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213 lines
5.1 KiB
HTML
213 lines
5.1 KiB
HTML
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<HTML>
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<!--
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-- Copyright (c) Jeremy Siek 2000
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--
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-- Permission to use, copy, modify, distribute and sell this software
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-- and its documentation for any purpose is hereby granted without fee,
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-- provided that the above copyright notice appears in all copies and
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-- that both that copyright notice and this permission notice appear
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-- in supporting documentation. Silicon Graphics makes no
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-- representations about the suitability of this software for any
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-- purpose. It is provided "as is" without express or implied warranty.
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-->
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<!--
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-- Copyright (c) 1996-1999
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-- Silicon Graphics Computer Systems, Inc.
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--
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-- Permission to use, copy, modify, distribute and sell this software
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-- and its documentation for any purpose is hereby granted without fee,
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-- provided that the above copyright notice appears in all copies and
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-- that both that copyright notice and this permission notice appear
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-- in supporting documentation. Silicon Graphics makes no
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-- representations about the suitability of this software for any
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-- purpose. It is provided "as is" without express or implied warranty.
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--
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-- Copyright (c) 1994
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-- Hewlett-Packard Company
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--
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-- Permission to use, copy, modify, distribute and sell this software
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-- and its documentation for any purpose is hereby granted without fee,
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-- provided that the above copyright notice appears in all copies and
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-- that both that copyright notice and this permission notice appear
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-- in supporting documentation. Hewlett-Packard Company makes no
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-- representations about the suitability of this software for any
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-- purpose. It is provided "as is" without express or implied warranty.
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--
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-->
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<Head>
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<Title>LessThanComparable</Title>
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</Head>
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<BODY BGCOLOR="#ffffff" LINK="#0000ee" TEXT="#000000" VLINK="#551a8b"
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ALINK="#ff0000">
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<IMG SRC="../../c++boost.gif"
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ALT="C++ Boost" width="277" height="86">
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<!--end header-->
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<BR Clear>
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<H1>LessThanComparable</H1>
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<h3>Description</h3>
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A type is LessThanComparable if it is ordered: it must
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be possible to compare two objects of that type using <tt>operator<</tt>, and
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<tt>operator<</tt> must be a strict weak ordering relation.
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<h3>Refinement of</h3>
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<h3>Associated types</h3>
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<h3>Notation</h3>
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<Table>
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<TR>
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<TD VAlign=top>
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<tt>X</tt>
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</TD>
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<TD VAlign=top>
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A type that is a model of LessThanComparable
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</TD>
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</TR>
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<TR>
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<TD VAlign=top>
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<tt>x</tt>, <tt>y</tt>, <tt>z</tt>
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</TD>
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<TD VAlign=top>
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Object of type <tt>X</tt>
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</TD>
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</tr>
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</table>
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<h3>Definitions</h3>
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Consider the relation <tt>!(x < y) && !(y < x)</tt>. If this relation is
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transitive (that is, if <tt>!(x < y) && !(y < x) && !(y < z) && !(z < y)</tt>
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implies <tt>!(x < z) && !(z < x)</tt>), then it satisfies the mathematical
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definition of an equivalence relation. In this case, <tt>operator<</tt>
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is a <i>strict weak ordering</i>.
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<P>
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If <tt>operator<</tt> is a strict weak ordering, and if each equivalence class
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has only a single element, then <tt>operator<</tt> is a <i>total ordering</i>.
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<h3>Valid expressions</h3>
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<Table border>
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<TR>
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<TH>
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Name
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</TH>
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<TH>
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Expression
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</TH>
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<TH>
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Type requirements
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</TH>
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<TH>
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Return type
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</TH>
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</TR>
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<TR>
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<TD VAlign=top>
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Less
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</TD>
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<TD VAlign=top>
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<tt>x < y</tt>
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</TD>
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<TD VAlign=top>
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</TD>
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<TD VAlign=top>
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Convertible to <tt>bool</tt>
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</TD>
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</TR>
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</table>
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<h3>Expression semantics</h3>
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<Table border>
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<TR>
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<TH>
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Name
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</TH>
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<TH>
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Expression
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</TH>
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<TH>
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Precondition
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</TH>
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<TH>
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Semantics
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</TH>
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<TH>
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Postcondition
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</TH>
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</TR>
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<TR>
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<TD VAlign=top>
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Less
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</TD>
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<TD VAlign=top>
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<tt>x < y</tt>
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</TD>
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<TD VAlign=top>
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<tt>x</tt> and <tt>y</tt> are in the domain of <tt><</tt>
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</TD>
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<TD VAlign=top>
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</TD>
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</table>
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<h3>Complexity guarantees</h3>
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<h3>Invariants</h3>
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<Table border>
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<TR>
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<TD VAlign=top>
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Irreflexivity
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</TD>
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<TD VAlign=top>
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<tt>x < x</tt> must be false.
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</TD>
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</TR>
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<TR>
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<TD VAlign=top>
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Antisymmetry
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</TD>
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<TD VAlign=top>
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<tt>x < y</tt> implies !(y < x) <A href="#2">[2]</A>
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</TD>
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</TR>
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<TR>
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<TD VAlign=top>
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Transitivity
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</TD>
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<TD VAlign=top>
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<tt>x < y</tt> and <tt>y < z</tt> implies <tt>x < z</tt> <A href="#3">[3]</A>
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</TD>
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</tr>
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</table>
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<h3>Models</h3>
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<UL>
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<LI>
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int
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</UL>
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<h3>Notes</h3>
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<P><A name="1">[1]</A>
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Only <tt>operator<</tt> is fundamental; the other inequality operators
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are essentially syntactic sugar.
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<P><A name="2">[2]</A>
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Antisymmetry is a theorem, not an axiom: it follows from
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irreflexivity and transitivity.
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<P><A name="3">[3]</A>
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Because of irreflexivity and transitivity, <tt>operator<</tt> always
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satisfies the definition of a <i>partial ordering</i>. The definition of
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a <i>strict weak ordering</i> is stricter, and the definition of a
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<i>total ordering</i> is stricter still.
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<h3>See also</h3>
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<A href="http://www.sgi.com/Technology/STL/EqualityComparable.html">EqualityComparable</A>, <A href="http://www.sgi.com/Technology/STL/StrictWeakOrdering.html">StrictWeakOrdering</A>
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<br>
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<HR>
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<TABLE>
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<TR valign=top>
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<TD nowrap>Copyright © 2000</TD><TD>
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<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>)
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</TD></TR></TABLE>
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</BODY>
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</HTML>
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