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.




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 U PO_of_chain)}.




End Specific_orders.
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