1<html> 2<head> 3<meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> 4<title>Sets</title> 5<link rel="stylesheet" href="../../../../../../doc/src/boostbook.css" type="text/css"> 6<meta name="generator" content="DocBook XSL Stylesheets V1.79.1"> 7<link rel="home" href="../../index.html" title="Chapter 1. Boost.Icl"> 8<link rel="up" href="../semantics.html" title="Semantics"> 9<link rel="prev" href="../semantics.html" title="Semantics"> 10<link rel="next" href="maps.html" title="Maps"> 11</head> 12<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> 13<table cellpadding="2" width="100%"><tr> 14<td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../boost.png"></td> 15<td align="center"><a href="../../../../../../index.html">Home</a></td> 16<td align="center"><a href="../../../../../libraries.htm">Libraries</a></td> 17<td align="center"><a href="http://www.boost.org/users/people.html">People</a></td> 18<td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td> 19<td align="center"><a href="../../../../../../more/index.htm">More</a></td> 20</tr></table> 21<hr> 22<div class="spirit-nav"> 23<a accesskey="p" href="../semantics.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../semantics.html"><img src="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="maps.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> 24</div> 25<div class="section"> 26<div class="titlepage"><div><div><h3 class="title"> 27<a name="boost_icl.semantics.sets"></a><a class="link" href="sets.html" title="Sets">Sets</a> 28</h3></div></div></div> 29<p> 30 For all set types <code class="computeroutput"><span class="identifier">S</span></code> that 31 are models concept <code class="computeroutput"><span class="identifier">Set</span></code> (<a href="http://www.cplusplus.com/reference/stl/set/" target="_top"><code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">set</span></code> 32 </a>, <code class="computeroutput"><a class="link" href="../../boost/icl/interval_set.html" title="Class template interval_set">interval_set</a></code>, 33 <code class="computeroutput"><a class="link" href="../../boost/icl/separate_interval_set.html" title="Class template separate_interval_set">separate_interval_set</a></code> 34 and <code class="computeroutput"><a class="link" href="../../boost/icl/split_interval_set.html" title="Class template split_interval_set">split_interval_set</a></code>) 35 most of the well known mathematical <a href="http://en.wikipedia.org/wiki/Algebra_of_sets" target="_top">laws 36 on sets</a> were successfully checked via LaBatea. The next tables are 37 giving an overview over the checked laws ordered by operations. If possible, 38 the laws are formulated with the stronger lexicographical equality (<code class="computeroutput"><span class="keyword">operator</span> <span class="special">==</span></code>) 39 which implies the law's validity for the weaker element equality <code class="computeroutput"><span class="identifier">is_element_equal</span></code>. Throughout this chapter 40 we will denote element equality as <code class="computeroutput"><span class="special">=</span><span class="identifier">e</span><span class="special">=</span></code> instead 41 of <code class="computeroutput"><span class="identifier">is_element_equal</span></code> where 42 a short notation is advantageous. 43 </p> 44<h6> 45<a name="boost_icl.semantics.sets.h0"></a> 46 <span class="phrase"><a name="boost_icl.semantics.sets.laws_on_set_union"></a></span><a class="link" href="sets.html#boost_icl.semantics.sets.laws_on_set_union">Laws 47 on set union</a> 48 </h6> 49<p> 50 For the operation <span class="emphasis"><em><span class="bold"><strong>set union</strong></span></em></span> 51 available as <code class="computeroutput"><span class="keyword">operator</span> <span class="special">+,</span> 52 <span class="special">+=,</span> <span class="special">|,</span> <span class="special">|=</span></code> and the neutral element <code class="computeroutput"><span class="identifier">identity_element</span><span class="special"><</span><span class="identifier">S</span><span class="special">>::</span><span class="identifier">value</span><span class="special">()</span></code> 53 which is the empty set <code class="computeroutput"><span class="identifier">S</span><span class="special">()</span></code> these laws hold: 54</p> 55<pre class="programlisting"><span class="identifier">Associativity</span><span class="special"><</span><span class="identifier">S</span><span class="special">,+,==</span> <span class="special">>:</span> <span class="identifier">S</span> <span class="identifier">a</span><span class="special">,</span><span class="identifier">b</span><span class="special">,</span><span class="identifier">c</span><span class="special">;</span> <span class="identifier">a</span><span class="special">+(</span><span class="identifier">b</span><span class="special">+</span><span class="identifier">c</span><span class="special">)</span> <span class="special">==</span> <span class="special">(</span><span class="identifier">a</span><span class="special">+</span><span class="identifier">b</span><span class="special">)+</span><span class="identifier">c</span> 56<span class="identifier">Neutrality</span><span class="special"><</span><span class="identifier">S</span><span class="special">,+,==</span> <span class="special">></span> <span class="special">:</span> <span class="identifier">S</span> <span class="identifier">a</span><span class="special">;</span> <span class="identifier">a</span><span class="special">+</span><span class="identifier">S</span><span class="special">()</span> <span class="special">==</span> <span class="identifier">a</span> 57<span class="identifier">Commutativity</span><span class="special"><</span><span class="identifier">S</span><span class="special">,+,==</span> <span class="special">>:</span> <span class="identifier">S</span> <span class="identifier">a</span><span class="special">,</span><span class="identifier">b</span><span class="special">;</span> <span class="identifier">a</span><span class="special">+</span><span class="identifier">b</span> <span class="special">==</span> <span class="identifier">b</span><span class="special">+</span><span class="identifier">a</span> 58</pre> 59<p> 60 </p> 61<h6> 62<a name="boost_icl.semantics.sets.h1"></a> 63 <span class="phrase"><a name="boost_icl.semantics.sets.laws_on_set_intersection"></a></span><a class="link" href="sets.html#boost_icl.semantics.sets.laws_on_set_intersection">Laws 64 on set intersection</a> 65 </h6> 66<p> 67 For the operation <span class="emphasis"><em><span class="bold"><strong>set intersection</strong></span></em></span> 68 available as <code class="computeroutput"><span class="keyword">operator</span> <span class="special">&,</span> 69 <span class="special">&=</span></code> these laws were validated: 70 </p> 71<p> 72</p> 73<pre class="programlisting"><span class="identifier">Associativity</span><span class="special"><</span><span class="identifier">S</span><span class="special">,&,==</span> <span class="special">>:</span> <span class="identifier">S</span> <span class="identifier">a</span><span class="special">,</span><span class="identifier">b</span><span class="special">,</span><span class="identifier">c</span><span class="special">;</span> <span class="identifier">a</span><span class="special">&(</span><span class="identifier">b</span><span class="special">&</span><span class="identifier">c</span><span class="special">)</span> <span class="special">==</span> <span class="special">(</span><span class="identifier">a</span><span class="special">&</span><span class="identifier">b</span><span class="special">)&</span><span class="identifier">c</span> 74<span class="identifier">Commutativity</span><span class="special"><</span><span class="identifier">S</span><span class="special">,&,==</span> <span class="special">>:</span> <span class="identifier">S</span> <span class="identifier">a</span><span class="special">,</span><span class="identifier">b</span><span class="special">;</span> <span class="identifier">a</span><span class="special">&</span><span class="identifier">b</span> <span class="special">==</span> <span class="identifier">b</span><span class="special">&</span><span class="identifier">a</span> 75</pre> 76<p> 77 </p> 78<h6> 79<a name="boost_icl.semantics.sets.h2"></a> 80 <span class="phrase"><a name="boost_icl.semantics.sets.laws_on_set_difference"></a></span><a class="link" href="sets.html#boost_icl.semantics.sets.laws_on_set_difference">Laws 81 on set difference</a> 82 </h6> 83<p> 84 For set difference there are only these laws. It is not associative and not 85 commutative. It's neutrality is non symmetrical. 86 </p> 87<p> 88</p> 89<pre class="programlisting"><span class="identifier">RightNeutrality</span><span class="special"><</span><span class="identifier">S</span><span class="special">,-,==</span> <span class="special">></span> <span class="special">:</span> <span class="identifier">S</span> <span class="identifier">a</span><span class="special">;</span> <span class="identifier">a</span><span class="special">-</span><span class="identifier">S</span><span class="special">()</span> <span class="special">==</span> <span class="identifier">a</span> 90<span class="identifier">Inversion</span><span class="special"><</span><span class="identifier">S</span><span class="special">,-,==</span> <span class="special">>:</span> <span class="identifier">S</span> <span class="identifier">a</span><span class="special">;</span> <span class="identifier">a</span> <span class="special">-</span> <span class="identifier">a</span> <span class="special">==</span> <span class="identifier">S</span><span class="special">()</span> 91</pre> 92<p> 93 </p> 94<p> 95 Summarized in the next table are laws that use <code class="computeroutput"><span class="special">+</span></code>, 96 <code class="computeroutput"><span class="special">&</span></code> and <code class="computeroutput"><span class="special">-</span></code> 97 as a single operation. For all validated laws, the left and right hand sides 98 of the equations are lexicographically equal, as denoted by <code class="computeroutput"><span class="special">==</span></code> in the cells of the table. 99 </p> 100<p> 101</p> 102<pre class="programlisting"> <span class="special">+</span> <span class="special">&</span> <span class="special">-</span> 103<span class="identifier">Associativity</span> <span class="special">==</span> <span class="special">==</span> 104<span class="identifier">Neutrality</span> <span class="special">==</span> <span class="special">==</span> 105<span class="identifier">Commutativity</span> <span class="special">==</span> <span class="special">==</span> 106<span class="identifier">Inversion</span> <span class="special">==</span> 107</pre> 108<p> 109 </p> 110<h6> 111<a name="boost_icl.semantics.sets.h3"></a> 112 <span class="phrase"><a name="boost_icl.semantics.sets.distributivity_laws"></a></span><a class="link" href="sets.html#boost_icl.semantics.sets.distributivity_laws">Distributivity 113 Laws</a> 114 </h6> 115<p> 116 Laws, like distributivity, that use more than one operation can sometimes 117 be instantiated for different sequences of operators as can be seen below. 118 In the two instantiations of the distributivity laws operators <code class="computeroutput"><span class="special">+</span></code> and <code class="computeroutput"><span class="special">&</span></code> 119 are swapped. So we can have small operator signatures like <code class="computeroutput"><span class="special">+,&</span></code> and <code class="computeroutput"><span class="special">&,+</span></code> 120 to describe such instantiations, which will be used below. Not all instances 121 of distributivity laws hold for lexicographical equality. Therefore they 122 are denoted using a <span class="emphasis"><em>variable</em></span> equality <code class="computeroutput"><span class="special">=</span><span class="identifier">v</span><span class="special">=</span></code> 123 below. 124 </p> 125<p> 126</p> 127<pre class="programlisting"> <span class="identifier">Distributivity</span><span class="special"><</span><span class="identifier">S</span><span class="special">,+,&,=</span><span class="identifier">v</span><span class="special">=</span> <span class="special">></span> <span class="special">:</span> <span class="identifier">S</span> <span class="identifier">a</span><span class="special">,</span><span class="identifier">b</span><span class="special">,</span><span class="identifier">c</span><span class="special">;</span> <span class="identifier">a</span> <span class="special">+</span> <span class="special">(</span><span class="identifier">b</span> <span class="special">&</span> <span class="identifier">c</span><span class="special">)</span> <span class="special">=</span><span class="identifier">v</span><span class="special">=</span> <span class="special">(</span><span class="identifier">a</span> <span class="special">+</span> <span class="identifier">b</span><span class="special">)</span> <span class="special">&</span> <span class="special">(</span><span class="identifier">a</span> <span class="special">+</span> <span class="identifier">c</span><span class="special">)</span> 128 <span class="identifier">Distributivity</span><span class="special"><</span><span class="identifier">S</span><span class="special">,&,+,=</span><span class="identifier">v</span><span class="special">=</span> <span class="special">></span> <span class="special">:</span> <span class="identifier">S</span> <span class="identifier">a</span><span class="special">,</span><span class="identifier">b</span><span class="special">,</span><span class="identifier">c</span><span class="special">;</span> <span class="identifier">a</span> <span class="special">&</span> <span class="special">(</span><span class="identifier">b</span> <span class="special">+</span> <span class="identifier">c</span><span class="special">)</span> <span class="special">=</span><span class="identifier">v</span><span class="special">=</span> <span class="special">(</span><span class="identifier">a</span> <span class="special">&</span> <span class="identifier">b</span><span class="special">)</span> <span class="special">+</span> <span class="special">(</span><span class="identifier">a</span> <span class="special">&</span> <span class="identifier">c</span><span class="special">)</span> 129<span class="identifier">RightDistributivity</span><span class="special"><</span><span class="identifier">S</span><span class="special">,+,-,=</span><span class="identifier">v</span><span class="special">=</span> <span class="special">></span> <span class="special">:</span> <span class="identifier">S</span> <span class="identifier">a</span><span class="special">,</span><span class="identifier">b</span><span class="special">,</span><span class="identifier">c</span><span class="special">;</span> <span class="special">(</span><span class="identifier">a</span> <span class="special">+</span> <span class="identifier">b</span><span class="special">)</span> <span class="special">-</span> <span class="identifier">c</span> <span class="special">=</span><span class="identifier">v</span><span class="special">=</span> <span class="special">(</span><span class="identifier">a</span> <span class="special">-</span> <span class="identifier">c</span><span class="special">)</span> <span class="special">+</span> <span class="special">(</span><span class="identifier">b</span> <span class="special">-</span> <span class="identifier">c</span><span class="special">)</span> 130<span class="identifier">RightDistributivity</span><span class="special"><</span><span class="identifier">S</span><span class="special">,&,-,=</span><span class="identifier">v</span><span class="special">=</span> <span class="special">></span> <span class="special">:</span> <span class="identifier">S</span> <span class="identifier">a</span><span class="special">,</span><span class="identifier">b</span><span class="special">,</span><span class="identifier">c</span><span class="special">;</span> <span class="special">(</span><span class="identifier">a</span> <span class="special">&</span> <span class="identifier">b</span><span class="special">)</span> <span class="special">-</span> <span class="identifier">c</span> <span class="special">=</span><span class="identifier">v</span><span class="special">=</span> <span class="special">(</span><span class="identifier">a</span> <span class="special">-</span> <span class="identifier">c</span><span class="special">)</span> <span class="special">&</span> <span class="special">(</span><span class="identifier">b</span> <span class="special">-</span> <span class="identifier">c</span><span class="special">)</span> 131</pre> 132<p> 133 </p> 134<p> 135 The next table shows the relationship between law instances, <a class="link" href="../../index.html#boost_icl.introduction.interval_combining_styles" title="Interval Combining Styles">interval 136 combining style</a> and the used equality relation. 137 </p> 138<p> 139</p> 140<pre class="programlisting"> <span class="special">+,&</span> <span class="special">&,+</span> 141 <span class="identifier">Distributivity</span> <span class="identifier">joining</span> <span class="special">==</span> <span class="special">==</span> 142 <span class="identifier">separating</span> <span class="special">==</span> <span class="special">==</span> 143 <span class="identifier">splitting</span> <span class="special">=</span><span class="identifier">e</span><span class="special">=</span> <span class="special">=</span><span class="identifier">e</span><span class="special">=</span> 144 145 <span class="special">+,-</span> <span class="special">&,-</span> 146<span class="identifier">RightDistributivity</span> <span class="identifier">joining</span> <span class="special">==</span> <span class="special">==</span> 147 <span class="identifier">separating</span> <span class="special">==</span> <span class="special">==</span> 148 <span class="identifier">splitting</span> <span class="special">=</span><span class="identifier">e</span><span class="special">=</span> <span class="special">==</span> 149</pre> 150<p> 151 </p> 152<p> 153 The table gives an overview over 12 instantiations of the four distributivity 154 laws and shows the equalities which the instantiations holds for. For instance 155 <code class="computeroutput"><span class="identifier">RightDistributivity</span></code> with 156 operator signature <code class="computeroutput"><span class="special">+,-</span></code> instantiated 157 for <code class="computeroutput"><a class="link" href="../../boost/icl/split_interval_set.html" title="Class template split_interval_set">split_interval_sets</a></code> 158 holds only for element equality (denoted as <code class="computeroutput"><span class="special">=</span><span class="identifier">e</span><span class="special">=</span></code>): 159</p> 160<pre class="programlisting"><span class="identifier">RightDistributivity</span><span class="special"><</span><span class="identifier">S</span><span class="special">,+,-,=</span><span class="identifier">e</span><span class="special">=</span> <span class="special">></span> <span class="special">:</span> <span class="identifier">S</span> <span class="identifier">a</span><span class="special">,</span><span class="identifier">b</span><span class="special">,</span><span class="identifier">c</span><span class="special">;</span> <span class="special">(</span><span class="identifier">a</span> <span class="special">+</span> <span class="identifier">b</span><span class="special">)</span> <span class="special">-</span> <span class="identifier">c</span> <span class="special">=</span><span class="identifier">e</span><span class="special">=</span> <span class="special">(</span><span class="identifier">a</span> <span class="special">-</span> <span class="identifier">c</span><span class="special">)</span> <span class="special">+</span> <span class="special">(</span><span class="identifier">b</span> <span class="special">-</span> <span class="identifier">c</span><span class="special">)</span> 161</pre> 162<p> 163 The remaining five instantiations of <code class="computeroutput"><span class="identifier">RightDistributivity</span></code> 164 are valid for lexicographical equality (demoted as <code class="computeroutput"><span class="special">==</span></code>) 165 as well. 166 </p> 167<p> 168 <a class="link" href="../../index.html#boost_icl.introduction.interval_combining_styles" title="Interval Combining Styles">Interval 169 combining styles</a> correspond to containers according to 170</p> 171<pre class="programlisting"><span class="identifier">style</span> <span class="identifier">set</span> 172<span class="identifier">joining</span> <span class="identifier">interval_set</span> 173<span class="identifier">separating</span> <span class="identifier">separate_interval_set</span> 174<span class="identifier">splitting</span> <span class="identifier">split_interval_set</span> 175</pre> 176<p> 177 </p> 178<p> 179 Finally there are two laws that combine all three major set operations: De 180 Mogans Law and Symmetric Difference. 181 </p> 182<h6> 183<a name="boost_icl.semantics.sets.h4"></a> 184 <span class="phrase"><a name="boost_icl.semantics.sets.demorgan_s_law"></a></span><a class="link" href="sets.html#boost_icl.semantics.sets.demorgan_s_law">DeMorgan's 185 Law</a> 186 </h6> 187<p> 188 De Morgans Law is better known in an incarnation where the unary complement 189 operation <code class="computeroutput"><span class="special">~</span></code> is used. <code class="computeroutput"><span class="special">~(</span><span class="identifier">a</span><span class="special">+</span><span class="identifier">b</span><span class="special">)</span> <span class="special">==</span> 190 <span class="special">~</span><span class="identifier">a</span> <span class="special">*</span> <span class="special">~</span><span class="identifier">b</span></code>. 191 The version below is an adaption for the binary set difference <code class="computeroutput"><span class="special">-</span></code>, which is also called <span class="emphasis"><em><span class="bold"><strong>relative complement</strong></span></em></span>. 192</p> 193<pre class="programlisting"><span class="identifier">DeMorgan</span><span class="special"><</span><span class="identifier">S</span><span class="special">,+,&,=</span><span class="identifier">v</span><span class="special">=</span> <span class="special">></span> <span class="special">:</span> <span class="identifier">S</span> <span class="identifier">a</span><span class="special">,</span><span class="identifier">b</span><span class="special">,</span><span class="identifier">c</span><span class="special">;</span> <span class="identifier">a</span> <span class="special">-</span> <span class="special">(</span><span class="identifier">b</span> <span class="special">+</span> <span class="identifier">c</span><span class="special">)</span> <span class="special">=</span><span class="identifier">v</span><span class="special">=</span> <span class="special">(</span><span class="identifier">a</span> <span class="special">-</span> <span class="identifier">b</span><span class="special">)</span> <span class="special">&</span> <span class="special">(</span><span class="identifier">a</span> <span class="special">-</span> <span class="identifier">c</span><span class="special">)</span> 194<span class="identifier">DeMorgan</span><span class="special"><</span><span class="identifier">S</span><span class="special">,&,+,=</span><span class="identifier">v</span><span class="special">=</span> <span class="special">></span> <span class="special">:</span> <span class="identifier">S</span> <span class="identifier">a</span><span class="special">,</span><span class="identifier">b</span><span class="special">,</span><span class="identifier">c</span><span class="special">;</span> <span class="identifier">a</span> <span class="special">-</span> <span class="special">(</span><span class="identifier">b</span> <span class="special">&</span> <span class="identifier">c</span><span class="special">)</span> <span class="special">=</span><span class="identifier">v</span><span class="special">=</span> <span class="special">(</span><span class="identifier">a</span> <span class="special">-</span> <span class="identifier">b</span><span class="special">)</span> <span class="special">+</span> <span class="special">(</span><span class="identifier">a</span> <span class="special">-</span> <span class="identifier">c</span><span class="special">)</span> 195</pre> 196<p> 197 </p> 198<p> 199</p> 200<pre class="programlisting"> <span class="special">+,&</span> <span class="special">&,+</span> 201<span class="identifier">DeMorgan</span> <span class="identifier">joining</span> <span class="special">==</span> <span class="special">==</span> 202 <span class="identifier">separating</span> <span class="special">==</span> <span class="special">=</span><span class="identifier">e</span><span class="special">=</span> 203 <span class="identifier">splitting</span> <span class="special">==</span> <span class="special">=</span><span class="identifier">e</span><span class="special">=</span> 204</pre> 205<p> 206 </p> 207<p> 208 Again not all law instances are valid for lexicographical equality. The second 209 instantiations only holds for element equality, if the interval sets are 210 non joining. 211 </p> 212<h6> 213<a name="boost_icl.semantics.sets.h5"></a> 214 <span class="phrase"><a name="boost_icl.semantics.sets.symmetric_difference"></a></span><a class="link" href="sets.html#boost_icl.semantics.sets.symmetric_difference">Symmetric 215 Difference</a> 216 </h6> 217<p> 218</p> 219<pre class="programlisting"><span class="identifier">SymmetricDifference</span><span class="special"><</span><span class="identifier">S</span><span class="special">,==</span> <span class="special">></span> <span class="special">:</span> <span class="identifier">S</span> <span class="identifier">a</span><span class="special">,</span><span class="identifier">b</span><span class="special">,</span><span class="identifier">c</span><span class="special">;</span> <span class="special">(</span><span class="identifier">a</span> <span class="special">+</span> <span class="identifier">b</span><span class="special">)</span> <span class="special">-</span> <span class="special">(</span><span class="identifier">a</span> <span class="special">&</span> <span class="identifier">b</span><span class="special">)</span> <span class="special">==</span> <span class="special">(</span><span class="identifier">a</span> <span class="special">-</span> <span class="identifier">b</span><span class="special">)</span> <span class="special">+</span> <span class="special">(</span><span class="identifier">b</span> <span class="special">-</span> <span class="identifier">a</span><span class="special">)</span> 220</pre> 221<p> 222 </p> 223<p> 224 Finally Symmetric Difference holds for all of icl set types and lexicographical 225 equality. 226 </p> 227</div> 228<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> 229<td align="left"></td> 230<td align="right"><div class="copyright-footer">Copyright © 2007-2010 Joachim 231 Faulhaber<br>Copyright © 1999-2006 Cortex Software 232 GmbH<p> 233 Distributed under the Boost Software License, Version 1.0. (See accompanying 234 file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>) 235 </p> 236</div></td> 237</tr></table> 238<hr> 239<div class="spirit-nav"> 240<a accesskey="p" href="../semantics.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../semantics.html"><img src="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="maps.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> 241</div> 242</body> 243</html> 244
