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<Head>
<Title>LessThanComparable</Title>
</Head>
<BODY BGCOLOR="#ffffff" LINK="#0000ee" TEXT="#000000" VLINK="#551a8b"
ALINK="#ff0000">
<IMG SRC="../../c++boost.gif"
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.
<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>
<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>
<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>
<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>
</BODY>
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