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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.