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<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. 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It is provided "as is" without express or implied warranty. -- --><Head><Title>LessThan Comparable</Title><!-- Generated by htmldoc --></HEAD><BODY TEXT="#000000" LINK="#006600" ALINK="#003300" VLINK="#7C7F87" BGCOLOR="#FFFFFF"><A HREF="/"><IMG SRC="/images/common/sgilogo_small.gif" ALT="SGI Logo" WIDTH="80" HEIGHT="72" BORDER="0"></A><P><!--end header--><BR Clear><H1>LessThan Comparable</H1><Table CellPadding=0 CellSpacing=0 width=100%><TR><TD Align=left><Img src = "utilities.gif" Alt="" WIDTH = "194" HEIGHT = "38" ></TD><TD Align=right><Img src = "concept.gif" Alt="" WIDTH = "194" HEIGHT = "38" ></TD></TR><TR><TD Align=left VAlign=top><b>Category</b>: utilities</TD><TD Align=right VAlign=top><b>Component type</b>: concept</TD></TR></Table><h3>Description</h3>A type is LessThanComparable if it is ordered: it mustbe possible to compare two objects of that type using <tt>operator<</tt>, and<tt>operator<</tt> must be a partial ordering.<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 istransitive (that is, if <tt>!(x < y) && !(y < x) && !(y < z) && !(z < y)</tt>implies <tt>!(x < z) && !(z < x)</tt>), then it satisfies the mathematicaldefinition 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 classhas 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><TR><TD VAlign=top>Greater</TD><TD VAlign=top><tt>x > y</tt></TD><TD VAlign=top> </TD><TD VAlign=top>Convertible to <tt>bool</tt></TD></TR><TR><TD VAlign=top>Less or equal</TD><TD VAlign=top><tt>x <= y</tt></TD><TD VAlign=top> </TD><TD VAlign=top>Convertible to <tt>bool</tt></TD></TR><TR><TD VAlign=top>Greater or equal</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><TD VAlign=top> </TD></TR><TR><TD VAlign=top>Greater</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>Equivalent to <tt>y < x</tt> <A href="#1">[1]</A></TD><TD VAlign=top> </TD></TR><TR><TD VAlign=top>Less or equal</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>Equivalent to <tt>!(y < x)</tt> <A href="#1">[1]</A></TD><TD VAlign=top> </TD></TR><TR><TD VAlign=top>Greater or equal</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>Equivalent to <tt>!(x < y)</tt> <A href="#1">[1]</A></TD><TD VAlign=top> </TD></tr></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 operatorsare essentially syntactic sugar.<P><A name="2">[2]</A>Antisymmetry is a theorem, not an axiom: it follows fromirreflexivity and transitivity.<P><A name="3">[3]</A>Because of irreflexivity and transitivity, <tt>operator<</tt> alwayssatisfies the definition of a <i>partial ordering</i>. The definition ofa <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="EqualityComparable.html">EqualityComparable</A>, <A href="StrictWeakOrdering.html">StrictWeakOrdering</A><!-- start footer --><!-- Footer Begins --><STYLE TYPE="text/css"><!--TD.footer, TD.footer A{ font-family: Arial, helvetica, sans-serif; font-size: 8pt;}A.home {font-family: Arial, helvetica, sans-serif;}--></STYLE><P><A CLASS="home" HREF="index.html">STL Home</A><P><TABLE WIDTH="600" CELLPADDING="0" CELLPADDING="0" BORDER="0"> <TR> <TD ALIGN="RIGHT" CLASS="footer"><A HREF="/company_info/terms.html" TARGET="_top">terms of use</A> | <A HREF="/company_info/privacy.html" TARGET="_top">privacy policy</A></TD> <TD ALIGN="CENTER" CLASS="footer"> | </TD> <TD ALIGN="LEFT" CLASS="footer"><A HREF="/cgi-bin/feedback/" TARGET="_top">contact us</A></TD> </TR><TR> <TD ALIGN="RIGHT" CLASS="footer">Copyright © 1993-2003 Silicon Graphics, Inc. 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