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<div class="chlinkprevnexttop"> <a href="chap0.html">[Top of Book]</a> <a href="chap0.html#contents">[Contents]</a> <a href="chap21.html">[Previous Chapter]</a> <a href="chap23.html">[Next Chapter]</a> </div>
<p id="mathjaxlink" class="pcenter"><a href="chap22_mj.html">[MathJax on]</a></p>
<p><a id="X7AC531DD79B6938E" name="X7AC531DD79B6938E"></a></p>
<div class="ChapSects"><a href="chap22.html#X7AC531DD79B6938E">22 <span class="Heading">Boolean Lists</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap22.html#X7E7832B0804221AE">22.1 <span class="Heading">IsBlist (Filter)</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap22.html#X7BE078187A08DCEA">22.1-1 IsBlist</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap22.html#X7CC745317FE54C14">22.2 <span class="Heading">Boolean Lists Representing Subsets</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap22.html#X7C597B2D87CA2E6E">22.2-1 BlistList</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap22.html#X874BEF63785AB439">22.2-2 ListBlist</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap22.html#X85AD5EF77EFD7451">22.2-3 SizeBlist</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap22.html#X7BA42D03796ED4B3">22.2-4 IsSubsetBlist</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap22.html#X8100080382AECFF9">22.3 <span class="Heading">Set Operations via Boolean Lists</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap22.html#X7970BD3883C42D91">22.3-1 <span class="Heading">UnionBlist</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap22.html#X86E1F8DE85E1EE1E">22.3-2 <span class="Heading">IntersectionBlist</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap22.html#X7D6FC2C58725708C">22.3-3 DifferenceBlist</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap22.html#X8634D25D7B4C6151">22.4 <span class="Heading">Function that Modify Boolean Lists</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap22.html#X79815EB77CC8A389">22.4-1 UniteBlist</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap22.html#X7C86C8D3853BE5EB">22.4-2 UniteBlistList</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap22.html#X84EB70D37EB275DF">22.4-3 IntersectBlist</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap22.html#X7AA138407D5A3BAC">22.4-4 SubtractBlist</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap22.html#X7C71B225841DFC0F">22.5 <span class="Heading">More about Boolean Lists</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap22.html#X8453ADDA810B4C03">22.5-1 IsBlistRep</a></span>
</div></div>
</div>
<h3>22 <span class="Heading">Boolean Lists</span></h3>
<p>This chapter describes boolean lists. A <em>boolean list</em> is a list that has no holes and contains only the boolean values <code class="keyw">true</code> and <code class="keyw">false</code> (see Chapter <a href="chap20.html#X787B4AB77A2F5E14"><span class="RefLink">20</span></a>). In function names we call boolean lists <em>blists</em> for brevity.</p>
<p><a id="X7E7832B0804221AE" name="X7E7832B0804221AE"></a></p>
<h4>22.1 <span class="Heading">IsBlist (Filter)</span></h4>
<p><a id="X7BE078187A08DCEA" name="X7BE078187A08DCEA"></a></p>
<h5>22.1-1 IsBlist</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsBlist</code>( <var class="Arg">obj</var> )</td><td class="tdright">( category )</td></tr></table></div>
<p>A boolean list ("blist") is a list that has no holes and contains only <code class="keyw">true</code> and <code class="keyw">false</code>. Boolean lists can be represented in an efficient compact form, see <a href="chap22.html#X7C71B225841DFC0F"><span class="RefLink">22.5</span></a> for details.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsBlist( [ true, true, false, false ] );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsBlist( [] );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsBlist( [false,,true] ); # has holes</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">IsBlist( [1,1,0,0] ); # contains not only boolean values</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">IsBlist( 17 ); # is not even a list</span>
false
</pre></div>
<p>Boolean lists are lists and all operations for lists are therefore applicable to boolean lists.</p>
<p>Boolean lists can be used in various ways, but maybe the most important application is their use for the description of <em>subsets</em> of finite sets. Suppose <span class="SimpleMath">set</span> is a finite set, represented as a list. Then a subset <span class="SimpleMath">sub</span> of <span class="SimpleMath">set</span> is represented by a boolean list <span class="SimpleMath">blist</span> of the same length as <span class="SimpleMath">set</span> such that <span class="SimpleMath">blist[i]</span> is <code class="keyw">true</code> if <span class="SimpleMath">set[i]</span> is in <span class="SimpleMath">sub</span>, and <code class="keyw">false</code> otherwise.</p>
<p><a id="X7CC745317FE54C14" name="X7CC745317FE54C14"></a></p>
<h4>22.2 <span class="Heading">Boolean Lists Representing Subsets</span></h4>
<p><a id="X7C597B2D87CA2E6E" name="X7C597B2D87CA2E6E"></a></p>
<h5>22.2-1 BlistList</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ BlistList</code>( <var class="Arg">list</var>, <var class="Arg">sub</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns a new boolean list that describes the list <var class="Arg">sub</var> as a sublist of the dense list <var class="Arg">list</var>. That is <code class="func">BlistList</code> returns a boolean list <span class="SimpleMath">blist</span> of the same length as <var class="Arg">list</var> such that <span class="SimpleMath">blist[i]</span> is <code class="keyw">true</code> if <var class="Arg">list</var><span class="SimpleMath">[i]</span> is in <var class="Arg">sub</var> and <code class="keyw">false</code> otherwise.</p>
<p><var class="Arg">list</var> need not be a proper set (see <a href="chap21.html#X80ABC25582343910"><span class="RefLink">21.19</span></a>), even though in this case <code class="func">BlistList</code> is most efficient. In particular <var class="Arg">list</var> may contain duplicates. <var class="Arg">sub</var> need not be a proper sublist of <var class="Arg">list</var>, i.e., <var class="Arg">sub</var> may contain elements that are not in <var class="Arg">list</var>. Those elements of course have no influence on the result of <code class="func">BlistList</code>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">BlistList( [1..10], [2,3,5,7] );</span>
[ false, true, true, false, true, false, true, false, false, false ]
<span class="GAPprompt">gap></span> <span class="GAPinput">BlistList( [1,2,3,4,5,2,8,6,4,10], [4,8,9,16] );</span>
[ false, false, false, true, false, false, true, false, true, false ]
</pre></div>
<p>See also <code class="func">UniteBlistList</code> (<a href="chap22.html#X7C86C8D3853BE5EB"><span class="RefLink">22.4-2</span></a>).</p>
<p><a id="X874BEF63785AB439" name="X874BEF63785AB439"></a></p>
<h5>22.2-2 ListBlist</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ListBlist</code>( <var class="Arg">list</var>, <var class="Arg">blist</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>returns the sublist <span class="SimpleMath">sub</span> of the list <var class="Arg">list</var>, which must have no holes, represented by the boolean list <var class="Arg">blist</var>, which must have the same length as <var class="Arg">list</var>.</p>
<p><span class="SimpleMath">sub</span> contains the element <var class="Arg">list</var><span class="SimpleMath">[i]</span> if <var class="Arg">blist</var><span class="SimpleMath">[i]</span> is <code class="keyw">true</code> and does not contain the element if <var class="Arg">blist</var><span class="SimpleMath">[i]</span> is <code class="keyw">false</code>. The order of the elements in <span class="SimpleMath">sub</span> is the same as the order of the corresponding elements in <var class="Arg">list</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">ListBlist([1..8],[false,true,true,true,true,false,true,true]);</span>
[ 2, 3, 4, 5, 7, 8 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">ListBlist( [1,2,3,4,5,2,8,6,4,10],</span>
<span class="GAPprompt">></span> <span class="GAPinput">[false,false,false,true,false,false,true,false,true,false] );</span>
[ 4, 8, 4 ]
</pre></div>
<p><a id="X85AD5EF77EFD7451" name="X85AD5EF77EFD7451"></a></p>
<h5>22.2-3 SizeBlist</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SizeBlist</code>( <var class="Arg">blist</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns the number of entries of the boolean list <var class="Arg">blist</var> that are <code class="keyw">true</code>. This is the size of the subset represented by the boolean list <var class="Arg">blist</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">SizeBlist( [ false, true, false, true, false ] );</span>
2
</pre></div>
<p><a id="X7BA42D03796ED4B3" name="X7BA42D03796ED4B3"></a></p>
<h5>22.2-4 IsSubsetBlist</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsSubsetBlist</code>( <var class="Arg">blist1</var>, <var class="Arg">blist2</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns <code class="keyw">true</code> if the boolean list <var class="Arg">blist2</var> is a subset of the boolean list <var class="Arg">blist1</var>, which must have equal length, and <code class="keyw">false</code> otherwise. <var class="Arg">blist2</var> is a subset of <var class="Arg">blist1</var> if <var class="Arg">blist1</var><span class="SimpleMath">[i] =</span> <var class="Arg">blist1</var><span class="SimpleMath">[i]</span> <code class="keyw">or</code> <var class="Arg">blist2</var><span class="SimpleMath">[i]</span> for all <span class="SimpleMath">i</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">blist1 := [ true, true, false, false ];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">blist2 := [ true, false, true, false ];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsSubsetBlist( blist1, blist2 );</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">blist2 := [ true, false, false, false ];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsSubsetBlist( blist1, blist2 );</span>
true
</pre></div>
<p><a id="X8100080382AECFF9" name="X8100080382AECFF9"></a></p>
<h4>22.3 <span class="Heading">Set Operations via Boolean Lists</span></h4>
<p><a id="X7970BD3883C42D91" name="X7970BD3883C42D91"></a></p>
<h5>22.3-1 <span class="Heading">UnionBlist</span></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ UnionBlist</code>( <var class="Arg">blist1</var>, <var class="Arg">blist2</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">‣ UnionBlist</code>( <var class="Arg">list</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>In the first form <code class="func">UnionBlist</code> returns the union of the boolean lists <var class="Arg">blist1</var>, <var class="Arg">blist2</var>, etc., which must have equal length. The <em>union</em> is a new boolean list that contains at position <span class="SimpleMath">i</span> the value <var class="Arg">blist1</var><span class="SimpleMath">[i]</span> <code class="keyw">or</code> <var class="Arg">blist2</var><span class="SimpleMath">[i]</span> <code class="keyw">or</code> <span class="SimpleMath">...</span>.</p>
<p>The second form takes the union of all blists (which as for the first form must have equal length) in the list <var class="Arg">list</var>.</p>
<p><a id="X86E1F8DE85E1EE1E" name="X86E1F8DE85E1EE1E"></a></p>
<h5>22.3-2 <span class="Heading">IntersectionBlist</span></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IntersectionBlist</code>( <var class="Arg">blist1</var>, <var class="Arg">blist2</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">‣ IntersectionBlist</code>( <var class="Arg">list</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>In the first form <code class="func">IntersectionBlist</code> returns the intersection of the boolean lists <var class="Arg">blist1</var>, <var class="Arg">blist2</var>, etc., which must have equal length. The <em>intersection</em> is a new blist that contains at position <span class="SimpleMath">i</span> the value <var class="Arg">blist1</var><span class="SimpleMath">[i]</span> <code class="keyw">and</code> <var class="Arg">blist2</var><span class="SimpleMath">[i]</span> <code class="keyw">and</code> <span class="SimpleMath">...</span>.</p>
<p>In the second form <var class="Arg">list</var> must be a list of boolean lists <var class="Arg">blist1</var>, <var class="Arg">blist2</var>, etc., which must have equal length, and <code class="func">IntersectionBlist</code> returns the intersection of those boolean lists.</p>
<p><a id="X7D6FC2C58725708C" name="X7D6FC2C58725708C"></a></p>
<h5>22.3-3 DifferenceBlist</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DifferenceBlist</code>( <var class="Arg">blist1</var>, <var class="Arg">blist2</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns the asymmetric set difference of the two boolean lists <var class="Arg">blist1</var> and <var class="Arg">blist2</var>, which must have equal length. The <em>asymmetric set difference</em> is a new boolean list that contains at position <span class="SimpleMath">i</span> the value <var class="Arg">blist1</var><span class="SimpleMath">[i]</span> <code class="keyw">and</code> <code class="keyw">not</code> <var class="Arg">blist2</var><span class="SimpleMath">[i]</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">blist1 := [ true, true, false, false ];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">blist2 := [ true, false, true, false ];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">UnionBlist( blist1, blist2 );</span>
[ true, true, true, false ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IntersectionBlist( blist1, blist2 );</span>
[ true, false, false, false ]
<span class="GAPprompt">gap></span> <span class="GAPinput">DifferenceBlist( blist1, blist2 );</span>
[ false, true, false, false ]
</pre></div>
<p><a id="X8634D25D7B4C6151" name="X8634D25D7B4C6151"></a></p>
<h4>22.4 <span class="Heading">Function that Modify Boolean Lists</span></h4>
<p><a id="X79815EB77CC8A389" name="X79815EB77CC8A389"></a></p>
<h5>22.4-1 UniteBlist</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ UniteBlist</code>( <var class="Arg">blist1</var>, <var class="Arg">blist2</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><code class="func">UniteBlist</code> unites the boolean list <var class="Arg">blist1</var> with the boolean list <var class="Arg">blist2</var>, which must have the same length. This is equivalent to assigning <var class="Arg">blist1</var><span class="SimpleMath">[i] :=</span> <var class="Arg">blist1</var><span class="SimpleMath">[i]</span> <code class="keyw">or</code> <var class="Arg">blist2</var><span class="SimpleMath">[i]</span> for all <span class="SimpleMath">i</span>.</p>
<p><code class="func">UniteBlist</code> returns nothing, it is only called to change <var class="Arg">blist1</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">blist1 := [ true, true, false, false ];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">blist2 := [ true, false, true, false ];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">UniteBlist( blist1, blist2 );</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">blist1;</span>
[ true, true, true, false ]
</pre></div>
<p>The function <code class="func">UnionBlist</code> (<a href="chap22.html#X7970BD3883C42D91"><span class="RefLink">22.3-1</span></a>) is the nondestructive counterpart to <code class="func">UniteBlist</code>.</p>
<p><a id="X7C86C8D3853BE5EB" name="X7C86C8D3853BE5EB"></a></p>
<h5>22.4-2 UniteBlistList</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ UniteBlistList</code>( <var class="Arg">list</var>, <var class="Arg">blist</var>, <var class="Arg">sub</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>works like <code class="code">UniteBlist(<var class="Arg">blist</var>,BlistList(<var class="Arg">list</var>,<var class="Arg">sub</var>))</code>. As no intermediate blist is created, the performance is better than the separate function calls.</p>
<p><a id="X84EB70D37EB275DF" name="X84EB70D37EB275DF"></a></p>
<h5>22.4-3 IntersectBlist</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IntersectBlist</code>( <var class="Arg">blist1</var>, <var class="Arg">blist2</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>intersects the boolean list <var class="Arg">blist1</var> with the boolean list <var class="Arg">blist2</var>, which must have the same length. This is equivalent to assigning <var class="Arg">blist1</var><span class="SimpleMath">[i]:=</span> <var class="Arg">blist1</var><span class="SimpleMath">[i]</span> <code class="keyw">and</code> <var class="Arg">blist2</var><span class="SimpleMath">[i]</span> for all <span class="SimpleMath">i</span>.</p>
<p><code class="func">IntersectBlist</code> returns nothing, it is only called to change <var class="Arg">blist1</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">blist1 := [ true, true, false, false ];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">blist2 := [ true, false, true, false ];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IntersectBlist( blist1, blist2 );</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">blist1;</span>
[ true, false, false, false ]
</pre></div>
<p>The function <code class="func">IntersectionBlist</code> (<a href="chap22.html#X86E1F8DE85E1EE1E"><span class="RefLink">22.3-2</span></a>) is the nondestructive counterpart to <code class="func">IntersectBlist</code>.</p>
<p><a id="X7AA138407D5A3BAC" name="X7AA138407D5A3BAC"></a></p>
<h5>22.4-4 SubtractBlist</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SubtractBlist</code>( <var class="Arg">blist1</var>, <var class="Arg">blist2</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>subtracts the boolean list <var class="Arg">blist2</var> from the boolean list <var class="Arg">blist1</var>, which must have equal length. This is equivalent to assigning <var class="Arg">blist1</var><span class="SimpleMath">[i]:=</span> <var class="Arg">blist1</var><span class="SimpleMath">[i]</span> <code class="keyw">and</code> <code class="keyw">not</code> <var class="Arg">blist2</var><span class="SimpleMath">[i]</span> for all <span class="SimpleMath">i</span>.</p>
<p><code class="func">SubtractBlist</code> returns nothing, it is only called to change <var class="Arg">blist1</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">blist1 := [ true, true, false, false ];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">blist2 := [ true, false, true, false ];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SubtractBlist( blist1, blist2 );</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">blist1;</span>
[ false, true, false, false ]
</pre></div>
<p>The function <code class="func">DifferenceBlist</code> (<a href="chap22.html#X7D6FC2C58725708C"><span class="RefLink">22.3-3</span></a>) is the nondestructive counterpart to <code class="func">SubtractBlist</code>.</p>
<p><a id="X7C71B225841DFC0F" name="X7C71B225841DFC0F"></a></p>
<h4>22.5 <span class="Heading">More about Boolean Lists</span></h4>
<p>We defined a boolean list as a list that has no holes and contains only <code class="keyw">true</code> and <code class="keyw">false</code>. There is a special internal representation for boolean lists that needs only 1 bit for each entry. This bit is set if the entry is <code class="keyw">true</code> and reset if the entry is <code class="keyw">false</code>. This representation is of course much more compact than the ordinary representation of lists, which needs 32 or 64 bits per entry.</p>
<p>Not every boolean list is represented in this compact representation. It would be too much work to test every time a list is changed, whether this list has become a boolean list. This section tells you under which circumstances a boolean list is represented in the compact representation, so you can write your functions in such a way that you make best use of the compact representation.</p>
<p>If a dense list containing only <code class="keyw">true</code> and <code class="keyw">false</code> is read, it is stored in the compact representation. Furthermore, the results of <code class="func">BlistList</code> (<a href="chap22.html#X7C597B2D87CA2E6E"><span class="RefLink">22.2-1</span></a>), <code class="func">UnionBlist</code> (<a href="chap22.html#X7970BD3883C42D91"><span class="RefLink">22.3-1</span></a>), <code class="func">IntersectionBlist</code> (<a href="chap22.html#X86E1F8DE85E1EE1E"><span class="RefLink">22.3-2</span></a>) and <code class="func">DifferenceBlist</code> (<a href="chap22.html#X7D6FC2C58725708C"><span class="RefLink">22.3-3</span></a>) are known to be boolean lists by construction, and thus are represented in the compact representation upon creation.</p>
<p>If an argument of <code class="func">IsSubsetBlist</code> (<a href="chap22.html#X7BA42D03796ED4B3"><span class="RefLink">22.2-4</span></a>), <code class="func">ListBlist</code> (<a href="chap22.html#X874BEF63785AB439"><span class="RefLink">22.2-2</span></a>), <code class="func">UnionBlist</code> (<a href="chap22.html#X7970BD3883C42D91"><span class="RefLink">22.3-1</span></a>), <code class="func">IntersectionBlist</code> (<a href="chap22.html#X86E1F8DE85E1EE1E"><span class="RefLink">22.3-2</span></a>), <code class="func">DifferenceBlist</code> (<a href="chap22.html#X7D6FC2C58725708C"><span class="RefLink">22.3-3</span></a>), <code class="func">UniteBlist</code> (<a href="chap22.html#X79815EB77CC8A389"><span class="RefLink">22.4-1</span></a>), <code class="func">IntersectBlist</code> (<a href="chap22.html#X84EB70D37EB275DF"><span class="RefLink">22.4-3</span></a>) and <code class="func">SubtractBlist</code> (<a href="chap22.html#X7AA138407D5A3BAC"><span class="RefLink">22.4-4</span></a>) is a list represented in the ordinary representation, it is tested to see if it is in fact a boolean list. If it is not, an error is signalled. If it is, the representation of the list is changed to the compact representation.</p>
<p>If you change a boolean list that is represented in the compact representation by assignment (see <a href="chap21.html#X8611EF768210625B"><span class="RefLink">21.4</span></a>) or <code class="func">Add</code> (<a href="chap21.html#X795EC9D67E34DAB0"><span class="RefLink">21.4-2</span></a>) in such a way that the list remains a boolean list it will remain represented in the compact representation. Note that changing a list that is not represented in the compact representation, whether it is a boolean list or not, in such a way that the resulting list becomes a boolean list, will never change the representation of the list.</p>
<p><a id="X8453ADDA810B4C03" name="X8453ADDA810B4C03"></a></p>
<h5>22.5-1 IsBlistRep</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsBlistRep</code>( <var class="Arg">obj</var> )</td><td class="tdright">( representation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ConvertToBlistRep</code>( <var class="Arg">blist</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: <code class="keyw">true</code> or <code class="keyw">false</code></p>
<p>The first function is a filter that returns <code class="keyw">true</code> if the object <var class="Arg">obj</var> is a boolean list in compact representation and <code class="keyw">false</code> otherwise, see <a href="chap22.html#X7C71B225841DFC0F"><span class="RefLink">22.5</span></a>.</p>
<p>The second function converts the object <var class="Arg">blist</var> to a boolean list in compact representation and returns <code class="keyw">true</code> if this is possible. Otherwise <var class="Arg">blist</var> is unchanged and <code class="keyw">false</code> is returned.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">l := [true, false, true];</span>
[ true, false, true ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IsBlistRep(l);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">l := [true, false, 1]; </span>
[ true, false, 1 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">l[3] := false;</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">IsBlistRep(l);</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">ConvertToBlistRep(l);</span>
true
</pre></div>
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