ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- CategoryTheory.CosimplicialObject.augment_left
@[simps]
def augment (X : CosimplicialObject C) (X₀ : C) (f : X₀ ⟶ X.obj [0])
    (w : ∀ (i : SimplexCategory) (g₁ g₂ : ([0] : SimplexCategory) ⟶ i),
      f ≫ X.map g₁ = f ≫ X.map g₂) : CosimplicialObject.Augmented C 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] (X : CategoryTheory.CosimplicialObject C) (X₀ : C) (f : X₀ ⟶ X.obj (SimplexCategory.mk 0)) (w : ∀ (i : SimplexCategory) (g₁ g₂ : SimplexCategory.mk 0 ⟶ i), f ≫ X.map g₁ = f ≫ X.map g₂)
Goal: (X.augment X₀ f w).left = X₀
-- CategoryTheory.CosimplicialObject.augment_hom_app
@[simps]
def augment (X : CosimplicialObject C) (X₀ : C) (f : X₀ ⟶ X.obj [0])
    (w : ∀ (i : SimplexCategory) (g₁ g₂ : ([0] : SimplexCategory) ⟶ i),
      f ≫ X.map g₁ = f ≫ X.map g₂) : CosimplicialObject.Augmented C 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] (X : CategoryTheory.CosimplicialObject C) (X₀ : C) (f : X₀ ⟶ X.obj (SimplexCategory.mk 0)) (w : ∀ (i : SimplexCategory) (g₁ g₂ : SimplexCategory.mk 0 ⟶ i), f ≫ X.map g₁ = f ≫ X.map g₂) (i : SimplexCategory)
Goal: (X.augment X₀ f w).hom.app i = f ≫ X.map ((SimplexCategory.mk 0).const i 0)
-- CategoryTheory.CosimplicialObject.augment_hom_zero
theorem augment_hom_zero (X : CosimplicialObject C) (X₀ : C) (f : X₀ ⟶ X.obj [0]) (w) :
    (X.augment X₀ f w).hom.app [0] = f 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] (X : CategoryTheory.CosimplicialObject C) (X₀ : C) (f : X₀ ⟶ X.obj (SimplexCategory.mk 0)) (w : ∀ (i : SimplexCategory) (g₁ g₂ : SimplexCategory.mk 0 ⟶ i), f ≫ X.map g₁ = f ≫ X.map g₂)
Goal: (X.augment X₀ f w).hom.app (SimplexCategory.mk 0) = f
-- CategoryTheory.simplicialCosimplicialEquiv_unitIso_inv_app
@[simps!]
def simplicialCosimplicialEquiv : (SimplicialObject C)ᵒᵖ ≌ CosimplicialObject Cᵒᵖ 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C] (X : (SimplexCategoryᵒᵖ ⥤ C)ᵒᵖ)
Goal: (CategoryTheory.simplicialCosimplicialEquiv C).unitIso.inv.app X = (Opposite.unop X).rightOpLeftOpIso.inv.op
-- CategoryTheory.simplicialCosimplicialEquiv_functor_obj_obj
@[simps!]
def simplicialCosimplicialEquiv : (SimplicialObject C)ᵒᵖ ≌ CosimplicialObject Cᵒᵖ 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C] (F : (SimplexCategoryᵒᵖ ⥤ C)ᵒᵖ) (X : SimplexCategory)
Goal: ((CategoryTheory.simplicialCosimplicialEquiv C).functor.obj F).obj X =
  Opposite.op ((Opposite.unop F).obj (Opposite.op X))
-- CategoryTheory.simplicialCosimplicialEquiv_counitIso_hom_app_app
@[simps!]
def simplicialCosimplicialEquiv : (SimplicialObject C)ᵒᵖ ≌ CosimplicialObject Cᵒᵖ 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C] (X : SimplexCategory ⥤ Cᵒᵖ)
Goal: ∀ (X_1 : SimplexCategory), ((CategoryTheory.simplicialCosimplicialEquiv C).counitIso.hom.app X).app X_1 = 𝟙 (X.obj X_1)
-- CategoryTheory.simplicialCosimplicialEquiv_inverse_obj
@[simps!]
def simplicialCosimplicialEquiv : (SimplicialObject C)ᵒᵖ ≌ CosimplicialObject Cᵒᵖ 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C] (F : SimplexCategory ⥤ Cᵒᵖ)
Goal: (CategoryTheory.simplicialCosimplicialEquiv C).inverse.obj F = Opposite.op F.leftOp
-- CategoryTheory.simplicialCosimplicialEquiv_inverse_map
@[simps!]
def simplicialCosimplicialEquiv : (SimplicialObject C)ᵒᵖ ≌ CosimplicialObject Cᵒᵖ 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C]
Goal: ∀ {X Y : SimplexCategory ⥤ Cᵒᵖ} (η : X ⟶ Y),
  (CategoryTheory.simplicialCosimplicialEquiv C).inverse.map η = (CategoryTheory.NatTrans.leftOp η).op
-- CategoryTheory.simplicialCosimplicialEquiv_unitIso_hom_app
@[simps!]
def simplicialCosimplicialEquiv : (SimplicialObject C)ᵒᵖ ≌ CosimplicialObject Cᵒᵖ 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C] (X : (SimplexCategoryᵒᵖ ⥤ C)ᵒᵖ)
Goal: (CategoryTheory.simplicialCosimplicialEquiv C).unitIso.hom.app X = (Opposite.unop X).rightOpLeftOpIso.hom.op
-- CategoryTheory.simplicialCosimplicialEquiv_functor_obj_map
@[simps!]
def simplicialCosimplicialEquiv : (SimplicialObject C)ᵒᵖ ≌ CosimplicialObject Cᵒᵖ 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C] (F : (SimplexCategoryᵒᵖ ⥤ C)ᵒᵖ)
Goal: ∀ {X Y : SimplexCategory} (f : X ⟶ Y),
  ((CategoryTheory.simplicialCosimplicialEquiv C).functor.obj F).map f = ((Opposite.unop F).map f.op).op
-- CategoryTheory.simplicialCosimplicialEquiv_counitIso_inv_app_app
@[simps!]
def simplicialCosimplicialEquiv : (SimplicialObject C)ᵒᵖ ≌ CosimplicialObject Cᵒᵖ 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C] (X : SimplexCategory ⥤ Cᵒᵖ)
Goal: ∀ (X_1 : SimplexCategory), ((CategoryTheory.simplicialCosimplicialEquiv C).counitIso.inv.app X).app X_1 = 𝟙 (X.obj X_1)
-- CategoryTheory.simplicialCosimplicialEquiv_functor_map_app
@[simps!]
def simplicialCosimplicialEquiv : (SimplicialObject C)ᵒᵖ ≌ CosimplicialObject Cᵒᵖ 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C]
Goal: ∀ {X Y : (SimplexCategoryᵒᵖ ⥤ C)ᵒᵖ} (η : X ⟶ Y) (X_1 : SimplexCategory),
  ((CategoryTheory.simplicialCosimplicialEquiv C).functor.map η).app X_1 = (η.unop.app (Opposite.op X_1)).op
-- CategoryTheory.cosimplicialSimplicialEquiv_unitIso_inv_app
@[simps!]
def cosimplicialSimplicialEquiv : (CosimplicialObject C)ᵒᵖ ≌ SimplicialObject Cᵒᵖ 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C] (X : (SimplexCategory ⥤ C)ᵒᵖ)
Goal: (CategoryTheory.cosimplicialSimplicialEquiv C).unitIso.inv.app X = (Opposite.unop X).opUnopIso.inv.op
-- CategoryTheory.cosimplicialSimplicialEquiv_counitIso_hom_app_app
@[simps!]
def cosimplicialSimplicialEquiv : (CosimplicialObject C)ᵒᵖ ≌ SimplicialObject Cᵒᵖ 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C] (X : SimplexCategoryᵒᵖ ⥤ Cᵒᵖ)
Goal: ∀ (X_1 : SimplexCategoryᵒᵖ),
  ((CategoryTheory.cosimplicialSimplicialEquiv C).counitIso.hom.app X).app X_1 = 𝟙 (X.obj X_1)
-- CategoryTheory.cosimplicialSimplicialEquiv_functor_obj_obj
@[simps!]
def cosimplicialSimplicialEquiv : (CosimplicialObject C)ᵒᵖ ≌ SimplicialObject Cᵒᵖ 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C] (F : (SimplexCategory ⥤ C)ᵒᵖ) (X : SimplexCategoryᵒᵖ)
Goal: ((CategoryTheory.cosimplicialSimplicialEquiv C).functor.obj F).obj X =
  Opposite.op ((Opposite.unop F).obj (Opposite.unop X))
-- CategoryTheory.cosimplicialSimplicialEquiv_functor_map_app
@[simps!]
def cosimplicialSimplicialEquiv : (CosimplicialObject C)ᵒᵖ ≌ SimplicialObject Cᵒᵖ 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C]
Goal: ∀ {X Y : (SimplexCategory ⥤ C)ᵒᵖ} (α : X ⟶ Y) (X_1 : SimplexCategoryᵒᵖ),
  ((CategoryTheory.cosimplicialSimplicialEquiv C).functor.map α).app X_1 = (α.unop.app (Opposite.unop X_1)).op
-- CategoryTheory.cosimplicialSimplicialEquiv_inverse_map
@[simps!]
def cosimplicialSimplicialEquiv : (CosimplicialObject C)ᵒᵖ ≌ SimplicialObject Cᵒᵖ 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C]
Goal: ∀ {X Y : SimplexCategoryᵒᵖ ⥤ Cᵒᵖ} (α : X ⟶ Y),
  (CategoryTheory.cosimplicialSimplicialEquiv C).inverse.map α =
    Quiver.Hom.op { app := fun X_1 => (α.app (Opposite.op X_1)).unop, naturality := ⋯ }
-- CategoryTheory.cosimplicialSimplicialEquiv_functor_obj_map
@[simps!]
def cosimplicialSimplicialEquiv : (CosimplicialObject C)ᵒᵖ ≌ SimplicialObject Cᵒᵖ 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C] (F : (SimplexCategory ⥤ C)ᵒᵖ)
Goal: ∀ {X Y : SimplexCategoryᵒᵖ} (f : X ⟶ Y),
  ((CategoryTheory.cosimplicialSimplicialEquiv C).functor.obj F).map f = ((Opposite.unop F).map f.unop).op
-- CategoryTheory.cosimplicialSimplicialEquiv_inverse_obj
@[simps!]
def cosimplicialSimplicialEquiv : (CosimplicialObject C)ᵒᵖ ≌ SimplicialObject Cᵒᵖ 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C] (F : SimplexCategoryᵒᵖ ⥤ Cᵒᵖ)
Goal: (CategoryTheory.cosimplicialSimplicialEquiv C).inverse.obj F = Opposite.op F.unop
-- CategoryTheory.cosimplicialSimplicialEquiv_unitIso_hom_app
@[simps!]
def cosimplicialSimplicialEquiv : (CosimplicialObject C)ᵒᵖ ≌ SimplicialObject Cᵒᵖ 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C] (X : (SimplexCategory ⥤ C)ᵒᵖ)
Goal: (CategoryTheory.cosimplicialSimplicialEquiv C).unitIso.hom.app X = (Opposite.unop X).opUnopIso.hom.op
-- CategoryTheory.cosimplicialSimplicialEquiv_counitIso_inv_app_app
@[simps!]
def cosimplicialSimplicialEquiv : (CosimplicialObject C)ᵒᵖ ≌ SimplicialObject Cᵒᵖ 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C] (X : SimplexCategoryᵒᵖ ⥤ Cᵒᵖ)
Goal: ∀ (X_1 : SimplexCategoryᵒᵖ),
  ((CategoryTheory.cosimplicialSimplicialEquiv C).counitIso.inv.app X).app X_1 = 𝟙 (X.obj X_1)
-- CategoryTheory.SimplicialObject.Augmented.rightOp_right_map
@[simps!]
def SimplicialObject.Augmented.rightOp (X : SimplicialObject.Augmented C) :
    CosimplicialObject.Augmented Cᵒᵖ 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] (X : CategoryTheory.SimplicialObject.Augmented C)
Goal: ∀ {X_1 Y : SimplexCategory} (f : X_1 ⟶ Y), X.rightOp.right.map f = (X.left.map f.op).op
-- CategoryTheory.SimplicialObject.Augmented.rightOp_left
@[simps!]
def SimplicialObject.Augmented.rightOp (X : SimplicialObject.Augmented C) :
    CosimplicialObject.Augmented Cᵒᵖ 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] (X : CategoryTheory.SimplicialObject.Augmented C)
Goal: X.rightOp.left = Opposite.op X.right
-- CategoryTheory.SimplicialObject.Augmented.rightOp_hom_app
@[simps!]
def SimplicialObject.Augmented.rightOp (X : SimplicialObject.Augmented C) :
    CosimplicialObject.Augmented Cᵒᵖ 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] (X : CategoryTheory.SimplicialObject.Augmented C)
Goal: ∀ (X_1 : SimplexCategory), X.rightOp.hom.app X_1 = (X.hom.app (Opposite.op X_1)).op
-- CategoryTheory.SimplicialObject.Augmented.rightOp_right_obj
@[simps!]
def SimplicialObject.Augmented.rightOp (X : SimplicialObject.Augmented C) :
    CosimplicialObject.Augmented Cᵒᵖ 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] (X : CategoryTheory.SimplicialObject.Augmented C)
Goal: ∀ (X_1 : SimplexCategory), X.rightOp.right.obj X_1 = Opposite.op (X.left.obj (Opposite.op X_1))
-- CategoryTheory.CosimplicialObject.Augmented.leftOp_left_map
@[simps!]
def CosimplicialObject.Augmented.leftOp (X : CosimplicialObject.Augmented Cᵒᵖ) :
    SimplicialObject.Augmented C 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] (X : CategoryTheory.CosimplicialObject.Augmented Cᵒᵖ)
Goal: ∀ {X_1 Y : SimplexCategoryᵒᵖ} (f : X_1 ⟶ Y), X.leftOp.left.map f = (X.right.map f.unop).unop
-- CategoryTheory.CosimplicialObject.Augmented.leftOp_hom_app
@[simps!]
def CosimplicialObject.Augmented.leftOp (X : CosimplicialObject.Augmented Cᵒᵖ) :
    SimplicialObject.Augmented C 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] (X : CategoryTheory.CosimplicialObject.Augmented Cᵒᵖ)
Goal: ∀ (X_1 : SimplexCategoryᵒᵖ), X.leftOp.hom.app X_1 = (X.hom.app (Opposite.unop X_1)).unop
-- CategoryTheory.CosimplicialObject.Augmented.leftOp_right
@[simps!]
def CosimplicialObject.Augmented.leftOp (X : CosimplicialObject.Augmented Cᵒᵖ) :
    SimplicialObject.Augmented C 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] (X : CategoryTheory.CosimplicialObject.Augmented Cᵒᵖ)
Goal: X.leftOp.right = Opposite.unop X.left
-- CategoryTheory.CosimplicialObject.Augmented.leftOp_left_obj
@[simps!]
def CosimplicialObject.Augmented.leftOp (X : CosimplicialObject.Augmented Cᵒᵖ) :
    SimplicialObject.Augmented C 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] (X : CategoryTheory.CosimplicialObject.Augmented Cᵒᵖ)
Goal: ∀ (X_1 : SimplexCategoryᵒᵖ), X.leftOp.left.obj X_1 = Opposite.unop (X.right.obj (Opposite.unop X_1))
-- CategoryTheory.SimplicialObject.Augmented.rightOpLeftOpIso_inv_right
@[simps!]
def SimplicialObject.Augmented.rightOpLeftOpIso (X : SimplicialObject.Augmented C) :
    X.rightOp.leftOp ≅ X 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] (X : CategoryTheory.SimplicialObject.Augmented C)
Goal: X.rightOpLeftOpIso.inv.right = 𝟙 X.right
-- CategoryTheory.SimplicialObject.Augmented.rightOpLeftOpIso_hom_left_app
@[simps!]
def SimplicialObject.Augmented.rightOpLeftOpIso (X : SimplicialObject.Augmented C) :
    X.rightOp.leftOp ≅ X 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] (X : CategoryTheory.SimplicialObject.Augmented C)
Goal: ∀ (X_1 : SimplexCategoryᵒᵖ), X.rightOpLeftOpIso.hom.left.app X_1 = 𝟙 (X.left.obj X_1)
-- CategoryTheory.SimplicialObject.Augmented.rightOpLeftOpIso_inv_left_app
@[simps!]
def SimplicialObject.Augmented.rightOpLeftOpIso (X : SimplicialObject.Augmented C) :
    X.rightOp.leftOp ≅ X 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] (X : CategoryTheory.SimplicialObject.Augmented C)
Goal: ∀ (X_1 : SimplexCategoryᵒᵖ), X.rightOpLeftOpIso.inv.left.app X_1 = 𝟙 (X.left.obj X_1)
-- CategoryTheory.SimplicialObject.Augmented.rightOpLeftOpIso_hom_right
@[simps!]
def SimplicialObject.Augmented.rightOpLeftOpIso (X : SimplicialObject.Augmented C) :
    X.rightOp.leftOp ≅ X 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] (X : CategoryTheory.SimplicialObject.Augmented C)
Goal: X.rightOpLeftOpIso.hom.right = 𝟙 X.right
-- CategoryTheory.CosimplicialObject.Augmented.leftOpRightOpIso_hom_left
@[simps!]
def CosimplicialObject.Augmented.leftOpRightOpIso (X : CosimplicialObject.Augmented Cᵒᵖ) :
    X.leftOp.rightOp ≅ X 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] (X : CategoryTheory.CosimplicialObject.Augmented Cᵒᵖ)
Goal: X.leftOpRightOpIso.hom.left = 𝟙 X.left
-- CategoryTheory.CosimplicialObject.Augmented.leftOpRightOpIso_inv_left
@[simps!]
def CosimplicialObject.Augmented.leftOpRightOpIso (X : CosimplicialObject.Augmented Cᵒᵖ) :
    X.leftOp.rightOp ≅ X 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] (X : CategoryTheory.CosimplicialObject.Augmented Cᵒᵖ)
Goal: X.leftOpRightOpIso.inv.left = 𝟙 X.left
-- CategoryTheory.CosimplicialObject.Augmented.leftOpRightOpIso_hom_right_app
@[simps!]
def CosimplicialObject.Augmented.leftOpRightOpIso (X : CosimplicialObject.Augmented Cᵒᵖ) :
    X.leftOp.rightOp ≅ X 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] (X : CategoryTheory.CosimplicialObject.Augmented Cᵒᵖ)
Goal: ∀ (X_1 : SimplexCategory), X.leftOpRightOpIso.hom.right.app X_1 = 𝟙 (X.right.obj X_1)
-- CategoryTheory.CosimplicialObject.Augmented.leftOpRightOpIso_inv_right_app
@[simps!]
def CosimplicialObject.Augmented.leftOpRightOpIso (X : CosimplicialObject.Augmented Cᵒᵖ) :
    X.leftOp.rightOp ≅ X 
Arguments: {C : Type u} [CategoryTheory.Category.{v, u} C] (X : CategoryTheory.CosimplicialObject.Augmented Cᵒᵖ)
Goal: ∀ (X_1 : SimplexCategory), X.leftOpRightOpIso.inv.right.app X_1 = 𝟙 (X.right.obj X_1)
-- CategoryTheory.simplicialToCosimplicialAugmented_obj
@[simps]
def simplicialToCosimplicialAugmented :
    (SimplicialObject.Augmented C)ᵒᵖ ⥤ CosimplicialObject.Augmented Cᵒᵖ 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C] (X : (CategoryTheory.SimplicialObject.Augmented C)ᵒᵖ)
Goal: (CategoryTheory.simplicialToCosimplicialAugmented C).obj X = (Opposite.unop X).rightOp
-- CategoryTheory.simplicialToCosimplicialAugmented_map_right
@[simps]
def simplicialToCosimplicialAugmented :
    (SimplicialObject.Augmented C)ᵒᵖ ⥤ CosimplicialObject.Augmented Cᵒᵖ 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C]
Goal: ∀ {X Y : (CategoryTheory.SimplicialObject.Augmented C)ᵒᵖ} (f : X ⟶ Y),
  ((CategoryTheory.simplicialToCosimplicialAugmented C).map f).right = CategoryTheory.NatTrans.rightOp f.unop.left
-- CategoryTheory.simplicialToCosimplicialAugmented_map_left
@[simps]
def simplicialToCosimplicialAugmented :
    (SimplicialObject.Augmented C)ᵒᵖ ⥤ CosimplicialObject.Augmented Cᵒᵖ 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C]
Goal: ∀ {X Y : (CategoryTheory.SimplicialObject.Augmented C)ᵒᵖ} (f : X ⟶ Y),
  ((CategoryTheory.simplicialToCosimplicialAugmented C).map f).left = f.unop.right.op
-- CategoryTheory.cosimplicialToSimplicialAugmented_obj
@[simps]
def cosimplicialToSimplicialAugmented :
    CosimplicialObject.Augmented Cᵒᵖ ⥤ (SimplicialObject.Augmented C)ᵒᵖ 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C] (X : CategoryTheory.CosimplicialObject.Augmented Cᵒᵖ)
Goal: (CategoryTheory.cosimplicialToSimplicialAugmented C).obj X = Opposite.op X.leftOp
-- CategoryTheory.cosimplicialToSimplicialAugmented_map
@[simps]
def cosimplicialToSimplicialAugmented :
    CosimplicialObject.Augmented Cᵒᵖ ⥤ (SimplicialObject.Augmented C)ᵒᵖ 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C]
Goal: ∀ {X Y : CategoryTheory.CosimplicialObject.Augmented Cᵒᵖ} (f : X ⟶ Y),
  (CategoryTheory.cosimplicialToSimplicialAugmented C).map f =
    Quiver.Hom.op { left := CategoryTheory.NatTrans.leftOp f.right, right := f.left.unop, w := ⋯ }
-- CategoryTheory.simplicialCosimplicialAugmentedEquiv_functor
@[simps! functor inverse]
def simplicialCosimplicialAugmentedEquiv :
    (SimplicialObject.Augmented C)ᵒᵖ ≌ CosimplicialObject.Augmented Cᵒᵖ 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C]
Goal: (CategoryTheory.simplicialCosimplicialAugmentedEquiv C).functor = CategoryTheory.simplicialToCosimplicialAugmented C
-- CategoryTheory.simplicialCosimplicialAugmentedEquiv_inverse
@[simps! functor inverse]
def simplicialCosimplicialAugmentedEquiv :
    (SimplicialObject.Augmented C)ᵒᵖ ≌ CosimplicialObject.Augmented Cᵒᵖ 
Arguments: (C : Type u) [CategoryTheory.Category.{v, u} C]
Goal: (CategoryTheory.simplicialCosimplicialAugmentedEquiv C).inverse = CategoryTheory.cosimplicialToSimplicialAugmented C
-- Matrix.IsAdjMatrix.symm
structure IsAdjMatrix [Zero α] [One α] (A : Matrix V V α) : Prop where
  zero_or_one : ∀ i j, A i j = 0 ∨ A i j = 1 := by aesop
  symm : A.IsSymm := by aesop
  apply_diag : ∀ i, A i i = 0 := by aesop
Arguments: {V : Type u_1} {α : Type u_2} [Zero α] [One α] {A : Matrix V V α} (self : A.IsAdjMatrix)
Goal: A.IsSymm
-- Matrix.IsAdjMatrix.zero_or_one
structure IsAdjMatrix [Zero α] [One α] (A : Matrix V V α) : Prop where
  zero_or_one : ∀ i j, A i j = 0 ∨ A i j = 1 := by aesop
  symm : A.IsSymm := by aesop
  apply_diag : ∀ i, A i i = 0 := by aesop
Arguments: {V : Type u_1} {α : Type u_2} [Zero α] [One α] {A : Matrix V V α} (self : A.IsAdjMatrix) (i j : V)
Goal: A i j = 0 ∨ A i j = 1
-- Matrix.IsAdjMatrix.apply_diag
structure IsAdjMatrix [Zero α] [One α] (A : Matrix V V α) : Prop where
  zero_or_one : ∀ i j, A i j = 0 ∨ A i j = 1 := by aesop
  symm : A.IsSymm := by aesop
  apply_diag : ∀ i, A i i = 0 := by aesop
Arguments: {V : Type u_1} {α : Type u_2} [Zero α] [One α] {A : Matrix V V α} (self : A.IsAdjMatrix) (i : V)
Goal: A i i = 0
-- Matrix.IsAdjMatrix.apply_diag_ne
@[simp]
theorem apply_diag_ne [MulZeroOneClass α] [Nontrivial α] (h : IsAdjMatrix A) (i : V) :
    ¬A i i = 1 
Arguments: {V : Type u_1} {α : Type u_2} {A : Matrix V V α} [MulZeroOneClass α] [Nontrivial α] (h : A.IsAdjMatrix) (i : V)
Goal: ¬A i i = 1
-- Matrix.IsAdjMatrix.apply_ne_one_iff
@[simp]
theorem apply_ne_one_iff [MulZeroOneClass α] [Nontrivial α] (h : IsAdjMatrix A) (i j : V) :
    ¬A i j = 1 ↔ A i j = 0 
Arguments: {V : Type u_1} {α : Type u_2} {A : Matrix V V α} [MulZeroOneClass α] [Nontrivial α] (h : A.IsAdjMatrix) (i j : V)
Goal: ¬A i j = 1 ↔ A i j = 0
-- Matrix.IsAdjMatrix.apply_ne_zero_iff
@[simp]
theorem apply_ne_zero_iff [MulZeroOneClass α] [Nontrivial α] (h : IsAdjMatrix A) (i j : V) :
    ¬A i j = 0 ↔ A i j = 1 
Arguments: {V : Type u_1} {α : Type u_2} {A : Matrix V V α} [MulZeroOneClass α] [Nontrivial α] (h : A.IsAdjMatrix) (i j : V)
Goal: ¬A i j = 0 ↔ A i j = 1
-- Matrix.IsAdjMatrix.toGraph_adj
@[simps]
def toGraph [MulZeroOneClass α] [Nontrivial α] (h : IsAdjMatrix A) : SimpleGraph V 
Arguments: {V : Type u_1} {α : Type u_2} {A : Matrix V V α} [MulZeroOneClass α] [Nontrivial α] (h : A.IsAdjMatrix) (i j : V)
Goal: h.toGraph.Adj i j = (A i j = 1)
-- Matrix.compl_apply_diag
@[simp]
theorem compl_apply_diag [Zero α] [One α] (i : V) : A.compl i i = 0 
Arguments: {V : Type u_1} {α : Type u_2} [DecidableEq α] [DecidableEq V] (A : Matrix V V α) [Zero α] [One α] (i : V)
Goal: A.compl i i = 0
-- Matrix.compl_apply
@[simp]
theorem compl_apply [Zero α] [One α] (i j : V) : A.compl i j = 0 ∨ A.compl i j = 1 
Arguments: {V : Type u_1} {α : Type u_2} [DecidableEq α] [DecidableEq V] (A : Matrix V V α) [Zero α] [One α] (i j : V)
Goal: A.compl i j = 0 ∨ A.compl i j = 1
-- Matrix.isSymm_compl
@[simp]
theorem isSymm_compl [Zero α] [One α] (h : A.IsSymm) : A.compl.IsSymm 
Arguments: {V : Type u_1} {α : Type u_2} [DecidableEq α] [DecidableEq V] (A : Matrix V V α) [Zero α] [One α] (h : A.IsSymm)
Goal: A.compl.IsSymm
-- Matrix.isAdjMatrix_compl
@[simp]
theorem isAdjMatrix_compl [Zero α] [One α] (h : A.IsSymm) : IsAdjMatrix A.compl 
Arguments: {V : Type u_1} {α : Type u_2} [DecidableEq α] [DecidableEq V] (A : Matrix V V α) [Zero α] [One α] (h : A.IsSymm)
Goal: A.compl.IsAdjMatrix
-- Matrix.IsAdjMatrix.compl
@[simp]
theorem compl [Zero α] [One α] (h : IsAdjMatrix A) : IsAdjMatrix A.compl 
Arguments: {V : Type u_1} {α : Type u_2} [DecidableEq α] [DecidableEq V] {A : Matrix V V α} [Zero α] [One α] (h : A.IsAdjMatrix)
Goal: A.compl.IsAdjMatrix
-- Matrix.IsAdjMatrix.toGraph_compl_eq
theorem toGraph_compl_eq [MulZeroOneClass α] [Nontrivial α] (h : IsAdjMatrix A) :
    h.compl.toGraph = h.toGraphᶜ 
Arguments: {V : Type u_1} {α : Type u_2} [DecidableEq α] [DecidableEq V] {A : Matrix V V α} [MulZeroOneClass α] [Nontrivial α] (h : A.IsAdjMatrix)
Goal: ⋯.toGraph = h.toGraphᶜ
-- SimpleGraph.adjMatrix_apply
@[simp]
theorem adjMatrix_apply (v w : V) [Zero α] [One α] :
    G.adjMatrix α v w = if G.Adj v w then 1 else 0 
Arguments: {V : Type u_1} {α : Type u_2} (G : SimpleGraph V) [DecidableRel G.Adj] (v w : V) [Zero α] [One α]
Goal: SimpleGraph.adjMatrix α G v w = if G.Adj v w then 1 else 0
-- SimpleGraph.transpose_adjMatrix
@[simp]
theorem transpose_adjMatrix [Zero α] [One α] : (G.adjMatrix α)ᵀ = G.adjMatrix α 
Arguments: {V : Type u_1} {α : Type u_2} (G : SimpleGraph V) [DecidableRel G.Adj] [Zero α] [One α]
Goal: (SimpleGraph.adjMatrix α G)ᵀ = SimpleGraph.adjMatrix α G
-- SimpleGraph.isSymm_adjMatrix
@[simp]
theorem isSymm_adjMatrix [Zero α] [One α] : (G.adjMatrix α).IsSymm 
Arguments: {V : Type u_1} {α : Type u_2} (G : SimpleGraph V) [DecidableRel G.Adj] [Zero α] [One α]
Goal: (SimpleGraph.adjMatrix α G).IsSymm
-- SimpleGraph.isAdjMatrix_adjMatrix
@[simp]
theorem isAdjMatrix_adjMatrix [Zero α] [One α] : (G.adjMatrix α).IsAdjMatrix 
Arguments: {V : Type u_1} (α : Type u_2) (G : SimpleGraph V) [DecidableRel G.Adj] [Zero α] [One α]
Goal: (SimpleGraph.adjMatrix α G).IsAdjMatrix
-- SimpleGraph.toGraph_adjMatrix_eq
theorem toGraph_adjMatrix_eq [MulZeroOneClass α] [Nontrivial α] :
    (G.isAdjMatrix_adjMatrix α).toGraph = G 
Arguments: {V : Type u_1} (α : Type u_2) (G : SimpleGraph V) [DecidableRel G.Adj] [MulZeroOneClass α] [Nontrivial α]
Goal: ⋯.toGraph = G
-- SimpleGraph.adjMatrix_dotProduct
@[simp]
theorem adjMatrix_dotProduct [NonAssocSemiring α] (v : V) (vec : V → α) :
    dotProduct (G.adjMatrix α v) vec = ∑ u ∈ G.neighborFinset v, vec u 
Arguments: {V : Type u_1} {α : Type u_2} (G : SimpleGraph V) [DecidableRel G.Adj] [Fintype V] [NonAssocSemiring α] (v : V) (vec : V → α)
Goal: SimpleGraph.adjMatrix α G v ⬝ᵥ vec = ∑ u ∈ G.neighborFinset v, vec u
-- SimpleGraph.dotProduct_adjMatrix
@[simp]
theorem dotProduct_adjMatrix [NonAssocSemiring α] (v : V) (vec : V → α) :
    dotProduct vec (G.adjMatrix α v) = ∑ u ∈ G.neighborFinset v, vec u 
Arguments: {V : Type u_1} {α : Type u_2} (G : SimpleGraph V) [DecidableRel G.Adj] [Fintype V] [NonAssocSemiring α] (v : V) (vec : V → α)
Goal: vec ⬝ᵥ SimpleGraph.adjMatrix α G v = ∑ u ∈ G.neighborFinset v, vec u
-- SimpleGraph.adjMatrix_mulVec_apply
@[simp]
theorem adjMatrix_mulVec_apply [NonAssocSemiring α] (v : V) (vec : V → α) :
    (G.adjMatrix α *ᵥ vec) v = ∑ u ∈ G.neighborFinset v, vec u 
Arguments: {V : Type u_1} {α : Type u_2} (G : SimpleGraph V) [DecidableRel G.Adj] [Fintype V] [NonAssocSemiring α] (v : V) (vec : V → α)
Goal: (SimpleGraph.adjMatrix α G *ᵥ vec) v = ∑ u ∈ G.neighborFinset v, vec u
-- SimpleGraph.adjMatrix_vecMul_apply
@[simp]
theorem adjMatrix_vecMul_apply [NonAssocSemiring α] (v : V) (vec : V → α) :
    (vec ᵥ* G.adjMatrix α) v = ∑ u ∈ G.neighborFinset v, vec u 
Arguments: {V : Type u_1} {α : Type u_2} (G : SimpleGraph V) [DecidableRel G.Adj] [Fintype V] [NonAssocSemiring α] (v : V) (vec : V → α)
Goal: (vec ᵥ* SimpleGraph.adjMatrix α G) v = ∑ u ∈ G.neighborFinset v, vec u