(* *********************************************************************)
(*                                                                     *)
(*              The Compcert verified compiler                         *)
(*                                                                     *)
(*          Xavier Leroy, INRIA Paris-Rocquencourt                     *)
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(*  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. *)
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.

(* Specifications: *)
Theorem unroll_Fix:
forall x, Fix x = F (fun (y: A) (P: R y x) => Fix y).
Theorem unroll_Fix:
forall x, Fix x = F (fun (y: A) (P: R y x) => Fix y).