(** This file is part of the Flocq formalization of floating-point arithmetic in Coq: http://flocq.gforge.inria.fr/  Copyright (C) 2011-2018 Sylvie Boldo #<br /># Copyright (C) 2011-2018 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. *)
(** Number of bits (radix 2) of a positive integer.  It serves as an upper bound on the number of digits to ensure termination. *)
Fixpoint digits2_Pnat (n : positive) : nat :=
match n with
| xH => O
| xO p => S (digits2_Pnat p)
| xI p => S (digits2_Pnat p)
end.

Theorem digits2_Pnat_correct :
forall n,
let d := digits2_Pnat n in
(Zpower_nat 2 d <= Zpos n < Zpower_nat 2 (S d))%Z.