mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.CategoryTheory.Triangulated.Pretriangulated

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive
hC : Pretriangulated C
T : Triangle C
hT : T ∈ distinguishedTriangles
h :
  -(CategoryTheory.shiftFunctor C (-1)).map T.mor₃ ≫ (shiftFunctorCompIsoId C 1 (-1) ⋯).hom.app T.obj₁ = 0 ↔ Mono T.mor₁
⊢ T.mor₃ = 0 ↔ (CategoryTheory.shiftFunctor C (-1)).map T.mor₃ = 0

Tactic:
]

Goal state after:
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive
hC : Pretriangulated C
T : Triangle C
hT : T ∈ distinguishedTriangles
h :
  -(CategoryTheory.shiftFunctor C (-1)).map T.mor₃ ≫ (shiftFunctorCompIsoId C 1 (-1) ⋯).hom.app T.obj₁ = 0 ↔ Mono T.mor₁
⊢ T.mor₃ = 0 ↔ (CategoryTheory.shiftFunctor C (-1)).map T.mor₃ = 0
Module: Mathlib.CategoryTheory.Triangulated.Pretriangulated

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive
hC : Pretriangulated C
T : Triangle C
hT : T ∈ distinguishedTriangles
h₁ : T.mor₁ = 0
h₂ : T.mor₂ = 0
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: case mpr
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive
hC : Pretriangulated C
T : Triangle C
hT : T ∈ distinguishedTriangles
h₁ : T.mor₁ = 0
h₂ : T.mor₂ = 0
f : T.obj₂ ⟶ T.obj₁
hf : 𝟙 T.obj₂ = f ≫ T.mor₁
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive
hC : Pretriangulated C
T : Triangle C
hT : T ∈ distinguishedTriangles
h₂ : IsZero T.obj₂
h₃ : IsZero T.obj₃
⊢ T.mor₁ = 0 ∧ T.mor₃ = 0

Tactic:
]

Goal state after:
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive
hC : Pretriangulated C
T : Triangle C
hT : T ∈ distinguishedTriangles
h₂ : IsZero T.obj₂
h₃ : IsZero T.obj₃
⊢ T.mor₁ = 0 ∧ T.mor₃ = 0
Module: Mathlib.CategoryTheory.Triangulated.Pretriangulated

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive
hC : Pretriangulated C
T : Triangle C
hT : T ∈ distinguishedTriangles
h₁ : IsZero T.obj₁
h₃ : IsZero T.obj₃
⊢ T.mor₁ = 0 ∧ T.mor₂ = 0

Tactic:
]

Goal state after:
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive
hC : Pretriangulated C
T : Triangle C
hT : T ∈ distinguishedTriangles
h₁ : IsZero T.obj₁
h₃ : IsZero T.obj₃
⊢ T.mor₁ = 0 ∧ T.mor₂ = 0
Module: Mathlib.CategoryTheory.Triangulated.Triangulated

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
X₁ X₂ X₃ Z₁₂ Z₂₃ Z₁₃ : C
u₁₂ : X₁ ⟶ X₂
u₂₃ : X₂ ⟶ X₃
u₁₃ : X₁ ⟶ X₃
comm : u₁₂ ≫ u₂₃ = u₁₃
v₁₂ : X₂ ⟶ Z₁₂
w₁₂ : Z₁₂ ⟶ (shiftFunctor C 1).obj X₁
h₁₂ : Triangle.mk u₁₂ v₁₂ w₁₂ ∈ distinguishedTriangles
v₂₃ : X₃ ⟶ Z₂₃
w₂₃ : Z₂₃ ⟶ (shiftFunctor C 1).obj X₂
h₂₃ : Triangle.mk u₂₃ v₂₃ w₂₃ ∈ distinguishedTriangles
v₁₃ : X₃ ⟶ Z₁₃
w₁₃ : Z₁₃ ⟶ (shiftFunctor C 1).obj X₁
h₁₃ : Triangle.mk u₁₃ v₁₃ w₁₃ ∈ distinguishedTriangles
h : Octahedron comm h₁₂ h₂₃ h₁₃
X₁' X₂' X₃' Z₁₂' Z₂₃' Z₁₃' : C
u₁₂' : X₁' ⟶ X₂'
u₂₃' : X₂' ⟶ X₃'
u₁₃' : X₁' ⟶ X₃'
comm' : u₁₂' ≫ u₂₃' = u₁₃'
e₁ : X₁ ≅ X₁'
e₂ : X₂ ≅ X₂'
e₃ : X₃ ≅ X₃'
comm₁₂ : u₁₂ ≫ e₂.hom = e₁.hom ≫ u₁₂'
comm₂₃ : u₂₃ ≫ e₃.hom = e₂.hom ≫ u₂₃'
v₁₂' : X₂' ⟶ Z₁₂'
w₁₂' : Z₁₂' ⟶ (shiftFunctor C 1).obj X₁'
h₁₂' : Triangle.mk u₁₂' v₁₂' w₁₂' ∈ distinguishedTriangles
v₂₃' : X₃' ⟶ Z₂₃'
w₂₃' : Z₂₃' ⟶ (shiftFunctor C 1).obj X₂'
h₂₃' : Triangle.mk u₂₃' v₂₃' w₂₃' ∈ distinguishedTriangles
v₁₃' : X₃' ⟶ Z₁₃'
w₁₃' : Z₁₃' ⟶ (shiftFunctor C 1).obj X₁'
h₁₃' : Triangle.mk u₁₃' v₁₃' w₁₃' ∈ distinguishedTriangles
H : Octahedron comm' h₁₂' h₂₃' h₁₃'
iso₁₂ : Triangle.mk u₁₂ v₁₂ w₁₂ ≅ Triangle.mk u₁₂' v₁₂' w₁₂' :=
  isoTriangleOfIso₁₂ (Triangle.mk u₁₂ v₁₂ w₁₂) (Triangle.mk u₁₂' v₁₂' w₁₂') h₁₂ h₁₂' e₁ e₂ comm₁₂
iso₂₃ : Triangle.mk u₂₃ v₂₃ w₂₃ ≅ Triangle.mk u₂₃' v₂₃' w₂₃' :=
  isoTriangleOfIso₁₂ (Triangle.mk u₂₃ v₂₃ w₂₃) (Triangle.mk u₂₃' v₂₃' w₂₃') h₂₃ h₂₃' e₂ e₃ comm₂₃
⊢ u₁₂ ≫ e₂.hom ≫ u₂₃' = u₁₂ ≫ e₂.hom ≫ u₂₃'

Tactic:
]

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

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive
hC : Pretriangulated C
T : Triangle C
hT : T ∈ distinguishedTriangles
h₂ : IsZero T.obj₂
h₁ : 𝟙 T.obj₁ = 0
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive
hC : Pretriangulated C
T : Triangle C
hT : T ∈ distinguishedTriangles
⊢ T.mor₁ = 0 ∧ T.mor₃ = 0 ↔ IsIso T.mor₂

Tactic:
]

Goal state after:
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive
hC : Pretriangulated C
T : Triangle C
hT : T ∈ distinguishedTriangles
⊢ T.mor₁ = 0 ∧ T.mor₃ = 0 ↔ IsIso T.mor₂
Module: Mathlib.CategoryTheory.Triangulated.Pretriangulated

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive
hC : Pretriangulated C
T : Triangle C
hT : T ∈ distinguishedTriangles
h₁ : T.mor₁ = 0
h₃ : T.mor₃ = 0
this : Epi T.mor₂
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive
hC : Pretriangulated C
T : Triangle C
hT : T ∈ distinguishedTriangles
h₁ : T.mor₁ = 0
h₃ : T.mor₃ = 0
this : Epi T.mor₂
f : T.obj₃ ⟶ T.obj₂
hf : 𝟙 T.obj₂ = T.mor₂ ≫ f
⊢ T.mor₂ = T.mor₂

Tactic:
]

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

Goal state before:
pp: case mpr
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive
hC : Pretriangulated C
T : Triangle C
hT : T ∈ distinguishedTriangles
a✝ : IsIso T.mor₂
⊢ Mono T.mor₂ ∧ Epi T.mor₂

Tactic:
]

Goal state after:
case mpr
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive
hC : Pretriangulated C
T : Triangle C
hT : T ∈ distinguishedTriangles
a✝ : IsIso T.mor₂
⊢ Mono T.mor₂ ∧ Epi T.mor₂
Module: Mathlib.CategoryTheory.Triangulated.Pretriangulated

Goal state before:
pp: case mpr
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive
hC : Pretriangulated C
T : Triangle C
hT : T ∈ distinguishedTriangles
a✝ : IsIso T.mor₂
⊢ Mono T.mor₂ ∧ Epi T.mor₂

Tactic:
constructor <;> infer_instance

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

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive
hC : Pretriangulated C
T : Triangle C
h : IsZero T.obj₃
hT : T ∈ distinguishedTriangles
⊢ IsZero T.obj₃

Tactic:
]

Goal state after:
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive
hC : Pretriangulated C
T : Triangle C
h : IsZero T.obj₃
hT : T ∈ distinguishedTriangles
⊢ IsZero T.obj₃
Module: Mathlib.CategoryTheory.Triangulated.Pretriangulated

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive
hC : Pretriangulated C
T : Triangle C
h : IsZero T.obj₁
⊢ IsIso T.rotate.mor₁ ↔ IsIso T.mor₂

Tactic:
]

Goal state after:
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive
hC : Pretriangulated C
T : Triangle C
h : IsZero T.obj₁
⊢ IsIso T.rotate.mor₁ ↔ IsIso T.mor₂
Module: Mathlib.CategoryTheory.Triangulated.Pretriangulated

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive
hC : Pretriangulated C
T : Triangle C
h : IsZero T.obj₂
⊢ IsIso T.rotate.mor₂ ↔ IsIso T.mor₃

Tactic:
]

Goal state after:
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive
hC : Pretriangulated C
T : Triangle C
h : IsZero T.obj₂
⊢ IsIso T.rotate.mor₂ ↔ IsIso T.mor₃
Module: Mathlib.CategoryTheory.Triangulated.Pretriangulated

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
X✝ Y✝ : C
f : X✝ ⟶ Y✝
hf : Epi f
Z : C
g : Y✝ ⟶ Z
h : Z ⟶ (shiftFunctor C 1).obj X✝
hT : Triangle.mk f g h ∈ distinguishedTriangles
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
X✝ Y✝ : C
f : X✝ ⟶ Y✝
hf : Mono f
X : C
g : X ⟶ X✝
h : Y✝ ⟶ (shiftFunctor C 1).obj X
hT : Triangle.mk g f h ∈ distinguishedTriangles
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
T T' : Triangle C
φ : T ⟶ T'
hT : T ∈ distinguishedTriangles
hT' : T' ∈ distinguishedTriangles
h₁ : IsIso φ.hom₁
h₃ : IsIso φ.hom₃
⊢ ∀ (P : C) (g : P ⟶ T.obj₂), g ≫ φ.hom₂ = 0 → g = 0

Tactic:
]

Goal state after:
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
T T' : Triangle C
φ : T ⟶ T'
hT : T ∈ distinguishedTriangles
hT' : T' ∈ distinguishedTriangles
h₁ : IsIso φ.hom₁
h₃ : IsIso φ.hom₃
⊢ ∀ (P : C) (g : P ⟶ T.obj₂), g ≫ φ.hom₂ = 0 → g = 0
Module: Mathlib.CategoryTheory.Triangulated.Pretriangulated

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
T T' : Triangle C
φ : T ⟶ T'
hT : T ∈ distinguishedTriangles
hT' : T' ∈ distinguishedTriangles
h₁ : IsIso φ.hom₁
h₃ : IsIso φ.hom₃
A : C
f : A ⟶ T.obj₂
hf : f ≫ φ.hom₂ = 0
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
T T' : Triangle C
φ : T ⟶ T'
hT : T ∈ distinguishedTriangles
hT' : T' ∈ distinguishedTriangles
h₁ : IsIso φ.hom₁
h₃ : IsIso φ.hom₃
A : C
g : A ⟶ T.obj₁
hf : g ≫ T.mor₁ ≫ φ.hom₂ = 0
⊢ g ≫ T.mor₁ = 0

Tactic:
]

Goal state after:
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
T T' : Triangle C
φ : T ⟶ T'
hT : T ∈ distinguishedTriangles
hT' : T' ∈ distinguishedTriangles
h₁ : IsIso φ.hom₁
h₃ : IsIso φ.hom₃
A : C
g : A ⟶ T.obj₁
hf : g ≫ T.mor₁ ≫ φ.hom₂ = 0
⊢ g ≫ T.mor₁ = 0
Module: Mathlib.CategoryTheory.Triangulated.Pretriangulated

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
T T' : Triangle C
φ : T ⟶ T'
hT : T ∈ distinguishedTriangles
hT' : T' ∈ distinguishedTriangles
h₁ : IsIso φ.hom₁
h₃ : IsIso φ.hom₃
A : C
g : A ⟶ T.obj₁
hf : g ≫ T.mor₁ ≫ φ.hom₂ = 0
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
T T' : Triangle C
φ : T ⟶ T'
hT : T ∈ distinguishedTriangles
hT' : T' ∈ distinguishedTriangles
h₁ : IsIso φ.hom₁
h₃ : IsIso φ.hom₃
A : C
g : A ⟶ T.obj₁
hf : g ≫ T.mor₁ ≫ φ.hom₂ = 0
h : A ⟶ T'.invRotate.obj₁
hh : g ≫ φ.hom₁ = h ≫ T'.invRotate.mor₁
eq : T.invRotate.mor₁ ≫ φ.hom₁ = (shiftFunctor C (-1)).map φ.hom₃ ≫ T'.invRotate.mor₁
⊢ h ≫ T'.invRotate.mor₁ = h ≫ T'.invRotate.mor₁

Tactic:
]

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

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
T T' : Triangle C
φ : T ⟶ T'
hT : T ∈ distinguishedTriangles
hT' : T' ∈ distinguishedTriangles
h₁ : IsIso φ.hom₁
h₃ : IsIso φ.hom₃
A : C
h : A ⟶ T'.invRotate.obj₁
k : A ⟶ T.invRotate.obj₁
hf : (k ≫ T.invRotate.mor₁) ≫ T.mor₁ ≫ φ.hom₂ = 0
hh : (k ≫ T.invRotate.mor₁) ≫ φ.hom₁ = h ≫ T'.invRotate.mor₁
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
T T' : Triangle C
φ : T ⟶ T'
hT : T ∈ distinguishedTriangles
hT' : T' ∈ distinguishedTriangles
h₁ : IsIso φ.hom₁
h₃ : IsIso φ.hom₃
this : Mono φ.hom₂
A : C
y₂ : A ⟶ T'.obj₂
x₃ : A ⟶ T.obj₃
hx₃ : x₃ ≫ φ.hom₃ = y₂ ≫ T'.mor₂
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
T T' : Triangle C
φ : T ⟶ T'
hT : T ∈ distinguishedTriangles
hT' : T' ∈ distinguishedTriangles
h₁ : IsIso φ.hom₁
h₃ : IsIso φ.hom₃
this : Mono φ.hom₂
A : C
y₂ : A ⟶ T'.obj₂
x₃ : A ⟶ T.obj₃
hx₃ : x₃ ≫ φ.hom₃ = y₂ ≫ T'.mor₂
x₂ : A ⟶ T.obj₂
hx₂ : x₃ = x₂ ≫ T.mor₂
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: case refine_2
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
T T' : Triangle C
φ : T ⟶ T'
hT : T ∈ distinguishedTriangles
hT' : T' ∈ distinguishedTriangles
h₁ : IsIso φ.hom₁
h₃ : IsIso φ.hom₃
this : Mono φ.hom₂
A : C
y₂ : A ⟶ T'.obj₂
x₃ : A ⟶ T.obj₃
hx₃ : x₃ ≫ φ.hom₃ = y₂ ≫ T'.mor₂
x₂ : A ⟶ T.obj₂
hx₂ : x₃ = x₂ ≫ T.mor₂
y₁ : A ⟶ T'.obj₁
hy₁ : y₂ - x₂ ≫ φ.hom₂ = y₁ ≫ T'.mor₁
x₁ : A ⟶ T.obj₁
hx₁ : x₁ ≫ φ.hom₁ = y₁
⊢ y₂ = y₂

Tactic:
]

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

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
T : Triangle C
hT : T ∈ distinguishedTriangles
hT₀ : T.mor₃ = 0
inr : T.obj₃ ⟶ T.obj₂
inr_snd : inr ≫ T.mor₂ = 𝟙 T.obj₃
fst : T.obj₂ ⟶ T.obj₁
total : fst ≫ T.mor₁ + T.mor₂ ≫ inr = 𝟙 T.obj₂
this : Mono T.mor₁
⊢ fst ≫ T.mor₁ = fst ≫ T.mor₁

Tactic:
]

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

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 : HasBinaryBiproducts C
inst✝¹ : Preadditive D
inst✝ : HasBinaryBiproducts D
K L : CochainComplex C ℤ
φ : K ⟶ L
p q : ℤ
hpq : p + -1 = q
⊢ p = p

Tactic:
]

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

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 : HasBinaryBiproducts C
inst✝¹ : Preadditive D
inst✝ : HasBinaryBiproducts D
K L : CochainComplex C ℤ
φ : K ⟶ L
K₁ L₁ K₂ L₂ K₃ L₃ : CochainComplex C ℤ
φ₁ : K₁ ⟶ L₁
φ₂ : K₂ ⟶ L₂
a : K₁ ⟶ K₂
b : L₁ ⟶ L₂
H : Homotopy (φ₁ ≫ b) (a ≫ φ₂)
⊢ (HomotopyCategory.quotient C (ComplexShape.up ℤ)).map (triangle φ₁).mor₃ ≫
      (Functor.commShiftIso (HomotopyCategory.quotient C (ComplexShape.up ℤ)) 1).hom.app K₁ ≫
        (CategoryTheory.shiftFunctor (HomotopyCategory C (ComplexShape.up ℤ)) 1).map
          ((HomotopyCategory.quotient C (ComplexShape.up ℤ)).map a) =
    (HomotopyCategory.quotient C (ComplexShape.up ℤ)).map (triangle φ₁).mor₃ ≫
      (HomotopyCategory.quotient C (ComplexShape.up ℤ)).map
          ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) 1).map a) ≫
        (Functor.commShiftIso (HomotopyCategory.quotient C (ComplexShape.up ℤ)) 1).hom.app K₂

Tactic:
]

Goal state after:
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 : HasBinaryBiproducts C
inst✝¹ : Preadditive D
inst✝ : HasBinaryBiproducts D
K L : CochainComplex C ℤ
φ : K ⟶ L
K₁ L₁ K₂ L₂ K₃ L₃ : CochainComplex C ℤ
φ₁ : K₁ ⟶ L₁
φ₂ : K₂ ⟶ L₂
a : K₁ ⟶ K₂
b : L₁ ⟶ L₂
H : Homotopy (φ₁ ≫ b) (a ≫ φ₂)
⊢ (HomotopyCategory.quotient C (ComplexShape.up ℤ)).map (triangle φ₁).mor₃ ≫
      (Functor.commShiftIso (HomotopyCategory.quotient C (ComplexShape.up ℤ)) 1).hom.app K₁ ≫
        (CategoryTheory.shiftFunctor (HomotopyCategory C (ComplexShape.up ℤ)) 1).map
          ((HomotopyCategory.quotient C (ComplexShape.up ℤ)).map a) =
    (HomotopyCategory.quotient C (ComplexShape.up ℤ)).map (triangle φ₁).mor₃ ≫
      (HomotopyCategory.quotient C (ComplexShape.up ℤ)).map
          ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) 1).map a) ≫
        (Functor.commShiftIso (HomotopyCategory.quotient C (ComplexShape.up ℤ)) 1).hom.app K₂
Module: Mathlib.CategoryTheory.Triangulated.Pretriangulated

Goal state before:
pp: C : Type u
inst✝⁸ : Category.{v, u} C
inst✝⁷ : HasZeroObject C
inst✝⁶ : HasShift C ℤ
inst✝⁵ : Preadditive C
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
J : Type u_1
T : J → Triangle C
hT : ∀ (j : J), T j ∈ distinguishedTriangles
inst✝³ : HasProduct fun j ↦ (T j).obj₁
inst✝² : HasProduct fun j ↦ (T j).obj₂
inst✝¹ : HasProduct fun j ↦ (T j).obj₃
inst✝ : HasProduct fun j ↦ (shiftFunctor C 1).obj (T j).obj₁
f₁ : (∏ᶜ fun j ↦ (T j).obj₁) ⟶ ∏ᶜ fun j ↦ (T j).obj₂ := Limits.Pi.map fun j ↦ (T j).mor₁
Z : C
f₂ : (∏ᶜ fun j ↦ (T j).obj₂) ⟶ Z
f₃ : Z ⟶ (shiftFunctor C 1).obj (∏ᶜ fun j ↦ (T j).obj₁)
T' : Triangle C := Triangle.mk f₁ f₂ f₃
hT' : T' ∈ distinguishedTriangles
φ : (j : J) → T' ⟶ T j :=
  fun j ↦
    completeDistinguishedTriangleMorphism T' (T j) hT' ⋯ (Pi.π (fun j ↦ (T j).obj₁) j) (Pi.π (fun j ↦ (T j).obj₂) j) ⋯
φ' : T' ⟶ productTriangle T := productTriangle.lift T φ
h₁ : φ'.hom₁ = 𝟙 T'.obj₁
h₂ : φ'.hom₂ = 𝟙 T'.obj₂
⊢ IsIso (𝟙 T'.obj₁)

Tactic:
]

Goal state after:
C : Type u
inst✝⁸ : Category.{v, u} C
inst✝⁷ : HasZeroObject C
inst✝⁶ : HasShift C ℤ
inst✝⁵ : Preadditive C
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
J : Type u_1
T : J → Triangle C
hT : ∀ (j : J), T j ∈ distinguishedTriangles
inst✝³ : HasProduct fun j ↦ (T j).obj₁
inst✝² : HasProduct fun j ↦ (T j).obj₂
inst✝¹ : HasProduct fun j ↦ (T j).obj₃
inst✝ : HasProduct fun j ↦ (shiftFunctor C 1).obj (T j).obj₁
f₁ : (∏ᶜ fun j ↦ (T j).obj₁) ⟶ ∏ᶜ fun j ↦ (T j).obj₂ := Limits.Pi.map fun j ↦ (T j).mor₁
Z : C
f₂ : (∏ᶜ fun j ↦ (T j).obj₂) ⟶ Z
f₃ : Z ⟶ (shiftFunctor C 1).obj (∏ᶜ fun j ↦ (T j).obj₁)
T' : Triangle C := Triangle.mk f₁ f₂ f₃
hT' : T' ∈ distinguishedTriangles
φ : (j : J) → T' ⟶ T j :=
  fun j ↦
    completeDistinguishedTriangleMorphism T' (T j) hT' ⋯ (Pi.π (fun j ↦ (T j).obj₁) j) (Pi.π (fun j ↦ (T j).obj₂) j) ⋯
φ' : T' ⟶ productTriangle T := productTriangle.lift T φ
h₁ : φ'.hom₁ = 𝟙 T'.obj₁
h₂ : φ'.hom₂ = 𝟙 T'.obj₂
⊢ IsIso (𝟙 T'.obj₁)
Module: Mathlib.Algebra.Homology.HomotopyCategory.Pretriangulated

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 : HasBinaryBiproducts C
inst✝¹ : Preadditive D
inst✝ : HasBinaryBiproducts D
K L : CochainComplex C ℤ
φ : K ⟶ L
K₁ L₁ K₂ L₂ K₃ L₃ : CochainComplex C ℤ
φ₁ : K₁ ⟶ L₁
φ₂ : K₂ ⟶ L₂
φ₃ : K₃ ⟶ L₃
a : K₁ ⟶ K₂
b : L₁ ⟶ L₂
comm : φ₁ ≫ b = a ≫ φ₂
⊢ φ = φ

Tactic:
]

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

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 : HasBinaryBiproducts C
inst✝¹ : Preadditive D
inst✝ : HasBinaryBiproducts D
K L : CochainComplex C ℤ
φ : K ⟶ L
K₁ L₁ K₂ L₂ K₃ L₃ : CochainComplex C ℤ
φ₁ : K₁ ⟶ L₁
φ₂ : K₂ ⟶ L₂
φ₃ : K₃ ⟶ L₃
a : K₁ ⟶ K₂
b : L₁ ⟶ L₂
comm : φ₁ ≫ b = a ≫ φ₂
a' : K₂ ⟶ K₃
b' : L₂ ⟶ L₃
comm' : φ₂ ≫ b' = a' ≫ φ₃
⊢ a ≫ a' ≫ φ₃ = a ≫ a' ≫ φ₃

Tactic:
]

Goal state after: