forked from boostorg/utility
Fixed license & copyright issues and converted to HTML 4.01
[SVN r36280]
This commit is contained in:
Andreas Huber
+196
-198
@@ -1,212 +1,210 @@
|
||||
<HTML>
|
||||
<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN">
|
||||
|
||||
<html>
|
||||
<!--
|
||||
-- Copyright (c) Jeremy Siek 2000
|
||||
--
|
||||
-- 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) 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.
|
||||
==
|
||||
-->
|
||||
<!--
|
||||
-- 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>
|
||||
<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">
|
||||
<!--end header-->
|
||||
<BR Clear>
|
||||
<H1>LessThanComparable</H1>
|
||||
|
||||
<h3>Description</h3>
|
||||
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.
|
||||
<head>
|
||||
<meta http-equiv="Content-Language" content="en-us">
|
||||
<meta http-equiv="Content-Type" content="text/html; charset=us-ascii">
|
||||
|
||||
<title>LessThanComparable</title>
|
||||
</head>
|
||||
|
||||
<h3>Refinement of</h3>
|
||||
<h3>Associated types</h3>
|
||||
<h3>Notation</h3>
|
||||
<Table>
|
||||
<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>
|
||||
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>
|
||||
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>.
|
||||
<h3>Valid expressions</h3>
|
||||
<Table border>
|
||||
<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>
|
||||
<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>
|
||||
|
||||
<h3>Expression semantics</h3>
|
||||
<Table border>
|
||||
<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>
|
||||
</table>
|
||||
<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>Complexity guarantees</h3>
|
||||
<h3>Invariants</h3>
|
||||
<Table border>
|
||||
<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="#2">[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="#3">[3]</A>
|
||||
</TD>
|
||||
</tr>
|
||||
</table>
|
||||
<h3>Models</h3>
|
||||
<UL>
|
||||
<LI>
|
||||
int
|
||||
</UL>
|
||||
<h3>Notes</h3>
|
||||
<P><A name="1">[1]</A>
|
||||
Only <tt>operator<</tt> is fundamental; the other inequality operators
|
||||
are essentially syntactic sugar.
|
||||
<P><A name="2">[2]</A>
|
||||
Antisymmetry is a theorem, not an axiom: it follows from
|
||||
irreflexivity and transitivity.
|
||||
<P><A name="3">[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.
|
||||
<h3>See also</h3>
|
||||
<A href="http://www.sgi.com/tech/stl/EqualityComparable.html">EqualityComparable</A>, <A href="http://www.sgi.com/tech/stl/StrictWeakOrdering.html">StrictWeakOrdering</A>
|
||||
<h3>Associated types</h3>
|
||||
|
||||
<h3>Notation</h3>
|
||||
|
||||
<table summary="">
|
||||
<tr>
|
||||
<td valign="top"><tt>X</tt></td>
|
||||
|
||||
<br>
|
||||
<HR>
|
||||
<TABLE>
|
||||
<TR valign=top>
|
||||
<TD nowrap>Copyright © 2000</TD><TD>
|
||||
<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>)
|
||||
</TD></TR></TABLE>
|
||||
<td valign="top">A type that is a model of LessThanComparable</td>
|
||||
</tr>
|
||||
|
||||
</BODY>
|
||||
</HTML>
|
||||
<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=
|
||||
"http://www.w3.org/Icons/valid-html401" 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>
|
||||
|
||||
Reference in New Issue
Block a user