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 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
X₂ : C
h₂ : P' X₂
e : T.obj₂ ≅ X₂
⊢ (P ⊓ P').isoClosure T.obj₂

Tactic:
⟨X₂, ⟨P.prop_of_iso e (P.ext_of_isTriangulatedClosed₂ T hT h₁.1 h₃.1), h₂⟩, ⟨e⟩⟩

Premises used: CategoryTheory.ObjectProperty.prop_of_iso, Nonempty.intro, And.left, And.intro, CategoryTheory.ObjectProperty.isoClosure, Exists.intro, 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
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
X₂ : C
h₂ : P' X₂
e : T.obj₂ ≅ X₂
⊢ (P ⊓ P').isoClosure T.obj₂

Tactic:
exact ⟨X₂, ⟨P.prop_of_iso e (P.ext_of_isTriangulatedClosed₂ T hT h₁.1 h₃.1), h₂⟩, ⟨e⟩⟩
-- 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 : P.IsTriangulated
inst_15 : P.IsClosedUnderIsomorphisms
X : C
p : P.LimitOfShape (Discrete WalkingPair) X
⊢ P X

Tactic:
P.prop_of_iso ?_ (P.ext_of_isTriangulatedClosed₂ _
      (binaryProductTriangle_distinguished _ _)
      (p.prop_diag_obj (.mk .left)) (p.prop_diag_obj (.mk .right)))

Premises used: CategoryTheory.ObjectProperty.LimitOfShape.prop_diag_obj, CategoryTheory.ObjectProperty.prop_of_iso, CategoryTheory.Limits.WalkingPair.right, CategoryTheory.Pretriangulated.binaryProductTriangle_distinguished, CategoryTheory.Discrete.mk, CategoryTheory.ObjectProperty.ext_of_isTriangulatedClosed₂, CategoryTheory.Limits.WalkingPair.left
-- 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 : P.IsTriangulated
inst_15 : P.IsClosedUnderIsomorphisms
X : C
p : P.LimitOfShape (Discrete WalkingPair) X
refine :
  Triangle.obj₂.{v_1, u_1}
      (binaryProductTriangle (p.toLimitPresentation (P := P).diag.1 { as := WalkingPair.left })
        (p.toLimitPresentation (P := P).diag.1 { as := WalkingPair.right })) ≅
    X
⊢ P X

Tactic:
refine P.prop_of_iso ?_ (P.ext_of_isTriangulatedClosed₂ _
      (binaryProductTriangle_distinguished _ _)
      (p.prop_diag_obj (.mk .left)) (p.prop_diag_obj (.mk .right)))

Premises used: CategoryTheory.ObjectProperty.LimitOfShape.prop_diag_obj, CategoryTheory.ObjectProperty.prop_of_iso, CategoryTheory.Limits.WalkingPair.right, CategoryTheory.Pretriangulated.binaryProductTriangle_distinguished, CategoryTheory.Discrete.mk, CategoryTheory.ObjectProperty.ext_of_isTriangulatedClosed₂, CategoryTheory.Limits.WalkingPair.left
-- 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 : P.IsTriangulated
inst_15 : P.IsClosedUnderIsomorphisms
X : C
p : P.LimitOfShape (Discrete WalkingPair) X
⊢ Triangle.obj₂.{v_1, u_1}
      (binaryProductTriangle (p.toLimitPresentation (P := P).diag.1 { as := WalkingPair.left })
        (p.toLimitPresentation (P := P).diag.1 { as := WalkingPair.right })) ≅
    X

Tactic:
exact IsLimit.conePointUniqueUpToIso (prodIsProd _ _)
      ((IsLimit.postcomposeHomEquiv (diagramIsoPair p.diag) _).2 p.isLimit)

Premises used: CategoryTheory.Limits.IsLimit.postcomposeHomEquiv, CategoryTheory.Limits.IsLimit.conePointUniqueUpToIso, CategoryTheory.ObjectProperty.LimitOfShape.toLimitPresentation, CategoryTheory.Limits.prodIsProd, CategoryTheory.Limits.LimitPresentation.isLimit, CategoryTheory.Limits.LimitPresentation.diag, Equiv.invFun, CategoryTheory.Limits.diagramIsoPair
-- 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 : P.IsTriangulated
inst_15 : P.IsClosedUnderIsomorphisms
⊢ P.IsClosedUnderFiniteProducts

Tactic:
.mk'

Premises used: CategoryTheory.ObjectProperty.IsClosedUnderFiniteProducts.mk'
-- 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 : P.IsTriangulated
rw : P.isoClosure.trW.IsStableUnderFiniteProducts
⊢ P.trW.IsStableUnderFiniteProducts

Tactic:
rw [← trW_isoClosure]

Premises used: CategoryTheory.ObjectProperty.trW_isoClosure
-- 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 : P.IsTriangulated
J : Type
h : Finite J
⊢ P.isoClosure.trW.IsStableUnderProductsOfShape J

Tactic:
MorphismProperty.IsStableUnderProductsOfShape.mk _ _ ?_

Premises used: CategoryTheory.MorphismProperty.IsStableUnderProductsOfShape.mk
-- 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 : P.IsTriangulated
J : Type
h : Finite J
refine :
  ∀ (X₁ X₂ : J → C) [inst_15 : HasProduct.{0, v_1, u_1} X₁] [inst_16 : HasProduct.{0, v_1, u_1} X₂]
    (f : (j : J) → X₁ j ⟶ X₂ j),
    (∀ (j : J), P.isoClosure.trW (f j)) →
      P.isoClosure.trW (X := piObj.{0, v_1, u_1} X₁) (Y := piObj.{0, v_1, u_1} X₂) (Limits.Pi.map f)
⊢ P.isoClosure.trW.IsStableUnderProductsOfShape J

Tactic:
refine MorphismProperty.IsStableUnderProductsOfShape.mk _ _ ?_

Premises used: CategoryTheory.MorphismProperty.IsStableUnderProductsOfShape.mk
-- 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 : P.IsTriangulated
J : Type
h : Finite J
X₁ : J → C
_ :
  ∀ (X₂ : J → C) [inst_15 : HasProduct.{0, v_1, u_1} X₁] [inst_16 : HasProduct.{0, v_1, u_1} X₂]
    (f : (j : J) → X₁ j ⟶ X₂ j),
    (∀ (j : J), P.isoClosure.trW (f j)) →
      P.isoClosure.trW (X := piObj.{0, v_1, u_1} X₁) (Y := piObj.{0, v_1, u_1} X₂) (Limits.Pi.map f)
⊢ ∀ (X₂ : J → C) [inst_15 : HasProduct.{0, v_1, u_1} X₁] [inst_16 : HasProduct.{0, v_1, u_1} X₂]
    (f : (j : J) → X₁ j ⟶ X₂ j),
    (∀ (j : J), P.isoClosure.trW (f j)) →
      P.isoClosure.trW (X := piObj.{0, v_1, u_1} X₁) (Y := piObj.{0, v_1, u_1} X₂) (Limits.Pi.map f)

Tactic:
_
-- 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 : P.IsTriangulated
J : Type
h : Finite J
X₁_1 X₂_1 : J → C
X₁ : HasProduct.{0, v_1, u_1} X₁_1
X₂ : HasProduct.{0, v_1, u_1} X₂_1
f : (j : J) → X₁_1 j ⟶ X₂_1 j
hf : ∀ (j : J), P.isoClosure.trW (f j)
⊢ P.isoClosure.trW (X := piObj.{0, v_1, u_1} X₁_1) (Y := piObj.{0, v_1, u_1} X₂_1) (Limits.Pi.map f)

Tactic:
trW.mk _ (productTriangle_distinguished _
      (fun j => (hf j).choose_spec.choose_spec.choose_spec.choose))
      (P.isoClosure.prop_pi _
        (fun j => (hf j).choose_spec.choose_spec.choose_spec.choose_spec))

Premises used: Exists.choose_spec, CategoryTheory.ObjectProperty.prop_pi, CategoryTheory.Limits.limit, CategoryTheory.Limits.IsLimit.map, CategoryTheory.ObjectProperty.isoClosure, CategoryTheory.Pretriangulated.productTriangle_distinguished, CategoryTheory.ObjectProperty.trW.mk, CategoryTheory.Limits.limMap, Exists.choose, CategoryTheory.Limits.Pi.map
-- 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 : P.IsTriangulated
J : Type
h : Finite J
X₁_1 X₂_1 : J → C
X₁ : HasProduct.{0, v_1, u_1} X₁_1
X₂ : HasProduct.{0, v_1, u_1} X₂_1
f : (j : J) → X₁_1 j ⟶ X₂_1 j
hf : ∀ (j : J), P.isoClosure.trW (f j)
⊢ P.isoClosure.trW (X := piObj.{0, v_1, u_1} X₁_1) (Y := piObj.{0, v_1, u_1} X₂_1) (Limits.Pi.map f)

Tactic:
exact trW.mk _ (productTriangle_distinguished _
      (fun j => (hf j).choose_spec.choose_spec.choose_spec.choose))
      (P.isoClosure.prop_pi _
        (fun j => (hf j).choose_spec.choose_spec.choose_spec.choose_spec))

Premises used: Exists.choose_spec, CategoryTheory.ObjectProperty.prop_pi, CategoryTheory.ObjectProperty.isoClosure, CategoryTheory.Pretriangulated.productTriangle_distinguished, CategoryTheory.ObjectProperty.trW.mk, Exists.choose
-- 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 : P.IsTriangulated
⊢ P.isoClosure.trW.IsStableUnderFiniteProducts

Tactic:
exact ⟨fun J _ => by
    refine MorphismProperty.IsStableUnderProductsOfShape.mk _ _ ?_
    intro _ _ X₁ X₂ f hf
    exact trW.mk _ (productTriangle_distinguished _
      (fun j => (hf j).choose_spec.choose_spec.choose_spec.choose))
      (P.isoClosure.prop_pi _
        (fun j => (hf j).choose_spec.choose_spec.choose_spec.choose_spec))⟩
-- 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 : P.IsTriangulated
X Y : P.FullSubcategory
f : X ⟶ Y
obtain :
  ∀ (Z' : C) (g' : P.ι.obj Y ⟶ Z') (h' : Z' ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (P.ι.obj X))
    (mem :
      Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
        (Triangle.mk (P.ι.map f) g' h')),
    ∃ (Z : P.FullSubcategory) (g : Y ⟶ Z) (h :
      Z ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor P.FullSubcategory (A := ℤ) 1) X),
      Membership.mem (γ := Set (Triangle.{v_1, u_1} P.FullSubcategory))
        (Set.preimage (β := Triangle.{v_1, u_1} C) (Functor.obj P.ι.mapTriangle)
          (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _))
        (Triangle.mk f g h)
⊢ ∃ (Z : P.FullSubcategory) (g : Y ⟶ Z) (h :
    Z ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor P.FullSubcategory (A := ℤ) 1) X),
    Membership.mem (γ := Set (Triangle.{v_1, u_1} P.FullSubcategory))
      (Set.preimage (β := Triangle.{v_1, u_1} C) (Functor.obj P.ι.mapTriangle)
        (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _))
      (Triangle.mk f g h)

Tactic:
obtain ⟨Z', g', h', mem⟩ := distinguished_cocone_triangle (P.ι.map f)

Premises used: CategoryTheory.ObjectProperty.ι, CategoryTheory.Functor.map, CategoryTheory.Pretriangulated.distinguished_cocone_triangle
-- 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 : P.IsTriangulated
X Y : P.FullSubcategory
f : X ⟶ Y
Z' : C
g' : P.ι.obj Y ⟶ Z'
h' : Z' ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (P.ι.obj X)
mem :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk (P.ι.map f) g' h')
obtain :
  ∀ (Z'' : C) (hZ'' : P Z'') (e : (Triangle.mk (P.ι.map f) g' h').obj₃ ≅ Z''),
    ∃ (Z : P.FullSubcategory) (g : Y ⟶ Z) (h :
      Z ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor P.FullSubcategory (A := ℤ) 1) X),
      Membership.mem (γ := Set (Triangle.{v_1, u_1} P.FullSubcategory))
        (Set.preimage (β := Triangle.{v_1, u_1} C) (Functor.obj P.ι.mapTriangle)
          (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _))
        (Triangle.mk f g h)
⊢ ∃ (Z : P.FullSubcategory) (g : Y ⟶ Z) (h :
    Z ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor P.FullSubcategory (A := ℤ) 1) X),
    Membership.mem (γ := Set (Triangle.{v_1, u_1} P.FullSubcategory))
      (Set.preimage (β := Triangle.{v_1, u_1} C) (Functor.obj P.ι.mapTriangle)
        (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _))
      (Triangle.mk f g h)

Tactic:
obtain ⟨Z'', hZ'', ⟨e⟩⟩ := P.ext_of_isTriangulatedClosed₃' _ mem X.2 Y.2

Premises used: CategoryTheory.ObjectProperty.ext_of_isTriangulatedClosed₃', CategoryTheory.ObjectProperty.FullSubcategory.property
-- 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 : P.IsTriangulated
X Y : P.FullSubcategory
f : X ⟶ Y
Z' : C
g' : P.ι.obj Y ⟶ Z'
h' : Z' ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (P.ι.obj X)
mem :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk (P.ι.map f) g' h')
Z'' : C
hZ'' : P Z''
e : (Triangle.mk (P.ι.map f) g' h').obj₃ ≅ Z''
⊢ ∃ (Z : P.FullSubcategory) (g : Y ⟶ Z) (h :
    Z ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor P.FullSubcategory (A := ℤ) 1) X),
    Membership.mem (γ := Set (Triangle.{v_1, u_1} P.FullSubcategory))
      (Set.preimage (β := Triangle.{v_1, u_1} C) (Functor.obj P.ι.mapTriangle)
        (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _))
      (Triangle.mk f g h)

Tactic:
⟨⟨Z'', hZ''⟩, P.fullyFaithfulι.preimage (g' ≫ e.hom),
      P.fullyFaithfulι.preimage (e.inv ≫ h' ≫ (P.ι.commShiftIso (1 : ℤ)).inv.app X),
      isomorphic_distinguished _ mem _ (Triangle.isoMk _ _ (Iso.refl _) (Iso.refl _) e.symm)⟩

Premises used: CategoryTheory.ObjectProperty.FullSubcategory.mk, CategoryTheory.ObjectProperty.ι, CategoryTheory.ObjectProperty.fullyFaithfulι, CategoryTheory.Iso.refl, CategoryTheory.Pretriangulated.Triangle.isoMk, CategoryTheory.Functor.FullyFaithful.preimage, CategoryTheory.Pretriangulated.isomorphic_distinguished, Exists.intro, CategoryTheory.Iso.symm, CategoryTheory.CategoryStruct.comp, OfNat.ofNat, CategoryTheory.Functor.CommShift.commShiftIso, CategoryTheory.Iso.hom, CategoryTheory.Iso.inv, CategoryTheory.NatTrans.app