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compcert.flocq.Calc.Fcalc_round.html
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<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
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<html xmlns="http://www.w3.org/1999/xhtml">
<head>
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
<link href="coqdoc.css" rel="stylesheet" type="text/css" />
<title>compcert.flocq.Calc.Fcalc_round</title>
</head>
<body>
<div id="page">
<div id="header">
</div>
<div id="main">
<h1 class="libtitle">Library compcert.flocq.Calc.Fcalc_round</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) 2010-2013 Sylvie Boldo
<br />
Copyright (C) 2010-2013 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="lab32"></a><h1 class="section">Helper function for computing the rounded value of a real number.</h1>
</div>
<div class="code">
<br/>
<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_bracket.html#"><span class="id" title="library">Fcalc_bracket</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/>
<br/>
<span class="id" title="keyword">Section</span> <a name="Fcalc_round"><span class="id" title="section">Fcalc_round</span></a>.<br/>
<br/>
<span class="id" title="keyword">Variable</span> <span class="id" title="keyword">beta</span> : <a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#radix"><span class="id" title="record">radix</span></a>.<br/>
<span class="id" title="keyword">Notation</span> <a name="bpow"><span class="id" title="abbreviation">bpow</span></a> <span class="id" title="var">e</span> := (<a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#bpow"><span class="id" title="definition">bpow</span></a> <span class="id" title="keyword">beta</span> <span class="id" title="var">e</span>).<br/>
<br/>
<span class="id" title="keyword">Section</span> <a name="Fcalc_round.Fcalc_round_fexp"><span class="id" title="section">Fcalc_round_fexp</span></a>.<br/>
<br/>
<span class="id" title="keyword">Variable</span> <a name="Fcalc_round.Fcalc_round_fexp.fexp"><span class="id" title="variable">fexp</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="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">Context</span> { <span class="id" title="var">valid_exp</span> : <a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#Valid_exp"><span class="id" title="class">Valid_exp</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#Fcalc_round.Fcalc_round_fexp.fexp"><span class="id" title="variable">fexp</span></a> }.<br/>
<span class="id" title="keyword">Notation</span> <a name="format"><span class="id" title="abbreviation">format</span></a> := (<a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#generic_format"><span class="id" title="definition">generic_format</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#Fcalc_round.Fcalc_round_fexp.fexp"><span class="id" title="variable">fexp</span></a>).<br/>
<br/>
</div>
<div class="doc">
Relates location and rounding.
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Theorem</span> <a name="inbetween_float_round"><span class="id" title="lemma">inbetween_float_round</span></a> :<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">rnd</span> <span class="id" title="var">choice</span>,<br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">(</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">m</span> <span class="id" title="var">l</span>, <a class="idref" href="compcert.flocq.Calc.Fcalc_bracket.html#inbetween_int"><span class="id" title="definition">inbetween_int</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#l"><span class="id" title="variable">l</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.Calc.Fcalc_round.html#rnd"><span class="id" title="variable">rnd</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.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.Calc.Fcalc_round.html#choice"><span class="id" title="variable">choice</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#l"><span class="id" title="variable">l</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.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">m</span> <span class="id" title="var">l</span>,<br/>
<span class="id" title="keyword">let</span> <span class="id" title="var">e</span> := <a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#canonic_exp"><span class="id" title="definition">canonic_exp</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#Fcalc_round.Fcalc_round_fexp.fexp"><span class="id" title="variable">fexp</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#x"><span class="id" title="variable">x</span></a> <span class="id" title="keyword">in</span><br/>
<a class="idref" href="compcert.flocq.Calc.Fcalc_bracket.html#inbetween_float"><span class="id" title="definition">inbetween_float</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#l"><span class="id" title="variable">l</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.Core.Fcore_generic_fmt.html#round"><span class="id" title="definition">round</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#Fcalc_round.Fcalc_round_fexp.fexp"><span class="id" title="variable">fexp</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#rnd"><span class="id" title="variable">rnd</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.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.Core.Fcore_defs.html#F2R"><span class="id" title="definition">F2R</span></a> (<a class="idref" href="compcert.flocq.Core.Fcore_defs.html#Float"><span class="id" title="constructor">Float</span></a> <span class="id" title="keyword">beta</span> (<a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#choice"><span class="id" title="variable">choice</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#l"><span class="id" title="variable">l</span></a>) <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#e"><span class="id" title="variable">e</span></a>).<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">rnd</span> <span class="id" title="var">choice</span> <span class="id" title="var">Hc</span> <span class="id" title="var">x</span> <span class="id" title="var">m</span> <span class="id" title="var">l</span> <span class="id" title="var">e</span> <span class="id" title="var">Hl</span>.<br/>
<span class="id" title="tactic">unfold</span> <a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#round"><span class="id" title="definition">round</span></a>, <a class="idref" href="compcert.flocq.Core.Fcore_defs.html#F2R"><span class="id" title="definition">F2R</span></a>. <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> (<span class="id" title="keyword">fun</span> <span class="id" title="var">m</span> ⇒ (<a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Z2R"><span class="id" title="definition">Z2R</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Reals.Rdefinitions.html#361afbb9b173fae6251865f2da226003"><span class="id" title="notation">×</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#bpow"><span class="id" title="abbreviation">bpow</span></a> <span class="id" title="var">e</span>)%<span class="id" title="var">R</span>)).<br/>
<span class="id" title="tactic">apply</span> <span class="id" title="var">Hc</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_bracket.html#inbetween_mult_reg"><span class="id" title="lemma">inbetween_mult_reg</span></a> <span class="id" title="keyword">with</span> (<a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#bpow"><span class="id" title="abbreviation">bpow</span></a> <span class="id" title="var">e</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="var">now</span> <span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#scaled_mantissa_mult_bpow"><span class="id" title="lemma">scaled_mantissa_mult_bpow</span></a>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Definition</span> <a name="cond_incr"><span class="id" title="definition">cond_incr</span></a> (<span class="id" title="var">b</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="keyword">if</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#b"><span class="id" title="variable">b</span></a> <span class="id" title="keyword">then</span> (<a class="idref" href="compcert.flocq.Calc.Fcalc_round.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#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a> 1)%<span class="id" title="var">Z</span> <span class="id" title="keyword">else</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</span></a>.<br/>
<br/>
<span class="id" title="keyword">Theorem</span> <a name="inbetween_float_round_sign"><span class="id" title="lemma">inbetween_float_round_sign</span></a> :<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">rnd</span> <span class="id" title="var">choice</span>,<br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">(</span></a> <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">m</span> <span class="id" title="var">l</span>, <a class="idref" href="compcert.flocq.Calc.Fcalc_bracket.html#inbetween_int"><span class="id" title="definition">inbetween_int</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Reals.Rbasic_fun.html#Rabs"><span class="id" title="definition">Rabs</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#l"><span class="id" title="variable">l</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.Calc.Fcalc_round.html#rnd"><span class="id" title="variable">rnd</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.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.Core.Fcore_Zaux.html#cond_Zopp"><span class="id" title="definition">cond_Zopp</span></a> (<a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Rlt_bool"><span class="id" title="definition">Rlt_bool</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#x"><span class="id" title="variable">x</span></a> 0) (<a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#choice"><span class="id" title="variable">choice</span></a> (<a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Rlt_bool"><span class="id" title="definition">Rlt_bool</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#x"><span class="id" title="variable">x</span></a> 0) <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#l"><span class="id" title="variable">l</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.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">m</span> <span class="id" title="var">l</span>,<br/>
<span class="id" title="keyword">let</span> <span class="id" title="var">e</span> := <a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#canonic_exp"><span class="id" title="definition">canonic_exp</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#Fcalc_round.Fcalc_round_fexp.fexp"><span class="id" title="variable">fexp</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#x"><span class="id" title="variable">x</span></a> <span class="id" title="keyword">in</span><br/>
<a class="idref" href="compcert.flocq.Calc.Fcalc_bracket.html#inbetween_float"><span class="id" title="definition">inbetween_float</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#e"><span class="id" title="variable">e</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Reals.Rbasic_fun.html#Rabs"><span class="id" title="definition">Rabs</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#l"><span class="id" title="variable">l</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.Core.Fcore_generic_fmt.html#round"><span class="id" title="definition">round</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#Fcalc_round.Fcalc_round_fexp.fexp"><span class="id" title="variable">fexp</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#rnd"><span class="id" title="variable">rnd</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.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.Core.Fcore_defs.html#F2R"><span class="id" title="definition">F2R</span></a> (<a class="idref" href="compcert.flocq.Core.Fcore_defs.html#Float"><span class="id" title="constructor">Float</span></a> <span class="id" title="keyword">beta</span> (<a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#cond_Zopp"><span class="id" title="definition">cond_Zopp</span></a> (<a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Rlt_bool"><span class="id" title="definition">Rlt_bool</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#x"><span class="id" title="variable">x</span></a> 0) (<a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#choice"><span class="id" title="variable">choice</span></a> (<a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Rlt_bool"><span class="id" title="definition">Rlt_bool</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#x"><span class="id" title="variable">x</span></a> 0) <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#l"><span class="id" title="variable">l</span></a>)) <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#e"><span class="id" title="variable">e</span></a>).<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">rnd</span> <span class="id" title="var">choice</span> <span class="id" title="var">Hc</span> <span class="id" title="var">x</span> <span class="id" title="var">m</span> <span class="id" title="var">l</span> <span class="id" title="var">e</span> <span class="id" title="var">Hx</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">m</span> ⇒ (<a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Z2R"><span class="id" title="definition">Z2R</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Reals.Rdefinitions.html#361afbb9b173fae6251865f2da226003"><span class="id" title="notation">×</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#bpow"><span class="id" title="abbreviation">bpow</span></a> <span class="id" title="var">e</span>)%<span class="id" title="var">R</span>)).<br/>
<span class="id" title="tactic">simpl</span>.<br/>
<span class="id" title="tactic">replace</span> (<a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Rlt_bool"><span class="id" title="definition">Rlt_bool</span></a> <span class="id" title="var">x</span> 0) <span class="id" title="keyword">with</span> (<a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Rlt_bool"><span class="id" title="definition">Rlt_bool</span></a> (<a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#scaled_mantissa"><span class="id" title="definition">scaled_mantissa</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#Fcalc_round.Fcalc_round_fexp.fexp"><span class="id" title="variable">fexp</span></a> <span class="id" title="var">x</span>) 0).<br/>
<span class="id" title="tactic">apply</span> <span class="id" title="var">Hc</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_bracket.html#inbetween_mult_reg"><span class="id" title="lemma">inbetween_mult_reg</span></a> <span class="id" title="keyword">with</span> (<a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#bpow"><span class="id" title="abbreviation">bpow</span></a> <span class="id" title="var">e</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">rewrite</span> <- (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Reals.Rbasic_fun.html#Rabs_right"><span class="id" title="lemma">Rabs_right</span></a> (<a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#bpow"><span class="id" title="abbreviation">bpow</span></a> <span class="id" title="var">e</span>)) <span class="id" title="tactic">at</span> 3.<br/>
<span class="id" title="tactic">rewrite</span> <- <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Reals.Rbasic_fun.html#Rabs_mult"><span class="id" title="lemma">Rabs_mult</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_generic_fmt.html#scaled_mantissa_mult_bpow"><span class="id" title="lemma">scaled_mantissa_mult_bpow</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#Rle_ge"><span class="id" title="lemma">Rle_ge</span></a>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#bpow_ge_0"><span class="id" title="lemma">bpow_ge_0</span></a>.<br/>
<span class="id" title="tactic">destruct</span> (<a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Rlt_bool_spec"><span class="id" title="lemma">Rlt_bool_spec</span></a> <span class="id" title="var">x</span> 0) <span class="id" title="keyword">as</span> [<span class="id" title="var">Zx</span>|<span class="id" title="var">Zx</span>] ; <span class="id" title="tactic">simpl</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Rlt_bool_true"><span class="id" title="lemma">Rlt_bool_true</span></a>.<br/>
<span class="id" title="tactic">rewrite</span> <- (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Reals.RIneq.html#Rmult_0_l"><span class="id" title="lemma">Rmult_0_l</span></a> (<a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#bpow"><span class="id" title="abbreviation">bpow</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>))).<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Reals.RIneq.html#Rmult_lt_compat_r"><span class="id" title="lemma">Rmult_lt_compat_r</span></a> <span class="id" title="keyword">with</span> (2 := <span class="id" title="var">Zx</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">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Rlt_bool_false"><span class="id" title="lemma">Rlt_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.Reals.RIneq.html#Rmult_le_pos"><span class="id" title="lemma">Rmult_le_pos</span></a> <span class="id" title="keyword">with</span> (1 := <span class="id" title="var">Zx</span>).<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#bpow_ge_0"><span class="id" title="lemma">bpow_ge_0</span></a>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
Relates location and rounding down.
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Theorem</span> <a name="inbetween_int_DN"><span class="id" title="lemma">inbetween_int_DN</span></a> :<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">m</span> <span class="id" title="var">l</span>,<br/>
<a class="idref" href="compcert.flocq.Calc.Fcalc_bracket.html#inbetween_int"><span class="id" title="definition">inbetween_int</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#l"><span class="id" title="variable">l</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.Core.Fcore_Raux.html#Zfloor"><span class="id" title="definition">Zfloor</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.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.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</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">m</span> <span class="id" title="var">l</span> <span class="id" title="var">Hl</span>.<br/>
<span class="id" title="tactic">refine</span> (<a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Zfloor_imp"><span class="id" title="lemma">Zfloor_imp</span></a> <span class="id" title="var">m</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="compcert.flocq.Calc.Fcalc_bracket.html#inbetween_bounds"><span class="id" title="lemma">inbetween_bounds</span></a> <span class="id" title="keyword">with</span> (2 := <span class="id" title="var">Hl</span>).<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">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zorder.html#Zlt_succ"><span class="id" title="abbreviation">Zlt_succ</span></a>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Theorem</span> <a name="inbetween_float_DN"><span class="id" title="lemma">inbetween_float_DN</span></a> :<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">m</span> <span class="id" title="var">l</span>,<br/>
<span class="id" title="keyword">let</span> <span class="id" title="var">e</span> := <a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#canonic_exp"><span class="id" title="definition">canonic_exp</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#Fcalc_round.Fcalc_round_fexp.fexp"><span class="id" title="variable">fexp</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#x"><span class="id" title="variable">x</span></a> <span class="id" title="keyword">in</span><br/>
<a class="idref" href="compcert.flocq.Calc.Fcalc_bracket.html#inbetween_float"><span class="id" title="definition">inbetween_float</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#l"><span class="id" title="variable">l</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.Core.Fcore_generic_fmt.html#round"><span class="id" title="definition">round</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#Fcalc_round.Fcalc_round_fexp.fexp"><span class="id" title="variable">fexp</span></a> <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Zfloor"><span class="id" title="definition">Zfloor</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.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.Core.Fcore_defs.html#F2R"><span class="id" title="definition">F2R</span></a> (<a class="idref" href="compcert.flocq.Core.Fcore_defs.html#Float"><span class="id" title="constructor">Float</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#e"><span class="id" title="variable">e</span></a>).<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#inbetween_float_round"><span class="id" title="lemma">inbetween_float_round</span></a> <span class="id" title="keyword">with</span> (<span class="id" title="var">choice</span> := <span class="id" title="keyword">fun</span> <span class="id" title="var">m</span> <span class="id" title="var">l</span> ⇒ <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</span></a>).<br/>
<span class="id" title="tactic">exact</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#inbetween_int_DN"><span class="id" title="lemma">inbetween_int_DN</span></a>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Definition</span> <a name="round_sign_DN"><span class="id" title="definition">round_sign_DN</span></a> <span class="id" title="var">s</span> <span class="id" title="var">l</span> :=<br/>
<span class="id" title="keyword">match</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#l"><span class="id" title="variable">l</span></a> <span class="id" title="keyword">with</span><br/>
| <a class="idref" href="compcert.flocq.Calc.Fcalc_bracket.html#loc_Exact"><span class="id" title="constructor">loc_Exact</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><br/>
| <span class="id" title="var">_</span> ⇒ <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#s"><span class="id" title="variable">s</span></a><br/>
<span class="id" title="keyword">end</span>.<br/>
<br/>
<span class="id" title="keyword">Theorem</span> <a name="inbetween_int_DN_sign"><span class="id" title="lemma">inbetween_int_DN_sign</span></a> :<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">m</span> <span class="id" title="var">l</span>,<br/>
<a class="idref" href="compcert.flocq.Calc.Fcalc_bracket.html#inbetween_int"><span class="id" title="definition">inbetween_int</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Reals.Rbasic_fun.html#Rabs"><span class="id" title="definition">Rabs</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#l"><span class="id" title="variable">l</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.Core.Fcore_Raux.html#Zfloor"><span class="id" title="definition">Zfloor</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.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.Core.Fcore_Zaux.html#cond_Zopp"><span class="id" title="definition">cond_Zopp</span></a> (<a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Rlt_bool"><span class="id" title="definition">Rlt_bool</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#x"><span class="id" title="variable">x</span></a> 0) (<a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#cond_incr"><span class="id" title="definition">cond_incr</span></a> (<a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#round_sign_DN"><span class="id" title="definition">round_sign_DN</span></a> (<a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Rlt_bool"><span class="id" title="definition">Rlt_bool</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#x"><span class="id" title="variable">x</span></a> 0) <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#l"><span class="id" title="variable">l</span></a>) <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</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">m</span> <span class="id" title="var">l</span> <span class="id" title="var">Hl</span>.<br/>
<span class="id" title="tactic">unfold</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Reals.Rbasic_fun.html#Rabs"><span class="id" title="definition">Rabs</span></a> <span class="id" title="keyword">in</span> <span class="id" title="var">Hl</span>.<br/>
<span class="id" title="tactic">destruct</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Reals.Rbasic_fun.html#Rcase_abs"><span class="id" title="lemma">Rcase_abs</span></a> <span class="id" title="var">x</span>) <span class="id" title="keyword">as</span> [<span class="id" title="var">Zx</span>|<span class="id" title="var">Zx</span>] .<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Rlt_bool_true"><span class="id" title="lemma">Rlt_bool_true</span></a> <span class="id" title="keyword">with</span> (1 := <span class="id" title="var">Zx</span>).<br/>
<span class="id" title="tactic">inversion_clear</span> <span class="id" title="var">Hl</span> ; <span class="id" title="tactic">simpl</span>.<br/>
<span class="id" title="tactic">rewrite</span> <- (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Reals.RIneq.html#Ropp_involutive"><span class="id" title="lemma">Ropp_involutive</span></a> <span class="id" title="var">x</span>).<br/>
<span class="id" title="tactic">rewrite</span> <span class="id" title="var">H</span>, <- <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Z2R_opp"><span class="id" title="lemma">Z2R_opp</span></a>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Zfloor_Z2R"><span class="id" title="lemma">Zfloor_Z2R</span></a>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Zfloor_imp"><span class="id" title="lemma">Zfloor_imp</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.Reals.RIneq.html#Rlt_le"><span class="id" title="lemma">Rlt_le</span></a>.<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Z2R_opp"><span class="id" title="lemma">Z2R_opp</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#Ropp_lt_cancel"><span class="id" title="lemma">Ropp_lt_cancel</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.Reals.RIneq.html#Ropp_involutive"><span class="id" title="lemma">Ropp_involutive</span></a>.<br/>
<span class="id" title="var">ring_simplify</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#0a3978eabdacce0a128118074de19651"><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">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> 1<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#0a3978eabdacce0a128118074de19651"><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>.<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Z2R_opp"><span class="id" title="lemma">Z2R_opp</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#Ropp_lt_cancel"><span class="id" title="lemma">Ropp_lt_cancel</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.Reals.RIneq.html#Ropp_involutive"><span class="id" title="lemma">Ropp_involutive</span></a>.<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Rlt_bool_false"><span class="id" title="lemma">Rlt_bool_false</span></a>.<br/>
<span class="id" title="tactic">inversion_clear</span> <span class="id" title="var">Hl</span> ; <span class="id" title="tactic">simpl</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="compcert.flocq.Core.Fcore_Raux.html#Zfloor_Z2R"><span class="id" title="lemma">Zfloor_Z2R</span></a>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Zfloor_imp"><span class="id" title="lemma">Zfloor_imp</span></a>.<br/>
<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.Reals.RIneq.html#Rlt_le"><span class="id" title="lemma">Rlt_le</span></a>.<br/>
<span class="id" title="tactic">apply</span> <span class="id" title="var">H</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.Reals.RIneq.html#Rge_le"><span class="id" title="lemma">Rge_le</span></a>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Theorem</span> <a name="inbetween_float_DN_sign"><span class="id" title="lemma">inbetween_float_DN_sign</span></a> :<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">m</span> <span class="id" title="var">l</span>,<br/>
<span class="id" title="keyword">let</span> <span class="id" title="var">e</span> := <a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#canonic_exp"><span class="id" title="definition">canonic_exp</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#Fcalc_round.Fcalc_round_fexp.fexp"><span class="id" title="variable">fexp</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#x"><span class="id" title="variable">x</span></a> <span class="id" title="keyword">in</span><br/>
<a class="idref" href="compcert.flocq.Calc.Fcalc_bracket.html#inbetween_float"><span class="id" title="definition">inbetween_float</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#e"><span class="id" title="variable">e</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Reals.Rbasic_fun.html#Rabs"><span class="id" title="definition">Rabs</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#l"><span class="id" title="variable">l</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.Core.Fcore_generic_fmt.html#round"><span class="id" title="definition">round</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#Fcalc_round.Fcalc_round_fexp.fexp"><span class="id" title="variable">fexp</span></a> <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Zfloor"><span class="id" title="definition">Zfloor</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.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.Core.Fcore_defs.html#F2R"><span class="id" title="definition">F2R</span></a> (<a class="idref" href="compcert.flocq.Core.Fcore_defs.html#Float"><span class="id" title="constructor">Float</span></a> <span class="id" title="keyword">beta</span> (<a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#cond_Zopp"><span class="id" title="definition">cond_Zopp</span></a> (<a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Rlt_bool"><span class="id" title="definition">Rlt_bool</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#x"><span class="id" title="variable">x</span></a> 0) (<a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#cond_incr"><span class="id" title="definition">cond_incr</span></a> (<a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#round_sign_DN"><span class="id" title="definition">round_sign_DN</span></a> (<a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Rlt_bool"><span class="id" title="definition">Rlt_bool</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#x"><span class="id" title="variable">x</span></a> 0) <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#l"><span class="id" title="variable">l</span></a>) <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</span></a>)) <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#e"><span class="id" title="variable">e</span></a>).<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#inbetween_float_round_sign"><span class="id" title="lemma">inbetween_float_round_sign</span></a> <span class="id" title="keyword">with</span> (<span class="id" title="var">choice</span> := <span class="id" title="keyword">fun</span> <span class="id" title="var">s</span> <span class="id" title="var">m</span> <span class="id" title="var">l</span> ⇒ <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#cond_incr"><span class="id" title="definition">cond_incr</span></a> (<a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#round_sign_DN"><span class="id" title="definition">round_sign_DN</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#l"><span class="id" title="variable">l</span></a>) <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</span></a>).<br/>
<span class="id" title="tactic">exact</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#inbetween_int_DN_sign"><span class="id" title="lemma">inbetween_int_DN_sign</span></a>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
Relates location and rounding up.
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Definition</span> <a name="round_UP"><span class="id" title="definition">round_UP</span></a> <span class="id" title="var">l</span> :=<br/>
<span class="id" title="keyword">match</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#l"><span class="id" title="variable">l</span></a> <span class="id" title="keyword">with</span><br/>
| <a class="idref" href="compcert.flocq.Calc.Fcalc_bracket.html#loc_Exact"><span class="id" title="constructor">loc_Exact</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><br/>
| <span class="id" title="var">_</span> ⇒ <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">end</span>.<br/>
<br/>
<span class="id" title="keyword">Theorem</span> <a name="inbetween_int_UP"><span class="id" title="lemma">inbetween_int_UP</span></a> :<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">m</span> <span class="id" title="var">l</span>,<br/>
<a class="idref" href="compcert.flocq.Calc.Fcalc_bracket.html#inbetween_int"><span class="id" title="definition">inbetween_int</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#l"><span class="id" title="variable">l</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.Core.Fcore_Raux.html#Zceil"><span class="id" title="definition">Zceil</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.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.Calc.Fcalc_round.html#cond_incr"><span class="id" title="definition">cond_incr</span></a> (<a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#round_UP"><span class="id" title="definition">round_UP</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#l"><span class="id" title="variable">l</span></a>) <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</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">m</span> <span class="id" title="var">l</span> <span class="id" title="var">Hl</span>.<br/>
<span class="id" title="tactic">assert</span> (<span class="id" title="var">Hl'</span>: <span class="id" title="var">l</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.Calc.Fcalc_bracket.html#loc_Exact"><span class="id" title="constructor">loc_Exact</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#7a45dffb109c3069e5c675be68643e60"><span class="id" title="notation">∨</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#7a45dffb109c3069e5c675be68643e60"><span class="id" title="notation">(</span></a><span class="id" title="var">l</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#32263a1c8b01baecdff9deb038955bc9"><span class="id" title="notation">≠</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_bracket.html#loc_Exact"><span class="id" title="constructor">loc_Exact</span></a> <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.Calc.Fcalc_round.html#round_UP"><span class="id" title="definition">round_UP</span></a> <span class="id" title="var">l</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="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.Logic.html#7a45dffb109c3069e5c675be68643e60"><span class="id" title="notation">)</span></a>).<br/>
<span class="id" title="tactic">case</span> <span class="id" title="var">l</span> ; <span class="id" title="tactic">try</span> (<span class="id" title="var">now</span> <span class="id" title="tactic">left</span>) ; <span class="id" title="var">now</span> <span class="id" title="tactic">right</span> ; <span class="id" title="tactic">split</span>.<br/>
<span class="id" title="tactic">destruct</span> <span class="id" title="var">Hl'</span> <span class="id" title="keyword">as</span> [<span class="id" title="var">Hl'</span>|(<span class="id" title="var">Hl1</span>, <span class="id" title="var">Hl2</span>)].<br/>
<span class="id" title="tactic">rewrite</span> <span class="id" title="var">Hl'</span>.<br/>
<span class="id" title="tactic">destruct</span> <span class="id" title="var">Hl</span> ; <span class="id" title="tactic">try</span> <span class="id" title="var">easy</span>.<br/>
<span class="id" title="tactic">rewrite</span> <span class="id" title="var">H</span>.<br/>
<span class="id" title="tactic">exact</span> (<a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Zceil_Z2R"><span class="id" title="lemma">Zceil_Z2R</span></a> <span class="id" title="var">_</span>).<br/>
<span class="id" title="tactic">rewrite</span> <span class="id" title="var">Hl2</span>.<br/>
<span class="id" title="tactic">simpl</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Zceil_imp"><span class="id" title="lemma">Zceil_imp</span></a>.<br/>
<span class="id" title="var">ring_simplify</span> (<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> 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> 1)%<span class="id" title="var">Z</span>.<br/>
<span class="id" title="tactic">refine</span> (<span class="id" title="keyword">let</span> <span class="id" title="var">H</span> := <span class="id" title="var">_</span> <span class="id" title="keyword">in</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#conj"><span class="id" title="constructor">conj</span></a> (<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> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#H"><span class="id" title="variable">H</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Reals.RIneq.html#Rlt_le"><span class="id" title="lemma">Rlt_le</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#proj2"><span class="id" title="lemma">proj2</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#H"><span class="id" title="variable">H</span></a>))).<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_bracket.html#inbetween_bounds_not_Eq"><span class="id" title="lemma">inbetween_bounds_not_Eq</span></a> <span class="id" title="keyword">with</span> (2 := <span class="id" title="var">Hl1</span>) (1 := <span class="id" title="var">Hl</span>).<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Theorem</span> <a name="inbetween_float_UP"><span class="id" title="lemma">inbetween_float_UP</span></a> :<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">m</span> <span class="id" title="var">l</span>,<br/>
<span class="id" title="keyword">let</span> <span class="id" title="var">e</span> := <a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#canonic_exp"><span class="id" title="definition">canonic_exp</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#Fcalc_round.Fcalc_round_fexp.fexp"><span class="id" title="variable">fexp</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#x"><span class="id" title="variable">x</span></a> <span class="id" title="keyword">in</span><br/>
<a class="idref" href="compcert.flocq.Calc.Fcalc_bracket.html#inbetween_float"><span class="id" title="definition">inbetween_float</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#l"><span class="id" title="variable">l</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.Core.Fcore_generic_fmt.html#round"><span class="id" title="definition">round</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#Fcalc_round.Fcalc_round_fexp.fexp"><span class="id" title="variable">fexp</span></a> <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Zceil"><span class="id" title="definition">Zceil</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.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.Core.Fcore_defs.html#F2R"><span class="id" title="definition">F2R</span></a> (<a class="idref" href="compcert.flocq.Core.Fcore_defs.html#Float"><span class="id" title="constructor">Float</span></a> <span class="id" title="keyword">beta</span> (<a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#cond_incr"><span class="id" title="definition">cond_incr</span></a> (<a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#round_UP"><span class="id" title="definition">round_UP</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#l"><span class="id" title="variable">l</span></a>) <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</span></a>) <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#e"><span class="id" title="variable">e</span></a>).<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#inbetween_float_round"><span class="id" title="lemma">inbetween_float_round</span></a> <span class="id" title="keyword">with</span> (<span class="id" title="var">choice</span> := <span class="id" title="keyword">fun</span> <span class="id" title="var">m</span> <span class="id" title="var">l</span> ⇒ <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#cond_incr"><span class="id" title="definition">cond_incr</span></a> (<a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#round_UP"><span class="id" title="definition">round_UP</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#l"><span class="id" title="variable">l</span></a>) <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</span></a>).<br/>
<span class="id" title="tactic">exact</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#inbetween_int_UP"><span class="id" title="lemma">inbetween_int_UP</span></a>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Definition</span> <a name="round_sign_UP"><span class="id" title="definition">round_sign_UP</span></a> <span class="id" title="var">s</span> <span class="id" title="var">l</span> :=<br/>
<span class="id" title="keyword">match</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#l"><span class="id" title="variable">l</span></a> <span class="id" title="keyword">with</span><br/>
| <a class="idref" href="compcert.flocq.Calc.Fcalc_bracket.html#loc_Exact"><span class="id" title="constructor">loc_Exact</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><br/>
| <span class="id" title="var">_</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#negb"><span class="id" title="definition">negb</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#s"><span class="id" title="variable">s</span></a><br/>
<span class="id" title="keyword">end</span>.<br/>
<br/>
<span class="id" title="keyword">Theorem</span> <a name="inbetween_int_UP_sign"><span class="id" title="lemma">inbetween_int_UP_sign</span></a> :<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">m</span> <span class="id" title="var">l</span>,<br/>
<a class="idref" href="compcert.flocq.Calc.Fcalc_bracket.html#inbetween_int"><span class="id" title="definition">inbetween_int</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Reals.Rbasic_fun.html#Rabs"><span class="id" title="definition">Rabs</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#l"><span class="id" title="variable">l</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.Core.Fcore_Raux.html#Zceil"><span class="id" title="definition">Zceil</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.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.Core.Fcore_Zaux.html#cond_Zopp"><span class="id" title="definition">cond_Zopp</span></a> (<a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Rlt_bool"><span class="id" title="definition">Rlt_bool</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#x"><span class="id" title="variable">x</span></a> 0) (<a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#cond_incr"><span class="id" title="definition">cond_incr</span></a> (<a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#round_sign_UP"><span class="id" title="definition">round_sign_UP</span></a> (<a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Rlt_bool"><span class="id" title="definition">Rlt_bool</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#x"><span class="id" title="variable">x</span></a> 0) <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#l"><span class="id" title="variable">l</span></a>) <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</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">m</span> <span class="id" title="var">l</span> <span class="id" title="var">Hl</span>.<br/>
<span class="id" title="tactic">unfold</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Reals.Rbasic_fun.html#Rabs"><span class="id" title="definition">Rabs</span></a> <span class="id" title="keyword">in</span> <span class="id" title="var">Hl</span>.<br/>
<span class="id" title="tactic">destruct</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Reals.Rbasic_fun.html#Rcase_abs"><span class="id" title="lemma">Rcase_abs</span></a> <span class="id" title="var">x</span>) <span class="id" title="keyword">as</span> [<span class="id" title="var">Zx</span>|<span class="id" title="var">Zx</span>] .<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Rlt_bool_true"><span class="id" title="lemma">Rlt_bool_true</span></a> <span class="id" title="keyword">with</span> (1 := <span class="id" title="var">Zx</span>).<br/>
<span class="id" title="tactic">simpl</span>.<br/>
<span class="id" title="tactic">unfold</span> <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Zceil"><span class="id" title="definition">Zceil</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">inversion_clear</span> <span class="id" title="var">Hl</span> ; <span class="id" title="tactic">simpl</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="compcert.flocq.Core.Fcore_Raux.html#Zfloor_Z2R"><span class="id" title="lemma">Zfloor_Z2R</span></a>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Zfloor_imp"><span class="id" title="lemma">Zfloor_imp</span></a>.<br/>
<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.Reals.RIneq.html#Rlt_le"><span class="id" title="lemma">Rlt_le</span></a>.<br/>
<span class="id" title="tactic">apply</span> <span class="id" title="var">H</span>.<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Rlt_bool_false"><span class="id" title="lemma">Rlt_bool_false</span></a>.<br/>
<span class="id" title="tactic">simpl</span>.<br/>
<span class="id" title="tactic">inversion_clear</span> <span class="id" title="var">Hl</span> ; <span class="id" title="tactic">simpl</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="compcert.flocq.Core.Fcore_Raux.html#Zceil_Z2R"><span class="id" title="lemma">Zceil_Z2R</span></a>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Zceil_imp"><span class="id" title="lemma">Zceil_imp</span></a>.<br/>
<span class="id" title="tactic">split</span>.<br/>
<span class="id" title="tactic">change</span> (<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> 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> 1)%<span class="id" title="var">Z</span> <span class="id" title="keyword">with</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#Zpred"><span class="id" title="abbreviation">Zpred</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#Zsucc"><span class="id" title="abbreviation">Zsucc</span></a> <span class="id" title="var">m</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#Zpred_succ"><span class="id" title="lemma">Zpred_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.Reals.RIneq.html#Rlt_le"><span class="id" title="lemma">Rlt_le</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.Reals.RIneq.html#Rge_le"><span class="id" title="lemma">Rge_le</span></a>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Theorem</span> <a name="inbetween_float_UP_sign"><span class="id" title="lemma">inbetween_float_UP_sign</span></a> :<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">m</span> <span class="id" title="var">l</span>,<br/>
<span class="id" title="keyword">let</span> <span class="id" title="var">e</span> := <a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#canonic_exp"><span class="id" title="definition">canonic_exp</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#Fcalc_round.Fcalc_round_fexp.fexp"><span class="id" title="variable">fexp</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#x"><span class="id" title="variable">x</span></a> <span class="id" title="keyword">in</span><br/>
<a class="idref" href="compcert.flocq.Calc.Fcalc_bracket.html#inbetween_float"><span class="id" title="definition">inbetween_float</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#e"><span class="id" title="variable">e</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Reals.Rbasic_fun.html#Rabs"><span class="id" title="definition">Rabs</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#l"><span class="id" title="variable">l</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.Core.Fcore_generic_fmt.html#round"><span class="id" title="definition">round</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#Fcalc_round.Fcalc_round_fexp.fexp"><span class="id" title="variable">fexp</span></a> <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Zceil"><span class="id" title="definition">Zceil</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.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.Core.Fcore_defs.html#F2R"><span class="id" title="definition">F2R</span></a> (<a class="idref" href="compcert.flocq.Core.Fcore_defs.html#Float"><span class="id" title="constructor">Float</span></a> <span class="id" title="keyword">beta</span> (<a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#cond_Zopp"><span class="id" title="definition">cond_Zopp</span></a> (<a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Rlt_bool"><span class="id" title="definition">Rlt_bool</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#x"><span class="id" title="variable">x</span></a> 0) (<a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#cond_incr"><span class="id" title="definition">cond_incr</span></a> (<a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#round_sign_UP"><span class="id" title="definition">round_sign_UP</span></a> (<a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Rlt_bool"><span class="id" title="definition">Rlt_bool</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#x"><span class="id" title="variable">x</span></a> 0) <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#l"><span class="id" title="variable">l</span></a>) <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</span></a>)) <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#e"><span class="id" title="variable">e</span></a>).<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#inbetween_float_round_sign"><span class="id" title="lemma">inbetween_float_round_sign</span></a> <span class="id" title="keyword">with</span> (<span class="id" title="var">choice</span> := <span class="id" title="keyword">fun</span> <span class="id" title="var">s</span> <span class="id" title="var">m</span> <span class="id" title="var">l</span> ⇒ <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#cond_incr"><span class="id" title="definition">cond_incr</span></a> (<a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#round_sign_UP"><span class="id" title="definition">round_sign_UP</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#s"><span class="id" title="variable">s</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#l"><span class="id" title="variable">l</span></a>) <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</span></a>).<br/>
<span class="id" title="tactic">exact</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#inbetween_int_UP_sign"><span class="id" title="lemma">inbetween_int_UP_sign</span></a>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
Relates location and rounding toward zero.
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Definition</span> <a name="round_ZR"><span class="id" title="definition">round_ZR</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">l</span> :=<br/>
<span class="id" title="keyword">match</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#l"><span class="id" title="variable">l</span></a> <span class="id" title="keyword">with</span><br/>
| <a class="idref" href="compcert.flocq.Calc.Fcalc_bracket.html#loc_Exact"><span class="id" title="constructor">loc_Exact</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><br/>
| <span class="id" title="var">_</span> ⇒ <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#s"><span class="id" title="variable">s</span></a><br/>
<span class="id" title="keyword">end</span>.<br/>
<br/>
<span class="id" title="keyword">Theorem</span> <a name="inbetween_int_ZR"><span class="id" title="lemma">inbetween_int_ZR</span></a> :<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">m</span> <span class="id" title="var">l</span>,<br/>
<a class="idref" href="compcert.flocq.Calc.Fcalc_bracket.html#inbetween_int"><span class="id" title="definition">inbetween_int</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#l"><span class="id" title="variable">l</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.Core.Fcore_Raux.html#Ztrunc"><span class="id" title="definition">Ztrunc</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.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.Calc.Fcalc_round.html#cond_incr"><span class="id" title="definition">cond_incr</span></a> (<a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#round_ZR"><span class="id" title="definition">round_ZR</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zbool.html#Zlt_bool"><span class="id" title="abbreviation">Zlt_bool</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</span></a> 0) <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#l"><span class="id" title="variable">l</span></a>) <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</span></a>.<br/>
<span class="id" title="keyword">Proof</span> <span class="id" title="keyword">with</span> <span class="id" title="tactic">auto</span> <span class="id" title="keyword">with</span> <span class="id" title="var">typeclass_instances</span>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">x</span> <span class="id" title="var">m</span> <span class="id" title="var">l</span> <span class="id" title="var">Hl</span>.<br/>
<span class="id" title="tactic">inversion_clear</span> <span class="id" title="var">Hl</span> <span class="id" title="keyword">as</span> [<span class="id" title="var">Hx</span>|<span class="id" title="var">l'</span> <span class="id" title="var">Hx</span> <span class="id" title="var">Hl'</span>].<br/>
<span class="id" title="tactic">rewrite</span> <span class="id" title="var">Hx</span>.<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#Zrnd_Z2R"><span class="id" title="method">Zrnd_Z2R</span></a>...<br/>
<span class="id" title="tactic">unfold</span> <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Ztrunc"><span class="id" title="definition">Ztrunc</span></a>.<br/>
<span class="id" title="tactic">assert</span> (<span class="id" title="var">Hm</span>: <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Zfloor"><span class="id" title="definition">Zfloor</span></a> <span class="id" title="var">x</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> <span class="id" title="var">m</span>).<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Zfloor_imp"><span class="id" title="lemma">Zfloor_imp</span></a>.<br/>
<span class="id" title="tactic">exact</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#conj"><span class="id" title="constructor">conj</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Reals.RIneq.html#Rlt_le"><span class="id" title="lemma">Rlt_le</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</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">Hx</span>)) (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#proj2"><span class="id" title="lemma">proj2</span></a> <span class="id" title="var">Hx</span>)).<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Zceil_floor_neq"><span class="id" title="lemma">Zceil_floor_neq</span></a>.<br/>
<span class="id" title="tactic">rewrite</span> <span class="id" title="var">Hm</span>.<br/>
<span class="id" title="tactic">unfold</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#cond_incr"><span class="id" title="definition">cond_incr</span></a>.<br/>
<span class="id" title="tactic">simpl</span>.<br/>
<span class="id" title="tactic">case</span> <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Rlt_bool_spec"><span class="id" title="lemma">Rlt_bool_spec</span></a> ; <span class="id" title="tactic">intros</span> <span class="id" title="var">Hx'</span> ;<br/>
<span class="id" title="tactic">case</span> <a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#Zlt_bool_spec"><span class="id" title="lemma">Zlt_bool_spec</span></a> ; <span class="id" title="tactic">intros</span> <span class="id" title="var">Hm'</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.Init.Logic.html#refl_equal"><span class="id" title="abbreviation">refl_equal</span></a>.<br/>
<span class="id" title="tactic">elim</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Reals.RIneq.html#Rlt_not_le"><span class="id" title="lemma">Rlt_not_le</span></a> <span class="id" title="keyword">with</span> (1 := <span class="id" title="var">Hx'</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#Rlt_le"><span class="id" title="lemma">Rlt_le</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#Rle_lt_trans"><span class="id" title="lemma">Rle_lt_trans</span></a> <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">Hx</span>).<br/>
<span class="id" title="var">now</span> <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> 0).<br/>
<span class="id" title="tactic">elim</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Reals.RIneq.html#Rle_not_lt"><span class="id" title="lemma">Rle_not_lt</span></a> <span class="id" title="keyword">with</span> (1 := <span class="id" title="var">Hx'</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#Rlt_le_trans"><span class="id" title="lemma">Rlt_le_trans</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#proj2"><span class="id" title="lemma">proj2</span></a> <span class="id" title="var">Hx</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> <span class="id" title="var">_</span> 0).<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_succ"><span class="id" title="lemma">Zlt_le_succ</span></a>.<br/>
<span class="id" title="tactic">rewrite</span> <span class="id" title="var">Hm</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.Reals.RIneq.html#Rlt_not_eq"><span class="id" title="lemma">Rlt_not_eq</span></a>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Theorem</span> <a name="inbetween_float_ZR"><span class="id" title="lemma">inbetween_float_ZR</span></a> :<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">m</span> <span class="id" title="var">l</span>,<br/>
<span class="id" title="keyword">let</span> <span class="id" title="var">e</span> := <a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#canonic_exp"><span class="id" title="definition">canonic_exp</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#Fcalc_round.Fcalc_round_fexp.fexp"><span class="id" title="variable">fexp</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#x"><span class="id" title="variable">x</span></a> <span class="id" title="keyword">in</span><br/>
<a class="idref" href="compcert.flocq.Calc.Fcalc_bracket.html#inbetween_float"><span class="id" title="definition">inbetween_float</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#l"><span class="id" title="variable">l</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.Core.Fcore_generic_fmt.html#round"><span class="id" title="definition">round</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#Fcalc_round.Fcalc_round_fexp.fexp"><span class="id" title="variable">fexp</span></a> <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Ztrunc"><span class="id" title="definition">Ztrunc</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.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.Core.Fcore_defs.html#F2R"><span class="id" title="definition">F2R</span></a> (<a class="idref" href="compcert.flocq.Core.Fcore_defs.html#Float"><span class="id" title="constructor">Float</span></a> <span class="id" title="keyword">beta</span> (<a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#cond_incr"><span class="id" title="definition">cond_incr</span></a> (<a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#round_ZR"><span class="id" title="definition">round_ZR</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zbool.html#Zlt_bool"><span class="id" title="abbreviation">Zlt_bool</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</span></a> 0) <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#l"><span class="id" title="variable">l</span></a>) <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</span></a>) <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#e"><span class="id" title="variable">e</span></a>).<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#inbetween_float_round"><span class="id" title="lemma">inbetween_float_round</span></a> <span class="id" title="keyword">with</span> (<span class="id" title="var">choice</span> := <span class="id" title="keyword">fun</span> <span class="id" title="var">m</span> <span class="id" title="var">l</span> ⇒ <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#cond_incr"><span class="id" title="definition">cond_incr</span></a> (<a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#round_ZR"><span class="id" title="definition">round_ZR</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zbool.html#Zlt_bool"><span class="id" title="abbreviation">Zlt_bool</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</span></a> 0) <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#l"><span class="id" title="variable">l</span></a>) <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</span></a>).<br/>
<span class="id" title="tactic">exact</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#inbetween_int_ZR"><span class="id" title="lemma">inbetween_int_ZR</span></a>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Theorem</span> <a name="inbetween_int_ZR_sign"><span class="id" title="lemma">inbetween_int_ZR_sign</span></a> :<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">m</span> <span class="id" title="var">l</span>,<br/>
<a class="idref" href="compcert.flocq.Calc.Fcalc_bracket.html#inbetween_int"><span class="id" title="definition">inbetween_int</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Reals.Rbasic_fun.html#Rabs"><span class="id" title="definition">Rabs</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#l"><span class="id" title="variable">l</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.Core.Fcore_Raux.html#Ztrunc"><span class="id" title="definition">Ztrunc</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.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.Core.Fcore_Zaux.html#cond_Zopp"><span class="id" title="definition">cond_Zopp</span></a> (<a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Rlt_bool"><span class="id" title="definition">Rlt_bool</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#x"><span class="id" title="variable">x</span></a> 0) <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</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">m</span> <span class="id" title="var">l</span> <span class="id" title="var">Hl</span>.<br/>
<span class="id" title="tactic">simpl</span>.<br/>
<span class="id" title="tactic">unfold</span> <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Ztrunc"><span class="id" title="definition">Ztrunc</span></a>.<br/>
<span class="id" title="tactic">destruct</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Reals.RIneq.html#Rlt_le_dec"><span class="id" title="lemma">Rlt_le_dec</span></a> <span class="id" title="var">x</span> 0) <span class="id" title="keyword">as</span> [<span class="id" title="var">Zx</span>|<span class="id" title="var">Zx</span>].<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Rlt_bool_true"><span class="id" title="lemma">Rlt_bool_true</span></a> <span class="id" title="keyword">with</span> (1 := <span class="id" title="var">Zx</span>).<br/>
<span class="id" title="tactic">simpl</span>.<br/>
<span class="id" title="tactic">unfold</span> <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Zceil"><span class="id" title="definition">Zceil</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">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Zfloor_imp"><span class="id" title="lemma">Zfloor_imp</span></a>.<br/>
<span class="id" title="tactic">rewrite</span> <- <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Reals.Rbasic_fun.html#Rabs_left"><span class="id" title="lemma">Rabs_left</span></a> <span class="id" title="keyword">with</span> (1 := <span class="id" title="var">Zx</span>).<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_bracket.html#inbetween_bounds"><span class="id" title="lemma">inbetween_bounds</span></a> <span class="id" title="keyword">with</span> (2 := <span class="id" title="var">Hl</span>).<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">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zorder.html#Zlt_succ"><span class="id" title="abbreviation">Zlt_succ</span></a>.<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Rlt_bool_false"><span class="id" title="lemma">Rlt_bool_false</span></a> <span class="id" title="keyword">with</span> (1 := <span class="id" title="var">Zx</span>).<br/>
<span class="id" title="tactic">simpl</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Zfloor_imp"><span class="id" title="lemma">Zfloor_imp</span></a>.<br/>
<span class="id" title="tactic">rewrite</span> <- <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Reals.Rbasic_fun.html#Rabs_pos_eq"><span class="id" title="lemma">Rabs_pos_eq</span></a> <span class="id" title="keyword">with</span> (1 := <span class="id" title="var">Zx</span>).<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_bracket.html#inbetween_bounds"><span class="id" title="lemma">inbetween_bounds</span></a> <span class="id" title="keyword">with</span> (2 := <span class="id" title="var">Hl</span>).<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">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zorder.html#Zlt_succ"><span class="id" title="abbreviation">Zlt_succ</span></a>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Theorem</span> <a name="inbetween_float_ZR_sign"><span class="id" title="lemma">inbetween_float_ZR_sign</span></a> :<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">m</span> <span class="id" title="var">l</span>,<br/>
<span class="id" title="keyword">let</span> <span class="id" title="var">e</span> := <a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#canonic_exp"><span class="id" title="definition">canonic_exp</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#Fcalc_round.Fcalc_round_fexp.fexp"><span class="id" title="variable">fexp</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#x"><span class="id" title="variable">x</span></a> <span class="id" title="keyword">in</span><br/>
<a class="idref" href="compcert.flocq.Calc.Fcalc_bracket.html#inbetween_float"><span class="id" title="definition">inbetween_float</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#e"><span class="id" title="variable">e</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Reals.Rbasic_fun.html#Rabs"><span class="id" title="definition">Rabs</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#l"><span class="id" title="variable">l</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.Core.Fcore_generic_fmt.html#round"><span class="id" title="definition">round</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#Fcalc_round.Fcalc_round_fexp.fexp"><span class="id" title="variable">fexp</span></a> <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Ztrunc"><span class="id" title="definition">Ztrunc</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.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.Core.Fcore_defs.html#F2R"><span class="id" title="definition">F2R</span></a> (<a class="idref" href="compcert.flocq.Core.Fcore_defs.html#Float"><span class="id" title="constructor">Float</span></a> <span class="id" title="keyword">beta</span> (<a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#cond_Zopp"><span class="id" title="definition">cond_Zopp</span></a> (<a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Rlt_bool"><span class="id" title="definition">Rlt_bool</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#x"><span class="id" title="variable">x</span></a> 0) <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</span></a>) <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#e"><span class="id" title="variable">e</span></a>).<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#inbetween_float_round_sign"><span class="id" title="lemma">inbetween_float_round_sign</span></a> <span class="id" title="keyword">with</span> (<span class="id" title="var">choice</span> := <span class="id" title="keyword">fun</span> <span class="id" title="var">s</span> <span class="id" title="var">m</span> <span class="id" title="var">l</span> ⇒ <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</span></a>).<br/>
<span class="id" title="tactic">exact</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#inbetween_int_ZR_sign"><span class="id" title="lemma">inbetween_int_ZR_sign</span></a>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
Relates location and rounding to nearest.
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Definition</span> <a name="round_N"><span class="id" title="definition">round_N</span></a> (<span class="id" title="var">p</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">l</span> :=<br/>
<span class="id" title="keyword">match</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#l"><span class="id" title="variable">l</span></a> <span class="id" title="keyword">with</span><br/>
| <a class="idref" href="compcert.flocq.Calc.Fcalc_bracket.html#loc_Exact"><span class="id" title="constructor">loc_Exact</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><br/>
| <a class="idref" href="compcert.flocq.Calc.Fcalc_bracket.html#loc_Inexact"><span class="id" title="constructor">loc_Inexact</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#Lt"><span class="id" title="constructor">Lt</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><br/>
| <a class="idref" href="compcert.flocq.Calc.Fcalc_bracket.html#loc_Inexact"><span class="id" title="constructor">loc_Inexact</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#Eq"><span class="id" title="constructor">Eq</span></a> ⇒ <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#p"><span class="id" title="variable">p</span></a><br/>
| <a class="idref" href="compcert.flocq.Calc.Fcalc_bracket.html#loc_Inexact"><span class="id" title="constructor">loc_Inexact</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#Gt"><span class="id" title="constructor">Gt</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">end</span>.<br/>
<br/>
<span class="id" title="keyword">Theorem</span> <a name="inbetween_int_N"><span class="id" title="lemma">inbetween_int_N</span></a> :<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">choice</span> <span class="id" title="var">x</span> <span class="id" title="var">m</span> <span class="id" title="var">l</span>,<br/>
<a class="idref" href="compcert.flocq.Calc.Fcalc_bracket.html#inbetween_int"><span class="id" title="definition">inbetween_int</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#l"><span class="id" title="variable">l</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.Core.Fcore_generic_fmt.html#Znearest"><span class="id" title="definition">Znearest</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#choice"><span class="id" title="variable">choice</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.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.Calc.Fcalc_round.html#cond_incr"><span class="id" title="definition">cond_incr</span></a> (<a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#round_N"><span class="id" title="definition">round_N</span></a> (<a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#choice"><span class="id" title="variable">choice</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</span></a>) <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#l"><span class="id" title="variable">l</span></a>) <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</span></a>.<br/>
<span class="id" title="keyword">Proof</span> <span class="id" title="keyword">with</span> <span class="id" title="tactic">auto</span> <span class="id" title="keyword">with</span> <span class="id" title="var">typeclass_instances</span>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">choice</span> <span class="id" title="var">x</span> <span class="id" title="var">m</span> <span class="id" title="var">l</span> <span class="id" title="var">Hl</span>.<br/>
<span class="id" title="tactic">inversion_clear</span> <span class="id" title="var">Hl</span> <span class="id" title="keyword">as</span> [<span class="id" title="var">Hx</span>|<span class="id" title="var">l'</span> <span class="id" title="var">Hx</span> <span class="id" title="var">Hl'</span>].<br/>
<span class="id" title="tactic">rewrite</span> <span class="id" title="var">Hx</span>.<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#Zrnd_Z2R"><span class="id" title="method">Zrnd_Z2R</span></a>...<br/>
<span class="id" title="tactic">unfold</span> <a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#Znearest"><span class="id" title="definition">Znearest</span></a>.<br/>
<span class="id" title="tactic">assert</span> (<span class="id" title="var">Hm</span>: <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Zfloor"><span class="id" title="definition">Zfloor</span></a> <span class="id" title="var">x</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> <span class="id" title="var">m</span>).<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Zfloor_imp"><span class="id" title="lemma">Zfloor_imp</span></a>.<br/>
<span class="id" title="tactic">exact</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#conj"><span class="id" title="constructor">conj</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Reals.RIneq.html#Rlt_le"><span class="id" title="lemma">Rlt_le</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</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">Hx</span>)) (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#proj2"><span class="id" title="lemma">proj2</span></a> <span class="id" title="var">Hx</span>)).<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Zceil_floor_neq"><span class="id" title="lemma">Zceil_floor_neq</span></a>.<br/>
<span class="id" title="tactic">rewrite</span> <span class="id" title="var">Hm</span>.<br/>
<span class="id" title="tactic">replace</span> (<a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Rcompare"><span class="id" title="definition">Rcompare</span></a> (<span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Reals.Rdefinitions.html#0f8d5ec6876eafaec2dca3581c24e3a0"><span class="id" title="notation">-</span></a> <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Z2R"><span class="id" title="definition">Z2R</span></a> <span class="id" title="var">m</span>) (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Reals.Rdefinitions.html#2f596a539cf733f1c05f7537bd65e225"><span class="id" title="notation">/</span></a>2)) <span class="id" title="keyword">with</span> <span class="id" title="var">l'</span>.<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">case</span> <span class="id" title="var">l'</span>.<br/>
<span class="id" title="tactic">rewrite</span> <- <span class="id" title="var">Hl'</span>.<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Z2R_plus"><span class="id" title="lemma">Z2R_plus</span></a>.<br/>
<span class="id" title="tactic">rewrite</span> <- (<a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Rcompare_plus_r"><span class="id" title="lemma">Rcompare_plus_r</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Reals.Rdefinitions.html#0575b33c544988081e4e30ee15f09d84"><span class="id" title="notation">-</span></a> <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Z2R"><span class="id" title="definition">Z2R</span></a> <span class="id" title="var">m</span>) <span class="id" title="var">x</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">simpl</span> (<a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Z2R"><span class="id" title="definition">Z2R</span></a> 1).<br/>
<span class="id" title="tactic">field</span>.<br/>
<span class="id" title="tactic">rewrite</span> <span class="id" title="var">Hm</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.Reals.RIneq.html#Rlt_not_eq"><span class="id" title="lemma">Rlt_not_eq</span></a>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Theorem</span> <a name="inbetween_int_N_sign"><span class="id" title="lemma">inbetween_int_N_sign</span></a> :<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">choice</span> <span class="id" title="var">x</span> <span class="id" title="var">m</span> <span class="id" title="var">l</span>,<br/>
<a class="idref" href="compcert.flocq.Calc.Fcalc_bracket.html#inbetween_int"><span class="id" title="definition">inbetween_int</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Reals.Rbasic_fun.html#Rabs"><span class="id" title="definition">Rabs</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#l"><span class="id" title="variable">l</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.Core.Fcore_generic_fmt.html#Znearest"><span class="id" title="definition">Znearest</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#choice"><span class="id" title="variable">choice</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.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.Core.Fcore_Zaux.html#cond_Zopp"><span class="id" title="definition">cond_Zopp</span></a> (<a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Rlt_bool"><span class="id" title="definition">Rlt_bool</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#x"><span class="id" title="variable">x</span></a> 0) (<a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#cond_incr"><span class="id" title="definition">cond_incr</span></a> (<a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#round_N"><span class="id" title="definition">round_N</span></a> (<span class="id" title="keyword">if</span> <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Rlt_bool"><span class="id" title="definition">Rlt_bool</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#x"><span class="id" title="variable">x</span></a> 0 <span class="id" title="keyword">then</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#negb"><span class="id" title="definition">negb</span></a> (<a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#choice"><span class="id" title="variable">choice</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><a class="idref" href="compcert.flocq.Calc.Fcalc_round.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#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#0a3978eabdacce0a128118074de19651"><span class="id" title="notation">)</span></a>)%<span class="id" title="var">Z</span>) <span class="id" title="keyword">else</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#choice"><span class="id" title="variable">choice</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</span></a>) <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#l"><span class="id" title="variable">l</span></a>) <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</span></a>).<br/>
<span class="id" title="keyword">Proof</span> <span class="id" title="keyword">with</span> <span class="id" title="tactic">auto</span> <span class="id" title="keyword">with</span> <span class="id" title="var">typeclass_instances</span>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">choice</span> <span class="id" title="var">x</span> <span class="id" title="var">m</span> <span class="id" title="var">l</span> <span class="id" title="var">Hl</span>.<br/>
<span class="id" title="tactic">simpl</span>.<br/>
<span class="id" title="tactic">unfold</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Reals.Rbasic_fun.html#Rabs"><span class="id" title="definition">Rabs</span></a> <span class="id" title="keyword">in</span> <span class="id" title="var">Hl</span>.<br/>
<span class="id" title="tactic">destruct</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Reals.Rbasic_fun.html#Rcase_abs"><span class="id" title="lemma">Rcase_abs</span></a> <span class="id" title="var">x</span>) <span class="id" title="keyword">as</span> [<span class="id" title="var">Zx</span>|<span class="id" title="var">Zx</span>].<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Rlt_bool_true"><span class="id" title="lemma">Rlt_bool_true</span></a> <span class="id" title="keyword">with</span> (1 := <span class="id" title="var">Zx</span>).<br/>
<span class="id" title="tactic">simpl</span>.<br/>
<span class="id" title="tactic">rewrite</span> <- (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Reals.RIneq.html#Ropp_involutive"><span class="id" title="lemma">Ropp_involutive</span></a> <span class="id" title="var">x</span>).<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#Znearest_opp"><span class="id" title="lemma">Znearest_opp</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">inversion_clear</span> <span class="id" title="var">Hl</span> <span class="id" title="keyword">as</span> [<span class="id" title="var">Hx</span>|<span class="id" title="var">l'</span> <span class="id" title="var">Hx</span> <span class="id" title="var">Hl'</span>].<br/>
<span class="id" title="tactic">rewrite</span> <span class="id" title="var">Hx</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#Zrnd_Z2R"><span class="id" title="method">Zrnd_Z2R</span></a>...<br/>
<span class="id" title="tactic">assert</span> (<span class="id" title="var">Hm</span>: <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Zfloor"><span class="id" title="definition">Zfloor</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Reals.Rdefinitions.html#0575b33c544988081e4e30ee15f09d84"><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.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <span class="id" title="var">m</span>).<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Zfloor_imp"><span class="id" title="lemma">Zfloor_imp</span></a>.<br/>
<span class="id" title="tactic">exact</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#conj"><span class="id" title="constructor">conj</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Reals.RIneq.html#Rlt_le"><span class="id" title="lemma">Rlt_le</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</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">Hx</span>)) (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#proj2"><span class="id" title="lemma">proj2</span></a> <span class="id" title="var">Hx</span>)).<br/>
<span class="id" title="tactic">unfold</span> <a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#Znearest"><span class="id" title="definition">Znearest</span></a>.<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Zceil_floor_neq"><span class="id" title="lemma">Zceil_floor_neq</span></a>.<br/>
<span class="id" title="tactic">rewrite</span> <span class="id" title="var">Hm</span>.<br/>
<span class="id" title="tactic">replace</span> (<a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Rcompare"><span class="id" title="definition">Rcompare</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Reals.Rdefinitions.html#0575b33c544988081e4e30ee15f09d84"><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.Reals.Rdefinitions.html#0f8d5ec6876eafaec2dca3581c24e3a0"><span class="id" title="notation">-</span></a> <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Z2R"><span class="id" title="definition">Z2R</span></a> <span class="id" title="var">m</span>) (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Reals.Rdefinitions.html#2f596a539cf733f1c05f7537bd65e225"><span class="id" title="notation">/</span></a>2)) <span class="id" title="keyword">with</span> <span class="id" title="var">l'</span>.<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">case</span> <span class="id" title="var">l'</span>.<br/>
<span class="id" title="tactic">rewrite</span> <- <span class="id" title="var">Hl'</span>.<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Z2R_plus"><span class="id" title="lemma">Z2R_plus</span></a>.<br/>
<span class="id" title="tactic">rewrite</span> <- (<a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Rcompare_plus_r"><span class="id" title="lemma">Rcompare_plus_r</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Reals.Rdefinitions.html#0575b33c544988081e4e30ee15f09d84"><span class="id" title="notation">-</span></a> <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Z2R"><span class="id" title="definition">Z2R</span></a> <span class="id" title="var">m</span>) (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Reals.Rdefinitions.html#0575b33c544988081e4e30ee15f09d84"><span class="id" title="notation">-</span></a><span class="id" title="var">x</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">simpl</span> (<a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Z2R"><span class="id" title="definition">Z2R</span></a> 1).<br/>
<span class="id" title="tactic">field</span>.<br/>
<span class="id" title="tactic">rewrite</span> <span class="id" title="var">Hm</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.Reals.RIneq.html#Rlt_not_eq"><span class="id" title="lemma">Rlt_not_eq</span></a>.<br/>
<span class="id" title="tactic">generalize</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Reals.RIneq.html#Rge_le"><span class="id" title="lemma">Rge_le</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">Zx</span>).<br/>
<span class="id" title="tactic">clear</span> <span class="id" title="var">Zx</span>. <span class="id" title="tactic">intros</span> <span class="id" title="var">Zx</span>.<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Rlt_bool_false"><span class="id" title="lemma">Rlt_bool_false</span></a> <span class="id" title="keyword">with</span> (1 := <span class="id" title="var">Zx</span>).<br/>
<span class="id" title="tactic">simpl</span>.<br/>
<span class="id" title="tactic">inversion_clear</span> <span class="id" title="var">Hl</span> <span class="id" title="keyword">as</span> [<span class="id" title="var">Hx</span>|<span class="id" title="var">l'</span> <span class="id" title="var">Hx</span> <span class="id" title="var">Hl'</span>].<br/>
<span class="id" title="tactic">rewrite</span> <span class="id" title="var">Hx</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#Zrnd_Z2R"><span class="id" title="method">Zrnd_Z2R</span></a>...<br/>
<span class="id" title="tactic">assert</span> (<span class="id" title="var">Hm</span>: <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Zfloor"><span class="id" title="definition">Zfloor</span></a> <span class="id" title="var">x</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> <span class="id" title="var">m</span>).<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Zfloor_imp"><span class="id" title="lemma">Zfloor_imp</span></a>.<br/>
<span class="id" title="tactic">exact</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#conj"><span class="id" title="constructor">conj</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Reals.RIneq.html#Rlt_le"><span class="id" title="lemma">Rlt_le</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</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">Hx</span>)) (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#proj2"><span class="id" title="lemma">proj2</span></a> <span class="id" title="var">Hx</span>)).<br/>
<span class="id" title="tactic">unfold</span> <a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#Znearest"><span class="id" title="definition">Znearest</span></a>.<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Zceil_floor_neq"><span class="id" title="lemma">Zceil_floor_neq</span></a>.<br/>
<span class="id" title="tactic">rewrite</span> <span class="id" title="var">Hm</span>.<br/>
<span class="id" title="tactic">replace</span> (<a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Rcompare"><span class="id" title="definition">Rcompare</span></a> (<span class="id" title="var">x</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Reals.Rdefinitions.html#0f8d5ec6876eafaec2dca3581c24e3a0"><span class="id" title="notation">-</span></a> <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Z2R"><span class="id" title="definition">Z2R</span></a> <span class="id" title="var">m</span>) (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Reals.Rdefinitions.html#2f596a539cf733f1c05f7537bd65e225"><span class="id" title="notation">/</span></a>2)) <span class="id" title="keyword">with</span> <span class="id" title="var">l'</span>.<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">case</span> <span class="id" title="var">l'</span>.<br/>
<span class="id" title="tactic">rewrite</span> <- <span class="id" title="var">Hl'</span>.<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Z2R_plus"><span class="id" title="lemma">Z2R_plus</span></a>.<br/>
<span class="id" title="tactic">rewrite</span> <- (<a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Rcompare_plus_r"><span class="id" title="lemma">Rcompare_plus_r</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Reals.Rdefinitions.html#0575b33c544988081e4e30ee15f09d84"><span class="id" title="notation">-</span></a> <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Z2R"><span class="id" title="definition">Z2R</span></a> <span class="id" title="var">m</span>) <span class="id" title="var">x</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">simpl</span> (<a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Z2R"><span class="id" title="definition">Z2R</span></a> 1).<br/>
<span class="id" title="tactic">field</span>.<br/>
<span class="id" title="tactic">rewrite</span> <span class="id" title="var">Hm</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.Reals.RIneq.html#Rlt_not_eq"><span class="id" title="lemma">Rlt_not_eq</span></a>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
Relates location and rounding to nearest even.
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Theorem</span> <a name="inbetween_int_NE"><span class="id" title="lemma">inbetween_int_NE</span></a> :<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">m</span> <span class="id" title="var">l</span>,<br/>
<a class="idref" href="compcert.flocq.Calc.Fcalc_bracket.html#inbetween_int"><span class="id" title="definition">inbetween_int</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#l"><span class="id" title="variable">l</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.Core.Fcore_rnd_ne.html#ZnearestE"><span class="id" title="abbreviation">ZnearestE</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.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.Calc.Fcalc_round.html#cond_incr"><span class="id" title="definition">cond_incr</span></a> (<a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#round_N"><span class="id" title="definition">round_N</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#negb"><span class="id" title="definition">negb</span></a> (<a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#Zeven"><span class="id" title="definition">Zeven</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</span></a>)) <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#l"><span class="id" title="variable">l</span></a>) <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</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">m</span> <span class="id" title="var">l</span> <span class="id" title="var">Hl</span>.<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#inbetween_int_N"><span class="id" title="lemma">inbetween_int_N</span></a> <span class="id" title="keyword">with</span> (<span class="id" title="var">choice</span> := <span class="id" title="keyword">fun</span> <span class="id" title="var">x</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#negb"><span class="id" title="definition">negb</span></a> (<a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#Zeven"><span class="id" title="definition">Zeven</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#x"><span class="id" title="variable">x</span></a>)).<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Theorem</span> <a name="inbetween_float_NE"><span class="id" title="lemma">inbetween_float_NE</span></a> :<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">m</span> <span class="id" title="var">l</span>,<br/>
<span class="id" title="keyword">let</span> <span class="id" title="var">e</span> := <a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#canonic_exp"><span class="id" title="definition">canonic_exp</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#Fcalc_round.Fcalc_round_fexp.fexp"><span class="id" title="variable">fexp</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#x"><span class="id" title="variable">x</span></a> <span class="id" title="keyword">in</span><br/>
<a class="idref" href="compcert.flocq.Calc.Fcalc_bracket.html#inbetween_float"><span class="id" title="definition">inbetween_float</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#l"><span class="id" title="variable">l</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.Core.Fcore_generic_fmt.html#round"><span class="id" title="definition">round</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#Fcalc_round.Fcalc_round_fexp.fexp"><span class="id" title="variable">fexp</span></a> <a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#ZnearestE"><span class="id" title="abbreviation">ZnearestE</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.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.Core.Fcore_defs.html#F2R"><span class="id" title="definition">F2R</span></a> (<a class="idref" href="compcert.flocq.Core.Fcore_defs.html#Float"><span class="id" title="constructor">Float</span></a> <span class="id" title="keyword">beta</span> (<a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#cond_incr"><span class="id" title="definition">cond_incr</span></a> (<a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#round_N"><span class="id" title="definition">round_N</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#negb"><span class="id" title="definition">negb</span></a> (<a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#Zeven"><span class="id" title="definition">Zeven</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</span></a>)) <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#l"><span class="id" title="variable">l</span></a>) <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</span></a>) <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#e"><span class="id" title="variable">e</span></a>).<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#inbetween_float_round"><span class="id" title="lemma">inbetween_float_round</span></a> <span class="id" title="keyword">with</span> (<span class="id" title="var">choice</span> := <span class="id" title="keyword">fun</span> <span class="id" title="var">m</span> <span class="id" title="var">l</span> ⇒ <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#cond_incr"><span class="id" title="definition">cond_incr</span></a> (<a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#round_N"><span class="id" title="definition">round_N</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#negb"><span class="id" title="definition">negb</span></a> (<a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#Zeven"><span class="id" title="definition">Zeven</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</span></a>)) <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#l"><span class="id" title="variable">l</span></a>) <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</span></a>).<br/>
<span class="id" title="tactic">exact</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#inbetween_int_NE"><span class="id" title="lemma">inbetween_int_NE</span></a>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Theorem</span> <a name="inbetween_int_NE_sign"><span class="id" title="lemma">inbetween_int_NE_sign</span></a> :<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">m</span> <span class="id" title="var">l</span>,<br/>
<a class="idref" href="compcert.flocq.Calc.Fcalc_bracket.html#inbetween_int"><span class="id" title="definition">inbetween_int</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Reals.Rbasic_fun.html#Rabs"><span class="id" title="definition">Rabs</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#l"><span class="id" title="variable">l</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.Core.Fcore_rnd_ne.html#ZnearestE"><span class="id" title="abbreviation">ZnearestE</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.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.Core.Fcore_Zaux.html#cond_Zopp"><span class="id" title="definition">cond_Zopp</span></a> (<a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Rlt_bool"><span class="id" title="definition">Rlt_bool</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#x"><span class="id" title="variable">x</span></a> 0) (<a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#cond_incr"><span class="id" title="definition">cond_incr</span></a> (<a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#round_N"><span class="id" title="definition">round_N</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#negb"><span class="id" title="definition">negb</span></a> (<a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#Zeven"><span class="id" title="definition">Zeven</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</span></a>)) <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#l"><span class="id" title="variable">l</span></a>) <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</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">m</span> <span class="id" title="var">l</span> <span class="id" title="var">Hl</span>.<br/>
<span class="id" title="var">erewrite</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#inbetween_int_N_sign"><span class="id" title="lemma">inbetween_int_N_sign</span></a> <span class="id" title="keyword">with</span> (<span class="id" title="var">choice</span> := <span class="id" title="keyword">fun</span> <span class="id" title="var">x</span> ⇒ <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#negb"><span class="id" title="definition">negb</span></a> (<a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#Zeven"><span class="id" title="definition">Zeven</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#x"><span class="id" title="variable">x</span></a>)).<br/>
2: <span class="id" title="tactic">eexact</span> <span class="id" title="var">Hl</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">case</span> <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Rlt_bool"><span class="id" title="definition">Rlt_bool</span></a>.<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#Zeven_opp"><span class="id" title="lemma">Zeven_opp</span></a>, <a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#Zeven_plus"><span class="id" title="lemma">Zeven_plus</span></a>.<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">case</span> (<a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#Zeven"><span class="id" title="definition">Zeven</span></a> <span class="id" title="var">m</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">Theorem</span> <a name="inbetween_float_NE_sign"><span class="id" title="lemma">inbetween_float_NE_sign</span></a> :<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">m</span> <span class="id" title="var">l</span>,<br/>
<span class="id" title="keyword">let</span> <span class="id" title="var">e</span> := <a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#canonic_exp"><span class="id" title="definition">canonic_exp</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#Fcalc_round.Fcalc_round_fexp.fexp"><span class="id" title="variable">fexp</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#x"><span class="id" title="variable">x</span></a> <span class="id" title="keyword">in</span><br/>
<a class="idref" href="compcert.flocq.Calc.Fcalc_bracket.html#inbetween_float"><span class="id" title="definition">inbetween_float</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#e"><span class="id" title="variable">e</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Reals.Rbasic_fun.html#Rabs"><span class="id" title="definition">Rabs</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#l"><span class="id" title="variable">l</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.Core.Fcore_generic_fmt.html#round"><span class="id" title="definition">round</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#Fcalc_round.Fcalc_round_fexp.fexp"><span class="id" title="variable">fexp</span></a> <a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#ZnearestE"><span class="id" title="abbreviation">ZnearestE</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.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.Core.Fcore_defs.html#F2R"><span class="id" title="definition">F2R</span></a> (<a class="idref" href="compcert.flocq.Core.Fcore_defs.html#Float"><span class="id" title="constructor">Float</span></a> <span class="id" title="keyword">beta</span> (<a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#cond_Zopp"><span class="id" title="definition">cond_Zopp</span></a> (<a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Rlt_bool"><span class="id" title="definition">Rlt_bool</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#x"><span class="id" title="variable">x</span></a> 0) (<a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#cond_incr"><span class="id" title="definition">cond_incr</span></a> (<a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#round_N"><span class="id" title="definition">round_N</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#negb"><span class="id" title="definition">negb</span></a> (<a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#Zeven"><span class="id" title="definition">Zeven</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</span></a>)) <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#l"><span class="id" title="variable">l</span></a>) <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</span></a>)) <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#e"><span class="id" title="variable">e</span></a>).<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#inbetween_float_round_sign"><span class="id" title="lemma">inbetween_float_round_sign</span></a> <span class="id" title="keyword">with</span> (<span class="id" title="var">choice</span> := <span class="id" title="keyword">fun</span> <span class="id" title="var">s</span> <span class="id" title="var">m</span> <span class="id" title="var">l</span> ⇒ <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#cond_incr"><span class="id" title="definition">cond_incr</span></a> (<a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#round_N"><span class="id" title="definition">round_N</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#negb"><span class="id" title="definition">negb</span></a> (<a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#Zeven"><span class="id" title="definition">Zeven</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</span></a>)) <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#l"><span class="id" title="variable">l</span></a>) <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</span></a>).<br/>
<span class="id" title="tactic">exact</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#inbetween_int_NE_sign"><span class="id" title="lemma">inbetween_int_NE_sign</span></a>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
Relates location and rounding to nearest away.
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Theorem</span> <a name="inbetween_int_NA"><span class="id" title="lemma">inbetween_int_NA</span></a> :<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">m</span> <span class="id" title="var">l</span>,<br/>
<a class="idref" href="compcert.flocq.Calc.Fcalc_bracket.html#inbetween_int"><span class="id" title="definition">inbetween_int</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#l"><span class="id" title="variable">l</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.Core.Fcore_generic_fmt.html#ZnearestA"><span class="id" title="abbreviation">ZnearestA</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.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.Calc.Fcalc_round.html#cond_incr"><span class="id" title="definition">cond_incr</span></a> (<a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#round_N"><span class="id" title="definition">round_N</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> 0 <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</span></a>) <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#l"><span class="id" title="variable">l</span></a>) <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</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">m</span> <span class="id" title="var">l</span> <span class="id" title="var">Hl</span>.<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#inbetween_int_N"><span class="id" title="lemma">inbetween_int_N</span></a> <span class="id" title="keyword">with</span> (<span class="id" title="var">choice</span> := <span class="id" title="keyword">fun</span> <span class="id" title="var">x</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.Calc.Fcalc_round.html#x"><span class="id" title="variable">x</span></a>).<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Theorem</span> <a name="inbetween_float_NA"><span class="id" title="lemma">inbetween_float_NA</span></a> :<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">m</span> <span class="id" title="var">l</span>,<br/>
<span class="id" title="keyword">let</span> <span class="id" title="var">e</span> := <a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#canonic_exp"><span class="id" title="definition">canonic_exp</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#Fcalc_round.Fcalc_round_fexp.fexp"><span class="id" title="variable">fexp</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#x"><span class="id" title="variable">x</span></a> <span class="id" title="keyword">in</span><br/>
<a class="idref" href="compcert.flocq.Calc.Fcalc_bracket.html#inbetween_float"><span class="id" title="definition">inbetween_float</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#l"><span class="id" title="variable">l</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.Core.Fcore_generic_fmt.html#round"><span class="id" title="definition">round</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#Fcalc_round.Fcalc_round_fexp.fexp"><span class="id" title="variable">fexp</span></a> <a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#ZnearestA"><span class="id" title="abbreviation">ZnearestA</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.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.Core.Fcore_defs.html#F2R"><span class="id" title="definition">F2R</span></a> (<a class="idref" href="compcert.flocq.Core.Fcore_defs.html#Float"><span class="id" title="constructor">Float</span></a> <span class="id" title="keyword">beta</span> (<a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#cond_incr"><span class="id" title="definition">cond_incr</span></a> (<a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#round_N"><span class="id" title="definition">round_N</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> 0 <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</span></a>) <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#l"><span class="id" title="variable">l</span></a>) <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</span></a>) <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#e"><span class="id" title="variable">e</span></a>).<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#inbetween_float_round"><span class="id" title="lemma">inbetween_float_round</span></a> <span class="id" title="keyword">with</span> (<span class="id" title="var">choice</span> := <span class="id" title="keyword">fun</span> <span class="id" title="var">m</span> <span class="id" title="var">l</span> ⇒ <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#cond_incr"><span class="id" title="definition">cond_incr</span></a> (<a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#round_N"><span class="id" title="definition">round_N</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> 0 <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</span></a>) <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#l"><span class="id" title="variable">l</span></a>) <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</span></a>).<br/>
<span class="id" title="tactic">exact</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#inbetween_int_NA"><span class="id" title="lemma">inbetween_int_NA</span></a>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Theorem</span> <a name="inbetween_int_NA_sign"><span class="id" title="lemma">inbetween_int_NA_sign</span></a> :<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">m</span> <span class="id" title="var">l</span>,<br/>
<a class="idref" href="compcert.flocq.Calc.Fcalc_bracket.html#inbetween_int"><span class="id" title="definition">inbetween_int</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Reals.Rbasic_fun.html#Rabs"><span class="id" title="definition">Rabs</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#l"><span class="id" title="variable">l</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.Core.Fcore_generic_fmt.html#ZnearestA"><span class="id" title="abbreviation">ZnearestA</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.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.Core.Fcore_Zaux.html#cond_Zopp"><span class="id" title="definition">cond_Zopp</span></a> (<a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Rlt_bool"><span class="id" title="definition">Rlt_bool</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#x"><span class="id" title="variable">x</span></a> 0) (<a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#cond_incr"><span class="id" title="definition">cond_incr</span></a> (<a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#round_N"><span class="id" title="definition">round_N</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="compcert.flocq.Calc.Fcalc_round.html#l"><span class="id" title="variable">l</span></a>) <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</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">m</span> <span class="id" title="var">l</span> <span class="id" title="var">Hl</span>.<br/>
<span class="id" title="var">erewrite</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#inbetween_int_N_sign"><span class="id" title="lemma">inbetween_int_N_sign</span></a> <span class="id" title="keyword">with</span> (<span class="id" title="var">choice</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).<br/>
2: <span class="id" title="tactic">eexact</span> <span class="id" title="var">Hl</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">assert</span> (<span class="id" title="var">Hm</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">m</span>)%<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#Zlt_succ_le"><span class="id" title="lemma">Zlt_succ_le</span></a>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#lt_Z2R"><span class="id" title="lemma">lt_Z2R</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#Rle_lt_trans"><span class="id" title="lemma">Rle_lt_trans</span></a> <span class="id" title="keyword">with</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Reals.Rbasic_fun.html#Rabs"><span class="id" title="definition">Rabs</span></a> <span class="id" title="var">x</span>).<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Reals.Rbasic_fun.html#Rabs_pos"><span class="id" title="lemma">Rabs_pos</span></a>.<br/>
<span class="id" title="tactic">refine</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#proj2"><span class="id" title="lemma">proj2</span></a> (<a class="idref" href="compcert.flocq.Calc.Fcalc_bracket.html#inbetween_bounds"><span class="id" title="lemma">inbetween_bounds</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">_</span> <span class="id" title="var">_</span> <span class="id" title="var">Hl</span>)).<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">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zorder.html#Zlt_succ"><span class="id" title="abbreviation">Zlt_succ</span></a>.<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> <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="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">case</span> <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Rlt_bool"><span class="id" title="definition">Rlt_bool</span></a>.<br/>
<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="truncate_aux"><span class="id" title="definition">truncate_aux</span></a> <span class="id" title="var">t</span> <span class="id" title="var">k</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">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> <span class="id" title="var">l</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.Calc.Fcalc_round.html#t"><span class="id" title="variable">t</span></a> <span class="id" title="keyword">in</span><br/>
<span class="id" title="keyword">let</span> <span class="id" title="var">p</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> <span class="id" title="keyword">beta</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#k"><span class="id" title="variable">k</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.Zdiv.html#Zdiv"><span class="id" title="abbreviation">Zdiv</span></a> <span class="id" title="var">m</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#p"><span class="id" title="variable">p</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">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="compcert.flocq.Calc.Fcalc_round.html#k"><span class="id" title="variable">k</span></a>)%<span class="id" title="var">Z</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.Calc.Fcalc_bracket.html#new_location"><span class="id" title="definition">new_location</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#p"><span class="id" title="variable">p</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> <span class="id" title="var">m</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#p"><span class="id" title="variable">p</span></a>) <span class="id" title="var">l</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/>
<br/>
<span class="id" title="keyword">Theorem</span> <a name="truncate_aux_comp"><span class="id" title="lemma">truncate_aux_comp</span></a> :<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">t</span> <span class="id" title="var">k1</span> <span class="id" title="var">k2</span>,<br/>
(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.Calc.Fcalc_round.html#k1"><span class="id" title="variable">k1</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#273beba4f3453dbb29192b3ac740bb71"><span class="id" title="notation"><</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#k2"><span class="id" title="variable">k2</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.Calc.Fcalc_round.html#truncate_aux"><span class="id" title="definition">truncate_aux</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#t"><span class="id" title="variable">t</span></a> (<a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#k1"><span class="id" title="variable">k1</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.Calc.Fcalc_round.html#k2"><span class="id" title="variable">k2</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.Calc.Fcalc_round.html#truncate_aux"><span class="id" title="definition">truncate_aux</span></a> (<a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#truncate_aux"><span class="id" title="definition">truncate_aux</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#t"><span class="id" title="variable">t</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#k1"><span class="id" title="variable">k1</span></a>) <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#k2"><span class="id" title="variable">k2</span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">intros</span> ((<span class="id" title="var">m</span>,<span class="id" title="var">e</span>),<span class="id" title="var">l</span>) <span class="id" title="var">k1</span> <span class="id" title="var">k2</span> <span class="id" title="var">Hk1</span> <span class="id" title="var">Hk2</span>.<br/>
<span class="id" title="tactic">unfold</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#truncate_aux"><span class="id" title="definition">truncate_aux</span></a>.<br/>
<span class="id" title="tactic">destruct</span> (<a class="idref" href="compcert.flocq.Calc.Fcalc_bracket.html#inbetween_float_ex"><span class="id" title="lemma">inbetween_float_ex</span></a> <span class="id" title="keyword">beta</span> <span class="id" title="var">m</span> <span class="id" title="var">e</span> <span class="id" title="var">l</span>) <span class="id" title="keyword">as</span> (<span class="id" title="var">x</span>,<span class="id" title="var">Hx</span>).<br/>
<span class="id" title="tactic">assert</span> (<span class="id" title="var">B1</span> := <a class="idref" href="compcert.flocq.Calc.Fcalc_bracket.html#inbetween_float_new_location"><span class="id" title="lemma">inbetween_float_new_location</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">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">Hk1</span> <span class="id" title="var">Hx</span>).<br/>
<span class="id" title="tactic">assert</span> (<span class="id" title="var">Hk3</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> <span class="id" title="var">k1</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">k2</span>)%<span class="id" title="var">Z</span>).<br/>
<span class="id" title="tactic">change</span> <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> <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> 0)%<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#Zplus_lt_compat"><span class="id" title="abbreviation">Zplus_lt_compat</span></a>.<br/>
<span class="id" title="tactic">assert</span> (<span class="id" title="var">B3</span> := <a class="idref" href="compcert.flocq.Calc.Fcalc_bracket.html#inbetween_float_new_location"><span class="id" title="lemma">inbetween_float_new_location</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">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">Hk3</span> <span class="id" title="var">Hx</span>).<br/>
<span class="id" title="tactic">assert</span> (<span class="id" title="var">B2</span> := <a class="idref" href="compcert.flocq.Calc.Fcalc_bracket.html#inbetween_float_new_location"><span class="id" title="lemma">inbetween_float_new_location</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">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">Hk2</span> <span class="id" title="var">B1</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_assoc"><span class="id" title="abbreviation">Zplus_assoc</span></a> <span class="id" title="keyword">in</span> <span class="id" title="var">B3</span>.<br/>
<span class="id" title="tactic">destruct</span> (<a class="idref" href="compcert.flocq.Calc.Fcalc_bracket.html#inbetween_float_unique"><span class="id" title="lemma">inbetween_float_unique</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">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">B2</span> <span class="id" title="var">B3</span>).<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">rewrite</span> <span class="id" title="var">H</span>, <span class="id" title="var">H0</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>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
Given a triple (mantissa, exponent, position), this function
computes a triple with a canonic exponent, assuming the
original triple had enough precision.
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Definition</span> <a name="truncate"><span class="id" title="definition">truncate</span></a> <span class="id" title="var">t</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">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> <span class="id" title="var">l</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.Calc.Fcalc_round.html#t"><span class="id" title="variable">t</span></a> <span class="id" title="keyword">in</span><br/>
<span class="id" title="keyword">let</span> <span class="id" title="var">k</span> := (<a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#Fcalc_round.Fcalc_round_fexp.fexp"><span class="id" title="variable">fexp</span></a> (<a class="idref" href="compcert.flocq.Core.Fcore_digits.html#Zdigits"><span class="id" title="definition">Zdigits</span></a> <span class="id" title="keyword">beta</span> <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">e</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> <span class="id" title="var">e</span>)%<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#Zlt_bool"><span class="id" title="abbreviation">Zlt_bool</span></a> 0 <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#k"><span class="id" title="variable">k</span></a> <span class="id" title="keyword">then</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#truncate_aux"><span class="id" title="definition">truncate_aux</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#t"><span class="id" title="variable">t</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#k"><span class="id" title="variable">k</span></a><br/>
<span class="id" title="keyword">else</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#t"><span class="id" title="variable">t</span></a>.<br/>
<br/>
<span class="id" title="keyword">Theorem</span> <a name="truncate_0"><span class="id" title="lemma">truncate_0</span></a> :<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">e</span> <span class="id" title="var">l</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">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> <span class="id" title="var">l'</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.Calc.Fcalc_round.html#truncate"><span class="id" title="definition">truncate</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>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="compcert.flocq.Calc.Fcalc_round.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> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#l"><span class="id" title="variable">l</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> <span class="id" title="keyword">in</span><br/>
<span class="id" title="var">m'</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="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#Z0"><span class="id" title="constructor">Z0</span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">e</span> <span class="id" title="var">l</span>.<br/>
<span class="id" title="tactic">unfold</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#truncate"><span class="id" title="definition">truncate</span></a>.<br/>
<span class="id" title="tactic">case</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zbool.html#Zlt_bool"><span class="id" title="abbreviation">Zlt_bool</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.ZArith.Zdiv.html#Zdiv_0_l"><span class="id" title="lemma">Zdiv_0_l</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="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Theorem</span> <a name="generic_format_truncate"><span class="id" title="lemma">generic_format_truncate</span></a> :<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">m</span> <span class="id" title="var">e</span> <span class="id" title="var">l</span>,<br/>
(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> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</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/>
<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">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> <span class="id" title="var">l'</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.Calc.Fcalc_round.html#truncate"><span class="id" title="definition">truncate</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.Calc.Fcalc_round.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.Calc.Fcalc_round.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> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#l"><span class="id" title="variable">l</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="keyword">in</span><br/>
<a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#format"><span class="id" title="abbreviation">format</span></a> (<a class="idref" href="compcert.flocq.Core.Fcore_defs.html#F2R"><span class="id" title="definition">F2R</span></a> (<a class="idref" href="compcert.flocq.Core.Fcore_defs.html#Float"><span class="id" title="constructor">Float</span></a> <span class="id" title="keyword">beta</span> <span class="id" title="var">m'</span> <span class="id" title="var">e'</span>)).<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">m</span> <span class="id" title="var">e</span> <span class="id" title="var">l</span> <span class="id" title="var">Hm</span>.<br/>
<span class="id" title="tactic">unfold</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#truncate"><span class="id" title="definition">truncate</span></a>.<br/>
<span class="id" title="tactic">set</span> (<span class="id" title="var">k</span> := (<a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#Fcalc_round.Fcalc_round_fexp.fexp"><span class="id" title="variable">fexp</span></a> (<a class="idref" href="compcert.flocq.Core.Fcore_digits.html#Zdigits"><span class="id" title="definition">Zdigits</span></a> <span class="id" title="keyword">beta</span> <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">e</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> <span class="id" title="var">e</span>)%<span class="id" title="var">Z</span>).<br/>
<span class="id" title="tactic">case</span> <a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#Zlt_bool_spec"><span class="id" title="lemma">Zlt_bool_spec</span></a> ; <span class="id" title="tactic">intros</span> <span class="id" title="var">Hk</span>.<br/>
<span class="id" title="tactic">unfold</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#truncate_aux"><span class="id" title="definition">truncate_aux</span></a>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#generic_format_F2R"><span class="id" title="lemma">generic_format_F2R</span></a>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">Hd</span>.<br/>
<span class="id" title="tactic">unfold</span> <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> <span class="id" title="keyword">with</span> (1 := <span class="id" title="var">Hd</span>).<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.flocq.Core.Fcore_digits.html#Zdigits_div_Zpower"><span class="id" title="lemma">Zdigits_div_Zpower</span></a> <span class="id" title="keyword">with</span> (1 := <span class="id" title="var">Hm</span>).<br/>
<span class="id" title="tactic">replace</span> (<a class="idref" href="compcert.flocq.Core.Fcore_digits.html#Zdigits"><span class="id" title="definition">Zdigits</span></a> <span class="id" title="keyword">beta</span> <span class="id" title="var">m</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> <span class="id" title="var">k</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><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> <span class="id" title="var">k</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> (<a class="idref" href="compcert.flocq.Core.Fcore_digits.html#Zdigits"><span class="id" title="definition">Zdigits</span></a> <span class="id" title="keyword">beta</span> <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">e</span>)%<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">unfold</span> <span class="id" title="var">k</span>.<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#Zle_refl"><span class="id" title="abbreviation">Zle_refl</span></a>.<br/>
<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="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zorder.html#Znot_gt_le"><span class="id" title="lemma">Znot_gt_le</span></a>.<br/>
<span class="id" title="var">contradict</span> <span class="id" title="var">Hd</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_small"><span class="id" title="lemma">Zdiv_small</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#conj"><span class="id" title="constructor">conj</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.Zabs.html#Zabs_eq"><span class="id" title="abbreviation">Zabs_eq</span></a> <span class="id" title="keyword">with</span> (1 := <span class="id" title="var">Hm</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>.<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_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#Zgt_lt"><span class="id" title="abbreviation">Zgt_lt</span></a>.<br/>
<span class="id" title="tactic">destruct</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zorder.html#Zle_lt_or_eq"><span class="id" title="lemma">Zle_lt_or_eq</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">Hm</span>) <span class="id" title="keyword">as</span> [<span class="id" title="var">Hm'</span>|<span class="id" title="var">Hm'</span>].<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#generic_format_F2R"><span class="id" title="lemma">generic_format_F2R</span></a>.<br/>
<span class="id" title="tactic">unfold</span> <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>.<br/>
2: <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="tactic">unfold</span> <span class="id" title="var">k</span> <span class="id" title="keyword">in</span> <span class="id" title="var">Hk</span>. <span class="id" title="tactic">clear</span> -<span class="id" title="var">Hk</span>.<br/>
<span class="id" title="tactic">omega</span>.<br/>
<span class="id" title="tactic">rewrite</span> <- <span class="id" title="var">Hm'</span>, <a class="idref" href="compcert.flocq.Core.Fcore_float_prop.html#F2R_0"><span class="id" title="lemma">F2R_0</span></a>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#generic_format_0"><span class="id" title="lemma">generic_format_0</span></a>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Theorem</span> <a name="truncate_correct_format"><span class="id" title="lemma">truncate_correct_format</span></a> :<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">m</span> <span class="id" title="var">e</span>,<br/>
<a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#32263a1c8b01baecdff9deb038955bc9"><span class="id" title="notation">≠</span></a> <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="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> <span class="id" title="var">x</span> := <a class="idref" href="compcert.flocq.Core.Fcore_defs.html#F2R"><span class="id" title="definition">F2R</span></a> (<a class="idref" href="compcert.flocq.Core.Fcore_defs.html#Float"><span class="id" title="constructor">Float</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#e"><span class="id" title="variable">e</span></a>) <span class="id" title="keyword">in</span><br/>
<a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#generic_format"><span class="id" title="definition">generic_format</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#Fcalc_round.Fcalc_round_fexp.fexp"><span class="id" title="variable">fexp</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.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#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/>
(<a class="idref" href="compcert.flocq.Calc.Fcalc_round.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#208bafb0e148fe7fb7dcd812c227f4ee"><span class="id" title="notation">≤</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#Fcalc_round.Fcalc_round_fexp.fexp"><span class="id" title="variable">fexp</span></a> (<a class="idref" href="compcert.flocq.Core.Fcore_digits.html#Zdigits"><span class="id" title="definition">Zdigits</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.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#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#e"><span class="id" title="variable">e</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/>
<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">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> <span class="id" title="var">l'</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.Calc.Fcalc_round.html#truncate"><span class="id" title="definition">truncate</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.Calc.Fcalc_round.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.Calc.Fcalc_round.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> <a class="idref" href="compcert.flocq.Calc.Fcalc_bracket.html#loc_Exact"><span class="id" title="constructor">loc_Exact</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="keyword">in</span><br/>
<a class="idref" href="compcert.flocq.Calc.Fcalc_round.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.Core.Fcore_defs.html#F2R"><span class="id" title="definition">F2R</span></a> (<a class="idref" href="compcert.flocq.Core.Fcore_defs.html#Float"><span class="id" title="constructor">Float</span></a> <span class="id" title="keyword">beta</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#d82a7d96d3659d805ffe732283716822"><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.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#canonic_exp"><span class="id" title="definition">canonic_exp</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#Fcalc_round.Fcalc_round_fexp.fexp"><span class="id" title="variable">fexp</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.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">m</span> <span class="id" title="var">e</span> <span class="id" title="var">Hm</span> <span class="id" title="var">x</span> <span class="id" title="var">Fx</span> <span class="id" title="var">He</span>.<br/>
<span class="id" title="tactic">assert</span> (<span class="id" title="var">Hc</span>: <a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#canonic_exp"><span class="id" title="definition">canonic_exp</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#Fcalc_round.Fcalc_round_fexp.fexp"><span class="id" title="variable">fexp</span></a> <span class="id" title="var">x</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.Calc.Fcalc_round.html#Fcalc_round.Fcalc_round_fexp.fexp"><span class="id" title="variable">fexp</span></a> (<a class="idref" href="compcert.flocq.Core.Fcore_digits.html#Zdigits"><span class="id" title="definition">Zdigits</span></a> <span class="id" title="keyword">beta</span> <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">e</span>)).<br/>
<span class="id" title="tactic">unfold</span> <a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#canonic_exp"><span class="id" title="definition">canonic_exp</span></a>, <span class="id" title="var">x</span>.<br/>
<span class="id" title="var">now</span> <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>.<br/>
<span class="id" title="tactic">unfold</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#truncate"><span class="id" title="definition">truncate</span></a>.<br/>
<span class="id" title="tactic">rewrite</span> <- <span class="id" title="var">Hc</span>.<br/>
<span class="id" title="tactic">set</span> (<span class="id" title="var">k</span> := (<a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#canonic_exp"><span class="id" title="definition">canonic_exp</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#Fcalc_round.Fcalc_round_fexp.fexp"><span class="id" title="variable">fexp</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#239a40728d107295b3cb2c790f57f9e9"><span class="id" title="notation">-</span></a> <span class="id" title="var">e</span>)%<span class="id" title="var">Z</span>).<br/>
<span class="id" title="tactic">case</span> <a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#Zlt_bool_spec"><span class="id" title="lemma">Zlt_bool_spec</span></a> ; <span class="id" title="tactic">intros</span> <span class="id" title="var">Hk</span>.<br/>
<span class="id" title="tactic">unfold</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#truncate_aux"><span class="id" title="definition">truncate_aux</span></a>.<br/>
<span class="id" title="tactic">rewrite</span> <span class="id" title="var">Fx</span> <span class="id" title="tactic">at</span> 1.<br/>
<span class="id" title="tactic">assert</span> (<span class="id" title="var">H</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> <span class="id" title="var">k</span>)%<span class="id" title="var">Z</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.Core.Fcore_generic_fmt.html#canonic_exp"><span class="id" title="definition">canonic_exp</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#Fcalc_round.Fcalc_round_fexp.fexp"><span class="id" title="variable">fexp</span></a> <span class="id" title="var">x</span>).<br/>
<span class="id" title="tactic">unfold</span> <span class="id" title="var">k</span>. <span class="id" title="tactic">ring</span>.<br/>
<span class="id" title="tactic">refine</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#conj"><span class="id" title="constructor">conj</span></a> <span class="id" title="var">_</span> <span class="id" title="var">H</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="compcert.flocq.Core.Fcore_float_prop.html#F2R_eq_compat"><span class="id" title="lemma">F2R_eq_compat</span></a>.<br/>
<span class="id" title="tactic">replace</span> (<a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#scaled_mantissa"><span class="id" title="definition">scaled_mantissa</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#Fcalc_round.Fcalc_round_fexp.fexp"><span class="id" title="variable">fexp</span></a> <span class="id" title="var">x</span>) <span class="id" title="keyword">with</span> (<a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Z2R"><span class="id" title="definition">Z2R</span></a> (<a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Zfloor"><span class="id" title="definition">Zfloor</span></a> (<a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#scaled_mantissa"><span class="id" title="definition">scaled_mantissa</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#Fcalc_round.Fcalc_round_fexp.fexp"><span class="id" title="variable">fexp</span></a> <span class="id" title="var">x</span>))).<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Ztrunc_Z2R"><span class="id" title="lemma">Ztrunc_Z2R</span></a>.<br/>
<span class="id" title="tactic">unfold</span> <a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#scaled_mantissa"><span class="id" title="definition">scaled_mantissa</span></a>.<br/>
<span class="id" title="tactic">rewrite</span> <- <span class="id" title="var">H</span>.<br/>
<span class="id" title="tactic">unfold</span> <span class="id" title="var">x</span>, <a class="idref" href="compcert.flocq.Core.Fcore_defs.html#F2R"><span class="id" title="definition">F2R</span></a>. <span class="id" title="tactic">simpl</span>.<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Reals.Raxioms.html#Rmult_assoc"><span class="id" title="axiom">Rmult_assoc</span></a>, <- <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#bpow_plus"><span class="id" title="lemma">bpow_plus</span></a>.<br/>
<span class="id" title="var">ring_simplify</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> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#0a3978eabdacce0a128118074de19651"><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> <span class="id" title="var">k</span><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">Z</span>.<br/>
<span class="id" title="tactic">clear</span> -<span class="id" title="var">Hm</span> <span class="id" title="var">Hk</span>.<br/>
<span class="id" title="tactic">destruct</span> <span class="id" title="var">k</span> <span class="id" title="keyword">as</span> [|<span class="id" title="var">k</span>|<span class="id" title="var">k</span>] ; <span class="id" title="tactic">try</span> <span class="id" title="var">easy</span>.<br/>
<span class="id" title="tactic">simpl</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Zfloor_div"><span class="id" title="lemma">Zfloor_div</span></a>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">H</span>.<br/>
<span class="id" title="tactic">generalize</span> (<a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#Zpower_pos_gt_0"><span class="id" title="lemma">Zpower_pos_gt_0</span></a> <span class="id" title="keyword">beta</span> <span class="id" title="var">k</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> (<a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#radix_prop"><span class="id" title="projection">radix_prop</span></a> <span class="id" title="keyword">beta</span>)).<br/>
<span class="id" title="tactic">omega</span>.<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#scaled_mantissa_generic"><span class="id" title="lemma">scaled_mantissa_generic</span></a> <span class="id" title="keyword">with</span> (1 := <span class="id" title="var">Fx</span>).<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Zfloor_Z2R"><span class="id" title="lemma">Zfloor_Z2R</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.Init.Logic.html#refl_equal"><span class="id" title="abbreviation">refl_equal</span></a>.<br/>
<span class="id" title="tactic">unfold</span> <span class="id" title="var">k</span> <span class="id" title="keyword">in</span> <span class="id" title="var">Hk</span>.<br/>
<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="truncate_correct_partial"><span class="id" title="lemma">truncate_correct_partial</span></a> :<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">m</span> <span class="id" title="var">e</span> <span class="id" title="var">l</span>,<br/>
(0 <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Reals.Rdefinitions.html#1cc22ba6849531267d3b25ca7b262449"><span class="id" title="notation"><</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#x"><span class="id" title="variable">x</span></a>)%<span class="id" title="var">R</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.Calc.Fcalc_bracket.html#inbetween_float"><span class="id" title="definition">inbetween_float</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#l"><span class="id" title="variable">l</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.Calc.Fcalc_round.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#208bafb0e148fe7fb7dcd812c227f4ee"><span class="id" title="notation">≤</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#Fcalc_round.Fcalc_round_fexp.fexp"><span class="id" title="variable">fexp</span></a> (<a class="idref" href="compcert.flocq.Core.Fcore_digits.html#Zdigits"><span class="id" title="definition">Zdigits</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.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#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#e"><span class="id" title="variable">e</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/>
<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">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> <span class="id" title="var">l'</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.Calc.Fcalc_round.html#truncate"><span class="id" title="definition">truncate</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.Calc.Fcalc_round.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.Calc.Fcalc_round.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> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#l"><span class="id" title="variable">l</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="keyword">in</span><br/>
<a class="idref" href="compcert.flocq.Calc.Fcalc_bracket.html#inbetween_float"><span class="id" title="definition">inbetween_float</span></a> <span class="id" title="keyword">beta</span> <span class="id" title="var">m'</span> <span class="id" title="var">e'</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#x"><span class="id" title="variable">x</span></a> <span class="id" title="var">l'</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> <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.Core.Fcore_generic_fmt.html#canonic_exp"><span class="id" title="definition">canonic_exp</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#Fcalc_round.Fcalc_round_fexp.fexp"><span class="id" title="variable">fexp</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.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">m</span> <span class="id" title="var">e</span> <span class="id" title="var">l</span> <span class="id" title="var">Hx</span> <span class="id" title="var">H1</span> <span class="id" title="var">H2</span>.<br/>
<span class="id" title="tactic">unfold</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#truncate"><span class="id" title="definition">truncate</span></a>.<br/>
<span class="id" title="tactic">set</span> (<span class="id" title="var">k</span> := (<a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#Fcalc_round.Fcalc_round_fexp.fexp"><span class="id" title="variable">fexp</span></a> (<a class="idref" href="compcert.flocq.Core.Fcore_digits.html#Zdigits"><span class="id" title="definition">Zdigits</span></a> <span class="id" title="keyword">beta</span> <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">e</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> <span class="id" title="var">e</span>)%<span class="id" title="var">Z</span>).<br/>
<span class="id" title="tactic">set</span> (<span class="id" title="var">p</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> <span class="id" title="keyword">beta</span> <span class="id" title="var">k</span>).<br/>
<span class="id" title="tactic">assert</span> (<span class="id" title="var">Hx'</span>: (<a class="idref" href="compcert.flocq.Core.Fcore_defs.html#F2R"><span class="id" title="definition">F2R</span></a> (<a class="idref" href="compcert.flocq.Core.Fcore_defs.html#Float"><span class="id" title="constructor">Float</span></a> <span class="id" title="keyword">beta</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.Reals.Rdefinitions.html#ed7870e49208be293c8dbcf28a708f81"><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.Reals.Rdefinitions.html#ed7870e49208be293c8dbcf28a708f81"><span class="id" title="notation"><</span></a> <a class="idref" href="compcert.flocq.Core.Fcore_defs.html#F2R"><span class="id" title="definition">F2R</span></a> (<a class="idref" href="compcert.flocq.Core.Fcore_defs.html#Float"><span class="id" title="constructor">Float</span></a> <span class="id" title="keyword">beta</span> (<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> 1) <span class="id" title="var">e</span>))%<span class="id" title="var">R</span>).<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_bracket.html#inbetween_float_bounds"><span class="id" title="lemma">inbetween_float_bounds</span></a> <span class="id" title="keyword">with</span> (1 := <span class="id" title="var">H1</span>).<br/>
<span class="id" title="tactic">assert</span> (<span class="id" title="var">Hm</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">m</span>)%<span class="id" title="var">Z</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#273beba4f3453dbb29192b3ac740bb71"><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> 1)%<span class="id" title="var">Z</span>. <span class="id" title="tactic">omega</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_float_prop.html#F2R_lt_reg"><span class="id" title="lemma">F2R_lt_reg</span></a> <span class="id" title="keyword">with</span> <span class="id" title="keyword">beta</span> <span class="id" title="var">e</span>.<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.flocq.Core.Fcore_float_prop.html#F2R_0"><span class="id" title="lemma">F2R_0</span></a>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Reals.Raxioms.html#Rlt_trans"><span class="id" title="axiom">Rlt_trans</span></a> <span class="id" title="keyword">with</span> (1 := <span class="id" title="var">Hx</span>).<br/>
<span class="id" title="tactic">apply</span> <span class="id" title="var">Hx'</span>.<br/>
<span class="id" title="tactic">assert</span> (<span class="id" title="var">He</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> <span class="id" title="var">k</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.Core.Fcore_generic_fmt.html#canonic_exp"><span class="id" title="definition">canonic_exp</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#Fcalc_round.Fcalc_round_fexp.fexp"><span class="id" title="variable">fexp</span></a> <span class="id" title="var">x</span>)%<span class="id" title="var">Z</span>).<br/>
<span class="id" title="tactic">unfold</span> <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">destruct</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zorder.html#Zle_lt_or_eq"><span class="id" title="lemma">Zle_lt_or_eq</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">Hm</span>) <span class="id" title="keyword">as</span> [<span class="id" title="var">Hm'</span>|<span class="id" title="var">Hm'</span>].<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.flocq.Core.Fcore_float_prop.html#ln_beta_F2R_bounds"><span class="id" title="lemma">ln_beta_F2R_bounds</span></a> <span class="id" title="keyword">with</span> (1 := <span class="id" title="var">Hm'</span>) (2 := <span class="id" title="var">Hx'</span>).<br/>
<span class="id" title="tactic">assert</span> (<span class="id" title="var">H</span>: <span class="id" title="var">m</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#32263a1c8b01baecdff9deb038955bc9"><span class="id" title="notation">≠</span></a> <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>).<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_not_eq"><span class="id" title="abbreviation">sym_not_eq</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_not_eq"><span class="id" title="abbreviation">Zlt_not_eq</span></a>.<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.flocq.Core.Fcore_float_prop.html#ln_beta_F2R"><span class="id" title="lemma">ln_beta_F2R</span></a> <span class="id" title="keyword">with</span> (1 := <span class="id" title="var">H</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> <span class="id" title="keyword">with</span> (1 := <span class="id" title="var">H</span>).<br/>
<span class="id" title="tactic">unfold</span> <span class="id" title="var">k</span>.<br/>
<span class="id" title="tactic">ring</span>.<br/>
<span class="id" title="tactic">rewrite</span> <- <span class="id" title="var">Hm'</span> <span class="id" title="keyword">in</span> <span class="id" title="var">H2</span>.<br/>
<span class="id" title="tactic">destruct</span> (<a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#ln_beta"><span class="id" title="definition">ln_beta</span></a> <span class="id" title="keyword">beta</span> <span class="id" title="var">x</span>) <span class="id" title="keyword">as</span> (<span class="id" title="var">ex</span>, <span class="id" title="var">Hex</span>).<br/>
<span class="id" title="tactic">simpl</span>.<br/>
<span class="id" title="tactic">specialize</span> (<span class="id" title="var">Hex</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Reals.RIneq.html#Rgt_not_eq"><span class="id" title="lemma">Rgt_not_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> <span class="id" title="var">k</span>.<br/>
<span class="id" title="var">ring_simplify</span>.<br/>
<span class="id" title="tactic">rewrite</span> <- <span class="id" title="var">Hm'</span>.<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#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.Calc.Fcalc_round.html#Fcalc_round.Fcalc_round_fexp.valid_exp"><span class="id" title="variable">valid_exp</span></a>.<br/>
<span class="id" title="tactic">exact</span> <span class="id" title="var">H2</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_trans"><span class="id" title="abbreviation">Zle_trans</span></a> <span class="id" title="keyword">with</span> <span class="id" title="var">e</span>.<br/>
<span class="id" title="tactic">eapply</span> <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#bpow_lt_bpow"><span class="id" title="lemma">bpow_lt_bpow</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#Rle_lt_trans"><span class="id" title="lemma">Rle_lt_trans</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">Hex</span>).<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Reals.Rbasic_fun.html#Rabs_pos_eq"><span class="id" title="lemma">Rabs_pos_eq</span></a>.<br/>
<span class="id" title="tactic">rewrite</span> <- <a class="idref" href="compcert.flocq.Core.Fcore_float_prop.html#F2R_bpow"><span class="id" title="lemma">F2R_bpow</span></a>.<br/>
<span class="id" title="tactic">rewrite</span> <- <span class="id" title="var">Hm'</span> <span class="id" title="keyword">in</span> <span class="id" title="var">Hx'</span>.<br/>
<span class="id" title="tactic">apply</span> <span class="id" title="var">Hx'</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.Reals.RIneq.html#Rlt_le"><span class="id" title="lemma">Rlt_le</span></a>.<br/>
<span class="id" title="tactic">exact</span> <span class="id" title="var">H2</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#Zlt_cases"><span class="id" title="lemma">Zlt_cases</span></a> 0 <span class="id" title="var">k</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#Zlt_bool"><span class="id" title="abbreviation">Zlt_bool</span></a> 0 <span class="id" title="var">k</span>) ; <span class="id" title="tactic">intros</span> <span class="id" title="var">Hk</span> ; <span class="id" title="tactic">unfold</span> <span class="id" title="var">k</span> <span class="id" title="keyword">in</span> <span class="id" title="var">Hk</span>.<br/>
<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="compcert.flocq.Calc.Fcalc_bracket.html#inbetween_float_new_location"><span class="id" title="lemma">inbetween_float_new_location</span></a>.<br/>
<span class="id" title="tactic">exact</span> <span class="id" title="var">He</span>.<br/>
<span class="id" title="tactic">split</span>.<br/>
<span class="id" title="tactic">exact</span> <span class="id" title="var">H1</span>.<br/>
<span class="id" title="tactic">rewrite</span> <- <span class="id" title="var">He</span>.<br/>
<span class="id" title="tactic">unfold</span> <span class="id" title="var">k</span>.<br/>
<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="truncate_correct"><span class="id" title="lemma">truncate_correct</span></a> :<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">m</span> <span class="id" title="var">e</span> <span class="id" title="var">l</span>,<br/>
(0 <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Reals.Rdefinitions.html#39ab57f76c1eb2e6d107f9799a31545a"><span class="id" title="notation">≤</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#x"><span class="id" title="variable">x</span></a>)%<span class="id" title="var">R</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.Calc.Fcalc_bracket.html#inbetween_float"><span class="id" title="definition">inbetween_float</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#l"><span class="id" title="variable">l</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.Calc.Fcalc_round.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#208bafb0e148fe7fb7dcd812c227f4ee"><span class="id" title="notation">≤</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#Fcalc_round.Fcalc_round_fexp.fexp"><span class="id" title="variable">fexp</span></a> (<a class="idref" href="compcert.flocq.Core.Fcore_digits.html#Zdigits"><span class="id" title="definition">Zdigits</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.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#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#e"><span class="id" title="variable">e</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#7a45dffb109c3069e5c675be68643e60"><span class="id" title="notation">∨</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#l"><span class="id" title="variable">l</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.Calc.Fcalc_bracket.html#loc_Exact"><span class="id" title="constructor">loc_Exact</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/>
<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">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> <span class="id" title="var">l'</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.Calc.Fcalc_round.html#truncate"><span class="id" title="definition">truncate</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.Calc.Fcalc_round.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.Calc.Fcalc_round.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> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#l"><span class="id" title="variable">l</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="keyword">in</span><br/>
<a class="idref" href="compcert.flocq.Calc.Fcalc_bracket.html#inbetween_float"><span class="id" title="definition">inbetween_float</span></a> <span class="id" title="keyword">beta</span> <span class="id" title="var">m'</span> <span class="id" title="var">e'</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#x"><span class="id" title="variable">x</span></a> <span class="id" title="var">l'</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><br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><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.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#canonic_exp"><span class="id" title="definition">canonic_exp</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#Fcalc_round.Fcalc_round_fexp.fexp"><span class="id" title="variable">fexp</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.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#7a45dffb109c3069e5c675be68643e60"><span class="id" title="notation">∨</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#7a45dffb109c3069e5c675be68643e60"><span class="id" title="notation">(</span></a><span class="id" title="var">l'</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.Calc.Fcalc_bracket.html#loc_Exact"><span class="id" title="constructor">loc_Exact</span></a> <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.Calc.Fcalc_round.html#format"><span class="id" title="abbreviation">format</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.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#7a45dffb109c3069e5c675be68643e60"><span class="id" title="notation">)</span></a><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><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">x</span> <span class="id" title="var">m</span> <span class="id" title="var">e</span> <span class="id" title="var">l</span> [<span class="id" title="var">Hx</span>|<span class="id" title="var">Hx</span>] <span class="id" title="var">H1</span> <span class="id" title="var">H2</span>.<br/>
<span class="id" title="tactic">destruct</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zorder.html#Zle_or_lt"><span class="id" title="abbreviation">Zle_or_lt</span></a> <span class="id" title="var">e</span> (<a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#Fcalc_round.Fcalc_round_fexp.fexp"><span class="id" title="variable">fexp</span></a> (<a class="idref" href="compcert.flocq.Core.Fcore_digits.html#Zdigits"><span class="id" title="definition">Zdigits</span></a> <span class="id" title="keyword">beta</span> <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">e</span>))) <span class="id" title="keyword">as</span> [<span class="id" title="var">H3</span>|<span class="id" title="var">H3</span>].<br/>
<span class="id" title="tactic">generalize</span> (<a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#truncate_correct_partial"><span class="id" title="lemma">truncate_correct_partial</span></a> <span class="id" title="var">x</span> <span class="id" title="var">m</span> <span class="id" title="var">e</span> <span class="id" title="var">l</span> <span class="id" title="var">Hx</span> <span class="id" title="var">H1</span> <span class="id" title="var">H3</span>).<br/>
<span class="id" title="tactic">destruct</span> (<a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#truncate"><span class="id" title="definition">truncate</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">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> <span class="id" title="var">l</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="keyword">as</span> ((<span class="id" title="var">m'</span>, <span class="id" title="var">e'</span>), <span class="id" title="var">l'</span>).<br/>
<span class="id" title="tactic">intros</span> (<span class="id" title="var">H4</span>, <span class="id" title="var">H5</span>).<br/>
<span class="id" title="tactic">split</span>.<br/>
<span class="id" title="tactic">exact</span> <span class="id" title="var">H4</span>.<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">left</span>.<br/>
<span class="id" title="tactic">destruct</span> <span class="id" title="var">H2</span> <span class="id" title="keyword">as</span> [<span class="id" title="var">H2</span>|<span class="id" title="var">H2</span>].<br/>
<span class="id" title="tactic">elim</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zorder.html#Zlt_irrefl"><span class="id" title="abbreviation">Zlt_irrefl</span></a> <span class="id" title="var">e</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#Zle_lt_trans"><span class="id" title="abbreviation">Zle_lt_trans</span></a> <span class="id" title="keyword">with</span> (1 := <span class="id" title="var">H2</span>).<br/>
<span class="id" title="tactic">rewrite</span> <span class="id" title="var">H2</span> <span class="id" title="keyword">in</span> <span class="id" title="var">H1</span> |- ×.<br/>
<span class="id" title="tactic">unfold</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#truncate"><span class="id" title="definition">truncate</span></a>.<br/>
<span class="id" title="tactic">generalize</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zbool.html#Zlt_cases"><span class="id" title="lemma">Zlt_cases</span></a> 0 (<a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#Fcalc_round.Fcalc_round_fexp.fexp"><span class="id" title="variable">fexp</span></a> (<a class="idref" href="compcert.flocq.Core.Fcore_digits.html#Zdigits"><span class="id" title="definition">Zdigits</span></a> <span class="id" title="keyword">beta</span> <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">e</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> <span class="id" title="var">e</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#Zlt_bool"><span class="id" title="abbreviation">Zlt_bool</span></a>.<br/>
<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.Init.Logic.html#False_ind"><span class="id" title="definition">False_ind</span></a>.<br/>
<span class="id" title="tactic">omega</span>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">_</span>.<br/>
<span class="id" title="tactic">split</span>.<br/>
<span class="id" title="tactic">exact</span> <span class="id" title="var">H1</span>.<br/>
<span class="id" title="tactic">right</span>.<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.Init.Logic.html#refl_equal"><span class="id" title="abbreviation">refl_equal</span></a>.<br/>
<span class="id" title="tactic">inversion_clear</span> <span class="id" title="var">H1</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="compcert.flocq.Core.Fcore_generic_fmt.html#generic_format_F2R"><span class="id" title="lemma">generic_format_F2R</span></a>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">Zm</span>.<br/>
<span class="id" title="tactic">unfold</span> <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> <span class="id" title="keyword">with</span> (1 := <span class="id" title="var">Zm</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">assert</span> (<span class="id" title="var">Hm</span>: <span class="id" title="var">m</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="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Numbers.BinNums.html#Z0"><span class="id" title="constructor">Z0</span></a>).<br/>
<span class="id" title="tactic">cut</span> (<span class="id" title="var">m</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> 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">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> 1)%<span class="id" title="var">Z</span>. <span class="id" title="tactic">omega</span>.<br/>
<span class="id" title="tactic">assert</span> (<span class="id" title="var">Hx'</span>: (<a class="idref" href="compcert.flocq.Core.Fcore_defs.html#F2R"><span class="id" title="definition">F2R</span></a> (<a class="idref" href="compcert.flocq.Core.Fcore_defs.html#Float"><span class="id" title="constructor">Float</span></a> <span class="id" title="keyword">beta</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.Reals.Rdefinitions.html#ed7870e49208be293c8dbcf28a708f81"><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.Reals.Rdefinitions.html#ed7870e49208be293c8dbcf28a708f81"><span class="id" title="notation"><</span></a> <a class="idref" href="compcert.flocq.Core.Fcore_defs.html#F2R"><span class="id" title="definition">F2R</span></a> (<a class="idref" href="compcert.flocq.Core.Fcore_defs.html#Float"><span class="id" title="constructor">Float</span></a> <span class="id" title="keyword">beta</span> (<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> 1) <span class="id" title="var">e</span>))%<span class="id" title="var">R</span>).<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_bracket.html#inbetween_float_bounds"><span class="id" title="lemma">inbetween_float_bounds</span></a> <span class="id" title="keyword">with</span> (1 := <span class="id" title="var">H1</span>).<br/>
<span class="id" title="tactic">rewrite</span> <- <span class="id" title="var">Hx</span> <span class="id" title="keyword">in</span> <span class="id" title="var">Hx'</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_float_prop.html#F2R_le_0_reg"><span class="id" title="lemma">F2R_le_0_reg</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">Hx'</span>).<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_float_prop.html#F2R_gt_0_reg"><span class="id" title="lemma">F2R_gt_0_reg</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#proj2"><span class="id" title="lemma">proj2</span></a> <span class="id" title="var">Hx'</span>).<br/>
<span class="id" title="tactic">rewrite</span> <span class="id" title="var">Hm</span>, <- <span class="id" title="var">Hx</span> <span class="id" title="keyword">in</span> <span class="id" title="var">H1</span> |- ×.<br/>
<span class="id" title="tactic">clear</span> -<span class="id" title="var">H1</span>.<br/>
<span class="id" title="tactic">case</span> <span class="id" title="var">H1</span>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">_</span>.<br/>
<span class="id" title="tactic">assert</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><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.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">,</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#truncate"><span class="id" title="definition">truncate</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.Numbers.BinNums.html#Z0"><span class="id" title="constructor">Z0</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">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.Calc.Fcalc_bracket.html#loc_Exact"><span class="id" title="constructor">loc_Exact</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.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="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="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.Calc.Fcalc_round.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> <a class="idref" href="compcert.flocq.Calc.Fcalc_bracket.html#loc_Exact"><span class="id" title="constructor">loc_Exact</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="tactic">unfold</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#truncate"><span class="id" title="definition">truncate</span></a>, <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#truncate_aux"><span class="id" title="definition">truncate_aux</span></a>.<br/>
<span class="id" title="tactic">case</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zbool.html#Zlt_bool"><span class="id" title="abbreviation">Zlt_bool</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_0_l"><span class="id" title="lemma">Zdiv_0_l</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zdiv.html#Zmod_0_l"><span class="id" title="lemma">Zmod_0_l</span></a>.<br/>
<span class="id" title="tactic">eexists</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">unfold</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_bracket.html#new_location"><span class="id" title="definition">new_location</span></a>.<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">case</span> <a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#Zeven"><span class="id" title="definition">Zeven</span></a>.<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">eexists</span>.<br/>
<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">e'</span>, <span class="id" title="var">H</span>).<br/>
<span class="id" title="tactic">rewrite</span> <span class="id" title="var">H</span>.<br/>
<span class="id" title="tactic">split</span>.<br/>
<span class="id" title="tactic">constructor</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_float_prop.html#F2R_0"><span class="id" title="lemma">F2R_0</span></a>.<br/>
<span class="id" title="tactic">right</span>.<br/>
<span class="id" title="tactic">repeat</span> <span class="id" title="tactic">split</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#generic_format_0"><span class="id" title="lemma">generic_format_0</span></a>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">l'</span> (<span class="id" title="var">H</span>, <span class="id" title="var">_</span>) <span class="id" title="var">_</span>.<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.flocq.Core.Fcore_float_prop.html#F2R_0"><span class="id" title="lemma">F2R_0</span></a> <span class="id" title="keyword">in</span> <span class="id" title="var">H</span>.<br/>
<span class="id" title="tactic">elim</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Reals.RIneq.html#Rlt_irrefl"><span class="id" title="lemma">Rlt_irrefl</span></a> <span class="id" title="keyword">with</span> (1 := <span class="id" title="var">H</span>).<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Section</span> <a name="Fcalc_round.Fcalc_round_fexp.round_dir"><span class="id" title="section">round_dir</span></a>.<br/>
<br/>
<span class="id" title="keyword">Variable</span> <a name="Fcalc_round.Fcalc_round_fexp.round_dir.rnd"><span class="id" title="variable">rnd</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Reals.Rdefinitions.html#R"><span class="id" title="axiom">R</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>.<br/>
<span class="id" title="keyword">Context</span> { <span class="id" title="var">valid_rnd</span> : <a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#Valid_rnd"><span class="id" title="class">Valid_rnd</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#Fcalc_round.Fcalc_round_fexp.round_dir.rnd"><span class="id" title="variable">rnd</span></a> }.<br/>
<br/>
<span class="id" title="keyword">Variable</span> <a name="Fcalc_round.Fcalc_round_fexp.round_dir.choice"><span class="id" title="variable">choice</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.Calc.Fcalc_bracket.html#location"><span class="id" title="inductive">location</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>.<br/>
<span class="id" title="keyword">Hypothesis</span> <a name="Fcalc_round.Fcalc_round_fexp.round_dir.inbetween_int_valid"><span class="id" title="variable">inbetween_int_valid</span></a> :<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">m</span> <span class="id" title="var">l</span>,<br/>
<a class="idref" href="compcert.flocq.Calc.Fcalc_bracket.html#inbetween_int"><span class="id" title="definition">inbetween_int</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#l"><span class="id" title="variable">l</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.Calc.Fcalc_round.html#Fcalc_round.Fcalc_round_fexp.round_dir.rnd"><span class="id" title="variable">rnd</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.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.Calc.Fcalc_round.html#Fcalc_round.Fcalc_round_fexp.round_dir.choice"><span class="id" title="variable">choice</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#l"><span class="id" title="variable">l</span></a>.<br/>
<br/>
<span class="id" title="keyword">Theorem</span> <a name="round_any_correct"><span class="id" title="lemma">round_any_correct</span></a> :<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">m</span> <span class="id" title="var">e</span> <span class="id" title="var">l</span>,<br/>
<a class="idref" href="compcert.flocq.Calc.Fcalc_bracket.html#inbetween_float"><span class="id" title="definition">inbetween_float</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#l"><span class="id" title="variable">l</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="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.Calc.Fcalc_round.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="compcert.flocq.Core.Fcore_generic_fmt.html#canonic_exp"><span class="id" title="definition">canonic_exp</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#Fcalc_round.Fcalc_round_fexp.fexp"><span class="id" title="variable">fexp</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.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#7a45dffb109c3069e5c675be68643e60"><span class="id" title="notation">∨</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#7a45dffb109c3069e5c675be68643e60"><span class="id" title="notation">(</span></a><a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#l"><span class="id" title="variable">l</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.Calc.Fcalc_bracket.html#loc_Exact"><span class="id" title="constructor">loc_Exact</span></a> <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.Calc.Fcalc_round.html#format"><span class="id" title="abbreviation">format</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.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#7a45dffb109c3069e5c675be68643e60"><span class="id" title="notation">)</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.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#round"><span class="id" title="definition">round</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#Fcalc_round.Fcalc_round_fexp.fexp"><span class="id" title="variable">fexp</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#Fcalc_round.Fcalc_round_fexp.round_dir.rnd"><span class="id" title="variable">rnd</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.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.Core.Fcore_defs.html#F2R"><span class="id" title="definition">F2R</span></a> (<a class="idref" href="compcert.flocq.Core.Fcore_defs.html#Float"><span class="id" title="constructor">Float</span></a> <span class="id" title="keyword">beta</span> (<a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#Fcalc_round.Fcalc_round_fexp.round_dir.choice"><span class="id" title="variable">choice</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#l"><span class="id" title="variable">l</span></a>) <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#e"><span class="id" title="variable">e</span></a>).<br/>
<span class="id" title="keyword">Proof</span> <span class="id" title="keyword">with</span> <span class="id" title="tactic">auto</span> <span class="id" title="keyword">with</span> <span class="id" title="var">typeclass_instances</span>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">x</span> <span class="id" title="var">m</span> <span class="id" title="var">e</span> <span class="id" title="var">l</span> <span class="id" title="var">Hin</span> [<span class="id" title="var">He</span>|(<span class="id" title="var">Hl</span>,<span class="id" title="var">Hf</span>)].<br/>
<span class="id" title="tactic">rewrite</span> <span class="id" title="var">He</span> <span class="id" title="keyword">in</span> <span class="id" title="var">Hin</span> |- ×.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#inbetween_float_round"><span class="id" title="lemma">inbetween_float_round</span></a> <span class="id" title="keyword">with</span> (2 := <span class="id" title="var">Hin</span>).<br/>
<span class="id" title="tactic">exact</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#Fcalc_round.Fcalc_round_fexp.round_dir.inbetween_int_valid"><span class="id" title="variable">inbetween_int_valid</span></a>.<br/>
<span class="id" title="tactic">rewrite</span> <span class="id" title="var">Hl</span> <span class="id" title="keyword">in</span> <span class="id" title="var">Hin</span>.<br/>
<span class="id" title="tactic">inversion_clear</span> <span class="id" title="var">Hin</span>.<br/>
<span class="id" title="tactic">rewrite</span> <span class="id" title="var">Hl</span>.<br/>
<span class="id" title="tactic">replace</span> (<a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#Fcalc_round.Fcalc_round_fexp.round_dir.choice"><span class="id" title="variable">choice</span></a> <span class="id" title="var">m</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_bracket.html#loc_Exact"><span class="id" title="constructor">loc_Exact</span></a>) <span class="id" title="keyword">with</span> <span class="id" title="var">m</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="compcert.flocq.Core.Fcore_generic_fmt.html#round_generic"><span class="id" title="lemma">round_generic</span></a>...<br/>
<span class="id" title="tactic">rewrite</span> <- (<a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#Zrnd_Z2R"><span class="id" title="method">Zrnd_Z2R</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#Fcalc_round.Fcalc_round_fexp.round_dir.rnd"><span class="id" title="variable">rnd</span></a> <span class="id" title="var">m</span>) <span class="id" title="tactic">at</span> 1.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#Fcalc_round.Fcalc_round_fexp.round_dir.inbetween_int_valid"><span class="id" title="variable">inbetween_int_valid</span></a>.<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">constructor</span>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
</div>
<div class="doc">
Truncating a triple is sufficient to round a real number.
</div>
<div class="code">
<br/>
<span class="id" title="keyword">Theorem</span> <a name="round_trunc_any_correct"><span class="id" title="lemma">round_trunc_any_correct</span></a> :<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">m</span> <span class="id" title="var">e</span> <span class="id" title="var">l</span>,<br/>
(0 <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Reals.Rdefinitions.html#39ab57f76c1eb2e6d107f9799a31545a"><span class="id" title="notation">≤</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#x"><span class="id" title="variable">x</span></a>)%<span class="id" title="var">R</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.Calc.Fcalc_bracket.html#inbetween_float"><span class="id" title="definition">inbetween_float</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#l"><span class="id" title="variable">l</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.Calc.Fcalc_round.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#208bafb0e148fe7fb7dcd812c227f4ee"><span class="id" title="notation">≤</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#Fcalc_round.Fcalc_round_fexp.fexp"><span class="id" title="variable">fexp</span></a> (<a class="idref" href="compcert.flocq.Core.Fcore_digits.html#Zdigits"><span class="id" title="definition">Zdigits</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.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#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#e"><span class="id" title="variable">e</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#7a45dffb109c3069e5c675be68643e60"><span class="id" title="notation">∨</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#l"><span class="id" title="variable">l</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.Calc.Fcalc_bracket.html#loc_Exact"><span class="id" title="constructor">loc_Exact</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.Core.Fcore_generic_fmt.html#round"><span class="id" title="definition">round</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#Fcalc_round.Fcalc_round_fexp.fexp"><span class="id" title="variable">fexp</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#Fcalc_round.Fcalc_round_fexp.round_dir.rnd"><span class="id" title="variable">rnd</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.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> <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">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> <span class="id" title="var">l'</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.Calc.Fcalc_round.html#truncate"><span class="id" title="definition">truncate</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.Calc.Fcalc_round.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.Calc.Fcalc_round.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> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#l"><span class="id" title="variable">l</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="keyword">in</span> <a class="idref" href="compcert.flocq.Core.Fcore_defs.html#F2R"><span class="id" title="definition">F2R</span></a> (<a class="idref" href="compcert.flocq.Core.Fcore_defs.html#Float"><span class="id" title="constructor">Float</span></a> <span class="id" title="keyword">beta</span> (<a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#Fcalc_round.Fcalc_round_fexp.round_dir.choice"><span class="id" title="variable">choice</span></a> <span class="id" title="var">m'</span> <span class="id" title="var">l'</span>) <span class="id" title="var">e'</span>).<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">m</span> <span class="id" title="var">e</span> <span class="id" title="var">l</span> <span class="id" title="var">Hx</span> <span class="id" title="var">Hl</span> <span class="id" title="var">He</span>.<br/>
<span class="id" title="tactic">generalize</span> (<a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#truncate_correct"><span class="id" title="lemma">truncate_correct</span></a> <span class="id" title="var">x</span> <span class="id" title="var">m</span> <span class="id" title="var">e</span> <span class="id" title="var">l</span> <span class="id" title="var">Hx</span> <span class="id" title="var">Hl</span> <span class="id" title="var">He</span>).<br/>
<span class="id" title="tactic">destruct</span> (<a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#truncate"><span class="id" title="definition">truncate</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">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> <span class="id" title="var">l</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="keyword">as</span> ((<span class="id" title="var">m'</span>, <span class="id" title="var">e'</span>), <span class="id" title="var">l'</span>).<br/>
<span class="id" title="tactic">intros</span> (<span class="id" title="var">H1</span>, <span class="id" title="var">H2</span>).<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#round_any_correct"><span class="id" title="lemma">round_any_correct</span></a>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#Fcalc_round.Fcalc_round_fexp.round_dir"><span class="id" title="section">round_dir</span></a>.<br/>
<br/>
<span class="id" title="keyword">Section</span> <a name="Fcalc_round.Fcalc_round_fexp.round_dir_sign"><span class="id" title="section">round_dir_sign</span></a>.<br/>
<br/>
<span class="id" title="keyword">Variable</span> <a name="Fcalc_round.Fcalc_round_fexp.round_dir_sign.rnd"><span class="id" title="variable">rnd</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Reals.Rdefinitions.html#R"><span class="id" title="axiom">R</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>.<br/>
<span class="id" title="keyword">Context</span> { <span class="id" title="var">valid_rnd</span> : <a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#Valid_rnd"><span class="id" title="class">Valid_rnd</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#Fcalc_round.Fcalc_round_fexp.round_dir_sign.rnd"><span class="id" title="variable">rnd</span></a> }.<br/>
<br/>
<span class="id" title="keyword">Variable</span> <a name="Fcalc_round.Fcalc_round_fexp.round_dir_sign.choice"><span class="id" title="variable">choice</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.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="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.Calc.Fcalc_bracket.html#location"><span class="id" title="inductive">location</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>.<br/>
<span class="id" title="keyword">Hypothesis</span> <a name="Fcalc_round.Fcalc_round_fexp.round_dir_sign.inbetween_int_valid"><span class="id" title="variable">inbetween_int_valid</span></a> :<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">m</span> <span class="id" title="var">l</span>,<br/>
<a class="idref" href="compcert.flocq.Calc.Fcalc_bracket.html#inbetween_int"><span class="id" title="definition">inbetween_int</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Reals.Rbasic_fun.html#Rabs"><span class="id" title="definition">Rabs</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#l"><span class="id" title="variable">l</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.Calc.Fcalc_round.html#Fcalc_round.Fcalc_round_fexp.round_dir_sign.rnd"><span class="id" title="variable">rnd</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.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.Core.Fcore_Zaux.html#cond_Zopp"><span class="id" title="definition">cond_Zopp</span></a> (<a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Rlt_bool"><span class="id" title="definition">Rlt_bool</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#x"><span class="id" title="variable">x</span></a> 0) (<a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#Fcalc_round.Fcalc_round_fexp.round_dir_sign.choice"><span class="id" title="variable">choice</span></a> (<a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Rlt_bool"><span class="id" title="definition">Rlt_bool</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#x"><span class="id" title="variable">x</span></a> 0) <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#l"><span class="id" title="variable">l</span></a>).<br/>
<br/>
<span class="id" title="keyword">Theorem</span> <a name="round_sign_any_correct"><span class="id" title="lemma">round_sign_any_correct</span></a> :<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">m</span> <span class="id" title="var">e</span> <span class="id" title="var">l</span>,<br/>
<a class="idref" href="compcert.flocq.Calc.Fcalc_bracket.html#inbetween_float"><span class="id" title="definition">inbetween_float</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#e"><span class="id" title="variable">e</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Reals.Rbasic_fun.html#Rabs"><span class="id" title="definition">Rabs</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#l"><span class="id" title="variable">l</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="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.Calc.Fcalc_round.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="compcert.flocq.Core.Fcore_generic_fmt.html#canonic_exp"><span class="id" title="definition">canonic_exp</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#Fcalc_round.Fcalc_round_fexp.fexp"><span class="id" title="variable">fexp</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.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#7a45dffb109c3069e5c675be68643e60"><span class="id" title="notation">∨</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#7a45dffb109c3069e5c675be68643e60"><span class="id" title="notation">(</span></a><a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#l"><span class="id" title="variable">l</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.Calc.Fcalc_bracket.html#loc_Exact"><span class="id" title="constructor">loc_Exact</span></a> <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.Calc.Fcalc_round.html#format"><span class="id" title="abbreviation">format</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.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#7a45dffb109c3069e5c675be68643e60"><span class="id" title="notation">)</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.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/>
<a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#round"><span class="id" title="definition">round</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#Fcalc_round.Fcalc_round_fexp.fexp"><span class="id" title="variable">fexp</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#Fcalc_round.Fcalc_round_fexp.round_dir_sign.rnd"><span class="id" title="variable">rnd</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.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.Core.Fcore_defs.html#F2R"><span class="id" title="definition">F2R</span></a> (<a class="idref" href="compcert.flocq.Core.Fcore_defs.html#Float"><span class="id" title="constructor">Float</span></a> <span class="id" title="keyword">beta</span> (<a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#cond_Zopp"><span class="id" title="definition">cond_Zopp</span></a> (<a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Rlt_bool"><span class="id" title="definition">Rlt_bool</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#x"><span class="id" title="variable">x</span></a> 0) (<a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#Fcalc_round.Fcalc_round_fexp.round_dir_sign.choice"><span class="id" title="variable">choice</span></a> (<a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Rlt_bool"><span class="id" title="definition">Rlt_bool</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#x"><span class="id" title="variable">x</span></a> 0) <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#m"><span class="id" title="variable">m</span></a> <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#l"><span class="id" title="variable">l</span></a>)) <a class="idref" href="compcert.flocq.Calc.Fcalc_round.html#e"><span class="id" title="variable">e</span></a>).<br/>
<span class="id" title="keyword">Proof</span> <span class="id" title="keyword">with</span> <span class="id" title="tactic">auto</span> <span class="id" title="keyword">with</span> <span class="id" title="var">typeclass_instances</span>.<br/>