ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- CategoryTheory.LocalizerMorphism.RightResolution.unop_w
@[simps]
def RightResolution.unop {X₂ : C₂ᵒᵖ} (L : Φ.op.RightResolution X₂) :
    Φ.LeftResolution X₂.unop 
Arguments: {C₁ : Type u_1} {C₂ : Type u_2} [CategoryTheory.Category.{u_5, u_1} C₁] [CategoryTheory.Category.{u_6, u_2} C₂] {W₁ : CategoryTheory.MorphismProperty C₁} {W₂ : CategoryTheory.MorphismProperty C₂} {Φ : CategoryTheory.LocalizerMorphism W₁ W₂} {X₂ : C₂ᵒᵖ} (L : Φ.op.RightResolution X₂)
Goal: L.unop.w = L.w.unop
-- CategoryTheory.LocalizerMorphism.RightResolution.unop_X₁
@[simps]
def RightResolution.unop {X₂ : C₂ᵒᵖ} (L : Φ.op.RightResolution X₂) :
    Φ.LeftResolution X₂.unop 
Arguments: {C₁ : Type u_1} {C₂ : Type u_2} [CategoryTheory.Category.{u_5, u_1} C₁] [CategoryTheory.Category.{u_6, u_2} C₂] {W₁ : CategoryTheory.MorphismProperty C₁} {W₂ : CategoryTheory.MorphismProperty C₂} {Φ : CategoryTheory.LocalizerMorphism W₁ W₂} {X₂ : C₂ᵒᵖ} (L : Φ.op.RightResolution X₂)
Goal: L.unop.X₁ = Opposite.unop L.X₁
-- CategoryTheory.LocalizerMorphism.nonempty_leftResolution_iff_op
lemma nonempty_leftResolution_iff_op (X₂ : C₂) :
    Nonempty (Φ.LeftResolution X₂) ↔ Nonempty (Φ.op.RightResolution (Opposite.op X₂)) 
Arguments: {C₁ : Type u_1} {C₂ : Type u_2} [CategoryTheory.Category.{u_6, u_1} C₁] [CategoryTheory.Category.{u_5, u_2} C₂] {W₁ : CategoryTheory.MorphismProperty C₁} {W₂ : CategoryTheory.MorphismProperty C₂} (Φ : CategoryTheory.LocalizerMorphism W₁ W₂) (X₂ : C₂)
Goal: Nonempty (Φ.LeftResolution X₂) ↔ Nonempty (Φ.op.RightResolution (Opposite.op X₂))
-- CategoryTheory.LocalizerMorphism.nonempty_rightResolution_iff_op
lemma nonempty_rightResolution_iff_op (X₂ : C₂) :
    Nonempty (Φ.RightResolution X₂) ↔ Nonempty (Φ.op.LeftResolution (Opposite.op X₂)) 
Arguments: {C₁ : Type u_1} {C₂ : Type u_2} [CategoryTheory.Category.{u_6, u_1} C₁] [CategoryTheory.Category.{u_5, u_2} C₂] {W₁ : CategoryTheory.MorphismProperty C₁} {W₂ : CategoryTheory.MorphismProperty C₂} (Φ : CategoryTheory.LocalizerMorphism W₁ W₂) (X₂ : C₂)
Goal: Nonempty (Φ.RightResolution X₂) ↔ Nonempty (Φ.op.LeftResolution (Opposite.op X₂))
-- CategoryTheory.LocalizerMorphism.hasLeftResolutions_iff_op
lemma hasLeftResolutions_iff_op : Φ.HasLeftResolutions ↔ Φ.op.HasRightResolutions 
Arguments: {C₁ : Type u_1} {C₂ : Type u_2} [CategoryTheory.Category.{u_5, u_1} C₁] [CategoryTheory.Category.{u_6, u_2} C₂] {W₁ : CategoryTheory.MorphismProperty C₁} {W₂ : CategoryTheory.MorphismProperty C₂} (Φ : CategoryTheory.LocalizerMorphism W₁ W₂)
Goal: Φ.HasLeftResolutions ↔ Φ.op.HasRightResolutions
-- CategoryTheory.LocalizerMorphism.hasRightResolutions_iff_op
lemma hasRightResolutions_iff_op : Φ.HasRightResolutions ↔ Φ.op.HasLeftResolutions 
Arguments: {C₁ : Type u_1} {C₂ : Type u_2} [CategoryTheory.Category.{u_5, u_1} C₁] [CategoryTheory.Category.{u_6, u_2} C₂] {W₁ : CategoryTheory.MorphismProperty C₁} {W₂ : CategoryTheory.MorphismProperty C₂} (Φ : CategoryTheory.LocalizerMorphism W₁ W₂)
Goal: Φ.HasRightResolutions ↔ Φ.op.HasLeftResolutions
-- CategoryTheory.LocalizerMorphism.LeftResolution.opFunctor_obj
@[simps]
def LeftResolution.opFunctor (X₂ : C₂) [W₁.IsMultiplicative] :
    (Φ.LeftResolution X₂)ᵒᵖ ⥤ Φ.op.RightResolution (Opposite.op X₂) 
Arguments: {C₁ : Type u_1} {C₂ : Type u_2} [CategoryTheory.Category.{u_5, u_1} C₁] [CategoryTheory.Category.{u_6, u_2} C₂] {W₁ : CategoryTheory.MorphismProperty C₁} {W₂ : CategoryTheory.MorphismProperty C₂} (Φ : CategoryTheory.LocalizerMorphism W₁ W₂) (X₂ : C₂) [W₁.IsMultiplicative] (L : (Φ.LeftResolution X₂)ᵒᵖ)
Goal: (CategoryTheory.LocalizerMorphism.LeftResolution.opFunctor Φ X₂).obj L = (Opposite.unop L).op
-- CategoryTheory.LocalizerMorphism.LeftResolution.opFunctor_map_f
@[simps]
def LeftResolution.opFunctor (X₂ : C₂) [W₁.IsMultiplicative] :
    (Φ.LeftResolution X₂)ᵒᵖ ⥤ Φ.op.RightResolution (Opposite.op X₂) 
Arguments: {C₁ : Type u_1} {C₂ : Type u_2} [CategoryTheory.Category.{u_5, u_1} C₁] [CategoryTheory.Category.{u_6, u_2} C₂] {W₁ : CategoryTheory.MorphismProperty C₁} {W₂ : CategoryTheory.MorphismProperty C₂} (Φ : CategoryTheory.LocalizerMorphism W₁ W₂) (X₂ : C₂) [W₁.IsMultiplicative]
Goal: ∀ {X Y : (Φ.LeftResolution X₂)ᵒᵖ} (φ : X ⟶ Y),
  ((CategoryTheory.LocalizerMorphism.LeftResolution.opFunctor Φ X₂).map φ).f = φ.unop.f.op
-- CategoryTheory.LocalizerMorphism.RightResolution.unopFunctor_map_f
@[simps]
def RightResolution.unopFunctor (X₂ : C₂ᵒᵖ) [W₁.IsMultiplicative] :
    (Φ.op.RightResolution X₂)ᵒᵖ ⥤ Φ.LeftResolution X₂.unop 
Arguments: {C₁ : Type u_1} {C₂ : Type u_2} [CategoryTheory.Category.{u_5, u_1} C₁] [CategoryTheory.Category.{u_6, u_2} C₂] {W₁ : CategoryTheory.MorphismProperty C₁} {W₂ : CategoryTheory.MorphismProperty C₂} (Φ : CategoryTheory.LocalizerMorphism W₁ W₂) (X₂ : C₂ᵒᵖ) [W₁.IsMultiplicative]
Goal: ∀ {X Y : (Φ.op.RightResolution X₂)ᵒᵖ} (φ : X ⟶ Y),
  ((CategoryTheory.LocalizerMorphism.RightResolution.unopFunctor Φ X₂).map φ).f = φ.unop.f.unop
-- CategoryTheory.LocalizerMorphism.RightResolution.unopFunctor_obj
@[simps]
def RightResolution.unopFunctor (X₂ : C₂ᵒᵖ) [W₁.IsMultiplicative] :
    (Φ.op.RightResolution X₂)ᵒᵖ ⥤ Φ.LeftResolution X₂.unop 
Arguments: {C₁ : Type u_1} {C₂ : Type u_2} [CategoryTheory.Category.{u_5, u_1} C₁] [CategoryTheory.Category.{u_6, u_2} C₂] {W₁ : CategoryTheory.MorphismProperty C₁} {W₂ : CategoryTheory.MorphismProperty C₂} (Φ : CategoryTheory.LocalizerMorphism W₁ W₂) (X₂ : C₂ᵒᵖ) [W₁.IsMultiplicative] (R : (Φ.op.RightResolution X₂)ᵒᵖ)
Goal: (CategoryTheory.LocalizerMorphism.RightResolution.unopFunctor Φ X₂).obj R = (Opposite.unop R).unop
-- CategoryTheory.LocalizerMorphism.LeftResolution.opEquivalence_functor
@[simps]
def LeftResolution.opEquivalence (X₂ : C₂) [W₁.IsMultiplicative] :
    (Φ.LeftResolution X₂)ᵒᵖ ≌ Φ.op.RightResolution (Opposite.op X₂) 
Arguments: {C₁ : Type u_1} {C₂ : Type u_2} [CategoryTheory.Category.{u_5, u_1} C₁] [CategoryTheory.Category.{u_6, u_2} C₂] {W₁ : CategoryTheory.MorphismProperty C₁} {W₂ : CategoryTheory.MorphismProperty C₂} (Φ : CategoryTheory.LocalizerMorphism W₁ W₂) (X₂ : C₂) [W₁.IsMultiplicative]
Goal: (CategoryTheory.LocalizerMorphism.LeftResolution.opEquivalence Φ X₂).functor =
  CategoryTheory.LocalizerMorphism.LeftResolution.opFunctor Φ X₂
-- CategoryTheory.LocalizerMorphism.LeftResolution.opEquivalence_counitIso
@[simps]
def LeftResolution.opEquivalence (X₂ : C₂) [W₁.IsMultiplicative] :
    (Φ.LeftResolution X₂)ᵒᵖ ≌ Φ.op.RightResolution (Opposite.op X₂) 
Arguments: {C₁ : Type u_1} {C₂ : Type u_2} [CategoryTheory.Category.{u_5, u_1} C₁] [CategoryTheory.Category.{u_6, u_2} C₂] {W₁ : CategoryTheory.MorphismProperty C₁} {W₂ : CategoryTheory.MorphismProperty C₂} (Φ : CategoryTheory.LocalizerMorphism W₁ W₂) (X₂ : C₂) [W₁.IsMultiplicative]
Goal: (CategoryTheory.LocalizerMorphism.LeftResolution.opEquivalence Φ X₂).counitIso =
  CategoryTheory.Iso.refl
    ((CategoryTheory.LocalizerMorphism.RightResolution.unopFunctor Φ (Opposite.op X₂)).rightOp.comp
      (CategoryTheory.LocalizerMorphism.LeftResolution.opFunctor Φ X₂))
-- CategoryTheory.LocalizerMorphism.LeftResolution.opEquivalence_inverse
@[simps]
def LeftResolution.opEquivalence (X₂ : C₂) [W₁.IsMultiplicative] :
    (Φ.LeftResolution X₂)ᵒᵖ ≌ Φ.op.RightResolution (Opposite.op X₂) 
Arguments: {C₁ : Type u_1} {C₂ : Type u_2} [CategoryTheory.Category.{u_5, u_1} C₁] [CategoryTheory.Category.{u_6, u_2} C₂] {W₁ : CategoryTheory.MorphismProperty C₁} {W₂ : CategoryTheory.MorphismProperty C₂} (Φ : CategoryTheory.LocalizerMorphism W₁ W₂) (X₂ : C₂) [W₁.IsMultiplicative]
Goal: (CategoryTheory.LocalizerMorphism.LeftResolution.opEquivalence Φ X₂).inverse =
  (CategoryTheory.LocalizerMorphism.RightResolution.unopFunctor Φ (Opposite.op X₂)).rightOp
-- CategoryTheory.LocalizerMorphism.LeftResolution.opEquivalence_unitIso
@[simps]
def LeftResolution.opEquivalence (X₂ : C₂) [W₁.IsMultiplicative] :
    (Φ.LeftResolution X₂)ᵒᵖ ≌ Φ.op.RightResolution (Opposite.op X₂) 
Arguments: {C₁ : Type u_1} {C₂ : Type u_2} [CategoryTheory.Category.{u_5, u_1} C₁] [CategoryTheory.Category.{u_6, u_2} C₂] {W₁ : CategoryTheory.MorphismProperty C₁} {W₂ : CategoryTheory.MorphismProperty C₂} (Φ : CategoryTheory.LocalizerMorphism W₁ W₂) (X₂ : C₂) [W₁.IsMultiplicative]
Goal: (CategoryTheory.LocalizerMorphism.LeftResolution.opEquivalence Φ X₂).unitIso =
  CategoryTheory.Iso.refl (CategoryTheory.Functor.id (Φ.LeftResolution X₂)ᵒᵖ)
-- CategoryTheory.LocalizerMorphism.essSurj_of_hasRightResolutions
lemma essSurj_of_hasRightResolutions [Φ.HasRightResolutions] : (Φ.functor ⋙ L₂).EssSurj 
Arguments: {C₁ : Type u_1} {C₂ : Type u_2} {D₂ : Type u_3} [CategoryTheory.Category.{u_5, u_1} C₁] [CategoryTheory.Category.{u_6, u_2} C₂] [CategoryTheory.Category.{u_7, u_3} D₂] {W₁ : CategoryTheory.MorphismProperty C₁} {W₂ : CategoryTheory.MorphismProperty C₂} (Φ : CategoryTheory.LocalizerMorphism W₁ W₂) (L₂ : CategoryTheory.Functor C₂ D₂) [L₂.IsLocalization W₂] [Φ.HasRightResolutions]
Goal: (Φ.functor.comp L₂).EssSurj
-- CategoryTheory.LocalizerMorphism.isIso_iff_of_hasRightResolutions
lemma isIso_iff_of_hasRightResolutions [Φ.HasRightResolutions] {F G : D₂ ⥤ H} (α : F ⟶ G) :
    IsIso α ↔ ∀ (X₁ : C₁), IsIso (α.app (L₂.obj (Φ.functor.obj X₁))) 
Arguments: {C₁ : Type u_1} {C₂ : Type u_2} {D₂ : Type u_3} {H : Type u_4} [CategoryTheory.Category.{u_5, u_1} C₁] [CategoryTheory.Category.{u_6, u_2} C₂] [CategoryTheory.Category.{u_7, u_3} D₂] [CategoryTheory.Category.{u_8, u_4} H] {W₁ : CategoryTheory.MorphismProperty C₁} {W₂ : CategoryTheory.MorphismProperty C₂} (Φ : CategoryTheory.LocalizerMorphism W₁ W₂) (L₂ : CategoryTheory.Functor C₂ D₂) [L₂.IsLocalization W₂] [Φ.HasRightResolutions] {F G : CategoryTheory.Functor D₂ H} (α : F ⟶ G)
Goal: CategoryTheory.IsIso α ↔ ∀ (X₁ : C₁), CategoryTheory.IsIso (α.app (L₂.obj (Φ.functor.obj X₁)))
-- CategoryTheory.LocalizerMorphism.essSurj_of_hasLeftResolutions
lemma essSurj_of_hasLeftResolutions [Φ.HasLeftResolutions] : (Φ.functor ⋙ L₂).EssSurj 
Arguments: {C₁ : Type u_1} {C₂ : Type u_2} {D₂ : Type u_3} [CategoryTheory.Category.{u_5, u_1} C₁] [CategoryTheory.Category.{u_6, u_2} C₂] [CategoryTheory.Category.{u_7, u_3} D₂] {W₁ : CategoryTheory.MorphismProperty C₁} {W₂ : CategoryTheory.MorphismProperty C₂} (Φ : CategoryTheory.LocalizerMorphism W₁ W₂) (L₂ : CategoryTheory.Functor C₂ D₂) [L₂.IsLocalization W₂] [Φ.HasLeftResolutions]
Goal: (Φ.functor.comp L₂).EssSurj
-- CategoryTheory.LocalizerMorphism.isIso_iff_of_hasLeftResolutions
lemma isIso_iff_of_hasLeftResolutions [Φ.HasLeftResolutions] {F G : D₂ ⥤ H} (α : F ⟶ G) :
    IsIso α ↔ ∀ (X₁ : C₁), IsIso (α.app (L₂.obj (Φ.functor.obj X₁))) 
Arguments: {C₁ : Type u_1} {C₂ : Type u_2} {D₂ : Type u_3} {H : Type u_4} [CategoryTheory.Category.{u_5, u_1} C₁] [CategoryTheory.Category.{u_6, u_2} C₂] [CategoryTheory.Category.{u_7, u_3} D₂] [CategoryTheory.Category.{u_8, u_4} H] {W₁ : CategoryTheory.MorphismProperty C₁} {W₂ : CategoryTheory.MorphismProperty C₂} (Φ : CategoryTheory.LocalizerMorphism W₁ W₂) (L₂ : CategoryTheory.Functor C₂ D₂) [L₂.IsLocalization W₂] [Φ.HasLeftResolutions] {F G : CategoryTheory.Functor D₂ H} (α : F ⟶ G)
Goal: CategoryTheory.IsIso α ↔ ∀ (X₁ : C₁), CategoryTheory.IsIso (α.app (L₂.obj (Φ.functor.obj X₁)))
-- CategoryTheory.LocalizerMorphism.IsRightDerivabilityStructure.guitartExact'
class IsRightDerivabilityStructure : Prop where
  hasRightResolutions : Φ.HasRightResolutions := by infer_instance
  guitartExact' : TwoSquare.GuitartExact ((Φ.catCommSq W₁.Q W₂.Q).iso).hom
Arguments: {C₁ : Type u₁} {C₂ : Type u₂} [CategoryTheory.Category.{v₁, u₁} C₁] [CategoryTheory.Category.{v₂, u₂} C₂] {W₁ : CategoryTheory.MorphismProperty C₁} {W₂ : CategoryTheory.MorphismProperty C₂} {Φ : CategoryTheory.LocalizerMorphism W₁ W₂} [self : Φ.IsRightDerivabilityStructure]
Goal: CategoryTheory.TwoSquare.GuitartExact
  (CategoryTheory.CatCommSq.iso Φ.functor W₁.Q W₂.Q (Φ.localizedFunctor W₁.Q W₂.Q)).hom
-- CategoryTheory.LocalizerMorphism.IsRightDerivabilityStructure.hasRightResolutions
class IsRightDerivabilityStructure : Prop where
  hasRightResolutions : Φ.HasRightResolutions := by infer_instance
  guitartExact' : TwoSquare.GuitartExact ((Φ.catCommSq W₁.Q W₂.Q).iso).hom
Arguments: {C₁ : Type u₁} {C₂ : Type u₂} [CategoryTheory.Category.{v₁, u₁} C₁] [CategoryTheory.Category.{v₂, u₂} C₂] {W₁ : CategoryTheory.MorphismProperty C₁} {W₂ : CategoryTheory.MorphismProperty C₂} {Φ : CategoryTheory.LocalizerMorphism W₁ W₂} [self : Φ.IsRightDerivabilityStructure]
Goal: Φ.HasRightResolutions
-- CategoryTheory.LocalizerMorphism.isRightDerivabilityStructure_iff
lemma isRightDerivabilityStructure_iff [Φ.HasRightResolutions] (e : Φ.functor ⋙ L₂ ≅ L₁ ⋙ F) :
    Φ.IsRightDerivabilityStructure ↔ TwoSquare.GuitartExact e.hom 
Arguments: {C₁ : Type u₁} {C₂ : Type u₂} [CategoryTheory.Category.{v₁, u₁} C₁] [CategoryTheory.Category.{v₂, u₂} C₂] {W₁ : CategoryTheory.MorphismProperty C₁} {W₂ : CategoryTheory.MorphismProperty C₂} (Φ : CategoryTheory.LocalizerMorphism W₁ W₂) {D₁ : Type u_1} {D₂ : Type u_2} [CategoryTheory.Category.{u_4, u_1} D₁] [CategoryTheory.Category.{u_3, u_2} D₂] (L₁ : CategoryTheory.Functor C₁ D₁) (L₂ : CategoryTheory.Functor C₂ D₂) [L₁.IsLocalization W₁] [L₂.IsLocalization W₂] (F : CategoryTheory.Functor D₁ D₂) [Φ.HasRightResolutions] (e : Φ.functor.comp L₂ ≅ L₁.comp F)
Goal: Φ.IsRightDerivabilityStructure ↔ CategoryTheory.TwoSquare.GuitartExact e.hom
-- CategoryTheory.LocalizerMorphism.guitartExact_of_isRightDerivabilityStructure'
lemma guitartExact_of_isRightDerivabilityStructure' [h : Φ.IsRightDerivabilityStructure]
    (e : Φ.functor ⋙ L₂ ≅ L₁ ⋙ F) : TwoSquare.GuitartExact e.hom 
Arguments: {C₁ : Type u₁} {C₂ : Type u₂} [CategoryTheory.Category.{v₁, u₁} C₁] [CategoryTheory.Category.{v₂, u₂} C₂] {W₁ : CategoryTheory.MorphismProperty C₁} {W₂ : CategoryTheory.MorphismProperty C₂} (Φ : CategoryTheory.LocalizerMorphism W₁ W₂) {D₁ : Type u_1} {D₂ : Type u_2} [CategoryTheory.Category.{u_4, u_1} D₁] [CategoryTheory.Category.{u_3, u_2} D₂] (L₁ : CategoryTheory.Functor C₁ D₁) (L₂ : CategoryTheory.Functor C₂ D₂) [L₁.IsLocalization W₁] [L₂.IsLocalization W₂] (F : CategoryTheory.Functor D₁ D₂) [h : Φ.IsRightDerivabilityStructure] (e : Φ.functor.comp L₂ ≅ L₁.comp F)
Goal: CategoryTheory.TwoSquare.GuitartExact e.hom
-- CategoryTheory.LocalizerMorphism.guitartExact_of_isRightDerivabilityStructure
lemma guitartExact_of_isRightDerivabilityStructure [Φ.IsRightDerivabilityStructure] :
    TwoSquare.GuitartExact ((Φ.catCommSq L₁ L₂).iso).hom 
Arguments: {C₁ : Type u₁} {C₂ : Type u₂} [CategoryTheory.Category.{v₁, u₁} C₁] [CategoryTheory.Category.{v₂, u₂} C₂] {W₁ : CategoryTheory.MorphismProperty C₁} {W₂ : CategoryTheory.MorphismProperty C₂} (Φ : CategoryTheory.LocalizerMorphism W₁ W₂) {D₁ : Type u_1} {D₂ : Type u_2} [CategoryTheory.Category.{u_3, u_1} D₁] [CategoryTheory.Category.{u_4, u_2} D₂] (L₁ : CategoryTheory.Functor C₁ D₁) (L₂ : CategoryTheory.Functor C₂ D₂) [L₁.IsLocalization W₁] [L₂.IsLocalization W₂] [Φ.IsRightDerivabilityStructure]
Goal: CategoryTheory.TwoSquare.GuitartExact (CategoryTheory.CatCommSq.iso Φ.functor L₁ L₂ (Φ.localizedFunctor L₁ L₂)).hom
-- CategoryTheory.Functor.empty_ext'
theorem empty_ext' (F G : Discrete.{w} PEmpty ⥤ C) : F = G 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] (F G : CategoryTheory.Functor (CategoryTheory.Discrete PEmpty.{w + 1}) C)
Goal: F = G
-- CategoryTheory.oppositeShiftFunctorZero_inv_app
lemma oppositeShiftFunctorZero_inv_app (X : OppositeShift C A) :
    (shiftFunctorZero (OppositeShift C A) A).inv.app X =
      ((shiftFunctorZero C A).hom.app X.unop).op 
Arguments: (C : Type u_1) [CategoryTheory.Category.{u_3, u_1} C] (A : Type u_2) [AddMonoid A] [CategoryTheory.HasShift C A] (X : CategoryTheory.OppositeShift C A)
Goal: (CategoryTheory.shiftFunctorZero (CategoryTheory.OppositeShift C A) A).inv.app X =
  ((CategoryTheory.shiftFunctorZero C A).hom.app (Opposite.unop X)).op
-- CategoryTheory.oppositeShiftFunctorZero_hom_app
lemma oppositeShiftFunctorZero_hom_app (X : OppositeShift C A) :
    (shiftFunctorZero (OppositeShift C A) A).hom.app X =
      ((shiftFunctorZero C A).inv.app X.unop).op 
Arguments: (C : Type u_1) [CategoryTheory.Category.{u_3, u_1} C] (A : Type u_2) [AddMonoid A] [CategoryTheory.HasShift C A] (X : CategoryTheory.OppositeShift C A)
Goal: (CategoryTheory.shiftFunctorZero (CategoryTheory.OppositeShift C A) A).hom.app X =
  ((CategoryTheory.shiftFunctorZero C A).inv.app (Opposite.unop X)).op
-- CategoryTheory.oppositeShiftFunctorAdd_inv_app
lemma oppositeShiftFunctorAdd_inv_app :
    (shiftFunctorAdd (OppositeShift C A) a b).inv.app X =
      ((shiftFunctorAdd C a b).hom.app X.unop).op 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] {A : Type u_2} [AddMonoid A] [CategoryTheory.HasShift C A] (X : CategoryTheory.OppositeShift C A) (a b : A)
Goal: (CategoryTheory.shiftFunctorAdd (CategoryTheory.OppositeShift C A) a b).inv.app X =
  ((CategoryTheory.shiftFunctorAdd C a b).hom.app (Opposite.unop X)).op
-- CategoryTheory.oppositeShiftFunctorAdd_hom_app
lemma oppositeShiftFunctorAdd_hom_app :
    (shiftFunctorAdd (OppositeShift C A) a b).hom.app X =
      ((shiftFunctorAdd C a b).inv.app X.unop).op 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] {A : Type u_2} [AddMonoid A] [CategoryTheory.HasShift C A] (X : CategoryTheory.OppositeShift C A) (a b : A)
Goal: (CategoryTheory.shiftFunctorAdd (CategoryTheory.OppositeShift C A) a b).hom.app X =
  ((CategoryTheory.shiftFunctorAdd C a b).inv.app (Opposite.unop X)).op
-- CategoryTheory.oppositeShiftFunctorAdd'_inv_app
lemma oppositeShiftFunctorAdd'_inv_app :
    (shiftFunctorAdd' (OppositeShift C A) a b c h).inv.app X =
      ((shiftFunctorAdd' C a b c h).hom.app X.unop).op 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] {A : Type u_2} [AddMonoid A] [CategoryTheory.HasShift C A] (X : CategoryTheory.OppositeShift C A) (a b c : A) (h : a + b = c)
Goal: (CategoryTheory.shiftFunctorAdd' (CategoryTheory.OppositeShift C A) a b c h).inv.app X =
  ((CategoryTheory.shiftFunctorAdd' C a b c h).hom.app (Opposite.unop X)).op
-- CategoryTheory.oppositeShiftFunctorAdd'_hom_app
lemma oppositeShiftFunctorAdd'_hom_app :
    (shiftFunctorAdd' (OppositeShift C A) a b c h).hom.app X =
      ((shiftFunctorAdd' C a b c h).inv.app X.unop).op 
Arguments: {C : Type u_1} [CategoryTheory.Category.{u_3, u_1} C] {A : Type u_2} [AddMonoid A] [CategoryTheory.HasShift C A] (X : CategoryTheory.OppositeShift C A) (a b c : A) (h : a + b = c)
Goal: (CategoryTheory.shiftFunctorAdd' (CategoryTheory.OppositeShift C A) a b c h).hom.app X =
  ((CategoryTheory.shiftFunctorAdd' C a b c h).inv.app (Opposite.unop X)).op
-- CategoryTheory.LocalizerMorphism.homMap_map
@[simp]
lemma homMap_map (f : X ⟶ Y) :
    Φ.homMap L₁ L₂ (L₁.map f) = L₂.map (Φ.functor.map f) 
Arguments: {C₁ : Type u_2} {C₂ : Type u_3} {D₁ : Type u_5} {D₂ : Type u_6} [CategoryTheory.Category.{u_8, u_2} C₁] [CategoryTheory.Category.{u_10, u_3} C₂] [CategoryTheory.Category.{u_11, u_5} D₁] [CategoryTheory.Category.{u_9, u_6} D₂] {W₁ : CategoryTheory.MorphismProperty C₁} {W₂ : CategoryTheory.MorphismProperty C₂} (Φ : CategoryTheory.LocalizerMorphism W₁ W₂) (L₁ : CategoryTheory.Functor C₁ D₁) [L₁.IsLocalization W₁] (L₂ : CategoryTheory.Functor C₂ D₂) [L₂.IsLocalization W₂] {X Y : C₁} (f : X ⟶ Y)
Goal: Φ.homMap L₁ L₂ (L₁.map f) = L₂.map (Φ.functor.map f)
-- CategoryTheory.LocalizerMorphism.homMap_id
@[simp]
lemma homMap_id  :
    Φ.homMap L₁ L₂ (𝟙 (L₁.obj X)) = 𝟙 (L₂.obj (Φ.functor.obj X)) 
Arguments: {C₁ : Type u_2} {C₂ : Type u_3} {D₁ : Type u_5} {D₂ : Type u_6} [CategoryTheory.Category.{u_10, u_2} C₁] [CategoryTheory.Category.{u_9, u_3} C₂] [CategoryTheory.Category.{u_11, u_5} D₁] [CategoryTheory.Category.{u_8, u_6} D₂] {W₁ : CategoryTheory.MorphismProperty C₁} {W₂ : CategoryTheory.MorphismProperty C₂} (Φ : CategoryTheory.LocalizerMorphism W₁ W₂) (L₁ : CategoryTheory.Functor C₁ D₁) [L₁.IsLocalization W₁] (L₂ : CategoryTheory.Functor C₂ D₂) [L₂.IsLocalization W₂] (X : C₁)
Goal: Φ.homMap L₁ L₂ (CategoryTheory.CategoryStruct.id (L₁.obj X)) =
  CategoryTheory.CategoryStruct.id (L₂.obj (Φ.functor.obj X))
-- CategoryTheory.LocalizerMorphism.homMap_comp
@[reassoc]
lemma homMap_comp (f : L₁.obj X ⟶ L₁.obj Y) (g : L₁.obj Y ⟶ L₁.obj Z) :
    Φ.homMap L₁ L₂ (f ≫ g) = Φ.homMap L₁ L₂ f ≫ Φ.homMap L₁ L₂ g 
Arguments: {C₁ : Type u_2} {C₂ : Type u_3} {D₁ : Type u_5} {D₂ : Type u_6} [CategoryTheory.Category.{u_9, u_2} C₁] [CategoryTheory.Category.{u_11, u_3} C₂] [CategoryTheory.Category.{u_8, u_5} D₁] [CategoryTheory.Category.{u_10, u_6} D₂] {W₁ : CategoryTheory.MorphismProperty C₁} {W₂ : CategoryTheory.MorphismProperty C₂} (Φ : CategoryTheory.LocalizerMorphism W₁ W₂) (L₁ : CategoryTheory.Functor C₁ D₁) [L₁.IsLocalization W₁] (L₂ : CategoryTheory.Functor C₂ D₂) [L₂.IsLocalization W₂] {X Y Z : C₁} (f : L₁.obj X ⟶ L₁.obj Y) (g : L₁.obj Y ⟶ L₁.obj Z)
Goal: Φ.homMap L₁ L₂ (CategoryTheory.CategoryStruct.comp f g) =
  CategoryTheory.CategoryStruct.comp (Φ.homMap L₁ L₂ f) (Φ.homMap L₁ L₂ g)
-- CategoryTheory.LocalizerMorphism.homMap_apply
@[reassoc]
lemma homMap_apply (G : D₁ ⥤ D₂) (e : Φ.functor ⋙ L₂ ≅ L₁ ⋙ G) (f : L₁.obj X ⟶ L₁.obj Y) :
    Φ.homMap L₁ L₂ f = e.hom.app X ≫ G.map f ≫ e.inv.app Y 
Arguments: {C₁ : Type u_2} {C₂ : Type u_3} {D₁ : Type u_5} {D₂ : Type u_6} [CategoryTheory.Category.{u_10, u_2} C₁] [CategoryTheory.Category.{u_11, u_3} C₂] [CategoryTheory.Category.{u_8, u_5} D₁] [CategoryTheory.Category.{u_9, u_6} D₂] {W₁ : CategoryTheory.MorphismProperty C₁} {W₂ : CategoryTheory.MorphismProperty C₂} (Φ : CategoryTheory.LocalizerMorphism W₁ W₂) (L₁ : CategoryTheory.Functor C₁ D₁) [L₁.IsLocalization W₁] (L₂ : CategoryTheory.Functor C₂ D₂) [L₂.IsLocalization W₂] {X Y : C₁} (G : CategoryTheory.Functor D₁ D₂) (e : Φ.functor.comp L₂ ≅ L₁.comp G) (f : L₁.obj X ⟶ L₁.obj Y)
Goal: Φ.homMap L₁ L₂ f =
  CategoryTheory.CategoryStruct.comp (e.hom.app X) (CategoryTheory.CategoryStruct.comp (G.map f) (e.inv.app Y))
-- CategoryTheory.LocalizerMorphism.id_homMap
@[simp]
lemma id_homMap (f : L₁.obj X ⟶ L₁.obj Y) :
    (id W₁).homMap L₁ L₁ f = f 
Arguments: {C₁ : Type u_2} {D₁ : Type u_5} [CategoryTheory.Category.{u_9, u_2} C₁] [CategoryTheory.Category.{u_8, u_5} D₁] {W₁ : CategoryTheory.MorphismProperty C₁} (L₁ : CategoryTheory.Functor C₁ D₁) [L₁.IsLocalization W₁] {X Y : C₁} (f : L₁.obj X ⟶ L₁.obj Y)
Goal: (CategoryTheory.LocalizerMorphism.id W₁).homMap L₁ L₁ f = f
-- CategoryTheory.LocalizerMorphism.homMap_homMap
@[simp]
lemma homMap_homMap (f : L₁.obj X ⟶ L₁.obj Y) :
    Ψ.homMap L₂ L₃ (Φ.homMap L₁ L₂ f) = (Φ.comp Ψ).homMap L₁ L₃ f 
Arguments: {C₁ : Type u_2} {C₂ : Type u_3} {C₃ : Type u_4} {D₁ : Type u_5} {D₂ : Type u_6} {D₃ : Type u_7} [CategoryTheory.Category.{u_9, u_2} C₁] [CategoryTheory.Category.{u_12, u_3} C₂] [CategoryTheory.Category.{u_11, u_4} C₃] [CategoryTheory.Category.{u_8, u_5} D₁] [CategoryTheory.Category.{u_13, u_6} D₂] [CategoryTheory.Category.{u_10, u_7} D₃] {W₁ : CategoryTheory.MorphismProperty C₁} {W₂ : CategoryTheory.MorphismProperty C₂} {W₃ : CategoryTheory.MorphismProperty C₃} (Φ : CategoryTheory.LocalizerMorphism W₁ W₂) (Ψ : CategoryTheory.LocalizerMorphism W₂ W₃) (L₁ : CategoryTheory.Functor C₁ D₁) [L₁.IsLocalization W₁] (L₂ : CategoryTheory.Functor C₂ D₂) [L₂.IsLocalization W₂] (L₃ : CategoryTheory.Functor C₃ D₃) [L₃.IsLocalization W₃] {X Y : C₁} (f : L₁.obj X ⟶ L₁.obj Y)
Goal: Ψ.homMap L₂ L₃ (Φ.homMap L₁ L₂ f) = (Φ.comp Ψ).homMap L₁ L₃ f
-- CategoryTheory.Localization.homEquiv_apply
@[simps (config := .lemmasOnly) apply]
noncomputable def homEquiv :
    (L₁.obj X ⟶ L₁.obj Y) ≃ (L₂.obj X ⟶ L₂.obj Y) 
Arguments: {C : Type u_1} {D₁ : Type u_5} {D₂ : Type u_6} [CategoryTheory.Category.{u_8, u_1} C] [CategoryTheory.Category.{u_9, u_5} D₁] [CategoryTheory.Category.{u_10, u_6} D₂] (W : CategoryTheory.MorphismProperty C) (L₁ : CategoryTheory.Functor C D₁) [L₁.IsLocalization W] (L₂ : CategoryTheory.Functor C D₂) [L₂.IsLocalization W] {X Y : C} (f : L₁.obj X ⟶ L₁.obj Y)
Goal: (CategoryTheory.Localization.homEquiv W L₁ L₂) f = (CategoryTheory.LocalizerMorphism.id W).homMap L₁ L₂ f