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
P Q : ObjectProperty.{v_1, u_1} C
h : P ≤ Q
X Y : C
f : X ⟶ Y
Z : C
a : Y ⟶ Z
b : Z ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) X
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk f a b)
hZ : P Z
⊢ ∃ (Z : C) (g : Y ⟶ Z) (h : Z ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) X) (_ :
    Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
      (Triangle.mk f g h)),
    Q Z

Tactic:
⟨Z, a, b, hT, h _ hZ⟩

Premises used: Exists.intro
-- Module: Mathlib.CategoryTheory.Triangulated.Subcategory
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : HasShift.{v_1, u_1, 0} C ℤ
inst_3 : Preadditive.{v_1, u_1} C
inst_4 : ∀ (n : ℤ), Functor.Additive.{v_1, u_1, v_1, u_1} (shiftFunctor C n)
inst_5 : Pretriangulated.{v_1, u_1} C
P Q : ObjectProperty.{v_1, u_1} C
h : P ≤ Q
X Y : C
f : X ⟶ Y
Z : C
a : Y ⟶ Z
b : Z ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) X
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk f a b)
hZ : P Z
⊢ ∃ (Z : C) (g : Y ⟶ Z) (h : Z ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) X) (_ :
    Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
      (Triangle.mk f g h)),
    Q Z

Tactic:
exact ⟨Z, a, b, hT, h _ hZ⟩
-- 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
X' X Y : C
e : X' ⟶ X
he : IsIso e
f : X ⟶ Y
Z : C
g : Y ⟶ Z
h : Z ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) X
mem :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk f g h)
mem' : P Z
⊢ P.trW (e ≫ f)

Tactic:
⟨Z, g, h ≫ inv e⟦(1 : ℤ)⟧', isomorphic_distinguished _ mem _ ?_, mem'⟩

Premises used: CategoryTheory.shiftFunctor, CategoryTheory.Functor.map, CategoryTheory.Pretriangulated.isomorphic_distinguished, CategoryTheory.ObjectProperty.trW, Exists.intro, CategoryTheory.CategoryStruct.comp, CategoryTheory.inv, OfNat.ofNat
-- 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
X' X Y : C
e : X' ⟶ X
he : IsIso e
f : X ⟶ Y
Z : C
g : Y ⟶ Z
h : Z ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) X
mem :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk f g h)
mem' : P Z
refine : Triangle.mk (e ≫ f) g (h ≫ Functor.map (shiftFunctor C (A := ℤ) 1) (inv e)) ≅ Triangle.mk f g h
⊢ P.trW (e ≫ f)

Tactic:
refine ⟨Z, g, h ≫ inv e⟦(1 : ℤ)⟧', isomorphic_distinguished _ mem _ ?_, mem'⟩
-- Module: Mathlib.CategoryTheory.Triangulated.Subcategory
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : HasShift.{v_1, u_1, 0} C ℤ
inst_3 : Preadditive.{v_1, u_1} C
inst_4 : ∀ (n : ℤ), Functor.Additive.{v_1, u_1, v_1, u_1} (shiftFunctor C n)
inst_5 : Pretriangulated.{v_1, u_1} C
D : Type u_2
inst_6 : Category.{v_2, u_2} D
inst_7 : Preadditive.{v_2, u_2} D
inst_8 : HasZeroObject.{v_2, u_2} D
inst_9 : HasShift.{v_2, u_2, 0} D ℤ
inst_10 : ∀ (n : ℤ), Functor.Additive.{v_2, u_2, v_2, u_2} (shiftFunctor D n)
inst_11 : Pretriangulated.{v_2, u_2} D
E : Type u_3
inst_12 : Category.{v_3, u_3} E
inst_13 : HasShift.{v_3, u_3, 0} E ℤ
P : ObjectProperty.{v_1, u_1} C
X' X Y : C
e : X' ⟶ X
he : IsIso e
f : X ⟶ Y
Z : C
g : Y ⟶ Z
h : Z ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) X
mem :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk f g h)
mem' : P Z
⊢ (Triangle.mk (e ≫ f) g (h ≫ Functor.map (shiftFunctor C (A := ℤ) 1) (inv e))).mor₁ ≫
      Iso.hom (Iso.refl (Triangle.mk (e ≫ f) g (h ≫ Functor.map (shiftFunctor C (A := ℤ) 1) (inv e))).obj₂) =
    (asIso e).hom ≫ (Triangle.mk f g h).mor₁

Tactic:
simp
-- 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
X' X Y : C
e : X' ⟶ X
he : IsIso e
f : X ⟶ Y
Z : C
g : Y ⟶ Z
h : Z ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) X
mem :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk f g h)
mem' : P Z
⊢ (Triangle.mk (e ≫ f) g (h ≫ Functor.map (shiftFunctor C (A := ℤ) 1) (inv e))).mor₂ ≫
      Iso.hom (Iso.refl (Triangle.mk (e ≫ f) g (h ≫ Functor.map (shiftFunctor C (A := ℤ) 1) (inv e))).obj₃) =
    Iso.hom (Iso.refl (Triangle.mk (e ≫ f) g (h ≫ Functor.map (shiftFunctor C (A := ℤ) 1) (inv e))).obj₂) ≫
      (Triangle.mk f g h).mor₂

Tactic:
simp
-- 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
X' X Y : C
e : X' ⟶ X
he : IsIso e
f : X ⟶ Y
Z : C
g : Y ⟶ Z
h : Z ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) X
mem :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk f g h)
mem' : P Z
refine :
  (Triangle.mk (e ≫ f) g (h ≫ Functor.map (shiftFunctor C (A := ℤ) 1) (inv e))).mor₃ ≫
      Functor.map (shiftFunctor C (A := ℤ) 1) (asIso e).hom =
    Iso.hom (Iso.refl (Triangle.mk (e ≫ f) g (h ≫ Functor.map (shiftFunctor C (A := ℤ) 1) (inv e))).obj₃) ≫
      (Triangle.mk f g h).mor₃
⊢ Triangle.mk (e ≫ f) g (h ≫ Functor.map (shiftFunctor C (A := ℤ) 1) (inv e)) ≅ Triangle.mk f g h

Tactic:
refine Triangle.isoMk _ _ (asIso e) (Iso.refl _) (Iso.refl _) (by simp) (by simp) ?_

Premises used: CategoryTheory.Iso.refl, CategoryTheory.Pretriangulated.Triangle.isoMk, CategoryTheory.asIso
-- 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
X' X Y : C
e : X' ⟶ X
he : IsIso e
f : X ⟶ Y
Z : C
g : Y ⟶ Z
h : Z ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) X
mem :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk f g h)
mem' : P Z
dsimp : (h ≫ Functor.map (shiftFunctor C (A := ℤ) 1) (inv e)) ≫ Functor.map (shiftFunctor C (A := ℤ) 1) e = 𝟙 Z ≫ h
⊢ (Triangle.mk (e ≫ f) g (h ≫ Functor.map (shiftFunctor C (A := ℤ) 1) (inv e))).mor₃ ≫
      Functor.map (shiftFunctor C (A := ℤ) 1) (asIso e).hom =
    Iso.hom (Iso.refl (Triangle.mk (e ≫ f) g (h ≫ Functor.map (shiftFunctor C (A := ℤ) 1) (inv e))).obj₃) ≫
      (Triangle.mk f g h).mor₃

Tactic:
dsimp
-- 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
X' X Y : C
e : X' ⟶ X
he : IsIso e
f : X ⟶ Y
Z : C
g : Y ⟶ Z
h : Z ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) X
mem :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk f g h)
mem' : P Z
⊢ (h ≫ Functor.map (shiftFunctor C (A := ℤ) 1) (inv e)) ≫ Functor.map (shiftFunctor C (A := ℤ) 1) e = 𝟙 Z ≫ h

Tactic:
simp only [Functor.map_inv, assoc, IsIso.inv_hom_id, comp_id, id_comp]

Premises used: CategoryTheory.Category.assoc, CategoryTheory.IsIso.inv_hom_id, CategoryTheory.Functor.map_inv, CategoryTheory.Category.comp_id, CategoryTheory.Category.id_comp
-- 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
X Y Y' : C
e : Y ⟶ Y'
he : IsIso e
f : X ⟶ Y
Z : C
g : Y ⟶ Z
h : Z ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) X
mem :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk f g h)
mem' : P Z
⊢ P.trW (f ≫ e)

Tactic:
⟨Z, inv e ≫ g, h, isomorphic_distinguished _ mem _ ?_, mem'⟩

Premises used: CategoryTheory.Pretriangulated.isomorphic_distinguished, CategoryTheory.ObjectProperty.trW, Exists.intro, CategoryTheory.CategoryStruct.comp, CategoryTheory.inv
-- 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
X Y Y' : C
e : Y ⟶ Y'
he : IsIso e
f : X ⟶ Y
Z : C
g : Y ⟶ Z
h : Z ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) X
mem :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk f g h)
mem' : P Z
refine : Triangle.mk (f ≫ e) (inv e ≫ g) h ≅ Triangle.mk f g h
⊢ P.trW (f ≫ e)

Tactic:
refine ⟨Z, inv e ≫ g, h, isomorphic_distinguished _ mem _ ?_, mem'⟩
-- Module: Mathlib.CategoryTheory.Triangulated.Subcategory
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : HasShift.{v_1, u_1, 0} C ℤ
inst_3 : Preadditive.{v_1, u_1} C
inst_4 : ∀ (n : ℤ), Functor.Additive.{v_1, u_1, v_1, u_1} (shiftFunctor C n)
inst_5 : Pretriangulated.{v_1, u_1} C
D : Type u_2
inst_6 : Category.{v_2, u_2} D
inst_7 : Preadditive.{v_2, u_2} D
inst_8 : HasZeroObject.{v_2, u_2} D
inst_9 : HasShift.{v_2, u_2, 0} D ℤ
inst_10 : ∀ (n : ℤ), Functor.Additive.{v_2, u_2, v_2, u_2} (shiftFunctor D n)
inst_11 : Pretriangulated.{v_2, u_2} D
E : Type u_3
inst_12 : Category.{v_3, u_3} E
inst_13 : HasShift.{v_3, u_3, 0} E ℤ
P : ObjectProperty.{v_1, u_1} C
X Y Y' : C
e : Y ⟶ Y'
he : IsIso e
f : X ⟶ Y
Z : C
g : Y ⟶ Z
h : Z ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) X
mem :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk f g h)
mem' : P Z
⊢ Triangle.mk (f ≫ e) (inv e ≫ g) h ≅ Triangle.mk f g h

Tactic:
exact Triangle.isoMk _ _ (Iso.refl _) (asIso e).symm (Iso.refl _)

Premises used: CategoryTheory.Iso.refl, CategoryTheory.Pretriangulated.Triangle.isoMk
-- 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.ContainsZero
rw : P.isoClosure.trW.ContainsIdentities
⊢ P.trW.ContainsIdentities

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.ContainsZero
⊢ P.isoClosure.trW.ContainsIdentities

Tactic:
⟨fun X => ⟨_, _, _, contractible_distinguished X, prop_zero _⟩⟩

Premises used: CategoryTheory.MorphismProperty.ContainsIdentities.mk, CategoryTheory.ObjectProperty.isoClosure, Exists.intro, CategoryTheory.ObjectProperty.prop_zero, CategoryTheory.Pretriangulated.contractible_distinguished, OfNat.ofNat
-- 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.ContainsZero
⊢ P.isoClosure.trW.ContainsIdentities

Tactic:
exact ⟨fun X => ⟨_, _, _, contractible_distinguished X, prop_zero _⟩⟩
-- 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
P : ObjectProperty.{v_1, u_1} C
inst_6 : P.ContainsZero
X Y : C
f : X ⟶ Y
inst_7 : IsIso f
⊢ P.trW f

Tactic:
(P.trW.arrow_mk_iso_iff ?_).1 (MorphismProperty.id_mem _ X)

Premises used: CategoryTheory.MorphismProperty.arrow_mk_iso_iff, CategoryTheory.ObjectProperty.trW, Iff.mp, CategoryTheory.MorphismProperty.id_mem
-- Module: Mathlib.CategoryTheory.Triangulated.Subcategory
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : HasShift.{v_1, u_1, 0} C ℤ
inst_3 : Preadditive.{v_1, u_1} C
inst_4 : ∀ (n : ℤ), Functor.Additive.{v_1, u_1, v_1, u_1} (shiftFunctor C n)
inst_5 : Pretriangulated.{v_1, u_1} C
P : ObjectProperty.{v_1, u_1} C
inst_6 : P.ContainsZero
X Y : C
f : X ⟶ Y
inst_7 : IsIso f
refine : Arrow.mk (𝟙 X) ≅ Arrow.mk f
⊢ P.trW f

Tactic:
refine (P.trW.arrow_mk_iso_iff ?_).1 (MorphismProperty.id_mem _ X)

Premises used: CategoryTheory.MorphismProperty.arrow_mk_iso_iff, CategoryTheory.ObjectProperty.trW, Iff.mp, CategoryTheory.MorphismProperty.id_mem
-- Module: Mathlib.CategoryTheory.Triangulated.Subcategory
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : HasShift.{v_1, u_1, 0} C ℤ
inst_3 : Preadditive.{v_1, u_1} C
inst_4 : ∀ (n : ℤ), Functor.Additive.{v_1, u_1, v_1, u_1} (shiftFunctor C n)
inst_5 : Pretriangulated.{v_1, u_1} C
P : ObjectProperty.{v_1, u_1} C
inst_6 : P.ContainsZero
X Y : C
f : X ⟶ Y
inst_7 : IsIso f
⊢ Arrow.mk (𝟙 X) ≅ Arrow.mk f

Tactic:
exact Arrow.isoMk (Iso.refl _) (asIso f)

Premises used: CategoryTheory.Iso.refl, CategoryTheory.asIso, CategoryTheory.Arrow.isoMk
-- 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
P : ObjectProperty.{v_1, u_1} C
X Y : C
f : X ⟶ Y
n : ℤˣ
⊢ P.trW (n • f) ↔ P.trW f

Tactic:
P.trW.arrow_mk_iso_iff (Arrow.isoMk (n • (Iso.refl _)) (Iso.refl _))

Premises used: CategoryTheory.Iso.refl, CategoryTheory.MorphismProperty.arrow_mk_iso_iff, CategoryTheory.ObjectProperty.trW, HSMul.hSMul, CategoryTheory.Arrow.isoMk
-- 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
P : ObjectProperty.{v_1, u_1} C
inst_6 : P.IsStableUnderShift ℤ
X₁ X₂ : C
f : X₁ ⟶ X₂
hf : P.trW f
n : ℤ
rw :
  P.trW
    (HSMul.hSMul (β :=
      Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C n) X₁ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C n) X₂)
      n.negOnePow (Functor.map (shiftFunctor C n) f))
⊢ P.trW (X := Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C n) X₁) (Functor.map (shiftFunctor C n) f)

Tactic:
rw [← smul_mem_trW_iff _ _ (n.negOnePow)]

Premises used: Int.negOnePow, CategoryTheory.ObjectProperty.smul_mem_trW_iff
-- 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
P : ObjectProperty.{v_1, u_1} C
inst_6 : P.IsStableUnderShift ℤ
X₁ X₂ : C
f : X₁ ⟶ X₂
hf : P.trW f
n : ℤ
obtain :
  ∀ (X₃ : C) (g : X₂ ⟶ X₃) (h : X₃ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) X₁)
    (hT :
      Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
        (Triangle.mk f g h))
    (mem : P X₃),
    P.trW
      (HSMul.hSMul (β :=
        Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C n) X₁ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C n) X₂)
        n.negOnePow (Functor.map (shiftFunctor C n) f))
⊢ P.trW
    (HSMul.hSMul (β :=
      Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C n) X₁ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C n) X₂)
      n.negOnePow (Functor.map (shiftFunctor C n) f))

Tactic:
obtain ⟨X₃, g, h, hT, mem⟩ := hf
-- 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
P : ObjectProperty.{v_1, u_1} C
inst_6 : P.IsStableUnderShift ℤ
X₁ X₂ : C
f : X₁ ⟶ X₂
n : ℤ
X₃ : C
g : X₂ ⟶ X₃
h : X₃ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) X₁
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk f g h)
mem : P X₃
⊢ P.trW
    (HSMul.hSMul (β :=
      Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C n) X₁ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C n) X₂)
      n.negOnePow (Functor.map (shiftFunctor C n) f))

Tactic:
⟨_, _, _, Pretriangulated.Triangle.shift_distinguished _ hT n, P.le_shift _ _ mem⟩

Premises used: CategoryTheory.Pretriangulated.Triangle.shift_distinguished, CategoryTheory.ObjectProperty.trW, Exists.intro, CategoryTheory.ObjectProperty.le_shift
-- 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
P : ObjectProperty.{v_1, u_1} C
inst_6 : P.IsStableUnderShift ℤ
X₁ X₂ : C
f : X₁ ⟶ X₂
n : ℤ
X₃ : C
g : X₂ ⟶ X₃
h : X₃ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) X₁
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk f g h)
mem : P X₃
⊢ P.trW
    (HSMul.hSMul (β :=
      Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C n) X₁ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C n) X₂)
      n.negOnePow (Functor.map (shiftFunctor C n) f))

Tactic:
exact ⟨_, _, _, Pretriangulated.Triangle.shift_distinguished _ hT n, P.le_shift _ _ mem⟩