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(************************************************************************)
(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
(*   \VV/  **************************************************************)
(*    //   *      This file is distributed under the terms of the       *)
(*         *       GNU Lesser General Public License Version 2.1        *)
(************************************************************************)

(*i $Id: Union.v,v 1.9.2.1 2004/07/16 19:31:19 herbelin Exp $ i*)

(** Author: Bruno Barras *)

Require Import Relation_Operators.
Require Import Relation_Definitions.
Require Import Transitive_Closure.

Section WfUnion.
  Variable A : Set.
  Variables R1 R2 : relation A.
  
 Notation Union := (union A R1 R2).

 Hint Resolve Acc_clos_trans wf_clos_trans.

Remark strip_commut :
 commut A R1 R2 ->
 forall x y:A,
   clos_trans A R1 y x ->
   forall z:A, R2 z y ->  exists2 y' : A, R2 y' x & clos_trans A R1 z y'.
Proof.
 induction 2 as [x y| x y z H0 IH1 H1 IH2]; intros.
 elim H with y x z; auto with sets; intros x0 H2 H3.
 exists x0; auto with sets.

 elim IH1 with z0; auto with sets; intros.
 elim IH2 with x0; auto with sets; intros.
 exists x1; auto with sets.
 apply t_trans with x0; auto with sets.
Qed.


  Lemma Acc_union :
   commut A R1 R2 ->
   (forall x:A, Acc R2 x -> Acc R1 x) -> forall a:A, Acc R2 a -> Acc Union a.
Proof.
 induction 3 as [x H1 H2].
 apply Acc_intro; intros.
 elim H3; intros; auto with sets.
 cut (clos_trans A R1 y x); auto with sets.
 elimtype (Acc (clos_trans A R1) y); intros.
 apply Acc_intro; intros.
 elim H8; intros.
 apply H6; auto with sets.
 apply t_trans with x0; auto with sets.

 elim strip_commut with x x0 y0; auto with sets; intros.
 apply Acc_inv_trans with x1; auto with sets.
 unfold union in |- *.
 elim H11; auto with sets; intros.
 apply t_trans with y1; auto with sets.

 apply (Acc_clos_trans A).
 apply Acc_inv with x; auto with sets.
 apply H0.
 apply Acc_intro; auto with sets.
Qed.


  Theorem wf_union :
   commut A R1 R2 -> well_founded R1 -> well_founded R2 -> well_founded Union.
Proof.
 unfold well_founded in |- *.
 intros.
 apply Acc_union; auto with sets.
Qed.

End WfUnion.