Quellcodebibliothek Statistik Leitseite products/sources/formale Sprachen/GAP/pkg/datastructures/doc/   (Algebra von RWTH Aachen Version 4.15.1©)  Datei vom 14.9.2025 mit Größe 21 kB image not shown  

Quelle  chap10_mj.html   Sprache: HTML

 
 products/sources/formale Sprachen/GAP/pkg/datastructures/doc/chap10_mj.html


<?xml version="1.0" encoding="UTF-8"?>

<!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" xml:lang="en">
<head>
<script type="text/javascript"
  src="https://cdn.jsdelivr.net/npm/mathjax@2/MathJax.js?config=TeX-AMS-MML_HTMLorMML">
</script>
<title>GAP (datastructures) - Chapter 10: Ordered Set Datastructures</title>
<meta http-equiv="content-type" content="text/html; charset=UTF-8" />
<meta name="generator" content="GAPDoc2HTML" />
<link rel="stylesheet" type="text/css" href="manual.css" />
<script src="manual.js" type="text/javascript"></script>
<script type="text/javascript">overwriteStyle();</script>
</head>
<body class="chap10"  onload="jscontent()">


<div class="chlinktop"><span class="chlink1">Goto Chapter: </span><a href="chap0_mj.html">Top</a>  <a href="chap1_mj.html">1</a>  <a href="chap2_mj.html">2</a>  <a href="chap3_mj.html">3</a>  <a href="chap4_mj.html">4</a>  <a href="chap5_mj.html">5</a>  <a href="chap6_mj.html">6</a>  <a href="chap7_mj.html">7</a>  <a href="chap8_mj.html">8</a>  <a href="chap9_mj.html">9</a>  <a href="chap10_mj.html">10</a>  <a href="chap11_mj.html">11</a>  <a href="chap12_mj.html">12</a>  <a href="chapBib_mj.html">Bib</a>  <a href="chapInd_mj.html">Ind</a>  </div>

<div class="chlinkprevnexttop"> <a href="chap0_mj.html">[Top of Book]</a>   <a href="chap0_mj.html#contents">[Contents]</a>    <a href="chap9_mj.html">[Previous Chapter]</a>    <a href="chap11_mj.html">[Next Chapter]</a>   </div>

<p id="mathjaxlink" class="pcenter"><a href="chap10.html">[MathJax off]</a></p>
<p><a id="X8134174379D8CD1E" name="X8134174379D8CD1E"></a></p>
<div class="ChapSects"><a href="chap10_mj.html#X8134174379D8CD1E">10 <span class="Heading">Ordered Set Datastructures</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap10_mj.html#X86A9B6F87E619FFF">10.1 <span class="Heading">Usage</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap10_mj.html#X7C5F33687C53BEF0">10.2 <span class="Heading">API</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10_mj.html#X829BF6468120FCF4">10.2-1 IsOrderedSetDS</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10_mj.html#X83DF983D824ADA52">10.2-2 IsStandardOrderedSetDS</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10_mj.html#X7C0A0ACD833B09E8">10.2-3 OrderedSetDS</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10_mj.html#X784F06667B9CAE61">10.2-4 OrderedSetDS</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10_mj.html#X82EBF9967A1BC1F4">10.2-5 OrderedSetDS</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10_mj.html#X7B3E75657C8758DA">10.2-6 OrderedSetDS</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10_mj.html#X7EE9AD66864EFD95">10.2-7 OrderedSetDS</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10_mj.html#X84341B117B22B200">10.2-8 OrderedSetDS</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10_mj.html#X87CC3ADD8273BDDC">10.2-9 OrderedSetDS</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10_mj.html#X785F29D484E84A9B">10.2-10 AddSet</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10_mj.html#X79CA0BD27822C1FB">10.2-11 RemoveSet</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10_mj.html#X7DD7225D7A6F92C6"><code>10.2-12 \in</code></a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10_mj.html#X85F4EDA0841AADFD">10.2-13 LessFunction</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10_mj.html#X7B2E5BB28435AE50">10.2-14 Size</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10_mj.html#X83A682A08318DBA0">10.2-15 IteratorSorted</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap10_mj.html#X8373B442852AB64B">10.3 <span class="Heading">Default methods</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10_mj.html#X805D48307D3FBC57">10.3-1 Iterator</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10_mj.html#X7BA094DE791ADBC1">10.3-2 AsSSortedList</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10_mj.html#X7CEA1C9283A69F26">10.3-3 AsSortedList</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10_mj.html#X84A6178D7F4FEF35">10.3-4 AsList</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10_mj.html#X7B0571977A8480EA">10.3-5 EnumeratorSorted</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10_mj.html#X786CD5B57A439204">10.3-6 Enumerator</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10_mj.html#X7DD174FE79F86562">10.3-7 IsEmpty</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10_mj.html#X78B39A307B3B8879">10.3-8 Length</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10_mj.html#X81CAD5FC819FD0AA">10.3-9 Position</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10_mj.html#X7B74B4107A33E68A">10.3-10 PositionSortedOp</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10_mj.html#X7FDCE67179997482">10.3-11 PositionSortedOp</a></span>
</div></div>
</div>

<h3>10 <span class="Heading">Ordered Set Datastructures</span></h3>

<p>In this chapter we deal with datastructures designed to represent sets of objects which have an intrinsic ordering. Such datastructures should support fast (possibly amortised) <span class="SimpleMath">\(O(\log n)\)</span> addition, deletion and membership test operations and allow efficient iteration through all the objects in the datastructure in the order determined by the given comparison function. Since they represent a set, adding an object equal to one already present has no effect.</p>

<p>We refer to these as ordered set <em>datastructure</em> because the differ from the <strong class="pkg">GAP</strong> notion of a set in a number of ways:</p>


<ul>
<li><p>They all lie in a common family <code class="code">OrderedSetDSFamily</code> and pay no attention to the families of the objects stored in them.</p>

</li>
<li><p>Equality of these structures is by identity, not equality of the represented set</p>

</li>
<li><p>The ordering of the objects in the set does not have to be default <strong class="pkg">GAP</strong> ordering "less than", but is determined by the attribute <code class="func">LessFunction</code> (<a href="chap10_mj.html#X85F4EDA0841AADFD"><span class="RefLink">10.2-13</span></a>)</p>

</li>
</ul>
<p>Three implementations of ordered set data structures are currently included: skiplists, binary search trees and (as a specialisation of binary search trees) AVL trees. AVL trees seem to be the fastest in general, and memory usage is similar. More details to come</p>

<p><a id="X86A9B6F87E619FFF" name="X86A9B6F87E619FFF"></a></p>

<h4>10.1 <span class="Heading">Usage</span></h4>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">s := OrderedSetDS(IsSkipListRep, {x,y} -> String(x) < String(y));</span>
<skiplist 0 entries>
<span class="GAPprompt">gap></span> <span class="GAPinput">Addset(s, 1);</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">AddSet(s, 2);</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">AddSet(s, 10);</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">AddSet(s, (1,2,3));</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">RemoveSet(s, (1,2,3));</span>
1
<span class="GAPprompt">gap></span> <span class="GAPinput">AsListSorted(s);</span>
[ 1, 10, 2 ]

<span class="GAPprompt">gap></span> <span class="GAPinput">b := OrderedSetDS(IsBinarySearchTreeRep, Primes);</span>
<bst size 168>
<span class="GAPprompt">gap></span> <span class="GAPinput">91 in b;</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">97 in b;</span>
true
</pre></div>

<p><a id="X7C5F33687C53BEF0" name="X7C5F33687C53BEF0"></a></p>

<h4>10.2 <span class="Heading">API</span></h4>

<p>Every implementation of an ordered set datastructure must follow the API set out below</p>

<p><a id="X829BF6468120FCF4" name="X829BF6468120FCF4"></a></p>

<h5>10.2-1 IsOrderedSetDS</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsOrderedSetDS</code>( <var class="Arg">arg</var> )</td><td class="tdright">( filter )</td></tr></table></div>
<p>Returns: <code class="keyw">true</code> or <code class="keyw">false</code></p>

<p>Category of ordered set.</p>

<p><a id="X83DF983D824ADA52" name="X83DF983D824ADA52"></a></p>

<h5>10.2-2 IsStandardOrderedSetDS</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsStandardOrderedSetDS</code>( <var class="Arg">arg</var> )</td><td class="tdright">( filter )</td></tr></table></div>
<p>Returns: <code class="keyw">true</code> or <code class="keyw">false</code></p>

<p>Subcategory of ordered sets where the ordering is <strong class="pkg">GAP</strong>'s default <



<p><a id="X7C0A0ACD833B09E8" name="X7C0A0ACD833B09E8"></a></p>

<h5>10.2-3 OrderedSetDS</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ OrderedSetDS</code>( <var class="Arg">filter</var>[, <var class="Arg">lessThan</var>[, <var class="Arg">initialEntries</var>[, <var class="Arg">randomSource</var>]]] )</td><td class="tdright">( constructor )</td></tr></table></div>
<p>Returns: an ordered set datastructure</p>

<p>The family that contains all ordered set datastructures. Constructors for ordered sets</p>

<p>The argument <var class="Arg">filter</var> is a filter that the resulting ordered set object will have.</p>

<p>The optional argument <var class="Arg">lessThan</var> must be a binary function that returns <code class="keyw">true</code> if its first argument is less than its second argument, and <code class="keyw">false</code> otherwise. The default <var class="Arg">lessThan</var> is <strong class="pkg">GAP</strong>'s built in <.



<p>The optional argument <var class="Arg"> initialEntries</var> gives a collection of elements that the ordered set is initialised with, and defaults to the empty set.</p>

<p>The optional argument <var class="Arg">randomSource</var> is useful in a number of possible implementations that use randomised methods to achieve good amortised complexity with high probability and simple data structures. It defaults to the global Mersenne twister.</p>

<p><a id="X784F06667B9CAE61" name="X784F06667B9CAE61"></a></p>

<h5>10.2-4 OrderedSetDS</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ OrderedSetDS</code>( <var class="Arg">arg1</var>, <var class="Arg">arg2</var>, <var class="Arg">arg3</var> )</td><td class="tdright">( constructor )</td></tr></table></div>
<p><a id="X82EBF9967A1BC1F4" name="X82EBF9967A1BC1F4"></a></p>

<h5>10.2-5 OrderedSetDS</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ OrderedSetDS</code>( <var class="Arg">arg1</var>, <var class="Arg">arg2</var>, <var class="Arg">arg3</var> )</td><td class="tdright">( constructor )</td></tr></table></div>
<p><a id="X7B3E75657C8758DA" name="X7B3E75657C8758DA"></a></p>

<h5>10.2-6 OrderedSetDS</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ OrderedSetDS</code>( <var class="Arg">arg1</var>, <var class="Arg">arg2</var>, <var class="Arg">arg3</var> )</td><td class="tdright">( constructor )</td></tr></table></div>
<p><a id="X7EE9AD66864EFD95" name="X7EE9AD66864EFD95"></a></p>

<h5>10.2-7 OrderedSetDS</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ OrderedSetDS</code>( <var class="Arg">arg1</var>, <var class="Arg">arg2</var> )</td><td class="tdright">( constructor )</td></tr></table></div>
<p><a id="X84341B117B22B200" name="X84341B117B22B200"></a></p>

<h5>10.2-8 OrderedSetDS</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ OrderedSetDS</code>( <var class="Arg">arg1</var>, <var class="Arg">arg2</var> )</td><td class="tdright">( constructor )</td></tr></table></div>
<p><a id="X87CC3ADD8273BDDC" name="X87CC3ADD8273BDDC"></a></p>

<h5>10.2-9 OrderedSetDS</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ OrderedSetDS</code>( <var class="Arg">arg</var> )</td><td class="tdright">( constructor )</td></tr></table></div>
<p><a id="X785F29D484E84A9B" name="X785F29D484E84A9B"></a></p>

<h5>10.2-10 AddSet</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AddSet</code>( <var class="Arg">set</var>, <var class="Arg">object</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Other constructors cover making an ordered set from another ordered set, from an iterator, from a function and an iterator, or from a function, an iterator and a random source.</p>

<p>Adds <var class="Arg">object</var> to <var class="Arg">set</var>. Does nothing if <var class="Arg">object</var><code class="code">in</code><var class="Arg">set</var>set.</p>

<p><a id="X79CA0BD27822C1FB" name="X79CA0BD27822C1FB"></a></p>

<h5>10.2-11 RemoveSet</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RemoveSet</code>( <var class="Arg">set</var>, <var class="Arg">object</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: <code class="keyw">0</code> or <code class="keyw">1</code></p>

<p>Removes <var class="Arg">object</var> from <var class="Arg">set</var> if present, and returns the number of copies of <var class="Arg">object</var> that were in <var class="Arg">set</var>, that is <code class="keyw">0</code> or <code class="keyw">1</code>. This for consistency with multisets.</p>

<p><a id="X7DD7225D7A6F92C6" name="X7DD7225D7A6F92C6"></a></p>

<h5><code>10.2-12 \in</code></h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ \in</code>( <var class="Arg">object</var>, <var class="Arg">set</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>All objects in IsOrderedSetDS must implement \in, which returns <var class="Arg">true</var> if <var class="Arg">object</var> is present in <var class="Arg">set</var> and <code class="keyw">false</code> otherwise.</p>

<p><a id="X85F4EDA0841AADFD" name="X85F4EDA0841AADFD"></a></p>

<h5>10.2-13 LessFunction</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LessFunction</code>( <var class="Arg">set</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>The binary function to perform the comparison for elements of the set.</p>

<p><a id="X7B2E5BB28435AE50" name="X7B2E5BB28435AE50"></a></p>

<h5>10.2-14 Size</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Size</code>( <var class="Arg">set</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>The number of objects in the set</p>

<p><a id="X83A682A08318DBA0" name="X83A682A08318DBA0"></a></p>

<h5>10.2-15 IteratorSorted</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IteratorSorted</code>( <var class="Arg">set</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: iterator</p>

<p>Returns an iterator of <var class="Arg">set</var> that can be used to iterate through the elements of <var class="Arg">set</var> in the order imposed by <code class="func">LessFunction</code> (<a href="chap10_mj.html#X85F4EDA0841AADFD"><span class="RefLink">10.2-13</span></a>).</p>

<p><a id="X8373B442852AB64B" name="X8373B442852AB64B"></a></p>

<h4>10.3 <span class="Heading">Default methods</span></h4>

<p>Default methods based on <code class="func">IteratorSorted</code> (<a href="/home/runner/gap/doc/ref/chap30_mj.html#X8688C20B828FC129"><span class="RefLink">Reference: IteratorSorted</span></a>) are installed for the following operations and attributes, but can be overridden for data structures that support better algorithms.</p>

<p><a id="X805D48307D3FBC57" name="X805D48307D3FBC57"></a></p>

<h5>10.3-1 Iterator</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Iterator</code>( <var class="Arg">arg</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p><a id="X7BA094DE791ADBC1" name="X7BA094DE791ADBC1"></a></p>

<h5>10.3-2 AsSSortedList</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AsSSortedList</code>( <var class="Arg">arg</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p><a id="X7CEA1C9283A69F26" name="X7CEA1C9283A69F26"></a></p>

<h5>10.3-3 AsSortedList</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AsSortedList</code>( <var class="Arg">arg</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p><a id="X84A6178D7F4FEF35" name="X84A6178D7F4FEF35"></a></p>

<h5>10.3-4 AsList</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AsList</code>( <var class="Arg">arg</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p><a id="X7B0571977A8480EA" name="X7B0571977A8480EA"></a></p>

<h5>10.3-5 EnumeratorSorted</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ EnumeratorSorted</code>( <var class="Arg">arg</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p><a id="X786CD5B57A439204" name="X786CD5B57A439204"></a></p>

<h5>10.3-6 Enumerator</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Enumerator</code>( <var class="Arg">arg</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p><a id="X7DD174FE79F86562" name="X7DD174FE79F86562"></a></p>

<h5>10.3-7 IsEmpty</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsEmpty</code>( <var class="Arg">arg</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: <code class="keyw">true</code> or <code class="keyw">false</code></p>

<p><a id="X78B39A307B3B8879" name="X78B39A307B3B8879"></a></p>

<h5>10.3-8 Length</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Length</code>( <var class="Arg">arg</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p><a id="X81CAD5FC819FD0AA" name="X81CAD5FC819FD0AA"></a></p>

<h5>10.3-9 Position</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Position</code>( <var class="Arg">arg1</var>, <var class="Arg">arg2</var>, <var class="Arg">arg3</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p><a id="X7B74B4107A33E68A" name="X7B74B4107A33E68A"></a></p>

<h5>10.3-10 PositionSortedOp</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PositionSortedOp</code>( <var class="Arg">arg1</var>, <var class="Arg">arg2</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p><a id="X7FDCE67179997482" name="X7FDCE67179997482"></a></p>

<h5>10.3-11 PositionSortedOp</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PositionSortedOp</code>( <var class="Arg">arg1</var>, <var class="Arg">arg2</var>, <var class="Arg">arg3</var> )</td><td class="tdright">( operation )</td></tr></table></div>

<div class="chlinkprevnextbot"> <a href="chap0_mj.html">[Top of Book]</a>   <a href="chap0_mj.html#contents">[Contents]</a>    <a href="chap9_mj.html">[Previous Chapter]</a>    <a href="chap11_mj.html">[Next Chapter]</a>   </div>


<div class="chlinkbot"><span class="chlink1">Goto Chapter: </span><a href="chap0_mj.html">Top</a>  <a href="chap1_mj.html">1</a>  <a href="chap2_mj.html">2</a>  <a href="chap3_mj.html">3</a>  <a href="chap4_mj.html">4</a>  <a href="chap5_mj.html">5</a>  <a href="chap6_mj.html">6</a>  <a href="chap7_mj.html">7</a>  <a href="chap8_mj.html">8</a>  <a href="chap9_mj.html">9</a>  <a href="chap10_mj.html">10</a>  <a href="chap11_mj.html">11</a>  <a href="chap12_mj.html">12</a>  <a href="chapBib_mj.html">Bib</a>  <a href="chapInd_mj.html">Ind</a>  </div>

<hr />
<p class="foot">generated by <a href="https://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc">GAPDoc2HTML</a></p>
</body>
</html>

94%


¤ Dauer der Verarbeitung: 0.14 Sekunden  (vorverarbeitet)  ¤

*© Formatika GbR, Deutschland




Wurzel

Suchen

Beweissystem der NASA

Beweissystem Isabelle

NIST Cobol Testsuite

Cephes Mathematical Library

Wiener Entwicklungsmethode

Haftungshinweis

Die Informationen auf dieser Webseite wurden nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit, noch Qualität der bereit gestellten Informationen zugesichert.

Bemerkung:

Die farbliche Syntaxdarstellung ist noch experimentell.