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

Goal state before:
pp: V : Type u₁
inst✝ : Quiver V
P : MorphismProperty (Paths V)
id : ∀ {v : V}, P (𝟙 ((of V).obj v))
comp : ∀ {u v w : V} (p : u ⟶ v) (q : (of V).obj v ⟶ (of V).obj w), P q → P ((of V).map p ≫ q)
X✝ Y✝ : Paths V
f✝ : X✝ ⟶ Y✝
⊢ P f✝ ↔ ⊤ f✝

Tactic:
constructor

Goal state after:
case mp
V : Type u₁
inst✝ : Quiver V
P : MorphismProperty (Paths V)
id : ∀ {v : V}, P (𝟙 ((of V).obj v))
comp : ∀ {u v w : V} (p : u ⟶ v) (q : (of V).obj v ⟶ (of V).obj w), P q → P ((of V).map p ≫ q)
X✝ Y✝ : Paths V
f✝ : X✝ ⟶ Y✝
⊢ P f✝ → ⊤ f✝

case mpr
V : Type u₁
inst✝ : Quiver V
P : MorphismProperty (Paths V)
id : ∀ {v : V}, P (𝟙 ((of V).obj v))
comp : ∀ {u v w : V} (p : u ⟶ v) (q : (of V).obj v ⟶ (of V).obj w), P q → P ((of V).map p ≫ q)
X✝ Y✝ : Paths V
f✝ : X✝ ⟶ Y✝
⊢ ⊤ f✝ → P f✝
Module: Mathlib.Algebra.Homology.HomologySequence

Goal state before:
pp: case refine_1
C : Type u_1
ι : Type u_2
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Abelian C
c : ComplexShape ι
S : ShortComplex (HomologicalComplex C c)
hS : S.Exact
inst✝³ : Mono S.f
i : ι
inst✝² : S.X₁.HasHomology i
inst✝¹ : S.X₂.HasHomology i
inst✝ : S.X₃.HasHomology i
this : Mono (S.map (eval C c i)).f
hi : IsLimit (KernelFork.ofι (S.map (eval C c i)).f ⋯)
A : C
k : A ⟶ S.X₂.cycles i
hk : k ≫ cyclesMap S.g i = 0
H : { l // l ≫ S.f.f i = k ≫ S.X₂.iCycles i }
⊢ ↑H ≫ S.X₁.d i (c.next i) ≫ S.f.f (c.next i) = 0

Tactic:
← Hom.comm,

Goal state after:
case refine_1
C : Type u_1
ι : Type u_2
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Abelian C
c : ComplexShape ι
S : ShortComplex (HomologicalComplex C c)
hS : S.Exact
inst✝³ : Mono S.f
i : ι
inst✝² : S.X₁.HasHomology i
inst✝¹ : S.X₂.HasHomology i
inst✝ : S.X₃.HasHomology i
this : Mono (S.map (eval C c i)).f
hi : IsLimit (KernelFork.ofι (S.map (eval C c i)).f ⋯)
A : C
k : A ⟶ S.X₂.cycles i
hk : k ≫ cyclesMap S.g i = 0
H : { l // l ≫ S.f.f i = k ≫ S.X₂.iCycles i }
⊢ ↑H ≫ S.f.f i ≫ S.X₂.d i (c.next i) = 0
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
⊢ Function.RightInverse (fun X ↦ X.as.obj) W.Q.obj

Tactic:
by
    rintro ⟨⟨X⟩⟩
    rfl

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

Goal state before:
pp: 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
⊢ P f = ⊤ f

Tactic:
ext

Goal state after:
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
⊢ P f ↔ ⊤ f
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
E : Type u_3
inst✝ : Category.{v_3, u_3} E
⊢ Localization.Construction.lift W.Q ⋯ = 𝟭 W.Localization

Tactic:
apply Localization.Construction.uniq

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
W : MorphismProperty C
E : Type u_3
inst✝ : Category.{v_3, u_3} E
⊢ W.Q ⋙ Localization.Construction.lift W.Q ⋯ = W.Q ⋙ 𝟭 W.Localization
Module: Mathlib.CategoryTheory.Localization.Construction

Goal state before:
pp: 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
⊢ P f ↔ ⊤ f

Tactic:
constructor

Goal state after:
case mp
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
⊢ P f → ⊤ f

case mpr
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
⊢ ⊤ f → P f
Module: Mathlib.CategoryTheory.Localization.Predicate

Goal state before:
pp: C : Type u_1
inst✝ : Category.{v_1, u_1} C
W : MorphismProperty C
X Y : C
f : X ⟶ Y
hf : W f
⊢ wIso f hf = isoOfHom W.Q W f hf

Tactic:
ext

Goal state after:
C : Type u_1
inst✝ : Category.{v_1, u_1} C
W : MorphismProperty C
X Y : C
f : X ⟶ Y
hf : W f
⊢ (wIso f hf).hom = (isoOfHom W.Q W f hf).hom
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 : D ⥤ E
⊢ ((whiskeringLeft W.Localization D E).obj (equivalenceFromModel L W).functor ⋙
          (Construction.whiskeringLeftEquivalence W E).functor).obj
      F =
    (whiskeringLeftFunctor L W E).obj F

Tactic:
ext

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
E : Type u_3
inst✝¹ : Category.{v_3, u_3} E
inst✝ : L.IsLocalization W
F : D ⥤ E
⊢ (((whiskeringLeft W.Localization D E).obj (equivalenceFromModel L W).functor ⋙
            (Construction.whiskeringLeftEquivalence W E).functor).obj
        F).obj =
    ((whiskeringLeftFunctor L W E).obj F).obj
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
X✝ Y✝ : D ⥤ E
τ : X✝ ⟶ Y✝
⊢ ((whiskeringLeft W.Localization D E).obj (equivalenceFromModel L W).functor ⋙
            (Construction.whiskeringLeftEquivalence W E).functor).map
        τ ≫
      (eqToIso ⋯).hom =
    (eqToIso ⋯).hom ≫ (whiskeringLeftFunctor L W E).map τ

Tactic:
ext

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
E : Type u_3
inst✝¹ : Category.{v_3, u_3} E
inst✝ : L.IsLocalization W
X✝ Y✝ : D ⥤ E
τ : X✝ ⟶ Y✝
x✝ : C
⊢ (((whiskeringLeft W.Localization D E).obj (equivalenceFromModel L W).functor ⋙
                  (Construction.whiskeringLeftEquivalence W E).functor).map
              τ ≫
            (eqToIso ⋯).hom).hom.app
      x✝ =
    ((eqToIso ⋯).hom ≫ (whiskeringLeftFunctor L W E).map τ).hom.app x✝
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
X✝ Y✝ : D ⥤ E
τ : X✝ ⟶ Y✝
x✝ : C
⊢ τ.app (L.obj (W.Q.obj x✝).as.obj) ≫ (eqToHom ⋯).hom.app x✝ =
    (eqToHom ⋯).hom.app x✝ ≫ ((whiskeringLeftFunctor L W E).map τ).hom.app x✝

Tactic:
ObjectProperty.eqToHom_hom,

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
E : Type u_3
inst✝¹ : Category.{v_3, u_3} E
inst✝ : L.IsLocalization W
X✝ Y✝ : D ⥤ E
τ : X✝ ⟶ Y✝
x✝ : C
⊢ τ.app (L.obj (W.Q.obj x✝).as.obj) ≫ (eqToHom ⋯).app x✝ =
    (eqToHom ⋯).hom.app x✝ ≫ ((whiskeringLeftFunctor L W E).map τ).hom.app x✝
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
X✝ Y✝ : D ⥤ E
τ : X✝ ⟶ Y✝
x✝ : C
⊢ τ.app (L.obj (W.Q.obj x✝).as.obj) ≫ (eqToHom ⋯).app x✝ =
    (eqToHom ⋯).hom.app x✝ ≫ ((whiskeringLeftFunctor L W E).map τ).hom.app x✝

Tactic:
ObjectProperty.eqToHom_hom,

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
E : Type u_3
inst✝¹ : Category.{v_3, u_3} E
inst✝ : L.IsLocalization W
X✝ Y✝ : D ⥤ E
τ : X✝ ⟶ Y✝
x✝ : C
⊢ τ.app (L.obj (W.Q.obj x✝).as.obj) ≫ (eqToHom ⋯).app x✝ =
    (eqToHom ⋯).app x✝ ≫ ((whiskeringLeftFunctor L W E).map τ).hom.app x✝
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
X₁ X₂ X₃ Z₁₂ Z₂₃ Z₁₃ : C
u₁₂ : X₁ ⟶ X₂
u₂₃ : X₂ ⟶ X₃
e₁ : X₁' ≅ X₁
e₂ : X₂' ≅ X₂
e₃ : X₃' ≅ X₃
comm₁₂ : u₁₂' ≫ e₂.hom = e₁.hom ≫ u₁₂
comm₂₃ : u₂₃' ≫ e₃.hom = e₂.hom ≫ u₂₃
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 (u₁₂ ≫ u₂₃) v₁₃ w₁₃ ∈ distinguishedTriangles
H : Nonempty (Octahedron ⋯ h₁₂ h₂₃ h₁₃)
⊢ Nonempty (Octahedron comm' h₁₂' h₂₃' h₁₃')

Tactic:
exact ⟨Octahedron.ofIso u₁₂' u₂₃' u₁₃' comm' h₁₂' h₂₃' h₁₃'
      u₁₂ u₂₃ _ rfl e₁ e₂ e₃ comm₁₂ comm₂₃ v₁₂ w₁₂ h₁₂ v₂₃ w₂₃ h₂₃ v₁₃ w₁₃ h₁₃ H.some⟩

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:
ext

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
⊢ ((equivalence L₁ W₁ L₂ W₂ G G' F F' α β).counitIso.app (L₂.obj X)).hom =
    ((Lifting.iso L₂ W₂ (F ⋙ G') (F' ⋙ G')).app X ≪≫ β.app X).hom
Module: Mathlib.CategoryTheory.Localization.Construction

Goal state before:
pp: case refine_1
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 : C ⥤ D
hG : W.IsInvertedBy G
⊢ (inverse W D ⋙ functor W D).obj { obj := G, property := hG } =
    (𝟭 (W.FunctorsInverting D)).obj { obj := G, property := hG }

Tactic:
ext

Goal state after:
case refine_1
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 : C ⥤ D
hG : W.IsInvertedBy G
⊢ ((inverse W D ⋙ functor W D).obj { obj := G, property := hG }).obj =
    ((𝟭 (W.FunctorsInverting D)).obj { obj := G, property := hG }).obj
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:
ext

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
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.hom = (isoOfHom L W w.unop hw).hom
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 }
⊢ (inverse W D ⋙ functor W D).map f = eqToHom ⋯ ≫ (𝟭 (W.FunctorsInverting D)).map f ≫ eqToHom ⋯

Tactic:
ext

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
⊢ ((inverse W D ⋙ functor W D).map f).hom.app x✝ =
    (eqToHom ⋯ ≫ (𝟭 (W.FunctorsInverting D)).map f ≫ eqToHom ⋯).hom.app x✝
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:
ext

Goal state after:
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
x✝ : C
⊢ ((WhiskeringLeftEquivalence.functor W D).map ((WhiskeringLeftEquivalence.unitIso W D).hom.app F) ≫
            (WhiskeringLeftEquivalence.counitIso W D).hom.app ((WhiskeringLeftEquivalence.functor W D).obj F)).hom.app
      x✝ =
    (𝟙 ((WhiskeringLeftEquivalence.functor W D).obj F)).hom.app x✝
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
⊢ W.Q.map f = W.Q.map g ↔ L.map f = L.map g

Tactic:
constructor

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
⊢ W.Q.map f = W.Q.map g → L.map f = L.map g

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
⊢ L.map f = L.map g → W.Q.map f = W.Q.map g
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
⊢ LeftFractionRel φ.fst ψ.fst ∧ LeftFractionRel φ.snd ψ.snd ↔ LeftFraction₂Rel φ ψ

Tactic:
constructor

Goal state after:
case mp
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
⊢ LeftFractionRel φ.fst ψ.fst ∧ LeftFractionRel φ.snd ψ.snd → LeftFraction₂Rel φ ψ

case mpr
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
⊢ LeftFraction₂Rel φ ψ → LeftFractionRel φ.fst ψ.fst ∧ LeftFractionRel φ.snd ψ.snd
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✝⁸ : Category.{v₁, u₁} C₁
inst✝⁷ : Category.{v₂, u₂} C₂
inst✝⁶ : Category.{v₃, u₃} C₃
inst✝⁵ : Category.{v₄, u₄} D₁
inst✝⁴ : Category.{v₅, u₅} D₂
W₁ : MorphismProperty C₁
W₂ : MorphismProperty C₂
W₃ : MorphismProperty C₃
Φ : LocalizerMorphism W₁ W₂
L₁ : C₁ ⥤ D₁
inst✝³ : L₁.IsLocalization W₁
L₂ : C₂ ⥤ D₂
inst✝² : L₂.IsLocalization W₂
G : D₁ ⥤ D₂
Ψ : LocalizerMorphism W₂ W₃
inst✝¹ : Φ.IsInduced
inst✝ : Ψ.IsInduced
⊢ W₃.inverseImage (Φ.comp Ψ).functor = W₁

Tactic:
by
    simp [← Φ.inverseImage_eq, ← Ψ.inverseImage_eq]

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

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁸ : Category.{v_1, u_1} C
inst✝⁷ : Category.{v_2, u_2} D
inst✝⁶ : Preadditive C
L : C ⥤ D
W : MorphismProperty C
inst✝⁵ : L.IsLocalization W
inst✝⁴ : W.HasLeftCalculusOfFractions
E : Type u_3
inst✝³ : Category.{v_3, u_3} E
inst✝² : Preadditive E
inst✝¹ : Preadditive D
inst✝ : L.Additive
G : D ⥤ E
⊢ G.Additive ↔ (L ⋙ G).Additive

Tactic:
constructor

Goal state after:
case mp
C : Type u_1
D : Type u_2
inst✝⁸ : Category.{v_1, u_1} C
inst✝⁷ : Category.{v_2, u_2} D
inst✝⁶ : Preadditive C
L : C ⥤ D
W : MorphismProperty C
inst✝⁵ : L.IsLocalization W
inst✝⁴ : W.HasLeftCalculusOfFractions
E : Type u_3
inst✝³ : Category.{v_3, u_3} E
inst✝² : Preadditive E
inst✝¹ : Preadditive D
inst✝ : L.Additive
G : D ⥤ E
⊢ G.Additive → (L ⋙ G).Additive

case mpr
C : Type u_1
D : Type u_2
inst✝⁸ : Category.{v_1, u_1} C
inst✝⁷ : Category.{v_2, u_2} D
inst✝⁶ : Preadditive C
L : C ⥤ D
W : MorphismProperty C
inst✝⁵ : L.IsLocalization W
inst✝⁴ : W.HasLeftCalculusOfFractions
E : Type u_3
inst✝³ : Category.{v_3, u_3} E
inst✝² : Preadditive E
inst✝¹ : Preadditive D
inst✝ : L.Additive
G : D ⥤ E
⊢ (L ⋙ G).Additive → G.Additive
Module: Mathlib.CategoryTheory.Shift.Localization

Goal state before:
pp: C : Type u₁
D : Type u₂
inst✝¹⁰ : Category.{v₁, u₁} C
inst✝⁹ : Category.{v₂, u₂} D
E : Type u₃
inst✝⁸ : Category.{v₃, u₃} E
L : C ⥤ D
W : MorphismProperty C
inst✝⁷ : L.IsLocalization W
A : Type w
inst✝⁶ : AddMonoid A
inst✝⁵ : HasShift C A
F : C ⥤ E
F' : D ⥤ E
inst✝⁴ : Lifting L W F F'
inst✝³ : HasShift D A
inst✝² : HasShift E A
inst✝¹ : L.CommShift A
inst✝ : F.CommShift A
this : F'.CommShift A := L.commShiftOfLocalization W A F F'
⊢ CommShift (Lifting.iso L W F F').hom A

Tactic:
constructor

Goal state after:
C : Type u₁
D : Type u₂
inst✝¹⁰ : Category.{v₁, u₁} C
inst✝⁹ : Category.{v₂, u₂} D
E : Type u₃
inst✝⁸ : Category.{v₃, u₃} E
L : C ⥤ D
W : MorphismProperty C
inst✝⁷ : L.IsLocalization W
A : Type w
inst✝⁶ : AddMonoid A
inst✝⁵ : HasShift C A
F : C ⥤ E
F' : D ⥤ E
inst✝⁴ : Lifting L W F F'
inst✝³ : HasShift D A
inst✝² : HasShift E A
inst✝¹ : L.CommShift A
inst✝ : F.CommShift A
this : F'.CommShift A := ⋯
⊢ autoParam
    (∀ (a : A),
      (Functor.commShiftIso (L ⋙ F') a).hom ≫ Functor.whiskerRight (Lifting.iso L W F F').hom (shiftFunctor E a) =
        (shiftFunctor C a).whiskerLeft (Lifting.iso L W F F').hom ≫ (Functor.commShiftIso F a).hom)
    CommShift.shift_comm._autoParam
Module: Mathlib.CategoryTheory.Localization.Triangulated

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝¹¹ : Category.{v_1, u_1} C
inst✝¹⁰ : Category.{v_2, u_2} D
L : C ⥤ D
inst✝⁹ : HasShift C ℤ
inst✝⁸ : Preadditive C
inst✝⁷ : HasZeroObject C
inst✝⁶ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁵ : Pretriangulated C
inst✝⁴ : HasShift D ℤ
inst✝³ : L.CommShift ℤ
inst✝² : Preadditive D
inst✝¹ : L.Additive
inst✝ : ∀ (n : ℤ), (shiftFunctor D n).Additive
T : Triangle D
⊢ T ∈ L.essImageDistTriang ↔ T.rotate ∈ L.essImageDistTriang

Tactic:
constructor

Goal state after:
case mp
C : Type u_1
D : Type u_2
inst✝¹¹ : Category.{v_1, u_1} C
inst✝¹⁰ : Category.{v_2, u_2} D
L : C ⥤ D
inst✝⁹ : HasShift C ℤ
inst✝⁸ : Preadditive C
inst✝⁷ : HasZeroObject C
inst✝⁶ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁵ : Pretriangulated C
inst✝⁴ : HasShift D ℤ
inst✝³ : L.CommShift ℤ
inst✝² : Preadditive D
inst✝¹ : L.Additive
inst✝ : ∀ (n : ℤ), (shiftFunctor D n).Additive
T : Triangle D
⊢ T ∈ L.essImageDistTriang → T.rotate ∈ L.essImageDistTriang

case mpr
C : Type u_1
D : Type u_2
inst✝¹¹ : Category.{v_1, u_1} C
inst✝¹⁰ : Category.{v_2, u_2} D
L : C ⥤ D
inst✝⁹ : HasShift C ℤ
inst✝⁸ : Preadditive C
inst✝⁷ : HasZeroObject C
inst✝⁶ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁵ : Pretriangulated C
inst✝⁴ : HasShift D ℤ
inst✝³ : L.CommShift ℤ
inst✝² : Preadditive D
inst✝¹ : L.Additive
inst✝ : ∀ (n : ℤ), (shiftFunctor D n).Additive
T : Triangle D
⊢ T.rotate ∈ L.essImageDistTriang → T ∈ L.essImageDistTriang
Module: Mathlib.SetTheory.Cardinal.HasCardinalLT

Goal state before:
pp: X : Type u
Y : Type u'
κ : Cardinal.{w}
hκ : Cardinal.aleph0 ≤ κ
⊢ HasCardinalLT (X ⊕ Y) κ ↔ HasCardinalLT X κ ∧ HasCardinalLT Y κ

Tactic:
constructor

Goal state after:
case mp
X : Type u
Y : Type u'
κ : Cardinal.{w}
hκ : Cardinal.aleph0 ≤ κ
⊢ HasCardinalLT (X ⊕ Y) κ → HasCardinalLT X κ ∧ HasCardinalLT Y κ

case mpr
X : Type u
Y : Type u'
κ : Cardinal.{w}
hκ : Cardinal.aleph0 ≤ κ
⊢ HasCardinalLT X κ ∧ HasCardinalLT Y κ → HasCardinalLT (X ⊕ Y) κ
Module: Mathlib.CategoryTheory.Localization.CalculusOfFractions.Preadditive

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁹ : Category.{v_1, u_1} C
inst✝⁸ : Category.{v_2, u_2} D
inst✝⁷ : Preadditive C
L : C ⥤ D
W : MorphismProperty C
inst✝⁶ : L.IsLocalization W
E : Type u_3
inst✝⁵ : Category.{v_3, u_3} E
F : D ⥤ E
inst✝⁴ : W.HasLeftCalculusOfFractions
inst✝³ : Preadditive D
inst✝² : Preadditive E
inst✝¹ : L.Additive
inst✝ : F.Additive
h : ∀ ⦃X₁ X₂ : C⦄ (f : X₁ ⟶ X₂), F.map (L.map f) = 0 → L.map f = 0
X₁ X₂ : C
f g : X₁ ⟶ X₂
hfg : F.map (L.map f) = F.map (L.map g)
⊢ F.map (L.map f) - F.map (L.map g) = 0

Tactic:
hfg,

Goal state after:
C : Type u_1
D : Type u_2
inst✝⁹ : Category.{v_1, u_1} C
inst✝⁸ : Category.{v_2, u_2} D
inst✝⁷ : Preadditive C
L : C ⥤ D
W : MorphismProperty C
inst✝⁶ : L.IsLocalization W
E : Type u_3
inst✝⁵ : Category.{v_3, u_3} E
F : D ⥤ E
inst✝⁴ : W.HasLeftCalculusOfFractions
inst✝³ : Preadditive D
inst✝² : Preadditive E
inst✝¹ : L.Additive
inst✝ : F.Additive
h : ∀ ⦃X₁ X₂ : C⦄ (f : X₁ ⟶ X₂), F.map (L.map f) = 0 → L.map f = 0
X₁ X₂ : C
f g : X₁ ⟶ X₂
hfg : F.map (L.map f) = F.map (L.map g)
⊢ F.map (L.map g) - F.map (L.map g) = 0
Module: Mathlib.CategoryTheory.ObjectProperty.ColimitsClosure

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

Tactic:
by
  refine le_antisymm ?_ ?_
  · apply colimitsClosure_le
    rw [← op_monotone_iff, op_unop]
    apply le_limitsClosure
  · rw [← op_monotone_iff, op_unop]
    apply limitsClosure_le
    rw [op_monotone_iff]
    apply le_colimitsClosure

Goal state after: