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| <div class="section boost_icl_semantics_collectors__maps_of_sets" lang="en"> |
| <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 type="1"> |
| <li> |
| 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> |
| 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> |
| <a name="boost_icl.semantics.collectors__maps_of_sets.laws_on_set_union__set_intersection_and_set_difference"></a><h6> |
| <a name="id1090371"></a> |
| <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> |
| <a name="boost_icl.semantics.collectors__maps_of_sets.distributivity_laws"></a><h6> |
| <a name="id1091034"></a> |
| <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> |
| <a name="boost_icl.semantics.collectors__maps_of_sets.demorgan_s_law_and_symmetric_difference"></a><h6> |
| <a name="id1091906"></a> |
| <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> |
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