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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(** Author: Bruno Barras *)
Section Inverse_Image.
Variables A B : Type.
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.
Proof.
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.
Proof.
intros; apply (Acc_lemma (f x)); trivial.
Qed.
Theorem wf_inverse_image : well_founded R -> well_founded Rof.
Proof.
red; 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.
Proof.
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.
Proof.
red; constructor; intros.
case H0; intros.
apply (Acc_inverse_rel x); auto.
Qed.
End Inverse_Image.
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