<!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.Prop.Fprop_div_sqrt_error</title>
</head>
<body>
<div id="page">
<div id="header">
</div>
<div id="main">
<h1 class="libtitle">Library Flocq.Prop.Fprop_div_sqrt_error</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="lab18"></a><h1 class="section">Remainder of the division and square root are in the FLX 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.html#"><span class="id" title="library">Fcore</span></a>.<br/>
<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="Flocq.Calc.Fcalc_ops.html#"><span class="id" title="library">Fcalc_ops</span></a>.<br/>
<span class="id" title="keyword">Require</span> <span class="id" title="keyword">Import</span> <a class="idref" href="Flocq.Prop.Fprop_relative.html#"><span class="id" title="library">Fprop_relative</span></a>.<br/>
<br/>
<span class="id" title="keyword">Section</span> <a name="Fprop_divsqrt_error"><span class="id" title="section">Fprop_divsqrt_error</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/>
<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="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="Fprop_divsqrt_error.prec"><span class="id" title="variable">prec</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">Theorem</span> <a name="generic_format_plus_prec"><span class="id" title="lemma">generic_format_plus_prec</span></a>:<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">fexp</span>, <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">(</span></a><span class="id" title="keyword">∀</span> <span class="id" title="var">e</span>, (<a class="idref" href="Flocq.Prop.Fprop_div_sqrt_error.html#fexp"><span class="id" title="variable">fexp</span></a> <a class="idref" href="Flocq.Prop.Fprop_div_sqrt_error.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.ZArith.BinInt.html#208bafb0e148fe7fb7dcd812c227f4ee"><span class="id" title="notation">≤</span></a> <a class="idref" href="Flocq.Prop.Fprop_div_sqrt_error.html#e"><span class="id" title="variable">e</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.ZArith.BinInt.html#239a40728d107295b3cb2c790f57f9e9"><span class="id" title="notation">-</span></a> <a class="idref" href="Flocq.Prop.Fprop_div_sqrt_error.html#Fprop_divsqrt_error.prec"><span class="id" title="variable">prec</span></a>)%<span class="id" title="var">Z</span><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="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#d43e996736952df71ebeeae74d10a287"><span class="id" title="notation">→</span></a><br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">x</span> <span class="id" title="var">y</span> (<span class="id" title="var">fx</span> <span class="id" title="var">fy</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>),<br/>
(<a class="idref" href="Flocq.Prop.Fprop_div_sqrt_error.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.Prop.Fprop_div_sqrt_error.html#fx"><span class="id" title="variable">fx</span></a>)%<span class="id" title="var">R</span> <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.Prop.Fprop_div_sqrt_error.html#y"><span class="id" title="variable">y</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.Prop.Fprop_div_sqrt_error.html#fy"><span class="id" title="variable">fy</span></a>)%<span class="id" title="var">R</span> <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="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rbasic_fun.html#Rabs"><span class="id" title="definition">Rabs</span></a> (<a class="idref" href="Flocq.Prop.Fprop_div_sqrt_error.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#7de2d195108bdf12ab090711a555a032"><span class="id" title="notation">+</span></a><a class="idref" href="Flocq.Prop.Fprop_div_sqrt_error.html#y"><span class="id" title="variable">y</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#1cc22ba6849531267d3b25ca7b262449"><span class="id" title="notation"><</span></a> <a class="idref" href="Flocq.Prop.Fprop_div_sqrt_error.html#bpow"><span class="id" title="abbreviation">bpow</span></a> (<a class="idref" href="Flocq.Prop.Fprop_div_sqrt_error.html#Fprop_divsqrt_error.prec"><span class="id" title="variable">prec</span></a><a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><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.Prop.Fprop_div_sqrt_error.html#fx"><span class="id" title="variable">fx</span></a>))%<span class="id" title="var">R</span> <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="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rbasic_fun.html#Rabs"><span class="id" title="definition">Rabs</span></a> (<a class="idref" href="Flocq.Prop.Fprop_div_sqrt_error.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#7de2d195108bdf12ab090711a555a032"><span class="id" title="notation">+</span></a><a class="idref" href="Flocq.Prop.Fprop_div_sqrt_error.html#y"><span class="id" title="variable">y</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#1cc22ba6849531267d3b25ca7b262449"><span class="id" title="notation"><</span></a> <a class="idref" href="Flocq.Prop.Fprop_div_sqrt_error.html#bpow"><span class="id" title="abbreviation">bpow</span></a> (<a class="idref" href="Flocq.Prop.Fprop_div_sqrt_error.html#Fprop_divsqrt_error.prec"><span class="id" title="variable">prec</span></a><a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.ZArith.BinInt.html#a3448b794f7a26d611ad36492b5d640b"><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.Prop.Fprop_div_sqrt_error.html#fy"><span class="id" title="variable">fy</span></a>))%<span class="id" title="var">R</span><br/>
<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.Prop.Fprop_div_sqrt_error.html#fexp"><span class="id" title="variable">fexp</span></a> (<a class="idref" href="Flocq.Prop.Fprop_div_sqrt_error.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#7de2d195108bdf12ab090711a555a032"><span class="id" title="notation">+</span></a><a class="idref" href="Flocq.Prop.Fprop_div_sqrt_error.html#y"><span class="id" title="variable">y</span></a>)%<span class="id" title="var">R</span>.<br/>
<br/>
<span class="id" title="keyword">Theorem</span> <a name="ex_Fexp_canonic"><span class="id" title="lemma">ex_Fexp_canonic</span></a>: <span class="id" title="keyword">∀</span> <span class="id" title="var">fexp</span>, <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.Prop.Fprop_div_sqrt_error.html#fexp"><span class="id" title="variable">fexp</span></a> <a class="idref" href="Flocq.Prop.Fprop_div_sqrt_error.html#x"><span class="id" title="variable">x</span></a><br/>
<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="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Logic.html#84eb6d2849dbf3581b1c0c05add5f2d8"><span class="id" title="notation">∃</span></a> <span class="id" title="var">fx</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> (<a class="idref" href="Flocq.Prop.Fprop_div_sqrt_error.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.Prop.Fprop_div_sqrt_error.html#fx"><span class="id" title="variable">fx</span></a>)%<span class="id" title="var">R</span> <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.Prop.Fprop_div_sqrt_error.html#fx"><span class="id" title="variable">fx</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_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="Flocq.Prop.Fprop_div_sqrt_error.html#fexp"><span class="id" title="variable">fexp</span></a> <a class="idref" href="Flocq.Prop.Fprop_div_sqrt_error.html#x"><span class="id" title="variable">x</span></a>.<br/>
<br/>
<span class="id" title="keyword">Context</span> { <span class="id" title="var">prec_gt_0_</span> : <a class="idref" href="Flocq.Core.Fcore_FLX.html#Prec_gt_0"><span class="id" title="class">Prec_gt_0</span></a> <a class="idref" href="Flocq.Prop.Fprop_div_sqrt_error.html#Fprop_divsqrt_error.prec"><span class="id" title="variable">prec</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="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_FLX.html#FLX_exp"><span class="id" title="definition">FLX_exp</span></a> <a class="idref" href="Flocq.Prop.Fprop_div_sqrt_error.html#Fprop_divsqrt_error.prec"><span class="id" title="variable">prec</span></a>)).<br/>
<span class="id" title="keyword">Notation</span> <a name="cexp"><span class="id" title="abbreviation">cexp</span></a> := (<a class="idref" href="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="Flocq.Core.Fcore_FLX.html#FLX_exp"><span class="id" title="definition">FLX_exp</span></a> <a class="idref" href="Flocq.Prop.Fprop_div_sqrt_error.html#Fprop_divsqrt_error.prec"><span class="id" title="variable">prec</span></a>)).<br/>
<br/>
<span class="id" title="keyword">Variable</span> <a name="Fprop_divsqrt_error.choice"><span class="id" title="variable">choice</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> <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="http://coq.inria.fr/distrib/current/stdlib/Coq.Init.Datatypes.html#bool"><span class="id" title="inductive">bool</span></a>.<br/>
<br/>
</div>
<div class="doc">
Remainder of the division in FLX
</div>
<div class="code">
<span class="id" title="keyword">Theorem</span> <a name="div_error_FLX"><span class="id" title="lemma">div_error_FLX</span></a> :<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">rnd</span> { <span class="id" title="var">Zrnd</span> : <a class="idref" href="Flocq.Core.Fcore_generic_fmt.html#Valid_rnd"><span class="id" title="class">Valid_rnd</span></a> <a class="idref" href="Flocq.Prop.Fprop_div_sqrt_error.html#rnd"><span class="id" title="variable">rnd</span></a> } <span class="id" title="var">x</span> <span class="id" title="var">y</span>,<br/>
<a class="idref" href="Flocq.Prop.Fprop_div_sqrt_error.html#format"><span class="id" title="abbreviation">format</span></a> <a class="idref" href="Flocq.Prop.Fprop_div_sqrt_error.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.Prop.Fprop_div_sqrt_error.html#format"><span class="id" title="abbreviation">format</span></a> <a class="idref" href="Flocq.Prop.Fprop_div_sqrt_error.html#y"><span class="id" title="variable">y</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><br/>
<a class="idref" href="Flocq.Prop.Fprop_div_sqrt_error.html#format"><span class="id" title="abbreviation">format</span></a> (<a class="idref" href="Flocq.Prop.Fprop_div_sqrt_error.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#0f8d5ec6876eafaec2dca3581c24e3a0"><span class="id" title="notation">-</span></a> <a class="idref" href="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="Flocq.Core.Fcore_FLX.html#FLX_exp"><span class="id" title="definition">FLX_exp</span></a> <a class="idref" href="Flocq.Prop.Fprop_div_sqrt_error.html#Fprop_divsqrt_error.prec"><span class="id" title="variable">prec</span></a>) <a class="idref" href="Flocq.Prop.Fprop_div_sqrt_error.html#rnd"><span class="id" title="variable">rnd</span></a> (<a class="idref" href="Flocq.Prop.Fprop_div_sqrt_error.html#x"><span class="id" title="variable">x</span></a><a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#0c0207ff801f7c795ce431f130ae670a"><span class="id" title="notation">/</span></a><a class="idref" href="Flocq.Prop.Fprop_div_sqrt_error.html#y"><span class="id" title="variable">y</span></a>) <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#361afbb9b173fae6251865f2da226003"><span class="id" title="notation">×</span></a> <a class="idref" href="Flocq.Prop.Fprop_div_sqrt_error.html#y"><span class="id" title="variable">y</span></a>)%<span class="id" title="var">R</span>.<br/>
<br/>
</div>
<div class="doc">
Remainder of the square in FLX (with p>1) and rounding to nearest
</div>
<div class="code">
<span class="id" title="keyword">Variable</span> <a name="Fprop_divsqrt_error.Hp1"><span class="id" title="variable">Hp1</span></a> : <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.ZArith.BinInt.html#Zlt"><span class="id" title="abbreviation">Zlt</span></a> 1 <a class="idref" href="Flocq.Prop.Fprop_div_sqrt_error.html#Fprop_divsqrt_error.prec"><span class="id" title="variable">prec</span></a>.<br/>
<br/>
<span class="id" title="keyword">Theorem</span> <a name="sqrt_error_FLX_N"><span class="id" title="lemma">sqrt_error_FLX_N</span></a> :<br/>
<span class="id" title="keyword">∀</span> <span class="id" title="var">x</span>, <a class="idref" href="Flocq.Prop.Fprop_div_sqrt_error.html#format"><span class="id" title="abbreviation">format</span></a> <a class="idref" href="Flocq.Prop.Fprop_div_sqrt_error.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><br/>
<a class="idref" href="Flocq.Prop.Fprop_div_sqrt_error.html#format"><span class="id" title="abbreviation">format</span></a> (<a class="idref" href="Flocq.Prop.Fprop_div_sqrt_error.html#x"><span class="id" title="variable">x</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.Rdefinitions.html#0f8d5ec6876eafaec2dca3581c24e3a0"><span class="id" title="notation">-</span></a> <a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.RIneq.html#Rsqr"><span class="id" title="definition">Rsqr</span></a> (<a class="idref" href="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="Flocq.Core.Fcore_FLX.html#FLX_exp"><span class="id" title="definition">FLX_exp</span></a> <a class="idref" href="Flocq.Prop.Fprop_div_sqrt_error.html#Fprop_divsqrt_error.prec"><span class="id" title="variable">prec</span></a>) (<a class="idref" href="Flocq.Core.Fcore_generic_fmt.html#Znearest"><span class="id" title="definition">Znearest</span></a> <a class="idref" href="Flocq.Prop.Fprop_div_sqrt_error.html#Fprop_divsqrt_error.choice"><span class="id" title="variable">choice</span></a>) (<a class="idref" href="http://coq.inria.fr/distrib/current/stdlib/Coq.Reals.R_sqrt.html#sqrt"><span class="id" title="definition">sqrt</span></a> <a class="idref" href="Flocq.Prop.Fprop_div_sqrt_error.html#x"><span class="id" title="variable">x</span></a>)))%<span class="id" title="var">R</span>.<br/>
<br/>
<span class="id" title="keyword">End</span> <a class="idref" href="Flocq.Prop.Fprop_div_sqrt_error.html#Fprop_divsqrt_error"><span class="id" title="section">Fprop_divsqrt_error</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>
