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compcert.flocq.Appli.Fappli_IEEE_bits.html
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<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
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<title>compcert.flocq.Appli.Fappli_IEEE_bits</title>
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<body>
<div id="page">
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<div id="main">
<h1 class="libtitle">Library compcert.flocq.Appli.Fappli_IEEE_bits</h1>
<div class="code">
</div>
<div class="doc">
This file is part of the Flocq formalization of floating-point
arithmetic in Coq: http://flocq.gforge.inria.fr/
<div class="paragraph"> </div>
Copyright (C) 2011-2013 Sylvie Boldo
<br />
Copyright (C) 2011-2013 Guillaume Melquiond
<div class="paragraph"> </div>
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.
<div class="paragraph"> </div>
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.
<div class="paragraph"> </div>
<a name="lab38"></a><h1 class="section">IEEE-754 encoding of binary floating-point data</h1>
</div>
<div class="code">
<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="compcert.flocq.Core.Fcore.html#"><span class="id" title="library">Fcore</span></a>.<br/>
<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="compcert.flocq.Core.Fcore_digits.html#"><span class="id" title="library">Fcore_digits</span></a>.<br/>
<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_digits.html#"><span class="id" title="library">Fcalc_digits</span></a>.<br/>
<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE.html#"><span class="id" title="library">Fappli_IEEE</span></a>.<br/>
<br/>
<span class="id" title="keyword">Section</span> <a name="Binary_Bits"><span class="id" title="section">Binary_Bits</span></a>.<br/>
<br/>
<span class="id" title="keyword">Implicit Arguments</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Specif.html#exist"><span class="id" title="constructor">exist</span></a> [[<span class="id" title="var">A</span>] [<span class="id" title="var">P</span>]].<br/>
<span class="id" title="keyword">Implicit Arguments</span> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE.html#B754_zero"><span class="id" title="constructor">B754_zero</span></a> [[<span class="id" title="var">prec</span>] [<span class="id" title="var">emax</span>]].<br/>
<span class="id" title="keyword">Implicit Arguments</span> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE.html#B754_infinity"><span class="id" title="constructor">B754_infinity</span></a> [[<span class="id" title="var">prec</span>] [<span class="id" title="var">emax</span>]].<br/>
<span class="id" title="keyword">Implicit Arguments</span> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE.html#B754_nan"><span class="id" title="constructor">B754_nan</span></a> [[<span class="id" title="var">prec</span>] [<span class="id" title="var">emax</span>]].<br/>
<span class="id" title="keyword">Implicit Arguments</span> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE.html#B754_finite"><span class="id" title="constructor">B754_finite</span></a> [[<span class="id" title="var">prec</span>] [<span class="id" title="var">emax</span>]].<br/>
<br/>
</div>
<div class="doc">
Number of bits for the fraction and exponent
</div>
<div class="code">
<span class="id" title="keyword">Variable</span> <a name="Binary_Bits.mw"><span class="id" title="variable">mw</span></a> <a name="Binary_Bits.ew"><span class="id" title="variable">ew</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>.<br/>
<span class="id" title="keyword">Hypothesis</span> <a name="Binary_Bits.Hmw"><span class="id" title="variable">Hmw</span></a> : (0 <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#273beba4f3453dbb29192b3ac740bb71"><span class="id" title="notation"><</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.mw"><span class="id" title="variable">mw</span></a>)%<span class="id" title="var">Z</span>.<br/>
<span class="id" title="keyword">Hypothesis</span> <a name="Binary_Bits.Hew"><span class="id" title="variable">Hew</span></a> : (0 <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#273beba4f3453dbb29192b3ac740bb71"><span class="id" title="notation"><</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.ew"><span class="id" title="variable">ew</span></a>)%<span class="id" title="var">Z</span>.<br/>
<br/>
<span class="id" title="keyword">Let</span> <a name="Binary_Bits.emax"><span class="id" title="variable">emax</span></a> := <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zpow_def.html#Zpower"><span class="id" title="abbreviation">Zpower</span></a> 2 (<a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.ew"><span class="id" title="variable">ew</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#239a40728d107295b3cb2c790f57f9e9"><span class="id" title="notation">-</span></a> 1).<br/>
<span class="id" title="keyword">Let</span> <a name="Binary_Bits.prec"><span class="id" title="variable">prec</span></a> := (<a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.mw"><span class="id" title="variable">mw</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a> 1)%<span class="id" title="var">Z</span>.<br/>
<span class="id" title="keyword">Let</span> <a name="Binary_Bits.emin"><span class="id" title="variable">emin</span></a> := (3 <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#239a40728d107295b3cb2c790f57f9e9"><span class="id" title="notation">-</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.emax"><span class="id" title="variable">emax</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#239a40728d107295b3cb2c790f57f9e9"><span class="id" title="notation">-</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.prec"><span class="id" title="variable">prec</span></a>)%<span class="id" title="var">Z</span>.<br/>
<span class="id" title="keyword">Let</span> <a name="Binary_Bits.binary_float"><span class="id" title="variable">binary_float</span></a> := <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE.html#binary_float"><span class="id" title="inductive">binary_float</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.prec"><span class="id" title="variable">prec</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.emax"><span class="id" title="variable">emax</span></a>.<br/>
<br/>
<span class="id" title="keyword">Let</span> <a name="Binary_Bits.Hprec"><span class="id" title="variable">Hprec</span></a> : (0 <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#273beba4f3453dbb29192b3ac740bb71"><span class="id" title="notation"><</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.prec"><span class="id" title="variable">prec</span></a>)%<span class="id" title="var">Z</span>.<br/>
<span class="id" title="tactic">unfold</span> <span class="id" title="var">prec</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zorder.html#Zle_lt_succ"><span class="id" title="lemma">Zle_lt_succ</span></a>.<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zorder.html#Zlt_le_weak"><span class="id" title="abbreviation">Zlt_le_weak</span></a>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Let</span> <a name="Binary_Bits.Hm_gt_0"><span class="id" title="variable">Hm_gt_0</span></a> : (0 <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#273beba4f3453dbb29192b3ac740bb71"><span class="id" title="notation"><</span></a> 2<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#186f1fc9207177b1a4e8373ac9655977"><span class="id" title="notation">^</span></a><a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.mw"><span class="id" title="variable">mw</span></a>)%<span class="id" title="var">Z</span>.<br/>
<span class="id" title="tactic">apply</span> (<a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#Zpower_gt_0"><span class="id" title="lemma">Zpower_gt_0</span></a> <a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#radix2"><span class="id" title="definition">radix2</span></a>).<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zorder.html#Zlt_le_weak"><span class="id" title="abbreviation">Zlt_le_weak</span></a>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Let</span> <a name="Binary_Bits.He_gt_0"><span class="id" title="variable">He_gt_0</span></a> : (0 <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#273beba4f3453dbb29192b3ac740bb71"><span class="id" title="notation"><</span></a> 2<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#186f1fc9207177b1a4e8373ac9655977"><span class="id" title="notation">^</span></a><a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.ew"><span class="id" title="variable">ew</span></a>)%<span class="id" title="var">Z</span>.<br/>
<span class="id" title="tactic">apply</span> (<a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#Zpower_gt_0"><span class="id" title="lemma">Zpower_gt_0</span></a> <a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#radix2"><span class="id" title="definition">radix2</span></a>).<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zorder.html#Zlt_le_weak"><span class="id" title="abbreviation">Zlt_le_weak</span></a>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Hypothesis</span> <a name="Binary_Bits.Hmax"><span class="id" title="variable">Hmax</span></a> : (<a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.prec"><span class="id" title="variable">prec</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#273beba4f3453dbb29192b3ac740bb71"><span class="id" title="notation"><</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.emax"><span class="id" title="variable">emax</span></a>)%<span class="id" title="var">Z</span>.<br/>
<br/>
<span class="id" title="keyword">Definition</span> <a name="join_bits"><span class="id" title="definition">join_bits</span></a> (<span class="id" title="var">s</span> : <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>) <span class="id" title="var">m</span> <span class="id" title="var">e</span> :=<br/>
(<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#Z.shiftl"><span class="id" title="definition">Z.shiftl</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">(</span></a><span class="id" title="keyword">if</span> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#s"><span class="id" title="variable">s</span></a> <span class="id" title="keyword">then</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zpow_def.html#Zpower"><span class="id" title="abbreviation">Zpower</span></a> 2 <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.ew"><span class="id" title="variable">ew</span></a> <span class="id" title="keyword">else</span> 0<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#e"><span class="id" title="variable">e</span></a>) <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.mw"><span class="id" title="variable">mw</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#m"><span class="id" title="variable">m</span></a>)%<span class="id" title="var">Z</span>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="join_bits_range"><span class="id" title="lemma">join_bits_range</span></a> :<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">s</span> <span class="id" title="var">m</span> <span class="id" title="var">e</span>,<br/>
(0 <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#cb2fd1306e4fe5d333fd9fb7e2c4a1d2"><span class="id" title="notation">≤</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#cb2fd1306e4fe5d333fd9fb7e2c4a1d2"><span class="id" title="notation"><</span></a> 2<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#186f1fc9207177b1a4e8373ac9655977"><span class="id" title="notation">^</span></a><a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.mw"><span class="id" title="variable">mw</span></a>)%<span class="id" title="var">Z</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/>
(0 <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#cb2fd1306e4fe5d333fd9fb7e2c4a1d2"><span class="id" title="notation">≤</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#cb2fd1306e4fe5d333fd9fb7e2c4a1d2"><span class="id" title="notation"><</span></a> 2<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#186f1fc9207177b1a4e8373ac9655977"><span class="id" title="notation">^</span></a><a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.ew"><span class="id" title="variable">ew</span></a>)%<span class="id" title="var">Z</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/>
(0 <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#cb2fd1306e4fe5d333fd9fb7e2c4a1d2"><span class="id" title="notation">≤</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#join_bits"><span class="id" title="definition">join_bits</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#cb2fd1306e4fe5d333fd9fb7e2c4a1d2"><span class="id" title="notation"><</span></a> 2 <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#186f1fc9207177b1a4e8373ac9655977"><span class="id" title="notation">^</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#186f1fc9207177b1a4e8373ac9655977"><span class="id" title="notation">(</span></a><a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.mw"><span class="id" title="variable">mw</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.ew"><span class="id" title="variable">ew</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#186f1fc9207177b1a4e8373ac9655977"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">Z</span>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">s</span> <span class="id" title="var">m</span> <span class="id" title="var">e</span> <span class="id" title="var">Hm</span> <span class="id" title="var">He</span>.<br/>
<span class="id" title="tactic">unfold</span> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#join_bits"><span class="id" title="definition">join_bits</span></a>.<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#Z.shiftl_mul_pow2"><span class="id" title="lemma">Z.shiftl_mul_pow2</span></a> <span class="id" title="tactic">by</span> <span class="id" title="var">now</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zorder.html#Zlt_le_weak"><span class="id" title="abbreviation">Zlt_le_weak</span></a>.<br/>
<span class="id" title="tactic">split</span>.<br/>
- <span class="id" title="tactic">apply</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zorder.html#Zplus_le_compat"><span class="id" title="abbreviation">Zplus_le_compat</span></a> 0 <span class="id" title="var">_</span> 0) <span class="id" title="keyword">with</span> (2 := <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#proj1"><span class="id" title="lemma">proj1</span></a> <span class="id" title="var">Hm</span>).<br/>
<span class="id" title="tactic">rewrite</span> <- (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#Zmult_0_l"><span class="id" title="abbreviation">Zmult_0_l</span></a> (2<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#186f1fc9207177b1a4e8373ac9655977"><span class="id" title="notation">^</span></a><a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.mw"><span class="id" title="variable">mw</span></a>)).<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zorder.html#Zmult_le_compat_r"><span class="id" title="lemma">Zmult_le_compat_r</span></a>.<br/>
<span class="id" title="tactic">case</span> <span class="id" title="var">s</span>.<br/>
<span class="id" title="tactic">clear</span> -<span class="id" title="var">He</span> ; <span class="id" title="tactic">omega</span>.<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#Zmult_0_l"><span class="id" title="abbreviation">Zmult_0_l</span></a>.<br/>
<span class="id" title="tactic">clear</span> -<span class="id" title="var">Hm</span> ; <span class="id" title="tactic">omega</span>.<br/>
- <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zorder.html#Zlt_le_trans"><span class="id" title="abbreviation">Zlt_le_trans</span></a> <span class="id" title="keyword">with</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#88e34ed7a73f02772126289867960764"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">(</span></a><span class="id" title="keyword">if</span> <span class="id" title="var">s</span> <span class="id" title="keyword">then</span> 2 <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#186f1fc9207177b1a4e8373ac9655977"><span class="id" title="notation">^</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.ew"><span class="id" title="variable">ew</span></a> <span class="id" title="keyword">else</span> 0<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a> <span class="id" title="var">e</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#88e34ed7a73f02772126289867960764"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#88e34ed7a73f02772126289867960764"><span class="id" title="notation">×</span></a> 2 <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#186f1fc9207177b1a4e8373ac9655977"><span class="id" title="notation">^</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.mw"><span class="id" title="variable">mw</span></a>)%<span class="id" title="var">Z</span>.<br/>
<span class="id" title="tactic">rewrite</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#Zmult_plus_distr_l"><span class="id" title="abbreviation">Zmult_plus_distr_l</span></a> <span class="id" title="var">_</span> 1).<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zorder.html#Zplus_lt_compat_l"><span class="id" title="lemma">Zplus_lt_compat_l</span></a>.<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#Zmult_1_l"><span class="id" title="abbreviation">Zmult_1_l</span></a>.<br/>
<span class="id" title="tactic">rewrite</span> <- (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#Zplus_assoc"><span class="id" title="abbreviation">Zplus_assoc</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.mw"><span class="id" title="variable">mw</span></a>), (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#Zplus_comm"><span class="id" title="abbreviation">Zplus_comm</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.mw"><span class="id" title="variable">mw</span></a>), <a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#Zpower_plus"><span class="id" title="lemma">Zpower_plus</span></a>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zorder.html#Zmult_le_compat_r"><span class="id" title="lemma">Zmult_le_compat_r</span></a>.<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#Zpower_plus"><span class="id" title="lemma">Zpower_plus</span></a>.<br/>
<span class="id" title="tactic">change</span> (2<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#186f1fc9207177b1a4e8373ac9655977"><span class="id" title="notation">^</span></a>1)%<span class="id" title="var">Z</span> <span class="id" title="keyword">with</span> 2%<span class="id" title="var">Z</span>.<br/>
<span class="id" title="tactic">case</span> <span class="id" title="var">s</span> ; <span class="id" title="tactic">clear</span> -<span class="id" title="var">He</span> ; <span class="id" title="tactic">omega</span>.<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zorder.html#Zlt_le_weak"><span class="id" title="abbreviation">Zlt_le_weak</span></a>.<br/>
<span class="id" title="var">easy</span>.<br/>
<span class="id" title="tactic">clear</span> -<span class="id" title="var">Hm</span> ; <span class="id" title="tactic">omega</span>.<br/>
<span class="id" title="tactic">clear</span> -<span class="id" title="var">Hew</span> ; <span class="id" title="tactic">omega</span>.<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zorder.html#Zlt_le_weak"><span class="id" title="abbreviation">Zlt_le_weak</span></a>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Definition</span> <a name="split_bits"><span class="id" title="definition">split_bits</span></a> <span class="id" title="var">x</span> :=<br/>
<span class="id" title="keyword">let</span> <span class="id" title="var">mm</span> := <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zpow_def.html#Zpower"><span class="id" title="abbreviation">Zpower</span></a> 2 <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.mw"><span class="id" title="variable">mw</span></a> <span class="id" title="keyword">in</span><br/>
<span class="id" title="keyword">let</span> <span class="id" title="var">em</span> := <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zpow_def.html#Zpower"><span class="id" title="abbreviation">Zpower</span></a> 2 <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.ew"><span class="id" title="variable">ew</span></a> <span class="id" title="keyword">in</span><br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zbool.html#Zle_bool"><span class="id" title="abbreviation">Zle_bool</span></a> (<a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#mm"><span class="id" title="variable">mm</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#88e34ed7a73f02772126289867960764"><span class="id" title="notation">×</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#em"><span class="id" title="variable">em</span></a>) <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zdiv.html#Zmod"><span class="id" title="abbreviation">Zmod</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#mm"><span class="id" title="variable">mm</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zdiv.html#Zmod"><span class="id" title="abbreviation">Zmod</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zdiv.html#Zdiv"><span class="id" title="abbreviation">Zdiv</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#mm"><span class="id" title="variable">mm</span></a>) <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#em"><span class="id" title="variable">em</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>%<span class="id" title="var">Z</span>.<br/>
<br/>
<span class="id" title="keyword">Theorem</span> <a name="split_join_bits"><span class="id" title="lemma">split_join_bits</span></a> :<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">s</span> <span class="id" title="var">m</span> <span class="id" title="var">e</span>,<br/>
(0 <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#cb2fd1306e4fe5d333fd9fb7e2c4a1d2"><span class="id" title="notation">≤</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#cb2fd1306e4fe5d333fd9fb7e2c4a1d2"><span class="id" title="notation"><</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zpow_def.html#Zpower"><span class="id" title="abbreviation">Zpower</span></a> 2 <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.mw"><span class="id" title="variable">mw</span></a>)%<span class="id" title="var">Z</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/>
(0 <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#cb2fd1306e4fe5d333fd9fb7e2c4a1d2"><span class="id" title="notation">≤</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#cb2fd1306e4fe5d333fd9fb7e2c4a1d2"><span class="id" title="notation"><</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zpow_def.html#Zpower"><span class="id" title="abbreviation">Zpower</span></a> 2 <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.ew"><span class="id" title="variable">ew</span></a>)%<span class="id" title="var">Z</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#split_bits"><span class="id" title="definition">split_bits</span></a> (<a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#join_bits"><span class="id" title="definition">join_bits</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#e"><span class="id" title="variable">e</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#s"><span class="id" title="variable">s</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#e"><span class="id" title="variable">e</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">s</span> <span class="id" title="var">m</span> <span class="id" title="var">e</span> <span class="id" title="var">Hm</span> <span class="id" title="var">He</span>.<br/>
<span class="id" title="tactic">unfold</span> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#split_bits"><span class="id" title="definition">split_bits</span></a>, <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#join_bits"><span class="id" title="definition">join_bits</span></a>.<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#Z.shiftl_mul_pow2"><span class="id" title="lemma">Z.shiftl_mul_pow2</span></a> <span class="id" title="tactic">by</span> <span class="id" title="var">now</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zorder.html#Zlt_le_weak"><span class="id" title="abbreviation">Zlt_le_weak</span></a>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#f_equal2"><span class="id" title="lemma">f_equal2</span></a>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#f_equal2"><span class="id" title="lemma">f_equal2</span></a>.<br/>
<span class="id" title="tactic">case</span> <span class="id" title="var">s</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#Zle_bool_true"><span class="id" title="lemma">Zle_bool_true</span></a>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zorder.html#Zle_0_minus_le"><span class="id" title="lemma">Zle_0_minus_le</span></a>.<br/>
<span class="id" title="var">ring_simplify</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zorder.html#Zplus_le_0_compat"><span class="id" title="abbreviation">Zplus_le_0_compat</span></a>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zorder.html#Zmult_le_0_compat"><span class="id" title="abbreviation">Zmult_le_0_compat</span></a>.<br/>
<span class="id" title="tactic">apply</span> <span class="id" title="var">He</span>.<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zorder.html#Zlt_le_weak"><span class="id" title="abbreviation">Zlt_le_weak</span></a>.<br/>
<span class="id" title="tactic">apply</span> <span class="id" title="var">Hm</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#Zle_bool_false"><span class="id" title="lemma">Zle_bool_false</span></a>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zorder.html#Zplus_lt_reg_l"><span class="id" title="lemma">Zplus_lt_reg_l</span></a> <span class="id" title="keyword">with</span> (2<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#186f1fc9207177b1a4e8373ac9655977"><span class="id" title="notation">^</span></a><a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.mw"><span class="id" title="variable">mw</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#88e34ed7a73f02772126289867960764"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#88e34ed7a73f02772126289867960764"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#0a3978eabdacce0a128118074de19651"><span class="id" title="notation">-</span></a><span class="id" title="var">e</span><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#88e34ed7a73f02772126289867960764"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">Z</span>.<br/>
<span class="id" title="tactic">replace</span> (2 <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#186f1fc9207177b1a4e8373ac9655977"><span class="id" title="notation">^</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.mw"><span class="id" title="variable">mw</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#88e34ed7a73f02772126289867960764"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#0a3978eabdacce0a128118074de19651"><span class="id" title="notation">-</span></a> <span class="id" title="var">e</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#88e34ed7a73f02772126289867960764"><span class="id" title="notation">(</span></a>0 <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a> <span class="id" title="var">e</span><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#88e34ed7a73f02772126289867960764"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#88e34ed7a73f02772126289867960764"><span class="id" title="notation">×</span></a> 2 <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#186f1fc9207177b1a4e8373ac9655977"><span class="id" title="notation">^</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.mw"><span class="id" title="variable">mw</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a> <span class="id" title="var">m</span><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">Z</span> <span class="id" title="keyword">with</span> (<span class="id" title="var">m</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#88e34ed7a73f02772126289867960764"><span class="id" title="notation">×</span></a> 1)%<span class="id" title="var">Z</span> <span class="id" title="tactic">by</span> <span class="id" title="tactic">ring</span>.<br/>
<span class="id" title="tactic">rewrite</span> <- <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#Zmult_plus_distr_r"><span class="id" title="abbreviation">Zmult_plus_distr_r</span></a>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zorder.html#Zlt_le_trans"><span class="id" title="abbreviation">Zlt_le_trans</span></a> <span class="id" title="keyword">with</span> (2<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#186f1fc9207177b1a4e8373ac9655977"><span class="id" title="notation">^</span></a><a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.mw"><span class="id" title="variable">mw</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#88e34ed7a73f02772126289867960764"><span class="id" title="notation">×</span></a> 1)%<span class="id" title="var">Z</span>.<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zorder.html#Zmult_lt_compat_r"><span class="id" title="lemma">Zmult_lt_compat_r</span></a>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zorder.html#Zmult_le_compat_l"><span class="id" title="lemma">Zmult_le_compat_l</span></a>.<br/>
<span class="id" title="tactic">clear</span> -<span class="id" title="var">He</span>. <span class="id" title="tactic">omega</span>.<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zorder.html#Zlt_le_weak"><span class="id" title="abbreviation">Zlt_le_weak</span></a>.<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#Zplus_comm"><span class="id" title="abbreviation">Zplus_comm</span></a>.<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zdiv.html#Z_mod_plus_full"><span class="id" title="lemma">Z_mod_plus_full</span></a>.<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zdiv.html#Zmod_small"><span class="id" title="lemma">Zmod_small</span></a>.<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zdiv.html#Z_div_plus_full_l"><span class="id" title="lemma">Z_div_plus_full_l</span></a>.<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zdiv.html#Zdiv_small"><span class="id" title="lemma">Zdiv_small</span></a> <span class="id" title="keyword">with</span> (1 := <span class="id" title="var">Hm</span>).<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#Zplus_0_r"><span class="id" title="abbreviation">Zplus_0_r</span></a>.<br/>
<span class="id" title="tactic">case</span> <span class="id" title="var">s</span>.<br/>
<span class="id" title="tactic">replace</span> (2<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#186f1fc9207177b1a4e8373ac9655977"><span class="id" title="notation">^</span></a><a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.ew"><span class="id" title="variable">ew</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a> <span class="id" title="var">e</span>)%<span class="id" title="var">Z</span> <span class="id" title="keyword">with</span> (<span class="id" title="var">e</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#88e34ed7a73f02772126289867960764"><span class="id" title="notation">×</span></a> 2<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#186f1fc9207177b1a4e8373ac9655977"><span class="id" title="notation">^</span></a><a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.ew"><span class="id" title="variable">ew</span></a>)%<span class="id" title="var">Z</span> <span class="id" title="tactic">by</span> <span class="id" title="tactic">ring</span>.<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zdiv.html#Z_mod_plus_full"><span class="id" title="lemma">Z_mod_plus_full</span></a>.<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zdiv.html#Zmod_small"><span class="id" title="lemma">Zmod_small</span></a>.<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zdiv.html#Zmod_small"><span class="id" title="lemma">Zmod_small</span></a>.<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#Zgt_not_eq"><span class="id" title="lemma">Zgt_not_eq</span></a>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Theorem</span> <a name="join_split_bits"><span class="id" title="lemma">join_split_bits</span></a> :<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>,<br/>
(0 <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#cb2fd1306e4fe5d333fd9fb7e2c4a1d2"><span class="id" title="notation">≤</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#cb2fd1306e4fe5d333fd9fb7e2c4a1d2"><span class="id" title="notation"><</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zpow_def.html#Zpower"><span class="id" title="abbreviation">Zpower</span></a> 2 (<a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.mw"><span class="id" title="variable">mw</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.ew"><span class="id" title="variable">ew</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a> 1))%<span class="id" title="var">Z</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/>
<span class="id" title="keyword">let</span> '<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><span class="id" title="var">s</span><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <span class="id" title="var">m</span><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <span class="id" title="var">e</span><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a> := <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#split_bits"><span class="id" title="definition">split_bits</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#x"><span class="id" title="variable">x</span></a> <span class="id" title="keyword">in</span><br/>
<a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#join_bits"><span class="id" title="definition">join_bits</span></a> <span class="id" title="var">s</span> <span class="id" title="var">m</span> <span class="id" title="var">e</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#x"><span class="id" title="variable">x</span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">x</span> <span class="id" title="var">Hx</span>.<br/>
<span class="id" title="tactic">unfold</span> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#split_bits"><span class="id" title="definition">split_bits</span></a>, <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#join_bits"><span class="id" title="definition">join_bits</span></a>.<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#Z.shiftl_mul_pow2"><span class="id" title="lemma">Z.shiftl_mul_pow2</span></a> <span class="id" title="tactic">by</span> <span class="id" title="var">now</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zorder.html#Zlt_le_weak"><span class="id" title="abbreviation">Zlt_le_weak</span></a>.<br/>
<span class="id" title="tactic">pattern</span> <span class="id" title="var">x</span> <span class="id" title="tactic">at</span> 4 ; <span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zdiv.html#Z_div_mod_eq_full"><span class="id" title="abbreviation">Z_div_mod_eq_full</span></a> <span class="id" title="keyword">with</span> <span class="id" title="var">x</span> (2<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#186f1fc9207177b1a4e8373ac9655977"><span class="id" title="notation">^</span></a><a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.mw"><span class="id" title="variable">mw</span></a>)%<span class="id" title="var">Z</span>.<br/>
<span class="id" title="tactic">apply</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#f_equal"><span class="id" title="lemma">f_equal</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">v</span> ⇒ (<a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#v"><span class="id" title="variable">v</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a> <span class="id" title="var">_</span>)%<span class="id" title="var">Z</span>)).<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#Zmult_comm"><span class="id" title="abbreviation">Zmult_comm</span></a>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#f_equal"><span class="id" title="lemma">f_equal</span></a>.<br/>
<span class="id" title="tactic">pattern</span> (<span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#66073ffd934badf1f6e703023820601f"><span class="id" title="notation">/</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#66073ffd934badf1f6e703023820601f"><span class="id" title="notation">(</span></a>2<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#186f1fc9207177b1a4e8373ac9655977"><span class="id" title="notation">^</span></a><a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.mw"><span class="id" title="variable">mw</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#66073ffd934badf1f6e703023820601f"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">Z</span> <span class="id" title="tactic">at</span> 2 ; <span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zdiv.html#Z_div_mod_eq_full"><span class="id" title="abbreviation">Z_div_mod_eq_full</span></a> <span class="id" title="keyword">with</span> (<span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#66073ffd934badf1f6e703023820601f"><span class="id" title="notation">/</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#66073ffd934badf1f6e703023820601f"><span class="id" title="notation">(</span></a>2<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#186f1fc9207177b1a4e8373ac9655977"><span class="id" title="notation">^</span></a><a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.mw"><span class="id" title="variable">mw</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#66073ffd934badf1f6e703023820601f"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">Z</span> (2<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#186f1fc9207177b1a4e8373ac9655977"><span class="id" title="notation">^</span></a><a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.ew"><span class="id" title="variable">ew</span></a>)%<span class="id" title="var">Z</span>.<br/>
<span class="id" title="tactic">apply</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#f_equal"><span class="id" title="lemma">f_equal</span></a> (<span class="id" title="keyword">fun</span> <span class="id" title="var">v</span> ⇒ (<a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#v"><span class="id" title="variable">v</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a> <span class="id" title="var">_</span>)%<span class="id" title="var">Z</span>)).<br/>
<span class="id" title="tactic">replace</span> (<span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#66073ffd934badf1f6e703023820601f"><span class="id" title="notation">/</span></a> 2 <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#186f1fc9207177b1a4e8373ac9655977"><span class="id" title="notation">^</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.mw"><span class="id" title="variable">mw</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#66073ffd934badf1f6e703023820601f"><span class="id" title="notation">/</span></a> 2 <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#186f1fc9207177b1a4e8373ac9655977"><span class="id" title="notation">^</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.ew"><span class="id" title="variable">ew</span></a>)%<span class="id" title="var">Z</span> <span class="id" title="keyword">with</span> (<span class="id" title="keyword">if</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zbool.html#Zle_bool"><span class="id" title="abbreviation">Zle_bool</span></a> (2 <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#186f1fc9207177b1a4e8373ac9655977"><span class="id" title="notation">^</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.mw"><span class="id" title="variable">mw</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#88e34ed7a73f02772126289867960764"><span class="id" title="notation">×</span></a> 2 <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#186f1fc9207177b1a4e8373ac9655977"><span class="id" title="notation">^</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.ew"><span class="id" title="variable">ew</span></a>) <span class="id" title="var">x</span> <span class="id" title="keyword">then</span> 1 <span class="id" title="keyword">else</span> 0)%<span class="id" title="var">Z</span>.<br/>
<span class="id" title="tactic">case</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zbool.html#Zle_bool"><span class="id" title="abbreviation">Zle_bool</span></a>.<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#Zmult_1_r"><span class="id" title="abbreviation">Zmult_1_r</span></a>.<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#Zmult_0_r"><span class="id" title="abbreviation">Zmult_0_r</span></a>.<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zdiv.html#Zdiv_Zdiv"><span class="id" title="lemma">Zdiv_Zdiv</span></a>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#sym_eq"><span class="id" title="abbreviation">sym_eq</span></a>.<br/>
<span class="id" title="tactic">case</span> <a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#Zle_bool_spec"><span class="id" title="lemma">Zle_bool_spec</span></a> ; <span class="id" title="tactic">intros</span> <span class="id" title="var">Hs</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zorder.html#Zle_antisym"><span class="id" title="abbreviation">Zle_antisym</span></a>.<br/>
<span class="id" title="tactic">cut</span> (<span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#66073ffd934badf1f6e703023820601f"><span class="id" title="notation">/</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#66073ffd934badf1f6e703023820601f"><span class="id" title="notation">(</span></a>2<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#186f1fc9207177b1a4e8373ac9655977"><span class="id" title="notation">^</span></a><a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.mw"><span class="id" title="variable">mw</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#88e34ed7a73f02772126289867960764"><span class="id" title="notation">×</span></a> 2<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#186f1fc9207177b1a4e8373ac9655977"><span class="id" title="notation">^</span></a><a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.ew"><span class="id" title="variable">ew</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#66073ffd934badf1f6e703023820601f"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#273beba4f3453dbb29192b3ac740bb71"><span class="id" title="notation"><</span></a> 2)%<span class="id" title="var">Z</span>. <span class="id" title="tactic">clear</span> ; <span class="id" title="tactic">omega</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zdiv.html#Zdiv_lt_upper_bound"><span class="id" title="lemma">Zdiv_lt_upper_bound</span></a>.<br/>
<span class="id" title="tactic">try</span> <span class="id" title="tactic">apply</span> <span class="id" title="var">Hx</span>. <span class="id" title="var">now</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zorder.html#Zmult_lt_0_compat"><span class="id" title="abbreviation">Zmult_lt_0_compat</span></a>.<br/>
<span class="id" title="tactic">rewrite</span> <- <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zpower.html#Zpower_exp"><span class="id" title="lemma">Zpower_exp</span></a> ; <span class="id" title="tactic">try</span> ( <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zorder.html#Zle_ge"><span class="id" title="abbreviation">Zle_ge</span></a> ; <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zorder.html#Zlt_le_weak"><span class="id" title="abbreviation">Zlt_le_weak</span></a> ; <span class="id" title="tactic">assumption</span> ).<br/>
<span class="id" title="tactic">change</span> 2%<span class="id" title="var">Z</span> <span class="id" title="tactic">at</span> 1 <span class="id" title="keyword">with</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zpow_def.html#Zpower"><span class="id" title="abbreviation">Zpower</span></a> 2 1).<br/>
<span class="id" title="tactic">rewrite</span> <- <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zpower.html#Zpower_exp"><span class="id" title="lemma">Zpower_exp</span></a>.<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#Zplus_comm"><span class="id" title="abbreviation">Zplus_comm</span></a>.<br/>
<span class="id" title="tactic">discriminate</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zorder.html#Zle_ge"><span class="id" title="abbreviation">Zle_ge</span></a>.<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zorder.html#Zplus_le_0_compat"><span class="id" title="abbreviation">Zplus_le_0_compat</span></a> ; <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zorder.html#Zlt_le_weak"><span class="id" title="abbreviation">Zlt_le_weak</span></a>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zdiv.html#Zdiv_le_lower_bound"><span class="id" title="lemma">Zdiv_le_lower_bound</span></a>.<br/>
<span class="id" title="tactic">try</span> <span class="id" title="tactic">apply</span> <span class="id" title="var">Hx</span>. <span class="id" title="var">now</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zorder.html#Zmult_lt_0_compat"><span class="id" title="abbreviation">Zmult_lt_0_compat</span></a>.<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#Zmult_1_l"><span class="id" title="abbreviation">Zmult_1_l</span></a>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zdiv.html#Zdiv_small"><span class="id" title="lemma">Zdiv_small</span></a>.<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">split</span>.<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zorder.html#Zlt_le_weak"><span class="id" title="abbreviation">Zlt_le_weak</span></a>.<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zorder.html#Zlt_le_weak"><span class="id" title="abbreviation">Zlt_le_weak</span></a>.<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#Zgt_not_eq"><span class="id" title="lemma">Zgt_not_eq</span></a>.<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#Zgt_not_eq"><span class="id" title="lemma">Zgt_not_eq</span></a>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Theorem</span> <a name="split_bits_inj"><span class="id" title="lemma">split_bits_inj</span></a> :<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span>,<br/>
(0 <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#cb2fd1306e4fe5d333fd9fb7e2c4a1d2"><span class="id" title="notation">≤</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#cb2fd1306e4fe5d333fd9fb7e2c4a1d2"><span class="id" title="notation"><</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zpow_def.html#Zpower"><span class="id" title="abbreviation">Zpower</span></a> 2 (<a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.mw"><span class="id" title="variable">mw</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.ew"><span class="id" title="variable">ew</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a> 1))%<span class="id" title="var">Z</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/>
(0 <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#cb2fd1306e4fe5d333fd9fb7e2c4a1d2"><span class="id" title="notation">≤</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#cb2fd1306e4fe5d333fd9fb7e2c4a1d2"><span class="id" title="notation"><</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zpow_def.html#Zpower"><span class="id" title="abbreviation">Zpower</span></a> 2 (<a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.mw"><span class="id" title="variable">mw</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.ew"><span class="id" title="variable">ew</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a> 1))%<span class="id" title="var">Z</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#split_bits"><span class="id" title="definition">split_bits</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#split_bits"><span class="id" title="definition">split_bits</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#y"><span class="id" title="variable">y</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#y"><span class="id" title="variable">y</span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> <span class="id" title="var">Hx</span> <span class="id" title="var">Hy</span>.<br/>
<span class="id" title="tactic">generalize</span> (<a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#join_split_bits"><span class="id" title="lemma">join_split_bits</span></a> <span class="id" title="var">x</span> <span class="id" title="var">Hx</span>) (<a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#join_split_bits"><span class="id" title="lemma">join_split_bits</span></a> <span class="id" title="var">y</span> <span class="id" title="var">Hy</span>).<br/>
<span class="id" title="tactic">destruct</span> (<a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#split_bits"><span class="id" title="definition">split_bits</span></a> <span class="id" title="var">x</span>) <span class="id" title="keyword">as</span> ((<span class="id" title="var">sx</span>, <span class="id" title="var">mx</span>), <span class="id" title="var">ex</span>).<br/>
<span class="id" title="tactic">destruct</span> (<a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#split_bits"><span class="id" title="definition">split_bits</span></a> <span class="id" title="var">y</span>) <span class="id" title="keyword">as</span> ((<span class="id" title="var">sy</span>, <span class="id" title="var">my</span>), <span class="id" title="var">ey</span>).<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">Jx</span> <span class="id" title="var">Jy</span> <span class="id" title="var">H</span>. <span class="id" title="var">revert</span> <span class="id" title="var">Jx</span> <span class="id" title="var">Jy</span>.<br/>
<span class="id" title="tactic">inversion_clear</span> <span class="id" title="var">H</span>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">Jx</span> <span class="id" title="var">Jy</span>.<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">rewrite</span> <- <span class="id" title="var">Jx</span>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Definition</span> <a name="bits_of_binary_float"><span class="id" title="definition">bits_of_binary_float</span></a> (<span class="id" title="var">x</span> : <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.binary_float"><span class="id" title="variable">binary_float</span></a>) :=<br/>
<span class="id" title="keyword">match</span> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#x"><span class="id" title="variable">x</span></a> <span class="id" title="keyword">with</span><br/>
| <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE.html#B754_zero"><span class="id" title="constructor">B754_zero</span></a> <span class="id" title="var">sx</span> ⇒ <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#join_bits"><span class="id" title="definition">join_bits</span></a> <span class="id" title="var">sx</span> 0 0<br/>
| <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE.html#B754_infinity"><span class="id" title="constructor">B754_infinity</span></a> <span class="id" title="var">sx</span> ⇒ <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#join_bits"><span class="id" title="definition">join_bits</span></a> <span class="id" title="var">sx</span> 0 (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zpow_def.html#Zpower"><span class="id" title="abbreviation">Zpower</span></a> 2 <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.ew"><span class="id" title="variable">ew</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#239a40728d107295b3cb2c790f57f9e9"><span class="id" title="notation">-</span></a> 1)<br/>
| <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE.html#B754_nan"><span class="id" title="constructor">B754_nan</span></a> <span class="id" title="var">sx</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Specif.html#exist"><span class="id" title="constructor">exist</span></a> <span class="id" title="var">plx</span> <span class="id" title="var">_</span>) ⇒ <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#join_bits"><span class="id" title="definition">join_bits</span></a> <span class="id" title="var">sx</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#Zpos"><span class="id" title="constructor">Zpos</span></a> <span class="id" title="var">plx</span>) (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zpow_def.html#Zpower"><span class="id" title="abbreviation">Zpower</span></a> 2 <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.ew"><span class="id" title="variable">ew</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#239a40728d107295b3cb2c790f57f9e9"><span class="id" title="notation">-</span></a> 1)<br/>
| <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE.html#B754_finite"><span class="id" title="constructor">B754_finite</span></a> <span class="id" title="var">sx</span> <span class="id" title="var">mx</span> <span class="id" title="var">ex</span> <span class="id" title="var">_</span> ⇒<br/>
<span class="id" title="keyword">let</span> <span class="id" title="var">m</span> := (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#Zpos"><span class="id" title="constructor">Zpos</span></a> <span class="id" title="var">mx</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#239a40728d107295b3cb2c790f57f9e9"><span class="id" title="notation">-</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zpow_def.html#Zpower"><span class="id" title="abbreviation">Zpower</span></a> 2 <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.mw"><span class="id" title="variable">mw</span></a>)%<span class="id" title="var">Z</span> <span class="id" title="keyword">in</span><br/>
<span class="id" title="keyword">if</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zbool.html#Zle_bool"><span class="id" title="abbreviation">Zle_bool</span></a> 0 <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#m"><span class="id" title="variable">m</span></a> <span class="id" title="keyword">then</span><br/>
<a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#join_bits"><span class="id" title="definition">join_bits</span></a> <span class="id" title="var">sx</span> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#m"><span class="id" title="variable">m</span></a> (<span class="id" title="var">ex</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#239a40728d107295b3cb2c790f57f9e9"><span class="id" title="notation">-</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.emin"><span class="id" title="variable">emin</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a> 1)<br/>
<span class="id" title="keyword">else</span><br/>
<a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#join_bits"><span class="id" title="definition">join_bits</span></a> <span class="id" title="var">sx</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#Zpos"><span class="id" title="constructor">Zpos</span></a> <span class="id" title="var">mx</span>) 0<br/>
<span class="id" title="keyword">end</span>.<br/>
<br/>
<span class="id" title="keyword">Definition</span> <a name="split_bits_of_binary_float"><span class="id" title="definition">split_bits_of_binary_float</span></a> (<span class="id" title="var">x</span> : <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.binary_float"><span class="id" title="variable">binary_float</span></a>) :=<br/>
<span class="id" title="keyword">match</span> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#x"><span class="id" title="variable">x</span></a> <span class="id" title="keyword">with</span><br/>
| <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE.html#B754_zero"><span class="id" title="constructor">B754_zero</span></a> <span class="id" title="var">sx</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><span class="id" title="var">sx</span><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>%<span class="id" title="var">Z</span><br/>
| <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE.html#B754_infinity"><span class="id" title="constructor">B754_infinity</span></a> <span class="id" title="var">sx</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><span class="id" title="var">sx</span><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zpow_def.html#Zpower"><span class="id" title="abbreviation">Zpower</span></a> 2 <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.ew"><span class="id" title="variable">ew</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#239a40728d107295b3cb2c790f57f9e9"><span class="id" title="notation">-</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>%<span class="id" title="var">Z</span><br/>
| <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE.html#B754_nan"><span class="id" title="constructor">B754_nan</span></a> <span class="id" title="var">sx</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Specif.html#exist"><span class="id" title="constructor">exist</span></a> <span class="id" title="var">plx</span> <span class="id" title="var">_</span>) ⇒ <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><span class="id" title="var">sx</span><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#Zpos"><span class="id" title="constructor">Zpos</span></a> <span class="id" title="var">plx</span><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zpow_def.html#Zpower"><span class="id" title="abbreviation">Zpower</span></a> 2 <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.ew"><span class="id" title="variable">ew</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#239a40728d107295b3cb2c790f57f9e9"><span class="id" title="notation">-</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>%<span class="id" title="var">Z</span><br/>
| <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE.html#B754_finite"><span class="id" title="constructor">B754_finite</span></a> <span class="id" title="var">sx</span> <span class="id" title="var">mx</span> <span class="id" title="var">ex</span> <span class="id" title="var">_</span> ⇒<br/>
<span class="id" title="keyword">let</span> <span class="id" title="var">m</span> := (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#Zpos"><span class="id" title="constructor">Zpos</span></a> <span class="id" title="var">mx</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#239a40728d107295b3cb2c790f57f9e9"><span class="id" title="notation">-</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zpow_def.html#Zpower"><span class="id" title="abbreviation">Zpower</span></a> 2 <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.mw"><span class="id" title="variable">mw</span></a>)%<span class="id" title="var">Z</span> <span class="id" title="keyword">in</span><br/>
<span class="id" title="keyword">if</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zbool.html#Zle_bool"><span class="id" title="abbreviation">Zle_bool</span></a> 0 <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#m"><span class="id" title="variable">m</span></a> <span class="id" title="keyword">then</span><br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><span class="id" title="var">sx</span><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#m"><span class="id" title="variable">m</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <span class="id" title="var">ex</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#239a40728d107295b3cb2c790f57f9e9"><span class="id" title="notation">-</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.emin"><span class="id" title="variable">emin</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>%<span class="id" title="var">Z</span><br/>
<span class="id" title="keyword">else</span><br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><span class="id" title="var">sx</span><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#Zpos"><span class="id" title="constructor">Zpos</span></a> <span class="id" title="var">mx</span><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> 0<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>%<span class="id" title="var">Z</span><br/>
<span class="id" title="keyword">end</span>.<br/>
<br/>
<span class="id" title="keyword">Theorem</span> <a name="split_bits_of_binary_float_correct"><span class="id" title="lemma">split_bits_of_binary_float_correct</span></a> :<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>,<br/>
<a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#split_bits"><span class="id" title="definition">split_bits</span></a> (<a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#bits_of_binary_float"><span class="id" title="definition">bits_of_binary_float</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#split_bits_of_binary_float"><span class="id" title="definition">split_bits_of_binary_float</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#x"><span class="id" title="variable">x</span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">intros</span> [<span class="id" title="var">sx</span>|<span class="id" title="var">sx</span>|<span class="id" title="var">sx</span> [<span class="id" title="var">plx</span> <span class="id" title="var">Hplx</span>]|<span class="id" title="var">sx</span> <span class="id" title="var">mx</span> <span class="id" title="var">ex</span> <span class="id" title="var">Hx</span>] ;<br/>
<span class="id" title="tactic">try</span> ( <span class="id" title="tactic">simpl</span> ; <span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#split_join_bits"><span class="id" title="lemma">split_join_bits</span></a> ; <span class="id" title="tactic">split</span> ; <span class="id" title="tactic">try</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zorder.html#Zle_refl"><span class="id" title="abbreviation">Zle_refl</span></a> ; <span class="id" title="tactic">try</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zorder.html#Zlt_pred"><span class="id" title="abbreviation">Zlt_pred</span></a> ; <span class="id" title="tactic">trivial</span> ; <span class="id" title="tactic">omega</span> ).<br/>
<span class="id" title="tactic">simpl</span>. <span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#split_join_bits"><span class="id" title="lemma">split_join_bits</span></a>; <span class="id" title="tactic">split</span>; <span class="id" title="tactic">try</span> (<span class="id" title="var">zify</span>; <span class="id" title="tactic">omega</span>).<br/>
<span class="id" title="tactic">destruct</span> (<a class="idref" href="compcert.flocq.Core.Fcore_digits.html#digits2_Pnat_correct"><span class="id" title="lemma">digits2_Pnat_correct</span></a> <span class="id" title="var">plx</span>).<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.flocq.Core.Fcore_digits.html#Zpos_digits2_pos"><span class="id" title="lemma">Zpos_digits2_pos</span></a>, <- <a class="idref" href="compcert.flocq.Core.Fcore_digits.html#Z_of_nat_S_digits2_Pnat"><span class="id" title="lemma">Z_of_nat_S_digits2_Pnat</span></a> <span class="id" title="keyword">in</span> <span class="id" title="var">Hplx</span>.<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zpower.html#Zpower_nat_Z"><span class="id" title="lemma">Zpower_nat_Z</span></a> <span class="id" title="keyword">in</span> <span class="id" title="var">H0</span>.<br/>
<span class="id" title="tactic">eapply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zorder.html#Zlt_le_trans"><span class="id" title="abbreviation">Zlt_le_trans</span></a>. <span class="id" title="tactic">apply</span> <span class="id" title="var">H0</span>.<br/>
<span class="id" title="tactic">change</span> 2%<span class="id" title="var">Z</span> <span class="id" title="keyword">with</span> (<a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#radix_val"><span class="id" title="projection">radix_val</span></a> <a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#radix2"><span class="id" title="definition">radix2</span></a>). <span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#Zpower_le"><span class="id" title="lemma">Zpower_le</span></a>.<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#Z.ltb_lt"><span class="id" title="lemma">Z.ltb_lt</span></a> <span class="id" title="keyword">in</span> <span class="id" title="var">Hplx</span>.<br/>
<span class="id" title="tactic">unfold</span> <span class="id" title="var">prec</span> <span class="id" title="keyword">in</span> ×. <span class="id" title="var">zify</span>; <span class="id" title="tactic">omega</span>.<br/>
<span class="id" title="tactic">unfold</span> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#bits_of_binary_float"><span class="id" title="definition">bits_of_binary_float</span></a>, <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#split_bits_of_binary_float"><span class="id" title="definition">split_bits_of_binary_float</span></a>.<br/>
<span class="id" title="tactic">assert</span> (<span class="id" title="var">Hf</span>: (<a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.emin"><span class="id" title="variable">emin</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#208bafb0e148fe7fb7dcd812c227f4ee"><span class="id" title="notation">≤</span></a> <span class="id" title="var">ex</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">∧</span></a> <a class="idref" href="compcert.flocq.Core.Fcore_digits.html#Zdigits"><span class="id" title="definition">Zdigits</span></a> <a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#radix2"><span class="id" title="definition">radix2</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#Zpos"><span class="id" title="constructor">Zpos</span></a> <span class="id" title="var">mx</span>) <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#208bafb0e148fe7fb7dcd812c227f4ee"><span class="id" title="notation">≤</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.prec"><span class="id" title="variable">prec</span></a>)%<span class="id" title="var">Z</span>).<br/>
<span class="id" title="tactic">destruct</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#andb_prop"><span class="id" title="lemma">andb_prop</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">Hx</span>) <span class="id" title="keyword">as</span> (<span class="id" title="var">Hx'</span>, <span class="id" title="var">_</span>).<br/>
<span class="id" title="tactic">unfold</span> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE.html#canonic_mantissa"><span class="id" title="definition">canonic_mantissa</span></a> <span class="id" title="keyword">in</span> <span class="id" title="var">Hx'</span>.<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.flocq.Core.Fcore_digits.html#Zpos_digits2_pos"><span class="id" title="lemma">Zpos_digits2_pos</span></a> <span class="id" title="keyword">in</span> <span class="id" title="var">Hx'</span>.<br/>
<span class="id" title="tactic">generalize</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zbool.html#Zeq_bool_eq"><span class="id" title="lemma">Zeq_bool_eq</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">Hx'</span>).<br/>
<span class="id" title="tactic">unfold</span> <a class="idref" href="compcert.flocq.Core.Fcore_FLT.html#FLT_exp"><span class="id" title="definition">FLT_exp</span></a>.<br/>
<span class="id" title="tactic">unfold</span> <span class="id" title="var">emin</span>.<br/>
<span class="id" title="tactic">clear</span> ; <span class="id" title="var">zify</span> ; <span class="id" title="tactic">omega</span>.<br/>
<span class="id" title="tactic">case</span> <a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#Zle_bool_spec"><span class="id" title="lemma">Zle_bool_spec</span></a> ; <span class="id" title="tactic">intros</span> <span class="id" title="var">H</span> ;<br/>
[ <span class="id" title="tactic">apply</span> → <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#Z.le_0_sub"><span class="id" title="lemma">Z.le_0_sub</span></a> <span class="id" title="keyword">in</span> <span class="id" title="var">H</span> | <span class="id" title="tactic">apply</span> → <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#Z.lt_sub_0"><span class="id" title="lemma">Z.lt_sub_0</span></a> <span class="id" title="keyword">in</span> <span class="id" title="var">H</span> ] ;<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#split_join_bits"><span class="id" title="lemma">split_join_bits</span></a> ; <span class="id" title="tactic">try</span> <span class="id" title="var">now</span> <span class="id" title="tactic">split</span>.<br/>
<span class="id" title="tactic">split</span>.<br/>
<span class="id" title="tactic">clear</span> -<span class="id" title="var">He_gt_0</span> <span class="id" title="var">H</span> ; <span class="id" title="tactic">omega</span>.<br/>
<span class="id" title="tactic">cut</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#Zpos"><span class="id" title="constructor">Zpos</span></a> <span class="id" title="var">mx</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#273beba4f3453dbb29192b3ac740bb71"><span class="id" title="notation"><</span></a> 2 <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#88e34ed7a73f02772126289867960764"><span class="id" title="notation">×</span></a> 2<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#186f1fc9207177b1a4e8373ac9655977"><span class="id" title="notation">^</span></a><a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.mw"><span class="id" title="variable">mw</span></a>)%<span class="id" title="var">Z</span>. <span class="id" title="tactic">clear</span> ; <span class="id" title="tactic">omega</span>.<br/>
<span class="id" title="tactic">replace</span> (2 <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#88e34ed7a73f02772126289867960764"><span class="id" title="notation">×</span></a> 2<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#186f1fc9207177b1a4e8373ac9655977"><span class="id" title="notation">^</span></a><a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.mw"><span class="id" title="variable">mw</span></a>)%<span class="id" title="var">Z</span> <span class="id" title="keyword">with</span> (2<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#186f1fc9207177b1a4e8373ac9655977"><span class="id" title="notation">^</span></a><a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.prec"><span class="id" title="variable">prec</span></a>)%<span class="id" title="var">Z</span>.<br/>
<span class="id" title="tactic">apply</span> (<a class="idref" href="compcert.flocq.Core.Fcore_digits.html#Zpower_gt_Zdigits"><span class="id" title="lemma">Zpower_gt_Zdigits</span></a> <a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#radix2"><span class="id" title="definition">radix2</span></a> <span class="id" title="var">_</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#Zpos"><span class="id" title="constructor">Zpos</span></a> <span class="id" title="var">mx</span>)).<br/>
<span class="id" title="tactic">apply</span> <span class="id" title="var">Hf</span>.<br/>
<span class="id" title="tactic">unfold</span> <span class="id" title="var">prec</span>.<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#Zplus_comm"><span class="id" title="abbreviation">Zplus_comm</span></a>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zpower.html#Zpower_exp"><span class="id" title="lemma">Zpower_exp</span></a> ; <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zorder.html#Zle_ge"><span class="id" title="abbreviation">Zle_ge</span></a>.<br/>
<span class="id" title="tactic">discriminate</span>.<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zorder.html#Zlt_le_weak"><span class="id" title="abbreviation">Zlt_le_weak</span></a>.<br/>
<span class="id" title="tactic">split</span>.<br/>
<span class="id" title="tactic">generalize</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#proj1"><span class="id" title="lemma">proj1</span></a> <span class="id" title="var">Hf</span>).<br/>
<span class="id" title="tactic">clear</span> ; <span class="id" title="tactic">omega</span>.<br/>
<span class="id" title="tactic">destruct</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#andb_prop"><span class="id" title="lemma">andb_prop</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">Hx</span>) <span class="id" title="keyword">as</span> (<span class="id" title="var">_</span>, <span class="id" title="var">Hx'</span>).<br/>
<span class="id" title="tactic">unfold</span> <span class="id" title="var">emin</span>.<br/>
<span class="id" title="tactic">replace</span> (2<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#186f1fc9207177b1a4e8373ac9655977"><span class="id" title="notation">^</span></a><a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.ew"><span class="id" title="variable">ew</span></a>)%<span class="id" title="var">Z</span> <span class="id" title="keyword">with</span> (2 <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#88e34ed7a73f02772126289867960764"><span class="id" title="notation">×</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.emax"><span class="id" title="variable">emax</span></a>)%<span class="id" title="var">Z</span>.<br/>
<span class="id" title="tactic">generalize</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zbool.html#Zle_bool_imp_le"><span class="id" title="lemma">Zle_bool_imp_le</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">Hx'</span>).<br/>
<span class="id" title="tactic">clear</span> ; <span class="id" title="tactic">omega</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#sym_eq"><span class="id" title="abbreviation">sym_eq</span></a>.<br/>
<span class="id" title="tactic">rewrite</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#Zsucc_pred"><span class="id" title="lemma">Zsucc_pred</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.ew"><span class="id" title="variable">ew</span></a>).<br/>
<span class="id" title="tactic">unfold</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#Z.succ"><span class="id" title="definition">Zsucc</span></a>.<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#Zplus_comm"><span class="id" title="abbreviation">Zplus_comm</span></a>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zpower.html#Zpower_exp"><span class="id" title="lemma">Zpower_exp</span></a> ; <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zorder.html#Zle_ge"><span class="id" title="abbreviation">Zle_ge</span></a>.<br/>
<span class="id" title="tactic">discriminate</span>.<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zorder.html#Zlt_0_le_0_pred"><span class="id" title="lemma">Zlt_0_le_0_pred</span></a>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Theorem</span> <a name="bits_of_binary_float_range"><span class="id" title="lemma">bits_of_binary_float_range</span></a>:<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, (0 <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#cb2fd1306e4fe5d333fd9fb7e2c4a1d2"><span class="id" title="notation">≤</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#bits_of_binary_float"><span class="id" title="definition">bits_of_binary_float</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#cb2fd1306e4fe5d333fd9fb7e2c4a1d2"><span class="id" title="notation"><</span></a> 2<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#186f1fc9207177b1a4e8373ac9655977"><span class="id" title="notation">^(</span></a><a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.mw"><span class="id" title="variable">mw</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a><a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.ew"><span class="id" title="variable">ew</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a>1<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#186f1fc9207177b1a4e8373ac9655977"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">Z</span>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">unfold</span> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#bits_of_binary_float"><span class="id" title="definition">bits_of_binary_float</span></a>.<br/>
<span class="id" title="tactic">intros</span> [<span class="id" title="var">sx</span>|<span class="id" title="var">sx</span>|<span class="id" title="var">sx</span> [<span class="id" title="var">pl</span> <span class="id" title="var">pl_range</span>]|<span class="id" title="var">sx</span> <span class="id" title="var">mx</span> <span class="id" title="var">ex</span> <span class="id" title="var">H</span>].<br/>
- <span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#join_bits_range"><span class="id" title="lemma">join_bits_range</span></a> ; <span class="id" title="var">now</span> <span class="id" title="tactic">split</span>.<br/>
- <span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#join_bits_range"><span class="id" title="lemma">join_bits_range</span></a>.<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">split</span>.<br/>
<span class="id" title="tactic">clear</span> -<span class="id" title="var">He_gt_0</span> ; <span class="id" title="tactic">omega</span>.<br/>
- <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#Z.ltb_lt"><span class="id" title="lemma">Z.ltb_lt</span></a> <span class="id" title="keyword">in</span> <span class="id" title="var">pl_range</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#join_bits_range"><span class="id" title="lemma">join_bits_range</span></a>.<br/>
<span class="id" title="tactic">split</span>.<br/>
<span class="id" title="var">easy</span>.<br/>
<span class="id" title="tactic">apply</span> (<a class="idref" href="compcert.flocq.Core.Fcore_digits.html#Zpower_gt_Zdigits"><span class="id" title="lemma">Zpower_gt_Zdigits</span></a> <a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#radix2"><span class="id" title="definition">radix2</span></a> <span class="id" title="var">_</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#Zpos"><span class="id" title="constructor">Zpos</span></a> <span class="id" title="var">pl</span>)).<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#Z.lt_succ_r"><span class="id" title="lemma">Z.lt_succ_r</span></a>.<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">rewrite</span> <- <a class="idref" href="compcert.flocq.Core.Fcore_digits.html#Zdigits2_Zdigits"><span class="id" title="lemma">Zdigits2_Zdigits</span></a>.<br/>
<span class="id" title="tactic">clear</span> -<span class="id" title="var">He_gt_0</span> ; <span class="id" title="tactic">omega</span>.<br/>
- <span class="id" title="tactic">unfold</span> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE.html#bounded"><span class="id" title="definition">bounded</span></a> <span class="id" title="keyword">in</span> <span class="id" title="var">H</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Bool.Bool.html#andb_true_iff"><span class="id" title="lemma">Bool.andb_true_iff</span></a> <span class="id" title="keyword">in</span> <span class="id" title="var">H</span> ; <span class="id" title="tactic">destruct</span> <span class="id" title="var">H</span> <span class="id" title="keyword">as</span> [<span class="id" title="var">A</span> <span class="id" title="var">B</span>].<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#Z.leb_le"><span class="id" title="lemma">Z.leb_le</span></a> <span class="id" title="keyword">in</span> <span class="id" title="var">B</span>.<br/>
<span class="id" title="tactic">unfold</span> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE.html#canonic_mantissa"><span class="id" title="definition">canonic_mantissa</span></a>, <a class="idref" href="compcert.flocq.Core.Fcore_FLT.html#FLT_exp"><span class="id" title="definition">FLT_exp</span></a> <span class="id" title="keyword">in</span> <span class="id" title="var">A</span>. <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zbool.html#Zeq_bool_eq"><span class="id" title="lemma">Zeq_bool_eq</span></a> <span class="id" title="keyword">in</span> <span class="id" title="var">A</span>.<br/>
<span class="id" title="tactic">case</span> <a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#Zle_bool_spec"><span class="id" title="lemma">Zle_bool_spec</span></a> ; <span class="id" title="tactic">intros</span> <span class="id" title="var">H</span>.<br/>
+ <span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#join_bits_range"><span class="id" title="lemma">join_bits_range</span></a>.<br/>
× <span class="id" title="tactic">split</span>.<br/>
<span class="id" title="tactic">clear</span> -<span class="id" title="var">H</span> ; <span class="id" title="tactic">omega</span>.<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.flocq.Core.Fcore_digits.html#Zpos_digits2_pos"><span class="id" title="lemma">Zpos_digits2_pos</span></a> <span class="id" title="keyword">in</span> <span class="id" title="var">A</span>.<br/>
<span class="id" title="tactic">cut</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#Zpos"><span class="id" title="constructor">Zpos</span></a> <span class="id" title="var">mx</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#273beba4f3453dbb29192b3ac740bb71"><span class="id" title="notation"><</span></a> 2 <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#186f1fc9207177b1a4e8373ac9655977"><span class="id" title="notation">^</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.prec"><span class="id" title="variable">prec</span></a>)%<span class="id" title="var">Z</span>.<br/>
<span class="id" title="tactic">unfold</span> <span class="id" title="var">prec</span>.<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#Zpower_plus"><span class="id" title="lemma">Zpower_plus</span></a> <span class="id" title="tactic">by</span> (<span class="id" title="tactic">clear</span> -<span class="id" title="var">Hmw</span> ; <span class="id" title="tactic">omega</span>).<br/>
<span class="id" title="tactic">change</span> (2<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#186f1fc9207177b1a4e8373ac9655977"><span class="id" title="notation">^</span></a>1)%<span class="id" title="var">Z</span> <span class="id" title="keyword">with</span> 2%<span class="id" title="var">Z</span>.<br/>
<span class="id" title="tactic">clear</span> ; <span class="id" title="tactic">omega</span>.<br/>
<span class="id" title="tactic">apply</span> (<a class="idref" href="compcert.flocq.Core.Fcore_digits.html#Zpower_gt_Zdigits"><span class="id" title="lemma">Zpower_gt_Zdigits</span></a> <a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#radix2"><span class="id" title="definition">radix2</span></a> <span class="id" title="var">_</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#Zpos"><span class="id" title="constructor">Zpos</span></a> <span class="id" title="var">mx</span>)).<br/>
<span class="id" title="tactic">clear</span> -<span class="id" title="var">A</span> ; <span class="id" title="var">zify</span> ; <span class="id" title="tactic">omega</span>.<br/>
× <span class="id" title="tactic">split</span>.<br/>
<span class="id" title="tactic">unfold</span> <span class="id" title="var">emin</span> ; <span class="id" title="tactic">clear</span> -<span class="id" title="var">A</span> ; <span class="id" title="var">zify</span> ; <span class="id" title="tactic">omega</span>.<br/>
<span class="id" title="tactic">replace</span> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.ew"><span class="id" title="variable">ew</span></a> <span class="id" title="keyword">with</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">(</span></a><a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.ew"><span class="id" title="variable">ew</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#239a40728d107295b3cb2c790f57f9e9"><span class="id" title="notation">-</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a> 1)%<span class="id" title="var">Z</span> <span class="id" title="tactic">by</span> <span class="id" title="tactic">ring</span>.<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#Zpower_plus"><span class="id" title="lemma">Zpower_plus</span></a> <span class="id" title="tactic">by</span> (<span class="id" title="tactic">clear</span> - <span class="id" title="var">Hew</span> ; <span class="id" title="tactic">omega</span>).<br/>
<span class="id" title="tactic">unfold</span> <span class="id" title="var">emin</span>, <span class="id" title="var">emax</span> <span class="id" title="keyword">in</span> ×.<br/>
<span class="id" title="tactic">change</span> (2<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#186f1fc9207177b1a4e8373ac9655977"><span class="id" title="notation">^</span></a>1)%<span class="id" title="var">Z</span> <span class="id" title="keyword">with</span> 2%<span class="id" title="var">Z</span>.<br/>
<span class="id" title="tactic">clear</span> -<span class="id" title="var">B</span> ; <span class="id" title="tactic">omega</span>.<br/>
+ <span class="id" title="tactic">apply</span> → <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#Z.lt_sub_0"><span class="id" title="lemma">Z.lt_sub_0</span></a> <span class="id" title="keyword">in</span> <span class="id" title="var">H</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#join_bits_range"><span class="id" title="lemma">join_bits_range</span></a> ; <span class="id" title="var">now</span> <span class="id" title="tactic">split</span>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Definition</span> <a name="binary_float_of_bits_aux"><span class="id" title="definition">binary_float_of_bits_aux</span></a> <span class="id" title="var">x</span> :=<br/>
<span class="id" title="keyword">let</span> '<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><span class="id" title="var">sx</span><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <span class="id" title="var">mx</span><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <span class="id" title="var">ex</span><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a> := <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#split_bits"><span class="id" title="definition">split_bits</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#x"><span class="id" title="variable">x</span></a> <span class="id" title="keyword">in</span><br/>
<span class="id" title="keyword">if</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zbool.html#Zeq_bool"><span class="id" title="definition">Zeq_bool</span></a> <span class="id" title="var">ex</span> 0 <span class="id" title="keyword">then</span><br/>
<span class="id" title="keyword">match</span> <span class="id" title="var">mx</span> <span class="id" title="keyword">with</span><br/>
| <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#Z0"><span class="id" title="constructor">Z0</span></a> ⇒ <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE.html#F754_zero"><span class="id" title="constructor">F754_zero</span></a> <span class="id" title="var">sx</span><br/>
| <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#Zpos"><span class="id" title="constructor">Zpos</span></a> <span class="id" title="var">px</span> ⇒ <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE.html#F754_finite"><span class="id" title="constructor">F754_finite</span></a> <span class="id" title="var">sx</span> <span class="id" title="var">px</span> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.emin"><span class="id" title="variable">emin</span></a><br/>
| <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#Zneg"><span class="id" title="constructor">Zneg</span></a> <span class="id" title="var">_</span> ⇒ <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE.html#F754_nan"><span class="id" title="constructor">F754_nan</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#xH"><span class="id" title="constructor">xH</span></a> <br/>
<span class="id" title="keyword">end</span><br/>
<span class="id" title="keyword">else</span> <span class="id" title="keyword">if</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zbool.html#Zeq_bool"><span class="id" title="definition">Zeq_bool</span></a> <span class="id" title="var">ex</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zpow_def.html#Zpower"><span class="id" title="abbreviation">Zpower</span></a> 2 <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.ew"><span class="id" title="variable">ew</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#239a40728d107295b3cb2c790f57f9e9"><span class="id" title="notation">-</span></a> 1) <span class="id" title="keyword">then</span><br/>
<span class="id" title="keyword">match</span> <span class="id" title="var">mx</span> <span class="id" title="keyword">with</span><br/>
| <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#Z0"><span class="id" title="constructor">Z0</span></a> ⇒ <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE.html#F754_infinity"><span class="id" title="constructor">F754_infinity</span></a> <span class="id" title="var">sx</span><br/>
| <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#Zpos"><span class="id" title="constructor">Zpos</span></a> <span class="id" title="var">plx</span> ⇒ <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE.html#F754_nan"><span class="id" title="constructor">F754_nan</span></a> <span class="id" title="var">sx</span> <span class="id" title="var">plx</span><br/>
| <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#Zneg"><span class="id" title="constructor">Zneg</span></a> <span class="id" title="var">_</span> ⇒ <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE.html#F754_nan"><span class="id" title="constructor">F754_nan</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#xH"><span class="id" title="constructor">xH</span></a> <br/>
<span class="id" title="keyword">end</span><br/>
<span class="id" title="keyword">else</span><br/>
<span class="id" title="keyword">match</span> (<span class="id" title="var">mx</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zpow_def.html#Zpower"><span class="id" title="abbreviation">Zpower</span></a> 2 <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.mw"><span class="id" title="variable">mw</span></a>)%<span class="id" title="var">Z</span> <span class="id" title="keyword">with</span><br/>
| <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#Zpos"><span class="id" title="constructor">Zpos</span></a> <span class="id" title="var">px</span> ⇒ <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE.html#F754_finite"><span class="id" title="constructor">F754_finite</span></a> <span class="id" title="var">sx</span> <span class="id" title="var">px</span> (<span class="id" title="var">ex</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.emin"><span class="id" title="variable">emin</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#239a40728d107295b3cb2c790f57f9e9"><span class="id" title="notation">-</span></a> 1)<br/>
| <span class="id" title="var">_</span> ⇒ <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE.html#F754_nan"><span class="id" title="constructor">F754_nan</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#xH"><span class="id" title="constructor">xH</span></a> <br/>
<span class="id" title="keyword">end</span>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="binary_float_of_bits_aux_correct"><span class="id" title="lemma">binary_float_of_bits_aux_correct</span></a> :<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>,<br/>
<a class="idref" href="compcert.flocq.Appli.Fappli_IEEE.html#valid_binary"><span class="id" title="definition">valid_binary</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.prec"><span class="id" title="variable">prec</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.emax"><span class="id" title="variable">emax</span></a> (<a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#binary_float_of_bits_aux"><span class="id" title="definition">binary_float_of_bits_aux</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">x</span>.<br/>
<span class="id" title="tactic">unfold</span> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#binary_float_of_bits_aux"><span class="id" title="definition">binary_float_of_bits_aux</span></a>, <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#split_bits"><span class="id" title="definition">split_bits</span></a>.<br/>
<span class="id" title="tactic">case</span> <a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#Zeq_bool_spec"><span class="id" title="lemma">Zeq_bool_spec</span></a> ; <span class="id" title="tactic">intros</span> <span class="id" title="var">He1</span>.<br/>
<span class="id" title="var">case_eq</span> (<span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#ef49f19e36a5751007de81142fab2595"><span class="id" title="notation">mod</span></a> 2<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#186f1fc9207177b1a4e8373ac9655977"><span class="id" title="notation">^</span></a><a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.mw"><span class="id" title="variable">mw</span></a>)%<span class="id" title="var">Z</span> ; <span class="id" title="tactic">try</span> <span class="id" title="var">easy</span>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">px</span> <span class="id" title="var">Hm</span>.<br/>
<span class="id" title="tactic">assert</span> (<a class="idref" href="compcert.flocq.Core.Fcore_digits.html#Zdigits"><span class="id" title="definition">Zdigits</span></a> <a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#radix2"><span class="id" title="definition">radix2</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#Zpos"><span class="id" title="constructor">Zpos</span></a> <span class="id" title="var">px</span>) <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#208bafb0e148fe7fb7dcd812c227f4ee"><span class="id" title="notation">≤</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.mw"><span class="id" title="variable">mw</span></a>)%<span class="id" title="var">Z</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_digits.html#Zdigits_le_Zpower"><span class="id" title="lemma">Zdigits_le_Zpower</span></a>.<br/>
<span class="id" title="tactic">simpl</span>.<br/>
<span class="id" title="tactic">rewrite</span> <- <span class="id" title="var">Hm</span>.<br/>
<span class="id" title="tactic">eapply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zdiv.html#Z_mod_lt"><span class="id" title="lemma">Z_mod_lt</span></a>.<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zorder.html#Zlt_gt"><span class="id" title="abbreviation">Zlt_gt</span></a>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE.html#bounded_canonic_lt_emax"><span class="id" title="lemma">bounded_canonic_lt_emax</span></a> ; <span class="id" title="tactic">try</span> <span class="id" title="tactic">assumption</span>.<br/>
<span class="id" title="tactic">unfold</span> <a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#canonic"><span class="id" title="definition">canonic</span></a>, <a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#canonic_exp"><span class="id" title="definition">canonic_exp</span></a>.<br/>
<span class="id" title="tactic">fold</span> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.emin"><span class="id" title="variable">emin</span></a>.<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_digits.html#ln_beta_F2R_Zdigits"><span class="id" title="lemma">ln_beta_F2R_Zdigits</span></a>. 2: <span class="id" title="tactic">discriminate</span>.<br/>
<span class="id" title="tactic">unfold</span> <a class="idref" href="compcert.flocq.Core.Fcore_defs.html#Fexp"><span class="id" title="projection">Fexp</span></a>, <a class="idref" href="compcert.flocq.Core.Fcore_FLT.html#FLT_exp"><span class="id" title="definition">FLT_exp</span></a>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#sym_eq"><span class="id" title="abbreviation">sym_eq</span></a>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zmax.html#Zmax_right"><span class="id" title="abbreviation">Zmax_right</span></a>.<br/>
<span class="id" title="tactic">clear</span> -<span class="id" title="var">H</span> <span class="id" title="var">Hprec</span>.<br/>
<span class="id" title="tactic">unfold</span> <span class="id" title="var">prec</span> ; <span class="id" title="tactic">omega</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Reals.RIneq.html#Rnot_le_lt"><span class="id" title="lemma">Rnot_le_lt</span></a>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">H0</span>.<br/>
<span class="id" title="tactic">refine</span> (<span class="id" title="var">_</span> (<a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#ln_beta_le"><span class="id" title="lemma">ln_beta_le</span></a> <a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#radix2"><span class="id" title="definition">radix2</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">H0</span>)).<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#ln_beta_bpow"><span class="id" title="lemma">ln_beta_bpow</span></a>.<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_digits.html#ln_beta_F2R_Zdigits"><span class="id" title="lemma">ln_beta_F2R_Zdigits</span></a>. 2: <span class="id" title="tactic">discriminate</span>.<br/>
<span class="id" title="tactic">unfold</span> <span class="id" title="var">emin</span>, <span class="id" title="var">prec</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zorder.html#Zlt_not_le"><span class="id" title="lemma">Zlt_not_le</span></a>.<br/>
<span class="id" title="tactic">cut</span> (0 <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#273beba4f3453dbb29192b3ac740bb71"><span class="id" title="notation"><</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.emax"><span class="id" title="variable">emax</span></a>)%<span class="id" title="var">Z</span>. <span class="id" title="tactic">clear</span> -<span class="id" title="var">H</span> <span class="id" title="var">Hew</span> ; <span class="id" title="tactic">omega</span>.<br/>
<span class="id" title="tactic">apply</span> (<a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#Zpower_gt_0"><span class="id" title="lemma">Zpower_gt_0</span></a> <a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#radix2"><span class="id" title="definition">radix2</span></a>).<br/>
<span class="id" title="tactic">clear</span> -<span class="id" title="var">Hew</span> ; <span class="id" title="tactic">omega</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#bpow_gt_0"><span class="id" title="lemma">bpow_gt_0</span></a>.<br/>
<span class="id" title="tactic">simpl</span>. <span class="id" title="tactic">intros</span>. <span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#Z.ltb_lt"><span class="id" title="lemma">Z.ltb_lt</span></a>. <span class="id" title="tactic">unfold</span> <span class="id" title="var">prec</span>. <span class="id" title="var">zify</span>; <span class="id" title="tactic">omega</span>.<br/>
<span class="id" title="tactic">case</span> <a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#Zeq_bool_spec"><span class="id" title="lemma">Zeq_bool_spec</span></a> ; <span class="id" title="tactic">intros</span> <span class="id" title="var">He2</span>.<br/>
<span class="id" title="var">case_eq</span> (<span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#ef49f19e36a5751007de81142fab2595"><span class="id" title="notation">mod</span></a> 2 <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#186f1fc9207177b1a4e8373ac9655977"><span class="id" title="notation">^</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.mw"><span class="id" title="variable">mw</span></a>)%<span class="id" title="var">Z</span>; <span class="id" title="tactic">try</span> <span class="id" title="var">easy</span>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">plx</span> <span class="id" title="var">Eqplx</span>. <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#Z.ltb_lt"><span class="id" title="lemma">Z.ltb_lt</span></a>.<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.flocq.Core.Fcore_digits.html#Zpos_digits2_pos"><span class="id" title="lemma">Zpos_digits2_pos</span></a>.<br/>
<span class="id" title="tactic">assert</span> (<span class="id" title="keyword">∀</span> <span class="id" title="var">a</span> <span class="id" title="var">b</span>, <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#208bafb0e148fe7fb7dcd812c227f4ee"><span class="id" title="notation">≤</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#b"><span class="id" title="variable">b</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#a"><span class="id" title="variable">a</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#273beba4f3453dbb29192b3ac740bb71"><span class="id" title="notation"><</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#b"><span class="id" title="variable">b</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a>1)%<span class="id" title="var">Z</span> <span class="id" title="tactic">by</span> (<span class="id" title="tactic">intros</span>; <span class="id" title="tactic">omega</span>). <span class="id" title="tactic">apply</span> <span class="id" title="var">H</span>. <span class="id" title="tactic">clear</span> <span class="id" title="var">H</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_digits.html#Zdigits_le_Zpower"><span class="id" title="lemma">Zdigits_le_Zpower</span></a>. <span class="id" title="tactic">simpl</span>.<br/>
<span class="id" title="tactic">rewrite</span> <- <span class="id" title="var">Eqplx</span>. <span class="id" title="var">edestruct</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zdiv.html#Z_mod_lt"><span class="id" title="lemma">Z_mod_lt</span></a>; <span class="id" title="tactic">eauto</span>.<br/>
<span class="id" title="tactic">change</span> 2%<span class="id" title="var">Z</span> <span class="id" title="keyword">with</span> (<a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#radix_val"><span class="id" title="projection">radix_val</span></a> <a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#radix2"><span class="id" title="definition">radix2</span></a>).<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#Z.lt_gt"><span class="id" title="lemma">Z.lt_gt</span></a>, <a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#Zpower_gt_0"><span class="id" title="lemma">Zpower_gt_0</span></a>. <span class="id" title="tactic">omega</span>.<br/>
<span class="id" title="tactic">simpl</span>. <span class="id" title="tactic">intros</span>. <span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#Z.ltb_lt"><span class="id" title="lemma">Z.ltb_lt</span></a>. <span class="id" title="tactic">unfold</span> <span class="id" title="var">prec</span>. <span class="id" title="var">zify</span>; <span class="id" title="tactic">omega</span>.<br/>
<span class="id" title="var">case_eq</span> (<span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#ef49f19e36a5751007de81142fab2595"><span class="id" title="notation">mod</span></a> 2<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#186f1fc9207177b1a4e8373ac9655977"><span class="id" title="notation">^</span></a><a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.mw"><span class="id" title="variable">mw</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a> 2<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#186f1fc9207177b1a4e8373ac9655977"><span class="id" title="notation">^</span></a><a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.mw"><span class="id" title="variable">mw</span></a>)%<span class="id" title="var">Z</span> ; <span class="id" title="tactic">try</span> <span class="id" title="var">easy</span>.<br/>
<span class="id" title="tactic">simpl</span>. <span class="id" title="tactic">intros</span>. <span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#Z.ltb_lt"><span class="id" title="lemma">Z.ltb_lt</span></a>. <span class="id" title="tactic">unfold</span> <span class="id" title="var">prec</span>. <span class="id" title="var">zify</span>; <span class="id" title="tactic">omega</span>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">px</span> <span class="id" title="var">Hm</span>.<br/>
<span class="id" title="tactic">assert</span> (<a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.prec"><span class="id" title="variable">prec</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="compcert.flocq.Core.Fcore_digits.html#Zdigits"><span class="id" title="definition">Zdigits</span></a> <a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#radix2"><span class="id" title="definition">radix2</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#Zpos"><span class="id" title="constructor">Zpos</span></a> <span class="id" title="var">px</span>)).<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_digits.html#Zdigits_ln_beta"><span class="id" title="lemma">Zdigits_ln_beta</span></a>. 2: <span class="id" title="tactic">discriminate</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#sym_eq"><span class="id" title="abbreviation">sym_eq</span></a>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#ln_beta_unique"><span class="id" title="lemma">ln_beta_unique</span></a>.<br/>
<span class="id" title="tactic">rewrite</span> <- <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Z2R_abs"><span class="id" title="lemma">Z2R_abs</span></a>.<br/>
<span class="id" title="tactic">unfold</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#Z.abs"><span class="id" title="definition">Zabs</span></a>.<br/>
<span class="id" title="tactic">replace</span> (<a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.prec"><span class="id" title="variable">prec</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#239a40728d107295b3cb2c790f57f9e9"><span class="id" title="notation">-</span></a> 1)%<span class="id" title="var">Z</span> <span class="id" title="keyword">with</span> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.mw"><span class="id" title="variable">mw</span></a> <span class="id" title="tactic">by</span> ( <span class="id" title="tactic">unfold</span> <span class="id" title="var">prec</span> ; <span class="id" title="tactic">ring</span> ).<br/>
<span class="id" title="tactic">rewrite</span> <- <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Z2R_Zpower"><span class="id" title="lemma">Z2R_Zpower</span></a> <span class="id" title="keyword">with</span> (1 := <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zorder.html#Zlt_le_weak"><span class="id" title="abbreviation">Zlt_le_weak</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.Hmw"><span class="id" title="variable">Hmw</span></a>).<br/>
<span class="id" title="tactic">rewrite</span> <- <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Z2R_Zpower"><span class="id" title="lemma">Z2R_Zpower</span></a>. 2: <span class="id" title="var">now</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zorder.html#Zlt_le_weak"><span class="id" title="abbreviation">Zlt_le_weak</span></a>.<br/>
<span class="id" title="tactic">rewrite</span> <- <span class="id" title="var">Hm</span>.<br/>
<span class="id" title="tactic">split</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Z2R_le"><span class="id" title="lemma">Z2R_le</span></a>.<br/>
<span class="id" title="tactic">change</span> (<a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#radix2"><span class="id" title="definition">radix2</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#186f1fc9207177b1a4e8373ac9655977"><span class="id" title="notation">^</span></a><a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.mw"><span class="id" title="variable">mw</span></a>)%<span class="id" title="var">Z</span> <span class="id" title="keyword">with</span> (0 <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a> 2<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#186f1fc9207177b1a4e8373ac9655977"><span class="id" title="notation">^</span></a><a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.mw"><span class="id" title="variable">mw</span></a>)%<span class="id" title="var">Z</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zorder.html#Zplus_le_compat_r"><span class="id" title="lemma">Zplus_le_compat_r</span></a>.<br/>
<span class="id" title="tactic">eapply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zdiv.html#Z_mod_lt"><span class="id" title="lemma">Z_mod_lt</span></a>.<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zorder.html#Zlt_gt"><span class="id" title="abbreviation">Zlt_gt</span></a>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Z2R_lt"><span class="id" title="lemma">Z2R_lt</span></a>.<br/>
<span class="id" title="tactic">unfold</span> <span class="id" title="var">prec</span>.<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zpower.html#Zpower_exp"><span class="id" title="lemma">Zpower_exp</span></a>. 2: <span class="id" title="var">now</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zorder.html#Zle_ge"><span class="id" title="abbreviation">Zle_ge</span></a> ; <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zorder.html#Zlt_le_weak"><span class="id" title="abbreviation">Zlt_le_weak</span></a>. 2: <span class="id" title="tactic">discriminate</span>.<br/>
<span class="id" title="tactic">rewrite</span> <- <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#Zplus_diag_eq_mult_2"><span class="id" title="lemma">Zplus_diag_eq_mult_2</span></a>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zorder.html#Zplus_lt_compat_r"><span class="id" title="lemma">Zplus_lt_compat_r</span></a>.<br/>
<span class="id" title="tactic">eapply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zdiv.html#Z_mod_lt"><span class="id" title="lemma">Z_mod_lt</span></a>.<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zorder.html#Zlt_gt"><span class="id" title="abbreviation">Zlt_gt</span></a>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE.html#bounded_canonic_lt_emax"><span class="id" title="lemma">bounded_canonic_lt_emax</span></a> ; <span class="id" title="tactic">try</span> <span class="id" title="tactic">assumption</span>.<br/>
<span class="id" title="tactic">unfold</span> <a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#canonic"><span class="id" title="definition">canonic</span></a>, <a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#canonic_exp"><span class="id" title="definition">canonic_exp</span></a>.<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_digits.html#ln_beta_F2R_Zdigits"><span class="id" title="lemma">ln_beta_F2R_Zdigits</span></a>. 2: <span class="id" title="tactic">discriminate</span>.<br/>
<span class="id" title="tactic">unfold</span> <a class="idref" href="compcert.flocq.Core.Fcore_defs.html#Fexp"><span class="id" title="projection">Fexp</span></a>, <a class="idref" href="compcert.flocq.Core.Fcore_FLT.html#FLT_exp"><span class="id" title="definition">FLT_exp</span></a>.<br/>
<span class="id" title="tactic">rewrite</span> <- <span class="id" title="var">H</span>.<br/>
<span class="id" title="tactic">set</span> (<span class="id" title="var">ex</span> := (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#ef49f19e36a5751007de81142fab2595"><span class="id" title="notation">(</span></a><span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#66073ffd934badf1f6e703023820601f"><span class="id" title="notation">/</span></a> 2<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#186f1fc9207177b1a4e8373ac9655977"><span class="id" title="notation">^</span></a><a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.mw"><span class="id" title="variable">mw</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#ef49f19e36a5751007de81142fab2595"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#ef49f19e36a5751007de81142fab2595"><span class="id" title="notation">mod</span></a> 2<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#186f1fc9207177b1a4e8373ac9655977"><span class="id" title="notation">^</span></a><a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.ew"><span class="id" title="variable">ew</span></a>)%<span class="id" title="var">Z</span>).<br/>
<span class="id" title="tactic">replace</span> (<a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.prec"><span class="id" title="variable">prec</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">(</span></a><span class="id" title="var">ex</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.emin"><span class="id" title="variable">emin</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#239a40728d107295b3cb2c790f57f9e9"><span class="id" title="notation">-</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#239a40728d107295b3cb2c790f57f9e9"><span class="id" title="notation">-</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.prec"><span class="id" title="variable">prec</span></a>)%<span class="id" title="var">Z</span> <span class="id" title="keyword">with</span> (<span class="id" title="var">ex</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.emin"><span class="id" title="variable">emin</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#239a40728d107295b3cb2c790f57f9e9"><span class="id" title="notation">-</span></a> 1)%<span class="id" title="var">Z</span> <span class="id" title="tactic">by</span> <span class="id" title="tactic">ring</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#sym_eq"><span class="id" title="abbreviation">sym_eq</span></a>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zmax.html#Zmax_left"><span class="id" title="lemma">Zmax_left</span></a>.<br/>
<span class="id" title="var">revert</span> <span class="id" title="var">He1</span>.<br/>
<span class="id" title="tactic">fold</span> <span class="id" title="var">ex</span>.<br/>
<span class="id" title="tactic">cut</span> (0 <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#208bafb0e148fe7fb7dcd812c227f4ee"><span class="id" title="notation">≤</span></a> <span class="id" title="var">ex</span>)%<span class="id" title="var">Z</span>.<br/>
<span class="id" title="tactic">unfold</span> <span class="id" title="var">emin</span>.<br/>
<span class="id" title="tactic">clear</span> ; <span class="id" title="tactic">intros</span> <span class="id" title="var">H1</span> <span class="id" title="var">H2</span> ; <span class="id" title="tactic">omega</span>.<br/>
<span class="id" title="tactic">eapply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zdiv.html#Z_mod_lt"><span class="id" title="lemma">Z_mod_lt</span></a>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zorder.html#Zlt_gt"><span class="id" title="abbreviation">Zlt_gt</span></a>.<br/>
<span class="id" title="tactic">apply</span> (<a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#Zpower_gt_0"><span class="id" title="lemma">Zpower_gt_0</span></a> <a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#radix2"><span class="id" title="definition">radix2</span></a>).<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zorder.html#Zlt_le_weak"><span class="id" title="abbreviation">Zlt_le_weak</span></a>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Reals.RIneq.html#Rnot_le_lt"><span class="id" title="lemma">Rnot_le_lt</span></a>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">H0</span>.<br/>
<span class="id" title="tactic">refine</span> (<span class="id" title="var">_</span> (<a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#ln_beta_le"><span class="id" title="lemma">ln_beta_le</span></a> <a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#radix2"><span class="id" title="definition">radix2</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">H0</span>)).<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#ln_beta_bpow"><span class="id" title="lemma">ln_beta_bpow</span></a>.<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_digits.html#ln_beta_F2R_Zdigits"><span class="id" title="lemma">ln_beta_F2R_Zdigits</span></a>. 2: <span class="id" title="tactic">discriminate</span>.<br/>
<span class="id" title="tactic">rewrite</span> <- <span class="id" title="var">H</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zorder.html#Zlt_not_le"><span class="id" title="lemma">Zlt_not_le</span></a>.<br/>
<span class="id" title="tactic">unfold</span> <span class="id" title="var">emin</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zorder.html#Zplus_lt_reg_r"><span class="id" title="lemma">Zplus_lt_reg_r</span></a> <span class="id" title="keyword">with</span> (<a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.emax"><span class="id" title="variable">emax</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#239a40728d107295b3cb2c790f57f9e9"><span class="id" title="notation">-</span></a> 1)%<span class="id" title="var">Z</span>.<br/>
<span class="id" title="var">ring_simplify</span>.<br/>
<span class="id" title="var">revert</span> <span class="id" title="var">He2</span>.<br/>
<span class="id" title="tactic">set</span> (<span class="id" title="var">ex</span> := (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#ef49f19e36a5751007de81142fab2595"><span class="id" title="notation">(</span></a><span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#66073ffd934badf1f6e703023820601f"><span class="id" title="notation">/</span></a> 2<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#186f1fc9207177b1a4e8373ac9655977"><span class="id" title="notation">^</span></a><a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.mw"><span class="id" title="variable">mw</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#ef49f19e36a5751007de81142fab2595"><span class="id" title="notation">)</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#ef49f19e36a5751007de81142fab2595"><span class="id" title="notation">mod</span></a> 2<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#186f1fc9207177b1a4e8373ac9655977"><span class="id" title="notation">^</span></a><a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.ew"><span class="id" title="variable">ew</span></a>)%<span class="id" title="var">Z</span>).<br/>
<span class="id" title="tactic">cut</span> (<span class="id" title="var">ex</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#273beba4f3453dbb29192b3ac740bb71"><span class="id" title="notation"><</span></a> 2<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#186f1fc9207177b1a4e8373ac9655977"><span class="id" title="notation">^</span></a><a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.ew"><span class="id" title="variable">ew</span></a>)%<span class="id" title="var">Z</span>.<br/>
<span class="id" title="tactic">replace</span> (2<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#186f1fc9207177b1a4e8373ac9655977"><span class="id" title="notation">^</span></a><a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.ew"><span class="id" title="variable">ew</span></a>)%<span class="id" title="var">Z</span> <span class="id" title="keyword">with</span> (2 <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#88e34ed7a73f02772126289867960764"><span class="id" title="notation">×</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.emax"><span class="id" title="variable">emax</span></a>)%<span class="id" title="var">Z</span>.<br/>
<span class="id" title="tactic">clear</span> ; <span class="id" title="tactic">intros</span> <span class="id" title="var">H1</span> <span class="id" title="var">H2</span> ; <span class="id" title="tactic">omega</span>.<br/>
<span class="id" title="tactic">replace</span> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.ew"><span class="id" title="variable">ew</span></a> <span class="id" title="keyword">with</span> (1 <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">(</span></a><a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.ew"><span class="id" title="variable">ew</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#239a40728d107295b3cb2c790f57f9e9"><span class="id" title="notation">-</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">Z</span> <span class="id" title="tactic">by</span> <span class="id" title="tactic">ring</span>.<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zpower.html#Zpower_exp"><span class="id" title="lemma">Zpower_exp</span></a>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#refl_equal"><span class="id" title="abbreviation">refl_equal</span></a>.<br/>
<span class="id" title="tactic">discriminate</span>.<br/>
<span class="id" title="tactic">clear</span> -<span class="id" title="var">Hew</span> ; <span class="id" title="tactic">omega</span>.<br/>
<span class="id" title="tactic">eapply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zdiv.html#Z_mod_lt"><span class="id" title="lemma">Z_mod_lt</span></a>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zorder.html#Zlt_gt"><span class="id" title="abbreviation">Zlt_gt</span></a>.<br/>
<span class="id" title="tactic">apply</span> (<a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#Zpower_gt_0"><span class="id" title="lemma">Zpower_gt_0</span></a> <a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#radix2"><span class="id" title="definition">radix2</span></a>).<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zorder.html#Zlt_le_weak"><span class="id" title="abbreviation">Zlt_le_weak</span></a>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#bpow_gt_0"><span class="id" title="lemma">bpow_gt_0</span></a>.<br/>
<span class="id" title="tactic">simpl</span>. <span class="id" title="tactic">intros</span>. <span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#Z.ltb_lt"><span class="id" title="lemma">Z.ltb_lt</span></a>. <span class="id" title="tactic">unfold</span> <span class="id" title="var">prec</span>. <span class="id" title="var">zify</span>; <span class="id" title="tactic">omega</span>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Definition</span> <a name="binary_float_of_bits"><span class="id" title="definition">binary_float_of_bits</span></a> <span class="id" title="var">x</span> :=<br/>
<a class="idref" href="compcert.flocq.Appli.Fappli_IEEE.html#FF2B"><span class="id" title="definition">FF2B</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.prec"><span class="id" title="variable">prec</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.emax"><span class="id" title="variable">emax</span></a> <span class="id" title="var">_</span> (<a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#binary_float_of_bits_aux_correct"><span class="id" title="lemma">binary_float_of_bits_aux_correct</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#x"><span class="id" title="variable">x</span></a>).<br/>
<br/>
<span class="id" title="keyword">Theorem</span> <a name="binary_float_of_bits_of_binary_float"><span class="id" title="lemma">binary_float_of_bits_of_binary_float</span></a> :<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>,<br/>
<a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#binary_float_of_bits"><span class="id" title="definition">binary_float_of_bits</span></a> (<a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#bits_of_binary_float"><span class="id" title="definition">bits_of_binary_float</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#x"><span class="id" title="variable">x</span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">x</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE.html#B2FF_inj"><span class="id" title="lemma">B2FF_inj</span></a>.<br/>
<span class="id" title="tactic">unfold</span> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#binary_float_of_bits"><span class="id" title="definition">binary_float_of_bits</span></a>.<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE.html#B2FF_FF2B"><span class="id" title="lemma">B2FF_FF2B</span></a>.<br/>
<span class="id" title="tactic">unfold</span> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#binary_float_of_bits_aux"><span class="id" title="definition">binary_float_of_bits_aux</span></a>.<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#split_bits_of_binary_float_correct"><span class="id" title="lemma">split_bits_of_binary_float_correct</span></a>.<br/>
<span class="id" title="tactic">destruct</span> <span class="id" title="var">x</span> <span class="id" title="keyword">as</span> [<span class="id" title="var">sx</span>|<span class="id" title="var">sx</span>|<span class="id" title="var">sx</span> [<span class="id" title="var">plx</span> <span class="id" title="var">Hplx</span>]|<span class="id" title="var">sx</span> <span class="id" title="var">mx</span> <span class="id" title="var">ex</span> <span class="id" title="var">Bx</span>].<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#refl_equal"><span class="id" title="abbreviation">refl_equal</span></a>.<br/>
<span class="id" title="tactic">simpl</span>.<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#Zeq_bool_false"><span class="id" title="lemma">Zeq_bool_false</span></a>.<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#Zeq_bool_true"><span class="id" title="lemma">Zeq_bool_true</span></a>.<br/>
<span class="id" title="tactic">cut</span> (1 <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#273beba4f3453dbb29192b3ac740bb71"><span class="id" title="notation"><</span></a> 2<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#186f1fc9207177b1a4e8373ac9655977"><span class="id" title="notation">^</span></a><a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.ew"><span class="id" title="variable">ew</span></a>)%<span class="id" title="var">Z</span>. <span class="id" title="tactic">clear</span> ; <span class="id" title="tactic">omega</span>.<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">apply</span> (<a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#Zpower_gt_1"><span class="id" title="lemma">Zpower_gt_1</span></a> <a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#radix2"><span class="id" title="definition">radix2</span></a>).<br/>
<span class="id" title="tactic">simpl</span>.<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#Zeq_bool_false"><span class="id" title="lemma">Zeq_bool_false</span></a>.<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#Zeq_bool_true"><span class="id" title="lemma">Zeq_bool_true</span></a>; <span class="id" title="tactic">auto</span>.<br/>
<span class="id" title="tactic">cut</span> (1 <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#273beba4f3453dbb29192b3ac740bb71"><span class="id" title="notation"><</span></a> 2<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#186f1fc9207177b1a4e8373ac9655977"><span class="id" title="notation">^</span></a><a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.ew"><span class="id" title="variable">ew</span></a>)%<span class="id" title="var">Z</span>. <span class="id" title="tactic">clear</span> ; <span class="id" title="tactic">omega</span>.<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">apply</span> (<a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#Zpower_gt_1"><span class="id" title="lemma">Zpower_gt_1</span></a> <a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#radix2"><span class="id" title="definition">radix2</span></a>).<br/>
<span class="id" title="tactic">unfold</span> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#split_bits_of_binary_float"><span class="id" title="definition">split_bits_of_binary_float</span></a>.<br/>
<span class="id" title="tactic">case</span> <a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#Zle_bool_spec"><span class="id" title="lemma">Zle_bool_spec</span></a> ; <span class="id" title="tactic">intros</span> <span class="id" title="var">Hm</span>.<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#Zeq_bool_false"><span class="id" title="lemma">Zeq_bool_false</span></a>.<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#Zeq_bool_false"><span class="id" title="lemma">Zeq_bool_false</span></a>.<br/>
<span class="id" title="var">now</span> <span class="id" title="var">ring_simplify</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#Zpos"><span class="id" title="constructor">Zpos</span></a> <span class="id" title="var">mx</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#239a40728d107295b3cb2c790f57f9e9"><span class="id" title="notation">-</span></a> 2 <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#186f1fc9207177b1a4e8373ac9655977"><span class="id" title="notation">^</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.mw"><span class="id" title="variable">mw</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a> 2 <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#186f1fc9207177b1a4e8373ac9655977"><span class="id" title="notation">^</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.mw"><span class="id" title="variable">mw</span></a>)%<span class="id" title="var">Z</span> (<span class="id" title="var">ex</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#239a40728d107295b3cb2c790f57f9e9"><span class="id" title="notation">-</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.emin"><span class="id" title="variable">emin</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.emin"><span class="id" title="variable">emin</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#239a40728d107295b3cb2c790f57f9e9"><span class="id" title="notation">-</span></a> 1)%<span class="id" title="var">Z</span>.<br/>
<span class="id" title="tactic">destruct</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#andb_prop"><span class="id" title="lemma">andb_prop</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">Bx</span>) <span class="id" title="keyword">as</span> (<span class="id" title="var">_</span>, <span class="id" title="var">H1</span>).<br/>
<span class="id" title="tactic">generalize</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zbool.html#Zle_bool_imp_le"><span class="id" title="lemma">Zle_bool_imp_le</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">H1</span>).<br/>
<span class="id" title="tactic">unfold</span> <span class="id" title="var">emin</span>.<br/>
<span class="id" title="tactic">replace</span> (2<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#186f1fc9207177b1a4e8373ac9655977"><span class="id" title="notation">^</span></a><a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.ew"><span class="id" title="variable">ew</span></a>)%<span class="id" title="var">Z</span> <span class="id" title="keyword">with</span> (2 <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#88e34ed7a73f02772126289867960764"><span class="id" title="notation">×</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.emax"><span class="id" title="variable">emax</span></a>)%<span class="id" title="var">Z</span>.<br/>
<span class="id" title="tactic">clear</span> ; <span class="id" title="tactic">omega</span>.<br/>
<span class="id" title="tactic">replace</span> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.ew"><span class="id" title="variable">ew</span></a> <span class="id" title="keyword">with</span> (1 <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">(</span></a><a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.ew"><span class="id" title="variable">ew</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#239a40728d107295b3cb2c790f57f9e9"><span class="id" title="notation">-</span></a> 1<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">Z</span> <span class="id" title="tactic">by</span> <span class="id" title="tactic">ring</span>.<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zpower.html#Zpower_exp"><span class="id" title="lemma">Zpower_exp</span></a>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#refl_equal"><span class="id" title="abbreviation">refl_equal</span></a>.<br/>
<span class="id" title="tactic">discriminate</span>.<br/>
<span class="id" title="tactic">clear</span> -<span class="id" title="var">Hew</span> ; <span class="id" title="tactic">omega</span>.<br/>
<span class="id" title="tactic">destruct</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#andb_prop"><span class="id" title="lemma">andb_prop</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">Bx</span>) <span class="id" title="keyword">as</span> (<span class="id" title="var">H1</span>, <span class="id" title="var">_</span>).<br/>
<span class="id" title="tactic">generalize</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zbool.html#Zeq_bool_eq"><span class="id" title="lemma">Zeq_bool_eq</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">H1</span>).<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.flocq.Core.Fcore_digits.html#Zpos_digits2_pos"><span class="id" title="lemma">Zpos_digits2_pos</span></a>.<br/>
<span class="id" title="tactic">unfold</span> <a class="idref" href="compcert.flocq.Core.Fcore_FLT.html#FLT_exp"><span class="id" title="definition">FLT_exp</span></a>, <span class="id" title="var">emin</span>.<br/>
<span class="id" title="tactic">generalize</span> (<a class="idref" href="compcert.flocq.Core.Fcore_digits.html#Zdigits"><span class="id" title="definition">Zdigits</span></a> <a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#radix2"><span class="id" title="definition">radix2</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#Zpos"><span class="id" title="constructor">Zpos</span></a> <span class="id" title="var">mx</span>)).<br/>
<span class="id" title="tactic">clear</span>.<br/>
<span class="id" title="tactic">intros</span> ; <span class="id" title="var">zify</span> ; <span class="id" title="tactic">omega</span>.<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#Zeq_bool_true"><span class="id" title="lemma">Zeq_bool_true</span></a>. 2: <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#refl_equal"><span class="id" title="abbreviation">refl_equal</span></a>.<br/>
<span class="id" title="tactic">simpl</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#f_equal"><span class="id" title="lemma">f_equal</span></a>.<br/>
<span class="id" title="tactic">destruct</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#andb_prop"><span class="id" title="lemma">andb_prop</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">Bx</span>) <span class="id" title="keyword">as</span> (<span class="id" title="var">H1</span>, <span class="id" title="var">_</span>).<br/>
<span class="id" title="tactic">generalize</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zbool.html#Zeq_bool_eq"><span class="id" title="lemma">Zeq_bool_eq</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">H1</span>).<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.flocq.Core.Fcore_digits.html#Zpos_digits2_pos"><span class="id" title="lemma">Zpos_digits2_pos</span></a>.<br/>
<span class="id" title="tactic">unfold</span> <a class="idref" href="compcert.flocq.Core.Fcore_FLT.html#FLT_exp"><span class="id" title="definition">FLT_exp</span></a>, <span class="id" title="var">emin</span>, <span class="id" title="var">prec</span>.<br/>
<span class="id" title="tactic">apply</span> → <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#Z.lt_sub_0"><span class="id" title="lemma">Z.lt_sub_0</span></a> <span class="id" title="keyword">in</span> <span class="id" title="var">Hm</span>.<br/>
<span class="id" title="tactic">generalize</span> (<a class="idref" href="compcert.flocq.Core.Fcore_digits.html#Zdigits_le_Zpower"><span class="id" title="lemma">Zdigits_le_Zpower</span></a> <a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#radix2"><span class="id" title="definition">radix2</span></a> <span class="id" title="var">_</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#Zpos"><span class="id" title="constructor">Zpos</span></a> <span class="id" title="var">mx</span>) <span class="id" title="var">Hm</span>).<br/>
<span class="id" title="tactic">generalize</span> (<a class="idref" href="compcert.flocq.Core.Fcore_digits.html#Zdigits"><span class="id" title="definition">Zdigits</span></a> <a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#radix2"><span class="id" title="definition">radix2</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#Zpos"><span class="id" title="constructor">Zpos</span></a> <span class="id" title="var">mx</span>)).<br/>
<span class="id" title="tactic">clear</span>.<br/>
<span class="id" title="tactic">intros</span> ; <span class="id" title="var">zify</span> ; <span class="id" title="tactic">omega</span>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Theorem</span> <a name="bits_of_binary_float_of_bits"><span class="id" title="lemma">bits_of_binary_float_of_bits</span></a> :<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>,<br/>
(0 <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#cb2fd1306e4fe5d333fd9fb7e2c4a1d2"><span class="id" title="notation">≤</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#cb2fd1306e4fe5d333fd9fb7e2c4a1d2"><span class="id" title="notation"><</span></a> 2<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#186f1fc9207177b1a4e8373ac9655977"><span class="id" title="notation">^(</span></a><a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.mw"><span class="id" title="variable">mw</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a><a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.ew"><span class="id" title="variable">ew</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a>1<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#186f1fc9207177b1a4e8373ac9655977"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">Z</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#bits_of_binary_float"><span class="id" title="definition">bits_of_binary_float</span></a> (<a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#binary_float_of_bits"><span class="id" title="definition">binary_float_of_bits</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#x"><span class="id" title="variable">x</span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">x</span> <span class="id" title="var">Hx</span>.<br/>
<span class="id" title="tactic">unfold</span> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#binary_float_of_bits"><span class="id" title="definition">binary_float_of_bits</span></a>, <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#bits_of_binary_float"><span class="id" title="definition">bits_of_binary_float</span></a>.<br/>
<span class="id" title="tactic">set</span> (<span class="id" title="var">Cx</span> := <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#binary_float_of_bits_aux_correct"><span class="id" title="lemma">binary_float_of_bits_aux_correct</span></a> <span class="id" title="var">x</span>).<br/>
<span class="id" title="var">clearbody</span> <span class="id" title="var">Cx</span>.<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE.html#match_FF2B"><span class="id" title="lemma">match_FF2B</span></a>.<br/>
<span class="id" title="var">revert</span> <span class="id" title="var">Cx</span>.<br/>
<span class="id" title="tactic">generalize</span> (<a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#join_split_bits"><span class="id" title="lemma">join_split_bits</span></a> <span class="id" title="var">x</span> <span class="id" title="var">Hx</span>).<br/>
<span class="id" title="tactic">unfold</span> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#binary_float_of_bits_aux"><span class="id" title="definition">binary_float_of_bits_aux</span></a>.<br/>
<span class="id" title="var">case_eq</span> (<a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#split_bits"><span class="id" title="definition">split_bits</span></a> <span class="id" title="var">x</span>).<br/>
<span class="id" title="tactic">intros</span> (<span class="id" title="var">sx</span>, <span class="id" title="var">mx</span>) <span class="id" title="var">ex</span> <span class="id" title="var">Sx</span>.<br/>
<span class="id" title="tactic">assert</span> (<span class="id" title="var">Bm</span>: (0 <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#cb2fd1306e4fe5d333fd9fb7e2c4a1d2"><span class="id" title="notation">≤</span></a> <span class="id" title="var">mx</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#cb2fd1306e4fe5d333fd9fb7e2c4a1d2"><span class="id" title="notation"><</span></a> 2<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#186f1fc9207177b1a4e8373ac9655977"><span class="id" title="notation">^</span></a><a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.mw"><span class="id" title="variable">mw</span></a>)%<span class="id" title="var">Z</span>).<br/>
<span class="id" title="tactic">inversion_clear</span> <span class="id" title="var">Sx</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zdiv.html#Z_mod_lt"><span class="id" title="lemma">Z_mod_lt</span></a>.<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zorder.html#Zlt_gt"><span class="id" title="abbreviation">Zlt_gt</span></a>.<br/>
<span class="id" title="tactic">case</span> <a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#Zeq_bool_spec"><span class="id" title="lemma">Zeq_bool_spec</span></a> ; <span class="id" title="tactic">intros</span> <span class="id" title="var">He1</span>.<br/>
<span class="id" title="var">case_eq</span> <span class="id" title="var">mx</span>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">Hm</span> <span class="id" title="var">Jx</span> <span class="id" title="var">_</span>.<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">rewrite</span> <span class="id" title="var">He1</span> <span class="id" title="keyword">in</span> <span class="id" title="var">Jx</span>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">px</span> <span class="id" title="var">Hm</span> <span class="id" title="var">Jx</span> <span class="id" title="var">_</span>.<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#Zle_bool_false"><span class="id" title="lemma">Zle_bool_false</span></a>.<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">rewrite</span> <- <span class="id" title="var">He1</span>.<br/>
<span class="id" title="tactic">apply</span> <- <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#Z.lt_sub_0"><span class="id" title="lemma">Z.lt_sub_0</span></a>.<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">rewrite</span> <- <span class="id" title="var">Hm</span>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">px</span> <span class="id" title="var">Hm</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#False_ind"><span class="id" title="definition">False_ind</span></a>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zorder.html#Zle_not_lt"><span class="id" title="lemma">Zle_not_lt</span></a> <span class="id" title="keyword">with</span> (1 := <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#proj1"><span class="id" title="lemma">proj1</span></a> <span class="id" title="var">Bm</span>).<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">rewrite</span> <span class="id" title="var">Hm</span>.<br/>
<span class="id" title="tactic">case</span> <a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#Zeq_bool_spec"><span class="id" title="lemma">Zeq_bool_spec</span></a> ; <span class="id" title="tactic">intros</span> <span class="id" title="var">He2</span>.<br/>
<span class="id" title="var">case_eq</span> <span class="id" title="var">mx</span>; <span class="id" title="tactic">intros</span> <span class="id" title="var">Hm</span>.<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">rewrite</span> <span class="id" title="var">He2</span>.<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">rewrite</span> <span class="id" title="var">He2</span>.<br/>
<span class="id" title="tactic">intros</span>. <span class="id" title="var">zify</span>; <span class="id" title="tactic">omega</span>.<br/>
<span class="id" title="var">case_eq</span> (<span class="id" title="var">mx</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a> 2 <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#186f1fc9207177b1a4e8373ac9655977"><span class="id" title="notation">^</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.mw"><span class="id" title="variable">mw</span></a>)%<span class="id" title="var">Z</span>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">Hm</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#False_ind"><span class="id" title="definition">False_ind</span></a>.<br/>
<span class="id" title="tactic">clear</span> -<span class="id" title="var">Bm</span> <span class="id" title="var">Hm</span> ; <span class="id" title="tactic">omega</span>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">p</span> <span class="id" title="var">Hm</span> <span class="id" title="var">Jx</span> <span class="id" title="var">Cx</span>.<br/>
<span class="id" title="tactic">rewrite</span> <- <span class="id" title="var">Hm</span>.<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#Zle_bool_true"><span class="id" title="lemma">Zle_bool_true</span></a>.<br/>
<span class="id" title="var">now</span> <span class="id" title="var">ring_simplify</span> (<span class="id" title="var">mx</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a> 2<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#186f1fc9207177b1a4e8373ac9655977"><span class="id" title="notation">^</span></a><a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.mw"><span class="id" title="variable">mw</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#239a40728d107295b3cb2c790f57f9e9"><span class="id" title="notation">-</span></a> 2<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#186f1fc9207177b1a4e8373ac9655977"><span class="id" title="notation">^</span></a><a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.mw"><span class="id" title="variable">mw</span></a>)%<span class="id" title="var">Z</span> (<span class="id" title="var">ex</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.emin"><span class="id" title="variable">emin</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#239a40728d107295b3cb2c790f57f9e9"><span class="id" title="notation">-</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#239a40728d107295b3cb2c790f57f9e9"><span class="id" title="notation">-</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits.emin"><span class="id" title="variable">emin</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a> 1)%<span class="id" title="var">Z</span>.<br/>
<span class="id" title="var">now</span> <span class="id" title="var">ring_simplify</span>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">p</span> <span class="id" title="var">Hm</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#False_ind"><span class="id" title="definition">False_ind</span></a>.<br/>
<span class="id" title="tactic">clear</span> -<span class="id" title="var">Bm</span> <span class="id" title="var">Hm</span> ; <span class="id" title="var">zify</span> ; <span class="id" title="tactic">omega</span>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#Binary_Bits"><span class="id" title="section">Binary_Bits</span></a>.<br/>
<br/>
</div>
<div class="doc">
Specialization for IEEE single precision operations
</div>
<div class="code">
<span class="id" title="keyword">Section</span> <a name="B32_Bits"><span class="id" title="section">B32_Bits</span></a>.<br/>
<br/>
<span class="id" title="keyword">Implicit Arguments</span> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE.html#B754_nan"><span class="id" title="constructor">B754_nan</span></a> [[<span class="id" title="var">prec</span>] [<span class="id" title="var">emax</span>]].<br/>
<br/>
<span class="id" title="keyword">Definition</span> <a name="binary32"><span class="id" title="definition">binary32</span></a> := <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE.html#binary_float"><span class="id" title="inductive">binary_float</span></a> 24 128.<br/>
<br/>
<span class="id" title="keyword">Let</span> <a name="B32_Bits.Hprec"><span class="id" title="variable">Hprec</span></a> : (0 <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#273beba4f3453dbb29192b3ac740bb71"><span class="id" title="notation"><</span></a> 24)%<span class="id" title="var">Z</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#refl_equal"><span class="id" title="abbreviation">refl_equal</span></a>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Let</span> <a name="B32_Bits.Hprec_emax"><span class="id" title="variable">Hprec_emax</span></a> : (24 <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#273beba4f3453dbb29192b3ac740bb71"><span class="id" title="notation"><</span></a> 128)%<span class="id" title="var">Z</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#refl_equal"><span class="id" title="abbreviation">refl_equal</span></a>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Definition</span> <a name="default_nan_pl32"><span class="id" title="definition">default_nan_pl32</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">×</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE.html#nan_pl"><span class="id" title="definition">nan_pl</span></a> 24 :=<br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Specif.html#exist"><span class="id" title="constructor">exist</span></a> <span class="id" title="var">_</span> (<a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#iter_nat"><span class="id" title="definition">iter_nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#xO"><span class="id" title="constructor">xO</span></a> 22 <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#xH"><span class="id" title="constructor">xH</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#refl_equal"><span class="id" title="abbreviation">refl_equal</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/>
<br/>
<span class="id" title="keyword">Definition</span> <a name="unop_nan_pl32"><span class="id" title="definition">unop_nan_pl32</span></a> (<span class="id" title="var">f</span> : <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#binary32"><span class="id" title="definition">binary32</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">×</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE.html#nan_pl"><span class="id" title="definition">nan_pl</span></a> 24 :=<br/>
<span class="id" title="keyword">match</span> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#f"><span class="id" title="variable">f</span></a> <span class="id" title="keyword">with</span><br/>
| <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE.html#B754_nan"><span class="id" title="constructor">B754_nan</span></a> <span class="id" title="var">s</span> <span class="id" title="var">pl</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><span class="id" title="var">s</span><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <span class="id" title="var">pl</span><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a><br/>
| <span class="id" title="var">_</span> ⇒ <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#default_nan_pl32"><span class="id" title="definition">default_nan_pl32</span></a><br/>
<span class="id" title="keyword">end</span>.<br/>
<br/>
<span class="id" title="keyword">Definition</span> <a name="binop_nan_pl32"><span class="id" title="definition">binop_nan_pl32</span></a> (<span class="id" title="var">f1</span> <span class="id" title="var">f2</span> : <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#binary32"><span class="id" title="definition">binary32</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">×</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE.html#nan_pl"><span class="id" title="definition">nan_pl</span></a> 24 :=<br/>
<span class="id" title="keyword">match</span> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#f1"><span class="id" title="variable">f1</span></a>, <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#f2"><span class="id" title="variable">f2</span></a> <span class="id" title="keyword">with</span><br/>
| <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE.html#B754_nan"><span class="id" title="constructor">B754_nan</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">pl1</span>, <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><span class="id" title="var">s1</span><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <span class="id" title="var">pl1</span><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a><br/>
| <span class="id" title="var">_</span>, <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE.html#B754_nan"><span class="id" title="constructor">B754_nan</span></a> <span class="id" title="var">s2</span> <span class="id" title="var">pl2</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><span class="id" title="var">s2</span><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <span class="id" title="var">pl2</span><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a><br/>
| <span class="id" title="var">_</span>, <span class="id" title="var">_</span> ⇒ <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#default_nan_pl32"><span class="id" title="definition">default_nan_pl32</span></a><br/>
<span class="id" title="keyword">end</span>.<br/>
<br/>
<span class="id" title="keyword">Definition</span> <a name="b32_opp"><span class="id" title="definition">b32_opp</span></a> := <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE.html#Bopp"><span class="id" title="definition">Bopp</span></a> 24 128 <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#pair"><span class="id" title="constructor">pair</span></a>.<br/>
<span class="id" title="keyword">Definition</span> <a name="b32_plus"><span class="id" title="definition">b32_plus</span></a> := <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE.html#Bplus"><span class="id" title="definition">Bplus</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#B32_Bits.Hprec"><span class="id" title="variable">Hprec</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#B32_Bits.Hprec_emax"><span class="id" title="variable">Hprec_emax</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#binop_nan_pl32"><span class="id" title="definition">binop_nan_pl32</span></a>.<br/>
<span class="id" title="keyword">Definition</span> <a name="b32_minus"><span class="id" title="definition">b32_minus</span></a> := <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE.html#Bminus"><span class="id" title="definition">Bminus</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#B32_Bits.Hprec"><span class="id" title="variable">Hprec</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#B32_Bits.Hprec_emax"><span class="id" title="variable">Hprec_emax</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#binop_nan_pl32"><span class="id" title="definition">binop_nan_pl32</span></a>.<br/>
<span class="id" title="keyword">Definition</span> <a name="b32_mult"><span class="id" title="definition">b32_mult</span></a> := <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE.html#Bmult"><span class="id" title="definition">Bmult</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#B32_Bits.Hprec"><span class="id" title="variable">Hprec</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#B32_Bits.Hprec_emax"><span class="id" title="variable">Hprec_emax</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#binop_nan_pl32"><span class="id" title="definition">binop_nan_pl32</span></a>.<br/>
<span class="id" title="keyword">Definition</span> <a name="b32_div"><span class="id" title="definition">b32_div</span></a> := <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE.html#Bdiv"><span class="id" title="definition">Bdiv</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#B32_Bits.Hprec"><span class="id" title="variable">Hprec</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#B32_Bits.Hprec_emax"><span class="id" title="variable">Hprec_emax</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#binop_nan_pl32"><span class="id" title="definition">binop_nan_pl32</span></a>.<br/>
<span class="id" title="keyword">Definition</span> <a name="b32_sqrt"><span class="id" title="definition">b32_sqrt</span></a> := <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE.html#Bsqrt"><span class="id" title="definition">Bsqrt</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#B32_Bits.Hprec"><span class="id" title="variable">Hprec</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#B32_Bits.Hprec_emax"><span class="id" title="variable">Hprec_emax</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#unop_nan_pl32"><span class="id" title="definition">unop_nan_pl32</span></a>.<br/>
<br/>
<span class="id" title="keyword">Definition</span> <a name="b32_of_bits"><span class="id" title="definition">b32_of_bits</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#binary32"><span class="id" title="definition">binary32</span></a> := <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#binary_float_of_bits"><span class="id" title="definition">binary_float_of_bits</span></a> 23 8 (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#refl_equal"><span class="id" title="abbreviation">refl_equal</span></a> <span class="id" title="var">_</span>) (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#refl_equal"><span class="id" title="abbreviation">refl_equal</span></a> <span class="id" title="var">_</span>) (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#refl_equal"><span class="id" title="abbreviation">refl_equal</span></a> <span class="id" title="var">_</span>).<br/>
<span class="id" title="keyword">Definition</span> <a name="bits_of_b32"><span class="id" title="definition">bits_of_b32</span></a> : <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#binary32"><span class="id" title="definition">binary32</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a> := <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#bits_of_binary_float"><span class="id" title="definition">bits_of_binary_float</span></a> 23 8.<br/>
<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#B32_Bits"><span class="id" title="section">B32_Bits</span></a>.<br/>
<br/>
</div>
<div class="doc">
Specialization for IEEE double precision operations
</div>
<div class="code">
<span class="id" title="keyword">Section</span> <a name="B64_Bits"><span class="id" title="section">B64_Bits</span></a>.<br/>
<br/>
<span class="id" title="keyword">Implicit Arguments</span> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE.html#B754_nan"><span class="id" title="constructor">B754_nan</span></a> [[<span class="id" title="var">prec</span>] [<span class="id" title="var">emax</span>]].<br/>
<br/>
<span class="id" title="keyword">Definition</span> <a name="binary64"><span class="id" title="definition">binary64</span></a> := <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE.html#binary_float"><span class="id" title="inductive">binary_float</span></a> 53 1024.<br/>
<br/>
<span class="id" title="keyword">Let</span> <a name="B64_Bits.Hprec"><span class="id" title="variable">Hprec</span></a> : (0 <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#273beba4f3453dbb29192b3ac740bb71"><span class="id" title="notation"><</span></a> 53)%<span class="id" title="var">Z</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#refl_equal"><span class="id" title="abbreviation">refl_equal</span></a>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Let</span> <a name="B64_Bits.Hprec_emax"><span class="id" title="variable">Hprec_emax</span></a> : (53 <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#273beba4f3453dbb29192b3ac740bb71"><span class="id" title="notation"><</span></a> 1024)%<span class="id" title="var">Z</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#refl_equal"><span class="id" title="abbreviation">refl_equal</span></a>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Definition</span> <a name="default_nan_pl64"><span class="id" title="definition">default_nan_pl64</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">×</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE.html#nan_pl"><span class="id" title="definition">nan_pl</span></a> 53 :=<br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#false"><span class="id" title="constructor">false</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Specif.html#exist"><span class="id" title="constructor">exist</span></a> <span class="id" title="var">_</span> (<a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#iter_nat"><span class="id" title="definition">iter_nat</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#xO"><span class="id" title="constructor">xO</span></a> 51 <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#xH"><span class="id" title="constructor">xH</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#refl_equal"><span class="id" title="abbreviation">refl_equal</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>)<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a>.<br/>
<br/>
<span class="id" title="keyword">Definition</span> <a name="unop_nan_pl64"><span class="id" title="definition">unop_nan_pl64</span></a> (<span class="id" title="var">f</span> : <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#binary64"><span class="id" title="definition">binary64</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">×</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE.html#nan_pl"><span class="id" title="definition">nan_pl</span></a> 53 :=<br/>
<span class="id" title="keyword">match</span> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#f"><span class="id" title="variable">f</span></a> <span class="id" title="keyword">with</span><br/>
| <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE.html#B754_nan"><span class="id" title="constructor">B754_nan</span></a> <span class="id" title="var">s</span> <span class="id" title="var">pl</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><span class="id" title="var">s</span><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <span class="id" title="var">pl</span><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a><br/>
| <span class="id" title="var">_</span> ⇒ <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#default_nan_pl64"><span class="id" title="definition">default_nan_pl64</span></a><br/>
<span class="id" title="keyword">end</span>.<br/>
<br/>
<span class="id" title="keyword">Definition</span> <a name="binop_nan_pl64"><span class="id" title="definition">binop_nan_pl64</span></a> (<span class="id" title="var">pl1</span> <span class="id" title="var">pl2</span> : <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#binary64"><span class="id" title="definition">binary64</span></a>) : <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#d19c7eafd0e2d195d10df94b392087b5"><span class="id" title="notation">×</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE.html#nan_pl"><span class="id" title="definition">nan_pl</span></a> 53 :=<br/>
<span class="id" title="keyword">match</span> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#pl1"><span class="id" title="variable">pl1</span></a>, <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#pl2"><span class="id" title="variable">pl2</span></a> <span class="id" title="keyword">with</span><br/>
| <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE.html#B754_nan"><span class="id" title="constructor">B754_nan</span></a> <span class="id" title="var">s1</span> <span class="id" title="var">pl1</span>, <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><span class="id" title="var">s1</span><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#pl1"><span class="id" title="variable">pl1</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a><br/>
| <span class="id" title="var">_</span>, <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE.html#B754_nan"><span class="id" title="constructor">B754_nan</span></a> <span class="id" title="var">s2</span> <span class="id" title="var">pl2</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">(</span></a><span class="id" title="var">s2</span><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">,</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#pl2"><span class="id" title="variable">pl2</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#44400027531d4bc3f586a1997dc874c0"><span class="id" title="notation">)</span></a><br/>
| <span class="id" title="var">_</span>, <span class="id" title="var">_</span> ⇒ <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#default_nan_pl64"><span class="id" title="definition">default_nan_pl64</span></a><br/>
<span class="id" title="keyword">end</span>.<br/>
<br/>
<span class="id" title="keyword">Definition</span> <a name="b64_opp"><span class="id" title="definition">b64_opp</span></a> := <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE.html#Bopp"><span class="id" title="definition">Bopp</span></a> 53 1024 <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#pair"><span class="id" title="constructor">pair</span></a>.<br/>
<span class="id" title="keyword">Definition</span> <a name="b64_plus"><span class="id" title="definition">b64_plus</span></a> := <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE.html#Bplus"><span class="id" title="definition">Bplus</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#B64_Bits.Hprec"><span class="id" title="variable">Hprec</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#B64_Bits.Hprec_emax"><span class="id" title="variable">Hprec_emax</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#binop_nan_pl64"><span class="id" title="definition">binop_nan_pl64</span></a>.<br/>
<span class="id" title="keyword">Definition</span> <a name="b64_minus"><span class="id" title="definition">b64_minus</span></a> := <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE.html#Bminus"><span class="id" title="definition">Bminus</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#B64_Bits.Hprec"><span class="id" title="variable">Hprec</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#B64_Bits.Hprec_emax"><span class="id" title="variable">Hprec_emax</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#binop_nan_pl64"><span class="id" title="definition">binop_nan_pl64</span></a>.<br/>
<span class="id" title="keyword">Definition</span> <a name="b64_mult"><span class="id" title="definition">b64_mult</span></a> := <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE.html#Bmult"><span class="id" title="definition">Bmult</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#B64_Bits.Hprec"><span class="id" title="variable">Hprec</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#B64_Bits.Hprec_emax"><span class="id" title="variable">Hprec_emax</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#binop_nan_pl64"><span class="id" title="definition">binop_nan_pl64</span></a>.<br/>
<span class="id" title="keyword">Definition</span> <a name="b64_div"><span class="id" title="definition">b64_div</span></a> := <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE.html#Bdiv"><span class="id" title="definition">Bdiv</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#B64_Bits.Hprec"><span class="id" title="variable">Hprec</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#B64_Bits.Hprec_emax"><span class="id" title="variable">Hprec_emax</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#binop_nan_pl64"><span class="id" title="definition">binop_nan_pl64</span></a>.<br/>
<span class="id" title="keyword">Definition</span> <a name="b64_sqrt"><span class="id" title="definition">b64_sqrt</span></a> := <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE.html#Bsqrt"><span class="id" title="definition">Bsqrt</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#B64_Bits.Hprec"><span class="id" title="variable">Hprec</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#B64_Bits.Hprec_emax"><span class="id" title="variable">Hprec_emax</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#unop_nan_pl64"><span class="id" title="definition">unop_nan_pl64</span></a>.<br/>
<br/>
<span class="id" title="keyword">Definition</span> <a name="b64_of_bits"><span class="id" title="definition">b64_of_bits</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#binary64"><span class="id" title="definition">binary64</span></a> := <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#binary_float_of_bits"><span class="id" title="definition">binary_float_of_bits</span></a> 52 11 (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#refl_equal"><span class="id" title="abbreviation">refl_equal</span></a> <span class="id" title="var">_</span>) (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#refl_equal"><span class="id" title="abbreviation">refl_equal</span></a> <span class="id" title="var">_</span>) (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#refl_equal"><span class="id" title="abbreviation">refl_equal</span></a> <span class="id" title="var">_</span>).<br/>
<span class="id" title="keyword">Definition</span> <a name="bits_of_b64"><span class="id" title="definition">bits_of_b64</span></a> : <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#binary64"><span class="id" title="definition">binary64</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a> := <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#bits_of_binary_float"><span class="id" title="definition">bits_of_binary_float</span></a> 52 11.<br/>
<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="compcert.flocq.Appli.Fappli_IEEE_bits.html#B64_Bits"><span class="id" title="section">B64_Bits</span></a>.<br/>
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