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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: Disjoint_Union.v,v 1.1.2.1 2004/07/16 19:31:41 herbelin Exp $ i*)
+
+(** Author: Cristina Cornes
+    From : Constructing Recursion Operators in Type Theory                 
+           L. Paulson  JSC (1986) 2, 325-355 *) 
+
+Require Relation_Operators.
+
+Section Wf_Disjoint_Union.
+Variable A,B:Set.
+Variable leA: A->A->Prop.
+Variable leB: B->B->Prop.
+
+Notation Le_AsB := (le_AsB A B leA leB).
+
+Lemma acc_A_sum: (x:A)(Acc A leA x)->(Acc A+B Le_AsB (inl A B x)).
+Proof.
+ NewInduction 1.
+ Apply Acc_intro;Intros y H2.
+ Inversion_clear H2.
+ Auto with sets.
+Qed.
+
+Lemma acc_B_sum: (well_founded A leA) ->(x:B)(Acc B leB x)
+                        ->(Acc A+B Le_AsB (inr A B x)).
+Proof.
+ NewInduction 2.
+ Apply Acc_intro;Intros y H3.
+ Inversion_clear H3;Auto with sets.
+ Apply acc_A_sum;Auto with sets.
+Qed.
+
+
+Lemma wf_disjoint_sum:
+ (well_founded A leA) 
+    -> (well_founded B leB) -> (well_founded A+B Le_AsB).
+Proof.
+ Intros.
+ Unfold well_founded .
+ NewDestruct a as [a|b].
+ Apply (acc_A_sum a).
+ Apply (H a).
+
+ Apply (acc_B_sum H b).
+ Apply (H0 b).
+Qed.
+
+End Wf_Disjoint_Union.