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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 1.1.2.1 2004/07/16 19:31:42 herbelin Exp $ i*)
+
+(** Author: Bruno Barras *)
+
+Require Relation_Operators.
+Require Relation_Definitions.
+Require Transitive_Closure.
+
+Section WfUnion.
+  Variable A: Set.
+  Variable R1,R2: (relation A).
+  
+ Notation Union := (union A R1 R2).
+
+ Hints Resolve Acc_clos_trans wf_clos_trans.
+
+Remark strip_commut: 
+      (commut A R1 R2)->(x,y:A)(clos_trans A R1 y x)->(z:A)(R2 z y)
+            ->(EX y':A | (R2 y' x) & (clos_trans A R1 z y')).
+Proof.
+ NewInduction 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)->((x:A)(Acc A R2 x)->(Acc A R1 x))
+                         ->(a:A)(Acc A R2 a)->(Acc A Union a).
+Proof.
+ NewInduction 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 A (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 .
+ 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 A R1)->(well_founded A R2)
+                  ->(well_founded A Union).
+Proof.
+ Unfold well_founded .
+ Intros.
+ Apply Acc_union;Auto with sets.
+Qed.
+
+End WfUnion.