| <!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN"> |
| |
| <html> |
| <!-- |
| == 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, |
| == provided that the above copyright notice appears in all copies and |
| == that both that copyright notice and this permission notice appear |
| == in supporting documentation. Silicon Graphics makes no |
| == representations about the suitability of this software for any |
| == purpose. It is provided "as is" without express or implied warranty. |
| == |
| == Copyright (c) 1994 |
| == Hewlett-Packard Company |
| == |
| == 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 |
| == that both that copyright notice and this permission notice appear |
| == in supporting documentation. Hewlett-Packard Company makes no |
| == representations about the suitability of this software for any |
| == purpose. It is provided "as is" without express or implied warranty. |
| == |
| --> |
| |
| <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> |
