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+<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN">
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+<html>
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+ == Copyright (c) 1996-1999
+ == Silicon Graphics Computer Systems, Inc.
+ ==
+ == Permission to use, copy, modify, distribute and sell this software
+ == and its documentation for any purpose is hereby granted without fee,
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+ == purpose. It is provided "as is" without express or implied warranty.
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+ == Copyright (c) 1994
+ == Hewlett-Packard Company
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+ == Permission to use, copy, modify, distribute and sell this software
+ == and its documentation for any purpose is hereby granted without fee,
+ == provided that the above copyright notice appears in all copies and
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+ == in supporting documentation. Hewlett-Packard Company makes no
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+
+<head>
+ <meta http-equiv="Content-Language" content="en-us">
+ <meta http-equiv="Content-Type" content="text/html; charset=us-ascii">
+
+ <title>LessThanComparable</title>
+</head>
+
+<body bgcolor="#FFFFFF" link="#0000EE" text="#000000" vlink="#551A8B" alink=
+"#FF0000">
+ <img src="../../boost.png" alt="C++ Boost" width="277" height=
+ "86"><br clear="none">
+
+ <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>
+</body>
+</html>