mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Algebra.Homology.BifunctorHomotopy

Goal state before:
pp: case neg
C₁ : Type u_1
C₂ : Type u_2
D : Type u_3
I₁ : Type u_4
I₂ : Type u_5
J : Type u_6
inst✝¹¹ : Category.{v_1, u_1} C₁
inst✝¹⁰ : Category.{v_2, u_2} C₂
inst✝⁹ : Category.{v_3, u_3} D
inst✝⁸ : Preadditive C₁
inst✝⁷ : Preadditive C₂
inst✝⁶ : Preadditive D
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
K₁ L₁ : HomologicalComplex C₁ c₁
f₁ f₁' : K₁ ⟶ L₁
h₁ : Homotopy f₁ f₁'
K₂ L₂ : HomologicalComplex C₂ c₂
f₂ f₂' : K₂ ⟶ L₂
h₂ : Homotopy f₂ f₂'
F : C₁ ⥤ C₂ ⥤ D
inst✝⁵ : F.Additive
inst✝⁴ : ∀ (X₁ : C₁), (F.obj X₁).Additive
c : ComplexShape J
inst✝³ : DecidableEq J
inst✝² : TotalComplexShape c₁ c₂ c
inst✝¹ : K₁.HasMapBifunctor K₂ F c
inst✝ : L₁.HasMapBifunctor L₂ F c
this✝² : TotalComplexShape c₂ c₁ c := TotalComplexShape.symm c₁ c₂ c
this✝¹ : TotalComplexShapeSymmetry c₁ c₂ c := TotalComplexShape.symmSymmetry c₁ c₂ c
this✝ : F.flip.Additive
this : ∀ (X₁ : C₂), (F.flip.obj X₁).Additive
H : Homotopy (mapBifunctorMap f₁ f₂ F c) (mapBifunctorMap f₁ f₂' F c) :=
  (Homotopy.ofEq ⋯).trans
    ((((mapBifunctorMapHomotopy₁ h₂ f₁ F.flip c).compRight (L₁.mapBifunctorFlipIso L₂ F c).hom).compLeft
          (K₁.mapBifunctorFlipIso K₂ F c).inv).trans
      (Homotopy.ofEq ⋯))
j j' : J
i₁ : I₁
i₂ : I₂
hj : c₁.π c₂ c (i₁, i₂) = j
hj' : ¬c₁.π c₂ c (i₁, c₂.prev i₂) = j'
⊢ c₁.ε₂ c₂ c (i₁, c₂.prev i₂) • (F.map (f₁.f i₁)).app (K₂.X i₂) ≫ 0 =
    c₁.σ c₂ c i₁ i₂ •
      c₂.ε₁ c₁ c (c₂.prev i₂, i₁) •
        (F.map (f₁.f i₁)).app (K₂.X i₂) ≫
          (F.obj (L₁.X i₁)).map (h₂.hom i₂ (c₂.prev i₂)) ≫ 0 ≫ (L₁.mapBifunctorFlipIso L₂ F c).hom.f j'

Tactic:
comp_zero,

Goal state after:
case neg
C₁ : Type u_1
C₂ : Type u_2
D : Type u_3
I₁ : Type u_4
I₂ : Type u_5
J : Type u_6
inst✝¹¹ : Category.{v_1, u_1} C₁
inst✝¹⁰ : Category.{v_2, u_2} C₂
inst✝⁹ : Category.{v_3, u_3} D
inst✝⁸ : Preadditive C₁
inst✝⁷ : Preadditive C₂
inst✝⁶ : Preadditive D
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
K₁ L₁ : HomologicalComplex C₁ c₁
f₁ f₁' : K₁ ⟶ L₁
h₁ : Homotopy f₁ f₁'
K₂ L₂ : HomologicalComplex C₂ c₂
f₂ f₂' : K₂ ⟶ L₂
h₂ : Homotopy f₂ f₂'
F : C₁ ⥤ C₂ ⥤ D
inst✝⁵ : F.Additive
inst✝⁴ : ∀ (X₁ : C₁), (F.obj X₁).Additive
c : ComplexShape J
inst✝³ : DecidableEq J
inst✝² : TotalComplexShape c₁ c₂ c
inst✝¹ : K₁.HasMapBifunctor K₂ F c
inst✝ : L₁.HasMapBifunctor L₂ F c
this✝² : TotalComplexShape c₂ c₁ c := TotalComplexShape.symm c₁ c₂ c
this✝¹ : TotalComplexShapeSymmetry c₁ c₂ c := TotalComplexShape.symmSymmetry c₁ c₂ c
this✝ : F.flip.Additive
this : ∀ (X₁ : C₂), (F.flip.obj X₁).Additive
H : Homotopy (mapBifunctorMap f₁ f₂ F c) (mapBifunctorMap f₁ f₂' F c) :=
  (Homotopy.ofEq ⋯).trans
    ((((mapBifunctorMapHomotopy₁ h₂ f₁ F.flip c).compRight (L₁.mapBifunctorFlipIso L₂ F c).hom).compLeft
          (K₁.mapBifunctorFlipIso K₂ F c).inv).trans
      (Homotopy.ofEq ⋯))
j j' : J
i₁ : I₁
i₂ : I₂
hj : c₁.π c₂ c (i₁, i₂) = j
hj' : ¬c₁.π c₂ c (i₁, c₂.prev i₂) = j'
⊢ c₁.ε₂ c₂ c (i₁, c₂.prev i₂) • 0 =
    c₁.σ c₂ c i₁ i₂ •
      c₂.ε₁ c₁ c (c₂.prev i₂, i₁) •
        (F.map (f₁.f i₁)).app (K₂.X i₂) ≫
          (F.obj (L₁.X i₁)).map (h₂.hom i₂ (c₂.prev i₂)) ≫ 0 ≫ (L₁.mapBifunctorFlipIso L₂ F c).hom.f j'
Module: Mathlib.Algebra.Homology.BifunctorHomotopy

Goal state before:
pp: case neg
C₁ : Type u_1
C₂ : Type u_2
D : Type u_3
I₁ : Type u_4
I₂ : Type u_5
J : Type u_6
inst✝¹¹ : Category.{v_1, u_1} C₁
inst✝¹⁰ : Category.{v_2, u_2} C₂
inst✝⁹ : Category.{v_3, u_3} D
inst✝⁸ : Preadditive C₁
inst✝⁷ : Preadditive C₂
inst✝⁶ : Preadditive D
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
K₁ L₁ : HomologicalComplex C₁ c₁
f₁ f₁' : K₁ ⟶ L₁
h₁ : Homotopy f₁ f₁'
K₂ L₂ : HomologicalComplex C₂ c₂
f₂ f₂' : K₂ ⟶ L₂
h₂ : Homotopy f₂ f₂'
F : C₁ ⥤ C₂ ⥤ D
inst✝⁵ : F.Additive
inst✝⁴ : ∀ (X₁ : C₁), (F.obj X₁).Additive
c : ComplexShape J
inst✝³ : DecidableEq J
inst✝² : TotalComplexShape c₁ c₂ c
inst✝¹ : K₁.HasMapBifunctor K₂ F c
inst✝ : L₁.HasMapBifunctor L₂ F c
this✝² : TotalComplexShape c₂ c₁ c := TotalComplexShape.symm c₁ c₂ c
this✝¹ : TotalComplexShapeSymmetry c₁ c₂ c := TotalComplexShape.symmSymmetry c₁ c₂ c
this✝ : F.flip.Additive
this : ∀ (X₁ : C₂), (F.flip.obj X₁).Additive
H : Homotopy (mapBifunctorMap f₁ f₂ F c) (mapBifunctorMap f₁ f₂' F c) :=
  (Homotopy.ofEq ⋯).trans
    ((((mapBifunctorMapHomotopy₁ h₂ f₁ F.flip c).compRight (L₁.mapBifunctorFlipIso L₂ F c).hom).compLeft
          (K₁.mapBifunctorFlipIso K₂ F c).inv).trans
      (Homotopy.ofEq ⋯))
j j' : J
i₁ : I₁
i₂ : I₂
hj : c₁.π c₂ c (i₁, i₂) = j
hj' : ¬c₁.π c₂ c (i₁, c₂.prev i₂) = j'
⊢ 0 =
    c₁.σ c₂ c i₁ i₂ •
      c₂.ε₁ c₁ c (c₂.prev i₂, i₁) • (F.map (f₁.f i₁)).app (K₂.X i₂) ≫ (F.obj (L₁.X i₁)).map (h₂.hom i₂ (c₂.prev i₂)) ≫ 0

Tactic:
comp_zero,

Goal state after:
case neg
C₁ : Type u_1
C₂ : Type u_2
D : Type u_3
I₁ : Type u_4
I₂ : Type u_5
J : Type u_6
inst✝¹¹ : Category.{v_1, u_1} C₁
inst✝¹⁰ : Category.{v_2, u_2} C₂
inst✝⁹ : Category.{v_3, u_3} D
inst✝⁸ : Preadditive C₁
inst✝⁷ : Preadditive C₂
inst✝⁶ : Preadditive D
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
K₁ L₁ : HomologicalComplex C₁ c₁
f₁ f₁' : K₁ ⟶ L₁
h₁ : Homotopy f₁ f₁'
K₂ L₂ : HomologicalComplex C₂ c₂
f₂ f₂' : K₂ ⟶ L₂
h₂ : Homotopy f₂ f₂'
F : C₁ ⥤ C₂ ⥤ D
inst✝⁵ : F.Additive
inst✝⁴ : ∀ (X₁ : C₁), (F.obj X₁).Additive
c : ComplexShape J
inst✝³ : DecidableEq J
inst✝² : TotalComplexShape c₁ c₂ c
inst✝¹ : K₁.HasMapBifunctor K₂ F c
inst✝ : L₁.HasMapBifunctor L₂ F c
this✝² : TotalComplexShape c₂ c₁ c := TotalComplexShape.symm c₁ c₂ c
this✝¹ : TotalComplexShapeSymmetry c₁ c₂ c := TotalComplexShape.symmSymmetry c₁ c₂ c
this✝ : F.flip.Additive
this : ∀ (X₁ : C₂), (F.flip.obj X₁).Additive
H : Homotopy (mapBifunctorMap f₁ f₂ F c) (mapBifunctorMap f₁ f₂' F c) :=
  (Homotopy.ofEq ⋯).trans
    ((((mapBifunctorMapHomotopy₁ h₂ f₁ F.flip c).compRight (L₁.mapBifunctorFlipIso L₂ F c).hom).compLeft
          (K₁.mapBifunctorFlipIso K₂ F c).inv).trans
      (Homotopy.ofEq ⋯))
j j' : J
i₁ : I₁
i₂ : I₂
hj : c₁.π c₂ c (i₁, i₂) = j
hj' : ¬c₁.π c₂ c (i₁, c₂.prev i₂) = j'
⊢ 0 = c₁.σ c₂ c i₁ i₂ • c₂.ε₁ c₁ c (c₂.prev i₂, i₁) • (F.map (f₁.f i₁)).app (K₂.X i₂) ≫ 0
Module: Mathlib.Algebra.Homology.BifunctorHomotopy

Goal state before:
pp: case neg
C₁ : Type u_1
C₂ : Type u_2
D : Type u_3
I₁ : Type u_4
I₂ : Type u_5
J : Type u_6
inst✝¹¹ : Category.{v_1, u_1} C₁
inst✝¹⁰ : Category.{v_2, u_2} C₂
inst✝⁹ : Category.{v_3, u_3} D
inst✝⁸ : Preadditive C₁
inst✝⁷ : Preadditive C₂
inst✝⁶ : Preadditive D
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
K₁ L₁ : HomologicalComplex C₁ c₁
f₁ f₁' : K₁ ⟶ L₁
h₁ : Homotopy f₁ f₁'
K₂ L₂ : HomologicalComplex C₂ c₂
f₂ f₂' : K₂ ⟶ L₂
h₂ : Homotopy f₂ f₂'
F : C₁ ⥤ C₂ ⥤ D
inst✝⁵ : F.Additive
inst✝⁴ : ∀ (X₁ : C₁), (F.obj X₁).Additive
c : ComplexShape J
inst✝³ : DecidableEq J
inst✝² : TotalComplexShape c₁ c₂ c
inst✝¹ : K₁.HasMapBifunctor K₂ F c
inst✝ : L₁.HasMapBifunctor L₂ F c
this✝² : TotalComplexShape c₂ c₁ c := TotalComplexShape.symm c₁ c₂ c
this✝¹ : TotalComplexShapeSymmetry c₁ c₂ c := TotalComplexShape.symmSymmetry c₁ c₂ c
this✝ : F.flip.Additive
this : ∀ (X₁ : C₂), (F.flip.obj X₁).Additive
H : Homotopy (mapBifunctorMap f₁ f₂ F c) (mapBifunctorMap f₁ f₂' F c) :=
  (Homotopy.ofEq ⋯).trans
    ((((mapBifunctorMapHomotopy₁ h₂ f₁ F.flip c).compRight (L₁.mapBifunctorFlipIso L₂ F c).hom).compLeft
          (K₁.mapBifunctorFlipIso K₂ F c).inv).trans
      (Homotopy.ofEq ⋯))
j j' : J
i₁ : I₁
i₂ : I₂
hj : c₁.π c₂ c (i₁, i₂) = j
hj' : ¬c₁.π c₂ c (i₁, c₂.prev i₂) = j'
⊢ 0 = c₁.σ c₂ c i₁ i₂ • c₂.ε₁ c₁ c (c₂.prev i₂, i₁) • (F.map (f₁.f i₁)).app (K₂.X i₂) ≫ 0

Tactic:
comp_zero,

Goal state after:
case neg
C₁ : Type u_1
C₂ : Type u_2
D : Type u_3
I₁ : Type u_4
I₂ : Type u_5
J : Type u_6
inst✝¹¹ : Category.{v_1, u_1} C₁
inst✝¹⁰ : Category.{v_2, u_2} C₂
inst✝⁹ : Category.{v_3, u_3} D
inst✝⁸ : Preadditive C₁
inst✝⁷ : Preadditive C₂
inst✝⁶ : Preadditive D
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
K₁ L₁ : HomologicalComplex C₁ c₁
f₁ f₁' : K₁ ⟶ L₁
h₁ : Homotopy f₁ f₁'
K₂ L₂ : HomologicalComplex C₂ c₂
f₂ f₂' : K₂ ⟶ L₂
h₂ : Homotopy f₂ f₂'
F : C₁ ⥤ C₂ ⥤ D
inst✝⁵ : F.Additive
inst✝⁴ : ∀ (X₁ : C₁), (F.obj X₁).Additive
c : ComplexShape J
inst✝³ : DecidableEq J
inst✝² : TotalComplexShape c₁ c₂ c
inst✝¹ : K₁.HasMapBifunctor K₂ F c
inst✝ : L₁.HasMapBifunctor L₂ F c
this✝² : TotalComplexShape c₂ c₁ c := TotalComplexShape.symm c₁ c₂ c
this✝¹ : TotalComplexShapeSymmetry c₁ c₂ c := TotalComplexShape.symmSymmetry c₁ c₂ c
this✝ : F.flip.Additive
this : ∀ (X₁ : C₂), (F.flip.obj X₁).Additive
H : Homotopy (mapBifunctorMap f₁ f₂ F c) (mapBifunctorMap f₁ f₂' F c) :=
  (Homotopy.ofEq ⋯).trans
    ((((mapBifunctorMapHomotopy₁ h₂ f₁ F.flip c).compRight (L₁.mapBifunctorFlipIso L₂ F c).hom).compLeft
          (K₁.mapBifunctorFlipIso K₂ F c).inv).trans
      (Homotopy.ofEq ⋯))
j j' : J
i₁ : I₁
i₂ : I₂
hj : c₁.π c₂ c (i₁, i₂) = j
hj' : ¬c₁.π c₂ c (i₁, c₂.prev i₂) = j'
⊢ 0 = c₁.σ c₂ c i₁ i₂ • c₂.ε₁ c₁ c (c₂.prev i₂, i₁) • 0
Module: Mathlib.Algebra.Homology.TotalComplexShift

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : Preadditive C
K L : HomologicalComplex₂ C (up ℤ) (up ℤ)
f : K ⟶ L
y : ℤ
inst✝¹ : K.HasTotal (up ℤ)
inst✝ : L.HasTotal (up ℤ)
n i₁ i₂ : ℤ
h : i₁ + i₂ = n
⊢ ((shiftFunctor₂ C y).obj K).ιTotal (up ℤ) i₁ i₂ n h ≫
      (total.map ((shiftFunctor₂ C y).map f) (up ℤ)).f n ≫ (L.totalShift₂Iso y).hom.f n =
    ((shiftFunctor₂ C y).obj K).ιTotal (up ℤ) i₁ i₂ n h ≫ (K.totalShift₂Iso y).hom.f n ≫ (total.map f (up ℤ)).f (n + y)

Tactic:
rw [ιTotal_map_assoc, L.ι_totalShift₂Iso_hom_f y i₁ i₂ n h _ rfl _ rfl,
    K.ι_totalShift₂Iso_hom_f_assoc y i₁ i₂ n h _ rfl _ rfl]

Goal state after:
C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : Preadditive C
K L : HomologicalComplex₂ C (up ℤ) (up ℤ)
f : K ⟶ L
y : ℤ
inst✝¹ : K.HasTotal (up ℤ)
inst✝ : L.HasTotal (up ℤ)
n i₁ i₂ : ℤ
h : i₁ + i₂ = n
⊢ (((shiftFunctor₂ C y).map f).f i₁).f i₂ ≫
      ((i₁ * y).negOnePow •
        (L.shiftFunctor₂XXIso i₁ i₂ y (i₂ + y) ⋯).hom ≫
          L.ιTotal (up ℤ) i₁ (i₂ + y) (n + y) ⋯ ≫
            (CochainComplex.shiftFunctorObjXIso (L.total (up ℤ)) y n (n + y) ⋯).inv) =
    ((i₁ * y).negOnePow •
        (K.shiftFunctor₂XXIso i₁ i₂ y (i₂ + y) ⋯).hom ≫
          K.ιTotal (up ℤ) i₁ (i₂ + y) (n + y) ⋯ ≫
            (CochainComplex.shiftFunctorObjXIso (K.total (up ℤ)) y n (n + y) ⋯).inv) ≫
      (total.map f (up ℤ)).f (n + y)
Module: Mathlib.Algebra.Homology.BifunctorShift

Goal state before:
pp: C₁ : Type u_1
C₂ : Type u_2
D : Type u_3
inst✝⁹ : Category.{v_1, u_1} C₁
inst✝⁸ : Category.{v_2, u_2} C₂
inst✝⁷ : Category.{v_3, u_3} D
inst✝⁶ : HasZeroMorphisms C₁
inst✝⁵ : Preadditive C₂
inst✝⁴ : Preadditive D
K₁ : CochainComplex C₁ ℤ
K₂ L₂ : CochainComplex C₂ ℤ
f₂ : K₂ ⟶ L₂
F : C₁ ⥤ C₂ ⥤ D
inst✝³ : F.PreservesZeroMorphisms
inst✝² : ∀ (X₁ : C₁), (F.obj X₁).Additive
y : ℤ
inst✝¹ : K₁.HasMapBifunctor K₂ F
inst✝ : K₁.HasMapBifunctor L₂ F
⊢ mapBifunctorMap (𝟙 K₁) ((CategoryTheory.shiftFunctor (HomologicalComplex C₂ (ComplexShape.up ℤ)) y).map f₂) F
        (ComplexShape.up ℤ) ≫
      (K₁.mapBifunctorShift₂Iso L₂ F y).hom =
    (K₁.mapBifunctorShift₂Iso K₂ F y).hom ≫
      (CategoryTheory.shiftFunctor (CochainComplex D ℤ) y).map (mapBifunctorMap (𝟙 K₁) f₂ F (ComplexShape.up ℤ))

Tactic:
ext n p q h
  simp [ι_mapBifunctorShift₂Iso_hom_f _ _ _ _ _ _ _ _ (q + y) (n + y) rfl rfl,
    ι_mapBifunctorShift₂Iso_hom_f_assoc _ _ _ _ _ _ _ _ (q + y) (n + y) rfl rfl]

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

Goal state before:
pp: case neg
C₁ : Type u_1
C₂ : Type u_2
C₁₂ : Type u_3
C₃ : Type u_5
C₄ : Type u_6
inst✝²⁰ : Category.{v_1, u_1} C₁
inst✝¹⁹ : Category.{v_2, u_2} C₂
inst✝¹⁸ : Category.{v_3, u_5} C₃
inst✝¹⁷ : Category.{v_4, u_6} C₄
inst✝¹⁶ : Category.{v_5, u_3} C₁₂
inst✝¹⁵ : HasZeroMorphisms C₁
inst✝¹⁴ : HasZeroMorphisms C₂
inst✝¹³ : HasZeroMorphisms C₃
inst✝¹² : Preadditive C₁₂
inst✝¹¹ : Preadditive C₄
F₁₂ : C₁ ⥤ C₂ ⥤ C₁₂
G : C₁₂ ⥤ C₃ ⥤ C₄
inst✝¹⁰ : F₁₂.PreservesZeroMorphisms
inst✝⁹ : ∀ (X₁ : C₁), (F₁₂.obj X₁).PreservesZeroMorphisms
inst✝⁸ : G.Additive
inst✝⁷ : ∀ (X₁₂ : C₁₂), (G.obj X₁₂).PreservesZeroMorphisms
ι₁ : Type u_7
ι₂ : Type u_8
ι₃ : Type u_9
ι₁₂ : Type u_10
ι₄ : Type u_12
inst✝⁶ : DecidableEq ι₄
c₁ : ComplexShape ι₁
c₂ : ComplexShape ι₂
c₃ : ComplexShape ι₃
K₁ : HomologicalComplex C₁ c₁
K₂ : HomologicalComplex C₂ c₂
K₃ : HomologicalComplex C₃ c₃
c₁₂ : ComplexShape ι₁₂
c₄ : ComplexShape ι₄
inst✝⁵ : TotalComplexShape c₁ c₂ c₁₂
inst✝⁴ : TotalComplexShape c₁₂ c₃ c₄
inst✝³ : K₁.HasMapBifunctor K₂ F₁₂ c₁₂
inst✝² : DecidableEq ι₁₂
inst✝¹ : (K₁.mapBifunctor K₂ F₁₂ c₁₂).HasMapBifunctor K₃ G c₄
j j' : ι₄
inst✝ : HasGoodTrifunctor₁₂Obj F₁₂ G K₁ K₂ K₃ c₁₂ c₄
i₁ : ι₁
i₂ : ι₂
i₃ : ι₃
h : c₁.r c₂ c₃ c₁₂ c₄ (i₁, i₂, i₃) = j
i₁₂ : ι₁₂ := c₁.π c₂ c₁₂ (i₁, i₂)
h₁ : ¬c₁₂.Rel i₁₂ (c₁₂.next i₁₂)
⊢ (G.map (K₁.ιMapBifunctor K₂ F₁₂ c₁₂ i₁ i₂ (c₁.π c₂ c₁₂ (i₁, i₂)) ⋯)).app (K₃.X i₃) ≫ 0 =
    d₁ F₁₂ G K₁ K₂ K₃ c₁₂ c₄ i₁ i₂ i₃ j' + d₂ F₁₂ G K₁ K₂ K₃ c₁₂ c₄ i₁ i₂ i₃ j'

Tactic:
comp_zero,

Goal state after:
case neg
C₁ : Type u_1
C₂ : Type u_2
C₁₂ : Type u_3
C₃ : Type u_5
C₄ : Type u_6
inst✝²⁰ : Category.{v_1, u_1} C₁
inst✝¹⁹ : Category.{v_2, u_2} C₂
inst✝¹⁸ : Category.{v_3, u_5} C₃
inst✝¹⁷ : Category.{v_4, u_6} C₄
inst✝¹⁶ : Category.{v_5, u_3} C₁₂
inst✝¹⁵ : HasZeroMorphisms C₁
inst✝¹⁴ : HasZeroMorphisms C₂
inst✝¹³ : HasZeroMorphisms C₃
inst✝¹² : Preadditive C₁₂
inst✝¹¹ : Preadditive C₄
F₁₂ : C₁ ⥤ C₂ ⥤ C₁₂
G : C₁₂ ⥤ C₃ ⥤ C₄
inst✝¹⁰ : F₁₂.PreservesZeroMorphisms
inst✝⁹ : ∀ (X₁ : C₁), (F₁₂.obj X₁).PreservesZeroMorphisms
inst✝⁸ : G.Additive
inst✝⁷ : ∀ (X₁₂ : C₁₂), (G.obj X₁₂).PreservesZeroMorphisms
ι₁ : Type u_7
ι₂ : Type u_8
ι₃ : Type u_9
ι₁₂ : Type u_10
ι₄ : Type u_12
inst✝⁶ : DecidableEq ι₄
c₁ : ComplexShape ι₁
c₂ : ComplexShape ι₂
c₃ : ComplexShape ι₃
K₁ : HomologicalComplex C₁ c₁
K₂ : HomologicalComplex C₂ c₂
K₃ : HomologicalComplex C₃ c₃
c₁₂ : ComplexShape ι₁₂
c₄ : ComplexShape ι₄
inst✝⁵ : TotalComplexShape c₁ c₂ c₁₂
inst✝⁴ : TotalComplexShape c₁₂ c₃ c₄
inst✝³ : K₁.HasMapBifunctor K₂ F₁₂ c₁₂
inst✝² : DecidableEq ι₁₂
inst✝¹ : (K₁.mapBifunctor K₂ F₁₂ c₁₂).HasMapBifunctor K₃ G c₄
j j' : ι₄
inst✝ : HasGoodTrifunctor₁₂Obj F₁₂ G K₁ K₂ K₃ c₁₂ c₄
i₁ : ι₁
i₂ : ι₂
i₃ : ι₃
h : c₁.r c₂ c₃ c₁₂ c₄ (i₁, i₂, i₃) = j
i₁₂ : ι₁₂ := c₁.π c₂ c₁₂ (i₁, i₂)
h₁ : ¬c₁₂.Rel i₁₂ (c₁₂.next i₁₂)
⊢ 0 = d₁ F₁₂ G K₁ K₂ K₃ c₁₂ c₄ i₁ i₂ i₃ j' + d₂ F₁₂ G K₁ K₂ K₃ c₁₂ c₄ i₁ i₂ i₃ j'
Module: Mathlib.Algebra.Homology.Embedding.ExtendHomotopy

Goal state before:
pp: case neg
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 : HasZeroObject C
inst✝¹ : Preadditive C
K L : HomologicalComplex C c
f g : K ⟶ L
h : Homotopy f g
e : c.Embedding c'
inst✝ : e.IsRelIff
j' : ι'
i : ι
hij' : ¬c'.Rel j' (e.f i)
hj' : ¬∃ j, e.f j = j'
⊢ extend.hom e h.hom (e.f i) j' = 0

Tactic:
exact extend.hom_eq_zero₂ _ _ _ _ (by tauto)

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

Goal state before:
pp: case neg
C₁ : Type u_1
C₂ : Type u_2
C₂₃ : Type u_4
C₃ : Type u_5
C₄ : Type u_6
inst✝²² : Category.{v_1, u_1} C₁
inst✝²¹ : Category.{v_2, u_2} C₂
inst✝²⁰ : Category.{v_3, u_5} C₃
inst✝¹⁹ : Category.{v_4, u_6} C₄
inst✝¹⁸ : Category.{v_6, u_4} C₂₃
inst✝¹⁷ : HasZeroMorphisms C₁
inst✝¹⁶ : HasZeroMorphisms C₂
inst✝¹⁵ : HasZeroMorphisms C₃
inst✝¹⁴ : Preadditive C₂₃
inst✝¹³ : Preadditive C₄
F : C₁ ⥤ C₂₃ ⥤ C₄
G₂₃ : C₂ ⥤ C₃ ⥤ C₂₃
inst✝¹² : G₂₃.PreservesZeroMorphisms
inst✝¹¹ : ∀ (X₂ : C₂), (G₂₃.obj X₂).PreservesZeroMorphisms
inst✝¹⁰ : F.PreservesZeroMorphisms
inst✝⁹ : ∀ (X₁ : C₁), (F.obj X₁).Additive
ι₁ : Type u_7
ι₂ : Type u_8
ι₃ : Type u_9
ι₁₂ : Type u_10
ι₂₃ : Type u_11
ι₄ : Type u_12
inst✝⁸ : DecidableEq ι₄
c₁ : ComplexShape ι₁
c₂ : ComplexShape ι₂
c₃ : ComplexShape ι₃
K₁ : HomologicalComplex C₁ c₁
K₂ : HomologicalComplex C₂ c₂
K₃ : HomologicalComplex C₃ c₃
c₁₂ : ComplexShape ι₁₂
c₂₃ : ComplexShape ι₂₃
c₄ : ComplexShape ι₄
inst✝⁷ : TotalComplexShape c₁ c₂ c₁₂
inst✝⁶ : TotalComplexShape c₁₂ c₃ c₄
inst✝⁵ : TotalComplexShape c₂ c₃ c₂₃
inst✝⁴ : TotalComplexShape c₁ c₂₃ c₄
inst✝³ : K₂.HasMapBifunctor K₃ G₂₃ c₂₃
inst✝² : c₁.Associative c₂ c₃ c₁₂ c₂₃ c₄
inst✝¹ : DecidableEq ι₂₃
inst✝ : K₁.HasMapBifunctor (K₂.mapBifunctor K₃ G₂₃ c₂₃) F c₄
i₁ : ι₁
i₂ : ι₂
i₃ : ι₃
j j' : ι₄
h : c₁.r c₂ c₃ c₁₂ c₄ (i₁, i₂, i₃) = j
h₁ : c₁.Rel i₁ (c₁.next i₁)
h₂ : ¬c₁.π c₂₃ c₄ (c₁.next i₁, c₂.π c₃ c₂₃ (i₂, i₃)) = j'
⊢ 0 =
    c₁.ε₁ c₂₃ c₄ (i₁, c₂.π c₃ c₂₃ (i₂, i₃)) • (F.map (K₁.d i₁ (c₁.next i₁))).app ((G₂₃.obj (K₂.X i₂)).obj (K₃.X i₃)) ≫ 0

Tactic:
comp_zero,

Goal state after:
case neg
C₁ : Type u_1
C₂ : Type u_2
C₂₃ : Type u_4
C₃ : Type u_5
C₄ : Type u_6
inst✝²² : Category.{v_1, u_1} C₁
inst✝²¹ : Category.{v_2, u_2} C₂
inst✝²⁰ : Category.{v_3, u_5} C₃
inst✝¹⁹ : Category.{v_4, u_6} C₄
inst✝¹⁸ : Category.{v_6, u_4} C₂₃
inst✝¹⁷ : HasZeroMorphisms C₁
inst✝¹⁶ : HasZeroMorphisms C₂
inst✝¹⁵ : HasZeroMorphisms C₃
inst✝¹⁴ : Preadditive C₂₃
inst✝¹³ : Preadditive C₄
F : C₁ ⥤ C₂₃ ⥤ C₄
G₂₃ : C₂ ⥤ C₃ ⥤ C₂₃
inst✝¹² : G₂₃.PreservesZeroMorphisms
inst✝¹¹ : ∀ (X₂ : C₂), (G₂₃.obj X₂).PreservesZeroMorphisms
inst✝¹⁰ : F.PreservesZeroMorphisms
inst✝⁹ : ∀ (X₁ : C₁), (F.obj X₁).Additive
ι₁ : Type u_7
ι₂ : Type u_8
ι₃ : Type u_9
ι₁₂ : Type u_10
ι₂₃ : Type u_11
ι₄ : Type u_12
inst✝⁸ : DecidableEq ι₄
c₁ : ComplexShape ι₁
c₂ : ComplexShape ι₂
c₃ : ComplexShape ι₃
K₁ : HomologicalComplex C₁ c₁
K₂ : HomologicalComplex C₂ c₂
K₃ : HomologicalComplex C₃ c₃
c₁₂ : ComplexShape ι₁₂
c₂₃ : ComplexShape ι₂₃
c₄ : ComplexShape ι₄
inst✝⁷ : TotalComplexShape c₁ c₂ c₁₂
inst✝⁶ : TotalComplexShape c₁₂ c₃ c₄
inst✝⁵ : TotalComplexShape c₂ c₃ c₂₃
inst✝⁴ : TotalComplexShape c₁ c₂₃ c₄
inst✝³ : K₂.HasMapBifunctor K₃ G₂₃ c₂₃
inst✝² : c₁.Associative c₂ c₃ c₁₂ c₂₃ c₄
inst✝¹ : DecidableEq ι₂₃
inst✝ : K₁.HasMapBifunctor (K₂.mapBifunctor K₃ G₂₃ c₂₃) F c₄
i₁ : ι₁
i₂ : ι₂
i₃ : ι₃
j j' : ι₄
h : c₁.r c₂ c₃ c₁₂ c₄ (i₁, i₂, i₃) = j
h₁ : c₁.Rel i₁ (c₁.next i₁)
h₂ : ¬c₁.π c₂₃ c₄ (c₁.next i₁, c₂.π c₃ c₂₃ (i₂, i₃)) = j'
⊢ 0 = c₁.ε₁ c₂₃ c₄ (i₁, c₂.π c₃ c₂₃ (i₂, i₃)) • 0
Module: Mathlib.CategoryTheory.Triangulated.TStructure.Basic

Goal state before:
pp: C : Type u_1
inst✝⁷ : Category.{v_1, u_1} C
inst✝⁶ : Preadditive C
inst✝⁵ : HasZeroObject C
inst✝⁴ : HasShift C ℤ
inst✝³ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝² : Pretriangulated C
t : TStructure C
X : C
n₀ n₁ : ℤ
h : n₀ < n₁
inst✝¹ : t.IsLE X n₀
inst✝ : t.IsGE X n₁
⊢ 𝟙 X = 0

Tactic:
exact t.zero _ n₀ n₁ h

Goal state after:
Module: Mathlib.Algebra.Homology.DerivedCategory.TStructure

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasDerivedCategory C
X Y : DerivedCategory C
f : X ⟶ Y
K : CochainComplex C ℤ
e₁ : X ≅ Q.obj K
h✝¹ : K.IsStrictlyLE 0
L : CochainComplex C ℤ
e₂ : Y ≅ Q.obj L
h✝ : L.IsStrictlyGE 1
⊢ (e₁.inv ≫ f) ≫ e₂.hom = (e₁.inv ≫ 0) ≫ e₂.hom

Tactic:
comp_zero,

Goal state after:
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasDerivedCategory C
X Y : DerivedCategory C
f : X ⟶ Y
K : CochainComplex C ℤ
e₁ : X ≅ Q.obj K
h✝¹ : K.IsStrictlyLE 0
L : CochainComplex C ℤ
e₂ : Y ≅ Q.obj L
h✝ : L.IsStrictlyGE 1
⊢ (e₁.inv ≫ f) ≫ e₂.hom = 0 ≫ e₂.hom
Module: Mathlib.Algebra.Homology.DerivedCategory.TStructure

Goal state before:
pp: case mpr
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasDerivedCategory C
X : DerivedCategory C
n : ℤ
hX : ∀ i < n, IsZero ((homologyFunctor C i).obj X)
this : (Q.objPreimage X).IsGE n
⊢ X.IsGE n

Tactic:
exact ⟨(Q.objPreimage X).truncGE n, (Q.objObjPreimageIso X).symm ≪≫
      asIso (Q.map ((Q.objPreimage X).πTruncGE n)), inferInstance⟩

Goal state after:
Module: Mathlib.Algebra.Homology.DerivedCategory.TStructure

Goal state before:
pp: case mpr
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasDerivedCategory C
X : DerivedCategory C
n : ℤ
⊢ (∀ (i : ℤ), n < i → IsZero ((homologyFunctor C i).obj X)) → X.IsLE n

Tactic:
· intro hX
    have : (Q.objPreimage X).IsLE n := by
      rw [CochainComplex.isLE_iff]
      intro i hi
      rw [HomologicalComplex.exactAt_iff_isZero_homology]
      apply (hX i hi).of_iso
      exact (homologyFunctorFactors C i).symm.app _ ≪≫
        (homologyFunctor C i).mapIso (Q.objObjPreimageIso X)
    exact ⟨(Q.objPreimage X).truncLE n, (Q.objObjPreimageIso X).symm ≪≫
      (asIso (Q.map ((Q.objPreimage X).ιTruncLE n))).symm, inferInstance⟩

Goal state after:
Module: Mathlib.AlgebraicTopology.ModelCategory.Cylinder

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : ModelCategory C
A : C
P : Cylinder A
h : (cofibrations C).MapFactorizationData (trivialFibrations C) (codiag A)
⊢ Cofibration (ofFactorizationData h).i

Tactic:
by simpa using inferInstanceAs (Cofibration h.i)

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

Goal state before:
pp: case refine_2
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasBinaryBiproducts C
K : CochainComplex C ℤ
n : ℤ
hn : K.IsStrictlyGE n
i : ℤ
hi : autoParam (i < n - 1) isStrictlyGE_iff._auto_1
j : ℤ
hj : i + 1 = j
⊢ IsZero (K.X j)

Tactic:
exact K.isZero_of_isStrictlyGE n _ (by lia)

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

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Category.{v_2, u_2} D
inst✝³ : Preadditive C
inst✝² : Preadditive D
A : Type u_3
inst✝¹ : Category.{v_3, u_3} A
inst✝ : Abelian A
K : CochainComplex C ℤ
h : CochainComplex.plus C K
⊢ plus C ((CochainComplex.Plus.ι C ⋙ HomotopyCategory.quotient C (ComplexShape.up ℤ)).obj { obj := K, property := h })

Tactic:
simpa [plus_quotient_obj_iff]

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

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁷ : Category.{v_1, u_1} C
inst✝⁶ : Category.{v_2, u_2} D
inst✝⁵ : Preadditive C
inst✝⁴ : Preadditive D
A : Type u_3
inst✝³ : Category.{v_3, u_3} A
inst✝² : Abelian A
inst✝¹ : HasZeroObject C
inst✝ : HasBinaryBiproducts C
⊢ ∀ ⦃X Y : CochainComplex.Plus C⦄ (f₀ f₁ : X ⟶ Y),
    (quotient C).map f₀ = (quotient C).map f₁ →
      ∃ P x, (homotopyEquivalences C (ComplexShape.up ℤ)).inverseImage (CochainComplex.Plus.ι C) P.π

Tactic:
by
    rintro K L f₀ f₁ hf
    obtain ⟨f₀, rfl⟩ := ObjectProperty.homMk_surjective f₀
    obtain ⟨f₁, rfl⟩ := ObjectProperty.homMk_surjective f₁
    replace hf := homotopyOfEq f₀ f₁ ((HomotopyCategory.Plus.ι _).congr_map hf)
    exact ⟨K.precylinder, Precylinder.LeftHomotopy.fullSubcategoryEquiv.symm
      { h := cylinder.desc _ _ hf }, ⟨cylinder.homotopyEquiv _ (fun n ↦ ⟨n - 1, by simp⟩), rfl⟩⟩

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
t : TStructure C
T T' : Triangle C
f₁ f₂ : T ⟶ T'
hT : T ∈ distinguishedTriangles
hT' : T' ∈ distinguishedTriangles
a b : ℤ
h₀ : t.IsLE T.obj₁ a
h₁ : t.IsGE T'.obj₃ b
H : f₁.hom₂ = f₂.hom₂
hab : a ≤ b
this : ∀ (f : T ⟶ T'), f.hom₂ = 0 → f = 0
⊢ f₁ = f₂

Tactic:
by rw [← sub_eq_zero]; cat_disch

Goal state after:
Module: Mathlib.Algebra.Homology.Factorizations.Basic

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
K L : CochainComplex C ℤ
f : K ⟶ L
hL : IsZero L
n : ℤ
⊢ epiWithInjectiveKernel (f.f n) ↔ Injective (K.X n)

Tactic:
rw [epiWithInjectiveKernel_iff_of_isZero]
    exact (HomologicalComplex.eval _ _ n).map_isZero hL

Goal state after: