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/**************************************************************************/
/* */
/* The Why platform for program certification */
/* Copyright (C) 2002-2008 */
/* Romain BARDOU */
/* Jean-Franois COUCHOT */
/* Mehdi DOGGUY */
/* Jean-Christophe FILLITRE */
/* Thierry HUBERT */
/* Claude MARCH */
/* Yannick MOY */
/* Christine PAULIN */
/* Yann RGIS-GIANAS */
/* Nicolas ROUSSET */
/* Xavier URBAIN */
/* */
/* This software is free software; you can redistribute it and/or */
/* modify it under the terms of the GNU General Public */
/* License version 2, as published by the Free Software Foundation. */
/* */
/* This software 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 GNU General Public License version 2 for more details */
/* (enclosed in the file GPL). */
/* */
/**************************************************************************/
// the i-th bit of x as 0 or 2^i
//@ logic int bit(int x, int i)
// the number of bits 1 in x
//@ logic int nbits(int x)
//@ axiom nbits_nonneg : \forall int x; nbits(x) >= 0
//@ axiom nbits_zero : nbits(0) == 0
// the lowest bit 1 in x
//@ logic int lowest_bit(int x)
/*@ axiom lowest_bit_def :
@ \forall int x; x != 0 =>
@ \exists int i; (i >= 0 &&
@ lowest_bit(x) == bit(x,i) &&
@ bit(x,i) != 0 &&
@ \forall int j; 0 <= j < i => bit(x,j) == 0)
@*/
/*@ axiom lowest_bit_zero :
@ \forall int x; lowest_bit(x) == 0 <=> x == 0
@*/
/*@ axiom lowest_bit_trick :
@ \forall int x; x & -x == lowest_bit(x)
@*/
/*@ axiom remove_one_bit :
@ \forall int x, int i;
@ bit(x,i) != 0 => nbits(x - bit(x,i)) == nbits(x) - 1
@*/
/*@ ensures \result == nbits(x) */
int count_bits(int x) {
int d, c;
/*@ invariant c + nbits(x) == nbits(\at(x,init))
@ variant nbits(x)
@*/
for (c = 0; d = x&-x; x -= d) c++;
return c;
}
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