File: Mathlib/Order/Category/BoolAlg.lean

full_name: BoolAlg
code: /-- The category of boolean algebras. -/
def BoolAlg 
goal: Type (u_1 + 1)

full_name: BoolAlg.instCoeSortType
code: instance : CoeSort BoolAlg Type* 
goal: CoeSort BoolAlg (Type u_1)

full_name: BoolAlg.instBooleanAlgebra
code: instance instBooleanAlgebra (X : BoolAlg) : BooleanAlgebra X 
goal: BooleanAlgebra ↑X

full_name: BoolAlg.of
code: /-- Construct a bundled `BoolAlg` from a `BooleanAlgebra`. -/
def of (α : Type*) [BooleanAlgebra α] : BoolAlg 
goal: BoolAlg

full_name: BoolAlg.coe_of
code: @[simp]
theorem coe_of (α : Type*) [BooleanAlgebra α] : ↥(of α) = α 
goal: ↑(BoolAlg.of α) = α

full_name: BoolAlg.instInhabited
code: instance : Inhabited BoolAlg 
goal: Inhabited BoolAlg

full_name: BoolAlg.toBddDistLat
code: /-- Turn a `BoolAlg` into a `BddDistLat` by forgetting its complement operation. -/
def toBddDistLat (X : BoolAlg) : BddDistLat 
goal: BddDistLat

full_name: BoolAlg.coe_toBddDistLat
code: @[simp]
theorem coe_toBddDistLat (X : BoolAlg) : ↥X.toBddDistLat = ↥X 
goal: ↑X.toBddDistLat.toDistLat = ↑X

full_name: BoolAlg.instLargeCategory
code: instance : LargeCategory.{u} BoolAlg 
goal: CategoryTheory.LargeCategory BoolAlg

full_name: BoolAlg.instConcreteCategory
code: instance : ConcreteCategory BoolAlg 
goal: CategoryTheory.ConcreteCategory BoolAlg

full_name: BoolAlg.hasForgetToBddDistLat
code: instance hasForgetToBddDistLat : HasForget₂ BoolAlg BddDistLat 
goal: CategoryTheory.HasForget₂ BoolAlg BddDistLat

full_name: BoolAlg.hasForgetToHeytAlg
code: @[simps]
instance hasForgetToHeytAlg : HasForget₂ BoolAlg HeytAlg 
goal: CategoryTheory.HasForget₂ BoolAlg HeytAlg

full_name: BoolAlg.hasForgetToHeytAlg_forget₂_map_toFun
code: @[simps]
instance hasForgetToHeytAlg : HasForget₂ BoolAlg HeytAlg 
goal: (CategoryTheory.HasForget₂.forget₂.map f) a = f a

full_name: BoolAlg.hasForgetToHeytAlg_forget₂_obj_α
code: @[simps]
instance hasForgetToHeytAlg : HasForget₂ BoolAlg HeytAlg 
goal: ↑(CategoryTheory.HasForget₂.forget₂.obj X) = ↑X

full_name: BoolAlg.hasForgetToHeytAlg_forget₂_obj_str
code: @[simps]
instance hasForgetToHeytAlg : HasForget₂ BoolAlg HeytAlg 
goal: (CategoryTheory.HasForget₂.forget₂.obj X).str = inferInstance

full_name: BoolAlg.Iso.mk_inv_toLatticeHom_toSupHom_toFun
code: /-- Constructs an equivalence between Boolean algebras from an order isomorphism between them. -/
@[simps]
def Iso.mk {α β : BoolAlg.{u}} (e : α ≃o β) : α ≅ β 
goal: (BoolAlg.Iso.mk e).inv.toSupHom a = e.symm a

full_name: BoolAlg.Iso.mk_hom_toLatticeHom_toSupHom_toFun
code: /-- Constructs an equivalence between Boolean algebras from an order isomorphism between them. -/
@[simps]
def Iso.mk {α β : BoolAlg.{u}} (e : α ≃o β) : α ≅ β 
goal: (BoolAlg.Iso.mk e).hom.toSupHom a = e a

full_name: BoolAlg.Iso.mk
code: /-- Constructs an equivalence between Boolean algebras from an order isomorphism between them. -/
@[simps]
def Iso.mk {α β : BoolAlg.{u}} (e : α ≃o β) : α ≅ β 
goal: α ≅ β

full_name: BoolAlg.dual
code: /-- `OrderDual` as a functor. -/
@[simps]
def dual : BoolAlg ⥤ BoolAlg 
goal: BoolAlg ⥤ BoolAlg

full_name: BoolAlg.dual_obj
code: /-- `OrderDual` as a functor. -/
@[simps]
def dual : BoolAlg ⥤ BoolAlg 
goal: BoolAlg.dual.obj X = BoolAlg.of (↑X)ᵒᵈ

full_name: BoolAlg.dual_map
code: /-- `OrderDual` as a functor. -/
@[simps]
def dual : BoolAlg ⥤ BoolAlg 
goal: BoolAlg.dual.map a = BoundedLatticeHom.dual a

full_name: BoolAlg.dualEquiv
code: /-- The equivalence between `BoolAlg` and itself induced by `OrderDual` both ways. -/
@[simps functor inverse]
def dualEquiv : BoolAlg ≌ BoolAlg 
goal: BoolAlg ≌ BoolAlg

full_name: BoolAlg.dualEquiv_functor
code: /-- The equivalence between `BoolAlg` and itself induced by `OrderDual` both ways. -/
@[simps functor inverse]
def dualEquiv : BoolAlg ≌ BoolAlg 
goal: BoolAlg.dualEquiv.functor = BoolAlg.dual

full_name: BoolAlg.dualEquiv_inverse
code: /-- The equivalence between `BoolAlg` and itself induced by `OrderDual` both ways. -/
@[simps functor inverse]
def dualEquiv : BoolAlg ≌ BoolAlg 
goal: BoolAlg.dualEquiv.inverse = BoolAlg.dual

full_name: boolAlg_dual_comp_forget_to_bddDistLat
code: theorem boolAlg_dual_comp_forget_to_bddDistLat :
    BoolAlg.dual ⋙ forget₂ BoolAlg BddDistLat =
    forget₂ BoolAlg BddDistLat ⋙ BddDistLat.dual 
goal: BoolAlg.dual ⋙ CategoryTheory.forget₂ BoolAlg BddDistLat = CategoryTheory.forget₂ BoolAlg BddDistLat ⋙ BddDistLat.dual

full_name: typeToBoolAlgOp
code: /-- The powerset functor. `Set` as a contravariant functor. -/
@[simps]
def typeToBoolAlgOp : Type u ⥤ BoolAlgᵒᵖ 
goal: Type u ⥤ BoolAlgᵒᵖ

full_name: typeToBoolAlgOp_obj
code: /-- The powerset functor. `Set` as a contravariant functor. -/
@[simps]
def typeToBoolAlgOp : Type u ⥤ BoolAlgᵒᵖ 
goal: typeToBoolAlgOp.obj X = Opposite.op (BoolAlg.of (Set X))

full_name: typeToBoolAlgOp_map
code: /-- The powerset functor. `Set` as a contravariant functor. -/
@[simps]
def typeToBoolAlgOp : Type u ⥤ BoolAlgᵒᵖ 
goal: typeToBoolAlgOp.map f =
  Quiver.Hom.op
    (let __src := { toFun := ⇑(CompleteLatticeHom.setPreimage f), map_sup' := ⋯, map_inf' := ⋯ };
    { toFun := ⇑(CompleteLatticeHom.setPreimage f), map_sup' := ⋯, map_inf' := ⋯, map_top' := ⋯, map_bot' := ⋯ })