File: Mathlib/Order/Category/PartOrd.lean

full_name: PartOrd
code: /-- The category of partially ordered types. -/
def PartOrd 
goal: Type (u_1 + 1)

full_name: PartOrd.instParentProjectionPreorderPartialOrderToPreorder
code: instance : BundledHom.ParentProjection @PartialOrder.toPreorder 
goal: CategoryTheory.BundledHom.ParentProjection @PartialOrder.toPreorder

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

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

full_name: PartOrd.of
code: /-- Construct a bundled PartOrd from the underlying type and typeclass. -/
def of (α : Type*) [PartialOrder α] : PartOrd 
goal: PartOrd

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

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

full_name: PartOrd.instPartialOrderα
code: instance (α : PartOrd) : PartialOrder α 
goal: PartialOrder ↑α

full_name: PartOrd.hasForgetToPreord
code: instance hasForgetToPreord : HasForget₂ PartOrd Preord 
goal: CategoryTheory.HasForget₂ PartOrd Preord

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

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

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

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

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

full_name: PartOrd.dual_map
code: /-- `OrderDual` as a functor. -/
@[simps]
def dual : PartOrd ⥤ PartOrd 
goal: ∀ {X Y : PartOrd} (a : ↑X →o ↑Y), PartOrd.dual.map a = OrderHom.dual a

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

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

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

full_name: partOrd_dual_comp_forget_to_preord
code: theorem partOrd_dual_comp_forget_to_preord :
    PartOrd.dual ⋙ forget₂ PartOrd Preord =
      forget₂ PartOrd Preord ⋙ Preord.dual 
goal: PartOrd.dual ⋙ CategoryTheory.forget₂ PartOrd Preord = CategoryTheory.forget₂ PartOrd Preord ⋙ Preord.dual

full_name: preordToPartOrd
code: /-- `Antisymmetrization` as a functor. It is the free functor. -/
def preordToPartOrd : Preord.{u} ⥤ PartOrd 
goal: Preord ⥤ PartOrd

full_name: preordToPartOrdForgetAdjunction
code: /-- `preordToPartOrd` is left adjoint to the forgetful functor, meaning it is the free
functor from `Preord` to `PartOrd`. -/
def preordToPartOrdForgetAdjunction :
    preordToPartOrd.{u} ⊣ forget₂ PartOrd Preord 
goal: preordToPartOrd ⊣ CategoryTheory.forget₂ PartOrd Preord

full_name: preordToPartOrdCompToDualIsoToDualCompPreordToPartOrd_inv_app_coe
code: /-- `PreordToPartOrd` and `OrderDual` commute. -/
@[simps! inv_app_coe, simps! (config := .lemmasOnly) hom_app_coe]
def preordToPartOrdCompToDualIsoToDualCompPreordToPartOrd :
    preordToPartOrd.{u} ⋙ PartOrd.dual ≅ Preord.dual ⋙ preordToPartOrd 
goal: (preordToPartOrdCompToDualIsoToDualCompPreordToPartOrd.inv.app X) a = (OrderIso.dualAntisymmetrization ↑X).symm a

full_name: preordToPartOrdCompToDualIsoToDualCompPreordToPartOrd_hom_app_coe
code: /-- `PreordToPartOrd` and `OrderDual` commute. -/
@[simps! inv_app_coe, simps! (config := .lemmasOnly) hom_app_coe]
def preordToPartOrdCompToDualIsoToDualCompPreordToPartOrd :
    preordToPartOrd.{u} ⋙ PartOrd.dual ≅ Preord.dual ⋙ preordToPartOrd 
goal: (preordToPartOrdCompToDualIsoToDualCompPreordToPartOrd.hom.app X) a = (OrderIso.dualAntisymmetrization ↑X) a

full_name: preordToPartOrdCompToDualIsoToDualCompPreordToPartOrd
code: /-- `PreordToPartOrd` and `OrderDual` commute. -/
@[simps! inv_app_coe, simps! (config := .lemmasOnly) hom_app_coe]
def preordToPartOrdCompToDualIsoToDualCompPreordToPartOrd :
    preordToPartOrd.{u} ⋙ PartOrd.dual ≅ Preord.dual ⋙ preordToPartOrd 
goal: preordToPartOrd ⋙ PartOrd.dual ≅ Preord.dual ⋙ preordToPartOrd