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(***********************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA-Rocquencourt & LRI-CNRS-Orsay *)
(* \VV/ *************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(***********************************************************************)
(*i $Id$ i*)
(** Author: Bruno Barras *)
Require Relation_Definitions.
Require Relation_Operators.
Section Wf_Transitive_Closure.
Variable A: Set.
Variable R: (relation A).
Notation trans_clos := (clos_trans A R).
Lemma incl_clos_trans: (inclusion A R trans_clos).
Red;Auto with sets.
Qed.
Lemma Acc_clos_trans: (x:A)(Acc A R x)->(Acc A trans_clos x).
Induction 1.
Intros x0 H0 H1.
Apply Acc_intro.
Intros y H2.
Generalize H1 .
Elim H2;Auto with sets.
Intros x1 y0 z H3 H4 H5 H6 H7.
Apply Acc_inv with y0 ;Auto with sets.
Qed.
Hints Resolve Acc_clos_trans.
Lemma Acc_inv_trans: (x,y:A)(trans_clos y x)->(Acc A R x)->(Acc A R y).
Proof.
Induction 1;Auto with sets.
Intros x0 y0 H0 H1.
Apply Acc_inv with y0 ;Auto with sets.
Qed.
Theorem wf_clos_trans: (well_founded A R) ->(well_founded A trans_clos).
Proof.
Unfold well_founded;Auto with sets.
Qed.
End Wf_Transitive_Closure.
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