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<div class="titlepage"><div><div><h2 class="title" style="clear: both">
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<a name="boost_icl.semantics"></a><a class="link" href="semantics.html" title="Semantics">Semantics</a>
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</h2></div></div></div>
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<div class="toc"><dl class="toc">
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<dt><span class="section"><a href="semantics.html#boost_icl.semantics.orderings_and_equivalences">Orderings
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and Equivalences</a></span></dt>
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<dt><span class="section"><a href="semantics/sets.html">Sets</a></span></dt>
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<dt><span class="section"><a href="semantics/maps.html">Maps</a></span></dt>
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<dt><span class="section"><a href="semantics/collectors__maps_of_sets.html">Collectors:
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Maps of Sets</a></span></dt>
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<dt><span class="section"><a href="semantics/quantifiers__maps_of_numbers.html">Quantifiers:
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Maps of Numbers</a></span></dt>
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<dt><span class="section"><a href="semantics/concept_induction.html">Concept Induction</a></span></dt>
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</dl></div>
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<p>
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<span class="quote">“<span class="quote">Beauty is the ultimate defense against complexity</span>”</span> -- <a href="http://en.wikipedia.org/wiki/David_Gelernter" target="_top">David Gelernter</a>
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</p>
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<p>
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In the <span class="bold"><strong>icl</strong></span> we follow the notion, that the
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semantics of a <span class="emphasis"><em><span class="bold"><strong>concept</strong></span></em></span>
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or <span class="emphasis"><em><span class="bold"><strong>abstract data type</strong></span></em></span>
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can be expressed by <span class="emphasis"><em><span class="bold"><strong>laws</strong></span></em></span>.
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We formulate laws over interval containers that can be evaluated for a given
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instantiation of the variables contained in the law. The following pseudocode
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gives a shorthand notation of such a law.
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</p>
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<pre class="programlisting"><span class="identifier">Commutativity</span><span class="special"><</span><span class="identifier">T</span><span class="special">,+>:</span>
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<span class="identifier">T</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="special">;</span>
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</pre>
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<p>
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This can of course be coded as a proper c++ class template which has been done
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for the validation of the <span class="bold"><strong>icl</strong></span>. For sake of
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simplicity we will use pseudocode here.
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</p>
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<p>
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The laws that describe the semantics of the <span class="bold"><strong>icl's</strong></span>
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class templates were validated using the Law based Test Automaton <span class="emphasis"><em><span class="bold"><strong>LaBatea</strong></span></em></span>, a tool that generates instances
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for the law's variables and then tests it's validity. Since the <span class="bold"><strong>icl</strong></span>
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deals with sets, maps and relations, that are well known objects from mathematics,
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the laws that we are using are mostly <span class="emphasis"><em>recycled</em></span> ones. Also
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some of those laws are grouped in notions like e.g. <span class="emphasis"><em>orderings</em></span>
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or <span class="emphasis"><em>algebras</em></span>.
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</p>
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<div class="section">
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<div class="titlepage"><div><div><h3 class="title">
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<a name="boost_icl.semantics.orderings_and_equivalences"></a><a class="link" href="semantics.html#boost_icl.semantics.orderings_and_equivalences" title="Orderings and Equivalences">Orderings
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and Equivalences</a>
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</h3></div></div></div>
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<h5>
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<a name="boost_icl.semantics.orderings_and_equivalences.h0"></a>
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<span class="phrase"><a name="boost_icl.semantics.orderings_and_equivalences.lexicographical_ordering_and_equality"></a></span><a class="link" href="semantics.html#boost_icl.semantics.orderings_and_equivalences.lexicographical_ordering_and_equality">Lexicographical
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Ordering and Equality</a>
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</h5>
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<p>
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On all set and map containers of the icl, there is an <code class="computeroutput"><span class="keyword">operator</span>
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<span class="special"><</span></code> that implements a <a href="https://boost.org/sgi/stl/StrictWeakOrdering.html" target="_top">strict
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weak ordering</a>. The semantics of <code class="computeroutput"><span class="keyword">operator</span>
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<span class="special"><</span></code> is the same as for an stl's
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<a href="https://boost.org/sgi/stl/SortedAssociativeContainer.html" target="_top">SortedAssociativeContainer</a>,
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specifically <a href="https://boost.org/sgi/stl/set.html" target="_top">stl::set</a>
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and <a href="https://boost.org/sgi/stl/Map.html" target="_top">stl::map</a>:
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</p>
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<pre class="programlisting"><span class="identifier">Irreflexivity</span><span class="special"><</span><span class="identifier">T</span><span class="special">,<</span> <span class="special">></span> <span class="special">:</span> <span class="identifier">T</span> <span class="identifier">a</span><span class="special">;</span> <span class="special">!(</span><span class="identifier">a</span><span class="special"><</span><span class="identifier">a</span><span class="special">)</span>
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<span class="identifier">Asymmetry</span><span class="special"><</span><span class="identifier">T</span><span class="special">,<</span> <span class="special">></span> <span class="special">:</span> <span class="identifier">T</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="identifier">implies</span> <span class="special">!(</span><span class="identifier">b</span><span class="special"><</span><span class="identifier">a</span><span class="special">)</span>
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<span class="identifier">Transitivity</span><span class="special"><</span><span class="identifier">T</span><span class="special">,<</span> <span class="special">></span> <span class="special">:</span> <span class="identifier">T</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">b</span><span class="special"><</span><span class="identifier">c</span> <span class="identifier">implies</span> <span class="identifier">a</span><span class="special"><</span><span class="identifier">c</span>
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</pre>
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<p>
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</p>
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<p>
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<code class="computeroutput"><span class="identifier">Operator</span> <span class="special"><</span></code>
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depends on the icl::container's template parameter <code class="computeroutput"><span class="identifier">Compare</span></code>
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that implements a <span class="emphasis"><em>strict weak ordering</em></span> for the container's
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<code class="computeroutput"><span class="identifier">domain_type</span></code>. For a given
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<code class="computeroutput"><span class="identifier">Compare</span></code> ordering, <code class="computeroutput"><span class="keyword">operator</span> <span class="special"><</span></code>
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implements a lexicographical comparison on icl::containers, that uses the
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<code class="computeroutput"><span class="identifier">Compare</span></code> order to establish
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a unique sequence of values in the container.
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</p>
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<p>
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The induced equivalence of <code class="computeroutput"><span class="keyword">operator</span>
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<span class="special"><</span></code> is lexicographical equality which
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is implemented as <code class="computeroutput"><span class="keyword">operator</span> <span class="special">==</span></code>.
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</p>
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<pre class="programlisting"><span class="comment">//equivalence induced by strict weak ordering <</span>
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<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> <span class="identifier">implies</span> <span class="identifier">a</span> <span class="special">==</span> <span class="identifier">b</span><span class="special">;</span>
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</pre>
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<p>
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Again this follows the semantics of the <span class="bold"><strong>stl</strong></span>.
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Lexicographical equality is stronger than the equality of elements. Two containers
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that contain the same elements can be lexicographically unequal, if their
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elements are differently sorted. Lexicographical comparison belongs to the
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<span class="emphasis"><em><span class="bold"><strong>segmental</strong></span></em></span> aspect. Of
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all the different sequences that are valid for unordered sets and maps, one
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such sequence is selected by the <code class="computeroutput"><span class="identifier">Compare</span></code>
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order of elements. Based on this selection a unique iteration is possible.
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</p>
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<h5>
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<a name="boost_icl.semantics.orderings_and_equivalences.h1"></a>
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<span class="phrase"><a name="boost_icl.semantics.orderings_and_equivalences.subset_ordering_and_element_equality"></a></span><a class="link" href="semantics.html#boost_icl.semantics.orderings_and_equivalences.subset_ordering_and_element_equality">Subset
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Ordering and Element Equality</a>
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</h5>
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<p>
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On the fundamental aspect only membership of elements matters, not their
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sequence. So there are functions <code class="computeroutput"><span class="identifier">contained_in</span></code>
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and <code class="computeroutput"><span class="identifier">element_equal</span></code> that implement
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the subset relation and the equality on elements. Yet, <code class="computeroutput"><span class="identifier">contained_in</span></code>
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and <code class="computeroutput"><span class="identifier">is_element_equal</span></code> functions
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are not really working on the level of elements. They also work on the basis
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of the containers templates <code class="computeroutput"><span class="identifier">Compare</span></code>
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parameter. In practical terms we need to distinguish between lexicographical
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equality <code class="computeroutput"><span class="keyword">operator</span> <span class="special">==</span></code>
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and equality of elements <code class="computeroutput"><span class="identifier">is_element_equal</span></code>,
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if we work with interval splitting interval containers:
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</p>
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<pre class="programlisting"><span class="identifier">split_interval_set</span><span class="special"><</span><span class="identifier">time</span><span class="special">></span> <span class="identifier">w1</span><span class="special">,</span> <span class="identifier">w2</span><span class="special">;</span> <span class="comment">//Pseudocode</span>
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<span class="identifier">w1</span> <span class="special">=</span> <span class="special">{[</span><span class="identifier">Mon</span> <span class="special">..</span> <span class="identifier">Sun</span><span class="special">)};</span> <span class="comment">//split_interval_set containing a week</span>
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<span class="identifier">w2</span> <span class="special">=</span> <span class="special">{[</span><span class="identifier">Mon</span> <span class="special">..</span> <span class="identifier">Fri</span><span class="special">)[</span><span class="identifier">Sat</span> <span class="special">..</span> <span class="identifier">Sun</span><span class="special">)};</span> <span class="comment">//Same week split in work and week end parts.</span>
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<span class="identifier">w1</span> <span class="special">==</span> <span class="identifier">w2</span><span class="special">;</span> <span class="comment">//false: Different segmentation</span>
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<span class="identifier">is_element_equal</span><span class="special">(</span><span class="identifier">w1</span><span class="special">,</span><span class="identifier">w2</span><span class="special">);</span> <span class="comment">//true: Same elements contained </span>
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</pre>
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<p>
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</p>
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<p>
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For a constant <code class="computeroutput"><span class="identifier">Compare</span></code> order
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on key elements, member function <code class="computeroutput"><span class="identifier">contained_in</span></code>
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that is defined for all icl::containers implements a <a href="http://en.wikipedia.org/wiki/Partially_ordered_set" target="_top">partial
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order</a> on icl::containers.
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</p>
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<p>
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</p>
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<pre class="programlisting"><span class="identifier">with</span> <span class="special"><=</span> <span class="keyword">for</span> <span class="identifier">contained_in</span><span class="special">,</span>
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<span class="special">=</span><span class="identifier">e</span><span class="special">=</span> <span class="keyword">for</span> <span class="identifier">is_element_equal</span><span class="special">:</span>
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<span class="identifier">Reflexivity</span><span class="special"><</span><span class="identifier">T</span><span class="special">,<=</span> <span class="special">></span> <span class="special">:</span> <span class="identifier">T</span> <span class="identifier">a</span><span class="special">;</span> <span class="identifier">a</span> <span class="special"><=</span> <span class="identifier">a</span>
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<span class="identifier">Antisymmetry</span><span class="special"><</span><span class="identifier">T</span><span class="special">,<=,=</span><span class="identifier">e</span><span class="special">=></span> <span class="special">:</span> <span class="identifier">T</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">implies</span> <span class="identifier">a</span> <span class="special">=</span><span class="identifier">e</span><span class="special">=</span> <span class="identifier">b</span>
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<span class="identifier">Transitivity</span><span class="special"><</span><span class="identifier">T</span><span class="special">,<=</span> <span class="special">></span> <span class="special">:</span> <span class="identifier">T</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">b</span> <span class="special"><=</span> <span class="identifier">c</span> <span class="identifier">implies</span> <span class="identifier">a</span> <span class="special"><=</span> <span class="identifier">c</span>
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</pre>
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<p>
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</p>
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<p>
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The induced equivalence is the equality of elements that is implemented via
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function <code class="computeroutput"><span class="identifier">is_element_equal</span></code>.
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</p>
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<pre class="programlisting"><span class="comment">//equivalence induced by the partial ordering contained_in on icl::container a,b</span>
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<span class="identifier">a</span><span class="special">.</span><span class="identifier">contained_in</span><span class="special">(</span><span class="identifier">b</span><span class="special">)</span> <span class="special">&&</span> <span class="identifier">b</span><span class="special">.</span><span class="identifier">contained_in</span><span class="special">(</span><span class="identifier">a</span><span class="special">)</span> <span class="identifier">implies</span> <span class="identifier">is_element_equal</span><span class="special">(</span><span class="identifier">a</span><span class="special">,</span> <span class="identifier">b</span><span class="special">);</span>
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</pre>
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<p>
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</p>
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</div>
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</div>
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<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
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<td align="left"></td>
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<td align="right"><div class="copyright-footer">Copyright © 2007-2010 Joachim
|
||
Faulhaber<br>Copyright © 1999-2006 Cortex Software
|
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GmbH<p>
|
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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>)
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</p>
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</div></td>
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