What majority decisions are possible \\ Sh816
Saharon Shelah \thanks {\null\newline Partially supported by the United States-Israel Binational Science Foundation \null\newline I would like to thank Alice Leonhardt for the beautiful typing. \null\newline First Typed - 02/Jan/16 \null\newline Latest Revision - 03/Mar/6} \endthanks
816.tex
% 816, processed by citealice (2002-10-18) on Thu Mar  6 16:51:43 CST 2003

\ifx\shlhetal\undefinedcontrolsequence\let\shlhetal\relax\fi
% To: "Saharon Shelah's Office" <shlhetal@math.huji.ac.il>,   Andrzej Roslanowski <roslanow@member.ams.org>
% Subject: sh816
% Date: Thu, 6 Mar 2003 13:29:52 -0500 (EST)
% From: Alice Leonhardt <leonhard@math.rutgers.edu>
% Mime-Version: 1.0
% Content-Description: revisions
% X-sliced-and-diced-by: 'savemail' 1.3b, Feb 2003


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\def\multint@{\int\ifnum\intno@=\z@\intdots@                                %1
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 \int}                                                                      %5
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\def\ints@@{\iflimtoken@                                                    %1
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  \else\multint@\nolimits\fi                                                %3
  \eat@}                                                                    %4
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\def\cot{\qopname@{cot}}
\def\coth{\qopname@{coth}}
\def\csc{\qopname@{csc}}
\def\deg{\qopname@{deg}}
\def\det{\qopnamewl@{det}}
\def\dim{\qopname@{dim}}
\def\exp{\qopname@{exp}}
\def\gcd{\qopnamewl@{gcd}}
\def\hom{\qopname@{hom}}
\def\inf{\qopnamewl@{inf}}
\def\injlim{\qopnamewl@{inj\,lim}}
\def\ker{\qopname@{ker}}
\def\lg{\qopname@{lg}}
\def\lim{\qopnamewl@{lim}}
\def\liminf{\qopnamewl@{lim\,inf}}
\def\limsup{\qopnamewl@{lim\,sup}}
\def\ln{\qopname@{ln}}
\def\log{\qopname@{log}}
\def\max{\qopnamewl@{max}}
\def\min{\qopnamewl@{min}}
\def\Pr{\qopnamewl@{Pr}}
\def\projlim{\qopnamewl@{proj\,lim}}
\def\sec{\qopname@{sec}}
\def\sin{\qopname@{sin}}
\def\sinh{\qopname@{sinh}}
\def\sup{\qopnamewl@{sup}}
\def\tan{\qopname@{tan}}
\def\tanh{\qopname@{tanh}}
\def\varinjlim{\mathop{\vtop{\ialign{##\crcr
 \hfil\rm lim\hfil\crcr\noalign{\nointerlineskip}\rightarrowfill\crcr
 \noalign{\nointerlineskip\kern-\ex@}\crcr}}}}
\def\varprojlim{\mathop{\vtop{\ialign{##\crcr
 \hfil\rm lim\hfil\crcr\noalign{\nointerlineskip}\leftarrowfill\crcr
 \noalign{\nointerlineskip\kern-\ex@}\crcr}}}}
\def\varliminf{\mathop{\underline{\vrule height\z@ depth.2exwidth\z@
 \hbox{\rm lim}}}}
\def\varlimsup{\mathop{\overline{\hbox{\rm lim}}}}
\newdimen\buffer@
\buffer@\fontdimen13 \tenex
\newdimen\buffer
\buffer\buffer@
\def\ChangeBuffer#1{\fontdimen13 \tenex#1\relax
 \global\buffer\fontdimen13 \tenex}
\def\ResetBuffer{\fontdimen13 \tenex\buffer@\global\buffer\buffer@}
\def\shave#1{\mathop{\hbox{$\m@th\fontdimen13 \tenex\z@                     %1
 \displaystyle{#1}$}}\fontdimen13 \tenex\buffer}
\def\topshave#1{\mathop{\topsmash{\displaystyle{#1}}\vphantom{\shave{#1}}}}
\def\botshave#1{\mathop{\botsmash{\displaystyle{#1}}\vphantom{\shave{#1}}}}
\message{multilevel sub/superscripts,}
\Invalid@\\
\def\Let@{\relax\iffalse{\fi\let\\=\cr\iffalse}\fi}
\Invalid@\vspace
\def\vspace@{\def\vspace##1{\crcr\noalign{\vskip##1\relax}}}
\def\multilimits@{\bgroup\vspace@\Let@
 \baselineskip\fontdimen10 \scriptfont\tw@
 \advance\baselineskip\fontdimen12 \scriptfont\tw@
 \lineskip\thr@@\fontdimen8 \scriptfont\thr@@
 \lineskiplimit\lineskip
 \vbox\bgroup\ialign\bgroup\hfil$\m@th\scriptstyle{##}$\hfil\crcr}
\def\Sb{_\multilimits@}
\def\endSb{\crcr\egroup\egroup\egroup}
\def\Sp{^\multilimits@}
\let\endSp\endSb
\def\spreadlines#1{\RIfMIfI@\onlydmatherr@\spreadlines\else
 \openup#1\relax\fi\else\onlydmatherr@\spreadlines\fi}
\def\Mathstrut@{\copy\Mathstrutbox@}
\newbox\Mathstrutbox@
\setbox\Mathstrutbox@\null
\setboxz@h{$\m@th($}
\ht\Mathstrutbox@\ht\z@
\dp\Mathstrutbox@\dp\z@
\message{matrices,}
\newdimen\spreadmlines@
\def\spreadmatrixlines#1{\RIfMIfI@
 \onlydmatherr@\spreadmatrixlines\else
 \spreadmlines@#1\relax\fi\else\onlydmatherr@\spreadmatrixlines\fi}
\def\matrix{\null\,\vcenter\bgroup\Let@\vspace@
 \normalbaselines\openup\spreadmlines@\ialign
 \bgroup\hfil$\m@th##$\hfil&&\quad\hfil$\m@th##$\hfil\crcr
 \Mathstrut@\crcr\noalign{\kern-\baselineskip}}
\def\endmatrix{\crcr\Mathstrut@\crcr\noalign{\kern-\baselineskip}\egroup
 \egroup\,}
\def\format{\crcr\egroup\iffalse{\fi\ifnum`}=0 \fi\format@}
\newtoks\hashtoks@
\hashtoks@{#}
\def\format@#1\\{\def\preamble@{#1}%
 \def\l{$\m@th\the\hashtoks@$\hfil}%
 \def\c{\hfil$\m@th\the\hashtoks@$\hfil}%
 \def\r{\hfil$\m@th\the\hashtoks@$}%
 \edef\preamble@@{\preamble@}\ifnum`{=0 \fi\iffalse}\fi
 \ialign\bgroup\span\preamble@@\crcr}
\def\smallmatrix{\null\,\vcenter\bgroup\vspace@\Let@
 \baselineskip9\ex@\lineskip\ex@
 \ialign\bgroup\hfil$\m@th\scriptstyle{##}$\hfil&&\thickspace\hfil
 $\m@th\scriptstyle{##}$\hfil\crcr}
\def\endsmallmatrix{\crcr\egroup\egroup\,}
\def\pmatrix{\left(\matrix}
\def\endpmatrix{\endmatrix\right)}
\def\bmatrix{\left[\matrix}
\def\endbmatrix{\endmatrix\right]}
\def\vmatrix{\left|\matrix}
\def\endvmatrix{\endmatrix\right|}
\def\Vmatrix{\left\|\matrix}
\def\endVmatrix{\endmatrix\right\|}
\def\hdots{\mathinner{\ldotp\ldotp\ldotp}}
\newmuskip\dotsspace@
\dotsspace@1.5mu
\def\strip@#1 {#1}
\def\spacehdots#1\for#2{\multispan{#2}\xleaders
 \hbox{$\m@th\mkern\strip@#1 \dotsspace@.\mkern\strip@#1 \dotsspace@$}\hfill}
\def\hdotsfor#1{\spacehdots\@ne\for{#1}}
\def\multispan@#1{\omit\mscount#1\unskip\loop\ifnum\mscount>\@ne\sp@n\repeat}
\def\spaceinnerhdots#1\for#2\after#3{\multispan@{\strip@#2 }#3\xleaders
 \hbox{$\m@th\mkern\strip@#1 \dotsspace@.\mkern\strip@#1 \dotsspace@$}\hfill}
\def\innerhdotsfor#1\after#2{\spaceinnerhdots\@ne\for#1\after{#2}}
\def\cases{\bgroup\spreadmlines@\jot\left\{\,\matrix\format\l&\quad\l\\}
\def\endcases{\endmatrix\right.\egroup}
\message{multiline displays,}
\newif\ifinany@
\newif\ifinalign@
\newif\ifingather@
\def\strut@{\copy\strutbox@}
\newbox\strutbox@
\setbox\strutbox@\hbox{\vrule height8\p@ depth3\p@ width\z@}
\def\topaligned{\null\,\vtop\aligned@}
\def\botaligned{\null\,\vbox\aligned@}
\def\aligned{\null\,\vcenter\aligned@}
\def\aligned@{\bgroup\vspace@\Let@
 \ifinany@\else\openup\jot\fi\ialign
 \bgroup\hfil\strut@$\m@th\displaystyle{##}$&
 $\m@th\displaystyle{{}##}$\hfil\crcr}
\def\endaligned{\crcr\egroup\egroup}
\let\endtopaligned\endaligned
\let\endbotaligned\endaligned
\def\alignedat#1{\null\,\vcenter\bgroup\doat@{#1}\vspace@\Let@
 \ifinany@\else\openup\jot\fi\ialign\bgroup\span\preamble@@\crcr}
\newcount\atcount@
\def\doat@#1{\toks@{\hfil\strut@$\m@th
 \displaystyle{\the\hashtoks@}$&$\m@th\displaystyle
 {{}\the\hashtoks@}$\hfil}%                                                 %1
 \atcount@#1\relax\advance\atcount@\m@ne                                    %2
 \loop\ifnum\atcount@>\z@\toks@=\expandafter{\the\toks@&\hfil$\m@th
 \displaystyle{\the\hashtoks@}$&$\m@th
 \displaystyle{{}\the\hashtoks@}$\hfil}\advance
  \atcount@\m@ne\repeat                                                     %3
 \xdef\preamble@{\the\toks@}\xdef\preamble@@{\preamble@}}
\let\endalignedat\endaligned
\def\gathered{\null\,\vcenter\bgroup\vspace@\Let@
 \ifinany@\else\openup\jot\fi\ialign
 \bgroup\hfil\strut@$\m@th\displaystyle{##}$\hfil\crcr}
\def\endgathered{\crcr\egroup\egroup}
\newif\iftagsleft@
\def\TagsOnLeft{\global\tagsleft@true}
\def\TagsOnRight{\global\tagsleft@false}
\TagsOnLeft
\newif\ifmathtags@
\def\TagsAsMath{\global\mathtags@true}
\def\TagsAsText{\global\mathtags@false}
\TagsAsText
\def\tagform@#1{\hbox{\rm(\ignorespaces#1\unskip)}}
\def\thetag{\leavevmode\tagform@}
\def\tag#1$${\iftagsleft@\leqno\else\eqno\fi                                %1
 \maketag@#1\maketag@                                                       %2
 $$}                                                                        %3
\def\maketag@{\FN@\maketag@@}
\def\maketag@@{\ifx\next"\expandafter\maketag@@@\else\expandafter\maketag@@@@
 \fi}
\def\maketag@@@"#1"#2\maketag@{\hbox{\rm#1}}                                %1
\def\maketag@@@@#1\maketag@{\ifmathtags@\tagform@{$\m@th#1$}\else
 \tagform@{#1}\fi}
\interdisplaylinepenalty\@M
\def\allowdisplaybreaks{\RIfMIfI@
 \onlydmatherr@\allowdisplaybreaks\else
 \interdisplaylinepenalty\z@\fi\else\onlydmatherr@\allowdisplaybreaks\fi}
\Invalid@\allowdisplaybreak
\Invalid@\displaybreak
\Invalid@\intertext
\def\allowdisplaybreak@{\def\allowdisplaybreak{\crcr\noalign{\allowbreak}}}
\def\displaybreak@{\def\displaybreak{\crcr\noalign{\break}}}
\def\intertext@{\def\intertext##1{\crcr\noalign{%
 \penalty\postdisplaypenalty \vskip\belowdisplayskip
 \vbox{\normalbaselines\noindent##1}%
 \penalty\predisplaypenalty \vskip\abovedisplayskip}}}
\newskip\centering@
\centering@\z@ plus\@m\p@
\def\align{\relax\ifingather@\DN@{\csname align (in
  \string\gather)\endcsname}\else
 \ifmmode\ifinner\DN@{\onlydmatherr@\align}\else
  \let\next@\align@\fi
 \else\DN@{\onlydmatherr@\align}\fi\fi\next@}
\newhelp\andhelp@
{An extra & here is so disastrous that you should probably exit^^J
and fix things up.}
\newif\iftag@
\newcount\and@
\def\align@{\inalign@true\inany@true
 \vspace@\allowdisplaybreak@\displaybreak@\intertext@
 \def\tag{\global\tag@true\ifnum\and@=\z@\DN@{&&}\else
          \DN@{&}\fi\next@}%
 \iftagsleft@\DN@{\csname align \endcsname}\else
  \DN@{\csname align \space\endcsname}\fi\next@}
\def\Tag@{\iftag@\else\errhelp\andhelp@\err@{Extra & on this line}\fi}
\newdimen\lwidth@
\newdimen\rwidth@
\newdimen\maxlwidth@
\newdimen\maxrwidth@
\newdimen\totwidth@
\def\measure@#1\endalign{\lwidth@\z@\rwidth@\z@\maxlwidth@\z@\maxrwidth@\z@
 \global\and@\z@                                                            %1
 \setbox@ne\vbox                                                            %2
  {\everycr{\noalign{\global\tag@false\global\and@\z@}}\Let@                %3
  \halign{\setboxz@h{$\m@th\displaystyle{\@lign##}$}%                       %4
   \global\lwidth@\wdz@                                                     %5
   \ifdim\lwidth@>\maxlwidth@\global\maxlwidth@\lwidth@\fi                  %6
   \global\advance\and@\@ne                                                 %7
   &\setboxz@h{$\m@th\displaystyle{{}\@lign##}$}\global\rwidth@\wdz@        %8
   \ifdim\rwidth@>\maxrwidth@\global\maxrwidth@\rwidth@\fi                  %9
   \global\advance\and@\@ne                                                %10
   &\Tag@
   \eat@{##}\crcr#1\crcr}}%                                                %11
 \totwidth@\maxlwidth@\advance\totwidth@\maxrwidth@}                       %12
\def\displ@y@{\global\dt@ptrue\openup\jot
 \everycr{\noalign{\global\tag@false\global\and@\z@\ifdt@p\global\dt@pfalse
 \vskip-\lineskiplimit\vskip\normallineskiplimit\else
 \penalty\interdisplaylinepenalty\fi}}}
\def\black@#1{\noalign{\ifdim#1>\displaywidth
 \dimen@\prevdepth\nointerlineskip                                          %1
 \vskip-\ht\strutbox@\vskip-\dp\strutbox@                                   %2
 \vbox{\noindent\hbox to#1{\strut@\hfill}}%                                 %3
 \prevdepth\dimen@                                                          %4
 \fi}}
\expandafter\def\csname align \space\endcsname#1\endalign
 {\measure@#1\endalign\global\and@\z@                                       %1
 \ifingather@\everycr{\noalign{\global\and@\z@}}\else\displ@y@\fi           %2
 \Let@\tabskip\centering@                                                   %3
 \halign to\displaywidth
  {\hfil\strut@\setboxz@h{$\m@th\displaystyle{\@lign##}$}%                  %4
  \global\lwidth@\wdz@\boxz@\global\advance\and@\@ne                        %5
  \tabskip\z@skip                                                           %6
  &\setboxz@h{$\m@th\displaystyle{{}\@lign##}$}%                            %7
  \global\rwidth@\wdz@\boxz@\hfill\global\advance\and@\@ne                  %8
  \tabskip\centering@                                                       %9
  &\setboxz@h{\@lign\strut@\maketag@##\maketag@}%                          %10
  \dimen@\displaywidth\advance\dimen@-\totwidth@
  \divide\dimen@\tw@\advance\dimen@\maxrwidth@\advance\dimen@-\rwidth@     %11
  \ifdim\dimen@<\tw@\wdz@\llap{\vtop{\normalbaselines\null\boxz@}}%        %12
  \else\llap{\boxz@}\fi                                                    %13
  \tabskip\z@skip                                                          %14
  \crcr#1\crcr                                                             %15
  \black@\totwidth@}}                                                      %16
\newdimen\lineht@
\expandafter\def\csname align \endcsname#1\endalign{\measure@#1\endalign
 \global\and@\z@
 \ifdim\totwidth@>\displaywidth\let\displaywidth@\totwidth@\else
  \let\displaywidth@\displaywidth\fi                                        %1
 \ifingather@\everycr{\noalign{\global\and@\z@}}\else\displ@y@\fi
 \Let@\tabskip\centering@\halign to\displaywidth
  {\hfil\strut@\setboxz@h{$\m@th\displaystyle{\@lign##}$}%
  \global\lwidth@\wdz@\global\lineht@\ht\z@                                 %2
  \boxz@\global\advance\and@\@ne
  \tabskip\z@skip&\setboxz@h{$\m@th\displaystyle{{}\@lign##}$}%
  \global\rwidth@\wdz@\ifdim\ht\z@>\lineht@\global\lineht@\ht\z@\fi         %3
  \boxz@\hfil\global\advance\and@\@ne
  \tabskip\centering@&\kern-\displaywidth@                                  %4
  \setboxz@h{\@lign\strut@\maketag@##\maketag@}%
  \dimen@\displaywidth\advance\dimen@-\totwidth@
  \divide\dimen@\tw@\advance\dimen@\maxlwidth@\advance\dimen@-\lwidth@
  \ifdim\dimen@<\tw@\wdz@
   \rlap{\vbox{\normalbaselines\boxz@\vbox to\lineht@{}}}\else
   \rlap{\boxz@}\fi
  \tabskip\displaywidth@\crcr#1\crcr\black@\totwidth@}}
\expandafter\def\csname align (in \string\gather)\endcsname
  #1\endalign{\vcenter{\align@#1\endalign}}
\Invalid@\endalign
\newif\ifxat@
\def\alignat{\RIfMIfI@\DN@{\onlydmatherr@\alignat}\else
 \DN@{\csname alignat \endcsname}\fi\else
 \DN@{\onlydmatherr@\alignat}\fi\next@}
\newif\ifmeasuring@
\newbox\savealignat@
\expandafter\def\csname alignat \endcsname#1#2\endalignat                   %1
 {\inany@true\xat@false
 \def\tag{\global\tag@true\count@#1\relax\multiply\count@\tw@
  \xdef\tag@{}\loop\ifnum\count@>\and@\xdef\tag@{&\tag@}\advance\count@\m@ne
  \repeat\tag@}%
 \vspace@\allowdisplaybreak@\displaybreak@\intertext@
 \displ@y@\measuring@true                                                   %2
 \setbox\savealignat@\hbox{$\m@th\displaystyle\Let@
  \attag@{#1}%                                                              %3
  \vbox{\halign{\span\preamble@@\crcr#2\crcr}}$}%
 \measuring@false                                                           %4
 \Let@\attag@{#1}%                                                          %5
 \tabskip\centering@\halign to\displaywidth
  {\span\preamble@@\crcr#2\crcr                                             %6
  \black@{\wd\savealignat@}}}                                               %7
\Invalid@\endalignat
\def\xalignat{\RIfMIfI@
 \DN@{\onlydmatherr@\xalignat}\else
 \DN@{\csname xalignat \endcsname}\fi\else
 \DN@{\onlydmatherr@\xalignat}\fi\next@}
\expandafter\def\csname xalignat \endcsname#1#2\endxalignat
 {\inany@true\xat@true
 \def\tag{\global\tag@true\def\tag@{}\count@#1\relax\multiply\count@\tw@
  \loop\ifnum\count@>\and@\xdef\tag@{&\tag@}\advance\count@\m@ne\repeat\tag@}%
 \vspace@\allowdisplaybreak@\displaybreak@\intertext@
 \displ@y@\measuring@true\setbox\savealignat@\hbox{$\m@th\displaystyle\Let@
 \attag@{#1}\vbox{\halign{\span\preamble@@\crcr#2\crcr}}$}%
 \measuring@false\Let@
 \attag@{#1}\tabskip\centering@\halign to\displaywidth
 {\span\preamble@@\crcr#2\crcr\black@{\wd\savealignat@}}}
\def\attag@#1{\let\Maketag@\maketag@\let\TAG@\Tag@                          %1
 \let\Tag@=0\let\maketag@=0%                                                %2
 \ifmeasuring@\def\llap@##1{\setboxz@h{##1}\hbox to\tw@\wdz@{}}%
  \def\rlap@##1{\setboxz@h{##1}\hbox to\tw@\wdz@{}}\else
  \let\llap@\llap\let\rlap@\rlap\fi                                         %3
 \toks@{\hfil\strut@$\m@th\displaystyle{\@lign\the\hashtoks@}$\tabskip\z@skip
  \global\advance\and@\@ne&$\m@th\displaystyle{{}\@lign\the\hashtoks@}$\hfil
  \ifxat@\tabskip\centering@\fi\global\advance\and@\@ne}%                   %4
 \iftagsleft@
  \toks@@{\tabskip\centering@&\Tag@\kern-\displaywidth
   \rlap@{\@lign\maketag@\the\hashtoks@\maketag@}%
   \global\advance\and@\@ne\tabskip\displaywidth}\else
  \toks@@{\tabskip\centering@&\Tag@\llap@{\@lign\maketag@
   \the\hashtoks@\maketag@}\global\advance\and@\@ne\tabskip\z@skip}\fi      %5
 \atcount@#1\relax\advance\atcount@\m@ne
 \loop\ifnum\atcount@>\z@
 \toks@=\expandafter{\the\toks@&\hfil$\m@th\displaystyle{\@lign
  \the\hashtoks@}$\global\advance\and@\@ne
  \tabskip\z@skip&$\m@th\displaystyle{{}\@lign\the\hashtoks@}$\hfil\ifxat@
  \tabskip\centering@\fi\global\advance\and@\@ne}\advance\atcount@\m@ne
 \repeat                                                                    %6
 \xdef\preamble@{\the\toks@\the\toks@@}%                                    %7
 \xdef\preamble@@{\preamble@}%                                              %8
 \let\maketag@\Maketag@\let\Tag@\TAG@}                                      %9
\Invalid@\endxalignat
\def\xxalignat{\RIfMIfI@
 \DN@{\onlydmatherr@\xxalignat}\else\DN@{\csname xxalignat
  \endcsname}\fi\else
 \DN@{\onlydmatherr@\xxalignat}\fi\next@}
\expandafter\def\csname xxalignat \endcsname#1#2\endxxalignat{\inany@true
 \vspace@\allowdisplaybreak@\displaybreak@\intertext@
 \displ@y\setbox\savealignat@\hbox{$\m@th\displaystyle\Let@
 \xxattag@{#1}\vbox{\halign{\span\preamble@@\crcr#2\crcr}}$}%
 \Let@\xxattag@{#1}\tabskip\z@skip\halign to\displaywidth
 {\span\preamble@@\crcr#2\crcr\black@{\wd\savealignat@}}}
\def\xxattag@#1{\toks@{\tabskip\z@skip\hfil\strut@
 $\m@th\displaystyle{\the\hashtoks@}$&%
 $\m@th\displaystyle{{}\the\hashtoks@}$\hfil\tabskip\centering@&}%
 \atcount@#1\relax\advance\atcount@\m@ne\loop\ifnum\atcount@>\z@
 \toks@=\expandafter{\the\toks@&\hfil$\m@th\displaystyle{\the\hashtoks@}$%
  \tabskip\z@skip&$\m@th\displaystyle{{}\the\hashtoks@}$\hfil
  \tabskip\centering@}\advance\atcount@\m@ne\repeat
 \xdef\preamble@{\the\toks@\tabskip\z@skip}\xdef\preamble@@{\preamble@}}
\Invalid@\endxxalignat
\newdimen\gwidth@
\newdimen\gmaxwidth@
\def\gmeasure@#1\endgather{\gwidth@\z@\gmaxwidth@\z@\setbox@ne\vbox{\Let@
 \halign{\setboxz@h{$\m@th\displaystyle{##}$}\global\gwidth@\wdz@
 \ifdim\gwidth@>\gmaxwidth@\global\gmaxwidth@\gwidth@\fi
 &\eat@{##}\crcr#1\crcr}}}
\def\gather{\RIfMIfI@\DN@{\onlydmatherr@\gather}\else
 \ingather@true\inany@true\def\tag{&}%
 \vspace@\allowdisplaybreak@\displaybreak@\intertext@
 \displ@y\Let@
 \iftagsleft@\DN@{\csname gather \endcsname}\else
  \DN@{\csname gather \space\endcsname}\fi\fi
 \else\DN@{\onlydmatherr@\gather}\fi\next@}
\expandafter\def\csname gather \space\endcsname#1\endgather
 {\gmeasure@#1\endgather\tabskip\centering@
 \halign to\displaywidth{\hfil\strut@\setboxz@h{$\m@th\displaystyle{##}$}%
 \global\gwidth@\wdz@\boxz@\hfil&
 \setboxz@h{\strut@{\maketag@##\maketag@}}%
 \dimen@\displaywidth\advance\dimen@-\gwidth@
 \ifdim\dimen@>\tw@\wdz@\llap{\boxz@}\else
 \llap{\vtop{\normalbaselines\null\boxz@}}\fi
 \tabskip\z@skip\crcr#1\crcr\black@\gmaxwidth@}}
\newdimen\glineht@
\expandafter\def\csname gather \endcsname#1\endgather{\gmeasure@#1\endgather
 \ifdim\gmaxwidth@>\displaywidth\let\gdisplaywidth@\gmaxwidth@\else
 \let\gdisplaywidth@\displaywidth\fi\tabskip\centering@\halign to\displaywidth
 {\hfil\strut@\setboxz@h{$\m@th\displaystyle{##}$}%
 \global\gwidth@\wdz@\global\glineht@\ht\z@\boxz@\hfil&\kern-\gdisplaywidth@
 \setboxz@h{\strut@{\maketag@##\maketag@}}%
 \dimen@\displaywidth\advance\dimen@-\gwidth@
 \ifdim\dimen@>\tw@\wdz@\rlap{\boxz@}\else
 \rlap{\vbox{\normalbaselines\boxz@\vbox to\glineht@{}}}\fi
 \tabskip\gdisplaywidth@\crcr#1\crcr\black@\gmaxwidth@}}
\newif\ifctagsplit@
\def\CenteredTagsOnSplits{\global\ctagsplit@true}
\def\TopOrBottomTagsOnSplits{\global\ctagsplit@false}
\TopOrBottomTagsOnSplits
\def\split{\relax\ifinany@\let\next@\insplit@\else
 \ifmmode\ifinner\def\next@{\onlydmatherr@\split}\else
 \let\next@\outsplit@\fi\else
 \def\next@{\onlydmatherr@\split}\fi\fi\next@}
\def\insplit@{\global\setbox\z@\vbox\bgroup\vspace@\Let@\ialign\bgroup
 \hfil\strut@$\m@th\displaystyle{##}$&$\m@th\displaystyle{{}##}$\hfill\crcr}
\def\endsplit{\crcr\egroup\egroup\iftagsleft@\expandafter\lendsplit@\else
 \expandafter\rendsplit@\fi}
\def\rendsplit@{\global\setbox9 \vbox
 {\unvcopy\z@\global\setbox8 \lastbox\unskip}%                              %1
 \setbox@ne\hbox{\unhcopy8 \unskip\global\setbox\tw@\lastbox
 \unskip\global\setbox\thr@@\lastbox}%                                      %2
 \global\setbox7 \hbox{\unhbox\tw@\unskip}%                                 %3
 \ifinalign@\ifctagsplit@                                                   %4
  \gdef\split@{\hbox to\wd\thr@@{}&
   \vcenter{\vbox{\moveleft\wd\thr@@\boxz@}}}%                              %5
 \else\gdef\split@{&\vbox{\moveleft\wd\thr@@\box9}\crcr
  \box\thr@@&\box7}\fi                                                      %6
 \else                                                                      %7
  \ifctagsplit@\gdef\split@{\vcenter{\boxz@}}\else
  \gdef\split@{\box9\crcr\hbox{\box\thr@@\box7}}\fi
 \fi
 \split@}                                                                   %8
\def\lendsplit@{\global\setbox9\vtop{\unvcopy\z@}%                          %1
 \setbox@ne\vbox{\unvcopy\z@\global\setbox8\lastbox}%                       %2
 \setbox@ne\hbox{\unhcopy8\unskip\setbox\tw@\lastbox
  \unskip\global\setbox\thr@@\lastbox}%                                     %3
 \ifinalign@\ifctagsplit@                                                   %4
  \gdef\split@{\hbox to\wd\thr@@{}&
  \vcenter{\vbox{\moveleft\wd\thr@@\box9}}}%                                %5
  \else                                                                     %6
  \gdef\split@{\hbox to\wd\thr@@{}&\vbox{\moveleft\wd\thr@@\box9}}\fi
 \else
  \ifctagsplit@\gdef\split@{\vcenter{\box9}}\else
  \gdef\split@{\box9}\fi
 \fi\split@}
\def\outsplit@#1$${\align\insplit@#1\endalign$$}
\newdimen\multlinegap@
\multlinegap@1em
\newdimen\multlinetaggap@
\multlinetaggap@1em
\def\MultlineGap#1{\global\multlinegap@#1\relax}
\def\multlinegap#1{\RIfMIfI@\onlydmatherr@\multlinegap\else
 \multlinegap@#1\relax\fi\else\onlydmatherr@\multlinegap\fi}
\def\nomultlinegap{\multlinegap{\z@}}
\def\multline{\RIfMIfI@
 \DN@{\onlydmatherr@\multline}\else
 \DN@{\multline@}\fi\else
 \DN@{\onlydmatherr@\multline}\fi\next@}
\newif\iftagin@
\def\tagin@#1{\tagin@false\in@\tag{#1}\ifin@\tagin@true\fi}
\def\multline@#1$${\inany@true\vspace@\allowdisplaybreak@\displaybreak@
 \tagin@{#1}\iftagsleft@\DN@{\multline@l#1$$}\else
 \DN@{\multline@r#1$$}\fi\next@}
\newdimen\mwidth@
\def\rmmeasure@#1\endmultline{%
 \def\shoveleft##1{##1}\def\shoveright##1{##1}%                             %1
 \setbox@ne\vbox{\Let@\halign{\setboxz@h
  {$\m@th\@lign\displaystyle{}##$}\global\mwidth@\wdz@
  \crcr#1\crcr}}}
\newdimen\mlineht@
\newif\ifzerocr@
\newif\ifonecr@
\def\lmmeasure@#1\endmultline{\global\zerocr@true\global\onecr@false
 \everycr{\noalign{\ifonecr@\global\onecr@false\fi
  \ifzerocr@\global\zerocr@false\global\onecr@true\fi}}%                    %1
  \def\shoveleft##1{##1}\def\shoveright##1{##1}%
 \setbox@ne\vbox{\Let@\halign{\setboxz@h
  {$\m@th\@lign\displaystyle{}##$}\ifonecr@\global\mwidth@\wdz@
  \global\mlineht@\ht\z@\fi\crcr#1\crcr}}}
\newbox\mtagbox@
\newdimen\ltwidth@
\newdimen\rtwidth@
\def\multline@l#1$${\iftagin@\DN@{\lmultline@@#1$$}\else
 \DN@{\setbox\mtagbox@\null\ltwidth@\z@\rtwidth@\z@
  \lmultline@@@#1$$}\fi\next@}
\def\lmultline@@#1\endmultline\tag#2$${%
 \setbox\mtagbox@\hbox{\maketag@#2\maketag@}%                               %1
 \lmmeasure@#1\endmultline\dimen@\mwidth@\advance\dimen@\wd\mtagbox@
 \advance\dimen@\multlinetaggap@                                            %2
 \ifdim\dimen@>\displaywidth\ltwidth@\z@\else\ltwidth@\wd\mtagbox@\fi       %3
 \lmultline@@@#1\endmultline$$}
\def\lmultline@@@{\displ@y
 \def\shoveright##1{##1\hfilneg\hskip\multlinegap@}%
 \def\shoveleft##1{\setboxz@h{$\m@th\displaystyle{}##1$}%
  \setbox@ne\hbox{$\m@th\displaystyle##1$}%
  \hfilneg
  \iftagin@
   \ifdim\ltwidth@>\z@\hskip\ltwidth@\hskip\multlinetaggap@\fi
  \else\hskip\multlinegap@\fi\hskip.5\wd@ne\hskip-.5\wdz@##1}%              %1
  \halign\bgroup\Let@\hbox to\displaywidth
   {\strut@$\m@th\displaystyle\hfil{}##\hfil$}\crcr
   \hfilneg                                                                 %2
   \iftagin@                                                                %3
    \ifdim\ltwidth@>\z@                                                     %4
     \box\mtagbox@\hskip\multlinetaggap@                                    %5
    \else
     \rlap{\vbox{\normalbaselines\hbox{\strut@\box\mtagbox@}%
     \vbox to\mlineht@{}}}\fi                                               %6
   \else\hskip\multlinegap@\fi}                                             %7
\def\multline@r#1$${\iftagin@\DN@{\rmultline@@#1$$}\else
 \DN@{\setbox\mtagbox@\null\ltwidth@\z@\rtwidth@\z@
  \rmultline@@@#1$$}\fi\next@}
\def\rmultline@@#1\endmultline\tag#2$${\ltwidth@\z@
 \setbox\mtagbox@\hbox{\maketag@#2\maketag@}%
 \rmmeasure@#1\endmultline\dimen@\mwidth@\advance\dimen@\wd\mtagbox@
 \advance\dimen@\multlinetaggap@
 \ifdim\dimen@>\displaywidth\rtwidth@\z@\else\rtwidth@\wd\mtagbox@\fi
 \rmultline@@@#1\endmultline$$}
\def\rmultline@@@{\displ@y
 \def\shoveright##1{##1\hfilneg\iftagin@\ifdim\rtwidth@>\z@
  \hskip\rtwidth@\hskip\multlinetaggap@\fi\else\hskip\multlinegap@\fi}%
 \def\shoveleft##1{\setboxz@h{$\m@th\displaystyle{}##1$}%
  \setbox@ne\hbox{$\m@th\displaystyle##1$}%
  \hfilneg\hskip\multlinegap@\hskip.5\wd@ne\hskip-.5\wdz@##1}%
 \halign\bgroup\Let@\hbox to\displaywidth
  {\strut@$\m@th\displaystyle\hfil{}##\hfil$}\crcr
 \hfilneg\hskip\multlinegap@}
\def\endmultline{\iftagsleft@\expandafter\lendmultline@\else
 \expandafter\rendmultline@\fi}
\def\lendmultline@{\hfilneg\hskip\multlinegap@\crcr\egroup}
\def\rendmultline@{\iftagin@                                                %1
 \ifdim\rtwidth@>\z@                                                        %2
  \hskip\multlinetaggap@\box\mtagbox@                                       %3
 \else\llap{\vtop{\normalbaselines\null\hbox{\strut@\box\mtagbox@}}}\fi     %4
 \else\hskip\multlinegap@\fi                                                %5
 \hfilneg\crcr\egroup}
\def\bmod{\mskip-\medmuskip\mkern5mu\mathbin{\fam\z@ mod}\penalty900
 \mkern5mu\mskip-\medmuskip}
\def\pmod#1{\allowbreak\ifinner\mkern8mu\else\mkern18mu\fi
 ({\fam\z@ mod}\,\,#1)}
\def\pod#1{\allowbreak\ifinner\mkern8mu\else\mkern18mu\fi(#1)}
\def\mod#1{\allowbreak\ifinner\mkern12mu\else\mkern18mu\fi{\fam\z@ mod}\,\,#1}
\message{continued fractions,}
\newcount\cfraccount@
\def\cfrac{\bgroup\bgroup\advance\cfraccount@\@ne\strut
 \iffalse{\fi\def\\{\over\displaystyle}\iffalse}\fi}
\def\lcfrac{\bgroup\bgroup\advance\cfraccount@\@ne\strut
 \iffalse{\fi\def\\{\hfill\over\displaystyle}\iffalse}\fi}
\def\rcfrac{\bgroup\bgroup\advance\cfraccount@\@ne\strut\hfill
 \iffalse{\fi\def\\{\over\displaystyle}\iffalse}\fi}
\def\gloop@#1\repeat{\gdef\body{#1}\iterate}
\def\endcfrac{\gloop@\ifnum\cfraccount@>\z@\global\advance\cfraccount@\m@ne
 \egroup\hskip-\nulldelimiterspace\egroup\repeat}
\message{compound symbols,}
\def\binrel@#1{\setboxz@h{\thinmuskip0mu
  \medmuskip\m@ne mu\thickmuskip\@ne mu$#1\m@th$}%
 \setbox@ne\hbox{\thinmuskip0mu\medmuskip\m@ne mu\thickmuskip
  \@ne mu${}#1{}\m@th$}%
 \setbox\tw@\hbox{\hskip\wd@ne\hskip-\wdz@}}
\def\overset#1\to#2{\binrel@{#2}\ifdim\wd\tw@<\z@
 \mathbin{\mathop{\kern\z@#2}\limits^{#1}}\else\ifdim\wd\tw@>\z@
 \mathrel{\mathop{\kern\z@#2}\limits^{#1}}\else
 {\mathop{\kern\z@#2}\limits^{#1}}{}\fi\fi}
\def\underset#1\to#2{\binrel@{#2}\ifdim\wd\tw@<\z@
 \mathbin{\mathop{\kern\z@#2}\limits_{#1}}\else\ifdim\wd\tw@>\z@
 \mathrel{\mathop{\kern\z@#2}\limits_{#1}}\else
 {\mathop{\kern\z@#2}\limits_{#1}}{}\fi\fi}
\def\oversetbrace#1\to#2{\overbrace{#2}^{#1}}
\def\undersetbrace#1\to#2{\underbrace{#2}_{#1}}
\def\sideset#1\and#2\to#3{%
 \setbox@ne\hbox{$\dsize{\vphantom{#3}}#1{#3}\m@th$}%
 \setbox\tw@\hbox{$\dsize{#3}#2\m@th$}%
 \hskip\wd@ne\hskip-\wd\tw@\mathop{\hskip\wd\tw@\hskip-\wd@ne
  {\vphantom{#3}}#1{#3}#2}}
\def\rightarrowfill@#1{\setboxz@h{$#1-\m@th$}\ht\z@\z@
  $#1\m@th\copy\z@\mkern-6mu\cleaders
  \hbox{$#1\mkern-2mu\box\z@\mkern-2mu$}\hfill
  \mkern-6mu\mathord\rightarrow$}
\def\leftarrowfill@#1{\setboxz@h{$#1-\m@th$}\ht\z@\z@
  $#1\m@th\mathord\leftarrow\mkern-6mu\cleaders
  \hbox{$#1\mkern-2mu\copy\z@\mkern-2mu$}\hfill
  \mkern-6mu\box\z@$}
\def\leftrightarrowfill@#1{\setboxz@h{$#1-\m@th$}\ht\z@\z@
  $#1\m@th\mathord\leftarrow\mkern-6mu\cleaders
  \hbox{$#1\mkern-2mu\box\z@\mkern-2mu$}\hfill
  \mkern-6mu\mathord\rightarrow$}
\def\overrightarrow{\mathpalette\overrightarrow@}
\def\overrightarrow@#1#2{\vbox{\ialign{##\crcr\rightarrowfill@#1\crcr
 \noalign{\kern-\ex@\nointerlineskip}$\m@th\hfil#1#2\hfil$\crcr}}}
\let\overarrow\overrightarrow
\def\overleftarrow{\mathpalette\overleftarrow@}
\def\overleftarrow@#1#2{\vbox{\ialign{##\crcr\leftarrowfill@#1\crcr
 \noalign{\kern-\ex@\nointerlineskip}$\m@th\hfil#1#2\hfil$\crcr}}}
\def\overleftrightarrow{\mathpalette\overleftrightarrow@}
\def\overleftrightarrow@#1#2{\vbox{\ialign{##\crcr\leftrightarrowfill@#1\crcr
 \noalign{\kern-\ex@\nointerlineskip}$\m@th\hfil#1#2\hfil$\crcr}}}
\def\underrightarrow{\mathpalette\underrightarrow@}
\def\underrightarrow@#1#2{\vtop{\ialign{##\crcr$\m@th\hfil#1#2\hfil$\crcr
 \noalign{\nointerlineskip}\rightarrowfill@#1\crcr}}}
\let\underarrow\underrightarrow
\def\underleftarrow{\mathpalette\underleftarrow@}
\def\underleftarrow@#1#2{\vtop{\ialign{##\crcr$\m@th\hfil#1#2\hfil$\crcr
 \noalign{\nointerlineskip}\leftarrowfill@#1\crcr}}}
\def\underleftrightarrow{\mathpalette\underleftrightarrow@}
\def\underleftrightarrow@#1#2{\vtop{\ialign{##\crcr$\m@th\hfil#1#2\hfil$\crcr
 \noalign{\nointerlineskip}\leftrightarrowfill@#1\crcr}}}
\message{various kinds of dots,}
\let\DOTSI\relax
\let\DOTSB\relax
\let\DOTSX\relax
\newif\ifmath@
{\uccode`7=`\\ \uccode`8=`m \uccode`9=`a \uccode`0=`t \uccode`!=`h
 \uppercase{\gdef\math@#1#2#3#4#5#6\math@{\global\math@false\ifx 7#1\ifx 8#2%
 \ifx 9#3\ifx 0#4\ifx !#5\xdef\meaning@{#6}\global\math@true\fi\fi\fi\fi\fi}}}
\newif\ifmathch@
{\uccode`7=`c \uccode`8=`h \uccode`9=`\"
 \uppercase{\gdef\mathch@#1#2#3#4#5#6\mathch@{\global\mathch@false
  \ifx 7#1\ifx 8#2\ifx 9#5\global\mathch@true\xdef\meaning@{9#6}\fi\fi\fi}}}
\newcount\classnum@
\def\getmathch@#1.#2\getmathch@{\classnum@#1 \divide\classnum@4096
 \ifcase\number\classnum@\or\or\gdef\thedots@{\dotsb@}\or
 \gdef\thedots@{\dotsb@}\fi}
\newif\ifmathbin@
{\uccode`4=`b \uccode`5=`i \uccode`6=`n
 \uppercase{\gdef\mathbin@#1#2#3{\relaxnext@
  \DNii@##1\mathbin@{\ifx\space@\next\global\mathbin@true\fi}%
 \global\mathbin@false\DN@##1\mathbin@{}%
 \ifx 4#1\ifx 5#2\ifx 6#3\DN@{\FN@\nextii@}\fi\fi\fi\next@}}}
\newif\ifmathrel@
{\uccode`4=`r \uccode`5=`e \uccode`6=`l
 \uppercase{\gdef\mathrel@#1#2#3{\relaxnext@
  \DNii@##1\mathrel@{\ifx\space@\next\global\mathrel@true\fi}%
 \global\mathrel@false\DN@##1\mathrel@{}%
 \ifx 4#1\ifx 5#2\ifx 6#3\DN@{\FN@\nextii@}\fi\fi\fi\next@}}}
\newif\ifmacro@
{\uccode`5=`m \uccode`6=`a \uccode`7=`c
 \uppercase{\gdef\macro@#1#2#3#4\macro@{\global\macro@false
  \ifx 5#1\ifx 6#2\ifx 7#3\global\macro@true
  \xdef\meaning@{\macro@@#4\macro@@}\fi\fi\fi}}}
\def\macro@@#1->#2\macro@@{#2}
\newif\ifDOTS@
\newcount\DOTSCASE@
{\uccode`6=`\\ \uccode`7=`D \uccode`8=`O \uccode`9=`T \uccode`0=`S
 \uppercase{\gdef\DOTS@#1#2#3#4#5{\global\DOTS@false\DN@##1\DOTS@{}%
  \ifx 6#1\ifx 7#2\ifx 8#3\ifx 9#4\ifx 0#5\let\next@\DOTS@@\fi\fi\fi\fi\fi
  \next@}}}
{\uccode`3=`B \uccode`4=`I \uccode`5=`X
 \uppercase{\gdef\DOTS@@#1{\relaxnext@
  \DNii@##1\DOTS@{\ifx\space@\next\global\DOTS@true\fi}%
  \DN@{\FN@\nextii@}%
  \ifx 3#1\global\DOTSCASE@\z@\else
  \ifx 4#1\global\DOTSCASE@\@ne\else
  \ifx 5#1\global\DOTSCASE@\tw@\else\DN@##1\DOTS@{}%
  \fi\fi\fi\next@}}}
\newif\ifnot@
{\uccode`5=`\\ \uccode`6=`n \uccode`7=`o \uccode`8=`t
 \uppercase{\gdef\not@#1#2#3#4{\relaxnext@
  \DNii@##1\not@{\ifx\space@\next\global\not@true\fi}%
 \global\not@false\DN@##1\not@{}%
 \ifx 5#1\ifx 6#2\ifx 7#3\ifx 8#4\DN@{\FN@\nextii@}\fi\fi\fi
 \fi\next@}}}
\newif\ifkeybin@
\def\keybin@{\keybin@true
 \ifx\next+\else\ifx\next=\else\ifx\next<\else\ifx\next>\else\ifx\next-\else
 \ifx\next*\else\ifx\next:\else\keybin@false\fi\fi\fi\fi\fi\fi\fi}
\def\dots{\RIfM@\expandafter\mdots@\else\expandafter\tdots@\fi}
\def\tdots@{\unskip\relaxnext@
 \DN@{$\m@th\mathinner{\ldotp\ldotp\ldotp}\,
   \ifx\next,\,$\else\ifx\next.\,$\else\ifx\next;\,$\else\ifx\next:\,$\else
   \ifx\next?\,$\else\ifx\next!\,$\else$ \fi\fi\fi\fi\fi\fi}%
 \ \FN@\next@}
\def\mdots@{\FN@\mdots@@}
\def\mdots@@{\gdef\thedots@{\dotso@}%                                       %1
 \ifx\next\boldkey\gdef\thedots@\boldkey{\boldkeydots@}\else                %2
 \ifx\next\boldsymbol\gdef\thedots@\boldsymbol{\boldsymboldots@}\else       %3
 \ifx,\next\gdef\thedots@{\dotsc}%                                          %4
 \else\ifx\not\next\gdef\thedots@{\dotsb@}%                                 %5
 \else\keybin@
 \ifkeybin@\gdef\thedots@{\dotsb@}%                                         %6
 \else\xdef\meaning@{\meaning\next..........}\xdef\meaning@@{\meaning@}%    %7
  \expandafter\math@\meaning@\math@
  \ifmath@
   \expandafter\mathch@\meaning@\mathch@
   \ifmathch@\expandafter\getmathch@\meaning@\getmathch@\fi                 %8
  \else\expandafter\macro@\meaning@@\macro@                                 %9
  \ifmacro@                                                                %10
   \expandafter\not@\meaning@\not@\ifnot@\gdef\thedots@{\dotsb@}%          %11
  \else\expandafter\DOTS@\meaning@\DOTS@
  \ifDOTS@
   \ifcase\number\DOTSCASE@\gdef\thedots@{\dotsb@}%
    \or\gdef\thedots@{\dotsi}\else\fi                                      %12
  \else\expandafter\math@\meaning@\math@                                   %13
  \ifmath@\expandafter\mathbin@\meaning@\mathbin@
  \ifmathbin@\gdef\thedots@{\dotsb@}%                                      %14
  \else\expandafter\mathrel@\meaning@\mathrel@
  \ifmathrel@\gdef\thedots@{\dotsb@}%                                      %15
  \fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi
 \thedots@}
\def\plainldots@{\mathinner{\ldotp\ldotp\ldotp}}
\def\plaincdots@{\mathinner{\cdotp\cdotp\cdotp}}
\def\dotsi{\!\plaincdots@}
\let\dotsb@\plaincdots@
\newif\ifextra@
\newif\ifrightdelim@
\def\rightdelim@{\global\rightdelim@true                                    %1
 \ifx\next)\else                                                            %2
 \ifx\next]\else
 \ifx\next\rbrack\else
 \ifx\next\}\else
 \ifx\next\rbrace\else
 \ifx\next\rangle\else
 \ifx\next\rceil\else
 \ifx\next\rfloor\else
 \ifx\next\rgroup\else
 \ifx\next\rmoustache\else
 \ifx\next\right\else
 \ifx\next\bigr\else
 \ifx\next\biggr\else
 \ifx\next\Bigr\else                                                        %3
 \ifx\next\Biggr\else\global\rightdelim@false
 \fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi}
\def\extra@{%
 \global\extra@false\rightdelim@\ifrightdelim@\global\extra@true            %1
 \else\ifx\next$\global\extra@true                                          %2
 \else\xdef\meaning@{\meaning\next..........}%                              %3
 \expandafter\macro@\meaning@\macro@\ifmacro@                               %4
 \expandafter\DOTS@\meaning@\DOTS@
 \ifDOTS@
 \ifnum\DOTSCASE@=\tw@\global\extra@true                                    %5
 \fi\fi\fi\fi\fi}
\newif\ifbold@
\def\dotso@{\relaxnext@
 \ifbold@
  \let\next\delayed@
  \DNii@{\extra@\plainldots@\ifextra@\,\fi}%
 \else
  \DNii@{\DN@{\extra@\plainldots@\ifextra@\,\fi}\FN@\next@}%
 \fi
 \nextii@}
\def\extrap@#1{%
 \ifx\next,\DN@{#1\,}\else
 \ifx\next;\DN@{#1\,}\else
 \ifx\next.\DN@{#1\,}\else\extra@
 \ifextra@\DN@{#1\,}\else
 \let\next@#1\fi\fi\fi\fi\next@}
\def\ldots{\DN@{\extrap@\plainldots@}%
 \FN@\next@}
\def\cdots{\DN@{\extrap@\plaincdots@}%
 \FN@\next@}
\let\dotso\ldots
\let\dotsb\cdots
\let\dotsm\dotsb
\def\dotsc{\relaxnext@
 \DN@{\ifx\next;\plainldots@\,\else
  \ifx\next.\plainldots@\,\else\extra@\plainldots@
  \ifextra@\,\fi\fi\fi}%
 \FN@\next@}
\def\cdot{\mathchar"2201 }
\def\longrightarrow{\DOTSB\relbar\joinrel\rightarrow}
\def\Longrightarrow{\DOTSB\Relbar\joinrel\Rightarrow}
\def\longleftarrow{\DOTSB\leftarrow\joinrel\relbar}
\def\Longleftarrow{\DOTSB\Leftarrow\joinrel\Relbar}
\def\longleftrightarrow{\DOTSB\leftarrow\joinrel\rightarrow}
\def\Longleftrightarrow{\DOTSB\Leftarrow\joinrel\Rightarrow}
\def\mapsto{\DOTSB\mapstochar\rightarrow}
\def\longmapsto{\DOTSB\mapstochar\longrightarrow}
\def\hookrightarrow{\DOTSB\lhook\joinrel\rightarrow}
\def\hookleftarrow{\DOTSB\leftarrow\joinrel\rhook}
\def\doteq{\DOTSB\buildrel\textstyle.\over=}
\def\iff{\DOTSB\;\Longleftrightarrow\;}
\message{special superscripts,}
\def\dddot#1{{\mathop{#1}\limits^{\vbox to-1.4\ex@{\kern-\tw@\ex@
 \hbox{\rm...}\vss}}}}
\def\ddddot#1{{\mathop{#1}\limits^{\vbox to-1.4\ex@{\kern-\tw@\ex@
 \hbox{\rm....}\vss}}}}
\def\sphat{^{\mathchoice{}{}%
 {\,\,\botsmash{\hbox{\lower4\ex@\hbox{$\m@th\widehat{\null}$}}}}%
 {\,\botsmash{\hbox{\lower3\ex@\hbox{$\m@th\hat{\null}$}}}}}}
\def\spcheck{^\vee}
\def\sptilde{^\sim}
\def\spacute{^{\!\botsmash{\hbox{\lower\@ne ex\hbox{\'{}}}}}}
\def\spgrave{^{\mathchoice{}{}{}{\!}%
 \botsmash{\hbox{\lower\@ne ex\hbox{\`{}}}}}}
\def\spdot{^{\hbox{\raise\ex@\hbox{\rm.}}}}
\def\spddot{^{\hbox{\raise\ex@\hbox{\rm..}}}}
\def\spdddot{^{\hbox{\raise\ex@\hbox{\rm...}}}}
\def\spddddot{^{\hbox{\raise\ex@\hbox{\rm....}}}}
\def\spbreve{^{\!\botsmash{\hbox{\lower4\ex@\hbox{\u{}}}}}}
\def\spbar{^-}
\def\spvec{^\rightarrow}
\message{\string\text,}
\def\textonlyfont@#1#2{\def#1{\RIfM@
 \Err@{Use \string#1\space only in text}\else#2\fi}}
\textonlyfont@\rm\tenrm
\textonlyfont@\it\tenit
\textonlyfont@\sl\tensl
\textonlyfont@\bf\tenbf
\def\oldnos#1{\RIfM@{\mathcode`\,="013B \fam\@ne#1}\else
 \leavevmode\hbox{$\m@th\mathcode`\,="013B \fam\@ne#1$}\fi}
\def\text{\RIfM@\expandafter\text@\else\expandafter\text@@\fi}
\def\text@@#1{\leavevmode\hbox{#1}}
\def\mathhexbox@#1#2#3{\text{$\m@th\mathchar"#1#2#3$}}
\def\dag{{\mathhexbox@279}}
\def\ddag{{\mathhexbox@27A}}
\def\S{{\mathhexbox@278}}
\def\P{{\mathhexbox@27B}}
\newif\iffirstchoice@
\firstchoice@true
\def\text@#1{\mathchoice
 {\hbox{\everymath{\displaystyle}\def\textfonti{\the\textfont\@ne}%
  \def\textfontii{\the\textfont\tw@}\textdef@@ T#1}}
 {\hbox{\firstchoice@false
  \everymath{\textstyle}\def\textfonti{\the\textfont\@ne}%
  \def\textfontii{\the\textfont\tw@}\textdef@@ T#1}}
 {\hbox{\firstchoice@false
  \everymath{\scriptstyle}\def\textfonti{\the\scriptfont\@ne}%
  \def\textfontii{\the\scriptfont\tw@}\textdef@@ S\rm#1}}
 {\hbox{\firstchoice@false
  \everymath{\scriptscriptstyle}\def\textfonti
  {\the\scriptscriptfont\@ne}%
  \def\textfontii{\the\scriptscriptfont\tw@}\textdef@@ s\rm#1}}}
\def\textdef@@#1{\textdef@#1\rm\textdef@#1\bf\textdef@#1\sl\textdef@#1\it}
\def\rmfam{0}
\def\textdef@#1#2{%
 \DN@{\csname\expandafter\eat@\string#2fam\endcsname}%
 \if S#1\edef#2{\the\scriptfont\next@\relax}%
 \else\if s#1\edef#2{\the\scriptscriptfont\next@\relax}%
 \else\edef#2{\the\textfont\next@\relax}\fi\fi}
\scriptfont\itfam\tenit \scriptscriptfont\itfam\tenit
\scriptfont\slfam\tensl \scriptscriptfont\slfam\tensl
\newif\iftopfolded@
\newif\ifbotfolded@
\def\topfoldedtext{\topfolded@true\botfolded@false\foldedtext@}
\def\botfoldedtext{\botfolded@true\topfolded@false\foldedtext@}
\def\foldedtext{\topfolded@false\botfolded@false\foldedtext@}
\Invalid@\foldedwidth
\def\foldedtext@{\relaxnext@
 \DN@{\ifx\next\foldedwidth\let\next@\nextii@\else
  \DN@{\nextii@\foldedwidth{.3\hsize}}\fi\next@}%
 \DNii@\foldedwidth##1##2{\setbox\z@\vbox
  {\normalbaselines\hsize##1\relax
  \tolerance1600 \noindent\ignorespaces##2}\ifbotfolded@\boxz@\else
  \iftopfolded@\vtop{\unvbox\z@}\else\vcenter{\boxz@}\fi\fi}%
 \FN@\next@}
\message{math font commands,}
\def\bold{\RIfM@\expandafter\bold@\else
 \expandafter\nonmatherr@\expandafter\bold\fi}
\def\bold@#1{{\bold@@{#1}}}
\def\bold@@#1{\fam\bffam\relax#1}
\def\slanted{\RIfM@\expandafter\slanted@\else
 \expandafter\nonmatherr@\expandafter\slanted\fi}
\def\slanted@#1{{\slanted@@{#1}}}
\def\slanted@@#1{\fam\slfam\relax#1}
\def\roman{\RIfM@\expandafter\roman@\else
 \expandafter\nonmatherr@\expandafter\roman\fi}
\def\roman@#1{{\roman@@{#1}}}
\def\roman@@#1{\fam\rmfam\relax#1}
\def\italic{\RIfM@\expandafter\italic@\else
 \expandafter\nonmatherr@\expandafter\italic\fi}
\def\italic@#1{{\italic@@{#1}}}
\def\italic@@#1{\fam\itfam\relax#1}
\def\Cal{\RIfM@\expandafter\Cal@\else
 \expandafter\nonmatherr@\expandafter\Cal\fi}
\def\Cal@#1{{\Cal@@{#1}}}
\def\Cal@@#1{\noaccents@\fam\tw@#1}
\mathchardef\Gamma="0000
\mathchardef\Delta="0001
\mathchardef\Theta="0002
\mathchardef\Lambda="0003
\mathchardef\Xi="0004
\mathchardef\Pi="0005
\mathchardef\Sigma="0006
\mathchardef\Upsilon="0007
\mathchardef\Phi="0008
\mathchardef\Psi="0009
\mathchardef\Omega="000A
\mathchardef\varGamma="0100
\mathchardef\varDelta="0101
\mathchardef\varTheta="0102
\mathchardef\varLambda="0103
\mathchardef\varXi="0104
\mathchardef\varPi="0105
\mathchardef\varSigma="0106
\mathchardef\varUpsilon="0107
\mathchardef\varPhi="0108
\mathchardef\varPsi="0109
\mathchardef\varOmega="010A
\let\alloc@@\alloc@
\def\hexnumber@#1{\ifcase#1 0\or 1\or 2\or 3\or 4\or 5\or 6\or 7\or 8\or
 9\or A\or B\or C\or D\or E\or F\fi}
\def\loadmsam{%
 \font@\tenmsa=msam10
 \font@\sevenmsa=msam7
 \font@\fivemsa=msam5
 \alloc@@8\fam\chardef\sixt@@n\msafam
 \textfont\msafam=\tenmsa
 \scriptfont\msafam=\sevenmsa
 \scriptscriptfont\msafam=\fivemsa
 \edef\next{\hexnumber@\msafam}%
 \mathchardef\dabar@"0\next39
 \edef\dashrightarrow{\mathrel{\dabar@\dabar@\mathchar"0\next4B}}%
 \edef\dashleftarrow{\mathrel{\mathchar"0\next4C\dabar@\dabar@}}%
 \let\dasharrow\dashrightarrow
 \edef\ulcorner{\delimiter"4\next70\next70 }%
 \edef\urcorner{\delimiter"5\next71\next71 }%
 \edef\llcorner{\delimiter"4\next78\next78 }%
 \edef\lrcorner{\delimiter"5\next79\next79 }%
 \edef\yen{{\noexpand\mathhexbox@\next55}}%
 \edef\checkmark{{\noexpand\mathhexbox@\next58}}%
 \edef\circledR{{\noexpand\mathhexbox@\next72}}%
 \edef\maltese{{\noexpand\mathhexbox@\next7A}}%
 \global\let\loadmsam\empty}%
\def\loadmsbm{%
 \font@\tenmsb=msbm10 \font@\sevenmsb=msbm7 \font@\fivemsb=msbm5
 \alloc@@8\fam\chardef\sixt@@n\msbfam
 \textfont\msbfam=\tenmsb
 \scriptfont\msbfam=\sevenmsb \scriptscriptfont\msbfam=\fivemsb
 \global\let\loadmsbm\empty
 }
\def\widehat#1{\ifx\undefined\msbfam \DN@{362}%
  \else \setboxz@h{$\m@th#1$}%
    \edef\next@{\ifdim\wdz@>\tw@ em%
        \hexnumber@\msbfam 5B%
      \else 362\fi}\fi
  \mathaccent"0\next@{#1}}
\def\widetilde#1{\ifx\undefined\msbfam \DN@{365}%
  \else \setboxz@h{$\m@th#1$}%
    \edef\next@{\ifdim\wdz@>\tw@ em%
        \hexnumber@\msbfam 5D%
      \else 365\fi}\fi
  \mathaccent"0\next@{#1}}
\message{\string\newsymbol,}
\def\newsymbol#1#2#3#4#5{\define#1{}%
  \count@#2\relax \advance\count@\m@ne % to push case 0 to the \else clause
 \ifcase\count@
   \ifx\undefined\msafam\loadmsam\fi \let\next@\msafam
 \or \ifx\undefined\msbfam\loadmsbm\fi \let\next@\msbfam
 \else  \Err@{\Invalid@@\string\newsymbol}\let\next@\tw@\fi
 \mathchardef#1="#3\hexnumber@\next@#4#5\space}
\def\UseAMSsymbols{\loadmsam\loadmsbm \input amssym.tex }
\def\Bbb{\RIfM@\expandafter\Bbb@\else
 \expandafter\nonmatherr@\expandafter\Bbb\fi}
\def\Bbb@#1{{\Bbb@@{#1}}}
\def\Bbb@@#1{\noaccents@\fam\msbfam\relax#1}
\message{bold Greek and bold symbols,}
\def\loadbold{%
 \font@\tencmmib=cmmib10 \font@\sevencmmib=cmmib7 \font@\fivecmmib=cmmib5
 \skewchar\tencmmib'177 \skewchar\sevencmmib'177 \skewchar\fivecmmib'177
 \alloc@@8\fam\chardef\sixt@@n\cmmibfam
 \textfont\cmmibfam\tencmmib
 \scriptfont\cmmibfam\sevencmmib \scriptscriptfont\cmmibfam\fivecmmib
 \font@\tencmbsy=cmbsy10 \font@\sevencmbsy=cmbsy7 \font@\fivecmbsy=cmbsy5
 \skewchar\tencmbsy'60 \skewchar\sevencmbsy'60 \skewchar\fivecmbsy'60
 \alloc@@8\fam\chardef\sixt@@n\cmbsyfam
 \textfont\cmbsyfam\tencmbsy
 \scriptfont\cmbsyfam\sevencmbsy \scriptscriptfont\cmbsyfam\fivecmbsy
 \let\loadbold\empty
}
\def\boldnotloaded#1{\Err@{\ifcase#1\or First\else Second\fi
       bold symbol font not loaded}}
\def\mathchari@#1#2#3{\ifx\undefined\cmmibfam
    \boldnotloaded@\@ne
  \else\mathchar"#1\hexnumber@\cmmibfam#2#3\space \fi}
\def\mathcharii@#1#2#3{\ifx\undefined\cmbsyfam
    \boldnotloaded\tw@
  \else \mathchar"#1\hexnumber@\cmbsyfam#2#3\space\fi}
\edef\bffam@{\hexnumber@\bffam}
\def\boldkey#1{\ifcat\noexpand#1A%
  \ifx\undefined\cmmibfam \boldnotloaded\@ne
  \else {\fam\cmmibfam#1}\fi
 \else
 \ifx#1!\mathchar"5\bffam@21 \else
 \ifx#1(\mathchar"4\bffam@28 \else\ifx#1)\mathchar"5\bffam@29 \else
 \ifx#1+\mathchar"2\bffam@2B \else\ifx#1:\mathchar"3\bffam@3A \else
 \ifx#1;\mathchar"6\bffam@3B \else\ifx#1=\mathchar"3\bffam@3D \else
 \ifx#1?\mathchar"5\bffam@3F \else\ifx#1[\mathchar"4\bffam@5B \else
 \ifx#1]\mathchar"5\bffam@5D \else
 \ifx#1,\mathchari@63B \else
 \ifx#1-\mathcharii@200 \else
 \ifx#1.\mathchari@03A \else
 \ifx#1/\mathchari@03D \else
 \ifx#1<\mathchari@33C \else
 \ifx#1>\mathchari@33E \else
 \ifx#1*\mathcharii@203 \else
 \ifx#1|\mathcharii@06A \else
 \ifx#10\bold0\else\ifx#11\bold1\else\ifx#12\bold2\else\ifx#13\bold3\else
 \ifx#14\bold4\else\ifx#15\bold5\else\ifx#16\bold6\else\ifx#17\bold7\else
 \ifx#18\bold8\else\ifx#19\bold9\else
  \Err@{\string\boldkey\space can't be used with #1}%
 \fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi
 \fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi}
\def\boldsymbol#1{%
 \DN@{\Err@{You can't use \string\boldsymbol\space with \string#1}#1}%
 \ifcat\noexpand#1A%
   \let\next@\relax
   \ifx\undefined\cmmibfam \boldnotloaded\@ne
   \else {\fam\cmmibfam#1}\fi
 \else
  \xdef\meaning@{\meaning#1.........}%
  \expandafter\math@\meaning@\math@
  \ifmath@
   \expandafter\mathch@\meaning@\mathch@
   \ifmathch@
    \expandafter\boldsymbol@@\meaning@\boldsymbol@@
   \fi
  \else
   \expandafter\macro@\meaning@\macro@
   \expandafter\delim@\meaning@\delim@
   \ifdelim@
    \expandafter\delim@@\meaning@\delim@@
   \else
    \boldsymbol@{#1}%
   \fi
  \fi
 \fi
 \next@}
\def\mathhexboxii@#1#2{\ifx\undefined\cmbsyfam
    \boldnotloaded\tw@
  \else \mathhexbox@{\hexnumber@\cmbsyfam}{#1}{#2}\fi}
\def\boldsymbol@#1{\let\next@\relax\let\next#1%
 \ifx\next\cdot\mathcharii@201 \else
 \ifx\next\prime{{\null\mathcharii@030 \null}}\else
 \ifx\next\lbrack\mathchar"4\bffam@5B \else
 \ifx\next\rbrack\mathchar"5\bffam@5D \else
 \ifx\next\{\mathcharii@466 \else
 \ifx\next\lbrace\mathcharii@466 \else
 \ifx\next\}\mathcharii@567 \else
 \ifx\next\rbrace\mathcharii@567 \else
 \ifx\next\surd{{\mathcharii@170}}\else
 \ifx\next\S{{\mathhexboxii@78}}\else
 \ifx\next\P{{\mathhexboxii@7B}}\else
 \ifx\next\dag{{\mathhexboxii@79}}\else
 \ifx\next\ddag{{\mathhexboxii@7A}}\else
 \DN@{\Err@{You can't use \string\boldsymbol\space with \string#1}#1}%
 \fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi}
\def\boldsymbol@@#1.#2\boldsymbol@@{\classnum@#1 \count@@@\classnum@        %1
 \divide\classnum@4096 \count@\classnum@                                    %2
 \multiply\count@4096 \advance\count@@@-\count@ \count@@\count@@@           %3
 \divide\count@@@\@cclvi \count@\count@@                                    %4
 \multiply\count@@@\@cclvi \advance\count@@-\count@@@                       %5
 \divide\count@@@\@cclvi                                                    %6
 \multiply\classnum@4096 \advance\classnum@\count@@                         %7
 \ifnum\count@@@=\z@                                                        %8
  \count@"\bffam@ \multiply\count@\@cclvi
  \advance\classnum@\count@
  \DN@{\mathchar\number\classnum@}%
 \else
  \ifnum\count@@@=\@ne                                                      %9
   \ifx\undefined\cmmibfam \DN@{\boldnotloaded\@ne}%
   \else \count@\cmmibfam \multiply\count@\@cclvi
     \advance\classnum@\count@
     \DN@{\mathchar\number\classnum@}\fi
  \else
   \ifnum\count@@@=\tw@                                                    %10
     \ifx\undefined\cmbsyfam
       \DN@{\boldnotloaded\tw@}%
     \else
       \count@\cmbsyfam \multiply\count@\@cclvi
       \advance\classnum@\count@
       \DN@{\mathchar\number\classnum@}%
     \fi
  \fi
 \fi
\fi}
\newif\ifdelim@
\newcount\delimcount@
{\uccode`6=`\\ \uccode`7=`d \uccode`8=`e \uccode`9=`l
 \uppercase{\gdef\delim@#1#2#3#4#5\delim@
  {\delim@false\ifx 6#1\ifx 7#2\ifx 8#3\ifx 9#4\delim@true
   \xdef\meaning@{#5}\fi\fi\fi\fi}}}
\def\delim@@#1"#2#3#4#5#6\delim@@{\if#32%
\let\next@\relax
 \ifx\undefined\cmbsyfam \boldnotloaded\@ne
 \else \mathcharii@#2#4#5\space \fi\fi}
\def\vert{\delimiter"026A30C }
\def\Vert{\delimiter"026B30D }
\let\|\Vert
\def\backslash{\delimiter"026E30F }
\def\boldkeydots@#1{\bold@true\let\next=#1\let\delayed@=#1\mdots@@
 \boldkey#1\bold@false}  % = required!
\def\boldsymboldots@#1{\bold@true\let\next#1\let\delayed@#1\mdots@@
 \boldsymbol#1\bold@false}
\message{Euler fonts,}
\def\loadeufm{\loadmathfont{eufm}}
\def\loadeufb{\loadmathfont{eufb}}
\def\loadeusm{\loadmathfont{eusm}}
\def\loadeusb{\loadmathfont{eusb}}
\def\loadeurm{\loadmathfont{eurm}}
\def\loadeurb{\loadmathfont{eurb}}
\def\frak{\mathfont@\frak}
\def\frakfam{\eufmfam}
\let\goth\frak
\def\loadmathfont#1{% 
   \expandafter\font@\csname ten#1\endcsname=#110
   \expandafter\font@\csname seven#1\endcsname=#17
   \expandafter\font@\csname five#1\endcsname=#15
   \edef\next{\noexpand\alloc@@8\fam\chardef\sixt@@n
     \expandafter\noexpand\csname#1fam\endcsname}%
   \next
   \textfont\csname#1fam\endcsname \csname ten#1\endcsname
   \scriptfont\csname#1fam\endcsname \csname seven#1\endcsname
   \scriptscriptfont\csname#1fam\endcsname \csname five#1\endcsname
   \expandafter\def\csname #1\expandafter\endcsname\expandafter{%
      \expandafter\mathfont@\csname#1\endcsname}%
 \expandafter\gdef\csname load#1\endcsname{}%
}
\def\mathfont@#1{\RIfM@\expandafter\mathfont@@\expandafter#1\else
  \expandafter\nonmatherr@\expandafter#1\fi}
\def\mathfont@@#1#2{{\mathfont@@@#1{#2}}}
\def\mathfont@@@#1#2{\noaccents@
   \fam\csname\expandafter\eat@\string#1fam\endcsname
   \relax#2}
\message{math accents,}
\def\accentclass@{7}
\def\noaccents@{\def\accentclass@{0}}
\def\makeacc@#1#2{\def#1{\mathaccent"\accentclass@#2 }}
\makeacc@\hat{05E}
\makeacc@\check{014}
\makeacc@\tilde{07E}
\makeacc@\acute{013}
\makeacc@\grave{012}
\makeacc@\dot{05F}
\makeacc@\ddot{07F}
\makeacc@\breve{015}
\makeacc@\bar{016}
\def\vec{\mathaccent"017E }
\newcount\skewcharcount@
\newcount\familycount@
\def\theskewchar@{\familycount@\@ne
 \global\skewcharcount@\the\skewchar\textfont\@ne                           %1
 \ifnum\fam>\m@ne\ifnum\fam<16
  \global\familycount@\the\fam\relax
  \global\skewcharcount@\the\skewchar\textfont\the\fam\relax\fi\fi          %2
 \ifnum\skewcharcount@>\m@ne
  \ifnum\skewcharcount@<128
  \multiply\familycount@256
  \global\advance\skewcharcount@\familycount@
  \global\advance\skewcharcount@28672
  \mathchar\skewcharcount@\else
  \global\skewcharcount@\m@ne\fi\else
 \global\skewcharcount@\m@ne\fi}                                            %3
\newcount\pointcount@
\def\getpoints@#1.#2\getpoints@{\pointcount@#1 }
\newdimen\accentdimen@
\newcount\accentmu@
\def\dimentomu@{\multiply\accentdimen@ 100
 \expandafter\getpoints@\the\accentdimen@\getpoints@
 \multiply\pointcount@18
 \divide\pointcount@\@m
 \global\accentmu@\pointcount@}
\def\Makeacc@#1#2{\def#1{\RIfM@\DN@{\mathaccent@
 {"\accentclass@#2 }}\else\DN@{\nonmatherr@{#1}}\fi\next@}}
\def\unbracefonts@{\let\Cal@\Cal@@\let\roman@\roman@@\let\bold@\bold@@
 \let\slanted@\slanted@@}
\def\mathaccent@#1#2{\ifnum\fam=\m@ne\xdef\thefam@{1}\else
 \xdef\thefam@{\the\fam}\fi                                                 %1
 \accentdimen@\z@                                                           %2
 \setboxz@h{\unbracefonts@$\m@th\fam\thefam@\relax#2$}%                     %3
 \ifdim\accentdimen@=\z@\DN@{\mathaccent#1{#2}}%                            %4
  \setbox@ne\hbox{\unbracefonts@$\m@th\fam\thefam@\relax#2\theskewchar@$}% %5a
  \setbox\tw@\hbox{$\m@th\ifnum\skewcharcount@=\m@ne\else
   \mathchar\skewcharcount@\fi$}%                                          %5b
  \global\accentdimen@\wd@ne\global\advance\accentdimen@-\wdz@
  \global\advance\accentdimen@-\wd\tw@                                     %5c
  \global\multiply\accentdimen@\tw@
  \dimentomu@\global\advance\accentmu@\@ne                                 %5d
 \else\DN@{{\mathaccent#1{#2\mkern\accentmu@ mu}%
    \mkern-\accentmu@ mu}{}}\fi                                             %6
 \next@}\Makeacc@\Hat{05E}
\Makeacc@\Check{014}
\Makeacc@\Tilde{07E}
\Makeacc@\Acute{013}
\Makeacc@\Grave{012}
\Makeacc@\Dot{05F}
\Makeacc@\Ddot{07F}
\Makeacc@\Breve{015}
\Makeacc@\Bar{016}
\def\Vec{\RIfM@\DN@{\mathaccent@{"017E }}\else
 \DN@{\nonmatherr@\Vec}\fi\next@}
\def\accentedsymbol#1#2{\csname newbox\expandafter\endcsname
  \csname\expandafter\eat@\string#1@box\endcsname
 \expandafter\setbox\csname\expandafter\eat@
  \string#1@box\endcsname\hbox{$\m@th#2$}\define
  #1{\copy\csname\expandafter\eat@\string#1@box\endcsname{}}}
\message{roots,}
\def\sqrt#1{\radical"270370 {#1}}
\let\underline@\underline
\let\overline@\overline
\def\underline#1{\underline@{#1}}
\def\overline#1{\overline@{#1}}
\Invalid@\leftroot
\Invalid@\uproot
\newcount\uproot@
\newcount\leftroot@
\def\root{\relaxnext@
  \DN@{\ifx\next\uproot\let\next@\nextii@\else
   \ifx\next\leftroot\let\next@\nextiii@\else
   \let\next@\plainroot@\fi\fi\next@}%
  \DNii@\uproot##1{\uproot@##1\relax\FN@\nextiv@}%
  \def\nextiv@{\ifx\next\space@\DN@. {\FN@\nextv@}\else
   \DN@.{\FN@\nextv@}\fi\next@.}%
  \def\nextv@{\ifx\next\leftroot\let\next@\nextvi@\else
   \let\next@\plainroot@\fi\next@}%
  \def\nextvi@\leftroot##1{\leftroot@##1\relax\plainroot@}%
   \def\nextiii@\leftroot##1{\leftroot@##1\relax\FN@\nextvii@}%
  \def\nextvii@{\ifx\next\space@
   \DN@. {\FN@\nextviii@}\else
   \DN@.{\FN@\nextviii@}\fi\next@.}%
  \def\nextviii@{\ifx\next\uproot\let\next@\nextix@\else
   \let\next@\plainroot@\fi\next@}%
  \def\nextix@\uproot##1{\uproot@##1\relax\plainroot@}%
  \bgroup\uproot@\z@\leftroot@\z@\FN@\next@}
\def\plainroot@#1\of#2{\setbox\rootbox\hbox{$\m@th\scriptscriptstyle{#1}$}%
 \mathchoice{\r@@t\displaystyle{#2}}{\r@@t\textstyle{#2}}
 {\r@@t\scriptstyle{#2}}{\r@@t\scriptscriptstyle{#2}}\egroup}
\def\r@@t#1#2{\setboxz@h{$\m@th#1\sqrt{#2}$}%
 \dimen@\ht\z@\advance\dimen@-\dp\z@
 \setbox@ne\hbox{$\m@th#1\mskip\uproot@ mu$}\advance\dimen@ 1.667\wd@ne
 \mkern-\leftroot@ mu\mkern5mu\raise.6\dimen@\copy\rootbox
 \mkern-10mu\mkern\leftroot@ mu\boxz@}
\def\boxed#1{\setboxz@h{$\m@th\displaystyle{#1}$}\dimen@.4\ex@
 \advance\dimen@3\ex@\advance\dimen@\dp\z@
 \hbox{\lower\dimen@\hbox{%
 \vbox{\hrule height.4\ex@
 \hbox{\vrule width.4\ex@\hskip3\ex@\vbox{\vskip3\ex@\boxz@\vskip3\ex@}%
 \hskip3\ex@\vrule width.4\ex@}\hrule height.4\ex@}%
 }}}
\message{commutative diagrams,}
\let\ampersand@\relax
\newdimen\minaw@
\minaw@11.11128\ex@
\newdimen\minCDaw@
\minCDaw@2.5pc
\def\minCDarrowwidth#1{\RIfMIfI@\onlydmatherr@\minCDarrowwidth
 \else\minCDaw@#1\relax\fi\else\onlydmatherr@\minCDarrowwidth\fi}
\newif\ifCD@
\def\CD{\bgroup\vspace@\relax\iffalse{\fi\let\ampersand@&\iffalse}\fi
 \CD@true\vcenter\bgroup\Let@\tabskip\z@skip\baselineskip20\ex@
 \lineskip3\ex@\lineskiplimit3\ex@\halign\bgroup
 &\hfill$\m@th##$\hfill\crcr}
\def\endCD{\crcr\egroup\egroup\egroup}
\newdimen\bigaw@
\atdef@>#1>#2>{\ampersand@                                                  %1
 \setboxz@h{$\m@th\ssize\;{#1}\;\;$}%                                       %2
 \setbox@ne\hbox{$\m@th\ssize\;{#2}\;\;$}%                                  %3
 \setbox\tw@\hbox{$\m@th#2$}%                                               %4
 \ifCD@\global\bigaw@\minCDaw@\else\global\bigaw@\minaw@\fi                 %5
 \ifdim\wdz@>\bigaw@\global\bigaw@\wdz@\fi
 \ifdim\wd@ne>\bigaw@\global\bigaw@\wd@ne\fi                                %6
 \ifCD@\enskip\fi                                                           %7
 \ifdim\wd\tw@>\z@
  \mathrel{\mathop{\hbox to\bigaw@{\rightarrowfill@\displaystyle}}%
    \limits^{#1}_{#2}}%                                                     %8
 \else\mathrel{\mathop{\hbox to\bigaw@{\rightarrowfill@\displaystyle}}%
    \limits^{#1}}\fi                                                        %9
 \ifCD@\enskip\fi                                                          %10
 \ampersand@}                                                              %11
\atdef@<#1<#2<{\ampersand@\setboxz@h{$\m@th\ssize\;\;{#1}\;$}%
 \setbox@ne\hbox{$\m@th\ssize\;\;{#2}\;$}\setbox\tw@\hbox{$\m@th#2$}%
 \ifCD@\global\bigaw@\minCDaw@\else\global\bigaw@\minaw@\fi
 \ifdim\wdz@>\bigaw@\global\bigaw@\wdz@\fi
 \ifdim\wd@ne>\bigaw@\global\bigaw@\wd@ne\fi
 \ifCD@\enskip\fi
 \ifdim\wd\tw@>\z@
  \mathrel{\mathop{\hbox to\bigaw@{\leftarrowfill@\displaystyle}}%
       \limits^{#1}_{#2}}\else
  \mathrel{\mathop{\hbox to\bigaw@{\leftarrowfill@\displaystyle}}%
       \limits^{#1}}\fi
 \ifCD@\enskip\fi\ampersand@}
\begingroup
 \catcode`\~=\active \lccode`\~=`\@
 \lowercase{%
  \global\atdef@)#1)#2){~>#1>#2>}
  \global\atdef@(#1(#2({~<#1<#2<}}
\endgroup
\atdef@ A#1A#2A{\llap{$\m@th\vcenter{\hbox
 {$\ssize#1$}}$}\Big\uparrow\rlap{$\m@th\vcenter{\hbox{$\ssize#2$}}$}&&}
\atdef@ V#1V#2V{\llap{$\m@th\vcenter{\hbox
 {$\ssize#1$}}$}\Big\downarrow\rlap{$\m@th\vcenter{\hbox{$\ssize#2$}}$}&&}
\atdef@={&\enskip\mathrel
 {\vbox{\hrule width\minCDaw@\vskip3\ex@\hrule width
 \minCDaw@}}\enskip&}
\atdef@|{\Big\Vert&&}
\atdef@\vert{\Big\Vert&&}
\def\pretend#1\haswidth#2{\setboxz@h{$\m@th\scriptstyle{#2}$}\hbox
 to\wdz@{\hfill$\m@th\scriptstyle{#1}$\hfill}}
\message{poor man's bold,}
\def\pmb{\RIfM@\expandafter\mathpalette\expandafter\pmb@\else
 \expandafter\pmb@@\fi}
\def\pmb@@#1{\leavevmode\setboxz@h{#1}%
   \dimen@-\wdz@
   \kern-.5\ex@\copy\z@
   \kern\dimen@\kern.25\ex@\raise.4\ex@\copy\z@
   \kern\dimen@\kern.25\ex@\box\z@
}
\def\binrel@@#1{\ifdim\wd2<\z@\mathbin{#1}\else\ifdim\wd\tw@>\z@
 \mathrel{#1}\else{#1}\fi\fi}
\newdimen\pmbraise@
%      Note: because of the use of \mathpalette, if \pmb@ is
%      applied to a single math italic character (or a single
%      character from some other slanted math font), the italic
%      correction will be added.  This will cause subscripts
%      to fall too far away from the character in some
%      cases, e.g., $\pmb{T}_1$ or $\pmb{\Cal T}_1$.
\def\pmb@#1#2{\setbox\thr@@\hbox{$\m@th#1{#2}$}%
 \setbox4\hbox{$\m@th#1\mkern.5mu$}\pmbraise@\wd4\relax
 \binrel@{#2}%
 \dimen@-\wd\thr@@
   \binrel@@{%
   \mkern-.8mu\copy\thr@@
   \kern\dimen@\mkern.4mu\raise\pmbraise@\copy\thr@@
   \kern\dimen@\mkern.4mu\box\thr@@
}}
\def\documentstyle#1{\W@{}\input #1.sty\relax}
\message{syntax check,}
\font\dummyft@=dummy
\fontdimen1 \dummyft@=\z@
\fontdimen2 \dummyft@=\z@
\fontdimen3 \dummyft@=\z@
\fontdimen4 \dummyft@=\z@
\fontdimen5 \dummyft@=\z@
\fontdimen6 \dummyft@=\z@
\fontdimen7 \dummyft@=\z@
\fontdimen8 \dummyft@=\z@
\fontdimen9 \dummyft@=\z@
\fontdimen10 \dummyft@=\z@
\fontdimen11 \dummyft@=\z@
\fontdimen12 \dummyft@=\z@
\fontdimen13 \dummyft@=\z@
\fontdimen14 \dummyft@=\z@
\fontdimen15 \dummyft@=\z@
\fontdimen16 \dummyft@=\z@
\fontdimen17 \dummyft@=\z@
\fontdimen18 \dummyft@=\z@
\fontdimen19 \dummyft@=\z@
\fontdimen20 \dummyft@=\z@
\fontdimen21 \dummyft@=\z@
\fontdimen22 \dummyft@=\z@
\def\fontlist@{\\{\tenrm}\\{\sevenrm}\\{\fiverm}\\{\teni}\\{\seveni}%
 \\{\fivei}\\{\tensy}\\{\sevensy}\\{\fivesy}\\{\tenex}\\{\tenbf}\\{\sevenbf}%
 \\{\fivebf}\\{\tensl}\\{\tenit}}
\def\font@#1=#2 {\rightappend@#1\to\fontlist@\font#1=#2 }
\def\dodummy@{{\def\\##1{\global\let##1\dummyft@}\fontlist@}}
\def\nopages@{\output{\setbox\z@\box\@cclv \deadcycles\z@}%
 \alloc@5\toks\toksdef\@cclvi\output}
\let\galleys\nopages@
\newif\ifsyntax@
\newcount\countxviii@
\def\syntax{\syntax@true\dodummy@\countxviii@\count18
 \loop\ifnum\countxviii@>\m@ne\textfont\countxviii@=\dummyft@
 \scriptfont\countxviii@=\dummyft@\scriptscriptfont\countxviii@=\dummyft@
 \advance\countxviii@\m@ne\repeat                                           %1
 \dummyft@\tracinglostchars\z@\nopages@\frenchspacing\hbadness\@M}
\def\first@#1#2\end{#1}
\def\printoptions{\W@{Do you want S(yntax check),
  G(alleys) or P(ages)?}%
 \message{Type S, G or P, followed by <return>: }%
 \begingroup % to localize the following change to \endlinechar:
 \endlinechar\m@ne % to prevent a space or \par in \ans@ from ^^M
 \read\m@ne to\ans@
%  Define \ans@ to uppercase itself, and default to P if the user
%  just pressed <return>.
 \edef\ans@{\uppercase{\def\noexpand\ans@{%
   \expandafter\first@\ans@ P\end}}}%
%  Cast the new definition of \ans@ outside the group
 \expandafter\endgroup\ans@
 \if\ans@ P% fine, no action needs to be taken
 \else \if\ans@ S\syntax
 \else \if\ans@ G\galleys
 \else\message{? Unknown option: \ans@; using the `pages' option.}%
 \fi\fi\fi}
\def\alloc@#1#2#3#4#5{\global\advance\count1#1by\@ne
 \ch@ck#1#4#2\allocationnumber=\count1#1
 \global#3#5=\allocationnumber
 \ifalloc@\wlog{\string#5=\string#2\the\allocationnumber}\fi}
\def\document{\def\alloclist@{}\def\fontlist@{}}
\let\enddocument\bye
\let\plainproclaim\proclaim
\let\plainfootnote\footnote
\let\proclaim\undefined
\let\footnote\undefined
\let\=\undefined
\let\>\undefined
\let\cal\undefined
\let\mit\undefined
\let\oldstyle\undefined
\catcode`\@=\active
\message{... finished}

%  *** end including amstex.tex *** 
% % \input mathdefs
%  *** start including mathdefs.tex *** 
\expandafter\ifx\csname mathdefs.tex\endcsname\relax
  \expandafter\gdef\csname mathdefs.tex\endcsname{}
\else \message{Hey!  Apparently you were trying to
  \string\input{mathdefs.tex} twice.   This does not make sense.} 
\errmessage{Please edit your file (probably \jobname.tex) and remove
any duplicate ``\string\input'' lines}\endinput\fi


%mathdefs.tex v1.3.2

%%% Changes from v1.0: footnote macros, warning for duplicated tags,
%%%   control sequences \( and \verbatimtags.
%%% From v1.2: \pretags, redefinition of \( using \ifinner, multi-part
%%%   equation numbering, control sequences \[, \references, and
%%%   \resetbracket. 
%%% From v1.3: \rm in \lastpart; write root of multi-part tag to .tgs 

%See file texdefs.doc for documentation.

\catcode`\X=12\catcode`\@=11

%Minor control sequences:
\def\n@wcount{\alloc@0\count\countdef\insc@unt}
\def\n@wwrite{\alloc@7\write\chardef\sixt@@n}
\def\n@wread{\alloc@6\read\chardef\sixt@@n}
\def\r@s@t{\relax}\def\v@idline{\par}\def\@mputate#1/{#1}
\def\l@c@l#1X{\firstpart.#1}\def\gl@b@l#1X{#1}\def\t@d@l#1X{{}}

%Creation of tag families and output of assignments and citations:
\def\crossrefs#1{\ifx\all#1\let\tr@ce=\all\else\def\tr@ce{#1,}\fi
   \n@wwrite\cit@tionsout\openout\cit@tionsout=\jobname.cit 
   \write\cit@tionsout{\tr@ce}\expandafter\setfl@gs\tr@ce,}
\def\setfl@gs#1,{\def\@{#1}\ifx\@\empty\let\next=\relax
   \else\let\next=\setfl@gs\expandafter\xdef
   \csname#1tr@cetrue\endcsname{}\fi\next}
\def\m@ketag#1#2{\expandafter\n@wcount\csname#2tagno\endcsname
     \csname#2tagno\endcsname=0\let\tail=\all\xdef\all{\tail#2,}
   \ifx#1\l@c@l\let\tail=\r@s@t\xdef\r@s@t{\csname#2tagno\endcsname=0\tail}\fi
   \expandafter\gdef\csname#2cite\endcsname##1{\expandafter
     \ifx\csname#2tag##1\endcsname\relax?\else\csname#2tag##1\endcsname\fi
     \expandafter\ifx\csname#2tr@cetrue\endcsname\relax\else
     \write\cit@tionsout{#2tag ##1 cited on page \folio.}\fi}
   \expandafter\gdef\csname#2page\endcsname##1{\expandafter
     \ifx\csname#2page##1\endcsname\relax?\else\csname#2page##1\endcsname\fi
     \expandafter\ifx\csname#2tr@cetrue\endcsname\relax\else
     \write\cit@tionsout{#2tag ##1 cited on page \folio.}\fi}
   \expandafter\gdef\csname#2tag\endcsname##1{\expandafter
      \ifx\csname#2check##1\endcsname\relax
      \expandafter\xdef\csname#2check##1\endcsname{}%
      \else\immediate\write16{Warning: #2tag ##1 used more than once.}\fi
      \multit@g{#1}{#2}##1/X%
      \write\t@gsout{#2tag ##1 assigned number \csname#2tag##1\endcsname\space
      on page \number\count0.}%
   \csname#2tag##1\endcsname}}


\def\multit@g#1#2#3/#4X{\def\t@mp{#4}\ifx\t@mp\empty%
      \global\advance\csname#2tagno\endcsname by 1 
      \expandafter\xdef\csname#2tag#3\endcsname
      {#1\number\csname#2tagno\endcsnameX}%
   \else\expandafter\ifx\csname#2last#3\endcsname\relax
      \expandafter\n@wcount\csname#2last#3\endcsname
      \global\advance\csname#2tagno\endcsname by 1 
      \expandafter\xdef\csname#2tag#3\endcsname
      {#1\number\csname#2tagno\endcsnameX}
      \write\t@gsout{#2tag #3 assigned number \csname#2tag#3\endcsname\space
      on page \number\count0.}\fi
   \global\advance\csname#2last#3\endcsname by 1
   \def\t@mp{\expandafter\xdef\csname#2tag#3/}%
   \expandafter\t@mp\@mputate#4\endcsname
   {\csname#2tag#3\endcsname\lastpart{\csname#2last#3\endcsname}}\fi}
\def\t@gs#1{\def\all{}\m@ketag#1e\m@ketag#1s\m@ketag\t@d@l p
\let\realscite\scite
\let\realstag\stag
   \m@ketag\gl@b@l r \n@wread\t@gsin
   \openin\t@gsin=\jobname.tgs \re@der \closein\t@gsin
   \n@wwrite\t@gsout\openout\t@gsout=\jobname.tgs }
\outer\def\localtags{\t@gs\l@c@l}
\outer\def\globaltags{\t@gs\gl@b@l}
\outer\def\newlocaltag#1{\m@ketag\l@c@l{#1}}
\outer\def\newglobaltag#1{\m@ketag\gl@b@l{#1}}

%Reading in tag information:
\newif\ifpr@ 
\def\m@kecs #1tag #2 assigned number #3 on page #4.%
   {\expandafter\gdef\csname#1tag#2\endcsname{#3}
   \expandafter\gdef\csname#1page#2\endcsname{#4}
   \ifpr@\expandafter\xdef\csname#1check#2\endcsname{}\fi}
\def\re@der{\ifeof\t@gsin\let\next=\relax\else
   \read\t@gsin to\t@gline\ifx\t@gline\v@idline\else
   \expandafter\m@kecs \t@gline\fi\let \next=\re@der\fi\next}
\def\pretags#1{\pr@true\pret@gs#1,,}
\def\pret@gs#1,{\def\@{#1}\ifx\@\empty\let\n@xtfile=\relax
   \else\let\n@xtfile=\pret@gs \openin\t@gsin=#1.tgs \message{#1} \re@der 
   \closein\t@gsin\fi \n@xtfile}

%Sections and subsections; local numbering:
\newcount\sectno\sectno=0\newcount\subsectno\subsectno=0
\newif\ifultr@local \def\ultralocal{\ultr@localtrue}
\def\firstpart{\number\sectno}
\def\lastpart#1{\ifcase#1 \or a\or b\or c\or d\or e\or f\or g\or h\or 
   i\or k\or l\or m\or n\or o\or p\or q\or r\or s\or t\or u\or v\or w\or 
   x\or y\or z \fi}
\def\closeup{\vskip-\bigskipamount}
\def\resetall{\global\advance\sectno by 1\subsectno=0
   \gdef\firstpart{\number\sectno}\r@s@t}
\def\resetsub{\global\advance\subsectno by 1
   \gdef\firstpart{\number\sectno.\number\subsectno}\r@s@t}
\def\newsection#1\par{\resetall\vskip0pt plus.3\vsize\penalty-250
   \vskip0pt plus-.3\vsize\bigskip\bigskip
   \message{#1}\leftline{\bf#1}\nobreak\bigskip}
\def\subsection#1\par{\ifultr@local\resetsub\fi
   \vskip0pt plus.2\vsize\penalty-250\vskip0pt plus-.2\vsize
   \bigskip\smallskip\message{#1}\leftline{\bf#1}\nobreak\medskip}

%jj tags:
% On Andrzej's request:  we want to be able 
% to show tags as in noverbatim, with verbatim in the margin,
% and cites as in verbatim, with nonverbatim in the margin
% mg -- July 2000

\newdimen\marginshift

\newdimen\margindelta
\newdimen\marginmax
\newdimen\marginmin

\def\margininit{       
\marginmax=3 true cm                  % how much room, approximately
				      
\margindelta=0.1 true cm              % distance between entries
\marginmin=0.1true cm                 % where will leftmost entry be
\marginshift=\marginmin
}    % we cannot execute this right now, since 
     % there may be a \magnification coming later in the 
     % main file.   So we call \margininit at the end of 
     % alice2jlem

\def\t@gsjj#1,{\def\@{#1}\ifx\@\empty\let\next=\relax\else\let\next=\t@gsjj
   \def\@@{p}\ifx\@\@@\else
   \expandafter\gdef\csname#1cite\endcsname##1{\citejj{##1}}
   \expandafter\gdef\csname#1page\endcsname##1{?}
   \expandafter\gdef\csname#1tag\endcsname##1{\tagjj{##1}}\fi\fi\next}
\newif\ifshowstuffinmargin
\showstuffinmarginfalse
\def\jjtags{\ifx\shlhetal\relax 
      % so this is a public version --> no-op 
  \else
\ifx\shlhetal\undefinedcontrolseq
      % again, this is a public version --> no-op 
\else
\showstuffinmargintrue
\ifx\all\relax\else\expandafter\t@gsjj\all,\fi\fi \fi
}

% \def\zeigentag#1{\def\xxy{#1}\show\xxy\show\realstag\realstag#1\mginpar{\hbox{$\langle$}#1\hbox{$\rangle$}}}
% \def\zeigencite#1{\hbox{$\langle$}#1\hbox{$\rangle$}\mginpar{\realscite#1}}


\def\tagjj#1{\realstag{#1}\mginpar{\zeigen{#1}}}
% \def\citejj#1{\zeigen{#1}\mginpar{\rechnen{#1}}}
\def\citejj#1{\rechnen{#1}\mginpar{\zeigen{#1}}}     % modified Sep 02, saharon's suggestion

% \def\rechnen#1{rechnen(\expandafter\ifx\csname stag#1\endcsname\relax ??\else
%                            \csname stag#1\endcsname\fi)endrech }
\def\rechnen#1{\expandafter\ifx\csname stag#1\endcsname\relax ??\else
                           \csname stag#1\endcsname\fi}

% \def\zeigentag#1{//#1--\csname stag#1\endcsname::}

\newdimen\theight



\def\marginfont{\sevenrm}



\def\trymarginbox#1{\setbox0=\hbox{\marginfont\hskip\marginshift #1}%
		\global\marginshift\wd0 
		\global\advance\marginshift\margindelta}


\def \mginpar#1{%
\ifvmode\setbox0\hbox to \hsize{\hfill\rlap{\marginfont\quad#1}}%
\ht0 0cm
\dp0 0cm
\box0\vskip-\baselineskip
\else 
             \vadjust{\trymarginbox{#1}%
		\ifdim\marginshift>\marginmax \global\marginshift\marginmin
			\trymarginbox{#1}%
                \fi
             \theight=\ht0
             \advance\theight by \dp0    \advance\theight by \lineskip
             \kern -\theight \vbox to \theight{\rightline{\rlap{\box0}}%
\vss}}\fi}


% \def\mginpar#1{mg-#1-mg }



%Verbatim tags:
\def\t@gsoff#1,{\def\@{#1}\ifx\@\empty\let\next=\relax\else\let\next=\t@gsoff
   \def\@@{p}\ifx\@\@@\else
   \expandafter\gdef\csname#1cite\endcsname##1{\zeigen{##1}}
   \expandafter\gdef\csname#1page\endcsname##1{?}
   \expandafter\gdef\csname#1tag\endcsname##1{\zeigen{##1}}\fi\fi\next}
\def\verbatimtags{\showstuffinmarginfalse
\ifx\all\relax\else\expandafter\t@gsoff\all,\fi}
%%% \def\zeigen#1{zeigen(\hbox{$\langle$}#1\hbox{$\rangle$})endz }
\def\zeigen#1{\hbox{$\langle$}#1\hbox{$\rangle$}}
\def\margincite#1{\ifshowstuffinmargin\mginpar{\rechnen{#1}}\fi}
\def\margintag#1{\ifshowstuffinmargin\mginpar{\zeigen{#1}}\fi}

%Equation numbering:
\def\(#1){\edef\dot@g{\ifmmode\ifinner(\hbox{\noexpand\etag{#1}})
   \else\noexpand\eqno(\hbox{\noexpand\etag{#1}})\fi
   \else(\noexpand\ecite{#1})\fi}\dot@g}

%Reference numbering:
\newif\ifbr@ck
\def\eat#1{}
\def\[#1]{\br@cktrue[\br@cket#1'X]}
\def\br@cket#1'#2X{\def\temp{#2}\ifx\temp\empty\let\next\eat
   \else\let\next\br@cket\fi
   \ifbr@ck\br@ckfalse\br@ck@t#1,X\else\br@cktrue#1\fi\next#2X}
\def\br@ck@t#1,#2X{\def\temp{#2}\ifx\temp\empty\let\neext\eat
   \else\let\neext\br@ck@t\def\temp{,}\fi
   \def\teemp{#1}\ifx\teemp\empty\else\rcite{#1}\fi\temp\neext#2X}
\def\resetbr@cket{\gdef\[##1]{[\rtag{##1}]}}
\def\references{\resetbr@cket\newsection References\par}

%Footnotes:
\newtoks\symb@ls\newtoks\s@mb@ls\newtoks\p@gelist\n@wcount\ftn@mber
    \ftn@mber=1\newif\ifftn@mbers\ftn@mbersfalse\newif\ifbyp@ge\byp@gefalse
\def\defm@rk{\ifftn@mbers\n@mberm@rk\else\symb@lm@rk\fi}
\def\n@mberm@rk{\xdef\m@rk{{\the\ftn@mber}}%
    \global\advance\ftn@mber by 1 }
\def\rot@te#1{\let\temp=#1\global#1=\expandafter\r@t@te\the\temp,X}
\def\r@t@te#1,#2X{{#2#1}\xdef\m@rk{{#1}}}
\def\b@@st#1{{$^{#1}$}}\def\str@p#1{#1}
\def\symb@lm@rk{\ifbyp@ge\rot@te\p@gelist\ifnum\expandafter\str@p\m@rk=1 
    \s@mb@ls=\symb@ls\fi\write\f@nsout{\number\count0}\fi \rot@te\s@mb@ls}
\def\byp@ge{\byp@getrue\n@wwrite\f@nsin\openin\f@nsin=\jobname.fns 
    \n@wcount\currentp@ge\currentp@ge=0\p@gelist={0}
    \re@dfns\closein\f@nsin\rot@te\p@gelist
    \n@wread\f@nsout\openout\f@nsout=\jobname.fns }
\def\m@kelist#1X#2{{#1,#2}}
\def\re@dfns{\ifeof\f@nsin\let\next=\relax\else\read\f@nsin to \f@nline
    \ifx\f@nline\v@idline\else\let\t@mplist=\p@gelist
    \ifnum\currentp@ge=\f@nline
    \global\p@gelist=\expandafter\m@kelist\the\t@mplistX0
    \else\currentp@ge=\f@nline
    \global\p@gelist=\expandafter\m@kelist\the\t@mplistX1\fi\fi
    \let\next=\re@dfns\fi\next}
\def\symbols#1{\symb@ls={#1}\s@mb@ls=\symb@ls} 
\def\bigsymbol{\textstyle}
\symbols{\bigsymbol\ast,\dagger,\ddagger,\sharp,\flat,\natural,\star}
\def\ftnumbers{\ftn@mberstrue} \def\ftsymbols{\ftn@mbersfalse}
\def\paginal{\byp@ge} \def\resetftnumbers{\ftn@mber=1}
\def\ftnote#1{\defm@rk\expandafter\expandafter\expandafter\footnote
    \expandafter\b@@st\m@rk{#1}}

%Miscellaneous macros:
\long\def\jump#1\endjump{}
\def\ssum{\mathop{\lower .1em\hbox{$\textstyle\Sigma$}}\nolimits}
\def\down#1{_{{}_{\scriptstyle #1}}}
\def\qed{\nobreak\kern 1em \vrule height .5em width .5em depth 0em}
\def\newneq{\hbox{\rlap{\hbox to 1\wd9{\hss$=$\hss}}\raise .1em 
   \hbox to 1\wd9{\hss$\scriptscriptstyle/$\hss}}}
\def\subsetne{\setbox9 = \hbox{$\subset$}\mathrel{\hbox{\rlap
   {\lower .4em \newneq}\raise .13em \hbox{$\subset$}}}}
\def\supsetne{\setbox9 = \hbox{$\subset$}\mathrel{\hbox{\rlap
   {\lower .4em \newneq}\raise .13em \hbox{$\supset$}}}}

%Blackboard bold:
\def\vbar{\mathchoice{\vrule height6.3ptdepth-.5ptwidth.8pt\kern-.8pt}
   {\vrule height6.3ptdepth-.5ptwidth.8pt\kern-.8pt}
   {\vrule height4.1ptdepth-.35ptwidth.6pt\kern-.6pt}
   {\vrule height3.1ptdepth-.25ptwidth.5pt\kern-.5pt}}
\def\f@dge{\mathchoice{}{}{\mkern.5mu}{\mkern.8mu}}
\def\b@c#1#2{{\rm \mkern#2mu\vbar\mkern-#2mu#1}}
\def\b@b#1{{\rm I\mkern-3.5mu #1}}
\def\b@a#1#2{{\rm #1\mkern-#2mu\f@dge #1}}
\def\bb#1{{\count4=`#1 \advance\count4by-64 \ifcase\count4\or\b@a A{11.5}\or
   \b@b B\or\b@c C{5}\or\b@b D\or\b@b E\or\b@b F \or\b@c G{5}\or\b@b H\or
   \b@b I\or\b@c J{3}\or\b@b K\or\b@b L \or\b@b M\or\b@b N\or\b@c O{5} \or
   \b@b P\or\b@c Q{5}\or\b@b R\or\b@a S{8}\or\b@a T{10.5}\or\b@c U{5}\or
   \b@a V{12}\or\b@a W{16.5}\or\b@a X{11}\or\b@a Y{11.7}\or\b@a Z{7.5}\fi}}

\catcode`\X=11 \catcode`\@=12

%  *** end including mathdefs.tex *** 
% % \input citeadd
%  *** start including citeadd.tex *** 
%   citeadd -- a few additions for 
% files from alice that were procesed with "citealice"

\expandafter\ifx\csname citeadd.tex\endcsname\relax
\expandafter\gdef\csname citeadd.tex\endcsname{}
\else \message{Hey!  Apparently you were trying to
\string\input{citeadd.tex} twice.   This does not make sense.} 
\errmessage{Please edit your file (probably \jobname.tex) and remove
any duplicate ``\string\input'' lines}\endinput\fi





\def\sciteu{\sciteerror{undefined}}
\def\sciteuphantom{\complainaboutcitation{undefined}}

\def\scitet{\sciteerror{ambiguous}}
\def\scitetphantom{\complainaboutcitation{ambiguous}}




\def\sciteerror#1#2{{\mathortextbf{\scite{#2}}}\complainaboutcitation{#1}{#2}}
\def\mathortextbf#1{\hbox{\bf #1}}
\def\complainaboutcitation#1#2{%
\vadjust{\line{\llap{---$\!\!>$ }\qquad scite$\{$#2$\}$ #1\hfil}}}





%  *** end including citeadd.tex *** 
\sectno=-1   % start with sect 0
\localtags
\jjtags
\NoBlackBoxes
\define\mr{\medskip\roster}
\define\sn{\smallskip\noindent}
\define\mn{\medskip\noindent}
\define\bn{\bigskip\noindent}
\define\ub{\underbar}
\define\wilog{\text{without loss of generality}}
\define\ermn{\endroster\medskip\noindent}
\define\dbca{\dsize\bigcap}
\define\dbcu{\dsize\bigcup}
\define \nl{\newline}
\magnification=\magstep 1
\documentstyle{amsppt}
% % \input alice2000
%  *** start including alice2000.tex *** 
% This file should be inputted whenever we use amsppt.sty and 
% old tex.  
%  Here we redefine \subjclass (use 1991 instead of 2000, otherwise 
% the following definition comes directly from 
%% 
%%              `amsppt.sty', generated 
%% on <1997/2/2> with the docstrip utility (2.2i).
%% 
%% The original source files were:
%% 
%% amsppt.doc 
%%% ====================================================================
%%% @AMSTeX-style-file{
%%%   filename  = "amsppt.sty",
%%%   version   = "2.1h",
%%%   date      = "1997/02/02",
%%%   time      = "09:27:44 EST",
%%%   checksum  = "56844 3264 16617 137829",
%%%   author    = "American Mathematical Society",
%%%   address   = "PO Box 6248, Providence, RI 02940-6248, USA",
%%%   telephone = "401-455-4080 or (in the USA) 800-321-4AMS",

{    % the braces make the catcode-change local. 
\catcode`@11

\ifx\alicetwothousandloaded@\relax
  \endinput\else\global\let\alicetwothousandloaded@\relax\fi


\gdef\subjclass{\let\savedef@\subjclass
 \def\subjclass##1\endsubjclass{\let\subjclass\savedef@
   \toks@{\def\usualspace{{\rm\enspace}}\eightpoint}%
   \toks@@{##1\unskip.}%
   \edef\thesubjclass@{\the\toks@
     \frills@{{\noexpand\rm2000 {\noexpand\it Mathematics Subject
       Classification}.\noexpand\enspace}}%
     \the\toks@@}}%
  \nofrillscheck\subjclass}
} 

%  *** end including alice2000.tex *** 
% % \input alice2jlem
%  *** start including alice2jlem.tex *** 
%% # Keywords  Input file to be used for texing Alice's files

\expandafter\ifx\csname alice2jlem.tex\endcsname\relax
  \expandafter\xdef\csname alice2jlem.tex\endcsname{\the\catcode`@}
\else \message{Hey!  Apparently you were trying to
\string\input{alice2jlem.tex}  twice.   This does not make sense.}
\errmessage{Please edit your file (probably \jobname.tex) and remove
any duplicate ``\string\input'' lines}\endinput\fi



% % \input bib4plain
%  *** start including bib4plain.tex *** 
\expandafter\ifx\csname bib4plain.tex\endcsname\relax
  \expandafter\gdef\csname bib4plain.tex\endcsname{}
\else \message{Hey!  Apparently you were trying to \string\input
  bib4plain.tex twice.   This does not make sense.}
\errmessage{Please edit your file (probably \jobname.tex) and remove
any duplicate ``\string\input'' lines}\endinput\fi


%  This file should be inputted if you want to use 
%  bibtex fom within plain TeX. 
\def\makeatletter{\catcode`\@11 }      % Not really need for standard
\def\makeatother{\catcode`\@12 }       % bibtex files, but these commands
\def\renewcommand{\newcommand}	       % are used in our literal-unsrt.bst
				       % and other files. 
%  The above lines were written by Martin.Goldstern@tuwien.ac.at
%  The rest is just the file btxmac.tex. 
%  (except for one small change below, regarding undefined citations)

%% @texfile{
%%   author = "Karl Berry and Oren Patashnik",
%%   version = "0.99j",
%%   date = "14 Mar 1992",
%%   filename = "btxmac.tex",
%%   address = "Please use electronic mail",
%%   checksum = "834    4503   33061",
%%   email = "opbibtex@cs.stanford.edu",
%%   codetable = "ISO/ASCII",
%%   supported = "yes",
%%   docstring = "Defines macros that make BibTeX work with plain TeX",
%% }
% BibTeX-for-TeX macros, version 0.99j, for BibTeX 0.99c, TeX 3.0 or later.
% Copyright (C) 1990--92 by Karl Berry and Oren Patashnik; all rights reserved.
% You may copy this file provided: that it's accompanied by the
% "BibTeXing" document, whose text is contained in the file `btxdoc.tex';
% that any documentation you write for these macros also gives a
% reference for "BibTeXing"; and that either you make absolutely no
% changes to your copy, or if you do make changes, (1) you name the file
% something other than `btxmac.tex' and you remove all occurrences of
% `btxmac.tex' from the file, (2) you put, somewhere in the first twenty
% lines of the file, your name, along with an electronic address at which
% others who might use the file may reach you, and (3) you remove each
% occurrence of Oren's name and electronic address from this file.  These
% restrictions help ensure that all standard versions of these macros are
% identical, and that Oren doesn't get deluged with inappropriate e-mail.
%
% This file, btxmac.tex, contains TeX macros that allow BibTeX, a
% bibliography program that was originally designed for use with LaTeX,
% to work with plain TeX.  Please report any bugs (outright goofs,
% improvable macros, misfeatures, or unclear documentation) to Oren
% Patashnik (opbibtex@cs.stanford.edu).  These macros will become frozen
% shortly after BibTeX version 1.00 is released.
%
% AMS-TEX WARNING: We tried very hard, for version .99i of these macros,
% to make them compatible with AmS-TeX.  We succeeded to the extent
% that, if you use one of the standard bibliography styles, you probably
% won't notice any problems with version 0.99i of btxmac.tex.  But
% ultimately we failed, in that the inherent incompatibilities between
% plain TeX and AmS-TeX kept making these macros break, for certain
% inputs or certain styles.  Examples:  (1) AmS-TeX treats at-signs as
% special, in ways that plain TeX and LaTeX don't, so that, for example,
% you can't have any `@' characters in an argument to the \cite command,
% the way you can in TeX or LaTeX; (2) AmS-TeX decided that plain TeX's
% and LaTeX's macron-accent control sequence `\=' should be undefined;
% so you'll need to define `\=' to be `\B' to get the xampl.bib example
% suggested below to work with AmS-TeX; (3) AmS-TeX redefines the tie
% character `~' of plain TeX, and AmS-TeX's `amsppt' style redefines
% plain TeX's `\nobreak' macro, so that if you use an author-date style
% like `apalike' and you have a multiple-author reference for which the
% author-date style automatically produces a citation in the text like
% `(Jones et~al., 1992)' you will throw AmS-TeX's `amsppt' style into
% an infinite loop, exceeding its input stack size.  In practice, such
% incompatibilities surface infrequently; but it is now clear to us that
% it's not worth the effort (perhaps it's not even possible) to make the
% btxmac.tex macros robust when used both with plain TeX and Ams-TeX.
% If the BibTeX/AmS-TeX results attainable with the current btxmac.tex
% macros are sufficient, fine.  But if there's a demand for more robust
% BibTeX/AmS-TeX behavior, then someone who's very familiar with the
% AmS-TeX package should probably make an amsbtxmc.tex version of the
% macros (remembering to follow the copyright restrictions above).
% Until then, if you're an AmS-TeX user, or a LaTeX or plain TeX user
% sharing files with an AmS-TeX user, beware.
% END OF AMS-TEX WARNING.
%
% To use these macros you should be familiar with how BibTeX interacts
% with LaTeX, since BibTeX's interaction with TeX is very similar; that
% interaction is explained in the LaTeX manual.  It also helps to
% have read "BibTeXing", the documentation that accompanies BibTeX.
% Then, if you want, you should redefine any of the macros that begin
% with `\bbl' or `\biblabel' or `\print' that you need to get formatting
% different from the default (the default settings are designed to
% accompany a bibliography style like BibTeX's standard style `plain').
% The macros you might want to change are described briefly a few
% paragraphs hence.  [To get started without reading any documentation,
% try running the nine-line .tex file below through TeX and BibTeX.
% Remember the general scheme: Running (La)TeX writes information on
% the .aux (auxiliary) file; then running BibTeX reads information from
% the .aux, .bst (style), and .bib (database) files, and writes
% information (the bibliography) on a .bbl file; then running (La)TeX
% incorporates the bibliography; then running (La)TeX once more fixes
% the remaining forward references into the bibliography.  Thus, to get
% everything incorporated into your output, you'll have to run (La)TeX,
% BibTeX, (La)TeX, (La)TeX.  (Standup, sitdown, fight, fight, fight.)]
%
% These macros can stand alone or they can be \input into a macro
% package, like Eplain, that is sufficiently compatible with plain TeX.
% To use these macros to format the 0.99 version of the xampl.bib file
% that's distributed with BibTeX (that version of the file has no
% self-identification), you'll need to define \mbox, which is a LaTeX
% command, to be \hbox, as in the example below.
%
% Here's a nine-line plain TeX file for trying out btxmac.tex; of course
% you'll have to remove the comment characters at the beginning of each
% line, and, depending on your system, you might have to take steps so
% that BibTeX can "see" the files xampl.bib and plain.bst (BibTeX will
% give you two empty-field warning messages that you should ignore).
%
%     \def\mbox#1{\leavevmode\hbox{#1}}
%     \input btxmac
%     \noindent This cites Aamport's gnominious article~\cite{article-full}.
%     \medskip
%     \leftline{\bf References}
%     \nocite{*}   % put all database entries into the reference list
%     \bibliography{xampl}   % specify the database files; here, just xampl.bib
%     \bibliographystyle{plain}   % specify plain.bst as the style file
%     \bye
%
%
%   HISTORY
%
% Karl Berry wrote the original version of these macros in 1989 and
% 1990, for use in his `Eplain' package.  Oren Patashnik modified them
% slightly in July 1990, as part of the official BibTeX distribution.
%
%    1-Aug-90  Version 0.99a, not released to the general public.
%   14-Aug-90  0.99b, first general release.
%   26-Aug-90  0.99c, made \@undefinedmessage work with other macro packages.
%    6-Sep-90  0.99d, allowed for general formatting of bibliography labels,
%                     for general formatting of (in-text) citations, and for
%                     changing certain catcodes while reading the .aux file.
%   14-Nov-90  0.99e, changed the way \@setletters works, made some \new...'s
%                     non-outer, and changed the way Eplain reads this file.
%   12-Dec-90  0.99f, made \@resetnumerals change the `,' and `.' catcodes; and
%                     added \biblabelextrahang, \@getoptionalarg, and \bblsc.
%   11-Mar-91  0.99g, made a few minor changes required by the way Eplain reads
%                     this file, but no functional changes.
%   24-Apr-91  0.99h, inhibited the reading and writing of the .aux file if it
%                     isn't used or if the \noauxfile macro is defined, and
%                     removed some .aux-file-opening detritus; printed the
%                     cite-key of undefined citations in \tt font; changed the
%                     catcode of `_' inside \cite; and called \@resetnumerals
%                     from inside a group.
%   29-Feb-92  0.99i, made these macros semi-compatible with AmS-TeX; removed
%                     \@resetnumerals, \@setletters, \@tokstostring, and
%                     friends; changed the way \cite handles catcodes; changed
%                     \@getoptionalarg, and had \bibitem and \newcommand use
%                     it; added \@futurenonspacelet and (to facilitate the use
%                     of multiple reference lists) \bblfilebasename; changed
%                     \biblabelprint to use the new macros \biblabelprecontents
%                     and \biblabelpostcontents, and to, by default, right-
%                     justify numeric labels; and renamed \biblabelextrahang to
%                     the more descriptive \biblabelextraspace.
%   14-Mar-92  0.99j, made 0.99i's use of `\\' local to btxmac.tex.
%
%
% The LaTeX-related commands defined in this file include (a) the four
% commands that a user types (\bibliography, \bibliographystyle, \cite,
% and \nocite); (b) the three commands that BibTeX looks for in the .aux
% file (\bibdata, \bibstyle, and \citation---there is a fourth command
% that BibTeX looks for, but that command is related to LaTeX's \include
% facility, so these macros ignore that command); and (c) a LaTeX
% command (\newcommand) that's written by one of the four standard
% bibliography styles (alpha).  The definitions here are not exactly the
% same as the corresponding LaTeX definitions (those eight LaTeX
% definitions depend on a significant fraction of LaTeX itself).  But
% the only substantial differences are with \newcommand, which here,
% without complaining, lets you redefine a preexisting control sequence
% (in LaTeX, \newcommand won't let you redefine a preexisting command),
% and which here doesn't make the control sequences it defines \long (in
% LaTeX, that happens automatically); there may also be other minor
% differences.  To summarize: Unless you know what you're doing, you
% shouldn't define any control sequences with these eight names:
%
% \bibdata
% \bibliography
% \bibliographystyle
% \bibstyle
% \citation
% \cite
% \newcommand
% \nocite
%
% There are three other commands written by one or more of the four
% standard (plain, abbrv, alpha, unsrt) or four semistandard (acm,
% apalike, ieeetr, siam) bibliography styles; they take effect only
% within the bibliography, and are redefinable, as explained later:
%
% \em
% \newblock
% \sc
%
% There's one control sequence you might want to use (but not redefine)
% in redefining \biblabelprint:
%
% \biblabelwidth
%
% There are fifteen other control sequences (explained later in more detail)
% that the macros of this file will use if you define them---you should
% define them after the \input btxmac command but before the \bibliography
% command.  The first six begin with `\bbl' and affect fonts, spacing,
% perhaps other characteristics of the bibliography, and which .bbl files
% get read; the next five begin with `\biblabel' and determine how labels
% are formatted in the bibliography; and the last four begin with `\print'
% and determine how the in-text citations are formatted:
%
% \bblem
% \bblfilebasename
% \bblhook
% \bblnewblock
% \bblrm
% \bblsc
% \biblabelcontents
% \biblabelprecontents
% \biblabelprint
% \biblabelpostcontents
% \biblabelextraspace
% \printbetweencitations
% \printcitefinish
% \printcitenote
% \printcitestart
%
% If it's defined before the \input btxmac command, the control sequence
% below inhibits the reading and writing of the .aux file(s), and the
% issuing of related warning messages.  Any definition will do.  This
% feature might help when you're working on draft stages of a document:
%
% \noauxfile
%
%
% Here's another control sequence (it's described later) that you
% probably won't want to redefine unless you are writing another macro
% package; if you do redefine it, however, do it before the \input btxmac
% command (and notice that it has an `@' in its name):
%
% \@undefinedmessage
%
% Any other control sequence in this file that might conflict with
% something you've defined will have an `@' in its name, so such conflicts
% are unlikely; but if you're worried about a specific control sequence
% name, do a text search of this file to look for it.
%
%
% So to start things off we turn `@' into a letter (category code 11),
% keeping track of the old category code for future restoration.
% (Simply resetting it to 12 when we leave these macros is
% insufficient.)  The use of `\cite' as a temporary control sequence is
% a kludge, but it's a reasonably simple way to accomplish what we need
% without possibly overwriting something (without an `@' in its name)
% that might already be defined.
%
\edef\cite{\the\catcode`@}%
\catcode`@ = 11
\let\@oldatcatcode = \cite
\chardef\@letter = 11
\chardef\@other = 12
%
%
% Next come some things that will be useful later.
%
% Make an outer definition into an inner one (due to Chris Thompson).
% The arguments should be the control sequence to be defined, and the
% new of the \outer control sequence, as characters; the control
% sequence #1 is defined to be just the same as \csname#2\endcsname, but
% not \outer.  For example, \@innerdef\innernewcount{newcount} would
% define \innernewcount to be a non-outer version of \newcount.
%
\def\@innerdef#1#2{\edef#1{\expandafter\noexpand\csname #2\endcsname}}%
%
% We use \@innerdef to make some of our allocations local, because
% Eplain includes our code inside a conditional.  We put @'s in the
% names to minimize the (already small) chance of conflicts.
%
\@innerdef\@innernewcount{newcount}%
\@innerdef\@innernewdimen{newdimen}%
\@innerdef\@innernewif{newif}%
\@innerdef\@innernewwrite{newwrite}%
%
%
% Swallow one parameter.
%
\def\@gobble#1{}%
%
%
% Use TeX 3.0's \inputlineno to get the line number, for better error
% messages, but if we're using an old version of TeX, don't do anything.
%
\ifx\inputlineno\@undefined
   \let\@linenumber = \empty % Pre-3.0.
\else
   \def\@linenumber{\the\inputlineno:\space}%
\fi
%
%
% The following macro \@futurenonspacelet (from the TeXbook) behaves
% essentially like \futurelet except that it discards any implicit or
% explicit space tokens that intervene before a nonspace is scanned:
%
\def\@futurenonspacelet#1{\def\cs{#1}%
   \afterassignment\@stepone\let\@nexttoken=
}%
\begingroup % The grouping here avoids stepping on an outside use of `\\'.
\def\\{\global\let\@stoken= }%
\\ % now \@stoken is a space token (\\ is a control symbol, so that
   % space after it is seen).
\endgroup
\def\@stepone{\expandafter\futurelet\cs\@steptwo}%
\def\@steptwo{\expandafter\ifx\cs\@stoken\let\@@next=\@stepthree
   \else\let\@@next=\@nexttoken\fi \@@next}%
\def\@stepthree{\afterassignment\@stepone\let\@@next= }%
%
%
% \@getoptionalarg\CS gets an optional argument from the input, enclosed
% in brackets, then expands \CS.  We set \@optionalarg to \empty if we
% don't find one, otherwise to the text of the argument.  This assumes
% the brackets don't have a funny category code.
%
\def\@getoptionalarg#1{%
   \let\@optionaltemp = #1%
   \let\@optionalnext = \relax
   \@futurenonspacelet\@optionalnext\@bracketcheck
}%
%
% The \expandafter's in this macro let us avoid the use of \aftergroup,
% which is somewhat more expensive.
%
\def\@bracketcheck{%
   \ifx [\@optionalnext
      \expandafter\@@getoptionalarg
   \else
      \let\@optionalarg = \empty
      % We can't do the \temp after the \fi, because then the \temp gets
      % in the way of reading the optional argument from the input, if
      % we do have one.
      \expandafter\@optionaltemp
   \fi
}%
%
\def\@@getoptionalarg[#1]{%
   \def\@optionalarg{#1}%
   \@optionaltemp
}%
%
%
% From LaTeX.
%
\def\@nnil{\@nil}%
\def\@fornoop#1\@@#2#3{}%
%
\def\@for#1:=#2\do#3{%
   \edef\@fortmp{#2}%
   \ifx\@fortmp\empty \else
      \expandafter\@forloop#2,\@nil,\@nil\@@#1{#3}%
   \fi
}%
%
\def\@forloop#1,#2,#3\@@#4#5{\def#4{#1}\ifx #4\@nnil \else
       #5\def#4{#2}\ifx #4\@nnil \else#5\@iforloop #3\@@#4{#5}\fi\fi
}%
%
\def\@iforloop#1,#2\@@#3#4{\def#3{#1}\ifx #3\@nnil
       \let\@nextwhile=\@fornoop \else
      #4\relax\let\@nextwhile=\@iforloop\fi\@nextwhile#2\@@#3{#4}%
}%
%
%
% This macro tests if a file \jobname.#1 exists, and sets \if@fileexists
% appropriately.  If an optional argument is given, it is used as the
% root part of the filename instead of \jobname.
%
\@innernewif\if@fileexists
%
\def\@testfileexistence{\@getoptionalarg\@finishtestfileexistence}%
\def\@finishtestfileexistence#1{%
   \begingroup
      \def\extension{#1}%
      \immediate\openin0 =
         \ifx\@optionalarg\empty\jobname\else\@optionalarg\fi
         \ifx\extension\empty \else .#1\fi
         \space
      \ifeof 0
         \global\@fileexistsfalse
      \else
         \global\@fileexiststrue
      \fi
      \immediate\closein0
   \endgroup
}%
%
%
%% [[[start of BibTeX-specific stuff]]]
%
% Now come the four main LaTeX commands and their associated .aux
% commands.  Just as in LaTeX, \bibliographystyle defines the BibTeX
% style name (.bst file, that is), and \bibliography defines the
% database (.bib) file(s).  The corresponding .aux-file commands are
% \bibstyle and \bibdata, which are there only for BibTeX's (but not
% LaTeX's) use.
%
\def\bibliographystyle#1{%
   \@readauxfile
   \@writeaux{\string\bibstyle{#1}}%
}%
\let\bibstyle = \@gobble
%
% As well as writing the \bibdata command to tell BibTeX which .bib
% files to read, we read the .bbl file that BibTeX (or a person,
% conceivably) has produced.  We use \bblfilebasename as the root of the
% filename to read; this defaults to \jobname.
%
\let\bblfilebasename = \jobname
\def\bibliography#1{%
   \@readauxfile
   \@writeaux{\string\bibdata{#1}}%
   \@testfileexistence[\bblfilebasename]{bbl}%
   \if@fileexists
      % We just output a non-discardable item (the `whatsit' with the
      % \bibdata command).  This means that the glue that will be
      % inserted next (\parskip or \baselineskip, most likely) will be a
      % legal breakpoint.  Most likely, this is after some kind of
      % heading, where we don't want to allow a page break.  So:
      \nobreak
      \@readbblfile
   \fi
}%
\let\bibdata = \@gobble
%
% The \nocite{label,label,...} command writes its argument to \@auxfile,
% unless instructed not to, but produces no text in the document.  Both
% the \nocite and \cite commands produce \citation commands in the .aux file.
%
\def\nocite#1{%
   \@readauxfile
   \@writeaux{\string\citation{#1}}%
}%
%
\@innernewif\if@notfirstcitation
%
% \cite[note]{label,label,...} produces the citations for the labels as
% well.  If the optional argument `note' is present, it's added after
% the labels.  Since \cite calls \nocite to do its .aux-file writing,
% \cite doesn't need to call \@readauxfile (\nocite does).
%
\def\cite{\@getoptionalarg\@cite}%
%
% Typeset the citations for the labels in #1, followed by the note, if
% it exists.  To change the citation's format in the text, redefine one
% or more `\print...' macros, whose defaults appear later in this file.
%
\def\@cite#1{%
   % Remember the optional argument, in case one of the macros we call
   % below ends up looking for an optional argument itself.  For
   % example, if a \cite[note] triggers reading the .aux file, then the
   % [note] would be clobbered, since \@testfileexistence looks for an
   % optional arg.
   \let\@citenotetext = \@optionalarg
   % Start printing the text, beginning with a left bracket by default.
   \printcitestart
   % It's complicated, but because \nocite puts a `whatsit' onto the list,
   % \nocite should follow \printcitestart.  It's conceivable, but very
   % unlikely, that this `whatsit' will cause a problem (glue that doesn't
   % disappear when you want it to is the most likely symptom), requiring
   % a change either to \printcitestart or to the label that the .bst file
   % produces.
   \nocite{#1}%
   \@notfirstcitationfalse
   \@for \@citation :=#1\do
   {%
      \expandafter\@onecitation\@citation\@@
   }%
   \ifx\empty\@citenotetext\else
      \printcitenote{\@citenotetext}%
   \fi
   \printcitefinish
}%
%
\newif\ifweareinprivate
\weareinprivatetrue
\ifx\shlhetal\undefinedcontrolseq\weareinprivatefalse\fi
\ifx\shlhetal\relax\weareinprivatefalse\fi
\def\@onecitation#1\@@{%
   \if@notfirstcitation
      \printbetweencitations
   \fi
   %
   \expandafter \ifx \csname\@citelabel{#1}\endcsname \relax
      \if@citewarning
         \message{\@linenumber Undefined citation `#1'.}%
      \fi
      % Give it a dummy definition:
     \ifweareinprivate
      \expandafter\gdef\csname\@citelabel{#1}\endcsname{%
% Change: marginal remark added, goldstrn@math.huji.ac.il, 
% goldstern@tuwien.ac.at, May 1996 mg
%  !!! change !!!
\strut 
\vadjust{\vskip-\dp\strutbox
\vbox to 0pt{\vss\parindent0cm \leftskip=\hsize 
\advance\leftskip3mm
\advance\hsize 4cm\strut\openup-4pt 
\rightskip 0cm plus 1cm minus 0.5cm ?  #1 ?\strut}}
         {\tt
            \escapechar = -1
            \nobreak\hskip0pt\pfeilsw\special{ps:100 0 0 setrgbcolor }%
            \expandafter\string\csname#1\endcsname
                     \special{ps:0 0 0 setrgbcolor }\pfeilso
            \nobreak\hskip0pt
         }%
      }%
     \else  % ifweareinprivate = false
      \expandafter\gdef\csname\@citelabel{#1}\endcsname{%
            {\tt\expandafter\string\csname#1\endcsname}
      }%
     \fi  % ifweareinprivate
   \fi
   % Now produce the text, whether it was undefined or not.
   \csname\@citelabel{#1}\endcsname
   \@notfirstcitationtrue
}%
%
% Given a label `foo', the macro `\b@foo' is supposed to
% hold the text that should be produced.
%
\def\@citelabel#1{b@#1}%
%
% So, how does a citation label get defined?  When we read the .bbl file
% (below), a \bibitem writes out a \@citedef command.  And when we read
% the \@citedef, we define \@citelabel{#1}, where #1 is the user's
% label.
%
\def\@citedef#1#2{\expandafter\gdef\csname\@citelabel{#1}\endcsname{#2}}%
%
%
% Reading the .bbl file also produces the typeset bibliography.  Please
% notice, however, that we do not produce the title for the references
% (e.g., `References'), as LaTeX does.  The formatting and spacing of
% that title, whether it should go into the headline, and so on, are all
% things determined by your format.  We cannot know those things in
% advance.  If you wish, you can define \bblhook to produce the title.
% Or just do it before the \bibliography command.
%
\def\@readbblfile{%
   % Define a counter to tell us which item number we are on, unless
   % we've already defined it (because the document has more than one
   % bibliography).
   \ifx\@itemnum\@undefined
      \@innernewcount\@itemnum
   \fi
   %
   \begingroup
      \def\begin##1##2{%
         % ##1 is just `thebibliography'.
         % ##2 is the widest label.
         % We set (new dimen) \biblabelwidth based on the widest label
         \setbox0 = \hbox{\biblabelcontents{##2}}%
         \biblabelwidth = \wd0
      }%
      \def\end##1{}% ##1 is `thebibliography' again.
      %
      % Here we have two possibilities:
      % \bibitem[typesetlabel]{citationlabel}
      % \bibitem{citationlabel}
      % If we have the second of these, the citations are numbered, starting
      % from one; we use our own count register \@itemnum for this.
      %
      \@itemnum = 0
      \def\bibitem{\@getoptionalarg\@bibitem}%
      \def\@bibitem{%
         \ifx\@optionalarg\empty
            \expandafter\@numberedbibitem
         \else
            \expandafter\@alphabibitem
         \fi
      }%
      \def\@alphabibitem##1{%
         % Need \xdef here for various reasons.
         \expandafter \xdef\csname\@citelabel{##1}\endcsname {\@optionalarg}%
         % Left-justify alpha labels, unless \biblabel{pre,post}contents
         % are already defined.
         \ifx\biblabelprecontents\@undefined
            \let\biblabelprecontents = \relax
         \fi
         \ifx\biblabelpostcontents\@undefined
            \let\biblabelpostcontents = \hss
         \fi
         \@finishbibitem{##1}%
      }%
      %
      \def\@numberedbibitem##1{%
         \advance\@itemnum by 1
         \expandafter \xdef\csname\@citelabel{##1}\endcsname{\number\@itemnum}%
         % Right-justify numeric labels, unless \biblabel{pre,post}contents
         % are already defined.
         \ifx\biblabelprecontents\@undefined
            \let\biblabelprecontents = \hss
         \fi
         \ifx\biblabelpostcontents\@undefined
            \let\biblabelpostcontents = \relax
         \fi
         \@finishbibitem{##1}%
      }%
      %
      \def\@finishbibitem##1{%
         \biblabelprint{\csname\@citelabel{##1}\endcsname}%
         \@writeaux{\string\@citedef{##1}{\csname\@citelabel{##1}\endcsname}}%
         \ignorespaces
      }%
      %
      % Do the printing (we're producing the bibliography, remember).
      %
      \let\em = \bblem
      \let\newblock = \bblnewblock
      \let\sc = \bblsc
      % Punctuation won't affect spacing;
      \frenchspacing
      % the penalties below are from LaTeX's [article,book,report].sty;
      \clubpenalty = 4000 \widowpenalty = 4000
      % the next two values come from LaTeX's \sloppy command;
      \tolerance = 10000 \hfuzz = .5pt
      \everypar = {\hangindent = \biblabelwidth
                      \advance\hangindent by \biblabelextraspace}%
      \bblrm
      % the \parskip is a guess at what looks good;
      \parskip = 1.5ex plus .5ex minus .5ex
      % and the space between label and text comes from LaTeX's \labelsep.
      \biblabelextraspace = .5em
      \bblhook
      %
      \input \bblfilebasename.bbl
   \endgroup
}%
%
% The widest label's width is useful for redefining \biblabelprint;
% you redefine \biblabelwidth, in effect, by redefining the
% \biblabelcontents macro that appears below.  And \biblabelextraspace,
% which is redefinable inside \bblhook, is added to \biblabelwidth to
% determine the amount of hanging indentation.
%
\@innernewdimen\biblabelwidth
\@innernewdimen\biblabelextraspace
%
% Now come the main macros that are related to the printing of the
% bibliography.  Since you might want to redefine them, they are given
% default definitions outside of \@readbblfile.
%
% The first one controls the printing of a bibliography entry's label.
% If you change it, make sure that it starts with something like
% \noindent or \indent or \leavevmode that puts TeX into horizontal mode
% (even if the label itself is empty); otherwise, the hanging
% indentation will get messed up in certain circumstances.
%
\def\biblabelprint#1{%
   \noindent
   \hbox to \biblabelwidth{%
      \biblabelprecontents
      \biblabelcontents{#1}%
      \biblabelpostcontents
   }%
   \kern\biblabelextraspace
}%
%
% If you are using numeric labels, and you want them left-justified
% (numeric labels by default are right-justified), do something like:
%     \def\biblabelprecontents{\relax}
%     \def\biblabelpostcontents{\hss}
%
% By default the labels are typeset in \bblrm, and enclosed in brackets.
%
\def\biblabelcontents#1{{\bblrm [#1]}}%
%
% The main text, too, is typeset using \bblrm, which is \rm by default.
%
\def\bblrm{\rm}%
%
% Emphasis for producing, e.g., titles, is done with \it by default.
%
\def\bblem{\it}%
%
% Some styles use a caps-and-small-caps font for author names.  LaTeX
% defines an \sc command but plain TeX doesn't, so we need one here.
% The definition below doesn't load the font unless it's needed, but it
% tries to load only the 10pt version, because it might not exist at
% other point sizes.
%
\def\bblsc{\ifx\@scfont\@undefined
              \font\@scfont = cmcsc10
           \fi
           \@scfont
}%
%
% The major parts of an entry are separated with \bblnewblock.  The
% numbers below are taken from LaTeX's `article' style.
%
\def\bblnewblock{\hskip .11em plus .33em minus .07em }%
%
% Here's where you stick any other bibliography-formatting goodies, or
% redefine the values above.
%
\let\bblhook = \empty
%
%
% Here are the four default definitions for formatting the in-text
% citations.  These are what you redefine (after your \input btxmac but
% before your \bibliography) to get parens instead of brackets, or
% superscripts, or footnotes, or whatever.
%
\def\printcitestart{[}%         left bracket
\def\printcitefinish{]}%        right bracket
\def\printbetweencitations{, }% comma, space
\def\printcitenote#1{, #1}%     comma, space, note (if it exists)
%
% That scheme is pretty flexible.  For example you could use
%     \def\printcitestart{\unskip $^\bgroup}
%     \def\printcitefinish{\egroup$}
%     \def\printbetweencitations{,}
%     \def\printcitenote#1{\hbox{\sevenrm\space (#1)}}
%     \font\eighttt = cmtt8
%     \scriptfont\ttfam = \eighttt
% to get superscripted in-text citations.  (The scriptfont stuff
% exists only to print an undefined citation; it's in cmtt8 because
% there is no cmtt7.)  To get something radically different, however,
% you'll have to define your own \cite command.
%
% When we read `\citation' from the .aux file, it means nothing.
%
\let\citation = \@gobble
%
%
% Now comes the stuff for dealing with LaTeX's \newcommand.  As
% mentioned earlier, this \newcommand will redefine a preexisting
% command; that's different from how LaTeX's \newcommand behaves.
%
\@innernewcount\@numparams
%
% \newcommand{\foo}[n]{text} defines the control sequence \foo to have
% n parameters, and replacement text `text'.
%
\def\newcommand#1{%
   \def\@commandname{#1}%
   \@getoptionalarg\@continuenewcommand
}%
%
% Figure out if this definition has parameters.
%
\def\@continuenewcommand{%
   % If no optional argument, we have zero parameters.  Otherwise, we
   % have that many.
   \@numparams = \ifx\@optionalarg\empty 0\else\@optionalarg \fi \relax
   \@newcommand
}%
%
% \@numparams is how many arguments this command has.  The name of the
% command is \@commandname.  The replacement text for the new macro is #1.
%
\def\@newcommand#1{%
   \def\@startdef{\expandafter\edef\@commandname}%
   \ifnum\@numparams=0
      \let\@paramdef = \empty
   \else
      \ifnum\@numparams>9
         \errmessage{\the\@numparams\space is too many parameters}%
      \else
         \ifnum\@numparams<0
            \errmessage{\the\@numparams\space is too few parameters}%
         \else
            \edef\@paramdef{%
               % This is disgusting, but \loop doesn't work inside \edef,
               % because \body isn't defined.
               \ifcase\@numparams
                  \empty  No arguments.
               \or ####1%
               \or ####1####2%
               \or ####1####2####3%
               \or ####1####2####3####4%
               \or ####1####2####3####4####5%
               \or ####1####2####3####4####5####6%
               \or ####1####2####3####4####5####6####7%
               \or ####1####2####3####4####5####6####7####8%
               \or ####1####2####3####4####5####6####7####8####9%
               \fi
            }%
         \fi
      \fi
   \fi
   \expandafter\@startdef\@paramdef{#1}%
}%
%
%% [[[end of BibTeX-specific stuff]]]
%
%
% Names of references (arguments given in the \cite and \nocite
% commands) and file names (arguments given in the \bibliography and
% \bibliographystyle commands) are recorded in \jobname.aux, called the
% \@auxfile in these macros.  Here's how they get read in.
%
\def\@readauxfile{%
   \if@auxfiledone \else % remember: \@auxfiledonetrue if \noauxfile is defined
      \global\@auxfiledonetrue
      \@testfileexistence{aux}%
      \if@fileexists
         \begingroup
            % Because we might be in horizontal mode when \@readauxfile
            % is called (if it's in response to a \cite or \nocite), we
            % want to ignore all the would-be spaces at the ends of
            % lines in the aux file.  Fortunately, it's highly unlikely
            % an end-of-line might actually be desired.
            % And because we don't change the category code of anything
            % but @, primitives like \gdef can't be used to define labels
            % in the aux file.  The solution adopted by btxmac.tex is to
            % write `\@citedef{LABEL}{DEFINITION}' to the aux file, and
            % use \csname on LABEL.
            \endlinechar = -1
            \catcode`@ = 11
            \input \jobname.aux
         \endgroup
      \else
         \message{\@undefinedmessage}%
         \global\@citewarningfalse
      \fi
      \immediate\openout\@auxfile = \jobname.aux
   \fi
}%
%
% The \@readauxfile macro does all that work the first time it's called.
% Since it's called once for every \cite, \nocite, \bibliography, and
% \bibliographystyle command that the user issues, we need to remember
% whether the work's been done.  It's considered done if we're not to do
% it---that is, if \noauxfile is defined.
%
\newif\if@auxfiledone
\ifx\noauxfile\@undefined \else \@auxfiledonetrue\fi
%
% It's conceivable you'd want to change how other characters are read;
% to do that, change their category code before doing \input btxmac.
%
%
% After reading the .aux file, \@readauxfile opens it for writing.
% The \@writeaux macro does the actual writing (as long as
% \noauxfile is undefined).
%
\@innernewwrite\@auxfile
\def\@writeaux#1{\ifx\noauxfile\@undefined \write\@auxfile{#1}\fi}%
%
%
% A macro package that uses btxmac.tex might define
% \@undefinedmessage (before doing an \input btxmac).
%
\ifx\@undefinedmessage\@undefined
   \def\@undefinedmessage{No .aux file; I won't give you warnings about
                          undefined citations.}%
\fi
%
% Even if citations are undefined, we want to complain only if
% \@citewarningtrue.  The default is to set \@citewarningtrue unless
% \noauxfile is defined.  Again, a macro package that uses
% btxmac.tex might want to redefine this.
%
\@innernewif\if@citewarning
\ifx\noauxfile\@undefined \@citewarningtrue\fi
%
%
% Finally, before leaving we restore @'s old category code.
%
\catcode`@ = \@oldatcatcode



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%  *** end including bib4plain.tex *** 
  % This will define \cite and make sure it works as in latex


\def\widestnumber#1#2{}
  % Our amstex-ppt style does not know about \widestnumber




\def\citewarning#1{\ifx\shlhetal\relax 
   % normal mode, do not write anything
    \else
   % private mode
    \par{#1}\par
    \fi
}




\def\rm{\fam0 \tenrm}


\def\fakesubhead#1\endsubhead{\bigskip\noindent{\bf#1}\par}


% % \input rsfs
%  *** start including rsfs.tex *** 

% # Keywords: Script or Calligraphic (Caligraphic) letters with the RSFS Font

% The story so far:    July 1998 -- Saharon would like to have a
% ``nicer'' calligraphic font. In particualr, the leters S and P in
% the usual calligraphic font do not look ``special'' enough. 
% 
% I found out that ``rsfs'' (``Ralph Smith Formal Script'') may be
% what he wants.   I installed the mf file, the .tfm file, as well as
% a few pk files in ~/TeX/rsfs.    Let's hope that this is enough.
% Using amstex, all you have to do is to \input rsfs.tex 
% Files prepared with citealice willdothis automatically. 
%
%  Note:  for some reason xdvi calls MakeTeXpk, then Maketexpk
%  complains about wrong resolution, but still writes commands to
%  missfont.log...  
%



% we redefine a macro inside amstex's \Cal command , so that it calls
% our nice font ``rsfs'' rather than the usual calligraphic font. 
% Note thisworks for amstex only.   
% In plain tex, would have to add definitions of \Cal
% in latex... we should insteaduse mathrsfs.sty
% 



\font\textrsfs=rsfs10
\font\scriptrsfs=rsfs7
\font\scriptscriptrsfs=rsfs5


\newfam\rsfsfam
\textfont\rsfsfam=\textrsfs
\scriptfont\rsfsfam=\scriptrsfs
\scriptscriptfont\rsfsfam=\scriptscriptrsfs


\edef\oldcatcodeofat{\the\catcode`\@}
\catcode`\@11

\def\Cal@@#1{\noaccents@ \fam \rsfsfam #1}

\catcode`\@\oldcatcodeofat

%  *** end including rsfs.tex *** 

\expandafter\ifx \csname margininit\endcsname \relax\else\margininit\fi

%  *** end including alice2jlem.tex *** 
\pageheight{8.5truein}
%\pageheight{48.5pc}
\topmatter
\title{What majority decisions are possible \\
 Sh816} \endtitle
\author {Saharon Shelah \thanks {\null\newline 
Partially supported by the United States-Israel Binational Science
Foundation \null\newline
I would like to thank 
Alice Leonhardt for the beautiful typing. \null\newline
 First Typed - 02/Jan/16 \null\newline 
 Latest Revision - 03/Mar/6} \endthanks} \endauthor 
 %formerly F545 
 %Done Dec. 2001 
 %Previous version - 03/Feb/28
\affil{Institute of Mathematics\\
 The Hebrew University\\
 Jerusalem, Israel
 \medskip
 Rutgers University\\
 Mathematics Department\\
 New Brunswick, NJ  USA} \endaffil
\endtopmatter
\document  
 
\newpage

\head {Annotated Content} \endhead  \resetall 
 % \resetall 
\bn
\S0 Introduction
\bn
\S1 Basic definitions and facts
\bn
\S2 When every majority choice is possible: a characterization
\bn
\S3 Balanced choice functions
\mr
\item "{${{}}$}" [We characterize what the majority choice can be on
pr-$c \ell({\Cal D})$ for ${\Cal D} \subseteq {\frak C}$ which is
balanced, i.e., does not fall under \S2.  We get the full answer.]
\endroster
\bn
\S4 Generalization
\mr
\item "{${{}}$}" [We consider allowing the original voters to abstain
(i.e., allow a draw) and more.]
\endroster
\newpage

\head {\S0 Introduction} \endhead  \resetall \sectno=0
 % \resetall 
\bigskip

Condorcet's ``paradox'' demonstrates that given three candidates A, B and C,
the majority rule may result in the 
society preferring A to B , B to C and C to A.
McGarvey \cite {McG53} proved a far-reaching extension of Condorcet's paradox:
For every asymmetric relation $R$ on a finite set $M$ of
candidates there is a strict-preferences
(linear orders, no ties) voter profile that has the relation $R$
as its strict simple majority relation. In other words,
for every assymetric relation (equivalently, a tournament)
$R$ on a set $M$ of $m$ elements
there are $n$ order relations on $M$, $R_1,R_2, \dots, R_n$
such that for every $a,b \in M$, $aRb$ if and only if
$$|\{ i: aR_ib\}|> n/2.$$ McGarvey's proof gave $n = m(m-1)$.
Stearns \cite {Ste59} found a construction with
$n = m$ and noticed that a simple 
counting argument implies that $n$ must be at least  $m/\log m$.
Erd\H{o}s and Moser \cite {ErMo64} were able to give a construction
with $n=O(m/\log m)$. Alon \cite {Alo02} showed that for some constant
$c_1>0$ we can find $R_1,\dots, R_n$
with $$|\{ i: aR_ib\}|> (1/2+ c_1/\sqrt {n})n,$$ and
that this is no longer the case if $c_1$ is
replaced with another constant $c_2>0$.
\sn
Gil Kalai asked to what extent the assertion of McGarvey's theorem
holds if we replace the set of order relations
by an arbitrary isomorphism class of choice functions on pairs of
elements (or equivalently tournaments).
Namely, under which conditions (A) below holds (i.e., question \scite{1.3}).
%but after hearing a preliminary try of mine find a counterexample.
%Earlier there was much work on the case ${\frak D} =$ the set of
%choice functions induced by a linear order on $X$.
\nl
Instead of choice functions we can speak on tournaments. \nl

The main result is (follows by \scite{2.1})
\demo{\stag{0.1} Conclusion}  Let $X$ be a finite set and ${\frak D}$
be a non empty family of choice functions for $\binom X 2$ closed under
permutation of $X$.  Then the following conditions are equivalent:
\mr
\item "{$(A)$}"  for any choice function $c$ on $\binom X2$ we can
find a finite set $J$ and $c_j \in {\frak D}$ for $j \in J$ such that
for any $x \ne y \in X$:
$$
c\{x,y\} = y \text{ \ub{iff} } |J|/2 < \{j \in J:c_j\{x,y\} \setminus y\}
$$
(so equality never occurs)
\sn
\item "{$(B)$}"  for some $c \in {\frak D}$ and $x \in X$ we have
$|\{y:c\{x,y\} = y\}| \ne (|X|-1)/2$.
\ermn
Gil Kalai further asks \nl
\margintag{0.q}\ub{\stag{0.q} Question}:  1) In \scite{0.1} can we bound $|J|$? 
\nl
2) What is the result of demanding a ``nontrivial majority"? (say 51\%?)
\mn
Under \scite{0.1} it seems reasonable to ask to characterize the two place
relation $\bold R$ on the family of choice for $\binom X2$, see
\scite{1.4}.  We then give a complete solution: what is the closure of
a set of choice functions by majority; in fact, there are just two
possibilities (see \scite{b.5}) we then discuss a generalization (in
\S4). \nl
I thank Gil for the simulating discussion and writing the historical
background. 
\enddemo
\bn
\margintag{0.2}\ub{\stag{0.2} Notation}:  Let $n,m,k,\ell,i,j$ denote natural numbers. \nl
Let $r,s,t,a,b$ denote real numbers.  \nl
Let $x,y,z,u,v,w$ denote members of the finite set $X$. \nl
Let $\binom Xk$ be the family of subsets of $X$ with exactly $k$
members. \nl
Let $c,d$ denote choice functions on $\binom X2$. \nl
Let conv$(A)$ be the convex hull of $A$, here for $A \subseteq \Bbb R
\times \Bbb R$. \nl
Let Per$(X)$ be the set of permutations of $X$.
\newpage

\head {\S1 Basic definitions and facts} \endhead  \resetall \sectno=1
 % \resetall 
\bigskip

\demo{\stag{1.1} Hypothesis}  Assume
\mr
\item "{$(a)$}"  $X$ is a (fixed) finite set with $\bold n \ge 3$
members, i.e. $\bold n = |X|$
\sn
\item "{$(b)$}"  ${\frak C} = {\frak C}_X$ is the set of choice
functions on $\binom X2$
\sn
\item "{$(c)$}"  ${\frak D}$ is fixed as a nonempty subset of
${\frak C}$ which is symmetric where
\endroster
\enddemo
\bigskip

\definition{\stag{1.1A} Definition}  1) ${\Cal C} \subseteq {\frak
C}$ is symmetric if it is closed under permutations of $X$ 
(i.e. for every $\pi \in \text{ Per}(X)$ the permutation $\hat \pi$ maps
${\Cal C}$ onto itself where $\pi$ induces $\hat \pi$, 
a permutation of ${\frak C}$, that is we can write $c_1 = c^\pi_2$ or $c_1
= \hat \pi c_2$ where: $x_1 = \pi(x_2),y_1 = \pi(y_2)$ implies
$c_1\{x_1,y_1\} = y_1 \Leftrightarrow c_2\{x_2,y_2\} = y_2$).
\enddefinition
\bigskip

\definition{\stag{1.2} Definition}  For ${\Cal D} \subseteq {\frak
C}$ let maj-$cl({\Cal D})$ be the set of $d \in {\frak C}$ such that
for some real numbers $r_c \in [0,1]_{\Bbb R}$ for $c \in {\Cal D}$
satisfying $\underset{c \in {\Cal D}} {}\to \Sigma r_c = 1$ we have
\footnote{note that there is no reason to assume that ${\Cal D}_2 =
\text{ maj}-c \ell({\Cal D}_1)$ implies ${\Cal D}_2 = \text{ maj}-c
\ell({\Cal D}_2)$} 


$$
d(\{x,y\}) = x \Leftrightarrow \frac 12 < \Sigma\{r_c:c\{x,y\} = x
\text{ and } c \in {\Cal D}\}
$$
(so the sum is never $\frac 12$).
\enddefinition
\bn
So Kalai's question was \nl
\margintag{1.3}\ub{\stag{1.3} Question}:  Do we have (for $|X|$ large enough),
maj-cl$({\Cal D}^*) = {\frak C}$?
\bigskip

\definition{\stag{1.4} Definition}  1) Let Dis $= \text{ Dis}(X) = \{\mu:\mu$ a
distribution on ${\frak C}_X\}$; of course, ``$\mu$ a distribution on
${\frak C}$" means $\mu$ is a function from ${\frak C}$ into $[0,1]_{\Bbb
R}$ such that $\Sigma\{\mu(c):c \in {\frak C}\} = 1$, and we let
$\mu({\Cal C}) = \Sigma\{\mu(c):c \in {\Cal C}\}$. \nl
2) For ${\Cal C} \subseteq {\frak C}$ and $\mu \in \text{Dis}({\frak C})$
let $\mu({\Cal C}) = \Sigma\{\mu(c):c \in {\Cal C}\}$ so $\mu(\{{\Cal C}\})
\ge 0,\mu({\frak C}) = 1$. \nl
3) For ${\Cal D} \subseteq {\frak C}$ let Dis$_{\Cal D} = \{\mu \in
\text{ Dis}:\mu({\Cal D}) =1\}$. \nl
4) Let pr$({\frak C}) = \{\bar t:\bar t = \langle t_{x,y}:x \ne y
\in X \rangle\}$ such that $t_{x,y} \in [0,1]_{\Bbb R},t_{y,x} =
1-t_{x,y}$, we may write $\bar t(x,y)$; pr stands for probability. \nl
5) For $T \subseteq \text{ pr}({\frak C})$ let pr-cl$(T)$ be the convex hull
of $T$. \nl
6) For $d \in {\frak C}$ let $\bar t[d] = \langle t_{x,y}[d]:x \ne y
\in X \rangle$ be defined by $[t_{x,y}[d] = 1 \Leftrightarrow d\{x,y\} = y
\Leftrightarrow t_{x,y}[d] \ne 0]$. \nl
7) Let pr-cl$({\Cal D})$ for ${\Cal D} \subseteq {\frak C}$ be
pr-cl$(\{\bar t[d]:d \in {\Cal D}\})$, prd$({\Cal D}) = \{\bar t[c]:c
\in {\Cal D}\}$. \nl
8) For ${\Cal C} \subseteq {\frak C}$ we let sym-$c \ell({\Cal C})$ be
the minimal ${\Cal D} \subseteq {\frak C}$ which is symmetric and
includes ${\Cal C}$.
\nl
9) For $T \subseteq \text{ pr}({\frak C})$ let maj$(T) = \{c \in {\frak
C}$: for some $\bar t \in T$ for any $x \ne y$ from $X$ we have
$c\{x,y\} = y \Leftrightarrow t_{x,y} > \frac 12 \Leftrightarrow
t_{x,y} \ge \{ \frac 12\}\}$. 
\enddefinition
\bigskip

\proclaim{\stag{1.8} Claim}  
1) For $d \in {\frak C}$ we have $\bar t[d] \in \,{\text{\rm pr\/}}({\frak
C})$. \nl
2) For ${\Cal D} \subseteq {\frak C}$ we have 
${\text{\rm Dis\/}}_{\Cal D} \subseteq { \text{\rm Dis\/}}$. \nl
3) ${\text{\rm prd\/}}({\frak C}) \subseteq { \text{\rm pr\/}}({\frak C})$ 
and if ${\Cal D} \subseteq {\frak C}$ then 
${\text{\rm prd\/}}({\Cal D}) \subseteq {\text{\rm pr-cl\/}}({\Cal D})
\subseteq { \text{\rm Dis\/}}$. \nl
4) If ${\Cal C} \subseteq {\frak C}$ then ${\Cal C} 
\subseteq { \text{\rm sym\/}}-c \ell({\Cal C}) \subseteq {\frak C}$. \nl
5) If $T \subseteq { \text{\rm pr\/}}({\frak C})$ \ub{then} 
${\text{\rm maj\/}}(T) \subseteq {\frak C}$. \nl
6) For ${\Cal D} \subseteq {\frak C}$ we have ${\text{\rm maj\/}}-c
\ell({\Cal D}) = {\text{\rm maj(pr-cl\/}}({\Cal D}))$.
\endproclaim
\bigskip

\demo{Proof}  Obvious.
\enddemo
\bigskip

\proclaim{\stag{1.9} Claim}  (G. Kalai) If ${\Cal C} \subseteq {\frak
C}$ and for every $c \in {\Cal C}$ and $x \in X$, the in-valency and
out-valency are equal, (i.e., ${\text{\rm val\/}}_c(x) = (|X|-1)/2$,
see below) \ub{then} every $d \in {\text{\rm maj-cl\/}}({\Cal C})$
satisfies:
\mr
\item "{$(*)$}"  if $\emptyset \ne Y \subsetneqq X$ then there are
edges from $Y$ to $X \backslash Y$ and from $X \backslash Y$ to $Y$, see
\scite{b.2}(1),(2). 
\endroster
\endproclaim
\bigskip

\definition{\stag{1.10} Definition}  1) For $d \in {\frak C}$ and $x
\in X$ let val$_d(x)$, the valency of $x$ for $d$, be $|\{y:y \in X,y
\ne x$ and $d\{x,y\} = y\}|$ so val$_d(x) \in \{0,\dotsc,|\bold n|-1\}$. \nl
We also call val$_d(x)$ the out-valency \footnote{natural under the
tournament interpretation} of $x$ in $d$ and $\bold n -
\text{ val}_d(x)-1$ the in-valency of $x$ in $d$. \nl
2) For $d \in {\frak C}$ let Val$(d) = \{\text{val}_d(x):x \in X\}$.
\nl
3) For $d \in {\frak C}$ and $\ell \in \{0,1\}$ let $V_\ell(d) =
\{(\text{val}_d(x)$,val$_d(y)):x \ne y \in X$ and $d\{x,y\} = y
\Leftrightarrow \ell = 1\}$. \nl
4) For $d \in {\frak C}$ and $\ell \in \{0,1\}$ let $V^*_\ell(d) =
\{\bar k-(\ell,1-\ell):\bar k \in V_\ell(d)\}$ and let $V^*(d) = V^*_0(d) \cup
V^*_1(d)$.
\nl
5) For $c \in {\frak C}$ let ${\text{\rm
dual\/}} (c) \in {\frak C}$ be ${\text{\rm dual\/}}(c)\{x,y\} 
\in \{x,y\} \backslash \{c\{x,y\}\}$; similarly $\bar t' = \text{ dual}
(\bar t)$ for $\bar t \in \text{ pr}({\frak C})$ means that $t'_{x,y}
= 1 - t_{x,y}$.
\enddefinition
\bigskip

\proclaim{\stag{1.11} Claim}  1) 
\mr
\item "{$(\alpha)$}"  $c_1 \in { \text {\rm sym-cl\/}}\{c_2\} 
\,{\text {\rm \ub{iff} dual\/}}(c_1) \in 
{ \text{\rm sym-cl\/}}\{{\text{\rm dual\/}}\{c_2\}\}$ 
\sn
\item "{$(\beta)$}"    $c_1 \in { \text{\rm maj-cl\/}}
({\text{\rm sym-cl\/}}\{c_2\}) 
\,{\text {\rm \ub{iff}\, dual\/}}(c_1) \in 
{\text{\rm maj-cl\/}}({\text{\rm sym-cl\/}}\{{\text{\rm dual\/}}(c_2)\})$.
\ermn
2) $(k_0,k_1) \in V_0(d) \Rightarrow (k_1,k_0) \in V_1(d)$. \nl
3) ``$c_2 \in { \text{\rm sym-cl\/}}\{c_2\}$" is an equivalence
relation on ${\frak C}$ and it implies $V_\ell(c_1) = V_\ell(c_2)$ for
$\ell=0,1$.  
\endproclaim
\bigskip

\demo{Proof}  Easy.
\enddemo
\newpage

\head {\S2 When every majority choice is possible: a characterization} \endhead  \resetall \sectno=2
 % \resetall 
\bigskip

The following is the main part of the solution
(probably $(c) \Leftrightarrow (g)$
should be stated as the main conclusion here).
\proclaim{\stag{2.1} Main Claim}  The following conditions on ${\frak D}
\subseteq {\frak C}$ which is symmetric and not empty, (i.e.,
${\frak D}$ is a set of choice
functions on $\binom X2$ closed under permutation on $X$, 
${\frak D} \ne \emptyset$) and for simplicity assuming for the time
being that ${\frak D} = { \text{\rm sym-cl\/}}(d^*)$ for 
any $d^* \in {\frak D}$, are equivalent, where $x,y$ vary
on distinct members of $X$:
\mr
\widestnumber\item{$(b)_{x,y}$}
\item "{$(a)$}"  ${\text{\rm maj-cl\/}}({\frak D}) = {\frak C}$
\sn
\item "{$(b)_{x,y}$}"  there is $\bar t \in { \text{\rm pr-cl\/}}({\frak D})
\subseteq {\text{\rm pr\/}}({\frak C})$ such that 
{\roster
\itemitem{ $(i)$ }  $t_{x,y} > \frac 12$
\sn
\itemitem{ $(ii)$ }  $\{x,y\} \ne \{u,v\} \in \binom Xr 
\Rightarrow t_{u,v} = \frac 12$
\endroster}
\item "{$(c)$}"  for any $c \in {\frak C}$ we can find a finite set
$J$ and sequence $\langle d_j:j \in J \rangle$ such that $d_j \in
{\frak D}$ and: if $u \ne v \in X$ then \nl
$c\{u,v\} = v \Leftrightarrow (\{j \in J:d_j\{u,v\} = j\}| > |J|/2$
\sn
\item "{$(d)$}"  $(\frac 12,\frac 12)$ belongs to 
${\text{\rm Pr\/}}_{> \frac 12}({\frak D})$, see definition below
\sn
\item "{$(e)$}"  $(\frac 12,\frac 12) \in {\text{\rm Pr\/}}_{\ne 1/2}({\frak
D})$
\sn
\item "{$(f)$}"  $(\frac{\bold n}2 - 1,\frac{\bold n}2 -1)$ can be
represented as $r^*_0 \times \bar s_0 + r^*_1 \times \bar s_1$ where 
{\roster
\itemitem{ $(*)(i)$ }  $r^*_0,r^*_1 
\in [0,1]_{\Bbb R} \backslash \{\frac 12\}$
\sn
\itemitem{ $(ii)$ }  $1 = r^*_0 + r^*_1$
\sn
\itemitem{ $(iii)$ }  for $\ell =0,1$ the pair 
$\bar s_\ell \text{ belongs to the convex hull of } V^*_\ell(d^*)
\text{ for some } d^* \in {\frak D}$, see Definition
\scite{1.10}(3), but recall that by a hypothesis of the claim, the
choice of $d^*$ is immaterial
\endroster}
\sn
\item "{$(g)$}"  for some $(d^* \in {\frak D}$ and) $x \in X$ we have
${\text{\rm val\/}}_{d^*}(x) \ne \frac{\bold n-1}2$. 
\endroster
\endproclaim
\bigskip

\demo{Proof}  $(b)_{x,y} \Leftrightarrow (b)_{x',y'}$:

(So $x,y,x',y' \in X$ and $x \ne y,x' \ne y'$).  
Trivial as ${\frak D}$ is closed under permutations of $X$.
\mn
\ub{$(b)_{x,y} \Rightarrow (a)$}:

Let $c \in {\frak C}^*$. \nl
Let $\{(u_i,v_i):i < i(*)\}$ list the pairs $(u,v)$ of distinct
members of $X$ such that $c\{u,v\} = v$ so $i(*) = \binom{|X|}{2}$.  For
each $i < i(*)$ as $(b)_{x,y} \Rightarrow (b)_{u_i,v_i}$ clearly
there is $\bar t^i \in \text{ pr-cl}({\frak D}^*)$
such that

$$
t^i_{u_i,v_i} > \frac 12 \text{ so } t^i_{v_i,u_i} = 1 - t_{u_i,v_i} <
\frac 12
$$

$$
\{u_i,v_i\} \ne \{u,v\} \in \binom X2 \Rightarrow t^i_{u,v} = \frac
12.
$$
\mn
Let $\bar t^* = \langle t^*_{u,v}:u \ne v \in X \rangle$ be defined
to

$$
t^*_{u,v} = \Sigma\{t^i_{u,v}:i < i(*)\}/i(*).
$$
\mn
As pr-cl$({\frak D})$ is convex and $i < i(*) \Rightarrow \bar t^i
\in \text{ pr-cl}({\frak D})$ clearly $\bar t \in \text{ pr-cl}
({\frak D})$.  Now for each $j < i(*),t^i_{u_j,v_j}$ is 
$\frac 12$ if $i \ne j$ and is $> \frac 12$ if $i=j$.   Hence
$t^*_{u_j,v_j}$ being the average of $\langle t^i_{u_j,v_j}:i < i(*)
\rangle$ is $> \frac 12$.  Hence $t^*_{v_j,u_j} = 1-t^*_{u_j,v_j} <
\frac 12$.  So $t^*_{u,v} > \frac 12$ \ub{iff} $t^*_{u,v} \ge \frac
12$ \ub{iff} $c\{u,v\} = v$ by the
choice of $\langle(u_i,v_i):i < i(*) \rangle$.  So $\bar t^*$ witness
$c \in$ maj-cl$({\frak D})$ as required in clause (a).
\mn
\ub{$(a) \Rightarrow (b)_{x,y}$}: 

By clause (a), for every $d \in {\frak C}$ there is $\langle r_c:c \in
{\frak D} \rangle$ as in Definition \scite{1.2}, so some
$\varepsilon_d > 0,d(\{x,y\}) = y \Rightarrow \frac 12 + \varepsilon_d
< \Sigma\{r_c:c \in {\frak D}$ and $c\{x,y\} = y\}$.  Hence
$\varepsilon = \text{ Min}\{\varepsilon_d:d \in {\frak D}\}$ 
is a real $>0$. \nl
Let $T = \{\bar t:\bar t \in \text{ pr-cl}({\frak D})$ and $t_{x,y}
\ge \frac 12 + \varepsilon$ and $u \ne v \in X \Rightarrow t_{u,v} \le
t_{x,y}\}$, so
\mr
\item "{$(*)_1$}"  $T \ne \emptyset$
\nl
[Why?  As ${\frak D}$ is symmetric clearly ${\frak D}_{x,y} = \{c \in {\frak
D}:c\{x,y\} = y\}$ is non empty, and $\bar t[d] \in T$ for every $d
\in {\frak D}_{x,y}$]
\sn
\item "{$(*)_2$}"    $T$ is convex and closed.
\ermn
[Why?  Trivial.] \nl
For $\bar t \in T$ define 
\mr
\item "{$\boxtimes$}"  err$(T) = \max\{|t_{u,v} - \frac 12|:u \ne
v \in X \text{ and } \{u,v\} \ne \{x,y\}\}$
\sn
\item "{$(*)_3$}"   if $\bar t \in T$, err$(\bar t) > 0$ \ub{then} we
can find $\bar t' \in T$ 
such that err$(\bar t') \le \frac 12 \text{ err}(t)$ and $t'_{x,y} \ge
(t_{x,y} + \frac 12 + \varepsilon)/2$.
\ermn
Why?  Choose $c \in {\frak C}$ such that $c\{x,y\} = y$ and

$$
u \ne v \in X \and \{u,v\} \ne \{x,y\} \and t_{u,v} > \frac 12
\Rightarrow c\{u,v\} = u
$$
\mn
(so if $t_{u,v} = t_{v,u} = \frac 12$ it does not matter).

So $c$ is ``a try to correct $\bar t$".

As we are assuming clause (a) and the choice of $\varepsilon_d$,
we can find $\bar r^* = \langle r^*_d:d
\in {\frak D} \rangle$ with $r^*_d \in [0,1]_{\Bbb R}$ and $1 =
\Sigma\{r^*_d:d \in {\frak D}\}$ such that  

$$
\frac 12 + \varepsilon_d < \Sigma\{r^*_d:d\{x,y\} = y\}
$$

$$
c\{u,v\} = v \Rightarrow \frac 12 < \Sigma\{r^*_d:d\{u,v\} = v\}
$$
hence

$$
c\{u,v\} = u \Rightarrow \frac 12 < \Sigma\{r^*_d:d\{u,v\} = u\}.
$$
\mn
By the choice of $\varepsilon,\frac 12 + \varepsilon < \Sigma\{r^*_d:d\{x,y\}
= y\}$. \nl
Let $\bar s = \langle s_{u,v}:u \ne v \in X \rangle$ be defined by
$s_{u,v} = \Sigma\{r^*_d:d\{u,v\} = v\}$, so
\mr
\item "{$\circledast(i)$}"   $\bar s \in \text{ pr-cl}({\frak D}^*)$
\sn
\item "{$(ii)$}"  $s_{x,y} > \frac 12 + \varepsilon$ (so $s_{y,x} <
\frac 12$)
\sn
\item "{$(iii)$}"  if $t_{u,v} > \frac 12$ and $u \ne v \in X,\{u,v\}
\ne \{x,y\}$ then $c\{u,v\} = u$ hence $s_{u,v} < \frac 12$
\sn
\item "{$(iv)$}"  if $t_{u,v} < \frac 12$ and $u \ne v \in X,\{u,v\}
\ne \{x,y\}$ then $c\{u,v\} = v$ hence $s_{u,v} > \frac 12$.
\ermn
Let $\bar t' = \frac 12 \bar t + \frac 12 \bar s$, i.e. $t'_{u,v} =
\frac 12(t_{u,v} + s_{u,v})$ so clearly
\mr
\item "{$\circledast(i)$}"  $\bar t' \in \text{ pr-cl}({\frak D})$
\sn
\item "{$(ii)$}"  $t'_{x,y} = (t_{x,y} + \frac 12 + \varepsilon)/2$ (hence 
$\bar t' \in T$)
\sn
\item "{$(iii)$}"  if $ u \ne v \in X,\{u,v\} \ne \{x,y\}$ then
$|t'_{u,v} - \frac 12| \le \frac 12 |t_{u,v} - \frac 12|$.
\ermn
So we are done proving $(*)_3$.

As $T$ is closed (and included in a $\{\bar t:\bar t = \langle
t_{u,v}:u \ne v \in X \rangle,0 \le t_{u,v} \le 1\})$, clearly there
is $t \in T$ such that $u \ne v \in X \and \{u,v\} \ne \{x,y\}
\Rightarrow t_{u,v} = \frac 12$ as required.
\mn
\ub{$(c) \Rightarrow (a)$}:

Let $d \in {\frak C}$ and let $\langle c_j:j \in J \rangle$ witness
clause (c). \nl
Let $r_c = |\{j \in J:c_j = c\}|/|J|$ now $\langle r_c:c \in {\frak
D} \rangle$ witness clause (a), i.e., witness that $d \in$ 
maj-cl$({\frak D}^*)$.
\mn
\ub{$(a) \Rightarrow (c)$}:

Let $d \in {\frak C}$ and let $\langle r_c:c \in {\frak D}
\rangle$ be as guaranteed for $d$ by clause (a).  Let $n(*) > 0$ be large
enough and for $c \in {\frak D}$ let $k_c \in \{0,\dotsc,n(*)-1\}$ be
such that $c \in {\frak D} \Rightarrow k_c \le n(*) \times r_c < k_c +
1$; note that $k_c$ exists as $r_c \in [0,1]_{\Bbb R}$.
As $\dsize \sum_c \frac{k_c}{n(*)} \le 1 \le \dsize \sum_c
\frac{k_c+1}{n(*)}$, we can choose $m_c \in \{k_c,k_c+1\}$ such that $r'_c =
\frac{m_c}{n(*)}$ satisfies $\Sigma\{r'_c:c \in {\frak D}^*\} = 1$.
Let $J = \{(c,m):c \in {\frak D}$ and $m \in \{1,\dotsc,m_c\}\}$ and
we let $\bold c_{(d,m)} = d$ for $(d,m) \in J$.  
Now the ``majority" of $\langle \bold c_t:t
\in J \rangle$, see Definition \scite{1.2}, choose $d^*$ so clause (c) holds.
\enddemo
\bn
Before we deal with clauses (d),(e),(f) and (g) of \scite{2.1}, we define
\definition{\stag{2.2} Definition}  1) For ${\Cal D} \subseteq 
{\frak C}$ and $A \subseteq [0,1]_{\Bbb R}$ let 
Pr$_A({\Cal D})$ be the set of pairs $(s_0,s_1)$ of real
numbers $\in [0,1]_{\Bbb R}$ such that 
for some $\bar t \in \text{ pr-cl}({\Cal D})$ and $x \ne y \in X$ and
$a \in A$ we have $\bar t = \bar t \langle x,y,a,s_0,s_1 \rangle$
where \nl
2) $\bar t = \bar t\langle x,y,a,s_0,s_1 \rangle$ where
$x \ne y \in X,a \in [0,1]_{\Bbb R}$ and $s_0,s_1 \in [0,1]_{\Bbb R}$ and
$\bar t = \langle t_{u,v}:u \ne y \in X \rangle \in \text{ pr}({\frak
C})$ is defined by 
\mr
\item "{$(\alpha)$}"  $t_{x,y} = a$
\sn
\item "{$(\beta)$}"  if $z \in X \backslash \{x,y\}$ \ub{then}
$t_{y,z} = s_1$ (hence $t_{z,y} = 1 - s_1$)
\sn
\item "{$(\gamma)$}"  if $z \in X \backslash \{x,y\}$ \ub{then}
$t_{x,z} = s_0$ (hence $t_{z,x} = 1-s_0$)
\sn
\item "{$(\delta)$}"  if $z_1 \ne z_2 \in X \backslash \{x,y\}$
\ub{then} $t_{z_1,z_2} = \frac 12$.
\ermn
3) In Pr$_A({\Cal D})$ we may replace $A$ by $1,0 \ne,\frac 12,> \frac
12,< \frac 12$ if $A$ is
$\{1\},\{0\},[0,1]_{\Bbb R} \backslash \{\frac 12\},(\frac 12,1]_{\Bbb
R},[0,\frac 12)_{\Bbb R}$ respectively. \nl
4) For $\ell \in \{0,1\}$ and 
${\Cal D} \subseteq {\frak C}^*$ let Prd$_\ell({\Cal D})$ be the set
of pairs $\bar s = (s_0,s_1)$ of real (actually rational) 
numbers $\in [0,1]_{\Bbb R}$
such that for some $c \in {\Cal D}$ and $x \ne y \in X$ we have $\bar
s = \bar s^{c,x,y} = (s^{c,x,y}_0,s^{c,x,y}_1)$ where
\mr
\widestnumber\item{$(iii)$}
\item "{$(i)$}"  $s^{c,x,y}_1 = 
|\{z:z \in X \backslash \{x,y\} \text{ and } c\{y,z\} =
z\}|/(\bold n - 2)$
\sn
\item "{$(ii)$}"  $s^{c,x,y}_0 = 
|\{z:z \in X \backslash \{x,y\} \text{ and } c\{x,z\} = z\}|/(\bold n
- 2)$
\sn
\item "{$(iii)$}"  $\ell = 1 
\Leftrightarrow \ell \ne 0 \Leftrightarrow c\{x,y\} = y$.
\ermn
5) Prd$({\Cal D})$ is Prd$_0({\Cal D}) \cup \text{ Prd}_1({\Cal D})$
and ${\Cal D}_{x,y} = \{c \in {\Cal D}:c\{x,y\} = y\}$.
\enddefinition
\bigskip

\proclaim{\stag{2.3} Claim}   Let ${\Cal D} \subseteq {\frak C}$
\mr
\item  ${\text{\rm Pr\/}}_{A_1}
({\Cal D}_1) \subseteq { \text{\rm Pr\/}}_{A_2}({\Cal D}_2)$
if $A_1 \subseteq A_2 \subseteq [0,1]_{\Bbb R}$ and ${\Cal D}_1
\subseteq {\Cal D}_2 \subseteq {\frak C}$
\sn
\item   ${\text{\rm Pr\/}}_A({\Cal D})$ is a convex subset of
$[0,1]_{\Bbb R} \times [0,1]_{\Bbb R}$
\sn
\item  ${\text{\rm Prd\/}}_\ell
({\Cal D})$ is finite and its convex hull is $\subseteq
{\text{\rm Pr\/}}_{\{\ell\}}({\Cal D})$, increasing with ${\Cal D}(\subseteq
{\frak C})$ for $\ell =0,1$
\sn
\item  For $x \ne y \in X$ and $c \in {\frak C}$ satisfying $c\{x,y\}
= y \Leftrightarrow \ell =1 \Leftrightarrow \ell \ne 0$ we have (see
Claim \scite{2.2}(4))
$$
s^{c,x,y}_0 = ({\text{\rm val\/}}_c(x) - \ell)/(\bold n - 2)
$$

$$
s^{c,x,y}_1 = ({\text{\rm val\/}}_c(y) - (1 - \ell))/(\bold n - 2).
$$
\sn
\item  ${\text{\rm Pr\/}}_{A_1 \cup A_2}
({\Cal D}) = { \text{\rm Pr\/}}_{A_1}({\Cal D}) \cup
{ \text{\rm Pr\/}}_{A_2}({\Cal D})$, in fact ${\text{\rm Pr\/}}_A({\Cal D}) =
\cup\{{\text{\rm Pr\/}}_{\{a\}}({\Cal D}):a \in A\}$.
\endroster
\endproclaim
\bigskip

\demo{Proof}  Immediate.
\enddemo
\bigskip

\proclaim{\stag{2.3a} Claim}  1) If $x \ne y \in X$ and $c \in {\frak
C}$ and $\ell \in \{0,1\}$ satisfies $(\ell = 1) \equiv (c(\{x,y\})
=y)$ \ub{then} $\bar t \langle x,y,\ell,s^{c,x,y}_0,s^{c,x,y}_1
\rangle = \frac{1}{|\Pi_{x,y}|} \Sigma\{\bar t[\hat \pi(c)]:\pi \in
\Pi_{x,y}\}$ where $\Pi_{x,y} =: \{\pi \in { \text{\rm Per\/}}(X):\pi(x) = x$
and $\pi(y) = y\}$ hence $|\Pi_{x,y}| = (\bold n -2)!$. \nl
2) If ${\Cal D} \subseteq {\frak C}$ is symmetric and $\bar t \in 
{\text{\rm pr-cl\/}}({\Cal D})$ and $\bar t^* = \Sigma\{\bar t^\pi:\pi
\in \Pi_{x,y}\}/|\Pi_{x,y}|$ where $\bar t^\pi = \langle t^\pi_{u,v}:u
\ne v \in X \rangle,t^\pi_{u,v} = t_{\pi^{-1}(u),\pi^{-1}(v)}$
\ub{then} $t^* \in { \text{\rm pr-cl\/}}({\Cal D})$ and $\bar t^* \in
{ \text{\rm Pr\/}}_{\{a\}}(s_0,s_1)$ where $a = t_{x,y},s_0 =
\Sigma\{t_{x,z}:z \in X \backslash \{x,y\}\}/(\bold n-2)!,s_1 =
\Sigma\{t_{y;z}:z \in X \backslash \{y,z\}/(\bold n-2)!$.
\endproclaim
\bigskip

\demo{Proof}  Easy.
\enddemo
\bigskip

\proclaim{\stag{2.4} Claim}  For any symmetric ${\Cal D} \subseteq
{\frak C}$ (i.e., closed under permutations of $X$): \nl
1) For $\ell \in \{0,1\}$, the set ${\text{\rm Pr\/}}_\ell
({\Cal D})$ is the convex
hull of ${\text{\rm Prd\/}}_\ell({\Cal D})$ in $\Bbb R \times \Bbb R$. \nl
2) ${\text{\rm Pr\/}}_{[0,1]}({\Cal D})$ is the convex hull of 
${\text{\rm Prd\/}}({\Cal D})$. \nl
3) Let $s^*_0,s^*_1 \in [0,1]_{\Bbb R}$.  \ub{Then} $\bar t^* = \bar
t\langle x,y,a,s^*_0,s^*_1 \rangle \in {\text{\rm pr-cl\/}}({\Cal
D})$ \ub{iff} we can find $\langle r_{\bar s,\ell}:\ell \in \{0,1\}$
and $\bar s \in {\text{\rm Prd\/}}_\ell({\Cal D})\rangle$ such that
$r_{\bar s,\ell} \in [0,1]_{\Bbb R}$ and $1 = \Sigma\{r_{\bar
s,\ell}:\ell \in \{0,1\},\bar s \in 
{\text{\rm Prd\/}}_\ell({\Cal D})\}$ and $\bar s = \Sigma\{r_{\bar
s,\ell} \times \bar s:\ell \in \{0,1\},\bar s \in { \text{\rm
Prd\/}}_\ell({\Cal D})\}$
and $(\frac 12,\frac 12) = \Sigma\{r_{\bar s,\ell} \times \bar
s:\ell \in \{0,1\},\bar s \in {\text{\rm Prd\/}}_\ell({\Cal D})\}$ and
$a = \Sigma\{r_{\bar s,1}:\bar s \in {\text{\rm Prd\/}}_1({\Cal D})\}$.
\endproclaim
\bigskip

\demo{Proof}  1) By \scite{2.3}(3) we have one inclusion.

For the other direction assume $(s^*_0,s^*_1) \in \text{
Pr}_\ell({\Cal D})$ and we should prove that it belongs to the convex
hull of Prd$_\ell({\Cal D})$.
Fix $x \ne y \in X$ and let $\bar t^* = \bar t \langle
x,y,\ell,s^*_0,s^*_1 \rangle$ so
$\bar t^* = \langle t^*_{u,v}:u \ne v \in X \rangle$ is defined as
follows $t^*_{x,y} = \ell,t^*_{u,v} = \frac 12$ if 
$u \ne v \in X \backslash \{x,y\},t^*_{y,z} =
s^*_1$ if $z \in X \backslash \{x,y\},t_{x,z} = s^*_0$ if $z \in X
\backslash \{x,y\}$.
As $(s^*_0,s^*_1) \in \text{ Pr}_\ell({\Cal D})$ by 
Definition\scite{2.2} we know that $\bar t^* \in \text{ pr-cl}({\Cal D})$ and
let $\bar r = \langle r_c:c \in {\Cal D} \rangle$ be such that 
\mr
\item "{$\circledast_0$}"  $x,y,\bar r$ witness that 
$\bar t^* \in \text{ pr-cl}({\Cal D})$, so
$r_c \ge 0$ and $1 = \Sigma\{r_c:c \in {\Cal D}\}$ and

$$
\bar t^* = \Sigma\{r_c \times \bar t[c]:c \in {\Cal D}\}.
$$
\ermn
As $t^*_{x,y} = \ell$, necessarily
\mr
\item "{$\boxtimes_1$}"   $r_c \ne 0 \Rightarrow c \in {\Cal
D}_{x,y} =: \{c \in {\Cal D}:(c\{x,y\} = y) \equiv (\ell=1)\}$.
\ermn
To make the rest of the proof also a proof of part (3) let $a = \ell$
(as the real number $a$ maybe $\ne 0,1$ we need $m \in \{0,1\}$ below).
\nl
Let $\Pi_{x,y} = \{\pi \in \text{ Per}(X):\pi(x) = x,\pi(y) = y\}$ and
recall that for $\pi \in \text{ Per}(X),\hat \pi$ is the 
permutation of ${\frak C}$ which $\pi$ induces so $\hat
\pi$ maps ${\Cal D}_{x,y}$ onto ${\Cal D}_{x,y}$.  Clearly
$|\Pi_{x,y}| = (\bold n - 2)!$
\nl
For $(s_0,s_1) \in \text{ Prd}_\ell({\Cal D}) \subseteq 
\{(\frac{m_1}{(\bold n-2)!},\frac{m_2}{(\bold n-2)!}):m_1,m_2 \in 
\{0,1\dotsc,(\bold n-2)!\}\}$ let $r^*_{(s_0,s_1)} =
\Sigma\{r_c:\bar s^{c,x,y} = \bar s\}$ and for $m \in \{0,1\}$ we 
let $\bar r^*_{(s_0,s_1),m}
= \Sigma\{r_c:\bar s^{c,x,y} = \bar s$ and $c\{x,y\} = y \equiv (m=1)\}$.

Clearly $\pi \in \Pi_{x,y} \Rightarrow \langle t^*_{\pi(u),\pi(v)}:u \ne
v \in X \rangle = \bar t^*$, just check the definition, hence
recalling \scite{2.3}(4) we have
\mr
\item "{$\boxtimes_2$}"  $\bar t^* = \dsize \frac{1}{| \Pi_{x,y}|} \dsize
\sum_{\pi \in \Pi_{x,y}} \langle t^*_{\pi(u),\pi(v)}:u \ne v \in X
\rangle = \frac{1}{|\Pi_{x,y}|} \dsize \sum_{\pi \in \Pi_{x,y}} \,
\dsize \sum_{c \in {\Cal D}} r_c \times \bar t^*[\hat \pi(c)] =$ \nl 
$\dsize \sum_{c \in {\Cal D}} \, r_c(\frac{1}{|\Pi_{x,y}|} \, \dsize
\sum_{\pi \in \Pi_{x,y}} \bar t^*[\hat \pi(c)]) =
\dsize \sum_{c \in {\Cal D}} r_c \times \bar t \langle
x,y,a,s^{c,x,y}_0,s^{c,x,y}_1 \rangle =$ \nl
$\dsize \sum_{m \in \{0,1\}} \,\, 
\dsize \sum_{(s_0,s_1) \in \text{ Pr}_\ell({\Cal D})} 
\, (\Sigma\{r_c:c \in {\Cal D},c\{x,y\} = y \equiv (k=1)$ \nl

\hskip110pt and $\bar s^{c,x,y} = (s_0,s_1)\}) \times
\bar t \langle x,y,m,\bar s_0,s_1) =$ \nl
$\dsize \sum_{m \in \{0,1\}} \, 
\dsize \sum_{(s_0,s_1) \in \text{ Pr}_\ell({\Cal D})}
r^*_{(s_0,s_1),m} \bar t\langle x,y,m,s_0,s_1 \rangle$
(we have used \scite{2.3a}).  
\ermn
Now concentrate again on the case $a = \ell \in \{0,1\}$, so
$r^*_{\bar s,1-\ell} = 0$ by $\boxtimes_1$ and $r^*_{\bar s,\ell} = r^*_s$.
So clearly
\mr
\item "{$\circledast_1$}"  $r^*_{\bar s} \ge 0$ \nl
[Why?  As the sum of non negative integers]
\sn
\item "{$\circledast_2$}"   $1 =
\Sigma\{r^*_{\bar s}:\bar s \in \text{ Prd}_\ell({\Cal D})\}$ \nl
[Why?  As by Definition \scite{2.2}(2), $c \in {\Cal D} \and r_c > 0
\Rightarrow c \in {\Cal D}_{x,y} \Rightarrow \bar s^{c,x,y} \in
\text{prd}_\ell({\Cal D})$ and the definition of $r^*_{\bar s}$]
\sn
\item "{$\circledast_3$}"  we have
{\roster
\itemitem{ $(\alpha)$ }  $z \in X \backslash \{x,y\} \Rightarrow s^*_1
= t^*_{y,z} = \Sigma\{r^*_{(s_0,s_1)} \times s_1:(s_0,s_1) \in \text{
Prd}_\ell({\Cal D})\}$,
\sn
\itemitem{ $(\beta)$ }  $z \in X \backslash \{y,z\} \Rightarrow s^*_0
= t^*_{x,z} = \Sigma\{r^*_{(s_0,s_1)} \times s_0:(s_0,s_1) \in \text{ Prd}_\ell
({\Cal D})\}$ \nl
[Why?  By $\boxtimes_2$.]
\endroster}
\ermn
So $\langle r^*_{\bar s}:\bar s \in \text{ Prd}_\ell({\Cal D}) \rangle$
witness that $(s^*_0,s^*_1) \in$ convex hull of Prd$_\ell({\Cal
D})$. \nl
2) Similar proof (and not used). \nl
3) One direction is as in \scite{2.3}(3).  For the other, by the
hypothesis, $\circledast_0$ in the proof of part (1) with $\ell$
replaced by $a$ holds.  So by the part of the proof of part (1) after
$\boxtimes_1,r^*_{\bar s,m}$ are defined and $\boxtimes_2$ holds.  So
\mr
\item "{$\boxdot_1$}"  $r^*_{\bar s,m} \ge 0$
\sn
\item "{$\boxdot_2$}"  $1= \Sigma\{r^*_{\bar s,m}:m \in \{0,1\}$ and
$\bar s \in$ Prd$_m({\Cal D})\}$
\sn
\item "{$\boxdot_3$}"  $(s^*_0,s^*_1) = \Sigma\{r^*_{\bar s,m} \times
\bar s:m \in \{0,1\},\bar s \in$ Prd$_m({\Cal D})\}$ \nl
[Why?  By $\boxtimes_2$.]
\sn
\item "{$\boxdot_4$}"  $a = \Sigma\{r^*_{\bar s,1}:\bar s \in$
Prd$_1({\Cal D})\}$.
\ermn
So we are done.   \hfill$\square_{\scite{2.4}}$\margincite{2.4}
\enddemo
\bn
\ub{Continuation of the proof of \scite{2.1}}:
\sn
\ub{$(d) \Leftrightarrow (b)_{x,y}$}:

Read the definition \scite{2.2}(1) 
(and the symmetry).  \hfill$\square_{\scite{2.1}}$\margincite{2.1}
\bn
\ub{$(d) \Rightarrow (e)$}:

By \scite{2.3}(1).
\bn
\ub{$(e) \Rightarrow (d)$}:

Why?  If clause (e) holds, for some $a \in [0,1]_{\Bbb R} \backslash
\{\frac 12\}$ and $x \ne y \in X$ we have $\bar t^* =: \bar t \langle
x,y,a,\frac 12,\frac 12 \rangle \in$  pr-cl$({\frak D})$.  
If $a > \frac 12$ this witness $(\frac 12,\frac 12) \in \text{ Pr}_{>
1/2}({\Cal D})$, so assume $a < \frac 12$.  But trivially $\bar t
\langle y,x,1-a,\frac 12,\frac 12 \rangle$ is equal to $t^*$ hence (as
in \scite{1.11}) is in pr-cl$({\frak D})$ and we are done.
\bn
\ub{$(e) \Leftrightarrow (f)$}:

Clearly (e) means that
\mr
\item "{$(*)_0$}"  there are $r_c \in [0,1]_{\Bbb R}$ for 
$c \in {\frak D}$ such that 
$1 = \Sigma\{r_c:c \in {\frak D}\}$ and
$a \in [0,1]_{\Bbb R} \backslash \{\frac
12\}$ such that $\bar t \langle x,y,a, \frac 12,\frac 12 \rangle = 
\Sigma\{r_c \times \bar t[c]:c \in {\frak D}\}$. 
\ermn
By \scite{2.4}(3) we know that $(*)_0$ is equivalent to
\mr
\item "{$(*)_1$}"  there are $r_{\bar s,\ell} \in [0,1]_{\Bbb R}$ for $\bar
s \in \text{ Prd}_\ell({\frak D}),\ell \in \{0,1\}$ such that 
{\roster
\itemitem{ $(i)$ }  $1 = \Sigma\{r_{\bar s,\ell}:\bar s \in \text{
Prd}_\ell({\frak D}),\ell \in \{0,1\}\}$
\sn
\itemitem{ $(ii)$ }  $(\frac 12,\frac 12) = 
\Sigma\{r_{\bar s,\ell} \times \bar s:\bar s \in
\text{ Prd}_\ell({\frak D}),\ell \in \{0,1\}\}$
\sn
\itemitem{ $(iii)$ }  $\frac 12 \ne a = \Sigma\{r_{\bar s,1}:
\bar s \in \text{ Prd}_1({\frak D})\}$.
\endroster}
\ermn
But by \scite{2.3}(4) and Definition \scite{2.2}(4) for $\ell \in \{0,1\}$:

$$
\text{Prd}_\ell({\frak D}) = \bigl\{ 
\bigl( \frac{\text{val}_c(x) - \ell}{\bold n -2}, \,
\frac{\text{val}_c(y)-(1-\ell)}{\bold n-2} \bigr): c \in {\frak D} \text{ and }
x \ne y \text{ and } (c\{x,y\} =y) \equiv (\ell =1)\bigr\}.
$$
\mn
Let $d^* \in {\frak D}$ recall that ${\frak D} = 
\text{ sym-cl}(\{d^*\})$ by a hypothesis of \scite{2.1} and 
$V_\ell(d^*) = \{(k_1,k_2):\text{ for
some } x_1 \ne x_2 \in X,k_1 = \text{ val}_d(x_1),k_2 = \text{
val}_d(x_2)$ and $d\{x_1,x_2\} = x_{\ell +1}\}$.  So $(*)_1$ means
(recalling the definition of Prd$_\ell({\frak D}))$
\mr
\item "{$(*)_2$}"  there is a sequence $\langle r_{\bar k,\ell}:\bar k
\in V_\ell(d^*)$ and $\ell \in \{0,1\} \rangle$ such that
{\roster
\itemitem{ $(i)$ }  $r_{\bar k,\ell} \in [0,1]_{\Bbb R}$ and
\sn
\itemitem{ $(ii)$ }  $1 = \Sigma\{r_{\bar k,\ell}:\bar k \in
V_\ell(d^*)\}$ and
\sn
\itemitem{ $(iii)$ }  $(\frac 12,\frac 12) = \Sigma \bigl\{
r_{(k_1,k_2),\ell} \times 
\bigl( \frac{k_1 - \ell}{\bold n-2}, \, 
\frac{k_2 -(1-\ell)}{\bold n-2} \bigr):\ell \in
\{0,1\}$ and \nl

\hskip65pt $(k_1,k_2) \in V_\ell(d^*)\}$
\sn
\itemitem{ $(iv)$ }  $\frac 12 \ne \Sigma\{r_{\bar k,1}:\bar k \in
V_1(d)\}$.
\endroster}
\ermn
Let us analyze $(*)_2$.  
Let $r^*_\ell = \Sigma\{r_{\bar k,\ell}:\bar k \in V_\ell(d^*)\}$ for
$\ell \in \{0,1\}$.  So
\mr
\item "{$\circledast_1$}"  $r^*_\ell \in 
[0,1]_{\Bbb R}$ and $1 = r^*_0 + r^*_1$.
\ermn
If $r^*_\ell = 0$ for some $\ell \in \{0,1\}$ 
we can finish easily.   So we assume
\mr
\item "{$\circledast_2$}"  $r^*_0,r^*_1 \ne 0$.
\ermn
Now clause $(iii)$ of $(*)_2$  means $(iii)_1 + (iii)_2$ where
\mr
\item "{$(iii)_1$}"  $\frac 12 = \frac{1}{\bold n-2} \bigl(
\Sigma\{r_{\bar k,\ell} \times k_1:\bar k \in V_\ell(d^*)$ and $\ell \in
\{0,1\}\} \bigr) -$ \nl

\hskip30pt $\frac{1}{\bold n-2} \, \Sigma\{r_{\bar k,\ell} \times \ell:
\bar k \in V_\ell(d^*),\ell \in \{0,1\}\} = $ \nl

\hskip30pt $\frac{1}{\bold n -2} \, \Sigma\{r_{\bar k,\ell} \times k_1:
\bar k \in V_\ell(d^*)$ and $\ell \in \{0,1\}\} - \frac{r^*_1}{\bold n-2}$, \nl
i.e., 
\sn
\item "{$(iii)'_1$}"  $\frac{\bold n}{2} - 
(1 - r^*_1) = \frac{\bold n -2}{2} + r^*_1 = 
\Sigma\{r_{\bar k,\ell} \times k_1:\bar k \in V_\ell(d^*)$ and $\ell \in
\{0,1\}\}$
\sn
\item "{$(iii)_2$}"  $\frac 12 = \frac{1}{\bold n -2} \,
\Sigma\{r_{\bar k,\ell} \times k_2:\bar k \in V_\ell(d^*)$ and $\ell \in
\{0,1\}\} - \frac{1}{\bold n -2} \, \Sigma\{r_{\bar k,\ell}
\times(1-\ell):$ \nl

\hskip30pt $\bar k \in V_\ell(d^*)$ and $\ell \in \{0,1\}\} =$ \nl

\hskip30pt $\frac{1}{\bold n -2} \, 
\Sigma\{r_{\bar k,\ell} \times k_2:\bar k \in
V_\ell(d^*),\ell \in \{0,1\}\} - \frac{r^*_0}{\bold n -2}$, \nl
i.e., 
\sn
\item "{$(iii)'_2$}"   $\frac{\bold n}{2} -(1 - r^*_0) = \frac{\bold
n}{2} + r^*_1 = 
\Sigma\{r_{\bar k,\ell} \times k_2:\bar k \in V_\ell(d^*)$ and $\ell \in
\{0,1\}\}$.
\ermn
Together $(iii)$ of $(*)_2$ is equivalent to 
\mr
\item "{$(iii)^+$}"  $\bigl(\frac{\bold n}{2} - (1-r^*_1),\frac{\bold
n}{2} - (1-r^*_0) \bigr) =
\Sigma\{r_{\bar k,\ell} \times \bar k:\bar k \in V_\ell(d^*)$ and $\ell \in
\{0,1\}\}$.
\ermn
Let $\bar s_\ell = \Sigma\{r_{\bar k,\ell} \times \bar k:
\bar k \in V_\ell(d^*)\} / r^*_\ell$, so $(*)_2$ is equivalent to
($V_\ell(d^*)$ is from Definition \scite{1.10})
\mr
\item "{$(*)_3$}"  there are $\bar s_0,\bar s_1,r^*_0,r^*_1$ such that
{\roster
\itemitem{ $(i)$ }  $\bar s_\ell \in \text{ convex}(V_\ell(d^*))$ for
$\ell = 0,1$
\sn
\itemitem{ $(ii)$ }  $r^*_0,r^*_1 \in [0,1]_{\Bbb R}$ and $r = r^*_0 +
r^*_1$
\sn
\itemitem{ $(iii)$ }  $(\frac{\bold n}{2} - (1-r^*_1),\frac{\bold n}{2} -
(1-r^*_0))$ is $r^*_0 \times \bar s_0 + r^*_1 \times \bar s_1$
\sn
\itemitem{ $(iv)$ }  $r^*_\ell \ne \frac 12$ (by clause $(iv)$ in $(*)_2$
above).
\endroster}
\ermn
Clearly $(*)_3(iii)$ is equivalent to
\mr
\item "{$(*)_4(iii)'$}"  $(\frac{\bold n}{2} - 1,\frac{\bold n}{2} -
1)$ is $\bigl(r^*_0 \times (\bar s_0 - (0,1)),r^*_1 \times 
(\bar s_1 - (1,0)\bigr)$.
\ermn
Recalling the definition of $V^*_\ell(d^*)$, see Definition
\scite{1.10}(4), clearly $(*)_4(iii)'$ means
\mr
\item "{$(*)_4(iii)''$}"  $(\frac{\bold n}{2} -1,\frac{\bold n}{2} -1)
\in V^*_\ell(d^*)$.
\ermn
But this is clause (f).
\sn
\ub{$(g) \Rightarrow (f)$}:  By \scite{2.5}, \scite{2.6}, \scite{2.7}
below (i.e., they show $(g) + \neg(f)$ lead to contradiction).
\bn
\ub{$(f) \Rightarrow (g)$}:  As $\neg(g) \Rightarrow \neg(f)$
trivially.

We have proved $(b)_{x,y} \Leftrightarrow (b)_{x',y'},(b)_{x,y} \Rightarrow
(a) \Rightarrow (b)_{x,y},(c) \Rightarrow (a) \Rightarrow (c),(d)
\Leftrightarrow (b)_{x,y} (d) \Rightarrow (e) \Rightarrow (d),(e)
\Leftrightarrow (f),(g) \Rightarrow (f) \Rightarrow (g)$, 
so we are done proving \scite{2.1}.  \hfill$\square_{\scite{2.1}}$\margincite{2.1} 
\bigskip

\proclaim{\stag{2.5} Claim}  Assume that clause (f) of \scite{2.1}
fails, $d = d^* \in {\frak D}$ but clause (g) of \scite{2.1} holds
(equivalently $\langle {\text{\rm val\/}}_d(x):x \in X \rangle$ is not
constant).  \ub{Then} the following hold
\mr
\item "{$\boxdot_1$}"  there are no $\bar s^0 \in {\text{\rm
conv\/}}(V^*_0(d^*)),\bar s^1 \in {\text{\rm conv\/}}(V^*_1(d^*))$
such that $(\frac{\bold n}{2} -1,\frac{\bold n}{2} -1)$ lie on
${\text{\rm conv\/}}\{\bar s^0,\bar s^1\}$ and for some $\ell \in
\{0,1\}$ this set contains an interior point of $V^*_\ell(d^*)$
\sn
\item "{$\boxdot_2$}"   the lines $L^*_0 = 
\{(\frac{\bold n}{2} -1,y):y \in \Bbb R\},L^*_1 = 
\{(x,\frac{\bold n}{2} -1):x \in \Bbb R\}$ divides the plane; and
${\text{\rm conv\/}}(V^*(d^*))$ is
{\roster
\itemitem{ $(i)$ }  included in one of the four closed half planes \ub{or}
\sn
\itemitem{ $(ii)$ }  is disjoint to at least one of the closed
quarters minus $\{(\frac{\bold n}{2} -1,\frac{\bold n}{2} -1)\}$. 
\endroster}
\ermn
\endproclaim
\bigskip

\remark{\stag{2.5A} Remark}  1) Recall $V^*(d^*) = 
V^*_0(d^*) \cup V^*_1(d^*)$ and $V^*_\ell(d^*) = 
\{\bar s - (\ell,1-\ell):\bar s \in V_1(d^*)\}$. \nl
2) So
\mr
\widestnumber\item{$(iii)$}
\item "{$(i)$}"  $(k_1,k_2) \in V^*_0(d^*) \Leftrightarrow (k_1,k_2) +
(0,1) \in V_0(d^*) \Leftrightarrow (k_1,k_2+1) \in
V_0(d^*)$
\nl
[Why?  By Definition \scite{1.10}(4).]
\sn
\item "{$(ii)$}"  $(k_2,k_1) \in V^*_1(d^*) 
\Leftrightarrow (k_2,k_1) + (1,0) \in V_1(d^*) 
\Leftrightarrow (k_2+1,k_1) \in V_1(d^*)$ \nl
(see \scite{1.11}(2)) hence
\sn
\item "{$(iii)$}"   $(k_1,k_2) \in V^*_0(d^*) \Leftrightarrow
(k_1,k_2-1) \in V_0(d^*) \Leftrightarrow (k_2-1,k_1) \in V_1(d^*)
\Leftrightarrow (k_2,k_1) \in V^*_1(d^*)$. \nl
[Why? By the above $(i) + (ii)$ and \scite{1.11}(2).]
\endroster
\endremark
\bigskip

\demo{Proof}  Toward contradiction assume that $\boxdot_2$ or
$\boxdot_1$ in the claim fails.  So necessarily
\mr
\item "{$(*)_0$}"  $(\frac{\bold n}{2}-1,\frac{\bold n}{2} -1) \notin
V^*(d^*)$ \nl
[Why?  If it belongs to  $V^*_\ell(d^*)$ let $r^*_\ell =1,r^*_{1 - \ell}
= 0$ and we get clause (f) of \scite{2.1} which we are assuming fails]
\sn
\item "{$(*)'_0$}"  $(\frac{\bold n}{2}-1,\frac{\bold n}{2} -1) \notin
\text{ conv}(V^*_\ell(d^*))$ \nl
[Why?  As in the proof of $(*)_0$.]
\sn
\item "{$(*)_1$}"  $(\frac{\bold n}{2}-1,\frac{\bold n}{2}-1)$ belongs
to the convex hull of $V^*_0(d^*) \cup V^*_1(d^*)$ hence of
conv$(V^*_0(d^*)) \cup \text{ conv}(V^*_1(d^*))$ \nl
[Why?  Otherwise $\boxdot_1$ trivially holds; also there is a line $L$ through
$(\frac{\bold n}{2}-1,\frac{\bold n}{2}-1)$ such that $V^*(d^*)
\backslash L$ lie in one half plane of $L$, so easily clause (ii) of
$\boxdot_2$ holds so $\boxdot_2$ holds.]
\ermn
Let $E = \{(\bar s_0,\bar s_1):\bar s_\ell \in \text{
conv}(V^*_\ell(d^*))$ and $(\frac{\bold n}{2}-1,\frac{\bold n}{2}-1)$
belongs to the convex hull of $\{\bar s_0,\bar s_1\}\}$
\mr
\item "{$(*)_2$}"  $E \ne \emptyset$ \nl
[Why?  By $(*)_1$]
\sn
\item "{$(*)_3$}"  if $r_0,r_1 \in [0,1]_{\Bbb R},1 = r_0 + r_1,
(\frac{\bold n}{2}-1,\frac{\bold n}{2} -1) = r_0 \times \bar s_0 + r_1
\times \bar s_1$ and $\bar s_\ell \in \text{ conv}(V^*_\ell(d^*))$ for
$\ell =0,1$  \ub{then} $r_0 = r_1
= \frac 12$ \nl
[Why?  Otherwise clause (f) holds.]
\sn
\item "{$(*)_4$}"  if $(\bar s_0,\bar s_1) \in E$ 
then $(\frac{\bold n}{2}-1,\frac{\bold n}{2}-1) = \frac 12(\bar s_0 +
\bar s_1)$ \nl
[Why?  By $(*)_3$ and the definition of $E$)
\sn
\item "{$(*)_5$}"  if $(\bar s_0,\bar s_1) \in E,i \in \{0,1\}$,
\ub{then} $\bar s_\ell$ is the unique member of conv$(V^*_\ell(d^*))$
which lies on the line through $\{\bar s_0,\bar s_1\}$ \nl
[Why?  Otherwise let $\bar s'_\ell$ be a counterexample.  If
$(\frac{\bold n}{2}-1,\frac{\bold n}{2}-1) \in \text{ conv}\{\bar
s_\ell,\bar s'_\ell\}$ then it belongs to conv$(V^*_\ell(d^*))$ 
hence clause (f) holds, contradiction.  So letting
$\bar s'_{1-\ell} = \bar s_{1-\ell}$ we know that $(\bar s'_0,\bar
s'_1) \in E$ by $(*)_4$ hence $\frac 12(\bar s'_0,\bar s'_1) =
(\frac{\bold n}{2}-1,\frac{\bold n}{2}-1) = \frac 12(\bar s_0 + \bar
s_1)$ by $(*)_4$ hence subtracting the two equations, $\bar s_{1 - \ell}$ is
cancelled and we get $\bar s'_\ell = \bar s_\ell$, contradiction]
\sn
\item "{$(*)_6$}"  $\boxdot_1$ holds (so by the assumption towards
contradiction $\boxdot_2$ fails). \nl
[Why?  Assume $\bar s_0,\bar s_1$ are as there hence (by the
definition of $E$), $(\bar s_0,s_1) \in E$, now by $(*)_5$ the set (the
line through $\bar s_0,\bar s_1$) $\cap \text{ conv}(V^*_\ell(d))$ is
equal to $\{\bar s_\ell\}$.  So the conclusion of $\boxdot_1$ for
$\bar s_0,\bar s_1$ holds as
$V^*_\ell(d^*)$ is convex for $\ell \in \{0,1\}$.]
\ermn
Easily
\mr
\item "{$(*)_7$}"   $E_\ell = 
\{\bar s_\ell:(\bar s_0,\bar s_1) \in E\}$ is a convex subset
of conv$(V^*_1(d^*)) \subseteq \Bbb R^2$.
\ermn
Also
\mr
\item "{$(*)_8$}"  $(\frac{\bold n}{2} -1,\frac{\bold n}{2}) \notin
E_\ell$ \nl
[Why?  By $(*)_5 + (*)'_0$.]
\endroster
\enddemo
\bn
Now we split the proof to three cases which trivially exhausts all the
possibilities. \nl
\ub{Case 1}:  $E$ is not a singleton. \nl
This implies by $(*)_4$  that $V^*_\ell(d^*)$ is not a singleton for
$\ell=0,1$.  As $|E| \ge 2$ by $(*)_4$
clearly $|E_1| \ge 2$.  Also by $(*)_5$ if 
$\bar s_1 \in E_1$ so $\bar s_1 \ne (\frac{\bold n}{2} -1,\frac{\bold
n}{2} -1)$ by $(*)_8$, then $\bar s_1$ is
the unique member of conv$(V^*_1(d_1)) \cap$ (the line through $\bar
s_1, (\frac{\bold n}{2}-1,\frac{\bold n}{2}-1))$.  Also $E_1$ is
convex (by $(*)_7$) so necessarily $E_1$ lies on a
line $L_1$ to which by $(*)_5$, the point
$(\frac{\bold n}{2}-1,\frac{\bold n}{2}-1)$ does not
belong. \nl
As $E_1 \subseteq V^*_1(d^*) \cap L_1$ is a convex set with $\ge 2$
members and $(*)_5$ it follows that conv$(V^*_1(d^*))$ is included 
in this line $L_1$ and as $V^*_0(d^*) = \{(k_2,k_1):(k_1,k_2) \in
V_1(d^*)\}$ (by clause $(iii)$ of \scite{2.5A}(2)) it follows that
conv$(V^*_0(d^*))$ is included in the line $L_0 = \{(a_0,a_1):(a_1,a_0) \in
L_1\}$ to which $(\frac{\bold n}{2} -1,\frac{\bold n}{2} -1)$ does not belong.

But $E_0 = \{\bar s_0:(\bar s_0,\bar s_1) \in E\}$ is necessarily an
interval of $L_0$ and by $(*)_4$ we have
\mr
\item "{$(*)_5$}"  $L_0 = \{(a_0,a_1):
2(\frac{\bold n}{2}-1,\frac{\bold n}{2} -1) - (a_0,a_1) \in L_1\}$.
\ermn
As $L_1$ is a line, for some reals $r_0,r_1,r_2$ we have

$$
L_1 = \{(a_0,a_1) \in \Bbb R^2:r_0 a_0 + r_1 a_1+r_2 = 0\}
$$
\mn
and

$$ 
(r_0,r_1) \ne (0,0).
$$
\mn
Hence by the definition of $L_0$ we have

$$
L_0 = \{(a_0,a_1) \in \Bbb R^2:r_1a_0+r_0 a_1+r_2 = 0\}
$$
\mn
and by $(*)_4$ the line $L_0$ includes the interval $\{2(\frac{\bold
n}{2} -1,\frac{\bold n}{2} -1) - \bar s_1:\bar s_1 \in E_1\}$ so

$$
L_0 = \{(a_0,a_1):(-r_0)a_0+(-r_1)a_1+r'_2 = 0\}
$$
\mn
where $r'_2 = 
2r_0(\frac{\bold n}{2}-1) + 2r_1(\frac{\bold n}{2} -1) + r_2$.

So for some $s \in \Bbb R$ we have $r_0 = -sr_1,r_1 = -sr_0,r_2 =
sr'_2$ but $(r_0,r_1) \ne (0,0)$
hence $s \in \{1,-1\}$ hence $r_0 \in \{r_1,-r_1\}$, so \wilog
\, $r_0=1,r_1 \in \{1,-1\}$.
\bn
\ub{Subcase 1A}:  $r_1 = -1$.

So $d^*\{x,y\} = y \Rightarrow (\text{val}_{d^*}(x),\text{val}_{d^*}(y)) \in
V_1(d^*) \Rightarrow (\text{val}_{d^*}(x)-1,
\text{ val}_{d^*}(x)) \in L_1 \Rightarrow \text{ val}_{d^*}(x) 
- \text{ val}_{d^*}(y) = -r_2 +1$, i.e., 
is constant, is the same for any such pair $(x,y)$.  
But the directed graph $(X,\{(u,v):c\{u,v\} =
v\})$ necessarily contains a cycle, so necessarily $r_2+1=0$, 
so the val$_{d^*}(x)$ is the same for all $x \in X$ hence is
necessarily $(\frac{\bold n}{2}-1)$, which is an ``allowable" case, in
particular, contradict clause (g) of \scite{2.1} which we are assuming.
\bn
\ub{Subcase 1B}:  $r_1 = 1$.

Clearly $x \ne y \in X \and d^*\{x,y\} = y \Rightarrow
(\text{val}_{d^*}(x),\text{val}_{d^*}(y)) \in V_1(d^*) \Rightarrow
(\text{val}_{d^*}(x)-1,\text{val}_{d^*}(y)) \in V_1(d^*) \Rightarrow 
(\text{val}_{d^*}(x)-1,\text{val}_{d^*}(y)) \in V^*_1(d^*) \Rightarrow
(\text{val}_{d^*}(x)-1,\text{val}_{d^*}(y)) \in L_1 \Rightarrow 
\text{ val}_{d^*}(x) + \text{ val}_{d^*}(x) = -r_2-1$.  As $\bold n \ge 3$
there are distinct $x_0,x_1,x_2 \in X$ so val$_{d^*}
(x_{\ell_1}) + \text{ val}_{d^*}(x_{\ell_2}) = -r_2-1$ for 
$\{\ell_1,\ell_2\} \in \{\{0,1\},\{0,2\},\{1,2\}\}$, the order is not
important as $r_1=r_2$ hence are equal and $-r_2-1$ is twice their
value.  So for $y \in X \backslash \{x_1\}$, we have val$_{d^*}(x_1) +
\text{ val}_{d^*}(y) = -r_2-1$ so val$_{d^*}(y) = 
\text{ val}_{d^*}(x_2)$, so we are done as in case 1A.
\bn
\ub{Case 2}:  $E = \{(\bar s^*_0,\bar s^*_1)\}$ and $\bar s^*_0 \ne
\bar s^*_1$.

Let $L$ be the line through $\{\bar s^*_0,\bar s^*_1\}$ so let the
real $r_0,r_1,r_2$ be such that $L= 
\{(a_0,a_1):r_0a_0+r_1a_1+r_2=0\}$ and $(r_0,r_1) \ne (0,0)$.

So by $(*)_5$ the set 
conv$(V^*_\ell(d^*))$ intersect $L$ in the singleton $\{\bar
s^*_\ell\}$.

If one (closed) half plane for the line $L$ contains $V^*_0(d^*) \cup
V^*_1(d^*)$ then $\boxdot_2$ of \scite{2.5} holds (if $L$ is not
parallel to the $x$-axis and the $y$-axis (i.e., $r_0,r_1 \ne 0$) then
$\boxdot_2(ii)$ holds, otherwise $\boxdot_2(i)$ holds.  So by $(*)_6$ 
we are done.  

If there are $\ell \in \{0,1\}$ and $\bar k',\bar k'' \in
V^*_\ell(d^*) \backslash L$ on different sides of $L$, also
$V^*_{1-\ell}(d^*)$ has a member outside $L$ (by clause $(iii)$ of
\scite{2.5A}(2) and $(*)_5$).
Hence (whether there is such $\ell$ or not, recalling the previous paragraph)
there are $\bar k_0 \in V^*_0(d^*)
\backslash \{\bar s^*_0\}$ and $\bar k_1 \in V^*_1(d^*) \backslash
\{\bar s^*_1\}$ and they are outside $L$ in different sides.  As
$(\frac{\bold n}{2}-1,\frac{\bold n}{2}-1)$ lie in the open interval
spanned by $\bar s^*_0$ and $\bar s^*_1$, necessarily
$(\frac{\bold n}{2}-1,\frac{\bold n}{2}-1)$ is an interior point of
$\{\bar s^*_0,\bar k_0,\bar s^*_1,\bar k_1\}$, contradiction to the
case assumption.
\bn
\ub{Case 3}:  $E = \{(\bar s^*_0,\bar s^*_1)\}$ and $\bar s^*_0 = \bar s^*_1$.

So $\bar s^*_\ell = (\frac{\bold n}{2} -1,\frac{\bold n}{2}-1)$, but
this contradicts $(*)'_0$ above. \hfill$\square_{\scite{2.5}}$\margincite{2.5}
\bigskip

\proclaim{\stag{2.6} Claim}  In \scite{2.5}, clause (i) of $\boxdot_2$ 
is impossible.
\endproclaim
\bigskip

\demo{Proof}  We know that $\langle \text{val}_{d^*}(x):
x \in X \rangle$ is not constant (as we assume clause (g) of
\scite{2.1} holds).  As the average of val$_{d^*}(x),x \in X$ is
$\frac{\bold n-1}{2} = \frac{\bold n}{2}-1$ clearly
\mr
\item "{$(*)_1$}"  for some points $x \in X$ we have val$_{d^*}(x)$ is 
$< \frac{\bold n-1}{2} = \frac{\bold n}{2} - \frac 12$ and for some it is 
$> \frac{\bold n}{2} - \frac 12$.
\ermn
The assumption leaves us four possibilities: which half plan and what
side, so we have 4 cases.  \hfill$\square_{\scite{2.6}}$\margincite{2.6}
\enddemo
\bn
\ub{Case 1}:  For no $(k_0,k_1) \in V^*(d^*)$ do we have $k_0 >
\frac{\bold n}{2}-1$.

It follows that

$$
(k_0,k_1) \in V_0(d^*) \Rightarrow (k_0,k_1-1) \in V^*_0(d^*)
\subseteq V^*(d^*) \Rightarrow k_0 \le \frac{\bold n}{2}-1.
$$
\mn
So if $x \in X$ and for some $y \in X \backslash \{x\}$ we have
$d^*\{x,y\} = x$ (this means just that, val$_{d^*}(x) < \bold n-1)$ then
val$_{d^*}(x) \le \frac{\bold n}{2}-1$, so
\mr
\item "{$(*)_2$}"  if $x \in X$ and val$_{d^*}(x) < \bold n-1$ \ub{then}
val$_{d^*}(x) \le \frac{\bold n}{2}-1$.
\ermn
There can be at most one $x \in X$ with val$_{d^*}(x) = \bold n -1$; if
there is none then we have:

if $x \in X$ then val$_{d^*}(x) \le \frac{\bold n}{2} - 1$.

But the average valency is $\frac{\bold n -1}{2}$ which is $>
\frac{\bold n}{2} -1$, contradiction.  So 
there is $x^* \in X$ such that val$_{d^*}(x) = \bold n-1$,
of course, it is unique.  Now $(X \backslash \{x^*\},\{(y,z):y \ne z
\in X \backslash \{x^*\},d^*\{y,z\} = z\})$ is a tournament with
$\bold n-1$ points each has out-valency $\le \frac{\bold n}{2}-1 = 
\frac{(\bold n-1)-1}{2}$, so necessarily $\bold n$ is even and every
node in the tournament has out-valency exactly $\frac{\bold n}{2} - 1 = 
\frac{(\bold n -1)-1}{2}$.
So $\bold n$ is even and as $\bold n \ge 3$ we can choose $y \ne z \in
X \backslash \{x^*\}$ and \wilog \, $d^*(y,z) = z$.  Now
$(\frac{\bold n}{2}-1,\frac{\bold
n}{2}-1),(\bold n-1,\frac{\bold n}{2}-1) \in V_1(d^*)$ as witnessed by
the pairs $(y,z),(x^*,y)$ respectively, hence
$(\frac{\bold n}{2}-2,\frac{\bold n}{2}-1),(\bold n-2,\frac{\bold
n}{2}-1) \in V^*_1(d)$, again by Definition \scite{1.11}, 
hence (as $\bold n \ge 3$ so $\bold n -2 \ge \frac{\bold n}{2} -1$) \,
$(\frac{\bold n}{2} -1,\frac{\bold n}{2}- 1) \in 
\text{ conv}(V^*_1(d^*))$, contradiction to ``clause (f) of
\scite{2.1} fails" assumed in \scite{2.5}.
\bn
\ub{Case 2}:  For no $(k_0,k_1) \in V^*(d^*)$ do we have $k_0 <
\frac{\bold n}{2}-1$.  \nl
So

$$
(k_0,k_1) \in V_0(d^*) \Rightarrow (k_0,k_1-1) \in V^*_0(d^*)
\subseteq V^*(d^*) \Rightarrow k_0 \ge \frac{\bold n}{2}-1.
$$
\mn
So if $x \in X$ and for some $y \in X \backslash \{x\}$ we have
$d^*\{x,y\} = x$ (equivalently val$_{d^*}(x) < \bold n -1)$
then $(\text{val}_{d^*}(x),\text{val}_{d^*}(y)) \in V_0(d^*)
\Rightarrow (\text{val}_{d^*}(x),\text{val}_{d^*}(y)-1) \in V^*_1(d^*)$ hence
val$_{d^*}(x) \ge \frac{\bold n}{2}-1$, but $\bold n -1 \ge
\frac{\bold n}{2} -1$ so in any case
\mr
\item "{$(*)_2$}"  if $x \in X$ then val$_{d^*}(x) \ge \frac{\bold
n}{2}-1$.
\ermn
So $x \in X \Rightarrow \text{ val}_{d^*}(x) \ge \frac{\bold n}{2}-1$.

If $\bold n$ is odd we have $x \in X \Rightarrow \text{ val}_{d^*}(x)
\ge \frac{\bold n}{2} - \frac 12 = \frac{\bold n -1}{2}$, impossible
by $(*)_1$ so $\bold n$ is even.  Let $k = \frac{\bold n}{2}-1$.  The
average val$_{d^*}(x)$ is necessarily $k + \frac 12$ hence $Y =: \{x \in
X:\text{val}_{d^*}(x) \le k$ (equivalently $=k)\}$ has at least 
$k+1 = \frac{\bold n}{2}$ members.  If $x \in
X$,val$_{d^*}(x) = k+1$ then $x \notin Y,|\{y:d\{x,y\} = y\}| = k+1 =
\frac{\bold n}{2} > |X \backslash Y \backslash \{x\}|$ 
so there is $y \in Y$ such that $d\{x,y\} = y$, hence $(k,k) =
(\text{val}_{d^*}(x)-1$, val$_{d^*}(y)) \in V_1(d^*)$ so clause (f) of
\scite{2.1} holds, contradiction. So
\mr
\item "{$(*)_3$}"  $x \in X \Rightarrow \text{ val}_{d^*}(x) \ne k+1$.
\ermn
Now $|Y| = \bold n$ is impossible by $(*)_1$.  Also if $|Y| = \bold
n-1$ let $z$ be the unique element of $X$ outside $Y$ so in the
tournament $(Y,\{(x,y):x \ne y$ are from $Y$ and $d^*\{x,y\} = y\})$
each $x$ has out-valency $\le$ val$_{d^*}(x) = k = \frac{(|Y|-1)}{2}$,
but this is the average so equality holds and we get contradiction as 
in Subcase 1b.  Hence
\mr
\item "{$(*)_4$}"  $|Y| \le \bold n -2$.
\ermn
Clearly we can find $x_1 \in Y$ such that $|\{y \in Y:y \ne x_1,d\{x,y_1\} =
y_1\}| \le \frac{|Y|-1}{2}$ (considering the average valency inside
$Y$) but $\frac{|Y|-1}{2} \le \frac{\bold n}{2} - \frac 32 = k - \frac
12$ hence there is $x_2 \in X \backslash Y$ such that
$d^*\{x_1,x_2\} = x_2$.  Now
let $m = \text{ val}_{d^*}(x_2)$ so $m > k+1$ and $(k-1,m) \in
V^*_1(d^*)$.  As $\bold n \ge 3,|Y| \ge \frac{\bold n}{2}$
obviously $|Y| \ge 2$ hence also $(k-1,k) \in V^*_1(d^*)$ and as easily
there is $x_3 \in Y$ such that $d\{x_2,x_3\} = x_3$ also $(m-1,k) \in
V^*_1(d^*)$ so $(\frac{\bold n}{2} -1,\frac{\bold n}{2} -1) = (k,k)
\in \text{ conv}\{(k-1,k),(m-1,k)\})$ recalling $m > k+1$, 
contradiction to ``not clause (f) of \scite{2.1}" with $r^*_0=0$.  
\bn
\ub{Case 3}: For no $(k_0,k_1) \in V^*(d^*)$ do we have $k_1 >
\frac{\bold n}{2} -1$.

So

$$
(k_0,k_1) \in V_1(d^*) \Rightarrow (k_0 -1,k_1) \in V^*_1(d^*)
\subseteq V^*(d^*) \Rightarrow k_1 \le \frac{\bold n}{2} -1.
$$
\mn
So if $y \in X$ and for some $x \in X \backslash \{y\}$ we have
$d\{x,y\} = y$ then $(\text{val}_{d^*}(x),\text{val}_{d^*}(y)) \in
V_1(d^*)$ hence val$_{d^*}(y) \le \frac{\bold n}{2} -1$.  But there is
such $x$ iff val$_{d^*}(y) \ne \bold n -1$, that is
\mr
\item "{$(*)_5$}"  if $y \in X$ and val$_{d^*}(y) \ne \bold n -1$ then
val$_{d^*}(y) \le \frac{\bold n}{2} -1$.
\ermn
We continue as in Case 1, (or dualize see \scite{1.11}(1)).
\bn
\ub{Case 4}:  For no $(k_0,k_1) \in V^*(d^*)$ do we have $k_1 <
\frac{\bold n}{2} -1$.  

So $(k_0,k_1) \in V_1(d^*) \Rightarrow (k_0-1,k_1) \in V^*_1(d^*)
\subseteq V^*(d^*) \Rightarrow k_1 \ge \frac{\bold n}{2} -1$.  So if
$y \in X$ and for some $x \in X \backslash \{y\}$ we have $d\{x,y\} =
y$ then $(\text{val}_{d^*}(x),\text{val}_{d^*}(y)) \in V_1(d^*)
\Rightarrow (\text{val}_{d^*}(x)-1,\text{val}_{d^*}(y)) \in V^*_1(d^*)
\subseteq V^*(d^*) \Rightarrow \text{ val}_{d^*}(y) \ge \frac{\bold
n}{2} -1$.  So if $y \in X$ and for some $x \in X \backslash \{y\}$
we have $d^*\{x,y\} = y$ then val$_{d^*}(y) \ge \frac{\bold n}{2}-1$
so if val$_{d^*}(y) < \bold n-1$ then val$_{d^*}(y) \ge \frac{\bold
n}{2} -1$, but if val$_{d^*}(y) \ge \bold n -1$ we get the same
conclusion, so
\mr
\item "{$(*)_6$}"  val$_{d^*}(y) \ge \frac{\bold n}{2} -1$
\ermn
and we can continue as in case 2.  \hfill$\square_{\scite{2.6}}$\margincite{2.6}
\bigskip

\proclaim{\stag{2.7} Claim}  In \scite{2.5}, clause (ii) of $\boxdot_2$
is impossible.
\endproclaim
\bigskip

\demo{Proof}  Note that as we are assuming clause (f) of \scite{2.1}
\mr
\item "{$(*)_0$}"  $(\frac{\bold n}{2} -1,\frac{\bold n}{2} -1) \notin
V^*(d^*)$.
\ermn
Again we have four cases.
\sn
\ub{Case 1}:  If $a_0 \ge \frac{\bold n}{2} -1,a_1 \ge \frac{\bold
n}{2} -1$ but $(a_0,a_1) \ne (\frac{\bold n}{2} -1,\frac{\bold n}{2} -1)$
then $(a_0,a_1) \notin \text{ conv}(V^*(d^*))$. \nl
So
\mr
\item "{$(*)_1$}"  for at most one $x \in X$ we have val$_{d^*}(x) \ge
\frac{\bold n}{2}$.\nl
[Why?  If $x \ne y \in X$ and val$_{d^*}(x) \ge \frac{\bold n}{2}$,
val$_{d^*}(x) \ge \frac{\bold n}{2}$ \ub{then}
$(\text{val}_{d^*}(x)-1,\text{val}_{d^*}(y)) \in V^*_0(d^*) \subseteq
V^*(d^*)$ or $(\text{val}_{d^*}(x),\text{val}_{d^*}(y)-1) \in
V^*_0(d^*) \subseteq V^*(d^*)$, a contradiction to the case assumption
in both cases.]
\ermn
If there is no $x \in X$ with val$_{d^*}(x) \ge \frac{\bold n}{2}$ 
then $x \in X \Rightarrow \text{ val}_{d^*}(x) <
\frac{\bold n}{2}$ and so $x \in X \Rightarrow \text{ val}_{d^*}(x)
\le \frac{\bold n-1}{2}$ but this is the average valency, so always equality
holds, contradicting an assumption of \scite{2.5}.
\nl
So assume
\mr
\item "{$(*)_2$}"  $x_0 \in X$, val$_{d^*}(x_0) \ge \frac{\bold
n}{2}$.
\ermn
Now
\mr
\item "{$(*)_3$}"  $y \in X \backslash \{x_0\} \Rightarrow \text{
val}_{d^*}(y) < \frac{\bold n}{2} -1$ (hence $\le \frac{\bold n}{2} -
\frac 32$) \nl
[Why?  If $d^*\{x_0,y\} = x_0$ then $(\text{val}_{d^*}(x_0)-1,
\text{val}_{d^*} (y)) \in V^*_1(d^*) \subseteq V^*(d^*)$, now
$\text{val}_{d^*}(x_0) -1 \ge \frac{\bold n}{2} -1$ hence by the case
assumption $+ (*)_0$ we have $\text{val}_{d^*}(y) < \frac{\bold n}{2}
-1$.  If $d^*(x_0,y) = y$ then
$(\text{val}_{d^*}(x_0)-1,\text{val}_{d^*}(y)) \in V^*_0(d^*)
\subseteq V^*(d^*)$ and the proof if similar.]
\ermn
So letting $Y = X \backslash \{x_0\}$ we have
$(Y,\{\{y,z\},y \ne z \in Y,d^*(y,z) =
z\})$ is a tournament each node has out-valency $\le \frac{\bold
n-3}{2} < \frac{(\bold n-1)-1}{2} = \frac{|Y|-1}{2}$, contradiction.
\enddemo
\bn
\ub{Case 2}:  If $a_1 \le \frac{\bold n}{2}-1$ and $a_2 \le
\frac{\bold n}{2}-1$ and $(a_1,a_2) \ne (\frac{\bold n}{2}
-1,\frac{\bold n}{2} -1)$ \ub{then} $(a_1,a_2) \notin \text{ conv}(V^*(d^*))$.
\nl
Clearly, as above in the proof of $(*)_3$
\mr
\item "{$(*)'_1$}"  there is at most one $x \in X$ with val$_{d^*}(x)
\le \frac{\bold n}{2}-1$.
\ermn
If there is none then $x \in X \Rightarrow \text{ val}_{d^*}(x) \ge
\frac{\bold n}{2}-1 + \frac 12 = \frac{\bold n-1}{2}$, so considering the
average of val$_{d^*}(y)$ equality always holds so clause (g) of
\scite{2.1} holds contradicting an assumption of \scite{2.5}.  So
assume
\mr
\item "{$(*)'_2$}"  $x_0 \in X$, val$_{d^*}(x_0) \le \frac{\bold n}{2}-1$
\ermn
and we can show, as above that
\mr
\item "{$(*)'_3$}"  if $y \in X \backslash \{x_0\}$
then val$_{d^*}(y) > (\frac{\bold n}{2}-1) +1 = \frac{\bold n}{2}$ so
val$_{d^*}(y) \ge \frac{\bold n-1}{2}$.
\ermn
The directed graph $(X \backslash \{x_0\},\{(y,z\}:d\{y,z\} = z\})$
has $\bold n-1$ nodes, and the out-valency of very node is 
$> \frac{\bold n}{2} -1$ hence is $\ge \frac{\bold n}{2} -1 + \frac 12
= \frac{\bold n}{2} - \frac 12 = \frac{\bold n-1}{2}$, 
(check by cases) easy contradiction as in case (1).
\bn
\ub{Case 3}:  If $a_1 \ge \frac{\bold n}{2}-1$ and $a_2 \le
\frac{\bold n}{2}-1$ and $(a_1,a_2) \ne (\frac{\bold n}{2}
-1,\frac{\bold n}{2} -1)$, \ub{then} $(a_1,a_2) \notin \text{
conv}(V^*(d^*))$.
\nl
So (as in the proof of $(*)_3$)
\mr
\item "{$\bigodot_1$}"  there cannot be $x_0,x_1 \in X$ such that
val$_{d^*}(x_0) \ge \frac{\bold n}{2}$ and val$_{d^*}(x_1) \le
\frac{\bold n}{2}-1$ (the $x_0 \ne x_1$ follows)
\ermn
so one of the following subcases hold.
\bn
\ub{Subcase 3A}:  $x \in X \Rightarrow \text{ val}_{d^*}(x) <
\frac{\bold n}{2}$.

So $x \in X \Rightarrow \text{ val}_{d^*}(x) \le \frac{\bold n-1}{2}$
and (looking at average valency) equality holds, contradicting clause
(g) of \scite{2.1} which we are assuming.
\bn
\ub{Subcase 3B}:  $x \in X \Rightarrow \text{ val}_{d^*}(x) >
\frac{\bold n}{2}-1$.

So $x \in X \Rightarrow \text{ val}_{d^*}(x) \ge \frac{\bold n-1}{2}$,
and we finish as above.
\bn
\ub{Case 4}: If $a_1 \le \frac{\bold n}{2} -1$ and $a_2 
\ge \frac{\bold n}{2}-1$ then $(a_1,a_2) \notin \text{ conv}(V(d^*))$.
\nl
Similar (or dualize the situation by \scite{1.11}(1)).
\hfill$\square_{\scite{2.7}}$\margincite{2.7}
\newpage

\head {\S3 Balanced choice functions} \endhead  \resetall \sectno=3
 % \resetall 
\bigskip

\definition{\stag{b.1} Definition}  1) $c \in {\frak C}$ is called
\ub{balanced} if $x \in X \Rightarrow \text{ val}_c(x) = (\bold n-1)/2$,
let ${\frak C}^{\text{bl}} = \{c \in {\frak C}:c$ is balanced$\}$. \nl
2) $\bar t \in \text{ pr}({\frak C})$ is called balanced \ub{if} $x
\ne y \in X \Rightarrow \Sigma\{t_{x,y}:y \in X \backslash \{x\}\} =
(\bold n-1)/2$.  Let $\text{pr}^{\text{bl}}({\frak C})$ be the set of balanced
$\bar t \in \text{ pr}({\frak C})$. \nl
3) $c \in {\frak C}$ is called pseudo-balanced \ub{if} for some
balanced $\bar t \in \text{ pr}({\frak C})$ we have

$$
c\{x,y\} = y \Rightarrow t_{x,y} > \frac 12.
$$
\mn
4) We call ${\Cal D} \subseteq {\frak C}$ balanced \ub{if} every $c
\in {\Cal D}$ is balanced, similarly $T \subseteq \text{ pr}({\frak
C})$ is called balanced if every $\bar t \in T$ is. \nl
5) If $x,y,z \in X$ are distinct, let $\bar t = \bar t^{<x,y,z>}$
be defined by: 

$t_{u,v}$ \nl

is $1$ if $(u,v) \in \{(x,y),(y,z),(z,x)\}$. \nl

is $0$ if $(u,v) \in \{(y,x),(z,y),(x,z)\}$ \nl

is $\frac 12$ if otherwise. \nl
6) For a sequence 
$\bar x = (x_0,\dotsc,x_{k-1})$ with $x_\ell \in X$ with no repetitions
and $a \in [0,1]_{\Bbb R}$ let $\bar t = 
\bar t_{\bar x,a} \in \text{ pr}({\frak C})$ be 
defined by $t_{x_i,x_j} = a,t_{x_j,x_i} = 1-a$ if
$j = i+1$ mod $k$ and $t_{x,y} = \frac 12$ otherwise.  If $a=1$ we may
omit it.
\enddefinition
\bigskip

\proclaim{\stag{3.1a} Claim}  If ${\frak D} \subseteq {\frak C}$ is non
empty, symmetric and not balanced, \ub{then} ${\text{\rm
maj-cl\/}}({\frak D}) = {\frak C}$.
\endproclaim
\bigskip

\demo{Proof}  Choose $d^* \in {\frak D}$ which is not balanced, and
let ${\frak D}' =: \text{ sym-cl}(\{d^*\})$, so ${\frak D}'$ is as in
\scite{2.1} and it satisfies clause (g) there hence it satisfies clause
(a) there.  This means that maj-cl$({\frak D}') = {\frak C}$ but
${\frak D}' \subseteq {\frak D} \subseteq {\frak C}$ hence ${\frak C}
= \text{ maj-cl}({\frak D}') \subseteq \text{ maj-cl}({\frak D})
\subseteq {\frak C}$ so we are done.  \hfill$\square_{\scite{3.1a}}$\margincite{3.1a}
\enddemo
\bn
\margintag{b.2}\ub{\stag{b.2} Fact}  1) If $c \in {\frak C}^{\text{bl}}$ then $\bar t[c]$
belongs to $\text{pr}^{\text{bl}}({\frak C})$; if $\bar t \in \text{
pr}({\frak C})$ is balanced and $c \in \text{ maj}\{\bar t\}$,
\ub{then} $c$ is pseudo-balanced; (also if $c$ is pseudo balanced then
it satisfies the condition from \scite{1.9}; and it follows from (4a)). \nl
2)  If ${\Cal D}$ is balanced, \ub{then} pr-cl$({\Cal D})$ is
balanced hence every member of maj-cl$({\Cal D})$ is
pseudo-balanced.
\bn
\margintag{b.2a}\ub{\stag{b.2a} Fact}:  1) If $c \in {\frak C}$ 
is pseudo-balanced, \ub{then} every edge
belongs to some directed cycle. \nl
2) Assume that $\bar t \in \text{ pr}({\frak C})$ is balanced, then
\mr
\item "{$(a)$}" if $t_{u,v} > \frac 12$ then we can find $k \ge 3$ and
$\langle x_0,\dotsc,x_{k-1} \rangle \in X$ with no repetitions such
that $(x_0,x_1) = (u,v)$ and $j=i+1$ mod $k \Rightarrow t_{x_i,x_j} >
\frac 12$
\sn
\item "{$(b)$}"  we can find $m(*)$ and $\bar x_m,a_m$ as in \scite{b.1}(6) 
and $r_m \ge 0$ for $m < m(*)$ satisfying $\Sigma\{r_m:m < m(*)\} =1$ such
that $\bar t^* = \Sigma\{r_m \bar t_{{\bar x}_m,a_m}:m < m(*)\}$ is
quite similar to $\bar t$: for some real $s \ge 1$ we have $u \ne v
\in X \Rightarrow t_{u,v} - \frac 12 = s(t^*_{u,v} - \frac 12)$
\sn
\item "{$(c)$}"  in (b); if all $t_{u,v}$ are rational then 
we can add $r_m = \frac{1}{m(*)},s = m(*)$.
\ermn
3) In fact (a), (b), (c) are equivalent.
\bigskip

\demo{Proof of \scite{b.2}}  1), 2) Check. 
\enddemo
\bigskip

\demo{Proof of \scite{b.2a}}  1) By (a) of part (2). \nl
2) \ub{Clause (a)}: by clause (b).
\mn
\ub{Clause (b)}:  We define a directed graph $G = G(\bar t)$ as
follows: the set of nodes is $X$, the set of edge $E = E^G = 
\{(u,v):u \ne v \in X,t_{u,v} > \frac
12\}$ and we define the function 
$\bold w = w^{\bold t}:E^{G(\bar t)} \rightarrow \Bbb R^{>
0}$ by $\bold w(u,v) = t_{u,v} - \frac 12$.  Now $(G,\bold w)$ or
$(E^G,\bold w)$ is called balanced in the sense that
\mr
\item "{$(*)_{E,\bold w)}$}"  for every node the in-valency is equal
to the out-valency, i.e., 
$\Sigma\{w(u):(u,x) \in E^G\} = \Sigma\{w(x,v):(x,v)
\in E^G\}$. 
\ermn
For such pairs $(E,\bold w)$ we shall prove that for some $(\bar
x_m,a_m)$ as in \scite{b.1}(6) for $m=0,\dotsc,m(*)-1$ we have $\bold
w^{\bold t} = \Sigma\{r_m \bold w^{{\bar t}_{{\bar x}_m,a_m}}:m <
m(*)\}$ for some $r_m \ge 0$ satisfying $1 = \Sigma\{r_m:m <m(*)\}$.
We prove this by induction on $\{(u,v) \in E:\bold w(u,v) > 0\}$.  For
any such pair $(E,\bold w)$ if $E \ne \emptyset$, by the equality 
$(*)_{(E,\bold w)}$ for any node, the in-valency is 
$> 0$ iff the out-valency is $>0$ hence
there is a directed cycle $\bar x = \langle x_0,\dotsc,x_{k-1}
\rangle$; so $a = \text{ Min}\{\bold w(x_0,x_1),\bold
w(x_1,x_2),\dotsc,\bold w(x_{k-2},x_{k-1}),\bold w(x_{k-1},x_0)\}$ is
$>0$.  Define $E' = E \backslash \{(x_i,x_j):j = i+1$ mod $k$ and
$\bold w(x_i,x_j) =a\},w'(u,v)$ is $w(u,v)-a$ if $(u,v) =
(x_i,x_j),j=i+1$ mod $k$ and $\bold w'(u,v) = \bold w(u,v)$ otherwise.
Applying the induction hypothesis to $(E',\bold w')$ and adding $\bar
t_{\bar x,a}$ we are done.  \nl
3) Should be clear.   \hfill$\square_{\scite{b.2}}$\margincite{b.2}
\enddemo
\bigskip

\proclaim{\stag{b.3} Claim}  Assume $|X| \ge 3,{\frak D} \subseteq
{\frak C}$ is symmetric, non empty and balanced.  \ub{Then}, for any
distinct $x,y,z \in Z$ we have $\bar t_{<x,y,z>} \in 
{\text{\rm pr-cl\/}}({\frak D})$. 
\endproclaim
\bigskip

\demo{Proof}  Let $d \in {\frak D}$, now as $(X,\{(u,v):d\{u,v\} =
v\})$ is a directed graph even a tournament with 
equal out-valance and in-valance
for every node, it has directed cycle and hence it has 
a triangle, i.e., $x,y,z \in X$ distinct such that
\mr
\item "{$(*)_1$}"  $d\{x,y\} = y,d\{y,z\} = z,d\{z,x\} = x$.
\ermn
Let $\Pi_{x,y,z} = \{\pi \in \text{ Per}(X):\pi \restriction
\{x,y,z\}$ is the identity$\}$.  Let $\bar t = \Sigma\{\bar
t[c^\pi]:\pi \in \Pi_{x,y,z}\}/|\Pi_{x,y,z}|$. \nl
Clearly $c^\pi \in {\frak D}$ for $\pi \in \Pi_{x,y,z}$ hence $\bar t
\in \text{ pr-cl}({\frak D})$.  Also by $(*)_1$ and the definition
of $\Pi_{x,y,z}$
\mr
\item "{$(*)_2$}"  $t_{x,y} = t_{y,z} = t_{z,y} =1$.
\ermn
Also

$$
\align
|\{w:&w \in X \backslash \{x,y,z\} \text{ and } d\{x,w\} = w\}| = \\
  &|\{w:w \in X \backslash \{x\} \text{ and } d\{x,w\} = w\}| - \{w:w
\in \{y,z\} \text{ and } d\{x,w\} = w\} = \\
  &(|X|-1)/2 -1 = (|X| -3)/2 = \\
  &|\{w:w \in X \backslash \{x,y,z\} \text{ and } d\{x,w\} = x\}|
\endalign
$$
\mn
hence
\mr
\item "{$(*)_3$}"  $t_{x,w} = 1/2$ for $w \in X \backslash \{x,y,z\}$.
\ermn
Similarly
\mr
\item "{$(*)_4$}"  $t_{y,w} = 1/2$ for $w \in X \backslash \{x,y,z\}$
\sn
\item "{$(*)_5$}"  $t_{z,w} = 1/2$ for $w \in X \backslash \{x,y,z\}$
\ermn
and even easier (and as in \S2)
\mr
\item "{$(*)_6$}"  $t_{u,v} = 1/2$ if $u \ne v \in X 
\backslash \{x,y,z\}$.
\ermn
So we are done.  \hfill$\square_{\scite{b.3}}$\margincite{b.3}
\enddemo
\bigskip

\proclaim{\stag{b.4} Claim}  Assume ${\frak D} \subseteq {\frak C}$ is
symmetric non empty and $c^* \in {\frak C}$ is pseudo balanced
\ub{then} $c^* \in {\text{\rm maj-cl\/}}({\frak D})$.
\endproclaim
\bigskip

\demo{Proof}  Without loss of generality ${\frak D}$ is balanced
(otherwise use \scite{3.1a}).  So by \scite{b.3}
\mr
\item "{$\circledast$}"  if $x,y,z \in X$ are distinct then 
$\bar t_{<x,y,z>} \in \text{ pr-cl}({\frak D})$.
\ermn
Let $\bar t^* = \bar t[c^*]$ and let $\langle \bar x^i:i < i(*) \rangle$
list the set cyc$(d)$ of tuples $\bar x = \langle x_\ell:\ell \le k
\rangle$ such that:

$$
k \ge 2,x_\ell \in X,\ell_1 < \ell_2 \le k \Rightarrow x_{\ell_1} \ne
x_{\ell_2}
$$

$$
c\{x_\ell,x_{\ell +1}\} = x_{\ell +1} \text{ for } \ell < k
$$

$$
c\{x_k,x_0\} = x_0
$$
\mn
Note
\mr
\item "{$\otimes$}"  for every $\bar x \in \text{ Cyc}(c^*)$ for some
$\bar t = \bar t^{\bar x} \in \text{ pr-cl}({\frak D})$ we have
{\roster
\itemitem{ $(a)$ }  $t_{u,v} = \frac 12 + \frac{1}{\ell g(\bar x)-2} 
\,\,\text{ \ub{if}}$ \nl

\hskip40pt $(u,v) \in \{(x_{\ell_1},x_{\ell_2}):\ell_1 < \ell g(\bar x)-1 \and
\ell_2 = \ell_1 +1 \text{ or}$ \nl

\hskip40pt $\ell_1 = \ell g(\bar x)-1 \and \ell_2=0\}$
\sn
\itemitem{ $(b)$ }  $t_{u,v} = \frac 12 
- \frac{1}{\ell g(\bar x)-2} \,\, \text{ \ub{if} }
(v,u) \text{ is as above}$
\sn
\itemitem{ $(c)$ }  $t_{u,v} = \frac 12 \text{ \ub{if} otherwise}$. 
\endroster}
\ermn
[Why?  If $\bar x = \langle x_\ell:\ell \le k \rangle$, let $\bar t$
be the arithmetic average of \nl
$\langle t^{<x_0,x_1,x_2>},
\bar t^{<x_0,x_2,x_3>},\dotsc,\bar t^{<x_0,x_{k-1},x_k>}\rangle$.] 
\sn
Now let

$$
\bar t = \Sigma\{ \frac{1}{i(*)} \bar t^{\bar x^i}:i < i(*)\}.
$$
\mn
(In fact we just need that $c\{y_0,y_1\} = y_1 \Rightarrow (y_0,y_1)$
appears in at least one cycle $\bar x^i,i < i(*)$).
\hfill$\square_{\scite{b.4}}$\margincite{b.4} 
\enddemo
\bn
So now we can give a complete answer.
\demo{\stag{b.5} Conclusion}  Assume
\mr
\item "{$(a)$}"  ${\frak D} \subseteq {\frak C}$ is symmetric non
empty
\sn
\item "{$(b)$}"  $c^* \in {\frak D}$.
\ermn
\ub{Then} $c^* \in \text{ maj-cl}({\frak D})$ iff ${\frak D}$ has
a non balanced member or $c^*$ is pseudo balanced.
\enddemo
\bigskip

\demo{Proof}  If ${\frak D}$ has no nonbalanced member and $c^*$ is
not pseudo-balanced, by \scite{b.2}(2) we know $c^* \notin$
maj-cl$({\frak D})$.  For the other direction, if 
$d^* \in {\frak D}$ is not balanced use \scite{b.2a} that is
$(a) \Leftrightarrow (g)$ 
of claim \scite{2.1} for sym-cl$\{d^*\}$.
Otherwise ${\frak D}$ is balanced, $c^*$ is pseudo balanced and 
we use \scite{b.4}.  \hfill$\square_{\scite{b.5}}$\margincite{b.5} 
\enddemo
\newpage

\head {\S4 More flexible voters} \endhead  \resetall \sectno=4
 % \resetall 
\bigskip

We may now relook and note that maj$(\bar t)$ for $\bar t \in \text{
pr}({\frak C})$ naturally allow a draw, in the case $t_{u,v} = \frac
12$.  We may allow the ``voters" to abstain, i.e., replace ${\frak C}$ by
${\frak C}_p$, the set of partial choice functions (so $\bar t = \bar
t[c]$ will be defined such that $t_{u,v} = \frac 12 \Leftrightarrow
(t\{u,v\}$ undefined)).  We may further ask what can be
$\{\text{maj}(c,\bar t):\bar t \in \text{ conv}(T)\}$ for $T$ a finite
subset of pr$({\frak C})$.  We shall continue this elsewhere. 
\newpage


% PRIVATE excursion starts here
     \shlhetal % number 1/1
% back from PRIVATE part number 1/1


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\newpage
    
REFERENCES.  
\bibliographystyle{lit-plain}
\bibliography{lista,listb,listx,listf,liste}

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816.bbl
\def\germ{\frak} \def\scr{\cal} \ifx\documentclass\undefinedcs
  \def\bf{\fam\bffam\tenbf}\def\rm{\fam0\tenrm}\fi % f**k-amstex!
  \def\defaultdefine#1#2{\expandafter\ifx\csname#1\endcsname\relax
  \expandafter\def\csname#1\endcsname{#2}\fi} \defaultdefine{Bbb}{\bf}
  \defaultdefine{frak}{\bf} \defaultdefine{mathfrak}{\frak}
  \defaultdefine{mathbb}{\bf} \defaultdefine{mathcal}{\cal}
  \defaultdefine{beth}{BETH}\defaultdefine{cal}{\bf} \def\bbfI{{\Bbb I}}
  \def\mbox{\hbox} \def\text{\hbox} \def\om{\omega} \def\Cal#1{{\bf #1}}
  \def\pcf{pcf} \defaultdefine{cf}{cf} \defaultdefine{reals}{{\Bbb R}}
  \defaultdefine{real}{{\Bbb R}} \def\restriction{{|}} \def\club{CLUB}
  \def\w{\omega} \def\exist{\exists} \def\se{{\germ se}} \def\bb{{\bf b}}
  \def\equivalence{\equiv} \let\lt< \let\gt> \def\implies{\Rightarrow}
\begin{thebibliography}{ErMo64}
\makeatletter \renewcommand{\@biblabel}[1]{[#1]} \makeatother
\def\eprintfootnotetext{References of the form {\tt math.XX/$\cdots$}
 refer to the {\tt xxx.lanl.gov} archive}
\ifx\documentstyle\undefinedcontrolsequence
   \def\anyfootnote{\footnote{*}}
   \else\def\anyfootnote{\footnote}\fi
\def\eprintfn{\ifEprint\anyfootnote{\eprintfootnotetext}\fi\Eprintfalse }
\newif\ifEprint  \Eprinttrue

\bibitem[Alo02]{Alo02}Noga Alon.
\newblock {Voting paradoxes and digraphs realizations}.
\newblock {\bf 29}:126--135, 2002.

\bibitem[ErMo64]{ErMo64}Paul Erd\H{o}s and Leo Moser.
\newblock {On the representation of directed graphs as unions of orderings}.
\newblock {\em Magyar Tud. Akad. Mat. Kutats Int. Kvzl.}, {\bf 9}:125--132,
  1964.

\bibitem[McG53]{McG53}David~C. McGarvey.
\newblock {A theorem on the construction of voting paradoxes}.
\newblock {\em Econometrica}, {\bf 21}:608--610, 1953.

\bibitem[Ste59]{Ste59}Richard Stearns.
\newblock {The voting problem}.
\newblock {\em American Mathematical Monthly}, {\bf 66}:761--763, 1959.

\end{thebibliography}
