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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: Inverse_Image.v,v 1.1.2.1 2004/07/16 19:31:41 herbelin Exp $ i*)
+
+(** Author: Bruno Barras *)
+
+Section Inverse_Image.
+
+  Variables A,B:Set.
+  Variable R : B->B->Prop.
+  Variable f:A->B.
+
+  Local Rof : A->A->Prop := [x,y:A](R (f x) (f y)).
+
+  Remark Acc_lemma : (y:B)(Acc B R y)->(x:A)(y=(f x))->(Acc A Rof x).
+    NewInduction 1 as [y _ IHAcc]; Intros x H.
+    Apply Acc_intro; Intros y0 H1.
+    Apply (IHAcc (f y0)); Try Trivial.
+    Rewrite H; Trivial.
+  Qed.
+
+  Lemma Acc_inverse_image : (x:A)(Acc B R (f x)) -> (Acc A Rof x).
+    Intros; Apply (Acc_lemma (f x)); Trivial.
+  Qed.
+
+  Theorem wf_inverse_image: (well_founded B R)->(well_founded A Rof).
+    Red; Intros; Apply Acc_inverse_image; Auto.
+  Qed.
+
+  Variable F : A -> B -> Prop.
+  Local RoF : A -> A -> Prop := [x,y]
+    (EX b : B | (F x b) & (c:B)(F y c)->(R b c)).
+
+Lemma Acc_inverse_rel :
+   (b:B)(Acc B R b)->(x:A)(F x b)->(Acc A RoF x).
+NewInduction 1 as [x _ IHAcc]; Intros x0 H2.
+Constructor; Intros y H3.
+NewDestruct H3.
+Apply (IHAcc x1); Auto.
+Save.
+
+
+Theorem wf_inverse_rel : 
+   (well_founded B R)->(well_founded A RoF).
+    Red; Constructor; Intros.
+    Case H0; Intros.
+    Apply (Acc_inverse_rel x); Auto.
+Save.
+
+End Inverse_Image.
+
+