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211 lines
5.9 KiB
HTML
211 lines
5.9 KiB
HTML
<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN">
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<html>
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<!--
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== Copyright (c) 1996-1999
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== Silicon Graphics Computer Systems, Inc.
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==
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== Permission to use, copy, modify, distribute and sell this software
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== and its documentation for any purpose is hereby granted without fee,
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== provided that the above copyright notice appears in all copies and
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== that both that copyright notice and this permission notice appear
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== in supporting documentation. Silicon Graphics makes no
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== representations about the suitability of this software for any
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== purpose. It is provided "as is" without express or implied warranty.
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==
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== Copyright (c) 1994
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== Hewlett-Packard Company
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==
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== Permission to use, copy, modify, distribute and sell this software
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== and its documentation for any purpose is hereby granted without fee,
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== provided that the above copyright notice appears in all copies and
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== that both that copyright notice and this permission notice appear
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== in supporting documentation. Hewlett-Packard Company makes no
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== representations about the suitability of this software for any
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== purpose. It is provided "as is" without express or implied warranty.
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==
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-->
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<head>
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<meta http-equiv="Content-Language" content="en-us">
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<meta http-equiv="Content-Type" content="text/html; charset=us-ascii">
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<title>LessThanComparable</title>
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</head>
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<body bgcolor="#FFFFFF" link="#0000EE" text="#000000" vlink="#551A8B" alink=
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"#FF0000">
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<img src="../../boost.png" alt="C++ Boost" width="277" height=
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"86"><br clear="none">
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<h1>LessThanComparable</h1>
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<h3>Description</h3>
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<p>A type is LessThanComparable if it is ordered: it must be possible to
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compare two objects of that type using <tt>operator<</tt>, and
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<tt>operator<</tt> must be a strict weak ordering relation.</p>
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<h3>Refinement of</h3>
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<h3>Associated types</h3>
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<h3>Notation</h3>
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<table summary="">
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<tr>
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<td valign="top"><tt>X</tt></td>
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<td valign="top">A type that is a model of LessThanComparable</td>
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</tr>
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<tr>
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<td valign="top"><tt>x</tt>, <tt>y</tt>, <tt>z</tt></td>
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<td valign="top">Object of type <tt>X</tt></td>
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</tr>
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</table>
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<h3>Definitions</h3>
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<p>Consider the relation <tt>!(x < y) && !(y < x)</tt>. If
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this relation is transitive (that is, if <tt>!(x < y) && !(y
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< x) && !(y < z) && !(z < y)</tt> implies <tt>!(x
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< z) && !(z < x)</tt>), then it satisfies the mathematical
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definition of an equivalence relation. In this case, <tt>operator<</tt>
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is a <i>strict weak ordering</i>.</p>
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<p>If <tt>operator<</tt> is a strict weak ordering, and if each
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equivalence class has only a single element, then <tt>operator<</tt> is
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a <i>total ordering</i>.</p>
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<h3>Valid expressions</h3>
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<table border summary="">
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<tr>
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<th>Name</th>
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<th>Expression</th>
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<th>Type requirements</th>
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<th>Return type</th>
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</tr>
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<tr>
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<td valign="top">Less</td>
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<td valign="top"><tt>x < y</tt></td>
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<td valign="top"> </td>
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<td valign="top">Convertible to <tt>bool</tt></td>
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</tr>
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</table>
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<h3>Expression semantics</h3>
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<table border summary="">
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<tr>
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<th>Name</th>
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<th>Expression</th>
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<th>Precondition</th>
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<th>Semantics</th>
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<th>Postcondition</th>
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</tr>
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<tr>
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<td valign="top">Less</td>
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<td valign="top"><tt>x < y</tt></td>
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<td valign="top"><tt>x</tt> and <tt>y</tt> are in the domain of
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<tt><</tt></td>
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<td valign="top"> </td>
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</tr>
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</table>
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<h3>Complexity guarantees</h3>
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<h3>Invariants</h3>
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<table border summary="">
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<tr>
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<td valign="top">Irreflexivity</td>
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<td valign="top"><tt>x < x</tt> must be false.</td>
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</tr>
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<tr>
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<td valign="top">Antisymmetry</td>
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<td valign="top"><tt>x < y</tt> implies !(y < x) <a href=
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"#n2">[2]</a></td>
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</tr>
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<tr>
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<td valign="top">Transitivity</td>
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<td valign="top"><tt>x < y</tt> and <tt>y < z</tt> implies <tt>x
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< z</tt> <a href="#n3">[3]</a></td>
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</tr>
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</table>
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<h3>Models</h3>
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<ul>
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<li>int</li>
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</ul>
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<h3>Notes</h3>
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<p><a name="n1" id="n1">[1]</a> Only <tt>operator<</tt> is fundamental;
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the other inequality operators are essentially syntactic sugar.</p>
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<p><a name="n2" id="n2">[2]</a> Antisymmetry is a theorem, not an axiom: it
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follows from irreflexivity and transitivity.</p>
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<p><a name="n3" id="n3">[3]</a> Because of irreflexivity and transitivity,
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<tt>operator<</tt> always satisfies the definition of a <i>partial
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ordering</i>. The definition of a <i>strict weak ordering</i> is stricter,
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and the definition of a <i>total ordering</i> is stricter still.</p>
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<h3>See also</h3>
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<p><a href=
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"http://www.sgi.com/tech/stl/EqualityComparable.html">EqualityComparable</a>,
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<a href=
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"http://www.sgi.com/tech/stl/StrictWeakOrdering.html">StrictWeakOrdering</a><br>
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</p>
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<hr>
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<p><a href="http://validator.w3.org/check?uri=referer"><img border="0" src=
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"../../doc/images/valid-html401.png" alt="Valid HTML 4.01 Transitional"
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height="31" width="88"></a></p>
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<p>Revised
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<!--webbot bot="Timestamp" s-type="EDITED" s-format="%d %B, %Y" startspan -->05
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December, 2006<!--webbot bot="Timestamp" endspan i-checksum="38516" --></p>
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<table summary="">
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<tr valign="top">
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<td nowrap><i>Copyright © 2000</i></td>
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<td><i><a href="http://www.lsc.nd.edu/~jsiek">Jeremy Siek</a>, Univ.of
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Notre Dame (<a href=
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"mailto:jsiek@lsc.nd.edu">jsiek@lsc.nd.edu</a>)</i></td>
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</tr>
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</table>
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<p><i>Distributed under the Boost Software License, Version 1.0. (See
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accompanying file <a href="../../LICENSE_1_0.txt">LICENSE_1_0.txt</a> or
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copy at <a href=
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"http://www.boost.org/LICENSE_1_0.txt">http://www.boost.org/LICENSE_1_0.txt</a>)</i></p>
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</body>
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</html>
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