# Library Coq.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.

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.

End Specific_orders.