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

Goal state before:
pp: C : Type u
inst✝⁸ : Category.{v, u} C
inst✝⁷ : HasZeroObject C
inst✝⁶ : HasShift C ℤ
inst✝⁵ : Preadditive C
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
J : Type u_1
T : J → Triangle C
hT : ∀ (j : J), T j ∈ distinguishedTriangles
inst✝³ : HasProduct fun j ↦ (T j).obj₁
inst✝² : HasProduct fun j ↦ (T j).obj₂
inst✝¹ : HasProduct fun j ↦ (T j).obj₃
inst✝ : HasProduct fun j ↦ (shiftFunctor C 1).obj (T j).obj₁
f₁ : (∏ᶜ fun j ↦ (T j).obj₁) ⟶ ∏ᶜ fun j ↦ (T j).obj₂ := Limits.Pi.map fun j ↦ (T j).mor₁
Z : C
f₂ : (∏ᶜ fun j ↦ (T j).obj₂) ⟶ Z
f₃ : Z ⟶ (shiftFunctor C 1).obj (∏ᶜ fun j ↦ (T j).obj₁)
T' : Triangle C := Triangle.mk f₁ f₂ f₃
hT' : T' ∈ distinguishedTriangles
φ : (j : J) → T' ⟶ T j :=
  fun j ↦
    completeDistinguishedTriangleMorphism T' (T j) hT' ⋯ (Pi.π (fun j ↦ (T j).obj₁) j) (Pi.π (fun j ↦ (T j).obj₂) j) ⋯
φ' : T' ⟶ productTriangle T := productTriangle.lift T φ
h₁ : φ'.hom₁ = 𝟙 T'.obj₁
h₂ : φ'.hom₂ = 𝟙 T'.obj₂
this✝ : IsIso φ'.hom₁
this : IsIso φ'.hom₂
A✝ A : C
f : A ⟶ T'.obj₃
hf : f ≫ φ'.hom₃ = 0
hf' : f ≫ T'.mor₃ = 0
g : A ⟶ T'.obj₂
hg : f = g ≫ T'.mor₂
j : J
⊢ g ≫ T'.mor₂ ≫ φ'.hom₃ ≫ Pi.π (fun j ↦ (T j).obj₃) j = 0

Tactic:
rw [← reassoc_of% hg, reassoc_of% hf, zero_comp]

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

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
T₁ T₂ : Triangle C
hT₁ : T₁ ∈ distinguishedTriangles
hT₂ : T₂ ∈ distinguishedTriangles
e₁ : T₁.obj₁ ≅ T₂.obj₁
e₃ : T₁.obj₃ ≅ T₂.obj₃
comm : T₁.mor₃ ≫ (shiftFunctor C 1).map e₁.hom = e₃.hom ≫ T₂.mor₃
⊢ ((shiftFunctor C (-1)).mapIso e₃).hom ≫ (Arrow.mk T₂.invRotate.mor₁).hom = (Arrow.mk T₁.invRotate.mor₁).hom ≫ e₁.hom

Tactic:
by
      have := (shiftFunctorCompIsoId C (1 : ℤ) (-1) (by simp)).hom.naturality e₁.hom
      dsimp at this ⊢
      simp only [comp_neg, neg_comp, assoc, neg_inj, ← Functor.map_comp_assoc, ← comm]
      simp [this]

Goal state after:
Module: Mathlib.CategoryTheory.CatCommSq

Goal state before:
pp: C₁ : Type u_1
C₂ : Type u_2
C₃ : Type u_3
C₄ : Type u_4
inst✝³ : Category.{v_1, u_1} C₁
inst✝² : Category.{v_2, u_2} C₂
inst✝¹ : Category.{v_3, u_3} C₃
inst✝ : Category.{v_4, u_4} C₄
T : C₁ ⥤ C₂
L : C₁ ≌ C₃
R : C₂ ≌ C₄
B : C₃ ⥤ C₄
h : CatCommSq T L.functor R.functor B
X : C₁
⊢ (iso T L.symm.inverse R.symm.inverse B).hom.app X = (iso T L.symm.inverse R.symm.inverse B).hom.app X

Tactic:
vInv_iso_hom_app

Goal state after:
C₁ : Type u_1
C₂ : Type u_2
C₃ : Type u_3
C₄ : Type u_4
inst✝³ : Category.{v_1, u_1} C₁
inst✝² : Category.{v_2, u_2} C₂
inst✝¹ : Category.{v_3, u_3} C₃
inst✝ : Category.{v_4, u_4} C₄
T : C₁ ⥤ C₂
L : C₁ ≌ C₃
R : C₂ ≌ C₄
B : C₃ ⥤ C₄
h : CatCommSq T L.functor R.functor B
X : C₁
⊢ R.symm.inverse.map (T.map (L.symm.counitIso.inv.app X)) ≫
      R.symm.inverse.map ((iso B L.symm.functor R.symm.functor T).inv.app (L.symm.inverse.obj X)) ≫
        R.symm.unitIso.inv.app (B.obj (L.symm.inverse.obj X)) =
    (iso T L.symm.inverse R.symm.inverse B).hom.app X
Module: Mathlib.CategoryTheory.Localization.CalculusOfFractions.Preadditive

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_2, u_2} D
inst✝² : Preadditive C
L : C ⥤ D
W : MorphismProperty C
inst✝¹ : L.IsLocalization W
inst✝ : W.HasLeftCalculusOfFractions
X Y : C
φ φ₀ : W.LeftFraction X Y
hφ₀ : neg' W (φ₀.map L ⋯) = φ₀.neg.map L ⋯
Y' : C
t₁ : φ₀.Y' ⟶ Y'
t₂ : φ.Y' ⟶ Y'
hst : φ₀.s ≫ t₁ = φ.s ≫ t₂
hft : φ₀.f ≫ t₁ = φ.f ≫ t₂
ht : W (φ₀.s ≫ t₁)
⊢ neg' W (φ₀.map L ⋯) = φ.neg.map L ⋯

Tactic:
have := inverts L W _ ht

Goal state after:
C : Type u_1
D : Type u_2
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_2, u_2} D
inst✝² : Preadditive C
L : C ⥤ D
W : MorphismProperty C
inst✝¹ : L.IsLocalization W
inst✝ : W.HasLeftCalculusOfFractions
X Y : C
φ φ₀ : W.LeftFraction X Y
hφ₀ : neg' W (φ₀.map L ⋯) = φ₀.neg.map L ⋯
Y' : C
t₁ : φ₀.Y' ⟶ Y'
t₂ : φ.Y' ⟶ Y'
hst : φ₀.s ≫ t₁ = φ.s ≫ t₂
hft : φ₀.f ≫ t₁ = φ.f ≫ t₂
ht : W (φ₀.s ≫ t₁)
this : IsIso (L.map (φ₀.s ≫ t₁))
⊢ neg' W (φ₀.map L ⋯) = φ.neg.map L ⋯
Module: Mathlib.CategoryTheory.Localization.CalculusOfFractions.Preadditive

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_2, u_2} D
inst✝² : Preadditive C
L : C ⥤ D
W : MorphismProperty C
inst✝¹ : L.IsLocalization W
inst✝ : W.HasLeftCalculusOfFractions
X Y : C
φ φ₀ : W.LeftFraction₂ X Y
hφ₁ : φ₀.fst.map L ⋯ = φ.fst.map L ⋯
hφ₂ : φ₀.snd.map L ⋯ = φ.snd.map L ⋯
hφ₀ : add' W (φ₀.fst.map L ⋯) (φ₀.snd.map L ⋯) = φ₀.add.map L ⋯
Z : C
t₁ : φ₀.Y' ⟶ Z
t₂ : φ.Y' ⟶ Z
hst : φ₀.s ≫ t₁ = φ.s ≫ t₂
hft : φ₀.f ≫ t₁ = φ.f ≫ t₂
hft' : φ₀.f' ≫ t₁ = φ.f' ≫ t₂
ht : W (φ₀.s ≫ t₁)
⊢ add' W (φ₀.fst.map L ⋯) (φ₀.snd.map L ⋯) = φ.add.map L ⋯

Tactic:
have := inverts L W _ ht

Goal state after:
C : Type u_1
D : Type u_2
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_2, u_2} D
inst✝² : Preadditive C
L : C ⥤ D
W : MorphismProperty C
inst✝¹ : L.IsLocalization W
inst✝ : W.HasLeftCalculusOfFractions
X Y : C
φ φ₀ : W.LeftFraction₂ X Y
hφ₁ : φ₀.fst.map L ⋯ = φ.fst.map L ⋯
hφ₂ : φ₀.snd.map L ⋯ = φ.snd.map L ⋯
hφ₀ : add' W (φ₀.fst.map L ⋯) (φ₀.snd.map L ⋯) = φ₀.add.map L ⋯
Z : C
t₁ : φ₀.Y' ⟶ Z
t₂ : φ.Y' ⟶ Z
hst : φ₀.s ≫ t₁ = φ.s ≫ t₂
hft : φ₀.f ≫ t₁ = φ.f ≫ t₂
hft' : φ₀.f' ≫ t₁ = φ.f' ≫ t₂
ht : W (φ₀.s ≫ t₁)
this : IsIso (L.map (φ₀.s ≫ t₁))
⊢ add' W (φ₀.fst.map L ⋯) (φ₀.snd.map L ⋯) = φ.add.map L ⋯
Module: Mathlib.CategoryTheory.Localization.LocalizerMorphism

Goal state before:
pp: C₁ : Type u₁
C₂ : Type u₂
D₁ : Type u₄
D₂ : Type u₅
inst✝⁶ : Category.{v₁, u₁} C₁
inst✝⁵ : Category.{v₂, u₂} C₂
inst✝⁴ : Category.{v₄, u₄} D₁
inst✝³ : Category.{v₅, u₅} D₂
W₁ : MorphismProperty C₁
W₂ : MorphismProperty C₂
Φ : LocalizerMorphism W₁ W₂
L₁ : C₁ ⥤ D₁
inst✝² : L₁.IsLocalization W₁
L₂ : C₂ ⥤ D₂
inst✝¹ : L₂.IsLocalization W₂
G : D₁ ⥤ D₂
inst✝ : CatCommSq Φ.functor L₁ L₂ G
hG : G.FullyFaithful
⊢ Nonempty (Φ.localizedFunctor W₁.Q W₂.Q).FullyFaithful

Tactic:
rw [Φ.nonempty_fullyFaithful_iff W₁.Q W₂.Q (Φ.localizedFunctor W₁.Q W₂.Q) L₁ L₂ G]

Goal state after:
C₁ : Type u₁
C₂ : Type u₂
D₁ : Type u₄
D₂ : Type u₅
inst✝⁶ : Category.{v₁, u₁} C₁
inst✝⁵ : Category.{v₂, u₂} C₂
inst✝⁴ : Category.{v₄, u₄} D₁
inst✝³ : Category.{v₅, u₅} D₂
W₁ : MorphismProperty C₁
W₂ : MorphismProperty C₂
Φ : LocalizerMorphism W₁ W₂
L₁ : C₁ ⥤ D₁
inst✝² : L₁.IsLocalization W₁
L₂ : C₂ ⥤ D₂
inst✝¹ : L₂.IsLocalization W₂
G : D₁ ⥤ D₂
inst✝ : CatCommSq Φ.functor L₁ L₂ G
hG : G.FullyFaithful
⊢ Nonempty G.FullyFaithful
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' : 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))
⊢ f = ψ.fst.map L ⋯

Tactic:
rw [← cancel_mono (L.map (φ'.s ≫ α.s)), LeftFraction.map_comp_map_s,
      hα, L.map_comp, hφ, LeftFraction.map_comp_map_s_assoc,
      L.map_comp]

Goal state after:
Module: Mathlib.CategoryTheory.Localization.CalculusOfFractions

Goal state before:
pp: case refine_1.refine_3
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✝
a : W.LeftFraction X Y
b₁ b₂ : W.LeftFraction Y Z✝
U : C
t₁ : b₁.Y' ⟶ U
t₂ : b₂.Y' ⟶ U
hst : b₁.s ≫ t₁ = b₂.s ≫ t₂
hft : b₁.f ≫ t₁ = b₂.f ≫ t₂
ht : W (b₁.s ≫ t₁)
z₁ : W.LeftFraction a.Y' b₁.Y'
fac₁ : b₁.f ≫ z₁.s = a.s ≫ z₁.f
z₂ : W.LeftFraction a.Y' b₂.Y'
fac₂ : b₂.f ≫ z₂.s = a.s ≫ z₂.f
w₁ : W.LeftFraction z₁.Y' U
fac₁' : t₁ ≫ w₁.s = z₁.s ≫ w₁.f
w₂ : W.LeftFraction z₂.Y' U
fac₂' : t₂ ≫ w₂.s = z₂.s ≫ w₂.f
u : W.LeftFraction w₁.Y' w₂.Y'
fac₃ : w₂.s ≫ u.s = w₁.s ≫ u.f
eq : a.s ≫ z₁.f ≫ w₁.f ≫ u.f = a.s ≫ z₂.f ≫ w₂.f ≫ u.s
Z : C
p : u.Y' ⟶ Z
hp : W p
fac₄ : z₁.f ≫ w₁.f ≫ u.f ≫ p = z₂.f ≫ w₂.f ≫ u.s ≫ p
⊢ W (b₁.s ≫ z₁.s ≫ w₁.f ≫ u.f ≫ p)

Tactic:
rw [← reassoc_of% fac₁', ← reassoc_of% fac₃, ← assoc]

Goal state after:
case refine_1.refine_3
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✝
a : W.LeftFraction X Y
b₁ b₂ : W.LeftFraction Y Z✝
U : C
t₁ : b₁.Y' ⟶ U
t₂ : b₂.Y' ⟶ U
hst : b₁.s ≫ t₁ = b₂.s ≫ t₂
hft : b₁.f ≫ t₁ = b₂.f ≫ t₂
ht : W (b₁.s ≫ t₁)
z₁ : W.LeftFraction a.Y' b₁.Y'
fac₁ : b₁.f ≫ z₁.s = a.s ≫ z₁.f
z₂ : W.LeftFraction a.Y' b₂.Y'
fac₂ : b₂.f ≫ z₂.s = a.s ≫ z₂.f
w₁ : W.LeftFraction z₁.Y' U
fac₁' : t₁ ≫ w₁.s = z₁.s ≫ w₁.f
w₂ : W.LeftFraction z₂.Y' U
fac₂' : t₂ ≫ w₂.s = z₂.s ≫ w₂.f
u : W.LeftFraction w₁.Y' w₂.Y'
fac₃ : w₂.s ≫ u.s = w₁.s ≫ u.f
eq : a.s ≫ z₁.f ≫ w₁.f ≫ u.f = a.s ≫ z₂.f ≫ w₂.f ≫ u.s
Z : C
p : u.Y' ⟶ Z
hp : W p
fac₄ : z₁.f ≫ w₁.f ≫ u.f ≫ p = z₂.f ≫ w₂.f ≫ u.s ≫ p
⊢ W ((b₁.s ≫ t₁) ≫ w₂.s ≫ u.s ≫ p)
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))
⊢ f = ψ.fst.map L ⋯

Tactic:
rw [← cancel_mono (L.map (β.s ≫ γ.s)), LeftFraction.map_comp_map_s, hα, hγ,
      L.map_comp, LeftFraction.map_comp_map_s_assoc, L.map_comp]

Goal state after:
Module: Mathlib.CategoryTheory.ObjectProperty.ColimitsClosure

Goal state before:
pp: case refine_2
C : Type u
inst✝¹ : Category.{v, u} C
P : ObjectProperty C
α : Type t
J : α → Type u'
inst✝ : (a : α) → Category.{v', u'} (J a)
⊢ (P.op.limitsClosure fun a ↦ (J a)ᵒᵖ) ≤ (P.colimitsClosure J).op

Tactic:
apply limitsClosure_le

Goal state after:
case refine_2
C : Type u
inst✝¹ : Category.{v, u} C
P : ObjectProperty C
α : Type t
J : α → Type u'
inst✝ : (a : α) → Category.{v', u'} (J a)
⊢ P.op ≤ (P.colimitsClosure J).op
Module: Mathlib.CategoryTheory.ObjectProperty.FiniteProducts

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : HasInitial C
P Q : ObjectProperty C
inst✝¹ : Q.IsClosedUnderBinaryCoproducts
inst✝ : Q.IsClosedUnderColimitsOfShape (Discrete PEmpty.{1})
h : P ≤ Q
this : Q.IsClosedUnderIsomorphisms := IsClosedUnderBinaryCoproducts.closedUnderIsomorphisms Q
⊢ P.binaryCoproductsClosure ≤ Q

Tactic:
exact colimitsClosure_le h

Goal state after:
Module: Mathlib.CategoryTheory.Localization.CalculusOfFractions

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
W : MorphismProperty C
inst✝ : W.HasLeftCalculusOfFractions
X Y : C
f : W.LeftFraction X Y
⊢ (Q W).map f.f ≫ Qinv f.s ⋯ = f.map (Q W) ⋯

Tactic:
← cancel_mono ((Q W).map f.s),

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
W : MorphismProperty C
inst✝ : W.HasLeftCalculusOfFractions
X Y : C
f : W.LeftFraction X Y
⊢ ((Q W).map f.f ≫ Qinv f.s ⋯) ≫ (Q W).map f.s = f.map (Q W) ⋯ ≫ (Q W).map f.s
Module: Mathlib.CategoryTheory.Localization.CalculusOfFractions

Goal state before:
pp: case mpr
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
W : MorphismProperty C
inst✝ : W.HasLeftCalculusOfFractions
X Y : C
f g : W.LeftFraction X Y
⊢ LeftFractionRel f g → (Hom.mk f).map (Q W) ⋯ = (Hom.mk g).map (Q W) ⋯

Tactic:
intro h
    congr 1
    exact Quot.sound h

Goal state after:
Module: Mathlib.CategoryTheory.MorphismProperty.Retract

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
P : MorphismProperty C
⊢ P.IsStableUnderRetracts ↔ P.retracts ≤ P

Tactic:
rw [isStableUnderRetracts_iff]
  constructor
  · intro h₁ X Y f ⟨_, _, _, h₂, h₃⟩
    exact h₁ h₂ h₃
  · intro h₁ _ _ _ _ _ _ h₂ h₃
    exact h₁ _ ⟨_, _, _, h₂, h₃⟩

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

Goal state before:
pp: case mp
C✝ : Type u_1
inst✝⁶ : Category.{v_1, u_1} C✝
inst✝⁵ : HasZeroObject C✝
inst✝⁴ : HasShift C✝ ℤ
inst✝³ : Preadditive C✝
inst✝² : ∀ (n : ℤ), (shiftFunctor C✝ n).Additive
inst✝¹ : Pretriangulated C✝
P Q R : ObjectProperty C✝
inst✝ : IsTriangulated C✝
X Y✝ C : C✝
f : Y✝ ⟶ X
g : X ⟶ C
h : C ⟶ (shiftFunctor C✝ 1).obj Y✝
hT : Triangle.mk f g h ∈ distinguishedTriangles
A B : C✝
f' : A ⟶ Y✝
g' : Y✝ ⟶ B
h' : B ⟶ (shiftFunctor C✝ 1).obj A
hT' : Triangle.mk f' g' h' ∈ distinguishedTriangles
hP : P A
hQ : Q B
hR : R C
Y : C✝
g'' : X ⟶ Y
h'' : Y ⟶ (shiftFunctor C✝ 1).obj A
hT'' : Triangle.mk (f' ≫ f) g'' h'' ∈ distinguishedTriangles
o : Octahedron ⋯ hT' hT hT'' := someOctahedron ⋯ hT' hT hT''
⊢ P.extensionProduct (Q.extensionProduct R) X

Tactic:
exact ⟨_, _, _, _, _, hT'', hP, ⟨_, _, _, _, _, o.mem, hQ, hR⟩⟩

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

Goal state before:
pp: C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : HasZeroObject C
inst✝⁴ : HasShift C ℤ
inst✝³ : Preadditive C
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
P₁ P₂ Q : ObjectProperty C
inst✝ : Q.IsTriangulatedClosed₂
h₁ : P₁ ≤ Q
h₂ : P₂ ≤ Q
⊢ P₁.extensionProduct P₂ ≤ Q.isoClosure

Tactic:
by
  intro _ ⟨_, _, _, _, _, hT, hY, hZ⟩
  exact ext_of_isTriangulatedClosed₂' Q _ hT (h₁ _ hY) (h₂ _ hZ)

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

Goal state before:
pp: ι : Type u_1
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
c : ComplexShape ι
K L : HomologicalComplex C c
f : K ⟶ L
i : ι
inst✝¹ : K.HasHomology i
inst✝ : L.HasHomology i
⊢ QuasiIsoAt f i ↔ IsIso (homologyMap f i)

Tactic:
by
  rw [quasiIsoAt_iff, ShortComplex.quasiIso_iff]
  rfl

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

Goal state before:
pp: C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
ι : Type u_2
c : ComplexShape ι
inst✝³ : Preadditive C
inst✝² : CategoryWithHomology C
inst✝¹ : (HomologicalComplex.quasiIso C c).HasLocalization
inst✝ : c.QFactorsThroughHomotopy C
K L : HomologicalComplex C c
φ : { as := K } ⟶ { as := L }
⊢ HomotopyCategory.quasiIso C c φ → IsIso (Qh.map φ)

Tactic:
obtain ⟨φ, rfl⟩ := (HomotopyCategory.quotient C c).map_surjective φ

Goal state after:
C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
ι : Type u_2
c : ComplexShape ι
inst✝³ : Preadditive C
inst✝² : CategoryWithHomology C
inst✝¹ : (HomologicalComplex.quasiIso C c).HasLocalization
inst✝ : c.QFactorsThroughHomotopy C
K L : HomologicalComplex C c
φ : K ⟶ L
⊢ HomotopyCategory.quasiIso C c ((HomotopyCategory.quotient C c).map φ) →
    IsIso (Qh.map ((HomotopyCategory.quotient C c).map φ))
Module: Mathlib.Algebra.Homology.Localization

Goal state before:
pp: ι : Type u_1
c : ComplexShape ι
hc : ∀ (j : ι), ∃ i, c.Rel i j
C : Type u_2
inst✝³ : Category.{v_1, u_2} C
inst✝² : Preadditive C
inst✝¹ : HasBinaryBiproducts C
inst✝ : CategoryWithHomology C
K L : HomologicalComplex C c
f g : K ⟶ L
h : Homotopy f g
⊢ AreEqualizedByLocalization (HomologicalComplex.quasiIso C c) f g

Tactic:
exact h.map_eq_of_inverts_homotopyEquivalences hc _
      (MorphismProperty.IsInvertedBy.of_le _ _ _
        (Localization.inverts _ (HomologicalComplex.quasiIso C _))
        (homotopyEquivalences_le_quasiIso C _))

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

Goal state before:
pp: case mpr
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasDerivedCategory C
T : Triangle (DerivedCategory C)
⊢ (∃ X Y f, Nonempty (T ≅ Q.mapTriangle.obj (CochainComplex.mappingCone.triangle f))) → T ∈ distinguishedTriangles

Tactic:
rintro ⟨X, Y, f, ⟨e⟩⟩
    refine isomorphic_distinguished _ (Qh.map_distinguished _ ?_) _
      (e ≪≫ (Functor.mapTriangleIso (quotientCompQhIso C)).symm.app _ ≪≫
      (Functor.mapTriangleCompIso (HomotopyCategory.quotient C _) Qh).app _)
    exact ⟨_, _, f, ⟨Iso.refl _⟩⟩

Goal state after:
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
X Y : HomotopyCategory C (ComplexShape.up ℤ)
f : X ⟶ Y
⊢ IsIso (Qh.map f) ↔ HomotopyCategory.quasiIso C (ComplexShape.up ℤ) f

Tactic:
by
  constructor
  · intro hf
    rw [HomotopyCategory.mem_quasiIso_iff]
    intro n
    rw [← NatIso.isIso_map_iff (homologyFunctorFactorsh C n) f]
    dsimp
    infer_instance
  · exact Localization.inverts Qh (HomotopyCategory.quasiIso _ _) _

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

Goal state before:
pp: case e_a
C : Type u_1
inst✝³ : Category.{v, u_1} C
inst✝² : Preadditive C
inst✝¹ : HasBinaryBiproducts C
X₁ X₂ X₃ : CochainComplex C ℤ
f : X₁ ⟶ X₂
g : X₂ ⟶ X₃
inst✝ : HasZeroObject C
⊢ (mappingCone.triangle g).mor₃ ≫ (CategoryTheory.shiftFunctor (CochainComplex C ℤ) 1).map (mappingCone.inr f) =
    (mappingConeCompHomotopyEquiv f g).hom ≫ (mappingCone.triangle (mappingCone.map f (f ≫ g) (𝟙 X₁) g ⋯)).mor₃

Tactic:
exact (mappingConeCompHomotopyEquiv_comm₂ f g).symm

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

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 : HasZeroObject C
inst✝ : HasZeroMorphisms C
K L M : HomologicalComplex C c
φ : K ⟶ L
φ' : L ⟶ M
e : c.Embedding c'
i' j' k' : ι'
x✝¹ : c'.Rel i' j'
x✝ : c'.Rel j' k'
⊢ extend.d K (e.r i') (e.r j') ≫ extend.d K (e.r j') (e.r k') = 0

Tactic:
by
    obtain hi' | ⟨i, hi⟩ := (e.r i').eq_none_or_eq_some
    · rw [extend.d_none_eq_zero K _ _ hi', zero_comp]
    · obtain hj' | ⟨j, hj⟩ := (e.r j').eq_none_or_eq_some
      · rw [extend.d_none_eq_zero K _ _ hj', comp_zero]
      · obtain hk' | ⟨k, hk⟩ := (e.r k').eq_none_or_eq_some
        · rw [extend.d_none_eq_zero' K _ _ hk', comp_zero]
        · rw [extend.d_eq K hi hj, extend.d_eq K hj hk, assoc, assoc,
            Iso.inv_hom_id_assoc, K.d_comp_d_assoc, zero_comp, comp_zero]

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

Goal state before:
pp: case inr
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 : HomologicalComplex C c
e : c.Embedding c'
i' j' : ι'
i : ι
hi : e.f i = i'
hi' : ¬c.Rel i (c.next i)
j : ι
hj : e.r j' = some j
⊢ (K.extend e).d i' j' = 0

Tactic:
rw [extend_d_eq K e hi (e.f_eq_of_r_eq_some hj), K.shape, zero_comp, comp_zero]

Goal state after:
case inr.a
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 : HomologicalComplex C c
e : c.Embedding c'
i' j' : ι'
i : ι
hi : e.f i = i'
hi' : ¬c.Rel i (c.next i)
j : ι
hj : e.r j' = some j
⊢ ¬c.Rel i j
Module: Mathlib.Algebra.Homology.Embedding.HomEquiv

Goal state before:
pp: 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' j' : ι'
hij' : c'.Rel i' j'
hi' : ¬∃ i, e.f i = i'
⊢ ∀ (i : ι), e.f i ≠ i'

Tactic:
simpa using hi'

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

Goal state before:
pp: 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
⊢ ∀ (i : ι), e.f i ≠ i'

Tactic:
simpa using hi'

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

Goal state before:
pp: 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
ψ : K ⟶ L.extend e
i' : ι'
hi' : ¬∃ i, e.f i = i'
⊢ ∀ (i : ι), e.f i ≠ i'

Tactic:
simpa using hi'

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

Goal state before:
pp: 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 K' : HomologicalComplex C c'
L : HomologicalComplex C c
inst✝ : e.IsRelIff
α : K' ⟶ K
ψ : K ⟶ L.extend e
i : ι
⊢ (e.homRestrict (α ≫ ψ)).f i = (restrictionMap α e ≫ e.homRestrict ψ).f i

Tactic:
simp [homRestrict_f _ _ rfl, restrictionXIso]

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

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
inst✝ : HasZeroObject C
K L M : HomologicalComplex C c
φ : K ⟶ L
φ' : L ⟶ M
e : c.Embedding c'
i j k : ι
i' j' k' : ι'
hj' : e.f j = j'
hi : c.prev j = i
hi' : c'.prev j' = i'
hk : c.next j = k
hk' : c'.next j' = k'
cocone : CokernelCofork (K.d i j)
hcocone : IsColimit cocone
cone : KernelFork (hcocone.desc (CokernelCofork.ofπ (K.d j k) ⋯))
hcone : IsLimit cone
⊢ Fork.ι cone ≫
      (isColimitCokernelCofork K e hj' hi hi' cocone hcocone).desc (CokernelCofork.ofπ ((K.extend e).d j' k') ⋯) =
    0

Tactic:
rw [← d_comp_desc_eq_zero_iff K e hj' hi hi' hk hk' cocone hcocone]
    exact cone.condition

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

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
inst✝ : HasZeroObject C
K : HomologicalComplex C c
e : c.Embedding c'
i j k : ι
i' j' k' : ι'
hj' : e.f j = j'
hi : c.prev j = i
hi' : c'.prev j' = i'
hk : c.next j = k
hk' : c'.next j' = k'
h : (K.sc' i j k).RightHomologyData
hk'' : e.f k = k'
this : (K.extendXIso e hj').hom ≫ h.p ≫ (rightHomologyData K e hj' hi hi' hk hk' h).g' = (K.extend e).d j' k'
⊢ (K.extendXIso e hj').hom ≫ h.p ≫ (rightHomologyData K e hj' hi hi' hk hk' h).g' =
    (K.extendXIso e hj').hom ≫ h.p ≫ h.g' ≫ (K.extendXIso e hk'').inv

Tactic:
rw [this, K.extend_d_eq e hj' hk'', h.p_g'_assoc, shortComplexFunctor'_obj_g]

Goal state after: