awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.LocalizingSubcategory (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
A B : ObjectProperty.{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
inst_6 : A.IsTriangulated
inst_7 : B.IsTriangulated
inst_8 : B.IsClosedUnderIsomorphisms
h : A.IsVerdierRightLocalizing B
X Y : C
s : X ⟶ Y
hX : A X
hs :
  ∃ (X_1 : C) (f : X_1 ⟶ X) (h : Y ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) X_1) (_ :
    Membership.mem (γ := Set (Triangle.{v_1, u_1} C))
      (@Pretriangulated.distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) (Triangle.mk f s h)),
    B X_1
⊢ ∃ (Z : C) (s' : X ⟶ Z) (b : Y ⟶ Z), A Z ∧ (A ⊓ B).trW s' ∧ s ≫ b = s'
have refine_1 : ∀ (W : C) (a : W ⟶ X) (b : Y ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) W)
  (hT :
    Membership.mem (γ := Set (Triangle.{v_1, u_1} C))
      (@Pretriangulated.distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) (Triangle.mk a s b))
  (hW : B W), ∃ (Z : C) (s' : X ⟶ Z) (b : Y ⟶ Z), A Z ∧ (A ⊓ B).trW s' ∧ s ≫ b = s'
Tactic: obtain ⟨W, a, b, hT, hW⟩ := hs
Cleared hypotheses: hs
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.LocalizingSubcategory (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
A B : ObjectProperty.{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
inst_6 : A.IsTriangulated
inst_7 : B.IsTriangulated
inst_8 : B.IsClosedUnderIsomorphisms
h : A.IsVerdierRightLocalizing B
X Y : C
s : X ⟶ Y
hX : A X
W : C
a : W ⟶ X
b : Y ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) W
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C))
    (@Pretriangulated.distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) (Triangle.mk a s b)
hW : B W
⊢ ∃ (Z : C) (s' : X ⟶ Z) (b : Y ⟶ Z), A Z ∧ (A ⊓ B).trW s' ∧ s ≫ b = s'
have refine_1 : ∀ (W' : C) (c : W ⟶ W') (d : W' ⟶ X) (h₁ : A W') (h₂ : B W') (fac : c ≫ d = a),
  ∃ (Z : C) (s' : X ⟶ Z) (b : Y ⟶ Z), A Z ∧ (A ⊓ B).trW s' ∧ s ≫ b = s'
Tactic: obtain ⟨W', c, d, h₁, h₂, fac⟩ := IsVerdierRightLocalizing.fac a hW hX
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.LocalizingSubcategory (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
A B : ObjectProperty.{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
inst_6 : A.IsTriangulated
inst_7 : B.IsTriangulated
inst_8 : B.IsClosedUnderIsomorphisms
h : A.IsVerdierRightLocalizing B
X Y : C
s : X ⟶ Y
hX : A X
W : C
a : W ⟶ X
b : Y ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) W
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C))
    (@Pretriangulated.distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) (Triangle.mk a s b)
hW : B W
W' : C
c : W ⟶ W'
d : W' ⟶ X
h₁ : A W'
h₂ : B W'
fac : c ≫ d = a
⊢ ∃ (Z : C) (s' : X ⟶ Z) (b : Y ⟶ Z), A Z ∧ (A ⊓ B).trW s' ∧ s ≫ b = s'
have refine_1 : ∀ (U : C) (hU : A U) (e : X ⟶ U) (f : U ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) W')
  (hT' :
    Membership.mem (γ := Set (Triangle.{v_1, u_1} C))
      (@Pretriangulated.distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) (Triangle.mk d e f)),
  ∃ (Z : C) (s' : X ⟶ Z) (b : Y ⟶ Z), A Z ∧ (A ⊓ B).trW s' ∧ s ≫ b = s'
Tactic: obtain ⟨U, hU, e, f, hT'⟩ := A.distinguished_cocone_triangle d h₁ hX
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.LocalizingSubcategory (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
A B : ObjectProperty.{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
inst_6 : A.IsTriangulated
inst_7 : B.IsTriangulated
inst_8 : B.IsClosedUnderIsomorphisms
h : A.IsVerdierRightLocalizing B
X Y : C
s : X ⟶ Y
hX : A X
W : C
a : W ⟶ X
b : Y ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) W
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C))
    (@Pretriangulated.distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) (Triangle.mk a s b)
hW : B W
W' : C
c : W ⟶ W'
d : W' ⟶ X
h₁ : A W'
h₂ : B W'
fac : c ≫ d = a
U : C
hU : A U
e : X ⟶ U
f : U ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) W'
hT' :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C))
    (@Pretriangulated.distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) (Triangle.mk d e f)
⊢ ∃ (Z : C) (s' : X ⟶ Z) (b : Y ⟶ Z), A Z ∧ (A ⊓ B).trW s' ∧ s ≫ b = s'
have refine_1 : ∀ (g : (Triangle.mk a s b).obj₃ ⟶ (Triangle.mk d e f).obj₃)
  (hg : (Triangle.mk a s b).mor₂ ≫ g = 𝟙 (Triangle.mk a s b).obj₂ ≫ (Triangle.mk d e f).mor₂),
  (Triangle.mk a s b).mor₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) c = g ≫ (Triangle.mk d e f).mor₃ →
    ∃ (Z : C) (s' : X ⟶ Z) (b : Y ⟶ Z), A Z ∧ (A ⊓ B).trW s' ∧ s ≫ b = s'
Tactic: obtain ⟨g, hg, _⟩ := complete_distinguished_triangle_morphism _ _ hT hT'
      c (𝟙 _) (by cat_disch)
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.LocalizingSubcategory (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
A B : ObjectProperty.{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
inst_6 : A.IsTriangulated
inst_7 : B.IsTriangulated
inst_8 : B.IsClosedUnderIsomorphisms
h : A.IsVerdierRightLocalizing B
X Y : C
s : X ⟶ Y
hX : A X
W : C
a : W ⟶ X
b : Y ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) W
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C))
    (@Pretriangulated.distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) (Triangle.mk a s b)
hW : B W
W' : C
c : W ⟶ W'
d : W' ⟶ X
h₁ : A W'
h₂ : B W'
fac : c ≫ d = a
U : C
hU : A U
e : X ⟶ U
f : U ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) W'
hT' :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C))
    (@Pretriangulated.distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) (Triangle.mk d e f)
g : (Triangle.mk a s b).obj₃ ⟶ (Triangle.mk d e f).obj₃
hg : (Triangle.mk a s b).mor₂ ≫ g = 𝟙 (Triangle.mk a s b).obj₂ ≫ (Triangle.mk d e f).mor₂
right : (Triangle.mk a s b).mor₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) c = g ≫ (Triangle.mk d e f).mor₃
⊢ ∃ (Z : C) (s' : X ⟶ Z) (b : Y ⟶ Z), A Z ∧ (A ⊓ B).trW s' ∧ s ≫ b = s'
have refine_1 : (A ⊓ B).trW e
Tactic: refine ⟨U, e, g, hU, ?_, by cat_disch⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.LocalizingSubcategory (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
A B : ObjectProperty.{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
inst_6 : A.IsTriangulated
inst_7 : B.IsTriangulated
inst_8 : B.IsClosedUnderIsomorphisms
h : A.IsVerdierRightLocalizing B
X Y : C
s : X ⟶ Y
hX : A X
W : C
a : W ⟶ X
b : Y ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) W
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C))
    (@Pretriangulated.distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) (Triangle.mk a s b)
hW : B W
W' : C
c : W ⟶ W'
d : W' ⟶ X
h₁ : A W'
h₂ : B W'
fac : c ≫ d = a
U : C
hU : A U
e : X ⟶ U
f : U ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) W'
hT' :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C))
    (@Pretriangulated.distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) (Triangle.mk d e f)
g : (Triangle.mk a s b).obj₃ ⟶ (Triangle.mk d e f).obj₃
hg : (Triangle.mk a s b).mor₂ ≫ g = 𝟙 (Triangle.mk a s b).obj₂ ≫ (Triangle.mk d e f).mor₂
right : (Triangle.mk a s b).mor₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) c = g ≫ (Triangle.mk d e f).mor₃
⊢ (A ⊓ B).trW e
have refine_1 : ∃ (X_1 : C) (f : X_1 ⟶ X) (h : U ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) X_1) (_ :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C))
    (@Pretriangulated.distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) (Triangle.mk f e h)),
  (A ⊓ B) X_1
Tactic: rw [ObjectProperty.trW_iff']
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.LocalizingSubcategory (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
A B : ObjectProperty.{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
inst_6 : A.IsTriangulated
inst_7 : B.IsTriangulated
inst_8 : B.IsClosedUnderIsomorphisms
hA : ∀ ⦃X Y : C⦄ (s : X ⟶ Y), A X → B.trW s → ∃ (Z : C) (s' : X ⟶ Z) (b : Y ⟶ Z), A Z ∧ (A ⊓ B).trW s' ∧ s ≫ b = s'
X Y : C
f : X ⟶ Y
hX : B X
hY : A Y
⊢ ∃ (Z : C) (a : X ⟶ Z) (b : Z ⟶ Y), A Z ∧ B Z ∧ a ≫ b = f
have refine_2 : ∀ (Z : C) (s : 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))
      (@Pretriangulated.distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) (Triangle.mk f s b)),
  ∃ (Z : C) (a : X ⟶ Z) (b : Z ⟶ Y), A Z ∧ B Z ∧ a ≫ b = f
Tactic: obtain ⟨Z, s, b, hT⟩ := Pretriangulated.distinguished_cocone_triangle f
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.LocalizingSubcategory (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
A B : ObjectProperty.{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
inst_6 : A.IsTriangulated
inst_7 : B.IsTriangulated
inst_8 : B.IsClosedUnderIsomorphisms
hA : ∀ ⦃X Y : C⦄ (s : X ⟶ Y), A X → B.trW s → ∃ (Z : C) (s' : X ⟶ Z) (b : Y ⟶ Z), A Z ∧ (A ⊓ B).trW s' ∧ s ≫ b = s'
X Y : C
f : X ⟶ Y
hX : B X
hY : A Y
Z : C
s : 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))
    (@Pretriangulated.distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) (Triangle.mk f s b)
⊢ B.trW s
have rw : ∃ (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))
    (@Pretriangulated.distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) (Triangle.mk f s h)),
  B X
Tactic: rw [trW_iff']
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.LocalizingSubcategory (hypothesis)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
A B : ObjectProperty.{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
inst_6 : A.IsTriangulated
inst_7 : B.IsTriangulated
inst_8 : B.IsClosedUnderIsomorphisms
hA : ∀ ⦃X Y : C⦄ (s : X ⟶ Y), A X → B.trW s → ∃ (Z : C) (s' : X ⟶ Z) (b : Y ⟶ Z), A Z ∧ (A ⊓ B).trW s' ∧ s ≫ b = s'
X Y : C
f : X ⟶ Y
hX : B X
hY : A Y
Z : C
s : 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))
    (@Pretriangulated.distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) (Triangle.mk f s b)
⊢ ∃ (Z : C) (a : X ⟶ Z) (b : Z ⟶ Y), A Z ∧ B Z ∧ a ≫ b = f
have hs : B.trW s
Tactic: have hs : B.trW s := by
      rw [trW_iff']
      exact ⟨_, _, _, hT, hX⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.LocalizingSubcategory (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
A B : ObjectProperty.{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
inst_6 : A.IsTriangulated
inst_7 : B.IsTriangulated
inst_8 : B.IsClosedUnderIsomorphisms
hA : ∀ ⦃X Y : C⦄ (s : X ⟶ Y), A X → B.trW s → ∃ (Z : C) (s' : X ⟶ Z) (b : Y ⟶ Z), A Z ∧ (A ⊓ B).trW s' ∧ s ≫ b = s'
X Y : C
f : X ⟶ Y
hX : B X
hY : A Y
Z : C
s : 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))
    (@Pretriangulated.distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) (Triangle.mk f s b)
hs : B.trW s
⊢ ∃ (Z : C) (a : X ⟶ Z) (b : Z ⟶ Y), A Z ∧ B Z ∧ a ≫ b = f
have refine_2 : ∀ (W : C) (s' : Y ⟶ W) (g : Z ⟶ W) (hW : A W) (hs' : (A ⊓ B).trW s') (fac : s ≫ g = s'),
  ∃ (Z : C) (a : X ⟶ Z) (b : Z ⟶ Y), A Z ∧ B Z ∧ a ≫ b = f
Tactic: obtain ⟨W, s', g, hW, hs', fac⟩ := hA s hY hs
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.LocalizingSubcategory (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
A B : ObjectProperty.{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
inst_6 : A.IsTriangulated
inst_7 : B.IsTriangulated
inst_8 : B.IsClosedUnderIsomorphisms
hA : ∀ ⦃X Y : C⦄ (s : X ⟶ Y), A X → B.trW s → ∃ (Z : C) (s' : X ⟶ Z) (b : Y ⟶ Z), A Z ∧ (A ⊓ B).trW s' ∧ s ≫ b = s'
X Y : C
f : X ⟶ Y
hX : B X
hY : A Y
Z : C
s : 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))
    (@Pretriangulated.distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) (Triangle.mk f s b)
hs : B.trW s
W : C
s' : Y ⟶ W
g : Z ⟶ W
hW : A W
hs' : (A ⊓ B).trW s'
fac : s ≫ g = s'
⊢ ∃ (Z : C) (a : X ⟶ Z) (b : Z ⟶ Y), A Z ∧ B Z ∧ a ≫ b = f
have refine_2 : ∀ (U : C) (hU : A U) (a : U ⟶ Y) (c : W ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) U)
  (hT' :
    Membership.mem (γ := Set (Triangle.{v_1, u_1} C))
      (@Pretriangulated.distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) (Triangle.mk a s' c)),
  ∃ (Z : C) (a : X ⟶ Z) (b : Z ⟶ Y), A Z ∧ B Z ∧ a ≫ b = f
Tactic: obtain ⟨U, hU, a, c, hT'⟩ := A.distinguished_cocone_triangle₁ s' hY hW
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.LocalizingSubcategory (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
A B : ObjectProperty.{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
inst_6 : A.IsTriangulated
inst_7 : B.IsTriangulated
inst_8 : B.IsClosedUnderIsomorphisms
hA : ∀ ⦃X Y : C⦄ (s : X ⟶ Y), A X → B.trW s → ∃ (Z : C) (s' : X ⟶ Z) (b : Y ⟶ Z), A Z ∧ (A ⊓ B).trW s' ∧ s ≫ b = s'
X Y : C
f : X ⟶ Y
hX : B X
hY : A Y
Z : C
s : 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))
    (@Pretriangulated.distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) (Triangle.mk f s b)
hs : B.trW s
W : C
s' : Y ⟶ W
g : Z ⟶ W
hW : A W
hs' : (A ⊓ B).trW s'
fac : s ≫ g = s'
U : C
hU : A U
a : U ⟶ Y
c : W ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) U
hT' :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C))
    (@Pretriangulated.distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) (Triangle.mk a s' c)
⊢ ∃ (Z : C) (a : X ⟶ Z) (b : Z ⟶ Y), A Z ∧ B Z ∧ a ≫ b = f
have refine_2 : ∀ (t : (Triangle.mk f s b).obj₁ ⟶ (Triangle.mk a s' c).obj₁)
  (ht : (Triangle.mk f s b).mor₁ ≫ 𝟙 Y = t ≫ (Triangle.mk a s' c).mor₁)
  (ht' : (Triangle.mk f s b).mor₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) t = g ≫ (Triangle.mk a s' c).mor₃),
  ∃ (Z : C) (a : X ⟶ Z) (b : Z ⟶ Y), A Z ∧ B Z ∧ a ≫ b = f
Tactic: obtain ⟨t, ht, ht'⟩ :=
      complete_distinguished_triangle_morphism₁ _ _ hT hT' (𝟙 Y) g (by cat_disch)
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.LocalizingSubcategory (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
A B : ObjectProperty.{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
inst_6 : A.IsTriangulated
inst_7 : B.IsTriangulated
inst_8 : B.IsClosedUnderIsomorphisms
⊢ A.IsVerdierLeftLocalizing B ↔
    ∀ ⦃X Y : C⦄ (s : X ⟶ Y), A Y → B.trW s → ∃ (Z : C) (s' : Z ⟶ Y) (a : Z ⟶ X), A Z ∧ (A ⊓ B).trW s' ∧ a ≫ s = s'
have rw : (∀ ⦃X Y : Cᵒᵖ⦄ (s : X ⟶ Y),
    A.op X → B.op.trW s → ∃ (Z : Cᵒᵖ) (s' : X ⟶ Z) (b : Y ⟶ Z), A.op Z ∧ (A.op ⊓ B.op).trW s' ∧ s ≫ b = s') ↔
  ∀ ⦃X Y : C⦄ (s : X ⟶ Y), A Y → B.trW s → ∃ (Z : C) (s' : Z ⟶ Y) (a : Z ⟶ X), A Z ∧ (A ⊓ B).trW s' ∧ a ≫ s = s'
Tactic: rw [← isVerdierRightLocalizing_op_iff, isVerdierRightLocalizing_iff]
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.LocalizingSubcategory (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
A B : ObjectProperty.{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
inst_6 : A.IsTriangulated
inst_7 : B.IsTriangulated
inst_8 : B.IsClosedUnderIsomorphisms
⊢ (∀ ⦃X Y : Cᵒᵖ⦄ (s : X ⟶ Y),
      A.op X → B.op.trW s → ∃ (Z : Cᵒᵖ) (s' : X ⟶ Z) (b : Y ⟶ Z), A.op Z ∧ (A.op ⊓ B.op).trW s' ∧ s ≫ b = s') ↔
    ∀ ⦃X Y : C⦄ (s : X ⟶ Y), A Y → B.trW s → ∃ (Z : C) (s' : Z ⟶ Y) (a : Z ⟶ X), A Z ∧ (A ⊓ B).trW s' ∧ a ≫ s = s'
have refine_1 : ∀
  (hA :
    ∀ ⦃X Y : Cᵒᵖ⦄ (s : X ⟶ Y),
      A.op X → B.op.trW s → ∃ (Z : Cᵒᵖ) (s' : X ⟶ Z) (b : Y ⟶ Z), A.op Z ∧ (A.op ⊓ B.op).trW s' ∧ s ≫ b = s')
  (X Y : C) (s : X ⟶ Y) (hY : A Y) (hs : B.trW s), ∃ (Z : C) (s' : Z ⟶ Y) (a : Z ⟶ X), A Z ∧ (A ⊓ B).trW s' ∧ a ≫ s = s'
Tactic: refine ⟨fun hA X Y s hY hs ↦ ?_, fun hA X Y s hX hs ↦ ?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.LocalizingSubcategory (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
A B : ObjectProperty.{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
inst_6 : A.IsTriangulated
inst_7 : B.IsTriangulated
inst_8 : B.IsClosedUnderIsomorphisms
refine_1 :
  ∀
    (hA :
      ∀ ⦃X Y : Cᵒᵖ⦄ (s : X ⟶ Y),
        A.op X → B.op.trW s → ∃ (Z : Cᵒᵖ) (s' : X ⟶ Z) (b : Y ⟶ Z), A.op Z ∧ (A.op ⊓ B.op).trW s' ∧ s ≫ b = s')
    (X Y : C) (s : X ⟶ Y) (hY : A Y) (hs : B.trW s),
    ∃ (Z : C) (s' : Z ⟶ Y) (a : Z ⟶ X), A Z ∧ (A ⊓ B).trW s' ∧ a ≫ s = s'
⊢ (∀ ⦃X Y : Cᵒᵖ⦄ (s : X ⟶ Y),
      A.op X → B.op.trW s → ∃ (Z : Cᵒᵖ) (s' : X ⟶ Z) (b : Y ⟶ Z), A.op Z ∧ (A.op ⊓ B.op).trW s' ∧ s ≫ b = s') ↔
    ∀ ⦃X Y : C⦄ (s : X ⟶ Y), A Y → B.trW s → ∃ (Z : C) (s' : Z ⟶ Y) (a : Z ⟶ X), A Z ∧ (A ⊓ B).trW s' ∧ a ≫ s = s'
have refine_2 : ∀ (hA : ∀ ⦃X Y : C⦄ (s : X ⟶ Y), A Y → B.trW s → ∃ (Z : C) (s' : Z ⟶ Y) (a : Z ⟶ X), A Z ∧ (A ⊓ B).trW s' ∧ a ≫ s = s')
  (X Y : Cᵒᵖ) (s : X ⟶ Y) (hX : A.op X) (hs : B.op.trW s),
  ∃ (Z : Cᵒᵖ) (s' : X ⟶ Z) (b : Y ⟶ Z), A.op Z ∧ (A.op ⊓ B.op).trW s' ∧ s ≫ b = s'
Tactic: refine ⟨fun hA X Y s hY hs ↦ ?_, fun hA X Y s hX hs ↦ ?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.LocalizingSubcategory (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
A B : ObjectProperty.{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
inst_6 : A.IsTriangulated
inst_7 : B.IsTriangulated
inst_8 : B.IsClosedUnderIsomorphisms
hA :
  ∀ ⦃X Y : Cᵒᵖ⦄ (s : X ⟶ Y),
    A.op X → B.op.trW s → ∃ (Z : Cᵒᵖ) (s' : X ⟶ Z) (b : Y ⟶ Z), A.op Z ∧ (A.op ⊓ B.op).trW s' ∧ s ≫ b = s'
X Y : C
s : X ⟶ Y
hY : A Y
hs : B.trW s
⊢ ∃ (Z : C) (s' : Z ⟶ Y) (a : Z ⟶ X), A Z ∧ (A ⊓ B).trW s' ∧ a ≫ s = s'
have refine_1 : ∀ (Z' : Cᵒᵖ) (s' : op Y ⟶ Z') (b : op X ⟶ Z') (hZ' : A.op Z') (hs' : (A.op ⊓ B.op).trW s') (fac : s.op ≫ b = s'),
  ∃ (Z : C) (s' : Z ⟶ Y) (a : Z ⟶ X), A Z ∧ (A ⊓ B).trW s' ∧ a ≫ s = s'
Tactic: obtain ⟨Z', s', b, hZ', hs', fac⟩ := hA s.op hY (by simpa [trW_op_iff])
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.LocalizingSubcategory (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
A B : ObjectProperty.{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
inst_6 : A.IsTriangulated
inst_7 : B.IsTriangulated
inst_8 : B.IsClosedUnderIsomorphisms
hA : ∀ ⦃X Y : C⦄ (s : X ⟶ Y), A Y → B.trW s → ∃ (Z : C) (s' : Z ⟶ Y) (a : Z ⟶ X), A Z ∧ (A ⊓ B).trW s' ∧ a ≫ s = s'
X Y : Cᵒᵖ
s : X ⟶ Y
hX : A.op X
hs : B.op.trW s
⊢ ∃ (Z : Cᵒᵖ) (s' : X ⟶ Z) (b : Y ⟶ Z), A.op Z ∧ (A.op ⊓ B.op).trW s' ∧ s ≫ b = s'
have refine_2 : ∀ (Z' : C) (s' : Z' ⟶ unop X) (b : Z' ⟶ unop Y) (hZ' : A Z') (hs' : (A ⊓ B).trW s') (fac : b ≫ s.unop = s'),
  ∃ (Z : Cᵒᵖ) (s' : X ⟶ Z) (b : Y ⟶ Z), A.op Z ∧ (A.op ⊓ B.op).trW s' ∧ s ≫ b = s'
Tactic: obtain ⟨Z', s', b, hZ', hs', fac⟩ := hA s.unop hX (trW_of_op _ hs)
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.LocalizingSubcategory (hypothesis)
Goal:
C : Type u_1
D : Type u_2
D₁ : Type u_3
D₂ : Type u_4
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
inst_2 : Category.{v_3, u_3} D₁
inst_3 : Category.{v_4, u_4} D₂
A B : ObjectProperty.{v_1, u_1} C
inst_4 : HasZeroObject.{v_1, u_1} C
inst_5 : HasShift.{v_1, u_1, 0} C ℤ
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 : A.IsTriangulated
X Y : A.FullSubcategory
f : X ⟶ Y
hf : (B.inverseImage A.ι).trW f
⊢ B.trW.inverseImage A.ι f
have hf₁ : ∃ (Z : A.FullSubcategory) (g : Y ⟶ Z) (h :
  Z ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor A.FullSubcategory (A := ℤ) 1) X) (_ :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} A.FullSubcategory))
    (@Pretriangulated.distinguishedTriangles _ _ sorry _ (Preadditive.fullSubcategory A) sorry _) (Triangle.mk f g h)),
  B.inverseImage A.ι Z
Tactic: simp only [MorphismProperty.inverseImage_iff, trW_iff] at hf ⊢
Cleared hypotheses: hf