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

Goal state before:
pp: 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 : ι₄
h : ¬c₂.Rel i₂ (c₂.next i₂)
⊢ (c₁.ε₂ c₂₃ c₄ (i₁, c₂.π c₃ c₂₃ (i₂, i₃)) * c₂.ε₁ c₃ c₂₃ (i₂, i₃)) •
      (F.obj (K₁.X i₁)).map ((G₂₃.map (K₂.d i₂ (c₂.next i₂))).app (K₃.X i₃)) ≫
        ιOrZero F G₂₃ K₁ K₂ K₃ c₁₂ c₂₃ c₄ i₁ (c₂.next i₂) i₃ j =
    0

Tactic:
shape _ _ _ h,

Goal state after:
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 : ι₄
h : ¬c₂.Rel i₂ (c₂.next i₂)
⊢ (c₁.ε₂ c₂₃ c₄ (i₁, c₂.π c₃ c₂₃ (i₂, i₃)) * c₂.ε₁ c₃ c₂₃ (i₂, i₃)) •
      (F.obj (K₁.X i₁)).map ((G₂₃.map 0).app (K₃.X i₃)) ≫ ιOrZero F G₂₃ K₁ K₂ K₃ c₁₂ c₂₃ c₄ i₁ (c₂.next i₂) i₃ j =
    0
Module: Mathlib.Algebra.Homology.BifunctorAssociator

Goal state before:
pp: 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 : ι₄
h : ¬c₃.Rel i₃ (c₃.next i₃)
⊢ (c₁.ε₂ c₂₃ c₄ (i₁, c₂.π c₃ c₂₃ (i₂, i₃)) * c₂.ε₂ c₃ c₂₃ (i₂, i₃)) •
      (F.obj (K₁.X i₁)).map ((G₂₃.obj (K₂.X i₂)).map (K₃.d i₃ (c₃.next i₃))) ≫
        ιOrZero F G₂₃ K₁ K₂ K₃ c₁₂ c₂₃ c₄ i₁ i₂ (c₃.next i₃) j =
    0

Tactic:
shape _ _ _ h,

Goal state after:
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 : ι₄
h : ¬c₃.Rel i₃ (c₃.next i₃)
⊢ (c₁.ε₂ c₂₃ c₄ (i₁, c₂.π c₃ c₂₃ (i₂, i₃)) * c₂.ε₂ c₃ c₂₃ (i₂, i₃)) •
      (F.obj (K₁.X i₁)).map ((G₂₃.obj (K₂.X i₂)).map 0) ≫ ιOrZero F G₂₃ K₁ K₂ K₃ c₁₂ c₂₃ c₄ i₁ i₂ (c₃.next i₃) j =
    0
Module: Mathlib.CategoryTheory.LiftingProperties.Limits

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
X Y Z W : C
f : X ⟶ Y
s : X ⟶ Z
g : Z ⟶ W
t : Y ⟶ W
h : IsPushout s f g t
Z' W' : C
g' : Z' ⟶ W'
inst✝ : HasLiftingProperty f g'
u : Z ⟶ Z'
v : W ⟶ W'
sq : CommSq u g g' v
⊢ (s ≫ u) ≫ g' = (s ≫ g) ≫ v

Tactic:
Category.assoc,

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
X Y Z W : C
f : X ⟶ Y
s : X ⟶ Z
g : Z ⟶ W
t : Y ⟶ W
h : IsPushout s f g t
Z' W' : C
g' : Z' ⟶ W'
inst✝ : HasLiftingProperty f g'
u : Z ⟶ Z'
v : W ⟶ W'
sq : CommSq u g g' v
⊢ s ≫ u ≫ g' = (s ≫ g) ≫ v
Module: Mathlib.CategoryTheory.LiftingProperties.Limits

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
X Y Z W : C
f : X ⟶ Y
s : X ⟶ Z
g : Z ⟶ W
t : Y ⟶ W
h : IsPushout s f g t
Z' W' : C
g' : Z' ⟶ W'
inst✝ : HasLiftingProperty f g'
u : Z ⟶ Z'
v : W ⟶ W'
sq : CommSq u g g' v
⊢ s ≫ u ≫ g' = (s ≫ g) ≫ v

Tactic:
Category.assoc,

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
X Y Z W : C
f : X ⟶ Y
s : X ⟶ Z
g : Z ⟶ W
t : Y ⟶ W
h : IsPushout s f g t
Z' W' : C
g' : Z' ⟶ W'
inst✝ : HasLiftingProperty f g'
u : Z ⟶ Z'
v : W ⟶ W'
sq : CommSq u g g' v
⊢ s ≫ u ≫ g' = s ≫ g ≫ v
Module: Mathlib.CategoryTheory.LiftingProperties.Limits

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
X Y Z W : C
f : X ⟶ Y
s : X ⟶ Z
g : Z ⟶ W
t : Y ⟶ W
h : IsPullback s f g t
X' Y' : C
f' : X' ⟶ Y'
inst✝ : HasLiftingProperty f' g
u : X' ⟶ X
v : Y' ⟶ Y
sq : CommSq u f' f v
⊢ (u ≫ s) ≫ g = f' ≫ v ≫ t

Tactic:
Category.assoc,

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
X Y Z W : C
f : X ⟶ Y
s : X ⟶ Z
g : Z ⟶ W
t : Y ⟶ W
h : IsPullback s f g t
X' Y' : C
f' : X' ⟶ Y'
inst✝ : HasLiftingProperty f' g
u : X' ⟶ X
v : Y' ⟶ Y
sq : CommSq u f' f v
⊢ u ≫ s ≫ g = f' ≫ v ≫ t
Module: Mathlib.CategoryTheory.LiftingProperties.Limits

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
X Y Z W : C
f : X ⟶ Y
s : X ⟶ Z
g : Z ⟶ W
t : Y ⟶ W
h : IsPullback s f g t
X' Y' : C
f' : X' ⟶ Y'
inst✝ : HasLiftingProperty f' g
u : X' ⟶ X
v : Y' ⟶ Y
sq : CommSq u f' f v
w : (u ≫ s) ≫ g = f' ≫ v ≫ t
⊢ (f' ≫ h.lift ⋯.lift v ⋯) ≫ s = u ≫ s

Tactic:
Category.assoc,

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
X Y Z W : C
f : X ⟶ Y
s : X ⟶ Z
g : Z ⟶ W
t : Y ⟶ W
h : IsPullback s f g t
X' Y' : C
f' : X' ⟶ Y'
inst✝ : HasLiftingProperty f' g
u : X' ⟶ X
v : Y' ⟶ Y
sq : CommSq u f' f v
w : (u ≫ s) ≫ g = f' ≫ v ≫ t
⊢ f' ≫ h.lift ⋯.lift v ⋯ ≫ s = u ≫ s
Module: Mathlib.CategoryTheory.LiftingProperties.Limits

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
X Y Z W : C
f : X ⟶ Y
s : X ⟶ Z
g : Z ⟶ W
t : Y ⟶ W
h : IsPullback s f g t
X' Y' : C
f' : X' ⟶ Y'
inst✝ : HasLiftingProperty f' g
u : X' ⟶ X
v : Y' ⟶ Y
sq : CommSq u f' f v
w : (u ≫ s) ≫ g = f' ≫ v ≫ t
⊢ (f' ≫ h.lift ⋯.lift v ⋯) ≫ s = u ≫ s

Tactic:
by rw [Category.assoc, h.lift_fst, CommSq.fac_left]

Goal state after:
Module: Mathlib.CategoryTheory.LiftingProperties.Limits

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
X Y Z W : C
f : X ⟶ Y
s : X ⟶ Z
g : Z ⟶ W
t : Y ⟶ W
h : IsPullback s f g t
X' Y' : C
f' : X' ⟶ Y'
inst✝ : HasLiftingProperty f' g
u : X' ⟶ X
v : Y' ⟶ Y
sq : CommSq u f' f v
w : (u ≫ s) ≫ g = f' ≫ v ≫ t
⊢ (f' ≫ h.lift ⋯.lift v ⋯) ≫ f = u ≫ f

Tactic:
Category.assoc,

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
X Y Z W : C
f : X ⟶ Y
s : X ⟶ Z
g : Z ⟶ W
t : Y ⟶ W
h : IsPullback s f g t
X' Y' : C
f' : X' ⟶ Y'
inst✝ : HasLiftingProperty f' g
u : X' ⟶ X
v : Y' ⟶ Y
sq : CommSq u f' f v
w : (u ≫ s) ≫ g = f' ≫ v ≫ t
⊢ f' ≫ h.lift ⋯.lift v ⋯ ≫ f = u ≫ f
Module: Mathlib.CategoryTheory.LiftingProperties.Limits

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
X Y Z W : C
f : X ⟶ Y
s : X ⟶ Z
g : Z ⟶ W
t : Y ⟶ W
h : IsPullback s f g t
X' Y' : C
f' : X' ⟶ Y'
inst✝ : HasLiftingProperty f' g
u : X' ⟶ X
v : Y' ⟶ Y
sq : CommSq u f' f v
w : (u ≫ s) ≫ g = f' ≫ v ≫ t
⊢ (f' ≫ h.lift ⋯.lift v ⋯) ≫ f = u ≫ f

Tactic:
by rw [Category.assoc, h.lift_snd, sq.w]

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
K : CochainComplex C ℤ
n : ℤ
⊢ (Q.obj K).IsGE n ↔ K.IsGE n

Tactic:
have eq := fun i ↦ ((homologyFunctorFactors C i).app K).isZero_iff
  simp only [Functor.comp_obj, HomologicalComplex.homologyFunctor_obj] at eq
  simp only [isGE_iff, CochainComplex.isGE_iff,
    HomologicalComplex.exactAt_iff_isZero_homology, eq]

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.Opposite.Triangulated

Goal state before:
pp: case refine_2
C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : HasShift C ℤ
inst✝⁴ : HasZeroObject C
inst✝³ : Preadditive C
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
inst✝ : IsTriangulated C
X₁ X₂ X₃ Z₁₂ Z₂₃ Z₁₃ : Cᵒᵖ
u₁₂ : X₁ ⟶ X₂
u₂₃ : X₂ ⟶ X₃
u₁₃ : X₁ ⟶ X₃
comm : u₁₂ ≫ u₂₃ = u₁₃
v₁₂ : X₂ ⟶ Z₁₂
w₁₂ : Z₁₂ ⟶ (shiftFunctor Cᵒᵖ 1).obj X₁
h₁₂ : Triangle.mk u₁₂ v₁₂ w₁₂ ∈ Pretriangulated.distinguishedTriangles
v₂₃ : X₃ ⟶ Z₂₃
w₂₃ : Z₂₃ ⟶ (shiftFunctor Cᵒᵖ 1).obj X₂
h₂₃ : Triangle.mk u₂₃ v₂₃ w₂₃ ∈ Pretriangulated.distinguishedTriangles
v₁₃ : X₃ ⟶ Z₁₃
w₁₃ : Z₁₃ ⟶ (shiftFunctor Cᵒᵖ 1).obj X₁
h₁₃ : Triangle.mk u₁₃ v₁₃ w₁₃ ∈ Pretriangulated.distinguishedTriangles
o : Triangulated.Octahedron' ⋯ ⋯ ⋯ ⋯ := Triangulated.someOctahedron' ⋯ ⋯ ⋯ ⋯
eq₂ :
  ((shiftFunctor C 1).map (w₂₃.unop ≫ o.m₁)).op ≫ (opShiftFunctorEquivalence C 1).unitIso.inv.app X₂ =
    (((shiftFunctor C 1).map w₁₃.unop).op ≫ (opShiftFunctorEquivalence C 1).unitIso.inv.app X₁) ≫ u₁₂
⊢ w₁₃ ≫ (shiftFunctor Cᵒᵖ 1).map u₁₂ = o.m₁.op ≫ w₂₃

Tactic:
Category.assoc,

Goal state after:
case refine_2
C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : HasShift C ℤ
inst✝⁴ : HasZeroObject C
inst✝³ : Preadditive C
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
inst✝ : IsTriangulated C
X₁ X₂ X₃ Z₁₂ Z₂₃ Z₁₃ : Cᵒᵖ
u₁₂ : X₁ ⟶ X₂
u₂₃ : X₂ ⟶ X₃
u₁₃ : X₁ ⟶ X₃
comm : u₁₂ ≫ u₂₃ = u₁₃
v₁₂ : X₂ ⟶ Z₁₂
w₁₂ : Z₁₂ ⟶ (shiftFunctor Cᵒᵖ 1).obj X₁
h₁₂ : Triangle.mk u₁₂ v₁₂ w₁₂ ∈ Pretriangulated.distinguishedTriangles
v₂₃ : X₃ ⟶ Z₂₃
w₂₃ : Z₂₃ ⟶ (shiftFunctor Cᵒᵖ 1).obj X₂
h₂₃ : Triangle.mk u₂₃ v₂₃ w₂₃ ∈ Pretriangulated.distinguishedTriangles
v₁₃ : X₃ ⟶ Z₁₃
w₁₃ : Z₁₃ ⟶ (shiftFunctor Cᵒᵖ 1).obj X₁
h₁₃ : Triangle.mk u₁₃ v₁₃ w₁₃ ∈ Pretriangulated.distinguishedTriangles
o : Triangulated.Octahedron' ⋯ ⋯ ⋯ ⋯ := Triangulated.someOctahedron' ⋯ ⋯ ⋯ ⋯
eq₂ :
  ((shiftFunctor C 1).map (w₂₃.unop ≫ o.m₁)).op ≫ (opShiftFunctorEquivalence C 1).unitIso.inv.app X₂ =
    ((shiftFunctor C 1).map w₁₃.unop).op ≫ (opShiftFunctorEquivalence C 1).unitIso.inv.app X₁ ≫ u₁₂
⊢ w₁₃ ≫ (shiftFunctor Cᵒᵖ 1).map u₁₂ = o.m₁.op ≫ w₂₃
Module: Mathlib.CategoryTheory.Triangulated.Opposite.Triangulated

Goal state before:
pp: C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : HasShift C ℤ
inst✝⁴ : HasZeroObject C
inst✝³ : Preadditive C
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
inst✝ : IsTriangulated C
X₁ X₂ X₃ Z₁₂ Z₂₃ Z₁₃ : Cᵒᵖ
u₁₂ : X₁ ⟶ X₂
u₂₃ : X₂ ⟶ X₃
u₁₃ : X₁ ⟶ X₃
comm : u₁₂ ≫ u₂₃ = u₁₃
v₁₂ : X₂ ⟶ Z₁₂
w₁₂ : Z₁₂ ⟶ (shiftFunctor Cᵒᵖ 1).obj X₁
h₁₂ : Triangle.mk u₁₂ v₁₂ w₁₂ ∈ Pretriangulated.distinguishedTriangles
v₂₃ : X₃ ⟶ Z₂₃
w₂₃ : Z₂₃ ⟶ (shiftFunctor Cᵒᵖ 1).obj X₂
h₂₃ : Triangle.mk u₂₃ v₂₃ w₂₃ ∈ Pretriangulated.distinguishedTriangles
v₁₃ : X₃ ⟶ Z₁₃
w₁₃ : Z₁₃ ⟶ (shiftFunctor Cᵒᵖ 1).obj X₁
h₁₃ : Triangle.mk u₁₃ v₁₃ w₁₃ ∈ Pretriangulated.distinguishedTriangles
o : Triangulated.Octahedron' ⋯ ⋯ ⋯ ⋯ := Triangulated.someOctahedron' ⋯ ⋯ ⋯ ⋯
this :
  Triangle.mk o.m₃.op o.m₁.op
      ((opShiftFunctorEquivalence C 1).counitIso.inv.app Z₂₃ ≫
        (shiftFunctor Cᵒᵖ 1).map
          ((((shiftFunctor C 1).map w₂₃.unop).op ≫ (opShiftFunctorEquivalence C 1).unitIso.inv.app X₂) ≫ v₁₂)) ∈
    Pretriangulated.distinguishedTriangles
⊢ w₂₃ ≫ (shiftFunctor Cᵒᵖ 1).map v₁₂ =
    (opShiftFunctorEquivalence C 1).counitIso.inv.app Z₂₃ ≫
      (shiftFunctor Cᵒᵖ 1).map
        ((((shiftFunctor C 1).map w₂₃.unop).op ≫ (opShiftFunctorEquivalence C 1).unitIso.inv.app X₂) ≫ v₁₂)

Tactic:
Category.assoc,

Goal state after:
C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : HasShift C ℤ
inst✝⁴ : HasZeroObject C
inst✝³ : Preadditive C
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
inst✝ : IsTriangulated C
X₁ X₂ X₃ Z₁₂ Z₂₃ Z₁₃ : Cᵒᵖ
u₁₂ : X₁ ⟶ X₂
u₂₃ : X₂ ⟶ X₃
u₁₃ : X₁ ⟶ X₃
comm : u₁₂ ≫ u₂₃ = u₁₃
v₁₂ : X₂ ⟶ Z₁₂
w₁₂ : Z₁₂ ⟶ (shiftFunctor Cᵒᵖ 1).obj X₁
h₁₂ : Triangle.mk u₁₂ v₁₂ w₁₂ ∈ Pretriangulated.distinguishedTriangles
v₂₃ : X₃ ⟶ Z₂₃
w₂₃ : Z₂₃ ⟶ (shiftFunctor Cᵒᵖ 1).obj X₂
h₂₃ : Triangle.mk u₂₃ v₂₃ w₂₃ ∈ Pretriangulated.distinguishedTriangles
v₁₃ : X₃ ⟶ Z₁₃
w₁₃ : Z₁₃ ⟶ (shiftFunctor Cᵒᵖ 1).obj X₁
h₁₃ : Triangle.mk u₁₃ v₁₃ w₁₃ ∈ Pretriangulated.distinguishedTriangles
o : Triangulated.Octahedron' ⋯ ⋯ ⋯ ⋯ := Triangulated.someOctahedron' ⋯ ⋯ ⋯ ⋯
this :
  Triangle.mk o.m₃.op o.m₁.op
      ((opShiftFunctorEquivalence C 1).counitIso.inv.app Z₂₃ ≫
        (shiftFunctor Cᵒᵖ 1).map
          ((((shiftFunctor C 1).map w₂₃.unop).op ≫ (opShiftFunctorEquivalence C 1).unitIso.inv.app X₂) ≫ v₁₂)) ∈
    Pretriangulated.distinguishedTriangles
⊢ w₂₃ ≫ (shiftFunctor Cᵒᵖ 1).map v₁₂ =
    (opShiftFunctorEquivalence C 1).counitIso.inv.app Z₂₃ ≫
      (shiftFunctor Cᵒᵖ 1).map
        (((shiftFunctor C 1).map w₂₃.unop).op ≫ (opShiftFunctorEquivalence C 1).unitIso.inv.app X₂ ≫ v₁₂)
Module: Mathlib.CategoryTheory.Triangulated.TStructure.TruncLEGT

Goal state before:
pp: C : Type u_1
inst✝⁵ : Category.{v_1, u_1} 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
a b : ℤ
h : a + 1 = b
⊢ (t.truncLEIsoTruncLT a b h).hom ≫ t.truncLTι b = t.truncLEι a

Tactic:
by
  subst h
  dsimp [truncLEIsoTruncLT, truncLEι]
  rw [Category.id_comp]

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

Goal state before:
pp: case refine_2
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : HasDerivedCategory C
K : CochainComplex C ℤ
hK : HomologicalComplex.Acyclic K
L : CochainComplex C ℤ
n : ℤ
hn : L.IsStrictlyGE n
φ : K ⟶ L
⊢ HomologicalComplex.Acyclic (K.truncGE n)

Tactic:
exact hK.truncGE _

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.LocalizingSubcategory

Goal state before:
pp: C : Type u_1
D : Type u_2
D₁ : Type u_3
D₂ : Type u_4
inst✝¹³ : Category.{v_1, u_1} C
inst✝¹² : Category.{v_2, u_2} D
inst✝¹¹ : Category.{v_3, u_3} D₁
inst✝¹⁰ : Category.{v_4, u_4} D₂
A B : ObjectProperty C
inst✝⁹ : HasZeroObject C
inst✝⁸ : HasShift C ℤ
inst✝⁷ : Preadditive C
inst✝⁶ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁵ : Pretriangulated C
inst✝⁴ : IsTriangulated C
inst✝³ : A.IsTriangulated
inst✝² : B.IsTriangulated
inst✝¹ : B.IsClosedUnderIsomorphisms
inst✝ : A.IsVerdierLeftLocalizing B
⊢ (A.triangulatedLocalizerMorphism B).IsLocalizedFullyFaithful

Tactic:
let L₁ := (B.inverseImage A.ι).trW.Q
  let L₂ := B.trW.Q
  let F : (B.inverseImage A.ι).trW.Localization ⥤ B.trW.Localization :=
    (A.triangulatedLocalizerMorphism B).localizedFunctor L₁ L₂
  let : CatCommSq (A.op.triangulatedLocalizerMorphism B.op).functor
    (A.opEquivalence.functor ⋙ L₁.op) L₂.op F.op :=
    ⟨Functor.isoWhiskerLeft A.opEquivalence.functor
      (NatIso.op (CatCommSq.iso (A.triangulatedLocalizerMorphism B).functor L₁ L₂ F).symm)⟩
  have : L₂.op.IsLocalization B.op.trW := by rw [trW_op]; infer_instance
  have : (A.opEquivalence.functor ⋙ L₁.op).IsLocalization (B.op.inverseImage A.op.ι).trW := by
    refine Functor.IsLocalization.of_equivalence_source L₁.op (B.inverseImage A.ι).trW.op
      _ _ A.opEquivalence.symm ?_ ?_
      ((Functor.associator _ _ _).symm ≪≫
        Functor.isoWhiskerRight A.opEquivalence.counitIso _ ≪≫ Functor.leftUnitor _)
    · rw [← trW_op, ← inverseImage_opEquivalence_inverse_trW_inverseImage_ι_op]
      intro _ _ f hf
      simp only [MorphismProperty.inverseImage_iff, Equivalence.symm_functor] at hf ⊢
      exact MorphismProperty.le_isoClosure _ _ hf
    · refine fun _ _ _ hf ↦ Localization.inverts L₁.op (B.inverseImage A.ι).trW.op _ ?_
      simpa [trW_inverseImage_ι_iff, ← op_inf, trW_op] using! hf
  exact LocalizerMorphism.IsLocalizedFullyFaithful.mk' (A.triangulatedLocalizerMorphism B)
    L₁ L₂ F (((A.op.triangulatedLocalizerMorphism B.op).fullyFaithful
    (A.opEquivalence.functor ⋙ L₁.op) L₂.op F.op).unop)

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
n₀ n₁ : ℤ
h : n₀ + 1 = n₁
X : C
hX : ∀ (Y : C) (f : Y ⟶ X), t.IsLE Y n₀ → f = 0
⊢ 𝟙 ((t.truncLT n₁).obj X) ≫ (t.truncLTι n₁).app X = 0 ≫ (t.truncLTι n₁).app X

Tactic:
by simpa using hX _ _ (by rw [← h]; infer_instance)

Goal state after:
Module: Mathlib.Algebra.Homology.ModelCategory.Lifting

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
A B X Y : CochainComplex C ℤ
t : A ⟶ X
i : A ⟶ B
p : X ⟶ Y
b : B ⟶ Y
sq : CommSq t i p b
hsq : (n : ℤ) → ⋯.LiftStruct
Q : CochainComplex C ℤ
π : B ⟶ Q
hπ : i ≫ π = 0
hQ : IsColimit (CokernelCofork.ofπ π hπ)
K : CochainComplex C ℤ
ι : K ⟶ X
hι : ι ≫ p = 0
hK : IsLimit (KernelFork.ofι ι hι)
this✝¹ : Epi π
this✝ : Mono ι
n : ℤ
this :
  (↑(cocycle₁' sq hsq)).v n (n + 1) ⋯ ≫ X.d (n + 1) (n + 2) +
      B.d n (n + 1) ≫ (↑(cocycle₁' sq hsq)).v (n + 1) (n + 2) ⋯ =
    0
⊢ π.f n ≫
      (((cochain₁ sq hsq hQ hK).v n (n + 1) ⋯ ≫ K.d (n + 1) (n + 2)) ≫ ι.f (n + 2) +
        (Q.d n (n + 1) ≫ (cochain₁ sq hsq hQ hK).v (n + 1) (n + 2) ⋯) ≫ ι.f (n + 2)) =
    π.f n ≫ 0 ≫ ι.f (n + 2)

Tactic:
Category.assoc,

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
A B X Y : CochainComplex C ℤ
t : A ⟶ X
i : A ⟶ B
p : X ⟶ Y
b : B ⟶ Y
sq : CommSq t i p b
hsq : (n : ℤ) → ⋯.LiftStruct
Q : CochainComplex C ℤ
π : B ⟶ Q
hπ : i ≫ π = 0
hQ : IsColimit (CokernelCofork.ofπ π hπ)
K : CochainComplex C ℤ
ι : K ⟶ X
hι : ι ≫ p = 0
hK : IsLimit (KernelFork.ofι ι hι)
this✝¹ : Epi π
this✝ : Mono ι
n : ℤ
this :
  (↑(cocycle₁' sq hsq)).v n (n + 1) ⋯ ≫ X.d (n + 1) (n + 2) +
      B.d n (n + 1) ≫ (↑(cocycle₁' sq hsq)).v (n + 1) (n + 2) ⋯ =
    0
⊢ π.f n ≫
      ((cochain₁ sq hsq hQ hK).v n (n + 1) ⋯ ≫ K.d (n + 1) (n + 2) ≫ ι.f (n + 2) +
        (Q.d n (n + 1) ≫ (cochain₁ sq hsq hQ hK).v (n + 1) (n + 2) ⋯) ≫ ι.f (n + 2)) =
    π.f n ≫ 0 ≫ ι.f (n + 2)
Module: Mathlib.Algebra.Homology.ModelCategory.Lifting

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
A B X Y : CochainComplex C ℤ
t : A ⟶ X
i : A ⟶ B
p : X ⟶ Y
b : B ⟶ Y
sq : CommSq t i p b
hsq : (n : ℤ) → ⋯.LiftStruct
Q : CochainComplex C ℤ
π : B ⟶ Q
hπ : i ≫ π = 0
hQ : IsColimit (CokernelCofork.ofπ π hπ)
K : CochainComplex C ℤ
ι : K ⟶ X
hι : ι ≫ p = 0
hK : IsLimit (KernelFork.ofι ι hι)
this✝¹ : Epi π
this✝ : Mono ι
n : ℤ
this :
  (↑(cocycle₁' sq hsq)).v n (n + 1) ⋯ ≫ X.d (n + 1) (n + 2) +
      B.d n (n + 1) ≫ (↑(cocycle₁' sq hsq)).v (n + 1) (n + 2) ⋯ =
    0
⊢ π.f n ≫
      ((cochain₁ sq hsq hQ hK).v n (n + 1) ⋯ ≫ K.d (n + 1) (n + 2) ≫ ι.f (n + 2) +
        (Q.d n (n + 1) ≫ (cochain₁ sq hsq hQ hK).v (n + 1) (n + 2) ⋯) ≫ ι.f (n + 2)) =
    π.f n ≫ 0 ≫ ι.f (n + 2)

Tactic:
Category.assoc,

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
A B X Y : CochainComplex C ℤ
t : A ⟶ X
i : A ⟶ B
p : X ⟶ Y
b : B ⟶ Y
sq : CommSq t i p b
hsq : (n : ℤ) → ⋯.LiftStruct
Q : CochainComplex C ℤ
π : B ⟶ Q
hπ : i ≫ π = 0
hQ : IsColimit (CokernelCofork.ofπ π hπ)
K : CochainComplex C ℤ
ι : K ⟶ X
hι : ι ≫ p = 0
hK : IsLimit (KernelFork.ofι ι hι)
this✝¹ : Epi π
this✝ : Mono ι
n : ℤ
this :
  (↑(cocycle₁' sq hsq)).v n (n + 1) ⋯ ≫ X.d (n + 1) (n + 2) +
      B.d n (n + 1) ≫ (↑(cocycle₁' sq hsq)).v (n + 1) (n + 2) ⋯ =
    0
⊢ π.f n ≫
      ((cochain₁ sq hsq hQ hK).v n (n + 1) ⋯ ≫ K.d (n + 1) (n + 2) ≫ ι.f (n + 2) +
        Q.d n (n + 1) ≫ (cochain₁ sq hsq hQ hK).v (n + 1) (n + 2) ⋯ ≫ ι.f (n + 2)) =
    π.f n ≫ 0 ≫ ι.f (n + 2)
Module: Mathlib.AlgebraicTopology.ModelCategory.Homotopy

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : ModelCategory C
X Y Z : C
f g : X ⟶ Y
inst✝¹ : IsCofibrant X
h✝ : LeftHomotopyRel f g
Q : PathObject Y
inst✝ : Q.IsGood
P : Cylinder X := ⋯.choose
h : ⋯.choose.LeftHomotopy f g
h' : ⋯.choose.IsGood
sq : CommSq (f ≫ Q.ι) P.i₀ Q.p (prod.lift (P.π ≫ f) h.h)
this : (sq.lift ≫ Q.p) ≫ prod.fst = prod.lift (P.π ≫ f) h.h ≫ prod.fst
⊢ (P.i₁ ≫ sq.lift) ≫ Q.p₀ = f

Tactic:
Category.assoc,

Goal state after:
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : ModelCategory C
X Y Z : C
f g : X ⟶ Y
inst✝¹ : IsCofibrant X
h✝ : LeftHomotopyRel f g
Q : PathObject Y
inst✝ : Q.IsGood
P : Cylinder X := ⋯.choose
h : ⋯.choose.LeftHomotopy f g
h' : ⋯.choose.IsGood
sq : CommSq (f ≫ Q.ι) P.i₀ Q.p (prod.lift (P.π ≫ f) h.h)
this : sq.lift ≫ Q.p ≫ prod.fst = prod.lift (P.π ≫ f) h.h ≫ prod.fst
⊢ (P.i₁ ≫ sq.lift) ≫ Q.p₀ = f
Module: Mathlib.AlgebraicTopology.ModelCategory.Homotopy

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : ModelCategory C
X Y Z : C
f g : X ⟶ Y
inst✝¹ : IsCofibrant X
h✝ : LeftHomotopyRel f g
Q : PathObject Y
inst✝ : Q.IsGood
P : Cylinder X := ⋯.choose
h : ⋯.choose.LeftHomotopy f g
h' : ⋯.choose.IsGood
sq : CommSq (f ≫ Q.ι) P.i₀ Q.p (prod.lift (P.π ≫ f) h.h)
this : (sq.lift ≫ Q.p) ≫ prod.snd = prod.lift (P.π ≫ f) h.h ≫ prod.snd
⊢ (P.i₁ ≫ sq.lift) ≫ Q.p₁ = g

Tactic:
Category.assoc,

Goal state after:
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : ModelCategory C
X Y Z : C
f g : X ⟶ Y
inst✝¹ : IsCofibrant X
h✝ : LeftHomotopyRel f g
Q : PathObject Y
inst✝ : Q.IsGood
P : Cylinder X := ⋯.choose
h : ⋯.choose.LeftHomotopy f g
h' : ⋯.choose.IsGood
sq : CommSq (f ≫ Q.ι) P.i₀ Q.p (prod.lift (P.π ≫ f) h.h)
this : sq.lift ≫ Q.p ≫ prod.snd = prod.lift (P.π ≫ f) h.h ≫ prod.snd
⊢ (P.i₁ ≫ sq.lift) ≫ Q.p₁ = g