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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.