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29<head>
30  <meta http-equiv="Content-Language" content="en-us">
31  <meta http-equiv="Content-Type" content="text/html; charset=us-ascii">
32
33  <title>LessThanComparable</title>
34</head>
35
36<body bgcolor="#FFFFFF" link="#0000EE" text="#000000" vlink="#551A8B" alink=
37"#FF0000">
38  <img src="../../boost.png" alt="C++ Boost" width="277" height=
39  "86"><br clear="none">
40
41  <h1>LessThanComparable</h1>
42
43  <h3>Description</h3>
44
45  <p>A type is LessThanComparable if it is ordered: it must be possible to
46  compare two objects of that type using <tt>operator&lt;</tt>, and
47  <tt>operator&lt;</tt> must be a strict weak ordering relation.</p>
48
49  <h3>Refinement of</h3>
50
51  <h3>Associated types</h3>
52
53  <h3>Notation</h3>
54
55  <table summary="">
56    <tr>
57      <td valign="top"><tt>X</tt></td>
58
59      <td valign="top">A type that is a model of LessThanComparable</td>
60    </tr>
61
62    <tr>
63      <td valign="top"><tt>x</tt>, <tt>y</tt>, <tt>z</tt></td>
64
65      <td valign="top">Object of type <tt>X</tt></td>
66    </tr>
67  </table>
68
69  <h3>Definitions</h3>
70
71  <p>Consider the relation <tt>!(x &lt; y) &amp;&amp; !(y &lt; x)</tt>. If
72  this relation is transitive (that is, if <tt>!(x &lt; y) &amp;&amp; !(y
73  &lt; x) &amp;&amp; !(y &lt; z) &amp;&amp; !(z &lt; y)</tt> implies <tt>!(x
74  &lt; z) &amp;&amp; !(z &lt; x)</tt>), then it satisfies the mathematical
75  definition of an equivalence relation. In this case, <tt>operator&lt;</tt>
76  is a <i>strict weak ordering</i>.</p>
77
78  <p>If <tt>operator&lt;</tt> is a strict weak ordering, and if each
79  equivalence class has only a single element, then <tt>operator&lt;</tt> is
80  a <i>total ordering</i>.</p>
81
82  <h3>Valid expressions</h3>
83
84  <table border summary="">
85    <tr>
86      <th>Name</th>
87
88      <th>Expression</th>
89
90      <th>Type requirements</th>
91
92      <th>Return type</th>
93    </tr>
94
95    <tr>
96      <td valign="top">Less</td>
97
98      <td valign="top"><tt>x &lt; y</tt></td>
99
100      <td valign="top">&nbsp;</td>
101
102      <td valign="top">Convertible to <tt>bool</tt></td>
103    </tr>
104  </table>
105
106  <h3>Expression semantics</h3>
107
108  <table border summary="">
109    <tr>
110      <th>Name</th>
111
112      <th>Expression</th>
113
114      <th>Precondition</th>
115
116      <th>Semantics</th>
117
118      <th>Postcondition</th>
119    </tr>
120
121    <tr>
122      <td valign="top">Less</td>
123
124      <td valign="top"><tt>x &lt; y</tt></td>
125
126      <td valign="top"><tt>x</tt> and <tt>y</tt> are in the domain of
127      <tt>&lt;</tt></td>
128
129      <td valign="top">&nbsp;</td>
130    </tr>
131  </table>
132
133  <h3>Complexity guarantees</h3>
134
135  <h3>Invariants</h3>
136
137  <table border summary="">
138    <tr>
139      <td valign="top">Irreflexivity</td>
140
141      <td valign="top"><tt>x &lt; x</tt> must be false.</td>
142    </tr>
143
144    <tr>
145      <td valign="top">Antisymmetry</td>
146
147      <td valign="top"><tt>x &lt; y</tt> implies !(y &lt; x) <a href=
148      "#n2">[2]</a></td>
149    </tr>
150
151    <tr>
152      <td valign="top">Transitivity</td>
153
154      <td valign="top"><tt>x &lt; y</tt> and <tt>y &lt; z</tt> implies <tt>x
155      &lt; z</tt> <a href="#n3">[3]</a></td>
156    </tr>
157  </table>
158
159  <h3>Models</h3>
160
161  <ul>
162    <li>int</li>
163  </ul>
164
165  <h3>Notes</h3>
166
167  <p><a name="n1" id="n1">[1]</a> Only <tt>operator&lt;</tt> is fundamental;
168  the other inequality operators are essentially syntactic sugar.</p>
169
170  <p><a name="n2" id="n2">[2]</a> Antisymmetry is a theorem, not an axiom: it
171  follows from irreflexivity and transitivity.</p>
172
173  <p><a name="n3" id="n3">[3]</a> Because of irreflexivity and transitivity,
174  <tt>operator&lt;</tt> always satisfies the definition of a <i>partial
175  ordering</i>. The definition of a <i>strict weak ordering</i> is stricter,
176  and the definition of a <i>total ordering</i> is stricter still.</p>
177
178  <h3>See also</h3>
179
180  <p><a href=
181  "http://www.sgi.com/tech/stl/EqualityComparable.html">EqualityComparable</a>,
182  <a href=
183  "http://www.sgi.com/tech/stl/StrictWeakOrdering.html">StrictWeakOrdering</a><br>
184  </p>
185  <hr>
186
187  <p><a href="http://validator.w3.org/check?uri=referer"><img border="0" src=
188  "http://www.w3.org/Icons/valid-html401" alt="Valid HTML 4.01 Transitional"
189  height="31" width="88"></a></p>
190
191  <p>Revised
192  <!--webbot bot="Timestamp" s-type="EDITED" s-format="%d %B, %Y" startspan -->05
193  December, 2006<!--webbot bot="Timestamp" endspan i-checksum="38516" --></p>
194
195  <table summary="">
196    <tr valign="top">
197      <td nowrap><i>Copyright &copy; 2000</i></td>
198
199      <td><i><a href="http://www.lsc.nd.edu/~jsiek">Jeremy Siek</a>, Univ.of
200      Notre Dame (<a href=
201      "mailto:jsiek@lsc.nd.edu">jsiek@lsc.nd.edu</a>)</i></td>
202    </tr>
203  </table>
204
205  <p><i>Distributed under the Boost Software License, Version 1.0. (See
206  accompanying file <a href="../../LICENSE_1_0.txt">LICENSE_1_0.txt</a> or
207  copy at <a href=
208  "http://www.boost.org/LICENSE_1_0.txt">http://www.boost.org/LICENSE_1_0.txt</a>)</i></p>
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