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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.
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