File: Mathlib/Order/Category/FinPartOrd.lean

full_name: FinPartOrd.toPartOrd
code: /-- The category of finite partial orders with monotone functions. -/
structure FinPartOrd where
  toPartOrd : PartOrd
  [isFintype : Fintype toPartOrd]
goal: PartOrd

full_name: FinPartOrd.isFintype
code: /-- The category of finite partial orders with monotone functions. -/
structure FinPartOrd where
  toPartOrd : PartOrd
  [isFintype : Fintype toPartOrd]
goal: Fintype ↑self.toPartOrd

full_name: FinPartOrd
code: /-- The category of finite partial orders with monotone functions. -/
structure FinPartOrd where
  toPartOrd : PartOrd
  [isFintype : Fintype toPartOrd]
goal: Type (u_1 + 1)

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

full_name: FinPartOrd.instPartialOrderαToPartOrd
code: instance (X : FinPartOrd) : PartialOrder X 
goal: PartialOrder ↑X.toPartOrd

full_name: FinPartOrd.of
code: /-- Construct a bundled `FinPartOrd` from `PartialOrder` + `Fintype`. -/
def of (α : Type*) [PartialOrder α] [Fintype α] : FinPartOrd 
goal: FinPartOrd

full_name: FinPartOrd.coe_of
code: @[simp]
theorem coe_of (α : Type*) [PartialOrder α] [Fintype α] : ↥(of α) = α 
goal: ↑(FinPartOrd.of α).toPartOrd = α

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

full_name: FinPartOrd.largeCategory
code: instance largeCategory : LargeCategory FinPartOrd 
goal: CategoryTheory.LargeCategory FinPartOrd

full_name: FinPartOrd.concreteCategory
code: instance concreteCategory : ConcreteCategory FinPartOrd 
goal: CategoryTheory.ConcreteCategory FinPartOrd

full_name: FinPartOrd.hasForgetToPartOrd
code: instance hasForgetToPartOrd : HasForget₂ FinPartOrd PartOrd 
goal: CategoryTheory.HasForget₂ FinPartOrd PartOrd

full_name: FinPartOrd.hasForgetToFintype
code: instance hasForgetToFintype : HasForget₂ FinPartOrd FintypeCat 
goal: CategoryTheory.HasForget₂ FinPartOrd FintypeCat

full_name: FinPartOrd.Iso.mk_inv
code: /-- Constructs an isomorphism of finite partial orders from an order isomorphism between them. -/
@[simps]
def Iso.mk {α β : FinPartOrd.{u}} (e : α ≃o β) : α ≅ β 
goal: (FinPartOrd.Iso.mk e).inv = ↑e.symm

full_name: FinPartOrd.Iso.mk_hom
code: /-- Constructs an isomorphism of finite partial orders from an order isomorphism between them. -/
@[simps]
def Iso.mk {α β : FinPartOrd.{u}} (e : α ≃o β) : α ≅ β 
goal: (FinPartOrd.Iso.mk e).hom = ↑e

full_name: FinPartOrd.Iso.mk
code: /-- Constructs an isomorphism of finite partial orders from an order isomorphism between them. -/
@[simps]
def Iso.mk {α β : FinPartOrd.{u}} (e : α ≃o β) : α ≅ β 
goal: α ≅ β

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

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

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

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

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

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

full_name: FinPartOrd_dual_comp_forget_to_partOrd
code: theorem FinPartOrd_dual_comp_forget_to_partOrd :
    FinPartOrd.dual ⋙ forget₂ FinPartOrd PartOrd =
      forget₂ FinPartOrd PartOrd ⋙ PartOrd.dual 
goal: FinPartOrd.dual ⋙ CategoryTheory.forget₂ FinPartOrd PartOrd = CategoryTheory.forget₂ FinPartOrd PartOrd ⋙ PartOrd.dual