Library Stdlib.Sets.Cpo

Require Export Ensembles.
Require Export Relations_1.
Require Export Partial_Order.

Section Bounds.
  Variable U : Type.
  Variable D : PO U.

  Let C := @Carrier_of U D.

  Let R := @Rel_of U D.

  Inductive Upper_Bound (B:Ensemble U) (x:U) : Prop :=
    Upper_Bound_definition :
    In U C x -> (forall y:U, In U B y -> R y x) -> Upper_Bound B x.

  Inductive Lower_Bound (B:Ensemble U) (x:U) : Prop :=
    Lower_Bound_definition :
    In U C x -> (forall y:U, In U B y -> R x y) -> Lower_Bound B x.

  Inductive Lub (B:Ensemble U) (x:U) : Prop :=
    Lub_definition :
    Upper_Bound B x -> (forall y:U, Upper_Bound B y -> R x y) -> Lub B x.

  Inductive Glb (B:Ensemble U) (x:U) : Prop :=
    Glb_definition :
    Lower_Bound B x -> (forall y:U, Lower_Bound B y -> R y x) -> Glb B x.

  Inductive Bottom (bot:U) : Prop :=
    Bottom_definition :
    In U C bot -> (forall y:U, In U C y -> R bot y) -> Bottom bot.

  Inductive Totally_ordered (B:Ensemble U) : Prop :=
    Totally_ordered_definition :
    (Included U B C ->
      forall x y:U, Included U (Couple U x y) B -> R x y \/ R y x) ->
    Totally_ordered B.

  Definition Compatible : Relation U :=
    fun x y:U =>
      In U C x ->
      In U C y -> exists z : _, In U C z /\ Upper_Bound (Couple U x y) z.

  Inductive Directed (X:Ensemble U) : Prop :=
    Definition_of_Directed :
    Included U X C ->
    Inhabited U X ->
    (forall x1 x2:U,
      Included U (Couple U x1 x2) X ->
      exists x3 : _, In U X x3 /\ Upper_Bound (Couple U x1 x2) x3) ->
    Directed X.

  Inductive Complete : Prop :=
    Definition_of_Complete :
    (exists bot : _, Bottom bot) ->
    (forall X:Ensemble U, Directed X -> exists bsup : _, Lub X bsup) ->
    Complete.

  Inductive Conditionally_complete : Prop :=
    Definition_of_Conditionally_complete :
    (forall X:Ensemble U,
      Included U X C ->
      (exists maj : _, Upper_Bound X maj) ->
      exists bsup : _, Lub X bsup) -> Conditionally_complete.
End Bounds.

#[global]
Hint Resolve Totally_ordered_definition Upper_Bound_definition
  Lower_Bound_definition Lub_definition Glb_definition Bottom_definition
  Definition_of_Complete Definition_of_Complete
  Definition_of_Conditionally_complete : core.

Section Specific_orders.
  Variable U : Type.

  Record Cpo : Type := Definition_of_cpo
    {PO_of_cpo : PO U; Cpo_cond : Complete U PO_of_cpo}.

  Record Chain : Type := Definition_of_chain
    {PO_of_chain : PO U;
    Chain_cond : Totally_ordered U PO_of_chain (@Carrier_of _ PO_of_chain)}.

End Specific_orders.