ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- CategoryTheory.Functor.shiftIso_add'_inv_app
lemma shiftIso_add'_inv_app (n m mn : M) (hnm : m + n = mn) (a a' a'' : M)
    (ha' : n + a = a') (ha'' : m + a' = a'') (X : C) :
    (F.shiftIso mn a a'' (by rw [← hnm, ← ha'', ← ha', add_assoc])).inv.app X =
      (shiftIso F m a' a'' ha'').inv.app X ≫
        (shiftIso F n a a' ha').inv.app ((shiftFunctor C m).obj X) ≫
        (shift F a).map ((shiftFunctorAdd' C m n mn hnm).inv.app X) 
Arguments: {C : Type u_1} {A : Type u_2} [CategoryTheory.Category.{u_6, u_1} C] [CategoryTheory.Category.{u_5, u_2} A] (F : C ⥤ A) {M : Type u_3} [AddMonoid M] [CategoryTheory.HasShift C M] [F.ShiftSequence M] (n m mn : M) (hnm : m + n = mn) (a a' a'' : M) (ha' : n + a = a') (ha'' : m + a' = a'') (X : C)
Goal: (F.shiftIso mn a a'' ⋯).inv.app X =
  (F.shiftIso m a' a'' ha'').inv.app X ≫
    (F.shiftIso n a a' ha').inv.app ((CategoryTheory.shiftFunctor C m).obj X) ≫
      (F.shift a).map ((CategoryTheory.shiftFunctorAdd' C m n mn hnm).inv.app X)
-- CategoryTheory.Functor.shiftIso_hom_app_comp
@[reassoc]
lemma shiftIso_hom_app_comp (n m mn : M) (hnm : m + n = mn)
    (a a' a'' : M) (ha' : n + a = a') (ha'' : m + a' = a'') (X : C) :
    (shiftIso F n a a' ha').hom.app ((shiftFunctor C m).obj X) ≫
      (shiftIso F m a' a'' ha'').hom.app X =
        (shift F a).map ((shiftFunctorAdd' C m n mn hnm).inv.app X) ≫
          (F.shiftIso mn a a'' (by rw [← hnm, ← ha'', ← ha', add_assoc])).hom.app X 
Arguments: {C : Type u_1} {A : Type u_2} [CategoryTheory.Category.{u_6, u_1} C] [CategoryTheory.Category.{u_5, u_2} A] (F : C ⥤ A) {M : Type u_3} [AddMonoid M] [CategoryTheory.HasShift C M] [F.ShiftSequence M] (n m mn : M) (hnm : m + n = mn) (a a' a'' : M) (ha' : n + a = a') (ha'' : m + a' = a'') (X : C)
Goal: (F.shiftIso n a a' ha').hom.app ((CategoryTheory.shiftFunctor C m).obj X) ≫ (F.shiftIso m a' a'' ha'').hom.app X =
  (F.shift a).map ((CategoryTheory.shiftFunctorAdd' C m n mn hnm).inv.app X) ≫ (F.shiftIso mn a a'' ⋯).hom.app X
-- CategoryTheory.Functor.shiftMap_comp
@[reassoc]
lemma shiftMap_comp {X Y Z : C} {n : M} (f : X ⟶ Y⟦n⟧) (g : Y ⟶ Z) (a a' : M) (ha' : n + a = a') :
    F.shiftMap (f ≫ g⟦n⟧') a a' ha' = F.shiftMap f a a' ha' ≫ (F.shift a').map g 
Arguments: {C : Type u_1} {A : Type u_2} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Category.{u_6, u_2} A] (F : C ⥤ A) {M : Type u_3} [AddMonoid M] [CategoryTheory.HasShift C M] [F.ShiftSequence M] {X Y Z : C} {n : M} (f : X ⟶ (CategoryTheory.shiftFunctor C n).obj Y) (g : Y ⟶ Z) (a a' : M) (ha' : n + a = a')
Goal: F.shiftMap (f ≫ (CategoryTheory.shiftFunctor C n).map g) a a' ha' = F.shiftMap f a a' ha' ≫ (F.shift a').map g
-- CategoryTheory.Functor.shiftMap_comp'
@[reassoc]
lemma shiftMap_comp' {X Y Z : C} {n : M} (f : X ⟶ Y) (g : Y ⟶ Z⟦n⟧) (a a' : M) (ha' : n + a = a') :
    F.shiftMap (f ≫ g) a a' ha' = (F.shift a).map f ≫ F.shiftMap g a a' ha' 
Arguments: {C : Type u_1} {A : Type u_2} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Category.{u_6, u_2} A] (F : C ⥤ A) {M : Type u_3} [AddMonoid M] [CategoryTheory.HasShift C M] [F.ShiftSequence M] {X Y Z : C} {n : M} (f : X ⟶ Y) (g : Y ⟶ (CategoryTheory.shiftFunctor C n).obj Z) (a a' : M) (ha' : n + a = a')
Goal: F.shiftMap (f ≫ g) a a' ha' = (F.shift a).map f ≫ F.shiftMap g a a' ha'
-- CategoryTheory.Functor.shiftIso_hom_app_comp_shiftMap
lemma shiftIso_hom_app_comp_shiftMap {X Y : C} {m : M} (f : X ⟶ Y⟦m⟧) (n mn : M) (hnm : m + n = mn)
    (a a' a'' : M) (ha' : n + a = a') (ha'' : m + a' = a'') :
    (F.shiftIso n a a' ha').hom.app X ≫ F.shiftMap f a' a'' ha'' =
      (F.shift a).map (f⟦n⟧') ≫ (F.shift a).map ((shiftFunctorAdd' C m n mn hnm).inv.app Y) ≫
        (F.shiftIso mn a a'' (by rw [← ha'', ← ha', ← hnm, add_assoc])).hom.app Y 
Arguments: {C : Type u_1} {A : Type u_2} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Category.{u_6, u_2} A] (F : C ⥤ A) {M : Type u_3} [AddMonoid M] [CategoryTheory.HasShift C M] [F.ShiftSequence M] {X Y : C} {m : M} (f : X ⟶ (CategoryTheory.shiftFunctor C m).obj Y) (n mn : M) (hnm : m + n = mn) (a a' a'' : M) (ha' : n + a = a') (ha'' : m + a' = a'')
Goal: (F.shiftIso n a a' ha').hom.app X ≫ F.shiftMap f a' a'' ha'' =
  (F.shift a).map ((CategoryTheory.shiftFunctor C n).map f) ≫
    (F.shift a).map ((CategoryTheory.shiftFunctorAdd' C m n mn hnm).inv.app Y) ≫ (F.shiftIso mn a a'' ⋯).hom.app Y
-- CategoryTheory.Functor.shiftIso_hom_app_comp_shiftMap_of_add_eq_zero
lemma shiftIso_hom_app_comp_shiftMap_of_add_eq_zero [F.ShiftSequence G]
    {X Y : C} {m : G} (f : X ⟶ Y⟦m⟧)
    (n : G) (hnm : n + m = 0) (a a' : G) (ha' : m + a = a') :
    (F.shiftIso n a' a (by rw [← ha', ← add_assoc, hnm, zero_add])).hom.app X ≫
      F.shiftMap f a a' ha' =
    (F.shift a').map (f⟦n⟧' ≫ (shiftFunctorCompIsoId C m n
      (by rw [← add_left_inj m, add_assoc, hnm, zero_add, add_zero])).hom.app Y) 
Arguments: {C : Type u_1} {A : Type u_2} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Category.{u_6, u_2} A] (F : C ⥤ A) {G : Type u_4} [AddGroup G] [CategoryTheory.HasShift C G] [F.ShiftSequence G] {X Y : C} {m : G} (f : X ⟶ (CategoryTheory.shiftFunctor C m).obj Y) (n : G) (hnm : n + m = 0) (a a' : G) (ha' : m + a = a')
Goal: (F.shiftIso n a' a ⋯).hom.app X ≫ F.shiftMap f a a' ha' =
  (F.shift a').map ((CategoryTheory.shiftFunctor C n).map f ≫ (CategoryTheory.shiftFunctorCompIsoId C m n ⋯).hom.app Y)
-- CategoryTheory.Functor.shiftMap_zero
@[simp]
lemma shiftMap_zero (X Y : C) (n a a' : M) (ha' : n + a = a') :
    F.shiftMap (0 : X ⟶ Y⟦n⟧) a a' ha' = 0 
Arguments: {C : Type u_1} {A : Type u_2} [CategoryTheory.Category.{u_6, u_1} C] [CategoryTheory.Category.{u_5, u_2} A] (F : C ⥤ A) {M : Type u_3} [AddMonoid M] [CategoryTheory.HasShift C M] [F.ShiftSequence M] [CategoryTheory.Limits.HasZeroMorphisms C] [CategoryTheory.Limits.HasZeroMorphisms A] [F.PreservesZeroMorphisms] [∀ (n : M), (CategoryTheory.shiftFunctor C n).PreservesZeroMorphisms] (X Y : C) (n a a' : M) (ha' : n + a = a')
Goal: F.shiftMap 0 a a' ha' = 0
-- CategoryTheory.Functor.IsLocalization.inverts
class IsLocalization : Prop where
    inverts : W.IsInvertedBy L
    isEquivalence : IsEquivalence (Localization.Construction.lift L inverts)
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] {L : CategoryTheory.Functor C D} {W : CategoryTheory.MorphismProperty C} [self : L.IsLocalization W]
Goal: W.IsInvertedBy L
-- CategoryTheory.Functor.IsLocalization.isEquivalence
class IsLocalization : Prop where
    inverts : W.IsInvertedBy L
    isEquivalence : IsEquivalence (Localization.Construction.lift L inverts)
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] {L : CategoryTheory.Functor C D} {W : CategoryTheory.MorphismProperty C} [self : L.IsLocalization W]
Goal: (CategoryTheory.Localization.Construction.lift L ⋯).IsEquivalence
-- CategoryTheory.Localization.StrictUniversalPropertyFixedTarget.inverts
structure StrictUniversalPropertyFixedTarget where
    inverts : W.IsInvertedBy L
    lift : ∀ (F : C ⥤ E) (_ : W.IsInvertedBy F), D ⥤ E
    fac : ∀ (F : C ⥤ E) (hF : W.IsInvertedBy F), L ⋙ lift F hF = F
    uniq : ∀ (F₁ F₂ : D ⥤ E) (_ : L ⋙ F₁ = L ⋙ F₂), F₁ = F₂
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] {L : CategoryTheory.Functor C D} {W : CategoryTheory.MorphismProperty C} {E : Type u_3} [CategoryTheory.Category.{u_6, u_3} E] (self : CategoryTheory.Localization.StrictUniversalPropertyFixedTarget L W E)
Goal: W.IsInvertedBy L
-- CategoryTheory.Localization.StrictUniversalPropertyFixedTarget.fac
structure StrictUniversalPropertyFixedTarget where
    inverts : W.IsInvertedBy L
    lift : ∀ (F : C ⥤ E) (_ : W.IsInvertedBy F), D ⥤ E
    fac : ∀ (F : C ⥤ E) (hF : W.IsInvertedBy F), L ⋙ lift F hF = F
    uniq : ∀ (F₁ F₂ : D ⥤ E) (_ : L ⋙ F₁ = L ⋙ F₂), F₁ = F₂
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] {L : CategoryTheory.Functor C D} {W : CategoryTheory.MorphismProperty C} {E : Type u_3} [CategoryTheory.Category.{u_6, u_3} E] (self : CategoryTheory.Localization.StrictUniversalPropertyFixedTarget L W E) (F : CategoryTheory.Functor C E) (hF : W.IsInvertedBy F)
Goal: L.comp (self.lift F hF) = F
-- CategoryTheory.Localization.StrictUniversalPropertyFixedTarget.uniq
structure StrictUniversalPropertyFixedTarget where
    inverts : W.IsInvertedBy L
    lift : ∀ (F : C ⥤ E) (_ : W.IsInvertedBy F), D ⥤ E
    fac : ∀ (F : C ⥤ E) (hF : W.IsInvertedBy F), L ⋙ lift F hF = F
    uniq : ∀ (F₁ F₂ : D ⥤ E) (_ : L ⋙ F₁ = L ⋙ F₂), F₁ = F₂
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] {L : CategoryTheory.Functor C D} {W : CategoryTheory.MorphismProperty C} {E : Type u_3} [CategoryTheory.Category.{u_6, u_3} E] (self : CategoryTheory.Localization.StrictUniversalPropertyFixedTarget L W E) (F₁ F₂ : CategoryTheory.Functor D E)
Goal: L.comp F₁ = L.comp F₂ → F₁ = F₂
-- CategoryTheory.Localization.strictUniversalPropertyFixedTargetQ_lift
@[simps]
def strictUniversalPropertyFixedTargetQ : StrictUniversalPropertyFixedTarget W.Q W E 
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] (G : CategoryTheory.Functor C E) (hG : W.IsInvertedBy G)
Goal: (CategoryTheory.Localization.strictUniversalPropertyFixedTargetQ W E).lift G hG =
  CategoryTheory.Localization.Construction.lift G hG
-- CategoryTheory.Localization.strictUniversalPropertyFixedTargetId_lift
@[simps]
def strictUniversalPropertyFixedTargetId (hW : W ≤ MorphismProperty.isomorphisms C) :
    StrictUniversalPropertyFixedTarget (𝟭 C) W E 
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] (hW : W ≤ CategoryTheory.MorphismProperty.isomorphisms C) (F : CategoryTheory.Functor C E)
Goal: ∀ (x : W.IsInvertedBy F), (CategoryTheory.Localization.strictUniversalPropertyFixedTargetId W E hW).lift F x = F
-- CategoryTheory.Functor.IsLocalization.mk'
theorem IsLocalization.mk' (h₁ : Localization.StrictUniversalPropertyFixedTarget L W D)
    (h₂ : Localization.StrictUniversalPropertyFixedTarget L W W.Localization) :
    IsLocalization L W 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] (L : CategoryTheory.Functor C D) (W : CategoryTheory.MorphismProperty C) (h₁ : CategoryTheory.Localization.StrictUniversalPropertyFixedTarget L W D) (h₂ : CategoryTheory.Localization.StrictUniversalPropertyFixedTarget L W W.Localization)
Goal: L.IsLocalization W
-- CategoryTheory.Functor.IsLocalization.for_id
theorem IsLocalization.for_id (hW : W ≤ MorphismProperty.isomorphisms C) : (𝟭 C).IsLocalization W 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_4, u_1} C] (W : CategoryTheory.MorphismProperty C) (hW : W ≤ CategoryTheory.MorphismProperty.isomorphisms C)
Goal: (CategoryTheory.Functor.id C).IsLocalization W
-- CategoryTheory.Localization.inverts
theorem inverts : W.IsInvertedBy L 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] (L : CategoryTheory.Functor C D) (W : CategoryTheory.MorphismProperty C) [L.IsLocalization W]
Goal: W.IsInvertedBy L
-- CategoryTheory.Localization.isoOfHom_hom
@[simps! hom]
def isoOfHom {X Y : C} (f : X ⟶ Y) (hf : W f) : L.obj X ≅ L.obj Y 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] (L : CategoryTheory.Functor C D) (W : CategoryTheory.MorphismProperty C) [L.IsLocalization W] {X Y : C} (f : X ⟶ Y) (hf : W f)
Goal: (CategoryTheory.Localization.isoOfHom L W f hf).hom = L.map f
-- CategoryTheory.Localization.isoOfHom_hom_inv_id
@[reassoc (attr := simp)]
lemma isoOfHom_hom_inv_id {X Y : C} (f : X ⟶ Y) (hf : W f) :
    L.map f ≫ (isoOfHom L W f hf).inv = 𝟙 _ 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] (L : CategoryTheory.Functor C D) (W : CategoryTheory.MorphismProperty C) [L.IsLocalization W] {X Y : C} (f : X ⟶ Y) (hf : W f)
Goal: CategoryTheory.CategoryStruct.comp (L.map f) (CategoryTheory.Localization.isoOfHom L W f hf).inv =
  CategoryTheory.CategoryStruct.id (L.obj X)
-- CategoryTheory.Localization.isoOfHom_inv_hom_id
@[reassoc (attr := simp)]
lemma isoOfHom_inv_hom_id {X Y : C} (f : X ⟶ Y) (hf : W f) :
    (isoOfHom L W f hf).inv ≫ L.map f = 𝟙 _ 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] (L : CategoryTheory.Functor C D) (W : CategoryTheory.MorphismProperty C) [L.IsLocalization W] {X Y : C} (f : X ⟶ Y) (hf : W f)
Goal: CategoryTheory.CategoryStruct.comp (CategoryTheory.Localization.isoOfHom L W f hf).inv (L.map f) =
  CategoryTheory.CategoryStruct.id (L.obj Y)
-- CategoryTheory.Localization.isoOfHom_id_inv
@[simp]
lemma isoOfHom_id_inv (X : C) (hX : W (𝟙 X)) :
    (isoOfHom L W (𝟙 X) hX).inv = 𝟙 _ 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] (L : CategoryTheory.Functor C D) (W : CategoryTheory.MorphismProperty C) [L.IsLocalization W] (X : C) (hX : W (CategoryTheory.CategoryStruct.id X))
Goal: (CategoryTheory.Localization.isoOfHom L W (CategoryTheory.CategoryStruct.id X) hX).inv =
  CategoryTheory.CategoryStruct.id (L.obj X)
-- CategoryTheory.Localization.Construction.wIso_eq_isoOfHom
lemma Construction.wIso_eq_isoOfHom {X Y : C} (f : X ⟶ Y) (hf : W f) :
    Construction.wIso f hf = isoOfHom W.Q W f hf 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_4, u_1} C] {W : CategoryTheory.MorphismProperty C} {X Y : C} (f : X ⟶ Y) (hf : W f)
Goal: CategoryTheory.Localization.Construction.wIso f hf = CategoryTheory.Localization.isoOfHom W.Q W f hf
-- CategoryTheory.Localization.Construction.wInv_eq_isoOfHom_inv
lemma Construction.wInv_eq_isoOfHom_inv {X Y : C} (f : X ⟶ Y) (hf : W f) :
    Construction.wInv f hf = (isoOfHom W.Q W f hf).inv 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_4, u_1} C] {W : CategoryTheory.MorphismProperty C} {X Y : C} (f : X ⟶ Y) (hf : W f)
Goal: CategoryTheory.Localization.Construction.wInv f hf = (CategoryTheory.Localization.isoOfHom W.Q W f hf).inv
-- CategoryTheory.Localization.essSurj
theorem essSurj (W) [L.IsLocalization W] : L.EssSurj 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] (L : CategoryTheory.Functor C D) (W : CategoryTheory.MorphismProperty C) [L.IsLocalization W]
Goal: L.EssSurj
-- CategoryTheory.Localization.whiskeringLeftFunctor'_eq
theorem whiskeringLeftFunctor'_eq :
    whiskeringLeftFunctor' L W E = Localization.whiskeringLeftFunctor L W E ⋙ inducedFunctor _ 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] (L : CategoryTheory.Functor C D) (W : CategoryTheory.MorphismProperty C) (E : Type u_3) [CategoryTheory.Category.{u_6, u_3} E] [L.IsLocalization W]
Goal: CategoryTheory.Localization.whiskeringLeftFunctor' L W E =
  (CategoryTheory.Localization.whiskeringLeftFunctor L W E).comp
    (CategoryTheory.inducedFunctor CategoryTheory.FullSubcategory.obj)
-- CategoryTheory.Localization.whiskeringLeftFunctor'_obj
@[simp]
theorem whiskeringLeftFunctor'_obj (F : D ⥤ E) : (whiskeringLeftFunctor' L W E).obj F = L ⋙ F 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_6, u_1} C] [CategoryTheory.Category.{u_4, u_2} D] (L : CategoryTheory.Functor C D) (W : CategoryTheory.MorphismProperty C) {E : Type u_3} [CategoryTheory.Category.{u_5, u_3} E] [L.IsLocalization W] (F : CategoryTheory.Functor D E)
Goal: (CategoryTheory.Localization.whiskeringLeftFunctor' L W E).obj F = L.comp F
-- CategoryTheory.Localization.full_whiskeringLeft
lemma full_whiskeringLeft (L : C ⥤ D) (W) [L.IsLocalization W] (E : Type*) [Category E] :
    ((whiskeringLeft C D E).obj L).Full 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Category.{u_6, u_2} D] (L : CategoryTheory.Functor C D) (W : CategoryTheory.MorphismProperty C) [L.IsLocalization W] (E : Type u_4) [CategoryTheory.Category.{u_7, u_4} E]
Goal: ((CategoryTheory.whiskeringLeft C D E).obj L).Full
-- CategoryTheory.Localization.faithful_whiskeringLeft
lemma faithful_whiskeringLeft (L : C ⥤ D) (W) [L.IsLocalization W] (E : Type*) [Category E] :
    ((whiskeringLeft C D E).obj L).Faithful 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Category.{u_6, u_2} D] (L : CategoryTheory.Functor C D) (W : CategoryTheory.MorphismProperty C) [L.IsLocalization W] (E : Type u_4) [CategoryTheory.Category.{u_7, u_4} E]
Goal: ((CategoryTheory.whiskeringLeft C D E).obj L).Faithful
-- CategoryTheory.Localization.natTrans_ext
theorem natTrans_ext (L : C ⥤ D) (W) [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' 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] {E : Type u_3} [CategoryTheory.Category.{u_6, u_3} E] (L : CategoryTheory.Functor C D) (W : CategoryTheory.MorphismProperty C) [L.IsLocalization W] {F₁ F₂ : CategoryTheory.Functor D E} (τ τ' : F₁ ⟶ F₂) (h : ∀ (X : C), τ.app (L.obj X) = τ'.app (L.obj X))
Goal: τ = τ'
-- CategoryTheory.Localization.liftNatTrans_app
@[simp]
theorem liftNatTrans_app (F₁ F₂ : C ⥤ E) (F₁' F₂' : D ⥤ E) [Lifting L W F₁ F₁'] [Lifting L W F₂ F₂']
    (τ : F₁ ⟶ F₂) (X : C) :
    (liftNatTrans L W F₁ F₂ F₁' F₂' τ).app (L.obj X) =
      (Lifting.iso L W F₁ F₁').hom.app X ≫ τ.app X ≫ (Lifting.iso L W F₂ F₂').inv.app X 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_6, u_2} D] (L : CategoryTheory.Functor C D) (W : CategoryTheory.MorphismProperty C) {E : Type u_3} [CategoryTheory.Category.{u_5, u_3} E] [L.IsLocalization W] (F₁ F₂ : CategoryTheory.Functor C E) (F₁' F₂' : CategoryTheory.Functor D E) [CategoryTheory.Localization.Lifting L W F₁ F₁'] [CategoryTheory.Localization.Lifting L W F₂ F₂'] (τ : F₁ ⟶ F₂) (X : C)
Goal: (CategoryTheory.Localization.liftNatTrans L W F₁ F₂ F₁' F₂' τ).app (L.obj X) =
  CategoryTheory.CategoryStruct.comp ((CategoryTheory.Localization.Lifting.iso L W F₁ F₁').hom.app X)
    (CategoryTheory.CategoryStruct.comp (τ.app X) ((CategoryTheory.Localization.Lifting.iso L W F₂ F₂').inv.app X))
-- CategoryTheory.Localization.comp_liftNatTrans
@[reassoc (attr := simp)]
theorem comp_liftNatTrans (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) :
    liftNatTrans L W F₁ F₂ F₁' F₂' τ ≫ liftNatTrans L W F₂ F₃ F₂' F₃' τ' =
      liftNatTrans L W F₁ F₃ F₁' F₃' (τ ≫ τ') 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_6, u_2} D] (L : CategoryTheory.Functor C D) (W : CategoryTheory.MorphismProperty C) {E : Type u_3} [CategoryTheory.Category.{u_5, u_3} E] [L.IsLocalization W] (F₁ F₂ F₃ : CategoryTheory.Functor C E) (F₁' F₂' F₃' : CategoryTheory.Functor D E) [h₁ : CategoryTheory.Localization.Lifting L W F₁ F₁'] [h₂ : CategoryTheory.Localization.Lifting L W F₂ F₂'] [h₃ : CategoryTheory.Localization.Lifting L W F₃ F₃'] (τ : F₁ ⟶ F₂) (τ' : F₂ ⟶ F₃)
Goal: CategoryTheory.CategoryStruct.comp (CategoryTheory.Localization.liftNatTrans L W F₁ F₂ F₁' F₂' τ)
    (CategoryTheory.Localization.liftNatTrans L W F₂ F₃ F₂' F₃' τ') =
  CategoryTheory.Localization.liftNatTrans L W F₁ F₃ F₁' F₃' (CategoryTheory.CategoryStruct.comp τ τ')
-- CategoryTheory.Localization.liftNatTrans_id
@[simp]
theorem liftNatTrans_id (F : C ⥤ E) (F' : D ⥤ E) [h : Lifting L W F F'] :
    liftNatTrans L W F F F' F' (𝟙 F) = 𝟙 F' 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_6, u_2} D] (L : CategoryTheory.Functor C D) (W : CategoryTheory.MorphismProperty C) {E : Type u_3} [CategoryTheory.Category.{u_5, u_3} E] [L.IsLocalization W] (F : CategoryTheory.Functor C E) (F' : CategoryTheory.Functor D E) [h : CategoryTheory.Localization.Lifting L W F F']
Goal: CategoryTheory.Localization.liftNatTrans L W F F F' F' (CategoryTheory.CategoryStruct.id F) =
  CategoryTheory.CategoryStruct.id F'
-- CategoryTheory.Localization.liftNatIso_inv
@[simps]
def liftNatIso (F₁ F₂ : C ⥤ E) (F₁' F₂' : D ⥤ E) [h₁ : Lifting L W F₁ F₁'] [h₂ : Lifting L W F₂ F₂']
    (e : F₁ ≅ F₂) : F₁' ≅ F₂' 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] (L : CategoryTheory.Functor C D) (W : CategoryTheory.MorphismProperty C) {E : Type u_3} [CategoryTheory.Category.{u_6, u_3} E] [L.IsLocalization W] (F₁ F₂ : CategoryTheory.Functor C E) (F₁' F₂' : CategoryTheory.Functor D E) [h₁ : CategoryTheory.Localization.Lifting L W F₁ F₁'] [h₂ : CategoryTheory.Localization.Lifting L W F₂ F₂'] (e : F₁ ≅ F₂)
Goal: (CategoryTheory.Localization.liftNatIso L W F₁ F₂ F₁' F₂' e).inv =
  CategoryTheory.Localization.liftNatTrans L W F₂ F₁ F₂' F₁' e.inv
-- CategoryTheory.Localization.liftNatIso_hom
@[simps]
def liftNatIso (F₁ F₂ : C ⥤ E) (F₁' F₂' : D ⥤ E) [h₁ : Lifting L W F₁ F₁'] [h₂ : Lifting L W F₂ F₂']
    (e : F₁ ≅ F₂) : F₁' ≅ F₂' 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] (L : CategoryTheory.Functor C D) (W : CategoryTheory.MorphismProperty C) {E : Type u_3} [CategoryTheory.Category.{u_6, u_3} E] [L.IsLocalization W] (F₁ F₂ : CategoryTheory.Functor C E) (F₁' F₂' : CategoryTheory.Functor D E) [h₁ : CategoryTheory.Localization.Lifting L W F₁ F₁'] [h₂ : CategoryTheory.Localization.Lifting L W F₂ F₂'] (e : F₁ ≅ F₂)
Goal: (CategoryTheory.Localization.liftNatIso L W F₁ F₂ F₁' F₂' e).hom =
  CategoryTheory.Localization.liftNatTrans L W F₁ F₂ F₁' F₂' e.hom
-- CategoryTheory.Localization.Lifting.compRight_iso'
@[simps]
instance compRight {E' : Type*} [Category E'] (F : C ⥤ E) (F' : D ⥤ E) [Lifting L W F F']
    (G : E ⥤ E') : Lifting L W (F ⋙ G) (F' ⋙ G) 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Category.{u_6, u_2} D] (L : CategoryTheory.Functor C D) (W : CategoryTheory.MorphismProperty C) {E : Type u_3} [CategoryTheory.Category.{u_7, u_3} E] {E' : Type u_4} [CategoryTheory.Category.{u_8, u_4} E'] (F : CategoryTheory.Functor C E) (F' : CategoryTheory.Functor D E) [CategoryTheory.Localization.Lifting L W F F'] (G : CategoryTheory.Functor E E')
Goal: CategoryTheory.Localization.Lifting.iso' W =
  CategoryTheory.isoWhiskerRight (CategoryTheory.Localization.Lifting.iso L W F F') G
-- CategoryTheory.Localization.Lifting.id_iso'
@[simps]
instance id : Lifting L W L (𝟭 D) 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] (L : CategoryTheory.Functor C D) (W : CategoryTheory.MorphismProperty C)
Goal: CategoryTheory.Localization.Lifting.iso' W = L.rightUnitor
-- CategoryTheory.Localization.Lifting.compLeft_iso'
@[simps]
instance compLeft (F : D ⥤ E) : Localization.Lifting L W (L ⋙ F) F 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] (L : CategoryTheory.Functor C D) (W : CategoryTheory.MorphismProperty C) {E : Type u_3} [CategoryTheory.Category.{u_6, u_3} E] (F : CategoryTheory.Functor D E)
Goal: CategoryTheory.Localization.Lifting.iso' W = CategoryTheory.Iso.refl (L.comp F)
-- CategoryTheory.Localization.Lifting.compLeft_iso
@[simp]
lemma compLeft_iso (W) (F : D ⥤ E) : Localization.Lifting.iso L W (L ⋙ F) F = Iso.refl _ 
Arguments: {C : Type u_1} {D : Type u_2} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] (L : CategoryTheory.Functor C D) {E : Type u_3} [CategoryTheory.Category.{u_6, u_3} E] (W : CategoryTheory.MorphismProperty C) (F : CategoryTheory.Functor D E)
Goal: CategoryTheory.Localization.Lifting.iso L W (L.comp F) F = CategoryTheory.Iso.refl (L.comp F)