awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.CategoryTheory.Triangulated.Subcategory
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : HasShift.{v_1, u_1, 0} C ℤ
inst_3 : Preadditive.{v_1, u_1} C
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
P₁ P₂ Q : ObjectProperty.{v_1, u_1} C
inst_6 : Q.IsTriangulatedClosed₂
h₁ : P₁ ≤ Q
h₂ : P₂ ≤ Q
i w w_1 : C
w_2 : w ⟶ i
w_3 : i ⟶ w_1
w_4 : w_1 ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) w
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk w_2 w_3 w_4)
hY : P₁ w
hZ : P₂ w_1
⊢ Q.isoClosure i

Tactic:
exact ext_of_isTriangulatedClosed₂' Q _ hT (h₁ _ hY) (h₂ _ hZ)

Premises used: CategoryTheory.ObjectProperty.ext_of_isTriangulatedClosed₂'
-- Module: Mathlib.CategoryTheory.Triangulated.Subcategory
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : HasShift.{v_1, u_1, 0} C ℤ
inst_3 : Preadditive.{v_1, u_1} C
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
P₁ P₂ Q : ObjectProperty.{v_1, u_1} C
inst_6 : Q.IsTriangulatedClosed₂
inst_7 : Q.IsClosedUnderIsomorphisms
h₁ : P₁ ≤ Q
h₂ : P₂ ≤ Q
i : C
this :
  ∀ (w w_1 : C) (w_2 : w ⟶ i) (w_3 : i ⟶ w_1)
    (w_4 : w_1 ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) w)
    (hT :
      Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
        (Triangle.mk w_2 w_3 w_4))
    (hY : P₁ w) (hZ : P₂ w_1), Q i
⊢ P₁.extensionProduct P₂ i ≤ Q i

Tactic:
⟨_, _, _, _, _, hT, hY, hZ⟩
-- Module: Mathlib.CategoryTheory.Triangulated.Subcategory
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : HasShift.{v_1, u_1, 0} C ℤ
inst_3 : Preadditive.{v_1, u_1} C
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
P₁ P₂ Q : ObjectProperty.{v_1, u_1} C
inst_6 : Q.IsTriangulatedClosed₂
inst_7 : Q.IsClosedUnderIsomorphisms
h₁ : P₁ ≤ Q
h₂ : P₂ ≤ Q
i w w_1 : C
w_2 : w ⟶ i
w_3 : i ⟶ w_1
w_4 : w_1 ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) w
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk w_2 w_3 w_4)
hY : P₁ w
hZ : P₂ w_1
⊢ Q i

Tactic:
ext_of_isTriangulatedClosed₂ Q _ hT (h₁ _ hY) (h₂ _ hZ)

Premises used: CategoryTheory.ObjectProperty.ext_of_isTriangulatedClosed₂
-- Module: Mathlib.CategoryTheory.Triangulated.Subcategory
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : HasShift.{v_1, u_1, 0} C ℤ
inst_3 : Preadditive.{v_1, u_1} C
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
P₁ P₂ Q : ObjectProperty.{v_1, u_1} C
inst_6 : Q.IsTriangulatedClosed₂
inst_7 : Q.IsClosedUnderIsomorphisms
h₁ : P₁ ≤ Q
h₂ : P₂ ≤ Q
i w w_1 : C
w_2 : w ⟶ i
w_3 : i ⟶ w_1
w_4 : w_1 ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) w
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk w_2 w_3 w_4)
hY : P₁ w
hZ : P₂ w_1
⊢ Q i

Tactic:
exact ext_of_isTriangulatedClosed₂ Q _ hT (h₁ _ hY) (h₂ _ hZ)

Premises used: CategoryTheory.ObjectProperty.ext_of_isTriangulatedClosed₂
-- Module: Mathlib.CategoryTheory.Triangulated.Subcategory
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : HasShift.{v_1, u_1, 0} C ℤ
inst_3 : Preadditive.{v_1, u_1} C
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
P Q : ObjectProperty.{v_1, u_1} C
X : C
this :
  ∀ (A B : C) (f₁ : A ⟶ X) (f₂ : X ⟶ B) (f₃ : B ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) A)
    (hT :
      Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
        (Triangle.mk f₁ f₂ f₃))
    (A' : C) (hP : P A') (a₁ : A ⟶ A') (b₁ : A' ⟶ A) (h₁ : a₁ ≫ b₁ = 𝟙 A) (B' : C) (hQ : Q B') (a₃ : B ⟶ B')
    (b₃ : B' ⟶ B) (h₃ : a₃ ≫ b₃ = 𝟙 B), (P.extensionProduct Q).retractClosure X
⊢ P.retractClosure.extensionProduct Q.retractClosure X ≤ (P.extensionProduct Q).retractClosure X

Tactic:
⟨A, B, f₁, f₂, f₃, hT, ⟨A', hP, ⟨a₁, b₁, h₁⟩⟩, ⟨B', hQ, ⟨a₃, b₃, h₃⟩⟩⟩
-- Module: Mathlib.CategoryTheory.Triangulated.Subcategory
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : HasShift.{v_1, u_1, 0} C ℤ
inst_3 : Preadditive.{v_1, u_1} C
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
P Q : ObjectProperty.{v_1, u_1} C
X A B : C
f₁ : A ⟶ X
f₂ : X ⟶ B
f₃ : B ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) A
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk f₁ f₂ f₃)
A' : C
hP : P A'
a₁ : A ⟶ A'
b₁ : A' ⟶ A
h₁ : a₁ ≫ b₁ = 𝟙 A
B' : C
hQ : Q B'
a₃ : B ⟶ B'
b₃ : B' ⟶ B
h₃ : a₃ ≫ b₃ = 𝟙 B
obtain :
  ∀ (X' : C) (g₁ : A' ⟶ X') (g₂ : X' ⟶ B')
    (hT' :
      Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
        (Triangle.mk g₁ g₂ (b₃ ≫ f₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) a₁))),
    (P.extensionProduct Q).retractClosure X
⊢ (P.extensionProduct Q).retractClosure X

Tactic:
obtain ⟨X', g₁, g₂, hT'⟩ := distinguished_cocone_triangle₂ (b₃ ≫ f₃ ≫ a₁⟦(1 : ℤ)⟧')

Premises used: CategoryTheory.shiftFunctor, CategoryTheory.Pretriangulated.distinguished_cocone_triangle₂, CategoryTheory.Functor.map, CategoryTheory.CategoryStruct.comp, OfNat.ofNat
-- Module: Mathlib.CategoryTheory.Triangulated.Subcategory
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : HasShift.{v_1, u_1, 0} C ℤ
inst_3 : Preadditive.{v_1, u_1} C
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
P Q : ObjectProperty.{v_1, u_1} C
X A B : C
f₁ : A ⟶ X
f₂ : X ⟶ B
f₃ : B ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) A
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk f₁ f₂ f₃)
A' : C
hP : P A'
a₁ : A ⟶ A'
b₁ : A' ⟶ A
h₁ : a₁ ≫ b₁ = 𝟙 A
B' : C
hQ : Q B'
a₃ : B ⟶ B'
b₃ : B' ⟶ B
h₃ : a₃ ≫ b₃ = 𝟙 B
X' : C
g₁ : A' ⟶ X'
g₂ : X' ⟶ B'
hT' :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk g₁ g₂ (b₃ ≫ f₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) a₁))
dsimp : f₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) a₁ = a₃ ≫ b₃ ≫ f₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) a₁
⊢ (Triangle.mk f₁ f₂ f₃).mor₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) a₁ =
    a₃ ≫ (Triangle.mk g₁ g₂ (b₃ ≫ f₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) a₁)).mor₃

Tactic:
dsimp
-- Module: Mathlib.CategoryTheory.Triangulated.Subcategory
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : HasShift.{v_1, u_1, 0} C ℤ
inst_3 : Preadditive.{v_1, u_1} C
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
P Q : ObjectProperty.{v_1, u_1} C
X A B : C
f₁ : A ⟶ X
f₂ : X ⟶ B
f₃ : B ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) A
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk f₁ f₂ f₃)
A' : C
hP : P A'
a₁ : A ⟶ A'
b₁ : A' ⟶ A
h₁ : a₁ ≫ b₁ = 𝟙 A
B' : C
hQ : Q B'
a₃ : B ⟶ B'
b₃ : B' ⟶ B
h₃ : a₃ ≫ b₃ = 𝟙 B
X' : C
g₁ : A' ⟶ X'
g₂ : X' ⟶ B'
hT' :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk g₁ g₂ (b₃ ≫ f₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) a₁))
⊢ f₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) a₁ = a₃ ≫ b₃ ≫ f₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) a₁

Tactic:
grind
-- Module: Mathlib.CategoryTheory.Triangulated.Subcategory
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : HasShift.{v_1, u_1, 0} C ℤ
inst_3 : Preadditive.{v_1, u_1} C
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
P Q : ObjectProperty.{v_1, u_1} C
X A B : C
f₁ : A ⟶ X
f₂ : X ⟶ B
f₃ : B ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) A
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk f₁ f₂ f₃)
A' : C
hP : P A'
a₁ : A ⟶ A'
b₁ : A' ⟶ A
h₁ : a₁ ≫ b₁ = 𝟙 A
B' : C
hQ : Q B'
a₃ : B ⟶ B'
b₃ : B' ⟶ B
h₃ : a₃ ≫ b₃ = 𝟙 B
X' : C
g₁ : A' ⟶ X'
g₂ : X' ⟶ B'
hT' :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk g₁ g₂ (b₃ ≫ f₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) a₁))
obtain :
  ∀ (a₂ : X ⟶ X')
    (ha₁₂ :
      (Triangle.mk f₁ f₂ f₃).mor₁ ≫ a₂ =
        a₁ ≫ (Triangle.mk g₁ g₂ (b₃ ≫ f₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) a₁)).mor₁)
    (ha₂₃ :
      (Triangle.mk f₁ f₂ f₃).mor₂ ≫ a₃ =
        a₂ ≫ (Triangle.mk g₁ g₂ (b₃ ≫ f₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) a₁)).mor₂),
    (P.extensionProduct Q).retractClosure X
⊢ (P.extensionProduct Q).retractClosure X

Tactic:
obtain ⟨a₂ : X ⟶ X', ha₁₂, ha₂₃⟩ :=
    complete_distinguished_triangle_morphism₂ _ _ hT hT' a₁ a₃ (by dsimp; grind)

Premises used: CategoryTheory.Pretriangulated.complete_distinguished_triangle_morphism₂
-- Module: Mathlib.CategoryTheory.Triangulated.Subcategory
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : HasShift.{v_1, u_1, 0} C ℤ
inst_3 : Preadditive.{v_1, u_1} C
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
P Q : ObjectProperty.{v_1, u_1} C
X A B : C
f₁ : A ⟶ X
f₂ : X ⟶ B
f₃ : B ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) A
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk f₁ f₂ f₃)
A' : C
hP : P A'
a₁ : A ⟶ A'
b₁ : A' ⟶ A
h₁ : a₁ ≫ b₁ = 𝟙 A
B' : C
hQ : Q B'
a₃ : B ⟶ B'
b₃ : B' ⟶ B
h₃ : a₃ ≫ b₃ = 𝟙 B
X' : C
g₁ : A' ⟶ X'
g₂ : X' ⟶ B'
hT' :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk g₁ g₂ (b₃ ≫ f₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) a₁))
a₂ : X ⟶ X'
ha₁₂ :
  (Triangle.mk f₁ f₂ f₃).mor₁ ≫ a₂ =
    a₁ ≫ (Triangle.mk g₁ g₂ (b₃ ≫ f₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) a₁)).mor₁
ha₂₃ :
  (Triangle.mk f₁ f₂ f₃).mor₂ ≫ a₃ =
    a₂ ≫ (Triangle.mk g₁ g₂ (b₃ ≫ f₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) a₁)).mor₂
dsimp : (b₃ ≫ f₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) a₁) ≫ Functor.map (shiftFunctor C (A := ℤ) 1) b₁ = b₃ ≫ f₃
⊢ (Triangle.mk g₁ g₂ (b₃ ≫ f₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) a₁)).mor₃ ≫
      Functor.map (shiftFunctor C (A := ℤ) 1) b₁ =
    b₃ ≫ (Triangle.mk f₁ f₂ f₃).mor₃

Tactic:
dsimp
-- Module: Mathlib.CategoryTheory.Triangulated.Subcategory
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : HasShift.{v_1, u_1, 0} C ℤ
inst_3 : Preadditive.{v_1, u_1} C
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
P Q : ObjectProperty.{v_1, u_1} C
X A B : C
f₁ : A ⟶ X
f₂ : X ⟶ B
f₃ : B ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) A
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk f₁ f₂ f₃)
A' : C
hP : P A'
a₁ : A ⟶ A'
b₁ : A' ⟶ A
h₁ : a₁ ≫ b₁ = 𝟙 A
B' : C
hQ : Q B'
a₃ : B ⟶ B'
b₃ : B' ⟶ B
h₃ : a₃ ≫ b₃ = 𝟙 B
X' : C
g₁ : A' ⟶ X'
g₂ : X' ⟶ B'
hT' :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk g₁ g₂ (b₃ ≫ f₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) a₁))
a₂ : X ⟶ X'
ha₁₂ :
  (Triangle.mk f₁ f₂ f₃).mor₁ ≫ a₂ =
    a₁ ≫ (Triangle.mk g₁ g₂ (b₃ ≫ f₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) a₁)).mor₁
ha₂₃ :
  (Triangle.mk f₁ f₂ f₃).mor₂ ≫ a₃ =
    a₂ ≫ (Triangle.mk g₁ g₂ (b₃ ≫ f₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) a₁)).mor₂
⊢ (b₃ ≫ f₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) a₁) ≫ Functor.map (shiftFunctor C (A := ℤ) 1) b₁ = b₃ ≫ f₃

Tactic:
grind
-- Module: Mathlib.CategoryTheory.Triangulated.Subcategory
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : HasShift.{v_1, u_1, 0} C ℤ
inst_3 : Preadditive.{v_1, u_1} C
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
P Q : ObjectProperty.{v_1, u_1} C
X A B : C
f₁ : A ⟶ X
f₂ : X ⟶ B
f₃ : B ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) A
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk f₁ f₂ f₃)
A' : C
hP : P A'
a₁ : A ⟶ A'
b₁ : A' ⟶ A
h₁ : a₁ ≫ b₁ = 𝟙 A
B' : C
hQ : Q B'
a₃ : B ⟶ B'
b₃ : B' ⟶ B
h₃ : a₃ ≫ b₃ = 𝟙 B
X' : C
g₁ : A' ⟶ X'
g₂ : X' ⟶ B'
hT' :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk g₁ g₂ (b₃ ≫ f₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) a₁))
a₂ : X ⟶ X'
ha₁₂ :
  (Triangle.mk f₁ f₂ f₃).mor₁ ≫ a₂ =
    a₁ ≫ (Triangle.mk g₁ g₂ (b₃ ≫ f₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) a₁)).mor₁
ha₂₃ :
  (Triangle.mk f₁ f₂ f₃).mor₂ ≫ a₃ =
    a₂ ≫ (Triangle.mk g₁ g₂ (b₃ ≫ f₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) a₁)).mor₂
obtain :
  ∀ (b₂ : X' ⟶ X)
    (hb₁₂ :
      (Triangle.mk g₁ g₂ (b₃ ≫ f₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) a₁)).mor₁ ≫ b₂ =
        b₁ ≫ (Triangle.mk f₁ f₂ f₃).mor₁)
    (hb₂₃ :
      (Triangle.mk g₁ g₂ (b₃ ≫ f₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) a₁)).mor₂ ≫ b₃ =
        b₂ ≫ (Triangle.mk f₁ f₂ f₃).mor₂),
    (P.extensionProduct Q).retractClosure X
⊢ (P.extensionProduct Q).retractClosure X

Tactic:
obtain ⟨b₂ : X' ⟶ X, hb₁₂, hb₂₃⟩ :=
    complete_distinguished_triangle_morphism₂ _ _ hT' hT b₁ b₃ (by dsimp; grind)

Premises used: CategoryTheory.Pretriangulated.complete_distinguished_triangle_morphism₂
-- Module: Mathlib.CategoryTheory.Triangulated.Subcategory
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : HasShift.{v_1, u_1, 0} C ℤ
inst_3 : Preadditive.{v_1, u_1} C
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
P Q : ObjectProperty.{v_1, u_1} C
X A B : C
f₁ : A ⟶ X
f₂ : X ⟶ B
f₃ : B ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) A
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk f₁ f₂ f₃)
A' : C
hP : P A'
a₁ : A ⟶ A'
b₁ : A' ⟶ A
h₁ : a₁ ≫ b₁ = 𝟙 A
B' : C
hQ : Q B'
a₃ : B ⟶ B'
b₃ : B' ⟶ B
h₃ : a₃ ≫ b₃ = 𝟙 B
X' : C
g₁ : A' ⟶ X'
g₂ : X' ⟶ B'
hT' :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk g₁ g₂ (b₃ ≫ f₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) a₁))
a₂ : X ⟶ X'
ha₁₂ : f₁ ≫ a₂ = a₁ ≫ g₁
ha₂₃ : f₂ ≫ a₃ = a₂ ≫ g₂
b₂ : X' ⟶ X
hb₁₂ : g₁ ≫ b₂ = b₁ ≫ f₁
hb₂₃ : g₂ ≫ b₃ = b₂ ≫ f₂
⊢ (P.extensionProduct Q).retractClosure X

Tactic:
⟨X', ⟨_, _, _, _, _, hT', hP, hQ⟩, ⟨?_⟩⟩

Premises used: Nonempty.intro, And.intro, Exists.intro, CategoryTheory.ObjectProperty.retractClosure
-- Module: Mathlib.CategoryTheory.Triangulated.Subcategory
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : HasShift.{v_1, u_1, 0} C ℤ
inst_3 : Preadditive.{v_1, u_1} C
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
P Q : ObjectProperty.{v_1, u_1} C
X A B : C
f₁ : A ⟶ X
f₂ : X ⟶ B
f₃ : B ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) A
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk f₁ f₂ f₃)
A' : C
hP : P A'
a₁ : A ⟶ A'
b₁ : A' ⟶ A
h₁ : a₁ ≫ b₁ = 𝟙 A
B' : C
hQ : Q B'
a₃ : B ⟶ B'
b₃ : B' ⟶ B
h₃ : a₃ ≫ b₃ = 𝟙 B
X' : C
g₁ : A' ⟶ X'
g₂ : X' ⟶ B'
hT' :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk g₁ g₂ (b₃ ≫ f₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) a₁))
a₂ : X ⟶ X'
ha₁₂ : f₁ ≫ a₂ = a₁ ≫ g₁
ha₂₃ : f₂ ≫ a₃ = a₂ ≫ g₂
b₂ : X' ⟶ X
hb₁₂ : g₁ ≫ b₂ = b₁ ≫ f₁
hb₂₃ : g₂ ≫ b₃ = b₂ ≫ f₂
refine : Retract X X'
⊢ (P.extensionProduct Q).retractClosure X

Tactic:
refine ⟨X', ⟨_, _, _, _, _, hT', hP, hQ⟩, ⟨?_⟩⟩
-- Module: Mathlib.CategoryTheory.Triangulated.Subcategory
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : HasShift.{v_1, u_1, 0} C ℤ
inst_3 : Preadditive.{v_1, u_1} C
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
P Q : ObjectProperty.{v_1, u_1} C
X A B : C
f₁ : A ⟶ X
f₂ : X ⟶ B
f₃ : B ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) A
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk f₁ f₂ f₃)
A' : C
hP : P A'
a₁ : A ⟶ A'
b₁ : A' ⟶ A
h₁ : a₁ ≫ b₁ = 𝟙 A
B' : C
hQ : Q B'
a₃ : B ⟶ B'
b₃ : B' ⟶ B
h₃ : a₃ ≫ b₃ = 𝟙 B
X' : C
g₁ : A' ⟶ X'
g₂ : X' ⟶ B'
hT' :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk g₁ g₂ (b₃ ≫ f₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) a₁))
a₂ : X ⟶ X'
ha₁₂ : f₁ ≫ a₂ = a₁ ≫ g₁
ha₂₃ : f₂ ≫ a₃ = a₂ ≫ g₂
b₂ : X' ⟶ X
hb₁₂ : g₁ ≫ b₂ = b₁ ≫ f₁
hb₂₃ : g₂ ≫ b₃ = b₂ ≫ f₂
dsimp : f₁ ≫ a₂ ≫ b₂ = 𝟙 A ≫ f₁
⊢ (Triangle.mk f₁ f₂ f₃).mor₁ ≫ a₂ ≫ b₂ = 𝟙 A ≫ (Triangle.mk f₁ f₂ f₃).mor₁

Tactic:
dsimp
-- Module: Mathlib.CategoryTheory.Triangulated.Subcategory
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : HasShift.{v_1, u_1, 0} C ℤ
inst_3 : Preadditive.{v_1, u_1} C
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
P Q : ObjectProperty.{v_1, u_1} C
X A B : C
f₁ : A ⟶ X
f₂ : X ⟶ B
f₃ : B ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) A
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk f₁ f₂ f₃)
A' : C
hP : P A'
a₁ : A ⟶ A'
b₁ : A' ⟶ A
h₁ : a₁ ≫ b₁ = 𝟙 A
B' : C
hQ : Q B'
a₃ : B ⟶ B'
b₃ : B' ⟶ B
h₃ : a₃ ≫ b₃ = 𝟙 B
X' : C
g₁ : A' ⟶ X'
g₂ : X' ⟶ B'
hT' :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk g₁ g₂ (b₃ ≫ f₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) a₁))
a₂ : X ⟶ X'
ha₁₂ : f₁ ≫ a₂ = a₁ ≫ g₁
ha₂₃ : f₂ ≫ a₃ = a₂ ≫ g₂
b₂ : X' ⟶ X
hb₁₂ : g₁ ≫ b₂ = b₁ ≫ f₁
hb₂₃ : g₂ ≫ b₃ = b₂ ≫ f₂
⊢ f₁ ≫ a₂ ≫ b₂ = 𝟙 A ≫ f₁

Tactic:
grind
-- Module: Mathlib.CategoryTheory.Triangulated.Subcategory
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : HasShift.{v_1, u_1, 0} C ℤ
inst_3 : Preadditive.{v_1, u_1} C
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
P Q : ObjectProperty.{v_1, u_1} C
X A B : C
f₁ : A ⟶ X
f₂ : X ⟶ B
f₃ : B ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) A
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk f₁ f₂ f₃)
A' : C
hP : P A'
a₁ : A ⟶ A'
b₁ : A' ⟶ A
h₁ : a₁ ≫ b₁ = 𝟙 A
B' : C
hQ : Q B'
a₃ : B ⟶ B'
b₃ : B' ⟶ B
h₃ : a₃ ≫ b₃ = 𝟙 B
X' : C
g₁ : A' ⟶ X'
g₂ : X' ⟶ B'
hT' :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk g₁ g₂ (b₃ ≫ f₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) a₁))
a₂ : X ⟶ X'
ha₁₂ : f₁ ≫ a₂ = a₁ ≫ g₁
ha₂₃ : f₂ ≫ a₃ = a₂ ≫ g₂
b₂ : X' ⟶ X
hb₁₂ : g₁ ≫ b₂ = b₁ ≫ f₁
hb₂₃ : g₂ ≫ b₃ = b₂ ≫ f₂
dsimp : f₂ ≫ 𝟙 B = (a₂ ≫ b₂) ≫ f₂
⊢ (Triangle.mk f₁ f₂ f₃).mor₂ ≫ 𝟙 B = (a₂ ≫ b₂) ≫ (Triangle.mk f₁ f₂ f₃).mor₂

Tactic:
dsimp
-- Module: Mathlib.CategoryTheory.Triangulated.Subcategory
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : HasShift.{v_1, u_1, 0} C ℤ
inst_3 : Preadditive.{v_1, u_1} C
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
P Q : ObjectProperty.{v_1, u_1} C
X A B : C
f₁ : A ⟶ X
f₂ : X ⟶ B
f₃ : B ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) A
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk f₁ f₂ f₃)
A' : C
hP : P A'
a₁ : A ⟶ A'
b₁ : A' ⟶ A
h₁ : a₁ ≫ b₁ = 𝟙 A
B' : C
hQ : Q B'
a₃ : B ⟶ B'
b₃ : B' ⟶ B
h₃ : a₃ ≫ b₃ = 𝟙 B
X' : C
g₁ : A' ⟶ X'
g₂ : X' ⟶ B'
hT' :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk g₁ g₂ (b₃ ≫ f₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) a₁))
a₂ : X ⟶ X'
ha₁₂ : f₁ ≫ a₂ = a₁ ≫ g₁
ha₂₃ : f₂ ≫ a₃ = a₂ ≫ g₂
b₂ : X' ⟶ X
hb₁₂ : g₁ ≫ b₂ = b₁ ≫ f₁
hb₂₃ : g₂ ≫ b₃ = b₂ ≫ f₂
⊢ f₂ ≫ 𝟙 B = (a₂ ≫ b₂) ≫ f₂

Tactic:
grind
-- Module: Mathlib.CategoryTheory.Triangulated.Subcategory
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : HasShift.{v_1, u_1, 0} C ℤ
inst_3 : Preadditive.{v_1, u_1} C
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
P Q : ObjectProperty.{v_1, u_1} C
X A B : C
f₁ : A ⟶ X
f₂ : X ⟶ B
f₃ : B ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) A
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk f₁ f₂ f₃)
A' : C
hP : P A'
a₁ : A ⟶ A'
b₁ : A' ⟶ A
h₁ : a₁ ≫ b₁ = 𝟙 A
B' : C
hQ : Q B'
a₃ : B ⟶ B'
b₃ : B' ⟶ B
h₃ : a₃ ≫ b₃ = 𝟙 B
X' : C
g₁ : A' ⟶ X'
g₂ : X' ⟶ B'
hT' :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk g₁ g₂ (b₃ ≫ f₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) a₁))
a₂ : X ⟶ X'
ha₁₂ : f₁ ≫ a₂ = a₁ ≫ g₁
ha₂₃ : f₂ ≫ a₃ = a₂ ≫ g₂
b₂ : X' ⟶ X
hb₁₂ : g₁ ≫ b₂ = b₁ ≫ f₁
hb₂₃ : g₂ ≫ b₃ = b₂ ≫ f₂
φ : Triangle.mk f₁ f₂ f₃ ⟶ Triangle.mk f₁ f₂ f₃ :=
  (Triangle.mk f₁ f₂ f₃).homMk (Triangle.mk f₁ f₂ f₃) (𝟙 A) (a₂ ≫ b₂) (𝟙 B) ⋯ ⋯ ⋯
⊢ IsIso (a₂ ≫ b₂)

Tactic:
haveI : IsIso (a₂ ≫ b₂) := isIso₂_of_isIso₁₃ φ hT hT (IsIso.id _) (IsIso.id _)
-- Module: Mathlib.CategoryTheory.Triangulated.Subcategory
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : HasShift.{v_1, u_1, 0} C ℤ
inst_3 : Preadditive.{v_1, u_1} C
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
P Q : ObjectProperty.{v_1, u_1} C
X A B : C
f₁ : A ⟶ X
f₂ : X ⟶ B
f₃ : B ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) A
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk f₁ f₂ f₃)
A' : C
hP : P A'
a₁ : A ⟶ A'
b₁ : A' ⟶ A
h₁ : a₁ ≫ b₁ = 𝟙 A
B' : C
hQ : Q B'
a₃ : B ⟶ B'
b₃ : B' ⟶ B
h₃ : a₃ ≫ b₃ = 𝟙 B
X' : C
g₁ : A' ⟶ X'
g₂ : X' ⟶ B'
hT' :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk g₁ g₂ (b₃ ≫ f₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) a₁))
a₂ : X ⟶ X'
ha₁₂ : f₁ ≫ a₂ = a₁ ≫ g₁
ha₂₃ : f₂ ≫ a₃ = a₂ ≫ g₂
b₂ : X' ⟶ X
hb₁₂ : g₁ ≫ b₂ = b₁ ≫ f₁
hb₂₃ : g₂ ≫ b₃ = b₂ ≫ f₂
φ : Triangle.mk f₁ f₂ f₃ ⟶ Triangle.mk f₁ f₂ f₃ :=
  (Triangle.mk f₁ f₂ f₃).homMk (Triangle.mk f₁ f₂ f₃) (𝟙 A) (a₂ ≫ b₂) (𝟙 B) ⋯ ⋯ ⋯
this : IsIso (a₂ ≫ b₂)
⊢ a₂ ≫ b₂ ≫ inv (a₂ ≫ b₂) = 𝟙 X

Tactic:
grind