awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Opposite.OpOp (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 : Preadditive.{v_1, u_1} C
inst_3 : HasZeroObject.{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_3 _ inst_2 inst_4 _) T
this :
  Eq (α :=
    Functor.obj.{v_1, v_1, u_1, u_1} (𝟭 C) T.obj₃ ⟶
      Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) (-1) ⋙ shiftFunctor C (A := ℤ) 1)
        (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) T.obj₁))
    (Functor.map (𝟭 C) T.mor₃ ≫
      NatTrans.app (Iso.inv (shiftFunctorCompIsoId C (-1) 1 ⋯))
        (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) T.obj₁))
    (NatTrans.app (Iso.inv (shiftFunctorCompIsoId C (-1) 1 ⋯)) T.obj₃ ≫
      Functor.map (shiftFunctor C (A := ℤ) (-1) ⋙ shiftFunctor C (A := ℤ) 1) T.mor₃)
⊢ (Triangle.mor₃ (Functor.obj (Functor.mapTriangle (opOp C)) T) ≫
        Functor.map (shiftFunctor Cᵒᵖᵒᵖ (A := ℤ) 1)
          (Iso.hom (Iso.refl (Triangle.obj₁ (Functor.obj (Functor.mapTriangle (opOp C)) T))))).unop =
    (Iso.hom (Iso.refl (Triangle.obj₃ (Functor.obj (Functor.mapTriangle (opOp C)) T))) ≫
        (Functor.obj (Equivalence.functor (triangleOpEquivalence Cᵒᵖ))
            (op (Functor.obj (Equivalence.functor (triangleOpEquivalence C)) (op T)))).mor₃).unop
have dsimp : Eq.{v_1 + 1} (α := unop (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor Cᵒᵖᵒᵖ (A := ℤ) 1) (op (op T.obj₁))) ⟶ op T.obj₃)
  (CategoryStruct.comp (Y := unop (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor Cᵒᵖᵒᵖ (A := ℤ) 1) (op (op T.obj₁))))
    (Quiver.Hom.unop (Functor.map (shiftFunctor Cᵒᵖᵒᵖ (A := ℤ) 1) (𝟙 (op (op T.obj₁)))) :
      unop (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor Cᵒᵖᵒᵖ (A := ℤ) 1) (op (op T.obj₁))) ⟶
        unop (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor Cᵒᵖᵒᵖ (A := ℤ) 1) (op (op T.obj₁))))
    (Quiver.Hom.unop (NatTrans.app (Iso.hom (Functor.commShiftIso (opOp C) 1)) T.obj₁) ≫ T.mor₃.op :
      unop (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor Cᵒᵖᵒᵖ (A := ℤ) 1) (op (op T.obj₁))) ⟶ op T.obj₃))
  (CategoryStruct.comp (Y :=
      unop
        (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor Cᵒᵖᵒᵖ (A := ℤ) 1)
          (op (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor Cᵒᵖ (A := ℤ) 1) (op T.obj₃)))))
      (Quiver.Hom.unop
        (Functor.map (shiftFunctor Cᵒᵖᵒᵖ (A := ℤ) 1)
          (CategoryStruct.comp (Y :=
            op
              (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor Cᵒᵖ (A := ℤ) 1)
                (op (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) T.obj₁))))
            (Quiver.Hom.op (Functor.map (shiftFunctor Cᵒᵖ (A := ℤ) 1) T.mor₃.op))
            (Quiver.Hom.op
              (NatTrans.app (Iso.inv (Equivalence.counitIso (opShiftFunctorEquivalence C 1))) (op T.obj₁))))))
      (Quiver.Hom.unop
        (NatTrans.app (Iso.inv (Equivalence.counitIso (opShiftFunctorEquivalence Cᵒᵖ 1))) (op (op T.obj₃)))) ≫
    𝟙 (op T.obj₃))
Tactic: dsimp at this ⊢
-- 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} Cᵒᵖ
⊢ Membership.mem (γ := Set (Triangle.{v_1, u_1} Cᵒᵖ)) (distinguishedTriangles C) T ↔
    ∃ (T' : Triangle.{v_1, u_1} C) (_ :
      Membership.mem (γ := Set (Triangle.{v_1, u_1} C))
        (@Pretriangulated.distinguishedTriangles _ _ inst_2 _ inst_3 inst_4 _) T'),
      Nonempty.{v_1 + 1} (T ≅ Functor.obj (Equivalence.functor (triangleOpEquivalence C)) (Opposite.op T'))
have rw : 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)) ↔
  ∃ (T' : Triangle.{v_1, u_1} C) (_ :
    Membership.mem (γ := Set (Triangle.{v_1, u_1} C))
      (@Pretriangulated.distinguishedTriangles _ _ inst_2 _ inst_3 inst_4 _) T'),
    Nonempty.{v_1 + 1} (T ≅ Functor.obj (Equivalence.functor (triangleOpEquivalence C)) (Opposite.op T'))
Tactic: rw [mem_distinguishedTriangles_iff]
-- 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} Cᵒᵖ
⊢ 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)) ↔
    ∃ (T' : Triangle.{v_1, u_1} C) (_ :
      Membership.mem (γ := Set (Triangle.{v_1, u_1} C))
        (@Pretriangulated.distinguishedTriangles _ _ inst_2 _ inst_3 inst_4 _) T'),
      Nonempty.{v_1 + 1} (T ≅ Functor.obj (Equivalence.functor (triangleOpEquivalence C)) (Opposite.op T'))
have mp : 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)) →
  ∃ (T' : Triangle.{v_1, u_1} C) (_ :
    Membership.mem (γ := Set (Triangle.{v_1, u_1} C))
      (@Pretriangulated.distinguishedTriangles _ _ inst_2 _ inst_3 inst_4 _) T'),
    Nonempty.{v_1 + 1} (T ≅ Functor.obj (Equivalence.functor (triangleOpEquivalence C)) (Opposite.op T'))
Tactic: constructor
-- 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} Cᵒᵖ
mp :
  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)) →
    ∃ (T' : Triangle.{v_1, u_1} C) (_ :
      Membership.mem (γ := Set (Triangle.{v_1, u_1} C))
        (@Pretriangulated.distinguishedTriangles _ _ inst_2 _ inst_3 inst_4 _) T'),
      Nonempty.{v_1 + 1} (T ≅ Functor.obj (Equivalence.functor (triangleOpEquivalence C)) (Opposite.op T'))
⊢ 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)) ↔
    ∃ (T' : Triangle.{v_1, u_1} C) (_ :
      Membership.mem (γ := Set (Triangle.{v_1, u_1} C))
        (@Pretriangulated.distinguishedTriangles _ _ inst_2 _ inst_3 inst_4 _) T'),
      Nonempty.{v_1 + 1} (T ≅ Functor.obj (Equivalence.functor (triangleOpEquivalence C)) (Opposite.op T'))
have mpr : (∃ (T' : Triangle.{v_1, u_1} C) (_ :
    Membership.mem (γ := Set (Triangle.{v_1, u_1} C))
      (@Pretriangulated.distinguishedTriangles _ _ inst_2 _ inst_3 inst_4 _) T'),
    Nonempty.{v_1 + 1} (T ≅ Functor.obj (Equivalence.functor (triangleOpEquivalence C)) (Opposite.op T'))) →
  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: constructor
-- 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} Cᵒᵖ
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'
e : T ≅ Functor.obj (Equivalence.functor (triangleOpEquivalence C)) (Opposite.op T')
⊢ 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))
have mpr : Opposite.unop (Functor.obj (Equivalence.inverse (triangleOpEquivalence C)) T) ≅ T'
Tactic: refine isomorphic_distinguished _ hT' _ ?_
-- 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 : Triangle.{v_1, u_1} Cᵒᵖ
hT₁ : Membership.mem (γ := Set (Triangle.{v_1, u_1} Cᵒᵖ)) (distinguishedTriangles C) T₁
T₂ : Triangle.{v_1, u_1} Cᵒᵖ
e : T₂ ≅ T₁
⊢ Membership.mem (γ := Set (Triangle.{v_1, u_1} Cᵒᵖ)) (distinguishedTriangles C) T₂
have simp : ∀
  (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))),
  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))
Tactic: simp only [mem_distinguishedTriangles_iff] at hT₁ ⊢
Cleared hypotheses: hT₁
-- 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 : Cᵒᵖ
⊢ IsZero.{v_1, u_1}
    (Triangle.obj₃.{v_1, u_1}
      (Functor.obj (Equivalence.functor (triangleOpEquivalence C))
        (Opposite.op (Triangle.invRotate (contractibleTriangle (Opposite.unop X))))))
have dsimp : IsZero.{v_1, u_1} (Opposite.op (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) (-1)) 0))
Tactic: dsimp
-- 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 : Cᵒᵖ
⊢ IsZero.{v_1, u_1} (Opposite.op (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) (-1)) 0))
have rw : Eq.{v_1 + 1} (α :=
  Opposite.op (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) (-1)) 0) ⟶
    Opposite.op (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) (-1)) 0))
  (𝟙 (Opposite.op (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) (-1)) 0))) 0
Tactic: rw [IsZero.iff_id_eq_zero]
-- 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 : Cᵒᵖ
⊢ Eq.{v_1 + 1} (α :=
    Opposite.op (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) (-1)) 0) ⟶
      Opposite.op (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) (-1)) 0))
    (𝟙 (Opposite.op (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) (-1)) 0))) 0
have change : Eq.{v_1 + 1} (α :=
  Opposite.op (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) (-1)) 0) ⟶
    Opposite.op (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) (-1)) 0))
  (Quiver.Hom.op (𝟙 (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) (-1)) 0))) 0
Tactic: change (𝟙 ((0 : C)⟦(-1 : ℤ)⟧)).op = 0
-- 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 : Cᵒᵖ
⊢ Eq.{v_1 + 1} (α :=
    Opposite.op (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) (-1)) 0) ⟶
      Opposite.op (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) (-1)) 0))
    (Quiver.Hom.op (𝟙 (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) (-1)) 0))) 0
have rw : Eq.{v_1 + 1} (α :=
  Opposite.op (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) (-1)) 0) ⟶
    Opposite.op (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) (-1)) 0))
  (Quiver.Hom.op (Functor.map (shiftFunctor C (A := ℤ) (-1)) (𝟙 0))) 0
Tactic: rw [← Functor.map_id, id_zero, Functor.map_zero, op_zero]
-- 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 : Cᵒᵖ
rw :
  Eq.{v_1 + 1} (α :=
    Opposite.op (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) (-1)) 0) ⟶
      Opposite.op (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) (-1)) 0))
    (Quiver.Hom.op (Functor.map (shiftFunctor C (A := ℤ) (-1)) (𝟙 0))) 0
⊢ Eq.{v_1 + 1} (α :=
    Opposite.op (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) (-1)) 0) ⟶
      Opposite.op (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) (-1)) 0))
    (Quiver.Hom.op (𝟙 (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) (-1)) 0))) 0
have rw₁ : Eq.{v_1 + 1} (α :=
  Opposite.op (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) (-1)) 0) ⟶
    Opposite.op (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) (-1)) 0))
  (Quiver.Hom.op (Functor.map (shiftFunctor C (A := ℤ) (-1)) 0)) 0
Tactic: rw [← Functor.map_id, id_zero, Functor.map_zero, op_zero]
-- 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 : Cᵒᵖ
rw :
  Eq.{v_1 + 1} (α :=
    Opposite.op (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) (-1)) 0) ⟶
      Opposite.op (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) (-1)) 0))
    (Quiver.Hom.op (Functor.map (shiftFunctor C (A := ℤ) (-1)) (𝟙 0))) 0
rw₁ :
  Eq.{v_1 + 1} (α :=
    Opposite.op (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) (-1)) 0) ⟶
      Opposite.op (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) (-1)) 0))
    (Quiver.Hom.op (Functor.map (shiftFunctor C (A := ℤ) (-1)) 0)) 0
⊢ Eq.{v_1 + 1} (α :=
    Opposite.op (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) (-1)) 0) ⟶
      Opposite.op (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) (-1)) 0))
    (Quiver.Hom.op (𝟙 (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) (-1)) 0))) 0
have rw₂ : Eq.{v_1 + 1} (α :=
  Opposite.op (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) (-1)) 0) ⟶
    Opposite.op (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) (-1)) 0))
  (Quiver.Hom.op 0) 0
Tactic: rw [← Functor.map_id, id_zero, Functor.map_zero, op_zero]
-- 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 : Cᵒᵖ
rw :
  Eq.{v_1 + 1} (α :=
    Opposite.op (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) (-1)) 0) ⟶
      Opposite.op (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) (-1)) 0))
    (Quiver.Hom.op (Functor.map (shiftFunctor C (A := ℤ) (-1)) (𝟙 0))) 0
rw₁ :
  Eq.{v_1 + 1} (α :=
    Opposite.op (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) (-1)) 0) ⟶
      Opposite.op (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) (-1)) 0))
    (Quiver.Hom.op (Functor.map (shiftFunctor C (A := ℤ) (-1)) 0)) 0
rw₂ :
  Eq.{v_1 + 1} (α :=
    Opposite.op (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) (-1)) 0) ⟶
      Opposite.op (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) (-1)) 0))
    (Quiver.Hom.op 0) 0
⊢ Eq.{v_1 + 1} (α :=
    Opposite.op (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) (-1)) 0) ⟶
      Opposite.op (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) (-1)) 0))
    (Quiver.Hom.op (𝟙 (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) (-1)) 0))) 0
have rw₃ : Eq.{v_1 + 1} (α :=
  Opposite.op (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) (-1)) 0) ⟶
    Opposite.op (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) (-1)) 0))
  0 0
Tactic: rw [← Functor.map_id, id_zero, Functor.map_zero, op_zero]
-- 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 : Cᵒᵖ
⊢ Membership.mem (γ := Set (Triangle.{v_1, u_1} Cᵒᵖ)) (distinguishedTriangles C) (contractibleTriangle X)
have rw : ∃ (T' : Triangle.{v_1, u_1} C) (_ :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C))
    (@Pretriangulated.distinguishedTriangles _ _ inst_2 _ inst_3 inst_4 _) T'),
  Nonempty.{v_1 + 1}
    (contractibleTriangle X ≅ Functor.obj (Equivalence.functor (triangleOpEquivalence C)) (Opposite.op T'))
Tactic: rw [mem_distinguishedTriangles_iff']
-- 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} Cᵒᵖ
⊢ Membership.mem (γ := Set (Triangle.{v_1, u_1} Cᵒᵖ)) (distinguishedTriangles C) T ↔
    Membership.mem (γ := Set (Triangle.{v_1, u_1} Cᵒᵖ)) (distinguishedTriangles C) T.rotate
have simp : 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)) ↔
  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.rotate)).rotate
Tactic: simp only [mem_distinguishedTriangles_iff, Pretriangulated.rotate_distinguished_triangle
    ((triangleOpEquivalence C).inverse.obj (T.rotate)).unop]
-- 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 : Cᵒᵖ) (g : Y ⟶ Z) (h : Z ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor Cᵒᵖ (A := ℤ) 1) X),
    Membership.mem (γ := Set (Triangle.{v_1, u_1} Cᵒᵖ)) (distinguishedTriangles C) (Triangle.mk f g h)
have obtain : ∀ (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)),
  ∃ (Z : Cᵒᵖ) (g : Y ⟶ Z) (h : Z ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor Cᵒᵖ (A := ℤ) 1) X),
    Membership.mem (γ := Set (Triangle.{v_1, u_1} Cᵒᵖ)) (distinguishedTriangles C) (Triangle.mk f g h)
Tactic: obtain ⟨Z, g, h, H⟩ := Pretriangulated.distinguished_cocone_triangle₁ f.unop
-- 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)
⊢ ∃ (Z : Cᵒᵖ) (g : Y ⟶ Z) (h : Z ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor Cᵒᵖ (A := ℤ) 1) X),
    Membership.mem (γ := Set (Triangle.{v_1, u_1} Cᵒᵖ)) (distinguishedTriangles C) (Triangle.mk f g h)
have refine : Membership.mem (γ := Set (Triangle.{v_1, u_1} Cᵒᵖ)) (distinguishedTriangles 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)))
Tactic: refine ⟨_, g.op, (opShiftFunctorEquivalence C 1).counitIso.inv.app (Opposite.op Z) ≫
    (shiftFunctor Cᵒᵖ (1 : ℤ)).map h.op, ?_⟩
-- 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} Cᵒᵖ)) (distinguishedTriangles 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 simp : 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)))))
Tactic: simp only [mem_distinguishedTriangles_iff]