(** 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. *)
(** * Functions for computing the number of digits of integers and related theorems. *)
Section Fcalc_digits.
Variable beta : radix.
Notation bpow e := (bpow beta e).

(* Specifications: *)
Theorem Zdigits_ln_beta :
forall n,
n <> Z0 ->
Zdigits beta n = ln_beta beta (Z2R n).
Theorem ln_beta_F2R_Zdigits :
forall m e, m <> Z0 ->
(ln_beta beta (F2R (Float beta m e)) = Zdigits beta m + e :> Z)%Z.
Theorem Zdigits_ln_beta :
forall n,
n <> Z0 ->
Zdigits beta n = ln_beta beta (Z2R n).
Theorem ln_beta_F2R_Zdigits :
forall m e, m <> Z0 ->
(ln_beta beta (F2R (Float beta m e)) = Zdigits beta m + e :> Z)%Z.