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

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝¹⁵ : Category.{v_1, u_1} C
inst✝¹⁴ : Category.{v_2, u_2} D
inst✝¹³ : HasShift C ℤ
inst✝¹² : HasShift D ℤ
inst✝¹¹ : HasZeroObject C
inst✝¹⁰ : HasZeroObject D
inst✝⁹ : Preadditive C
inst✝⁸ : Preadditive D
inst✝⁷ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁶ : ∀ (n : ℤ), (shiftFunctor D n).Additive
inst✝⁵ : Pretriangulated C
inst✝⁴ : Pretriangulated D
F : C ⥤ D
inst✝³ : F.CommShift ℤ
inst✝² : F.IsTriangulated
inst✝¹ : (F.mapComposableArrows 2).EssSurj
inst✝ : IsTriangulated C
Y₁ Y₂ Y₃ : D
⊢ ∀ {Z₁₂ Z₂₃ Z₁₃ : D} {u₁₂ : Y₁ ⟶ Y₂} {u₂₃ : Y₂ ⟶ Y₃} {u₁₃ : Y₁ ⟶ Y₃} (comm : u₁₂ ≫ u₂₃ = u₁₃) {v₁₂ : Y₂ ⟶ Z₁₂}
    {w₁₂ : Z₁₂ ⟶ (shiftFunctor D 1).obj Y₁} (h₁₂ : Triangle.mk u₁₂ v₁₂ w₁₂ ∈ distinguishedTriangles) {v₂₃ : Y₃ ⟶ Z₂₃}
    {w₂₃ : Z₂₃ ⟶ (shiftFunctor D 1).obj Y₂} (h₂₃ : Triangle.mk u₂₃ v₂₃ w₂₃ ∈ distinguishedTriangles) {v₁₃ : Y₃ ⟶ Z₁₃}
    {w₁₃ : Z₁₃ ⟶ (shiftFunctor D 1).obj Y₁} (h₁₃ : Triangle.mk u₁₃ v₁₃ w₁₃ ∈ distinguishedTriangles),
    Nonempty (Octahedron comm h₁₂ h₂₃ h₁₃)

Tactic:
Z₁₂

Goal state after:
C : Type u_1
D : Type u_2
inst✝¹⁵ : Category.{v_1, u_1} C
inst✝¹⁴ : Category.{v_2, u_2} D
inst✝¹³ : HasShift C ℤ
inst✝¹² : HasShift D ℤ
inst✝¹¹ : HasZeroObject C
inst✝¹⁰ : HasZeroObject D
inst✝⁹ : Preadditive C
inst✝⁸ : Preadditive D
inst✝⁷ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁶ : ∀ (n : ℤ), (shiftFunctor D n).Additive
inst✝⁵ : Pretriangulated C
inst✝⁴ : Pretriangulated D
F : C ⥤ D
inst✝³ : F.CommShift ℤ
inst✝² : F.IsTriangulated
inst✝¹ : (F.mapComposableArrows 2).EssSurj
inst✝ : IsTriangulated C
Y₁ Y₂ Y₃ Z₁₂ : D
⊢ ∀ {Z₂₃ Z₁₃ : D} {u₁₂ : Y₁ ⟶ Y₂} {u₂₃ : Y₂ ⟶ Y₃} {u₁₃ : Y₁ ⟶ Y₃} (comm : u₁₂ ≫ u₂₃ = u₁₃) {v₁₂ : Y₂ ⟶ Z₁₂}
    {w₁₂ : Z₁₂ ⟶ (shiftFunctor D 1).obj Y₁} (h₁₂ : Triangle.mk u₁₂ v₁₂ w₁₂ ∈ distinguishedTriangles) {v₂₃ : Y₃ ⟶ Z₂₃}
    {w₂₃ : Z₂₃ ⟶ (shiftFunctor D 1).obj Y₂} (h₂₃ : Triangle.mk u₂₃ v₂₃ w₂₃ ∈ distinguishedTriangles) {v₁₃ : Y₃ ⟶ Z₁₃}
    {w₁₃ : Z₁₃ ⟶ (shiftFunctor D 1).obj Y₁} (h₁₃ : Triangle.mk u₁₃ v₁₃ w₁₃ ∈ distinguishedTriangles),
    Nonempty (Octahedron comm h₁₂ h₂₃ h₁₃)
Module: Mathlib.CategoryTheory.Localization.Construction

Goal state before:
pp: case mpr.cons
C : Type uC
inst✝¹ : Category.{uC', uC} C
W : MorphismProperty C
P : MorphismProperty W.Localization
inst✝ : P.IsStableUnderComposition
hP₁ : ∀ ⦃X Y : C⦄ (f : X ⟶ Y), P (W.Q.map f)
hP₂ : ∀ ⦃X Y : C⦄ (w : X ⟶ Y) (hw : W w), P (wInv w hw)
X Y : W.Localization
f : X ⟶ Y
a✝ : ⊤ f
G : Paths (LocQuiver W) ⥤ W.Localization := Quotient.functor (relations W)
this : G.Full
X₁ X₂✝ X₂ X₃ : Paths (LocQuiver W)
p : Quiver.Path X₁ X₂
g : X₂ ⟶ X₃
hp : P (G.map p)
p' : X₁ ⟶ X₂ := p
⊢ P (G.map p' ≫ G.map g.toPath)

Tactic:
refine P.comp_mem _ _ hp ?_

Goal state after:
case mpr.cons
C : Type uC
inst✝¹ : Category.{uC', uC} C
W : MorphismProperty C
P : MorphismProperty W.Localization
inst✝ : P.IsStableUnderComposition
hP₁ : ∀ ⦃X Y : C⦄ (f : X ⟶ Y), P (W.Q.map f)
hP₂ : ∀ ⦃X Y : C⦄ (w : X ⟶ Y) (hw : W w), P (wInv w hw)
X Y : W.Localization
f : X ⟶ Y
a✝ : ⊤ f
G : Paths (LocQuiver W) ⥤ W.Localization := Quotient.functor (relations W)
this : G.Full
X₁ X₂✝ X₂ X₃ : Paths (LocQuiver W)
p : Quiver.Path X₁ X₂
g : X₂ ⟶ X₃
hp : P (G.map p)
p' : X₁ ⟶ X₂ := p
⊢ P (G.map g.toPath)
Module: Mathlib.CategoryTheory.Localization.Construction

Goal state before:
pp: case mpr.cons
C : Type uC
inst✝¹ : Category.{uC', uC} C
W : MorphismProperty C
P : MorphismProperty W.Localization
inst✝ : P.IsStableUnderComposition
hP₁ : ∀ ⦃X Y : C⦄ (f : X ⟶ Y), P (W.Q.map f)
hP₂ : ∀ ⦃X Y : C⦄ (w : X ⟶ Y) (hw : W w), P (wInv w hw)
X Y : W.Localization
f : X ⟶ Y
a✝ : ⊤ f
G : Paths (LocQuiver W) ⥤ W.Localization := Quotient.functor (relations W)
this : G.Full
X₁ X₂✝ X₂ X₃ : Paths (LocQuiver W)
p : Quiver.Path X₁ X₂
g : X₂ ⟶ X₃
hp : P (G.map p)
p' : X₁ ⟶ X₂ := p
⊢ P (G.map g.toPath)

Tactic:
rcases g with (g | ⟨g, hg⟩)

Goal state after:
case mpr.cons.inl
C : Type uC
inst✝¹ : Category.{uC', uC} C
W : MorphismProperty C
P : MorphismProperty W.Localization
inst✝ : P.IsStableUnderComposition
hP₁ : ∀ ⦃X Y : C⦄ (f : X ⟶ Y), P (W.Q.map f)
hP₂ : ∀ ⦃X Y : C⦄ (w : X ⟶ Y) (hw : W w), P (wInv w hw)
X Y : W.Localization
f : X ⟶ Y
a✝ : ⊤ f
G : Paths (LocQuiver W) ⥤ W.Localization := Quotient.functor (relations W)
this : G.Full
X₁ X₂✝ X₂ X₃ : Paths (LocQuiver W)
p : Quiver.Path X₁ X₂
hp : P (G.map p)
p' : X₁ ⟶ X₂ := p
g : X₂.obj ⟶ X₃.obj
⊢ P (G.map (Quiver.Hom.toPath (Sum.inl g)))

case mpr.cons.inr
C : Type uC
inst✝¹ : Category.{uC', uC} C
W : MorphismProperty C
P : MorphismProperty W.Localization
inst✝ : P.IsStableUnderComposition
hP₁ : ∀ ⦃X Y : C⦄ (f : X ⟶ Y), P (W.Q.map f)
hP₂ : ∀ ⦃X Y : C⦄ (w : X ⟶ Y) (hw : W w), P (wInv w hw)
X Y : W.Localization
f : X ⟶ Y
a✝ : ⊤ f
G : Paths (LocQuiver W) ⥤ W.Localization := Quotient.functor (relations W)
this : G.Full
X₁ X₂✝ X₂ X₃ : Paths (LocQuiver W)
p : Quiver.Path X₁ X₂
hp : P (G.map p)
p' : X₁ ⟶ X₂ := p
g : X₃.obj ⟶ X₂.obj
hg : W g
⊢ P (G.map (Quiver.Hom.toPath (Sum.inr ⟨g, hg⟩)))
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
⊢ R.functor.map (T.map (L.unitIso.hom.app X)) ≫
      R.functor.map (R.unitIso.hom.app (T.obj (L.inverse.obj (L.functor.obj X)))) ≫
        R.counit.app (R.functor.obj (T.obj (L.inverse.obj (L.functor.obj X)))) ≫
          (iso T L.functor R.functor B).hom.app (L.inverse.obj (L.functor.obj X)) ≫
            B.map (L.counitIso.hom.app (L.functor.obj X)) ≫ B.map (L.functor.map (L.unitIso.hom.app X)) =
    (iso T L.functor R.functor B).hom.app X ≫ B.map (L.functor.map (L.unitIso.hom.app X))

Tactic:
rw [← B.map_comp, L.counit_app_functor, ← L.functor.map_comp, ← NatTrans.comp_app,
    Iso.inv_hom_id, NatTrans.id_app, L.functor.map_id]

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.functor.map (T.map (L.unitIso.hom.app X)) ≫
      R.functor.map (R.unitIso.hom.app (T.obj (L.inverse.obj (L.functor.obj X)))) ≫
        R.counit.app (R.functor.obj (T.obj (L.inverse.obj (L.functor.obj X)))) ≫
          (iso T L.functor R.functor B).hom.app (L.inverse.obj (L.functor.obj X)) ≫
            B.map (𝟙 (L.functor.obj ((L.functor ⋙ L.inverse).obj X))) =
    (iso T L.functor R.functor B).hom.app X ≫ B.map (L.functor.map (L.unitIso.hom.app X))
Module: Mathlib.CategoryTheory.Localization.CalculusOfFractions

Goal state before:
pp: case refine_2.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
q : W.LeftFraction w₁.Y' w₂.Y'
fac₃ : z₂.s ≫ w₂.s ≫ q.s = z₁.s ≫ w₁.s ≫ q.f
⊢ LeftFractionRel p₁ p₂

Tactic:
have eq : a₁.s ≫ t₁ ≫ w₁.f ≫ q.f = a₁.s ≫ t₁ ≫ w₂.f ≫ q.s := by
        rw [← reassoc_of% fac₁', ← fac₃, reassoc_of% hst, reassoc_of% fac₂']

Goal state after:
case refine_2.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
q : W.LeftFraction w₁.Y' w₂.Y'
fac₃ : z₂.s ≫ w₂.s ≫ q.s = z₁.s ≫ w₁.s ≫ q.f
eq : a₁.s ≫ t₁ ≫ w₁.f ≫ q.f = a₁.s ≫ t₁ ≫ w₂.f ≫ q.s
⊢ LeftFractionRel p₁ p₂
Module: Mathlib.CategoryTheory.Localization.CalculusOfFractions

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
W : MorphismProperty C
inst✝¹ : W.HasLeftCalculusOfFractions
E : Type u_3
inst✝ : Category.{v_3, u_3} E
X Y : C
f : Hom W X Y
F : C ⥤ E
hF : W.IsInvertedBy F
a₁ a₂ : W.LeftFraction X Y
Z : C
t₁ : a₁.Y' ⟶ Z
t₂ : a₂.Y' ⟶ Z
hst : a₁.s ≫ t₁ = a₂.s ≫ t₂
hft : a₁.f ≫ t₁ = a₂.f ≫ t₂
h : W (a₁.s ≫ t₁)
this : IsIso (F.map (a₁.s ≫ t₁))
⊢ F.map a₁.f ≫ F.map t₁ = a₂.map F hF ≫ F.map a₁.s ≫ F.map t₁

Tactic:
← F.map_comp,

Goal state after:
C : Type u_1
D : Type u_2
inst✝³ : Category.{v_1, u_1} C
inst✝² : Category.{v_2, u_2} D
W : MorphismProperty C
inst✝¹ : W.HasLeftCalculusOfFractions
E : Type u_3
inst✝ : Category.{v_3, u_3} E
X Y : C
f : Hom W X Y
F : C ⥤ E
hF : W.IsInvertedBy F
a₁ a₂ : W.LeftFraction X Y
Z : C
t₁ : a₁.Y' ⟶ Z
t₂ : a₂.Y' ⟶ Z
hst : a₁.s ≫ t₁ = a₂.s ≫ t₂
hft : a₁.f ≫ t₁ = a₂.f ≫ t₂
h : W (a₁.s ≫ t₁)
this : IsIso (F.map (a₁.s ≫ t₁))
⊢ F.map (a₁.f ≫ t₁) = a₂.map F hF ≫ F.map a₁.s ≫ F.map t₁
Module: Mathlib.CategoryTheory.Localization.CalculusOfFractions

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
W : MorphismProperty C
inst✝¹ : W.HasLeftCalculusOfFractions
E : Type u_3
inst✝ : Category.{v_3, u_3} E
X Y : C
f : Hom W X Y
F : C ⥤ E
hF : W.IsInvertedBy F
a₁ a₂ : W.LeftFraction X Y
Z : C
t₁ : a₁.Y' ⟶ Z
t₂ : a₂.Y' ⟶ Z
hst : a₁.s ≫ t₁ = a₂.s ≫ t₂
hft : a₁.f ≫ t₁ = a₂.f ≫ t₂
h : W (a₁.s ≫ t₁)
this : IsIso (F.map (a₁.s ≫ t₁))
⊢ F.map (a₁.f ≫ t₁) = a₂.map F hF ≫ F.map a₁.s ≫ F.map t₁

Tactic:
← F.map_comp,

Goal state after:
C : Type u_1
D : Type u_2
inst✝³ : Category.{v_1, u_1} C
inst✝² : Category.{v_2, u_2} D
W : MorphismProperty C
inst✝¹ : W.HasLeftCalculusOfFractions
E : Type u_3
inst✝ : Category.{v_3, u_3} E
X Y : C
f : Hom W X Y
F : C ⥤ E
hF : W.IsInvertedBy F
a₁ a₂ : W.LeftFraction X Y
Z : C
t₁ : a₁.Y' ⟶ Z
t₂ : a₂.Y' ⟶ Z
hst : a₁.s ≫ t₁ = a₂.s ≫ t₂
hft : a₁.f ≫ t₁ = a₂.f ≫ t₂
h : W (a₁.s ≫ t₁)
this : IsIso (F.map (a₁.s ≫ t₁))
⊢ F.map (a₁.f ≫ t₁) = a₂.map F hF ≫ F.map (a₁.s ≫ t₁)
Module: Mathlib.CategoryTheory.Localization.CalculusOfFractions

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
W : MorphismProperty C
inst✝¹ : W.HasLeftCalculusOfFractions
E : Type u_3
inst✝ : Category.{v_3, u_3} E
F : C ⥤ E
hF : W.IsInvertedBy F
X Y Z : C
f : W.LeftFraction X Y
g : W.LeftFraction Y Z
z : W.LeftFraction f.Y' g.Y'
fac : g.f ≫ z.s = f.s ≫ z.f
this✝ : IsIso (F.map g.s)
this : IsIso (F.map z.s)
⊢ { Y' := z.Y', f := f.f ≫ z.f, s := g.s ≫ z.s, hs := ⋯ }.map F hF ≫ F.map g.s ≫ F.map z.s =
    f.map F hF ≫ F.map g.f ≫ F.map z.s

Tactic:
← F.map_comp,

Goal state after:
C : Type u_1
D : Type u_2
inst✝³ : Category.{v_1, u_1} C
inst✝² : Category.{v_2, u_2} D
W : MorphismProperty C
inst✝¹ : W.HasLeftCalculusOfFractions
E : Type u_3
inst✝ : Category.{v_3, u_3} E
F : C ⥤ E
hF : W.IsInvertedBy F
X Y Z : C
f : W.LeftFraction X Y
g : W.LeftFraction Y Z
z : W.LeftFraction f.Y' g.Y'
fac : g.f ≫ z.s = f.s ≫ z.f
this✝ : IsIso (F.map g.s)
this : IsIso (F.map z.s)
⊢ { Y' := z.Y', f := f.f ≫ z.f, s := g.s ≫ z.s, hs := ⋯ }.map F hF ≫ F.map (g.s ≫ z.s) =
    f.map F hF ≫ F.map g.f ≫ F.map z.s
Module: Mathlib.CategoryTheory.Localization.CalculusOfFractions

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
W : MorphismProperty C
inst✝¹ : W.HasLeftCalculusOfFractions
E : Type u_3
inst✝ : Category.{v_3, u_3} E
F : C ⥤ E
hF : W.IsInvertedBy F
X Y Z : C
f : W.LeftFraction X Y
g : W.LeftFraction Y Z
z : W.LeftFraction f.Y' g.Y'
fac : g.f ≫ z.s = f.s ≫ z.f
this✝ : IsIso (F.map g.s)
this : IsIso (F.map z.s)
⊢ { Y' := z.Y', f := f.f ≫ z.f, s := g.s ≫ z.s, hs := ⋯ }.map F hF ≫ F.map (g.s ≫ z.s) =
    f.map F hF ≫ F.map g.f ≫ F.map z.s

Tactic:
← F.map_comp,

Goal state after:
C : Type u_1
D : Type u_2
inst✝³ : Category.{v_1, u_1} C
inst✝² : Category.{v_2, u_2} D
W : MorphismProperty C
inst✝¹ : W.HasLeftCalculusOfFractions
E : Type u_3
inst✝ : Category.{v_3, u_3} E
F : C ⥤ E
hF : W.IsInvertedBy F
X Y Z : C
f : W.LeftFraction X Y
g : W.LeftFraction Y Z
z : W.LeftFraction f.Y' g.Y'
fac : g.f ≫ z.s = f.s ≫ z.f
this✝ : IsIso (F.map g.s)
this : IsIso (F.map z.s)
⊢ { Y' := z.Y', f := f.f ≫ z.f, s := g.s ≫ z.s, hs := ⋯ }.map F hF ≫ F.map (g.s ≫ z.s) = f.map F hF ≫ F.map (g.f ≫ z.s)
Module: Mathlib.CategoryTheory.Localization.CalculusOfFractions

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
W : MorphismProperty C
inst✝¹ : W.HasLeftCalculusOfFractions
E : Type u_3
inst✝ : Category.{v_3, u_3} E
F₁ F₂ : Localization W ⥤ E
h : Q W ⋙ F₁ = Q W ⋙ F₂
X Y : C
f : W.LeftFraction X Y
⊢ eqToHom ⋯ ≫ (Q W ⋙ F₂).map f.f ≫ eqToHom ⋯ ≫ F₁.map (Qinv f.s ⋯) =
    eqToHom ⋯ ≫ F₂.map ((Q W).map f.f) ≫ F₂.map (Qinv f.s ⋯) ≫ eqToHom ⋯

Tactic:
congr 2

Goal state after:
case e_a
C : Type u_1
inst✝² : Category.{v_1, u_1} C
W : MorphismProperty C
inst✝¹ : W.HasLeftCalculusOfFractions
E : Type u_3
inst✝ : Category.{v_3, u_3} E
F₁ F₂ : Localization W ⥤ E
h : Q W ⋙ F₁ = Q W ⋙ F₂
X Y : C
f : W.LeftFraction X Y
⊢ eqToHom ⋯ ≫ F₁.map (Qinv f.s ⋯) = F₂.map (Qinv f.s ⋯) ≫ eqToHom ⋯
Module: Mathlib.CategoryTheory.Localization.CalculusOfFractions

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
inst✝¹ : L.IsLocalization W
inst✝ : W.HasLeftCalculusOfFractions
f : Arrow D
this : L.EssSurj
X : C
eX : L.obj X ≅ f.left
⊢ L.mapArrow.essImage f

Tactic:
obtain ⟨Y, ⟨eY⟩⟩ : ∃ (Y : C), Nonempty (L.obj Y ≅ f.right) :=
      ⟨_, ⟨L.objObjPreimageIso f.right⟩⟩

Goal state after:
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
f : Arrow D
this : L.EssSurj
X : C
eX : L.obj X ≅ f.left
Y : C
eY : L.obj Y ≅ f.right
⊢ L.mapArrow.essImage f
Module: Mathlib.CategoryTheory.Shift.ShiftedHom

Goal state before:
pp: C : Type u_1
inst✝⁷ : Category.{v_1, u_1} C
D : Type u_2
inst✝⁶ : Category.{v_2, u_2} D
M : Type u_4
inst✝⁵ : AddMonoid M
inst✝⁴ : HasShift C M
inst✝³ : HasShift D M
X Y : C
a : M
f : ShiftedHom X Y a
F G : C ⥤ D
τ : F ⟶ G
inst✝² : F.CommShift M
inst✝¹ : G.CommShift M
inst✝ : NatTrans.CommShift τ M
⊢ (f.map F).comp (mk₀ 0 ⋯ (τ.app Y)) ⋯ = (mk₀ 0 ⋯ (τ.app X)).comp (f.map G) ⋯

Tactic:
comp_mk₀,

Goal state after:
C : Type u_1
inst✝⁷ : Category.{v_1, u_1} C
D : Type u_2
inst✝⁶ : Category.{v_2, u_2} D
M : Type u_4
inst✝⁵ : AddMonoid M
inst✝⁴ : HasShift C M
inst✝³ : HasShift D M
X Y : C
a : M
f : ShiftedHom X Y a
F G : C ⥤ D
τ : F ⟶ G
inst✝² : F.CommShift M
inst✝¹ : G.CommShift M
inst✝ : NatTrans.CommShift τ M
⊢ f.map F ≫ (shiftFunctor D a).map (τ.app Y) = (mk₀ 0 ⋯ (τ.app X)).comp (f.map G) ⋯
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
P Q : ObjectProperty C
inst✝ : P.ContainsZero
X : C
hX : Q X
⊢ P.extensionProduct Q X

Tactic:
← extensionProduct_isoClosure_left

Goal state after:
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 : ObjectProperty C
inst✝ : P.ContainsZero
X : C
hX : Q X
⊢ P.isoClosure.extensionProduct Q X
Module: Mathlib.CategoryTheory.Triangulated.Subcategory

Goal state before:
pp: case a
C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : Preadditive C
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
P Q : ObjectProperty C
⊢ P.extensionProduct Q.retractClosure ≤ P.retractClosure.extensionProduct Q.retractClosure

Tactic:
monotone_extensionProduct_left _ (le_retractClosure P)

Goal state after:
case a
C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : Preadditive C
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
P Q : ObjectProperty C
⊢ P.retractClosure.extensionProduct Q.retractClosure ≤ P.retractClosure.extensionProduct Q.retractClosure
Module: Mathlib.Algebra.Homology.QuasiIso

Goal state before:
pp: case mpr
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
hL : IsZero (L.sc i).homology
⊢ IsZero (K.sc i).homology → IsIso (ShortComplex.homologyMap ((shortComplexFunctor C c i).map f))

Tactic:
intro hK

Goal state after:
case mpr
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
hL : IsZero (L.sc i).homology
hK : IsZero (K.sc i).homology
⊢ IsIso (ShortComplex.homologyMap ((shortComplexFunctor C c i).map f))
Module: Mathlib.CategoryTheory.Triangulated.Subcategory

Goal state before:
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
eq : q ≫ r = 0
⊢ f₁ ≫ r = f₂ ≫ r

Tactic:
rw [← sub_eq_zero, ← sub_comp, hq, assoc, eq, comp_zero]

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

Goal state before:
pp: case mp
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝¹ : Category.{v_1, u_3} C
inst✝ : HasZeroMorphisms C
K : HomologicalComplex C c'
e : c.Embedding c'
⊢ IsZero K → K.IsStrictlySupported e ∧ K.IsStrictlySupportedOutside e

Tactic:
intro hK

Goal state after:
case mp
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝¹ : Category.{v_1, u_3} C
inst✝ : HasZeroMorphisms C
K : HomologicalComplex C c'
e : c.Embedding c'
hK : IsZero K
⊢ K.IsStrictlySupported e ∧ K.IsStrictlySupportedOutside e
Module: Mathlib.Algebra.Homology.HomotopyCategory.Triangulated

Goal state before:
pp: 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
⊢ (quotient C (ComplexShape.up ℤ)).map
        ((mappingCone.triangle g).mor₃ ≫ (CategoryTheory.shiftFunctor (CochainComplex C ℤ) 1).map (mappingCone.inr f)) ≫
      (Functor.commShiftIso (quotient C (ComplexShape.up ℤ)) 1).hom.app (mappingCone f) =
    (quotient C (ComplexShape.up ℤ)).map
        ((mappingConeCompHomotopyEquiv f g).hom ≫ (mappingCone.triangle (mappingCone.map f (f ≫ g) (𝟙 X₁) g ⋯)).mor₃) ≫
      (Functor.commShiftIso (quotient C (ComplexShape.up ℤ)) 1).hom.app (mappingCone f)

Tactic:
congr 2

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

Goal state before:
pp: case inr.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 L M : HomologicalComplex C c
φ : K ⟶ L
φ' : L ⟶ M
e : c.Embedding c'
j' : ι'
i j : ι
hj : e.r j' = some j
h : ¬c'.Rel (e.f i) j'
hi : e.r (e.f i) = some i
⊢ ¬c.Rel i j

Tactic:
obtain rfl := e.f_eq_of_r_eq_some hj

Goal state after:
case inr.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 L M : HomologicalComplex C c
φ : K ⟶ L
φ' : L ⟶ M
e : c.Embedding c'
i j : ι
hi : e.r (e.f i) = some i
hj : e.r (e.f j) = some j
h : ¬c'.Rel (e.f i) (e.f j)
⊢ ¬c.Rel i j
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'
inst✝ : e.IsRelIff
j k : ι
hk : c.Rel j k
a✝ : c'.Rel (c'.prev (e.f k)) (e.f k)
⊢ c'.Rel (e.f j) (e.f k)

Tactic:
by simpa only [e.rel_iff] using hk

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

Goal state before:
pp: case refine_5
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
X₁ X₂ X₃ : CochainComplex C ℤ
u₁₂ : X₁ ⟶ X₂
u₂₃ : X₂ ⟶ X₃
α : mappingCone.triangle u₁₂ ⟶ mappingCone.triangle (u₁₂ ≫ u₂₃) := mappingCone.triangleMap u₁₂ (u₁₂ ≫ u₂₃) (𝟙 X₁) u₂₃ ⋯
β : mappingCone.triangle (u₁₂ ≫ u₂₃) ⟶ mappingCone.triangle u₂₃ := mappingCone.triangleMap (u₁₂ ≫ u₂₃) u₂₃ u₁₂ (𝟙 X₃) ⋯
⊢ Triangle.mk ((quotient C (ComplexShape.up ℤ)).map α.hom₃) ((quotient C (ComplexShape.up ℤ)).map β.hom₃)
      (((quotient C (ComplexShape.up ℤ)).map (mappingCone.triangle u₂₃).mor₃ ≫
          (Functor.commShiftIso (quotient C (ComplexShape.up ℤ)) 1).hom.app (mappingCone.triangle u₂₃).obj₁) ≫
        (CategoryTheory.shiftFunctor (HomotopyCategory C (ComplexShape.up ℤ)) 1).map
          ((quotient C (ComplexShape.up ℤ)).map (mappingCone.triangle u₁₂).mor₂)) ≅
    mappingConeCompTriangleh u₁₂ u₂₃

Tactic:
exact Triangle.isoMk _ _ (Iso.refl _) (Iso.refl _) (Iso.refl _)
        (by dsimp [α, mappingConeCompTriangleh]; simp)
        (by dsimp [β, mappingConeCompTriangleh]; simp)
        (by dsimp [mappingConeCompTriangleh]; simp)

Goal state after: