(** This file is part of the Flocq formalization of floating-point arithmetic in Coq: http://flocq.gforge.inria.fr/  Copyright (C) 2010-2013 Sylvie Boldo #<br /># Copyright (C) 2010-2013 Guillaume Melquiond  This library is free software; you can redistribute it and/or modify it under the terms of the GNU Lesser General Public License as published by the Free Software Foundation; either version 3 of the License, or (at your option) any later version.  This library is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the COPYING file for more details. *)
(** * Floating-point format with abrupt underflow *)
Section RND_FTZ.
Variable beta : radix.
Notation bpow e := (bpow beta e).
Variable emin prec : Z.
Context { prec_gt_0_ : Prec_gt_0 prec }.
(* floating-point format with abrupt underflow *)
Definition FTZ_format (x : R) :=
exists f : float beta,
x = F2R f /\ (x <> R0 -> Zpower beta (prec - 1) <= Zabs (Fnum f) < Zpower beta prec)%Z /\
(emin <= Fexp f)%Z.
Definition FTZ_exp e := if Zlt_bool (e - prec) emin then (emin + prec - 1)%Z else (e - prec)%Z.
(** Properties of the FTZ format *)
Global Instance FTZ_exp_valid : Valid_exp FTZ_exp.

Theorem FTZ_format_FLXN :
forall x : R,
(bpow (emin + prec - 1) <= Rabs x)%R ->
FLXN_format beta prec x -> FTZ_format x.