ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- CategoryTheory.MorphismProperty.LeftFraction.Localization.Qiso_hom_inv_id
@[reassoc (attr := simp)]
lemma Qiso_hom_inv_id {X Y : C} (s : X ⟶ Y) (hs : W s) :
    (Q W).map s ≫ Qinv s hs = 𝟙 _ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] {W : CategoryTheory.MorphismProperty C} [W.HasLeftCalculusOfFractions] {X Y : C} (s : X ⟶ Y) (hs : W s)
Goal: CategoryTheory.CategoryStruct.comp ((CategoryTheory.MorphismProperty.LeftFraction.Localization.Q W).map s)
    (CategoryTheory.MorphismProperty.LeftFraction.Localization.Qinv s hs) =
  CategoryTheory.CategoryStruct.id ((CategoryTheory.MorphismProperty.LeftFraction.Localization.Q W).obj X)
-- CategoryTheory.MorphismProperty.LeftFraction.Localization.Qiso_inv_hom_id
@[reassoc (attr := simp)]
lemma Qiso_inv_hom_id {X Y : C} (s : X ⟶ Y) (hs : W s) :
    Qinv s hs  ≫ (Q W).map s = 𝟙 _ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] {W : CategoryTheory.MorphismProperty C} [W.HasLeftCalculusOfFractions] {X Y : C} (s : X ⟶ Y) (hs : W s)
Goal: CategoryTheory.CategoryStruct.comp (CategoryTheory.MorphismProperty.LeftFraction.Localization.Qinv s hs)
    ((CategoryTheory.MorphismProperty.LeftFraction.Localization.Q W).map s) =
  CategoryTheory.CategoryStruct.id ((CategoryTheory.MorphismProperty.LeftFraction.Localization.Q W).obj Y)
-- CategoryTheory.MorphismProperty.LeftFraction.Localization.Hom.map_mk
@[simp]
lemma Hom.map_mk {X Y : C} (f : LeftFraction W X Y)
    (F : C ⥤ E) (hF : W.IsInvertedBy F) :
  Hom.map (Hom.mk f) F hF = f.map F hF 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_4, u_1} C] {W : CategoryTheory.MorphismProperty C} {E : Type u_3} [CategoryTheory.Category.{u_5, u_3} E] {X Y : C} (f : W.LeftFraction X Y) (F : CategoryTheory.Functor C E) (hF : W.IsInvertedBy F)
Goal: (CategoryTheory.MorphismProperty.LeftFraction.Localization.Hom.mk f).map F hF = f.map F hF
-- CategoryTheory.MorphismProperty.LeftFraction.Localization.StrictUniversalPropertyFixedTarget.inverts
lemma inverts : W.IsInvertedBy (Q W) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_4, u_1} C] (W : CategoryTheory.MorphismProperty C) [W.HasLeftCalculusOfFractions]
Goal: W.IsInvertedBy (CategoryTheory.MorphismProperty.LeftFraction.Localization.Q W)
-- CategoryTheory.MorphismProperty.LeftFraction.Localization.StrictUniversalPropertyFixedTarget.fac
lemma fac (F : C ⥤ E) (hF : W.IsInvertedBy F) : Q W ⋙ lift F hF = F 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_4, u_1} C] {W : CategoryTheory.MorphismProperty C} [W.HasLeftCalculusOfFractions] {E : Type u_3} [CategoryTheory.Category.{u_5, u_3} E] (F : CategoryTheory.Functor C E) (hF : W.IsInvertedBy F)
Goal: (CategoryTheory.MorphismProperty.LeftFraction.Localization.Q W).comp
    (CategoryTheory.MorphismProperty.LeftFraction.Localization.StrictUniversalPropertyFixedTarget.lift F hF) =
  F
-- CategoryTheory.MorphismProperty.LeftFraction.Localization.StrictUniversalPropertyFixedTarget.uniq
lemma uniq (F₁ F₂ : Localization W ⥤ E) (h : Q W ⋙ F₁ = Q W ⋙ F₂) : F₁ = F₂ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_4, u_1} C] {W : CategoryTheory.MorphismProperty C} [W.HasLeftCalculusOfFractions] {E : Type u_3} [CategoryTheory.Category.{u_5, u_3} E] (F₁ F₂ : CategoryTheory.Functor (CategoryTheory.MorphismProperty.LeftFraction.Localization W) E) (h :
  (CategoryTheory.MorphismProperty.LeftFraction.Localization.Q W).comp F₁ =
    (CategoryTheory.MorphismProperty.LeftFraction.Localization.Q W).comp F₂)
Goal: F₁ = F₂
-- CategoryTheory.MorphismProperty.LeftFraction.Localization.homMk_eq
lemma homMk_eq {X Y : C} (f : LeftFraction W X Y) :
    homMk f = f.map (Q W) (Localization.inverts _ W) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] {W : CategoryTheory.MorphismProperty C} [W.HasLeftCalculusOfFractions] {X Y : C} (f : W.LeftFraction X Y)
Goal: CategoryTheory.MorphismProperty.LeftFraction.Localization.homMk f =
  f.map (CategoryTheory.MorphismProperty.LeftFraction.Localization.Q W) ⋯
-- CategoryTheory.MorphismProperty.LeftFraction.Localization.map_eq_iff
lemma map_eq_iff {X Y : C} (f g : LeftFraction W X Y) :
    f.map (LeftFraction.Localization.Q W) (Localization.inverts _ _) =
        g.map (LeftFraction.Localization.Q W) (Localization.inverts _ _) ↔
      LeftFractionRel f g 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] {W : CategoryTheory.MorphismProperty C} [W.HasLeftCalculusOfFractions] {X Y : C} (f g : W.LeftFraction X Y)
Goal: f.map (CategoryTheory.MorphismProperty.LeftFraction.Localization.Q W) ⋯ =
    g.map (CategoryTheory.MorphismProperty.LeftFraction.Localization.Q W) ⋯ ↔
  CategoryTheory.MorphismProperty.LeftFractionRel f g
-- CategoryTheory.MorphismProperty.LeftFraction.map_eq
lemma map_eq {X Y : C} (φ : W.LeftFraction X Y) (L : C ⥤ D) [L.IsLocalization W] :
    φ.map L (Localization.inverts L W) =
      L.map φ.f ≫ (Localization.isoOfHom L W φ.s φ.hs).inv 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Category.{u_4, u_2} D] {W : CategoryTheory.MorphismProperty C} {X Y : C} (φ : W.LeftFraction X Y) (L : CategoryTheory.Functor C D) [L.IsLocalization W]
Goal: φ.map L ⋯ = CategoryTheory.CategoryStruct.comp (L.map φ.f) (CategoryTheory.Localization.isoOfHom L W φ.s ⋯).inv
-- CategoryTheory.MorphismProperty.LeftFraction.map_compatibility
lemma map_compatibility {X Y : C}
    (φ : W.LeftFraction X Y) {E : Type*} [Category E]
    (L₁ : C ⥤ D) (L₂ : C ⥤ E) [L₁.IsLocalization W] [L₂.IsLocalization W] :
    (Localization.uniq L₁ L₂ W).functor.map (φ.map L₁ (Localization.inverts L₁ W)) =
      (Localization.compUniqFunctor L₁ L₂ W).hom.app X ≫
        φ.map L₂ (Localization.inverts L₂ W) ≫
        (Localization.compUniqFunctor L₁ L₂ W).inv.app Y 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_6, u_2} D] {W : CategoryTheory.MorphismProperty C} {X Y : C} (φ : W.LeftFraction X Y) {E : Type u_3} [CategoryTheory.Category.{u_5, u_3} E] (L₁ : CategoryTheory.Functor C D) (L₂ : CategoryTheory.Functor C E) [L₁.IsLocalization W] [L₂.IsLocalization W]
Goal: (CategoryTheory.Localization.uniq L₁ L₂ W).functor.map (φ.map L₁ ⋯) =
  CategoryTheory.CategoryStruct.comp ((CategoryTheory.Localization.compUniqFunctor L₁ L₂ W).hom.app X)
    (CategoryTheory.CategoryStruct.comp (φ.map L₂ ⋯) ((CategoryTheory.Localization.compUniqFunctor L₁ L₂ W).inv.app Y))
-- CategoryTheory.MorphismProperty.LeftFraction.map_eq_of_map_eq
lemma map_eq_of_map_eq {X Y : C}
    (φ₁ φ₂ : W.LeftFraction X Y) {E : Type*} [Category E]
    (L₁ : C ⥤ D) (L₂ : C ⥤ E) [L₁.IsLocalization W] [L₂.IsLocalization W]
    (h : φ₁.map L₁ (Localization.inverts L₁ W) = φ₂.map L₁ (Localization.inverts L₁ W)) :
    φ₁.map L₂ (Localization.inverts L₂ W) = φ₂.map L₂ (Localization.inverts L₂ W) 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_6, u_2} D] {W : CategoryTheory.MorphismProperty C} {X Y : C} (φ₁ φ₂ : W.LeftFraction X Y) {E : Type u_3} [CategoryTheory.Category.{u_5, u_3} E] (L₁ : CategoryTheory.Functor C D) (L₂ : CategoryTheory.Functor C E) [L₁.IsLocalization W] [L₂.IsLocalization W] (h : φ₁.map L₁ ⋯ = φ₂.map L₁ ⋯)
Goal: φ₁.map L₂ ⋯ = φ₂.map L₂ ⋯
-- CategoryTheory.MorphismProperty.LeftFraction.map_comp_map_eq_map
lemma map_comp_map_eq_map {X Y Z : C} (z₁ : W.LeftFraction X Y) (z₂ : W.LeftFraction Y Z)
    (z₃ : W.LeftFraction z₁.Y' z₂.Y') (h₃ : z₂.f ≫ z₃.s = z₁.s ≫ z₃.f)
    (L : C ⥤ D) [L.IsLocalization W] :
    z₁.map L (Localization.inverts L W) ≫ z₂.map L (Localization.inverts L W) =
      (z₁.comp₀ z₂ z₃).map L (Localization.inverts L W) 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Category.{u_4, u_2} D] {W : CategoryTheory.MorphismProperty C} [W.HasLeftCalculusOfFractions] {X Y Z : C} (z₁ : W.LeftFraction X Y) (z₂ : W.LeftFraction Y Z) (z₃ : W.LeftFraction z₁.Y' z₂.Y') (h₃ : CategoryTheory.CategoryStruct.comp z₂.f z₃.s = CategoryTheory.CategoryStruct.comp z₁.s z₃.f) (L : CategoryTheory.Functor C D) [L.IsLocalization W]
Goal: CategoryTheory.CategoryStruct.comp (z₁.map L ⋯) (z₂.map L ⋯) = (z₁.comp₀ z₂ z₃).map L ⋯
-- CategoryTheory.Localization.exists_leftFraction
lemma Localization.exists_leftFraction {X Y : C} (f : L.obj X ⟶ L.obj Y) :
    ∃ (φ : W.LeftFraction X Y), f = φ.map L (Localization.inverts L W) 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_3, u_2} D] (L : CategoryTheory.Functor C D) (W : CategoryTheory.MorphismProperty C) [L.IsLocalization W] [W.HasLeftCalculusOfFractions] {X Y : C} (f : L.obj X ⟶ L.obj Y)
Goal: ∃ φ, f = φ.map L ⋯
-- CategoryTheory.MorphismProperty.LeftFraction.map_eq_iff
lemma MorphismProperty.LeftFraction.map_eq_iff
    {X Y : C} (φ ψ : W.LeftFraction X Y) :
    φ.map L (Localization.inverts _ _) = ψ.map L (Localization.inverts _ _) ↔
      LeftFractionRel φ ψ 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Category.{u_4, u_2} D] (L : CategoryTheory.Functor C D) (W : CategoryTheory.MorphismProperty C) [L.IsLocalization W] [W.HasLeftCalculusOfFractions] {X Y : C} (φ ψ : W.LeftFraction X Y)
Goal: φ.map L ⋯ = ψ.map L ⋯ ↔ CategoryTheory.MorphismProperty.LeftFractionRel φ ψ
-- CategoryTheory.MorphismProperty.map_eq_iff_postcomp
lemma MorphismProperty.map_eq_iff_postcomp {X Y : C} (f₁ f₂ : X ⟶ Y) :
    L.map f₁ = L.map f₂ ↔ ∃ (Z : C) (s : Y ⟶ Z) (_ : W s), f₁ ≫ s = f₂ ≫ s 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Category.{u_4, u_2} D] (L : CategoryTheory.Functor C D) (W : CategoryTheory.MorphismProperty C) [L.IsLocalization W] [W.HasLeftCalculusOfFractions] {X Y : C} (f₁ f₂ : X ⟶ Y)
Goal: L.map f₁ = L.map f₂ ↔
  ∃ Z s, ∃ (_ : W s), CategoryTheory.CategoryStruct.comp f₁ s = CategoryTheory.CategoryStruct.comp f₂ s
-- CategoryTheory.Localization.essSurj_mapArrow
lemma Localization.essSurj_mapArrow :
    L.mapArrow.EssSurj 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Category.{u_4, u_2} D] (L : CategoryTheory.Functor C D) (W : CategoryTheory.MorphismProperty C) [L.IsLocalization W] [W.HasLeftCalculusOfFractions]
Goal: L.mapArrow.EssSurj
-- CategoryTheory.MorphismProperty.LeftFraction.op_f
@[simps]
def LeftFraction.op {X Y : C} (φ : W.LeftFraction X Y) :
    W.op.RightFraction (Opposite.op Y) (Opposite.op X) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] {W : CategoryTheory.MorphismProperty C} {X Y : C} (φ : W.LeftFraction X Y)
Goal: φ.op.f = φ.f.op
-- CategoryTheory.MorphismProperty.LeftFraction.op_X'
@[simps]
def LeftFraction.op {X Y : C} (φ : W.LeftFraction X Y) :
    W.op.RightFraction (Opposite.op Y) (Opposite.op X) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] {W : CategoryTheory.MorphismProperty C} {X Y : C} (φ : W.LeftFraction X Y)
Goal: φ.op.X' = Opposite.op φ.Y'
-- CategoryTheory.MorphismProperty.LeftFraction.op_s
@[simps]
def LeftFraction.op {X Y : C} (φ : W.LeftFraction X Y) :
    W.op.RightFraction (Opposite.op Y) (Opposite.op X) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] {W : CategoryTheory.MorphismProperty C} {X Y : C} (φ : W.LeftFraction X Y)
Goal: φ.op.s = φ.s.op
-- CategoryTheory.MorphismProperty.RightFraction.op_s
@[simps]
def RightFraction.op {X Y : C} (φ : W.RightFraction X Y) :
    W.op.LeftFraction (Opposite.op Y) (Opposite.op X) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] {W : CategoryTheory.MorphismProperty C} {X Y : C} (φ : W.RightFraction X Y)
Goal: φ.op.s = φ.s.op
-- CategoryTheory.MorphismProperty.RightFraction.op_Y'
@[simps]
def RightFraction.op {X Y : C} (φ : W.RightFraction X Y) :
    W.op.LeftFraction (Opposite.op Y) (Opposite.op X) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] {W : CategoryTheory.MorphismProperty C} {X Y : C} (φ : W.RightFraction X Y)
Goal: φ.op.Y' = Opposite.op φ.X'
-- CategoryTheory.MorphismProperty.RightFraction.op_f
@[simps]
def RightFraction.op {X Y : C} (φ : W.RightFraction X Y) :
    W.op.LeftFraction (Opposite.op Y) (Opposite.op X) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] {W : CategoryTheory.MorphismProperty C} {X Y : C} (φ : W.RightFraction X Y)
Goal: φ.op.f = φ.f.op
-- CategoryTheory.MorphismProperty.LeftFraction.unop_f
@[simps]
def LeftFraction.unop {W : MorphismProperty Cᵒᵖ}
    {X Y : Cᵒᵖ} (φ : W.LeftFraction X Y) :
    W.unop.RightFraction (Opposite.unop Y) (Opposite.unop X) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] {W : CategoryTheory.MorphismProperty Cᵒᵖ} {X Y : Cᵒᵖ} (φ : W.LeftFraction X Y)
Goal: φ.unop.f = φ.f.unop
-- CategoryTheory.MorphismProperty.LeftFraction.unop_s
@[simps]
def LeftFraction.unop {W : MorphismProperty Cᵒᵖ}
    {X Y : Cᵒᵖ} (φ : W.LeftFraction X Y) :
    W.unop.RightFraction (Opposite.unop Y) (Opposite.unop X) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] {W : CategoryTheory.MorphismProperty Cᵒᵖ} {X Y : Cᵒᵖ} (φ : W.LeftFraction X Y)
Goal: φ.unop.s = φ.s.unop
-- CategoryTheory.MorphismProperty.LeftFraction.unop_X'
@[simps]
def LeftFraction.unop {W : MorphismProperty Cᵒᵖ}
    {X Y : Cᵒᵖ} (φ : W.LeftFraction X Y) :
    W.unop.RightFraction (Opposite.unop Y) (Opposite.unop X) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] {W : CategoryTheory.MorphismProperty Cᵒᵖ} {X Y : Cᵒᵖ} (φ : W.LeftFraction X Y)
Goal: φ.unop.X' = Opposite.unop φ.Y'
-- CategoryTheory.MorphismProperty.RightFraction.unop_f
@[simps]
def RightFraction.unop {W : MorphismProperty Cᵒᵖ}
    {X Y : Cᵒᵖ} (φ : W.RightFraction X Y) :
    W.unop.LeftFraction (Opposite.unop Y) (Opposite.unop X) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] {W : CategoryTheory.MorphismProperty Cᵒᵖ} {X Y : Cᵒᵖ} (φ : W.RightFraction X Y)
Goal: φ.unop.f = φ.f.unop
-- CategoryTheory.MorphismProperty.RightFraction.unop_s
@[simps]
def RightFraction.unop {W : MorphismProperty Cᵒᵖ}
    {X Y : Cᵒᵖ} (φ : W.RightFraction X Y) :
    W.unop.LeftFraction (Opposite.unop Y) (Opposite.unop X) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] {W : CategoryTheory.MorphismProperty Cᵒᵖ} {X Y : Cᵒᵖ} (φ : W.RightFraction X Y)
Goal: φ.unop.s = φ.s.unop
-- CategoryTheory.MorphismProperty.RightFraction.unop_Y'
@[simps]
def RightFraction.unop {W : MorphismProperty Cᵒᵖ}
    {X Y : Cᵒᵖ} (φ : W.RightFraction X Y) :
    W.unop.LeftFraction (Opposite.unop Y) (Opposite.unop X) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] {W : CategoryTheory.MorphismProperty Cᵒᵖ} {X Y : Cᵒᵖ} (φ : W.RightFraction X Y)
Goal: φ.unop.Y' = Opposite.unop φ.X'
-- CategoryTheory.MorphismProperty.RightFraction.op_map
lemma RightFraction.op_map
    {X Y : C} (φ : W.RightFraction X Y) (L : C ⥤ D) (hL : W.IsInvertedBy L) :
    (φ.map L hL).op = φ.op.map L.op hL.op 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Category.{u_4, u_2} D] {W : CategoryTheory.MorphismProperty C} {X Y : C} (φ : W.RightFraction X Y) (L : CategoryTheory.Functor C D) (hL : W.IsInvertedBy L)
Goal: (φ.map L hL).op = φ.op.map L.op ⋯
-- CategoryTheory.MorphismProperty.LeftFraction.op_map
lemma LeftFraction.op_map
    {X Y : C} (φ : W.LeftFraction X Y) (L : C ⥤ D) (hL : W.IsInvertedBy L) :
    (φ.map L hL).op = φ.op.map L.op hL.op 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Category.{u_4, u_2} D] {W : CategoryTheory.MorphismProperty C} {X Y : C} (φ : W.LeftFraction X Y) (L : CategoryTheory.Functor C D) (hL : W.IsInvertedBy L)
Goal: (φ.map L hL).op = φ.op.map L.op ⋯
-- CategoryTheory.MorphismProperty.RightFractionRel.op
lemma RightFractionRel.op {X Y : C} {z₁ z₂ : W.RightFraction X Y}
    (h : RightFractionRel z₁ z₂) : LeftFractionRel z₁.op z₂.op 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] {W : CategoryTheory.MorphismProperty C} {X Y : C} {z₁ z₂ : W.RightFraction X Y} (h : CategoryTheory.MorphismProperty.RightFractionRel z₁ z₂)
Goal: CategoryTheory.MorphismProperty.LeftFractionRel z₁.op z₂.op
-- CategoryTheory.MorphismProperty.RightFractionRel.unop
lemma RightFractionRel.unop {W : MorphismProperty Cᵒᵖ} {X Y : Cᵒᵖ}
    {z₁ z₂ : W.RightFraction X Y}
    (h : RightFractionRel z₁ z₂) : LeftFractionRel z₁.unop z₂.unop 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] {W : CategoryTheory.MorphismProperty Cᵒᵖ} {X Y : Cᵒᵖ} {z₁ z₂ : W.RightFraction X Y} (h : CategoryTheory.MorphismProperty.RightFractionRel z₁ z₂)
Goal: CategoryTheory.MorphismProperty.LeftFractionRel z₁.unop z₂.unop
-- CategoryTheory.MorphismProperty.LeftFractionRel.op
lemma LeftFractionRel.op {X Y : C} {z₁ z₂ : W.LeftFraction X Y}
    (h : LeftFractionRel z₁ z₂) : RightFractionRel z₁.op z₂.op 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] {W : CategoryTheory.MorphismProperty C} {X Y : C} {z₁ z₂ : W.LeftFraction X Y} (h : CategoryTheory.MorphismProperty.LeftFractionRel z₁ z₂)
Goal: CategoryTheory.MorphismProperty.RightFractionRel z₁.op z₂.op
-- CategoryTheory.MorphismProperty.LeftFractionRel.unop
lemma LeftFractionRel.unop {W : MorphismProperty Cᵒᵖ} {X Y : Cᵒᵖ}
    {z₁ z₂ : W.LeftFraction X Y}
    (h : LeftFractionRel z₁ z₂) : RightFractionRel z₁.unop z₂.unop 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] {W : CategoryTheory.MorphismProperty Cᵒᵖ} {X Y : Cᵒᵖ} {z₁ z₂ : W.LeftFraction X Y} (h : CategoryTheory.MorphismProperty.LeftFractionRel z₁ z₂)
Goal: CategoryTheory.MorphismProperty.RightFractionRel z₁.unop z₂.unop
-- CategoryTheory.MorphismProperty.leftFractionRel_op_iff
lemma leftFractionRel_op_iff
    {X Y : C} (z₁ z₂ : W.RightFraction X Y) :
    LeftFractionRel z₁.op z₂.op ↔ RightFractionRel z₁ z₂ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] {W : CategoryTheory.MorphismProperty C} {X Y : C} (z₁ z₂ : W.RightFraction X Y)
Goal: CategoryTheory.MorphismProperty.LeftFractionRel z₁.op z₂.op ↔ CategoryTheory.MorphismProperty.RightFractionRel z₁ z₂
-- CategoryTheory.MorphismProperty.rightFractionRel_op_iff
lemma rightFractionRel_op_iff
    {X Y : C} (z₁ z₂ : W.LeftFraction X Y) :
    RightFractionRel z₁.op z₂.op ↔ LeftFractionRel z₁ z₂ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] {W : CategoryTheory.MorphismProperty C} {X Y : C} (z₁ z₂ : W.LeftFraction X Y)
Goal: CategoryTheory.MorphismProperty.RightFractionRel z₁.op z₂.op ↔ CategoryTheory.MorphismProperty.LeftFractionRel z₁ z₂
-- CategoryTheory.MorphismProperty.RightFractionRel.refl
lemma refl {X Y : C} (z : W.RightFraction X Y) : RightFractionRel z z 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] {W : CategoryTheory.MorphismProperty C} {X Y : C} (z : W.RightFraction X Y)
Goal: CategoryTheory.MorphismProperty.RightFractionRel z z
-- CategoryTheory.MorphismProperty.RightFractionRel.symm
lemma symm {X Y : C} {z₁ z₂ : W.RightFraction X Y} (h : RightFractionRel z₁ z₂) :
    RightFractionRel z₂ z₁ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] {W : CategoryTheory.MorphismProperty C} {X Y : C} {z₁ z₂ : W.RightFraction X Y} (h : CategoryTheory.MorphismProperty.RightFractionRel z₁ z₂)
Goal: CategoryTheory.MorphismProperty.RightFractionRel z₂ z₁
-- CategoryTheory.MorphismProperty.RightFractionRel.trans
lemma trans {X Y : C} {z₁ z₂ z₃ : W.RightFraction X Y}
    [HasRightCalculusOfFractions W]
    (h₁₂ : RightFractionRel z₁ z₂) (h₂₃ : RightFractionRel z₂ z₃) :
    RightFractionRel z₁ z₃ 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] {W : CategoryTheory.MorphismProperty C} {X Y : C} {z₁ z₂ z₃ : W.RightFraction X Y} [W.HasRightCalculusOfFractions] (h₁₂ : CategoryTheory.MorphismProperty.RightFractionRel z₁ z₂) (h₂₃ : CategoryTheory.MorphismProperty.RightFractionRel z₂ z₃)
Goal: CategoryTheory.MorphismProperty.RightFractionRel z₁ z₃
-- CategoryTheory.MorphismProperty.equivalenceRightFractionRel
lemma equivalenceRightFractionRel (X Y : C) [HasRightCalculusOfFractions W] :
    @_root_.Equivalence (W.RightFraction X Y) RightFractionRel 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] {W : CategoryTheory.MorphismProperty C} (X Y : C) [W.HasRightCalculusOfFractions]
Goal: Equivalence CategoryTheory.MorphismProperty.RightFractionRel
-- CategoryTheory.Localization.exists_rightFraction
lemma Localization.exists_rightFraction {X Y : C} (f : L.obj X ⟶ L.obj Y) :
    ∃ (φ : W.RightFraction X Y), f = φ.map L (Localization.inverts L W) 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_3, u_2} D] (L : CategoryTheory.Functor C D) (W : CategoryTheory.MorphismProperty C) [L.IsLocalization W] [W.HasRightCalculusOfFractions] {X Y : C} (f : L.obj X ⟶ L.obj Y)
Goal: ∃ φ, f = φ.map L ⋯
-- CategoryTheory.MorphismProperty.RightFraction.map_eq_iff
lemma MorphismProperty.RightFraction.map_eq_iff
    {X Y : C} (φ ψ : W.RightFraction X Y) :
    φ.map L (Localization.inverts _ _) = ψ.map L (Localization.inverts _ _) ↔
      RightFractionRel φ ψ 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Category.{u_4, u_2} D] (L : CategoryTheory.Functor C D) (W : CategoryTheory.MorphismProperty C) [L.IsLocalization W] [W.HasRightCalculusOfFractions] {X Y : C} (φ ψ : W.RightFraction X Y)
Goal: φ.map L ⋯ = ψ.map L ⋯ ↔ CategoryTheory.MorphismProperty.RightFractionRel φ ψ
-- CategoryTheory.MorphismProperty.map_eq_iff_precomp
lemma MorphismProperty.map_eq_iff_precomp {Y Z : C} (f₁ f₂ : Y ⟶ Z) :
    L.map f₁ = L.map f₂ ↔ ∃ (X : C) (s : X ⟶ Y) (_ : W s), s ≫ f₁ = s ≫ f₂ 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Category.{u_4, u_2} D] (L : CategoryTheory.Functor C D) (W : CategoryTheory.MorphismProperty C) [L.IsLocalization W] [W.HasRightCalculusOfFractions] {Y Z : C} (f₁ f₂ : Y ⟶ Z)
Goal: L.map f₁ = L.map f₂ ↔
  ∃ X s, ∃ (_ : W s), CategoryTheory.CategoryStruct.comp s f₁ = CategoryTheory.CategoryStruct.comp s f₂
-- CategoryTheory.Localization.essSurj_mapArrow_of_hasRightCalculusOfFractions
lemma Localization.essSurj_mapArrow_of_hasRightCalculusOfFractions :
    L.mapArrow.EssSurj 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Category.{u_4, u_2} D] (L : CategoryTheory.Functor C D) (W : CategoryTheory.MorphismProperty C) [L.IsLocalization W] [W.HasRightCalculusOfFractions]
Goal: L.mapArrow.EssSurj
-- CategoryTheory.Limits.CoproductDisjoint.mono_inr
class CoproductDisjoint (X₁ X₂ : C) where
  isInitialOfIsPullbackOfIsCoproduct :
    ∀ {X Z} {pX₁ : X₁ ⟶ X} {pX₂ : X₂ ⟶ X} {f : Z ⟶ X₁} {g : Z ⟶ X₂}
      (_cX : IsColimit (BinaryCofan.mk pX₁ pX₂)) {comm : f ≫ pX₁ = g ≫ pX₂},
      IsLimit (PullbackCone.mk _ _ comm) → IsInitial Z
  mono_inl : ∀ (X) (X₁ : X₁ ⟶ X) (X₂ : X₂ ⟶ X) (_cX : IsColimit (BinaryCofan.mk X₁ X₂)), Mono X₁
  mono_inr : ∀ (X) (X₁ : X₁ ⟶ X) (X₂ : X₂ ⟶ X) (_cX : IsColimit (BinaryCofan.mk X₁ X₂)), Mono X₂
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] {X₁ X₂ : C} [self : CategoryTheory.Limits.CoproductDisjoint X₁ X₂] (X : C)
Goal: ∀ (X₁_1 : X₁ ⟶ X) (X₂_1 : X₂ ⟶ X),
  CategoryTheory.Limits.IsColimit (CategoryTheory.Limits.BinaryCofan.mk X₁_1 X₂_1) → CategoryTheory.Mono X₂_1
-- CategoryTheory.Limits.CoproductDisjoint.mono_inl
class CoproductDisjoint (X₁ X₂ : C) where
  isInitialOfIsPullbackOfIsCoproduct :
    ∀ {X Z} {pX₁ : X₁ ⟶ X} {pX₂ : X₂ ⟶ X} {f : Z ⟶ X₁} {g : Z ⟶ X₂}
      (_cX : IsColimit (BinaryCofan.mk pX₁ pX₂)) {comm : f ≫ pX₁ = g ≫ pX₂},
      IsLimit (PullbackCone.mk _ _ comm) → IsInitial Z
  mono_inl : ∀ (X) (X₁ : X₁ ⟶ X) (X₂ : X₂ ⟶ X) (_cX : IsColimit (BinaryCofan.mk X₁ X₂)), Mono X₁
  mono_inr : ∀ (X) (X₁ : X₁ ⟶ X) (X₂ : X₂ ⟶ X) (_cX : IsColimit (BinaryCofan.mk X₁ X₂)), Mono X₂
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] {X₁ X₂ : C} [self : CategoryTheory.Limits.CoproductDisjoint X₁ X₂] (X : C)
Goal: ∀ (X₁_1 : X₁ ⟶ X) (X₂_1 : X₂ ⟶ X),
  CategoryTheory.Limits.IsColimit (CategoryTheory.Limits.BinaryCofan.mk X₁_1 X₂_1) → CategoryTheory.Mono X₁_1
-- CategoryTheory.Limits.initialMonoClass_of_disjoint_coproducts
theorem initialMonoClass_of_disjoint_coproducts [CoproductsDisjoint C] : InitialMonoClass C 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Limits.CoproductsDisjoint C]
Goal: CategoryTheory.Limits.InitialMonoClass C