awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.LocalizingSubcategory (goal)
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 : IsTriangulated.{v_1, u_1} C
inst_10 : A.IsTriangulated
inst_11 : B.IsTriangulated
inst_12 : B.IsClosedUnderIsomorphisms
inst_13 : A.IsVerdierRightLocalizing B
L₁ : Functor.{v_1, v_3, u_1, u_3} A.FullSubcategory D₁
L₂ : Functor.{v_1, v_4, u_1, u_4} C D₂
inst_14 : L₁.IsLocalization (B.inverseImage A.ι).trW
inst_15 : L₂.IsLocalization B.trW
F : Functor.{v_3, v_4, u_3, u_4} D₁ D₂ :=
  LocalizerMorphism.localizedFunctor (W₁ := (B.inverseImage A.ι).trW) (W₂ := B.trW) (A.triangulatedLocalizerMorphism B)
    L₁ L₂
this : L₁.EssSurj
e : A.ι ⋙ L₂ ≅ L₁ ⋙ F :=
  CatCommSq.iso
    (LocalizerMorphism.functor (W₁ := (B.inverseImage A.ι).trW) (W₂ := B.trW) (A.triangulatedLocalizerMorphism B)) L₁ L₂
    F
X₁ X₂ : A.FullSubcategory
φ : F.obj (L₁.obj X₁) ⟶ F.obj (L₁.obj X₂)
φ' : (A.ι ⋙ L₂).obj X₁ ⟶ (A.ι ⋙ L₂).obj X₂
hφ' : φ = e.inv.app X₁ ≫ φ' ≫ e.hom.app X₂
f : B.trW.LeftFraction (A.ι.obj X₁) (A.ι.obj X₂)
hf : φ' = f.map (W := B.trW) L₂ ⋯
X₃ : C
s' : A.ι.obj X₂ ⟶ X₃
a : f.Y' (W := B.trW) ⟶ X₃
hX₃ : A X₃
hs' : (A ⊓ B).trW s'
fac : f.s (W := B.trW) ≫ a = s'
⊢ (B.inverseImage A.ι).trW (X := X₂) (Y := FullSubcategory.mk (P := A) X₃ hX₃) (homMk (P := A) s')
have rwa : (A ⊓ B).trW (X := X₂.obj (P := A)) (Y := (FullSubcategory.mk (P := A) X₃ hX₃).obj (P := A))
  (InducedCategory.Hom.hom (homMk (P := A) s'))
Tactic: rwa [trW_inverseImage_ι_iff]
-- 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 : IsTriangulated.{v_1, u_1} C
inst_10 : A.IsTriangulated
inst_11 : B.IsTriangulated
inst_12 : B.IsClosedUnderIsomorphisms
inst_13 : A.IsVerdierRightLocalizing B
L₁ : Functor.{v_1, v_3, u_1, u_3} A.FullSubcategory D₁
L₂ : Functor.{v_1, v_4, u_1, u_4} C D₂
inst_14 : L₁.IsLocalization (B.inverseImage A.ι).trW
inst_15 : L₂.IsLocalization B.trW
F : Functor.{v_3, v_4, u_3, u_4} D₁ D₂ :=
  LocalizerMorphism.localizedFunctor (W₁ := (B.inverseImage A.ι).trW) (W₂ := B.trW) (A.triangulatedLocalizerMorphism B)
    L₁ L₂
this : L₁.EssSurj
e : A.ι ⋙ L₂ ≅ L₁ ⋙ F :=
  CatCommSq.iso
    (LocalizerMorphism.functor (W₁ := (B.inverseImage A.ι).trW) (W₂ := B.trW) (A.triangulatedLocalizerMorphism B)) L₁ L₂
    F
X₁ X₂ : A.FullSubcategory
φ : F.obj (L₁.obj X₁) ⟶ F.obj (L₁.obj X₂)
φ' : (A.ι ⋙ L₂).obj X₁ ⟶ (A.ι ⋙ L₂).obj X₂
hφ' : φ = e.inv.app X₁ ≫ φ' ≫ e.hom.app X₂
f : B.trW.LeftFraction (A.ι.obj X₁) (A.ι.obj X₂)
hf : φ' = f.map (W := B.trW) L₂ ⋯
X₃ : C
s' : A.ι.obj X₂ ⟶ X₃
a : f.Y' (W := B.trW) ⟶ X₃
hX₃ : A X₃
hs' : (A ⊓ B).trW s'
fac : f.s (W := B.trW) ≫ a = s'
g : (B.inverseImage A.ι).trW.LeftFraction X₁ X₂ :=
  MorphismProperty.LeftFraction.mk (W := (B.inverseImage A.ι).trW) (homMk (P := A) (f.f (W := B.trW) ≫ a))
    (homMk (P := A) s') ⋯
⊢ ∃ (f : L₁.obj X₁ ⟶ L₁.obj X₂), F.map f = φ
have this₁ : IsIso (L₁.map (g.s (W := (B.inverseImage A.ι).trW)))
Tactic: have := Localization.inverts L₁ _ _ g.hs
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.LocalizingSubcategory (goal)
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 : IsTriangulated.{v_1, u_1} C
inst_10 : A.IsTriangulated
inst_11 : B.IsTriangulated
inst_12 : B.IsClosedUnderIsomorphisms
inst_13 : A.IsVerdierRightLocalizing B
L₁ : Functor.{v_1, v_3, u_1, u_3} A.FullSubcategory D₁
L₂ : Functor.{v_1, v_4, u_1, u_4} C D₂
inst_14 : L₁.IsLocalization (B.inverseImage A.ι).trW
inst_15 : L₂.IsLocalization B.trW
F : Functor.{v_3, v_4, u_3, u_4} D₁ D₂ :=
  LocalizerMorphism.localizedFunctor (W₁ := (B.inverseImage A.ι).trW) (W₂ := B.trW) (A.triangulatedLocalizerMorphism B)
    L₁ L₂
this_1 : L₁.EssSurj
e : A.ι ⋙ L₂ ≅ L₁ ⋙ F :=
  CatCommSq.iso
    (LocalizerMorphism.functor (W₁ := (B.inverseImage A.ι).trW) (W₂ := B.trW) (A.triangulatedLocalizerMorphism B)) L₁ L₂
    F
X₁ X₂ : A.FullSubcategory
φ : F.obj (L₁.obj X₁) ⟶ F.obj (L₁.obj X₂)
φ' : (A.ι ⋙ L₂).obj X₁ ⟶ (A.ι ⋙ L₂).obj X₂
hφ' : φ = e.inv.app X₁ ≫ φ' ≫ e.hom.app X₂
f : B.trW.LeftFraction (A.ι.obj X₁) (A.ι.obj X₂)
hf : φ' = f.map (W := B.trW) L₂ ⋯
X₃ : C
s' : A.ι.obj X₂ ⟶ X₃
a : f.Y' (W := B.trW) ⟶ X₃
hX₃ : A X₃
hs' : (A ⊓ B).trW s'
fac : f.s (W := B.trW) ≫ a = s'
g : (B.inverseImage A.ι).trW.LeftFraction X₁ X₂ :=
  MorphismProperty.LeftFraction.mk (W := (B.inverseImage A.ι).trW) (homMk (P := A) (f.f (W := B.trW) ≫ a))
    (homMk (P := A) s') ⋯
this : IsIso (L₁.map (g.s (W := (B.inverseImage A.ι).trW)))
⊢ ∃ (f : L₁.obj X₁ ⟶ L₁.obj X₂), F.map f = φ
have refine : F.map (g.map (W := (B.inverseImage A.ι).trW) L₁ sorry) = φ
Tactic: refine ⟨g.map L₁ (Localization.inverts _ _), ?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.LocalizingSubcategory (goal)
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 : IsTriangulated.{v_1, u_1} C
inst_10 : A.IsTriangulated
inst_11 : B.IsTriangulated
inst_12 : B.IsClosedUnderIsomorphisms
inst_13 : A.IsVerdierRightLocalizing B
L₁ : Functor.{v_1, v_3, u_1, u_3} A.FullSubcategory D₁
L₂ : Functor.{v_1, v_4, u_1, u_4} C D₂
inst_14 : L₁.IsLocalization (B.inverseImage A.ι).trW
inst_15 : L₂.IsLocalization B.trW
F : Functor.{v_3, v_4, u_3, u_4} D₁ D₂ :=
  LocalizerMorphism.localizedFunctor (W₁ := (B.inverseImage A.ι).trW) (W₂ := B.trW) (A.triangulatedLocalizerMorphism B)
    L₁ L₂
this_1 : L₁.EssSurj
e : A.ι ⋙ L₂ ≅ L₁ ⋙ F :=
  CatCommSq.iso
    (LocalizerMorphism.functor (W₁ := (B.inverseImage A.ι).trW) (W₂ := B.trW) (A.triangulatedLocalizerMorphism B)) L₁ L₂
    F
X₁ X₂ : A.FullSubcategory
φ : F.obj (L₁.obj X₁) ⟶ F.obj (L₁.obj X₂)
φ' : (A.ι ⋙ L₂).obj X₁ ⟶ (A.ι ⋙ L₂).obj X₂
hφ' : φ = e.inv.app X₁ ≫ φ' ≫ e.hom.app X₂
f : B.trW.LeftFraction (A.ι.obj X₁) (A.ι.obj X₂)
hf : φ' = f.map (W := B.trW) L₂ ⋯
X₃ : C
s' : A.ι.obj X₂ ⟶ X₃
a : f.Y' (W := B.trW) ⟶ X₃
hX₃ : A X₃
hs' : (A ⊓ B).trW s'
fac : f.s (W := B.trW) ≫ a = s'
g : (B.inverseImage A.ι).trW.LeftFraction X₁ X₂ :=
  MorphismProperty.LeftFraction.mk (W := (B.inverseImage A.ι).trW) (homMk (P := A) (f.f (W := B.trW) ≫ a))
    (homMk (P := A) s') ⋯
this : IsIso (L₁.map (g.s (W := (B.inverseImage A.ι).trW)))
⊢ F.map (g.map (W := (B.inverseImage A.ι).trW) L₁ ⋯) = φ
have rw : F.map (L₁.map (g.f (W := (B.inverseImage A.ι).trW))) = φ ≫ F.map (L₁.map (g.s (W := (B.inverseImage A.ι).trW)))
Tactic: rw [← cancel_mono (F.map (L₁.map g.s)), ← Functor.map_comp,
    MorphismProperty.LeftFraction.map_comp_map_s]
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.LocalizingSubcategory (goal)
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 : IsTriangulated.{v_1, u_1} C
inst_10 : A.IsTriangulated
inst_11 : B.IsTriangulated
inst_12 : B.IsClosedUnderIsomorphisms
inst_13 : A.IsVerdierRightLocalizing B
L₁ : Functor.{v_1, v_3, u_1, u_3} A.FullSubcategory D₁
L₂ : Functor.{v_1, v_4, u_1, u_4} C D₂
inst_14 : L₁.IsLocalization (B.inverseImage A.ι).trW
inst_15 : L₂.IsLocalization B.trW
inst_16 : Preadditive.{v_3, u_3} D₁
inst_17 : Preadditive.{v_4, u_4} D₂
inst_18 : L₁.Additive
inst_19 : L₂.Additive
F : Functor.{v_3, v_4, u_3, u_4} D₁ D₂ :=
  LocalizerMorphism.localizedFunctor (W₁ := (B.inverseImage A.ι).trW) (W₂ := B.trW) (A.triangulatedLocalizerMorphism B)
    L₁ L₂
⊢ (LocalizerMorphism.localizedFunctor (W₁ := (B.inverseImage A.ι).trW) (W₂ := B.trW) (A.triangulatedLocalizerMorphism B)
      L₁ L₂).Additive
have rw : (L₁ ⋙
    LocalizerMorphism.localizedFunctor (W₁ := (B.inverseImage A.ι).trW) (W₂ := B.trW)
      (A.triangulatedLocalizerMorphism B) L₁ L₂).Additive
Tactic: rw [Localization.functor_additive_iff L₁ (B.inverseImage A.ι).trW]
-- 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 : IsTriangulated.{v_1, u_1} C
inst_10 : A.IsTriangulated
inst_11 : B.IsTriangulated
inst_12 : B.IsClosedUnderIsomorphisms
inst_13 : A.IsVerdierRightLocalizing B
L₁ : Functor.{v_1, v_3, u_1, u_3} A.FullSubcategory D₁
L₂ : Functor.{v_1, v_4, u_1, u_4} C D₂
inst_14 : L₁.IsLocalization (B.inverseImage A.ι).trW
inst_15 : L₂.IsLocalization B.trW
this_1 : Preadditive.{v_3, u_3} D₁ := Localization.preadditive L₁ (B.inverseImage A.ι).trW
this : Preadditive.{v_4, u_4} D₂ := Localization.preadditive L₂ B.trW
⊢ (LocalizerMorphism.localizedFunctor (W₁ := (B.inverseImage A.ι).trW) (W₂ := B.trW) (A.triangulatedLocalizerMorphism B)
      L₁ L₂).Faithful
have this₁ : L₁.Additive
Tactic: have := Localization.functor_additive L₁ (B.inverseImage A.ι).trW
-- 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 : IsTriangulated.{v_1, u_1} C
inst_10 : A.IsTriangulated
inst_11 : B.IsTriangulated
inst_12 : B.IsClosedUnderIsomorphisms
inst_13 : A.IsVerdierRightLocalizing B
L₁ : Functor.{v_1, v_3, u_1, u_3} A.FullSubcategory D₁
L₂ : Functor.{v_1, v_4, u_1, u_4} C D₂
inst_14 : L₁.IsLocalization (B.inverseImage A.ι).trW
inst_15 : L₂.IsLocalization B.trW
this_1 : Preadditive.{v_3, u_3} D₁ := Localization.preadditive L₁ (B.inverseImage A.ι).trW
this_2 : Preadditive.{v_4, u_4} D₂ := Localization.preadditive L₂ B.trW
this : L₁.Additive
⊢ (LocalizerMorphism.localizedFunctor (W₁ := (B.inverseImage A.ι).trW) (W₂ := B.trW) (A.triangulatedLocalizerMorphism B)
      L₁ L₂).Faithful
have this₁ : L₂.Additive
Tactic: have := Localization.functor_additive L₂ B.trW
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.LocalizingSubcategory (goal)
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 : IsTriangulated.{v_1, u_1} C
inst_10 : A.IsTriangulated
inst_11 : B.IsTriangulated
inst_12 : B.IsClosedUnderIsomorphisms
inst_13 : A.IsVerdierRightLocalizing B
L₁ : Functor.{v_1, v_3, u_1, u_3} A.FullSubcategory D₁
L₂ : Functor.{v_1, v_4, u_1, u_4} C D₂
inst_14 : L₁.IsLocalization (B.inverseImage A.ι).trW
inst_15 : L₂.IsLocalization B.trW
this_1 : Preadditive.{v_3, u_3} D₁ := Localization.preadditive L₁ (B.inverseImage A.ι).trW
this_2 : Preadditive.{v_4, u_4} D₂ := Localization.preadditive L₂ B.trW
this_3 : L₁.Additive
this : L₂.Additive
F : Functor.{v_3, v_4, u_3, u_4} D₁ D₂ :=
  LocalizerMorphism.localizedFunctor (W₁ := (B.inverseImage A.ι).trW) (W₂ := B.trW) (A.triangulatedLocalizerMorphism B)
    L₁ L₂
e : A.ι ⋙ L₂ ≅ L₁ ⋙ F :=
  CatCommSq.iso
    (LocalizerMorphism.functor (W₁ := (B.inverseImage A.ι).trW) (W₂ := B.trW) (A.triangulatedLocalizerMorphism B)) L₁ L₂
    F
⊢ (LocalizerMorphism.localizedFunctor (W₁ := (B.inverseImage A.ι).trW) (W₂ := B.trW) (A.triangulatedLocalizerMorphism B)
      L₁ L₂).Faithful
have refine : ∀ (X₁ X₂ : A.FullSubcategory) (f : X₁ ⟶ X₂) (hf : F.map (L₁.map f) = 0), L₁.map f = 0
Tactic: refine Functor.faithful_of_comp_cancel_zero_of_hasLeftCalculusOfFractions L₁
    (B.inverseImage A.ι).trW F (fun X₁ X₂ f hf ↦ ?_)
-- 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 : IsTriangulated.{v_1, u_1} C
inst_10 : A.IsTriangulated
inst_11 : B.IsTriangulated
inst_12 : B.IsClosedUnderIsomorphisms
inst_13 : A.IsVerdierRightLocalizing B
L₁ : Functor.{v_1, v_3, u_1, u_3} A.FullSubcategory D₁
L₂ : Functor.{v_1, v_4, u_1, u_4} C D₂
inst_14 : L₁.IsLocalization (B.inverseImage A.ι).trW
inst_15 : L₂.IsLocalization B.trW
this_1 : Preadditive.{v_3, u_3} D₁ := Localization.preadditive L₁ (B.inverseImage A.ι).trW
this_2 : Preadditive.{v_4, u_4} D₂ := Localization.preadditive L₂ B.trW
this_3 : L₁.Additive
this : L₂.Additive
F : Functor.{v_3, v_4, u_3, u_4} D₁ D₂ :=
  LocalizerMorphism.localizedFunctor (W₁ := (B.inverseImage A.ι).trW) (W₂ := B.trW) (A.triangulatedLocalizerMorphism B)
    L₁ L₂
e : A.ι ⋙ L₂ ≅ L₁ ⋙ F :=
  CatCommSq.iso
    (LocalizerMorphism.functor (W₁ := (B.inverseImage A.ι).trW) (W₂ := B.trW) (A.triangulatedLocalizerMorphism B)) L₁ L₂
    F
X₁ X₂ : A.FullSubcategory
f : X₁ ⟶ X₂
hf : F.map (L₁.map f) = 0
⊢ L₁.map f = 0
have hf₁ : L₂.map f.hom = L₂.map 0
Tactic: replace hf : L₂.map f.hom = L₂.map 0 := by
    simp [← dsimp% NatIso.naturality_2 e f, hf]
Cleared hypotheses: hf
-- 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 : IsTriangulated.{v_1, u_1} C
inst_10 : A.IsTriangulated
inst_11 : B.IsTriangulated
inst_12 : B.IsClosedUnderIsomorphisms
inst_13 : A.IsVerdierRightLocalizing B
L₁ : Functor.{v_1, v_3, u_1, u_3} A.FullSubcategory D₁
L₂ : Functor.{v_1, v_4, u_1, u_4} C D₂
inst_14 : L₁.IsLocalization (B.inverseImage A.ι).trW
inst_15 : L₂.IsLocalization B.trW
this_1 : Preadditive.{v_3, u_3} D₁ := Localization.preadditive L₁ (B.inverseImage A.ι).trW
this_2 : Preadditive.{v_4, u_4} D₂ := Localization.preadditive L₂ B.trW
this_3 : L₁.Additive
this : L₂.Additive
F : Functor.{v_3, v_4, u_3, u_4} D₁ D₂ :=
  LocalizerMorphism.localizedFunctor (W₁ := (B.inverseImage A.ι).trW) (W₂ := B.trW) (A.triangulatedLocalizerMorphism B)
    L₁ L₂
e : A.ι ⋙ L₂ ≅ L₁ ⋙ F :=
  CatCommSq.iso
    (LocalizerMorphism.functor (W₁ := (B.inverseImage A.ι).trW) (W₂ := B.trW) (A.triangulatedLocalizerMorphism B)) L₁ L₂
    F
X₁ X₂ : A.FullSubcategory
f : X₁ ⟶ X₂
hf : L₂.map f.hom = L₂.map 0
⊢ L₁.map f = 0
have hf₁ : ∃ (Z : C) (s : X₂.obj (P := A) ⟶ Z) (_ : B.trW s), f.hom ≫ s = 0 ≫ s
Tactic: rw [MorphismProperty.map_eq_iff_postcomp L₂ B.trW] at hf
Cleared hypotheses: hf
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.LocalizingSubcategory (goal)
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 : IsTriangulated.{v_1, u_1} C
inst_10 : A.IsTriangulated
inst_11 : B.IsTriangulated
inst_12 : B.IsClosedUnderIsomorphisms
inst_13 : A.IsVerdierRightLocalizing B
L₁ : Functor.{v_1, v_3, u_1, u_3} A.FullSubcategory D₁
L₂ : Functor.{v_1, v_4, u_1, u_4} C D₂
inst_14 : L₁.IsLocalization (B.inverseImage A.ι).trW
inst_15 : L₂.IsLocalization B.trW
this_1 : Preadditive.{v_3, u_3} D₁ := Localization.preadditive L₁ (B.inverseImage A.ι).trW
this_2 : Preadditive.{v_4, u_4} D₂ := Localization.preadditive L₂ B.trW
this_3 : L₁.Additive
this : L₂.Additive
F : Functor.{v_3, v_4, u_3, u_4} D₁ D₂ :=
  LocalizerMorphism.localizedFunctor (W₁ := (B.inverseImage A.ι).trW) (W₂ := B.trW) (A.triangulatedLocalizerMorphism B)
    L₁ L₂
e : A.ι ⋙ L₂ ≅ L₁ ⋙ F :=
  CatCommSq.iso
    (LocalizerMorphism.functor (W₁ := (B.inverseImage A.ι).trW) (W₂ := B.trW) (A.triangulatedLocalizerMorphism B)) L₁ L₂
    F
X₁ X₂ : A.FullSubcategory
f : X₁ ⟶ X₂
hf : ∃ (Z : C) (s : X₂.obj (P := A) ⟶ Z) (_ : B.trW s), f.hom ≫ s = 0 ≫ s
⊢ L₁.map f = 0
have obtain : ∀ (X₃ : C) (s : X₂.obj (P := A) ⟶ X₃) (hs : B.trW s) (fac : f.hom ≫ s = 0 ≫ s), L₁.map f = 0
Tactic: obtain ⟨X₃, s, hs, fac⟩ := hf
Cleared hypotheses: hf
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.LocalizingSubcategory (goal)
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 : IsTriangulated.{v_1, u_1} C
inst_10 : A.IsTriangulated
inst_11 : B.IsTriangulated
inst_12 : B.IsClosedUnderIsomorphisms
inst_13 : A.IsVerdierRightLocalizing B
L₁ : Functor.{v_1, v_3, u_1, u_3} A.FullSubcategory D₁
L₂ : Functor.{v_1, v_4, u_1, u_4} C D₂
inst_14 : L₁.IsLocalization (B.inverseImage A.ι).trW
inst_15 : L₂.IsLocalization B.trW
this_1 : Preadditive.{v_3, u_3} D₁ := Localization.preadditive L₁ (B.inverseImage A.ι).trW
this_2 : Preadditive.{v_4, u_4} D₂ := Localization.preadditive L₂ B.trW
this_3 : L₁.Additive
this : L₂.Additive
F : Functor.{v_3, v_4, u_3, u_4} D₁ D₂ :=
  LocalizerMorphism.localizedFunctor (W₁ := (B.inverseImage A.ι).trW) (W₂ := B.trW) (A.triangulatedLocalizerMorphism B)
    L₁ L₂
e : A.ι ⋙ L₂ ≅ L₁ ⋙ F :=
  CatCommSq.iso
    (LocalizerMorphism.functor (W₁ := (B.inverseImage A.ι).trW) (W₂ := B.trW) (A.triangulatedLocalizerMorphism B)) L₁ L₂
    F
X₁ X₂ : A.FullSubcategory
f : X₁ ⟶ X₂
X₃ : C
s : X₂.obj (P := A) ⟶ X₃
hs : B.trW s
fac : f.hom ≫ s = 0 ≫ s
⊢ L₁.map f = 0
have obtain : ∀ (X₄ : C) (t : X₂.obj (P := A) ⟶ X₄) (a : X₃ ⟶ X₄) (hX₄ : A X₄) (ht : (A ⊓ B).trW t) (fac' : s ≫ a = t), L₁.map f = 0
Tactic: obtain ⟨X₄, t, a, hX₄, ht, fac'⟩ :=
    IsVerdierRightLocalizing.fac' s X₂.property hs
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.LocalizingSubcategory (goal)
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 : IsTriangulated.{v_1, u_1} C
inst_10 : A.IsTriangulated
inst_11 : B.IsTriangulated
inst_12 : B.IsClosedUnderIsomorphisms
inst_13 : A.IsVerdierRightLocalizing B
L₁ : Functor.{v_1, v_3, u_1, u_3} A.FullSubcategory D₁
L₂ : Functor.{v_1, v_4, u_1, u_4} C D₂
inst_14 : L₁.IsLocalization (B.inverseImage A.ι).trW
inst_15 : L₂.IsLocalization B.trW
this_1 : Preadditive.{v_3, u_3} D₁ := Localization.preadditive L₁ (B.inverseImage A.ι).trW
this_2 : Preadditive.{v_4, u_4} D₂ := Localization.preadditive L₂ B.trW
this_3 : L₁.Additive
this : L₂.Additive
F : Functor.{v_3, v_4, u_3, u_4} D₁ D₂ :=
  LocalizerMorphism.localizedFunctor (W₁ := (B.inverseImage A.ι).trW) (W₂ := B.trW) (A.triangulatedLocalizerMorphism B)
    L₁ L₂
e : A.ι ⋙ L₂ ≅ L₁ ⋙ F :=
  CatCommSq.iso
    (LocalizerMorphism.functor (W₁ := (B.inverseImage A.ι).trW) (W₂ := B.trW) (A.triangulatedLocalizerMorphism B)) L₁ L₂
    F
X₁ X₂ : A.FullSubcategory
f : X₁ ⟶ X₂
X₃ : C
s : X₂.obj (P := A) ⟶ X₃
hs : B.trW s
fac : f.hom ≫ s = 0 ≫ s
X₄ : C
t : X₂.obj (P := A) ⟶ X₄
a : X₃ ⟶ X₄
hX₄ : A X₄
ht : (A ⊓ B).trW t
fac' : s ≫ a = t
t' : X₂ ⟶ FullSubcategory.mk (P := A) X₄ hX₄ := homMk (P := A) t
⊢ (B.inverseImage A.ι).trW t'
have rwa : (A ⊓ B).trW t'.hom
Tactic: rwa [trW_inverseImage_ι_iff]
-- 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 : IsTriangulated.{v_1, u_1} C
inst_10 : A.IsTriangulated
inst_11 : B.IsTriangulated
inst_12 : B.IsClosedUnderIsomorphisms
inst_13 : A.IsVerdierRightLocalizing B
L₁ : Functor.{v_1, v_3, u_1, u_3} A.FullSubcategory D₁
L₂ : Functor.{v_1, v_4, u_1, u_4} C D₂
inst_14 : L₁.IsLocalization (B.inverseImage A.ι).trW
inst_15 : L₂.IsLocalization B.trW
this_1 : Preadditive.{v_3, u_3} D₁ := Localization.preadditive L₁ (B.inverseImage A.ι).trW
this_2 : Preadditive.{v_4, u_4} D₂ := Localization.preadditive L₂ B.trW
this_3 : L₁.Additive
this : L₂.Additive
F : Functor.{v_3, v_4, u_3, u_4} D₁ D₂ :=
  LocalizerMorphism.localizedFunctor (W₁ := (B.inverseImage A.ι).trW) (W₂ := B.trW) (A.triangulatedLocalizerMorphism B)
    L₁ L₂
e : A.ι ⋙ L₂ ≅ L₁ ⋙ F :=
  CatCommSq.iso
    (LocalizerMorphism.functor (W₁ := (B.inverseImage A.ι).trW) (W₂ := B.trW) (A.triangulatedLocalizerMorphism B)) L₁ L₂
    F
X₁ X₂ : A.FullSubcategory
f : X₁ ⟶ X₂
X₃ : C
s : X₂.obj (P := A) ⟶ X₃
hs : B.trW s
fac : f.hom ≫ s = 0 ≫ s
X₄ : C
t : X₂.obj (P := A) ⟶ X₄
a : X₃ ⟶ X₄
hX₄ : A X₄
ht : (A ⊓ B).trW t
fac' : s ≫ a = t
t' : X₂ ⟶ FullSubcategory.mk (P := A) X₄ hX₄ := homMk (P := A) t
⊢ L₁.map f = 0
have this₁ : IsIso (L₁.map t')
Tactic: have := Localization.inverts L₁ (B.inverseImage A.ι).trW t'
    (by rwa [trW_inverseImage_ι_iff])