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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 1.7.2.1 2004/07/16 19:31:19 herbelin Exp $ i*)
(** Author: Bruno Barras *)
Require Import Relation_Definitions.
Require Import 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 in |- *; auto with sets.
Qed.
Lemma Acc_clos_trans : forall x:A, Acc R x -> Acc trans_clos x.
induction 1 as [x0 _ H1].
apply Acc_intro.
intros y H2.
induction H2; auto with sets.
apply Acc_inv with y; auto with sets.
Qed.
Hint Resolve Acc_clos_trans.
Lemma Acc_inv_trans : forall x y:A, trans_clos y x -> Acc R x -> Acc R y.
Proof.
induction 1 as [| x y]; auto with sets.
intro; apply Acc_inv with y; assumption.
Qed.
Theorem wf_clos_trans : well_founded R -> well_founded trans_clos.
Proof.
unfold well_founded in |- *; auto with sets.
Qed.
End Wf_Transitive_Closure.
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