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

Goal state before:
pp: case refine_1
C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : HasShift C ℤ
inst✝⁴ : HasZeroObject C
inst✝³ : Preadditive C
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
inst✝ : IsTriangulated C
X₁ X₂ X₃ Z₁₂ Z₂₃ Z₁₃ : Cᵒᵖ
u₁₂ : X₁ ⟶ X₂
u₂₃ : X₂ ⟶ X₃
u₁₃ : X₁ ⟶ X₃
comm : u₁₂ ≫ u₂₃ = u₁₃
v₁₂ : X₂ ⟶ Z₁₂
w₁₂ : Z₁₂ ⟶ (shiftFunctor Cᵒᵖ 1).obj X₁
h₁₂ : Triangle.mk u₁₂ v₁₂ w₁₂ ∈ Pretriangulated.distinguishedTriangles
v₂₃ : X₃ ⟶ Z₂₃
w₂₃ : Z₂₃ ⟶ (shiftFunctor Cᵒᵖ 1).obj X₂
h₂₃ : Triangle.mk u₂₃ v₂₃ w₂₃ ∈ Pretriangulated.distinguishedTriangles
v₁₃ : X₃ ⟶ Z₁₃
w₁₃ : Z₁₃ ⟶ (shiftFunctor Cᵒᵖ 1).obj X₁
h₁₃ : Triangle.mk u₁₃ v₁₃ w₁₃ ∈ Pretriangulated.distinguishedTriangles
o : Triangulated.Octahedron' ⋯ ⋯ ⋯ ⋯ := Triangulated.someOctahedron' ⋯ ⋯ ⋯ ⋯
eq₃ :
  (((shiftFunctor C 1).map o.m₃).op ≫ ((shiftFunctor C 1).map w₁₃.unop).op) ≫
      (opShiftFunctorEquivalence C 1).unitIso.inv.app X₁ =
    ((shiftFunctor C 1).map w₁₂.unop).op ≫ (opShiftFunctorEquivalence C 1).unitIso.inv.app X₁
⊢ o.m₃.op ≫ w₁₃ = w₁₂

Tactic:
← op_comp,

Goal state after:
case refine_1
C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : HasShift C ℤ
inst✝⁴ : HasZeroObject C
inst✝³ : Preadditive C
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
inst✝ : IsTriangulated C
X₁ X₂ X₃ Z₁₂ Z₂₃ Z₁₃ : Cᵒᵖ
u₁₂ : X₁ ⟶ X₂
u₂₃ : X₂ ⟶ X₃
u₁₃ : X₁ ⟶ X₃
comm : u₁₂ ≫ u₂₃ = u₁₃
v₁₂ : X₂ ⟶ Z₁₂
w₁₂ : Z₁₂ ⟶ (shiftFunctor Cᵒᵖ 1).obj X₁
h₁₂ : Triangle.mk u₁₂ v₁₂ w₁₂ ∈ Pretriangulated.distinguishedTriangles
v₂₃ : X₃ ⟶ Z₂₃
w₂₃ : Z₂₃ ⟶ (shiftFunctor Cᵒᵖ 1).obj X₂
h₂₃ : Triangle.mk u₂₃ v₂₃ w₂₃ ∈ Pretriangulated.distinguishedTriangles
v₁₃ : X₃ ⟶ Z₁₃
w₁₃ : Z₁₃ ⟶ (shiftFunctor Cᵒᵖ 1).obj X₁
h₁₃ : Triangle.mk u₁₃ v₁₃ w₁₃ ∈ Pretriangulated.distinguishedTriangles
o : Triangulated.Octahedron' ⋯ ⋯ ⋯ ⋯ := Triangulated.someOctahedron' ⋯ ⋯ ⋯ ⋯
eq₃ :
  ((shiftFunctor C 1).map w₁₃.unop ≫ (shiftFunctor C 1).map o.m₃).op ≫
      (opShiftFunctorEquivalence C 1).unitIso.inv.app X₁ =
    ((shiftFunctor C 1).map w₁₂.unop).op ≫ (opShiftFunctorEquivalence C 1).unitIso.inv.app X₁
⊢ o.m₃.op ≫ w₁₃ = w₁₂
Module: Mathlib.CategoryTheory.Triangulated.Opposite.Triangulated

Goal state before:
pp: case refine_2
C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : HasShift C ℤ
inst✝⁴ : HasZeroObject C
inst✝³ : Preadditive C
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
inst✝ : IsTriangulated C
X₁ X₂ X₃ Z₁₂ Z₂₃ Z₁₃ : Cᵒᵖ
u₁₂ : X₁ ⟶ X₂
u₂₃ : X₂ ⟶ X₃
u₁₃ : X₁ ⟶ X₃
comm : u₁₂ ≫ u₂₃ = u₁₃
v₁₂ : X₂ ⟶ Z₁₂
w₁₂ : Z₁₂ ⟶ (shiftFunctor Cᵒᵖ 1).obj X₁
h₁₂ : Triangle.mk u₁₂ v₁₂ w₁₂ ∈ Pretriangulated.distinguishedTriangles
v₂₃ : X₃ ⟶ Z₂₃
w₂₃ : Z₂₃ ⟶ (shiftFunctor Cᵒᵖ 1).obj X₂
h₂₃ : Triangle.mk u₂₃ v₂₃ w₂₃ ∈ Pretriangulated.distinguishedTriangles
v₁₃ : X₃ ⟶ Z₁₃
w₁₃ : Z₁₃ ⟶ (shiftFunctor Cᵒᵖ 1).obj X₁
h₁₃ : Triangle.mk u₁₃ v₁₃ w₁₃ ∈ Pretriangulated.distinguishedTriangles
o : Triangulated.Octahedron' ⋯ ⋯ ⋯ ⋯ := Triangulated.someOctahedron' ⋯ ⋯ ⋯ ⋯
eq₂ :
  (((shiftFunctor C 1).map o.m₁).op ≫ ((shiftFunctor C 1).map w₂₃.unop).op) ≫
      (opShiftFunctorEquivalence C 1).unitIso.inv.app X₂ =
    (((shiftFunctor C 1).map w₁₃.unop).op ≫ (opShiftFunctorEquivalence C 1).unitIso.inv.app X₁) ≫ u₁₂
⊢ w₁₃ ≫ (shiftFunctor Cᵒᵖ 1).map u₁₂ = o.m₁.op ≫ w₂₃

Tactic:
← op_comp,

Goal state after:
case refine_2
C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : HasShift C ℤ
inst✝⁴ : HasZeroObject C
inst✝³ : Preadditive C
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
inst✝ : IsTriangulated C
X₁ X₂ X₃ Z₁₂ Z₂₃ Z₁₃ : Cᵒᵖ
u₁₂ : X₁ ⟶ X₂
u₂₃ : X₂ ⟶ X₃
u₁₃ : X₁ ⟶ X₃
comm : u₁₂ ≫ u₂₃ = u₁₃
v₁₂ : X₂ ⟶ Z₁₂
w₁₂ : Z₁₂ ⟶ (shiftFunctor Cᵒᵖ 1).obj X₁
h₁₂ : Triangle.mk u₁₂ v₁₂ w₁₂ ∈ Pretriangulated.distinguishedTriangles
v₂₃ : X₃ ⟶ Z₂₃
w₂₃ : Z₂₃ ⟶ (shiftFunctor Cᵒᵖ 1).obj X₂
h₂₃ : Triangle.mk u₂₃ v₂₃ w₂₃ ∈ Pretriangulated.distinguishedTriangles
v₁₃ : X₃ ⟶ Z₁₃
w₁₃ : Z₁₃ ⟶ (shiftFunctor Cᵒᵖ 1).obj X₁
h₁₃ : Triangle.mk u₁₃ v₁₃ w₁₃ ∈ Pretriangulated.distinguishedTriangles
o : Triangulated.Octahedron' ⋯ ⋯ ⋯ ⋯ := Triangulated.someOctahedron' ⋯ ⋯ ⋯ ⋯
eq₂ :
  ((shiftFunctor C 1).map w₂₃.unop ≫ (shiftFunctor C 1).map o.m₁).op ≫
      (opShiftFunctorEquivalence C 1).unitIso.inv.app X₂ =
    (((shiftFunctor C 1).map w₁₃.unop).op ≫ (opShiftFunctorEquivalence C 1).unitIso.inv.app X₁) ≫ u₁₂
⊢ w₁₃ ≫ (shiftFunctor Cᵒᵖ 1).map u₁₂ = o.m₁.op ≫ w₂₃
Module: Mathlib.CategoryTheory.Triangulated.Opposite.Triangulated

Goal state before:
pp: case refine_2
C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : HasShift C ℤ
inst✝⁴ : HasZeroObject C
inst✝³ : Preadditive C
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
inst✝ : IsTriangulated C
X₁ X₂ X₃ Z₁₂ Z₂₃ Z₁₃ : Cᵒᵖ
u₁₂ : X₁ ⟶ X₂
u₂₃ : X₂ ⟶ X₃
u₁₃ : X₁ ⟶ X₃
comm : u₁₂ ≫ u₂₃ = u₁₃
v₁₂ : X₂ ⟶ Z₁₂
w₁₂ : Z₁₂ ⟶ (shiftFunctor Cᵒᵖ 1).obj X₁
h₁₂ : Triangle.mk u₁₂ v₁₂ w₁₂ ∈ Pretriangulated.distinguishedTriangles
v₂₃ : X₃ ⟶ Z₂₃
w₂₃ : Z₂₃ ⟶ (shiftFunctor Cᵒᵖ 1).obj X₂
h₂₃ : Triangle.mk u₂₃ v₂₃ w₂₃ ∈ Pretriangulated.distinguishedTriangles
v₁₃ : X₃ ⟶ Z₁₃
w₁₃ : Z₁₃ ⟶ (shiftFunctor Cᵒᵖ 1).obj X₁
h₁₃ : Triangle.mk u₁₃ v₁₃ w₁₃ ∈ Pretriangulated.distinguishedTriangles
o : Triangulated.Octahedron' ⋯ ⋯ ⋯ ⋯ := Triangulated.someOctahedron' ⋯ ⋯ ⋯ ⋯
eq₂ :
  ((shiftFunctor C 1).map (w₂₃.unop ≫ o.m₁)).op =
    ((shiftFunctor C 1).map w₁₃.unop).op ≫ ((shiftFunctor C 1).map ((shiftFunctor Cᵒᵖ 1).map u₁₂).unop).op
⊢ w₁₃ ≫ (shiftFunctor Cᵒᵖ 1).map u₁₂ = o.m₁.op ≫ w₂₃

Tactic:
← op_comp,

Goal state after:
case refine_2
C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : HasShift C ℤ
inst✝⁴ : HasZeroObject C
inst✝³ : Preadditive C
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
inst✝ : IsTriangulated C
X₁ X₂ X₃ Z₁₂ Z₂₃ Z₁₃ : Cᵒᵖ
u₁₂ : X₁ ⟶ X₂
u₂₃ : X₂ ⟶ X₃
u₁₃ : X₁ ⟶ X₃
comm : u₁₂ ≫ u₂₃ = u₁₃
v₁₂ : X₂ ⟶ Z₁₂
w₁₂ : Z₁₂ ⟶ (shiftFunctor Cᵒᵖ 1).obj X₁
h₁₂ : Triangle.mk u₁₂ v₁₂ w₁₂ ∈ Pretriangulated.distinguishedTriangles
v₂₃ : X₃ ⟶ Z₂₃
w₂₃ : Z₂₃ ⟶ (shiftFunctor Cᵒᵖ 1).obj X₂
h₂₃ : Triangle.mk u₂₃ v₂₃ w₂₃ ∈ Pretriangulated.distinguishedTriangles
v₁₃ : X₃ ⟶ Z₁₃
w₁₃ : Z₁₃ ⟶ (shiftFunctor Cᵒᵖ 1).obj X₁
h₁₃ : Triangle.mk u₁₃ v₁₃ w₁₃ ∈ Pretriangulated.distinguishedTriangles
o : Triangulated.Octahedron' ⋯ ⋯ ⋯ ⋯ := Triangulated.someOctahedron' ⋯ ⋯ ⋯ ⋯
eq₂ :
  ((shiftFunctor C 1).map (w₂₃.unop ≫ o.m₁)).op =
    ((shiftFunctor C 1).map ((shiftFunctor Cᵒᵖ 1).map u₁₂).unop ≫ (shiftFunctor C 1).map w₁₃.unop).op
⊢ w₁₃ ≫ (shiftFunctor Cᵒᵖ 1).map u₁₂ = o.m₁.op ≫ w₂₃
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✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_2, u_2} D
inst✝² : Category.{v_3, u_3} D₁
inst✝¹ : Category.{v_4, u_4} D₂
A B : ObjectProperty Cᵒᵖ
inst✝ : A.IsVerdierLeftLocalizing B
X✝ Y✝ : C
f : X✝ ⟶ Y✝
hX : B.unop X✝
hY : A.unop Y✝
Z : Cᵒᵖ
a : op Y✝ ⟶ Z
b : Z ⟶ op X✝
h₁ : A Z
h₂ : B Z
fac : a ≫ b = f.op
⊢ ∃ Z a b, A.unop Z ∧ B.unop Z ∧ a ≫ b = f

Tactic:
exact ⟨_, b.unop, a.unop, h₁, h₂, Quiver.Hom.op_inj fac⟩

Goal state after:
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✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_2, u_2} D
inst✝² : Category.{v_3, u_3} D₁
inst✝¹ : Category.{v_4, u_4} D₂
A B : ObjectProperty Cᵒᵖ
inst✝ : A.IsVerdierRightLocalizing B
X✝ Y✝ : C
f : X✝ ⟶ Y✝
hX : A.unop X✝
hY : B.unop Y✝
Z : Cᵒᵖ
a : op Y✝ ⟶ Z
b : Z ⟶ op X✝
h₁ : A Z
h₂ : B Z
fac : a ≫ b = f.op
⊢ ∃ Z a b, A.unop Z ∧ B.unop Z ∧ a ≫ b = f

Tactic:
exact ⟨_, b.unop, a.unop, h₁, h₂, Quiver.Hom.op_inj fac⟩

Goal state after:
Module: Mathlib.Algebra.Homology.DerivedCategory.Plus

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : Abelian C
inst✝¹ : HasDerivedCategory C
X : Plus C
n : ℤ
inst✝ : X.IsLE n
⊢ (ι.obj X).IsLE n

Tactic:
isLE_ι_obj_iff

Goal state after:
C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : Abelian C
inst✝¹ : HasDerivedCategory C
X : Plus C
n : ℤ
inst✝ : X.IsLE n
⊢ X.IsLE 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 := ⋯ }
⊢ ∃ f, F.map f = φ

Tactic:
have := Localization.inverts L₁ _ _ g.hs

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, F.map f = φ
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 = 0

Tactic:
← cancel_mono (L₁.map t'),

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 ≫ L₁.map t' = 0 ≫ L₁.map t'
Module: Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE

Goal state before:
pp: case refine_1
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
Y Z : C
g : Y ⟶ Z
n₀ n₁ : ℤ
hn : n₀ + 1 = n₁
hf : IsIso ((t.truncGE n₁).map g)
⊢ ∃ X f h, ∃ (_ : Triangle.mk f (g ≫ (t.truncGEπ n₁).app Z) h ∈ distinguishedTriangles), t.IsLE X n₀

Tactic:
refine ⟨_, (t.truncLTι n₁).app Y, inv ((t.truncGE n₁).map g) ≫ (t.truncGEδLT n₁).app Y,
      isomorphic_distinguished _ (t.triangleLTGE_distinguished n₁ Y) _ ?_,
      by subst hn; infer_instance⟩

Goal state after:
case refine_1
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
Y Z : C
g : Y ⟶ Z
n₀ n₁ : ℤ
hn : n₀ + 1 = n₁
hf : IsIso ((t.truncGE n₁).map g)
⊢ Triangle.mk ((t.truncLTι n₁).app Y) (g ≫ (t.truncGEπ n₁).app Z)
      (inv ((t.truncGE n₁).map g) ≫ (t.truncGEδLT n₁).app Y) ≅
    (t.triangleLTGE n₁).obj Y
Module: Mathlib.Algebra.Homology.Factorizations.CM5a

Goal state before:
pp: case inl
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
K L : CochainComplex C ℤ
inst✝ : EnoughInjectives C
n₀ n₁ : ℤ
hn₁ : n₀ + 1 = n₁
i : ℤ
hi✝ : i ≤ n₀
hi : i < n₀
⊢ QuasiIsoAt (π K L n₁) i

Tactic:
rw [quasiIsoAt_iff' _ (i - 1) i (i + 1) (by simp) (by simp)]
    let φ := (shortComplexFunctor' C _ (i - 1) i (i + 1)).map (π K L n₁)
    have : IsIso φ.τ₁ := isIso_π_f ..
    have : IsIso φ.τ₂ := isIso_π_f ..
    have : IsIso φ.τ₃ := isIso_π_f ..
    exact ShortComplex.quasiIso_of_epi_of_isIso_of_mono φ

Goal state after:
Module: Mathlib.AlgebraicTopology.ModelCategory.FibrantObjectHomotopy

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : ModelCategory C
⊢ (localizerMorphism C).IsLocalizedEquivalence

Tactic:
by
  let Hfib := (weakEquivalences (HoCat C)).Localization
  let Lfibπ : HoCat C ⥤ Hfib := (weakEquivalences (FibrantObject.HoCat C)).Q
  let Lfib : FibrantObject C ⥤ Hfib := toHoCat ⋙ Lfibπ
  let H := (weakEquivalences C).Localization
  let L : C ⥤ H := (weakEquivalences C).Q
  let F := (localizerMorphism C).localizedFunctor Lfib L
  let eF : ι ⋙ L ≅ Lfib ⋙ F := CatCommSq.iso (localizerMorphism C).functor Lfib L F
  let eF' : HoCat.toLocalization L ≅ Lfibπ ⋙ F :=
    CategoryTheory.Quotient.natIsoLift _
      (HoCat.toHoCatCompToLocalizationIso L ≪≫ eF ≪≫ associator _ _ _)
  let G : H ⥤ Hfib := (HoCat.localizerMorphismResolution C).localizedFunctor L Lfibπ
  let eG : HoCat.resolution ⋙ Lfibπ ≅ L ⋙ G :=
    CatCommSq.iso (HoCat.localizerMorphismResolution C).functor L Lfibπ G
  have : Localization.Lifting L (weakEquivalences C)
      (HoCat.resolution ⋙ HoCat.toLocalization L) (G ⋙ F) :=
    ⟨(associator _ _ _).symm ≪≫ isoWhiskerRight eG.symm _ ≪≫
      associator _ _ _ ≪≫ isoWhiskerLeft _ eF'.symm⟩
  have : Localization.Lifting Lfib (weakEquivalences (FibrantObject C))
        (ι ⋙ HoCat.resolution ⋙ Lfibπ) (F ⋙ G) :=
    ⟨(associator _ _ _).symm ≪≫ isoWhiskerRight eF.symm G ≪≫
      associator _ _ _ ≪≫ isoWhiskerLeft _ eG.symm⟩
  let E : Hfib ≌ H := CategoryTheory.Equivalence.mk F G
    (Localization.liftNatIso Lfib (weakEquivalences _) Lfib (ι ⋙ HoCat.resolution ⋙ Lfibπ) _ _
        (asIso (whiskerRight HoCat.ιCompResolutionNatTrans Lfibπ) ≪≫ associator _ _ _))
    (Localization.liftNatIso L (weakEquivalences _)
      (HoCat.resolution ⋙ HoCat.toLocalization L) L _ _
      (asIso (HoCat.resolutionCompToLocalizationNatTrans L)).symm)
  have : F.IsEquivalence := E.isEquivalence_functor
  exact LocalizerMorphism.IsLocalizedEquivalence.mk' (localizerMorphism C) Lfib L F

Goal state after:
Module: Mathlib.CategoryTheory.GuitartExact.VerticalComposition

Goal state before:
pp: C₁ : Type u_1
C₂ : Type u_2
C₃ : Type u_3
D₁ : Type u_4
D₂ : Type u_5
D₃ : Type u_6
inst✝⁷ : Category.{v_1, u_1} C₁
inst✝⁶ : Category.{v_2, u_2} C₂
inst✝⁵ : Category.{v_3, u_3} C₃
inst✝⁴ : Category.{v_4, u_4} D₁
inst✝³ : Category.{v_5, u_5} D₂
inst✝² : Category.{v_6, u_6} D₃
H₁ : C₁ ⥤ D₁
L₁ : C₁ ⥤ C₂
R₁ : D₁ ⥤ D₂
H₂ : C₂ ⥤ D₂
w : TwoSquare H₁ L₁ R₁ H₂
L₂ : C₂ ⥤ C₃
R₂ : D₂ ⥤ D₃
H₃ : C₃ ⥤ D₃
w' : TwoSquare H₂ L₂ R₂ H₃
L₁₂ : C₁ ⥤ C₃
R₁₂ : D₁ ⥤ D₃
eL : L₁ ⋙ L₂ ≅ L₁₂
eR : R₁ ⋙ R₂ ≅ R₁₂
inst✝¹ : R₁.EssSurj
inst✝ : w.GuitartExact
h : ((w ≫ᵥ w').whiskerVertical eL.hom eR.inv).GuitartExact
⊢ w'.GuitartExact

Tactic:
whiskerVertical_iff

Goal state after:
C₁ : Type u_1
C₂ : Type u_2
C₃ : Type u_3
D₁ : Type u_4
D₂ : Type u_5
D₃ : Type u_6
inst✝⁷ : Category.{v_1, u_1} C₁
inst✝⁶ : Category.{v_2, u_2} C₂
inst✝⁵ : Category.{v_3, u_3} C₃
inst✝⁴ : Category.{v_4, u_4} D₁
inst✝³ : Category.{v_5, u_5} D₂
inst✝² : Category.{v_6, u_6} D₃
H₁ : C₁ ⥤ D₁
L₁ : C₁ ⥤ C₂
R₁ : D₁ ⥤ D₂
H₂ : C₂ ⥤ D₂
w : TwoSquare H₁ L₁ R₁ H₂
L₂ : C₂ ⥤ C₃
R₂ : D₂ ⥤ D₃
H₃ : C₃ ⥤ D₃
w' : TwoSquare H₂ L₂ R₂ H₃
L₁₂ : C₁ ⥤ C₃
R₁₂ : D₁ ⥤ D₃
eL : L₁ ⋙ L₂ ≅ L₁₂
eR : R₁ ⋙ R₂ ≅ R₁₂
inst✝¹ : R₁.EssSurj
inst✝ : w.GuitartExact
h : (w ≫ᵥ w').GuitartExact
⊢ w'.GuitartExact
Module: Mathlib.CategoryTheory.GuitartExact.VerticalComposition

Goal state before:
pp: C₁ : Type u_1
C₂ : Type u_2
C₃ : Type u_3
D₁ : Type u_4
D₂ : Type u_5
D₃ : Type u_6
inst✝⁵ : Category.{v_1, u_1} C₁
inst✝⁴ : Category.{v_2, u_2} C₂
inst✝³ : Category.{v_3, u_3} C₃
inst✝² : Category.{v_4, u_4} D₁
inst✝¹ : Category.{v_5, u_5} D₂
inst✝ : Category.{v_6, u_6} D₃
H₁ : C₁ ⥤ D₁
L₁ : C₁ ⥤ C₂
R₁ : D₁ ⥤ D₂
H₂ : C₂ ⥤ D₂
w : TwoSquare H₁ L₁ R₁ H₂
H₃ : C₃ ⥤ D₃
E : C₂ ≌ C₃
E' : D₂ ≌ D₃
w' : H₂ ⋙ E'.functor ≅ E.functor ⋙ H₃
L₁₂ : C₁ ⥤ C₃
R₁₂ : D₁ ⥤ D₃
eL : L₁ ⋙ E.functor ≅ L₁₂
eR : R₁ ⋙ E'.functor ≅ R₁₂
⊢ ((w ≫ᵥ w'.hom).whiskerVertical eL.hom eR.inv).GuitartExact ↔ (w ≫ᵥ w'.hom).GuitartExact

Tactic:
whiskerVertical_iff

Goal state after:
C₁ : Type u_1
C₂ : Type u_2
C₃ : Type u_3
D₁ : Type u_4
D₂ : Type u_5
D₃ : Type u_6
inst✝⁵ : Category.{v_1, u_1} C₁
inst✝⁴ : Category.{v_2, u_2} C₂
inst✝³ : Category.{v_3, u_3} C₃
inst✝² : Category.{v_4, u_4} D₁
inst✝¹ : Category.{v_5, u_5} D₂
inst✝ : Category.{v_6, u_6} D₃
H₁ : C₁ ⥤ D₁
L₁ : C₁ ⥤ C₂
R₁ : D₁ ⥤ D₂
H₂ : C₂ ⥤ D₂
w : TwoSquare H₁ L₁ R₁ H₂
H₃ : C₃ ⥤ D₃
E : C₂ ≌ C₃
E' : D₂ ≌ D₃
w' : H₂ ⋙ E'.functor ≅ E.functor ⋙ H₃
L₁₂ : C₁ ⥤ C₃
R₁₂ : D₁ ⥤ D₃
eL : L₁ ⋙ E.functor ≅ L₁₂
eR : R₁ ⋙ E'.functor ≅ R₁₂
⊢ (w ≫ᵥ w'.hom).GuitartExact ↔ (w ≫ᵥ w'.hom).GuitartExact
Module: Mathlib.Algebra.Homology.Factorizations.CM5a

Goal state before:
pp: C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Abelian C
K L : CochainComplex C ℤ
f : K ⟶ L
inst✝² : EnoughInjectives C
n : ℤ
inst✝¹ : K.IsStrictlyGE (n + 1)
inst✝ : L.IsStrictlyGE n
⊢ ∃ K', ∃ (_ : K'.IsStrictlyGE n), ∃ ι π, Mono ι ∧ QuasiIso ι ∧ degreewiseEpiWithInjectiveKernel π ∧ ι ≫ π = f

Tactic:
have : K.IsStrictlyGE n := K.isStrictlyGE_of_ge n (n + 1) (by lia)

Goal state after:
C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Abelian C
K L : CochainComplex C ℤ
f : K ⟶ L
inst✝² : EnoughInjectives C
n : ℤ
inst✝¹ : K.IsStrictlyGE (n + 1)
inst✝ : L.IsStrictlyGE n
this : K.IsStrictlyGE n
⊢ ∃ K', ∃ (_ : K'.IsStrictlyGE n), ∃ ι π, Mono ι ∧ QuasiIso ι ∧ degreewiseEpiWithInjectiveKernel π ∧ ι ≫ π = f
Module: Mathlib.CategoryTheory.Functor.Derived.RightDerived

Goal state before:
pp: C : Type u_1
D : Type u_3
H : Type u_2
inst✝³ : Category.{v_1, u_1} C
inst✝² : Category.{v_3, u_3} D
inst✝¹ : Category.{v_5, u_2} H
F : C ⥤ H
L : C ⥤ D
W : MorphismProperty C
inst✝ : L.IsLocalization W
⊢ F.HasRightDerivedFunctor W ↔ L.HasLeftKanExtension F

Tactic:
have : HasRightDerivedFunctor F W ↔ HasLeftKanExtension W.Q F :=
    ⟨fun h => h.hasLeftKanExtension', fun h => ⟨h⟩⟩

Goal state after:
C : Type u_1
D : Type u_3
H : Type u_2
inst✝³ : Category.{v_1, u_1} C
inst✝² : Category.{v_3, u_3} D
inst✝¹ : Category.{v_5, u_2} H
F : C ⥤ H
L : C ⥤ D
W : MorphismProperty C
inst✝ : L.IsLocalization W
this : F.HasRightDerivedFunctor W ↔ W.Q.HasLeftKanExtension F
⊢ F.HasRightDerivedFunctor W ↔ L.HasLeftKanExtension F
Module: Mathlib.Algebra.Homology.Factorizations.CM5a

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : Abelian C
K : CochainComplex C ℤ
inst✝¹ : EnoughInjectives C
n : ℤ
inst✝ : K.IsStrictlyGE n
⊢ ∃ L i, ∃ (_ : QuasiIso i) (_ : ∀ (n : ℤ), Injective (L.X n)), L.IsStrictlyGE n

Tactic:
have : HasDerivedCategory C := MorphismProperty.HasLocalization.standard _
  obtain ⟨L, i, _, _, hL, _⟩ := exists_mono_quasiIso_injective K (n - 1) n (by simp)
  have : L.IsGE n := by
    have hK : K.IsGE n := inferInstance
    rw [← DerivedCategory.isGE_Q_obj_iff] at hK ⊢
    exact DerivedCategory.TStructure.t.isGE_of_iso (asIso (DerivedCategory.Q.map i)) n
  have : QuasiIso (L.πTruncGE n) := (L.quasiIso_πTruncGE_iff n).mpr inferInstance
  have : Injective (L.opcycles n) :=
    L.injective_opcycles (n - 1) n (L.exactAt_of_isGE n (n - 1))
  -- note: this `i ≫ L.πTruncGE n` is a mono in degrees > n, but it may not be in degree n
  refine ⟨L.truncGE n, i ≫ L.πTruncGE n, inferInstance, fun q ↦ ?_, inferInstance⟩
  obtain h | rfl | h := lt_trichotomy q n
  · exact (isZero_of_isStrictlyGE _ n _ h).injective
  · exact Injective.of_iso (L.truncGEXIsoOpcycles q).symm inferInstance
  · exact Injective.of_iso (L.truncGEXIso n q h).symm (hL q)

Goal state after:
Module: Mathlib.Algebra.Homology.HomotopyCategory.KProjective

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
K : CochainComplex C ℤ
⊢ K.IsKProjective ↔
    (HomotopyCategory.subcategoryAcyclic C).leftOrthogonal ((HomotopyCategory.quotient C (ComplexShape.up ℤ)).obj K)

Tactic:
refine ⟨fun _ L f hL ↦ ?_,
      fun hK ↦ ⟨fun {L} f hL ↦ ⟨HomotopyCategory.homotopyOfEq _ _ ?_⟩⟩⟩

Goal state after:
case refine_1
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
K : CochainComplex C ℤ
x✝ : K.IsKProjective
L : HomotopyCategory C (ComplexShape.up ℤ)
f : (HomotopyCategory.quotient C (ComplexShape.up ℤ)).obj K ⟶ L
hL : HomotopyCategory.subcategoryAcyclic C L
⊢ f = 0

case refine_2
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
K : CochainComplex C ℤ
hK : (HomotopyCategory.subcategoryAcyclic C).leftOrthogonal ((HomotopyCategory.quotient C (ComplexShape.up ℤ)).obj K)
L : CochainComplex C ℤ
f : K ⟶ L
hL : HomologicalComplex.Acyclic L
⊢ (HomotopyCategory.quotient C (ComplexShape.up ℤ)).map f = (HomotopyCategory.quotient C (ComplexShape.up ℤ)).map 0