awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Subcategory (goal)
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
n : ℕ
H : P.retractClosure.extensionProductIter n ≤ (P.extensionProductIter n).retractClosure
⊢ P.retractClosure.extensionProductIter (n + 1) ≤ (P.extensionProductIter (n + 1)).retractClosure
have succ : P.retractClosure.extensionProduct (P.retractClosure.extensionProductIter n) ≤
  (P.extensionProductIter (n + 1)).retractClosure
Tactic: grw [extensionProductIter_succ, extensionProductIter_succ, monotone_extensionProduct_right _ H,
      extensionProduct_retractClosure_retractClosure_le]
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Subcategory (goal)
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
n : ℕ
H : P.retractClosure.extensionProductIter n ≤ (P.extensionProductIter n).retractClosure
succ :
  P.retractClosure.extensionProduct (P.retractClosure.extensionProductIter n) ≤
    (P.extensionProductIter (n + 1)).retractClosure
⊢ P.retractClosure.extensionProductIter (n + 1) ≤ (P.extensionProductIter (n + 1)).retractClosure
have succ₁ : P.retractClosure.extensionProduct (P.retractClosure.extensionProductIter n) ≤
  (P.extensionProduct (P.extensionProductIter n)).retractClosure
Tactic: grw [extensionProductIter_succ, extensionProductIter_succ, monotone_extensionProduct_right _ H,
      extensionProduct_retractClosure_retractClosure_le]
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Subcategory (goal)
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
n : ℕ
H : P.retractClosure.extensionProductIter n ≤ (P.extensionProductIter n).retractClosure
succ :
  P.retractClosure.extensionProduct (P.retractClosure.extensionProductIter n) ≤
    (P.extensionProductIter (n + 1)).retractClosure
succ₁ :
  P.retractClosure.extensionProduct (P.retractClosure.extensionProductIter n) ≤
    (P.extensionProduct (P.extensionProductIter n)).retractClosure
⊢ P.retractClosure.extensionProductIter (n + 1) ≤ (P.extensionProductIter (n + 1)).retractClosure
have succ₂ : P.retractClosure.extensionProduct (P.extensionProductIter n).retractClosure ≤
  (P.extensionProduct (P.extensionProductIter n)).retractClosure
Tactic: grw [extensionProductIter_succ, extensionProductIter_succ, monotone_extensionProduct_right _ H,
      extensionProduct_retractClosure_retractClosure_le]
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Subcategory (goal)
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
n : ℕ
H : P.retractClosure.extensionProductIter n ≤ (P.extensionProductIter n).retractClosure
succ :
  P.retractClosure.extensionProduct (P.retractClosure.extensionProductIter n) ≤
    (P.extensionProductIter (n + 1)).retractClosure
succ₁ :
  P.retractClosure.extensionProduct (P.retractClosure.extensionProductIter n) ≤
    (P.extensionProduct (P.extensionProductIter n)).retractClosure
succ₂ :
  P.retractClosure.extensionProduct (P.extensionProductIter n).retractClosure ≤
    (P.extensionProduct (P.extensionProductIter n)).retractClosure
⊢ P.retractClosure.extensionProductIter (n + 1) ≤ (P.extensionProductIter (n + 1)).retractClosure
have succ₃ : (P.extensionProduct (P.extensionProductIter n)).retractClosure ≤
  (P.extensionProduct (P.extensionProductIter n)).retractClosure
Tactic: grw [extensionProductIter_succ, extensionProductIter_succ, monotone_extensionProduct_right _ H,
      extensionProduct_retractClosure_retractClosure_le]
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Subcategory (goal)
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
n : ℕ
⊢ P.retractClosure.extensionProductIter n ≤ (P.extensionProductIter n).retractClosure
have zero : P.retractClosure.extensionProductIter 0 ≤ (P.extensionProductIter 0).retractClosure
Tactic: induction n with
  | zero => simp
  | succ n H =>
    grw [extensionProductIter_succ, extensionProductIter_succ, monotone_extensionProduct_right _ H,
      extensionProduct_retractClosure_retractClosure_le]
Cleared hypotheses: n
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Subcategory (goal)
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
n : ℕ
zero : P.retractClosure.extensionProductIter 0 ≤ (P.extensionProductIter 0).retractClosure
⊢ P.retractClosure.extensionProductIter n ≤ (P.extensionProductIter n).retractClosure
have succ : ∀ (n : ℕ) (H : P.retractClosure.extensionProductIter n ≤ (P.extensionProductIter n).retractClosure),
  P.retractClosure.extensionProductIter (n + 1) ≤ (P.extensionProductIter (n + 1)).retractClosure
Tactic: induction n with
  | zero => simp
  | succ n H =>
    grw [extensionProductIter_succ, extensionProductIter_succ, monotone_extensionProduct_right _ H,
      extensionProduct_retractClosure_retractClosure_le]
Cleared hypotheses: n
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Subcategory (goal)
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
n : ℕ
⊢ (P.retractClosure.extensionProductIter n).retractClosure = (P.extensionProductIter n).retractClosure
have a : (P.retractClosure.extensionProductIter n).retractClosure ≤ (P.extensionProductIter n).retractClosure
Tactic: apply le_antisymm
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Subcategory (goal)
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
n : ℕ
a : (P.retractClosure.extensionProductIter n).retractClosure ≤ (P.extensionProductIter n).retractClosure
⊢ (P.retractClosure.extensionProductIter n).retractClosure = (P.extensionProductIter n).retractClosure
have a₁ : (P.extensionProductIter n).retractClosure ≤ (P.retractClosure.extensionProductIter n).retractClosure
Tactic: apply le_antisymm
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Subcategory (goal)
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
n : ℕ
⊢ (P.retractClosure.extensionProductIter n).retractClosure ≤ (P.extensionProductIter n).retractClosure
have a : P.retractClosure.extensionProductIter n ≤ (P.extensionProductIter n).retractClosure
Tactic: rw [retractClosure_le_iff]
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Subcategory (goal)
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
Y Z : C
g : Y ⟶ Z
⊢ (∃ (Z_1 : C) (g_1 : Z ⟶ Z_1) (h : Z_1 ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) Y) (_ :
      Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
        (Triangle.mk g g_1 h)),
      P Z_1) ↔
    ∃ (X : C) (f : X ⟶ Y) (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)),
      P X
have mp : (∃ (Z_1 : C) (g_1 : Z ⟶ Z_1) (h : Z_1 ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) Y) (_ :
    Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
      (Triangle.mk g g_1 h)),
    P Z_1) →
  ∃ (X : C) (f : X ⟶ Y) (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)),
    P X
Tactic: constructor
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Subcategory (goal)
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
Y Z : C
g : Y ⟶ Z
mp :
  (∃ (Z_1 : C) (g_1 : Z ⟶ Z_1) (h : Z_1 ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) Y) (_ :
      Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
        (Triangle.mk g g_1 h)),
      P Z_1) →
    ∃ (X : C) (f : X ⟶ Y) (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)),
      P X
⊢ (∃ (Z_1 : C) (g_1 : Z ⟶ Z_1) (h : Z_1 ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) Y) (_ :
      Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
        (Triangle.mk g g_1 h)),
      P Z_1) ↔
    ∃ (X : C) (f : X ⟶ Y) (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)),
      P X
have mpr : (∃ (X : C) (f : X ⟶ Y) (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)),
    P X) →
  ∃ (Z_1 : C) (g_1 : Z ⟶ Z_1) (h : Z_1 ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) Y) (_ :
    Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
      (Triangle.mk g g_1 h)),
    P Z_1
Tactic: constructor
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Subcategory (goal)
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
T : Triangle.{v_1, u_1} C
hT : Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) T
h : P T.obj₁
⊢ P.trW T.mor₂
have rw : ∃ (X : C) (f : X ⟶ T.obj₂) (h : T.obj₃ ⟶ 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 T.mor₂ h)),
  P X
Tactic: rw [trW_iff']
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Subcategory (goal)
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
⊢ P.isoClosure.trW = P.trW
have h : ∀ ⦃X Y : C⦄ (f : X ⟶ Y), P.isoClosure.trW f ↔ P.trW f
Tactic: ext X Y f
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Subcategory (goal)
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
⊢ P.isoClosure.trW f ↔ P.trW f
have h : P.isoClosure.trW f → P.trW f
Tactic: constructor
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Subcategory (goal)
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
h : P.isoClosure.trW f → P.trW f
⊢ P.isoClosure.trW f ↔ P.trW f
have h₁ : P.trW f → P.isoClosure.trW f
Tactic: constructor
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Subcategory (goal)
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
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)
Z' : C
hZ' : P Z'
e : Z ≅ Z'
⊢ P.trW f
have h₁ : Triangle.mk f (g ≫ e.hom) (e.inv ≫ h) ≅ Triangle.mk f g h
Tactic: refine ⟨Z', g ≫ e.hom, e.inv ≫ h, isomorphic_distinguished _ mem _ ?_, hZ'⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Subcategory (hypothesis)
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
hf : P.trW f
⊢ Q.trW f
have hf₁ : ∃ (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)),
  P Z
Tactic: rw [trW_iff] at hf ⊢
Cleared hypotheses: hf
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Subcategory (goal)
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
hf :
  ∃ (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)),
    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
have obtain : ∀ (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: obtain ⟨Z, a, b, hT, hZ⟩ := hf
Cleared hypotheses: hf
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Subcategory (goal)
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)
have refine : Triangle.mk (e ≫ f) g (h ≫ Functor.map (shiftFunctor C (A := ℤ) 1) (inv e)) ≅ Triangle.mk f g h
Tactic: refine ⟨Z, g, h ≫ inv e⟦(1 : ℤ)⟧', isomorphic_distinguished _ mem _ ?_, mem'⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Subcategory (goal)
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)) ≅ Triangle.mk f g h
have 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
Tactic: refine Triangle.isoMk _ _ (asIso e) (Iso.refl _) (Iso.refl _) (by simp) (by simp) ?_
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Subcategory (goal)
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))).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
have dsimp : (h ≫ Functor.map (shiftFunctor C (A := ℤ) 1) (inv e)) ≫ Functor.map (shiftFunctor C (A := ℤ) 1) e = 𝟙 Z ≫ h
Tactic: dsimp
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Subcategory (goal)
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)
have refine : Triangle.mk (f ≫ e) (inv e ≫ g) h ≅ Triangle.mk f g h
Tactic: refine ⟨Z, inv e ≫ g, h, isomorphic_distinguished _ mem _ ?_, mem'⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Subcategory (goal)
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.trW.ContainsIdentities
have rw : P.isoClosure.trW.ContainsIdentities
Tactic: rw [← trW_isoClosure]
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Subcategory (goal)
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
have refine : Arrow.mk (𝟙 X) ≅ Arrow.mk f
Tactic: refine (P.trW.arrow_mk_iso_iff ?_).1 (MorphismProperty.id_mem _ X)