫
      L.whiskerLeft ((Localization.lift F hF L).rightDerivedUnique RF (Localization.fac F hF L).inv α W).hom
⊢ IsIso
    ((Localization.fac F hF L).inv ≫
      L.whiskerLeft ((Localization.lift F hF L).rightDerivedUnique RF (Localization.fac F hF L).inv α W).hom)
Module: Mathlib.CategoryTheory.GuitartExact.HorizontalComposition

Goal state before:
pp: C₁ : Type u_1
C₂ : Type u_2
C₃ : Type u_3
D₁ : Type u_4
D₂ : Type u_5
D₃ : Type u_6
inst✝⁷ : Category.{v_1, u_1} C₁
inst✝⁶ : Category.{v_2, u_2} C₂
inst✝⁵ : Category.{v_3, u_3} C₃
inst✝⁴ : Category.{v_4, u_4} D₁
inst✝³ : Category.{v_5, u_5} D₂
inst✝² : Category.{v_6, u_6} D₃
V₁ : C₁ ⥤ D₁
T₁ : C₁ ⥤ C₂
B₁ : D₁ ⥤ D₂
V₂ : C₂ ⥤ D₂
w : TwoSquare T₁ V₁ V₂ B₁
T₂ : C₂ ⥤ C₃
B₂ : D₂ ⥤ D₃
V₃ : C₃ ⥤ D₃
w' : TwoSquare T₂ V₂ V₃ B₂
inst✝¹ : w.GuitartExact
inst✝ : w'.GuitartExact
this : (w ≫ₕ w').op = w.op ≫ᵥ w'.op
⊢ (w ≫ₕ w').op.GuitartExact

Tactic:
rw [this]

Goal state after:
C₁ : Type u_1
C₂ : Type u_2
C₃ : Type u_3
D₁ : Type u_4
D₂ : Type u_5
D₃ : Type u_6
inst✝⁷ : Category.{v_1, u_1} C₁
inst✝⁶ : Category.{v_2, u_2} C₂
inst✝⁵ : Category.{v_3, u_3} C₃
inst✝⁴ : Category.{v_4, u_4} D₁
inst✝³ : Category.{v_5, u_5} D₂
inst✝² : Category.{v_6, u_6} D₃
V₁ : C₁ ⥤ D₁
T₁ : C₁ ⥤ C₂
B₁ : D₁ ⥤ D₂
V₂ : C₂ ⥤ D₂
w : TwoSquare T₁ V₁ V₂ B₁
T₂ : C₂ ⥤ C₃
B₂ : D₂ ⥤ D₃
V₃ : C₃ ⥤ D₃
w' : TwoSquare T₂ V₂ V₃ B₂
inst✝¹ : w.GuitartExact
inst✝ : w'.GuitartExact
this : (w ≫ₕ w').op = w.op ≫ᵥ w'.op
⊢ (w.op ≫ᵥ w'.op).GuitartExact
Module: Mathlib.Algebra.Homology.Embedding.StupidTrunc

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝³ : Category.{v_1, u_3} C
inst✝² : HasZeroMorphisms C
inst✝¹ : HasZeroObject C
K : HomologicalComplex C c'
e : c.Embedding c'
inst✝ : e.IsRelIff
⊢ stupidTruncMap (𝟙 K) e = 𝟙 (K.stupidTrunc e)

Tactic:
simp [stupidTruncMap, stupidTrunc]

Goal state after:
Module: Mathlib.Algebra.Homology.Embedding.StupidTrunc

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝³ : Category.{v_1, u_3} C
inst✝² : HasZeroMorphisms C
inst✝¹ : HasZeroObject C
K L M : HomologicalComplex C c'
φ : K ⟶ L
φ' : L ⟶ M
e : c.Embedding c'
inst✝ : e.IsRelIff
⊢ stupidTruncMap (φ ≫ φ') e = stupidTruncMap φ e ≫ stupidTruncMap φ' e

Tactic:
simp [stupidTruncMap, stupidTrunc]

Goal state after:
Module: Mathlib.CategoryTheory.DifferentialObject

Goal state before:
pp: case a
S : Type u_1
inst✝⁶ : AddMonoidWithOne S
C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : HasZeroMorphisms C
inst✝³ : HasShift C S
D : Type u'
inst✝² : Category.{v', u'} D
inst✝¹ : HasZeroMorphisms D
inst✝ : HasShift D S
F : C ⥤ D
η : shiftFunctor C 1 ⋙ F ⟶ F ⋙ shiftFunctor D 1
hF : ∀ (c c' : C), F.map 0 = 0
X✝ Y✝ : DifferentialObject S C
f : X✝ ⟶ Y✝
| F.map (f.f ≫ Y✝.d)

pp: case a
S : Type u_1
inst✝⁶ : AddMonoidWithOne S
C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : HasZeroMorphisms C
inst✝³ : HasShift C S
D : Type u'
inst✝² : Category.{v', u'} D
inst✝¹ : HasZeroMorphisms D
inst✝ : HasShift D S
F : C ⥤ D
η : shiftFunctor C 1 ⋙ F ⟶ F ⋙ shiftFunctor D 1
hF : ∀ (c c' : C), F.map 0 = 0
X✝ Y✝ : DifferentialObject S C
f : X✝ ⟶ Y✝
| η.app Y✝.obj

Tactic:
F.map_comp

Goal state after:
case a
S : Type u_1
inst✝⁶ : AddMonoidWithOne S
C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : HasZeroMorphisms C
inst✝³ : HasShift C S
D : Type u'
inst✝² : Category.{v', u'} D
inst✝¹ : HasZeroMorphisms D
inst✝ : HasShift D S
F : C ⥤ D
η : shiftFunctor C 1 ⋙ F ⟶ F ⋙ shiftFunctor D 1
hF : ∀ (c c' : C), F.map 0 = 0
X✝ Y✝ : DifferentialObject S C
f : X✝ ⟶ Y✝
| F.map f.f ≫ F.map Y✝.d

case a
S : Type u_1
inst✝⁶ : AddMonoidWithOne S
C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : HasZeroMorphisms C
inst✝³ : HasShift C S
D : Type u'
inst✝² : Category.{v', u'} D
inst✝¹ : HasZeroMorphisms D
inst✝ : HasShift D S
F : C ⥤ D
η : shiftFunctor C 1 ⋙ F ⟶ F ⋙ shiftFunctor D 1
hF : ∀ (c c' : C), F.map 0 = 0
X✝ Y✝ : DifferentialObject S C
f : X✝ ⟶ Y✝
| η.app Y✝.obj
Module: Mathlib.Algebra.Homology.HomotopyCategory.HomComplexSingle

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preadditive C
inst✝ : HasZeroObject C
X : C
K : CochainComplex C ℤ
p q n : ℤ
h : p + n = q
⊢ fromSingleMk 0 h = 0

Tactic:
by
  simp [fromSingleMk]

Goal state after:
Module: Mathlib.Algebra.Homology.HomotopyCategory.HomComplexSingle

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preadditive C
inst✝ : HasZeroObject C
X : C
K : CochainComplex C ℤ
p q : ℤ
f : X ⟶ K.X q
n : ℤ
h : p + n = q
⊢ (fromSingleMk f h).v p q h = (HomologicalComplex.singleObjXSelf (ComplexShape.up ℤ) p X).hom ≫ f

Tactic:
by
  simp [fromSingleMk]

Goal state after:
Module: Mathlib.Algebra.Homology.HomotopyCategory.HomComplexSingle

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preadditive C
inst✝ : HasZeroObject C
X : C
K : CochainComplex C ℤ
p n : ℤ
α : Cocycle ((singleFunctor C p).obj X) K n
q : ℤ
h : p + n = q
q' : ℤ
hq' : q + 1 = q'
f : X ⟶ K.X q
hf : Cochain.fromSingleMk f h = ↑α
hα : (Cochain.fromSingleMk (f ≫ K.d q q') ⋯).v p q' ⋯ = Cochain.v 0 p q' ⋯
⊢ fromSingleMk f h q' hq' ⋯ = α

Tactic:
ext : 1; assumption

Goal state after:
Module: Mathlib.Algebra.Homology.HomotopyCategory.HomComplexSingle

Goal state before:
pp: case mp
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preadditive C
inst✝ : HasZeroObject C
X : C
K : CochainComplex C ℤ
p q : ℤ
f : X ⟶ K.X q
n : ℤ
h : p + n = q
q' : ℤ
hq' : q + 1 = q'
hf : f ≫ K.d q q' = 0
q'' : ℤ
hq'' : q'' + 1 = q
α : Cochain ((singleFunctor C p).obj X) K (n - 1)
hα : δ (n - 1) n α = ↑(fromSingleMk f h q' hq' hf)
⊢ ∃ g, g ≫ K.d q'' q = f

Tactic:
obtain ⟨g, hg⟩ := Cochain.fromSingleMk_surjective α q'' (by lia)

Goal state after:
case mp
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preadditive C
inst✝ : HasZeroObject C
X : C
K : CochainComplex C ℤ
p q : ℤ
f : X ⟶ K.X q
n : ℤ
h : p + n = q
q' : ℤ
hq' : q + 1 = q'
hf : f ≫ K.d q q' = 0
q'' : ℤ
hq'' : q'' + 1 = q
α : Cochain ((singleFunctor C p).obj X) K (n - 1)
hα : δ (n - 1) n α = ↑(fromSingleMk f h q' hq' hf)
g : X ⟶ K.X q''
hg : Cochain.fromSingleMk g ⋯ = α
⊢ ∃ g, g ≫ K.d q'' q = f
Module: Mathlib.Algebra.Homology.HomotopyCategory.HomComplexSingle

Goal state before:
pp: case refine_2
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preadditive C
inst✝ : HasZeroObject C
X : C
K : CochainComplex C ℤ
q n : ℤ
α : Cocycle K ((singleFunctor C q).obj X) n
p : ℤ
h : p + n = q
p' : ℤ
hp' : p' + 1 = p
f : K.X p ⟶ X
hf : Cochain.toSingleMk f h = ↑α
hα : Cochain.toSingleMk (K.d p' p ≫ f) ⋯ = 0
⊢ toSingleMk f h p' hp' ⋯ = α

Tactic:
ext : 1; assumption

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.Projective.Resolution

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasProjectiveResolutions C
X Y : C
f : X ⟶ Y
P : ProjectiveResolution X
Q : ProjectiveResolution Y
φ : P.complex ⟶ Q.complex
comm : φ.f 0 ≫ Q.π.f 0 = P.π.f 0 ≫ f
⊢ P.iso.inv ≫ (projectiveResolutions C).map f ≫ Q.iso.hom =
    P.iso.inv ≫ P.iso.hom ≫ (HomotopyCategory.quotient C (ComplexShape.down ℕ)).map φ

Tactic:
iso_inv_naturality_assoc f P Q φ comm,

Goal state after:
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasProjectiveResolutions C
X Y : C
f : X ⟶ Y
P : ProjectiveResolution X
Q : ProjectiveResolution Y
φ : P.complex ⟶ Q.complex
comm : φ.f 0 ≫ Q.π.f 0 = P.π.f 0 ≫ f
⊢ (HomotopyCategory.quotient C (ComplexShape.down ℕ)).map φ ≫ Q.iso.inv ≫ Q.iso.hom =
    P.iso.inv ≫ P.iso.hom ≫ (HomotopyCategory.quotient C (ComplexShape.down ℕ)).map φ
Module: Mathlib.CategoryTheory.Abelian.Projective.Resolution

Goal state before:
pp: case zero
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : EnoughProjectives C
Z : C
⊢ QuasiIsoAt (((ofComplex Z).toSingle₀Equiv Z).symm ⟨Projective.π Z, ⋯⟩) 0

Tactic:
ChainComplex.quasiIsoAt₀_iff,

Goal state after:
case zero
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : EnoughProjectives C
Z : C
⊢ ShortComplex.QuasiIso
    ((HomologicalComplex.shortComplexFunctor' C (ComplexShape.down ℕ) 1 0 0).map
      (((ofComplex Z).toSingle₀Equiv Z).symm ⟨Projective.π Z, ⋯⟩))
Module: Mathlib.CategoryTheory.GradedObject.Unitor

Goal state before:
pp: case pos
C : Type u_1
D : Type u_2
I : Type u_3
J : Type u_4
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Category.{v_2, u_2} D
inst✝³ : Zero I
inst✝² : DecidableEq I
inst✝¹ : HasInitial C
F : C ⥤ D ⥤ D
X : C
e : F.obj X ≅ 𝟭 D
inst✝ : ∀ (Y : D), PreservesColimit (Functor.empty C) (F.flip.obj Y)
p : I × J → J
hp : ∀ (j : J), p (0, j) = j
Y Y' : GradedObject J D
φ : Y ⟶ Y'
j : J
s : Cofan ((((mapBifunctor F I J).obj ((single₀ I).obj X)).obj Y).mapObjFun p j)
j' : J
h : (0, j') ∈ p ⁻¹' {j}
⊢ Cofan.inj (mapBifunctorLeftUnitorCofan F X e p hp Y j) ⟨(0, j'), h⟩ ≫
      e.inv.app (Y j) ≫ (F.map (singleObjApplyIso 0 X).inv).app (Y j) ≫ s.inj ⟨(0, j), ⋯⟩ =
    s.inj ⟨(0, j'), h⟩

Tactic:
simp only [Set.mem_preimage, hp, Set.mem_singleton_iff] at h

Goal state after:
case pos
C : Type u_1
D : Type u_2
I : Type u_3
J : Type u_4
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Category.{v_2, u_2} D
inst✝³ : Zero I
inst✝² : DecidableEq I
inst✝¹ : HasInitial C
F : C ⥤ D ⥤ D
X : C
e : F.obj X ≅ 𝟭 D
inst✝ : ∀ (Y : D), PreservesColimit (Functor.empty C) (F.flip.obj Y)
p : I × J → J
hp : ∀ (j : J), p (0, j) = j
Y Y' : GradedObject J D
φ : Y ⟶ Y'
j : J
s : Cofan ((((mapBifunctor F I J).obj ((single₀ I).obj X)).obj Y).mapObjFun p j)
j' : J
h✝ : (0, j') ∈ p ⁻¹' {j}
h : j' = j
⊢ Cofan.inj (mapBifunctorLeftUnitorCofan F X e p hp Y j) ⟨(0, j'), h✝⟩ ≫
      e.inv.app (Y j) ≫ (F.map (singleObjApplyIso 0 X).inv).app (Y j) ≫ s.inj ⟨(0, j), ⋯⟩ =
    s.inj ⟨(0, j'), h✝⟩
Module: Mathlib.CategoryTheory.GradedObject.Unitor

Goal state before:
pp: case pos
C : Type u_1
D : Type u_2
I : Type u_3
J : Type u_4
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Category.{v_2, u_2} D
inst✝³ : Zero I
inst✝² : DecidableEq I
inst✝¹ : HasInitial C
F : D ⥤ C ⥤ D
Y : C
e : F.flip.obj Y ≅ 𝟭 D
inst✝ : ∀ (X : D), PreservesColimit (Functor.empty C) (F.obj X)
p : J × I → J
hp : ∀ (j : J), p (j, 0) = j
X X' : GradedObject J D
φ : X ⟶ X'
j : J
s : Cofan ((((mapBifunctor F J I).obj X).obj ((single₀ I).obj Y)).mapObjFun p j)
j' : J
h : (j', 0) ∈ p ⁻¹' {j}
⊢ Cofan.inj (mapBifunctorRightUnitorCofan F Y e p hp X j) ⟨(j', 0), h⟩ ≫
      e.inv.app (X j) ≫ (F.obj (X j)).map (singleObjApplyIso 0 Y).inv ≫ s.inj ⟨(j, 0), ⋯⟩ =
    s.inj ⟨(j', 0), h⟩

Tactic:
simp only [Set.mem_preimage, hp, Set.mem_singleton_iff] at h

Goal state after:
case pos
C : Type u_1
D : Type u_2
I : Type u_3
J : Type u_4
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Category.{v_2, u_2} D
inst✝³ : Zero I
inst✝² : DecidableEq I
inst✝¹ : HasInitial C
F : D ⥤ C ⥤ D
Y : C
e : F.flip.obj Y ≅ 𝟭 D
inst✝ : ∀ (X : D), PreservesColimit (Functor.empty C) (F.obj X)
p : J × I → J
hp : ∀ (j : J), p (j, 0) = j
X X' : GradedObject J D
φ : X ⟶ X'
j : J
s : Cofan ((((mapBifunctor F J I).obj X).obj ((single₀ I).obj Y)).mapObjFun p j)
j' : J
h✝ : (j', 0) ∈ p ⁻¹' {j}
h : j' = j
⊢ Cofan.inj (mapBifunctorRightUnitorCofan F Y e p hp X j) ⟨(j', 0), h✝⟩ ≫
      e.inv.app (X j) ≫ (F.obj (X j)).map (singleObjApplyIso 0 Y).inv ≫ s.inj ⟨(j, 0), ⋯⟩ =
    s.inj ⟨(j', 0), h✝⟩
Module: Mathlib.Algebra.Homology.SpectralObject.Basic

Goal state before:
pp: C : Type u_1
ι : Type u_2
inst✝² : Category.{u_3, u_1} C
inst✝¹ : Category.{u_4, u_2} ι
inst✝ : Abelian C
X : SpectralObject C ι
i j k : ι
f : i ⟶ j
g : j ⟶ k
fg : i ⟶ k
h : f ≫ g = fg
n₀ n₁ : ℤ
hn₁ : n₀ + 1 = n₁
⊢ (X.sc₃ f g fg h n₀ n₁ hn₁).Exact

Tactic:
by
  subst h
  exact ((X.exact₃' n₀ n₁ hn₁ (mk₂ f g))).exact 0

Goal state after:
Module: Mathlib.Algebra.Homology.SpectralObject.Homology

Goal state before:
pp: C : Type u_1
ι : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : Category.{v_2, u_2} ι
X : SpectralObject C ι
i₀ i₁ i₂ i₃ i₄ i₅ : ι
f₁ : i₀ ⟶ i₁
f₂ : i₁ ⟶ i₂
f₃ : i₂ ⟶ i₃
f₄ : i₃ ⟶ i₄
f₅ : i₄ ⟶ i₅
f₃₄ : i₂ ⟶ i₄
h₃₄ : f₃ ≫ f₄ = f₃₄
n₀ n₁ n₂ n₃ : ℤ
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
hn₃ : n₂ + 1 = n₃
A : C
x₂ : A ⟶ (X.dCokernelSequence f₁ f₂ f₃ f₄ f₅ f₃₄ h₃₄ n₀ n₁ n₂ n₃ ⋯ ⋯ ⋯).X₂
hx₂ : x₂ ≫ X.map f₁ f₂ f₃ f₁ f₂ f₃₄ (fourδ₄Toδ₃ f₁ f₂ f₃ f₄ f₃₄ h₃₄) n₁ n₂ n₃ ⋯ ⋯ = 0
hx₂'✝ : x₂ ≫ X.map f₁ f₂ f₃ f₁ f₂ f₃₄ (fourδ₄Toδ₃ f₁ f₂ f₃ f₄ f₃₄ h₃₄) n₁ n₂ n₃ ⋯ ⋯ ≫ X.ιE f₁ f₂ f₃₄ n₁ n₂ n₃ ⋯ ⋯ = 0
hx₂' : x₂ ≫ X.ιE f₁ f₂ f₃ n₁ n₂ n₃ ⋯ ⋯ ≫ X.opcyclesMap f₂ f₃ f₂ f₃₄ (threeδ₃Toδ₂ f₂ f₃ f₄ f₃₄ ⋯) n₂ = 0
A₁ : C
π₁ : A₁ ⟶ A
w✝ : Epi π₁
x₁ : A₁ ⟶ ((X.sequenceΨ f₂ f₃ f₄ (f₂ ≫ f₃) ⋯ f₃₄ h₃₄ n₁ n₂ ⋯).sc ⋯ 1 ⋯).X₁
hx₁ : π₁ ≫ x₂ ≫ X.ιE f₁ f₂ f₃ n₁ n₂ n₃ ⋯ ⋯ = x₁ ≫ X.Ψ f₂ f₃ f₄ n₁ n₂ ⋯
⊢ x₁ ≫ X.Ψ f₂ f₃ f₄ n₁ n₂ ⋯ =
    x₁ ≫ X.πE f₃ f₄ f₅ n₀ n₁ n₂ ⋯ ⋯ ≫ X.d f₁ f₂ f₃ f₄ f₅ n₀ n₁ n₂ n₃ ⋯ ⋯ ⋯ ≫ X.ιE f₁ f₂ f₃ n₁ n₂ n₃ ⋯ ⋯

Tactic:
πE_d_ιE ..

Goal state after:
C : Type u_1
ι : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : Category.{v_2, u_2} ι
X : SpectralObject C ι
i₀ i₁ i₂ i₃ i₄ i₅ : ι
f₁ : i₀ ⟶ i₁
f₂ : i₁ ⟶ i₂
f₃ : i₂ ⟶ i₃
f₄ : i₃ ⟶ i₄
f₅ : i₄ ⟶ i₅
f₃₄ : i₂ ⟶ i₄
h₃₄ : f₃ ≫ f₄ = f₃₄
n₀ n₁ n₂ n₃ : ℤ
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
hn₃ : n₂ + 1 = n₃
A : C
x₂ : A ⟶ (X.dCokernelSequence f₁ f₂ f₃ f₄ f₅ f₃₄ h₃₄ n₀ n₁ n₂ n₃ ⋯ ⋯ ⋯).X₂
hx₂ : x₂ ≫ X.map f₁ f₂ f₃ f₁ f₂ f₃₄ (fourδ₄Toδ₃ f₁ f₂ f₃ f₄ f₃₄ h₃₄) n₁ n₂ n₃ ⋯ ⋯ = 0
hx₂'✝ : x₂ ≫ X.map f₁ f₂ f₃ f₁ f₂ f₃₄ (fourδ₄Toδ₃ f₁ f₂ f₃ f₄ f₃₄ h₃₄) n₁ n₂ n₃ ⋯ ⋯ ≫ X.ιE f₁ f₂ f₃₄ n₁ n₂ n₃ ⋯ ⋯ = 0
hx₂' : x₂ ≫ X.ιE f₁ f₂ f₃ n₁ n₂ n₃ ⋯ ⋯ ≫ X.opcyclesMap f₂ f₃ f₂ f₃₄ (threeδ₃Toδ₂ f₂ f₃ f₄ f₃₄ ⋯) n₂ = 0
A₁ : C
π₁ : A₁ ⟶ A
w✝ : Epi π₁
x₁ : A₁ ⟶ ((X.sequenceΨ f₂ f₃ f₄ (f₂ ≫ f₃) ⋯ f₃₄ h₃₄ n₁ n₂ ⋯).sc ⋯ 1 ⋯).X₁
hx₁ : π₁ ≫ x₂ ≫ X.ιE f₁ f₂ f₃ n₁ n₂ n₃ ⋯ ⋯ = x₁ ≫ X.Ψ f₂ f₃ f₄ n₁ n₂ ⋯
⊢ x₁ ≫ X.Ψ f₂ f₃ f₄ n₁ n₂ ⋯ = x₁ ≫ X.Ψ f₂ f₃ f₄ n₁ n₂ ⋯
Module: Mathlib.Algebra.Homology.SpectralObject.Page

Goal state before:
pp: C : Type u_1
ι : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} ι
inst✝ : Abelian C
X : SpectralObject C ι
i j k l : ι
f₁ : i ⟶ j
f₂ : j ⟶ k
f₃ : k ⟶ l
f₁₂ : i ⟶ k
h₁₂ : f₁ ≫ f₂ = f₁₂
f₂₃ : j ⟶ l
h₂₃ : f₂ ≫ f₃ = f₂₃
n₀ n₁ n₂ : ℤ
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
hi : IsLimit (KernelFork.ofι (X.kernelSequenceCycles f₁ f₂ n₁ n₂ hn₂).f ⋯) := ⋯.fIsKernel
this : hi.lift (KernelFork.ofι (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ ⋯ ⋯).f ⋯) = X.δToCycles f₁ f₂ f₃ n₀ n₁ ⋯
⊢ hi.lift (KernelFork.ofι (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).f ⋯) ≫ cokernel.π (X.δToCycles f₁ f₂ f₃ n₀ n₁ ⋯) =
    0

Tactic:
rw [this]

Goal state after:
C : Type u_1
ι : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} ι
inst✝ : Abelian C
X : SpectralObject C ι
i j k l : ι
f₁ : i ⟶ j
f₂ : j ⟶ k
f₃ : k ⟶ l
f₁₂ : i ⟶ k
h₁₂ : f₁ ≫ f₂ = f₁₂
f₂₃ : j ⟶ l
h₂₃ : f₂ ≫ f₃ = f₂₃
n₀ n₁ n₂ : ℤ
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
hi : IsLimit (KernelFork.ofι (X.kernelSequenceCycles f₁ f₂ n₁ n₂ hn₂).f ⋯) := ⋯.fIsKernel
this : hi.lift (KernelFork.ofι (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ ⋯ ⋯).f ⋯) = X.δToCycles f₁ f₂ f₃ n₀ n₁ ⋯
⊢ X.δToCycles f₁ f₂ f₃ n₀ n₁ ⋯ ≫ cokernel.π (X.δToCycles f₁ f₂ f₃ n₀ n₁ ⋯) = 0
Module: Mathlib.Algebra.Homology.SpectralObject.Page

Goal state before:
pp: C : Type u_1
ι : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} ι
inst✝ : Abelian C
X : SpectralObject C ι
i j k l : ι
f₁ : i ⟶ j
f₂ : j ⟶ k
f₃ : k ⟶ l
f₁₂ : i ⟶ k
h₁₂ : f₁ ≫ f₂ = f₁₂
f₂₃ : j ⟶ l
h₂₃ : f₂ ≫ f₃ = f₂₃
n₀ n₁ n₂ : ℤ
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
hp : IsColimit (CokernelCofork.ofπ (X.cokernelSequenceOpcycles f₂ f₃ n₀ n₁ hn₁).g ⋯) := ⋯.gIsCokernel
this : hp.desc (CokernelCofork.ofπ (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).g ⋯) = X.δFromOpcycles f₁ f₂ f₃ n₁ n₂ hn₂
⊢ kernel.ι (X.δFromOpcycles f₁ f₂ f₃ n₁ n₂ ⋯) ≫
      hp.desc (CokernelCofork.ofπ (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).g ⋯) =
    0

Tactic:
rw [this]

Goal state after:
C : Type u_1
ι : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} ι
inst✝ : Abelian C
X : SpectralObject C ι
i j k l : ι
f₁ : i ⟶ j
f₂ : j ⟶ k
f₃ : k ⟶ l
f₁₂ : i ⟶ k
h₁₂ : f₁ ≫ f₂ = f₁₂
f₂₃ : j ⟶ l
h₂₃ : f₂ ≫ f₃ = f₂₃
n₀ n₁ n₂ : ℤ
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
hp : IsColimit (CokernelCofork.ofπ (X.cokernelSequenceOpcycles f₂ f₃ n₀ n₁ hn₁).g ⋯) := ⋯.gIsCokernel
this : hp.desc (CokernelCofork.ofπ (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).g ⋯) = X.δFromOpcycles f₁ f₂ f₃ n₁ n₂ hn₂
⊢ kernel.ι (X.δFromOpcycles f₁ f₂ f₃ n₁ n₂ ⋯) ≫ X.δFromOpcycles f₁ f₂ f₃ n₁ n₂ hn₂ = 0
Module: Mathlib.Algebra.Homology.SpectralObject.Page

Goal state before:
pp: C : Type u_1
ι : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} ι
inst✝ : Abelian C
X : SpectralObject C ι
i j k l : ι
f₁ : i ⟶ j
f₂ : j ⟶ k
f₃ : k ⟶ l
f₁₂ : i ⟶ k
h₁₂ : f₁ ≫ f₂ = f₁₂
n₀ n₁ n₂ : ℤ
A : C
x : (X.H n₁).obj (mk₁ f₁₂) ⟶ A
h : (X.H n₁).map (twoδ₂Toδ₁ f₁ f₂ f₁₂ h₁₂) ≫ x = 0
hn₁ : n₀ + 1 = n₁
h' : X.δ f₁₂ f₃ n₀ n₁ hn₁ ≫ x = 0
hn₂ : n₁ + 1 = n₂
⊢ X.toCycles f₁ f₂ f₁₂ h₁₂ n₁ ≫ X.πE f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂ ≫ X.descE f₁ f₂ f₃ f₁₂ h₁₂ n₀ n₁ n₂ x h hn₁ h' hn₂ = x

Tactic:
dsimp only [descE]

Goal state after:
C : Type u_1
ι : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} ι
inst✝ : Abelian C
X : SpectralObject C ι
i j k l : ι
f₁ : i ⟶ j
f₂ : j ⟶ k
f₃ : k ⟶ l
f₁₂ : i ⟶ k
h₁₂ : f₁ ≫ f₂ = f₁₂
n₀ n₁ n₂ : ℤ
A : C
x : (X.H n₁).obj (mk₁ f₁₂) ⟶ A
h : (X.H n₁).map (twoδ₂Toδ₁ f₁ f₂ f₁₂ h₁₂) ≫ x = 0
hn₁ : n₀ + 1 = n₁
h' : X.δ f₁₂ f₃ n₀ n₁ hn₁ ≫ x = 0
hn₂ : n₁ + 1 = n₂
⊢ X.toCycles f₁ f₂ f₁₂ h₁₂ n₁ ≫ X.πE f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂ ≫ ⋯.desc x ⋯ = x