<!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>Flocq.Core.Fcore_FIX</title> </head> <body> <div id="page"> <div id="header"> </div> <div id="main"> <h1 class="libtitle">Library Flocq.Core.Fcore_FIX</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="lab4"></a><h1 class="section">Fixed-point format</h1> </div> <div class="code"> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="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="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="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="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="Flocq.Core.Fcore_ulp.html#"><span class="id" title="library">Fcore_ulp</span></a>.<br/> <span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="Flocq.Core.Fcore_rnd_ne.html#"><span class="id" title="library">Fcore_rnd_ne</span></a>.<br/> <br/> <span class="id" title="keyword">Section</span> <a name="RND_FIX"><span class="id" title="section">RND_FIX</span></a>.<br/> <br/> <span class="id" title="keyword">Variable</span> <span class="id" title="keyword">beta</span> : <a class="idref" href="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> := (<a class="idref" href="Flocq.Core.Fcore_Raux.html#bpow"><span class="id" title="definition">bpow</span></a> <span class="id" title="keyword">beta</span>).<br/> <br/> <span class="id" title="keyword">Variable</span> <a name="RND_FIX.emin"><span class="id" title="variable">emin</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>.<br/> <br/> <span class="id" title="keyword">Definition</span> <a name="FIX_format"><span class="id" title="definition">FIX_format</span></a> (<span class="id" title="var">x</span> : <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#R"><span class="id" title="axiom">R</span></a>) :=<br/> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">∃</span></a> <span class="id" title="var">f</span> : <a class="idref" href="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/current/stdlib/Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">,</span></a><br/> <a class="idref" href="Flocq.Core.Fcore_FIX.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="Flocq.Core.Fcore_defs.html#F2R"><span class="id" title="definition">F2R</span></a> <a class="idref" href="Flocq.Core.Fcore_FIX.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#d82a7d96d3659d805ffe732283716822"><span class="id" title="notation">∧</span></a> (<a class="idref" href="Flocq.Core.Fcore_defs.html#Fexp"><span class="id" title="projection">Fexp</span></a> <a class="idref" href="Flocq.Core.Fcore_FIX.html#f"><span class="id" title="variable">f</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="Flocq.Core.Fcore_FIX.html#RND_FIX.emin"><span class="id" title="variable">emin</span></a>)%<span class="id" title="var">Z</span>.<br/> <br/> <span class="id" title="keyword">Definition</span> <a name="FIX_exp"><span class="id" title="definition">FIX_exp</span></a> (<span class="id" title="var">e</span> : <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Numbers.BinNums.html#Z"><span class="id" title="inductive">Z</span></a>) := <a class="idref" href="Flocq.Core.Fcore_FIX.html#RND_FIX.emin"><span class="id" title="variable">emin</span></a>.<br/> <br/> </div> <div class="doc"> Properties of the FIX format </div> <div class="code"> <br/> <span class="id" title="keyword">Global Instance</span> <a name="FIX_exp_valid"><span class="id" title="instance">FIX_exp_valid</span></a> : <a class="idref" href="Flocq.Core.Fcore_generic_fmt.html#Valid_exp"><span class="id" title="class">Valid_exp</span></a> <a class="idref" href="Flocq.Core.Fcore_FIX.html#FIX_exp"><span class="id" title="definition">FIX_exp</span></a>.<br/> <br/> <span class="id" title="keyword">Theorem</span> <a name="generic_format_FIX"><span class="id" title="lemma">generic_format_FIX</span></a> :<br/> <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, <a class="idref" href="Flocq.Core.Fcore_FIX.html#FIX_format"><span class="id" title="definition">FIX_format</span></a> <a class="idref" href="Flocq.Core.Fcore_FIX.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="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="Flocq.Core.Fcore_FIX.html#FIX_exp"><span class="id" title="definition">FIX_exp</span></a> <a class="idref" href="Flocq.Core.Fcore_FIX.html#x"><span class="id" title="variable">x</span></a>.<br/> <br/> <span class="id" title="keyword">Theorem</span> <a name="FIX_format_generic"><span class="id" title="lemma">FIX_format_generic</span></a> :<br/> <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, <a class="idref" href="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="Flocq.Core.Fcore_FIX.html#FIX_exp"><span class="id" title="definition">FIX_exp</span></a> <a class="idref" href="Flocq.Core.Fcore_FIX.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a> <a class="idref" href="Flocq.Core.Fcore_FIX.html#FIX_format"><span class="id" title="definition">FIX_format</span></a> <a class="idref" href="Flocq.Core.Fcore_FIX.html#x"><span class="id" title="variable">x</span></a>.<br/> <br/> <span class="id" title="keyword">Theorem</span> <a name="FIX_format_satisfies_any"><span class="id" title="lemma">FIX_format_satisfies_any</span></a> :<br/> <a class="idref" href="Flocq.Core.Fcore_rnd.html#satisfies_any"><span class="id" title="inductive">satisfies_any</span></a> <a class="idref" href="Flocq.Core.Fcore_FIX.html#FIX_format"><span class="id" title="definition">FIX_format</span></a>.<br/> <br/> <span class="id" title="keyword">Global Instance</span> <a name="FIX_exp_monotone"><span class="id" title="instance">FIX_exp_monotone</span></a> : <a class="idref" href="Flocq.Core.Fcore_generic_fmt.html#Monotone_exp"><span class="id" title="class">Monotone_exp</span></a> <a class="idref" href="Flocq.Core.Fcore_FIX.html#FIX_exp"><span class="id" title="definition">FIX_exp</span></a>.<br/> <br/> <span class="id" title="keyword">Theorem</span> <a name="ulp_FIX"><span class="id" title="lemma">ulp_FIX</span></a>: <span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, <a class="idref" href="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="Flocq.Core.Fcore_FIX.html#FIX_exp"><span class="id" title="definition">FIX_exp</span></a> <a class="idref" href="Flocq.Core.Fcore_FIX.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#1c39bf18749e5cc609e83c0a0ba5a372"><span class="id" title="notation">=</span></a> <a class="idref" href="Flocq.Core.Fcore_FIX.html#bpow"><span class="id" title="abbreviation">bpow</span></a> <a class="idref" href="Flocq.Core.Fcore_FIX.html#RND_FIX.emin"><span class="id" title="variable">emin</span></a>.<br/> <br/> <span class="id" title="keyword">End</span> <a class="idref" href="Flocq.Core.Fcore_FIX.html#RND_FIX"><span class="id" title="section">RND_FIX</span></a>.<br/> </div> </div> <div id="footer"> <hr/><a href="index.html">Index</a><hr/>This page has been generated by <a href="http://coq.inria.fr/">coqdoc</a> </div> </div> </body> </html>
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