awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Opposite.Pretriangulated (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasShift.{v_1, u_1, 0} C ℤ
inst_2 : HasZeroObject.{v_1, u_1} C
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
X Y : Cᵒᵖ
f : X ⟶ Y
Z : C
g : Z ⟶ Opposite.unop Y
h : Opposite.unop X ⟶ 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))
    (@Pretriangulated.distinguishedTriangles _ _ inst_2 _ inst_3 inst_4 _) (Triangle.mk g f.unop h)
⊢ Membership.mem (γ := Set (Triangle.{v_1, u_1} C))
    (@Pretriangulated.distinguishedTriangles _ _ inst_2 _ inst_3 inst_4 _)
    (Opposite.unop
      (Functor.obj (Equivalence.inverse (triangleOpEquivalence C))
        (Triangle.mk f g.op
          (CategoryStruct.comp (Y :=
            Functor.obj.{v_1, v_1, u_1, u_1}
              (Equivalence.inverse (opShiftFunctorEquivalence C 1) ⋙
                Equivalence.functor (opShiftFunctorEquivalence C 1))
              (Opposite.op Z))
            (NatTrans.app (Iso.inv (Equivalence.counitIso (opShiftFunctorEquivalence C 1))) (Opposite.op Z))
            (Functor.map (shiftFunctor Cᵒᵖ (A := ℤ) 1) h.op)))))
have refine : Opposite.unop
    (Functor.obj (Equivalence.inverse (triangleOpEquivalence C))
      (Triangle.mk f g.op
        (CategoryStruct.comp (Y :=
          Functor.obj.{v_1, v_1, u_1, u_1}
            (Equivalence.inverse (opShiftFunctorEquivalence C 1) ⋙ Equivalence.functor (opShiftFunctorEquivalence C 1))
            (Opposite.op Z))
          (NatTrans.app (Iso.inv (Equivalence.counitIso (opShiftFunctorEquivalence C 1))) (Opposite.op Z))
          (Functor.map (shiftFunctor Cᵒᵖ (A := ℤ) 1) h.op)))) ≅
  Triangle.mk g f.unop h
Tactic: refine Pretriangulated.isomorphic_distinguished _ H _ ?_
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Opposite.Pretriangulated (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasShift.{v_1, u_1, 0} C ℤ
inst_2 : HasZeroObject.{v_1, u_1} C
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
T₁ T₂ : Triangle.{v_1, u_1} Cᵒᵖ
hT₁ : Membership.mem (γ := Set (Triangle.{v_1, u_1} Cᵒᵖ)) (distinguishedTriangles C) T₁
hT₂ : Membership.mem (γ := Set (Triangle.{v_1, u_1} Cᵒᵖ)) (distinguishedTriangles C) T₂
a : T₁.obj₁ ⟶ T₂.obj₁
b : T₁.obj₂ ⟶ T₂.obj₂
comm : T₁.mor₁ ≫ b = a ≫ T₂.mor₁
⊢ ∃ (c : T₁.obj₃ ⟶ T₂.obj₃),
    T₁.mor₂ ≫ c = b ≫ T₂.mor₂ ∧ T₁.mor₃ ≫ Functor.map (shiftFunctor Cᵒᵖ (A := ℤ) 1) a = c ≫ T₂.mor₃
have rw : ∀
  (hT₁ :
    Membership.mem (γ := Set (Triangle.{v_1, u_1} C))
      (@Pretriangulated.distinguishedTriangles _ _ inst_2 _ inst_3 inst_4 _)
      (Opposite.unop (Functor.obj (Equivalence.inverse (triangleOpEquivalence C)) T₁)))
  (hT₂ :
    Membership.mem (γ := Set (Triangle.{v_1, u_1} C))
      (@Pretriangulated.distinguishedTriangles _ _ inst_2 _ inst_3 inst_4 _)
      (Opposite.unop (Functor.obj (Equivalence.inverse (triangleOpEquivalence C)) T₂)))
  (a : T₁.obj₁ ⟶ T₂.obj₁) (b : T₁.obj₂ ⟶ T₂.obj₂) (comm : T₁.mor₁ ≫ b = a ≫ T₂.mor₁),
  ∃ (c : T₁.obj₃ ⟶ T₂.obj₃),
    T₁.mor₂ ≫ c = b ≫ T₂.mor₂ ∧ T₁.mor₃ ≫ Functor.map (shiftFunctor Cᵒᵖ (A := ℤ) 1) a = c ≫ T₂.mor₃
Tactic: rw [mem_distinguishedTriangles_iff] at hT₁ hT₂
Cleared hypotheses: hT₁, hT₂, a, b, comm
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Opposite.Pretriangulated (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasShift.{v_1, u_1, 0} C ℤ
inst_2 : HasZeroObject.{v_1, u_1} C
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
T₁ T₂ : Triangle.{v_1, u_1} Cᵒᵖ
hT₁ :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C))
    (@Pretriangulated.distinguishedTriangles _ _ inst_2 _ inst_3 inst_4 _)
    (Opposite.unop (Functor.obj (Equivalence.inverse (triangleOpEquivalence C)) T₁))
hT₂ :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C))
    (@Pretriangulated.distinguishedTriangles _ _ inst_2 _ inst_3 inst_4 _)
    (Opposite.unop (Functor.obj (Equivalence.inverse (triangleOpEquivalence C)) T₂))
a : T₁.obj₁ ⟶ T₂.obj₁
b : T₁.obj₂ ⟶ T₂.obj₂
comm : T₁.mor₁ ≫ b = a ≫ T₂.mor₁
⊢ ∃ (c : T₁.obj₃ ⟶ T₂.obj₃),
    T₁.mor₂ ≫ c = b ≫ T₂.mor₂ ∧ T₁.mor₃ ≫ Functor.map (shiftFunctor Cᵒᵖ (A := ℤ) 1) a = c ≫ T₂.mor₃
have obtain : ∀
  (c :
    (Opposite.unop (Functor.obj (Equivalence.inverse (triangleOpEquivalence C)) T₂)).obj₁ ⟶
      (Opposite.unop (Functor.obj (Equivalence.inverse (triangleOpEquivalence C)) T₁)).obj₁)
  (hc₁ :
    (Opposite.unop (Functor.obj (Equivalence.inverse (triangleOpEquivalence C)) T₂)).mor₁ ≫ b.unop =
      c ≫ (Opposite.unop (Functor.obj (Equivalence.inverse (triangleOpEquivalence C)) T₁)).mor₁)
  (hc₂ :
    (Opposite.unop (Functor.obj (Equivalence.inverse (triangleOpEquivalence C)) T₂)).mor₃ ≫
        Functor.map (shiftFunctor C (A := ℤ) 1) c =
      a.unop ≫ (Opposite.unop (Functor.obj (Equivalence.inverse (triangleOpEquivalence C)) T₁)).mor₃),
  ∃ (c : T₁.obj₃ ⟶ T₂.obj₃),
    T₁.mor₂ ≫ c = b ≫ T₂.mor₂ ∧ T₁.mor₃ ≫ Functor.map (shiftFunctor Cᵒᵖ (A := ℤ) 1) a = c ≫ T₂.mor₃
Tactic: obtain ⟨c, hc₁, hc₂⟩ :=
    Pretriangulated.complete_distinguished_triangle_morphism₁ _ _ hT₂ hT₁
      b.unop a.unop (Quiver.Hom.op_inj comm.symm)
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Opposite.Pretriangulated (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasShift.{v_1, u_1, 0} C ℤ
inst_2 : HasZeroObject.{v_1, u_1} C
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
T₁ T₂ : Triangle.{v_1, u_1} Cᵒᵖ
hT₁ :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C))
    (@Pretriangulated.distinguishedTriangles _ _ inst_2 _ inst_3 inst_4 _)
    (Opposite.unop (Functor.obj (Equivalence.inverse (triangleOpEquivalence C)) T₁))
hT₂ :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C))
    (@Pretriangulated.distinguishedTriangles _ _ inst_2 _ inst_3 inst_4 _)
    (Opposite.unop (Functor.obj (Equivalence.inverse (triangleOpEquivalence C)) T₂))
a : T₁.obj₁ ⟶ T₂.obj₁
b : T₁.obj₂ ⟶ T₂.obj₂
comm : T₁.mor₁ ≫ b = a ≫ T₂.mor₁
c : Opposite.unop T₂.obj₃ ⟶ Opposite.unop T₁.obj₃
hc₁ : T₂.mor₂.unop ≫ b.unop = c ≫ T₁.mor₂.unop
hc₂ :
  Eq (α :=
    Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1)
        (Opposite.unop (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor Cᵒᵖ (A := ℤ) 1) T₂.obj₁)) ⟶
      Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (Opposite.unop T₁.obj₃))
    (Functor.map (shiftFunctor C (A := ℤ) 1) T₂.mor₃.unop ≫ Functor.map (shiftFunctor C (A := ℤ) 1) c)
    (Quiver.Hom.unop (NatTrans.app (Iso.hom (Equivalence.unitIso (opShiftFunctorEquivalence C 1))) T₂.obj₁) ≫
      a.unop ≫
        CategoryStruct.comp (Y :=
          Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1)
            (Opposite.unop (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor Cᵒᵖ (A := ℤ) 1) T₁.obj₁)))
          (Quiver.Hom.unop (NatTrans.app (Iso.inv (Equivalence.unitIso (opShiftFunctorEquivalence C 1))) T₁.obj₁) :
            Opposite.unop T₁.obj₁ ⟶
              Opposite.unop
                (Opposite.op
                  (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1)
                    (Opposite.unop (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor Cᵒᵖ (A := ℤ) 1) T₁.obj₁)))))
          (Functor.map (shiftFunctor C (A := ℤ) 1) T₁.mor₃.unop :
            Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1)
                (Opposite.unop (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor Cᵒᵖ (A := ℤ) 1) T₁.obj₁)) ⟶
              Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (Opposite.unop T₁.obj₃)))
⊢ ∃ (c : T₁.obj₃ ⟶ T₂.obj₃),
    T₁.mor₂ ≫ c = b ≫ T₂.mor₂ ∧ T₁.mor₃ ≫ Functor.map (shiftFunctor Cᵒᵖ (A := ℤ) 1) a = c ≫ T₂.mor₃
have refine : Quiver.Hom.unop (T₁.mor₃ ≫ Functor.map (shiftFunctor Cᵒᵖ (A := ℤ) 1) a) = Quiver.Hom.unop (c.op ≫ T₂.mor₃)
Tactic: refine ⟨c.op, Quiver.Hom.unop_inj hc₁.symm, Quiver.Hom.unop_inj ?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Opposite.Pretriangulated (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasShift.{v_1, u_1, 0} C ℤ
inst_2 : HasZeroObject.{v_1, u_1} C
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
T₁ T₂ : Triangle.{v_1, u_1} Cᵒᵖ
hT₁ :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C))
    (@Pretriangulated.distinguishedTriangles _ _ inst_2 _ inst_3 inst_4 _)
    (Opposite.unop (Functor.obj (Equivalence.inverse (triangleOpEquivalence C)) T₁))
hT₂ :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C))
    (@Pretriangulated.distinguishedTriangles _ _ inst_2 _ inst_3 inst_4 _)
    (Opposite.unop (Functor.obj (Equivalence.inverse (triangleOpEquivalence C)) T₂))
a : T₁.obj₁ ⟶ T₂.obj₁
b : T₁.obj₂ ⟶ T₂.obj₂
comm : T₁.mor₁ ≫ b = a ≫ T₂.mor₁
c : Opposite.unop T₂.obj₃ ⟶ Opposite.unop T₁.obj₃
hc₁ : T₂.mor₂.unop ≫ b.unop = c ≫ T₁.mor₂.unop
hc₂ :
  Eq (α :=
    Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1)
        (Opposite.unop (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor Cᵒᵖ (A := ℤ) 1) T₂.obj₁)) ⟶
      Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (Opposite.unop T₁.obj₃))
    (Functor.map (shiftFunctor C (A := ℤ) 1) T₂.mor₃.unop ≫ Functor.map (shiftFunctor C (A := ℤ) 1) c)
    (Quiver.Hom.unop (NatTrans.app (Iso.hom (Equivalence.unitIso (opShiftFunctorEquivalence C 1))) T₂.obj₁) ≫
      a.unop ≫
        CategoryStruct.comp (Y :=
          Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1)
            (Opposite.unop (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor Cᵒᵖ (A := ℤ) 1) T₁.obj₁)))
          (Quiver.Hom.unop (NatTrans.app (Iso.inv (Equivalence.unitIso (opShiftFunctorEquivalence C 1))) T₁.obj₁) :
            Opposite.unop T₁.obj₁ ⟶
              Opposite.unop
                (Opposite.op
                  (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1)
                    (Opposite.unop (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor Cᵒᵖ (A := ℤ) 1) T₁.obj₁)))))
          (Functor.map (shiftFunctor C (A := ℤ) 1) T₁.mor₃.unop :
            Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1)
                (Opposite.unop (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor Cᵒᵖ (A := ℤ) 1) T₁.obj₁)) ⟶
              Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (Opposite.unop T₁.obj₃)))
⊢ Quiver.Hom.unop (T₁.mor₃ ≫ Functor.map (shiftFunctor Cᵒᵖ (A := ℤ) 1) a) = Quiver.Hom.unop (c.op ≫ T₂.mor₃)
have a₁ : Eq (α :=
  Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1)
      (Opposite.unop (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor Cᵒᵖ (A := ℤ) 1) T₂.obj₁)) ⟶
    Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (Opposite.unop T₁.obj₃))
  (Functor.map (shiftFunctor C (A := ℤ) 1) (Quiver.Hom.unop (T₁.mor₃ ≫ Functor.map (shiftFunctor Cᵒᵖ (A := ℤ) 1) a)))
  (Functor.map (shiftFunctor C (A := ℤ) 1) (Quiver.Hom.unop (c.op ≫ T₂.mor₃)))
Tactic: apply (shiftFunctor C (1 : ℤ)).map_injective
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Opposite.Pretriangulated (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasShift.{v_1, u_1, 0} C ℤ
inst_2 : HasZeroObject.{v_1, u_1} C
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
T : Triangle.{v_1, u_1} C
hT : Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_2 _ inst_3 inst_4 _) T
⊢ Membership.mem (γ := Set (Triangle.{v_1, u_1} Cᵒᵖ)) (@distinguishedTriangles _ _ ⋯ _ (instPreadditiveOpposite C) ⋯ _)
    (Functor.obj (Equivalence.functor (triangleOpEquivalence C)) (Opposite.op T))
have rw : ∃ (T' : Triangle.{v_1, u_1} C) (_ :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_2 _ inst_3 inst_4 _) T'),
  Nonempty.{v_1 + 1}
    (Functor.obj (Equivalence.functor (triangleOpEquivalence C)) (Opposite.op T) ≅
      Functor.obj (Equivalence.functor (triangleOpEquivalence C)) (Opposite.op T'))
Tactic: rw [mem_distTriang_op_iff']
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Opposite.Subcategory (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasShift.{v_1, u_1, 0} C ℤ
inst_2 : HasZeroObject.{v_1, u_1} C
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 : ObjectProperty.{v_1, u_1} Cᵒᵖ
inst_6 : IsStableUnderShift.{v_1, u_1, 0} P ℤ
n : ℤ
X : C
hX : P.unop X
⊢ P.unop.shift n X
have obtain : ∀ (m : ℤ), P.unop.shift (-m) X
Tactic: obtain ⟨m, rfl⟩ : ∃ m, n = -m := ⟨-n , by simp⟩
Cleared hypotheses: n
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Opposite.Subcategory (hypothesis)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasShift.{v_1, u_1, 0} C ℤ
inst_2 : HasZeroObject.{v_1, u_1} C
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 : ObjectProperty.{v_1, u_1} C
inst_6 : P.IsTriangulatedClosed₂
T : Triangle.{v_1, u_1} Cᵒᵖ
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} Cᵒᵖ))
    (@Pretriangulated.distinguishedTriangles _ _ ⋯ _ (instPreadditiveOpposite C) ⋯ _) T
h₁ : P.op T.obj₁
h₃ : P.op T.obj₃
⊢ isoClosure.{v_1, u_1} P.op T.obj₂
have hT₁ : Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@Pretriangulated.distinguishedTriangles _ _ inst_2 _ inst_3 inst_4 _)
  (Opposite.unop (Functor.obj (Equivalence.inverse (triangleOpEquivalence C)) T))
Tactic: rw [mem_distTriang_op_iff] at hT
Cleared hypotheses: hT, h₁, h₃
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Opposite.Subcategory (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasShift.{v_1, u_1, 0} C ℤ
inst_2 : HasZeroObject.{v_1, u_1} C
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 : ObjectProperty.{v_1, u_1} C
inst_6 : P.IsTriangulatedClosed₂
T : Triangle.{v_1, u_1} Cᵒᵖ
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C))
    (@Pretriangulated.distinguishedTriangles _ _ inst_2 _ inst_3 inst_4 _)
    (Opposite.unop (Functor.obj (Equivalence.inverse (triangleOpEquivalence C)) T))
h₁ : P.op T.obj₁
h₃ : P.op T.obj₃
⊢ isoClosure.{v_1, u_1} P.op T.obj₂
have obtain : ∀ (X : C) (hX : P X) (e : (Opposite.unop (Functor.obj (Equivalence.inverse (triangleOpEquivalence C)) T)).obj₂ ≅ X),
  isoClosure.{v_1, u_1} P.op T.obj₂
Tactic: obtain ⟨X, hX, ⟨e⟩⟩ := P.ext_of_isTriangulatedClosed₂' _ hT h₃ h₁
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Opposite.Subcategory (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasShift.{v_1, u_1, 0} C ℤ
inst_2 : HasZeroObject.{v_1, u_1} C
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 : ObjectProperty.{v_1, u_1} Cᵒᵖ
inst_6 : IsTriangulatedClosed₂.{v_1, u_1} P
T : Triangle.{v_1, u_1} C
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C))
    (@Pretriangulated.distinguishedTriangles _ _ inst_2 _ inst_3 inst_4 _) T
h₁ : P.unop T.obj₁
h₃ : P.unop T.obj₃
⊢ P.unop.isoClosure T.obj₂
have obtain : ∀ (X : Cᵒᵖ) (hX : P X) (e : (Functor.obj (Equivalence.functor (triangleOpEquivalence C)) (Opposite.op T)).obj₂ ≅ X),
  P.unop.isoClosure T.obj₂
Tactic: obtain ⟨X, hX, ⟨e⟩⟩ := P.ext_of_isTriangulatedClosed₂' _ (op_distinguished _ hT) h₃ h₁
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Opposite.Subcategory (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasShift.{v_1, u_1, 0} C ℤ
inst_2 : HasZeroObject.{v_1, u_1} C
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 : ObjectProperty.{v_1, u_1} C
inst_6 : P.IsTriangulated
X Y : C
f : X ⟶ Y
hf : trW.{v_1, u_1} P.op (X := Opposite.op Y) (Y := Opposite.op X) f.op
⊢ P.trW f
have obtain : ∀ (Z : Cᵒᵖ) (a : Opposite.op X ⟶ Z)
  (b : Z ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor Cᵒᵖ (A := ℤ) 1) (Opposite.op Y))
  (h₁ :
    Membership.mem (γ := Set (Triangle.{v_1, u_1} Cᵒᵖ))
      (@Pretriangulated.distinguishedTriangles _ _ sorry _ (instPreadditiveOpposite C) sorry _) (Triangle.mk f.op a b))
  (h₂ : P.op Z), P.trW f
Tactic: obtain ⟨Z, a, b, h₁, h₂⟩ := hf
Cleared hypotheses: hf
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Opposite.Subcategory (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasShift.{v_1, u_1, 0} C ℤ
inst_2 : HasZeroObject.{v_1, u_1} C
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 : ObjectProperty.{v_1, u_1} C
inst_6 : P.IsTriangulated
X Y : C
f : X ⟶ Y
Z : Cᵒᵖ
a : Opposite.op X ⟶ Z
b : Z ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor Cᵒᵖ (A := ℤ) 1) (Opposite.op Y)
h₁ :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} Cᵒᵖ))
    (@Pretriangulated.distinguishedTriangles _ _ ⋯ _ (instPreadditiveOpposite C) ⋯ _) (Triangle.mk f.op a b)
h₂ : P.op Z
⊢ P.trW f
have rw : ∃ (X_1 : C) (f_1 : X_1 ⟶ X) (h : Y ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) X_1) (_ :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C))
    (@Pretriangulated.distinguishedTriangles _ _ inst_2 _ inst_3 inst_4 _) (Triangle.mk f_1 f h)),
  P X_1
Tactic: rw [ObjectProperty.trW_iff']
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Opposite.Subcategory (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasShift.{v_1, u_1, 0} C ℤ
inst_2 : HasZeroObject.{v_1, u_1} C
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 : ObjectProperty.{v_1, u_1} Cᵒᵖ
inst_6 : ObjectProperty.IsTriangulated.{v_1, u_1} P
X Y : Cᵒᵖ
f : X ⟶ Y
hf : P.unop.trW f.unop
⊢ trW.{v_1, u_1} P (X := X) (Y := Y) f
have obtain : ∀ (Z : C) (a : Opposite.unop X ⟶ Z)
  (b : Z ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (Opposite.unop Y))
  (h₁ :
    Membership.mem (γ := Set (Triangle.{v_1, u_1} C))
      (@Pretriangulated.distinguishedTriangles _ _ inst_2 _ inst_3 inst_4 _) (Triangle.mk f.unop a b))
  (h₂ : P.unop Z), trW.{v_1, u_1} P (X := X) (Y := Y) f
Tactic: obtain ⟨Z, a, b, h₁, h₂⟩ := hf
Cleared hypotheses: hf
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Opposite.Subcategory (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasShift.{v_1, u_1, 0} C ℤ
inst_2 : HasZeroObject.{v_1, u_1} C
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 : ObjectProperty.{v_1, u_1} Cᵒᵖ
inst_6 : ObjectProperty.IsTriangulated.{v_1, u_1} P
X Y : Cᵒᵖ
f : X ⟶ Y
Z : C
a : Opposite.unop X ⟶ Z
b : Z ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (Opposite.unop Y)
h₁ :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C))
    (@Pretriangulated.distinguishedTriangles _ _ inst_2 _ inst_3 inst_4 _) (Triangle.mk f.unop a b)
h₂ : P.unop Z
⊢ trW.{v_1, u_1} P (X := X) (Y := Y) f
have rw : ∃ (X_1 : Cᵒᵖ) (f_1 : X_1 ⟶ X) (h : Y ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor Cᵒᵖ (A := ℤ) 1) X_1) (_ :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} Cᵒᵖ))
    (@Pretriangulated.distinguishedTriangles _ _ sorry _ (instPreadditiveOpposite C) sorry _) (Triangle.mk f_1 f h)),
  P X_1
Tactic: rw [ObjectProperty.trW_iff']
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Opposite.Subcategory (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasShift.{v_1, u_1, 0} C ℤ
inst_2 : HasZeroObject.{v_1, u_1} C
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 : ObjectProperty.{v_1, u_1} C
inst_6 : P.IsTriangulated
⊢ P.op.trW = P.trW.op
have h : ∀ ⦃X Y : Cᵒᵖ⦄ (f : X ⟶ Y), P.op.trW f ↔ P.trW.op f
Tactic: ext X Y f
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Opposite.Triangle (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasShift.{v_1, u_1, 0} C ℤ
T : Triangle.{v_1, u_1} Cᵒᵖ
⊢ Functor.obj (inverse C ⋙ functor C) T ≅ Functor.obj (𝟭 (Triangle.{v_1, u_1} Cᵒᵖ)) T
have refine_1 : (Functor.obj (inverse C ⋙ functor C) T).mor₁ ≫ Iso.hom (Iso.refl (Functor.obj (inverse C ⋙ functor C) T).obj₂) =
  Iso.hom (Iso.refl (Functor.obj (inverse C ⋙ functor C) T).obj₁) ≫ (Functor.obj (𝟭 (Triangle.{v_1, u_1} Cᵒᵖ)) T).mor₁
Tactic: refine Triangle.isoMk _ _ (Iso.refl _) (Iso.refl _) (Iso.refl _) ?_ ?_ ?_
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Opposite.Triangle (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasShift.{v_1, u_1, 0} C ℤ
T : Triangle.{v_1, u_1} Cᵒᵖ
refine_1 :
  (Functor.obj (inverse C ⋙ functor C) T).mor₁ ≫ Iso.hom (Iso.refl (Functor.obj (inverse C ⋙ functor C) T).obj₂) =
    Iso.hom (Iso.refl (Functor.obj (inverse C ⋙ functor C) T).obj₁) ≫ (Functor.obj (𝟭 (Triangle.{v_1, u_1} Cᵒᵖ)) T).mor₁
⊢ Functor.obj (inverse C ⋙ functor C) T ≅ Functor.obj (𝟭 (Triangle.{v_1, u_1} Cᵒᵖ)) T
have refine_2 : (Functor.obj (inverse C ⋙ functor C) T).mor₂ ≫ Iso.hom (Iso.refl (Functor.obj (inverse C ⋙ functor C) T).obj₃) =
  Iso.hom (Iso.refl (Functor.obj (inverse C ⋙ functor C) T).obj₂) ≫ (Functor.obj (𝟭 (Triangle.{v_1, u_1} Cᵒᵖ)) T).mor₂
Tactic: refine Triangle.isoMk _ _ (Iso.refl _) (Iso.refl _) (Iso.refl _) ?_ ?_ ?_
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Opposite.Triangle (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasShift.{v_1, u_1, 0} C ℤ
T : Triangle.{v_1, u_1} Cᵒᵖ
refine_1 :
  (Functor.obj (inverse C ⋙ functor C) T).mor₁ ≫ Iso.hom (Iso.refl (Functor.obj (inverse C ⋙ functor C) T).obj₂) =
    Iso.hom (Iso.refl (Functor.obj (inverse C ⋙ functor C) T).obj₁) ≫ (Functor.obj (𝟭 (Triangle.{v_1, u_1} Cᵒᵖ)) T).mor₁
refine_2 :
  (Functor.obj (inverse C ⋙ functor C) T).mor₂ ≫ Iso.hom (Iso.refl (Functor.obj (inverse C ⋙ functor C) T).obj₃) =
    Iso.hom (Iso.refl (Functor.obj (inverse C ⋙ functor C) T).obj₂) ≫ (Functor.obj (𝟭 (Triangle.{v_1, u_1} Cᵒᵖ)) T).mor₂
⊢ Functor.obj (inverse C ⋙ functor C) T ≅ Functor.obj (𝟭 (Triangle.{v_1, u_1} Cᵒᵖ)) T
have refine_3 : (Functor.obj (inverse C ⋙ functor C) T).mor₃ ≫
    Functor.map (shiftFunctor Cᵒᵖ (A := ℤ) 1) (Iso.hom (Iso.refl (Functor.obj (inverse C ⋙ functor C) T).obj₁)) =
  Iso.hom (Iso.refl (Functor.obj (inverse C ⋙ functor C) T).obj₃) ≫ (Functor.obj (𝟭 (Triangle.{v_1, u_1} Cᵒᵖ)) T).mor₃
Tactic: refine Triangle.isoMk _ _ (Iso.refl _) (Iso.refl _) (Iso.refl _) ?_ ?_ ?_