Library Flocq.Calc.Fcalc_digits

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
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.


Require Import Fcore_Raux.
Require Import Fcore_defs.
Require Import Fcore_float_prop.
Require Import Fcore_digits.

Section Fcalc_digits.

Variable beta : radix.
Notation bpow e := (bpow beta e).

Theorem Zdigits_ln_beta :
   n,
  n Z0
  Zdigits beta n = ln_beta beta (Z2R n).

Theorem ln_beta_F2R_Zdigits :
   m e, m Z0
  (ln_beta beta (F2R (Float beta m e)) = Zdigits beta m + e :> Z)%Z.

End Fcalc_digits.

Definition radix2 := Build_radix 2 (refl_equal _).

Theorem Z_of_nat_S_digits2_Pnat :
   m : positive,
  Z_of_nat (S (digits2_Pnat m)) = Zdigits radix2 (Zpos m).