awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory
Goal:
C : Type u_1
A : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : Preadditive.{v_1, u_1} C
inst_3 : HasShift.{v_1, u_1, 0} C ℤ
inst_4 : ∀ (n : ℤ), Functor.Additive.{v_1, u_1, v_1, u_1} (shiftFunctor C n)
inst_5 : Pretriangulated.{v_1, u_1} C
inst_6 : Category.{v_2, u_2} A
ι : Functor.{v_2, v_1, u_2, u_1} A C
hι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C n) (ι.obj Y)), n < 0 → f = 0
inst_7 : Preadditive.{v_2, u_2} A
inst_8 : ι.Full
inst_9 : ι.Faithful
inst_10 : HasFiniteProducts.{v_2, u_2} A
inst_11 : ι.Additive
X₁ X₂ X₃ : A
f₁ : X₁ ⟶ X₂
f₂ : X₂ ⟶ X₃
f₃ : ι.obj X₃ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj X₁)
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_2 inst_4 _)
    (Triangle.mk (ι.map f₁) (ι.map f₂) f₃)
refine_1 : IsZero.{v_1, u_1} (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj 0))
⊢ IsZero.{v_1, u_1}
    (Triangle.obj₁.{v_1, u_1} (Triangle.mk (X := (ι ⋙ shiftFunctor C (A := ℤ) 1).obj 0) 0 (𝟙 (ι.obj X₃)) 0))

Tactic:
dsimp
-- Module: Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory
Goal:
C : Type u_1
A : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : Preadditive.{v_1, u_1} C
inst_3 : HasShift.{v_1, u_1, 0} C ℤ
inst_4 : ∀ (n : ℤ), Functor.Additive.{v_1, u_1, v_1, u_1} (shiftFunctor C n)
inst_5 : Pretriangulated.{v_1, u_1} C
inst_6 : Category.{v_2, u_2} A
ι : Functor.{v_2, v_1, u_2, u_1} A C
hι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C n) (ι.obj Y)), n < 0 → f = 0
inst_7 : Preadditive.{v_2, u_2} A
inst_8 : ι.Full
inst_9 : ι.Faithful
inst_10 : HasFiniteProducts.{v_2, u_2} A
inst_11 : ι.Additive
X₁ X₂ X₃ : A
f₁ : X₁ ⟶ X₂
f₂ : X₂ ⟶ X₃
f₃ : ι.obj X₃ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj X₁)
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_2 inst_4 _)
    (Triangle.mk (ι.map f₁) (ι.map f₂) f₃)
refine_1 :
  Eq.{v_1 + 1} (α :=
    Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj 0) ⟶
      Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj 0))
    (𝟙 (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj 0))) 0
refine_1₁ :
  Eq.{v_1 + 1} (α :=
    Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj 0) ⟶
      Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj 0))
    (Functor.map (shiftFunctor C (A := ℤ) 1) (𝟙 (ι.obj 0))) 0
refine_1₂ :
  Eq (α :=
    Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj 0) ⟶
      Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj 0))
    (Functor.map (shiftFunctor C (A := ℤ) 1) (ι.map (𝟙 0))) 0
refine_1₃ :
  Eq (α :=
    Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj 0) ⟶
      Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj 0))
    (Functor.map (shiftFunctor C (A := ℤ) 1) (ι.map 0)) 0
refine_1₄ :
  Eq.{v_1 + 1} (α :=
    Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj 0) ⟶
      Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj 0))
    (Functor.map (shiftFunctor C (A := ℤ) 1) 0) 0
refine_1₅ refine_1₆ :
  Eq.{v_1 + 1} (α :=
    Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj 0) ⟶
      Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj 0))
    0 0
⊢ IsZero.{v_1, u_1} (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj 0))

Tactic:
rw [IsZero.iff_id_eq_zero, ← Functor.map_id, ← Functor.map_id, id_zero,
        Functor.map_zero, Functor.map_zero]

Premises used: CategoryTheory.Functor.map_zero, CategoryTheory.Limits.id_zero, CategoryTheory.Functor.map_id, CategoryTheory.Limits.IsZero.iff_id_eq_zero
-- Module: Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory
Goal:
C : Type u_1
A : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : Preadditive.{v_1, u_1} C
inst_3 : HasShift.{v_1, u_1, 0} C ℤ
inst_4 : ∀ (n : ℤ), Functor.Additive.{v_1, u_1, v_1, u_1} (shiftFunctor C n)
inst_5 : Pretriangulated.{v_1, u_1} C
inst_6 : Category.{v_2, u_2} A
ι : Functor.{v_2, v_1, u_2, u_1} A C
hι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C n) (ι.obj Y)), n < 0 → f = 0
inst_7 : Preadditive.{v_2, u_2} A
inst_8 : ι.Full
inst_9 : ι.Faithful
inst_10 : HasFiniteProducts.{v_2, u_2} A
inst_11 : ι.Additive
X₁ X₂ X₃ : A
f₁ : X₁ ⟶ X₂
f₂ : X₂ ⟶ X₃
f₃ : ι.obj X₃ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj X₁)
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_2 inst_4 _)
    (Triangle.mk (ι.map f₁) (ι.map f₂) f₃)
refine_2 : IsZero.{v_1, u_1} (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) (-1)) 0)
⊢ IsZero.{v_1, u_1} (Triangle.obj₁.{v_1, u_1} (Triangle.invRotate (contractibleTriangle (ι.obj X₃))))

Tactic:
dsimp
-- Module: Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory
Goal:
C : Type u_1
A : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : Preadditive.{v_1, u_1} C
inst_3 : HasShift.{v_1, u_1, 0} C ℤ
inst_4 : ∀ (n : ℤ), Functor.Additive.{v_1, u_1, v_1, u_1} (shiftFunctor C n)
inst_5 : Pretriangulated.{v_1, u_1} C
inst_6 : Category.{v_2, u_2} A
ι : Functor.{v_2, v_1, u_2, u_1} A C
hι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C n) (ι.obj Y)), n < 0 → f = 0
inst_7 : Preadditive.{v_2, u_2} A
inst_8 : ι.Full
inst_9 : ι.Faithful
inst_10 : HasFiniteProducts.{v_2, u_2} A
inst_11 : ι.Additive
X₁ X₂ X₃ : A
f₁ : X₁ ⟶ X₂
f₂ : X₂ ⟶ X₃
f₃ : ι.obj X₃ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj X₁)
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_2 inst_4 _)
    (Triangle.mk (ι.map f₁) (ι.map f₂) f₃)
refine_2 :
  Eq.{v_1 + 1} (α :=
    Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) (-1)) 0 ⟶
      Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) (-1)) 0)
    (𝟙 (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) (-1)) 0)) 0
refine_2₁ :
  Eq.{v_1 + 1} (α :=
    Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) (-1)) 0 ⟶
      Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) (-1)) 0)
    (Functor.map (shiftFunctor C (A := ℤ) (-1)) (𝟙 0)) 0
refine_2₂ :
  Eq.{v_1 + 1} (α :=
    Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) (-1)) 0 ⟶
      Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) (-1)) 0)
    (Functor.map (shiftFunctor C (A := ℤ) (-1)) 0) 0
refine_2₃ refine_2₄ :
  Eq.{v_1 + 1} (α :=
    Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) (-1)) 0 ⟶
      Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) (-1)) 0)
    0 0
⊢ IsZero.{v_1, u_1} (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) (-1)) 0)

Tactic:
rw [IsZero.iff_id_eq_zero, ← Functor.map_id, id_zero, Functor.map_zero]

Premises used: CategoryTheory.Functor.map_zero, CategoryTheory.Limits.id_zero, CategoryTheory.Functor.map_id, CategoryTheory.Limits.IsZero.iff_id_eq_zero
-- Module: Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory
Goal:
C : Type u_1
A : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : Preadditive.{v_1, u_1} C
inst_3 : HasShift.{v_1, u_1, 0} C ℤ
inst_4 : ∀ (n : ℤ), Functor.Additive.{v_1, u_1, v_1, u_1} (shiftFunctor C n)
inst_5 : Pretriangulated.{v_1, u_1} C
inst_6 : Category.{v_2, u_2} A
ι : Functor.{v_2, v_1, u_2, u_1} A C
hι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C n) (ι.obj Y)), n < 0 → f = 0
inst_7 : Preadditive.{v_2, u_2} A
inst_8 : ι.Full
inst_9 : ι.Faithful
inst_10 : HasFiniteProducts.{v_2, u_2} A
inst_11 : ι.Additive
X₁ X₂ X₃ : A
f₁ : X₁ ⟶ X₂
f₂ : X₂ ⟶ X₃
f₃ : ι.obj X₃ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj X₁)
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_2 inst_4 _)
    (Triangle.mk (ι.map f₁) (ι.map f₂) f₃)
⊢ Triangle.mor₁ (Triangle.mk (X := (ι ⋙ shiftFunctor C (A := ℤ) 1).obj 0) 0 (𝟙 (ι.obj X₃)) 0) ≫
      Iso.hom
        (Iso.refl
          (Triangle.obj₂.{v_1, u_1} (Triangle.mk (X := (ι ⋙ shiftFunctor C (A := ℤ) 1).obj 0) 0 (𝟙 (ι.obj X₃)) 0))) =
    Iso.hom (IsZero.iso ⋯ ⋯) ≫ Triangle.mor₁ (Triangle.invRotate (contractibleTriangle (ι.obj X₃)))

Tactic:
simp
-- Module: Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory
Goal:
C : Type u_1
A : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : Preadditive.{v_1, u_1} C
inst_3 : HasShift.{v_1, u_1, 0} C ℤ
inst_4 : ∀ (n : ℤ), Functor.Additive.{v_1, u_1, v_1, u_1} (shiftFunctor C n)
inst_5 : Pretriangulated.{v_1, u_1} C
inst_6 : Category.{v_2, u_2} A
ι : Functor.{v_2, v_1, u_2, u_1} A C
hι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C n) (ι.obj Y)), n < 0 → f = 0
inst_7 : Preadditive.{v_2, u_2} A
inst_8 : ι.Full
inst_9 : ι.Faithful
inst_10 : HasFiniteProducts.{v_2, u_2} A
inst_11 : ι.Additive
X₁ X₂ X₃ : A
f₁ : X₁ ⟶ X₂
f₂ : X₂ ⟶ X₃
f₃ : ι.obj X₃ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj X₁)
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_2 inst_4 _)
    (Triangle.mk (ι.map f₁) (ι.map f₂) f₃)
⊢ Triangle.mor₂ (Triangle.mk (X := (ι ⋙ shiftFunctor C (A := ℤ) 1).obj 0) 0 (𝟙 (ι.obj X₃)) 0) ≫
      Iso.hom
        (Iso.refl
          (Triangle.obj₃.{v_1, u_1} (Triangle.mk (X := (ι ⋙ shiftFunctor C (A := ℤ) 1).obj 0) 0 (𝟙 (ι.obj X₃)) 0))) =
    Iso.hom
        (Iso.refl
          (Triangle.obj₂.{v_1, u_1} (Triangle.mk (X := (ι ⋙ shiftFunctor C (A := ℤ) 1).obj 0) 0 (𝟙 (ι.obj X₃)) 0))) ≫
      Triangle.mor₂ (Triangle.invRotate (contractibleTriangle (ι.obj X₃)))

Tactic:
simp
-- Module: Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory
Goal:
C : Type u_1
A : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : Preadditive.{v_1, u_1} C
inst_3 : HasShift.{v_1, u_1, 0} C ℤ
inst_4 : ∀ (n : ℤ), Functor.Additive.{v_1, u_1, v_1, u_1} (shiftFunctor C n)
inst_5 : Pretriangulated.{v_1, u_1} C
inst_6 : Category.{v_2, u_2} A
ι : Functor.{v_2, v_1, u_2, u_1} A C
hι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C n) (ι.obj Y)), n < 0 → f = 0
inst_7 : Preadditive.{v_2, u_2} A
inst_8 : ι.Full
inst_9 : ι.Faithful
inst_10 : HasFiniteProducts.{v_2, u_2} A
inst_11 : ι.Additive
X₁ X₂ X₃ : A
f₁ : X₁ ⟶ X₂
f₂ : X₂ ⟶ X₃
f₃ : ι.obj X₃ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj X₁)
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_2 inst_4 _)
    (Triangle.mk (ι.map f₁) (ι.map f₂) f₃)
⊢ Triangle.mor₃ (Triangle.mk (X := (ι ⋙ shiftFunctor C (A := ℤ) 1).obj 0) 0 (𝟙 (ι.obj X₃)) 0) ≫
      Functor.map (shiftFunctor C (A := ℤ) 1) (Iso.hom (IsZero.iso ⋯ ⋯)) =
    Iso.hom
        (Iso.refl
          (Triangle.obj₃.{v_1, u_1} (Triangle.mk (X := (ι ⋙ shiftFunctor C (A := ℤ) 1).obj 0) 0 (𝟙 (ι.obj X₃)) 0))) ≫
      Triangle.mor₃ (Triangle.invRotate (contractibleTriangle (ι.obj X₃)))

Tactic:
simp
-- Module: Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory
Goal:
C : Type u_1
A : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : Preadditive.{v_1, u_1} C
inst_3 : HasShift.{v_1, u_1, 0} C ℤ
inst_4 : ∀ (n : ℤ), Functor.Additive.{v_1, u_1, v_1, u_1} (shiftFunctor C n)
inst_5 : Pretriangulated.{v_1, u_1} C
inst_6 : Category.{v_2, u_2} A
ι : Functor.{v_2, v_1, u_2, u_1} A C
hι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C n) (ι.obj Y)), n < 0 → f = 0
inst_7 : Preadditive.{v_2, u_2} A
inst_8 : ι.Full
inst_9 : ι.Faithful
inst_10 : HasFiniteProducts.{v_2, u_2} A
inst_11 : ι.Additive
X₁ X₂ X₃ : A
f₁ : X₁ ⟶ X₂
f₂ : X₂ ⟶ X₃
f₃ : ι.obj X₃ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj X₁)
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_2 inst_4 _)
    (Triangle.mk (ι.map f₁) (ι.map f₂) f₃)
⊢ Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_2 inst_4 _)
    (Triangle.mk (X := (ι ⋙ shiftFunctor C (A := ℤ) 1).obj 0) 0 (𝟙 (ι.obj X₃)) 0)

Tactic:
have hT' : Triangle.mk (0 : ((ι ⋙ shiftFunctor C (1 : ℤ)).obj 0) ⟶ _) (𝟙 (ι.obj X₃)) 0 ∈
      distTriang C := by
    refine isomorphic_distinguished _ (inv_rot_of_distTriang _
      (contractible_distinguished (ι.obj X₃))) _ ?_
    refine Triangle.isoMk _ _ (IsZero.iso ?_ ?_) (Iso.refl _) (Iso.refl _) ?_ ?_ ?_
    · dsimp
      rw [IsZero.iff_id_eq_zero, ← Functor.map_id, ← Functor.map_id, id_zero,
        Functor.map_zero, Functor.map_zero]
    · dsimp
      rw [IsZero.iff_id_eq_zero, ← Functor.map_id, id_zero, Functor.map_zero]
    all_goals simp
-- Module: Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory
Goal:
C : Type u_1
A : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : Preadditive.{v_1, u_1} C
inst_3 : HasShift.{v_1, u_1, 0} C ℤ
inst_4 : ∀ (n : ℤ), Functor.Additive.{v_1, u_1, v_1, u_1} (shiftFunctor C n)
inst_5 : Pretriangulated.{v_1, u_1} C
inst_6 : Category.{v_2, u_2} A
ι : Functor.{v_2, v_1, u_2, u_1} A C
hι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C n) (ι.obj Y)), n < 0 → f = 0
inst_7 : Preadditive.{v_2, u_2} A
inst_8 : ι.Full
inst_9 : ι.Faithful
inst_10 : HasFiniteProducts.{v_2, u_2} A
inst_11 : ι.Additive
X₁ X₂ X₃ : A
f₁ : X₁ ⟶ X₂
f₂ : X₂ ⟶ X₃
f₃ : ι.obj X₃ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj X₁)
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_2 inst_4 _)
    (Triangle.mk (ι.map f₁) (ι.map f₂) f₃)
hT' :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_2 inst_4 _)
    (Triangle.mk (X := (ι ⋙ shiftFunctor C (A := ℤ) 1).obj 0) 0 (𝟙 (ι.obj X₃)) 0)
refine :
  Iso (C := Cocone (parallelPair f₁ 0)) (CokernelCofork.ofπ (πQ (ι.map f₂) (𝟙 (ι.obj X₃))) ⋯) (CokernelCofork.ofπ f₂ ⋯)
⊢ IsColimit (F := parallelPair f₁ 0) (CokernelCofork.ofπ f₂ ⋯)

Tactic:
refine IsColimit.ofIsoColimit (AbelianSubcategory.isColimitCokernelCofork hι hT hT') ?_

Premises used: CategoryTheory.Triangulated.AbelianSubcategory.isColimitCokernelCofork, CategoryTheory.Limits.IsColimit.ofIsoColimit
-- Module: Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory
Goal:
C : Type u_1
A : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : Preadditive.{v_1, u_1} C
inst_3 : HasShift.{v_1, u_1, 0} C ℤ
inst_4 : ∀ (n : ℤ), Functor.Additive.{v_1, u_1, v_1, u_1} (shiftFunctor C n)
inst_5 : Pretriangulated.{v_1, u_1} C
inst_6 : Category.{v_2, u_2} A
ι : Functor.{v_2, v_1, u_2, u_1} A C
hι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C n) (ι.obj Y)), n < 0 → f = 0
inst_7 : Preadditive.{v_2, u_2} A
inst_8 : ι.Full
inst_9 : ι.Faithful
inst_10 : HasFiniteProducts.{v_2, u_2} A
inst_11 : ι.Additive
X₁ X₂ X₃ : A
f₁ : X₁ ⟶ X₂
f₂ : X₂ ⟶ X₃
f₃ : ι.obj X₃ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj X₁)
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_2 inst_4 _)
    (Triangle.mk (ι.map f₁) (ι.map f₂) f₃)
hT' :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_2 inst_4 _)
    (Triangle.mk (X := (ι ⋙ shiftFunctor C (A := ℤ) 1).obj 0) 0 (𝟙 (ι.obj X₃)) 0)
⊢ ι.map
      (Cofork.π (CokernelCofork.ofπ (πQ (ι.map f₂) (𝟙 (ι.obj X₃))) ⋯) ≫
        Iso.hom (Iso.refl (Cocone.pt (F := parallelPair f₁ 0) (CokernelCofork.ofπ (πQ (ι.map f₂) (𝟙 (ι.obj X₃))) ⋯)))) =
    ι.map (Cofork.π (CokernelCofork.ofπ f₂ ⋯))

Tactic:
simp
-- Module: Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory
Goal:
C : Type u_1
A : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : Preadditive.{v_1, u_1} C
inst_3 : HasShift.{v_1, u_1, 0} C ℤ
inst_4 : ∀ (n : ℤ), Functor.Additive.{v_1, u_1, v_1, u_1} (shiftFunctor C n)
inst_5 : Pretriangulated.{v_1, u_1} C
inst_6 : Category.{v_2, u_2} A
ι : Functor.{v_2, v_1, u_2, u_1} A C
hι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C n) (ι.obj Y)), n < 0 → f = 0
inst_7 : Preadditive.{v_2, u_2} A
inst_8 : ι.Full
inst_9 : ι.Faithful
inst_10 : HasFiniteProducts.{v_2, u_2} A
inst_11 : ι.Additive
X₁ X₂ X₃ : A
f₁ : X₁ ⟶ X₂
f₂ : X₂ ⟶ X₃
f₃ : ι.obj X₃ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj X₁)
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_2 inst_4 _)
    (Triangle.mk (ι.map f₁) (ι.map f₂) f₃)
hT' :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_2 inst_4 _)
    (Triangle.mk (X := (ι ⋙ shiftFunctor C (A := ℤ) 1).obj 0) 0 (𝟙 (ι.obj X₃)) 0)
⊢ Iso (C := Cocone (parallelPair f₁ 0)) (CokernelCofork.ofπ (πQ (ι.map f₂) (𝟙 (ι.obj X₃))) ⋯) (CokernelCofork.ofπ f₂ ⋯)

Tactic:
exact Cofork.ext (Iso.refl _) (ι.map_injective (by simp))

Premises used: CategoryTheory.Iso.refl, CategoryTheory.Limits.Cofork.ext, CategoryTheory.Functor.map_injective
-- Module: Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory
Goal:
C : Type u_1
A : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : Preadditive.{v_1, u_1} C
inst_3 : HasShift.{v_1, u_1, 0} C ℤ
inst_4 : ∀ (n : ℤ), Functor.Additive.{v_1, u_1, v_1, u_1} (shiftFunctor C n)
inst_5 : Pretriangulated.{v_1, u_1} C
inst_6 : Category.{v_2, u_2} A
ι : Functor.{v_2, v_1, u_2, u_1} A C
hι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C n) (ι.obj Y)), n < 0 → f = 0
inst_7 : Preadditive.{v_2, u_2} A
inst_8 : ι.Full
inst_9 : ι.Faithful
inst_10 : ι.Additive
hA : admissibleMorphism ι = ⊤
X₂ X₃ : A
π : X₂ ⟶ X₃
inst_11 : Epi π
obtain :
  ∀ (X₁ : C) (f₂ : ι.obj X₃ ⟶ X₁) (f₃ : X₁ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj X₂))
    (hT :
      Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_2 inst_4 _)
        (Triangle.mk (ι.map π) f₂ f₃)),
    ∃ (X₁ : A) (i : X₁ ⟶ X₂) (δ : ι.obj X₃ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj X₁)),
      Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_2 inst_4 _)
        (Triangle.mk (ι.map i) (ι.map π) δ)
⊢ ∃ (X₁ : A) (i : X₁ ⟶ X₂) (δ : ι.obj X₃ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj X₁)),
    Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_2 inst_4 _)
      (Triangle.mk (ι.map i) (ι.map π) δ)

Tactic:
obtain ⟨X₁, f₂, f₃, hT⟩ := distinguished_cocone_triangle (ι.map π)

Premises used: CategoryTheory.Functor.map, CategoryTheory.Pretriangulated.distinguished_cocone_triangle
-- Module: Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory
Goal:
C : Type u_1
A : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : Preadditive.{v_1, u_1} C
inst_3 : HasShift.{v_1, u_1, 0} C ℤ
inst_4 : ∀ (n : ℤ), Functor.Additive.{v_1, u_1, v_1, u_1} (shiftFunctor C n)
inst_5 : Pretriangulated.{v_1, u_1} C
inst_6 : Category.{v_2, u_2} A
ι : Functor.{v_2, v_1, u_2, u_1} A C
hι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C n) (ι.obj Y)), n < 0 → f = 0
inst_7 : Preadditive.{v_2, u_2} A
inst_8 : ι.Full
inst_9 : ι.Faithful
inst_10 : ι.Additive
hA : admissibleMorphism ι = ⊤
X₂ X₃ : A
π : X₂ ⟶ X₃
inst_11 : Epi π
X₁ : C
f₂ : ι.obj X₃ ⟶ X₁
f₃ : X₁ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj X₂)
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_2 inst_4 _)
    (Triangle.mk (ι.map π) f₂ f₃)
⊢ admissibleMorphism ι π

Tactic:
simp [hA]
-- Module: Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory
Goal:
C : Type u_1
A : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : Preadditive.{v_1, u_1} C
inst_3 : HasShift.{v_1, u_1, 0} C ℤ
inst_4 : ∀ (n : ℤ), Functor.Additive.{v_1, u_1, v_1, u_1} (shiftFunctor C n)
inst_5 : Pretriangulated.{v_1, u_1} C
inst_6 : Category.{v_2, u_2} A
ι : Functor.{v_2, v_1, u_2, u_1} A C
hι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C n) (ι.obj Y)), n < 0 → f = 0
inst_7 : Preadditive.{v_2, u_2} A
inst_8 : ι.Full
inst_9 : ι.Faithful
inst_10 : ι.Additive
hA : admissibleMorphism ι = ⊤
X₂ X₃ : A
π : X₂ ⟶ X₃
inst_11 : Epi π
X₁ : C
f₂ : ι.obj X₃ ⟶ X₁
f₃ : X₁ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj X₂)
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_2 inst_4 _)
    (Triangle.mk (ι.map π) f₂ f₃)
⊢ admissibleMorphism ι π

Tactic:
have : admissibleMorphism ι π := by simp [hA]
-- Module: Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory
Goal:
C : Type u_1
A : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : Preadditive.{v_1, u_1} C
inst_3 : HasShift.{v_1, u_1, 0} C ℤ
inst_4 : ∀ (n : ℤ), Functor.Additive.{v_1, u_1, v_1, u_1} (shiftFunctor C n)
inst_5 : Pretriangulated.{v_1, u_1} C
inst_6 : Category.{v_2, u_2} A
ι : Functor.{v_2, v_1, u_2, u_1} A C
hι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C n) (ι.obj Y)), n < 0 → f = 0
inst_7 : Preadditive.{v_2, u_2} A
inst_8 : ι.Full
inst_9 : ι.Faithful
inst_10 : ι.Additive
hA : admissibleMorphism ι = ⊤
X₂ X₃ : A
π : X₂ ⟶ X₃
inst_11 : Epi π
X₁ : C
f₂ : ι.obj X₃ ⟶ X₁
f₃ : X₁ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj X₂)
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_2 inst_4 _)
    (Triangle.mk (ι.map π) f₂ f₃)
this : admissibleMorphism ι π
obtain :
  ∀ (K Q : A) (α : Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj K) ⟶ X₁) (β : X₁ ⟶ ι.obj Q)
    (γ :
      ι.obj Q ⟶
        Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1)
          (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj K)))
    (hT' :
      Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_2 inst_4 _)
        (Triangle.mk α β γ)),
    ∃ (X₁ : A) (i : X₁ ⟶ X₂) (δ : ι.obj X₃ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj X₁)),
      Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_2 inst_4 _)
        (Triangle.mk (ι.map i) (ι.map π) δ)
⊢ ∃ (X₁ : A) (i : X₁ ⟶ X₂) (δ : ι.obj X₃ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj X₁)),
    Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_2 inst_4 _)
      (Triangle.mk (ι.map i) (ι.map π) δ)

Tactic:
obtain ⟨K, Q, α, β, γ, hT'⟩ := this f₂ f₃ hT