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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.
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