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
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 : ObjectProperty.{v_1, u_1} C
inst_14 : IsTriangulated.{v_1, u_1} C
inst_15 : P.IsTriangulated
Y Z_1 Z' : C
f₁ f₂ : Y ⟶ Z_1
s : Z_1 ⟶ Z'
hf₁ : f₁ ≫ s = f₂ ≫ s
Z : C
g : Z ⟶ Z_1
h : Z' ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) Z
H :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk g s h)
mem : P Z
hf₂ : (f₁ - f₂) ≫ s = 0
q : Y ⟶ (Triangle.mk g s h).obj₁
hq : f₁ - f₂ = q ≫ (Triangle.mk g s h).mor₁
Y' : C
r : Y' ⟶ Y
t : (Triangle.mk g s h).obj₁ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) Y'
mem' :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk r q t)
⊢ ∃ (X' : C) (t : X' ⟶ Y) (_ : P.trW t), t ≫ f₁ = t ≫ f₂

Tactic:
⟨Y', r, ?_, ?_⟩

Premises used: 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
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 : ObjectProperty.{v_1, u_1} C
inst_14 : IsTriangulated.{v_1, u_1} C
inst_15 : P.IsTriangulated
Y Z_1 Z' : C
f₁ f₂ : Y ⟶ Z_1
s : Z_1 ⟶ Z'
hf₁ : f₁ ≫ s = f₂ ≫ s
Z : C
g : Z ⟶ Z_1
h : Z' ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) Z
H :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk g s h)
mem : P Z
hf₂ : (f₁ - f₂) ≫ s = 0
q : Y ⟶ (Triangle.mk g s h).obj₁
hq : f₁ - f₂ = q ≫ (Triangle.mk g s h).mor₁
Y' : C
r : Y' ⟶ Y
t : (Triangle.mk g s h).obj₁ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) Y'
mem' :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk r q t)
refine_1 : P.trW r
refine_2 : r ≫ f₁ = r ≫ f₂
⊢ ∃ (X' : C) (t : X' ⟶ Y) (_ : P.trW t), t ≫ f₁ = t ≫ f₂

Tactic:
refine ⟨Y', r, ?_, ?_⟩
-- 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 : ObjectProperty.{v_1, u_1} C
inst_14 : IsTriangulated.{v_1, u_1} C
inst_15 : P.IsTriangulated
Y Z_1 Z' : C
f₁ f₂ : Y ⟶ Z_1
s : Z_1 ⟶ Z'
hf₁ : f₁ ≫ s = f₂ ≫ s
Z : C
g : Z ⟶ Z_1
h : Z' ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) Z
H :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk g s h)
mem : P Z
hf₂ : (f₁ - f₂) ≫ s = 0
q : Y ⟶ (Triangle.mk g s h).obj₁
hq : f₁ - f₂ = q ≫ (Triangle.mk g s h).mor₁
Y' : C
r : Y' ⟶ Y
t : (Triangle.mk g s h).obj₁ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) Y'
mem' :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk r q t)
⊢ P.trW r

Tactic:
⟨_, _, _, mem', mem⟩

Premises used: CategoryTheory.ObjectProperty.trW, 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
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 : ObjectProperty.{v_1, u_1} C
inst_14 : IsTriangulated.{v_1, u_1} C
inst_15 : P.IsTriangulated
Y Z_1 Z' : C
f₁ f₂ : Y ⟶ Z_1
s : Z_1 ⟶ Z'
hf₁ : f₁ ≫ s = f₂ ≫ s
Z : C
g : Z ⟶ Z_1
h : Z' ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) Z
H :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk g s h)
mem : P Z
hf₂ : (f₁ - f₂) ≫ s = 0
q : Y ⟶ (Triangle.mk g s h).obj₁
hq : f₁ - f₂ = q ≫ (Triangle.mk g s h).mor₁
Y' : C
r : Y' ⟶ Y
t : (Triangle.mk g s h).obj₁ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) Y'
mem' :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk r q t)
⊢ P.trW r

Tactic:
exact ⟨_, _, _, mem', mem⟩
-- 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 : ObjectProperty.{v_1, u_1} C
inst_14 : IsTriangulated.{v_1, u_1} C
inst_15 : P.IsTriangulated
Y Z_1 Z' : C
f₁ f₂ : Y ⟶ Z_1
s : Z_1 ⟶ Z'
hf₁ : f₁ ≫ s = f₂ ≫ s
Z : C
g : Z ⟶ Z_1
h : Z' ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) Z
H :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk g s h)
mem : P Z
hf₂ : (f₁ - f₂) ≫ s = 0
q : Y ⟶ (Triangle.mk g s h).obj₁
hq : f₁ - f₂ = q ≫ (Triangle.mk g s h).mor₁
Y' : C
r : Y' ⟶ Y
t : (Triangle.mk g s h).obj₁ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) Y'
mem' :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk r q t)
⊢ (Triangle.mk r q t).mor₁ ≫ (Triangle.mk r q t).mor₂ = 0

Tactic:
have eq := comp_distTriang_mor_zero₁₂ _ mem'
-- 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 : ObjectProperty.{v_1, u_1} C
inst_14 : IsTriangulated.{v_1, u_1} C
inst_15 : P.IsTriangulated
Y Z_1 Z' : C
f₁ f₂ : Y ⟶ Z_1
s : Z_1 ⟶ Z'
hf₁ : f₁ ≫ s = f₂ ≫ s
Z : C
g : Z ⟶ Z_1
h : Z' ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) Z
H :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk g s h)
mem : P Z
hf₂ : (f₁ - f₂) ≫ s = 0
q : Y ⟶ (Triangle.mk g s h).obj₁
hq : f₁ - f₂ = q ≫ (Triangle.mk g s h).mor₁
Y' : C
r : Y' ⟶ Y
t : (Triangle.mk g s h).obj₁ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) Y'
mem' :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk r q t)
eq : r ≫ q = 0
refine_2 : r ≫ f₁ - r ≫ f₂ = 0
refine_2₁ : r ≫ (f₁ - f₂) = 0
refine_2₂ : r ≫ q ≫ (Triangle.mk g s h).mor₁ = 0
refine_2₃ : Eq (α := Y' ⟶ Z_1) (0 ≫ (Triangle.mk g s h).mor₁) 0
refine_2₄ refine_2₅ : Eq.{v_1 + 1} (α := Y' ⟶ Z_1) 0 0
⊢ r ≫ f₁ = r ≫ f₂

Tactic:
rw [← sub_eq_zero, ← comp_sub, hq, reassoc_of% eq, zero_comp]

Premises used: sub_eq_zero, CategoryTheory.Limits.zero_comp, CategoryTheory.Preadditive.comp_sub
-- 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 : ObjectProperty.{v_1, u_1} C
inst_14 : IsTriangulated.{v_1, u_1} C
inst_15 : P.IsTriangulated
T₁ T₃ : Triangle.{v_1, u_1} C
mem₁ : Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) T₁
mem₃ : Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) T₃
a : T₁.obj₁ ⟶ T₃.obj₁
b : T₁.obj₂ ⟶ T₃.obj₂
Z₅ : C
g₅ : T₃.obj₁ ⟶ Z₅
h₅ : Z₅ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) T₁.obj₁
mem₅ :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk a g₅ h₅)
mem₅' : P Z₅
Z₄ : C
g₄ : T₃.obj₂ ⟶ Z₄
h₄ : Z₄ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) T₁.obj₂
mem₄ :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk b g₄ h₄)
mem₄' : P Z₄
comm : T₁.mor₁ ≫ b = a ≫ T₃.mor₁
obtain :
  ∀ (Z₂ : C) (g₂ : T₃.obj₂ ⟶ Z₂) (h₂ : Z₂ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) T₁.obj₁)
    (mem₂ :
      Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
        (Triangle.mk (T₁.mor₁ ≫ b) g₂ h₂)),
    ∃ (c : T₁.obj₃ ⟶ T₃.obj₃) (_ : P.trW c),
      T₁.mor₂ ≫ c = b ≫ T₃.mor₂ ∧ T₁.mor₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) a = c ≫ T₃.mor₃
⊢ ∃ (c : T₁.obj₃ ⟶ T₃.obj₃) (_ : P.trW c),
    T₁.mor₂ ≫ c = b ≫ T₃.mor₂ ∧ T₁.mor₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) a = c ≫ T₃.mor₃

Tactic:
obtain ⟨Z₂, g₂, h₂, mem₂⟩ := distinguished_cocone_triangle (T₁.mor₁ ≫ b)

Premises used: CategoryTheory.Pretriangulated.distinguished_cocone_triangle, CategoryTheory.CategoryStruct.comp, CategoryTheory.Pretriangulated.Triangle.mor₁
-- 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 : ObjectProperty.{v_1, u_1} C
inst_14 : IsTriangulated.{v_1, u_1} C
inst_15 : P.IsTriangulated
T₁ T₃ : Triangle.{v_1, u_1} C
mem₁ : Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) T₁
mem₃ : Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) T₃
a : T₁.obj₁ ⟶ T₃.obj₁
b : T₁.obj₂ ⟶ T₃.obj₂
Z₅ : C
g₅ : T₃.obj₁ ⟶ Z₅
h₅ : Z₅ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) T₁.obj₁
mem₅ :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk a g₅ h₅)
mem₅' : P Z₅
Z₄ : C
g₄ : T₃.obj₂ ⟶ Z₄
h₄ : Z₄ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) T₁.obj₂
mem₄ :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk b g₄ h₄)
mem₄' : P Z₄
comm : T₁.mor₁ ≫ b = a ≫ T₃.mor₁
Z₂ : C
g₂ : T₃.obj₂ ⟶ Z₂
h₂ : Z₂ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) T₁.obj₁
mem₂ :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk (T₁.mor₁ ≫ b) g₂ h₂)
have :
  ∀ (H : Octahedron ⋯ mem₁ mem₄ mem₂),
    ∃ (c : T₁.obj₃ ⟶ T₃.obj₃) (_ : P.trW c),
      T₁.mor₂ ≫ c = b ≫ T₃.mor₂ ∧ T₁.mor₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) a = c ≫ T₃.mor₃
⊢ ∃ (c : T₁.obj₃ ⟶ T₃.obj₃) (_ : P.trW c),
    T₁.mor₂ ≫ c = b ≫ T₃.mor₂ ∧ T₁.mor₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) a = c ≫ T₃.mor₃

Tactic:
have H := someOctahedron rfl mem₁ mem₄ mem₂
-- 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 : ObjectProperty.{v_1, u_1} C
inst_14 : IsTriangulated.{v_1, u_1} C
inst_15 : P.IsTriangulated
T₁ T₃ : Triangle.{v_1, u_1} C
mem₁ : Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) T₁
mem₃ : Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) T₃
a : T₁.obj₁ ⟶ T₃.obj₁
b : T₁.obj₂ ⟶ T₃.obj₂
Z₅ : C
g₅ : T₃.obj₁ ⟶ Z₅
h₅ : Z₅ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) T₁.obj₁
mem₅ :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk a g₅ h₅)
mem₅' : P Z₅
Z₄ : C
g₄ : T₃.obj₂ ⟶ Z₄
h₄ : Z₄ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) T₁.obj₂
mem₄ :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk b g₄ h₄)
mem₄' : P Z₄
comm : T₁.mor₁ ≫ b = a ≫ T₃.mor₁
Z₂ : C
g₂ : T₃.obj₂ ⟶ Z₂
h₂ : Z₂ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) T₁.obj₁
mem₂ :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk (T₁.mor₁ ≫ b) g₂ h₂)
H : Octahedron ⋯ mem₁ mem₄ mem₂
have :
  ∀ (H' : Octahedron ⋯ mem₅ mem₃ mem₂),
    ∃ (c : T₁.obj₃ ⟶ T₃.obj₃) (_ : P.trW c),
      T₁.mor₂ ≫ c = b ≫ T₃.mor₂ ∧ T₁.mor₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) a = c ≫ T₃.mor₃
⊢ ∃ (c : T₁.obj₃ ⟶ T₃.obj₃) (_ : P.trW c),
    T₁.mor₂ ≫ c = b ≫ T₃.mor₂ ∧ T₁.mor₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) a = c ≫ T₃.mor₃

Tactic:
have H' := someOctahedron comm.symm mem₅ mem₃ mem₂
-- 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 : ObjectProperty.{v_1, u_1} C
inst_14 : IsTriangulated.{v_1, u_1} C
inst_15 : P.IsTriangulated
T₁ T₃ : Triangle.{v_1, u_1} C
mem₁ : Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) T₁
mem₃ : Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) T₃
a : T₁.obj₁ ⟶ T₃.obj₁
b : T₁.obj₂ ⟶ T₃.obj₂
Z₅ : C
g₅ : T₃.obj₁ ⟶ Z₅
h₅ : Z₅ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) T₁.obj₁
mem₅ :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk a g₅ h₅)
mem₅' : P Z₅
Z₄ : C
g₄ : T₃.obj₂ ⟶ Z₄
h₄ : Z₄ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) T₁.obj₂
mem₄ :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk b g₄ h₄)
mem₄' : P Z₄
comm : T₁.mor₁ ≫ b = a ≫ T₃.mor₁
Z₂ : C
g₂ : T₃.obj₂ ⟶ Z₂
h₂ : Z₂ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) T₁.obj₁
mem₂ :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk (T₁.mor₁ ≫ b) g₂ h₂)
H : Octahedron ⋯ mem₁ mem₄ mem₂
H' : Octahedron ⋯ mem₅ mem₃ mem₂
φ : T₁ ⟶ T₃ := H.triangleMorphism₁ ≫ H'.triangleMorphism₂
⊢ T₁.mor₂ ≫ φ.hom₃ = b ≫ T₃.mor₂

Tactic:
simpa [φ] using φ.comm₂

Premises used: CategoryTheory.Pretriangulated.TriangleMorphism.comm₂
-- 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 : ObjectProperty.{v_1, u_1} C
inst_14 : IsTriangulated.{v_1, u_1} C
inst_15 : P.IsTriangulated
T₁ T₃ : Triangle.{v_1, u_1} C
mem₁ : Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) T₁
mem₃ : Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) T₃
a : T₁.obj₁ ⟶ T₃.obj₁
b : T₁.obj₂ ⟶ T₃.obj₂
Z₅ : C
g₅ : T₃.obj₁ ⟶ Z₅
h₅ : Z₅ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) T₁.obj₁
mem₅ :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk a g₅ h₅)
mem₅' : P Z₅
Z₄ : C
g₄ : T₃.obj₂ ⟶ Z₄
h₄ : Z₄ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) T₁.obj₂
mem₄ :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk b g₄ h₄)
mem₄' : P Z₄
comm : T₁.mor₁ ≫ b = a ≫ T₃.mor₁
Z₂ : C
g₂ : T₃.obj₂ ⟶ Z₂
h₂ : Z₂ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) T₁.obj₁
mem₂ :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk (T₁.mor₁ ≫ b) g₂ h₂)
H : Octahedron ⋯ mem₁ mem₄ mem₂
H' : Octahedron ⋯ mem₅ mem₃ mem₂
φ : T₁ ⟶ T₃ := H.triangleMorphism₁ ≫ H'.triangleMorphism₂
⊢ T₁.mor₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) a = φ.hom₃ ≫ T₃.mor₃

Tactic:
simpa [φ] using φ.comm₃

Premises used: CategoryTheory.Pretriangulated.TriangleMorphism.comm₃
-- 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 : ObjectProperty.{v_1, u_1} C
inst_14 : IsTriangulated.{v_1, u_1} C
inst_15 : P.IsTriangulated
T₁ T₃ : Triangle.{v_1, u_1} C
mem₁ : Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) T₁
mem₃ : Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) T₃
a : T₁.obj₁ ⟶ T₃.obj₁
b : T₁.obj₂ ⟶ T₃.obj₂
Z₅ : C
g₅ : T₃.obj₁ ⟶ Z₅
h₅ : Z₅ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) T₁.obj₁
mem₅ :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk a g₅ h₅)
mem₅' : P Z₅
Z₄ : C
g₄ : T₃.obj₂ ⟶ Z₄
h₄ : Z₄ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) T₁.obj₂
mem₄ :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk b g₄ h₄)
mem₄' : P Z₄
comm : T₁.mor₁ ≫ b = a ≫ T₃.mor₁
Z₂ : C
g₂ : T₃.obj₂ ⟶ Z₂
h₂ : Z₂ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) T₁.obj₁
mem₂ :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk (T₁.mor₁ ≫ b) g₂ h₂)
H : Octahedron ⋯ mem₁ mem₄ mem₂
H' : Octahedron ⋯ mem₅ mem₃ mem₂
φ : T₁ ⟶ T₃ := H.triangleMorphism₁ ≫ H'.triangleMorphism₂
⊢ ∃ (c : T₁.obj₃ ⟶ T₃.obj₃) (_ : P.trW c),
    T₁.mor₂ ≫ c = b ≫ T₃.mor₂ ∧ T₁.mor₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) a = c ≫ T₃.mor₃

Tactic:
exact ⟨φ.hom₃, P.trW.comp_mem _ _ (trW.mk P H.mem mem₄') (trW.mk' P H'.mem mem₅'),
    by simpa [φ] using φ.comm₂, by simpa [φ] using φ.comm₃⟩
-- 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 P' : ObjectProperty.{v_1, u_1} C
inst_14 : P.IsTriangulatedClosed₂
inst_15 : P.IsClosedUnderIsomorphisms
inst_16 : P'.IsTriangulatedClosed₂
T : Triangle.{v_1, u_1} C
hT : Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) T
h₁ : (P ⊓ P') T.obj₁
h₃ : (P ⊓ P') T.obj₃
obtain : ∀ (X₂ : C) (h₂ : P' X₂) (e : T.obj₂ ≅ X₂), (P ⊓ P').isoClosure T.obj₂
⊢ (P ⊓ P').isoClosure T.obj₂

Tactic:
obtain ⟨X₂, h₂, ⟨e⟩⟩ := P'.ext_of_isTriangulatedClosed₂' T hT h₁.2 h₃.2

Premises used: And.right, CategoryTheory.ObjectProperty.ext_of_isTriangulatedClosed₂'