awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Opposite.Functor (goal)
Goal:
C : Type u_1
D : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
inst_2 : HasShift.{v_1, u_1, 0} C ℤ
inst_3 : HasShift.{v_2, u_2, 0} D ℤ
F : Functor.{v_1, v_2, u_1, u_2} C D
inst_4 : F.CommShift ℤ
inst_5 : HasZeroObject.{v_1, u_1} C
inst_6 : Preadditive.{v_1, u_1} C
inst_7 : ∀ (n : ℤ), Additive.{v_1, u_1, v_1, u_1} (shiftFunctor C n)
inst_8 : Pretriangulated.{v_1, u_1} C
inst_9 : HasZeroObject.{v_2, u_2} D
inst_10 : Preadditive.{v_2, u_2} D
inst_11 : ∀ (n : ℤ), Additive.{v_2, u_2, v_2, u_2} (shiftFunctor D n)
inst_12 : Pretriangulated.{v_2, u_2} D
X Y : (Triangle.{v_1, u_1} C)ᵒᵖ
f : X ⟶ Y
⊢ Eq (α :=
    Triangle.obj₂ (obj (Functor.op F.mapTriangle ⋙ Equivalence.functor (triangleOpEquivalence D)) X) ⟶
      Triangle.obj₂ (obj (Equivalence.functor (triangleOpEquivalence C) ⋙ F.op.mapTriangle) Y))
    (TriangleMorphism.hom₂
      (map (Functor.op F.mapTriangle ⋙ Equivalence.functor (triangleOpEquivalence D)) f ≫
        Iso.hom
          ((fun (T : (Triangle.{v_1, u_1} C)ᵒᵖ) ↦
              (F.mapTriangleOpCompTriangleOpEquivalenceFunctorApp (unop T) :
                Iso (C := Triangle.{v_2, u_2} Dᵒᵖ)
                  (obj (Equivalence.functor (triangleOpEquivalence D)) (op (obj F.mapTriangle (unop T))))
                  (obj F.op.mapTriangle (obj (Equivalence.functor (triangleOpEquivalence C)) (op (unop T))))))
            Y)))
    (TriangleMorphism.hom₂
      (Iso.hom
          ((fun (T : (Triangle.{v_1, u_1} C)ᵒᵖ) ↦
              (F.mapTriangleOpCompTriangleOpEquivalenceFunctorApp (unop T) :
                Iso (C := Triangle.{v_2, u_2} Dᵒᵖ)
                  (obj (Equivalence.functor (triangleOpEquivalence D)) (op (obj F.mapTriangle (unop T))))
                  (obj F.op.mapTriangle (obj (Equivalence.functor (triangleOpEquivalence C)) (op (unop T))))))
            X) ≫
        map (Equivalence.functor (triangleOpEquivalence C) ⋙ F.op.mapTriangle) f))
have h₂ : Eq.{v_2 + 1} (α := op (F.obj (unop X).obj₂) ⟶ op (F.obj (unop Y).obj₂))
  ((F.map f.unop.hom₂).op ≫ 𝟙 (op (F.obj (unop Y).obj₂))) (𝟙 (op (F.obj (unop X).obj₂)) ≫ (F.map f.unop.hom₂).op)
Tactic: dsimp
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Opposite.Functor (goal)
Goal:
C : Type u_1
D : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
inst_2 : HasShift.{v_1, u_1, 0} C ℤ
inst_3 : HasShift.{v_2, u_2, 0} D ℤ
F : Functor.{v_1, v_2, u_1, u_2} C D
inst_4 : F.CommShift ℤ
inst_5 : HasZeroObject.{v_1, u_1} C
inst_6 : Preadditive.{v_1, u_1} C
inst_7 : ∀ (n : ℤ), Additive.{v_1, u_1, v_1, u_1} (shiftFunctor C n)
inst_8 : Pretriangulated.{v_1, u_1} C
inst_9 : HasZeroObject.{v_2, u_2} D
inst_10 : Preadditive.{v_2, u_2} D
inst_11 : ∀ (n : ℤ), Additive.{v_2, u_2, v_2, u_2} (shiftFunctor D n)
inst_12 : Pretriangulated.{v_2, u_2} D
X Y : (Triangle.{v_1, u_1} C)ᵒᵖ
f : X ⟶ Y
⊢ Eq (α :=
    Triangle.obj₃ (obj (Functor.op F.mapTriangle ⋙ Equivalence.functor (triangleOpEquivalence D)) X) ⟶
      Triangle.obj₃ (obj (Equivalence.functor (triangleOpEquivalence C) ⋙ F.op.mapTriangle) Y))
    (TriangleMorphism.hom₃
      (map (Functor.op F.mapTriangle ⋙ Equivalence.functor (triangleOpEquivalence D)) f ≫
        Iso.hom
          ((fun (T : (Triangle.{v_1, u_1} C)ᵒᵖ) ↦
              (F.mapTriangleOpCompTriangleOpEquivalenceFunctorApp (unop T) :
                Iso (C := Triangle.{v_2, u_2} Dᵒᵖ)
                  (obj (Equivalence.functor (triangleOpEquivalence D)) (op (obj F.mapTriangle (unop T))))
                  (obj F.op.mapTriangle (obj (Equivalence.functor (triangleOpEquivalence C)) (op (unop T))))))
            Y)))
    (TriangleMorphism.hom₃
      (Iso.hom
          ((fun (T : (Triangle.{v_1, u_1} C)ᵒᵖ) ↦
              (F.mapTriangleOpCompTriangleOpEquivalenceFunctorApp (unop T) :
                Iso (C := Triangle.{v_2, u_2} Dᵒᵖ)
                  (obj (Equivalence.functor (triangleOpEquivalence D)) (op (obj F.mapTriangle (unop T))))
                  (obj F.op.mapTriangle (obj (Equivalence.functor (triangleOpEquivalence C)) (op (unop T))))))
            X) ≫
        map (Equivalence.functor (triangleOpEquivalence C) ⋙ F.op.mapTriangle) f))
have h₃ : Eq.{v_2 + 1} (α := op (F.obj (unop X).obj₁) ⟶ op (F.obj (unop Y).obj₁))
  ((F.map f.unop.hom₁).op ≫ 𝟙 (op (F.obj (unop Y).obj₁))) (𝟙 (op (F.obj (unop X).obj₁)) ≫ (F.map f.unop.hom₁).op)
Tactic: dsimp
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Opposite.Functor (goal)
Goal:
C : Type u_1
D : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
inst_2 : HasShift.{v_1, u_1, 0} C ℤ
inst_3 : HasShift.{v_2, u_2, 0} D ℤ
F : Functor.{v_1, v_2, u_1, u_2} C D
inst_4 : F.CommShift ℤ
inst_5 : HasZeroObject.{v_1, u_1} C
inst_6 : Preadditive.{v_1, u_1} C
inst_7 : ∀ (n : ℤ), Functor.Additive.{v_1, u_1, v_1, u_1} (shiftFunctor C n)
inst_8 : Pretriangulated.{v_1, u_1} C
inst_9 : HasZeroObject.{v_2, u_2} D
inst_10 : Preadditive.{v_2, u_2} D
inst_11 : ∀ (n : ℤ), Functor.Additive.{v_2, u_2, v_2, u_2} (shiftFunctor D n)
inst_12 : Pretriangulated.{v_2, u_2} D
inst_13 : F.IsTriangulated
T : Triangle.{v_1, u_1} Cᵒᵖ
dT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} Cᵒᵖ))
    (@Pretriangulated.distinguishedTriangles _ _ ⋯ _ (instPreadditiveOpposite C) ⋯ _) T
⊢ Membership.mem (γ := Set (Triangle.{v_2, u_2} Dᵒᵖ))
    (@Pretriangulated.distinguishedTriangles _ _ ⋯ _ (instPreadditiveOpposite D) ⋯ _) (obj F.op.mapTriangle T)
have rw : Membership.mem (γ := Set (Triangle.{v_2, u_2} D))
  (@Pretriangulated.distinguishedTriangles _ _ inst_9 _ inst_10 inst_11 _)
  (unop (obj (Equivalence.inverse (triangleOpEquivalence D)) (obj F.op.mapTriangle T)))
Tactic: rw [mem_distTriang_op_iff]
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Opposite.Functor (hypothesis)
Goal:
C : Type u_1
D : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
inst_2 : HasShift.{v_1, u_1, 0} C ℤ
inst_3 : HasShift.{v_2, u_2, 0} D ℤ
F : Functor.{v_1, v_2, u_1, u_2} C D
inst_4 : F.CommShift ℤ
inst_5 : HasZeroObject.{v_1, u_1} C
inst_6 : Preadditive.{v_1, u_1} C
inst_7 : ∀ (n : ℤ), Additive.{v_1, u_1, v_1, u_1} (shiftFunctor C n)
inst_8 : Pretriangulated.{v_1, u_1} C
inst_9 : HasZeroObject.{v_2, u_2} D
inst_10 : Preadditive.{v_2, u_2} D
inst_11 : ∀ (n : ℤ), Additive.{v_2, u_2, v_2, u_2} (shiftFunctor D n)
inst_12 : Pretriangulated.{v_2, u_2} D
inst_13 : F.op.IsTriangulated
T : Triangle.{v_1, u_1} C
dT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C))
    (@Pretriangulated.distinguishedTriangles _ _ inst_5 _ inst_6 inst_7 _) T
⊢ Membership.mem (γ := Set (Triangle.{v_2, u_2} D))
    (@Pretriangulated.distinguishedTriangles _ _ inst_9 _ inst_10 inst_11 _) (obj F.mapTriangle T)
have this : Membership.mem (γ := Set (Triangle.{v_2, u_2} D))
    (@Pretriangulated.distinguishedTriangles _ _ inst_9 _ inst_10 inst_11 _)
    (unop
      (obj (Equivalence.functor (triangleOpEquivalence D) ⋙ Equivalence.inverse (triangleOpEquivalence D))
        (op (obj F.mapTriangle T)))) ↔
  Membership.mem (γ := Set (Triangle.{v_2, u_2} D))
    (@Pretriangulated.distinguishedTriangles _ _ inst_9 _ inst_10 inst_11 _)
    (unop (obj (𝟭 (Triangle.{v_2, u_2} D)ᵒᵖ) (op (obj F.mapTriangle T))))
Tactic: have := distinguished_iff_of_iso ((triangleOpEquivalence D).unitIso.app
      (Opposite.op (F.mapTriangle.obj T))).unop
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Opposite.Functor (hypothesis)
Goal:
C : Type u_1
D : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
inst_2 : HasShift.{v_1, u_1, 0} C ℤ
inst_3 : HasShift.{v_2, u_2, 0} D ℤ
F : Functor.{v_1, v_2, u_1, u_2} C D
inst_4 : F.CommShift ℤ
inst_5 : HasZeroObject.{v_1, u_1} C
inst_6 : Preadditive.{v_1, u_1} C
inst_7 : ∀ (n : ℤ), Additive.{v_1, u_1, v_1, u_1} (shiftFunctor C n)
inst_8 : Pretriangulated.{v_1, u_1} C
inst_9 : HasZeroObject.{v_2, u_2} D
inst_10 : Preadditive.{v_2, u_2} D
inst_11 : ∀ (n : ℤ), Additive.{v_2, u_2, v_2, u_2} (shiftFunctor D n)
inst_12 : Pretriangulated.{v_2, u_2} D
inst_13 : F.op.IsTriangulated
T : Triangle.{v_1, u_1} C
dT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C))
    (@Pretriangulated.distinguishedTriangles _ _ inst_5 _ inst_6 inst_7 _) T
this :
  Membership.mem (γ := Set (Triangle.{v_2, u_2} D))
      (@Pretriangulated.distinguishedTriangles _ _ inst_9 _ inst_10 inst_11 _)
      (unop
        (obj (Equivalence.functor (triangleOpEquivalence D) ⋙ Equivalence.inverse (triangleOpEquivalence D))
          (op (obj F.mapTriangle T)))) ↔
    Membership.mem (γ := Set (Triangle.{v_2, u_2} D))
      (@Pretriangulated.distinguishedTriangles _ _ inst_9 _ inst_10 inst_11 _)
      (unop (obj (𝟭 (Triangle.{v_2, u_2} D)ᵒᵖ) (op (obj F.mapTriangle T))))
⊢ Membership.mem (γ := Set (Triangle.{v_2, u_2} D))
    (@Pretriangulated.distinguishedTriangles _ _ inst_9 _ inst_10 inst_11 _) (obj F.mapTriangle T)
have this₁ : Membership.mem (γ := Set (Triangle.{v_2, u_2} D))
    (@Pretriangulated.distinguishedTriangles _ _ inst_9 _ inst_10 inst_11 _)
    (unop
      (obj (Equivalence.functor (triangleOpEquivalence D) ⋙ Equivalence.inverse (triangleOpEquivalence D))
        (op (obj F.mapTriangle T)))) ↔
  Membership.mem (γ := Set (Triangle.{v_2, u_2} D))
    (@Pretriangulated.distinguishedTriangles _ _ inst_9 _ inst_10 inst_11 _) (obj F.mapTriangle T)
Tactic: rw [Functor.id_obj, Opposite.unop_op (F.mapTriangle.obj T)] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Opposite.Functor (goal)
Goal:
C : Type u_1
D : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
inst_2 : HasShift.{v_1, u_1, 0} C ℤ
inst_3 : HasShift.{v_2, u_2, 0} D ℤ
F : Functor.{v_1, v_2, u_1, u_2} C D
inst_4 : F.CommShift ℤ
inst_5 : HasZeroObject.{v_1, u_1} C
inst_6 : Preadditive.{v_1, u_1} C
inst_7 : ∀ (n : ℤ), Additive.{v_1, u_1, v_1, u_1} (shiftFunctor C n)
inst_8 : Pretriangulated.{v_1, u_1} C
inst_9 : HasZeroObject.{v_2, u_2} D
inst_10 : Preadditive.{v_2, u_2} D
inst_11 : ∀ (n : ℤ), Additive.{v_2, u_2, v_2, u_2} (shiftFunctor D n)
inst_12 : Pretriangulated.{v_2, u_2} D
inst_13 : F.op.IsTriangulated
T : Triangle.{v_1, u_1} C
dT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C))
    (@Pretriangulated.distinguishedTriangles _ _ inst_5 _ inst_6 inst_7 _) T
this :
  Membership.mem (γ := Set (Triangle.{v_2, u_2} D))
      (@Pretriangulated.distinguishedTriangles _ _ inst_9 _ inst_10 inst_11 _)
      (unop
        (obj (Equivalence.functor (triangleOpEquivalence D) ⋙ Equivalence.inverse (triangleOpEquivalence D))
          (op (obj F.mapTriangle T)))) ↔
    Membership.mem (γ := Set (Triangle.{v_2, u_2} D))
      (@Pretriangulated.distinguishedTriangles _ _ inst_9 _ inst_10 inst_11 _) (obj F.mapTriangle T)
⊢ Membership.mem (γ := Set (Triangle.{v_2, u_2} D))
    (@Pretriangulated.distinguishedTriangles _ _ inst_9 _ inst_10 inst_11 _) (obj F.mapTriangle T)
have rw : Membership.mem (γ := Set (Triangle.{v_2, u_2} Dᵒᵖ))
  (@Pretriangulated.distinguishedTriangles _ _ sorry _ (instPreadditiveOpposite D) sorry _)
  (obj (Equivalence.functor (triangleOpEquivalence C) ⋙ F.op.mapTriangle) (op T))
Tactic: rw [← this, Functor.comp_obj, ← mem_distTriang_op_iff, ← Functor.op_obj, ← Functor.comp_obj,
      distinguished_iff_of_iso ((mapTriangleOpCompTriangleOpEquivalenceFunctor F).app
      (Opposite.op T))]
-- 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
X : C
n m : ℤ
hnm : n + m = 0
⊢ Eq.{v_1 + 1} (α :=
    Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C n ⋙ opOp C) X ⟶
      Functor.obj.{v_1, v_1, u_1, u_1} (opOp C ⋙ shiftFunctor Cᵒᵖᵒᵖ n) X)
    (NatTrans.app (Iso.hom (iso C n)) X)
    (Quiver.Hom.op (NatTrans.app (Iso.hom (shiftFunctorOpIso C m n ⋯)) (op X)) ≫
      NatTrans.app (Iso.inv (shiftFunctorOpIso Cᵒᵖ n m hnm)) (op (op X)))
have obtain : ∀ (hnm : n + -n = 0),
  Eq.{v_1 + 1} (α :=
    Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C n ⋙ opOp C) X ⟶
      Functor.obj.{v_1, v_1, u_1, u_1} (opOp C ⋙ shiftFunctor Cᵒᵖᵒᵖ n) X)
    (NatTrans.app (Iso.hom (iso C n)) X)
    (Quiver.Hom.op (NatTrans.app (Iso.hom (shiftFunctorOpIso C (-n) n sorry)) (op X)) ≫
      NatTrans.app (Iso.inv (shiftFunctorOpIso Cᵒᵖ n (-n) hnm)) (op (op X)))
Tactic: obtain rfl : m = -n := by lia
Cleared hypotheses: m, hnm
-- 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
X : C
n m : ℤ
hnm : n + m = 0
⊢ Eq.{v_1 + 1} (α :=
    Functor.obj.{v_1, v_1, u_1, u_1} (opOp C ⋙ shiftFunctor Cᵒᵖᵒᵖ n) X ⟶
      Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C n ⋙ opOp C) X)
    (NatTrans.app (Iso.inv (iso C n)) X)
    (NatTrans.app (Iso.hom (shiftFunctorOpIso Cᵒᵖ n m hnm)) (op (op X)) ≫
      Quiver.Hom.op (NatTrans.app (Iso.inv (shiftFunctorOpIso C m n ⋯)) (op X)))
have obtain : ∀ (hnm : n + -n = 0),
  Eq.{v_1 + 1} (α :=
    Functor.obj.{v_1, v_1, u_1, u_1} (opOp C ⋙ shiftFunctor Cᵒᵖᵒᵖ n) X ⟶
      Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C n ⋙ opOp C) X)
    (NatTrans.app (Iso.inv (iso C n)) X)
    (NatTrans.app (Iso.hom (shiftFunctorOpIso Cᵒᵖ n (-n) hnm)) (op (op X)) ≫
      Quiver.Hom.op (NatTrans.app (Iso.inv (shiftFunctorOpIso C (-n) n sorry)) (op X)))
Tactic: obtain rfl : m = -n := by lia
Cleared hypotheses: m, hnm
-- 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
X : Cᵒᵖᵒᵖ
n m : ℤ
hnm : n + m = 0
⊢ Eq.{v_1 + 1} (α :=
    Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor Cᵒᵖᵒᵖ n ⋙ unopUnop C) X ⟶
      Functor.obj.{v_1, v_1, u_1, u_1} (unopUnop C ⋙ shiftFunctor C n) X)
    (NatTrans.app (Iso.hom (iso C n)) X)
    (Quiver.Hom.unop (Quiver.Hom.unop (NatTrans.app (Iso.hom (shiftFunctorOpIso Cᵒᵖ n m hnm)) X)) ≫
      Quiver.Hom.unop (NatTrans.app (Iso.inv (shiftFunctorOpIso C m n ⋯)) (unop X)))
have obtain : ∀ (hnm : n + -n = 0),
  Eq.{v_1 + 1} (α :=
    Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor Cᵒᵖᵒᵖ n ⋙ unopUnop C) X ⟶
      Functor.obj.{v_1, v_1, u_1, u_1} (unopUnop C ⋙ shiftFunctor C n) X)
    (NatTrans.app (Iso.hom (iso C n)) X)
    (Quiver.Hom.unop (Quiver.Hom.unop (NatTrans.app (Iso.hom (shiftFunctorOpIso Cᵒᵖ n (-n) hnm)) X)) ≫
      Quiver.Hom.unop (NatTrans.app (Iso.inv (shiftFunctorOpIso C (-n) n sorry)) (unop X)))
Tactic: obtain rfl : m = -n := by lia
Cleared hypotheses: m, hnm
-- 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
X : Cᵒᵖᵒᵖ
n m : ℤ
hnm : n + m = 0
⊢ Eq.{v_1 + 1} (α :=
    Functor.obj.{v_1, v_1, u_1, u_1} (unopUnop C ⋙ shiftFunctor C n) X ⟶
      Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor Cᵒᵖᵒᵖ n ⋙ unopUnop C) X)
    (NatTrans.app (Iso.inv (iso C n)) X)
    (Quiver.Hom.unop (NatTrans.app (Iso.hom (shiftFunctorOpIso C m n ⋯)) (unop X)) ≫
      Quiver.Hom.unop (Quiver.Hom.unop (NatTrans.app (Iso.inv (shiftFunctorOpIso Cᵒᵖ n m hnm)) X)))
have obtain : ∀ (hnm : n + -n = 0),
  Eq.{v_1 + 1} (α :=
    Functor.obj.{v_1, v_1, u_1, u_1} (unopUnop C ⋙ shiftFunctor C n) X ⟶
      Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor Cᵒᵖᵒᵖ n ⋙ unopUnop C) X)
    (NatTrans.app (Iso.inv (iso C n)) X)
    (Quiver.Hom.unop (NatTrans.app (Iso.hom (shiftFunctorOpIso C (-n) n sorry)) (unop X)) ≫
      Quiver.Hom.unop (Quiver.Hom.unop (NatTrans.app (Iso.inv (shiftFunctorOpIso Cᵒᵖ n (-n) hnm)) X)))
Tactic: obtain rfl : m = -n := by lia
Cleared hypotheses: m, hnm
-- 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
X : C
⊢ Eq.{v_1 + 1} (α :=
    Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 0 ⋙ opOp C) X ⟶
      Functor.obj.{v_1, v_1, u_1, u_1} (opOp C ⋙ shiftFunctor Cᵒᵖᵒᵖ (A := ℤ) 0) X)
    (NatTrans.app (Iso.hom (iso C 0)) X) (NatTrans.app (Iso.hom (Functor.CommShift.isoZero (opOp C) ℤ)) X)
have w : Eq.{v_1 + 1} (α :=
  unop (unop (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 0 ⋙ opOp C) X)) ⟶
    unop (unop (Functor.obj.{v_1, v_1, u_1, u_1} (opOp C ⋙ shiftFunctor Cᵒᵖᵒᵖ (A := ℤ) 0) X)))
  (Quiver.Hom.unop (Quiver.Hom.unop (NatTrans.app (Iso.hom (iso C 0)) X)))
  (Quiver.Hom.unop (Quiver.Hom.unop (NatTrans.app (Iso.hom (Functor.CommShift.isoZero (opOp C) ℤ)) X)))
Tactic: refine Quiver.Hom.unop_inj (Quiver.Hom.unop_inj ?_)
-- 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
p q : ℤ
⊢ Eq (α :=
    Iso (C := Functor.{v_1, v_1, u_1, u_1} C Cᵒᵖᵒᵖ) (shiftFunctor C (p + q) ⋙ opOp C)
      (opOp C ⋙ shiftFunctor Cᵒᵖᵒᵖ (p + q)))
    (iso C (p + q)) (Functor.CommShift.isoAdd (iso C p) (iso C q))
have w : ∀ (X : C),
  Eq.{v_1 + 1} (α :=
    Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (p + q) ⋙ opOp C) X ⟶
      Functor.obj.{v_1, v_1, u_1, u_1} (opOp C ⋙ shiftFunctor Cᵒᵖᵒᵖ (p + q)) X)
    (NatTrans.app (Iso.hom (iso C (p + q))) X) (NatTrans.app (Iso.hom (Functor.CommShift.isoAdd (iso C p) (iso C q))) X)
Tactic: ext X
-- 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
p q : ℤ
X : C
⊢ Eq.{v_1 + 1} (α :=
    Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (p + q) ⋙ opOp C) X ⟶
      Functor.obj.{v_1, v_1, u_1, u_1} (opOp C ⋙ shiftFunctor Cᵒᵖᵒᵖ (p + q)) X)
    (NatTrans.app (Iso.hom (iso C (p + q))) X) (NatTrans.app (Iso.hom (Functor.CommShift.isoAdd (iso C p) (iso C q))) X)
have w : Eq.{v_1 + 1} (α :=
  unop (unop (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (p + q) ⋙ opOp C) X)) ⟶
    unop (unop (Functor.obj.{v_1, v_1, u_1, u_1} (opOp C ⋙ shiftFunctor Cᵒᵖᵒᵖ (p + q)) X)))
  (Quiver.Hom.unop (Quiver.Hom.unop (NatTrans.app (Iso.hom (iso C (p + q))) X)))
  (Quiver.Hom.unop (Quiver.Hom.unop (NatTrans.app (Iso.hom (Functor.CommShift.isoAdd (iso C p) (iso C q))) X)))
Tactic: refine Quiver.Hom.unop_inj (Quiver.Hom.unop_inj ?_)
-- 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
⊢ Eq (α :=
    Iso (C := Functor.{v_1, v_1, u_1, u_1} Cᵒᵖᵒᵖ C) (shiftFunctor Cᵒᵖᵒᵖ (A := ℤ) 0 ⋙ unopUnop C)
      (unopUnop C ⋙ shiftFunctor C (A := ℤ) 0))
    (iso C 0) (Functor.CommShift.isoZero (unopUnop C) ℤ)
have w : ∀ (X : Cᵒᵖᵒᵖ),
  Eq.{v_1 + 1} (α :=
    Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor Cᵒᵖᵒᵖ (A := ℤ) 0 ⋙ unopUnop C) X ⟶
      Functor.obj.{v_1, v_1, u_1, u_1} (unopUnop C ⋙ shiftFunctor C (A := ℤ) 0) X)
    (NatTrans.app (Iso.hom (iso C 0)) X) (NatTrans.app (Iso.hom (Functor.CommShift.isoZero (unopUnop C) ℤ)) X)
Tactic: ext X
-- 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
p q : ℤ
⊢ Eq (α :=
    Iso (C := Functor.{v_1, v_1, u_1, u_1} Cᵒᵖᵒᵖ C) (shiftFunctor Cᵒᵖᵒᵖ (p + q) ⋙ unopUnop C)
      (unopUnop C ⋙ shiftFunctor C (p + q)))
    (iso C (p + q)) (Functor.CommShift.isoAdd (iso C p) (iso C q))
have w : ∀ (X : Cᵒᵖᵒᵖ),
  Eq.{v_1 + 1} (α :=
    Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor Cᵒᵖᵒᵖ (p + q) ⋙ unopUnop C) X ⟶
      Functor.obj.{v_1, v_1, u_1, u_1} (unopUnop C ⋙ shiftFunctor C (p + q)) X)
    (NatTrans.app (Iso.hom (iso C (p + q))) X) (NatTrans.app (Iso.hom (Functor.CommShift.isoAdd (iso C p) (iso C q))) X)
Tactic: ext X
-- 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
n : ℤ
⊢ Eq (α :=
    Quiver.Hom (V := Functor.{v_1, v_1, u_1, u_1} C C)
      (shiftFunctor C n ⋙ Equivalence.inverse (opOpEquivalence C) ⋙ Equivalence.functor (opOpEquivalence C))
      (𝟭 C ⋙ shiftFunctor C n))
    (Iso.hom
        (Functor.commShiftIso (Equivalence.inverse (opOpEquivalence C) ⋙ Equivalence.functor (opOpEquivalence C)) n) ≫
      Functor.whiskerRight (Iso.hom (Equivalence.counitIso (opOpEquivalence C))) (shiftFunctor C n))
    (Functor.whiskerLeft (shiftFunctor C n) (Iso.hom (Equivalence.counitIso (opOpEquivalence C))) ≫
      Iso.hom (Functor.commShiftIso (𝟭 C) n))
have w : ∀ (X : C),
  Eq.{v_1 + 1} (α :=
    Functor.obj.{v_1, v_1, u_1, u_1}
        (shiftFunctor C n ⋙ Equivalence.inverse (opOpEquivalence C) ⋙ Equivalence.functor (opOpEquivalence C)) X ⟶
      Functor.obj.{v_1, v_1, u_1, u_1} (𝟭 C ⋙ shiftFunctor C n) X)
    (NatTrans.app
      (Iso.hom
          (Functor.commShiftIso (Equivalence.inverse (opOpEquivalence C) ⋙ Equivalence.functor (opOpEquivalence C)) n) ≫
        Functor.whiskerRight (Iso.hom (Equivalence.counitIso (opOpEquivalence C))) (shiftFunctor C n))
      X)
    (NatTrans.app
      (Functor.whiskerLeft (shiftFunctor C n) (Iso.hom (Equivalence.counitIso (opOpEquivalence C))) ≫
        Iso.hom (Functor.commShiftIso (𝟭 C) n))
      X)
Tactic: ext X
-- 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
⊢ Membership.mem (γ := Set (Triangle.{v_1, u_1} Cᵒᵖᵒᵖ))
    (@distinguishedTriangles _ _ ⋯ _ (instPreadditiveOpposite Cᵒᵖ) ⋯ _) (Functor.obj (Functor.mapTriangle (opOp C)) T)
have refine : (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
Tactic: refine isomorphic_distinguished _ (op_distinguished _ (op_distinguished _ hT)) _
      (Triangle.isoMk _ _ (Iso.refl _) (Iso.refl _) (Iso.refl _) (by simp) (by simp)
      (Quiver.Hom.unop_inj ?_))
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Opposite.OpOp (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 : 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
⊢ (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 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 sorry))
      (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 sorry)) T.obj₃ ≫
    Functor.map (shiftFunctor C (A := ℤ) (-1) ⋙ shiftFunctor C (A := ℤ) 1) T.mor₃)
Tactic: have := (shiftFunctorCompIsoId C (-1) 1 (neg_add_cancel 1)).inv.naturality T.mor₃