123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081828384858687888990919293949596979899100101102103104105106107108109110111112113114115116117118119120121122123124125126127128129130131132133134135136137138139140141142143144145146147148149150151152153154155156157158159160161162163164165166167168169170171172173174175176177178179180181182183184185186187188189190191192193194195196197198199200201202203204205206207208209210211212213214215216217218219220221222223224225226227228229230231 |
- <html>
- <head>
- <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
- <title>Collectors: Maps of Sets</title>
- <link rel="stylesheet" href="../../../../../../doc/src/boostbook.css" type="text/css">
- <meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
- <link rel="home" href="../../index.html" title="Chapter 1. Boost.Icl">
- <link rel="up" href="../semantics.html" title="Semantics">
- <link rel="prev" href="maps.html" title="Maps">
- <link rel="next" href="quantifiers__maps_of_numbers.html" title="Quantifiers: Maps of Numbers">
- </head>
- <body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF">
- <table cellpadding="2" width="100%"><tr>
- <td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../boost.png"></td>
- <td align="center"><a href="../../../../../../index.html">Home</a></td>
- <td align="center"><a href="../../../../../libraries.htm">Libraries</a></td>
- <td align="center"><a href="http://www.boost.org/users/people.html">People</a></td>
- <td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td>
- <td align="center"><a href="../../../../../../more/index.htm">More</a></td>
- </tr></table>
- <hr>
- <div class="spirit-nav">
- <a accesskey="p" href="maps.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../semantics.html"><img src="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="quantifiers__maps_of_numbers.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
- </div>
- <div class="section">
- <div class="titlepage"><div><div><h3 class="title">
- <a name="boost_icl.semantics.collectors__maps_of_sets"></a><a class="link" href="collectors__maps_of_sets.html" title="Collectors: Maps of Sets">Collectors:
- Maps of Sets</a>
- </h3></div></div></div>
- <p>
- Icl <code class="computeroutput"><span class="identifier">Collectors</span></code>, behave like
- <code class="computeroutput"><span class="identifier">Sets</span></code>. This can be understood
- easily, if we consider, that every map of sets can be transformed to an equivalent
- set of pairs. For instance in the pseudocode below map <code class="computeroutput"><span class="identifier">m</span></code>
- </p>
- <pre class="programlisting"><span class="identifier">icl</span><span class="special">::</span><span class="identifier">map</span><span class="special"><</span><span class="keyword">int</span><span class="special">,</span><span class="identifier">set</span><span class="special"><</span><span class="keyword">int</span><span class="special">></span> <span class="special">></span> <span class="identifier">m</span> <span class="special">=</span> <span class="special">{(</span><span class="number">1</span><span class="special">->{</span><span class="number">1</span><span class="special">,</span><span class="number">2</span><span class="special">}),</span> <span class="special">(</span><span class="number">2</span><span class="special">->{</span><span class="number">1</span><span class="special">})};</span>
- </pre>
- <p>
- is equivalent to set <code class="computeroutput"><span class="identifier">s</span></code>
- </p>
- <pre class="programlisting"><span class="identifier">icl</span><span class="special">::</span><span class="identifier">set</span><span class="special"><</span><span class="identifier">pair</span><span class="special"><</span><span class="keyword">int</span><span class="special">,</span><span class="keyword">int</span><span class="special">></span> <span class="special">></span> <span class="identifier">s</span> <span class="special">=</span> <span class="special">{(</span><span class="number">1</span><span class="special">,</span><span class="number">1</span><span class="special">),(</span><span class="number">1</span><span class="special">,</span><span class="number">2</span><span class="special">),</span> <span class="comment">//representing 1->{1,2}</span>
- <span class="special">(</span><span class="number">2</span><span class="special">,</span><span class="number">1</span><span class="special">)</span> <span class="special">};</span> <span class="comment">//representing 2->{1}</span>
- </pre>
- <p>
- </p>
- <p>
- Also the results of add, subtract and other operations on <code class="computeroutput"><span class="identifier">map</span>
- <span class="identifier">m</span></code> and <code class="computeroutput"><span class="identifier">set</span>
- <span class="identifier">s</span></code> preserves the equivalence of
- the containers <span class="emphasis"><em><span class="bold"><strong>almost</strong></span></em></span>
- perfectly:
- </p>
- <pre class="programlisting"><span class="identifier">m</span> <span class="special">+=</span> <span class="special">(</span><span class="number">1</span><span class="special">,</span><span class="number">3</span><span class="special">);</span>
- <span class="identifier">m</span> <span class="special">==</span> <span class="special">{(</span><span class="number">1</span><span class="special">->{</span><span class="number">1</span><span class="special">,</span><span class="number">2</span><span class="special">,</span><span class="number">3</span><span class="special">}),</span> <span class="special">(</span><span class="number">2</span><span class="special">->{</span><span class="number">1</span><span class="special">})};</span> <span class="comment">//aggregated on collision of key value 1</span>
- <span class="identifier">s</span> <span class="special">+=</span> <span class="special">(</span><span class="number">1</span><span class="special">,</span><span class="number">3</span><span class="special">);</span>
- <span class="identifier">s</span> <span class="special">==</span> <span class="special">{(</span><span class="number">1</span><span class="special">,</span><span class="number">1</span><span class="special">),(</span><span class="number">1</span><span class="special">,</span><span class="number">2</span><span class="special">),(</span><span class="number">1</span><span class="special">,</span><span class="number">3</span><span class="special">),</span> <span class="comment">//representing 1->{1,2,3}</span>
- <span class="special">(</span><span class="number">2</span><span class="special">,</span><span class="number">1</span><span class="special">)</span> <span class="special">};</span> <span class="comment">//representing 2->{1}</span>
- </pre>
- <p>
- </p>
- <p>
- The equivalence of <code class="computeroutput"><span class="identifier">m</span></code> and
- <code class="computeroutput"><span class="identifier">s</span></code> is only violated if an
- empty set occurres in <code class="computeroutput"><span class="identifier">m</span></code> by
- subtraction of a value pair:
- </p>
- <pre class="programlisting"><span class="identifier">m</span> <span class="special">-=</span> <span class="special">(</span><span class="number">2</span><span class="special">,</span><span class="number">1</span><span class="special">);</span>
- <span class="identifier">m</span> <span class="special">==</span> <span class="special">{(</span><span class="number">1</span><span class="special">->{</span><span class="number">1</span><span class="special">,</span><span class="number">2</span><span class="special">,</span><span class="number">3</span><span class="special">}),</span> <span class="special">(</span><span class="number">2</span><span class="special">->{})};</span> <span class="comment">//aggregated on collision of key value 2</span>
- <span class="identifier">s</span> <span class="special">-=</span> <span class="special">(</span><span class="number">2</span><span class="special">,</span><span class="number">1</span><span class="special">);</span>
- <span class="identifier">s</span> <span class="special">==</span> <span class="special">{(</span><span class="number">1</span><span class="special">,</span><span class="number">1</span><span class="special">),(</span><span class="number">1</span><span class="special">,</span><span class="number">2</span><span class="special">),(</span><span class="number">1</span><span class="special">,</span><span class="number">3</span><span class="special">)</span> <span class="comment">//representing 1->{1,2,3}</span>
- <span class="special">};</span> <span class="comment">//2->{} is not represented in s</span>
- </pre>
- <p>
- </p>
- <p>
- This problem can be dealt with in two ways.
- </p>
- <div class="orderedlist"><ol class="orderedlist" type="1">
- <li class="listitem">
- Deleting value pairs form the Collector, if it's associated value becomes
- a neutral value or <code class="computeroutput"><span class="identifier">identity_element</span></code>.
- </li>
- <li class="listitem">
- Using a different equality, called distinct equality in the laws to validate.
- Distinct equality only accounts for value pairs that that carry values
- unequal to the <code class="computeroutput"><span class="identifier">identity_element</span></code>.
- </li>
- </ol></div>
- <p>
- Solution (1) led to the introduction of map traits, particularly trait <span class="emphasis"><em><span class="bold"><strong>partial_absorber</strong></span></em></span>, which is the default
- setting in all icl's map templates.
- </p>
- <p>
- Solution (2), is applied to check the semantics of icl::Maps for the partial_enricher
- trait that does not delete value pairs that carry identity elements. Distinct
- equality is implemented by a non member function called <code class="computeroutput"><span class="identifier">is_distinct_equal</span></code>.
- Throughout this chapter distinct equality in pseudocode and law denotations
- is denoted as <code class="computeroutput"><span class="special">=</span><span class="identifier">d</span><span class="special">=</span></code> operator.
- </p>
- <p>
- The validity of the sets of laws that make up <code class="computeroutput"><span class="identifier">Set</span></code>
- semantics should now be quite evident. So the following text shows the laws
- that are validated for all <code class="computeroutput"><span class="identifier">Collector</span></code>
- types <code class="computeroutput"><span class="identifier">C</span></code>. Which are <code class="computeroutput"><a class="link" href="../../boost/icl/map.html" title="Class template map">icl::map</a></code><code class="computeroutput"><span class="special"><</span><span class="identifier">D</span><span class="special">,</span><span class="identifier">S</span><span class="special">,</span><span class="identifier">T</span><span class="special">></span></code>,
- <code class="computeroutput"><a class="link" href="../../boost/icl/interval_map.html" title="Class template interval_map">interval_map</a></code><code class="computeroutput"><span class="special"><</span><span class="identifier">D</span><span class="special">,</span><span class="identifier">S</span><span class="special">,</span><span class="identifier">T</span><span class="special">></span></code> and <code class="computeroutput"><a class="link" href="../../boost/icl/split_interval_map.html" title="Class template split_interval_map">split_interval_map</a></code><code class="computeroutput"><span class="special"><</span><span class="identifier">D</span><span class="special">,</span><span class="identifier">S</span><span class="special">,</span><span class="identifier">T</span><span class="special">></span></code> where <code class="computeroutput"><span class="identifier">CodomainT</span></code>
- type <code class="computeroutput"><span class="identifier">S</span></code> is a model of <code class="computeroutput"><span class="identifier">Set</span></code> and <code class="computeroutput"><span class="identifier">Trait</span></code>
- type <code class="computeroutput"><span class="identifier">T</span></code> is either <code class="computeroutput"><a class="link" href="../../boost/icl/partial_absorber.html" title="Struct partial_absorber">partial_absorber</a></code> or <code class="computeroutput"><a class="link" href="../../boost/icl/partial_enricher.html" title="Struct partial_enricher">partial_enricher</a></code>.
- </p>
- <h6>
- <a name="boost_icl.semantics.collectors__maps_of_sets.h0"></a>
- <span class="phrase"><a name="boost_icl.semantics.collectors__maps_of_sets.laws_on_set_union__set_intersection_and_set_difference"></a></span><a class="link" href="collectors__maps_of_sets.html#boost_icl.semantics.collectors__maps_of_sets.laws_on_set_union__set_intersection_and_set_difference">Laws
- on set union, set intersection and set difference</a>
- </h6>
- <p>
- </p>
- <pre class="programlisting"><span class="identifier">Associativity</span><span class="special"><</span><span class="identifier">C</span><span class="special">,+,==</span> <span class="special">>:</span> <span class="identifier">C</span> <span class="identifier">a</span><span class="special">,</span><span class="identifier">b</span><span class="special">,</span><span class="identifier">c</span><span class="special">;</span> <span class="identifier">a</span><span class="special">+(</span><span class="identifier">b</span><span class="special">+</span><span class="identifier">c</span><span class="special">)</span> <span class="special">==</span> <span class="special">(</span><span class="identifier">a</span><span class="special">+</span><span class="identifier">b</span><span class="special">)+</span><span class="identifier">c</span>
- <span class="identifier">Neutrality</span><span class="special"><</span><span class="identifier">C</span><span class="special">,+,==</span> <span class="special">></span> <span class="special">:</span> <span class="identifier">C</span> <span class="identifier">a</span><span class="special">;</span> <span class="identifier">a</span><span class="special">+</span><span class="identifier">C</span><span class="special">()</span> <span class="special">==</span> <span class="identifier">a</span>
- <span class="identifier">Commutativity</span><span class="special"><</span><span class="identifier">C</span><span class="special">,+,==</span> <span class="special">>:</span> <span class="identifier">C</span> <span class="identifier">a</span><span class="special">,</span><span class="identifier">b</span><span class="special">;</span> <span class="identifier">a</span><span class="special">+</span><span class="identifier">b</span> <span class="special">==</span> <span class="identifier">b</span><span class="special">+</span><span class="identifier">a</span>
- <span class="identifier">Associativity</span><span class="special"><</span><span class="identifier">C</span><span class="special">,&,==</span> <span class="special">>:</span> <span class="identifier">C</span> <span class="identifier">a</span><span class="special">,</span><span class="identifier">b</span><span class="special">,</span><span class="identifier">c</span><span class="special">;</span> <span class="identifier">a</span><span class="special">&(</span><span class="identifier">b</span><span class="special">&</span><span class="identifier">c</span><span class="special">)</span> <span class="special">==(</span><span class="identifier">a</span><span class="special">&</span><span class="identifier">b</span><span class="special">)&</span><span class="identifier">c</span>
- <span class="identifier">Commutativity</span><span class="special"><</span><span class="identifier">C</span><span class="special">,&,==</span> <span class="special">>:</span> <span class="identifier">C</span> <span class="identifier">a</span><span class="special">,</span><span class="identifier">b</span><span class="special">;</span> <span class="identifier">a</span><span class="special">&</span><span class="identifier">b</span> <span class="special">==</span> <span class="identifier">b</span><span class="special">&</span><span class="identifier">a</span>
- <span class="identifier">RightNeutrality</span><span class="special"><</span><span class="identifier">C</span><span class="special">,-,==</span> <span class="special">>:</span> <span class="identifier">C</span> <span class="identifier">a</span><span class="special">;</span> <span class="identifier">a</span><span class="special">-</span><span class="identifier">C</span><span class="special">()</span> <span class="special">==</span> <span class="identifier">a</span>
- <span class="identifier">Inversion</span><span class="special"><</span><span class="identifier">C</span><span class="special">,-,=</span><span class="identifier">v</span><span class="special">=</span> <span class="special">></span> <span class="special">:</span> <span class="identifier">C</span> <span class="identifier">a</span><span class="special">;</span> <span class="identifier">a</span> <span class="special">-</span> <span class="identifier">a</span> <span class="special">=</span><span class="identifier">v</span><span class="special">=</span> <span class="identifier">C</span><span class="special">()</span>
- </pre>
- <p>
- </p>
- <p>
- All the fundamental laws could be validated for all icl Maps in their instantiation
- as Maps of Sets or Collectors. As expected, Inversion only holds for distinct
- equality, if the map is not a <code class="computeroutput"><span class="identifier">partial_absorber</span></code>.
- </p>
- <p>
- </p>
- <pre class="programlisting"> <span class="special">+</span> <span class="special">&</span> <span class="special">-</span>
- <span class="identifier">Associativity</span> <span class="special">==</span> <span class="special">==</span>
- <span class="identifier">Neutrality</span> <span class="special">==</span> <span class="special">==</span>
- <span class="identifier">Commutativity</span> <span class="special">==</span> <span class="special">==</span>
- <span class="identifier">Inversion</span> <span class="identifier">partial_absorber</span> <span class="special">==</span>
- <span class="identifier">partial_enricher</span> <span class="special">=</span><span class="identifier">d</span><span class="special">=</span>
- </pre>
- <p>
- </p>
- <h6>
- <a name="boost_icl.semantics.collectors__maps_of_sets.h1"></a>
- <span class="phrase"><a name="boost_icl.semantics.collectors__maps_of_sets.distributivity_laws"></a></span><a class="link" href="collectors__maps_of_sets.html#boost_icl.semantics.collectors__maps_of_sets.distributivity_laws">Distributivity
- Laws</a>
- </h6>
- <p>
- </p>
- <pre class="programlisting"> <span class="identifier">Distributivity</span><span class="special"><</span><span class="identifier">C</span><span class="special">,+,&,=</span><span class="identifier">v</span><span class="special">=</span> <span class="special">></span> <span class="special">:</span> <span class="identifier">C</span> <span class="identifier">a</span><span class="special">,</span><span class="identifier">b</span><span class="special">,</span><span class="identifier">c</span><span class="special">;</span> <span class="identifier">a</span> <span class="special">+</span> <span class="special">(</span><span class="identifier">b</span> <span class="special">&</span> <span class="identifier">c</span><span class="special">)</span> <span class="special">=</span><span class="identifier">v</span><span class="special">=</span> <span class="special">(</span><span class="identifier">a</span> <span class="special">+</span> <span class="identifier">b</span><span class="special">)</span> <span class="special">&</span> <span class="special">(</span><span class="identifier">a</span> <span class="special">+</span> <span class="identifier">c</span><span class="special">)</span>
- <span class="identifier">Distributivity</span><span class="special"><</span><span class="identifier">C</span><span class="special">,&,+,=</span><span class="identifier">v</span><span class="special">=</span> <span class="special">></span> <span class="special">:</span> <span class="identifier">C</span> <span class="identifier">a</span><span class="special">,</span><span class="identifier">b</span><span class="special">,</span><span class="identifier">c</span><span class="special">;</span> <span class="identifier">a</span> <span class="special">&</span> <span class="special">(</span><span class="identifier">b</span> <span class="special">+</span> <span class="identifier">c</span><span class="special">)</span> <span class="special">=</span><span class="identifier">v</span><span class="special">=</span> <span class="special">(</span><span class="identifier">a</span> <span class="special">&</span> <span class="identifier">b</span><span class="special">)</span> <span class="special">+</span> <span class="special">(</span><span class="identifier">a</span> <span class="special">&</span> <span class="identifier">c</span><span class="special">)</span>
- <span class="identifier">RightDistributivity</span><span class="special"><</span><span class="identifier">C</span><span class="special">,+,-,=</span><span class="identifier">v</span><span class="special">=</span> <span class="special">></span> <span class="special">:</span> <span class="identifier">C</span> <span class="identifier">a</span><span class="special">,</span><span class="identifier">b</span><span class="special">,</span><span class="identifier">c</span><span class="special">;</span> <span class="special">(</span><span class="identifier">a</span> <span class="special">+</span> <span class="identifier">b</span><span class="special">)</span> <span class="special">-</span> <span class="identifier">c</span> <span class="special">=</span><span class="identifier">v</span><span class="special">=</span> <span class="special">(</span><span class="identifier">a</span> <span class="special">-</span> <span class="identifier">c</span><span class="special">)</span> <span class="special">+</span> <span class="special">(</span><span class="identifier">b</span> <span class="special">-</span> <span class="identifier">c</span><span class="special">)</span>
- <span class="identifier">RightDistributivity</span><span class="special"><</span><span class="identifier">C</span><span class="special">,&,-,=</span><span class="identifier">v</span><span class="special">=</span> <span class="special">></span> <span class="special">:</span> <span class="identifier">C</span> <span class="identifier">a</span><span class="special">,</span><span class="identifier">b</span><span class="special">,</span><span class="identifier">c</span><span class="special">;</span> <span class="special">(</span><span class="identifier">a</span> <span class="special">&</span> <span class="identifier">b</span><span class="special">)</span> <span class="special">-</span> <span class="identifier">c</span> <span class="special">=</span><span class="identifier">v</span><span class="special">=</span> <span class="special">(</span><span class="identifier">a</span> <span class="special">-</span> <span class="identifier">c</span><span class="special">)</span> <span class="special">&</span> <span class="special">(</span><span class="identifier">b</span> <span class="special">-</span> <span class="identifier">c</span><span class="special">)</span>
- </pre>
- <p>
- </p>
- <p>
- Results for the distributivity laws are almost identical to the validation
- of sets except that for a <code class="computeroutput"><span class="identifier">partial_enricher</span>
- <span class="identifier">map</span></code> the law <code class="computeroutput"><span class="special">(</span><span class="identifier">a</span> <span class="special">&</span> <span class="identifier">b</span><span class="special">)</span> <span class="special">-</span>
- <span class="identifier">c</span> <span class="special">==</span>
- <span class="special">(</span><span class="identifier">a</span> <span class="special">-</span> <span class="identifier">c</span><span class="special">)</span>
- <span class="special">&</span> <span class="special">(</span><span class="identifier">b</span> <span class="special">-</span> <span class="identifier">c</span><span class="special">)</span></code> holds for lexicographical equality.
- </p>
- <p>
- </p>
- <pre class="programlisting"> <span class="special">+,&</span> <span class="special">&,+</span>
- <span class="identifier">Distributivity</span> <span class="identifier">joining</span> <span class="special">==</span> <span class="special">==</span>
- <span class="identifier">splitting</span> <span class="identifier">partial_absorber</span> <span class="special">=</span><span class="identifier">e</span><span class="special">=</span> <span class="special">=</span><span class="identifier">e</span><span class="special">=</span>
- <span class="identifier">partial_enricher</span> <span class="special">=</span><span class="identifier">e</span><span class="special">=</span> <span class="special">==</span>
- <span class="special">+,-</span> <span class="special">&,-</span>
- <span class="identifier">RightDistributivity</span> <span class="identifier">joining</span> <span class="special">==</span> <span class="special">==</span>
- <span class="identifier">splitting</span> <span class="special">=</span><span class="identifier">e</span><span class="special">=</span> <span class="special">==</span>
- </pre>
- <p>
- </p>
- <h6>
- <a name="boost_icl.semantics.collectors__maps_of_sets.h2"></a>
- <span class="phrase"><a name="boost_icl.semantics.collectors__maps_of_sets.demorgan_s_law_and_symmetric_difference"></a></span><a class="link" href="collectors__maps_of_sets.html#boost_icl.semantics.collectors__maps_of_sets.demorgan_s_law_and_symmetric_difference">DeMorgan's
- Law and Symmetric Difference</a>
- </h6>
- <p>
- </p>
- <pre class="programlisting"><span class="identifier">DeMorgan</span><span class="special"><</span><span class="identifier">C</span><span class="special">,+,&,=</span><span class="identifier">v</span><span class="special">=</span> <span class="special">></span> <span class="special">:</span> <span class="identifier">C</span> <span class="identifier">a</span><span class="special">,</span><span class="identifier">b</span><span class="special">,</span><span class="identifier">c</span><span class="special">;</span> <span class="identifier">a</span> <span class="special">-</span> <span class="special">(</span><span class="identifier">b</span> <span class="special">+</span> <span class="identifier">c</span><span class="special">)</span> <span class="special">=</span><span class="identifier">v</span><span class="special">=</span> <span class="special">(</span><span class="identifier">a</span> <span class="special">-</span> <span class="identifier">b</span><span class="special">)</span> <span class="special">&</span> <span class="special">(</span><span class="identifier">a</span> <span class="special">-</span> <span class="identifier">c</span><span class="special">)</span>
- <span class="identifier">DeMorgan</span><span class="special"><</span><span class="identifier">C</span><span class="special">,&,+,=</span><span class="identifier">v</span><span class="special">=</span> <span class="special">></span> <span class="special">:</span> <span class="identifier">C</span> <span class="identifier">a</span><span class="special">,</span><span class="identifier">b</span><span class="special">,</span><span class="identifier">c</span><span class="special">;</span> <span class="identifier">a</span> <span class="special">-</span> <span class="special">(</span><span class="identifier">b</span> <span class="special">&</span> <span class="identifier">c</span><span class="special">)</span> <span class="special">=</span><span class="identifier">v</span><span class="special">=</span> <span class="special">(</span><span class="identifier">a</span> <span class="special">-</span> <span class="identifier">b</span><span class="special">)</span> <span class="special">+</span> <span class="special">(</span><span class="identifier">a</span> <span class="special">-</span> <span class="identifier">c</span><span class="special">)</span>
- </pre>
- <p>
- </p>
- <p>
- </p>
- <pre class="programlisting"> <span class="special">+,&</span> <span class="special">&,+</span>
- <span class="identifier">DeMorgan</span> <span class="identifier">joining</span> <span class="special">==</span> <span class="special">==</span>
- <span class="identifier">splitting</span> <span class="special">==</span> <span class="special">=</span><span class="identifier">e</span><span class="special">=</span>
- </pre>
- <p>
- </p>
- <p>
- </p>
- <pre class="programlisting"><span class="identifier">SymmetricDifference</span><span class="special"><</span><span class="identifier">C</span><span class="special">,==</span> <span class="special">></span> <span class="special">:</span> <span class="identifier">C</span> <span class="identifier">a</span><span class="special">,</span><span class="identifier">b</span><span class="special">,</span><span class="identifier">c</span><span class="special">;</span> <span class="special">(</span><span class="identifier">a</span> <span class="special">+</span> <span class="identifier">b</span><span class="special">)</span> <span class="special">-</span> <span class="special">(</span><span class="identifier">a</span> <span class="special">*</span> <span class="identifier">b</span><span class="special">)</span> <span class="special">==</span> <span class="special">(</span><span class="identifier">a</span> <span class="special">-</span> <span class="identifier">b</span><span class="special">)</span> <span class="special">+</span> <span class="special">(</span><span class="identifier">b</span> <span class="special">-</span> <span class="identifier">a</span><span class="special">)</span>
- </pre>
- <p>
- </p>
- <p>
- Reviewing the validity tables above shows, that the sets of valid laws for
- <code class="computeroutput"><span class="identifier">icl</span> <span class="identifier">Sets</span></code>
- and <code class="computeroutput"><span class="identifier">icl</span> <span class="identifier">Maps</span>
- <span class="identifier">of</span> <span class="identifier">Sets</span></code>
- that are <span class="emphasis"><em>identity absorbing</em></span> are exactly the same. As
- expected, only for Maps of Sets that represent empty sets as associated values,
- called <span class="emphasis"><em>identity enrichers</em></span>, there are marginal semantic
- differences.
- </p>
- </div>
- <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
- <td align="left"></td>
- <td align="right"><div class="copyright-footer">Copyright © 2007-2010 Joachim
- Faulhaber<br>Copyright © 1999-2006 Cortex Software
- GmbH<p>
- Distributed under the Boost Software License, Version 1.0. (See accompanying
- file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
- </p>
- </div></td>
- </tr></table>
- <hr>
- <div class="spirit-nav">
- <a accesskey="p" href="maps.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../semantics.html"><img src="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="quantifiers__maps_of_numbers.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
- </div>
- </body>
- </html>
|