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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: Transitive_Closure.v,v 1.1.2.1 2004/07/16 19:31:42 herbelin Exp $ 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).
NewInduction 1 as [x0 _ H1].
Apply Acc_intro.
Intros y H2.
NewInduction H2;Auto with sets.
Apply Acc_inv with y ;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.
NewInduction 1 as [|x y];Auto with sets.
Intro; Apply Acc_inv with y; Assumption.
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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