(** This file is part of the Flocq formalization of floating-point arithmetic in Coq: http://flocq.gforge.inria.fr/  Copyright (C) 2011-2013 Sylvie Boldo #<br /># Copyright (C) 2011-2013 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. *)
(** * IEEE-754 encoding of binary floating-point data *)
Section Binary_Bits.
Arguments exist [A] [P] _ _.
Arguments B754_zero [prec] [emax] _.
Arguments B754_infinity [prec] [emax] _.
Arguments B754_nan [prec] [emax] _ _.
Arguments B754_finite [prec] [emax] _ _ _ _.
(** Number of bits for the fraction and exponent *)
Variable mw ew : Z.
Hypothesis Hmw : (0 < mw)%Z.
Hypothesis Hew : (0 < ew)%Z.
Let emax := Zpower 2 (ew - 1).
Let prec := (mw + 1)%Z.
Let emin := (3 - emax - prec)%Z.
Let binary_float := binary_float prec emax.
Let Hprec : (0 < prec)%Z.
unfold prec.
apply Zle_lt_succ.
now apply Zlt_le_weak.
Qed.
Let Hm_gt_0 : (0 < 2^mw)%Z.
apply (Zpower_gt_0 radix2).
now apply Zlt_le_weak.
Qed.
Let He_gt_0 : (0 < 2^ew)%Z.
apply (Zpower_gt_0 radix2).
now apply Zlt_le_weak.
Qed.
Hypothesis Hmax : (prec < emax)%Z.
Definition join_bits (s : bool) m e :=
(Z.shiftl ((if s then Zpower 2 ew else 0) + e) mw + m)%Z.

Lemma join_bits_range :
forall s m e,
(0 <= m < 2^mw)%Z ->
(0 <= e < 2^ew)%Z ->
(0 <= join_bits s m e < 2 ^ (mw + ew + 1))%Z.