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

Goal state before:
pp: 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
F₁ F₂ : W.Localization ⥤ D
τ : W.Q ⋙ F₁ ⟶ W.Q ⋙ F₂
this : MorphismProperty.naturalityProperty (NatTransExtension.app τ) = ⊤
⊢ ∀ ⦃X Y : W.Localization⦄ (f : X ⟶ Y), F₁.map f ≫ NatTransExtension.app τ Y = NatTransExtension.app τ X ≫ F₂.map f

Tactic:
by

Goal state after:
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).Faithful

Tactic:
by

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

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Category.{v_2, u_2} D
L✝ : C ⥤ D
W✝ : MorphismProperty C
E✝ : Type u_3
inst✝³ : Category.{v_3, u_3} E✝
inst✝² : L✝.IsLocalization W✝
L : C ⥤ D
W : MorphismProperty C
inst✝¹ : L.IsLocalization W
E : Type u_4
inst✝ : Category.{v_4, u_4} E
⊢ ((whiskeringLeft C D E).obj L).FullyFaithful

Tactic:
by

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.functor L R B.functor
⊢ hInv T.symm R L B.symm (hInv T L R B h) = h

Tactic:
by

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

Goal state before:
pp: 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
F₁ F₂ : W.Localization ⥤ D
τ : W.Q ⋙ F₁ ⟶ W.Q ⋙ F₂
⊢ ∀ ⦃X Y : W.Localization⦄ (f : X ⟶ Y), F₁.map f ≫ NatTransExtension.app τ Y = NatTransExtension.app τ X ≫ F₂.map f

Tactic:
by

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

Goal state before:
pp: C : Type uC
inst✝¹ : Category.{uC', uC} C
W : MorphismProperty C
D : Type uD
inst✝ : Category.{uD', uD} D
F G : W.Localization ⥤ D
τ : W.Q ⋙ F ⟶ W.Q ⋙ G
⊢ W.Q.whiskerLeft (natTransExtension τ) = τ

Tactic:
by

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

Goal state before:
pp: 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
⊢ Octahedron' comm h₁₂ h₂₃ h₁₃

Tactic:
by

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

Goal state before:
pp: C : Type uC
inst✝¹ : Category.{uC', uC} C
W : MorphismProperty C
D : Type uD
inst✝ : Category.{uD', uD} D
F G : W.Localization ⥤ D
τ : W.Q ⋙ F ⟶ W.Q ⋙ G
⊢ 𝟙 W.Q ◫ natTransExtension τ = τ

Tactic:
by

Goal state after:
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
E : Type u_3
inst✝¹ : Category.{v_3, u_3} E
L : C ⥤ D
W : MorphismProperty C
inst✝ : L.IsLocalization W
F₁ F₂ : D ⥤ E
τ τ' : F₁ ⟶ F₂
h : ∀ (X : C), τ.app (L.obj X) = τ'.app (L.obj X)
⊢ τ = τ'

Tactic:
by

Goal state after:
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
F₁ F₂ F₃ : C ⥤ E
F₁' F₂' F₃' : D ⥤ E
h₁ : Lifting L W F₁ F₁'
h₂ : Lifting L W F₂ F₂'
h₃ : Lifting L W F₃ F₃'
τ : F₁ ⟶ F₂
τ' : F₂ ⟶ F₃
X : C
⊢ (liftNatTrans L W F₁ F₂ F₁' F₂' τ ≫ liftNatTrans L W F₂ F₃ F₂' F₃' τ').app (L.obj X) =
    (liftNatTrans L W F₁ F₃ F₁' F₃' (τ ≫ τ')).app (L.obj X)

Tactic:
by

Goal state after:
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
F : C ⥤ E
F' : D ⥤ E
h : Lifting L W F F'
X : C
⊢ (liftNatTrans L W F F F' F' (𝟙 F)).app (L.obj X) = (𝟙 F').app (L.obj X)

Tactic:
by

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

Goal state before:
pp: C : Type uC
inst✝¹ : Category.{uC', uC} C
W : MorphismProperty C
D : Type uD
inst✝ : Category.{uD', uD} D
F G : W.Localization ⥤ D
τ₁ τ₂ : F ⟶ G
h : 𝟙 W.Q ◫ τ₁ = 𝟙 W.Q ◫ τ₂
⊢ τ₁ = τ₂

Tactic:
by

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
W : MorphismProperty C
L₁ L₂ : C ⥤ D
e : L₁ ≅ L₂
inst✝ : L₁.IsLocalization W
⊢ L₂.IsLocalization W

Tactic:
by

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

Goal state before:
pp: 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
X✝ Y✝ : W.FunctorsInverting D
τ : X✝ ⟶ Y✝
⊢ W.Q ⋙ lift X✝.obj ⋯ = X✝.obj

Tactic:
by

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

Goal state before:
pp: 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
X✝ Y✝ : W.FunctorsInverting D
τ : X✝ ⟶ Y✝
⊢ Y✝.obj = W.Q ⋙ lift Y✝.obj ⋯

Tactic:
by

Goal state after:
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_4
inst✝¹ : Category.{v_4, u_4} E
L' : C ⥤ E
eq : D ≌ E
inst✝ : L.IsLocalization W
e : L ⋙ eq.functor ≅ L'
⊢ W.IsInvertedBy L'

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
h :
  ∀ ⦃X₁' X₂' X₃' : C⦄ (u₁₂' : X₁' ⟶ X₂') (u₂₃' : X₂' ⟶ X₃'),
    ∃ X₁ X₂ X₃ Z₁₂ Z₂₃ Z₁₃ u₁₂ u₂₃ e₁ e₂ e₃,
      ∃ (_ : u₁₂' ≫ e₂.hom = e₁.hom ≫ u₁₂) (_ : u₂₃' ≫ e₃.hom = e₂.hom ≫ u₂₃),
        ∃ v₁₂ w₁₂,
          ∃ (h₁₂ : Triangle.mk u₁₂ v₁₂ w₁₂ ∈ distinguishedTriangles),
            ∃ v₂₃ w₂₃,
              ∃ (h₂₃ : Triangle.mk u₂₃ v₂₃ w₂₃ ∈ distinguishedTriangles),
                ∃ v₁₃ w₁₃,
                  ∃ (h₁₃ : Triangle.mk (u₁₂ ≫ u₂₃) v₁₃ w₁₃ ∈ distinguishedTriangles),
                    Nonempty (Octahedron ⋯ h₁₂ h₂₃ h₁₃)
X₁' X₂' X₃' Z₁₂' Z₂₃' Z₁₃' : C
u₁₂' : X₁' ⟶ X₂'
u₂₃' : X₂' ⟶ X₃'
u₁₃' : X₁' ⟶ X₃'
comm' : u₁₂' ≫ u₂₃' = u₁₃'
v₁₂' : X₂' ⟶ Z₁₂'
w₁₂' : Z₁₂' ⟶ (shiftFunctor C 1).obj X₁'
h₁₂' : Triangle.mk u₁₂' v₁₂' w₁₂' ∈ distinguishedTriangles
v₂₃' : X₃' ⟶ Z₂₃'
w₂₃' : Z₂₃' ⟶ (shiftFunctor C 1).obj X₂'
h₂₃' : Triangle.mk u₂₃' v₂₃' w₂₃' ∈ distinguishedTriangles
v₁₃' : X₃' ⟶ Z₁₃'
w₁₃' : Z₁₃' ⟶ (shiftFunctor C 1).obj X₁'
h₁₃' : Triangle.mk u₁₃' v₁₃' w₁₃' ∈ distinguishedTriangles
⊢ Nonempty (Octahedron comm' h₁₂' h₂₃' h₁₃')

Tactic:
by

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

Goal state before:
pp: 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✝
G : W.FunctorsInverting D
⊢ 𝟙 W.Q ◫ natTransExtension (eqToHom ⋯ ≫ (𝟙 G).hom ≫ eqToHom ⋯) = 𝟙 W.Q ◫ 𝟙 (lift G.obj ⋯)

Tactic:
by

Goal state after:
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_4
inst✝¹ : Category.{v_4, u_4} E
L' : C ⥤ E
eq : D ≌ E
inst✝ : L.IsLocalization W
e : L ⋙ eq.functor ≅ L'
⊢ L'.IsLocalization W

Tactic:
by

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

Goal state before:
pp: 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
X✝ Y✝ Z✝ : W.FunctorsInverting D
τ₁ : X✝ ⟶ Y✝
τ₂ : Y✝ ⟶ Z✝
⊢ 𝟙 W.Q ◫ natTransExtension (eqToHom ⋯ ≫ (τ₁ ≫ τ₂).hom ≫ eqToHom ⋯) =
    𝟙 W.Q ◫ (natTransExtension (eqToHom ⋯ ≫ τ₁.hom ≫ eqToHom ⋯) ≫ natTransExtension (eqToHom ⋯ ≫ τ₂.hom ≫ eqToHom ⋯))

Tactic:
by

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
W : MorphismProperty C
hW : W ≤ MorphismProperty.isomorphisms C
inst✝ : L.IsEquivalence
⊢ L.IsLocalization W

Tactic:
by

Goal state after:
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
⊢ (equivalence L₁ W₁ L₂ W₂ G G' F F' α β).counitIso.app (L₂.obj X) =
    (Lifting.iso L₂ W₂ (F ⋙ G') (F' ⋙ G')).app X ≪≫ β.app X

Tactic:
by

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

Goal state before:
pp: C₁ : Type u_1
C₂ : Type u_2
D : Type u_3
inst✝³ : Category.{v_1, u_1} C₁
inst✝² : Category.{v_2, u_2} C₂
inst✝¹ : Category.{v_3, u_3} D
L₁ : C₁ ⥤ D
W₁ : MorphismProperty C₁
L₂ : C₂ ⥤ D
W₂ : MorphismProperty C₂
E : C₁ ≌ C₂
hW₁ : W₁ ≤ W₂.isoClosure.inverseImage E.functor
hW₂ : W₂.IsInvertedBy L₂
inst✝ : L₁.IsLocalization W₁
iso : E.functor ⋙ L₂ ≅ L₁
x✝¹ x✝ : C₁
f : x✝¹ ⟶ x✝
hf : W₁ f
⊢ IsIso ((E.functor ⋙ W₂.Q).map f)

Tactic:
by

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

Goal state before:
pp: C₁ : Type u_1
C₂ : Type u_2
D : Type u_3
inst✝³ : Category.{v_1, u_1} C₁
inst✝² : Category.{v_2, u_2} C₂
inst✝¹ : Category.{v_3, u_3} D
L₁ : C₁ ⥤ D
W₁ : MorphismProperty C₁
L₂ : C₂ ⥤ D
W₂ : MorphismProperty C₂
E : C₁ ≌ C₂
hW₁ : W₁ ≤ W₂.isoClosure.inverseImage E.functor
hW₂ : W₂.IsInvertedBy L₂
inst✝ : L₁.IsLocalization W₁
iso : E.functor ⋙ L₂ ≅ L₁
h : W₁.IsInvertedBy (E.functor ⋙ W₂.Q)
⊢ L₂ ⋙ lift (E.functor ⋙ W₂.Q) h L₁ ≅ W₂.Q

Tactic:
by

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 G₂ : W.Localization ⥤ D
τ : G₁ ⟶ G₂
⊢ (𝟭 (W.Localization ⥤ D)).map τ = eqToHom ⋯ ≫ (functor W D ⋙ inverse W D).map τ ≫ eqToHom ⋯

Tactic:
apply natTrans_hcomp_injective
        ext X
        simp

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

Goal state before:
pp: C₁ : Type u_1
C₂ : Type u_2
D : Type u_3
inst✝³ : Category.{v_1, u_1} C₁
inst✝² : Category.{v_2, u_2} C₂
inst✝¹ : Category.{v_3, u_3} D
L₁ : C₁ ⥤ D
W₁ : MorphismProperty C₁
L₂ : C₂ ⥤ D
W₂ : MorphismProperty C₂
E : C₁ ≌ C₂
hW₁ : W₁ ≤ W₂.isoClosure.inverseImage E.functor
hW₂ : W₂.IsInvertedBy L₂
inst✝ : L₁.IsLocalization W₁
iso : E.functor ⋙ L₂ ≅ L₁
⊢ L₂.IsLocalization W₂

Tactic:
by

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

Goal state before:
pp: 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
⊢ 𝟭 (W.Localization ⥤ D) = functor W D ⋙ inverse W D

Tactic:
by

Goal state after:
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✝⁵ : Category.{v_1, u_1} C₁
inst✝⁴ : Category.{v_2, u_2} C₂
inst✝³ : Category.{v_4, u_4} D₁
inst✝² : Category.{v_5, u_5} D₂
L₁ : C₁ ⥤ D₁
W₁ : MorphismProperty C₁
inst✝¹ : L₁.IsLocalization W₁
L₂ : C₂ ⥤ D₂
W₂ : MorphismProperty C₂
E : C₁ ≌ C₂
E' : D₁ ≌ D₂
inst✝ : CatCommSq E.functor L₁ L₂ E'.functor
hW₁ : W₁ ≤ W₂.isoClosure.inverseImage E.functor
hW₂ : W₂.IsInvertedBy L₂
⊢ L₂.IsLocalization W₂

Tactic:
by

Goal state after:
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
F : Cᵒᵖ ⥤ E
hF : W.op.IsInvertedBy F
⊢ L.op ⋙ (h.lift F.rightOp ⋯).leftOp = F

Tactic:
by

Goal state after:
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
⊢ F₁ = F₂

Tactic:
by

Goal state after:
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
⊢ F₁ = F₂

Tactic:
by
      rw [← F₁.rightOp_leftOp_eq, ← F₂.rightOp_leftOp_eq, this]

Goal state after:
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
⊢ F₁ = F₂

Tactic:
by

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

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✝
L : Cᵒᵖ ⥤ Dᵒᵖ
W : MorphismProperty Cᵒᵖ
inst✝ : L.IsLocalization W
this : CatCommSq (opOpEquivalence C).functor L.op L.unop (opOpEquivalence D).functor
x✝² x✝¹ : C
x✝ : x✝² ⟶ x✝¹
hf : W.unop x✝
⊢ IsIso (L.unop.map x✝)

Tactic:
by

Goal state after:
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
inst✝ : L.IsLocalization W
X Y : Cᵒᵖ
w : X ⟶ Y
hw : W.op w
⊢ (isoOfHom L.op W.op w hw).unop = isoOfHom L W w.unop hw

Tactic:
by

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
⊢ vInv B L.symm R.symm T (vInv T L R B h) = h

Tactic:
by

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

Goal state before:
pp: 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
⊢ inverse W D ⋙ functor W D = 𝟭 (W.FunctorsInverting D)

Tactic:
by

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

Goal state before:
pp: 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
F : W.Localization ⥤ D
⊢ (WhiskeringLeftEquivalence.functor W D).map ((WhiskeringLeftEquivalence.unitIso W D).hom.app F) ≫
      (WhiskeringLeftEquivalence.counitIso W D).hom.app ((WhiskeringLeftEquivalence.functor W D).obj F) =
    𝟙 ((WhiskeringLeftEquivalence.functor W D).obj F)

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
D₁ : Type u_4
D₂ : Type u_5
inst✝³ : Category.{v_4, u_4} D₁
inst✝² : Category.{v_5, u_5} D₂
L₁ : C ⥤ D₁
L₂ : C ⥤ D₂
W' : MorphismProperty C
inst✝¹ : L₁.IsLocalization W'
inst✝ : L₂.IsLocalization W'
⊢ (uniq L₁ L₂ W').symm = uniq L₂ L₁ W'

Tactic:
by

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

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.HasRightCalculusOfFractions
n : ℕ
Y : ComposableArrows D n
⊢ (L.mapComposableArrows n).essImage Y

Tactic:
by

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

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
n : ℕ
⊢ (L.mapComposableArrows n).EssSurj

Tactic:
by

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:
by

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
⊢ P = ⊤

Tactic:
by

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 ⊤
x✝¹ x✝ : C
f : x✝¹ ⟶ x✝
⊢ ⊤ f

Tactic:
by

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 ⊤
⊢ IsGroupoid D

Tactic:
by

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
W : MorphismProperty C
X Y : C
f g : X ⟶ Y
inst✝ : L.IsLocalization W
⊢ AreEqualizedByLocalization W f g ↔ L.map f = L.map g

Tactic:
by

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

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Category.{v_2, u_2} D
W : MorphismProperty C
X Y : C
f g : X ⟶ Y
h : AreEqualizedByLocalization W f g
F : C ⥤ D
hF : W.IsInvertedBy F
⊢ F.map f = F.map g

Tactic:
by

Goal state after:
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.fst z₂.fst

Tactic:
by

Goal state after:
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:
by

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

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Category.{v_2, u_2} D
W : MorphismProperty C
X Y : C
φ : W.LeftFraction X Y
L : C ⥤ D
hL : W.IsInvertedBy L
⊢ φ.map L hL ≫ L.map φ.s = L.map φ.f

Tactic:
by

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

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
W : MorphismProperty C
X Y : C
f : X ⟶ Y
L : C ⥤ D
hL : W.IsInvertedBy L
inst✝ : W.ContainsIdentities
⊢ (ofHom W f).map L hL = L.map f

Tactic:
by

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

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Category.{v_2, u_2} D
W : MorphismProperty C
X Y : C
s : Y ⟶ X
hs : W s
L : C ⥤ D
hL : W.IsInvertedBy L
⊢ (ofInv s hs).map L hL ≫ L.map s = 𝟙 (L.obj X)

Tactic:
by

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

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Category.{v_2, u_2} D
W : MorphismProperty C
X Y : C
s : Y ⟶ X
hs : W s
L : C ⥤ D
hL : W.IsInvertedBy L
⊢ L.map s ≫ (ofInv s hs).map L hL = 𝟙 (L.obj Y)

Tactic:
by

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

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Category.{v_2, u_2} D
W : MorphismProperty C
X Y : C
φ : W.RightFraction X Y
L : C ⥤ D
hL : W.IsInvertedBy L
⊢ L.map φ.s ≫ φ.map L hL = L.map φ.f

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝ : Category.{v_1, u_1} C
W : MorphismProperty C
X Y : C
z : W.LeftFraction X Y
⊢ W (z.s ≫ 𝟙 z.Y')

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝ : Category.{v_1, u_1} C
W : MorphismProperty C
X Y : C
z₁ z₂ : W.LeftFraction X Y
Z : C
t₁ : z₁.Y' ⟶ Z
t₂ : z₂.Y' ⟶ Z
hst : z₁.s ≫ t₁ = z₂.s ≫ t₂
hft : z₁.f ≫ t₁ = z₂.f ≫ t₂
ht : W (z₁.s ≫ t₁)
⊢ W (z₂.s ≫ t₂)

Tactic:
by

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

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

Tactic:
by

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
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
⊢ z₂.s ≫ u₂ ≫ v₅ = z₂.s ≫ t₂ ≫ v₄

Tactic:
by

Goal state after: