ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- CategoryTheory.uncurry_expComparison
theorem uncurry_expComparison (A B : C) :
    CartesianClosed.uncurry ((expComparison F A).app B) =
      inv (prodComparison F _ _) ≫ F.map ((exp.ev _).app _) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] {D : Type u'} [CategoryTheory.Category.{v, u'} D] [CategoryTheory.Limits.HasFiniteProducts C] [CategoryTheory.Limits.HasFiniteProducts D] (F : CategoryTheory.Functor C D) [CategoryTheory.CartesianClosed C] [CategoryTheory.CartesianClosed D] [CategoryTheory.Limits.PreservesLimitsOfShape (CategoryTheory.Discrete CategoryTheory.Limits.WalkingPair) F] (A B : C)
Goal: CategoryTheory.CartesianClosed.uncurry ((CategoryTheory.expComparison F A).app B) =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.inv (CategoryTheory.Limits.prodComparison F A (A ⟹ B)))
    (F.map ((CategoryTheory.exp.ev A).app B))
-- CategoryTheory.expComparison_whiskerLeft
theorem expComparison_whiskerLeft {A A' : C} (f : A' ⟶ A) :
    expComparison F A ≫ whiskerLeft _ (pre (F.map f)) =
      whiskerRight (pre f) _ ≫ expComparison F A' 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] {D : Type u'} [CategoryTheory.Category.{v, u'} D] [CategoryTheory.Limits.HasFiniteProducts C] [CategoryTheory.Limits.HasFiniteProducts D] (F : CategoryTheory.Functor C D) [CategoryTheory.CartesianClosed C] [CategoryTheory.CartesianClosed D] [CategoryTheory.Limits.PreservesLimitsOfShape (CategoryTheory.Discrete CategoryTheory.Limits.WalkingPair) F] {A A' : C} (f : A' ⟶ A)
Goal: CategoryTheory.CategoryStruct.comp (CategoryTheory.expComparison F A)
    (CategoryTheory.whiskerLeft F (CategoryTheory.pre (F.map f))) =
  CategoryTheory.CategoryStruct.comp (CategoryTheory.whiskerRight (CategoryTheory.pre f) F)
    (CategoryTheory.expComparison F A')
-- CategoryTheory.CartesianClosedFunctor.comparison_iso
class CartesianClosedFunctor : Prop where
  comparison_iso : ∀ A, IsIso (expComparison F A)
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] {D : Type u'} [CategoryTheory.Category.{v, u'} D] [CategoryTheory.Limits.HasFiniteProducts C] [CategoryTheory.Limits.HasFiniteProducts D] {F : CategoryTheory.Functor C D} [CategoryTheory.CartesianClosed C] [CategoryTheory.CartesianClosed D] [CategoryTheory.Limits.PreservesLimitsOfShape (CategoryTheory.Discrete CategoryTheory.Limits.WalkingPair) F] [self : CategoryTheory.CartesianClosedFunctor F] (A : C)
Goal: CategoryTheory.IsIso (CategoryTheory.expComparison F A)
-- CategoryTheory.frobeniusMorphism_mate
theorem frobeniusMorphism_mate (h : L ⊣ F) (A : C) :
    conjugateEquiv (h.comp (exp.adjunction A)) ((exp.adjunction (F.obj A)).comp h)
        (frobeniusMorphism F h A) =
      expComparison F A 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] {D : Type u'} [CategoryTheory.Category.{v, u'} D] [CategoryTheory.Limits.HasFiniteProducts C] [CategoryTheory.Limits.HasFiniteProducts D] (F : CategoryTheory.Functor C D) {L : CategoryTheory.Functor D C} [CategoryTheory.CartesianClosed C] [CategoryTheory.CartesianClosed D] [CategoryTheory.Limits.PreservesLimitsOfShape (CategoryTheory.Discrete CategoryTheory.Limits.WalkingPair) F] (h : L ⊣ F) (A : C)
Goal: (CategoryTheory.conjugateEquiv (h.comp (CategoryTheory.exp.adjunction A))
      ((CategoryTheory.exp.adjunction (F.obj A)).comp h))
    (CategoryTheory.frobeniusMorphism F h A) =
  CategoryTheory.expComparison F A
-- CategoryTheory.frobeniusMorphism_iso_of_expComparison_iso
theorem frobeniusMorphism_iso_of_expComparison_iso (h : L ⊣ F) (A : C)
    [i : IsIso (expComparison F A)] : IsIso (frobeniusMorphism F h A) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] {D : Type u'} [CategoryTheory.Category.{v, u'} D] [CategoryTheory.Limits.HasFiniteProducts C] [CategoryTheory.Limits.HasFiniteProducts D] (F : CategoryTheory.Functor C D) {L : CategoryTheory.Functor D C} [CategoryTheory.CartesianClosed C] [CategoryTheory.CartesianClosed D] [CategoryTheory.Limits.PreservesLimitsOfShape (CategoryTheory.Discrete CategoryTheory.Limits.WalkingPair) F] (h : L ⊣ F) (A : C) [i : CategoryTheory.IsIso (CategoryTheory.expComparison F A)]
Goal: CategoryTheory.IsIso (CategoryTheory.frobeniusMorphism F h A)
-- CategoryTheory.expComparison_iso_of_frobeniusMorphism_iso
theorem expComparison_iso_of_frobeniusMorphism_iso (h : L ⊣ F) (A : C)
    [i : IsIso (frobeniusMorphism F h A)] : IsIso (expComparison F A) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] {D : Type u'} [CategoryTheory.Category.{v, u'} D] [CategoryTheory.Limits.HasFiniteProducts C] [CategoryTheory.Limits.HasFiniteProducts D] (F : CategoryTheory.Functor C D) {L : CategoryTheory.Functor D C} [CategoryTheory.CartesianClosed C] [CategoryTheory.CartesianClosed D] [CategoryTheory.Limits.PreservesLimitsOfShape (CategoryTheory.Discrete CategoryTheory.Limits.WalkingPair) F] (h : L ⊣ F) (A : C) [i : CategoryTheory.IsIso (CategoryTheory.frobeniusMorphism F h A)]
Goal: CategoryTheory.IsIso (CategoryTheory.expComparison F A)
-- CategoryTheory.cartesianClosedFunctorOfLeftAdjointPreservesBinaryProducts
theorem cartesianClosedFunctorOfLeftAdjointPreservesBinaryProducts (h : L ⊣ F) [F.Full] [F.Faithful]
    [PreservesLimitsOfShape (Discrete WalkingPair) L] : CartesianClosedFunctor F 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] {D : Type u'} [CategoryTheory.Category.{v, u'} D] [CategoryTheory.Limits.HasFiniteProducts C] [CategoryTheory.Limits.HasFiniteProducts D] (F : CategoryTheory.Functor C D) {L : CategoryTheory.Functor D C} [CategoryTheory.CartesianClosed C] [CategoryTheory.CartesianClosed D] [CategoryTheory.Limits.PreservesLimitsOfShape (CategoryTheory.Discrete CategoryTheory.Limits.WalkingPair) F] (h : L ⊣ F) [F.Full] [F.Faithful] [CategoryTheory.Limits.PreservesLimitsOfShape (CategoryTheory.Discrete CategoryTheory.Limits.WalkingPair) L]
Goal: CategoryTheory.CartesianClosedFunctor F
-- CategoryTheory.Functor.closedIhom_obj_obj
@[simps!]
def closedIhom (F : D ⥤ C) : (D ⥤ C) ⥤ D ⥤ C 
Arguments: {D : Type u_1} {C : Type u_2} [CategoryTheory.Groupoid D] [CategoryTheory.Category.{u_4, u_2} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.MonoidalClosed C] (F Y : D ⥤ C) (X : D)
Goal: (F.closedIhom.obj Y).obj X = (CategoryTheory.ihom (F.obj X)).obj (Y.obj X)
-- CategoryTheory.Functor.closedIhom_obj_map
@[simps!]
def closedIhom (F : D ⥤ C) : (D ⥤ C) ⥤ D ⥤ C 
Arguments: {D : Type u_1} {C : Type u_2} [CategoryTheory.Groupoid D] [CategoryTheory.Category.{u_4, u_2} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.MonoidalClosed C] (F Y : D ⥤ C)
Goal: ∀ {X Y_1 : D} (f : X ⟶ Y_1),
  (F.closedIhom.obj Y).map f =
    (CategoryTheory.MonoidalClosed.pre (CategoryTheory.inv (F.map f))).app (Y.obj X) ≫
      (CategoryTheory.ihom (F.obj Y_1)).map (Y.map f)
-- CategoryTheory.Functor.closedIhom_map_app
@[simps!]
def closedIhom (F : D ⥤ C) : (D ⥤ C) ⥤ D ⥤ C 
Arguments: {D : Type u_1} {C : Type u_2} [CategoryTheory.Groupoid D] [CategoryTheory.Category.{u_4, u_2} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.MonoidalClosed C] (F : D ⥤ C)
Goal: ∀ {X Y : D ⥤ C} (g : X ⟶ Y) (X_1 : D), (F.closedIhom.map g).app X_1 = (CategoryTheory.ihom (F.obj X_1)).map (g.app X_1)
-- CategoryTheory.Functor.closedUnit_app_app
@[simps]
def closedUnit (F : D ⥤ C) : 𝟭 (D ⥤ C) ⟶ tensorLeft F ⋙ closedIhom F 
Arguments: {D : Type u_1} {C : Type u_2} [CategoryTheory.Groupoid D] [CategoryTheory.Category.{u_4, u_2} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.MonoidalClosed C] (F G : D ⥤ C) (X : D)
Goal: (F.closedUnit.app G).app X = (CategoryTheory.ihom.coev (F.obj X)).app (G.obj X)
-- CategoryTheory.Functor.closedCounit_app_app
@[simps]
def closedCounit (F : D ⥤ C) : closedIhom F ⋙ tensorLeft F ⟶ 𝟭 (D ⥤ C) 
Arguments: {D : Type u_1} {C : Type u_2} [CategoryTheory.Groupoid D] [CategoryTheory.Category.{u_4, u_2} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.MonoidalClosed C] (F G : D ⥤ C) (X : D)
Goal: (F.closedCounit.app G).app X = (CategoryTheory.ihom.ev (F.obj X)).app (G.obj X)
-- CategoryTheory.Functor.monoidalClosed_closed_adj
@[simps! closed_adj]
instance monoidalClosed : MonoidalClosed (D ⥤ C) 
Arguments: {D : Type u_1} {C : Type u_2} [CategoryTheory.Groupoid D] [CategoryTheory.Category.{u_4, u_2} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.MonoidalClosed C] (X : D ⥤ C)
Goal: CategoryTheory.Closed.adj =
  CategoryTheory.Adjunction.mkOfUnitCounit
    { unit := X.closedUnit, counit := X.closedCounit, left_triangle := ⋯, right_triangle := ⋯ }
-- CategoryTheory.Functor.ihom_map
theorem ihom_map (F : D ⥤ C) {G H : D ⥤ C} (f : G ⟶ H) : (ihom F).map f = (closedIhom F).map f 
Arguments: {D : Type u_1} {C : Type u_2} [CategoryTheory.Groupoid D] [CategoryTheory.Category.{u_4, u_2} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.MonoidalClosed C] (F : D ⥤ C) {G H : D ⥤ C} (f : G ⟶ H)
Goal: (CategoryTheory.ihom F).map f = F.closedIhom.map f
-- CategoryTheory.Functor.ihom_ev_app
theorem ihom_ev_app (F G : D ⥤ C) : (ihom.ev F).app G = (closedCounit F).app G 
Arguments: {D : Type u_1} {C : Type u_2} [CategoryTheory.Groupoid D] [CategoryTheory.Category.{u_4, u_2} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.MonoidalClosed C] (F G : D ⥤ C)
Goal: (CategoryTheory.ihom.ev F).app G = F.closedCounit.app G
-- CategoryTheory.Functor.ihom_coev_app
theorem ihom_coev_app (F G : D ⥤ C) : (ihom.coev F).app G = (closedUnit F).app G 
Arguments: {D : Type u_1} {C : Type u_2} [CategoryTheory.Groupoid D] [CategoryTheory.Category.{u_4, u_2} C] [CategoryTheory.MonoidalCategory C] [CategoryTheory.MonoidalClosed C] (F G : D ⥤ C)
Goal: (CategoryTheory.ihom.coev F).app G = F.closedUnit.app G
-- CategoryTheory.Equivalence.precoherent
theorem precoherent : Precoherent D 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_4, u_1} C] {D : Type u_2} [CategoryTheory.Category.{u_3, u_2} D] (e : C ≌ D) [CategoryTheory.Precoherent C]
Goal: CategoryTheory.Precoherent D
-- CategoryTheory.Equivalence.sheafCongrPrecoherent_counitIso_inv_app_val_app
@[simps!]
def sheafCongrPrecoherent : haveI := e.precoherent
    Sheaf (coherentTopology C) A ≌ Sheaf (coherentTopology D) A 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_4, u_1} C] {D : Type u_2} [CategoryTheory.Category.{u_5, u_2} D] (e : C ≌ D) [CategoryTheory.Precoherent C] (A : Type u_3) [CategoryTheory.Category.{u_6, u_3} A] (X : CategoryTheory.Sheaf (CategoryTheory.coherentTopology D) A)
Goal: ∀ (X_1 : Dᵒᵖ),
  ((e.sheafCongrPrecoherent A).counitIso.inv.app X).val.app X_1 = X.val.map (e.counitIso.hom.app (Opposite.unop X_1)).op
-- CategoryTheory.Equivalence.sheafCongrPrecoherent_inverse_obj_val_map
@[simps!]
def sheafCongrPrecoherent : haveI := e.precoherent
    Sheaf (coherentTopology C) A ≌ Sheaf (coherentTopology D) A 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_4, u_1} C] {D : Type u_2} [CategoryTheory.Category.{u_5, u_2} D] (e : C ≌ D) [CategoryTheory.Precoherent C] (A : Type u_3) [CategoryTheory.Category.{u_6, u_3} A] (F : CategoryTheory.Sheaf (CategoryTheory.coherentTopology D) A)
Goal: ∀ {X Y : Cᵒᵖ} (f : X ⟶ Y), ((e.sheafCongrPrecoherent A).inverse.obj F).val.map f = F.val.map (e.functor.map f.unop).op
-- CategoryTheory.Equivalence.sheafCongrPrecoherent_inverse_obj_val_obj
@[simps!]
def sheafCongrPrecoherent : haveI := e.precoherent
    Sheaf (coherentTopology C) A ≌ Sheaf (coherentTopology D) A 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_4, u_1} C] {D : Type u_2} [CategoryTheory.Category.{u_5, u_2} D] (e : C ≌ D) [CategoryTheory.Precoherent C] (A : Type u_3) [CategoryTheory.Category.{u_6, u_3} A] (F : CategoryTheory.Sheaf (CategoryTheory.coherentTopology D) A) (X : Cᵒᵖ)
Goal: ((e.sheafCongrPrecoherent A).inverse.obj F).val.obj X = F.val.obj (Opposite.op (e.functor.obj (Opposite.unop X)))
-- CategoryTheory.Equivalence.sheafCongrPrecoherent_unitIso_inv_app_val_app
@[simps!]
def sheafCongrPrecoherent : haveI := e.precoherent
    Sheaf (coherentTopology C) A ≌ Sheaf (coherentTopology D) A 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_4, u_1} C] {D : Type u_2} [CategoryTheory.Category.{u_5, u_2} D] (e : C ≌ D) [CategoryTheory.Precoherent C] (A : Type u_3) [CategoryTheory.Category.{u_6, u_3} A] (X : CategoryTheory.Sheaf (CategoryTheory.coherentTopology C) A)
Goal: ∀ (X_1 : Cᵒᵖ),
  ((e.sheafCongrPrecoherent A).unitIso.inv.app X).val.app X_1 = X.val.map (e.unitIso.hom.app (Opposite.unop X_1)).op
-- CategoryTheory.Equivalence.sheafCongrPrecoherent_functor_map_val_app
@[simps!]
def sheafCongrPrecoherent : haveI := e.precoherent
    Sheaf (coherentTopology C) A ≌ Sheaf (coherentTopology D) A 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_4, u_1} C] {D : Type u_2} [CategoryTheory.Category.{u_5, u_2} D] (e : C ≌ D) [CategoryTheory.Precoherent C] (A : Type u_3) [CategoryTheory.Category.{u_6, u_3} A]
Goal: ∀ {X Y : CategoryTheory.Sheaf (CategoryTheory.coherentTopology C) A} (f : X ⟶ Y) (X_1 : Dᵒᵖ),
  ((e.sheafCongrPrecoherent A).functor.map f).val.app X_1 = f.val.app (Opposite.op (e.inverse.obj (Opposite.unop X_1)))
-- CategoryTheory.Equivalence.sheafCongrPrecoherent_counitIso_hom_app_val_app
@[simps!]
def sheafCongrPrecoherent : haveI := e.precoherent
    Sheaf (coherentTopology C) A ≌ Sheaf (coherentTopology D) A 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_4, u_1} C] {D : Type u_2} [CategoryTheory.Category.{u_5, u_2} D] (e : C ≌ D) [CategoryTheory.Precoherent C] (A : Type u_3) [CategoryTheory.Category.{u_6, u_3} A] (X : CategoryTheory.Sheaf (CategoryTheory.coherentTopology D) A)
Goal: ∀ (X_1 : Dᵒᵖ),
  ((e.sheafCongrPrecoherent A).counitIso.hom.app X).val.app X_1 = X.val.map (e.counitIso.inv.app (Opposite.unop X_1)).op
-- CategoryTheory.Equivalence.sheafCongrPrecoherent_unitIso_hom_app_val_app
@[simps!]
def sheafCongrPrecoherent : haveI := e.precoherent
    Sheaf (coherentTopology C) A ≌ Sheaf (coherentTopology D) A 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_4, u_1} C] {D : Type u_2} [CategoryTheory.Category.{u_5, u_2} D] (e : C ≌ D) [CategoryTheory.Precoherent C] (A : Type u_3) [CategoryTheory.Category.{u_6, u_3} A] (X : CategoryTheory.Sheaf (CategoryTheory.coherentTopology C) A)
Goal: ∀ (X_1 : Cᵒᵖ),
  ((e.sheafCongrPrecoherent A).unitIso.hom.app X).val.app X_1 = X.val.map (e.unitIso.inv.app (Opposite.unop X_1)).op
-- CategoryTheory.Equivalence.sheafCongrPrecoherent_functor_obj_val_map
@[simps!]
def sheafCongrPrecoherent : haveI := e.precoherent
    Sheaf (coherentTopology C) A ≌ Sheaf (coherentTopology D) A 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_4, u_1} C] {D : Type u_2} [CategoryTheory.Category.{u_5, u_2} D] (e : C ≌ D) [CategoryTheory.Precoherent C] (A : Type u_3) [CategoryTheory.Category.{u_6, u_3} A] (F : CategoryTheory.Sheaf (CategoryTheory.coherentTopology C) A)
Goal: ∀ {X Y : Dᵒᵖ} (f : X ⟶ Y), ((e.sheafCongrPrecoherent A).functor.obj F).val.map f = F.val.map (e.inverse.map f.unop).op
-- CategoryTheory.Equivalence.sheafCongrPrecoherent_functor_obj_val_obj
@[simps!]
def sheafCongrPrecoherent : haveI := e.precoherent
    Sheaf (coherentTopology C) A ≌ Sheaf (coherentTopology D) A 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_4, u_1} C] {D : Type u_2} [CategoryTheory.Category.{u_5, u_2} D] (e : C ≌ D) [CategoryTheory.Precoherent C] (A : Type u_3) [CategoryTheory.Category.{u_6, u_3} A] (F : CategoryTheory.Sheaf (CategoryTheory.coherentTopology C) A) (X : Dᵒᵖ)
Goal: ((e.sheafCongrPrecoherent A).functor.obj F).val.obj X = F.val.obj (Opposite.op (e.inverse.obj (Opposite.unop X)))
-- CategoryTheory.Equivalence.sheafCongrPrecoherent_inverse_map_val_app
@[simps!]
def sheafCongrPrecoherent : haveI := e.precoherent
    Sheaf (coherentTopology C) A ≌ Sheaf (coherentTopology D) A 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_4, u_1} C] {D : Type u_2} [CategoryTheory.Category.{u_5, u_2} D] (e : C ≌ D) [CategoryTheory.Precoherent C] (A : Type u_3) [CategoryTheory.Category.{u_6, u_3} A]
Goal: ∀ {X Y : CategoryTheory.Sheaf (CategoryTheory.coherentTopology D) A} (f : X ⟶ Y) (X_1 : Cᵒᵖ),
  ((e.sheafCongrPrecoherent A).inverse.map f).val.app X_1 = f.val.app (Opposite.op (e.functor.obj (Opposite.unop X_1)))
-- CategoryTheory.Equivalence.precoherent_isSheaf_iff
theorem precoherent_isSheaf_iff (F : Cᵒᵖ ⥤ A) : haveI := e.precoherent
    IsSheaf (coherentTopology C) F ↔ IsSheaf (coherentTopology D) (e.inverse.op ⋙ F) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_4, u_1} C] {D : Type u_2} [CategoryTheory.Category.{u_6, u_2} D] (e : C ≌ D) [CategoryTheory.Precoherent C] (A : Type u_3) [CategoryTheory.Category.{u_5, u_3} A] (F : CategoryTheory.Functor Cᵒᵖ A)
Goal: CategoryTheory.Presheaf.IsSheaf (CategoryTheory.coherentTopology C) F ↔
  CategoryTheory.Presheaf.IsSheaf (CategoryTheory.coherentTopology D) (e.inverse.op.comp F)
-- CategoryTheory.Equivalence.precoherent_isSheaf_iff_of_essentiallySmall
theorem precoherent_isSheaf_iff_of_essentiallySmall [EssentiallySmall C] (F : Cᵒᵖ ⥤ A) :
    IsSheaf (coherentTopology C) F ↔
      IsSheaf (coherentTopology (SmallModel C)) ((equivSmallModel C).inverse.op ⋙ F) 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_5, u_1} C] [CategoryTheory.Precoherent C] (A : Type u_3) [CategoryTheory.Category.{u_6, u_3} A] [CategoryTheory.EssentiallySmall.{u_4, u_5, u_1} C] (F : CategoryTheory.Functor Cᵒᵖ A)
Goal: CategoryTheory.Presheaf.IsSheaf (CategoryTheory.coherentTopology C) F ↔
  CategoryTheory.Presheaf.IsSheaf (CategoryTheory.coherentTopology (CategoryTheory.SmallModel.{u_4, u_5, u_1} C))
    ((CategoryTheory.equivSmallModel C).inverse.op.comp F)
-- CategoryTheory.Equivalence.preregular
theorem preregular : Preregular D 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_4, u_1} C] {D : Type u_2} [CategoryTheory.Category.{u_3, u_2} D] (e : C ≌ D) [CategoryTheory.Preregular C]
Goal: CategoryTheory.Preregular D
-- CategoryTheory.Equivalence.sheafCongrPreregular_functor_obj_val_obj
@[simps!]
def sheafCongrPreregular : haveI := e.preregular
    Sheaf (regularTopology C) A ≌ Sheaf (regularTopology D) A 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_4, u_1} C] {D : Type u_2} [CategoryTheory.Category.{u_5, u_2} D] (e : C ≌ D) [CategoryTheory.Preregular C] (A : Type u_3) [CategoryTheory.Category.{u_6, u_3} A] (F : CategoryTheory.Sheaf (CategoryTheory.regularTopology C) A) (X : Dᵒᵖ)
Goal: ((e.sheafCongrPreregular A).functor.obj F).val.obj X = F.val.obj (Opposite.op (e.inverse.obj (Opposite.unop X)))
-- CategoryTheory.Equivalence.sheafCongrPreregular_unitIso_hom_app_val_app
@[simps!]
def sheafCongrPreregular : haveI := e.preregular
    Sheaf (regularTopology C) A ≌ Sheaf (regularTopology D) A 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_4, u_1} C] {D : Type u_2} [CategoryTheory.Category.{u_5, u_2} D] (e : C ≌ D) [CategoryTheory.Preregular C] (A : Type u_3) [CategoryTheory.Category.{u_6, u_3} A] (X : CategoryTheory.Sheaf (CategoryTheory.regularTopology C) A)
Goal: ∀ (X_1 : Cᵒᵖ),
  ((e.sheafCongrPreregular A).unitIso.hom.app X).val.app X_1 = X.val.map (e.unitIso.inv.app (Opposite.unop X_1)).op
-- CategoryTheory.Equivalence.sheafCongrPreregular_functor_obj_val_map
@[simps!]
def sheafCongrPreregular : haveI := e.preregular
    Sheaf (regularTopology C) A ≌ Sheaf (regularTopology D) A 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_4, u_1} C] {D : Type u_2} [CategoryTheory.Category.{u_5, u_2} D] (e : C ≌ D) [CategoryTheory.Preregular C] (A : Type u_3) [CategoryTheory.Category.{u_6, u_3} A] (F : CategoryTheory.Sheaf (CategoryTheory.regularTopology C) A)
Goal: ∀ {X Y : Dᵒᵖ} (f : X ⟶ Y), ((e.sheafCongrPreregular A).functor.obj F).val.map f = F.val.map (e.inverse.map f.unop).op
-- CategoryTheory.Equivalence.sheafCongrPreregular_inverse_obj_val_map
@[simps!]
def sheafCongrPreregular : haveI := e.preregular
    Sheaf (regularTopology C) A ≌ Sheaf (regularTopology D) A 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_4, u_1} C] {D : Type u_2} [CategoryTheory.Category.{u_5, u_2} D] (e : C ≌ D) [CategoryTheory.Preregular C] (A : Type u_3) [CategoryTheory.Category.{u_6, u_3} A] (F : CategoryTheory.Sheaf (CategoryTheory.regularTopology D) A)
Goal: ∀ {X Y : Cᵒᵖ} (f : X ⟶ Y), ((e.sheafCongrPreregular A).inverse.obj F).val.map f = F.val.map (e.functor.map f.unop).op
-- CategoryTheory.Equivalence.sheafCongrPreregular_counitIso_inv_app_val_app
@[simps!]
def sheafCongrPreregular : haveI := e.preregular
    Sheaf (regularTopology C) A ≌ Sheaf (regularTopology D) A 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_4, u_1} C] {D : Type u_2} [CategoryTheory.Category.{u_5, u_2} D] (e : C ≌ D) [CategoryTheory.Preregular C] (A : Type u_3) [CategoryTheory.Category.{u_6, u_3} A] (X : CategoryTheory.Sheaf (CategoryTheory.regularTopology D) A)
Goal: ∀ (X_1 : Dᵒᵖ),
  ((e.sheafCongrPreregular A).counitIso.inv.app X).val.app X_1 = X.val.map (e.counitIso.hom.app (Opposite.unop X_1)).op
-- CategoryTheory.Equivalence.sheafCongrPreregular_functor_map_val_app
@[simps!]
def sheafCongrPreregular : haveI := e.preregular
    Sheaf (regularTopology C) A ≌ Sheaf (regularTopology D) A 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_4, u_1} C] {D : Type u_2} [CategoryTheory.Category.{u_5, u_2} D] (e : C ≌ D) [CategoryTheory.Preregular C] (A : Type u_3) [CategoryTheory.Category.{u_6, u_3} A]
Goal: ∀ {X Y : CategoryTheory.Sheaf (CategoryTheory.regularTopology C) A} (f : X ⟶ Y) (X_1 : Dᵒᵖ),
  ((e.sheafCongrPreregular A).functor.map f).val.app X_1 = f.val.app (Opposite.op (e.inverse.obj (Opposite.unop X_1)))
-- CategoryTheory.Equivalence.sheafCongrPreregular_inverse_obj_val_obj
@[simps!]
def sheafCongrPreregular : haveI := e.preregular
    Sheaf (regularTopology C) A ≌ Sheaf (regularTopology D) A 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_4, u_1} C] {D : Type u_2} [CategoryTheory.Category.{u_5, u_2} D] (e : C ≌ D) [CategoryTheory.Preregular C] (A : Type u_3) [CategoryTheory.Category.{u_6, u_3} A] (F : CategoryTheory.Sheaf (CategoryTheory.regularTopology D) A) (X : Cᵒᵖ)
Goal: ((e.sheafCongrPreregular A).inverse.obj F).val.obj X = F.val.obj (Opposite.op (e.functor.obj (Opposite.unop X)))
-- CategoryTheory.Equivalence.sheafCongrPreregular_inverse_map_val_app
@[simps!]
def sheafCongrPreregular : haveI := e.preregular
    Sheaf (regularTopology C) A ≌ Sheaf (regularTopology D) A 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_4, u_1} C] {D : Type u_2} [CategoryTheory.Category.{u_5, u_2} D] (e : C ≌ D) [CategoryTheory.Preregular C] (A : Type u_3) [CategoryTheory.Category.{u_6, u_3} A]
Goal: ∀ {X Y : CategoryTheory.Sheaf (CategoryTheory.regularTopology D) A} (f : X ⟶ Y) (X_1 : Cᵒᵖ),
  ((e.sheafCongrPreregular A).inverse.map f).val.app X_1 = f.val.app (Opposite.op (e.functor.obj (Opposite.unop X_1)))
-- CategoryTheory.Equivalence.sheafCongrPreregular_unitIso_inv_app_val_app
@[simps!]
def sheafCongrPreregular : haveI := e.preregular
    Sheaf (regularTopology C) A ≌ Sheaf (regularTopology D) A 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_4, u_1} C] {D : Type u_2} [CategoryTheory.Category.{u_5, u_2} D] (e : C ≌ D) [CategoryTheory.Preregular C] (A : Type u_3) [CategoryTheory.Category.{u_6, u_3} A] (X : CategoryTheory.Sheaf (CategoryTheory.regularTopology C) A)
Goal: ∀ (X_1 : Cᵒᵖ),
  ((e.sheafCongrPreregular A).unitIso.inv.app X).val.app X_1 = X.val.map (e.unitIso.hom.app (Opposite.unop X_1)).op
-- CategoryTheory.Equivalence.sheafCongrPreregular_counitIso_hom_app_val_app
@[simps!]
def sheafCongrPreregular : haveI := e.preregular
    Sheaf (regularTopology C) A ≌ Sheaf (regularTopology D) A 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_4, u_1} C] {D : Type u_2} [CategoryTheory.Category.{u_5, u_2} D] (e : C ≌ D) [CategoryTheory.Preregular C] (A : Type u_3) [CategoryTheory.Category.{u_6, u_3} A] (X : CategoryTheory.Sheaf (CategoryTheory.regularTopology D) A)
Goal: ∀ (X_1 : Dᵒᵖ),
  ((e.sheafCongrPreregular A).counitIso.hom.app X).val.app X_1 = X.val.map (e.counitIso.inv.app (Opposite.unop X_1)).op