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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2015 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(** Author: Cristina Cornes.
From: Constructing Recursion Operators in Type Theory
L. Paulson JSC (1986) 2, 325-355 *)
Require Import Eqdep.
Section WellOrdering.
Variable A : Type.
Variable B : A -> Type.
Inductive WO : Type :=
sup : forall (a:A) (f:B a -> WO), WO.
Inductive le_WO : WO -> WO -> Prop :=
le_sup : forall (a:A) (f:B a -> WO) (v:B a), le_WO (f v) (sup a f).
Theorem wf_WO : well_founded le_WO.
Proof.
unfold well_founded; intro.
apply Acc_intro.
elim a.
intros.
inversion H0.
apply Acc_intro.
generalize H4; generalize H1; generalize f0; generalize v.
rewrite H3.
intros.
apply (H v0 y0).
cut (f = f1).
intros E; rewrite E; auto.
symmetry .
apply (inj_pair2 A (fun a0:A => B a0 -> WO) a0 f1 f H5).
Qed.
End WellOrdering.
Section Characterisation_wf_relations.
(** Wellfounded relations are the inverse image of wellordering types *)
(* in course of development *)
Variable A : Type.
Variable leA : A -> A -> Prop.
Definition B (a:A) := {x : A | leA x a}.
Definition wof : well_founded leA -> A -> WO A B.
Proof.
intros.
apply (well_founded_induction_type H (fun a:A => WO A B)); auto.
intros x H1.
apply (sup A B x).
unfold B at 1.
destruct 1 as [x0].
apply (H1 x0); auto.
Qed.
End Characterisation_wf_relations.
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