mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.CategoryTheory.Triangulated.LocalizingSubcategory

Goal state before:
pp: C : Type u_1
D : Type u_2
D₁ : Type u_3
D₂ : Type u_4
inst✝¹⁵ : Category.{v_1, u_1} C
inst✝¹⁴ : Category.{v_2, u_2} D
inst✝¹³ : Category.{v_3, u_3} D₁
inst✝¹² : Category.{v_4, u_4} D₂
A B : ObjectProperty C
inst✝¹¹ : HasZeroObject C
inst✝¹⁰ : HasShift C ℤ
inst✝⁹ : Preadditive C
inst✝⁸ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁷ : Pretriangulated C
inst✝⁶ : IsTriangulated C
inst✝⁵ : A.IsTriangulated
inst✝⁴ : B.IsTriangulated
inst✝³ : B.IsClosedUnderIsomorphisms
inst✝² : A.IsVerdierRightLocalizing B
L₁ : A.FullSubcategory ⥤ D₁
L₂ : C ⥤ D₂
inst✝¹ : L₁.IsLocalization (B.inverseImage A.ι).trW
inst✝ : L₂.IsLocalization B.trW
F : D₁ ⥤ D₂ := (A.triangulatedLocalizerMorphism B).localizedFunctor L₁ L₂
this : L₁.EssSurj
e : A.ι ⋙ L₂ ≅ L₁ ⋙ F := CatCommSq.iso (A.triangulatedLocalizerMorphism B).functor 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 L₂ ⋯
X₃ : C
s' : A.ι.obj X₂ ⟶ X₃
a : f.Y' ⟶ X₃
hX₃ : A X₃
hs' : (A ⊓ B).trW s'
fac : f.s ≫ a = s'
⊢ (A ⊓ B).trW (homMk s').hom

Tactic:
]

Goal state after:
C : Type u_1
D : Type u_2
D₁ : Type u_3
D₂ : Type u_4
inst✝¹⁵ : Category.{v_1, u_1} C
inst✝¹⁴ : Category.{v_2, u_2} D
inst✝¹³ : Category.{v_3, u_3} D₁
inst✝¹² : Category.{v_4, u_4} D₂
A B : ObjectProperty C
inst✝¹¹ : HasZeroObject C
inst✝¹⁰ : HasShift C ℤ
inst✝⁹ : Preadditive C
inst✝⁸ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁷ : Pretriangulated C
inst✝⁶ : IsTriangulated C
inst✝⁵ : A.IsTriangulated
inst✝⁴ : B.IsTriangulated
inst✝³ : B.IsClosedUnderIsomorphisms
inst✝² : A.IsVerdierRightLocalizing B
L₁ : A.FullSubcategory ⥤ D₁
L₂ : C ⥤ D₂
inst✝¹ : L₁.IsLocalization (B.inverseImage A.ι).trW
inst✝ : L₂.IsLocalization B.trW
F : D₁ ⥤ D₂ := (A.triangulatedLocalizerMorphism B).localizedFunctor L₁ L₂
this : L₁.EssSurj
e : A.ι ⋙ L₂ ≅ L₁ ⋙ F := CatCommSq.iso (A.triangulatedLocalizerMorphism B).functor 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 L₂ ⋯
X₃ : C
s' : A.ι.obj X₂ ⟶ X₃
a : f.Y' ⟶ X₃
hX₃ : A X₃
hs' : (A ⊓ B).trW s'
fac : f.s ≫ a = s'
⊢ (A ⊓ B).trW (homMk s').hom
Module: Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
t : TStructure C
n : ℤ
X Y : C
φ : X ⟶ Y
A : C
a b : ℤ
h : a ≤ b
H : ∃ f, f.hom₂ = 𝟙 (triangle t a A).obj₂
⊢ 𝟙 (triangle t a A).obj₂ ≫ (triangle t b A).mor₂ = 𝟙 (triangle t a A).obj₂ ≫ (triangle t b A).mor₂

Tactic:
]

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
t : TStructure C
n : ℤ
x✝³ x✝² : C
x✝¹ x✝ : x✝³ ⟶ x✝²
⊢ Triangle.π₁.map ((TruncAux.triangleFunctor t n).map x✝¹ + (TruncAux.triangleFunctor t n).map x✝) =
    Triangle.π₁.map ((TruncAux.triangleFunctor t n).map x✝¹) + Triangle.π₁.map ((TruncAux.triangleFunctor t n).map x✝)

Tactic:
]

Goal state after:
C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
t : TStructure C
n : ℤ
x✝³ x✝² : C
x✝¹ x✝ : x✝³ ⟶ x✝²
⊢ Triangle.π₁.map ((TruncAux.triangleFunctor t n).map x✝¹ + (TruncAux.triangleFunctor t n).map x✝) =
    Triangle.π₁.map ((TruncAux.triangleFunctor t n).map x✝¹) + Triangle.π₁.map ((TruncAux.triangleFunctor t n).map x✝)
Module: Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
t : TStructure C
n : ℤ
x✝³ x✝² : C
x✝¹ x✝ : x✝³ ⟶ x✝²
⊢ Triangle.π₃.map ((TruncAux.triangleFunctor t n).map x✝¹ + (TruncAux.triangleFunctor t n).map x✝) =
    Triangle.π₃.map ((TruncAux.triangleFunctor t n).map x✝¹) + Triangle.π₃.map ((TruncAux.triangleFunctor t n).map x✝)

Tactic:
]

Goal state after:
C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
t : TStructure C
n : ℤ
x✝³ x✝² : C
x✝¹ x✝ : x✝³ ⟶ x✝²
⊢ Triangle.π₃.map ((TruncAux.triangleFunctor t n).map x✝¹ + (TruncAux.triangleFunctor t n).map x✝) =
    Triangle.π₃.map ((TruncAux.triangleFunctor t n).map x✝¹) + Triangle.π₃.map ((TruncAux.triangleFunctor t n).map x✝)
Module: Mathlib.Algebra.Homology.Factorizations.Basic

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
K L : CochainComplex C ℤ
f : K ⟶ L
hL : IsZero L
n : ℤ
⊢ Injective (K.X n) ↔ Injective (K.X n)

pp: case hY
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
K L : CochainComplex C ℤ
f : K ⟶ L
hL : IsZero L
n : ℤ
⊢ IsZero (L.X n)

Tactic:
]

Goal state after:
case hY
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
K L : CochainComplex C ℤ
f : K ⟶ L
hL : IsZero L
n : ℤ
⊢ IsZero (L.X n)
Module: Mathlib.CategoryTheory.Triangulated.LocalizingSubcategory

Goal state before:
pp: C : Type u_1
D : Type u_2
D₁ : Type u_3
D₂ : Type u_4
inst✝¹⁵ : Category.{v_1, u_1} C
inst✝¹⁴ : Category.{v_2, u_2} D
inst✝¹³ : Category.{v_3, u_3} D₁
inst✝¹² : Category.{v_4, u_4} D₂
A B : ObjectProperty C
inst✝¹¹ : HasZeroObject C
inst✝¹⁰ : HasShift C ℤ
inst✝⁹ : Preadditive C
inst✝⁸ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁷ : Pretriangulated C
inst✝⁶ : IsTriangulated C
inst✝⁵ : A.IsTriangulated
inst✝⁴ : B.IsTriangulated
inst✝³ : B.IsClosedUnderIsomorphisms
inst✝² : A.IsVerdierRightLocalizing B
L₁ : A.FullSubcategory ⥤ D₁
L₂ : C ⥤ D₂
inst✝¹ : L₁.IsLocalization (B.inverseImage A.ι).trW
inst✝ : L₂.IsLocalization B.trW
F : D₁ ⥤ D₂ := (A.triangulatedLocalizerMorphism B).localizedFunctor L₁ L₂
this✝ : L₁.EssSurj
e : A.ι ⋙ L₂ ≅ L₁ ⋙ F := CatCommSq.iso (A.triangulatedLocalizerMorphism B).functor 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 L₂ ⋯
X₃ : C
s' : A.ι.obj X₂ ⟶ X₃
a : f.Y' ⟶ X₃
hX₃ : A X₃
hs' : (A ⊓ B).trW s'
fac : f.s ≫ a = s'
g : (B.inverseImage A.ι).trW.LeftFraction X₁ X₂ :=
  { Y' := { obj := X₃, property := hX₃ }, f := homMk (f.f ≫ a), s := homMk s', hs := ⋯ }
this : IsIso (L₁.map g.s)
⊢ F.map (L₁.map g.f) = φ ≫ F.map (L₁.map g.s)

Tactic:
]

Goal state after:
C : Type u_1
D : Type u_2
D₁ : Type u_3
D₂ : Type u_4
inst✝¹⁵ : Category.{v_1, u_1} C
inst✝¹⁴ : Category.{v_2, u_2} D
inst✝¹³ : Category.{v_3, u_3} D₁
inst✝¹² : Category.{v_4, u_4} D₂
A B : ObjectProperty C
inst✝¹¹ : HasZeroObject C
inst✝¹⁰ : HasShift C ℤ
inst✝⁹ : Preadditive C
inst✝⁸ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁷ : Pretriangulated C
inst✝⁶ : IsTriangulated C
inst✝⁵ : A.IsTriangulated
inst✝⁴ : B.IsTriangulated
inst✝³ : B.IsClosedUnderIsomorphisms
inst✝² : A.IsVerdierRightLocalizing B
L₁ : A.FullSubcategory ⥤ D₁
L₂ : C ⥤ D₂
inst✝¹ : L₁.IsLocalization (B.inverseImage A.ι).trW
inst✝ : L₂.IsLocalization B.trW
F : D₁ ⥤ D₂ := (A.triangulatedLocalizerMorphism B).localizedFunctor L₁ L₂
this✝ : L₁.EssSurj
e : A.ι ⋙ L₂ ≅ L₁ ⋙ F := CatCommSq.iso (A.triangulatedLocalizerMorphism B).functor 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 L₂ ⋯
X₃ : C
s' : A.ι.obj X₂ ⟶ X₃
a : f.Y' ⟶ X₃
hX₃ : A X₃
hs' : (A ⊓ B).trW s'
fac : f.s ≫ a = s'
g : (B.inverseImage A.ι).trW.LeftFraction X₁ X₂ :=
  { Y' := { obj := X₃, property := hX₃ }, f := homMk (f.f ≫ a), s := homMk s', hs := ⋯ }
this : IsIso (L₁.map g.s)
⊢ F.map (L₁.map g.f) = φ ≫ F.map (L₁.map g.s)
Module: Mathlib.CategoryTheory.Triangulated.LocalizingSubcategory

Goal state before:
pp: C : Type u_1
D : Type u_2
D₁ : Type u_3
D₂ : Type u_4
inst✝¹⁹ : Category.{v_1, u_1} C
inst✝¹⁸ : Category.{v_2, u_2} D
inst✝¹⁷ : Category.{v_3, u_3} D₁
inst✝¹⁶ : Category.{v_4, u_4} D₂
A B : ObjectProperty C
inst✝¹⁵ : HasZeroObject C
inst✝¹⁴ : HasShift C ℤ
inst✝¹³ : Preadditive C
inst✝¹² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹¹ : Pretriangulated C
inst✝¹⁰ : IsTriangulated C
inst✝⁹ : A.IsTriangulated
inst✝⁸ : B.IsTriangulated
inst✝⁷ : B.IsClosedUnderIsomorphisms
inst✝⁶ : A.IsVerdierRightLocalizing B
L₁ : A.FullSubcategory ⥤ D₁
L₂ : C ⥤ D₂
inst✝⁵ : L₁.IsLocalization (B.inverseImage A.ι).trW
inst✝⁴ : L₂.IsLocalization B.trW
inst✝³ : Preadditive D₁
inst✝² : Preadditive D₂
inst✝¹ : L₁.Additive
inst✝ : L₂.Additive
F : D₁ ⥤ D₂ := (A.triangulatedLocalizerMorphism B).localizedFunctor L₁ L₂
⊢ (L₁ ⋙ (A.triangulatedLocalizerMorphism B).localizedFunctor L₁ L₂).Additive

Tactic:
]

Goal state after:
C : Type u_1
D : Type u_2
D₁ : Type u_3
D₂ : Type u_4
inst✝¹⁹ : Category.{v_1, u_1} C
inst✝¹⁸ : Category.{v_2, u_2} D
inst✝¹⁷ : Category.{v_3, u_3} D₁
inst✝¹⁶ : Category.{v_4, u_4} D₂
A B : ObjectProperty C
inst✝¹⁵ : HasZeroObject C
inst✝¹⁴ : HasShift C ℤ
inst✝¹³ : Preadditive C
inst✝¹² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹¹ : Pretriangulated C
inst✝¹⁰ : IsTriangulated C
inst✝⁹ : A.IsTriangulated
inst✝⁸ : B.IsTriangulated
inst✝⁷ : B.IsClosedUnderIsomorphisms
inst✝⁶ : A.IsVerdierRightLocalizing B
L₁ : A.FullSubcategory ⥤ D₁
L₂ : C ⥤ D₂
inst✝⁵ : L₁.IsLocalization (B.inverseImage A.ι).trW
inst✝⁴ : L₂.IsLocalization B.trW
inst✝³ : Preadditive D₁
inst✝² : Preadditive D₂
inst✝¹ : L₁.Additive
inst✝ : L₂.Additive
F : D₁ ⥤ D₂ := (A.triangulatedLocalizerMorphism B).localizedFunctor L₁ L₂
⊢ (L₁ ⋙ (A.triangulatedLocalizerMorphism B).localizedFunctor L₁ L₂).Additive
Module: Mathlib.CategoryTheory.Abelian.EpiWithInjectiveKernel

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
X Y : C
f : X ⟶ Y
inst✝ : IsIso f
⊢ ∃ I,
    ∃ (_ : Injective I),
      ∃ f_1, ∃ (w : f_1 ≫ f = 0), Nonempty { X₁ := I, X₂ := X, X₃ := Y, f := f_1, g := f, zero := w }.Splitting

Tactic:
]

Goal state after:
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
X Y : C
f : X ⟶ Y
inst✝ : IsIso f
⊢ ∃ I,
    ∃ (_ : Injective I),
      ∃ f_1, ∃ (w : f_1 ≫ f = 0), Nonempty { X₁ := I, X₂ := X, X₃ := Y, f := f_1, g := f, zero := w }.Splitting
Module: Mathlib.CategoryTheory.Abelian.EpiWithInjectiveKernel

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
X Y Z : C
g₁ : X ⟶ Y
g₂ : Y ⟶ Z
hg₁ :
  ∃ I,
    ∃ (_ : Injective I),
      ∃ f, ∃ (w : f ≫ g₁ = 0), Nonempty { X₁ := I, X₂ := X, X₃ := Y, f := f, g := g₁, zero := w }.Splitting
hg₂ :
  ∃ I,
    ∃ (_ : Injective I),
      ∃ f, ∃ (w : f ≫ g₂ = 0), Nonempty { X₁ := I, X₂ := Y, X₃ := Z, f := f, g := g₂, zero := w }.Splitting
⊢ ∃ I,
    ∃ (_ : Injective I),
      ∃ f, ∃ (w : f ≫ g₁ ≫ g₂ = 0), Nonempty { X₁ := I, X₂ := X, X₃ := Z, f := f, g := g₁ ≫ g₂, zero := w }.Splitting

Tactic:
]

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
X Y Z : C
g₁ : X ⟶ Y
g₂ : Y ⟶ Z
hg₁ :
  ∃ I,
    ∃ (_ : Injective I),
      ∃ f, ∃ (w : f ≫ g₁ = 0), Nonempty { X₁ := I, X₂ := X, X₃ := Y, f := f, g := g₁, zero := w }.Splitting
hg₂ :
  ∃ I,
    ∃ (_ : Injective I),
      ∃ f, ∃ (w : f ≫ g₂ = 0), Nonempty { X₁ := I, X₂ := Y, X₃ := Z, f := f, g := g₂, zero := w }.Splitting
⊢ ∃ I,
    ∃ (_ : Injective I),
      ∃ f, ∃ (w : f ≫ g₁ ≫ g₂ = 0), Nonempty { X₁ := I, X₂ := X, X₃ := Z, f := f, g := g₁ ≫ g₂, zero := w }.Splitting
Module: Mathlib.CategoryTheory.Triangulated.LocalizingSubcategory

Goal state before:
pp: C : Type u_1
D : Type u_2
D₁ : Type u_3
D₂ : Type u_4
inst✝¹⁵ : Category.{v_1, u_1} C
inst✝¹⁴ : Category.{v_2, u_2} D
inst✝¹³ : Category.{v_3, u_3} D₁
inst✝¹² : Category.{v_4, u_4} D₂
A B : ObjectProperty C
inst✝¹¹ : HasZeroObject C
inst✝¹⁰ : HasShift C ℤ
inst✝⁹ : Preadditive C
inst✝⁸ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁷ : Pretriangulated C
inst✝⁶ : IsTriangulated C
inst✝⁵ : A.IsTriangulated
inst✝⁴ : B.IsTriangulated
inst✝³ : B.IsClosedUnderIsomorphisms
inst✝² : A.IsVerdierRightLocalizing B
L₁ : A.FullSubcategory ⥤ D₁
L₂ : C ⥤ D₂
inst✝¹ : L₁.IsLocalization (B.inverseImage A.ι).trW
inst✝ : L₂.IsLocalization B.trW
this✝² : Preadditive D₁ := Localization.preadditive L₁ (B.inverseImage A.ι).trW
this✝¹ : Preadditive D₂ := Localization.preadditive L₂ B.trW
this✝ : L₁.Additive
this : L₂.Additive
F : D₁ ⥤ D₂ := (A.triangulatedLocalizerMorphism B).localizedFunctor L₁ L₂
e : A.ι ⋙ L₂ ≅ L₁ ⋙ F := CatCommSq.iso (A.triangulatedLocalizerMorphism B).functor L₁ L₂ F
X₁ X₂ : A.FullSubcategory
f : X₁ ⟶ X₂
hf : ∃ Z s, ∃ (_ : B.trW s), f.hom ≫ s = 0 ≫ s
⊢ L₁.map f = 0

Tactic:
]

Goal state after:
C : Type u_1
D : Type u_2
D₁ : Type u_3
D₂ : Type u_4
inst✝¹⁵ : Category.{v_1, u_1} C
inst✝¹⁴ : Category.{v_2, u_2} D
inst✝¹³ : Category.{v_3, u_3} D₁
inst✝¹² : Category.{v_4, u_4} D₂
A B : ObjectProperty C
inst✝¹¹ : HasZeroObject C
inst✝¹⁰ : HasShift C ℤ
inst✝⁹ : Preadditive C
inst✝⁸ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁷ : Pretriangulated C
inst✝⁶ : IsTriangulated C
inst✝⁵ : A.IsTriangulated
inst✝⁴ : B.IsTriangulated
inst✝³ : B.IsClosedUnderIsomorphisms
inst✝² : A.IsVerdierRightLocalizing B
L₁ : A.FullSubcategory ⥤ D₁
L₂ : C ⥤ D₂
inst✝¹ : L₁.IsLocalization (B.inverseImage A.ι).trW
inst✝ : L₂.IsLocalization B.trW
this✝² : Preadditive D₁ := Localization.preadditive L₁ (B.inverseImage A.ι).trW
this✝¹ : Preadditive D₂ := Localization.preadditive L₂ B.trW
this✝ : L₁.Additive
this : L₂.Additive
F : D₁ ⥤ D₂ := (A.triangulatedLocalizerMorphism B).localizedFunctor L₁ L₂
e : A.ι ⋙ L₂ ≅ L₁ ⋙ F := CatCommSq.iso (A.triangulatedLocalizerMorphism B).functor L₁ L₂ F
X₁ X₂ : A.FullSubcategory
f : X₁ ⟶ X₂
hf : ∃ Z s, ∃ (_ : B.trW s), f.hom ≫ s = 0 ≫ s
⊢ L₁.map f = 0
Module: Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
t : TStructure C
n₀ : ℤ
X : C
x✝ : t.IsLE X n₀
e : contractibleTriangle X ≅ (t.triangleLTGE (n₀ + 1)).obj X
he : e.hom.hom₂ = (Iso.refl X).hom
⊢ (𝟙 ((t.triangleLTGE (n₀ + 1)).obj X)).hom₂ = 𝟙 X ≫ (Iso.refl X).hom

Tactic:
]

Goal state after:
C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
t : TStructure C
n₀ : ℤ
X : C
x✝ : t.IsLE X n₀
e : contractibleTriangle X ≅ (t.triangleLTGE (n₀ + 1)).obj X
he : e.hom.hom₂ = (Iso.refl X).hom
⊢ (𝟙 ((t.triangleLTGE (n₀ + 1)).obj X)).hom₂ = 𝟙 X ≫ (Iso.refl X).hom
Module: Mathlib.CategoryTheory.Triangulated.LocalizingSubcategory

Goal state before:
pp: C : Type u_1
D : Type u_2
D₁ : Type u_3
D₂ : Type u_4
inst✝¹⁵ : Category.{v_1, u_1} C
inst✝¹⁴ : Category.{v_2, u_2} D
inst✝¹³ : Category.{v_3, u_3} D₁
inst✝¹² : Category.{v_4, u_4} D₂
A B : ObjectProperty C
inst✝¹¹ : HasZeroObject C
inst✝¹⁰ : HasShift C ℤ
inst✝⁹ : Preadditive C
inst✝⁸ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁷ : Pretriangulated C
inst✝⁶ : IsTriangulated C
inst✝⁵ : A.IsTriangulated
inst✝⁴ : B.IsTriangulated
inst✝³ : B.IsClosedUnderIsomorphisms
inst✝² : A.IsVerdierRightLocalizing B
L₁ : A.FullSubcategory ⥤ D₁
L₂ : C ⥤ D₂
inst✝¹ : L₁.IsLocalization (B.inverseImage A.ι).trW
inst✝ : L₂.IsLocalization B.trW
this✝² : Preadditive D₁ := Localization.preadditive L₁ (B.inverseImage A.ι).trW
this✝¹ : Preadditive D₂ := Localization.preadditive L₂ B.trW
this✝ : L₁.Additive
this : L₂.Additive
F : D₁ ⥤ D₂ := (A.triangulatedLocalizerMorphism B).localizedFunctor L₁ L₂
e : A.ι ⋙ L₂ ≅ L₁ ⋙ F := CatCommSq.iso (A.triangulatedLocalizerMorphism B).functor L₁ L₂ F
X₁ X₂ : A.FullSubcategory
f : X₁ ⟶ X₂
X₃ : C
s : X₂.obj ⟶ X₃
hs : B.trW s
fac : f.hom ≫ s = 0 ≫ s
X₄ : C
t : X₂.obj ⟶ X₄
a : X₃ ⟶ X₄
hX₄ : A X₄
ht : (A ⊓ B).trW t
fac' : s ≫ a = t
t' : X₂ ⟶ { obj := X₄, property := hX₄ } := homMk t
⊢ (A ⊓ B).trW t'.hom

Tactic:
]

Goal state after:
C : Type u_1
D : Type u_2
D₁ : Type u_3
D₂ : Type u_4
inst✝¹⁵ : Category.{v_1, u_1} C
inst✝¹⁴ : Category.{v_2, u_2} D
inst✝¹³ : Category.{v_3, u_3} D₁
inst✝¹² : Category.{v_4, u_4} D₂
A B : ObjectProperty C
inst✝¹¹ : HasZeroObject C
inst✝¹⁰ : HasShift C ℤ
inst✝⁹ : Preadditive C
inst✝⁸ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁷ : Pretriangulated C
inst✝⁶ : IsTriangulated C
inst✝⁵ : A.IsTriangulated
inst✝⁴ : B.IsTriangulated
inst✝³ : B.IsClosedUnderIsomorphisms
inst✝² : A.IsVerdierRightLocalizing B
L₁ : A.FullSubcategory ⥤ D₁
L₂ : C ⥤ D₂
inst✝¹ : L₁.IsLocalization (B.inverseImage A.ι).trW
inst✝ : L₂.IsLocalization B.trW
this✝² : Preadditive D₁ := Localization.preadditive L₁ (B.inverseImage A.ι).trW
this✝¹ : Preadditive D₂ := Localization.preadditive L₂ B.trW
this✝ : L₁.Additive
this : L₂.Additive
F : D₁ ⥤ D₂ := (A.triangulatedLocalizerMorphism B).localizedFunctor L₁ L₂
e : A.ι ⋙ L₂ ≅ L₁ ⋙ F := CatCommSq.iso (A.triangulatedLocalizerMorphism B).functor L₁ L₂ F
X₁ X₂ : A.FullSubcategory
f : X₁ ⟶ X₂
X₃ : C
s : X₂.obj ⟶ X₃
hs : B.trW s
fac : f.hom ≫ s = 0 ≫ s
X₄ : C
t : X₂.obj ⟶ X₄
a : X₃ ⟶ X₄
hX₄ : A X₄
ht : (A ⊓ B).trW t
fac' : s ≫ a = t
t' : X₂ ⟶ { obj := X₄, property := hX₄ } := homMk t
⊢ (A ⊓ B).trW t'.hom
Module: Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
t : TStructure C
n₀ : ℤ
X : C
x✝ : t.IsLE X n₀
e : contractibleTriangle X ≅ (t.triangleLTGE (n₀ + 1)).obj X
he : e.hom.hom₂ = (Iso.refl X).hom
he' : e.inv.hom₂ = 𝟙 X
this : (t.truncLTι (n₀ + 1)).app X = e.inv.hom₁
⊢ IsIso e.inv.hom₁

Tactic:
]

Goal state after:
C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
t : TStructure C
n₀ : ℤ
X : C
x✝ : t.IsLE X n₀
e : contractibleTriangle X ≅ (t.triangleLTGE (n₀ + 1)).obj X
he : e.hom.hom₂ = (Iso.refl X).hom
he' : e.inv.hom₂ = 𝟙 X
this : (t.truncLTι (n₀ + 1)).app X = e.inv.hom₁
⊢ IsIso e.inv.hom₁
Module: Mathlib.CategoryTheory.Triangulated.LocalizingSubcategory

Goal state before:
pp: C : Type u_1
D : Type u_2
D₁ : Type u_3
D₂ : Type u_4
inst✝¹⁵ : Category.{v_1, u_1} C
inst✝¹⁴ : Category.{v_2, u_2} D
inst✝¹³ : Category.{v_3, u_3} D₁
inst✝¹² : Category.{v_4, u_4} D₂
A B : ObjectProperty C
inst✝¹¹ : HasZeroObject C
inst✝¹⁰ : HasShift C ℤ
inst✝⁹ : Preadditive C
inst✝⁸ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁷ : Pretriangulated C
inst✝⁶ : IsTriangulated C
inst✝⁵ : A.IsTriangulated
inst✝⁴ : B.IsTriangulated
inst✝³ : B.IsClosedUnderIsomorphisms
inst✝² : A.IsVerdierRightLocalizing B
L₁ : A.FullSubcategory ⥤ D₁
L₂ : C ⥤ D₂
inst✝¹ : L₁.IsLocalization (B.inverseImage A.ι).trW
inst✝ : L₂.IsLocalization B.trW
this✝³ : Preadditive D₁ := Localization.preadditive L₁ (B.inverseImage A.ι).trW
this✝² : Preadditive D₂ := Localization.preadditive L₂ B.trW
this✝¹ : L₁.Additive
this✝ : L₂.Additive
F : D₁ ⥤ D₂ := (A.triangulatedLocalizerMorphism B).localizedFunctor L₁ L₂
e : A.ι ⋙ L₂ ≅ L₁ ⋙ F := CatCommSq.iso (A.triangulatedLocalizerMorphism B).functor L₁ L₂ F
X₁ X₂ : A.FullSubcategory
f : X₁ ⟶ X₂
X₃ : C
s : X₂.obj ⟶ X₃
hs : B.trW s
fac : f.hom ≫ s = 0 ≫ s
X₄ : C
t : X₂.obj ⟶ X₄
a : X₃ ⟶ X₄
hX₄ : A X₄
ht : (A ⊓ B).trW t
fac' : s ≫ a = t
t' : X₂ ⟶ { obj := X₄, property := hX₄ } := homMk t
this : IsIso (L₁.map t')
⊢ L₁.map (f ≫ t') = L₁.map 0

Tactic:
]

Goal state after:
C : Type u_1
D : Type u_2
D₁ : Type u_3
D₂ : Type u_4
inst✝¹⁵ : Category.{v_1, u_1} C
inst✝¹⁴ : Category.{v_2, u_2} D
inst✝¹³ : Category.{v_3, u_3} D₁
inst✝¹² : Category.{v_4, u_4} D₂
A B : ObjectProperty C
inst✝¹¹ : HasZeroObject C
inst✝¹⁰ : HasShift C ℤ
inst✝⁹ : Preadditive C
inst✝⁸ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁷ : Pretriangulated C
inst✝⁶ : IsTriangulated C
inst✝⁵ : A.IsTriangulated
inst✝⁴ : B.IsTriangulated
inst✝³ : B.IsClosedUnderIsomorphisms
inst✝² : A.IsVerdierRightLocalizing B
L₁ : A.FullSubcategory ⥤ D₁
L₂ : C ⥤ D₂
inst✝¹ : L₁.IsLocalization (B.inverseImage A.ι).trW
inst✝ : L₂.IsLocalization B.trW
this✝³ : Preadditive D₁ := Localization.preadditive L₁ (B.inverseImage A.ι).trW
this✝² : Preadditive D₂ := Localization.preadditive L₂ B.trW
this✝¹ : L₁.Additive
this✝ : L₂.Additive
F : D₁ ⥤ D₂ := (A.triangulatedLocalizerMorphism B).localizedFunctor L₁ L₂
e : A.ι ⋙ L₂ ≅ L₁ ⋙ F := CatCommSq.iso (A.triangulatedLocalizerMorphism B).functor L₁ L₂ F
X₁ X₂ : A.FullSubcategory
f : X₁ ⟶ X₂
X₃ : C
s : X₂.obj ⟶ X₃
hs : B.trW s
fac : f.hom ≫ s = 0 ≫ s
X₄ : C
t : X₂.obj ⟶ X₄
a : X₃ ⟶ X₄
hX₄ : A X₄
ht : (A ⊓ B).trW t
fac' : s ≫ a = t
t' : X₂ ⟶ { obj := X₄, property := hX₄ } := homMk t
this : IsIso (L₁.map t')
⊢ L₁.map (f ≫ t') = L₁.map 0
Module: Mathlib.CategoryTheory.Triangulated.LocalizingSubcategory

Goal state before:
pp: C : Type u_1
D : Type u_2
D₁ : Type u_3
D₂ : Type u_4
inst✝¹³ : Category.{v_1, u_1} C
inst✝¹² : Category.{v_2, u_2} D
inst✝¹¹ : Category.{v_3, u_3} D₁
inst✝¹⁰ : Category.{v_4, u_4} D₂
A B : ObjectProperty C
inst✝⁹ : HasZeroObject C
inst✝⁸ : HasShift C ℤ
inst✝⁷ : Preadditive C
inst✝⁶ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁵ : Pretriangulated C
inst✝⁴ : IsTriangulated C
inst✝³ : A.IsTriangulated
inst✝² : B.IsTriangulated
inst✝¹ : B.IsClosedUnderIsomorphisms
inst✝ : A.IsVerdierLeftLocalizing B
L₁ : A.FullSubcategory ⥤ (B.inverseImage A.ι).trW.Localization := (B.inverseImage A.ι).trW.Q
L₂ : C ⥤ B.trW.Localization := B.trW.Q
F : (B.inverseImage A.ι).trW.Localization ⥤ B.trW.Localization :=
  (A.triangulatedLocalizerMorphism B).localizedFunctor L₁ L₂
this : CatCommSq (A.op.triangulatedLocalizerMorphism B.op).functor (A.opEquivalence.functor ⋙ L₁.op) L₂.op F.op :=
  {
    iso :=
      A.opEquivalence.functor.isoWhiskerLeft
        (NatIso.op (CatCommSq.iso (A.triangulatedLocalizerMorphism B).functor L₁ L₂ F).symm) }
⊢ L₂.op.IsLocalization B.trW.op

Tactic:
]

Goal state after:
C : Type u_1
D : Type u_2
D₁ : Type u_3
D₂ : Type u_4
inst✝¹³ : Category.{v_1, u_1} C
inst✝¹² : Category.{v_2, u_2} D
inst✝¹¹ : Category.{v_3, u_3} D₁
inst✝¹⁰ : Category.{v_4, u_4} D₂
A B : ObjectProperty C
inst✝⁹ : HasZeroObject C
inst✝⁸ : HasShift C ℤ
inst✝⁷ : Preadditive C
inst✝⁶ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁵ : Pretriangulated C
inst✝⁴ : IsTriangulated C
inst✝³ : A.IsTriangulated
inst✝² : B.IsTriangulated
inst✝¹ : B.IsClosedUnderIsomorphisms
inst✝ : A.IsVerdierLeftLocalizing B
L₁ : A.FullSubcategory ⥤ (B.inverseImage A.ι).trW.Localization := (B.inverseImage A.ι).trW.Q
L₂ : C ⥤ B.trW.Localization := B.trW.Q
F : (B.inverseImage A.ι).trW.Localization ⥤ B.trW.Localization :=
  (A.triangulatedLocalizerMorphism B).localizedFunctor L₁ L₂
this : CatCommSq (A.op.triangulatedLocalizerMorphism B.op).functor (A.opEquivalence.functor ⋙ L₁.op) L₂.op F.op :=
  {
    iso :=
      A.opEquivalence.functor.isoWhiskerLeft
        (NatIso.op (CatCommSq.iso (A.triangulatedLocalizerMorphism B).functor L₁ L₂ F).symm) }
⊢ L₂.op.IsLocalization B.trW.op
Module: Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
t : TStructure C
n : ℤ
X : C
h : t.IsGE X n
e : (contractibleTriangle X).invRotate ≅ (t.triangleLTGE n).obj X
he : e.hom.hom₂ = 𝟙 X
this : 𝟙 X ≫ e.hom.hom₃ = e.hom.hom₂ ≫ (t.truncGEπ n).app X
⊢ 𝟙 X ≫ e.hom.hom₃ = 𝟙 X ≫ e.hom.hom₃

Tactic:
]

Goal state after: