ajout de theories/Wellfounded

git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@900 85f007b7-540e-0410-9357-904b9bb8a0f7

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diff --git a/theories/Wellfounded/Transitive_Closure.v b/theories/Wellfounded/Transitive_Closure.v
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+
+(* $Id$ *)
+
+(****************************************************************************)
+(*                              Bruno Barras                                *)
+(****************************************************************************)
+
+Require Relation_Definitions.
+Require Relation_Operators.
+
+Section Wf_Transitive_Closure.
+  Variable A: Set.
+  Variable R: (relation A).
+
+  Syntactic Definition 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.