ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- CategoryTheory.locallySmall_congr
theorem locallySmall_congr {C : Type u} [Category.{v} C] {D : Type u'} [Category.{v'} D]
    (e : C ≌ D) : LocallySmall.{w} C ↔ LocallySmall.{w} D 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] {D : Type u'} [CategoryTheory.Category.{v', u'} D] (e : C ≌ D)
Goal: CategoryTheory.LocallySmall.{w, v, u} C ↔ CategoryTheory.LocallySmall.{w, v', u'} D
-- CategoryTheory.locallySmall_max
theorem locallySmall_max {C : Type u} [Category.{v} C] : LocallySmall.{max v w} C 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C]
Goal: CategoryTheory.LocallySmall.{max v w, v, u} C
-- CategoryTheory.ShrinkHoms.to_from
@[simp]
theorem to_from (X : C') : fromShrinkHoms (toShrinkHoms X) = X 
Arguments: {C' : Type u_1} (X : C')
Goal: (CategoryTheory.ShrinkHoms.toShrinkHoms X).fromShrinkHoms = X
-- CategoryTheory.ShrinkHoms.from_to
@[simp]
theorem from_to (X : ShrinkHoms C') : toShrinkHoms (fromShrinkHoms X) = X 
Arguments: {C' : Type u_1} (X : CategoryTheory.ShrinkHoms.{u_1} C')
Goal: CategoryTheory.ShrinkHoms.toShrinkHoms X.fromShrinkHoms = X
-- CategoryTheory.ShrinkHoms.instCategory_comp
@[simps]
noncomputable instance : Category.{w} (ShrinkHoms C) 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C] [CategoryTheory.LocallySmall.{w, v, u} C]
Goal: ∀ {X Y Z : CategoryTheory.ShrinkHoms.{u} C} (f : X ⟶ Y) (g : Y ⟶ Z),
  f ≫ g =
    (equivShrink (X.fromShrinkHoms ⟶ Z.fromShrinkHoms))
      ((equivShrink (X.fromShrinkHoms ⟶ Y.fromShrinkHoms)).symm f ≫
        (equivShrink (Y.fromShrinkHoms ⟶ Z.fromShrinkHoms)).symm g)
-- CategoryTheory.ShrinkHoms.instCategory_id
@[simps]
noncomputable instance : Category.{w} (ShrinkHoms C) 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C] [CategoryTheory.LocallySmall.{w, v, u} C] (X : CategoryTheory.ShrinkHoms.{u} C)
Goal: 𝟙 X = (equivShrink (X.fromShrinkHoms ⟶ X.fromShrinkHoms)) (𝟙 X.fromShrinkHoms)
-- CategoryTheory.ShrinkHoms.functor_obj
@[simps]
noncomputable def functor : C ⥤ ShrinkHoms C 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C] [CategoryTheory.LocallySmall.{w, v, u} C] (X : C)
Goal: (CategoryTheory.ShrinkHoms.functor C).obj X = CategoryTheory.ShrinkHoms.toShrinkHoms X
-- CategoryTheory.ShrinkHoms.functor_map
@[simps]
noncomputable def functor : C ⥤ ShrinkHoms C 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C] [CategoryTheory.LocallySmall.{w, v, u} C] {X Y : C} (f : X ⟶ Y)
Goal: (CategoryTheory.ShrinkHoms.functor C).map f = (equivShrink (X ⟶ Y)) f
-- CategoryTheory.ShrinkHoms.inverse_obj
@[simps]
noncomputable def inverse : ShrinkHoms C ⥤ C 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C] [CategoryTheory.LocallySmall.{w, v, u} C] (X : CategoryTheory.ShrinkHoms.{u} C)
Goal: (CategoryTheory.ShrinkHoms.inverse C).obj X = X.fromShrinkHoms
-- CategoryTheory.ShrinkHoms.inverse_map
@[simps]
noncomputable def inverse : ShrinkHoms C ⥤ C 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C] [CategoryTheory.LocallySmall.{w, v, u} C] {X Y : CategoryTheory.ShrinkHoms.{u} C} (f : X ⟶ Y)
Goal: (CategoryTheory.ShrinkHoms.inverse C).map f = (equivShrink (X.fromShrinkHoms ⟶ Y.fromShrinkHoms)).symm f
-- CategoryTheory.ShrinkHoms.equivalence_unitIso_hom_app
@[simps!]
noncomputable def equivalence : C ≌ ShrinkHoms C 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C] [CategoryTheory.LocallySmall.{w, v, u} C] (X : C)
Goal: (CategoryTheory.ShrinkHoms.equivalence C).unitIso.hom.app X = 𝟙 X
-- CategoryTheory.ShrinkHoms.equivalence_counitIso_hom_app
@[simps!]
noncomputable def equivalence : C ≌ ShrinkHoms C 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C] [CategoryTheory.LocallySmall.{w, v, u} C] (X : CategoryTheory.ShrinkHoms.{u} C)
Goal: (CategoryTheory.ShrinkHoms.equivalence C).counitIso.hom.app X =
  (equivShrink (X.fromShrinkHoms ⟶ X.fromShrinkHoms)) (𝟙 X.fromShrinkHoms)
-- CategoryTheory.ShrinkHoms.equivalence_counitIso_inv_app
@[simps!]
noncomputable def equivalence : C ≌ ShrinkHoms C 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C] [CategoryTheory.LocallySmall.{w, v, u} C] (X : CategoryTheory.ShrinkHoms.{u} C)
Goal: (CategoryTheory.ShrinkHoms.equivalence C).counitIso.inv.app X =
  (equivShrink (X.fromShrinkHoms ⟶ X.fromShrinkHoms)) (𝟙 X.fromShrinkHoms)
-- CategoryTheory.ShrinkHoms.equivalence_inverse_obj
@[simps!]
noncomputable def equivalence : C ≌ ShrinkHoms C 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C] [CategoryTheory.LocallySmall.{w, v, u} C] (X : CategoryTheory.ShrinkHoms.{u} C)
Goal: (CategoryTheory.ShrinkHoms.equivalence C).inverse.obj X = X.fromShrinkHoms
-- CategoryTheory.ShrinkHoms.equivalence_functor_obj
@[simps!]
noncomputable def equivalence : C ≌ ShrinkHoms C 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C] [CategoryTheory.LocallySmall.{w, v, u} C] (X : C)
Goal: (CategoryTheory.ShrinkHoms.equivalence C).functor.obj X = CategoryTheory.ShrinkHoms.toShrinkHoms X
-- CategoryTheory.ShrinkHoms.equivalence_functor_map
@[simps!]
noncomputable def equivalence : C ≌ ShrinkHoms C 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C] [CategoryTheory.LocallySmall.{w, v, u} C] {X Y : C} (f : X ⟶ Y)
Goal: (CategoryTheory.ShrinkHoms.equivalence C).functor.map f = (equivShrink (X ⟶ Y)) f
-- CategoryTheory.ShrinkHoms.equivalence_inverse_map
@[simps!]
noncomputable def equivalence : C ≌ ShrinkHoms C 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C] [CategoryTheory.LocallySmall.{w, v, u} C] {X Y : CategoryTheory.ShrinkHoms.{u} C} (f : X ⟶ Y)
Goal: (CategoryTheory.ShrinkHoms.equivalence C).inverse.map f = (equivShrink (X.fromShrinkHoms ⟶ Y.fromShrinkHoms)).symm f
-- CategoryTheory.ShrinkHoms.equivalence_unitIso_inv_app
@[simps!]
noncomputable def equivalence : C ≌ ShrinkHoms C 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C] [CategoryTheory.LocallySmall.{w, v, u} C] (X : C)
Goal: (CategoryTheory.ShrinkHoms.equivalence C).unitIso.inv.app X = 𝟙 X
-- CategoryTheory.essentiallySmall_iff
theorem essentiallySmall_iff (C : Type u) [Category.{v} C] :
    EssentiallySmall.{w} C ↔ Small.{w} (Skeleton C) ∧ LocallySmall.{w} C 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C]
Goal: CategoryTheory.EssentiallySmall.{w, v, u} C ↔
  Small.{w, u} (CategoryTheory.Skeleton C) ∧ CategoryTheory.LocallySmall.{w, v, u} C
-- CategoryTheory.essentiallySmall_of_small_of_locallySmall
theorem essentiallySmall_of_small_of_locallySmall [Small.{w} C] [LocallySmall.{w} C] :
    EssentiallySmall.{w} C 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C] [Small.{w, u} C] [CategoryTheory.LocallySmall.{w, v, u} C]
Goal: CategoryTheory.EssentiallySmall.{w, v, u} C
-- CategoryTheory.essentiallySmall_iff_of_thin
theorem essentiallySmall_iff_of_thin {C : Type u} [Category.{v} C] [Quiver.IsThin C] :
    EssentiallySmall.{w} C ↔ Small.{w} (Skeleton C) 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] [Quiver.IsThin C]
Goal: CategoryTheory.EssentiallySmall.{w, v, u} C ↔ Small.{w, u} (CategoryTheory.Skeleton C)
-- CategoryTheory.Functor.IsRightDerivedFunctor.isLeftKanExtension'
class IsRightDerivedFunctor [L.IsLocalization W] : Prop where
  isLeftKanExtension' : RF.IsLeftKanExtension α
Arguments: {C : Type u_1} {D : Type u_2} {H : Type u_3} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] [CategoryTheory.Category.{u_6, u_3} H] {RF : CategoryTheory.Functor D H} {F : CategoryTheory.Functor C H} {L : CategoryTheory.Functor C D} {α : F ⟶ L.comp RF} (W : CategoryTheory.MorphismProperty C) [L.IsLocalization W] [self : RF.IsRightDerivedFunctor α W]
Goal: RF.IsLeftKanExtension α
-- CategoryTheory.Functor.IsRightDerivedFunctor.isLeftKanExtension
lemma IsRightDerivedFunctor.isLeftKanExtension
    [L.IsLocalization W] [RF.IsRightDerivedFunctor α W] :
    RF.IsLeftKanExtension α 
Arguments: {C : Type u_1} {D : Type u_2} {H : Type u_5} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Category.{u_4, u_2} D] [CategoryTheory.Category.{u_6, u_5} H] (RF : CategoryTheory.Functor D H) {F : CategoryTheory.Functor C H} {L : CategoryTheory.Functor C D} (α : F ⟶ L.comp RF) (W : CategoryTheory.MorphismProperty C) [L.IsLocalization W] [RF.IsRightDerivedFunctor α W]
Goal: RF.IsLeftKanExtension α
-- CategoryTheory.Functor.isRightDerivedFunctor_iff_isLeftKanExtension
lemma isRightDerivedFunctor_iff_isLeftKanExtension [L.IsLocalization W] :
    RF.IsRightDerivedFunctor α W ↔ RF.IsLeftKanExtension α 
Arguments: {C : Type u_1} {D : Type u_2} {H : Type u_5} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Category.{u_4, u_2} D] [CategoryTheory.Category.{u_6, u_5} H] (RF : CategoryTheory.Functor D H) {F : CategoryTheory.Functor C H} {L : CategoryTheory.Functor C D} (α : F ⟶ L.comp RF) (W : CategoryTheory.MorphismProperty C) [L.IsLocalization W]
Goal: RF.IsRightDerivedFunctor α W ↔ RF.IsLeftKanExtension α
-- CategoryTheory.Functor.isRightDerivedFunctor_iff_of_iso
lemma isRightDerivedFunctor_iff_of_iso (α' : F ⟶ L ⋙ RF') (W : MorphismProperty C)
    [L.IsLocalization W] (e : RF ≅ RF') (comm : α ≫ whiskerLeft L e.hom = α') :
    RF.IsRightDerivedFunctor α W ↔ RF'.IsRightDerivedFunctor α' W 
Arguments: {C : Type u_1} {D : Type u_6} {H : Type u_3} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_6} D] [CategoryTheory.Category.{u_2, u_3} H] {RF RF' : CategoryTheory.Functor D H} {F : CategoryTheory.Functor C H} {L : CategoryTheory.Functor C D} (α : F ⟶ L.comp RF) (α' : F ⟶ L.comp RF') (W : CategoryTheory.MorphismProperty C) [L.IsLocalization W] (e : RF ≅ RF') (comm : CategoryTheory.CategoryStruct.comp α (CategoryTheory.whiskerLeft L e.hom) = α')
Goal: RF.IsRightDerivedFunctor α W ↔ RF'.IsRightDerivedFunctor α' W
-- CategoryTheory.Functor.rightDerived_fac
@[reassoc (attr := simp)]
lemma rightDerived_fac (G : D ⥤ H) (β : F ⟶ L ⋙ G) :
    α ≫ whiskerLeft L (RF.rightDerivedDesc α W G β) = β 
Arguments: {C : Type u_5} {D : Type u_3} {H : Type u_4} [CategoryTheory.Category.{u_6, u_5} C] [CategoryTheory.Category.{u_1, u_3} D] [CategoryTheory.Category.{u_2, u_4} H] (RF : CategoryTheory.Functor D H) {F : CategoryTheory.Functor C H} {L : CategoryTheory.Functor C D} (α : F ⟶ L.comp RF) (W : CategoryTheory.MorphismProperty C) [L.IsLocalization W] [RF.IsRightDerivedFunctor α W] (G : CategoryTheory.Functor D H) (β : F ⟶ L.comp G)
Goal: CategoryTheory.CategoryStruct.comp α (CategoryTheory.whiskerLeft L (RF.rightDerivedDesc α W G β)) = β
-- CategoryTheory.Functor.rightDerived_fac_app
@[reassoc (attr := simp)]
lemma rightDerived_fac_app (G : D ⥤ H) (β : F ⟶ L ⋙ G) (X : C) :
    α.app X ≫ (RF.rightDerivedDesc α W G β).app (L.obj X) = β.app X 
Arguments: {C : Type u_5} {D : Type u_3} {H : Type u_4} [CategoryTheory.Category.{u_6, u_5} C] [CategoryTheory.Category.{u_1, u_3} D] [CategoryTheory.Category.{u_2, u_4} H] (RF : CategoryTheory.Functor D H) {F : CategoryTheory.Functor C H} {L : CategoryTheory.Functor C D} (α : F ⟶ L.comp RF) (W : CategoryTheory.MorphismProperty C) [L.IsLocalization W] [RF.IsRightDerivedFunctor α W] (G : CategoryTheory.Functor D H) (β : F ⟶ L.comp G) (X : C)
Goal: CategoryTheory.CategoryStruct.comp (α.app X) ((RF.rightDerivedDesc α W G β).app (L.obj X)) = β.app X
-- CategoryTheory.Functor.rightDerived_ext
lemma rightDerived_ext (G : D ⥤ H) (γ₁ γ₂ : RF ⟶ G)
    (hγ : α ≫ whiskerLeft L γ₁ = α ≫ whiskerLeft L γ₂) : γ₁ = γ₂ 
Arguments: {C : Type u_5} {D : Type u_3} {H : Type u_4} [CategoryTheory.Category.{u_6, u_5} C] [CategoryTheory.Category.{u_1, u_3} D] [CategoryTheory.Category.{u_2, u_4} H] (RF : CategoryTheory.Functor D H) {F : CategoryTheory.Functor C H} {L : CategoryTheory.Functor C D} (α : F ⟶ L.comp RF) (W : CategoryTheory.MorphismProperty C) [L.IsLocalization W] [RF.IsRightDerivedFunctor α W] (G : CategoryTheory.Functor D H) (γ₁ γ₂ : RF ⟶ G) (hγ :
  CategoryTheory.CategoryStruct.comp α (CategoryTheory.whiskerLeft L γ₁) =
    CategoryTheory.CategoryStruct.comp α (CategoryTheory.whiskerLeft L γ₂))
Goal: γ₁ = γ₂
-- CategoryTheory.Functor.rightDerivedNatTrans_fac
@[reassoc (attr := simp)]
lemma rightDerivedNatTrans_fac (τ : F ⟶ F') :
    α ≫ whiskerLeft L (rightDerivedNatTrans RF RF' α α' W τ) = τ ≫ α' 
Arguments: {C : Type u_1} {D : Type u_6} {H : Type u_3} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_6} D] [CategoryTheory.Category.{u_2, u_3} H] (RF RF' : CategoryTheory.Functor D H) {F F' : CategoryTheory.Functor C H} {L : CategoryTheory.Functor C D} (α : F ⟶ L.comp RF) (α' : F' ⟶ L.comp RF') (W : CategoryTheory.MorphismProperty C) [L.IsLocalization W] [RF.IsRightDerivedFunctor α W] (τ : F ⟶ F')
Goal: CategoryTheory.CategoryStruct.comp α (CategoryTheory.whiskerLeft L (RF.rightDerivedNatTrans RF' α α' W τ)) =
  CategoryTheory.CategoryStruct.comp τ α'
-- CategoryTheory.Functor.rightDerivedNatTrans_app
@[reassoc (attr := simp)]
lemma rightDerivedNatTrans_app (τ : F ⟶ F') (X : C) :
    α.app X ≫ (rightDerivedNatTrans RF RF' α α' W τ).app (L.obj X) =
    τ.app X ≫ α'.app X 
Arguments: {C : Type u_1} {D : Type u_6} {H : Type u_3} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_6} D] [CategoryTheory.Category.{u_2, u_3} H] (RF RF' : CategoryTheory.Functor D H) {F F' : CategoryTheory.Functor C H} {L : CategoryTheory.Functor C D} (α : F ⟶ L.comp RF) (α' : F' ⟶ L.comp RF') (W : CategoryTheory.MorphismProperty C) [L.IsLocalization W] [RF.IsRightDerivedFunctor α W] (τ : F ⟶ F') (X : C)
Goal: CategoryTheory.CategoryStruct.comp (α.app X) ((RF.rightDerivedNatTrans RF' α α' W τ).app (L.obj X)) =
  CategoryTheory.CategoryStruct.comp (τ.app X) (α'.app X)
-- CategoryTheory.Functor.rightDerivedNatTrans_id
@[simp]
lemma rightDerivedNatTrans_id :
    rightDerivedNatTrans RF RF α α W (𝟙 F) = 𝟙 RF 
Arguments: {C : Type u_5} {D : Type u_1} {H : Type u_3} [CategoryTheory.Category.{u_6, u_5} C] [CategoryTheory.Category.{u_4, u_1} D] [CategoryTheory.Category.{u_2, u_3} H] (RF : CategoryTheory.Functor D H) {F : CategoryTheory.Functor C H} {L : CategoryTheory.Functor C D} (α : F ⟶ L.comp RF) (W : CategoryTheory.MorphismProperty C) [L.IsLocalization W] [RF.IsRightDerivedFunctor α W]
Goal: RF.rightDerivedNatTrans RF α α W (CategoryTheory.CategoryStruct.id F) = CategoryTheory.CategoryStruct.id RF
-- CategoryTheory.Functor.rightDerivedNatTrans_comp
@[reassoc (attr := simp)]
lemma rightDerivedNatTrans_comp (τ : F ⟶ F') (τ' : F' ⟶ F'') :
    rightDerivedNatTrans RF RF' α α' W τ ≫ rightDerivedNatTrans RF' RF'' α' α'' W τ' =
    rightDerivedNatTrans RF RF'' α α'' W (τ ≫ τ') 
Arguments: {C : Type u_1} {D : Type u_5} {H : Type u_3} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_6, u_5} D] [CategoryTheory.Category.{u_2, u_3} H] (RF RF' RF'' : CategoryTheory.Functor D H) {F F' F'' : CategoryTheory.Functor C H} {L : CategoryTheory.Functor C D} (α : F ⟶ L.comp RF) (α' : F' ⟶ L.comp RF') (α'' : F'' ⟶ L.comp RF'') (W : CategoryTheory.MorphismProperty C) [L.IsLocalization W] [RF.IsRightDerivedFunctor α W] [RF'.IsRightDerivedFunctor α' W] (τ : F ⟶ F') (τ' : F' ⟶ F'')
Goal: CategoryTheory.CategoryStruct.comp (RF.rightDerivedNatTrans RF' α α' W τ) (RF'.rightDerivedNatTrans RF'' α' α'' W τ') =
  RF.rightDerivedNatTrans RF'' α α'' W (CategoryTheory.CategoryStruct.comp τ τ')
-- CategoryTheory.Functor.rightDerivedNatIso_hom
@[simps]
noncomputable def rightDerivedNatIso (τ : F ≅ F') :
    RF ≅ RF' 
Arguments: {C : Type u_1} {D : Type u_2} {H : Type u_3} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] [CategoryTheory.Category.{u_6, u_3} H] (RF RF' : CategoryTheory.Functor D H) {F F' : CategoryTheory.Functor C H} {L : CategoryTheory.Functor C D} (α : F ⟶ L.comp RF) (α' : F' ⟶ L.comp RF') (W : CategoryTheory.MorphismProperty C) [L.IsLocalization W] [RF.IsRightDerivedFunctor α W] [RF'.IsRightDerivedFunctor α' W] (τ : F ≅ F')
Goal: (RF.rightDerivedNatIso RF' α α' W τ).hom = RF.rightDerivedNatTrans RF' α α' W τ.hom
-- CategoryTheory.Functor.rightDerivedNatIso_inv
@[simps]
noncomputable def rightDerivedNatIso (τ : F ≅ F') :
    RF ≅ RF' 
Arguments: {C : Type u_1} {D : Type u_2} {H : Type u_3} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] [CategoryTheory.Category.{u_6, u_3} H] (RF RF' : CategoryTheory.Functor D H) {F F' : CategoryTheory.Functor C H} {L : CategoryTheory.Functor C D} (α : F ⟶ L.comp RF) (α' : F' ⟶ L.comp RF') (W : CategoryTheory.MorphismProperty C) [L.IsLocalization W] [RF.IsRightDerivedFunctor α W] [RF'.IsRightDerivedFunctor α' W] (τ : F ≅ F')
Goal: (RF.rightDerivedNatIso RF' α α' W τ).inv = RF'.rightDerivedNatTrans RF α' α W τ.inv
-- CategoryTheory.Functor.isRightDerivedFunctor_iff_isIso_rightDerivedDesc
lemma isRightDerivedFunctor_iff_isIso_rightDerivedDesc (G : D ⥤ H) (β : F ⟶ L ⋙ G) :
    G.IsRightDerivedFunctor β W ↔ IsIso (RF.rightDerivedDesc α W G β) 
Arguments: {C : Type u_5} {D : Type u_3} {H : Type u_4} [CategoryTheory.Category.{u_6, u_5} C] [CategoryTheory.Category.{u_1, u_3} D] [CategoryTheory.Category.{u_2, u_4} H] (RF : CategoryTheory.Functor D H) {F : CategoryTheory.Functor C H} {L : CategoryTheory.Functor C D} (α : F ⟶ L.comp RF) (W : CategoryTheory.MorphismProperty C) [L.IsLocalization W] [RF.IsRightDerivedFunctor α W] (G : CategoryTheory.Functor D H) (β : F ⟶ L.comp G)
Goal: G.IsRightDerivedFunctor β W ↔ CategoryTheory.IsIso (RF.rightDerivedDesc α W G β)
-- CategoryTheory.Functor.HasRightDerivedFunctor.hasLeftKanExtension'
class HasRightDerivedFunctor : Prop where
  hasLeftKanExtension' : HasLeftKanExtension W.Q F
Arguments: {C : Type u_1} {H : Type u_2} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Category.{u_4, u_2} H] {F : CategoryTheory.Functor C H} {W : CategoryTheory.MorphismProperty C} [self : F.HasRightDerivedFunctor W]
Goal: W.Q.HasLeftKanExtension F
-- CategoryTheory.Functor.hasRightDerivedFunctor_iff
lemma hasRightDerivedFunctor_iff :
    F.HasRightDerivedFunctor W ↔ HasLeftKanExtension L F 
Arguments: {C : Type u_1} {D : Type u_5} {H : Type u_2} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Category.{u_6, u_5} D] [CategoryTheory.Category.{u_4, u_2} H] (F : CategoryTheory.Functor C H) (L : CategoryTheory.Functor C D) (W : CategoryTheory.MorphismProperty C) [L.IsLocalization W]
Goal: F.HasRightDerivedFunctor W ↔ L.HasLeftKanExtension F
-- CategoryTheory.Functor.hasRightDerivedFunctor_iff_of_iso
lemma hasRightDerivedFunctor_iff_of_iso :
    HasRightDerivedFunctor F W ↔ HasRightDerivedFunctor F' W 
Arguments: {C : Type u_1} {H : Type u_2} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Category.{u_4, u_2} H] {F F' : CategoryTheory.Functor C H} (e : F ≅ F') (W : CategoryTheory.MorphismProperty C)
Goal: F.HasRightDerivedFunctor W ↔ F'.HasRightDerivedFunctor W
-- CategoryTheory.Functor.HasRightDerivedFunctor.hasLeftKanExtension
lemma HasRightDerivedFunctor.hasLeftKanExtension [HasRightDerivedFunctor F W] :
    HasLeftKanExtension L F 
Arguments: {C : Type u_1} {D : Type u_5} {H : Type u_2} [CategoryTheory.Category.{u_3, u_1} C] [CategoryTheory.Category.{u_6, u_5} D] [CategoryTheory.Category.{u_4, u_2} H] (F : CategoryTheory.Functor C H) (L : CategoryTheory.Functor C D) (W : CategoryTheory.MorphismProperty C) [L.IsLocalization W] [F.HasRightDerivedFunctor W]
Goal: L.HasLeftKanExtension F
-- CategoryTheory.Functor.HasRightDerivedFunctor.mk'
lemma HasRightDerivedFunctor.mk' [RF.IsRightDerivedFunctor α W] :
    HasRightDerivedFunctor F W 
Arguments: {C : Type u_1} {D : Type u_2} {H : Type u_3} [CategoryTheory.Category.{u_4, u_1} C] [CategoryTheory.Category.{u_5, u_2} D] [CategoryTheory.Category.{u_6, u_3} H] (RF : CategoryTheory.Functor D H) {F : CategoryTheory.Functor C H} {L : CategoryTheory.Functor C D} (α : F ⟶ L.comp RF) {W : CategoryTheory.MorphismProperty C} [L.IsLocalization W] [RF.IsRightDerivedFunctor α W]
Goal: F.HasRightDerivedFunctor W
-- AddOpposite.unop_op
@[to_additive (attr := simp)]
theorem unop_op (x : α) : unop (op x) = x 
Arguments: {α : Type u_1} (x : α)
Goal: AddOpposite.unop (AddOpposite.op x) = x
-- MulOpposite.unop_op
@[to_additive (attr := simp)]
theorem unop_op (x : α) : unop (op x) = x 
Arguments: {α : Type u_1} (x : α)
Goal: MulOpposite.unop (MulOpposite.op x) = x
-- MulOpposite.op_unop
@[to_additive (attr := simp)]
theorem op_unop (x : αᵐᵒᵖ) : op (unop x) = x 
Arguments: {α : Type u_1} (x : αᵐᵒᵖ)
Goal: MulOpposite.op (MulOpposite.unop x) = x
-- AddOpposite.op_unop
@[to_additive (attr := simp)]
theorem op_unop (x : αᵐᵒᵖ) : op (unop x) = x 
Arguments: {α : Type u_1} (x : αᵃᵒᵖ)
Goal: AddOpposite.op (AddOpposite.unop x) = x
-- MulOpposite.op_comp_unop
@[to_additive (attr := simp)]
theorem op_comp_unop : (op : α → αᵐᵒᵖ) ∘ unop = id 
Arguments: {α : Type u_1}
Goal: MulOpposite.op ∘ MulOpposite.unop = id
-- AddOpposite.op_comp_unop
@[to_additive (attr := simp)]
theorem op_comp_unop : (op : α → αᵐᵒᵖ) ∘ unop = id 
Arguments: {α : Type u_1}
Goal: AddOpposite.op ∘ AddOpposite.unop = id
-- AddOpposite.unop_comp_op
@[to_additive (attr := simp)]
theorem unop_comp_op : (unop : αᵐᵒᵖ → α) ∘ op = id 
Arguments: {α : Type u_1}
Goal: AddOpposite.unop ∘ AddOpposite.op = id
-- MulOpposite.unop_comp_op
@[to_additive (attr := simp)]
theorem unop_comp_op : (unop : αᵐᵒᵖ → α) ∘ op = id 
Arguments: {α : Type u_1}
Goal: MulOpposite.unop ∘ MulOpposite.op = id
-- MulOpposite.opEquiv_apply
@[to_additive (attr := simps (config := .asFn) apply symm_apply)
  "The canonical bijection between `α` and `αᵃᵒᵖ`."]
def opEquiv : α ≃ αᵐᵒᵖ 
Arguments: {α : Type u_1}
Goal: ⇑MulOpposite.opEquiv = MulOpposite.op
-- AddOpposite.opEquiv_apply
@[to_additive (attr := simps (config := .asFn) apply symm_apply)
  "The canonical bijection between `α` and `αᵃᵒᵖ`."]
def opEquiv : α ≃ αᵐᵒᵖ 
Arguments: {α : Type u_1}
Goal: ⇑AddOpposite.opEquiv = AddOpposite.op
-- AddOpposite.opEquiv_symm_apply
@[to_additive (attr := simps (config := .asFn) apply symm_apply)
  "The canonical bijection between `α` and `αᵃᵒᵖ`."]
def opEquiv : α ≃ αᵐᵒᵖ 
Arguments: {α : Type u_1}
Goal: ⇑AddOpposite.opEquiv.symm = AddOpposite.unop
-- MulOpposite.opEquiv_symm_apply
@[to_additive (attr := simps (config := .asFn) apply symm_apply)
  "The canonical bijection between `α` and `αᵃᵒᵖ`."]
def opEquiv : α ≃ αᵐᵒᵖ 
Arguments: {α : Type u_1}
Goal: ⇑MulOpposite.opEquiv.symm = MulOpposite.unop
-- MulOpposite.op_bijective
@[to_additive]
theorem op_bijective : Bijective (op : α → αᵐᵒᵖ) 
Arguments: {α : Type u_1}
Goal: Function.Bijective MulOpposite.op
-- AddOpposite.op_bijective
@[to_additive]
theorem op_bijective : Bijective (op : α → αᵐᵒᵖ) 
Arguments: {α : Type u_1}
Goal: Function.Bijective AddOpposite.op
-- AddOpposite.unop_bijective
@[to_additive]
theorem unop_bijective : Bijective (unop : αᵐᵒᵖ → α) 
Arguments: {α : Type u_1}
Goal: Function.Bijective AddOpposite.unop
-- MulOpposite.unop_bijective
@[to_additive]
theorem unop_bijective : Bijective (unop : αᵐᵒᵖ → α) 
Arguments: {α : Type u_1}
Goal: Function.Bijective MulOpposite.unop
-- MulOpposite.op_injective
@[to_additive]
theorem op_injective : Injective (op : α → αᵐᵒᵖ) 
Arguments: {α : Type u_1}
Goal: Function.Injective MulOpposite.op
-- AddOpposite.op_injective
@[to_additive]
theorem op_injective : Injective (op : α → αᵐᵒᵖ) 
Arguments: {α : Type u_1}
Goal: Function.Injective AddOpposite.op
-- AddOpposite.op_surjective
@[to_additive]
theorem op_surjective : Surjective (op : α → αᵐᵒᵖ) 
Arguments: {α : Type u_1}
Goal: Function.Surjective AddOpposite.op
-- MulOpposite.op_surjective
@[to_additive]
theorem op_surjective : Surjective (op : α → αᵐᵒᵖ) 
Arguments: {α : Type u_1}
Goal: Function.Surjective MulOpposite.op
-- AddOpposite.unop_injective
@[to_additive]
theorem unop_injective : Injective (unop : αᵐᵒᵖ → α) 
Arguments: {α : Type u_1}
Goal: Function.Injective AddOpposite.unop
-- MulOpposite.unop_injective
@[to_additive]
theorem unop_injective : Injective (unop : αᵐᵒᵖ → α) 
Arguments: {α : Type u_1}
Goal: Function.Injective MulOpposite.unop
-- AddOpposite.unop_surjective
@[to_additive]
theorem unop_surjective : Surjective (unop : αᵐᵒᵖ → α) 
Arguments: {α : Type u_1}
Goal: Function.Surjective AddOpposite.unop
-- MulOpposite.unop_surjective
@[to_additive]
theorem unop_surjective : Surjective (unop : αᵐᵒᵖ → α) 
Arguments: {α : Type u_1}
Goal: Function.Surjective MulOpposite.unop
-- AddOpposite.op_inj
@[to_additive (attr := simp)]
theorem op_inj {x y : α} : op x = op y ↔ x = y 
Arguments: {α : Type u_1} {x y : α}
Goal: AddOpposite.op x = AddOpposite.op y ↔ x = y