mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.CategoryTheory.Localization.CalculusOfFractions.Fractions

Goal state before:
pp: case refine_2
C : Type u_1
D : Type u_2
inst✝³ : Category.{v_1, u_1} C
inst✝² : Category.{v_2, u_2} D
L : C ⥤ D
W : MorphismProperty C
inst✝¹ : L.IsLocalization W
inst✝ : W.HasLeftCalculusOfFractions
X Y : C
f f' : L.obj X ⟶ L.obj Y
φ : W.LeftFraction X Y
hφ : f = φ.map L ⋯
φ' : W.LeftFraction X Y
hφ' : f' = φ'.map L ⋯
α : W.LeftFraction φ.Y' φ'.Y'
hα : { X' := Y, s := φ.s, hs := ⋯, f := φ'.s }.f ≫ α.s = { X' := Y, s := φ.s, hs := ⋯, f := φ'.s }.s ≫ α.f
ψ : W.LeftFraction₂ X Y := { Y' := α.Y', f := φ.f ≫ α.f, f' := φ'.f ≫ α.s, s := φ'.s ≫ α.s, hs := ⋯ }
this : IsIso (L.map (φ'.s ≫ α.s))
⊢ φ'.map L ⋯ ≫ L.map φ'.s ≫ L.map α.s = ψ.snd.map L ⋯ ≫ L.map (φ'.s ≫ α.s)

Tactic:
LeftFraction.map_comp_map_s_assoc,

Goal state after:
case refine_2
C : Type u_1
D : Type u_2
inst✝³ : Category.{v_1, u_1} C
inst✝² : Category.{v_2, u_2} D
L : C ⥤ D
W : MorphismProperty C
inst✝¹ : L.IsLocalization W
inst✝ : W.HasLeftCalculusOfFractions
X Y : C
f f' : L.obj X ⟶ L.obj Y
φ : W.LeftFraction X Y
hφ : f = φ.map L ⋯
φ' : W.LeftFraction X Y
hφ' : f' = φ'.map L ⋯
α : W.LeftFraction φ.Y' φ'.Y'
hα : { X' := Y, s := φ.s, hs := ⋯, f := φ'.s }.f ≫ α.s = { X' := Y, s := φ.s, hs := ⋯, f := φ'.s }.s ≫ α.f
ψ : W.LeftFraction₂ X Y := { Y' := α.Y', f := φ.f ≫ α.f, f' := φ'.f ≫ α.s, s := φ'.s ≫ α.s, hs := ⋯ }
this : IsIso (L.map (φ'.s ≫ α.s))
⊢ L.map φ'.f ≫ L.map α.s = ψ.snd.map L ⋯ ≫ L.map (φ'.s ≫ α.s)
Module: Mathlib.CategoryTheory.Localization.CalculusOfFractions.Fractions

Goal state before:
pp: case refine_1
C : Type u_1
D : Type u_2
inst✝³ : Category.{v_1, u_1} C
inst✝² : Category.{v_2, u_2} D
L : C ⥤ D
W : MorphismProperty C
inst✝¹ : L.IsLocalization W
inst✝ : W.HasLeftCalculusOfFractions
X Y : C
f f' f'' : L.obj X ⟶ L.obj Y
α : W.LeftFraction₂ X Y
hα : f = α.fst.map L ⋯
hα' : f' = α.snd.map L ⋯
β : W.LeftFraction X Y
hβ : f'' = β.map L ⋯
γ : W.LeftFraction α.Y' β.Y'
hγ : β.s ≫ γ.s = α.s ≫ γ.f
ψ : W.LeftFraction₃ X Y := { Y' := γ.Y', f := α.f ≫ γ.f, f' := α.f' ≫ γ.f, f'' := β.f ≫ γ.s, s := β.s ≫ γ.s, hs := ⋯ }
this : IsIso (L.map (β.s ≫ γ.s))
⊢ α.fst.map L ⋯ ≫ L.map α.s ≫ L.map γ.f = L.map ψ.fst.f

Tactic:
LeftFraction.map_comp_map_s_assoc,

Goal state after:
case refine_1
C : Type u_1
D : Type u_2
inst✝³ : Category.{v_1, u_1} C
inst✝² : Category.{v_2, u_2} D
L : C ⥤ D
W : MorphismProperty C
inst✝¹ : L.IsLocalization W
inst✝ : W.HasLeftCalculusOfFractions
X Y : C
f f' f'' : L.obj X ⟶ L.obj Y
α : W.LeftFraction₂ X Y
hα : f = α.fst.map L ⋯
hα' : f' = α.snd.map L ⋯
β : W.LeftFraction X Y
hβ : f'' = β.map L ⋯
γ : W.LeftFraction α.Y' β.Y'
hγ : β.s ≫ γ.s = α.s ≫ γ.f
ψ : W.LeftFraction₃ X Y := { Y' := γ.Y', f := α.f ≫ γ.f, f' := α.f' ≫ γ.f, f'' := β.f ≫ γ.s, s := β.s ≫ γ.s, hs := ⋯ }
this : IsIso (L.map (β.s ≫ γ.s))
⊢ L.map α.fst.f ≫ L.map γ.f = L.map ψ.fst.f
Module: Mathlib.CategoryTheory.Localization.CalculusOfFractions

Goal state before:
pp: case refine_2
C : Type u_1
D : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} D
W : MorphismProperty C
inst✝ : W.HasLeftCalculusOfFractions
X Y Z : C
z₁✝ : Hom W X Y
z₂✝ : Hom W Y Z
a₁ a₂ : W.LeftFraction X Y
b : W.LeftFraction Y Z
U : C
t₁ : a₁.Y' ⟶ U
t₂ : a₂.Y' ⟶ U
hst : a₁.s ≫ t₁ = a₂.s ≫ t₂
hft : a₁.f ≫ t₁ = a₂.f ≫ t₂
ht : W (a₁.s ≫ t₁)
z₁ : W.LeftFraction a₁.Y' b.Y'
fac₁ : b.f ≫ z₁.s = a₁.s ≫ z₁.f
z₂ : W.LeftFraction a₂.Y' b.Y'
fac₂ : b.f ≫ z₂.s = a₂.s ≫ z₂.f
w₁ : W.LeftFraction U z₁.Y'
w₂ : W.LeftFraction U z₂.Y'
p₁ : W.LeftFraction X Z := { Y' := w₁.Y', f := a₁.f ≫ t₁ ≫ w₁.f, s := b.s ≫ z₁.s ≫ w₁.s, hs := ⋯ }
p₂ : W.LeftFraction X Z := { Y' := w₂.Y', f := a₂.f ≫ t₂ ≫ w₂.f, s := b.s ≫ z₂.s ≫ w₂.s, hs := ⋯ }
fac₁' : b.f ≫ z₁.s ≫ w₁.s = a₁.s ≫ t₁ ≫ w₁.f
fac₂' : b.f ≫ z₂.s ≫ w₂.s = a₂.s ≫ t₂ ≫ w₂.f
⊢ mk (a₁.comp₀ b z₁) = mk (a₂.comp₀ b z₂)

Tactic:
apply Quot.sound

Goal state after:
case refine_2
C : Type u_1
D : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} D
W : MorphismProperty C
inst✝ : W.HasLeftCalculusOfFractions
X Y Z : C
z₁✝ : Hom W X Y
z₂✝ : Hom W Y Z
a₁ a₂ : W.LeftFraction X Y
b : W.LeftFraction Y Z
U : C
t₁ : a₁.Y' ⟶ U
t₂ : a₂.Y' ⟶ U
hst : a₁.s ≫ t₁ = a₂.s ≫ t₂
hft : a₁.f ≫ t₁ = a₂.f ≫ t₂
ht : W (a₁.s ≫ t₁)
z₁ : W.LeftFraction a₁.Y' b.Y'
fac₁ : b.f ≫ z₁.s = a₁.s ≫ z₁.f
z₂ : W.LeftFraction a₂.Y' b.Y'
fac₂ : b.f ≫ z₂.s = a₂.s ≫ z₂.f
w₁ : W.LeftFraction U z₁.Y'
w₂ : W.LeftFraction U z₂.Y'
p₁ : W.LeftFraction X Z := ⋯
p₂ : W.LeftFraction X Z := ⋯
fac₁' : b.f ≫ z₁.s ≫ w₁.s = a₁.s ≫ t₁ ≫ w₁.f
fac₂' : b.f ≫ z₂.s ≫ w₂.s = a₂.s ≫ t₂ ≫ w₂.f
⊢ LeftFractionRel (a₁.comp₀ b z₁) (a₂.comp₀ b z₂)
Module: Mathlib.CategoryTheory.Localization.CalculusOfFractions.Fractions

Goal state before:
pp: case refine_2
C : Type u_1
D : Type u_2
inst✝³ : Category.{v_1, u_1} C
inst✝² : Category.{v_2, u_2} D
L : C ⥤ D
W : MorphismProperty C
inst✝¹ : L.IsLocalization W
inst✝ : W.HasLeftCalculusOfFractions
X Y : C
f f' f'' : L.obj X ⟶ L.obj Y
α : W.LeftFraction₂ X Y
hα : f = α.fst.map L ⋯
hα' : f' = α.snd.map L ⋯
β : W.LeftFraction X Y
hβ : f'' = β.map L ⋯
γ : W.LeftFraction α.Y' β.Y'
hγ : β.s ≫ γ.s = α.s ≫ γ.f
ψ : W.LeftFraction₃ X Y := { Y' := γ.Y', f := α.f ≫ γ.f, f' := α.f' ≫ γ.f, f'' := β.f ≫ γ.s, s := β.s ≫ γ.s, hs := ⋯ }
this : IsIso (L.map (β.s ≫ γ.s))
⊢ α.snd.map L ⋯ ≫ L.map α.s ≫ L.map γ.f = L.map ψ.snd.f

Tactic:
LeftFraction.map_comp_map_s_assoc,

Goal state after:
case refine_2
C : Type u_1
D : Type u_2
inst✝³ : Category.{v_1, u_1} C
inst✝² : Category.{v_2, u_2} D
L : C ⥤ D
W : MorphismProperty C
inst✝¹ : L.IsLocalization W
inst✝ : W.HasLeftCalculusOfFractions
X Y : C
f f' f'' : L.obj X ⟶ L.obj Y
α : W.LeftFraction₂ X Y
hα : f = α.fst.map L ⋯
hα' : f' = α.snd.map L ⋯
β : W.LeftFraction X Y
hβ : f'' = β.map L ⋯
γ : W.LeftFraction α.Y' β.Y'
hγ : β.s ≫ γ.s = α.s ≫ γ.f
ψ : W.LeftFraction₃ X Y := { Y' := γ.Y', f := α.f ≫ γ.f, f' := α.f' ≫ γ.f, f'' := β.f ≫ γ.s, s := β.s ≫ γ.s, hs := ⋯ }
this : IsIso (L.map (β.s ≫ γ.s))
⊢ L.map α.snd.f ≫ L.map γ.f = L.map ψ.snd.f
Module: Mathlib.CategoryTheory.Localization.CalculusOfFractions.Fractions

Goal state before:
pp: case refine_3
C : Type u_1
D : Type u_2
inst✝³ : Category.{v_1, u_1} C
inst✝² : Category.{v_2, u_2} D
L : C ⥤ D
W : MorphismProperty C
inst✝¹ : L.IsLocalization W
inst✝ : W.HasLeftCalculusOfFractions
X Y : C
f f' f'' : L.obj X ⟶ L.obj Y
α : W.LeftFraction₂ X Y
hα : f = α.fst.map L ⋯
hα' : f' = α.snd.map L ⋯
β : W.LeftFraction X Y
hβ : f'' = β.map L ⋯
γ : W.LeftFraction α.Y' β.Y'
hγ : β.s ≫ γ.s = α.s ≫ γ.f
ψ : W.LeftFraction₃ X Y := { Y' := γ.Y', f := α.f ≫ γ.f, f' := α.f' ≫ γ.f, f'' := β.f ≫ γ.s, s := β.s ≫ γ.s, hs := ⋯ }
this : IsIso (L.map (β.s ≫ γ.s))
⊢ β.map L ⋯ ≫ L.map β.s ≫ L.map γ.s = ψ.thd.map L ⋯ ≫ L.map (β.s ≫ γ.s)

Tactic:
LeftFraction.map_comp_map_s_assoc,

Goal state after:
case refine_3
C : Type u_1
D : Type u_2
inst✝³ : Category.{v_1, u_1} C
inst✝² : Category.{v_2, u_2} D
L : C ⥤ D
W : MorphismProperty C
inst✝¹ : L.IsLocalization W
inst✝ : W.HasLeftCalculusOfFractions
X Y : C
f f' f'' : L.obj X ⟶ L.obj Y
α : W.LeftFraction₂ X Y
hα : f = α.fst.map L ⋯
hα' : f' = α.snd.map L ⋯
β : W.LeftFraction X Y
hβ : f'' = β.map L ⋯
γ : W.LeftFraction α.Y' β.Y'
hγ : β.s ≫ γ.s = α.s ≫ γ.f
ψ : W.LeftFraction₃ X Y := { Y' := γ.Y', f := α.f ≫ γ.f, f' := α.f' ≫ γ.f, f'' := β.f ≫ γ.s, s := β.s ≫ γ.s, hs := ⋯ }
this : IsIso (L.map (β.s ≫ γ.s))
⊢ L.map β.f ≫ L.map γ.s = ψ.thd.map L ⋯ ≫ L.map (β.s ≫ γ.s)
Module: Mathlib.CategoryTheory.Localization.Triangulated

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝¹¹ : Category.{v_1, u_1} C
inst✝¹⁰ : Category.{v_2, u_2} D
L : C ⥤ D
inst✝⁹ : HasShift C ℤ
inst✝⁸ : Preadditive C
inst✝⁷ : HasZeroObject C
inst✝⁶ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁵ : Pretriangulated C
inst✝⁴ : HasShift D ℤ
inst✝³ : L.CommShift ℤ
W : MorphismProperty C
inst✝² : L.IsLocalization W
inst✝¹ : W.HasLeftCalculusOfFractions
inst✝ : W.IsCompatibleWithTriangulation
T₁ T₂ : Triangle C
hT₁ : T₁ ∈ distinguishedTriangles
hT₂ : T₂ ∈ distinguishedTriangles
a : L.obj T₁.obj₁ ⟶ L.obj T₂.obj₁
b : L.obj T₁.obj₂ ⟶ L.obj T₂.obj₂
fac✝ : L.map T₁.mor₁ ≫ b = a ≫ L.map T₂.mor₁
α : W.LeftFraction T₁.obj₁ T₂.obj₁
hα : a = α.map L ⋯
β : W.LeftFraction α.Y' T₂.obj₂
hβ : T₂.mor₁ ≫ β.s = α.s ≫ β.f
γ : W.LeftFraction T₁.obj₂ β.Y'
hγ : b ≫ L.map β.s = γ.map L ⋯
Z₂ : C
σ : γ.Y' ⟶ Z₂
hσ : W σ
fac : α.f ≫ β.f ≫ γ.s ≫ σ = T₁.mor₁ ≫ γ.f ≫ σ
Y₃ : C
g : Z₂ ⟶ Y₃
h : Y₃ ⟶ (shiftFunctor C 1).obj α.Y'
T₃ : Triangle C := Triangle.mk (β.f ≫ γ.s ≫ σ) g h
hT₃ : T₃ ∈ distinguishedTriangles
hβγσ : W (β.s ≫ γ.s ≫ σ)
⊢ ∃ φ, φ.hom₁ = a ∧ φ.hom₂ = b

Tactic:
obtain ⟨ψ₃, hψ₃, hψ₁, hψ₂⟩ := MorphismProperty.compatible_with_triangulation
    T₂ T₃ hT₂ hT₃ α.s (β.s ≫ γ.s ≫ σ) α.hs hβγσ (by dsimp [T₃]; rw [reassoc_of% hβ])

Goal state after:
C : Type u_1
D : Type u_2
inst✝¹¹ : Category.{v_1, u_1} C
inst✝¹⁰ : Category.{v_2, u_2} D
L : C ⥤ D
inst✝⁹ : HasShift C ℤ
inst✝⁸ : Preadditive C
inst✝⁷ : HasZeroObject C
inst✝⁶ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁵ : Pretriangulated C
inst✝⁴ : HasShift D ℤ
inst✝³ : L.CommShift ℤ
W : MorphismProperty C
inst✝² : L.IsLocalization W
inst✝¹ : W.HasLeftCalculusOfFractions
inst✝ : W.IsCompatibleWithTriangulation
T₁ T₂ : Triangle C
hT₁ : T₁ ∈ distinguishedTriangles
hT₂ : T₂ ∈ distinguishedTriangles
a : L.obj T₁.obj₁ ⟶ L.obj T₂.obj₁
b : L.obj T₁.obj₂ ⟶ L.obj T₂.obj₂
fac✝ : L.map T₁.mor₁ ≫ b = a ≫ L.map T₂.mor₁
α : W.LeftFraction T₁.obj₁ T₂.obj₁
hα : a = α.map L ⋯
β : W.LeftFraction α.Y' T₂.obj₂
hβ : T₂.mor₁ ≫ β.s = α.s ≫ β.f
γ : W.LeftFraction T₁.obj₂ β.Y'
hγ : b ≫ L.map β.s = γ.map L ⋯
Z₂ : C
σ : γ.Y' ⟶ Z₂
hσ : W σ
fac : α.f ≫ β.f ≫ γ.s ≫ σ = T₁.mor₁ ≫ γ.f ≫ σ
Y₃ : C
g : Z₂ ⟶ Y₃
h : Y₃ ⟶ (shiftFunctor C 1).obj α.Y'
T₃ : Triangle C := Triangle.mk (β.f ≫ γ.s ≫ σ) g h
hT₃ : T₃ ∈ distinguishedTriangles
hβγσ : W (β.s ≫ γ.s ≫ σ)
ψ₃ : T₂.obj₃ ⟶ T₃.obj₃
hψ₃ : W ψ₃
hψ₁ : T₂.mor₂ ≫ ψ₃ = (β.s ≫ γ.s ≫ σ) ≫ T₃.mor₂
hψ₂ : T₂.mor₃ ≫ (shiftFunctor C 1).map α.s = ψ₃ ≫ T₃.mor₃
⊢ ∃ φ, φ.hom₁ = a ∧ φ.hom₂ = b
Module: Mathlib.CategoryTheory.Shift.ShiftedHom

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
M : Type u_4
inst✝¹ : AddMonoid M
inst✝ : HasShift C M
X Y : C
a : M
f : ShiftedHom X Y a
m₀ : M
hm₀ : m₀ = 0
⊢ f.comp (mk₀ m₀ hm₀ (𝟙 Y)) ⋯ = f

Tactic:
by
  simp [comp_mk₀]

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.Subcategory

Goal state before:
pp: case refine_1
C : Type u_1
inst✝¹⁵ : Category.{v_1, u_1} C
inst✝¹⁴ : HasZeroObject C
inst✝¹³ : HasShift C ℤ
inst✝¹² : Preadditive C
inst✝¹¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹⁰ : Pretriangulated C
D : Type u_2
inst✝⁹ : Category.{v_2, u_2} D
inst✝⁸ : Preadditive D
inst✝⁷ : HasZeroObject D
inst✝⁶ : HasShift D ℤ
inst✝⁵ : ∀ (n : ℤ), (shiftFunctor D n).Additive
inst✝⁴ : Pretriangulated D
E : Type u_3
inst✝³ : Category.{v_3, u_3} E
inst✝² : HasShift E ℤ
P : ObjectProperty C
inst✝¹ : IsTriangulated C
inst✝ : P.IsTriangulated
X' X Y : C
f₁ f₂ : X ⟶ Y
s : X' ⟶ X
Z : C
g : X ⟶ Z
h : Z ⟶ (shiftFunctor C 1).obj X'
H : Triangle.mk s g h ∈ distinguishedTriangles
mem : P Z
hf₁ : s ≫ f₁ = s ≫ f₂
hf₂ : s ≫ (f₁ - f₂) = 0
q : (Triangle.mk s g h).obj₃ ⟶ Y
hq : f₁ - f₂ = (Triangle.mk s g h).mor₂ ≫ q
Y' : C
r : Y ⟶ Y'
t : Y' ⟶ (shiftFunctor C 1).obj (Triangle.mk s g h).obj₃
mem' : Triangle.mk q r t ∈ distinguishedTriangles
⊢ P.trW r

pp: case refine_2
C : Type u_1
inst✝¹⁵ : Category.{v_1, u_1} C
inst✝¹⁴ : HasZeroObject C
inst✝¹³ : HasShift C ℤ
inst✝¹² : Preadditive C
inst✝¹¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹⁰ : Pretriangulated C
D : Type u_2
inst✝⁹ : Category.{v_2, u_2} D
inst✝⁸ : Preadditive D
inst✝⁷ : HasZeroObject D
inst✝⁶ : HasShift D ℤ
inst✝⁵ : ∀ (n : ℤ), (shiftFunctor D n).Additive
inst✝⁴ : Pretriangulated D
E : Type u_3
inst✝³ : Category.{v_3, u_3} E
inst✝² : HasShift E ℤ
P : ObjectProperty C
inst✝¹ : IsTriangulated C
inst✝ : P.IsTriangulated
X' X Y : C
f₁ f₂ : X ⟶ Y
s : X' ⟶ X
Z : C
g : X ⟶ Z
h : Z ⟶ (shiftFunctor C 1).obj X'
H : Triangle.mk s g h ∈ distinguishedTriangles
mem : P Z
hf₁ : s ≫ f₁ = s ≫ f₂
hf₂ : s ≫ (f₁ - f₂) = 0
q : (Triangle.mk s g h).obj₃ ⟶ Y
hq : f₁ - f₂ = (Triangle.mk s g h).mor₂ ≫ q
Y' : C
r : Y ⟶ Y'
t : Y' ⟶ (shiftFunctor C 1).obj (Triangle.mk s g h).obj₃
mem' : Triangle.mk q r t ∈ distinguishedTriangles
⊢ f₁ ≫ r = f₂ ≫ r

Tactic:
· exact ⟨_, _, _, rot_of_distTriang _ mem', P.le_shift _ _ mem⟩

Goal state after:
case refine_2
C : Type u_1
inst✝¹⁵ : Category.{v_1, u_1} C
inst✝¹⁴ : HasZeroObject C
inst✝¹³ : HasShift C ℤ
inst✝¹² : Preadditive C
inst✝¹¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹⁰ : Pretriangulated C
D : Type u_2
inst✝⁹ : Category.{v_2, u_2} D
inst✝⁸ : Preadditive D
inst✝⁷ : HasZeroObject D
inst✝⁶ : HasShift D ℤ
inst✝⁵ : ∀ (n : ℤ), (shiftFunctor D n).Additive
inst✝⁴ : Pretriangulated D
E : Type u_3
inst✝³ : Category.{v_3, u_3} E
inst✝² : HasShift E ℤ
P : ObjectProperty C
inst✝¹ : IsTriangulated C
inst✝ : P.IsTriangulated
X' X Y : C
f₁ f₂ : X ⟶ Y
s : X' ⟶ X
Z : C
g : X ⟶ Z
h : Z ⟶ (shiftFunctor C 1).obj X'
H : Triangle.mk s g h ∈ distinguishedTriangles
mem : P Z
hf₁ : s ≫ f₁ = s ≫ f₂
hf₂ : s ≫ (f₁ - f₂) = 0
q : (Triangle.mk s g h).obj₃ ⟶ Y
hq : f₁ - f₂ = (Triangle.mk s g h).mor₂ ≫ q
Y' : C
r : Y ⟶ Y'
t : Y' ⟶ (shiftFunctor C 1).obj (Triangle.mk s g h).obj₃
mem' : Triangle.mk q r t ∈ distinguishedTriangles
⊢ f₁ ≫ r = f₂ ≫ r
Module: Mathlib.Algebra.Homology.DerivedCategory.HomologySequence

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasDerivedCategory C
K L : CochainComplex C ℤ
n : ℤ
f : K ⟶ (shiftFunctor (CochainComplex C ℤ) n).obj L
a a' : ℤ
h : n + a = a'
⊢ (homologyFunctor C 0).shiftMap (ShiftedHom.map f Q) a a' h =
    (homologyFunctorFactors C a).hom.app K ≫
      (HomologicalComplex.homologyFunctor C (ComplexShape.up ℤ) 0).shiftMap f a a' h ≫
        (homologyFunctorFactors C a').inv.app L

Tactic:
rw [← ShiftedHom.map_naturality_1 f (quotientCompQhIso C),
    ShiftedHom.mk₀_comp, ShiftedHom.comp_mk₀,
    Functor.shiftMap_comp', Functor.shiftMap_comp,
    ShiftedHom.comp_map, shiftMap_homologyFunctor_map_Qh ..,
    homologyFunctorFactorsh_hom_app_quotient_obj,
    homologyFunctorFactorsh_inv_app_quotient_obj,
    HomotopyCategory.homologyFunctor_shiftMap]

Goal state after:
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasDerivedCategory C
K L : CochainComplex C ℤ
n : ℤ
f : K ⟶ (shiftFunctor (CochainComplex C ℤ) n).obj L
a a' : ℤ
h : n + a = a'
⊢ ((homologyFunctor C 0).shift a).map ((quotientCompQhIso C).inv.app K) ≫
      (((homologyFunctor C a).map ((quotientCompQhIso C).hom.app K) ≫
            (homologyFunctorFactors C a).hom.app K ≫
              (HomotopyCategory.homologyFunctorFactors C (ComplexShape.up ℤ) a).inv.app K) ≫
          ((HomotopyCategory.homologyFunctorFactors C (ComplexShape.up ℤ) a).hom.app K ≫
              (HomologicalComplex.homologyFunctor C (ComplexShape.up ℤ) 0).shiftMap f a a' ⋯ ≫
                (HomotopyCategory.homologyFunctorFactors C (ComplexShape.up ℤ) a').inv.app L) ≫
            (HomotopyCategory.homologyFunctorFactors C (ComplexShape.up ℤ) a').hom.app L ≫
              (homologyFunctorFactors C a').inv.app L ≫ (homologyFunctor C a').map ((quotientCompQhIso C).inv.app L)) ≫
        ((homologyFunctor C 0).shift a').map ((quotientCompQhIso C).hom.app L) =
    (homologyFunctorFactors C a).hom.app K ≫
      (HomologicalComplex.homologyFunctor C (ComplexShape.up ℤ) 0).shiftMap f a a' h ≫
        (homologyFunctorFactors C a').inv.app L
Module: Mathlib.Algebra.Homology.Embedding.IsSupported

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝² : Category.{v_1, u_3} C
inst✝¹ : HasZeroMorphisms C
K L : HomologicalComplex C c'
e' : K ≅ L
φ : K ⟶ L
e : c.Embedding c'
inst✝ : K.IsStrictlySupported e
i' : ι'
hi' : ∀ (i : ι), e.f i ≠ i'
⊢ K.ExactAt i'

Tactic:
rw [exactAt_iff]
    exact ShortComplex.exact_of_isZero_X₂ _ (K.isZero_X_of_isStrictlySupported e i' hi')

Goal state after:
Module: Mathlib.Algebra.Homology.Embedding.Extend

Goal state before:
pp: case neg
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝² : Category.{v_1, u_3} C
inst✝¹ : HasZeroObject C
inst✝ : HasZeroMorphisms C
K L M : HomologicalComplex C c
φ : K ⟶ L
φ' : L ⟶ M
e : c.Embedding c'
i' j' : ι'
x✝ : c'.Rel i' j'
hi : ¬∃ i, e.f i = i'
⊢ extend.mapX φ (e.r i') ≫ (L.extend e).d i' j' = (K.extend e).d i' j' ≫ extend.mapX φ (e.r j')

Tactic:
have hi' := e.r_eq_none i' (fun i'' hi'' => hi ⟨i'', hi''⟩)

Goal state after:
case neg
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝² : Category.{v_1, u_3} C
inst✝¹ : HasZeroObject C
inst✝ : HasZeroMorphisms C
K L M : HomologicalComplex C c
φ : K ⟶ L
φ' : L ⟶ M
e : c.Embedding c'
i' j' : ι'
x✝ : c'.Rel i' j'
hi : ¬∃ i, e.f i = i'
hi' : e.r i' = none
⊢ extend.mapX φ (e.r i') ≫ (L.extend e).d i' j' = (K.extend e).d i' j' ≫ extend.mapX φ (e.r j')
Module: Mathlib.Algebra.Homology.Embedding.Boundary

Goal state before:
pp: case neg
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
e : c.Embedding c'
inst✝ : e.IsTruncGE
j k : ι
hjk : c.next j = k
hj : ¬c'.Rel (e.f j) (c'.next (e.f j))
⊢ e.f j = e.f k

Tactic:
← hjk,

Goal state after:
case neg
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
e : c.Embedding c'
inst✝ : e.IsTruncGE
j k : ι
hjk : c.next j = k
hj : ¬c'.Rel (e.f j) (c'.next (e.f j))
⊢ e.f j = e.f (c.next j)
Module: Mathlib.Algebra.Homology.Embedding.HomEquiv

Goal state before:
pp: case pos
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
e : c.Embedding c'
C : Type u_3
inst✝³ : Category.{v_1, u_3} C
inst✝² : HasZeroMorphisms C
inst✝¹ : HasZeroObject C
K : HomologicalComplex C c'
L : HomologicalComplex C c
inst✝ : e.IsRelIff
φ : K.restriction e ⟶ L
hφ : e.HasLift φ
i' : ι'
hi' : ¬∃ i, e.f i = i'
j : ι
hij' : c'.Rel i' (e.f j)
this : (L.extend e).d i' (e.f j) = 0
⊢ 0 = K.d i' (e.f j) ≫ f φ (e.f j)

Tactic:
rw [f_eq φ rfl]

Goal state after:
case pos
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
e : c.Embedding c'
C : Type u_3
inst✝³ : Category.{v_1, u_3} C
inst✝² : HasZeroMorphisms C
inst✝¹ : HasZeroObject C
K : HomologicalComplex C c'
L : HomologicalComplex C c
inst✝ : e.IsRelIff
φ : K.restriction e ⟶ L
hφ : e.HasLift φ
i' : ι'
hi' : ¬∃ i, e.f i = i'
j : ι
hij' : c'.Rel i' (e.f j)
this : (L.extend e).d i' (e.f j) = 0
⊢ 0 = K.d i' (e.f j) ≫ (K.restrictionXIso e ⋯).inv ≫ φ.f j ≫ (L.extendXIso e ⋯).inv
Module: Mathlib.Algebra.Homology.Embedding.TruncGE

Goal state before:
pp: case neg
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝³ : Category.{v_1, u_3} C
inst✝² : HasZeroMorphisms C
K : HomologicalComplex C c'
e : c.Embedding c'
inst✝¹ : e.IsTruncGE
inst✝ : ∀ (i' : ι'), K.HasHomology i'
i : ι
hi : ¬e.BoundaryGE i
⊢ (truncGE'Map (𝟙 K) e).f i = (𝟙 (K.truncGE' e)).f i

Tactic:
simp [truncGE'Map_f_eq _ _ hi rfl]

Goal state after:
Module: Mathlib.Algebra.Homology.Embedding.TruncGE

Goal state before:
pp: case neg
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝⁵ : Category.{v_1, u_3} C
inst✝⁴ : HasZeroMorphisms C
K L M : HomologicalComplex C c'
φ : K ⟶ L
φ' : L ⟶ M
e : c.Embedding c'
inst✝³ : e.IsTruncGE
inst✝² : ∀ (i' : ι'), K.HasHomology i'
inst✝¹ : ∀ (i' : ι'), L.HasHomology i'
inst✝ : ∀ (i' : ι'), M.HasHomology i'
i : ι
hi : ¬e.BoundaryGE i
⊢ (truncGE'Map (φ ≫ φ') e).f i = (truncGE'Map φ e ≫ truncGE'Map φ' e).f i

Tactic:
simp [truncGE'Map_f_eq _ _ hi rfl]

Goal state after:
Module: Mathlib.Algebra.Homology.Embedding.TruncGEHomology

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝⁵ : Category.{v_1, u_3} C
inst✝⁴ : HasZeroMorphisms C
K L : HomologicalComplex C c'
φ : K ⟶ L
e : c.Embedding c'
inst✝³ : e.IsTruncGE
inst✝² : ∀ (i' : ι'), K.HasHomology i'
inst✝¹ : ∀ (i' : ι'), L.HasHomology i'
inst✝ : HasZeroObject C
i j k : ι
j' : ι'
hj' : e.f j = j'
hi : c.prev j = i
hk : c.next j = k
hj : e.BoundaryGE j
⊢ (K.truncGE'XIsoOpcycles e hj' hj).inv ≫
      (extend.rightHomologyData (K.truncGE' e) e hj' hi ⋯ hk ⋯ (truncGE'.homologyData K e i j k hk hj' hj).right).g' =
    (ShortComplex.RightHomologyData.canonical (K.sc j')).g' ≫ ((shortComplexFunctor C c' j').map (K.πTruncGE e)).τ₃

Tactic:
by
    have hk' : e.f k = c'.next j' := by rw [← hj', e.next_f hk]
    dsimp
    rw [extend.rightHomologyData_g' _ _ _ _ _ _ _ _ hk', πTruncGE,
        e.liftExtend_f _ _ hk', truncGE'.homologyData_right_g']
    by_cases hjk : c.Rel j k
    · simp [K.truncGE'_d_eq_fromOpcycles e hjk hj' hk' hj,
        K.restrictionToTruncGE'_f_eq_iso_hom_iso_inv e hk' (e.not_boundaryGE_next hjk)]
      rfl
    · obtain rfl : k = j := by rw [← c.next_eq_self j (by simpa only [hk] using hjk), hk]
      rw [shape _ _ _ hjk, zero_comp, comp_zero,
        K.restrictionToTruncGE'_f_eq_iso_hom_pOpcycles_iso_inv e hk' hj]
      simp only [restriction_X, restrictionXIso, eqToIso.inv, eqToIso.hom, assoc,
        eqToHom_trans_assoc, eqToHom_refl, id_comp]
      change 0 = K.fromOpcycles _ _ ≫ _
      rw [← cancel_epi (K.pOpcycles _), comp_zero, p_fromOpcycles_assoc,
        d_pOpcycles_assoc, zero_comp]

Goal state after:
Module: Mathlib.Algebra.Homology.Embedding.TruncGEHomology

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝⁴ : Category.{v_1, u_3} C
inst✝³ : HasZeroMorphisms C
K : HomologicalComplex C c'
e : c.Embedding c'
inst✝² : e.IsTruncGE
inst✝¹ : ∀ (i' : ι'), K.HasHomology i'
inst✝ : HasZeroObject C
⊢ QuasiIso (K.πTruncGE e) ↔ K.IsSupported e

Tactic:
constructor
  · intro
    refine ⟨fun i' hi' => ?_⟩
    rw [exactAt_iff_of_quasiIsoAt (K.πTruncGE e) i']
    exact (K.truncGE e).exactAt_of_isSupported e i' hi'
  · intro
    rw [quasiIso_iff]
    intro i'
    by_cases hi' : ∃ i, e.f i = i'
    · obtain ⟨i, rfl⟩ := hi'
      infer_instance
    · rw [quasiIsoAt_iff_exactAt (K.πTruncGE e) i']
      all_goals exact exactAt_of_isSupported _ e i' (by simpa using hi')

Goal state after:
Module: Mathlib.Algebra.Homology.HomologySequenceLemmas

Goal state before:
pp: C : Type u_1
ι : Type u_2
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
c : ComplexShape ι
S S₁ S₂ : ShortComplex (HomologicalComplex C c)
φ : S₁ ⟶ S₂
hS₁ : S₁.ShortExact
hS₂ : S₂.ShortExact
i : ι
⊢ (composableArrows₂ S₁ i).map' 1 2 _proof_246✝ _proof_247✝ ≫ homologyMap φ.τ₃ i =
    homologyMap φ.τ₂ i ≫ (composableArrows₂ S₂ i).map' 1 2 _proof_246✝ _proof_247✝

Tactic:
dsimp [Precomp.map]
    simp only [← homologyMap_comp, φ.comm₂₃]

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.DiagramLemmas.Four

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
R₁ R₂ : ComposableArrows C 2
φ : R₁ ⟶ R₂
hR₁ : R₁.map' 0 2 mono_of_epi_of_mono_of_mono'._proof_2 mono_of_epi_of_epi_mono'._proof_1 = 0
hR₁' : Epi (R₁.map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6 mono_of_epi_of_epi_mono'._proof_1)
hR₂ : R₂.Exact
h₀ : Epi (app' φ 0 mono_of_epi_of_mono_of_mono'._proof_2)
h₁ : Mono (app' φ 1 mono_of_epi_of_mono_of_mono'._proof_6)
⊢ Mono (app' φ 2 mono_of_epi_of_epi_mono'._proof_1)

Tactic:
let ψ : mk₃ (R₁.map' 0 1) (R₁.map' 1 2) (0 : _ ⟶ R₁.obj' 0) ⟶
    mk₃ (R₂.map' 0 1) (R₂.map' 1 2) (0 : _ ⟶ R₁.obj' 0) := homMk₃ (app' φ 0) (app' φ 1)
      (app' φ 2) (𝟙 _) (naturality' φ 0 1) (naturality' φ 1 2) (by simp)
  refine mono_of_epi_of_mono_of_mono' ψ ?_ (exact₂_mk _ (by simp) ?_)
    (hR₂.exact 0).exact_toComposableArrows h₀ h₁ (by dsimp [ψ]; infer_instance)
  · dsimp
    rw [← Functor.map_comp]
    exact hR₁
  · rw [ShortComplex.exact_iff_epi _ (by simp)]
    exact hR₁'

Goal state after: