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<div class="chlinkprevnexttop"> <a href="chap0.html">[Top of Book]</a> <a href="chap0.html#contents">[Contents]</a> <a href="chap29.html">[Previous Chapter]</a> <a href="chap31.html">[Next Chapter]</a> </div>
<p id="mathjaxlink" class="pcenter"><a href="chap30_mj.html">[MathJax on]</a></p>
<p><a id="X8050A8037984E5B6" name="X8050A8037984E5B6"></a></p>
<div class="ChapSects"><a href="chap30.html#X8050A8037984E5B6">30 <span class="Heading">Collections</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap30.html#X8084F03A78ABD4F8">30.1 <span class="Heading">IsCollection (Filter)</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap30.html#X79C9FC7F86E2738C">30.1-1 IsCollection</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap30.html#X85D8D8F684B02DDF">30.2 <span class="Heading">Collection Families</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap30.html#X84E5A67E87D8DD66">30.2-1 CollectionsFamily</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap30.html#X856AC2DF7F7CBAAF">30.2-2 IsCollectionFamily</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap30.html#X864BB3748546F63F">30.2-3 ElementsFamily</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap30.html#X78C38017804B2EA7">30.2-4 CategoryCollections</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap30.html#X7C3722DF8736FFDB">30.3 <span class="Heading">Lists and Collections</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap30.html#X877128A77826DD69">30.3-1 IsListOrCollection</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap30.html#X7EF8910F82B45EC7">30.3-2 Enumerator</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap30.html#X80CD7DDC7D0C60D5">30.3-3 EnumeratorSorted</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap30.html#X85E149177AC547C3">30.3-4 EnumeratorByFunctions</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap30.html#X7F12F40E87F3C3A7">30.3-5 List</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap30.html#X82CE157A7FAD8036">30.3-6 SortedList</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap30.html#X7E399AC97FD98217">30.3-7 SSortedList</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap30.html#X8289FCCC8274C89D">30.3-8 AsList</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap30.html#X7BCA5C6181391007">30.3-9 AsSortedList</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap30.html#X856D927378C33548">30.3-10 AsSSortedList</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap30.html#X79B130FC7906FB4C">30.3-11 Elements</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap30.html#X79AD18737E70B414">30.4 <span class="Heading">Attributes and Properties for Collections</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap30.html#X7969C48780C5C1BC">30.4-1 IsEmpty</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap30.html#X808A4061809A6E67">30.4-2 IsFinite</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap30.html#X7E3402D6799D3C24">30.4-3 IsTrivial</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap30.html#X7F192373850B85B9">30.4-4 IsNonTrivial</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap30.html#X78EF6A137E8F66B0">30.4-5 IsWholeFamily</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap30.html#X858ADA3B7A684421">30.4-6 Size</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap30.html#X865507568182424E">30.4-7 Representative</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap30.html#X8026085680270D37">30.4-8 RepresentativeSmallest</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap30.html#X7F8FEA3278239ADE">30.5 <span class="Heading">Operations for Collections</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap30.html#X79CA175481F8105F">30.5-1 IsSubset</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap30.html#X851069107CACF98E">30.5-2 <span class="Heading">Intersection</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap30.html#X799F0E2F7A502DBA">30.5-3 <span class="Heading">Union</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap30.html#X825AC0F07E010B07">30.5-4 Difference</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap30.html#X82D39CF980FDBFFA">30.6 <span class="Heading">Membership Test for Collections</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap30.html#X84B7FA8C7C94400F"><code>30.6-1 \in</code></a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap30.html#X8151A51884B7EE2C">30.7 <span class="Heading">Random Elements</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap30.html#X7FF906E57D6936F8">30.7-1 Random</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap30.html#X811B5BD47DC5356B">30.7-2 PseudoRandom</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap30.html#X7EBA01EB83BC65A9">30.7-3 RandomList</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap30.html#X85A3F00985453F95">30.8 <span class="Heading">Iterators</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap30.html#X83ADF8287ED0668E">30.8-1 Iterator</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap30.html#X8688C20B828FC129">30.8-2 IteratorSorted</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap30.html#X87168A827E5B28E4">30.8-3 IsIterator</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap30.html#X8055FC557B5D899E">30.8-4 IsDoneIterator</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap30.html#X879F62F77D1D1179">30.8-5 NextIterator</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap30.html#X858A28667D137C4B">30.8-6 IteratorList</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap30.html#X7DB80BE68271247E">30.8-7 TrivialIterator</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap30.html#X82677D8F817D6701">30.8-8 IteratorByFunctions</a></span>
</div></div>
</div>
<h3>30 <span class="Heading">Collections</span></h3>
<p>A <em>collection</em> in <strong class="pkg">GAP</strong> consists of elements in the same family (see <a href="chap13.html#X846063757EC05986"><span class="RefLink">13.1</span></a>). The most important kinds of collections are <em>homogeneous lists</em> (see <a href="chap21.html#X7B256AE5780F140A"><span class="RefLink">21</span></a>) and <em>domains</em> (see <a href="chap12.html#X7BAF69417BB925F6"><span class="RefLink">12.4</span></a>). Note that a list is never a domain, and a domain is never a list. A list is a collection if and only if it is nonempty and homogeneous.</p>
<p>Basic operations for collections are <code class="func">Size</code> (<a href="chap30.html#X858ADA3B7A684421"><span class="RefLink">30.4-6</span></a>) and <code class="func">Enumerator</code> (<a href="chap30.html#X7EF8910F82B45EC7"><span class="RefLink">30.3-2</span></a>); for <em>finite</em> collections, <code class="func">Enumerator</code> (<a href="chap30.html#X7EF8910F82B45EC7"><span class="RefLink">30.3-2</span></a>) admits to delegate the other operations for collections (see <a href="chap30.html#X79AD18737E70B414"><span class="RefLink">30.4</span></a> and <a href="chap30.html#X7F8FEA3278239ADE"><span class="RefLink">30.5</span></a>) to functions for lists (see <a href="chap21.html#X7B256AE5780F140A"><span class="RefLink">21</span></a>). Obviously, special methods depending on the arguments are needed for the computation of e.g. the intersection of two <em>infinite</em> domains.</p>
<p><a id="X8084F03A78ABD4F8" name="X8084F03A78ABD4F8"></a></p>
<h4>30.1 <span class="Heading">IsCollection (Filter)</span></h4>
<p><a id="X79C9FC7F86E2738C" name="X79C9FC7F86E2738C"></a></p>
<h5>30.1-1 IsCollection</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsCollection</code>( <var class="Arg">obj</var> )</td><td class="tdright">( category )</td></tr></table></div>
<p>tests whether an object is a collection.</p>
<p>Some of the functions for lists and collections are described in the chapter about lists, mainly in Section <a href="chap21.html#X7DF510F7848CBBFD"><span class="RefLink">21.20</span></a>. In the current chapter, we describe those functions for which the "collection aspect" seems to be more important than the "list aspect".</p>
<p><a id="X85D8D8F684B02DDF" name="X85D8D8F684B02DDF"></a></p>
<h4>30.2 <span class="Heading">Collection Families</span></h4>
<p><a id="X84E5A67E87D8DD66" name="X84E5A67E87D8DD66"></a></p>
<h5>30.2-1 CollectionsFamily</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CollectionsFamily</code>( <var class="Arg">Fam</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>For a family <var class="Arg">Fam</var>, <code class="func">CollectionsFamily</code> returns the family of all collections over <var class="Arg">Fam</var>, that is, of all dense lists and domains that consist of objects in <var class="Arg">Fam</var>.</p>
<p>The <code class="func">NewFamily</code> (<a href="chap79.html#X7FB4123E7E22137D"><span class="RefLink">79.7-1</span></a>) call in the standard method of <code class="func">CollectionsFamily</code> is executed with second argument <code class="func">IsCollection</code> (<a href="chap30.html#X79C9FC7F86E2738C"><span class="RefLink">30.1-1</span></a>), since every object in the collections family must be a collection, and with third argument the collections categories of the involved categories in the implied filter of <var class="Arg">Fam</var>.</p>
<p>Note that families (see <a href="chap13.html#X846063757EC05986"><span class="RefLink">13.1</span></a>) are used to describe relations between objects. Important such relations are that between an element <span class="SimpleMath">e</span> and each collection of elements that lie in the same family as <span class="SimpleMath">e</span>, and that between two collections whose elements lie in the same family. Therefore, all collections of elements in the family <var class="Arg">Fam</var> form the new family <code class="code">CollectionsFamily( <var class="Arg">Fam</var> )</code>.</p>
<p><a id="X856AC2DF7F7CBAAF" name="X856AC2DF7F7CBAAF"></a></p>
<h5>30.2-2 IsCollectionFamily</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsCollectionFamily</code>( <var class="Arg">obj</var> )</td><td class="tdright">( category )</td></tr></table></div>
<p>is <code class="keyw">true</code> if <var class="Arg">Fam</var> is a family of collections, and <code class="keyw">false</code> otherwise.</p>
<p><a id="X864BB3748546F63F" name="X864BB3748546F63F"></a></p>
<h5>30.2-3 ElementsFamily</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ElementsFamily</code>( <var class="Arg">Fam</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>If <var class="Arg">Fam</var> is a collections family (see <code class="func">IsCollectionFamily</code> (<a href="chap30.html#X856AC2DF7F7CBAAF"><span class="RefLink">30.2-2</span></a>)) then <code class="func">ElementsFamily</code> returns the family from which <var class="Arg">Fam</var> was created by <code class="func">CollectionsFamily</code> (<a href="chap30.html#X84E5A67E87D8DD66"><span class="RefLink">30.2-1</span></a>). The way a collections family is created, it always has its elements family stored. If <var class="Arg">Fam</var> is not a collections family then an error is signalled.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">fam:= FamilyObj( (1,2) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">collfam:= CollectionsFamily( fam );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">fam = collfam; fam = ElementsFamily( collfam );</span>
false
true
<span class="GAPprompt">gap></span> <span class="GAPinput">collfam = FamilyObj( [ (1,2,3) ] );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">collfam = FamilyObj( Group( () ) );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">collfam = CollectionsFamily( collfam );</span>
false
</pre></div>
<p><a id="X78C38017804B2EA7" name="X78C38017804B2EA7"></a></p>
<h5>30.2-4 CategoryCollections</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CategoryCollections</code>( <var class="Arg">filter</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Let <var class="Arg">filter</var> be a filter that is <code class="keyw">true</code> for all elements of a family <var class="Arg">Fam</var>, by the construction of <var class="Arg">Fam</var>. Then <code class="func">CategoryCollections</code> returns the <em>collections category</em> of <var class="Arg">filter</var>. This is a category that is <code class="keyw">true</code> for all elements in <code class="code">CollectionsFamily( <var class="Arg">Fam</var> )</code>.</p>
<p>For example, the construction of <code class="func">PermutationsFamily</code> (<a href="chap42.html#X819628B083B3939B"><span class="RefLink">42.1-3</span></a>) guarantees that each of its elements lies in the filter <code class="func">IsPerm</code> (<a href="chap42.html#X7AA69C6686FC49EA"><span class="RefLink">42.1-1</span></a>), and each collection of permutations (permutation group or dense list of permutations) lies in the category <code class="code">CategoryCollections( IsPerm )</code>. <code class="code">CategoryCollections( IsPerm )</code>. Note that this works only if the collections category is created <em>before</em> the collections family. So it is necessary to construct interesting collections categories immediately after the underlying category has been created.</p>
<p><a id="X7C3722DF8736FFDB" name="X7C3722DF8736FFDB"></a></p>
<h4>30.3 <span class="Heading">Lists and Collections</span></h4>
<p>The following functions take a <em>list or collection</em> as argument, and return a corresponding <em>list</em>. They differ in whether or not the result is mutable or immutable (see <a href="chap12.html#X7F0C119682196D65"><span class="RefLink">12.6</span></a>), guaranteed to be sorted, or guaranteed to admit list access in constant time (see <code class="func">IsConstantTimeAccessList</code> (<a href="chap21.html#X7C84E16A85C99C8C"><span class="RefLink">21.1-6</span></a>)).</p>
<p><a id="X877128A77826DD69" name="X877128A77826DD69"></a></p>
<h5>30.3-1 IsListOrCollection</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsListOrCollection</code>( <var class="Arg">obj</var> )</td><td class="tdright">( category )</td></tr></table></div>
<p>Several functions are defined for both lists and collections, for example <code class="func">Intersection</code> (<a href="chap30.html#X851069107CACF98E"><span class="RefLink">30.5-2</span></a>), <code class="func">Iterator</code> (<a href="chap30.html#X83ADF8287ED0668E"><span class="RefLink">30.8-1</span></a>), and <code class="func">Random</code> (<a href="chap30.html#X7FF906E57D6936F8"><span class="RefLink">30.7-1</span></a>). <code class="func">IsListOrCollection</code> is a supercategory of <code class="func">IsList</code> (<a href="chap21.html#X7C4CC4EA8299701E"><span class="RefLink">21.1-1</span></a>) and <code class="func">IsCollection</code> (<a href="chap30.html#X79C9FC7F86E2738C"><span class="RefLink">30.1-1</span></a>) (that is, all lists and collections lie in this category), which is used to describe the arguments of functions such as the ones listed above.</p>
<p><a id="X7EF8910F82B45EC7" name="X7EF8910F82B45EC7"></a></p>
<h5>30.3-2 Enumerator</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Enumerator</code>( <var class="Arg">listorcoll</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p><code class="func">Enumerator</code> returns an immutable list <span class="SimpleMath">enum</span>. If the argument is a list (which may contain holes), then <code class="code">Length( </code><span class="SimpleMath">enum</span><code class="code"> )</code> is the length of this list, and <span class="SimpleMath">enum</span> contains the elements (and holes) of this list in the same order. If the argument is a collection that is not a list, then <code class="code">Length( </code><span class="SimpleMath">enum</span><code class="code"> )</code> is the number of different elements of <var class="Arg">C</var>, and <span class="SimpleMath">enum</span> contains the different elements of the collection in an unspecified order, which may change for repeated calls of <code class="func">Enumerator</code>. <span class="SimpleMath">enum[pos]</span> may not execute in constant time (see <code class="func">IsConstantTimeAccessList</code> (<a href="chap21.html#X7C84E16A85C99C8C"><span class="RefLink">21.1-6</span></a>)), and the size of <span class="SimpleMath">enum</span> in memory is as small as is feasible.</p>
<p>For lists, the default method is <code class="func">Immutable</code> (<a href="chap12.html#X7F0ABF2C870B0CBB"><span class="RefLink">12.6-3</span></a>). For collections that are not lists, there is no default method.</p>
<p><a id="X80CD7DDC7D0C60D5" name="X80CD7DDC7D0C60D5"></a></p>
<h5>30.3-3 EnumeratorSorted</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ EnumeratorSorted</code>( <var class="Arg">listorcoll</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p><code class="func">EnumeratorSorted</code> returns an immutable list <span class="SimpleMath">enum</span>. The argument must be a collection or a list <var class="Arg">listorcoll</var> which may contain holes but whose elements lie in the same family (see <a href="chap13.html#X846063757EC05986"><span class="RefLink">13.1</span></a>). <code class="code">Length( </code><span class="SimpleMath">enum</span><code class="code"> )</code> is the number of different elements of the argument, and <span class="SimpleMath">enum</span> contains the different elements in sorted order, w.r.t. <code class="code"><</code>. <span class="SimpleMath">enum[pos]</span> may not execute in constant time (see <code class="func">IsConstantTimeAccessList</code> (<a href="chap21.html#X7C84E16A85C99C8C"><span class="RefLink">21.1-6</span></a>)), and the size of <span class="SimpleMath">enum</span> in memory is as small as is feasible.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Enumerator( [ 1, 3,, 2 ] );</span>
[ 1, 3,, 2 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">enum:= Enumerator( Rationals );; elm:= enum[ 10^6 ];</span>
-69/907
<span class="GAPprompt">gap></span> <span class="GAPinput">Position( enum, elm );</span>
1000000
<span class="GAPprompt">gap></span> <span class="GAPinput">IsMutable( enum ); IsSortedList( enum );</span>
false
false
<span class="GAPprompt">gap></span> <span class="GAPinput">IsConstantTimeAccessList( enum );</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">EnumeratorSorted( [ 1, 3,, 2 ] );</span>
[ 1, 2, 3 ]
</pre></div>
<p><a id="X85E149177AC547C3" name="X85E149177AC547C3"></a></p>
<h5>30.3-4 EnumeratorByFunctions</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ EnumeratorByFunctions</code>( <var class="Arg">D</var>, <var class="Arg">record</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ EnumeratorByFunctions</code>( <var class="Arg">Fam</var>, <var class="Arg">record</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="func">EnumeratorByFunctions</code> returns an immutable, dense, and duplicate-free list <span class="SimpleMath">enum</span> for which <code class="func">IsBound</code> (<a href="chap21.html#X79EC565A7DCEC938"><span class="RefLink">21.5-1</span></a>), element access via <code class="func">\[\]</code> (<a href="chap21.html#X8297BBCD79642BE6"><span class="RefLink">21.2-1</span></a>), <code class="func">Length</code> (<a href="chap21.html#X780769238600AFD1"><span class="RefLink">21.17-5</span></a>), and <code class="func">Position</code> (<a href="chap21.html#X79975EC6783B4293"><span class="RefLink">21.16-1</span></a>) are computed via prescribed functions.</p>
<p>Let <var class="Arg">record</var> be a record with at least the following components.</p>
<dl>
<dt><strong class="Mark"><code class="code">ElementNumber</code></strong></dt>
<dd><p>a function taking two arguments <var class="Arg">enum</var> and <var class="Arg">pos</var>, which returns <code class="code"><var class="Arg">enum</var>[ <var class="Arg">pos</var> ]</code> (see <a href="chap21.html#X7B202D147A5C2884"><span class="RefLink">21.2</span></a>); it can be assumed that the argument <var class="Arg">pos</var> is a positive integer, but <var class="Arg">pos</var> may be larger than the length of <var class="Arg">enum</var> (in which case an error must be signalled); note that the result must be immutable since <var class="Arg">enum</var> itself is immutable,</p>
</dd>
<dt><strong class="Mark"><code class="code">NumberElement</code></strong></dt>
<dd><p>a function taking two arguments <var class="Arg">enum</var> and <var class="Arg">elm</var>, which returns <code class="code">Position( <var class="Arg">enum</var>, <var class="Arg">elm</var> )</code> (see <code class="func">Position</code> (<a href="chap21.html#X79975EC6783B4293"><span class="RefLink">21.16-1</span></a>)); it cannot be assumed that <var class="Arg">elm</var> is really contained in <var class="Arg">enum</var> (and <code class="keyw">fail</code> must be returned if not); note that for the three argument version of <code class="func">Position</code> (<a href="chap21.html#X79975EC6783B4293"><span class="RefLink">21.16-1</span></a>), the method that is available for duplicate-free lists suffices.</p>
</dd>
</dl>
<p>Further (data) components may be contained in <var class="Arg">record</var> which can be used by these function.</p>
<p>If the first argument is a domain <var class="Arg">D</var> then <var class="Arg">enum</var> lists the elements of <var class="Arg">D</var> (in general <var class="Arg">enum</var> is <em>not</em> sorted), and methods for <code class="func">Length</code> (<a href="chap21.html#X780769238600AFD1"><span class="RefLink">21.17-5</span></a>), <code class="func">IsBound</code> (<a href="chap21.html#X79EC565A7DCEC938"><span class="RefLink">21.5-1</span></a>), and <code class="func">PrintObj</code> (<a href="chap6.html#X815BF22186FD43C9"><span class="RefLink">6.3-5</span></a>) may use <var class="Arg">D</var>.</p>
<p>If one wants to describe the result without creating a domain then the elements are given implicitly by the functions in <var class="Arg">record</var>, and the first argument must be a family <var class="Arg">Fam</var> which will become the family of <var class="Arg">enum</var>; if <var class="Arg">enum</var> is not homogeneous then <var class="Arg">Fam</var> must be <code class="code">ListsFamily</code>, otherwise it must be the collections family of any element in <var class="Arg">enum</var>. In this case, additionally the following component in <var class="Arg">record</var> is needed.</p>
<dl>
<dt><strong class="Mark"><code class="code">Length</code></strong></dt>
<dd><p>a function taking the argument <var class="Arg">enum</var>, which returns the length of <var class="Arg">enum</var> (see <code class="func">Length</code> (<a href="chap21.html#X780769238600AFD1"><span class="RefLink">21.17-5</span></a>)).</p>
</dd>
</dl>
<p>The following components are optional; they are used if they are present but default methods are installed for the case that they are missing.</p>
<dl>
<dt><strong class="Mark"><code class="code">IsBound\[\]</code></strong></dt>
<dd><p>a function taking two arguments <var class="Arg">enum</var> and <var class="Arg">k</var>, which returns <code class="code">IsBound( <var class="Arg">enum</var>[ <var class="Arg">k</var> ] )</code> (see <a href="chap21.html#X7B202D147A5C2884"><span class="RefLink">21.2</span></a>); if this component is missing then <code class="func">Length</code> (<a href="chap21.html#X780769238600AFD1"><span class="RefLink">21.17-5</span></a>) is used for computing the result,</p>
</dd>
<dt><strong class="Mark"><code class="code">Membership</code></strong></dt>
<dd><p>a function taking two arguments <var class="Arg">elm</var> and <var class="Arg">enum</var>, which returns <code class="keyw">true</code> is <var class="Arg">elm</var> is an element of <var class="Arg">enum</var>, and <code class="keyw">false</code> otherwise (see <a href="chap21.html#X7B202D147A5C2884"><span class="RefLink">21.2</span></a>); if this component is missing then <code class="code">NumberElement</code> is used for computing the result,</p>
</dd>
<dt><strong class="Mark"><code class="code">AsList</code></strong></dt>
<dd><p>a function taking one argument <var class="Arg">enum</var>, which returns a list with the property that the access to each of its elements will take roughly the same time (see <code class="func">IsConstantTimeAccessList</code> (<a href="chap21.html#X7C84E16A85C99C8C"><span class="RefLink">21.1-6</span></a>)); if this component is missing then <code class="func">ConstantTimeAccessList</code> (<a href="chap21.html#X7B55FB967CDEF468"><span class="RefLink">21.17-6</span></a>) is used for computing the result,</p>
</dd>
<dt><strong class="Mark"><code class="code">ViewObj</code> and <code class="code">PrintObj</code></strong></dt>
<dd><p>two functions that print what one wants to be printed when <code class="code">View( <var class="Arg">enum</var> )</code> or <code class="code">Print( <var class="Arg">enum</var> )</code> is called (see <a href="chap6.html#X8074A8387C9DB9A8"><span class="RefLink">6.3</span></a>), if the <code class="code">ViewObj</code> component is missing then the <code class="code">PrintObj</code> method is used as a default.</p>
</dd>
</dl>
<p>If the result is known to have additional properties such as being strictly sorted (see <code class="func">IsSSortedList</code> (<a href="chap21.html#X80CDAF45782E8DCB"><span class="RefLink">21.17-4</span></a>)) then it can be useful to set these properties after the construction of the enumerator, before it is used for the first time. And in the case that a new sorted enumerator of a domain is implemented via <code class="func">EnumeratorByFunctions</code>, and this construction is installed as a method for the operation <code class="func">Enumerator</code> (<a href="chap30.html#X7EF8910F82B45EC7"><span class="RefLink">30.3-2</span></a>), then it should be installed also as a method for <code class="func">EnumeratorSorted</code> (<a href="chap30.html#X80CD7DDC7D0C60D5"><span class="RefLink">30.3-3</span></a>).</p>
<p>Note that it is <em>not</em> checked that <code class="func">EnumeratorByFunctions</code> really returns a dense and duplicate-free list. <code class="func">EnumeratorByFunctions</code> does <em>not</em> make a shallow copy of <var class="Arg">record</var>, this record is changed in place, see <a href="chap79.html#X82E86CF37B123FD4"><span class="RefLink">79.9</span></a>.</p>
<p>It would be easy to implement a slightly generalized setup for enumerators that need not be duplicate-free (where the three argument version of <code class="func">Position</code> (<a href="chap21.html#X79975EC6783B4293"><span class="RefLink">21.16-1</span></a>) is supported), but the resulting overhead for the methods seems not to be justified.</p>
<p><a id="X7F12F40E87F3C3A7" name="X7F12F40E87F3C3A7"></a></p>
<h5>30.3-5 List</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ List</code>( <var class="Arg">C</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>For a collection <var class="Arg">C</var> (see <a href="chap30.html#X8050A8037984E5B6"><span class="RefLink">30</span></a>) that is not a list, <code class="func">List</code> returns a new mutable list <var class="Arg">new</var> such that <code class="code">Length( <var class="Arg">new</var> )</code> is the number of different elements of <var class="Arg">C</var>, and <var class="Arg">new</var> contains the different elements of <var class="Arg">C</var> in an unspecified order which may change for repeated calls. <code class="code"><var class="Arg">new</var>[<var class="Arg">pos</var>]</code> executes in constant time (see <code class="func">IsConstantTimeAccessList</code> (<a href="chap21.html#X7C84E16A85C99C8C"><span class="RefLink">21.1-6</span></a>)), and the size of <var class="Arg">new</var> is proportional to its length. The generic method for this case is <code class="code">ShallowCopy( Enumerator( <var class="Arg">C</var> ) )</code>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">l:= List( Group( (1,2,3) ) );</span>
[ (), (1,3,2), (1,2,3) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IsMutable( l ); IsSortedList( l ); IsConstantTimeAccessList( l );</span>
true
false
true
</pre></div>
<p>(See also <code class="func">List</code> (<a href="chap21.html#X86CB7DCE8510F977"><span class="RefLink">21.20-19</span></a>).)</p>
<p><a id="X82CE157A7FAD8036" name="X82CE157A7FAD8036"></a></p>
<h5>30.3-6 SortedList</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SortedList</code>( <var class="Arg">listorcoll</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p><code class="func">SortedList</code> returns a new mutable and dense list <var class="Arg">new</var>. The argument must be a collection or list <var class="Arg">listorcoll</var> which may contain holes but whose elements lie in the same family (see <a href="chap13.html#X846063757EC05986"><span class="RefLink">13.1</span></a>). <code class="code">Length( <var class="Arg">new</var> )</code> is the number of elements of <var class="Arg">listorcoll</var>, and <var class="Arg">new</var> contains the elements in sorted order, w.r.t. <code class="code"><=</code>. <code class="code"><var class="Arg">new</var>[<var class="Arg">pos</var>]</code> executes in constant time (see <code class="func">IsConstantTimeAccessList</code> (<a href="chap21.html#X7C84E16A85C99C8C"><span class="RefLink">21.1-6</span></a>)), and the size of <var class="Arg">new</var> in memory is proportional to its length.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">l:= SortedList( Group( (1,2,3) ) );</span>
[ (), (1,2,3), (1,3,2) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IsMutable( l ); IsSortedList( l ); IsConstantTimeAccessList( l );</span>
true
true
true
<span class="GAPprompt">gap></span> <span class="GAPinput">SortedList( [ 1, 2, 1,, 3, 2 ] );</span>
[ 1, 1, 2, 2, 3 ]
</pre></div>
<p><a id="X7E399AC97FD98217" name="X7E399AC97FD98217"></a></p>
<h5>30.3-7 SSortedList</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SSortedList</code>( <var class="Arg">listorcoll</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Set</code>( <var class="Arg">C</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p><code class="func">SSortedList</code> ("strictly sorted list") returns a new dense, mutable, and duplicate free list <var class="Arg">new</var>. The argument must be a collection or list <var class="Arg">listorcoll</var> which may contain holes but whose elements lie in the same family (see <a href="chap13.html#X846063757EC05986"><span class="RefLink">13.1</span></a>). <code class="code">Length( <var class="Arg">new</var> )</code> is the number of different elements of <var class="Arg">listorcoll</var>, and <var class="Arg">new</var> contains the different elements in strictly sorted order, w.r.t. <code class="func">\<</code> (<a href="chap31.html#X7EF67D047F03CA6F"><span class="RefLink">31.11-1</span></a>). <code class="code"><var class="Arg">new</var>[<var class="Arg">pos</var>]</code> executes in constant time (see <code class="func">IsConstantTimeAccessList</code> (<a href="chap21.html#X7C84E16A85C99C8C"><span class="RefLink">21.1-6</span></a>)), and the size of <var class="Arg">new</var> in memory is proportional to its length.</p>
<p><code class="func">Set</code> is simply a synonym for <code class="func">SSortedList</code>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">l:= SSortedList( Group( (1,2,3) ) );</span>
[ (), (1,2,3), (1,3,2) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IsMutable( l ); IsSSortedList( l ); IsConstantTimeAccessList( l );</span>
true
true
true
<span class="GAPprompt">gap></span> <span class="GAPinput">SSortedList( [ 1, 2, 1,, 3, 2 ] );</span>
[ 1, 2, 3 ]
</pre></div>
<p><a id="X8289FCCC8274C89D" name="X8289FCCC8274C89D"></a></p>
<h5>30.3-8 AsList</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AsList</code>( <var class="Arg">listorcoll</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p><code class="func">AsList</code> returns a immutable list <var class="Arg">imm</var>. If the argument is a list (which may contain holes), then <code class="code">Length( <var class="Arg">imm</var> )</code> is the <code class="func">Length</code> (<a href="chap21.html#X780769238600AFD1"><span class="RefLink">21.17-5</span></a>) value of this list, and <var class="Arg">imm</var> contains the elements (and holes) of of the list in the same order. If the argument is a collection that is not a list, then <code class="code">Length( <var class="Arg">imm</var> )</code> is the number of different elements of this collection, and <var class="Arg">imm</var> contains the different elements of the collection in an unspecified order, which may change for repeated calls of <code class="func">AsList</code>. <code class="code"><var class="Arg">imm</var>[<var class="Arg">pos</var>]</code> executes in constant time (see <code class="func">IsConstantTimeAccessList</code> (<a href="chap21.html#X7C84E16A85C99C8C"><span class="RefLink">21.1-6</span></a>)), and the size of <var class="Arg">imm</var> in memory is proportional to its length.</p>
<p>If you expect to do many element tests in the resulting list, it might be worth to use a sorted list instead, using <code class="func">AsSSortedList</code> (<a href="chap30.html#X856D927378C33548"><span class="RefLink">30.3-10</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">l:= AsList( [ 1, 3, 3,, 2 ] );</span>
[ 1, 3, 3,, 2 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IsMutable( l ); IsSortedList( l ); IsConstantTimeAccessList( l );</span>
false
false
true
<span class="GAPprompt">gap></span> <span class="GAPinput">AsList( Group( (1,2,3), (1,2) ) );</span>
[ (), (2,3), (1,2), (1,2,3), (1,3,2), (1,3) ]
</pre></div>
<p><a id="X7BCA5C6181391007" name="X7BCA5C6181391007"></a></p>
<h5>30.3-9 AsSortedList</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AsSortedList</code>( <var class="Arg">listorcoll</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p><code class="func">AsSortedList</code> returns a dense and immutable list <var class="Arg">imm</var>. The argument must be a collection or list <var class="Arg">listorcoll</var> which may contain holes but whose elements lie in the same family (see <a href="chap13.html#X846063757EC05986"><span class="RefLink">13.1</span></a>). <code class="code">Length( <var class="Arg">imm</var> )</code> is the number of elements of the argument, and <var class="Arg">imm</var> contains the elements in sorted order, w.r.t. <code class="code"><=</code>. <code class="code"><var class="Arg">new</var>[<var class="Arg">pos</var>]</code> executes in constant time (see <code class="func">IsConstantTimeAccessList</code> (<a href="chap21.html#X7C84E16A85C99C8C"><span class="RefLink">21.1-6</span></a>)), and the size of <var class="Arg">imm</var> in memory is proportional to its length.</p>
<p>The only difference to the operation <code class="func">SortedList</code> (<a href="chap30.html#X82CE157A7FAD8036"><span class="RefLink">30.3-6</span></a>) is that <code class="func">AsSortedList</code> returns an <em>immutable</em> list.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">l:= AsSortedList( [ 1, 3, 3,, 2 ] );</span>
[ 1, 2, 3, 3 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IsMutable( l ); IsSortedList( l ); IsConstantTimeAccessList( l );</span>
false
true
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsSSortedList( l );</span>
false
</pre></div>
<p><a id="X856D927378C33548" name="X856D927378C33548"></a></p>
<h5>30.3-10 AsSSortedList</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AsSSortedList</code>( <var class="Arg">listorcoll</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AsSet</code>( <var class="Arg">listorcoll</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p><code class="func">AsSSortedList</code> ("as strictly sorted list") returns a dense, immutable, and duplicate free list <var class="Arg">imm</var>. The argument must be a collection or list <var class="Arg">listorcoll</var> which may contain holes but whose elements lie in the same family (see <a href="chap13.html#X846063757EC05986"><span class="RefLink">13.1</span></a>). <code class="code">Length( <var class="Arg">imm</var> )</code> is the number of different elements of <var class="Arg">listorcoll</var>, and <var class="Arg">imm</var> contains the different elements in strictly sorted order, w.r.t. <code class="func">\<</code> (<a href="chap31.html#X7EF67D047F03CA6F"><span class="RefLink">31.11-1</span></a>). <code class="code"><var class="Arg">imm</var>[<var class="Arg">pos</var>]</code> executes in constant time (see <code class="func">IsConstantTimeAccessList</code> (<a href="chap21.html#X7C84E16A85C99C8C"><span class="RefLink">21.1-6</span></a>)), and the size of <var class="Arg">imm</var> in memory is proportional to its length.</p>
<p>Because the comparisons required for sorting can be very expensive for some kinds of objects, you should use <code class="func">AsList</code> (<a href="chap30.html#X8289FCCC8274C89D"><span class="RefLink">30.3-8</span></a>) instead if you do not require the result to be sorted.</p>
<p>The only difference to the operation <code class="func">SSortedList</code> (<a href="chap30.html#X7E399AC97FD98217"><span class="RefLink">30.3-7</span></a>) is that <code class="func">AsSSortedList</code> returns an <em>immutable</em> list.</p>
<p><code class="func">AsSet</code> is simply a synonym for <code class="func">AsSSortedList</code>.</p>
<p>In general a function that returns a set of elements is free, in fact encouraged, to return a domain instead of the proper set of its elements. This allows one to keep a given structure, and moreover the representation by a domain object is usually more space efficient. <code class="func">AsSSortedList</code> must of course <em>not</em> do this, its only purpose is to create the proper set of elements.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">l:= AsSSortedList( l );</span>
[ 1, 2, 3 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IsMutable( l ); IsSSortedList( l ); IsConstantTimeAccessList( l );</span>
false
true
true
<span class="GAPprompt">gap></span> <span class="GAPinput">AsSSortedList( Group( (1,2,3), (1,2) ) );</span>
[ (), (2,3), (1,2), (1,2,3), (1,3,2), (1,3) ]
</pre></div>
<p><a id="X79B130FC7906FB4C" name="X79B130FC7906FB4C"></a></p>
<h5>30.3-11 Elements</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Elements</code>( <var class="Arg">C</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="func">Elements</code> does the same as <code class="func">AsSSortedList</code> (<a href="chap30.html#X856D927378C33548"><span class="RefLink">30.3-10</span></a>), that is, the return value is a strictly sorted list of the elements in the list or collection <var class="Arg">C</var>.</p>
<p><code class="func">Elements</code> is only supported for backwards compatibility. In many situations, the sortedness of the "element list" for a collection is in fact not needed, and one can save a lot of time by asking for a list that is <em>not</em> necessarily sorted, using <code class="func">AsList</code> (<a href="chap30.html#X8289FCCC8274C89D"><span class="RefLink">30.3-8</span></a>). If one is really interested in the strictly sorted list of elements in <var class="Arg">C</var> then one should use <code class="func">AsSet</code> (<a href="chap30.html#X856D927378C33548"><span class="RefLink">30.3-10</span></a>) or <code class="func">AsSSortedList</code> (<a href="chap30.html#X856D927378C33548"><span class="RefLink">30.3-10</span></a>) instead.</p>
<p><a id="X79AD18737E70B414" name="X79AD18737E70B414"></a></p>
<h4>30.4 <span class="Heading">Attributes and Properties for Collections</span></h4>
<p><a id="X7969C48780C5C1BC" name="X7969C48780C5C1BC"></a></p>
<h5>30.4-1 IsEmpty</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsEmpty</code>( <var class="Arg">listorcoll</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p><code class="func">IsEmpty</code> returns <code class="keyw">true</code> if the collection or list <var class="Arg">listorcoll</var> is <em>empty</em> (that is it contains no elements), and <code class="keyw">false</code> otherwise.</p>
<p><a id="X808A4061809A6E67" name="X808A4061809A6E67"></a></p>
<h5>30.4-2 IsFinite</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsFinite</code>( <var class="Arg">C</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p><code class="func">IsFinite</code> returns <code class="keyw">true</code> if the collection <var class="Arg">C</var> is finite, and <code class="keyw">false</code> otherwise.</p>
<p>The default method for <code class="func">IsFinite</code> checks the size (see <code class="func">Size</code> (<a href="chap30.html#X858ADA3B7A684421"><span class="RefLink">30.4-6</span></a>)) of <var class="Arg">C</var>.</p>
<p>Methods for <code class="func">IsFinite</code> may call <code class="func">Size</code> (<a href="chap30.html#X858ADA3B7A684421"><span class="RefLink">30.4-6</span></a>), but methods for <code class="func">Size</code> (<a href="chap30.html#X858ADA3B7A684421"><span class="RefLink">30.4-6</span></a>) must <em>not</em> call <code class="func">IsFinite</code>.</p>
<p><a id="X7E3402D6799D3C24" name="X7E3402D6799D3C24"></a></p>
<h5>30.4-3 IsTrivial</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsTrivial</code>( <var class="Arg">C</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p><code class="func">IsTrivial</code> returns <code class="keyw">true</code> if the collection <var class="Arg">C</var> consists of exactly one element.</p>
<p><a id="X7F192373850B85B9" name="X7F192373850B85B9"></a></p>
<h5>30.4-4 IsNonTrivial</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsNonTrivial</code>( <var class="Arg">C</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p><code class="func">IsNonTrivial</code> returns <code class="keyw">true</code> if the collection <var class="Arg">C</var> is empty or consists of at least two elements (see <code class="func">IsTrivial</code> (<a href="chap30.html#X7E3402D6799D3C24"><span class="RefLink">30.4-3</span></a>)).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsEmpty( [] ); IsEmpty( [ 1 .. 100 ] ); IsEmpty( Group( (1,2,3) ) );</span>
true
false
false
<span class="GAPprompt">gap></span> <span class="GAPinput">IsFinite( [ 1 .. 100 ] ); IsFinite( Integers );</span>
true
false
<span class="GAPprompt">gap></span> <span class="GAPinput">IsTrivial( Integers ); IsTrivial( Group( () ) );</span>
false
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsNonTrivial( Integers ); IsNonTrivial( Group( () ) );</span>
true
false
</pre></div>
<p><a id="X78EF6A137E8F66B0" name="X78EF6A137E8F66B0"></a></p>
<h5>30.4-5 IsWholeFamily</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsWholeFamily</code>( <var class="Arg">C</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p><code class="func">IsWholeFamily</code> returns <code class="keyw">true</code> if the collection <var class="Arg">C</var> contains the whole family (see <a href="chap13.html#X846063757EC05986"><span class="RefLink">13.1</span></a>) of its elements.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsWholeFamily( Integers )</span>
<span class="GAPprompt">></span> <span class="GAPinput"> ; # all rationals and cyclotomics lie in the family</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">IsWholeFamily( Integers mod 3 )</span>
<span class="GAPprompt">></span> <span class="GAPinput"> ; # all finite field elements in char. 3 lie in this family</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">IsWholeFamily( Integers mod 4 );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsWholeFamily( FreeGroup( 2 ) );</span>
true
</pre></div>
<p><a id="X858ADA3B7A684421" name="X858ADA3B7A684421"></a></p>
<h5>30.4-6 Size</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Size</code>( <var class="Arg">listorcoll</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p><code class="func">Size</code> returns the size of the list or collection <var class="Arg">listorcoll</var>, which is either an integer or <code class="func">infinity</code> (<a href="chap18.html#X8511B8DF83324C27"><span class="RefLink">18.2-1</span></a>). If the argument is a list then the result is its length (see <code class="func">Length</code> (<a href="chap21.html#X780769238600AFD1"><span class="RefLink">21.17-5</span></a>)).</p>
<p>The default method for <code class="func">Size</code> checks the length of an enumerator of <var class="Arg">listorcoll</var>.</p>
<p>Methods for <code class="func">IsFinite</code> (<a href="chap30.html#X808A4061809A6E67"><span class="RefLink">30.4-2</span></a>) may call <code class="func">Size</code>, but methods for <code class="func">Size</code> must not call <code class="func">IsFinite</code> (<a href="chap30.html#X808A4061809A6E67"><span class="RefLink">30.4-2</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Size( [1,2,3] ); Size( Group( () ) ); Size( Integers );</span>
3
1
infinity
</pre></div>
<p><a id="X865507568182424E" name="X865507568182424E"></a></p>
<h5>30.4-7 Representative</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Representative</code>( <var class="Arg">C</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p><code class="func">Representative</code> returns a <em>representative</em> of the collection <var class="Arg">C</var>.</p>
<p>Note that <code class="func">Representative</code> is free in choosing a representative if there are several elements in <var class="Arg">C</var>. It is not even guaranteed that <code class="func">Representative</code> returns the same representative if it is called several times for one collection. The main difference between <code class="func">Representative</code> and <code class="func">Random</code> (<a href="chap30.html#X7FF906E57D6936F8"><span class="RefLink">30.7-1</span></a>) is that <code class="func">Representative</code> is free to choose a value that is cheap to compute, while <code class="func">Random</code> (<a href="chap30.html#X7FF906E57D6936F8"><span class="RefLink">30.7-1</span></a>) must make an effort to randomly distribute its answers.</p>
<p>If <var class="Arg">C</var> is a domain then there are methods for <code class="func">Representative</code> that try to fetch an element from any known generator list of <var class="Arg">C</var>, see <a href="chap31.html#X7E651AC287AFDCC1"><span class="RefLink">31</span></a>. Note that <code class="func">Representative</code> does not try to <em>compute</em> generators of <var class="Arg">C</var>, thus <code class="func">Representative</code> may give up and signal an error if <var class="Arg">C</var> has no generators stored at all.</p>
<p><a id="X8026085680270D37" name="X8026085680270D37"></a></p>
<h5>30.4-8 RepresentativeSmallest</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RepresentativeSmallest</code>( <var class="Arg">C</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns the smallest element in the collection <var class="Arg">C</var>, w.r.t. the ordering <code class="func">\<</code> (<a href="chap31.html#X7EF67D047F03CA6F"><span class="RefLink">31.11-1</span></a>). While the operation defaults to comparing all elements, better methods are installed for some collections.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Representative( Rationals );</span>
0
<span class="GAPprompt">gap></span> <span class="GAPinput">Representative( [ -1, -2 .. -100 ] );</span>
-1
<span class="GAPprompt">gap></span> <span class="GAPinput">RepresentativeSmallest( [ -1, -2 .. -100 ] );</span>
-100
</pre></div>
<p><a id="X7F8FEA3278239ADE" name="X7F8FEA3278239ADE"></a></p>
<h4>30.5 <span class="Heading">Operations for Collections</span></h4>
<p><a id="X79CA175481F8105F" name="X79CA175481F8105F"></a></p>
<h5>30.5-1 IsSubset</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsSubset</code>( <var class="Arg">C1</var>, <var class="Arg">C2</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p><code class="func">IsSubset</code> returns <code class="keyw">true</code> if <var class="Arg">C2</var>, which must be a collection, is a <em>subset</em> of <var class="Arg">C1</var>, which also must be a collection, and <code class="keyw">false</code> otherwise.</p>
<p><var class="Arg">C2</var> is considered a subset of <var class="Arg">C1</var> if and only if each element of <var class="Arg">C2</var> is also an element of <var class="Arg">C1</var>. That is <code class="func">IsSubset</code> behaves as if implemented as <code class="code">IsSubsetSet( AsSSortedList( <var class="Arg">C1</var> ), AsSSortedList( <var class="Arg">C2</var> ) )</code>, except that it will also sometimes, but not always, work for infinite collections, and that it will usually work much faster than the above definition. Either argument may also be a proper set (see <a href="chap21.html#X80ABC25582343910"><span class="RefLink">21.19</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsSubset( Rationals, Integers );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsSubset( Integers, [ 1, 2, 3 ] );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsSubset( Group( (1,2,3,4) ), [ (1,2,3) ] );</span>
false
</pre></div>
<p><a id="X851069107CACF98E" name="X851069107CACF98E"></a></p>
<h5>30.5-2 <span class="Heading">Intersection</span></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Intersection</code>( <var class="Arg">C1</var>, <var class="Arg">C2</var>, <var class="Arg">...</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Intersection</code>( <var class="Arg">list</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Intersection2</code>( <var class="Arg">C1</var>, <var class="Arg">C2</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>In the first form <code class="func">Intersection</code> returns the intersection of the collections <var class="Arg">C1</var>, <var class="Arg">C2</var>, etc. In the second form <var class="Arg">list</var> must be a <em>nonempty</em> list of collections and <code class="func">Intersection</code> returns the intersection of those collections. Each argument or element of <var class="Arg">list</var> respectively may also be a homogeneous list that is not a proper set, in which case <code class="func">Intersection</code> silently applies <code class="func">Set</code> (<a href="chap30.html#X7E399AC97FD98217"><span class="RefLink">30.3-7</span></a>) to it first.</p>
<p>The result of <code class="func">Intersection</code> is the set of elements that lie in every of the collections <var class="Arg">C1</var>, <var class="Arg">C2</var>, etc. If the result is a list then it is mutable and new, i.e., not identical to any of <var class="Arg">C1</var>, <var class="Arg">C2</var>, etc.</p>
<p>Methods can be installed for the operation <code class="func">Intersection2</code> that takes only two arguments. <code class="func">Intersection</code> calls <code class="func">Intersection2</code>.</p>
<p>Methods for <code class="func">Intersection2</code> should try to maintain as much structure as possible, for example the intersection of two permutation groups is again a permutation group.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput"># this is one of the rare cases where the intersection of two</span>
<span class="GAPprompt">gap></span> <span class="GAPinput"># infinite domains works ('CF' is a shorthand for 'CyclotomicField'):</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Intersection( CyclotomicField(9), CyclotomicField(12) );</span>
CF(3)
<span class="GAPprompt">gap></span> <span class="GAPinput">D12 := Group( (2,6)(3,5), (1,2)(3,6)(4,5) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Intersection( D12, Group( (1,2), (1,2,3,4,5) ) );</span>
Group([ (1,5)(2,4) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">Intersection( D12, [ (1,3)(4,6), (1,2)(3,4) ] )</span>
<span class="GAPprompt">></span> <span class="GAPinput"> ; # note that the second argument is not a proper set</span>
[ (1,3)(4,6) ]
<span class="GAPprompt">gap></span> <span class="GAPinput"># although the result is mathematically a group it is returned as a</span>
<span class="GAPprompt">gap></span> <span class="GAPinput"># proper set because the second argument is not regarded as a group:</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Intersection( D12, [ (), (1,2)(3,4), (1,3)(4,6), (1,4)(5,6) ] );</span>
[ (), (1,3)(4,6) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Intersection( Group( () ), [1,2,3] );</span>
[ ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Intersection( [2,4,6,8,10], [3,6,9,12,15], [5,10,15,20,25] )</span>
<span class="GAPprompt">></span> <span class="GAPinput"> ; # two or more lists or collections as arguments are legal</span>
[ ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Intersection( [ [1,2,4], [2,3,4], [1,3,4] ] )</span>
<span class="GAPprompt">></span> <span class="GAPinput"> ; # or one list of lists or collections</span>
[ 4 ]
</pre></div>
<p><a id="X799F0E2F7A502DBA" name="X799F0E2F7A502DBA"></a></p>
<h5>30.5-3 <span class="Heading">Union</span></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Union</code>( <var class="Arg">C1</var>, <var class="Arg">C2</var>, <var class="Arg">...</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Union</code>( <var class="Arg">list</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Union2</code>( <var class="Arg">C1</var>, <var class="Arg">C2</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>In the first form <code class="func">Union</code> returns the union of the collections <var class="Arg">C1</var>, <var class="Arg">C2</var>, etc. In the second form <var class="Arg">list</var> must be a list of collections and <code class="func">Union</code> returns the union of those collections. Each argument or element of <var class="Arg">list</var> respectively may also be a homogeneous list that is not a proper set, in which case <code class="func">Union</code> silently applies <code class="func">Set</code> (<a href="chap30.html#X7E399AC97FD98217"><span class="RefLink">30.3-7</span></a>) to it first.</p>
<p>The result of <code class="func">Union</code> is the set of elements that lie in any of the collections <var class="Arg">C1</var>, <var class="Arg">C2</var>, etc. If the result is a list then it is mutable and new, i.e., not identical to any of <var class="Arg">C1</var>, <var class="Arg">C2</var>, etc.</p>
<p>Methods can be installed for the operation <code class="func">Union2</code> that takes only two arguments. <code class="func">Union</code> calls <code class="func">Union2</code>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Union( [ (1,2,3), (1,2,3,4) ], Group( (1,2,3), (1,2) ) );</span>
[ (), (2,3), (1,2), (1,2,3), (1,2,3,4), (1,3,2), (1,3) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Union( [2,4,6,8,10], [3,6,9,12,15], [5,10,15,20,25] )</span>
<span class="GAPprompt">></span> <span class="GAPinput"> ; # two or more lists or collections as arguments are legal</span>
[ 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 20, 25 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Union( [ [1,2,4], [2,3,4], [1,3,4] ] )</span>
<span class="GAPprompt">></span> <span class="GAPinput"> ; # or one list of lists or collections</span>
[ 1, 2, 3, 4 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Union( [ ] );</span>
[ ]
</pre></div>
<p><a id="X825AC0F07E010B07" name="X825AC0F07E010B07"></a></p>
<h5>30.5-4 Difference</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Difference</code>( <var class="Arg">C1</var>, <var class="Arg">C2</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p><code class="func">Difference</code> returns the set difference of the collections <var class="Arg">C1</var> and <var class="Arg">C2</var>. Either argument may also be a homogeneous list that is not a proper set, in which case <code class="func">Difference</code> silently applies <code class="func">Set</code> (<a href="chap30.html#X7E399AC97FD98217"><span class="RefLink">30.3-7</span></a>) to it first.</p>
<p>The result of <code class="func">Difference</code> is the set of elements that lie in <var class="Arg">C1</var> but not in <var class="Arg">C2</var>. Note that <var class="Arg">C2</var> need not be a subset of <var class="Arg">C1</var>. The elements of <var class="Arg">C2</var>, however, that are not elements of <var class="Arg">C1</var> play no role for the result. If the result is a list then it is mutable and new, i.e., not identical to <var class="Arg">C1</var> or <var class="Arg">C2</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Difference( [ (1,2,3), (1,2,3,4) ], Group( (1,2,3), (1,2) ) );</span>
[ (1,2,3,4) ]
</pre></div>
<p><a id="X82D39CF980FDBFFA" name="X82D39CF980FDBFFA"></a></p>
<h4>30.6 <span class="Heading">Membership Test for Collections</span></h4>
<p><a id="X84B7FA8C7C94400F" name="X84B7FA8C7C94400F"></a></p>
<h5><code>30.6-1 \in</code></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ \in</code>( <var class="Arg">obj</var>, <var class="Arg">C</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>returns <code class="keyw">true</code> if the object <var class="Arg">obj</var> lies in the collection <var class="Arg">C</var>, and <code class="keyw">false</code> otherwise.</p>
<p>The infix version of the command</p>
<p><var class="Arg">obj</var> <code class="keyw">in</code> <var class="Arg">C</var></p>
<p>calls the operation <code class="func">\in</code>, for which methods can be installed.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">13 in Integers; [ 1, 2 ] in Integers;</span>
true
false
<span class="GAPprompt">gap></span> <span class="GAPinput">g:= Group( (1,2) );; (1,2) in g; (1,2,3) in g;</span>
true
false
</pre></div>
<p><a id="X8151A51884B7EE2C" name="X8151A51884B7EE2C"></a></p>
<h4>30.7 <span class="Heading">Random Elements</span></h4>
<p>The method used by <strong class="pkg">GAP</strong> to obtain random elements may depend on the type object.</p>
<p>Most methods which produce random elements in <strong class="pkg">GAP</strong> use a global random number generator (see <code class="func">GlobalMersenneTwister</code> (<a href="chap14.html#X7F772E2686B35865"><span class="RefLink">14.7-4</span></a>)). This random number generator is (deliberately) initialized to the same values when <strong class="pkg">GAP</strong> is started, so different runs of <strong class="pkg">GAP</strong> with the same input will always produce the same result, even if random calculations are involved.</p>
<p>See <code class="func">Reset</code> (<a href="chap14.html#X819E3E3080297347"><span class="RefLink">14.7-3</span></a>) for a description of how to reset the random number generator to a previous state.</p>
<p><a id="X7FF906E57D6936F8" name="X7FF906E57D6936F8"></a></p>
<h5>30.7-1 Random</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Random</code>( <var class="Arg">listorcoll</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Random</code>( <var class="Arg">from</var>, <var class="Arg">to</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p><code class="func">Random</code> returns a (pseudo-)random element of the list or collection <var class="Arg">listorcoll</var>.</p>
<p>As lists or ranges are restricted in length (<span class="SimpleMath">2^28-1</span> or <span class="SimpleMath">2^60-1</span> depending on your system), the second form returns a random integer in the range <var class="Arg">from</var> to <var class="Arg">to</var> (inclusive) for arbitrary integers <var class="Arg">from</var> and <var class="Arg">to</var>.</p>
<p>The distribution of elements returned by <code class="func">Random</code> depends on the argument. For a list the distribution is uniform (all elements are equally likely). The same holds usually for finite collections that are not lists. For infinite collections some reasonable distribution is used.</p>
<p>See the chapters of the various collections to find out which distribution is being used.</p>
<p>For some collections ensuring a reasonable distribution can be difficult and require substantial runtime (for example for large finite groups). If speed is more important than a guaranteed distribution, the operation <code class="func">PseudoRandom</code> (<a href="chap30.html#X811B5BD47DC5356B"><span class="RefLink">30.7-2</span></a>) should be used instead.</p>
<p>Note that <code class="func">Random</code> is of course <em>not</em> an attribute.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Random([1..6]);</span>
6
<span class="GAPprompt">gap></span> <span class="GAPinput">Random(1, 2^100);</span>
866227015645295902682304086250
<span class="GAPprompt">gap></span> <span class="GAPinput">g:= Group( (1,2,3) );; Random( g ); Random( g );</span>
(1,3,2)
()
<span class="GAPprompt">gap></span> <span class="GAPinput">Random(Rationals);</span>
-4
</pre></div>
<p><a id="X811B5BD47DC5356B" name="X811B5BD47DC5356B"></a></p>
<h5>30.7-2 PseudoRandom</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PseudoRandom</code>( <var class="Arg">listorcoll</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p><code class="func">PseudoRandom</code> returns a pseudo random element of the list or collection <var class="Arg">listorcoll</var>, which can be roughly described as follows. For a list, <code class="func">PseudoRandom</code> returns the same as <code class="func">Random</code> (<a href="chap30.html#X7FF906E57D6936F8"><span class="RefLink">30.7-1</span></a>). For collections that are not lists, the elements returned by <code class="func">PseudoRandom</code> are <em>not</em> necessarily equally distributed, even for finite collections; the idea is that <code class="func">Random</code> (<a href="chap30.html#X7FF906E57D6936F8"><span class="RefLink">30.7-1</span></a>) returns elements according to a reasonable distribution, <code class="func">PseudoRandom</code> returns elements that are cheap to compute but need not satisfy this strong condition, and <code class="func">Representative</code> (<a href="chap30.html#X865507568182424E"><span class="RefLink">30.4-7</span></a>) returns arbitrary elements, probably the same element for each call.</p>
<p><a id="X7EBA01EB83BC65A9" name="X7EBA01EB83BC65A9"></a></p>
<h5>30.7-3 RandomList</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RandomList</code>( <var class="Arg">list</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>For a dense list <var class="Arg">list</var>, <code class="func">RandomList</code> returns a (pseudo-)random element with equal distribution.</p>
<p>This function uses the <code class="func">GlobalMersenneTwister</code> (<a href="chap14.html#X7F772E2686B35865"><span class="RefLink">14.7-4</span></a>) to produce the random elements (a source of high quality random numbers).</p>
<p><a id="X85A3F00985453F95" name="X85A3F00985453F95"></a></p>
<h4>30.8 <span class="Heading">Iterators</span></h4>
<p><a id="X83ADF8287ED0668E" name="X83ADF8287ED0668E"></a></p>
<h5>30.8-1 Iterator</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Iterator</code>( <var class="Arg">listorcoll</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsStandardIterator</code>( <var class="Arg">listorcoll</var> )</td><td class="tdright">( filter )</td></tr></table></div>
<p>Iterators provide a possibility to loop over the elements of a (countable) collection or list <var class="Arg">listorcoll</var>, without repetition. For many collections <span class="SimpleMath">C</span>, an iterator of <span class="SimpleMath">C</span> need not store all elements of <span class="SimpleMath">C</span>, for example it is possible to construct an iterator of some infinite domains, such as the field of rational numbers.</p>
<p><code class="func">Iterator</code> returns a mutable <em>iterator</em> <span class="SimpleMath">iter</span> for its argument. If this argument is a list (which may contain holes), then <span class="SimpleMath">iter</span> iterates over the elements (but not the holes) of this list in the same order (see <code class="func">IteratorList</code> (<a href="chap30.html#X858A28667D137C4B"><span class="RefLink">30.8-6</span></a>) for details). If this argument is a collection but not a list then <span class="SimpleMath">iter</span> iterates over the elements of this collection in an unspecified order, which may change for repeated calls of <code class="func">Iterator</code>. Because iterators returned by <code class="func">Iterator</code> are mutable (see <a href="chap12.html#X7F0C119682196D65"><span class="RefLink">12.6</span></a>), each call of <code class="func">Iterator</code> for the same argument returns a <em>new</em> iterator. Therefore <code class="func">Iterator</code> is not an attribute (see <a href="chap13.html#X7C701DBF7BAE649A"><span class="RefLink">13.5</span></a>).</p>
<p>The only operations for iterators are <code class="func">IsDoneIterator</code> (<a href="chap30.html#X8055FC557B5D899E"><span class="RefLink">30.8-4</span></a>), <code class="func">NextIterator</code> (<a href="chap30.html#X879F62F77D1D1179"><span class="RefLink">30.8-5</span></a>), and <code class="func">ShallowCopy</code> (<a href="chap12.html#X846BC7107C352031"><span class="RefLink">12.7-1</span></a>). In particular, it is only possible to access the next element of the iterator with <code class="func">NextIterator</code> (<a href="chap30.html#X879F62F77D1D1179"><span class="RefLink">30.8-5</span></a>) if there is one, and this can be checked with <code class="func">IsDoneIterator</code> (<a href="chap30.html#X8055FC557B5D899E"><span class="RefLink">30.8-4</span></a>) For an iterator <span class="SimpleMath">iter</span>, <code class="func">ShallowCopy</code> (<a href="chap12.html#X846BC7107C352031"><span class="RefLink">12.7-1</span></a>) returns a mutable iterator <span class="SimpleMath">new</span> that iterates over the remaining elements independent of <span class="SimpleMath">iter</span>; the results of <code class="func">IsDoneIterator</code> (<a href="chap30.html#X8055FC557B5D899E"><span class="RefLink">30.8-4</span></a>) for <span class="SimpleMath">iter</span> and <span class="SimpleMath">new</span> are equal, and if <span class="SimpleMath">iter</span> is mutable then also the results of <code class="func">NextIterator</code> (<a href="chap30.html#X879F62F77D1D1179"><span class="RefLink">30.8-5</span></a>) for <span class="SimpleMath">iter</span> and <span class="SimpleMath">new</span> are equal; note that <code class="code">=</code> is not defined for iterators, so the equality of two iterators cannot be checked with <code class="code">=</code>.</p>
<p>When <code class="func">Iterator</code> is called for a <em>mutable</em> collection <span class="SimpleMath">C</span> then it is not defined whether <span class="SimpleMath">iter</span> respects changes to <span class="SimpleMath">C</span> occurring after the construction of <span class="SimpleMath">iter</span>, except if the documentation explicitly promises a certain behaviour. The latter is the case if the argument is a mutable list (see <code class="func">IteratorList</code> (<a href="chap30.html#X858A28667D137C4B"><span class="RefLink">30.8-6</span></a>) for subtleties in this case).</p>
<p>It is possible to have <code class="keyw">for</code>-loops run over mutable iterators instead of lists.</p>
<p>In some situations, one can construct iterators with a special succession of elements, see <code class="func">IteratorByBasis</code> (<a href="chap61.html#X855625D47979005D"><span class="RefLink">61.6-6</span></a>) for the possibility to loop over the elements of a vector space w.r.t. a given basis.</p>
<p>For lists, <code class="func">Iterator</code> is implemented by <code class="func">IteratorList</code> (<a href="chap30.html#X858A28667D137C4B"><span class="RefLink">30.8-6</span></a>). For collections <span class="SimpleMath">C</span> that are not lists, the default method is <code class="code">IteratorList( Enumerator( </code><span class="SimpleMath">C</span><code class="code"> ) )</code>. Better methods depending on <span class="SimpleMath">C</span> should be provided if possible.</p>
<p>For random access to the elements of a (possibly infinite) collection, <em>enumerators</em> are used. See <a href="chap21.html#X7EA3ACE27E43D174"><span class="RefLink">21.23</span></a> for the facility to compute a list from <span class="SimpleMath">C</span>, which provides a (partial) mapping from <span class="SimpleMath">C</span> to the positive integers.</p>
<p>The filter <code class="func">IsStandardIterator</code> means that the iterator is implemented as a component object and has components <code class="code">IsDoneIterator</code> and <code class="code">NextIterator</code> which are bound to the methods of the operations of the same name for this iterator.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">iter:= Iterator( GF(5) );</span>
<iterator>
<span class="GAPprompt">gap></span> <span class="GAPinput">l:= [];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">for i in iter do Add( l, i ); od; l;</span>
[ 0*Z(5), Z(5)^0, Z(5), Z(5)^2, Z(5)^3 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">iter:= Iterator( [ 1, 2, 3, 4 ] );; l:= [];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">for i in iter do</span>
<span class="GAPprompt">></span> <span class="GAPinput"> new:= ShallowCopy( iter );</span>
<span class="GAPprompt">></span> <span class="GAPinput"> for j in new do Add( l, j ); od;</span>
<span class="GAPprompt">></span> <span class="GAPinput"> od; l;</span>
[ 2, 3, 4, 3, 4, 4 ]
</pre></div>
<p><a id="X8688C20B828FC129" name="X8688C20B828FC129"></a></p>
<h5>30.8-2 IteratorSorted</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IteratorSorted</code>( <var class="Arg">listorcoll</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p><code class="func">IteratorSorted</code> returns a mutable iterator. The argument must be a collection or a list that is not necessarily dense but whose elements lie in the same family (see <a href="chap13.html#X846063757EC05986"><span class="RefLink">13.1</span></a>). It loops over the different elements in sorted order.</p>
<p>For a collection <span class="SimpleMath">C</span> that is not a list, the generic method is <code class="code">IteratorList( EnumeratorSorted( </code><var class="Arg">C</var><code class="code"> ) )</code>.</p>
<p><a id="X87168A827E5B28E4" name="X87168A827E5B28E4"></a></p>
<h5>30.8-3 IsIterator</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsIterator</code>( <var class="Arg">obj</var> )</td><td class="tdright">( category )</td></tr></table></div>
<p>Every iterator lies in the category <code class="code">IsIterator</code>.</p>
<p><a id="X8055FC557B5D899E" name="X8055FC557B5D899E"></a></p>
<h5>30.8-4 IsDoneIterator</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsDoneIterator</code>( <var class="Arg">iter</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>If <var class="Arg">iter</var> is an iterator for the list or collection <span class="SimpleMath">C</span> then <code class="code">IsDoneIterator( <var class="Arg">iter</var> )</code> is <code class="keyw">true</code> if all elements of <span class="SimpleMath">C</span> have been returned already by <code class="code">NextIterator( <var class="Arg">iter</var> )</code>, and <code class="keyw">false</code> otherwise.</p>
<p><a id="X879F62F77D1D1179" name="X879F62F77D1D1179"></a></p>
<h5>30.8-5 NextIterator</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NextIterator</code>( <var class="Arg">iter</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Let <var class="Arg">iter</var> be a mutable iterator for the list or collection <span class="SimpleMath">C</span>. If <code class="code">IsDoneIterator( <var class="Arg">iter</var> )</code> is <code class="keyw">false</code> then <code class="func">NextIterator</code> is applicable to <var class="Arg">iter</var>, and the result is the next element of <span class="SimpleMath">C</span>, according to the succession defined by <var class="Arg">iter</var>.</p>
<p>If <code class="code">IsDoneIterator( <var class="Arg">iter</var> )</code> is <code class="keyw">true</code> then it is not defined what happens when <code class="func">NextIterator</code> is called for <var class="Arg">iter</var>; that is, it may happen that an error is signalled or that something meaningless is returned, or even that <strong class="pkg">GAP</strong> crashes.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">iter:= Iterator( [ 1, 2, 3, 4 ] );</span>
<iterator>
<span class="GAPprompt">gap></span> <span class="GAPinput">sum:= 0;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">while not IsDoneIterator( iter ) do</span>
<span class="GAPprompt">></span> <span class="GAPinput"> sum:= sum + NextIterator( iter );</span>
<span class="GAPprompt">></span> <span class="GAPinput"> od;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsDoneIterator( iter ); sum;</span>
true
10
<span class="GAPprompt">gap></span> <span class="GAPinput">ir:= Iterator( Rationals );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">l:= [];; for i in [1..20] do Add( l, NextIterator( ir ) ); od; l;</span>
[ 0, 1, -1, 1/2, 2, -1/2, -2, 1/3, 2/3, 3/2, 3, -1/3, -2/3, -3/2, -3,
1/4, 3/4, 4/3, 4, -1/4 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">for i in ir do</span>
<span class="GAPprompt">></span> <span class="GAPinput"> if DenominatorRat( i ) > 10 then break; fi;</span>
<span class="GAPprompt">></span> <span class="GAPinput"> od;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">i;</span>
1/11
</pre></div>
<p><a id="X858A28667D137C4B" name="X858A28667D137C4B"></a></p>
<h5>30.8-6 IteratorList</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IteratorList</code>( <var class="Arg">list</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="func">IteratorList</code> returns a new iterator that allows iteration over the elements of the list <var class="Arg">list</var> (which may have holes) in the same order.</p>
<p>If <var class="Arg">list</var> is mutable then it is in principle possible to change <var class="Arg">list</var> after the call of <code class="func">IteratorList</code>. In this case all changes concerning positions that have not yet been reached in the iteration will also affect the iterator. For example, if <var class="Arg">list</var> is enlarged then the iterator will iterate also over the new elements at the end of the changed list.</p>
<p><em>Note</em> that changes of <var class="Arg">list</var> will also affect all shallow copies of <var class="Arg">list</var>.</p>
<p><a id="X7DB80BE68271247E" name="X7DB80BE68271247E"></a></p>
<h5>30.8-7 TrivialIterator</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TrivialIterator</code>( <var class="Arg">elm</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>is a mutable iterator for the collection <code class="code">[ <var class="Arg">elm</var> ]</code> that consists of exactly one element <var class="Arg">elm</var> (see <code class="func">IsTrivial</code> (<a href="chap30.html#X7E3402D6799D3C24"><span class="RefLink">30.4-3</span></a>)).</p>
<p><a id="X82677D8F817D6701" name="X82677D8F817D6701"></a></p>
<h5>30.8-8 IteratorByFunctions</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IteratorByFunctions</code>( <var class="Arg">record</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="func">IteratorByFunctions</code> returns a (mutable) iterator <var class="Arg">iter</var> for which <code class="func">NextIterator</code> (<a href="chap30.html#X879F62F77D1D1179"><span class="RefLink">30.8-5</span></a>), <code class="func">IsDoneIterator</code> (<a href="chap30.html#X8055FC557B5D899E"><span class="RefLink">30.8-4</span></a>), and <code class="func">ShallowCopy</code> (<a href="chap12.html#X846BC7107C352031"><span class="RefLink">12.7-1</span></a>) are computed via prescribed functions.</p>
<p>Let <var class="Arg">record</var> be a record with at least the following components.</p>
<dl>
<dt><strong class="Mark"><code class="code">NextIterator</code></strong></dt>
<dd><p>a function taking one argument <var class="Arg">iter</var>, which returns the next element of <var class="Arg">iter</var> (see <code class="func">NextIterator</code> (<a href="chap30.html#X879F62F77D1D1179"><span class="RefLink">30.8-5</span></a>)); for that, the components of <var class="Arg">iter</var> are changed,</p>
</dd>
<dt><strong class="Mark"><code class="code">IsDoneIterator</code></strong></dt>
<dd><p>a function taking one argument <var class="Arg">iter</var>, which returns the <code class="func">IsDoneIterator</code> (<a href="chap30.html#X8055FC557B5D899E"><span class="RefLink">30.8-4</span></a>) value of <var class="Arg">iter</var>,</p>
</dd>
<dt><strong class="Mark"><code class="code">ShallowCopy</code></strong></dt>
<dd><p>a function taking one argument <var class="Arg">iter</var>, which returns a record for which <code class="func">IteratorByFunctions</code> can be called in order to create a new iterator that is independent of <var class="Arg">iter</var> but behaves like <var class="Arg">iter</var> w.r.t. the operations <code class="func">NextIterator</code> (<a href="chap30.html#X879F62F77D1D1179"><span class="RefLink">30.8-5</span></a>) and <code class="func">IsDoneIterator</code> (<a href="chap30.html#X8055FC557B5D899E"><span class="RefLink">30.8-4</span></a>).</p>
</dd>
<dt><strong class="Mark"><code class="code">ViewObj</code> and <code class="code">PrintObj</code></strong></dt>
<dd><p>two functions that print what one wants to be printed when <code class="code">View( <var class="Arg">iter</var> )</code> or <code class="code">Print( <var class="Arg">item</var> )</code> is called (see <a href="chap6.html#X8074A8387C9DB9A8"><span class="RefLink">6.3</span></a>), if the <code class="code">ViewObj</code> component is missing then the <code class="code">PrintObj</code> method is used as a default.</p>
</dd>
</dl>
<p>Further (data) components may be contained in <var class="Arg">record</var> which can be used by these function.</p>
<p><code class="func">IteratorByFunctions</code> does <em>not</em> make a shallow copy of <var class="Arg">record</var>, this record is changed in place (see Section <a href="chap79.html#X82E86CF37B123FD4"><span class="RefLink">79.9</span></a>).</p>
<p>Iterators constructed with <code class="func">IteratorByFunctions</code> are in the filter <code class="func">IsStandardIterator</code> (<a href="chap30.html#X83ADF8287ED0668E"><span class="RefLink">30.8-1</span></a>).</p>
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