(** 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. *)
(** * What is a real number belonging to a format, and many properties. *)
Section Generic.
Variable beta : radix.
Notation bpow e := (bpow beta e).
Section Format.
Variable fexp : Z -> Z.
(** To be a good fexp *)
Class Valid_exp :=
valid_exp :
forall k : Z,
( (fexp k < k)%Z -> (fexp (k + 1) <= k)%Z ) /\
( (k <= fexp k)%Z ->
(fexp (fexp k + 1) <= fexp k)%Z /\
forall l : Z, (l <= fexp k)%Z -> fexp l = fexp k ).
Context { valid_exp_ : Valid_exp }.

(* Specifications: *)
Theorem valid_exp_large :
forall k l,
(fexp k < k)%Z -> (k <= l)%Z ->
(fexp l < l)%Z.
Theorem valid_exp_large' :
forall k l,
(fexp k < k)%Z -> (l <= k)%Z ->
(fexp l < k)%Z.
Theorem valid_exp_large :
forall k l,
(fexp k < k)%Z -> (k <= l)%Z ->
(fexp l < l)%Z.
Theorem valid_exp_large' :
forall k l,
(fexp k < k)%Z -> (l <= k)%Z ->
(fexp l < k)%Z.