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
P Q : ObjectProperty.{v_1, u_1} C
inst_6 : P.ContainsZero
X : C
hX : Q X
obtain : ∀ (Z : C) (hZ : IsZero.{v_1, u_1} Z) (hP : P Z), P.isoClosure.extensionProduct Q X
⊢ P.isoClosure.extensionProduct Q X

Tactic:
obtain ⟨Z, hZ, hP⟩ := P.exists_prop_of_containsZero

Premises used: CategoryTheory.ObjectProperty.exists_prop_of_containsZero
-- 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
inst_6 : P.ContainsZero
X : C
hX : Q X
Z : C
hZ : IsZero.{v_1, u_1} Z
hP : P Z
⊢ P.isoClosure.extensionProduct Q X

Tactic:
⟨_, _, _, _, _, inv_rot_of_distTriang _ (contractible_distinguished X), ?_, hX⟩

Premises used: And.intro, CategoryTheory.Pretriangulated.inv_rot_of_distTriang, Exists.intro, CategoryTheory.ObjectProperty.extensionProduct, CategoryTheory.Pretriangulated.contractible_distinguished
-- 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
inst_6 : P.ContainsZero
X : C
hX : Q X
Z : C
hZ : IsZero.{v_1, u_1} Z
hP : P Z
refine :
  P.isoClosure
    (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) (-1))
      (Triangle.obj₃.{v_1, u_1} (contractibleTriangle X)))
⊢ P.isoClosure.extensionProduct Q X

Tactic:
refine ⟨_, _, _, _, _, inv_rot_of_distTriang _ (contractible_distinguished X), ?_, hX⟩
-- 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
inst_6 : P.ContainsZero
X : C
hX : Q X
Z : C
hZ : IsZero.{v_1, u_1} Z
hP : P Z
⊢ P.isoClosure
    (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) (-1))
      (Triangle.obj₃.{v_1, u_1} (contractibleTriangle X)))

Tactic:
⟨Z, hP, ⟨IsZero.iso (Functor.map_isZero _ (isZero_zero C)) hZ⟩⟩

Premises used: CategoryTheory.Limits.isZero_zero, CategoryTheory.Functor.map_isZero, Nonempty.intro, CategoryTheory.Limits.IsZero.iso, CategoryTheory.ObjectProperty.isoClosure, Exists.intro
-- 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
inst_6 : P.ContainsZero
X : C
hX : Q X
Z : C
hZ : IsZero.{v_1, u_1} Z
hP : P Z
⊢ P.isoClosure
    (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) (-1))
      (Triangle.obj₃.{v_1, u_1} (contractibleTriangle X)))

Tactic:
exact ⟨Z, hP, ⟨IsZero.iso (Functor.map_isZero _ (isZero_zero C)) 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
D : Type u_2
inst_6 : Category.{v_2, u_2} D
inst_7 : Preadditive.{v_2, u_2} D
inst_8 : HasZeroObject.{v_2, u_2} D
inst_9 : HasShift.{v_2, u_2, 0} D ℤ
inst_10 : ∀ (n : ℤ), Functor.Additive.{v_2, u_2, v_2, u_2} (shiftFunctor D n)
inst_11 : Pretriangulated.{v_2, u_2} D
E : Type u_3
inst_12 : Category.{v_3, u_3} E
inst_13 : HasShift.{v_3, u_3, 0} E ℤ
P Q R : ObjectProperty.{v_1, u_1} C
inst_14 : P.IsStableUnderShift ℤ
inst_15 : Q.IsStableUnderShift ℤ
a : ℤ
X : C
this :
  ∀ (Y Z : C) (f : Y ⟶ X) (g : X ⟶ Z) (h : Z ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) Y)
    (hT :
      Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
        (Triangle.mk f g h))
    (hP : P Y) (hQ : Q Z), (P.extensionProduct Q).shift a X
⊢ P.extensionProduct Q X ≤ (P.extensionProduct Q).shift a X

Tactic:
⟨Y, Z, f, g, h, 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
D : Type u_2
inst_6 : Category.{v_2, u_2} D
inst_7 : Preadditive.{v_2, u_2} D
inst_8 : HasZeroObject.{v_2, u_2} D
inst_9 : HasShift.{v_2, u_2, 0} D ℤ
inst_10 : ∀ (n : ℤ), Functor.Additive.{v_2, u_2, v_2, u_2} (shiftFunctor D n)
inst_11 : Pretriangulated.{v_2, u_2} D
E : Type u_3
inst_12 : Category.{v_3, u_3} E
inst_13 : HasShift.{v_3, u_3, 0} E ℤ
P Q R : ObjectProperty.{v_1, u_1} C
inst_14 : P.IsStableUnderShift ℤ
inst_15 : Q.IsStableUnderShift ℤ
a : ℤ
X Y Z : C
f : Y ⟶ X
g : X ⟶ Z
h : Z ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) Y
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk f g h)
hP : P Y
hQ : Q Z
⊢ (P.extensionProduct Q).shift a X

Tactic:
⟨_, _, _, _, _, Triangle.shift_distinguished _ hT a, ?_, ?_⟩

Premises used: CategoryTheory.Pretriangulated.Triangle.shift_distinguished, CategoryTheory.ObjectProperty.shift, And.intro, Exists.intro, CategoryTheory.ObjectProperty.extensionProduct
-- 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
D : Type u_2
inst_6 : Category.{v_2, u_2} D
inst_7 : Preadditive.{v_2, u_2} D
inst_8 : HasZeroObject.{v_2, u_2} D
inst_9 : HasShift.{v_2, u_2, 0} D ℤ
inst_10 : ∀ (n : ℤ), Functor.Additive.{v_2, u_2, v_2, u_2} (shiftFunctor D n)
inst_11 : Pretriangulated.{v_2, u_2} D
E : Type u_3
inst_12 : Category.{v_3, u_3} E
inst_13 : HasShift.{v_3, u_3, 0} E ℤ
P Q R : ObjectProperty.{v_1, u_1} C
inst_14 : P.IsStableUnderShift ℤ
inst_15 : Q.IsStableUnderShift ℤ
a : ℤ
X Y Z : C
f : Y ⟶ X
g : X ⟶ Z
h : Z ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) Y
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk f g h)
hP : P Y
hQ : Q Z
refine_1 : P (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C { as := a : Discrete ℤ }.as) (Triangle.mk f g h).obj₁)
refine_2 : Q (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C { as := a : Discrete ℤ }.as) (Triangle.mk f g h).obj₃)
⊢ (P.extensionProduct Q).shift a X

Tactic:
refine ⟨_, _, _, _, _, Triangle.shift_distinguished _ hT a, ?_, ?_⟩
-- 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
D : Type u_2
inst_6 : Category.{v_2, u_2} D
inst_7 : Preadditive.{v_2, u_2} D
inst_8 : HasZeroObject.{v_2, u_2} D
inst_9 : HasShift.{v_2, u_2, 0} D ℤ
inst_10 : ∀ (n : ℤ), Functor.Additive.{v_2, u_2, v_2, u_2} (shiftFunctor D n)
inst_11 : Pretriangulated.{v_2, u_2} D
E : Type u_3
inst_12 : Category.{v_3, u_3} E
inst_13 : HasShift.{v_3, u_3, 0} E ℤ
P Q R : ObjectProperty.{v_1, u_1} C
inst_14 : P.IsStableUnderShift ℤ
inst_15 : Q.IsStableUnderShift ℤ
a : ℤ
X Y Z : C
f : Y ⟶ X
g : X ⟶ Z
h : Z ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) Y
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk f g h)
hP : P Y
hQ : Q Z
refine_1 : P (Triangle.mk f g h).obj₁
⊢ P (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C { as := a : Discrete ℤ }.as) (Triangle.mk f g h).obj₁)

Tactic:
apply IsStableUnderShiftBy.le_shift

Premises used: CategoryTheory.ObjectProperty.IsStableUnderShiftBy.le_shift
-- 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
D : Type u_2
inst_6 : Category.{v_2, u_2} D
inst_7 : Preadditive.{v_2, u_2} D
inst_8 : HasZeroObject.{v_2, u_2} D
inst_9 : HasShift.{v_2, u_2, 0} D ℤ
inst_10 : ∀ (n : ℤ), Functor.Additive.{v_2, u_2, v_2, u_2} (shiftFunctor D n)
inst_11 : Pretriangulated.{v_2, u_2} D
E : Type u_3
inst_12 : Category.{v_3, u_3} E
inst_13 : HasShift.{v_3, u_3, 0} E ℤ
P Q R : ObjectProperty.{v_1, u_1} C
inst_14 : P.IsStableUnderShift ℤ
inst_15 : Q.IsStableUnderShift ℤ
a : ℤ
X Y Z : C
f : Y ⟶ X
g : X ⟶ Z
h : Z ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) Y
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk f g h)
hP : P Y
hQ : Q Z
⊢ P (Triangle.mk f g h).obj₁

Tactic:
assumption
-- 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
D : Type u_2
inst_6 : Category.{v_2, u_2} D
inst_7 : Preadditive.{v_2, u_2} D
inst_8 : HasZeroObject.{v_2, u_2} D
inst_9 : HasShift.{v_2, u_2, 0} D ℤ
inst_10 : ∀ (n : ℤ), Functor.Additive.{v_2, u_2, v_2, u_2} (shiftFunctor D n)
inst_11 : Pretriangulated.{v_2, u_2} D
E : Type u_3
inst_12 : Category.{v_3, u_3} E
inst_13 : HasShift.{v_3, u_3, 0} E ℤ
P Q R : ObjectProperty.{v_1, u_1} C
inst_14 : P.IsStableUnderShift ℤ
inst_15 : Q.IsStableUnderShift ℤ
a : ℤ
X Y Z : C
f : Y ⟶ X
g : X ⟶ Z
h : Z ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) Y
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk f g h)
hP : P Y
hQ : Q Z
refine_2 : Q (Triangle.mk f g h).obj₃
⊢ Q (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C { as := a : Discrete ℤ }.as) (Triangle.mk f g h).obj₃)

Tactic:
apply IsStableUnderShiftBy.le_shift

Premises used: CategoryTheory.ObjectProperty.IsStableUnderShiftBy.le_shift
-- 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
D : Type u_2
inst_6 : Category.{v_2, u_2} D
inst_7 : Preadditive.{v_2, u_2} D
inst_8 : HasZeroObject.{v_2, u_2} D
inst_9 : HasShift.{v_2, u_2, 0} D ℤ
inst_10 : ∀ (n : ℤ), Functor.Additive.{v_2, u_2, v_2, u_2} (shiftFunctor D n)
inst_11 : Pretriangulated.{v_2, u_2} D
E : Type u_3
inst_12 : Category.{v_3, u_3} E
inst_13 : HasShift.{v_3, u_3, 0} E ℤ
P Q R : ObjectProperty.{v_1, u_1} C
inst_14 : P.IsStableUnderShift ℤ
inst_15 : Q.IsStableUnderShift ℤ
a : ℤ
X Y Z : C
f : Y ⟶ X
g : X ⟶ Z
h : Z ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) Y
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk f g h)
hP : P Y
hQ : Q Z
⊢ Q (Triangle.mk f g h).obj₃

Tactic:
assumption
-- 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 R : ObjectProperty.{v_1, u_1} C
inst_6 : IsTriangulated.{v_1, u_1} C
h : ∀ (X : C), (P.extensionProduct Q).extensionProduct R X ↔ P.extensionProduct (Q.extensionProduct R) X
⊢ (P.extensionProduct Q).extensionProduct R = P.extensionProduct (Q.extensionProduct R)

Tactic:
ext X
-- 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 R : ObjectProperty.{v_1, u_1} C
inst_6 : IsTriangulated.{v_1, u_1} C
X : C
h : (P.extensionProduct Q).extensionProduct R X → P.extensionProduct (Q.extensionProduct R) X
h₁ : P.extensionProduct (Q.extensionProduct R) X → (P.extensionProduct Q).extensionProduct R X
⊢ (P.extensionProduct Q).extensionProduct R X ↔ P.extensionProduct (Q.extensionProduct R) X

Tactic:
constructor
-- Module: Mathlib.CategoryTheory.Triangulated.Subcategory
Goal:
C_1 : Type u_1
inst : Category.{v_1, u_1} C_1
inst_1 : HasZeroObject.{v_1, u_1} C_1
inst_2 : HasShift.{v_1, u_1, 0} C_1 ℤ
inst_3 : Preadditive.{v_1, u_1} C_1
inst_4 : ∀ (n : ℤ), Functor.Additive.{v_1, u_1, v_1, u_1} (shiftFunctor C_1 n)
inst_5 : Pretriangulated.{v_1, u_1} C_1
P Q R : ObjectProperty.{v_1, u_1} C_1
inst_6 : IsTriangulated.{v_1, u_1} C_1
X Y C : C_1
f : Y ⟶ X
g : X ⟶ C
h : C ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C_1 (A := ℤ) 1) Y
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C_1)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk f g h)
A B : C_1
f' : A ⟶ Y
g' : Y ⟶ B
h' : B ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C_1 (A := ℤ) 1) A
hT' :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C_1)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk f' g' h')
hP : P A
hQ : Q B
hR : R C
h₁ :
  ∀ (Y_1 : C_1) (g'' : X ⟶ Y_1) (h'' : Y_1 ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C_1 (A := ℤ) 1) A)
    (hT'' :
      Membership.mem (γ := Set (Triangle.{v_1, u_1} C_1)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
        (Triangle.mk (f' ≫ f) g'' h'')),
    P.extensionProduct (Q.extensionProduct R) X
⊢ P.extensionProduct (Q.extensionProduct R) X

Tactic:
obtain ⟨Y, g'', h'', hT''⟩ := distinguished_cocone_triangle (f' ≫ f)

Premises used: CategoryTheory.Pretriangulated.distinguished_cocone_triangle, CategoryTheory.CategoryStruct.comp
-- Module: Mathlib.CategoryTheory.Triangulated.Subcategory
Goal:
C_1 : Type u_1
inst : Category.{v_1, u_1} C_1
inst_1 : HasZeroObject.{v_1, u_1} C_1
inst_2 : HasShift.{v_1, u_1, 0} C_1 ℤ
inst_3 : Preadditive.{v_1, u_1} C_1
inst_4 : ∀ (n : ℤ), Functor.Additive.{v_1, u_1, v_1, u_1} (shiftFunctor C_1 n)
inst_5 : Pretriangulated.{v_1, u_1} C_1
P Q R : ObjectProperty.{v_1, u_1} C_1
inst_6 : IsTriangulated.{v_1, u_1} C_1
X Y_1 C : C_1
f : Y_1 ⟶ X
g : X ⟶ C
h : C ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C_1 (A := ℤ) 1) Y_1
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C_1)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk f g h)
A B : C_1
f' : A ⟶ Y_1
g' : Y_1 ⟶ B
h' : B ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C_1 (A := ℤ) 1) A
hT' :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C_1)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk f' g' h')
hP : P A
hQ : Q B
hR : R C
Y : C_1
g'' : X ⟶ Y
h'' : Y ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C_1 (A := ℤ) 1) A
hT'' :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C_1)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk (f' ≫ f) g'' h'')
o : Octahedron ⋯ hT' hT hT'' := someOctahedron ⋯ hT' hT hT''
⊢ P.extensionProduct (Q.extensionProduct R) X

Tactic:
⟨_, _, _, _, _, hT'', hP, ⟨_, _, _, _, _, o.mem, hQ, hR⟩⟩

Premises used: CategoryTheory.Triangulated.Octahedron.mem, And.intro, Exists.intro, CategoryTheory.ObjectProperty.extensionProduct
-- Module: Mathlib.CategoryTheory.Triangulated.Subcategory
Goal:
C_1 : Type u_1
inst : Category.{v_1, u_1} C_1
inst_1 : HasZeroObject.{v_1, u_1} C_1
inst_2 : HasShift.{v_1, u_1, 0} C_1 ℤ
inst_3 : Preadditive.{v_1, u_1} C_1
inst_4 : ∀ (n : ℤ), Functor.Additive.{v_1, u_1, v_1, u_1} (shiftFunctor C_1 n)
inst_5 : Pretriangulated.{v_1, u_1} C_1
P Q R : ObjectProperty.{v_1, u_1} C_1
inst_6 : IsTriangulated.{v_1, u_1} C_1
X Y_1 C : C_1
f : Y_1 ⟶ X
g : X ⟶ C
h : C ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C_1 (A := ℤ) 1) Y_1
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C_1)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk f g h)
A B : C_1
f' : A ⟶ Y_1
g' : Y_1 ⟶ B
h' : B ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C_1 (A := ℤ) 1) A
hT' :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C_1)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk f' g' h')
hP : P A
hQ : Q B
hR : R C
Y : C_1
g'' : X ⟶ Y
h'' : Y ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C_1 (A := ℤ) 1) A
hT'' :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C_1)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk (f' ≫ f) g'' h'')
o : Octahedron ⋯ hT' hT hT'' := someOctahedron ⋯ hT' hT hT''
⊢ P.extensionProduct (Q.extensionProduct R) X

Tactic:
exact ⟨_, _, _, _, _, hT'', hP, ⟨_, _, _, _, _, o.mem, hQ, hR⟩⟩
-- Module: Mathlib.CategoryTheory.Triangulated.Subcategory
Goal:
C_1 : Type u_1
inst : Category.{v_1, u_1} C_1
inst_1 : HasZeroObject.{v_1, u_1} C_1
inst_2 : HasShift.{v_1, u_1, 0} C_1 ℤ
inst_3 : Preadditive.{v_1, u_1} C_1
inst_4 : ∀ (n : ℤ), Functor.Additive.{v_1, u_1, v_1, u_1} (shiftFunctor C_1 n)
inst_5 : Pretriangulated.{v_1, u_1} C_1
P Q R : ObjectProperty.{v_1, u_1} C_1
inst_6 : IsTriangulated.{v_1, u_1} C_1
X A Z : C_1
f : A ⟶ X
g : X ⟶ Z
h : Z ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C_1 (A := ℤ) 1) A
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C_1)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk f g h)
hP : P A
B C : C_1
f' : B ⟶ Z
g' : Z ⟶ C
h' : C ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C_1 (A := ℤ) 1) B
hT' :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C_1)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk f' g' h')
hQ : Q B
hR : R C
h₁ :
  ∀ (Y : C_1) (f'' : Y ⟶ X) (h'' : C ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C_1 (A := ℤ) 1) Y)
    (hT'' :
      Membership.mem (γ := Set (Triangle.{v_1, u_1} C_1)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
        (Triangle.mk f'' (g ≫ g') h'')),
    (P.extensionProduct Q).extensionProduct R X
⊢ (P.extensionProduct Q).extensionProduct R X

Tactic:
obtain ⟨Y, f'', h'', hT''⟩ := distinguished_cocone_triangle₁ (g ≫ g')

Premises used: CategoryTheory.Pretriangulated.distinguished_cocone_triangle₁, CategoryTheory.CategoryStruct.comp
-- Module: Mathlib.CategoryTheory.Triangulated.Subcategory
Goal:
C_1 : Type u_1
inst : Category.{v_1, u_1} C_1
inst_1 : HasZeroObject.{v_1, u_1} C_1
inst_2 : HasShift.{v_1, u_1, 0} C_1 ℤ
inst_3 : Preadditive.{v_1, u_1} C_1
inst_4 : ∀ (n : ℤ), Functor.Additive.{v_1, u_1, v_1, u_1} (shiftFunctor C_1 n)
inst_5 : Pretriangulated.{v_1, u_1} C_1
P Q R : ObjectProperty.{v_1, u_1} C_1
inst_6 : IsTriangulated.{v_1, u_1} C_1
X A Z : C_1
f : A ⟶ X
g : X ⟶ Z
h : Z ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C_1 (A := ℤ) 1) A
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C_1)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk f g h)
hP : P A
B C : C_1
f' : B ⟶ Z
g' : Z ⟶ C
h' : C ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C_1 (A := ℤ) 1) B
hT' :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C_1)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk f' g' h')
hQ : Q B
hR : R C
Y : C_1
f'' : Y ⟶ X
h'' : C ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C_1 (A := ℤ) 1) Y
hT'' :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C_1)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk f'' (g ≫ g') h'')
o : Octahedron' ⋯ hT hT' hT'' := someOctahedron' ⋯ hT hT' hT''
⊢ (P.extensionProduct Q).extensionProduct R X

Tactic:
⟨_, _, _, _, _, hT'', ⟨_, _, _, _, _, o.mem, hP, hQ⟩, hR⟩

Premises used: CategoryTheory.Triangulated.Octahedron'.mem, And.intro, Exists.intro, CategoryTheory.ObjectProperty.extensionProduct
-- Module: Mathlib.CategoryTheory.Triangulated.Subcategory
Goal:
C_1 : Type u_1
inst : Category.{v_1, u_1} C_1
inst_1 : HasZeroObject.{v_1, u_1} C_1
inst_2 : HasShift.{v_1, u_1, 0} C_1 ℤ
inst_3 : Preadditive.{v_1, u_1} C_1
inst_4 : ∀ (n : ℤ), Functor.Additive.{v_1, u_1, v_1, u_1} (shiftFunctor C_1 n)
inst_5 : Pretriangulated.{v_1, u_1} C_1
P Q R : ObjectProperty.{v_1, u_1} C_1
inst_6 : IsTriangulated.{v_1, u_1} C_1
X A Z : C_1
f : A ⟶ X
g : X ⟶ Z
h : Z ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C_1 (A := ℤ) 1) A
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C_1)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk f g h)
hP : P A
B C : C_1
f' : B ⟶ Z
g' : Z ⟶ C
h' : C ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C_1 (A := ℤ) 1) B
hT' :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C_1)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk f' g' h')
hQ : Q B
hR : R C
Y : C_1
f'' : Y ⟶ X
h'' : C ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C_1 (A := ℤ) 1) Y
hT'' :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C_1)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk f'' (g ≫ g') h'')
o : Octahedron' ⋯ hT hT' hT'' := someOctahedron' ⋯ hT hT' hT''
⊢ (P.extensionProduct Q).extensionProduct R X

Tactic:
exact ⟨_, _, _, _, _, hT'', ⟨_, _, _, _, _, o.mem, hP, hQ⟩, hR⟩
-- 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 : 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.isoClosure i
⊢ P₁.extensionProduct P₂ i ≤ Q.isoClosure 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
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:
ext_of_isTriangulatedClosed₂' Q _ hT (h₁ _ hY) (h₂ _ hZ)

Premises used: CategoryTheory.ObjectProperty.ext_of_isTriangulatedClosed₂'