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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+
+(* $Id$ *)
+
+(****************************************************************************)
+(*                            Bruno Barras                                  *)
+(****************************************************************************)
+
+Require Relation_Operators.
+Require Relation_Definitions.
+Require Transitive_Closure.
+
+Section WfUnion.
+  Variable A: Set.
+  Variable R1,R2: (relation A).
+  
+ Syntactic Definition Union := (union A R1 R2).
+
+ Hints Resolve Acc_clos_trans wf_clos_trans.
+
+Remark strip_commut: 
+      (commut A R1 R2)->(x,y:A)(clos_trans A R1 y x)->(z:A)(R2 z y)
+            ->(EX y':A | (R2 y' x) & (clos_trans A R1 z y')).
+Proof.
+ Induction 2;Intros.
+ Elim H with y0 x0 z ;Auto with sets;Intros.
+ Exists x1;Auto with sets.
+
+ Elim H2 with z0 ;Auto with sets;Intros.
+ Elim H4 with x1 ;Auto with sets;Intros.
+ Exists x2;Auto with sets.
+ Apply t_trans with x1 ;Auto with sets.
+Save.
+
+
+  Lemma Acc_union:  (commut A R1 R2)->((x:A)(Acc A R2 x)->(Acc A R1 x))
+                         ->(a:A)(Acc A R2 a)->(Acc A Union a).
+Proof.
+ Induction 3.
+ Intros.
+ Apply Acc_intro;Intros.
+ Elim H4;Intros;Auto with sets.
+ Cut (clos_trans A  R1 y x);Auto with sets.
+ ElimType (Acc A (clos_trans A R1) y);Intros.
+ Apply Acc_intro;Intros.
+ Elim H9;Intros.
+ Apply H7;Auto with sets.
+ Apply t_trans with x0 ;Auto with sets.
+
+ Elim strip_commut with x x0 y0 ;Auto with sets;Intros.
+ Apply Acc_inv_trans with x1 ;Auto with sets.
+ Unfold union .
+ Elim H12;Auto with sets;Intros.
+ Apply t_trans with y1 ;Auto with sets.
+
+ Apply (Acc_clos_trans A).
+ Apply Acc_inv with x ;Auto with sets.
+ Apply H0.
+ Apply Acc_intro;Auto with sets.
+Save.
+
+
+  Theorem wf_union:  (commut A R1 R2)->(well_founded A R1)->(well_founded A R2)
+                  ->(well_founded A Union).
+Proof.
+ Unfold well_founded .
+ Intros.
+ Apply Acc_union;Auto with sets.
+Save.
+
+End WfUnion.