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<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<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.Core.Fcore_rnd_ne</title>
</head>
<body>
<div id="page">
<div id="header">
</div>
<div id="main">
<h1 class="libtitle">Library compcert.flocq.Core.Fcore_rnd_ne</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="lab26"></a><h1 class="section">Rounding to nearest, ties to even: existence, unicity...</h1>
</div>
<div class="code">
<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#"><span class="id" title="library">Fcore_Raux</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_defs.html#"><span class="id" title="library">Fcore_defs</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_rnd.html#"><span class="id" title="library">Fcore_rnd</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_generic_fmt.html#"><span class="id" title="library">Fcore_generic_fmt</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_float_prop.html#"><span class="id" title="library">Fcore_float_prop</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_ulp.html#"><span class="id" title="library">Fcore_ulp</span></a>.<br/>
<br/>
<span class="id" title="keyword">Notation</span> <a name="ZnearestE"><span class="id" title="abbreviation">ZnearestE</span></a> := (<a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#Znearest"><span class="id" title="definition">Znearest</span></a> (<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.Core.Fcore_rnd_ne.html#x"><span class="id" title="variable">x</span></a>))).<br/>
<br/>
<span class="id" title="keyword">Section</span> <a name="Fcore_rnd_NE"><span class="id" title="section">Fcore_rnd_NE</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/>
<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">Variable</span> <a name="Fcore_rnd_NE.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/>
<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.Core.Fcore_rnd_ne.html#Fcore_rnd_NE.fexp"><span class="id" title="variable">fexp</span></a> }.<br/>
<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.Core.Fcore_rnd_ne.html#Fcore_rnd_NE.fexp"><span class="id" title="variable">fexp</span></a>).<br/>
<span class="id" title="keyword">Notation</span> <a name="canonic"><span class="id" title="abbreviation">canonic</span></a> := (<a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#canonic"><span class="id" title="definition">canonic</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#Fcore_rnd_NE.fexp"><span class="id" title="variable">fexp</span></a>).<br/>
<br/>
<span class="id" title="keyword">Definition</span> <a name="NE_prop"><span class="id" title="definition">NE_prop</span></a> (<span class="id" title="var">_</span> : <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>) <span class="id" title="var">f</span> :=<br/>
<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">g</span> : <a class="idref" href="compcert.flocq.Core.Fcore_defs.html#float"><span class="id" title="record">float</span></a> <span class="id" title="keyword">beta</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.Core.Fcore_rnd_ne.html#f"><span class="id" title="variable">f</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_rnd_ne.html#g"><span class="id" title="variable">g</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.Core.Fcore_rnd_ne.html#canonic"><span class="id" title="abbreviation">canonic</span></a> <a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#g"><span class="id" title="variable">g</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.Core.Fcore_Zaux.html#Zeven"><span class="id" title="definition">Zeven</span></a> (<a class="idref" href="compcert.flocq.Core.Fcore_defs.html#Fnum"><span class="id" title="projection">Fnum</span></a> <a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#g"><span class="id" title="variable">g</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Datatypes.html#true"><span class="id" title="constructor">true</span></a>.<br/>
<br/>
<span class="id" title="keyword">Definition</span> <a name="Rnd_NE_pt"><span class="id" title="definition">Rnd_NE_pt</span></a> :=<br/>
<a class="idref" href="compcert.flocq.Core.Fcore_defs.html#Rnd_NG_pt"><span class="id" title="definition">Rnd_NG_pt</span></a> <a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#format"><span class="id" title="abbreviation">format</span></a> <a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#NE_prop"><span class="id" title="definition">NE_prop</span></a>.<br/>
<br/>
<span class="id" title="keyword">Definition</span> <a name="DN_UP_parity_pos_prop"><span class="id" title="definition">DN_UP_parity_pos_prop</span></a> :=<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">xd</span> <span class="id" title="var">xu</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.Core.Fcore_rnd_ne.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="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#611abc97cba304de784fa909dbdea1fa"><span class="id" title="notation">¬</span></a> <a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#format"><span class="id" title="abbreviation">format</span></a> <a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.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.Core.Fcore_rnd_ne.html#canonic"><span class="id" title="abbreviation">canonic</span></a> <a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#xd"><span class="id" title="variable">xd</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#canonic"><span class="id" title="abbreviation">canonic</span></a> <a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#xu"><span class="id" title="variable">xu</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_defs.html#F2R"><span class="id" title="definition">F2R</span></a> <a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#xd"><span class="id" title="variable">xd</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#round"><span class="id" title="definition">round</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#Fcore_rnd_NE.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.Core.Fcore_rnd_ne.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.Core.Fcore_defs.html#F2R"><span class="id" title="definition">F2R</span></a> <a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#xu"><span class="id" title="variable">xu</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#round"><span class="id" title="definition">round</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#Fcore_rnd_NE.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.Core.Fcore_rnd_ne.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.Core.Fcore_Zaux.html#Zeven"><span class="id" title="definition">Zeven</span></a> (<a class="idref" href="compcert.flocq.Core.Fcore_defs.html#Fnum"><span class="id" title="projection">Fnum</span></a> <a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#xu"><span class="id" title="variable">xu</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#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.Core.Fcore_defs.html#Fnum"><span class="id" title="projection">Fnum</span></a> <a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#xd"><span class="id" title="variable">xd</span></a>)).<br/>
<br/>
<span class="id" title="keyword">Definition</span> <a name="DN_UP_parity_prop"><span class="id" title="definition">DN_UP_parity_prop</span></a> :=<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">xd</span> <span class="id" title="var">xu</span>,<br/>
<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#611abc97cba304de784fa909dbdea1fa"><span class="id" title="notation">¬</span></a> <a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#format"><span class="id" title="abbreviation">format</span></a> <a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.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.Core.Fcore_rnd_ne.html#canonic"><span class="id" title="abbreviation">canonic</span></a> <a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#xd"><span class="id" title="variable">xd</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#canonic"><span class="id" title="abbreviation">canonic</span></a> <a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#xu"><span class="id" title="variable">xu</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_defs.html#F2R"><span class="id" title="definition">F2R</span></a> <a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#xd"><span class="id" title="variable">xd</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#round"><span class="id" title="definition">round</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#Fcore_rnd_NE.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.Core.Fcore_rnd_ne.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.Core.Fcore_defs.html#F2R"><span class="id" title="definition">F2R</span></a> <a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#xu"><span class="id" title="variable">xu</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#round"><span class="id" title="definition">round</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#Fcore_rnd_NE.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.Core.Fcore_rnd_ne.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.Core.Fcore_Zaux.html#Zeven"><span class="id" title="definition">Zeven</span></a> (<a class="idref" href="compcert.flocq.Core.Fcore_defs.html#Fnum"><span class="id" title="projection">Fnum</span></a> <a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#xu"><span class="id" title="variable">xu</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#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.Core.Fcore_defs.html#Fnum"><span class="id" title="projection">Fnum</span></a> <a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#xd"><span class="id" title="variable">xd</span></a>)).<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="DN_UP_parity_aux"><span class="id" title="lemma">DN_UP_parity_aux</span></a> :<br/>
<a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#DN_UP_parity_pos_prop"><span class="id" title="definition">DN_UP_parity_pos_prop</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#DN_UP_parity_prop"><span class="id" title="definition">DN_UP_parity_prop</span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">Hpos</span> <span class="id" title="var">x</span> <span class="id" title="var">xd</span> <span class="id" title="var">xu</span> <span class="id" title="var">Hfx</span> <span class="id" title="var">Hd</span> <span class="id" title="var">Hu</span> <span class="id" title="var">Hxd</span> <span class="id" title="var">Hxu</span>.<br/>
<span class="id" title="tactic">destruct</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Reals.Raxioms.html#total_order_T"><span class="id" title="axiom">total_order_T</span></a> 0 <span class="id" title="var">x</span>) <span class="id" title="keyword">as</span> [[<span class="id" title="var">Hx</span>|<span class="id" title="var">Hx</span>]|<span class="id" title="var">Hx</span>].<br/>
<span class="id" title="tactic">exact</span> (<span class="id" title="var">Hpos</span> <span class="id" title="var">x</span> <span class="id" title="var">xd</span> <span class="id" title="var">xu</span> <span class="id" title="var">Hx</span> <span class="id" title="var">Hfx</span> <span class="id" title="var">Hd</span> <span class="id" title="var">Hu</span> <span class="id" title="var">Hxd</span> <span class="id" title="var">Hxu</span>).<br/>
<span class="id" title="tactic">elim</span> <span class="id" title="var">Hfx</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#generic_format_0"><span class="id" title="lemma">generic_format_0</span></a>.<br/>
<span class="id" title="tactic">assert</span> (<span class="id" title="var">Hx'</span>: (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="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>)%<span class="id" title="var">R</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#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>, <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Reals.RIneq.html#Ropp_0"><span class="id" title="lemma">Ropp_0</span></a>.<br/>
<span class="id" title="tactic">destruct</span> <span class="id" title="var">xd</span> <span class="id" title="keyword">as</span> (<span class="id" title="var">md</span>, <span class="id" title="var">ed</span>).<br/>
<span class="id" title="tactic">destruct</span> <span class="id" title="var">xu</span> <span class="id" title="keyword">as</span> (<span class="id" title="var">mu</span>, <span class="id" title="var">eu</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.Bool.Bool.html#negb_involutive"><span class="id" title="lemma">Bool.negb_involutive</span></a> (<a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#Zeven"><span class="id" title="definition">Zeven</span></a> <span class="id" title="var">mu</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">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#sym_eq"><span class="id" title="abbreviation">sym_eq</span></a>.<br/>
<span class="id" title="tactic">rewrite</span> <- (<a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#Zeven_opp"><span class="id" title="lemma">Zeven_opp</span></a> <span class="id" title="var">mu</span>), <- (<a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#Zeven_opp"><span class="id" title="lemma">Zeven_opp</span></a> <span class="id" title="var">md</span>).<br/>
<span class="id" title="tactic">change</span> (<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.Core.Fcore_defs.html#Fnum"><span class="id" title="projection">Fnum</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="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">md</span>) <span class="id" title="var">ed</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#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.Core.Fcore_defs.html#Fnum"><span class="id" title="projection">Fnum</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="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">mu</span>) <span class="id" title="var">eu</span>)))).<br/>
<span class="id" title="tactic">apply</span> (<span class="id" title="var">Hpos</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>)%<span class="id" title="var">R</span> <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">intros</span> <span class="id" title="var">H</span>.<br/>
<span class="id" title="tactic">apply</span> <span class="id" title="var">Hfx</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>.<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#generic_format_opp"><span class="id" title="lemma">generic_format_opp</span></a>.<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#canonic_opp"><span class="id" title="lemma">canonic_opp</span></a>.<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#canonic_opp"><span class="id" title="lemma">canonic_opp</span></a>.<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#round_DN_opp"><span class="id" title="lemma">round_DN_opp</span></a>, <a class="idref" href="compcert.flocq.Core.Fcore_float_prop.html#F2R_Zopp"><span class="id" title="lemma">F2R_Zopp</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.Init.Logic.html#f_equal"><span class="id" title="lemma">f_equal</span></a>.<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#round_UP_opp"><span class="id" title="lemma">round_UP_opp</span></a>, <a class="idref" href="compcert.flocq.Core.Fcore_float_prop.html#F2R_Zopp"><span class="id" title="lemma">F2R_Zopp</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.Init.Logic.html#f_equal"><span class="id" title="lemma">f_equal</span></a>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Class</span> <a name="Exists_NE"><span class="id" title="record">Exists_NE</span></a> :=<br/>
<a name="exists_NE"><span class="id" title="projection">exists_NE</span></a> : <a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#Zeven"><span class="id" title="definition">Zeven</span></a> <span class="id" title="keyword">beta</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#false"><span class="id" title="constructor">false</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="keyword">∀</span> <span class="id" title="var">e</span>,<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>(<a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#Fcore_rnd_NE.fexp"><span class="id" title="variable">fexp</span></a> <a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.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#273beba4f3453dbb29192b3ac740bb71"><span class="id" title="notation"><</span></a> <a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.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> (<a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#Fcore_rnd_NE.fexp"><span class="id" title="variable">fexp</span></a> (<a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.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#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#273beba4f3453dbb29192b3ac740bb71"><span class="id" title="notation"><</span></a> <a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.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#d82a7d96d3659d805ffe732283716822"><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> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">(</span></a>(<a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.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.Core.Fcore_rnd_ne.html#Fcore_rnd_NE.fexp"><span class="id" title="variable">fexp</span></a> <a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.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> <a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#Fcore_rnd_NE.fexp"><span class="id" title="variable">fexp</span></a> (<a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#Fcore_rnd_NE.fexp"><span class="id" title="variable">fexp</span></a> <a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.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#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a> 1) <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_rnd_ne.html#Fcore_rnd_NE.fexp"><span class="id" title="variable">fexp</span></a> <a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.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#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">)</span></a>.<br/>
<br/>
<span class="id" title="keyword">Context</span> { <span class="id" title="var">exists_NE_</span> : <a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#Exists_NE"><span class="id" title="class">Exists_NE</span></a> }.<br/>
<br/>
<span class="id" title="keyword">Theorem</span> <a name="DN_UP_parity_generic_pos"><span class="id" title="lemma">DN_UP_parity_generic_pos</span></a> :<br/>
<a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#DN_UP_parity_pos_prop"><span class="id" title="definition">DN_UP_parity_pos_prop</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">xd</span> <span class="id" title="var">xu</span> <span class="id" title="var">H0x</span> <span class="id" title="var">Hfx</span> <span class="id" title="var">Hd</span> <span class="id" title="var">Hu</span> <span class="id" title="var">Hxd</span> <span class="id" title="var">Hxu</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">Hexa</span>).<br/>
<span class="id" title="tactic">specialize</span> (<span class="id" title="var">Hexa</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">H0x</span>)).<br/>
<span class="id" title="tactic">generalize</span> <span class="id" title="var">Hexa</span>. <span class="id" title="tactic">intros</span> <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> <span class="id" title="var">_</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> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">H0x</span>)) <span class="id" title="keyword">in</span> <span class="id" title="var">Hex</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">ex</span> (<a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#Fcore_rnd_NE.fexp"><span class="id" title="variable">fexp</span></a> <span class="id" title="var">ex</span>)) <span class="id" title="keyword">as</span> [<span class="id" title="var">Hxe</span>|<span class="id" title="var">Hxe</span>].<br/>
<span class="id" title="tactic">assert</span> (<span class="id" title="var">Hd3</span> : <a class="idref" href="compcert.flocq.Core.Fcore_defs.html#Fnum"><span class="id" title="projection">Fnum</span></a> <span class="id" title="var">xd</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">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_float_prop.html#F2R_eq_0_reg"><span class="id" title="lemma">F2R_eq_0_reg</span></a> <span class="id" title="keyword">with</span> <span class="id" title="keyword">beta</span> (<a class="idref" href="compcert.flocq.Core.Fcore_defs.html#Fexp"><span class="id" title="projection">Fexp</span></a> <span class="id" title="var">xd</span>).<br/>
<span class="id" title="tactic">change</span> (<a class="idref" href="compcert.flocq.Core.Fcore_defs.html#F2R"><span class="id" title="definition">F2R</span></a> <span class="id" title="var">xd</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.Reals.Rdefinitions.html#R0"><span class="id" title="axiom">R0</span></a>).<br/>
<span class="id" title="tactic">rewrite</span> <span class="id" title="var">Hxd</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#round_DN_small_pos"><span class="id" title="lemma">round_DN_small_pos</span></a> <span class="id" title="keyword">with</span> (1 := <span class="id" title="var">Hex</span>) (2 := <span class="id" title="var">Hxe</span>).<br/>
<span class="id" title="tactic">assert</span> (<span class="id" title="var">Hu3</span> : <span class="id" title="var">xu</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_defs.html#Float"><span class="id" title="constructor">Float</span></a> <span class="id" title="keyword">beta</span> (1 <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#88e34ed7a73f02772126289867960764"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.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.Core.Fcore_rnd_ne.html#Fcore_rnd_NE.fexp"><span class="id" title="variable">fexp</span></a> <span class="id" title="var">ex</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#239a40728d107295b3cb2c790f57f9e9"><span class="id" title="notation">-</span></a> <a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#Fcore_rnd_NE.fexp"><span class="id" title="variable">fexp</span></a> (<a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#Fcore_rnd_NE.fexp"><span class="id" title="variable">fexp</span></a> <span class="id" title="var">ex</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a> 1))) (<a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#Fcore_rnd_NE.fexp"><span class="id" title="variable">fexp</span></a> (<a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#Fcore_rnd_NE.fexp"><span class="id" title="variable">fexp</span></a> <span class="id" title="var">ex</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a> 1))).<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#canonic_unicity"><span class="id" title="lemma">canonic_unicity</span></a> <span class="id" title="keyword">with</span> (1 := <span class="id" title="var">Hu</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> <a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#Fcore_rnd_NE.fexp"><span class="id" title="variable">fexp</span></a>).<br/>
<span class="id" title="tactic">rewrite</span> <- <a class="idref" href="compcert.flocq.Core.Fcore_float_prop.html#F2R_change_exp"><span class="id" title="lemma">F2R_change_exp</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_float_prop.html#F2R_bpow"><span class="id" title="lemma">F2R_bpow</span></a>, <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#ln_beta_bpow"><span class="id" title="lemma">ln_beta_bpow</span></a>.<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#Fcore_rnd_NE.valid_exp"><span class="id" title="variable">valid_exp</span></a>.<br/>
<span class="id" title="tactic">rewrite</span> <- <a class="idref" href="compcert.flocq.Core.Fcore_float_prop.html#F2R_change_exp"><span class="id" title="lemma">F2R_change_exp</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">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#sym_eq"><span class="id" title="abbreviation">sym_eq</span></a>.<br/>
<span class="id" title="tactic">rewrite</span> <span class="id" title="var">Hxu</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_generic_fmt.html#round_UP_small_pos"><span class="id" title="lemma">round_UP_small_pos</span></a> <span class="id" title="keyword">with</span> (1 := <span class="id" title="var">Hex</span>) (2 := <span class="id" title="var">Hxe</span>).<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#Fcore_rnd_NE.valid_exp"><span class="id" title="variable">valid_exp</span></a>.<br/>
<span class="id" title="tactic">rewrite</span> <span class="id" title="var">Hd3</span>, <span class="id" title="var">Hu3</span>.<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#Zmult_1_l"><span class="id" title="abbreviation">Zmult_1_l</span></a>.<br/>
<span class="id" title="tactic">simpl</span>.<br/>
<span class="id" title="tactic">destruct</span> <a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#Fcore_rnd_NE.exists_NE_"><span class="id" title="variable">exists_NE_</span></a> <span class="id" title="keyword">as</span> [<span class="id" title="var">H</span>|<span class="id" title="var">H</span>].<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#Zeven_Zpower_odd"><span class="id" title="lemma">Zeven_Zpower_odd</span></a> <span class="id" title="keyword">with</span> (2 := <span class="id" title="var">H</span>).<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zorder.html#Zle_minus_le_0"><span class="id" title="lemma">Zle_minus_le_0</span></a>.<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#Fcore_rnd_NE.valid_exp"><span class="id" title="variable">valid_exp</span></a>.<br/>
<span class="id" title="tactic">rewrite</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">H</span> <span class="id" title="var">ex</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#Zminus_diag"><span class="id" title="abbreviation">Zminus_diag</span></a>.<br/>
<span class="id" title="tactic">exact</span> <span class="id" title="var">Hxe</span>.<br/>
<span class="id" title="tactic">assert</span> (<span class="id" title="var">Hd4</span>: (<a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#bpow"><span class="id" title="abbreviation">bpow</span></a> (<span class="id" title="var">ex</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#239a40728d107295b3cb2c790f57f9e9"><span class="id" title="notation">-</span></a> 1) <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="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.Core.Fcore_defs.html#F2R"><span class="id" title="definition">F2R</span></a> <span class="id" title="var">xd</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_rnd_ne.html#bpow"><span class="id" title="abbreviation">bpow</span></a> <span class="id" title="var">ex</span>)%<span class="id" title="var">R</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> <span class="id" title="var">Hxd</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_generic_fmt.html#round_DN_pt"><span class="id" title="lemma">round_DN_pt</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#Fcore_rnd_NE.fexp"><span class="id" title="variable">fexp</span></a> <span class="id" title="var">x</span>).<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#generic_format_bpow"><span class="id" title="lemma">generic_format_bpow</span></a>.<br/>
<span class="id" title="var">ring_simplify</span> (<span class="id" title="var">ex</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#239a40728d107295b3cb2c790f57f9e9"><span class="id" title="notation">-</span></a> 1 <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a> 1)%<span class="id" title="var">Z</span>.<br/>
<span class="id" title="tactic">omega</span>.<br/>
<span class="id" title="tactic">apply</span> <span class="id" title="var">Hex</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#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#proj2"><span class="id" title="lemma">proj2</span></a> <span class="id" title="var">Hex</span>).<br/>
<span class="id" title="tactic">apply</span> (<a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#round_DN_pt"><span class="id" title="lemma">round_DN_pt</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#Fcore_rnd_NE.fexp"><span class="id" title="variable">fexp</span></a> <span class="id" title="var">x</span>).<br/>
<span class="id" title="tactic">rewrite</span> <span class="id" title="var">Hxd</span>.<br/>
<span class="id" title="tactic">apply</span> (<a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#round_DN_pt"><span class="id" title="lemma">round_DN_pt</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#Fcore_rnd_NE.fexp"><span class="id" title="variable">fexp</span></a> <span class="id" title="var">x</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="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">assert</span> (<span class="id" title="var">Hxe2</span> : (<a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#Fcore_rnd_NE.fexp"><span class="id" title="variable">fexp</span></a> (<span class="id" title="var">ex</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a> 1) <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#208bafb0e148fe7fb7dcd812c227f4ee"><span class="id" title="notation">≤</span></a> <span class="id" title="var">ex</span>)%<span class="id" title="var">Z</span>) <span class="id" title="tactic">by</span> <span class="id" title="var">now</span> <span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#Fcore_rnd_NE.valid_exp"><span class="id" title="variable">valid_exp</span></a>.<br/>
<span class="id" title="tactic">assert</span> (<span class="id" title="var">Hud</span>: (<a class="idref" href="compcert.flocq.Core.Fcore_defs.html#F2R"><span class="id" title="definition">F2R</span></a> <span class="id" title="var">xu</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_defs.html#F2R"><span class="id" title="definition">F2R</span></a> <span class="id" title="var">xd</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Reals.Rdefinitions.html#7de2d195108bdf12ab090711a555a032"><span class="id" title="notation">+</span></a> <a class="idref" href="compcert.flocq.Core.Fcore_ulp.html#ulp"><span class="id" title="definition">ulp</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#Fcore_rnd_NE.fexp"><span class="id" title="variable">fexp</span></a> <span class="id" title="var">x</span>)%<span class="id" title="var">R</span>).<br/>
<span class="id" title="tactic">rewrite</span> <span class="id" title="var">Hxu</span>, <span class="id" title="var">Hxd</span>.<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_ulp.html#round_UP_DN_ulp"><span class="id" title="lemma">round_UP_DN_ulp</span></a>.<br/>
<span class="id" title="tactic">destruct</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Reals.Raxioms.html#total_order_T"><span class="id" title="axiom">total_order_T</span></a> (<a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#bpow"><span class="id" title="abbreviation">bpow</span></a> <span class="id" title="var">ex</span>) (<a class="idref" href="compcert.flocq.Core.Fcore_defs.html#F2R"><span class="id" title="definition">F2R</span></a> <span class="id" title="var">xu</span>)) <span class="id" title="keyword">as</span> [[<span class="id" title="var">Hu2</span>|<span class="id" title="var">Hu2</span>]|<span class="id" title="var">Hu2</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_not_le"><span class="id" title="lemma">Rlt_not_le</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">Hu2</span>).<br/>
<span class="id" title="tactic">rewrite</span> <span class="id" title="var">Hxu</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#round_bounded_large_pos"><span class="id" title="lemma">round_bounded_large_pos</span></a>...<br/>
<span class="id" title="tactic">assert</span> (<span class="id" title="var">Hu3</span>: <span class="id" title="var">xu</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_defs.html#Float"><span class="id" title="constructor">Float</span></a> <span class="id" title="keyword">beta</span> (1 <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#88e34ed7a73f02772126289867960764"><span class="id" title="notation">×</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zpow_def.html#Zpower"><span class="id" title="abbreviation">Zpower</span></a> <span class="id" title="keyword">beta</span> (<span class="id" title="var">ex</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#239a40728d107295b3cb2c790f57f9e9"><span class="id" title="notation">-</span></a> <a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#Fcore_rnd_NE.fexp"><span class="id" title="variable">fexp</span></a> (<span class="id" title="var">ex</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a> 1))) (<a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#Fcore_rnd_NE.fexp"><span class="id" title="variable">fexp</span></a> (<span class="id" title="var">ex</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a> 1))).<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#canonic_unicity"><span class="id" title="lemma">canonic_unicity</span></a> <span class="id" title="keyword">with</span> (1 := <span class="id" title="var">Hu</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> <a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#Fcore_rnd_NE.fexp"><span class="id" title="variable">fexp</span></a>).<br/>
<span class="id" title="tactic">rewrite</span> <- <a class="idref" href="compcert.flocq.Core.Fcore_float_prop.html#F2R_change_exp"><span class="id" title="lemma">F2R_change_exp</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_float_prop.html#F2R_bpow"><span class="id" title="lemma">F2R_bpow</span></a>, <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#ln_beta_bpow"><span class="id" title="lemma">ln_beta_bpow</span></a>.<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#Fcore_rnd_NE.valid_exp"><span class="id" title="variable">valid_exp</span></a>.<br/>
<span class="id" title="tactic">rewrite</span> <- <span class="id" title="var">Hu2</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#sym_eq"><span class="id" title="abbreviation">sym_eq</span></a>.<br/>
<span class="id" title="tactic">rewrite</span> <- <a class="idref" href="compcert.flocq.Core.Fcore_float_prop.html#F2R_change_exp"><span class="id" title="lemma">F2R_change_exp</span></a>.<br/>
<span class="id" title="tactic">apply</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">exact</span> <span class="id" title="var">Hxe2</span>.<br/>
<span class="id" title="tactic">assert</span> (<span class="id" title="var">Hd3</span>: <span class="id" title="var">xd</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_defs.html#Float"><span class="id" title="constructor">Float</span></a> <span class="id" title="keyword">beta</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">ex</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#239a40728d107295b3cb2c790f57f9e9"><span class="id" title="notation">-</span></a> <a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#Fcore_rnd_NE.fexp"><span class="id" title="variable">fexp</span></a> <span class="id" title="var">ex</span>) <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#239a40728d107295b3cb2c790f57f9e9"><span class="id" title="notation">-</span></a> 1) (<a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#Fcore_rnd_NE.fexp"><span class="id" title="variable">fexp</span></a> <span class="id" title="var">ex</span>)).<br/>
<span class="id" title="tactic">assert</span> (<span class="id" title="var">H</span>: <a class="idref" href="compcert.flocq.Core.Fcore_defs.html#F2R"><span class="id" title="definition">F2R</span></a> <span class="id" title="var">xd</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_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="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">ex</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#239a40728d107295b3cb2c790f57f9e9"><span class="id" title="notation">-</span></a> <a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#Fcore_rnd_NE.fexp"><span class="id" title="variable">fexp</span></a> <span class="id" title="var">ex</span>) <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#239a40728d107295b3cb2c790f57f9e9"><span class="id" title="notation">-</span></a> 1) (<a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#Fcore_rnd_NE.fexp"><span class="id" title="variable">fexp</span></a> <span class="id" title="var">ex</span>))).<br/>
<span class="id" title="tactic">unfold</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="compcert.flocq.Core.Fcore_Raux.html#Z2R_minus"><span class="id" title="lemma">Z2R_minus</span></a>.<br/>
<span class="id" title="tactic">unfold</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Reals.Rdefinitions.html#Rminus"><span class="id" title="definition">Rminus</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_plus_distr_r"><span class="id" title="lemma">Rmult_plus_distr_r</span></a>.<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Z2R_Zpower"><span class="id" title="lemma">Z2R_Zpower</span></a>, <- <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">ex</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#239a40728d107295b3cb2c790f57f9e9"><span class="id" title="notation">-</span></a> <a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#Fcore_rnd_NE.fexp"><span class="id" title="variable">fexp</span></a> <span class="id" title="var">ex</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a> <a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#Fcore_rnd_NE.fexp"><span class="id" title="variable">fexp</span></a> <span class="id" title="var">ex</span>)%<span class="id" title="var">Z</span>.<br/>
<span class="id" title="tactic">rewrite</span> <span class="id" title="var">Hu2</span>, <span class="id" title="var">Hud</span>.<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.flocq.Core.Fcore_ulp.html#ulp_neq_0"><span class="id" title="lemma">ulp_neq_0</span></a>;[<span class="id" title="tactic">idtac</span>|<span class="id" title="var">now</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Reals.RIneq.html#Rgt_not_eq"><span class="id" title="lemma">Rgt_not_eq</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.Core.Fcore_Raux.html#ln_beta_unique"><span class="id" title="lemma">ln_beta_unique</span></a> <span class="id" title="keyword">with</span> <span class="id" title="keyword">beta</span> <span class="id" title="var">x</span> <span class="id" title="var">ex</span>.<br/>
<span class="id" title="tactic">unfold</span> <a class="idref" href="compcert.flocq.Core.Fcore_defs.html#F2R"><span class="id" title="definition">F2R</span></a>.<br/>
<span class="id" title="tactic">simpl</span>. <span class="id" title="tactic">ring</span>.<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Reals.Rbasic_fun.html#Rabs_pos_eq"><span class="id" title="lemma">Rabs_pos_eq</span></a>.<br/>
<span class="id" title="tactic">exact</span> <span class="id" title="var">Hex</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> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zorder.html#Zle_minus_le_0"><span class="id" title="lemma">Zle_minus_le_0</span></a>.<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.Zorder.html#Zlt_le_weak"><span class="id" title="abbreviation">Zlt_le_weak</span></a>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#canonic_unicity"><span class="id" title="lemma">canonic_unicity</span></a> <span class="id" title="keyword">with</span> (1 := <span class="id" title="var">Hd</span>) (3 := <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#f_equal"><span class="id" title="lemma">f_equal</span></a> <a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#Fcore_rnd_NE.fexp"><span class="id" title="variable">fexp</span></a>).<br/>
<span class="id" title="tactic">rewrite</span> <- <span class="id" title="var">H</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#sym_eq"><span class="id" title="abbreviation">sym_eq</span></a>.<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#ln_beta_unique"><span class="id" title="lemma">ln_beta_unique</span></a>.<br/>
<span class="id" title="tactic">rewrite</span> <span class="id" title="var">Hd3</span>, <span class="id" title="var">Hu3</span>.<br/>
<span class="id" title="tactic">unfold</span> <a class="idref" href="compcert.flocq.Core.Fcore_defs.html#Fnum"><span class="id" title="projection">Fnum</span></a>.<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#Zeven_mult"><span class="id" title="lemma">Zeven_mult</span></a>. <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.ZArith.BinInt.html#Z.sub"><span class="id" title="definition">Zminus</span></a> <span class="id" title="tactic">at</span> 2.<br/>
<span class="id" title="tactic">rewrite</span> <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="tactic">rewrite</span> <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#eqb_sym"><span class="id" title="lemma">eqb_sym</span></a>. <span class="id" title="tactic">simpl</span>.<br/>
<span class="id" title="tactic">fold</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> (<span class="id" title="keyword">beta</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#186f1fc9207177b1a4e8373ac9655977"><span class="id" title="notation">^</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#186f1fc9207177b1a4e8373ac9655977"><span class="id" title="notation">(</span></a><span class="id" title="var">ex</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#239a40728d107295b3cb2c790f57f9e9"><span class="id" title="notation">-</span></a> <a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#Fcore_rnd_NE.fexp"><span class="id" title="variable">fexp</span></a> <span class="id" title="var">ex</span><a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#186f1fc9207177b1a4e8373ac9655977"><span class="id" title="notation">)</span></a>))).<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Bool.Bool.html#negb_involutive"><span class="id" title="lemma">Bool.negb_involutive</span></a>.<br/>
<span class="id" title="tactic">rewrite</span> (<a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#Zeven_Zpower"><span class="id" title="lemma">Zeven_Zpower</span></a> <span class="id" title="keyword">beta</span> (<span class="id" title="var">ex</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#239a40728d107295b3cb2c790f57f9e9"><span class="id" title="notation">-</span></a> <a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#Fcore_rnd_NE.fexp"><span class="id" title="variable">fexp</span></a> <span class="id" title="var">ex</span>)). 2: <span class="id" title="tactic">omega</span>.<br/>
<span class="id" title="tactic">destruct</span> <a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#Fcore_rnd_NE.exists_NE_"><span class="id" title="variable">exists_NE_</span></a>.<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_Zaux.html#Zeven_Zpower_odd"><span class="id" title="lemma">Zeven_Zpower_odd</span></a> <span class="id" title="keyword">with</span> (2 := <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.ZArith.Zorder.html#Zle_minus_le_0"><span class="id" title="lemma">Zle_minus_le_0</span></a>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#Zeven_Zpower"><span class="id" title="lemma">Zeven_Zpower</span></a>.<br/>
<span class="id" title="tactic">specialize</span> (<span class="id" title="var">H</span> <span class="id" title="var">ex</span>).<br/>
<span class="id" title="tactic">omega</span>.<br/>
<span class="id" title="var">revert</span> <span class="id" title="var">Hud</span>.<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.flocq.Core.Fcore_ulp.html#ulp_neq_0"><span class="id" title="lemma">ulp_neq_0</span></a>;[<span class="id" title="tactic">idtac</span>|<span class="id" title="var">now</span> <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Reals.RIneq.html#Rgt_not_eq"><span class="id" title="lemma">Rgt_not_eq</span></a>].<br/>
<span class="id" title="tactic">unfold</span> <a class="idref" href="compcert.flocq.Core.Fcore_defs.html#F2R"><span class="id" title="definition">F2R</span></a>.<br/>
<span class="id" title="tactic">rewrite</span> <span class="id" title="var">Hd</span>, <span class="id" title="var">Hu</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.Core.Fcore_Raux.html#ln_beta_unique"><span class="id" title="lemma">ln_beta_unique</span></a> <span class="id" title="keyword">with</span> <span class="id" title="keyword">beta</span> (<a class="idref" href="compcert.flocq.Core.Fcore_defs.html#F2R"><span class="id" title="definition">F2R</span></a> <span class="id" title="var">xu</span>) <span class="id" title="var">ex</span>.<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#ln_beta_unique"><span class="id" title="lemma">ln_beta_unique</span></a> <span class="id" title="keyword">with</span> (1 := <span class="id" title="var">Hd4</span>).<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#ln_beta_unique"><span class="id" title="lemma">ln_beta_unique</span></a> <span class="id" title="keyword">with</span> (1 := <span class="id" title="var">Hexa</span>).<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">H</span>.<br/>
<span class="id" title="tactic">replace</span> (<a class="idref" href="compcert.flocq.Core.Fcore_defs.html#Fnum"><span class="id" title="projection">Fnum</span></a> <span class="id" title="var">xu</span>) <span class="id" title="keyword">with</span> (<a class="idref" href="compcert.flocq.Core.Fcore_defs.html#Fnum"><span class="id" title="projection">Fnum</span></a> <span class="id" title="var">xd</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>.<br/>
<span class="id" title="tactic">rewrite</span> <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">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#eqb_sym"><span class="id" title="lemma">eqb_sym</span></a>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#sym_eq"><span class="id" title="abbreviation">sym_eq</span></a>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#eq_Z2R"><span class="id" title="lemma">eq_Z2R</span></a>.<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">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Rmult_eq_reg_r"><span class="id" title="lemma">Rmult_eq_reg_r</span></a> <span class="id" title="keyword">with</span> (<a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#bpow"><span class="id" title="abbreviation">bpow</span></a> (<a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#Fcore_rnd_NE.fexp"><span class="id" title="variable">fexp</span></a> <span class="id" title="var">ex</span>)).<br/>
<span class="id" title="tactic">rewrite</span> <span class="id" title="var">H</span>.<br/>
<span class="id" title="tactic">simpl</span>. <span class="id" title="tactic">ring</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Reals.RIneq.html#Rgt_not_eq"><span class="id" title="lemma">Rgt_not_eq</span></a>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#bpow_gt_0"><span class="id" title="lemma">bpow_gt_0</span></a>.<br/>
<span class="id" title="tactic">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">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#Rle_trans"><span class="id" title="lemma">Rle_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> <span class="id" title="var">Hxu</span>.<br/>
<span class="id" title="tactic">apply</span> (<a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#round_UP_pt"><span class="id" title="lemma">round_UP_pt</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#Fcore_rnd_NE.fexp"><span class="id" title="variable">fexp</span></a> <span class="id" title="var">x</span>).<br/>
<span class="id" title="tactic">exact</span> <span class="id" title="var">Hu2</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#Rlt_le_trans"><span class="id" title="lemma">Rlt_le_trans</span></a> <span class="id" title="keyword">with</span> (1 := <span class="id" title="var">H0x</span>).<br/>
<span class="id" title="tactic">rewrite</span> <span class="id" title="var">Hxu</span>.<br/>
<span class="id" title="tactic">apply</span> (<a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#round_UP_pt"><span class="id" title="lemma">round_UP_pt</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#Fcore_rnd_NE.fexp"><span class="id" title="variable">fexp</span></a> <span class="id" title="var">x</span>).<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Theorem</span> <a name="DN_UP_parity_generic"><span class="id" title="lemma">DN_UP_parity_generic</span></a> :<br/>
<a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#DN_UP_parity_prop"><span class="id" title="definition">DN_UP_parity_prop</span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#DN_UP_parity_aux"><span class="id" title="lemma">DN_UP_parity_aux</span></a>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#DN_UP_parity_generic_pos"><span class="id" title="lemma">DN_UP_parity_generic_pos</span></a>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Theorem</span> <a name="Rnd_NE_pt_total"><span class="id" title="lemma">Rnd_NE_pt_total</span></a> :<br/>
<a class="idref" href="compcert.flocq.Core.Fcore_defs.html#round_pred_total"><span class="id" title="definition">round_pred_total</span></a> <a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#Rnd_NE_pt"><span class="id" title="definition">Rnd_NE_pt</span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_rnd.html#satisfies_any_imp_NG"><span class="id" title="lemma">satisfies_any_imp_NG</span></a>.<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#generic_format_satisfies_any"><span class="id" title="lemma">generic_format_satisfies_any</span></a>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">x</span> <span class="id" title="var">d</span> <span class="id" title="var">u</span> <span class="id" title="var">Hf</span> <span class="id" title="var">Hd</span> <span class="id" title="var">Hu</span>.<br/>
<span class="id" title="tactic">generalize</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#proj1"><span class="id" title="lemma">proj1</span></a> <span class="id" title="var">Hd</span>).<br/>
<span class="id" title="tactic">unfold</span> <a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#generic_format"><span class="id" title="definition">generic_format</span></a>.<br/>
<span class="id" title="tactic">set</span> (<span class="id" title="var">ed</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.Core.Fcore_rnd_ne.html#Fcore_rnd_NE.fexp"><span class="id" title="variable">fexp</span></a> <span class="id" title="var">d</span>).<br/>
<span class="id" title="tactic">set</span> (<span class="id" title="var">md</span> := <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.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.Core.Fcore_rnd_ne.html#Fcore_rnd_NE.fexp"><span class="id" title="variable">fexp</span></a> <span class="id" title="var">d</span>)).<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">Hd1</span>.<br/>
<span class="id" title="var">case_eq</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">md</span>) ; [ <span class="id" title="tactic">intros</span> <span class="id" title="var">He</span> | <span class="id" title="tactic">intros</span> <span class="id" title="var">Ho</span> ].<br/>
<span class="id" title="tactic">right</span>.<br/>
<span class="id" title="tactic">∃</span> (<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">md</span> <span class="id" title="var">ed</span>).<br/>
<span class="id" title="tactic">unfold</span> <a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#canonic"><span class="id" title="definition">Fcore_generic_fmt.canonic</span></a>.<br/>
<span class="id" title="tactic">rewrite</span> <- <span class="id" title="var">Hd1</span>.<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">repeat</span> <span class="id" title="tactic">split</span>.<br/>
<span class="id" title="tactic">left</span>.<br/>
<span class="id" title="tactic">generalize</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#proj1"><span class="id" title="lemma">proj1</span></a> <span class="id" title="var">Hu</span>).<br/>
<span class="id" title="tactic">unfold</span> <a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#generic_format"><span class="id" title="definition">generic_format</span></a>.<br/>
<span class="id" title="tactic">set</span> (<span class="id" title="var">eu</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.Core.Fcore_rnd_ne.html#Fcore_rnd_NE.fexp"><span class="id" title="variable">fexp</span></a> <span class="id" title="var">u</span>).<br/>
<span class="id" title="tactic">set</span> (<span class="id" title="var">mu</span> := <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.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.Core.Fcore_rnd_ne.html#Fcore_rnd_NE.fexp"><span class="id" title="variable">fexp</span></a> <span class="id" title="var">u</span>)).<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">Hu1</span>.<br/>
<span class="id" title="tactic">rewrite</span> <span class="id" title="var">Hu1</span>.<br/>
<span class="id" title="tactic">eexists</span> ; <span class="id" title="tactic">repeat</span> <span class="id" title="tactic">split</span>.<br/>
<span class="id" title="tactic">unfold</span> <a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#canonic"><span class="id" title="definition">Fcore_generic_fmt.canonic</span></a>.<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">rewrite</span> <- <span class="id" title="var">Hu1</span>.<br/>
<span class="id" title="tactic">rewrite</span> (<a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#DN_UP_parity_generic"><span class="id" title="lemma">DN_UP_parity_generic</span></a> <span class="id" title="var">x</span> (<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">md</span> <span class="id" title="var">ed</span>) (<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">mu</span> <span class="id" title="var">eu</span>)).<br/>
<span class="id" title="tactic">simpl</span>.<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">rewrite</span> <span class="id" title="var">Ho</span>.<br/>
<span class="id" title="tactic">exact</span> <span class="id" title="var">Hf</span>.<br/>
<span class="id" title="tactic">unfold</span> <a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#canonic"><span class="id" title="definition">Fcore_generic_fmt.canonic</span></a>.<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">rewrite</span> <- <span class="id" title="var">Hd1</span>.<br/>
<span class="id" title="tactic">unfold</span> <a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#canonic"><span class="id" title="definition">Fcore_generic_fmt.canonic</span></a>.<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">rewrite</span> <- <span class="id" title="var">Hu1</span>.<br/>
<span class="id" title="tactic">rewrite</span> <- <span class="id" title="var">Hd1</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_rnd.html#Rnd_DN_pt_unicity"><span class="id" title="lemma">Rnd_DN_pt_unicity</span></a> <span class="id" title="keyword">with</span> (1 := <span class="id" title="var">Hd</span>).<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#round_DN_pt"><span class="id" title="lemma">round_DN_pt</span></a>.<br/>
<span class="id" title="tactic">rewrite</span> <- <span class="id" title="var">Hu1</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_rnd.html#Rnd_UP_pt_unicity"><span class="id" title="lemma">Rnd_UP_pt_unicity</span></a> <span class="id" title="keyword">with</span> (1 := <span class="id" title="var">Hu</span>).<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#round_UP_pt"><span class="id" title="lemma">round_UP_pt</span></a>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Theorem</span> <a name="Rnd_NE_pt_monotone"><span class="id" title="lemma">Rnd_NE_pt_monotone</span></a> :<br/>
<a class="idref" href="compcert.flocq.Core.Fcore_defs.html#round_pred_monotone"><span class="id" title="definition">round_pred_monotone</span></a> <a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#Rnd_NE_pt"><span class="id" title="definition">Rnd_NE_pt</span></a>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_rnd.html#Rnd_NG_pt_monotone"><span class="id" title="lemma">Rnd_NG_pt_monotone</span></a>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">x</span> <span class="id" title="var">d</span> <span class="id" title="var">u</span> <span class="id" title="var">Hd</span> <span class="id" title="var">Hdn</span> <span class="id" title="var">Hu</span> <span class="id" title="var">Hun</span> (<span class="id" title="var">cd</span>, (<span class="id" title="var">Hd1</span>, <span class="id" title="var">Hd2</span>)) (<span class="id" title="var">cu</span>, (<span class="id" title="var">Hu1</span>, <span class="id" title="var">Hu2</span>)).<br/>
<span class="id" title="tactic">destruct</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Reals.RIneq.html#Req_dec"><span class="id" title="lemma">Req_dec</span></a> <span class="id" title="var">x</span> <span class="id" title="var">d</span>) <span class="id" title="keyword">as</span> [<span class="id" title="var">Hx</span>|<span class="id" title="var">Hx</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="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_rnd.html#Rnd_UP_pt_idempotent"><span class="id" title="lemma">Rnd_UP_pt_idempotent</span></a> <span class="id" title="keyword">with</span> (1 := <span class="id" title="var">Hu</span>).<br/>
<span class="id" title="tactic">rewrite</span> <span class="id" title="var">Hx</span>.<br/>
<span class="id" title="tactic">apply</span> <span class="id" title="var">Hd</span>.<br/>
<span class="id" title="tactic">rewrite</span> (<a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#DN_UP_parity_aux"><span class="id" title="lemma">DN_UP_parity_aux</span></a> <a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#DN_UP_parity_generic_pos"><span class="id" title="lemma">DN_UP_parity_generic_pos</span></a> <span class="id" title="var">x</span> <span class="id" title="var">cd</span> <span class="id" title="var">cu</span>) <span class="id" title="keyword">in</span> <span class="id" title="var">Hu2</span> ; <span class="id" title="tactic">try</span> <span class="id" title="var">easy</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.Init.Logic.html#proj2"><span class="id" title="lemma">proj2</span></a> <span class="id" title="var">Hd2</span>) <span class="id" title="keyword">in</span> <span class="id" title="var">Hu2</span>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">Hf</span>.<br/>
<span class="id" title="tactic">apply</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#sym_eq"><span class="id" title="abbreviation">sym_eq</span></a>.<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_rnd.html#Rnd_DN_pt_idempotent"><span class="id" title="lemma">Rnd_DN_pt_idempotent</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> <- <span class="id" title="var">Hd1</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_rnd.html#Rnd_DN_pt_unicity"><span class="id" title="lemma">Rnd_DN_pt_unicity</span></a> <span class="id" title="keyword">with</span> (1 := <span class="id" title="var">Hd</span>).<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#round_DN_pt"><span class="id" title="lemma">round_DN_pt</span></a>.<br/>
<span class="id" title="tactic">rewrite</span> <- <span class="id" title="var">Hu1</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_rnd.html#Rnd_UP_pt_unicity"><span class="id" title="lemma">Rnd_UP_pt_unicity</span></a> <span class="id" title="keyword">with</span> (1 := <span class="id" title="var">Hu</span>).<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#round_UP_pt"><span class="id" title="lemma">round_UP_pt</span></a>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Theorem</span> <a name="Rnd_NE_pt_round"><span class="id" title="lemma">Rnd_NE_pt_round</span></a> :<br/>
<a class="idref" href="compcert.flocq.Core.Fcore_defs.html#round_pred"><span class="id" title="definition">round_pred</span></a> <a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#Rnd_NE_pt"><span class="id" title="definition">Rnd_NE_pt</span></a>.<br/>
<span class="id" title="tactic">split</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#Rnd_NE_pt_total"><span class="id" title="lemma">Rnd_NE_pt_total</span></a>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#Rnd_NE_pt_monotone"><span class="id" title="lemma">Rnd_NE_pt_monotone</span></a>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="round_NE_pt_pos"><span class="id" title="lemma">round_NE_pt_pos</span></a> :<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>,<br/>
(0 <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Reals.Rdefinitions.html#1cc22ba6849531267d3b25ca7b262449"><span class="id" title="notation"><</span></a> <a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.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.Core.Fcore_rnd_ne.html#Rnd_NE_pt"><span class="id" title="definition">Rnd_NE_pt</span></a> <a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#x"><span class="id" title="variable">x</span></a> (<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.Core.Fcore_rnd_ne.html#Fcore_rnd_NE.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.Core.Fcore_rnd_ne.html#x"><span class="id" title="variable">x</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">Hx</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.Core.Fcore_generic_fmt.html#round_N_pt"><span class="id" title="lemma">round_N_pt</span></a>.<br/>
<span class="id" title="tactic">unfold</span> <a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#NE_prop"><span class="id" title="definition">NE_prop</span></a>.<br/>
<span class="id" title="tactic">set</span> (<span class="id" title="var">mx</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.Core.Fcore_rnd_ne.html#Fcore_rnd_NE.fexp"><span class="id" title="variable">fexp</span></a> <span class="id" title="var">x</span>).<br/>
<span class="id" title="tactic">set</span> (<span class="id" title="var">xr</span> := <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.Core.Fcore_rnd_ne.html#Fcore_rnd_NE.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> <span class="id" title="var">x</span>).<br/>
<span class="id" title="tactic">destruct</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Reals.RIneq.html#Req_dec"><span class="id" title="lemma">Req_dec</span></a> (<span class="id" title="var">mx</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> (<a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Zfloor"><span class="id" title="definition">Zfloor</span></a> <span class="id" title="var">mx</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">as</span> [<span class="id" title="var">Hm</span>|<span class="id" title="var">Hm</span>].<br/>
<span class="id" title="tactic">left</span>.<br/>
<span class="id" title="tactic">∃</span> (<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_Raux.html#Ztrunc"><span class="id" title="definition">Ztrunc</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.Core.Fcore_rnd_ne.html#Fcore_rnd_NE.fexp"><span class="id" title="variable">fexp</span></a> <span class="id" title="var">xr</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.Core.Fcore_rnd_ne.html#Fcore_rnd_NE.fexp"><span class="id" title="variable">fexp</span></a> <span class="id" title="var">xr</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_generic_fmt.html#round_N_pt"><span class="id" title="lemma">round_N_pt</span></a>...<br/>
<span class="id" title="tactic">split</span>.<br/>
<span class="id" title="tactic">unfold</span> <a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#canonic"><span class="id" title="definition">Fcore_generic_fmt.canonic</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>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#round_N_pt"><span class="id" title="lemma">round_N_pt</span></a>...<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.Rgeom.html#xr"><span class="id" title="definition">xr</span></a>, <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_generic_fmt.html#Znearest"><span class="id" title="definition">Znearest</span></a>.<br/>
<span class="id" title="tactic">fold</span> <span class="id" title="var">mx</span>.<br/>
<span class="id" title="tactic">rewrite</span> <span class="id" title="var">Hm</span>.<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Rcompare_Eq"><span class="id" title="lemma">Rcompare_Eq</span></a>. 2: <span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#refl_equal"><span class="id" title="abbreviation">refl_equal</span></a>.<br/>
<span class="id" title="var">case_eq</span> (<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.Core.Fcore_Raux.html#Zfloor"><span class="id" title="definition">Zfloor</span></a> <span class="id" title="var">mx</span>)) ; <span class="id" title="tactic">intros</span> <span class="id" title="var">Hmx</span>.<br/>
<span class="id" title="tactic">change</span> (<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.Core.Fcore_Raux.html#Ztrunc"><span class="id" title="definition">Ztrunc</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.Core.Fcore_rnd_ne.html#Fcore_rnd_NE.fexp"><span class="id" title="variable">fexp</span></a> (<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.Core.Fcore_rnd_ne.html#Fcore_rnd_NE.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> <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="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="tactic">destruct</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Reals.RIneq.html#Rle_or_lt"><span class="id" title="lemma">Rle_or_lt</span></a> (<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.Core.Fcore_rnd_ne.html#Fcore_rnd_NE.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> <span class="id" title="var">x</span>) 0) <span class="id" title="keyword">as</span> [<span class="id" title="var">Hr</span>|<span class="id" title="var">Hr</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#Rle_antisym"><span class="id" title="lemma">Rle_antisym</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">Hr</span>).<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> <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>.<br/>
<span class="id" title="tactic">change</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Reals.Rdefinitions.html#R0"><span class="id" title="axiom">R0</span></a> <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> 0).<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#Ztrunc_Z2R"><span class="id" title="lemma">Ztrunc_Z2R</span></a> 0).<br/>
<span class="id" title="tactic">rewrite</span> <- (<a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#round_0"><span class="id" title="lemma">round_0</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#Fcore_rnd_NE.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>).<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#round_le"><span class="id" title="lemma">round_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#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_generic_fmt.html#scaled_mantissa_DN"><span class="id" title="lemma">scaled_mantissa_DN</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_Raux.html#Ztrunc_Z2R"><span class="id" title="lemma">Ztrunc_Z2R</span></a>.<br/>
<span class="id" title="tactic">change</span> (<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.Core.Fcore_Raux.html#Ztrunc"><span class="id" title="definition">Ztrunc</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.Core.Fcore_rnd_ne.html#Fcore_rnd_NE.fexp"><span class="id" title="variable">fexp</span></a> (<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.Core.Fcore_rnd_ne.html#Fcore_rnd_NE.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> <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="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="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">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">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="var">_</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> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">Hx</span>)) <span class="id" title="keyword">in</span> <span class="id" title="var">Hex</span>.<br/>
<span class="id" title="tactic">destruct</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.ZArith_dec.html#Z_lt_le_dec"><span class="id" title="lemma">Z_lt_le_dec</span></a> (<a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#Fcore_rnd_NE.fexp"><span class="id" title="variable">fexp</span></a> <span class="id" title="var">ex</span>) <span class="id" title="var">ex</span>) <span class="id" title="keyword">as</span> [<span class="id" title="var">He</span>|<span class="id" title="var">He</span>].<br/>
<span class="id" title="tactic">assert</span> (<span class="id" title="var">Hu</span> := <a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#round_bounded_large_pos"><span class="id" title="lemma">round_bounded_large_pos</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Zceil"><span class="id" title="definition">Zceil</span></a> <span class="id" title="var">_</span> <span class="id" title="var">_</span> <span class="id" title="var">He</span> <span class="id" title="var">Hex</span>).<br/>
<span class="id" title="tactic">assert</span> (<span class="id" title="var">Hfc</span>: <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Zceil"><span class="id" title="definition">Zceil</span></a> <span class="id" title="var">mx</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_Raux.html#Zfloor"><span class="id" title="definition">Zfloor</span></a> <span class="id" title="var">mx</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a> 1)%<span class="id" title="var">Z</span>).<br/>
<span class="id" title="tactic">apply</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">intros</span> <span class="id" title="var">H</span>.<br/>
<span class="id" title="tactic">rewrite</span> <span class="id" title="var">H</span> <span class="id" title="keyword">in</span> <span class="id" title="var">Hm</span>.<br/>
<span class="id" title="tactic">unfold</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Reals.Rdefinitions.html#Rminus"><span class="id" title="definition">Rminus</span></a> <span class="id" title="keyword">in</span> <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.Reals.Raxioms.html#Rplus_opp_r"><span class="id" title="axiom">Rplus_opp_r</span></a> <span class="id" title="keyword">in</span> <span class="id" title="var">Hm</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> 0).<br/>
<span class="id" title="tactic">rewrite</span> <span class="id" title="var">Hm</span> <span class="id" title="tactic">at</span> 2.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Reals.RIneq.html#Rinv_0_lt_compat"><span class="id" title="lemma">Rinv_0_lt_compat</span></a>.<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">apply</span> (<a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Z2R_lt"><span class="id" title="lemma">Z2R_lt</span></a> 0 2).<br/>
<span class="id" title="tactic">destruct</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">Hu</span>) <span class="id" title="keyword">as</span> [<span class="id" title="var">Hu'</span>|<span class="id" title="var">Hu'</span>].<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> <a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#canonic_exp_fexp_pos"><span class="id" title="lemma">canonic_exp_fexp_pos</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#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> <span class="id" title="var">Hu</span>) <span class="id" title="var">Hu'</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">rewrite</span> <a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#canonic_exp_fexp_pos"><span class="id" title="lemma">canonic_exp_fexp_pos</span></a> <span class="id" title="keyword">with</span> (1 := <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.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>, <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#Zplus_opp_r"><span class="id" title="abbreviation">Zplus_opp_r</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Reals.RIneq.html#Rmult_1_r"><span class="id" title="lemma">Rmult_1_r</span></a>.<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">fold</span> <span class="id" title="var">mx</span>.<br/>
<span class="id" title="tactic">rewrite</span> <span class="id" title="var">Hfc</span>.<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#Zeven_plus"><span class="id" title="lemma">Zeven_plus</span></a>, <span class="id" title="var">Hmx</span>.<br/>
<span class="id" title="tactic">rewrite</span> <span class="id" title="var">Hu'</span>.<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>, <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.Core.Fcore_Raux.html#ln_beta_bpow"><span class="id" title="lemma">ln_beta_bpow</span></a>.<br/>
<span class="id" title="tactic">rewrite</span> <- <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#bpow_plus"><span class="id" title="lemma">bpow_plus</span></a>, <- <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Z2R_Zpower"><span class="id" title="lemma">Z2R_Zpower</span></a>.<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="var">case_eq</span> (<a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#Zeven"><span class="id" title="definition">Zeven</span></a> <span class="id" title="keyword">beta</span>) ; <span class="id" title="tactic">intros</span> <span class="id" title="var">Hr</span>.<br/>
<span class="id" title="tactic">destruct</span> <a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#Fcore_rnd_NE.exists_NE_"><span class="id" title="variable">exists_NE_</span></a> <span class="id" title="keyword">as</span> [<span class="id" title="var">Hs</span>|<span class="id" title="var">Hs</span>].<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">rewrite</span> <span class="id" title="var">Hs</span> <span class="id" title="keyword">in</span> <span class="id" title="var">Hr</span>.<br/>
<span class="id" title="tactic">destruct</span> (<span class="id" title="var">Hs</span> <span class="id" title="var">ex</span>) <span class="id" title="keyword">as</span> (<span class="id" title="var">H</span>,<span class="id" title="var">_</span>).<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#Zeven_Zpower"><span class="id" title="lemma">Zeven_Zpower</span></a>.<br/>
<span class="id" title="tactic">exact</span> <span class="id" title="var">Hr</span>.<br/>
<span class="id" title="tactic">omega</span>.<br/>
<span class="id" title="tactic">assert</span> (<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.Core.Fcore_Raux.html#Zfloor"><span class="id" title="definition">Zfloor</span></a> <span class="id" title="var">mx</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>). 2: <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="keyword">in</span> <span class="id" title="var">Hmx</span>.<br/>
<span class="id" title="tactic">replace</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">mx</span>) <span class="id" title="keyword">with</span> (<a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Zceil"><span class="id" title="definition">Zceil</span></a> <span class="id" title="var">mx</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><span class="id" title="notation">+</span></a> -1)%<span class="id" title="var">Z</span> <span class="id" title="tactic">by</span> <span class="id" title="tactic">omega</span>.<br/>
<span class="id" title="tactic">rewrite</span> <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="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#eqb_true"><span class="id" title="lemma">eqb_true</span></a>.<br/>
<span class="id" title="tactic">unfold</span> <span class="id" title="var">mx</span>.<br/>
<span class="id" title="tactic">replace</span> (<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.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.Core.Fcore_rnd_ne.html#Fcore_rnd_NE.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="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">ex</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#239a40728d107295b3cb2c790f57f9e9"><span class="id" title="notation">-</span></a> <a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#Fcore_rnd_NE.fexp"><span class="id" title="variable">fexp</span></a> <span class="id" title="var">ex</span>)).<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#Zeven_Zpower_odd"><span class="id" title="lemma">Zeven_Zpower_odd</span></a> <span class="id" title="keyword">with</span> (2 := <span class="id" title="var">Hr</span>).<br/>
<span class="id" title="var">easy</span>.<br/>
<span class="id" title="tactic">omega</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#eq_Z2R"><span class="id" title="lemma">eq_Z2R</span></a>.<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Z2R_Zpower"><span class="id" title="lemma">Z2R_Zpower</span></a>. 2: <span class="id" title="tactic">omega</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Rmult_eq_reg_r"><span class="id" title="lemma">Rmult_eq_reg_r</span></a> <span class="id" title="keyword">with</span> (<a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#bpow"><span class="id" title="abbreviation">bpow</span></a> (<a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#Fcore_rnd_NE.fexp"><span class="id" title="variable">fexp</span></a> <span class="id" title="var">ex</span>)).<br/>
<span class="id" title="tactic">unfold</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#Z.sub"><span class="id" title="definition">Zminus</span></a>.<br/>
<span class="id" title="tactic">rewrite</span> <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="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>, <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.ZArith.BinInt.html#Zplus_opp_l"><span class="id" title="abbreviation">Zplus_opp_l</span></a>, <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Reals.RIneq.html#Rmult_1_r"><span class="id" title="lemma">Rmult_1_r</span></a>.<br/>
<span class="id" title="tactic">pattern</span> (<a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#Fcore_rnd_NE.fexp"><span class="id" title="variable">fexp</span></a> <span class="id" title="var">ex</span>) ; <span class="id" title="tactic">rewrite</span> <- <a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#canonic_exp_fexp_pos"><span class="id" title="lemma">canonic_exp_fexp_pos</span></a> <span class="id" title="keyword">with</span> (1 := <span class="id" title="var">Hex</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.Init.Logic.html#sym_eq"><span class="id" title="abbreviation">sym_eq</span></a>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Reals.RIneq.html#Rgt_not_eq"><span class="id" title="lemma">Rgt_not_eq</span></a>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#bpow_gt_0"><span class="id" title="lemma">bpow_gt_0</span></a>.<br/>
<span class="id" title="tactic">generalize</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#proj1"><span class="id" title="lemma">proj1</span></a> (<a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#Fcore_rnd_NE.valid_exp"><span class="id" title="variable">valid_exp</span></a> <span class="id" title="var">ex</span>) <span class="id" title="var">He</span>).<br/>
<span class="id" title="tactic">omega</span>.<br/>
<span class="id" title="tactic">assert</span> (<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.Core.Fcore_Raux.html#Zfloor"><span class="id" title="definition">Zfloor</span></a> <span class="id" title="var">mx</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>). 2: <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="keyword">in</span> <span class="id" title="var">Hmx</span>.<br/>
<span class="id" title="tactic">unfold</span> <span class="id" title="var">mx</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> <a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#canonic_exp_fexp_pos"><span class="id" title="lemma">canonic_exp_fexp_pos</span></a> <span class="id" title="keyword">with</span> (1 := <span class="id" title="var">Hex</span>).<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#mantissa_DN_small_pos"><span class="id" title="lemma">mantissa_DN_small_pos</span></a>.<br/>
<span class="id" title="tactic">right</span>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">g</span> <span class="id" title="var">Hg</span>.<br/>
<span class="id" title="tactic">destruct</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Reals.RIneq.html#Req_dec"><span class="id" title="lemma">Req_dec</span></a> <span class="id" title="var">x</span> <span class="id" title="var">g</span>) <span class="id" title="keyword">as</span> [<span class="id" title="var">Hxg</span>|<span class="id" title="var">Hxg</span>].<br/>
<span class="id" title="tactic">rewrite</span> <- <span class="id" title="var">Hxg</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_generic_fmt.html#round_generic"><span class="id" title="lemma">round_generic</span></a>...<br/>
<span class="id" title="tactic">rewrite</span> <span class="id" title="var">Hxg</span>.<br/>
<span class="id" title="tactic">apply</span> <span class="id" title="var">Hg</span>.<br/>
<span class="id" title="tactic">set</span> (<span class="id" title="var">d</span> := <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.Core.Fcore_rnd_ne.html#Fcore_rnd_NE.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> <span class="id" title="var">x</span>).<br/>
<span class="id" title="tactic">set</span> (<span class="id" title="var">u</span> := <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.Core.Fcore_rnd_ne.html#Fcore_rnd_NE.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> <span class="id" title="var">x</span>).<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_rnd.html#Rnd_N_pt_unicity"><span class="id" title="lemma">Rnd_N_pt_unicity</span></a> <span class="id" title="keyword">with</span> (<span class="id" title="var">d</span> := <span class="id" title="var">d</span>) (<span class="id" title="var">u</span> := <span class="id" title="var">u</span>) (4 := <span class="id" title="var">Hg</span>).<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#round_DN_pt"><span class="id" title="lemma">round_DN_pt</span></a>.<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#round_UP_pt"><span class="id" title="lemma">round_UP_pt</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_generic_fmt.html#round_N_pt"><span class="id" title="lemma">round_N_pt</span></a>.<br/>
<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> <span class="id" title="keyword">beta</span> <a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#Fcore_rnd_NE.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">d</span>, <span class="id" title="var">u</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>. <span class="id" title="tactic">fold</span> <span class="id" title="var">mx</span>.<br/>
<span class="id" title="tactic">rewrite</span> <- 2!<a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Rmult_minus_distr_r"><span class="id" title="lemma">Rmult_minus_distr_r</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="compcert.flocq.Core.Fcore_Raux.html#Rmult_eq_reg_r"><span class="id" title="lemma">Rmult_eq_reg_r</span></a> <span class="id" title="keyword">in</span> <span class="id" title="var">H</span>.<br/>
<span class="id" title="tactic">apply</span> <span class="id" title="var">Hm</span>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Rcompare_Eq_inv"><span class="id" title="lemma">Rcompare_Eq_inv</span></a>.<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#Rcompare_floor_ceil_mid"><span class="id" title="lemma">Rcompare_floor_ceil_mid</span></a>.<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Rcompare_Eq"><span class="id" title="lemma">Rcompare_Eq</span></a>.<br/>
<span class="id" title="var">contradict</span> <span class="id" title="var">Hxg</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_rnd.html#Rnd_N_pt_idempotent"><span class="id" title="lemma">Rnd_N_pt_idempotent</span></a> <span class="id" title="keyword">with</span> (1 := <span class="id" title="var">Hg</span>).<br/>
<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> <span class="id" title="keyword">beta</span> <a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#Fcore_rnd_NE.fexp"><span class="id" title="variable">fexp</span></a> <span class="id" title="var">x</span>).<br/>
<span class="id" title="tactic">fold</span> <span class="id" title="var">mx</span>.<br/>
<span class="id" title="tactic">rewrite</span> <- <span class="id" title="var">Hxg</span>.<br/>
<span class="id" title="tactic">change</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> <span class="id" title="var">mx</span>) <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.Core.Fcore_rnd_ne.html#bpow"><span class="id" title="abbreviation">bpow</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.Core.Fcore_rnd_ne.html#Fcore_rnd_NE.fexp"><span class="id" title="variable">fexp</span></a> <span class="id" title="var">x</span>))%<span class="id" title="var">R</span> <span class="id" title="keyword">with</span> <span class="id" title="var">d</span>.<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">eapply</span> <a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#round_DN_pt"><span class="id" title="lemma">round_DN_pt</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#Rgt_not_eq"><span class="id" title="lemma">Rgt_not_eq</span></a>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#bpow_gt_0"><span class="id" title="lemma">bpow_gt_0</span></a>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Theorem</span> <a name="round_NE_opp"><span class="id" title="lemma">round_NE_opp</span></a> :<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>,<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.Core.Fcore_rnd_ne.html#Fcore_rnd_NE.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="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_rnd_ne.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Reals.Rdefinitions.html#0575b33c544988081e4e30ee15f09d84"><span class="id" title="notation">-</span></a> <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.Core.Fcore_rnd_ne.html#Fcore_rnd_NE.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.Core.Fcore_rnd_ne.html#x"><span class="id" title="variable">x</span></a>)%<span class="id" title="var">R</span>.<br/>
<span class="id" title="keyword">Proof</span>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">x</span>.<br/>
<span class="id" title="tactic">unfold</span> <a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#round"><span class="id" title="definition">round</span></a>. <span class="id" title="tactic">simpl</span>.<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#scaled_mantissa_opp"><span class="id" title="lemma">scaled_mantissa_opp</span></a>, <a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#canonic_exp_opp"><span class="id" title="lemma">canonic_exp_opp</span></a>.<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">rewrite</span> <- <a class="idref" href="compcert.flocq.Core.Fcore_float_prop.html#F2R_Zopp"><span class="id" title="lemma">F2R_Zopp</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> (<span class="id" title="keyword">fun</span> <span class="id" title="var">v</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="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.Core.Fcore_rnd_ne.html#v"><span class="id" title="variable">v</span></a>) <span class="id" title="var">_</span>))).<br/>
<span class="id" title="tactic">set</span> (<span class="id" title="var">m</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.Core.Fcore_rnd_ne.html#Fcore_rnd_NE.fexp"><span class="id" title="variable">fexp</span></a> <span class="id" title="var">x</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">case</span> <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Rcompare"><span class="id" title="definition">Rcompare</span></a> ; <span class="id" title="tactic">trivial</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">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="keyword">if</span> <a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#b"><span class="id" title="variable">b</span></a> <span class="id" title="keyword">then</span> <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#Zceil"><span class="id" title="definition">Zceil</span></a> <span class="id" title="var">m</span> <span class="id" title="keyword">else</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">m</span>)).<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Bool.Bool.html#negb_involutive"><span class="id" title="lemma">Bool.negb_involutive</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>.<br/>
<span class="id" title="tactic">rewrite</span> <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">rewrite</span> <a class="idref" href="compcert.flocq.Core.Fcore_Raux.html#eqb_sym"><span class="id" title="lemma">eqb_sym</span></a>.<br/>
<span class="id" title="keyword">Qed</span>.<br/>
<br/>
<span class="id" title="keyword">Lemma</span> <a name="round_NE_abs"><span class="id" title="lemma">round_NE_abs</span></a>:<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> : <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>,<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.Core.Fcore_rnd_ne.html#Fcore_rnd_NE.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="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.Core.Fcore_rnd_ne.html#x"><span class="id" title="variable">x</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Reals.Rbasic_fun.html#Rabs"><span class="id" title="definition">Rabs</span></a> (<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.Core.Fcore_rnd_ne.html#Fcore_rnd_NE.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.Core.Fcore_rnd_ne.html#x"><span class="id" title="variable">x</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>.<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">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="tactic">at</span> 2.<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">Hx</span>|<span class="id" title="var">Hx</span>].<br/>
<span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#round_NE_opp"><span class="id" title="lemma">round_NE_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.Rbasic_fun.html#Rabs_left1"><span class="id" title="lemma">Rabs_left1</span></a>.<br/>
<span class="id" title="tactic">rewrite</span> <- (<a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#round_0"><span class="id" title="lemma">round_0</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#Fcore_rnd_NE.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>).<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#round_le"><span class="id" title="lemma">round_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#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.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_generic_fmt.html#round_0"><span class="id" title="lemma">round_0</span></a> <span class="id" title="keyword">beta</span> <a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#Fcore_rnd_NE.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>).<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#round_le"><span class="id" title="lemma">round_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="round_NE_pt"><span class="id" title="lemma">round_NE_pt</span></a> :<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>,<br/>
<a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#Rnd_NE_pt"><span class="id" title="definition">Rnd_NE_pt</span></a> <a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#x"><span class="id" title="variable">x</span></a> (<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.Core.Fcore_rnd_ne.html#Fcore_rnd_NE.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.Core.Fcore_rnd_ne.html#x"><span class="id" title="variable">x</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>.<br/>
<span class="id" title="tactic">destruct</span> (<a class="idref" href="http://coq.inria.fr/distrib/8.6/stdlib/Coq.Reals.Raxioms.html#total_order_T"><span class="id" title="axiom">total_order_T</span></a> <span class="id" title="var">x</span> 0) <span class="id" title="keyword">as</span> [[<span class="id" title="var">Hx</span>|<span class="id" title="var">Hx</span>]|<span class="id" title="var">Hx</span>].<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_rnd.html#Rnd_NG_pt_sym"><span class="id" title="lemma">Rnd_NG_pt_sym</span></a>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#generic_format_opp"><span class="id" title="lemma">generic_format_opp</span></a>.<br/>
<span class="id" title="tactic">unfold</span> <a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#NE_prop"><span class="id" title="definition">NE_prop</span></a>.<br/>
<span class="id" title="tactic">intros</span> <span class="id" title="var">_</span> <span class="id" title="var">f</span> ((<span class="id" title="var">mg</span>,<span class="id" title="var">eg</span>),(<span class="id" title="var">H1</span>,(<span class="id" title="var">H2</span>,<span class="id" title="var">H3</span>))).<br/>
<span class="id" title="tactic">∃</span> (<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="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">mg</span>) <span class="id" title="var">eg</span>).<br/>
<span class="id" title="tactic">repeat</span> <span class="id" title="tactic">split</span>.<br/>
<span class="id" title="tactic">rewrite</span> <span class="id" title="var">H1</span>.<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.flocq.Core.Fcore_float_prop.html#F2R_Zopp"><span class="id" title="lemma">F2R_Zopp</span></a>.<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#canonic_opp"><span class="id" title="lemma">canonic_opp</span></a>.<br/>
<span class="id" title="tactic">simpl</span>.<br/>
<span class="id" title="var">now</span> <span class="id" title="tactic">rewrite</span> <a class="idref" href="compcert.flocq.Core.Fcore_Zaux.html#Zeven_opp"><span class="id" title="lemma">Zeven_opp</span></a>.<br/>
<span class="id" title="tactic">rewrite</span> <- <a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#round_NE_opp"><span class="id" title="lemma">round_NE_opp</span></a>.<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#round_NE_pt_pos"><span class="id" title="lemma">round_NE_pt_pos</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#Ropp_0_gt_lt_contravar"><span class="id" title="lemma">Ropp_0_gt_lt_contravar</span></a>.<br/>
<span class="id" title="tactic">rewrite</span> <span class="id" title="var">Hx</span>, <a class="idref" href="compcert.flocq.Core.Fcore_generic_fmt.html#round_0"><span class="id" title="lemma">round_0</span></a>...<br/>
<span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_rnd.html#Rnd_NG_pt_refl"><span class="id" title="lemma">Rnd_NG_pt_refl</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="var">now</span> <span class="id" title="tactic">apply</span> <a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#round_NE_pt_pos"><span class="id" title="lemma">round_NE_pt_pos</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.Core.Fcore_rnd_ne.html#Fcore_rnd_NE"><span class="id" title="section">Fcore_rnd_NE</span></a>.<br/>
<br/>
</div>
<div class="doc">
Notations for backward-compatibility with Flocq 1.4.
</div>
<div class="code">
<span class="id" title="keyword">Notation</span> <a name="rndNE"><span class="id" title="abbreviation">rndNE</span></a> := <a class="idref" href="compcert.flocq.Core.Fcore_rnd_ne.html#ZnearestE"><span class="id" title="abbreviation">ZnearestE</span></a> (<span class="id" title="var">only</span> <span class="id" title="var">parsing</span>).<br/>
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