mathlib-initiative/mathlib-tactics huggingface dataset:
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) ≫ 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:
L.counit_app_functor,

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 (L.functor.map (L.unitInv.app X) ≫ 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))
Module: Mathlib.CategoryTheory.Localization.Equivalence

Goal state before:
pp: C₁ : Type u_1
C₂ : Type u_2
D₁ : Type u_4
D₂ : Type u_5
inst✝⁷ : Category.{v_1, u_1} C₁
inst✝⁶ : Category.{v_2, u_2} C₂
inst✝⁵ : Category.{v_4, u_4} D₁
inst✝⁴ : Category.{v_5, u_5} D₂
L₁ : C₁ ⥤ D₁
W₁ : MorphismProperty C₁
inst✝³ : L₁.IsLocalization W₁
L₂ : C₂ ⥤ D₂
W₂ : MorphismProperty C₂
inst✝² : L₂.IsLocalization W₂
G : C₁ ⥤ D₂
G' : D₁ ⥤ D₂
inst✝¹ : Lifting L₁ W₁ G G'
F : C₂ ⥤ D₁
F' : D₂ ⥤ D₁
inst✝ : Lifting L₂ W₂ F F'
α : G ⋙ F' ≅ L₁
β : F ⋙ G' ≅ L₂
X : C₂
⊢ (Lifting.iso L₂ W₂ (F ⋙ G') (F' ⋙ G')).hom.app X ≫ β.hom.app X ≫ (Lifting.iso L₂ W₂ L₂ (𝟭 D₂)).inv.app X =
    G'.map ((Lifting.iso L₂ W₂ F F').hom.app X) ≫ β.hom.app X

Tactic:
]

Goal state after:
C₁ : Type u_1
C₂ : Type u_2
D₁ : Type u_4
D₂ : Type u_5
inst✝⁷ : Category.{v_1, u_1} C₁
inst✝⁶ : Category.{v_2, u_2} C₂
inst✝⁵ : Category.{v_4, u_4} D₁
inst✝⁴ : Category.{v_5, u_5} D₂
L₁ : C₁ ⥤ D₁
W₁ : MorphismProperty C₁
inst✝³ : L₁.IsLocalization W₁
L₂ : C₂ ⥤ D₂
W₂ : MorphismProperty C₂
inst✝² : L₂.IsLocalization W₂
G : C₁ ⥤ D₂
G' : D₁ ⥤ D₂
inst✝¹ : Lifting L₁ W₁ G G'
F : C₂ ⥤ D₁
F' : D₂ ⥤ D₁
inst✝ : Lifting L₂ W₂ F F'
α : G ⋙ F' ≅ L₁
β : F ⋙ G' ≅ L₂
X : C₂
⊢ (Lifting.iso L₂ W₂ (F ⋙ G') (F' ⋙ G')).hom.app X ≫ β.hom.app X ≫ (Lifting.iso L₂ W₂ L₂ (𝟭 D₂)).inv.app X =
    G'.map ((Lifting.iso L₂ W₂ F F').hom.app X) ≫ β.hom.app X
Module: Mathlib.CategoryTheory.Localization.Equivalence

Goal state before:
pp: C₁ : Type u_1
C₂ : Type u_2
D₁ : Type u_4
D₂ : Type u_5
inst✝⁷ : Category.{v_1, u_1} C₁
inst✝⁶ : Category.{v_2, u_2} C₂
inst✝⁵ : Category.{v_4, u_4} D₁
inst✝⁴ : Category.{v_5, u_5} D₂
L₁ : C₁ ⥤ D₁
W₁ : MorphismProperty C₁
inst✝³ : L₁.IsLocalization W₁
L₂ : C₂ ⥤ D₂
W₂ : MorphismProperty C₂
inst✝² : L₂.IsLocalization W₂
G : C₁ ⥤ D₂
G' : D₁ ⥤ D₂
inst✝¹ : Lifting L₁ W₁ G G'
F : C₂ ⥤ D₁
F' : D₂ ⥤ D₁
inst✝ : Lifting L₂ W₂ F F'
α : G ⋙ F' ≅ L₁
β : F ⋙ G' ≅ L₂
X : C₂
⊢ G'.map ((Lifting.iso L₂ W₂ F F').hom.app X) ≫ β.hom.app X = G'.map ((Lifting.iso L₂ W₂ F F').hom.app X) ≫ β.hom.app X

Tactic:
]

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
⊢ 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))

Tactic:
]

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.Opposite

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} D
L : C ⥤ D
W : MorphismProperty C
E : Type u_3
inst✝ : Category.{v_3, u_3} E
h : StrictUniversalPropertyFixedTarget L W Eᵒᵖ
F₁ F₂ : Dᵒᵖ ⥤ E
eq : L.op ⋙ F₁ = L.op ⋙ F₂
this : F₁.rightOp = F₂.rightOp
⊢ F₂.rightOp.leftOp = F₂.rightOp.leftOp

Tactic:
]

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

Goal state before:
pp: case refine_2
C : Type uC
inst✝¹ : Category.{uC', uC} C
W : MorphismProperty C
D : Type uD
inst✝ : Category.{uD', uD} D
G : C ⥤ D
hG : W.IsInvertedBy G
G₁ : C ⥤ D
hG₁ : W.IsInvertedBy G₁
G₂ : C ⥤ D
hG₂ : W.IsInvertedBy G₂
f : { obj := G₁, property := hG₁ } ⟶ { obj := G₂, property := hG₂ }
x✝ : C
⊢ (eqToHom ⋯ ≫ f.hom ≫ eqToHom ⋯).app x✝ = (eqToHom ⋯).app x✝ ≫ f.hom.app x✝ ≫ (eqToHom ⋯).app x✝

Tactic:
]

Goal state after:
case refine_2
C : Type uC
inst✝¹ : Category.{uC', uC} C
W : MorphismProperty C
D : Type uD
inst✝ : Category.{uD', uD} D
G : C ⥤ D
hG : W.IsInvertedBy G
G₁ : C ⥤ D
hG₁ : W.IsInvertedBy G₁
G₂ : C ⥤ D
hG₂ : W.IsInvertedBy G₂
f : { obj := G₁, property := hG₁ } ⟶ { obj := G₂, property := hG₂ }
x✝ : C
⊢ (eqToHom ⋯ ≫ f.hom ≫ eqToHom ⋯).app x✝ = (eqToHom ⋯).app x✝ ≫ f.hom.app x✝ ≫ (eqToHom ⋯).app x✝
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
⊢ e.hom.app Y = e.hom.app Y

Tactic:
]

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
hP : P.inverseImage (uniq W.Q L W).functor = ⊤
⊢ ⊤ = ⊤

Tactic:
]

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

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} D
L : C ⥤ D
inst✝ : L.IsLocalization ⊤
⊢ MorphismProperty.isomorphisms D = ⊤

Tactic:
]

Goal state after:
C : Type u_1
D : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} D
L : C ⥤ D
inst✝ : L.IsLocalization ⊤
⊢ MorphismProperty.isomorphisms D = ⊤
Module: Mathlib.CategoryTheory.Localization.Predicate

Goal state before:
pp: case mp
C : Type u_1
D : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} D
L : C ⥤ D
W : MorphismProperty C
X Y : C
f g : X ⟶ Y
inst✝ : L.IsLocalization W
h : W.Q.map f = W.Q.map g
e : W.Q ⋙ (Localization.uniq W.Q L W).functor ≅ L := Localization.compUniqFunctor W.Q L W
⊢ e.inv.app X ≫ (W.Q ⋙ (Localization.uniq W.Q L W).functor).map f ≫ e.hom.app Y =
    e.inv.app X ≫ (W.Q ⋙ (Localization.uniq W.Q L W).functor).map g ≫ e.hom.app Y

Tactic:
]

Goal state after:
case mp
C : Type u_1
D : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} D
L : C ⥤ D
W : MorphismProperty C
X Y : C
f g : X ⟶ Y
inst✝ : L.IsLocalization W
h : W.Q.map f = W.Q.map g
e : W.Q ⋙ (Localization.uniq W.Q L W).functor ≅ L := Localization.compUniqFunctor W.Q L W
⊢ e.inv.app X ≫ (W.Q ⋙ (Localization.uniq W.Q L W).functor).map f ≫ e.hom.app Y =
    e.inv.app X ≫ (W.Q ⋙ (Localization.uniq W.Q L W).functor).map g ≫ e.hom.app Y
Module: Mathlib.CategoryTheory.Localization.Predicate

Goal state before:
pp: case mp
C : Type u_1
D : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} D
L : C ⥤ D
W : MorphismProperty C
X Y : C
f g : X ⟶ Y
inst✝ : L.IsLocalization W
h : W.Q.map f = W.Q.map g
e : W.Q ⋙ (Localization.uniq W.Q L W).functor ≅ L := Localization.compUniqFunctor W.Q L W
⊢ e.inv.app X ≫ (Localization.uniq W.Q L W).functor.map (W.Q.map g) ≫ e.hom.app Y =
    e.inv.app X ≫ (Localization.uniq W.Q L W).functor.map (W.Q.map g) ≫ e.hom.app Y

Tactic:
]

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

Goal state before:
pp: case mpr
C : Type u_1
D : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} D
L : C ⥤ D
W : MorphismProperty C
X Y : C
f g : X ⟶ Y
inst✝ : L.IsLocalization W
h : L.map f = L.map g
e : L ⋙ (Localization.uniq L W.Q W).functor ≅ W.Q := Localization.compUniqFunctor L W.Q W
⊢ e.inv.app X ≫ (L ⋙ (Localization.uniq L W.Q W).functor).map f ≫ e.hom.app Y =
    e.inv.app X ≫ (L ⋙ (Localization.uniq L W.Q W).functor).map g ≫ e.hom.app Y

Tactic:
]

Goal state after:
case mpr
C : Type u_1
D : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} D
L : C ⥤ D
W : MorphismProperty C
X Y : C
f g : X ⟶ Y
inst✝ : L.IsLocalization W
h : L.map f = L.map g
e : L ⋙ (Localization.uniq L W.Q W).functor ≅ W.Q := Localization.compUniqFunctor L W.Q W
⊢ e.inv.app X ≫ (L ⋙ (Localization.uniq L W.Q W).functor).map f ≫ e.hom.app Y =
    e.inv.app X ≫ (L ⋙ (Localization.uniq L W.Q W).functor).map g ≫ e.hom.app Y
Module: Mathlib.CategoryTheory.Localization.Predicate

Goal state before:
pp: case mpr
C : Type u_1
D : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} D
L : C ⥤ D
W : MorphismProperty C
X Y : C
f g : X ⟶ Y
inst✝ : L.IsLocalization W
h : L.map f = L.map g
e : L ⋙ (Localization.uniq L W.Q W).functor ≅ W.Q := Localization.compUniqFunctor L W.Q W
⊢ e.inv.app X ≫ (Localization.uniq L W.Q W).functor.map (L.map g) ≫ e.hom.app Y =
    e.inv.app X ≫ (Localization.uniq L W.Q W).functor.map (L.map g) ≫ e.hom.app Y

Tactic:
]

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

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
X Y : C
φ ψ : W.LeftFraction₂ X Y
Z : C
t₁ : φ.Y' ⟶ Z
t₂ : ψ.Y' ⟶ Z
hst : φ.s ≫ t₁ = ψ.s ≫ t₂
hft : φ.f ≫ t₁ = ψ.f ≫ t₂
ht : W (φ.s ≫ t₁)
Z' : C
t₁' : φ.Y' ⟶ Z'
t₂' : ψ.Y' ⟶ Z'
hst' : φ.s ≫ t₁' = ψ.s ≫ t₂'
hft' : φ.f' ≫ t₁' = ψ.f' ≫ t₂'
ht' : W (φ.s ≫ t₁')
α : W.LeftFraction Z Z'
hα : φ.s ≫ t₁' ≫ α.s = φ.s ≫ t₁ ≫ α.f
Z'' : C
u : α.Y' ⟶ Z''
hu : W u
fac : (t₁' ≫ α.s) ≫ u = (t₁ ≫ α.f) ≫ u
⊢ φ.s ≫ t₁' ≫ α.s ≫ u = φ.s ≫ t₁' ≫ α.s ≫ u

Tactic:
]

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

Goal state before:
pp: case refine_1
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
W : MorphismProperty C
X Y : C
z₁ z₂ z₃ : W.LeftFraction X Y
inst✝ : W.HasLeftCalculusOfFractions
Z₄ : C
t₁ : z₁.Y' ⟶ Z₄
t₂ : z₂.Y' ⟶ Z₄
hst : z₁.s ≫ t₁ = z₂.s ≫ t₂
hft : z₁.f ≫ t₁ = z₂.f ≫ t₂
ht : W (z₁.s ≫ t₁)
Z₅ : C
u₂ : z₂.Y' ⟶ Z₅
u₃ : z₃.Y' ⟶ Z₅
hsu : z₂.s ≫ u₂ = z₃.s ≫ u₃
hfu : z₂.f ≫ u₂ = z₃.f ≫ u₃
hu : W (z₂.s ≫ u₂)
Y'✝ : C
v₄ : Z₄ ⟶ Y'✝
v₅ : Z₅ ⟶ Y'✝
hv₅ : W v₅
fac : z₃.s ≫ u₃ ≫ v₅ = z₁.s ≫ t₁ ≫ v₄
eq : z₂.s ≫ u₂ ≫ v₅ = z₂.s ≫ t₂ ≫ v₄
Z₇ : C
w : Y'✝ ⟶ Z₇
hw : W w
fac' : u₂ ≫ v₅ ≫ w = t₂ ≫ v₄ ≫ w
⊢ z₁.s ≫ t₁ ≫ v₄ ≫ w = z₁.s ≫ t₁ ≫ v₄ ≫ w

Tactic:
]

Goal state after:
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)
⊢ F.map φ.f + F.map φ.f' = F.map φ.fst.f + F.map φ.snd.f

Tactic:
]

Goal state after:
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
hφ : LeftFractionRel φ₀ φ
hφ₀ : neg' W (φ₀.map L ⋯) = φ₀.neg.map L ⋯
⊢ neg' W (φ₀.map L ⋯) = φ.neg.map L ⋯

Tactic:
]

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
hφ : LeftFractionRel φ₀ φ
hφ₀ : neg' W (φ₀.map L ⋯) = φ₀.neg.map L ⋯
⊢ 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.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 ⋯) ≫ L.map (φ₀.s ≫ t₁) = φ.neg.map L ⋯ ≫ L.map (φ₀.s ≫ t₁)

Tactic:
]

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 ⋯) ≫ L.map (φ₀.s ≫ t₁) = φ.neg.map L ⋯ ≫ L.map (φ₀.s ≫ t₁)
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)
this : IsIso (L.map (φ₀.s ≫ t₁))
⊢ neg' W (φ₀.map L ⋯) ≫ L.map φ₀.s ≫ L.map t₁ = φ.neg.map L ⋯ ≫ L.map (φ₀.s ≫ t₁)

Tactic:
]

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 ⋯) ≫ L.map φ₀.s ≫ L.map t₁ = φ.neg.map L ⋯ ≫ L.map (φ₀.s ≫ t₁)
Module: Mathlib.CategoryTheory.Localization.CalculusOfFractions

Goal state before:
pp: case refine_2
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
W : MorphismProperty C
X Y : C
z₁ z₂ z₃ : W.LeftFraction X Y
inst✝ : W.HasLeftCalculusOfFractions
Z₄ : C
t₁ : z₁.Y' ⟶ Z₄
t₂ : z₂.Y' ⟶ Z₄
hst : z₁.s ≫ t₁ = z₂.s ≫ t₂
hft : z₁.f ≫ t₁ = z₂.f ≫ t₂
ht : W (z₁.s ≫ t₁)
Z₅ : C
u₂ : z₂.Y' ⟶ Z₅
u₃ : z₃.Y' ⟶ Z₅
hsu : z₂.s ≫ u₂ = z₃.s ≫ u₃
hfu : z₂.f ≫ u₂ = z₃.f ≫ u₃
hu : W (z₂.s ≫ u₂)
Y'✝ : C
v₄ : Z₄ ⟶ Y'✝
v₅ : Z₅ ⟶ Y'✝
hv₅ : W v₅
fac : z₃.s ≫ u₃ ≫ v₅ = z₁.s ≫ t₁ ≫ v₄
eq : z₂.s ≫ u₂ ≫ v₅ = z₂.s ≫ t₂ ≫ v₄
Z₇ : C
w : Y'✝ ⟶ Z₇
hw : W w
fac' : u₂ ≫ v₅ ≫ w = t₂ ≫ v₄ ≫ w
⊢ z₃.f ≫ u₃ ≫ v₅ ≫ w = z₃.f ≫ u₃ ≫ v₅ ≫ w

Tactic:
]

Goal state after:
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)
this : IsIso (L.map (φ₀.s ≫ t₁))
⊢ L.map (-φ.f ≫ t₂) = L.map (-φ.f ≫ t₂)

Tactic:
]

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

Goal state before:
pp: C₁ : Type u₁
C₂ : Type u₂
C₃ : Type u₃
D₁ : Type u₄
D₂ : Type u₅
inst✝⁴ : Category.{v₁, u₁} C₁
inst✝³ : Category.{v₂, u₂} C₂
inst✝² : Category.{v₃, u₃} C₃
inst✝¹ : Category.{v₄, u₄} D₁
inst✝ : Category.{v₅, u₅} D₂
W₁ : MorphismProperty C₁
W₂ : MorphismProperty C₂
W₃ : MorphismProperty C₃
F : C₁ ⥤ C₂
hW : W₁ = W₂.inverseImage F
⊢ W₂.inverseImage F ≤ W₂.inverseImage F

Tactic:
]

Goal state after:
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)
this : IsIso (L.map (φ₀.s ≫ t₁))
⊢ φ₀.add.map L ⋯ ≫ L.map (φ₀.s ≫ t₁) = φ.add.map L ⋯ ≫ L.map (φ₀.s ≫ t₁)

Tactic:
]

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))
⊢ φ₀.add.map L ⋯ ≫ L.map (φ₀.s ≫ t₁) = φ.add.map L ⋯ ≫ L.map (φ₀.s ≫ t₁)
Module: Mathlib.CategoryTheory.Localization.LocalizerMorphism

Goal state before:
pp: C₁ : Type u₁
C₂ : Type u₂
D₁ : Type u₄
D₂ : Type u₅
inst✝⁷ : Category.{v₁, u₁} C₁
inst✝⁶ : Category.{v₂, u₂} C₂
inst✝⁵ : Category.{v₄, u₄} D₁
inst✝⁴ : Category.{v₅, u₅} D₂
W₁ : MorphismProperty C₁
W₂ : MorphismProperty C₂
Φ : LocalizerMorphism W₁ W₂
L₁ : C₁ ⥤ D₁
inst✝³ : L₁.IsLocalization W₁
L₂ : C₂ ⥤ D₂
inst✝² : L₂.IsLocalization W₂
G : D₁ ⥤ D₂
inst✝¹ : CatCommSq Φ.functor L₁ L₂ G
inst✝ : G.IsEquivalence
⊢ G.IsEquivalence

Tactic:
]

Goal state after:
C₁ : Type u₁
C₂ : Type u₂
D₁ : Type u₄
D₂ : Type u₅
inst✝⁷ : Category.{v₁, u₁} C₁
inst✝⁶ : Category.{v₂, u₂} C₂
inst✝⁵ : Category.{v₄, u₄} D₁
inst✝⁴ : Category.{v₅, u₅} D₂
W₁ : MorphismProperty C₁
W₂ : MorphismProperty C₂
Φ : LocalizerMorphism W₁ W₂
L₁ : C₁ ⥤ D₁
inst✝³ : L₁.IsLocalization W₁
L₂ : C₂ ⥤ D₂
inst✝² : L₂.IsLocalization W₂
G : D₁ ⥤ D₂
inst✝¹ : CatCommSq Φ.functor L₁ L₂ G
inst✝ : G.IsEquivalence
⊢ G.IsEquivalence
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
h : Φ.IsLocalizedEquivalence
inst✝ : CatCommSq Φ.functor L₁ L₂ G
⊢ (Φ.localizedFunctor W₁.Q W₂.Q).IsEquivalence

Tactic:
]

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
h : Φ.IsLocalizedEquivalence
inst✝ : CatCommSq Φ.functor L₁ L₂ G
⊢ (Φ.localizedFunctor W₁.Q W₂.Q).IsEquivalence
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)
this : IsIso (L.map (φ₀.s ≫ t₁))
⊢ φ₀.add.map L ⋯ ≫ L.map (φ₀.s ≫ t₁) = φ.add.map L ⋯ ≫ L.map (φ.s ≫ t₂)

Tactic:
]

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))
⊢ φ₀.add.map L ⋯ ≫ L.map (φ₀.s ≫ t₁) = φ.add.map L ⋯ ≫ L.map (φ.s ≫ t₂)
Module: Mathlib.CategoryTheory.Localization.CalculusOfFractions.Fractions

Goal state before:
pp: case mp.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
φ ψ : W.LeftFraction₂ X Y
Z : C
t₁ : φ.Y' ⟶ Z
t₂ : ψ.Y' ⟶ Z
hst : φ.s ≫ t₁ = ψ.s ≫ t₂
hft : φ.f ≫ t₁ = ψ.f ≫ t₂
ht : W (φ.s ≫ t₁)
Z' : C
t₁' : φ.Y' ⟶ Z'
t₂' : ψ.Y' ⟶ Z'
hst' : φ.s ≫ t₁' = ψ.s ≫ t₂'
hft' : φ.f' ≫ t₁' = ψ.f' ≫ t₂'
ht' : W (φ.s ≫ t₁')
α : W.LeftFraction Z Z'
hα : φ.s ≫ t₁' ≫ α.s = φ.s ≫ t₁ ≫ α.f
Z'' : C
u : α.Y' ⟶ Z''
hu : W u
hα' : ψ.s ≫ t₂ ≫ α.f ≫ u = ψ.s ≫ t₂' ≫ α.s ≫ u
Z''' : C
u' : Z'' ⟶ Z'''
hu' : W u'
fac : t₁' ≫ α.s ≫ u = t₁ ≫ α.f ≫ u
fac' : t₂ ≫ α.f ≫ u ≫ u' = t₂' ≫ α.s ≫ u ≫ u'
⊢ ψ.s ≫ t₂' ≫ α.s ≫ u ≫ u' = ψ.s ≫ t₂' ≫ α.s ≫ u ≫ u'

Tactic:
]

Goal state after: