mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.CategoryTheory.PathCategory.Basic

Goal state before:
pp: C : Type u₁
inst✝ : Category.{v₁, u₁} C
X : Quotient (pathsHomRel C)
⊢ (quotientPathsTo C).map
        ((NatIso.ofComponents
                (fun X ↦
                  Quotient.casesOn (motive := fun t ↦
                    X = t → ((𝟭 (Quotient (pathsHomRel C))).obj X ≅ (quotientPathsTo C ⋙ toQuotientPaths C).obj X)) X
                    (fun as h ↦ ⋯ ▸ Iso.refl ((𝟭 (Quotient (pathsHomRel C))).obj { as := as })) ⋯)
                ⋯).hom.app
          X) ≫
      (NatIso.ofComponents (fun x ↦ Iso.refl ((toQuotientPaths C ⋙ quotientPathsTo C).obj x)) ⋯).hom.app
        ((quotientPathsTo C).obj X) =
    𝟙 ((quotientPathsTo C).obj X)

Tactic:
by
    cases X
    simp only [Functor.id_obj,
               quotientPathsTo_obj, Functor.comp_obj, toQuotientPaths_obj_as,
               NatIso.ofComponents_hom_app, Iso.refl_hom, quotientPathsTo_map, Category.comp_id]
    rfl

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

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 : ShortComplex (HomologicalComplex C c)
hS : S.ShortExact
i j : ι
hij : c.Rel i j
⊢ HomologicalComplex.homologyMap 0 i = 0

Tactic:
HomologicalComplex.homologyMap_zero

Goal state after:
C : Type u_1
ι : Type u_2
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
c : ComplexShape ι
S : ShortComplex (HomologicalComplex C c)
hS : S.ShortExact
i j : ι
hij : c.Rel i j
⊢ 0 = 0
Module: Mathlib.CategoryTheory.Triangulated.Triangulated

Goal state before:
pp: case refine_4
C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : Preadditive C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
inst✝ : IsTriangulated C
X₁ X₂ X₃ Z₁₂ Z₂₃ Z₁₃ : C
u₁₂ : X₁ ⟶ X₂
u₂₃ : X₂ ⟶ X₃
u₁₃ : X₁ ⟶ X₃
comm : u₁₂ ≫ u₂₃ = u₁₃
v₁₂ : Z₁₂ ⟶ X₁
w₁₂ : X₂ ⟶ (shiftFunctor C 1).obj Z₁₂
h₁₂ : Triangle.mk v₁₂ u₁₂ w₁₂ ∈ distinguishedTriangles
v₂₃ : Z₂₃ ⟶ X₂
w₂₃ : X₃ ⟶ (shiftFunctor C 1).obj Z₂₃
h₂₃ : Triangle.mk v₂₃ u₂₃ w₂₃ ∈ distinguishedTriangles
v₁₃ : Z₁₃ ⟶ X₁
w₁₃ : X₃ ⟶ (shiftFunctor C 1).obj Z₁₃
h₁₃ : Triangle.mk v₁₃ u₁₃ w₁₃ ∈ distinguishedTriangles
o : Octahedron comm ⋯ ⋯ ⋯ := someOctahedron comm ⋯ ⋯ ⋯
m₁ : Z₁₂ ⟶ Z₁₃ := (shiftShiftNeg Z₁₂ 1).inv ≫ (shiftFunctor C (-1)).map o.m₁ ≫ (shiftShiftNeg Z₁₃ 1).hom
m₃ : Z₁₃ ⟶ Z₂₃ := (shiftShiftNeg Z₁₃ 1).inv ≫ (shiftFunctor C (-1)).map o.m₃ ≫ (shiftShiftNeg Z₂₃ 1).hom
eq₁ : w₁₂ ≫ o.m₁ = u₂₃ ≫ w₁₃
eq₂ : o.m₁ ≫ (shiftFunctor C 1).map v₁₃ = (shiftFunctor C 1).map v₁₂
eq₃ : w₁₃ ≫ o.m₃ = w₂₃
eq₄ : (shiftFunctor C 1).map v₁₃ ≫ (shiftFunctor C 1).map u₁₂ = o.m₃ ≫ (shiftFunctor C 1).map v₂₃
⊢ (shiftFunctorCompIsoId C 1 (-1) ⋯).inv.app Z₁₃ ≫
      (shiftFunctor C (-1)).map ((shiftFunctor C 1).map (v₁₃ ≫ u₁₂)) ≫ (shiftFunctorCompIsoId C 1 (-1) ⋯).hom.app X₂ =
    v₁₃ ≫ u₁₂

Tactic:
shiftFunctorCompIsoId_naturality_1

Goal state after:
case refine_4
C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : Preadditive C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
inst✝ : IsTriangulated C
X₁ X₂ X₃ Z₁₂ Z₂₃ Z₁₃ : C
u₁₂ : X₁ ⟶ X₂
u₂₃ : X₂ ⟶ X₃
u₁₃ : X₁ ⟶ X₃
comm : u₁₂ ≫ u₂₃ = u₁₃
v₁₂ : Z₁₂ ⟶ X₁
w₁₂ : X₂ ⟶ (shiftFunctor C 1).obj Z₁₂
h₁₂ : Triangle.mk v₁₂ u₁₂ w₁₂ ∈ distinguishedTriangles
v₂₃ : Z₂₃ ⟶ X₂
w₂₃ : X₃ ⟶ (shiftFunctor C 1).obj Z₂₃
h₂₃ : Triangle.mk v₂₃ u₂₃ w₂₃ ∈ distinguishedTriangles
v₁₃ : Z₁₃ ⟶ X₁
w₁₃ : X₃ ⟶ (shiftFunctor C 1).obj Z₁₃
h₁₃ : Triangle.mk v₁₃ u₁₃ w₁₃ ∈ distinguishedTriangles
o : Octahedron comm ⋯ ⋯ ⋯ := someOctahedron comm ⋯ ⋯ ⋯
m₁ : Z₁₂ ⟶ Z₁₃ := (shiftShiftNeg Z₁₂ 1).inv ≫ (shiftFunctor C (-1)).map o.m₁ ≫ (shiftShiftNeg Z₁₃ 1).hom
m₃ : Z₁₃ ⟶ Z₂₃ := (shiftShiftNeg Z₁₃ 1).inv ≫ (shiftFunctor C (-1)).map o.m₃ ≫ (shiftShiftNeg Z₂₃ 1).hom
eq₁ : w₁₂ ≫ o.m₁ = u₂₃ ≫ w₁₃
eq₂ : o.m₁ ≫ (shiftFunctor C 1).map v₁₃ = (shiftFunctor C 1).map v₁₂
eq₃ : w₁₃ ≫ o.m₃ = w₂₃
eq₄ : (shiftFunctor C 1).map v₁₃ ≫ (shiftFunctor C 1).map u₁₂ = o.m₃ ≫ (shiftFunctor C 1).map v₂₃
⊢ v₁₃ ≫ u₁₂ = v₁₃ ≫ u₁₂
Module: Mathlib.CategoryTheory.Localization.Predicate

Goal state before:
pp: 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
E : Type u_3
inst✝¹ : Category.{v_3, u_3} E
inst✝ : L.IsLocalization W
⊢ (whiskeringLeftFunctor' L W E).Full

Tactic:
by
  rw [whiskeringLeftFunctor'_eq]
  apply @Functor.Full.comp _ _ _ _ _ _ _ _ ?_ ?_
  · infer_instance
  apply InducedCategory.full

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

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
L : C ⥤ D
W : MorphismProperty C
inst✝² : L.IsLocalization W
P : MorphismProperty D
inst✝¹ : P.RespectsIso
inst✝ : P.IsMultiplicative
h₁ : ∀ ⦃X Y : C⦄ (f : X ⟶ Y), P (L.map f)
h₂ : ∀ ⦃X Y : C⦄ (f : X ⟶ Y) (hf : W f), P (isoOfHom L W f hf).inv
e : W.Q ⋙ (uniq W.Q L W).functor ≅ L := compUniqFunctor W.Q L W
X Y : C
f : X ⟶ Y
hf : W f
⊢ P.inverseImage (uniq W.Q L W).functor (Construction.wInv f hf)

Tactic:
refine (P.arrow_mk_iso_iff (Arrow.isoMk (e.app _) (e.app _) ?_)).2 (h₂ f hf)

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
L : C ⥤ D
W : MorphismProperty C
inst✝² : L.IsLocalization W
P : MorphismProperty D
inst✝¹ : P.RespectsIso
inst✝ : P.IsMultiplicative
h₁ : ∀ ⦃X Y : C⦄ (f : X ⟶ Y), P (L.map f)
h₂ : ∀ ⦃X Y : C⦄ (f : X ⟶ Y) (hf : W f), P (isoOfHom L W f hf).inv
e : W.Q ⋙ (uniq W.Q L W).functor ≅ L := compUniqFunctor W.Q L W
X Y : C
f : X ⟶ Y
hf : W f
⊢ (e.app Y).hom ≫ (Arrow.mk (isoOfHom L W f hf).inv).hom =
    (Arrow.mk ((uniq W.Q L W).functor.map (Construction.wInv f hf))).hom ≫ (e.app X).hom
Module: Mathlib.CategoryTheory.Localization.CalculusOfFractions.Fractions

Goal state before:
pp: C : Type u_1
inst✝ : Category.{v_1, u_1} C
W : MorphismProperty C
X Y : C
z₁ z₂ : W.LeftFraction₂ X Y
h : LeftFraction₂Rel z₁ z₂
⊢ LeftFractionRel z₁.snd z₂.snd

Tactic:
obtain ⟨Z, t₁, t₂, hst, _, hft', ht⟩ := h

Goal state after:
C : Type u_1
inst✝ : Category.{v_1, u_1} C
W : MorphismProperty C
X Y : C
z₁ z₂ : W.LeftFraction₂ X Y
Z : C
t₁ : z₁.Y' ⟶ Z
t₂ : z₂.Y' ⟶ Z
hst : z₁.s ≫ t₁ = z₂.s ≫ t₂
w✝ : z₁.f ≫ t₁ = z₂.f ≫ t₂
hft' : z₁.f' ≫ t₁ = z₂.f' ≫ t₂
ht : W (z₁.s ≫ t₁)
⊢ LeftFractionRel z₁.snd z₂.snd
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
W : MorphismProperty C
X Y : C
φ : W.LeftFraction₂ X Y
F : C ⥤ D
hF : W.IsInvertedBy F
inst✝¹ : Preadditive D
inst✝ : F.Additive
this : IsIso (F.map φ.s)
⊢ φ.add.map F hF = φ.fst.map F hF + φ.snd.map F hF

Tactic:
← cancel_mono (F.map φ.s),

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
W : MorphismProperty C
X Y : C
φ : W.LeftFraction₂ X Y
F : C ⥤ D
hF : W.IsInvertedBy F
inst✝¹ : Preadditive D
inst✝ : F.Additive
this : IsIso (F.map φ.s)
⊢ φ.add.map F hF ≫ F.map φ.s = (φ.fst.map F hF + φ.snd.map F hF) ≫ F.map φ.s
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
f₁ f₂ : L.obj X ⟶ L.obj Y
α : W.LeftFraction₂ X Y
h₁ : f₁ = α.fst.map L ⋯
h₂ : f₂ = α.snd.map L ⋯
⊢ α.add.map L ⋯ = add' W f₂ f₁

Tactic:
add'_eq W f₂ f₁ α.symm h₂ h₁,

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
f₁ f₂ : L.obj X ⟶ L.obj Y
α : W.LeftFraction₂ X Y
h₁ : f₁ = α.fst.map L ⋯
h₂ : f₂ = α.snd.map L ⋯
⊢ α.add.map L ⋯ = α.symm.add.map L ⋯
Module: Mathlib.CategoryTheory.Localization.CalculusOfFractions

Goal state before:
pp: case refine_1
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 : { X' := Y, s := a.s, hs := ⋯, f := b₁.f }.f ≫ z₁.s = { X' := Y, s := a.s, hs := ⋯, f := b₁.f }.s ≫ z₁.f
z₂ : W.LeftFraction a.Y' b₂.Y'
fac₂ : { X' := Y, s := a.s, hs := ⋯, f := b₂.f }.f ≫ z₂.s = { X' := Y, s := a.s, hs := ⋯, f := b₂.f }.s ≫ z₂.f
⊢ a.comp b₁ = a.comp b₂

Tactic:
obtain ⟨w₁, fac₁'⟩ := exists_leftFraction (RightFraction.mk z₁.s z₁.hs t₁)

Goal state after:
case refine_1
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 : { X' := Y, s := a.s, hs := ⋯, f := b₁.f }.f ≫ z₁.s = { X' := Y, s := a.s, hs := ⋯, f := b₁.f }.s ≫ z₁.f
z₂ : W.LeftFraction a.Y' b₂.Y'
fac₂ : { X' := Y, s := a.s, hs := ⋯, f := b₂.f }.f ≫ z₂.s = { X' := Y, s := a.s, hs := ⋯, f := b₂.f }.s ≫ z₂.f
w₁ : W.LeftFraction z₁.Y' U
fac₁' : { X' := b₁.Y', s := z₁.s, hs := ⋯, f := t₁ }.f ≫ w₁.s = { X' := b₁.Y', s := z₁.s, hs := ⋯, f := t₁ }.s ≫ w₁.f
⊢ a.comp b₁ = a.comp b₂
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 Z : C
f₁ f₂ : L.obj X ⟶ L.obj Y
g : L.obj Y ⟶ L.obj Z
⊢ add' W f₁ f₂ ≫ g = add' W (f₁ ≫ g) (f₂ ≫ g)

Tactic:
obtain ⟨α, h₁, h₂⟩ := exists_leftFraction₂ L W f₁ f₂
  obtain ⟨β, hβ⟩ := exists_leftFraction L W g
  obtain ⟨γ, hγ⟩ := (RightFraction.mk _ α.hs β.f).exists_leftFraction
  dsimp at hγ
  rw [add'_eq W f₁ f₂ α h₁ h₂, add'_eq W (f₁ ≫ g) (f₂ ≫ g)
    (LeftFraction₂.mk (α.f ≫ γ.f) (α.f' ≫ γ.f) (β.s ≫ γ.s)
    (W.comp_mem _ _ β.hs γ.hs))]; rotate_left
  · rw [h₁, hβ]
    exact LeftFraction.map_comp_map_eq_map _ _ _ hγ _
  · rw [h₂, hβ]
    exact LeftFraction.map_comp_map_eq_map _ _ _ hγ _
  rw [hβ, LeftFraction.map_comp_map_eq_map _ _ γ hγ]
  dsimp [LeftFraction₂.add]
  rw [add_comp]

Goal state after:
Module: Mathlib.Order.TransfiniteIteration

Goal state before:
pp: I : Type u
inst✝⁴ : CompleteLattice I
φ : I → I
i₀ : I
J : Type w
inst✝³ : LinearOrder J
inst✝² : OrderBot J
inst✝¹ : SuccOrder J
inst✝ : WellFoundedLT J
hφ' : ∀ (i : I), i ≠ ⊤ → i < φ i
φtop : φ ⊤ = ⊤
H : ¬Function.Injective fun j ↦ transfiniteIterate φ j i₀
hφ : ∀ (i : I), i ≤ φ i
j₁ j₂ : J
hj : j₁ < j₂
eq : transfiniteIterate φ j₁ i₀ = transfiniteIterate φ j₂ i₀
this : ∀ (j : J), transfiniteIterate φ j i₀ ≠ ⊤
hj₁ : ¬IsMax j₁
⊢ transfiniteIterate φ (Order.succ j₁) i₀ ≤ transfiniteIterate φ j₂ i₀

Tactic:
exact monotone_transfiniteIterate _ _ hφ (Order.succ_le_of_lt hj)

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

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
n : ℤ
T : Pretriangulated.Triangle (HomotopyCategory C (ComplexShape.up ℤ))
hT : T ∈ CategoryTheory.Pretriangulated.distinguishedTriangles
S : ShortComplex (CochainComplex C ℤ)
σ : (n : ℤ) → (S.map (HomologicalComplex.eval C (ComplexShape.up ℤ) n)).Splitting
e : T ≅ CochainComplex.trianglehOfDegreewiseSplit S σ
hS : S.ShortExact
⊢ ∃ T' e, ((Pretriangulated.shortComplexOfDistTriangle T' ⋯).map (homologyFunctor C (ComplexShape.up ℤ) n)).Exact

Tactic:
exact ⟨_, e, (ShortComplex.exact_iff_of_iso
      (S.mapNatIso (homologyFunctorFactors C (ComplexShape.up ℤ) n))).2 (hS.homology_exact₂ n)⟩

Goal state after:
Module: Mathlib.CategoryTheory.Shift.InducedShiftSequence

Goal state before:
pp: C : Type u_1
D : Type u_2
A : Type u_3
inst✝⁹ : Category.{v_1, u_1} C
inst✝⁸ : Category.{v_2, u_2} D
inst✝⁷ : Category.{v_3, u_3} A
L : C ⥤ D
F : D ⥤ A
G : C ⥤ A
e : L ⋙ F ≅ G
M : Type u_4
inst✝⁶ : AddMonoid M
inst✝⁵ : HasShift C M
inst✝⁴ : G.ShiftSequence M
F' : M → D ⥤ A
e' : (m : M) → L ⋙ F' m ≅ G.shift m
inst✝³ : ((whiskeringLeft C D A).obj L).Full
inst✝² : ((whiskeringLeft C D A).obj L).Faithful
inst✝¹ : HasShift D M
inst✝ : L.CommShift M
X : C
⊢ (F.isoShiftZero M).hom.app (L.obj X) = (e' 0).hom.app X ≫ (G.isoShiftZero M).hom.app X ≫ e.inv.app X

Tactic:
apply induced.isoZero_hom_app_obj

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

Goal state before:
pp: case mpr
C : Type u
inst✝ : Category.{v, u} C
P : MorphismProperty C
h₁ : P.retracts ≤ P
X✝ Y✝ Z✝ W✝ : C
f✝ : X✝ ⟶ Y✝
g✝ : Z✝ ⟶ W✝
h₂ : RetractArrow f✝ g✝
h₃ : P g✝
⊢ P f✝

Tactic:
exact h₁ _ ⟨_, _, _, h₂, h₃⟩

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:
rw [quasiIsoAt_iff, ShortComplex.quasiIso_iff]

Goal state after:
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
⊢ IsIso (ShortComplex.homologyMap ((shortComplexFunctor C c i).map f)) ↔ IsIso (homologyMap f i)
Module: Mathlib.CategoryTheory.Localization.Composition

Goal state before:
pp: C₁ : Type u₁
C₂ : Type u₂
C₃ : Type u₃
inst✝⁴ : Category.{v₁, u₁} C₁
inst✝³ : Category.{v₂, u₂} C₂
inst✝² : Category.{v₃, u₃} C₃
L₁ : C₁ ⥤ C₂
L₂ : C₂ ⥤ C₃
W₁ : MorphismProperty C₁
W₂ : MorphismProperty C₂
inst✝¹ : L₁.IsLocalization W₁
inst✝ : L₂.IsLocalization W₂
W₃ : MorphismProperty C₁
hW₃ : W₃.IsInvertedBy (L₁ ⋙ L₂)
hW₁₃ : W₁ ≤ W₃
hW₂₃ : W₂ ≤ W₃.map L₁
E₂ : C₂ ≌ W₁.Localization := uniq L₁ W₁.Q W₁
W₂' : MorphismProperty W₁.Localization := W₂.map E₂.functor
Φ : LocalizerMorphism W₂ W₂' := { functor := E₂.functor, map := ⋯ }
this : Φ.IsLocalizedEquivalence
E₃ : C₃ ≌ W₂'.Localization := (Φ.localizedFunctor L₂ W₂'.Q).asEquivalence
iso : (W₁.Q ⋙ W₂'.Q) ⋙ E₃.inverse ≅ L₁ ⋙ L₂ :=
  Trans.trans
    (Trans.trans
      (W₁.Q.associator W₂'.Q E₃.inverse ≪≫
        isoWhiskerRight (compUniqFunctor L₁ W₁.Q W₁).symm (W₂'.Q ⋙ E₃.inverse) ≪≫
          L₁.associator (uniq L₁ W₁.Q W₁).functor (W₂'.Q ⋙ E₃.inverse))
      (L₁.isoWhiskerLeft
        ((E₂.functor.associator W₂'.Q E₃.inverse).symm ≪≫
          isoWhiskerRight (CatCommSq.iso Φ.functor L₂ W₂'.Q (Φ.localizedFunctor L₂ W₂'.Q)) E₃.inverse ≪≫
            L₂.associator (Φ.localizedFunctor L₂ W₂'.Q) E₃.inverse)))
    (L₁.isoWhiskerLeft (L₂.isoWhiskerLeft E₃.unitIso.symm ≪≫ L₂.rightUnitor))
hW₃' : W₃.IsInvertedBy (W₁.Q ⋙ W₂'.Q)
⊢ (fun x ↦ x.map E₂.functor) (W₃.map L₁) ≤ W₃.map W₁.Q

Tactic:
simpa only [W₃.map_map]
      using le_of_eq (W₃.map_eq_of_iso (compUniqFunctor L₁ W₁.Q W₁))

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 : Preadditive C
c : ComplexShape ι
K L : HomologicalComplex C c
e : HomotopyEquiv K L
n : ι
inst✝¹ : K.HasHomology n
inst✝ : L.HasHomology n
⊢ QuasiIsoAt e.hom n

Tactic:
rw [quasiIsoAt_iff, ShortComplex.quasiIso_iff]

Goal state after:
ι : Type u_1
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
c : ComplexShape ι
K L : HomologicalComplex C c
e : HomotopyEquiv K L
n : ι
inst✝¹ : K.HasHomology n
inst✝ : L.HasHomology n
⊢ IsIso (ShortComplex.homologyMap ((shortComplexFunctor C c n).map e.hom))
Module: Mathlib.CategoryTheory.Triangulated.Subcategory

Goal state before:
pp: 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
⊢ s ≫ (f₁ - f₂) = 0

Tactic:
comp_sub,

Goal state after:
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
⊢ s ≫ f₁ - s ≫ f₂ = 0
Module: Mathlib.Algebra.Homology.HomotopyCategory.MappingCocone

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
K L : CochainComplex C ℤ
φ : K ⟶ L
inst✝ : HasHomotopyCofiber φ
p q : ℤ
hpq : p + -1 = q
⊢ (inl φ).v p p ⋯ ≫ (snd φ).v p q hpq = 0

Tactic:
obtain rfl : q = p + -1 := by lia

Goal state after:
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
K L : CochainComplex C ℤ
φ : K ⟶ L
inst✝ : HasHomotopyCofiber φ
p : ℤ
hpq : p + -1 = p + -1
⊢ (inl φ).v p p ⋯ ≫ (snd φ).v p (p + -1) hpq = 0
Module: Mathlib.Algebra.Homology.HomotopyCategory.MappingCocone

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
K L : CochainComplex C ℤ
φ : K ⟶ L
inst✝ : HasHomotopyCofiber φ
p q : ℤ
hpq : p + -1 = q
⊢ (fst φ).f p ≫ (inl φ).v p p ⋯ + (snd φ).v p q hpq ≫ (↑(inr φ)).v q p ⋯ = 𝟙 ((mappingCocone φ).X p)

Tactic:
obtain rfl : q = p + -1 := by lia

Goal state after:
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
K L : CochainComplex C ℤ
φ : K ⟶ L
inst✝ : HasHomotopyCofiber φ
p : ℤ
hpq : p + -1 = p + -1
⊢ (fst φ).f p ≫ (inl φ).v p p ⋯ + (snd φ).v p (p + -1) hpq ≫ (↑(inr φ)).v (p + -1) p ⋯ = 𝟙 ((mappingCocone φ).X p)
Module: Mathlib.Algebra.Homology.Embedding.Basic

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
p : ℤ
⊢ (embeddingUpIntLE p).IsRelIff

Tactic:
by dsimp [embeddingUpIntLE]; infer_instance

Goal state after:
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
D : Type u_4
inst✝⁵ : Category.{v_1, u_3} C
inst✝⁴ : Category.{v_2, u_4} D
inst✝³ : HasZeroMorphisms C
inst✝² : HasZeroMorphisms D
K : HomologicalComplex C c'
F : C ⥤ D
inst✝¹ : F.PreservesZeroMorphisms
e : c.Embedding c'
inst✝ : K.IsStrictlySupported e
i' : ι'
hi' : ∀ (i : ι), e.f i ≠ i'
⊢ IsZero (((F.mapHomologicalComplex c').obj K).X i')

Tactic:
rw [IsZero.iff_id_eq_zero]
    dsimp
    rw [← F.map_id, (K.isZero_X_of_isStrictlySupported e i' hi').eq_of_src (𝟙 _) 0, F.map_zero]

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

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
e : c.Embedding c'
j : ι
hj : e.BoundaryGE j
inst✝ : e.IsTruncLE
⊢ False

Tactic:
obtain ⟨i, hi⟩ := e.mem_prev hj.1
  exact hj.2 i (by simpa only [hi] using hj.1)

Goal state after:
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' j' : ι'
hij' : c'.Rel i' j'
i : ι
hi : e.f i = i'
j : ι
hj : e.f j = j'
⊢ ((K.restrictionXIso e hi).inv ≫ φ.f i ≫ (L.extendXIso e hi).inv) ≫ (L.extend e).d i' j' =
    K.d i' j' ≫ (K.restrictionXIso e hj).inv ≫ φ.f j ≫ (L.extendXIso e hj).inv

Tactic:
L.extend_d_eq e hi hj

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' j' : ι'
hij' : c'.Rel i' j'
i : ι
hi : e.f i = i'
j : ι
hj : e.f j = j'
⊢ ((K.restrictionXIso e hi).inv ≫ φ.f i ≫ (L.extendXIso e hi).inv) ≫
      (L.extendXIso e hi).hom ≫ L.d i j ≫ (L.extendXIso e hj).inv =
    K.d i' j' ≫ (K.restrictionXIso e hj).inv ≫ φ.f j ≫ (L.extendXIso e hj).inv
Module: Mathlib.Algebra.Homology.Embedding.TruncGEHomology

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✝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
j : ι
j' : ι'
hj' : e.f j = j'
hj : ¬e.BoundaryGE j
φ : (shortComplexFunctor C c' j').obj K ⟶ (shortComplexFunctor C c' j').obj (K.truncGE e) :=
  (shortComplexFunctor C c' j').map (K.πTruncGE e)
this✝ : Epi φ.τ₁
this : IsIso φ.τ₂
⊢ ShortComplex.QuasiIso ((shortComplexFunctor C c' j').map (K.πTruncGE e))

Tactic:
have : IsIso φ.τ₃ := by
      dsimp [φ, πTruncGE]
      have : c'.next j' = e.f (c.next j) := by simpa only [← hj'] using e.next_f rfl
      rw [e.isIso_liftExtend_f_iff _ _ this.symm]
      exact K.isIso_restrictionToTruncGE' e _ (e.not_boundaryGE_next' hj rfl)

Goal state after:
case neg
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
j : ι
j' : ι'
hj' : e.f j = j'
hj : ¬e.BoundaryGE j
φ : (shortComplexFunctor C c' j').obj K ⟶ (shortComplexFunctor C c' j').obj (K.truncGE e) :=
  (shortComplexFunctor C c' j').map (K.πTruncGE e)
this✝¹ : Epi φ.τ₁
this✝ : IsIso φ.τ₂
this : IsIso φ.τ₃
⊢ ShortComplex.QuasiIso ((shortComplexFunctor C c' j').map (K.πTruncGE e))
Module: Mathlib.Algebra.Homology.Embedding.TruncLE

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝⁶ : Category.{v_1, u_3} C
inst✝⁵ : HasZeroMorphisms C
K L M : HomologicalComplex C c'
φ : K ⟶ L
φ' : L ⟶ M
e : c.Embedding c'
inst✝⁴ : e.IsTruncLE
inst✝³ : ∀ (i' : ι'), K.HasHomology i'
inst✝² : ∀ (i' : ι'), L.HasHomology i'
inst✝¹ : ∀ (i' : ι'), M.HasHomology i'
inst✝ : HasZeroObject C
⊢ (K.truncLE e).IsStrictlySupported e

Tactic:
rw [← isStrictlySupported_op_iff]
  exact inferInstanceAs ((K.op.truncGE e.op).IsStrictlySupported e.op)

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✝⁷ : Category.{v_1, u_3} C
inst✝⁶ : HasZeroMorphisms C
K L M : HomologicalComplex C c'
φ : K ⟶ L
φ' : L ⟶ M
e : c.Embedding c'
inst✝⁵ : e.IsTruncGE
inst✝⁴ : ∀ (i' : ι'), K.HasHomology i'
inst✝³ : ∀ (i' : ι'), L.HasHomology i'
inst✝² : ∀ (i' : ι'), M.HasHomology i'
inst✝¹ : HasZeroObject C
ι'' : Type u_4
c'' : ComplexShape ι''
e' : c''.Embedding c'
inst✝ : K.IsStrictlySupported e'
i' : ι'
hi' : ∀ (i : ι''), e'.f i ≠ i'
hi'' : ¬∃ i, e.f i = i'
⊢ IsZero ((K.truncGE e).X i')

Tactic:
exact (K.truncGE e).isZero_X_of_isStrictlySupported e _ (by simpa using hi'')

Goal state after: