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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 5920 2004-07-16 20:01:26Z herbelin $ i*)

(** Author: Bruno Barras *)

Section Inverse_Image.

  Variables A B : Set.
  Variable R : B -> B -> Prop.
  Variable f : A -> B.

  Let Rof (x y:A) : Prop := R (f x) (f y).

  Remark Acc_lemma : forall y:B, Acc R y -> forall x:A, y = f x -> Acc Rof x.
    induction 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 : forall x:A, Acc R (f x) -> Acc Rof x.
    intros; apply (Acc_lemma (f x)); trivial.
  Qed.

  Theorem wf_inverse_image : well_founded R -> well_founded Rof.
    red in |- *; intros; apply Acc_inverse_image; auto.
  Qed.

  Variable F : A -> B -> Prop.
  Let RoF (x y:A) : Prop :=
     exists2 b : B, F x b & (forall c:B, F y c -> R b c).

Lemma Acc_inverse_rel : forall b:B, Acc R b -> forall x:A, F x b -> Acc RoF x.
induction 1 as [x _ IHAcc]; intros x0 H2.
constructor; intros y H3.
destruct H3.
apply (IHAcc x1); auto.
Qed.


Theorem wf_inverse_rel : well_founded R -> well_founded RoF.
    red in |- *; constructor; intros.
    case H0; intros.
    apply (Acc_inverse_rel x); auto.
Qed.

End Inverse_Image.