(***********************************************************************)
(*  v      *   The Coq Proof Assistant  /  The Coq Development Team    *)
(* <O___,, *        INRIA-Rocquencourt  &  LRI-CNRS-Orsay              *)
(*   \VV/  *************************************************************)
(*    //   *      This file is distributed under the terms of the      *)
(*         *       GNU Lesser General Public License Version 2.1       *)
(***********************************************************************)
(****************************************************************************)
(*                                                                          *)
(*                         Naive Set Theory in Coq                          *)
(*                                                                          *)
(*                     INRIA                        INRIA                   *)
(*              Rocquencourt                        Sophia-Antipolis        *)
(*                                                                          *)
(*                                 Coq V6.1                                 *)
(*									    *)
(*			         Gilles Kahn 				    *)
(*				 Gerard Huet				    *)
(*									    *)
(*									    *)
(*                                                                          *)
(* Acknowledgments: This work was started in July 1993 by F. Prost. Thanks  *)
(* to the Newton Institute for providing an exceptional work environment    *)
(* in Summer 1995. Several developments by E. Ledinot were an inspiration.  *)
(****************************************************************************)

(*i $Id$ i*)

Require Export Ensembles.
Require Export Relations_1.

Section Partial_orders.
Variable U : Type.

Definition Carrier := Ensemble U.

Definition Rel := Relation U.

Record PO : Type := Definition_of_PO
  {Carrier_of : Ensemble U;
   Rel_of : Relation U;
   PO_cond1 : Inhabited U Carrier_of;
   PO_cond2 : Order U Rel_of}.
Variable p : PO.

Definition Strict_Rel_of : Rel := fun x y:U => Rel_of p x y /\ x <> y.

Inductive covers (y x:U) : Prop :=
    Definition_of_covers :
      Strict_Rel_of x y ->
      ~ (exists z : _, Strict_Rel_of x z /\ Strict_Rel_of z y) ->
      covers y x.

End Partial_orders.

Hint Unfold Carrier_of Rel_of Strict_Rel_of: sets v62.
Hint Resolve Definition_of_covers: sets v62.


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

Lemma Strict_Rel_Transitive_with_Rel :
 forall x y z:U,
   Strict_Rel_of U D x y -> Rel_of U D y z -> Strict_Rel_of U D x z.
unfold Strict_Rel_of at 1 in |- *.
red in |- *.
elim D; simpl in |- *.
intros C R H' H'0; elim H'0.
intros H'1 H'2 H'3 x y z H'4 H'5; split.
apply H'2 with (y := y); tauto.
red in |- *; intro H'6.
elim H'4; intros H'7 H'8; apply H'8; clear H'4.
apply H'3; auto.
rewrite H'6; tauto.
Qed.

Lemma Strict_Rel_Transitive_with_Rel_left :
 forall x y z:U,
   Rel_of U D x y -> Strict_Rel_of U D y z -> Strict_Rel_of U D x z.
unfold Strict_Rel_of at 1 in |- *.
red in |- *.
elim D; simpl in |- *.
intros C R H' H'0; elim H'0.
intros H'1 H'2 H'3 x y z H'4 H'5; split.
apply H'2 with (y := y); tauto.
red in |- *; intro H'6.
elim H'5; intros H'7 H'8; apply H'8; clear H'5.
apply H'3; auto.
rewrite <- H'6; auto.
Qed.

Lemma Strict_Rel_Transitive : Transitive U (Strict_Rel_of U D).
red in |- *.
intros x y z H' H'0.
apply Strict_Rel_Transitive_with_Rel with (y := y);
 [ intuition | unfold Strict_Rel_of in H', H'0; intuition ].
Qed.
End Partial_order_facts.