(* *********************************************************************)
(*                                                                     *)
(*              The Compcert verified compiler                         *)
(*                                                                     *)
(*          Xavier Leroy, INRIA Paris-Rocquencourt                     *)
(*                                                                     *)
(*  Copyright Institut National de Recherche en Informatique et en     *)
(*  Automatique.  All rights reserved.  This file is distributed       *)
(*  under the terms of the GNU General Public License as published by  *)
(*  the Free Software Foundation, either version 2 of the License, or  *)
(*  (at your option) any later version.  This file is also distributed *)
(*  under the terms of the INRIA Non-Commercial License Agreement.     *)
(*                                                                     *)
(* *********************************************************************)

(** Defining recursive functions by Noetherian induction.  This is a simplified
  interface to the [Wf] module of Coq's standard library, where the functions
  to be defined have non-dependent types, and function extensionality is assumed. *)

Require Import Axioms.
Require Import Init.Wf.
Require Import Wf_nat.
Set Implicit Arguments.
Section FIX.
Variables A B: Type.
Variable R: A -> A -> Prop.
Hypothesis Rwf: well_founded R.
Variable F: forall (x: A), (forall (y: A), R y x -> B) -> B.
Definition Fix (x: A) : B := Wf.Fix Rwf (fun (x: A) => B) F x.
Theorem unroll_Fix:
  forall x, Fix x = F (fun (y: A) (P: R y x) => Fix y).
Proof.
Admitted.
Proof.
unfold Fix.
intros.
apply Wf.Fix_eq with (P := fun (x: A) => B).
intros.
assert (f = g).
{
apply functional_extensionality_dep.
intros.
apply functional_extensionality.
intros.
auto.
}
{
subst g.
auto.
}
Qed.
End FIX.
(** Same, with a nonnegative measure instead of a well-founded ordering *)

Section FIXM.
Variables A B: Type.
Variable measure: A -> nat.
Variable F: forall (x: A), (forall (y: A), measure y < measure x -> B) -> B.
Definition Fixm (x: A) : B := Wf.Fix (well_founded_ltof A measure) (fun (x: A) => B) F x.
Theorem unroll_Fixm:
  forall x, Fixm x = F (fun (y: A) (P: measure y < measure x) => Fixm y).
Proof.
Admitted.
Proof.
unfold Fixm.
intros.
apply Wf.Fix_eq with (P := fun (x: A) => B).
intros.
assert (f = g).
{
apply functional_extensionality_dep.
intros.
apply functional_extensionality.
intros.
auto.
}
{
subst g.
auto.
}
Qed.
End FIXM.