awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory
Goal:
C : Type u_1
A : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : Preadditive.{v_1, u_1} C
inst_3 : HasShift.{v_1, u_1, 0} C ℤ
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
inst_6 : Category.{v_2, u_2} A
ι : Functor.{v_2, v_1, u_2, u_1} A C
hι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C n) (ι.obj Y)), n < 0 → f = 0
X₁ X₂ : A
f₁ : X₁ ⟶ X₂
X₃ : C
f₂ : ι.obj X₂ ⟶ X₃
f₃ : X₃ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj X₁)
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_2 inst_4 _)
    (Triangle.mk (ι.map f₁) f₂ f₃)
K Q : A
α : Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj K) ⟶ X₃
β : X₃ ⟶ ι.obj Q
γ :
  ι.obj Q ⟶
    Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1)
      (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj K))
hT' :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_2 inst_4 _)
    (Triangle.mk α β γ)
inst_7 : ι.Full
inst_8 : Preadditive.{v_2, u_2} A
inst_9 : ι.Faithful
B : A
k : B ⟶ K
hk : (Functor.map (shiftFunctor C (A := ℤ) 1) (ι.map k) ≫ α) ≫ f₃ = 0
l : Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj B) ⟶ (Triangle.mk (ι.map f₁) f₂ f₃).obj₂
hl : Functor.map (shiftFunctor C (A := ℤ) 1) (ι.map k) ≫ α = l ≫ (Triangle.mk (ι.map f₁) f₂ f₃).mor₂
⊢ LT.lt (α := ℤ) 0 1

Tactic:
lia
-- Module: Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory
Goal:
C : Type u_1
A : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : Preadditive.{v_1, u_1} C
inst_3 : HasShift.{v_1, u_1, 0} C ℤ
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
inst_6 : Category.{v_2, u_2} A
ι : Functor.{v_2, v_1, u_2, u_1} A C
hι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C n) (ι.obj Y)), n < 0 → f = 0
X₁ X₂ : A
f₁ : X₁ ⟶ X₂
X₃ : C
f₂ : ι.obj X₂ ⟶ X₃
f₃ : X₃ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj X₁)
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_2 inst_4 _)
    (Triangle.mk (ι.map f₁) f₂ f₃)
K Q : A
α : Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj K) ⟶ X₃
β : X₃ ⟶ ι.obj Q
γ :
  ι.obj Q ⟶
    Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1)
      (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj K))
hT' :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_2 inst_4 _)
    (Triangle.mk α β γ)
inst_7 : ι.Full
inst_8 : Preadditive.{v_2, u_2} A
inst_9 : ι.Faithful
B : A
k : B ⟶ K
hk : (Functor.map (shiftFunctor C (A := ℤ) 1) (ι.map k) ≫ α) ≫ f₃ = 0
l : Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj B) ⟶ (Triangle.mk (ι.map f₁) f₂ f₃).obj₂
hl : Functor.map (shiftFunctor C (A := ℤ) 1) (ι.map k) ≫ α = l ≫ (Triangle.mk (ι.map f₁) f₂ f₃).mor₂
⊢ Functor.map (shiftFunctor C (A := ℤ) 1) (ι.map k) ≫ α = 0

Tactic:
rw [eq_zero_of_hom_shift_pos hι l (by lia), zero_comp] at hl

Premises used: CategoryTheory.Triangulated.AbelianSubcategory.eq_zero_of_hom_shift_pos, CategoryTheory.Limits.zero_comp
-- Module: Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory
Goal:
C : Type u_1
A : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : Preadditive.{v_1, u_1} C
inst_3 : HasShift.{v_1, u_1, 0} C ℤ
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
inst_6 : Category.{v_2, u_2} A
ι : Functor.{v_2, v_1, u_2, u_1} A C
hι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C n) (ι.obj Y)), n < 0 → f = 0
X₁ X₂ : A
f₁ : X₁ ⟶ X₂
X₃ : C
f₂ : ι.obj X₂ ⟶ X₃
f₃ : X₃ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj X₁)
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_2 inst_4 _)
    (Triangle.mk (ι.map f₁) f₂ f₃)
K Q : A
α : Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj K) ⟶ X₃
β : X₃ ⟶ ι.obj Q
γ :
  ι.obj Q ⟶
    Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1)
      (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj K))
hT' :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_2 inst_4 _)
    (Triangle.mk α β γ)
inst_7 : ι.Full
inst_8 : Preadditive.{v_2, u_2} A
inst_9 : ι.Faithful
B : A
k : B ⟶ K
hk : (Functor.map (shiftFunctor C (A := ℤ) 1) (ι.map k) ≫ α) ≫ f₃ = 0
l : Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj B) ⟶ (Triangle.mk (ι.map f₁) f₂ f₃).obj₂
hl : Functor.map (shiftFunctor C (A := ℤ) 1) (ι.map k) ≫ α = 0
⊢ Eq (α :=
    Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1)
        (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj B)) ⟶
      Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (Triangle.mk α β γ).obj₂)
    (Functor.map (shiftFunctor C (A := ℤ) 1) (Functor.map (shiftFunctor C (A := ℤ) 1) (ι.map k)) ≫
      Functor.map (shiftFunctor C (A := ℤ) 1) (Triangle.mk α β γ).mor₁)
    0

Tactic:
simp [← Functor.map_comp, hl]

Premises used: CategoryTheory.Functor.map_comp
-- Module: Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory
Goal:
C : Type u_1
A : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : Preadditive.{v_1, u_1} C
inst_3 : HasShift.{v_1, u_1, 0} C ℤ
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
inst_6 : Category.{v_2, u_2} A
ι : Functor.{v_2, v_1, u_2, u_1} A C
hι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C n) (ι.obj Y)), n < 0 → f = 0
X₁ X₂ : A
f₁ : X₁ ⟶ X₂
X₃ : C
f₂ : ι.obj X₂ ⟶ X₃
f₃ : X₃ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj X₁)
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_2 inst_4 _)
    (Triangle.mk (ι.map f₁) f₂ f₃)
K Q : A
α : Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj K) ⟶ X₃
β : X₃ ⟶ ι.obj Q
γ :
  ι.obj Q ⟶
    Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1)
      (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj K))
hT' :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_2 inst_4 _)
    (Triangle.mk α β γ)
inst_7 : ι.Full
inst_8 : Preadditive.{v_2, u_2} A
inst_9 : ι.Faithful
B : A
k : B ⟶ K
hk : (Functor.map (shiftFunctor C (A := ℤ) 1) (ι.map k) ≫ α) ≫ f₃ = 0
l : Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj B) ⟶ (Triangle.mk (ι.map f₁) f₂ f₃).obj₂
hl : Functor.map (shiftFunctor C (A := ℤ) 1) (ι.map k) ≫ α = 0
a :
  ∀
    (m :
      Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1)
          (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj B)) ⟶
        (Triangle.mk α β γ).obj₃)
    (hm :
      Functor.map (shiftFunctor C (A := ℤ) 1) (Functor.map (shiftFunctor C (A := ℤ) 1) (ι.map k)) =
        m ≫ (Triangle.mk α β γ).mor₃),
    Eq (α :=
      Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj B) ⟶
        Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj K))
      (Functor.map (shiftFunctor C (A := ℤ) 1) (ι.map k)) 0
⊢ Eq (α :=
    Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj B) ⟶
      Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj K))
    (Functor.map (shiftFunctor C (A := ℤ) 1) (ι.map k)) 0

Tactic:
obtain ⟨m, hm⟩ := Triangle.coyoneda_exact₁ _ hT' ((ι.map k)⟦(1 : ℤ)⟧'⟦(1 : ℤ)⟧')
    (by simp [← Functor.map_comp, hl])

Premises used: CategoryTheory.shiftFunctor, CategoryTheory.Functor.map, OfNat.ofNat, CategoryTheory.Pretriangulated.Triangle.coyoneda_exact₁
-- Module: Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory
Goal:
C : Type u_1
A : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : Preadditive.{v_1, u_1} C
inst_3 : HasShift.{v_1, u_1, 0} C ℤ
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
inst_6 : Category.{v_2, u_2} A
ι : Functor.{v_2, v_1, u_2, u_1} A C
hι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C n) (ι.obj Y)), n < 0 → f = 0
X₁ X₂ : A
f₁ : X₁ ⟶ X₂
X₃ : C
f₂ : ι.obj X₂ ⟶ X₃
f₃ : X₃ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj X₁)
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_2 inst_4 _)
    (Triangle.mk (ι.map f₁) f₂ f₃)
K Q : A
α : Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj K) ⟶ X₃
β : X₃ ⟶ ι.obj Q
γ :
  ι.obj Q ⟶
    Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1)
      (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj K))
hT' :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_2 inst_4 _)
    (Triangle.mk α β γ)
inst_7 : ι.Full
inst_8 : Preadditive.{v_2, u_2} A
inst_9 : ι.Faithful
B : A
k : B ⟶ K
hk : (Functor.map (shiftFunctor C (A := ℤ) 1) (ι.map k) ≫ α) ≫ f₃ = 0
l : Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj B) ⟶ (Triangle.mk (ι.map f₁) f₂ f₃).obj₂
hl : Functor.map (shiftFunctor C (A := ℤ) 1) (ι.map k) ≫ α = 0
m :
  Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1)
      (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj B)) ⟶
    (Triangle.mk α β γ).obj₃
hm :
  Functor.map (shiftFunctor C (A := ℤ) 1) (Functor.map (shiftFunctor C (A := ℤ) 1) (ι.map k)) =
    m ≫ (Triangle.mk α β γ).mor₃
⊢ HAdd.hAdd (α := ℤ) 1 1 = 2

Tactic:
lia
-- Module: Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory
Goal:
C : Type u_1
A : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : Preadditive.{v_1, u_1} C
inst_3 : HasShift.{v_1, u_1, 0} C ℤ
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
inst_6 : Category.{v_2, u_2} A
ι : Functor.{v_2, v_1, u_2, u_1} A C
hι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C n) (ι.obj Y)), n < 0 → f = 0
X₁ X₂ : A
f₁ : X₁ ⟶ X₂
X₃ : C
f₂ : ι.obj X₂ ⟶ X₃
f₃ : X₃ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj X₁)
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_2 inst_4 _)
    (Triangle.mk (ι.map f₁) f₂ f₃)
K Q : A
α : Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj K) ⟶ X₃
β : X₃ ⟶ ι.obj Q
γ :
  ι.obj Q ⟶
    Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1)
      (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj K))
hT' :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_2 inst_4 _)
    (Triangle.mk α β γ)
inst_7 : ι.Full
inst_8 : Preadditive.{v_2, u_2} A
inst_9 : ι.Faithful
B : A
k : B ⟶ K
hk : (Functor.map (shiftFunctor C (A := ℤ) 1) (ι.map k) ≫ α) ≫ f₃ = 0
l : Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj B) ⟶ (Triangle.mk (ι.map f₁) f₂ f₃).obj₂
hl : Functor.map (shiftFunctor C (A := ℤ) 1) (ι.map k) ≫ α = 0
m :
  Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1)
      (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj B)) ⟶
    (Triangle.mk α β γ).obj₃
hm :
  Functor.map (shiftFunctor C (A := ℤ) 1) (Functor.map (shiftFunctor C (A := ℤ) 1) (ι.map k)) =
    m ≫ (Triangle.mk α β γ).mor₃
rw : NatTrans.app (Iso.hom (shiftFunctorAdd' C 1 1 2 mono_ιK._proof_2)) (ι.obj B) ≫ m = 0
⊢ m = 0

Tactic:
rw [← cancel_epi ((shiftFunctorAdd' C (1 : ℤ) 1 2 (by lia)).hom.app _), comp_zero]

Premises used: CategoryTheory.Limits.comp_zero, CategoryTheory.shiftFunctorAdd', OfNat.ofNat, CategoryTheory.cancel_epi, CategoryTheory.Iso.hom, CategoryTheory.NatTrans.app
-- Module: Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory
Goal:
C : Type u_1
A : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : Preadditive.{v_1, u_1} C
inst_3 : HasShift.{v_1, u_1, 0} C ℤ
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
inst_6 : Category.{v_2, u_2} A
ι : Functor.{v_2, v_1, u_2, u_1} A C
hι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C n) (ι.obj Y)), n < 0 → f = 0
X₁ X₂ : A
f₁ : X₁ ⟶ X₂
X₃ : C
f₂ : ι.obj X₂ ⟶ X₃
f₃ : X₃ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj X₁)
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_2 inst_4 _)
    (Triangle.mk (ι.map f₁) f₂ f₃)
K Q : A
α : Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj K) ⟶ X₃
β : X₃ ⟶ ι.obj Q
γ :
  ι.obj Q ⟶
    Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1)
      (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj K))
hT' :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_2 inst_4 _)
    (Triangle.mk α β γ)
inst_7 : ι.Full
inst_8 : Preadditive.{v_2, u_2} A
inst_9 : ι.Faithful
B : A
k : B ⟶ K
hk : (Functor.map (shiftFunctor C (A := ℤ) 1) (ι.map k) ≫ α) ≫ f₃ = 0
l : Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj B) ⟶ (Triangle.mk (ι.map f₁) f₂ f₃).obj₂
hl : Functor.map (shiftFunctor C (A := ℤ) 1) (ι.map k) ≫ α = 0
m :
  Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1)
      (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj B)) ⟶
    (Triangle.mk α β γ).obj₃
hm :
  Functor.map (shiftFunctor C (A := ℤ) 1) (Functor.map (shiftFunctor C (A := ℤ) 1) (ι.map k)) =
    m ≫ (Triangle.mk α β γ).mor₃
⊢ LT.lt (α := ℤ) 0 2

Tactic:
lia
-- Module: Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory
Goal:
C : Type u_1
A : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : Preadditive.{v_1, u_1} C
inst_3 : HasShift.{v_1, u_1, 0} C ℤ
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
inst_6 : Category.{v_2, u_2} A
ι : Functor.{v_2, v_1, u_2, u_1} A C
hι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C n) (ι.obj Y)), n < 0 → f = 0
X₁ X₂ : A
f₁ : X₁ ⟶ X₂
X₃ : C
f₂ : ι.obj X₂ ⟶ X₃
f₃ : X₃ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj X₁)
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_2 inst_4 _)
    (Triangle.mk (ι.map f₁) f₂ f₃)
K Q : A
α : Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj K) ⟶ X₃
β : X₃ ⟶ ι.obj Q
γ :
  ι.obj Q ⟶
    Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1)
      (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj K))
hT' :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_2 inst_4 _)
    (Triangle.mk α β γ)
inst_7 : ι.Full
inst_8 : Preadditive.{v_2, u_2} A
inst_9 : ι.Faithful
B : A
k : B ⟶ K
hk : (Functor.map (shiftFunctor C (A := ℤ) 1) (ι.map k) ≫ α) ≫ f₃ = 0
l : Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj B) ⟶ (Triangle.mk (ι.map f₁) f₂ f₃).obj₂
hl : Functor.map (shiftFunctor C (A := ℤ) 1) (ι.map k) ≫ α = 0
m :
  Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1)
      (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj B)) ⟶
    (Triangle.mk α β γ).obj₃
hm :
  Functor.map (shiftFunctor C (A := ℤ) 1) (Functor.map (shiftFunctor C (A := ℤ) 1) (ι.map k)) =
    m ≫ (Triangle.mk α β γ).mor₃
⊢ NatTrans.app (Iso.hom (shiftFunctorAdd' C 1 1 2 mono_ιK._proof_2)) (ι.obj B) ≫ m = 0

Tactic:
exact eq_zero_of_hom_shift_pos hι _ (by lia)

Premises used: CategoryTheory.Triangulated.AbelianSubcategory.eq_zero_of_hom_shift_pos
-- Module: Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory
Goal:
C : Type u_1
A : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : Preadditive.{v_1, u_1} C
inst_3 : HasShift.{v_1, u_1, 0} C ℤ
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
inst_6 : Category.{v_2, u_2} A
ι : Functor.{v_2, v_1, u_2, u_1} A C
hι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C n) (ι.obj Y)), n < 0 → f = 0
X₁ X₂ : A
f₁ : X₁ ⟶ X₂
X₃ : C
f₂ : ι.obj X₂ ⟶ X₃
f₃ : X₃ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj X₁)
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_2 inst_4 _)
    (Triangle.mk (ι.map f₁) f₂ f₃)
K Q : A
α : Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj K) ⟶ X₃
β : X₃ ⟶ ι.obj Q
γ :
  ι.obj Q ⟶
    Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1)
      (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj K))
hT' :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_2 inst_4 _)
    (Triangle.mk α β γ)
inst_7 : ι.Full
inst_8 : Preadditive.{v_2, u_2} A
inst_9 : ι.Faithful
B : A
k : B ⟶ K
hk : (Functor.map (shiftFunctor C (A := ℤ) 1) (ι.map k) ≫ α) ≫ f₃ = 0
l : Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj B) ⟶ (Triangle.mk (ι.map f₁) f₂ f₃).obj₂
hl : Functor.map (shiftFunctor C (A := ℤ) 1) (ι.map k) ≫ α = 0
m :
  Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1)
      (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj B)) ⟶
    (Triangle.mk α β γ).obj₃
hm :
  Functor.map (shiftFunctor C (A := ℤ) 1) (Functor.map (shiftFunctor C (A := ℤ) 1) (ι.map k)) =
    m ≫ (Triangle.mk α β γ).mor₃
⊢ Functor.map (shiftFunctor C (A := ℤ) 1) (Functor.map (shiftFunctor C (A := ℤ) 1) (ι.map k)) =
    0 ≫ (Triangle.mk α β γ).mor₃

Tactic:
obtain rfl : m = 0 := by
    rw [← cancel_epi ((shiftFunctorAdd' C (1 : ℤ) 1 2 (by lia)).hom.app _), comp_zero]
    exact eq_zero_of_hom_shift_pos hι _ (by lia)

Premises used: rfl
-- Module: Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory
Goal:
C : Type u_1
A : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : Preadditive.{v_1, u_1} C
inst_3 : HasShift.{v_1, u_1, 0} C ℤ
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
inst_6 : Category.{v_2, u_2} A
ι : Functor.{v_2, v_1, u_2, u_1} A C
hι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C n) (ι.obj Y)), n < 0 → f = 0
X₁ X₂ : A
f₁ : X₁ ⟶ X₂
X₃ : C
f₂ : ι.obj X₂ ⟶ X₃
f₃ : X₃ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj X₁)
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_2 inst_4 _)
    (Triangle.mk (ι.map f₁) f₂ f₃)
K Q : A
α : Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj K) ⟶ X₃
β : X₃ ⟶ ι.obj Q
γ :
  ι.obj Q ⟶
    Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1)
      (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj K))
hT' :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_2 inst_4 _)
    (Triangle.mk α β γ)
inst_7 : ι.Full
inst_8 : Preadditive.{v_2, u_2} A
inst_9 : ι.Faithful
B : A
k : B ⟶ K
hk : (Functor.map (shiftFunctor C (A := ℤ) 1) (ι.map k) ≫ α) ≫ f₃ = 0
l : Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj B) ⟶ (Triangle.mk (ι.map f₁) f₂ f₃).obj₂
hl : Functor.map (shiftFunctor C (A := ℤ) 1) (ι.map k) ≫ α = 0
hm :
  Functor.map (shiftFunctor C (A := ℤ) 1) (Functor.map (shiftFunctor C (A := ℤ) 1) (ι.map k)) =
    0 ≫ (Triangle.mk α β γ).mor₃
⊢ Eq (α :=
    Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1)
        (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj B)) ⟶
      Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (Triangle.mk α β γ).obj₁)
    (Functor.map (shiftFunctor C (A := ℤ) 1) (Functor.map (shiftFunctor C (A := ℤ) 1) (ι.map k))) 0

Tactic:
rw [zero_comp] at hm

Premises used: CategoryTheory.Limits.zero_comp
-- Module: Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory
Goal:
C : Type u_1
A : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : Preadditive.{v_1, u_1} C
inst_3 : HasShift.{v_1, u_1, 0} C ℤ
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
inst_6 : Category.{v_2, u_2} A
ι : Functor.{v_2, v_1, u_2, u_1} A C
hι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C n) (ι.obj Y)), n < 0 → f = 0
X₁ X₂ : A
f₁ : X₁ ⟶ X₂
X₃ : C
f₂ : ι.obj X₂ ⟶ X₃
f₃ : X₃ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj X₁)
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_2 inst_4 _)
    (Triangle.mk (ι.map f₁) f₂ f₃)
K Q : A
α : Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj K) ⟶ X₃
β : X₃ ⟶ ι.obj Q
γ :
  ι.obj Q ⟶
    Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1)
      (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj K))
hT' :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_2 inst_4 _)
    (Triangle.mk α β γ)
inst_7 : ι.Full
inst_8 : Preadditive.{v_2, u_2} A
inst_9 : ι.Faithful
B : A
k : B ⟶ K
hk : (Functor.map (shiftFunctor C (A := ℤ) 1) (ι.map k) ≫ α) ≫ f₃ = 0
l : Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj B) ⟶ (Triangle.mk (ι.map f₁) f₂ f₃).obj₂
hl : Functor.map (shiftFunctor C (A := ℤ) 1) (ι.map k) ≫ α = 0
hm :
  Eq (α :=
    Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1)
        (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj B)) ⟶
      Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (Triangle.mk α β γ).obj₁)
    (Functor.map (shiftFunctor C (A := ℤ) 1) (Functor.map (shiftFunctor C (A := ℤ) 1) (ι.map k))) 0
rw :
  Eq.{v_1 + 1} (α :=
    Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1)
        (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj B)) ⟶
      Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1)
        (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj K)))
    0 (Functor.map (shiftFunctor C (A := ℤ) 1) 0)
rw₁ rw₂ :
  Eq.{v_1 + 1} (α :=
    Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1)
        (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj B)) ⟶
      Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1)
        (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj K)))
    0 0
⊢ Eq (α :=
    Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1)
        (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj B)) ⟶
      Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1)
        (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj K)))
    (Functor.map (shiftFunctor C (A := ℤ) 1) (Functor.map (shiftFunctor C (A := ℤ) 1) (ι.map k)))
    (Functor.map (shiftFunctor C (A := ℤ) 1) 0)

Tactic:
rw [hm, Functor.map_zero]

Premises used: CategoryTheory.Functor.map_zero