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- <title>LessThanComparable</title>
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-
- <h1>LessThanComparable</h1>
-
- <h3>Description</h3>
-
- <p>A type is LessThanComparable if it is ordered: it must be possible to
- compare two objects of that type using <tt>operator<</tt>, and
- <tt>operator<</tt> must be a strict weak ordering relation.</p>
-
- <h3>Refinement of</h3>
-
- <h3>Associated types</h3>
-
- <h3>Notation</h3>
-
- <table summary="">
- <tr>
- <td valign="top"><tt>X</tt></td>
-
- <td valign="top">A type that is a model of LessThanComparable</td>
- </tr>
-
- <tr>
- <td valign="top"><tt>x</tt>, <tt>y</tt>, <tt>z</tt></td>
-
- <td valign="top">Object of type <tt>X</tt></td>
- </tr>
- </table>
-
- <h3>Definitions</h3>
-
- <p>Consider the relation <tt>!(x < y) && !(y < x)</tt>. If
- this relation is transitive (that is, if <tt>!(x < y) && !(y
- < x) && !(y < z) && !(z < y)</tt> implies <tt>!(x
- < z) && !(z < x)</tt>), then it satisfies the mathematical
- definition of an equivalence relation. In this case, <tt>operator<</tt>
- is a <i>strict weak ordering</i>.</p>
-
- <p>If <tt>operator<</tt> is a strict weak ordering, and if each
- equivalence class has only a single element, then <tt>operator<</tt> is
- a <i>total ordering</i>.</p>
-
- <h3>Valid expressions</h3>
-
- <table border summary="">
- <tr>
- <th>Name</th>
-
- <th>Expression</th>
-
- <th>Type requirements</th>
-
- <th>Return type</th>
- </tr>
-
- <tr>
- <td valign="top">Less</td>
-
- <td valign="top"><tt>x < y</tt></td>
-
- <td valign="top"> </td>
-
- <td valign="top">Convertible to <tt>bool</tt></td>
- </tr>
- </table>
-
- <h3>Expression semantics</h3>
-
- <table border summary="">
- <tr>
- <th>Name</th>
-
- <th>Expression</th>
-
- <th>Precondition</th>
-
- <th>Semantics</th>
-
- <th>Postcondition</th>
- </tr>
-
- <tr>
- <td valign="top">Less</td>
-
- <td valign="top"><tt>x < y</tt></td>
-
- <td valign="top"><tt>x</tt> and <tt>y</tt> are in the domain of
- <tt><</tt></td>
-
- <td valign="top"> </td>
- </tr>
- </table>
-
- <h3>Complexity guarantees</h3>
-
- <h3>Invariants</h3>
-
- <table border summary="">
- <tr>
- <td valign="top">Irreflexivity</td>
-
- <td valign="top"><tt>x < x</tt> must be false.</td>
- </tr>
-
- <tr>
- <td valign="top">Antisymmetry</td>
-
- <td valign="top"><tt>x < y</tt> implies !(y < x) <a href=
- "#n2">[2]</a></td>
- </tr>
-
- <tr>
- <td valign="top">Transitivity</td>
-
- <td valign="top"><tt>x < y</tt> and <tt>y < z</tt> implies <tt>x
- < z</tt> <a href="#n3">[3]</a></td>
- </tr>
- </table>
-
- <h3>Models</h3>
-
- <ul>
- <li>int</li>
- </ul>
-
- <h3>Notes</h3>
-
- <p><a name="n1" id="n1">[1]</a> Only <tt>operator<</tt> is fundamental;
- the other inequality operators are essentially syntactic sugar.</p>
-
- <p><a name="n2" id="n2">[2]</a> Antisymmetry is a theorem, not an axiom: it
- follows from irreflexivity and transitivity.</p>
-
- <p><a name="n3" id="n3">[3]</a> Because of irreflexivity and transitivity,
- <tt>operator<</tt> always satisfies the definition of a <i>partial
- ordering</i>. The definition of a <i>strict weak ordering</i> is stricter,
- and the definition of a <i>total ordering</i> is stricter still.</p>
-
- <h3>See also</h3>
-
- <p><a href=
- "http://www.sgi.com/tech/stl/EqualityComparable.html">EqualityComparable</a>,
- <a href=
- "http://www.sgi.com/tech/stl/StrictWeakOrdering.html">StrictWeakOrdering</a><br>
- </p>
- <hr>
-
- <p><a href="http://validator.w3.org/check?uri=referer"><img border="0" src=
- "../../doc/images/valid-html401.png" alt="Valid HTML 4.01 Transitional"
- height="31" width="88"></a></p>
-
- <p>Revised
- <!--webbot bot="Timestamp" s-type="EDITED" s-format="%d %B, %Y" startspan -->05
- December, 2006<!--webbot bot="Timestamp" endspan i-checksum="38516" --></p>
-
- <table summary="">
- <tr valign="top">
- <td nowrap><i>Copyright © 2000</i></td>
-
- <td><i><a href="http://www.lsc.nd.edu/~jsiek">Jeremy Siek</a>, Univ.of
- Notre Dame (<a href=
- "mailto:jsiek@lsc.nd.edu">jsiek@lsc.nd.edu</a>)</i></td>
- </tr>
- </table>
-
- <p><i>Distributed under the Boost Software License, Version 1.0. (See
- accompanying file <a href="../../LICENSE_1_0.txt">LICENSE_1_0.txt</a> or
- copy at <a href=
- "http://www.boost.org/LICENSE_1_0.txt">http://www.boost.org/LICENSE_1_0.txt</a>)</i></p>
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