Two cardinals models with gap one revisited
Saharon Shelah \thanks {\null\newline The author would like to thank the Israel Science Foundation for partial support of this research (Grant No. 242/03). Publication 824. \null\newline I would like to thank Alice Leonhardt for the beautiful typing.} \endthanks
cf
824.tex
% 824, processed by citealice (2003-12-22) on Sun Feb 29 12:05:04 IST 2004

\ifx\shlhetal\undefinedcontrolsequence\let\shlhetal\relax\fi
% To: "Saharon Shelah's Office" <shlhetal@math.huji.ac.il>, Saharon Shelah <shelah@math.huji.ac.il>, <shani@math.huji.ac.il>
% Subject: added ack
% Date: Fri, 27 Feb 2004 10:16:35 -0500 (EST)
% From: Alice Leonhardt <leonhard@math.rutgers.edu>
% Mime-Version: 1.0
% Content-Description: addition ISF
% 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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\def\multintlimits@{\intop\ifnum\intno@=\z@\intdots@\else\intkern@\fi
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\def\ints@@{\iflimtoken@                                                    %1
 \def\ints@@@{\iflimits@\negintic@\mathop{\intic@\multintlimits@}\limits    %2
  \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}
%  changed, april 2003, mg: we now have always the verbatim tag in the margin
\def\margincite#1{\ifshowstuffinmargin\mginpar{\zeigen{#1}}\fi}

\def\margintag#1{\ifshowstuffinmargin\mginpar{\zeigen{#1}}\fi}

\def\marginplain#1{\ifshowstuffinmargin\mginpar{{#1}}\fi}
\def\marginbf#1{\marginplain{{\bf \ \ #1}}}


%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


% Sep 2003, mg:

% % \input 300stuff
%  *** start including 300stuff.tex *** 

% definitions needed to process the 300x papers  (plus 88 etc) 

\let\thischap\jobname



\def\partof#1{\csname returnthe#1part\endcsname}
\def\chapof#1{\csname returnthe#1chap\endcsname}

\def\setchapter#1,#2,#3.{% 
  \expandafter\def\csname returnthe#1part\endcsname{#2}%
  \expandafter\def\csname returnthe#1chap\endcsname{#3}%
}

\setchapter 300a,A,I.
\setchapter 300b,A,II.
\setchapter 300c,A,III.
\setchapter 300d,A,IV.
\setchapter 300e,A,V.
\setchapter 300f,A,VI.
\setchapter 300g,A,VII.
\setchapter   88,B,I.
\setchapter  600,B,II.
\setchapter  705,B,III.


\def\cprefix#1{%     \cprefix{300b} generates A.II.  or II.  or "nothing"
% \leavevmode\vrule width 2cm height 1cm depth 0cm
\edef\theotherpart{\partof{#1}}\edef\theotherchap{\chapof{#1}}%
\ifx\theotherpart\thispart
   \ifx\theotherchap\thischap % nothing
    \else % same part, different chap
     \theotherchap%
    \fi
   \else % different part
     \theotherpart.\theotherchap\fi}


 % \sectioncite[\S4]{300b}  ->  II.\S4   or A.II.\S4
\def\sectioncite[#1]#2{%
     \cprefix{#2}#1}

\def\chaptercite#1{Chapter \cprefix{#1}}

% now define \thispart
\edef\thispart{\partof{\thischap}}
\edef\thischap{\chapof{\thischap}}


%  *** end including 300stuff.tex *** 

\def\spuriousreset{}

%  *** 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



\def\pfeilso{\leavevmode
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           \vrule width 1pt height8.3pt depth 0pt\relax
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\def\pfeilnw{\leavevmode
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\def\pfeilsw{ \leavevmode 
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\def\pfeilno{ \leavevmode 
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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



\long\def\red#1\endred{}
\long\def\green#1\endgreen{}
\long\def\blue#1\endblue{}

\def\endred{ \unmatched endred! }
\def\endgreen{ \unmatched endgreen! }
\def\endblue{ \unmatched endblue! }

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 \let\outerhead\head
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%  *** end including alice2jlem.tex *** 
\pageheight{8.5truein}
%\pageheight{48.5pc}
\topmatter
\title{Two cardinals models with gap one revisited} \endtitle
\rightheadtext{Two cardinal models}
\author {Saharon Shelah \thanks {\null\newline 
The author would like to thank the Israel Science Foundation for
partial support of this research (Grant No. 242/03). Publication 824.
\null\newline
I would like to thank Alice Leonhardt for the beautiful typing.}
\endthanks} \endauthor  

 %Previous version - 03/Dec/15

\affil{Institute of Mathematics\\
 The Hebrew University\\
 Jerusalem, Israel
 \medskip
 Rutgers University\\
 Mathematics Department\\
 New Brunswick, NJ  USA} \endaffil

\abstract   We succeed to say something on the identities of
$(\mu^+,\mu)$ when $\mu > \theta > \text{ cf}(\mu),\mu$ strong limit
$\theta$-compact or even $\mu$ limit of compact cardinals.  
This hopefully will help to prove that
\mr
\item "{$(a)$}"  the pair $(\mu^+,\mu)$ is compact and
\sn
\item "{$(b)$}"  the consistency
of ``some pair $(\mu^+,\mu)$ is not compact", however, this has 
not been proved. 
\endroster
\endabstract
\endtopmatter
\document  
 

\newpage

\head {Annotated Content} \endhead  \resetall 
 \spuriousreset
\bn
\S0 Introduction
\mr
\item "{${{}}$}"  [We give the basic definitions.]
\endroster
\bn
\S1 2-simplicity for gap one
\mr
\item "{${{}}$}"  [We prove that if $\mu = 2^{< \mu}$ then the family
of identities of $(\mu^+,\mu)$ is 2-simple.  So this applies to $\mu$
singular strong limit but also, e.g., to triples
$(\mu^+,\mu,\kappa),\mu = 2^{< \mu} > \kappa$.]
\endroster
\bn
\S2 Successor of strong limit above supercompact:2-identities 
\mr
\item "{${{}}$}"  [Consider a pair $(\mu^+,\mu)$ with $\mu$ strong
limit singular $> \theta > \text{ cf}(\mu),\theta$ a compact
cardinal. We point out quite simply 2-identities which belong to
ID$_2(\mu^+,\mu)$ but not to ID$_2(\aleph_1,\aleph_0)$.]
\endroster
\newpage

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

There has been much work on $\kappa$-compactness of pairs
$(\lambda,\mu)$ of cardinals, i.e., when: \ub{if} $T$ is a set of
first order sentences of cardinality $\le \kappa$ and every finite
subset has a $(\lambda,\mu)$ mod $M$ (i.e., $\|M\| = \lambda,|P^M| =
\mu$ for a fixed unary $P$).  \ub{Then} $T$ has a
$(\lambda,\mu)$-model.  

A particularly important case is $\lambda = \mu^+$ in which case this
can be represented as a problem on the $\kappa$-compactness of the
logic $\Bbb L(\bold Q^{\text{card}}_\lambda)$, i.e., 
$(\bold Q^{\text{card}}_{\ge \lambda} x)\varphi$ says that there are at least
$\lambda$ element $x$ satisfying $\varphi_i$.  We deal here only with
this case.  See Furkhen \cite{Fu65}, Morley and Vaught \cite{MoVa62},
Keisler \cite{Ke70}, Mitchel \cite{M1}; for more history see
\cite{Sh:604}. 

Now two cardinal theorems can be translated to partition problems: see
\cite{Sh:8}, \cite{Sh:E17}, lately Shelah and Vaananan
\cite{ShVa:790}.

Restricting ourselves to pairs $(\mu^+,\mu)$, the identities of
$(\aleph_1,\aleph_0)$ were sorted out in \cite{Sh:74}, but we do not
know of the identities of any really different pair $(\mu^+,\mu)$,
i.e., one for which $(\aleph_1,\aleph_0) \nrightarrow (\mu^+,\mu)$.
We know of some such pairs is suitable set theory.  By Mitchel
$(\aleph_2,\aleph_1)$ after suitably collapsing of a Mahlo strongly
inaccessible to $\aleph_2$.  The other, when there is a compact
cardinal in $(\text{cf}(\mu),\mu)$ by Litman and Shelah.  So it would
be nice to know (taking the extreme case).
\bn
\margintag{0.0}\ub{\stag{0.0} Question}:  Assume $\mu$ is a singular cardinal the
limit of compact and even supercompact cardinals.
\nl
1) What are the identities of $(\mu^+,\mu)$?
\nl
2) Is $(\mu^+,\mu) \, \aleph_0$-compact (equivalently $\mu$-compact)?

Note that though we already know that there are some identities of
$(\mu^+,\mu)$ which are not identities of $(\aleph_1,\aleph_0)$ we
have no explicit example.  We give here a partial solution to
\scite{0.0}(1) by finding families of such identities.

Another problem is consistency of failure of compactness.
\nl
In \cite{Sh:604} we have dealt with the simplest case for pairs
$(\lambda,\mu)$ by a reasonable criterion: including no use of 
large cardinals.  From another
perspective the simplest case is the consistency of non compactness of
$\Bbb L(\bold Q),\bold Q$ one cardinality quantifier, and the simplest
one is $\bold Q = \exists^{\ge \mu^+}$.  So we are again drawn to pairs
$(\mu^+,\mu)$, that is gap one instead of gap 2 as in \cite{Sh:604}, 
so necessarily we need to use large cardinals as if, e.g., $\neg 0^\#$
then every such pair is compact.  
\bigskip

\definition{\stag{0.1} Definition}  1) A partial \ub{identity}
\footnote{identification in the terminology of \cite{Sh:8}} 
$\bold s$ is a pair $(a,e) = (\text{Dom}_{\bold s},e_{\bold s})$ where
$a$ is a finite set and $e$ is an equivalence relation on a subfamily
of the family of the finite subsets of $a$, having the property

$$
b\, e\, c \Rightarrow |b| = |c|.
$$
\mn
The equivalence class of $b$ with respect to $e$ will be denoted
$b/e$. \nl
1A) We say $\bold s$ is a full identity or identity if Dom$(e) = {\Cal
P}(a)$. \nl
1B) We say that partial identities $\bold s_1 = (a_1,e_1),\bold s_2 =
(a_2,e_2)$ are isomorphic if there is an isomorphism $h$ from $\bold
s_1$ onto $\bold s_2$ which mean that $h$ is a one-to-one function
from $a_1$ onto $a_2$ such that for every $b_1,c_1 \subseteq a_1$ we
have $(b_1 e_1 c_1) \equiv h(b_1) e_2 h(b_2)$ (so $h$ maps Dom$(e_1)$
onto Dom$(e_2)$).  We define similarly ``$h$ is an embedding of $\bold
s_1$ into $\bold s_2$. \nl
2)  We say that $\lambda \rightarrow (a,e)_\mu$, \ub{if} $(a,e)$ is an
identity or a partial identity and for every function
$f:\,[\lambda]^{<\aleph_0}\to\mu$, there is a one-to-one function
$h:a \rightarrow \lambda$ such that

$$
b\, e\, c \Rightarrow f(h''(b)) = f(h''(c)).
$$
\mn
(Instead Rang$(f) \subseteq \mu$ we may just require $|\text{Rang}(f)|
\le \mu$, this is equivalent). \nl
3)  We define

$$
\text{ID}(\lambda,\mu) =: \{(n,e):\,n < \omega \and (n,e)
\text{ is an identity and } \lambda \rightarrow (n,e)_\mu\}
$$
\mn
and for $f:\,[\lambda]^{<\aleph_0} \rightarrow X$ we let

$$
\align
\text{ID}(f) =: \{(n,e):&(n,e) \text{ is an identity such that for some
one-to-one function} \\
  &h \text{ from } n = \{0,\dotsc,n-1\} \text{ to } \lambda \text{ we
have} \\
  &(\forall b,c \subseteq n)(b\, e\, c \Rightarrow f(h''(b)) = f(h''(c)))\}.
\endalign
$$
\enddefinition
\bn
Clearly two-place functions are easier to understand; this motivates:
\definition{\stag{0.2} Definition}  1) A two-identity or $2$-identity
\footnote{it is not an identity as $e$ is an equivalence relation on
too small set but it is a partial identity} 
is a pair $(a,e)$ where $a$ is a finite set and $e$ 
is an equivalence relation on $[a]^2$. 
Let $\lambda \rightarrow (a,e)_\mu$ mean $\lambda \rightarrow
(a,e^+)_\mu$ where $be^+ c \leftrightarrow [(bec) \vee (b = c \subseteq
a)]$ for any $b,c \subseteq a$.
\nl
2) We defined

$$
\text{ID}_2(\lambda,\mu) =: 
\{(n,e):\,(n,e) \text{ is a } 2\text{-identity and }
\lambda \rightarrow(n,e)_\mu\}
$$
\mn
we define ID$_2(f)$ when $f:[\lambda]^2 \rightarrow X$ as

$$
\align
\biggl\{(n,e):&(n,e) \text{ is a two-identity such that for some } h,
\\
  & \text{ a one-to-one function from } 
\{0,\dotsc,n-1\} \text{ into } \lambda \\
  &\text{ we have } \{\ell_1,\ell_2\} e \{k_1,k_2\} \text{ implies that }
\ell_1 \ne \ell_2 \in \{0,\dotsc,n-1\}, \\
  &k_1 \ne k_2 \in \{0,\dotsc,n-1\} \text{ and }
f(\{h(\ell_1),h(\ell_2)\}) = f(\{h(k_1),h(k_2)\}) \biggr\}.
\endalign
$$
\mn
3) Let us define

$$
\align
\text{ID}^\circledast_2 =: 
\{&({}^n 2,e):\,({}^n 2,e) \text{ is a two-identity and if} \\
  &\{\eta_1,\eta_2\} \ne \{\nu_1,\nu_2\} \text{ are } \subseteq {}^n
2, \text{ then} \\
  &\{\eta_1,\eta_2\}e\{\nu_1,\nu_2\} \Rightarrow \eta_1 \cap \eta_2
= \nu_1 \cap \nu_2\}.
\endalign
$$
\mn
4) In parts (1) and (2) we may replace 2 by $k<\omega$ (only $k <
|a_{\bold s}|$ is interesting) and by $(\le k)$.
\enddefinition
\bn
\margintag{0.3}\ub{\stag{0.3} Discussion}:  By \cite{Sh:49}, 
under the assumption $\aleph_\omega < 2^{\aleph_0}$,
the families ID$_2(\aleph_\omega,\aleph_0)$ and $ID^\circledast_2$ 
coincide (up
to an isomorphism of identities). 
In Gilchrist and Shelah \cite{GcSh:491} and \cite{GcSh:583}
we considered the question of the equality between these
ID$_2(2^{\aleph_0},\aleph_0)$ and ID$^\circledast_2$
under the assumption $2^{\aleph_0} = \aleph_2$. We showed that
consistently the answer may be ``yes" and may be ``no".

Note that $(\aleph_n,\aleph_0) \nrightarrow (\aleph_\omega,\aleph_0)$ 
so ID$(\aleph_2,\aleph_0) \ne \text{ ID}(\aleph_\omega,\aleph_0)$, but for
identities for pairs (i.e. ID$_2$) the question is meaningful.
\bn
We can look more at ordered identities 
\definition{\stag{0.4} Definition}  1) An ord-identity or order
identity is an identity
$\bold s$ such that $a_s \subseteq$ Ord or just: $a$ is an ordered set. \nl
2) $\lambda \rightarrow_{or} (\bold s)_\mu$ if $\bold s$ is an
ord-identity and for every $\bold c:[\lambda]^{< \aleph_0} \rightarrow
\mu$ we have $\bold s \in \text{ OID}(\bold c)$, see below (equivalently
Dom$(\bold c) = [\lambda]^{< \aleph_0},|\text{Rang}(\bold c)| \le \mu$). \nl
3) For $\bold c:[\lambda]^{< \aleph_0} < \mu$ let
OID$(\bold c) = \{(a,e):a$ is a set of ordinals and there is an order
preserving function $f:a \rightarrow \lambda$ such that $b_1 e b_2
\Rightarrow \bold c(f''(b_1)) = \bold c(f''(b_2))\}$. \nl
4) OID$(\lambda,\mu) = \{(n,e):(n,e) \in \text{ OID}(\bold c)$ for every
$\bold c:[\lambda]^{< \aleph_0} \rightarrow \mu \text{ we say } (n,e)
\in \text{ OID}(\bold c)\}$. \nl 
5) Similarly OID$_2$, OID$_k$, OID$_{\le k}$.
\enddefinition
\bn
Of course,
\proclaim{\stag{0.5A} Claim}  1)  {\rm ID}$(\lambda,\mu)$ can be computed from
{\rm OID}$(\lambda,\mu)$. \nl
2) Let $a$ be a finite set of ordinals and $e$ a function.
If $(a,e)$ is an identity, $a$ a set of ordinals and $\lambda >
\mu$, \ub{then} $(a,e) \in { \text{\rm ID\/}}(\lambda,\mu)$ \ub{iff} for some
permutation $\pi$ of $a$ we have $(a,e^\pi) \in { \text{\rm OID\/}}
(\lambda,\mu)$ where $e^\pi = \{(b,c):(\pi''(b),\pi''(c)) \in e\}$. \nl
3) Let $A$ be a set of ordinals, $(a,e)$ an ord-identity and $\bold c$
a function with domain $[A]^{< \aleph_0}$.  \ub{Then} 
$(a,e) \in { \text{\rm ID\/}}(\bold c)$ \ub{iff} for 
some permutation $\pi$ of $a,(a,e^\pi) \in { \text{\rm OID\/}}(\bold c)$. \nl
4) Similarly for 2-identities and $k$-identities and $(\le
k)$-identities and partial identities.
\endproclaim
\bigskip

\proclaim{\stag{0.6} Claim}  For $n \in [1,\omega)$ and $\bold s$ an
ordered partial identity \ub{then} 
there is a first order sentence $\psi_{\bold s}$ such
that: $\psi_{\bold s}$ has a $(\mu^{+n},\mu)$-model \ub{iff} $\bold s
\notin { \text{\rm OID\/}}(\mu^{+n},\mu)$.
\endproclaim
\bigskip

\demo{Proof}  Easy as for some first order $\psi$ sentence if $M$ is a
$(\mu^{+n},\mu)$-model of $\psi$ \ub{then} $<^M$ is a linear order of
$M$ (of cardinality $\mu^{+n}$) which is $\mu^{+n}$-like (i.e. every
initial segment has cardinality).  \hfill$\square_{\scite{0.6}}$\margincite{0.6}
\enddemo
\bn
We define simplicity:
\definition{\stag{0.7} Definition}  1) For $k \le \aleph_0$, we say
$(\lambda,\mu)$ has $k$-simple identities \ub{when} $(a,e) \in
\text{ ID}(\lambda,\mu) \Rightarrow (a,e') \in \text{
ID}(\lambda,\mu)$ whenever:
\mr
\item "{$(*)_k$}"  $a \subseteq \omega,(a,e)$ is an identity of
$(\lambda,\mu)$ and $e'$ is defined by
$$
b e' c \text{ iff } |b|=|c| \and (\forall b'c')[b' \subseteq b \and
|b'| \le k \and c' = \text{ OP}_{c,b}(b') \rightarrow b' ec];
$$
recall OP$_{A,B}(\alpha) = \beta$ iff $\alpha \in A \and \beta \in B
\and \text{ otp}(\alpha \cap A) = \text{ otp}(\beta \cap B)$.
\ermn
2) We define ``$(\lambda,\mu)$ for $k$-simple ordered identities".
\enddefinition
\bn
We can ask \nl
\margintag{0.8}\ub{\stag{0.8} Question}:  1) Define reasonably a pair $(\lambda,\mu)$
such that consistently
\mr
\item "{$\circledast$}"  ID$(\lambda,\mu)$ is not recursive
\sn
\item "{$\circledast'$}"  ID$(\lambda,\mu)$ is not, in a reasonable way,
finitely generated.
\ermn
2) Similarly for ID$_2(\lambda,\mu)$. \nl
3) Restrict yourself to $(\mu^+,\mu)$.
\newpage

\head {\S1 2-simplicity for gap one} \endhead  \resetall \sectno=1
 \spuriousreset
\bigskip

\proclaim{\stag{t.6} Claim}  1) If $\mu$ is strong limit singular 
\ub{then} {\rm ID}$_2(\mu^+,\mu)$ is 2-simple. \nl
2) If $\mu = 2^{< \mu}$ and $c_0:[\mu^+]^{< \aleph_0}
\rightarrow \mu$ \ub{then} we can find $c^*:[\mu^+]^2 \rightarrow \mu$ such
that:
\mr
\item "{$(\alpha)$}"  if $n \in [2,\omega)$ and
$\alpha_0,\dotsc,\alpha_{n-1} < \mu^+$ are with no repetitions and
$\beta_0,\dotsc,\beta_{n-1} < \mu^+$ are with no repetitions and $\ell
< k < n \Rightarrow c^*\{\alpha_\ell,\alpha_k\} =
c^*\{\beta_\ell,\beta_k\}$ \ub{then} $c_0\{\alpha_0,\dotsc,\alpha_{n-1}\} 
= c_0\{\beta_0,\dotsc,\beta_{n-1}\}$ and even
$c^*\{\alpha_0,\dotsc,\alpha_{n-1}\} = c^*\{\beta_0,\dotsc,\beta_{n-1}\}$
\sn
\item "{$(\beta)$}"  if in addition $\alpha_0 < \alpha_1 < \ldots <
\alpha_{n-1}$ \ub{then} $\beta_0 < \beta_1 < \ldots < \beta_{n-3} <
\beta_{n-2},\beta_{n-3} < \beta_{n-1}$.
\endroster
\endproclaim
\bigskip

\remark{\stag{t.6A} Remark}  1) We may wonder what is the gain in
\scite{t.6}(2) as compared to \scite{t.6}(1), as if $\mu = 2^{< \mu}$
is regular then we know all relevant theory on $(\mu^+,\mu)$?  The
answer is that it clarifies identities of triples $(\mu^+,\mu,\kappa)$, e.g.
\mr
\item "{$(a)$}"  $(\mu^+,\mu,\kappa),\mu$ strong limit singular $>
\kappa \ge \text{ cf}(\mu)$
\sn
\item "{$(b)$}"  $(\mu^+,\mu,\kappa),\mu = \mu^{\beth_\omega(\kappa)}$.
\ermn
2) Replacing $\mu^+,2$ by $\mu^{+k},k+1 \ge 2$ is similar and easier.
\endremark
\bigskip

\demo{Proof}  1) By part (2). \nl
2)  By $\boxdot_1 - \boxdot_5$ below the claim is easy (see details in
the end).
\mr
\item "{$\boxdot_1$}"  There is $c_1:[\mu^+]^2 \rightarrow \mu$ such
that if $\alpha_0 < \alpha_1 <\alpha_2 < \mu^+$ and $\beta_0,\beta_1,\beta_2
< \mu^+$ are with no repetitions and $c_1\{\beta_\ell,\beta_k\} =
c_1\{\alpha_\ell,\alpha_k\}$ for $\ell < k < 3$ \ub{then} at least two of
the following holds $\beta_0 < \beta_1,\beta_0 < \beta_2,\beta_1 <
\beta_2$.
\ermn
[Why?  Let $\eta_\alpha \in {}^\mu 2$ for $\alpha < \mu^+$ be pairwise
distinct and for $\alpha \ne \beta < \mu^+$ let
$\varepsilon\{\alpha,\beta\} = \text{ Min}\{\varepsilon:\eta_\alpha
\restriction \varepsilon \ne \eta_\beta \restriction \varepsilon\}$
and define the function $c'_1$ with domain $[\mu^+]^2$ by
$c'_1\{\alpha,\beta\} = \{\eta_\alpha \restriction
\varepsilon\{\alpha,\beta\},\eta_\beta \restriction
\varepsilon\{\alpha,\beta\}\}$, now $|\text{Rang}(c'_1)| \le \mu$
holds because $\mu = 2^{< \mu}$.  For $\alpha \ne \beta$, let
$c''_1\{\alpha,\beta\}$ be 1 if $(\eta_\alpha <_{\text{lex}}
\eta_\beta) \equiv (\alpha < \beta)$ and 0 otherwise (the Sierpinski
colouring).  Lastly, define $c_1$ by
$c_1,c_1\{\alpha,\beta\} =
(c'_1\{\alpha,\beta\},c''_1\{\alpha,\beta\})$, it 
is a function with domain $[\mu^+]^2$ and
range of cardinality $\le \mu$ and easily it is as required.]
\mr
\item "{$\boxdot_2$}"  for every $c:[\mu^+]^{< \aleph_0} \rightarrow \mu$
there is $c_2:[\mu^+]^2 \rightarrow \mu$ such that: if $n \ge 2,\alpha_0 <
\alpha_1 < \ldots < \alpha_{n-1} < \mu^+,\beta_0 < \beta_1 < \ldots <
\beta_{n-1} < \mu^+$ and $\ell < k < n \Rightarrow
c_2\{\alpha_\ell,\alpha_k\} = c_2\{\beta_\ell,\beta_k\}$ then
$c\{\alpha_0,\dotsc,\alpha_{n-1}\} = c\{\beta_0,\dotsc,\beta_{n-1}\}$.
\ermn
[Why?  We are given $c:[\mu^+]^{< \aleph_0} \rightarrow \mu$ and for
each $\alpha < \mu^+$ let $f_\alpha$ be a one-to-one function 
from $\alpha$ onto the ordinal $|\alpha| \le \mu$ and we shall use
those $f_\alpha$'s also later.
\nl
We define an equivalence relation $E$ on $[\mu^+]^2$
\mr
\item "{$(*)$}"  for $\alpha_1 < \beta_1 < \mu^+$ and $\alpha_2 <
\beta_2 < \mu^+$ we have $\{\alpha_1,\beta_1\} E \{\alpha_2,\beta_2\}$
\ub{iff}
{\roster
\itemitem{ $(a)$ }   $f_{\beta_1}(\alpha_1) = 
f_{\beta_2}(\alpha_2)$ and
\sn
\itemitem{ $(b)$ }   for any $n < \omega$ and 
$\gamma_0 < \ldots < \gamma_{n-1} < f_{\beta_1}(\alpha_1)$ we have
\endroster}
$$
c\{\alpha_1,\beta_1,
f^{-1}_{\beta_1}(\gamma_0),\dotsc,f^{-1}_{\beta_1}(\gamma_{n-1})\}
= c\{\alpha_2,\beta_2,
f^{-1}_{\beta_2}(\gamma_0),\dotsc,f^{-1}_{\beta_1}(\gamma_{n-1})\}
$$
and similarly if we omit $\alpha_1,\alpha_2$ and/or $\beta_1,\beta_2$.
\ermn
So $[\mu^+]^2/E$ has cardinality $\le {}^{\mu >} 2 = \mu$ 
and let $c_2:[\mu^+] \rightarrow \mu$
be such that $c_2\{\alpha_1,\beta_1\} = c_2\{\alpha_2,\beta_2\}$ iff
$\{\alpha_1,\beta_1\}/E = \{\alpha_2,\beta_2\}/E$.  We now check that 
it is as required in $\boxdot_2$.
Let $n,\langle \alpha_\ell:\ell < n \rangle,\langle \beta_\ell:\ell <
n \rangle$ be as in $\boxdot_2$; so $\ell < k < n \Rightarrow
c_2\{\alpha_\ell,\alpha_n\} = c_2\{\beta_\ell,\beta_n\}$, hence by
$(*)(a)$ above (for $k=n-1$) we have $\ell < n-1 \Rightarrow
f_{\alpha_{n-1}}(\alpha_\ell) = f_{\beta_{n-1}}(\beta_\ell)$, call it
$\gamma_\ell$.  Let $\ell(*) <n(*)$ be such that $\gamma_\ell$ is maximal.
Now apply $(*)(b)$ with
$\alpha_{\ell(*)},\alpha_{n-1},\beta_{\ell(*)},
\beta_{n-2}$ here standing for
$\alpha_1,\beta_1,\alpha_2,\beta_2$ there and we get the desired result.]
\mr
\item "{$\boxdot_3$}"   In $\boxdot_2$, using $f_\alpha:\alpha
\rightarrow \mu$ as in its proof, we have
$c\{\alpha_0,\dotsc,\alpha_{n-1}\} = c\{\beta_0,\dotsc,\beta_{n-2}\}$
also when
{\roster
\itemitem{ $(*)$ }  $n \ge 2,\alpha_0 < \alpha_1 < \ldots <
\alpha_{n-3} < \alpha_{n-2} < \alpha_{n-1} < \mu^+,\beta_0 < \beta_1 <
\ldots < \beta_{n-3} < \beta_{n-1} < \beta_{n-2}$ and $\ell < n-2
\Rightarrow f_{\alpha_{n-1}}(\alpha_\ell) =
f_{\alpha_{n-2}}(\alpha_\ell)$ and $\ell < k < n \Rightarrow
c_2\{\alpha_\ell,\alpha_k\} = c_2\{\beta_\ell,\beta_k\}$.
\endroster}
\ermn
[Why?  Just the same proof.]
\mr
\item "{$\boxdot_4$}"  there is $c_4:[\mu^+] \rightarrow \mu$ such
that if $\alpha_0 < \alpha_1 < \alpha_2 < \mu^+$ and
$\beta_0,\beta_1,\beta_2 < \mu^+$ with no repetitions,
$c_4\{\beta_\ell,\beta_k\} = c_4\{\alpha_\ell,\alpha_k\}$ for $\ell <
k < 3$ then $\beta_0 < \beta_1 \and \beta_0 < \beta_2$.
\ermn
[Why?  For $\alpha < \beta < \mu^+$ we let $c'\{\alpha,\beta\} =
\{f_\beta(\gamma):\gamma < \alpha \and f_\beta(\gamma) <
f_\beta(\beta)\}$ and let $c_4\{\alpha,\beta\} =
(c'\{\alpha,\beta\},c_1\{\alpha,\beta\},f_\beta(\alpha))$ where $c_1$
is from $\boxdot_1$ and $\langle f_\gamma:\gamma < \mu^+ \rangle$ is
from the proof of $\boxdot_2$. 
Clearly $|\text{Rang}(c')| \le \dsize \sum_{\zeta < \mu} 2^{|\zeta|} =
\mu$ hence $|\text{Rang}(c_4)| \le \mu^3 = \mu$.
If $\alpha_\ell,\beta_\ell(\ell < 3)$ form a counterexample, then 
$c_1\{\alpha_\ell,\alpha_k\} = c_1\{\beta_\ell,\beta_k\}$ for $\ell <
k < 3$ hence by $\boxdot_1$ we have four cases according to which one of the
inequalities $\beta_\ell < \beta_k,\ell < k < 3$ fail.  So the proof of
$\boxdot_4$ splits to three cases.
\bn
\ub{Case 0}:  $\beta_0 < \beta_1 < \beta_2$.

Trivial: the desired conclusion holds.
\bn
\ub{Case 1}:  $\beta_1 < \beta_0$ so $\beta_1 < \beta_0 < \beta_2$.

Let $\zeta_\ell = f_{\alpha_2}(\alpha_\ell)$ for $\ell = 0,1$ hence
$\zeta_0 \ne \zeta_1$ as $f_{\alpha_2}$ is one to one and $\zeta_\ell
= f_{\beta_2}(\beta_\ell)$.  Now on the one hand if
$\zeta_0 < \zeta_1$ then $c'\{\alpha_1,\alpha_2\} 
\ne c'\{\beta_1,\beta_2\}$ (as 
$\zeta_0 \in c'\{\alpha_1,\alpha_2\},\zeta_0 \notin
c'\{\beta_1,\beta_2\})$, contradiction.  On the other hand 
if $\zeta_1 < \zeta_0$ then $c'\{\alpha_0,\alpha_2\} \ne
c'\{\beta_0,\beta_2\}$ (as $\zeta_1 \in c'\{\beta_0,\beta_2\},
\zeta_1 \notin c'\{\alpha_0,\alpha_2\}$), a contradiction, too.
\bn
\ub{Case 2}:  $\beta_2 < \beta_0$.

Then at least one of $\beta_1 < \beta_0,\beta_2 < \beta_1$ hold
contradicting $\boxdot_1$, (i.e., the case we are in).
\bn
\ub{Case 3}:  $\beta_2 < \beta_1$.

By $\boxdot_1$ we have $\beta_0 < \beta_2 < \beta_1$.

This is O.K. for $\boxdot_4$.]
\mr
\item "{$\boxdot_5$}"  for every $c:[\mu^+]^2 \rightarrow \mu$ there
is $c_5:[\mu^+]^2 \rightarrow \mu$ such that
{\roster
\itemitem{ $(a)$ }  $c_5\{\alpha_1,\beta_1\} = c_5\{\alpha_2,\beta_2\}
\Rightarrow c_2\{\alpha_1,\beta_1\} = c_2\{\alpha_2,\beta_2\}$ where
$c_2$ is from $\boxdot_2$ (so also $\boxdot_3$)
\sn
\itemitem{ $(b)$ }  there are no $\alpha_0 < \alpha_1 < \alpha_2 <
\mu^+$ and $\beta_0 < \beta_1 < \beta_2 < \mu^+$ such that
$f_{\alpha_2}(\alpha_0) \ne
f_{\alpha_1}(\alpha_0),c_5\{\alpha_0,\alpha_1\} =
c_5\{\beta_0,\beta_2\},c_5\{\alpha_0,\alpha_2\} = c_5\{\beta_0,\beta_1\}$
and $c_5\{\alpha_1,\alpha_2\} = c_5\{\beta_1,\beta_2\}$
\sn
\itemitem{ $(c)$ }  $c_5\{\alpha_1,\beta_1\} =
c_5\{\alpha_2,\beta_2\} \Rightarrow c_4\{\alpha_1,\beta_1\} =
c_4\{\alpha_2,\beta_2\}$ where $c_4$ is from $\boxdot_4$.
\sn
[Why?  Let $\kappa = \text{ cf}(\mu) \le \mu$ and $\mu = \dsize \sum_{i <
\kappa} \lambda_i$ be such that if $\mu$ is a limit cardinal then 
$\lambda_i$ is (strictly) increasing continuous and if $\mu$ is a successor
cardinal then $\mu = \lambda^+$ and $\lambda_i = \lambda$ for $i <
\kappa$.   We can find $d:[\mu^+]^2 \rightarrow \kappa$ and $\bar g$ such that
\sn
\itemitem{ $\circledast_0$ }  $\qquad (i) \quad$ for 
$\beta < \mu^+,i < \kappa$ the set
$A_{\beta,i} =: \{\alpha < \beta:d\{\alpha,\beta\} \le i\}$ has \nl

\hskip55pt cardinality $\le \lambda_i$ and
\sn
\itemitem{ ${{}}$ } $\qquad (ii) \quad$ if 
$\alpha < \beta < \gamma < \mu^+$ then
$d\{\alpha,\gamma\} \le \text{
max}\{d\{\alpha,\beta\},d\{\beta,\gamma\}\}$
\sn
\itemitem{ ${{}}$ } $\qquad (iii) \quad \bar g$ is a sequence 
$\langle g_\alpha:\alpha < \mu^+ \rangle$ 
\sn
\itemitem{ ${{}}$ }  $\qquad (iv) \quad g_\alpha:\alpha \rightarrow
\mu$ is one to one and \nl

\hskip60pt $\lambda^+_i < \mu \and i < \kappa \and \alpha < 
\beta \Rightarrow ((g_\beta(\alpha) < \lambda^+_i) \equiv$ \nl

\hskip55pt $(d\{\alpha,\beta\} \le i))$
\sn
\itemitem{ ${{}}$ }  $\qquad (v) \quad$ if $\alpha <
\beta,d\{\alpha,\beta\} = i$ and $\lambda^+_i = \mu$ then
$g_\beta(\alpha) < d\{\alpha,\beta\}$.
\endroster}
\ermn
[Why we can find them?  By induction on $\beta < \mu^+$
by induction on $i < \mu$ for $\alpha=f^{-1}_\beta(i)$ 
we choose $d\{\alpha,\beta\}$ and $g_\beta(\alpha)$ as required.]
\sn
Define the functions $c'_6$ and $c'_7$ 
with domain $[\mu^+]^2$ as follows: if $\alpha < \beta$ then \nl
$c'_6\{\alpha,\beta\} = \{(t,\zeta_0,\zeta_1):\zeta_0,\zeta_1 \le
g_\beta(\alpha),t<2$ and $t=0 \Rightarrow g^{-1}_\beta(\zeta_1) <
g_\beta(\zeta_2),t=1 \Rightarrow g_\beta(\zeta_1) > g_\beta(\zeta_2)\}$ and 
$c'_7\{\alpha,\beta\} = \{(t,\zeta,\xi):\zeta \in 
\lambda^+_{d\{\alpha,\beta\}} \cap \text{ Rang}(g_\alpha)$ and $\xi \in
\lambda^+_{d\{\alpha,\beta\}} \cap \text{ Rang}(g_\beta)$ and
$[\lambda^+_{d\{\alpha,\beta\}} = \mu \Rightarrow \zeta <
d\{\alpha,\beta\} \and \xi < d\{\alpha,\beta\}]$  
and $g^{-1}_\alpha(\zeta) < g^{-1}_\beta(\xi) \and t=0$ or
$g^{-1}_\alpha(\zeta) = g^{-1}_\beta(\xi)
\and t=1$ or $g^{-1}_\alpha(\zeta) > g^{-1}_\beta(\xi) \and t=2\}$.

Now for $\alpha < \beta < \mu^+$ we define $c'_5\{\alpha,\beta\} \in
\Pi\{\lambda^+_j:j \le d\{\alpha,\beta\}\}$, we do this by induction
on $\beta$ and for a fixed $\beta$ by induction $i=d\{\alpha,\beta\}$
and for a fixed $\beta$ and $i$ by induction on $\alpha$. \nl
Arriving to $\alpha < \beta$ so $\zeta < 
\lambda^+_{d\{\alpha,\beta\}}$, for each
$j \le d\{\alpha,\beta\}$, let $(c'_5\{\alpha,\beta\})(j)$ be the first
ordinal $\xi < \lambda^+_j$ such that:
\mr
\item "{$\circledast_1$}"  if $\gamma < \beta \and 
d\{\gamma,\beta\} \le j \and (d\{\gamma,\beta\} = d\{\alpha,\beta\} 
\Rightarrow \gamma < \alpha)$ then
$$
(c'_5\{\alpha,\gamma\})(j) < \xi.
$$
\ermn
Clearly possible.  The colouring we use is $c_5$ where for $\alpha <
\beta < \mu^+$ we let
$c_5\{\alpha,\beta\} = (d\{\alpha,\beta\},g_\beta(\alpha),f_\beta(\alpha),
c_2\{\alpha,\beta\},c'_5\{\alpha,\beta\},
c'_6\{\alpha,\beta\},c'_7\{\alpha,\beta\},c_4\{\alpha,\beta\})$,
recalling $c_4$ is from $\boxdot_4$ and $c_2$ is from $\boxdot_2$.
Obviously, $|\text{Rang}(c_5)| \le \mu$ and clauses (a) + (c) of $\boxdot_5$
holds.
So assume $\alpha_0 < \alpha_1 < \alpha_2,\beta_0 < \beta_1 < \beta_2$
form a counterexample to clause (b) of $\boxdot_5$ and we shall
eventually derive a contradiction.
\sn
Clearly
\mr
\item "{$\circledast_2$}"  $(i) \quad d\{\alpha_0,\alpha_2\} =
d\{\beta_0,\beta_1\},d\{\alpha_0,\alpha_1\} =
d\{\beta_0,\beta_2\},d\{\alpha_1,\alpha_2\} = d\{\beta_1,\beta_2\}$
\sn
\item "{${{}}$}"  $(ii) \quad$ similarly for $c',c'_0,c'_1,c_4$.
\ermn
By clause (ii) above we have $d\{\alpha_0,\alpha_2\} \le 
\text{ max}\{d\{\alpha_0,\alpha_1\},d\{\alpha_1,\alpha_2\}\}$, and applying
clause (ii) to $\beta_0 < \beta_1 < \beta_2$ and using $\circledast_2$ we have
$d\{\alpha_0,\alpha_1\} \le \text{ max}\{d\{\alpha_0,\alpha_2\},
d\{\alpha_1,\alpha_2\}$.  Hence
$d\{\alpha_0,\alpha_1\} =
d\{\alpha_0,\alpha_2\} > d\{\alpha_1,\alpha_2\}$ or $\dsize
\bigwedge^2_{\ell=1} [d\{\alpha_0,\alpha_\ell\} \le
d\{\alpha_1,\alpha_2\}]$;
we deal with those two cases separately.
\bn
\ub{Case 1}:  $\varepsilon = d\{\alpha_0,\alpha_1\} = d\{\alpha_0,\alpha_2\} >
d\{\alpha_1,\alpha_2\}$. \nl
So (see the definition of $c'_5$, with
$\alpha_0,\alpha_2,\alpha_1,\varepsilon$ here standing for
$\alpha,\beta,\gamma,j$ there recalling that
$\alpha_0 < \alpha_1 < \alpha_2$) we have $\lambda^+_\varepsilon >
(c'_5\{\alpha_0,\alpha_2\})(\varepsilon) > (c'_5\{\alpha_0,\alpha_1\})
(\varepsilon)$.
Similarly, $\lambda^+_\varepsilon > (c'_5\{\beta_0,\beta_2\})(\varepsilon) >
(c'_5\{\beta_0,\beta_1\})(\varepsilon)$.  
This contradicts $c'_5\{\alpha_0,\alpha_\ell\}
= c'_5\{\beta_0,\beta_{3 - \ell}\}$ for $\ell=1,2$.
\bn
\ub{Case 2}:  $d\{\alpha_0,\alpha_\ell\} \le
d\{\alpha_1,\alpha_2\}$ for $\ell=1,2$.

Let $\varepsilon = d\{\alpha_1,\alpha_2\}$.  Let $\zeta_\ell =
g_{\alpha_\ell}(\alpha_0)$ for $\ell = 1,2$ so $\zeta_\ell =
g_{\beta_{3 - \ell}}(\beta_0)$ for $\ell = 1,2$.
By the assumption toward contradiction, i.e., by a demand
in clause (b) of $\boxdot_5$ we have $\zeta_1 \ne
\zeta_2$.  Clearly $\zeta_\ell < \lambda^+_{d\{\alpha_0,\alpha_\ell\}}
\le \lambda^+_{d\{\alpha_1,\alpha_2\}} = \lambda^+_\varepsilon$ and
$\lambda^+_\varepsilon = \mu \Rightarrow \zeta_\ell <
d\{\alpha_0,\alpha_\ell\} \le d\{\alpha_1,\alpha_2\} \le \varepsilon$.

As $c'_7\{\alpha_1,\alpha_2\} = c'_7\{\beta_1,\beta_2\}$ and
$g^{-1}_{\alpha_1}(\zeta_1) = g^{-1}_{\alpha_2}(\zeta_2)$ clearly
$g^{-1}_{\beta_1}(\zeta_1) = g^{-1}_{\beta_2}(\zeta_2)$ and they are
well defined.

For $\ell=1,2$ as $c_5\{\alpha_0,\alpha_\ell\} = 
c_5\{\beta_0,\beta_{3-\ell}\}$ by the 
choice of $\zeta_\ell$ (that is $\zeta_\ell =
g_{\alpha_\ell}(\alpha_0))$ we have 
$g_{\beta_\ell}(\beta_0) = \zeta_{3-\ell}$ so 
$g^{-1}_{\beta_\ell}(\zeta_{3-\ell}) = \beta_0$ for $\ell=1,2$ hence
$g^{-1}_{\beta_1}(\zeta_2) = g^{-1}_{\beta_2}(\zeta_1)$.
As $c_5\{\alpha_1,\alpha_2\} = c_5\{\beta_1,\beta_2\}$ we have
$c'_7\{\alpha_1,\alpha_2\} = c'_7\{\beta_1,\beta_2\}$ but $\zeta_1,\zeta_2
\le g_{\alpha_2}(\alpha_1)$ hence
\mr
\item "{$\circledast_3$}"  $(g^{-1}_{\alpha_\ell}(\zeta_1) < 
g^{-1}_{\alpha_\ell}(\zeta_2)) \equiv
(g^{-1}_{\beta_\ell}(\zeta_1) < g^{-1}_{\beta_\ell}(\zeta_2))$ 
for $\ell = 1,2$.
\ermn
As $\zeta_1 \ne \zeta_2$ we have $g^{-1}_{\alpha_1}(\zeta_1) \ne
g^{-1}_{\alpha_1}(\zeta_2)$. 

By symmetry \wilog \, $\zeta_1 > \zeta_2$ so 
$g^{-1}_{\beta_1}(\zeta_1) < g^{-1}_{\beta_1}(\zeta_2)$ \ub{iff} (by
equalities above) $g^{-1}_{\beta_2}(\zeta_2) < g^{-1}_{\beta_2}(\zeta_1)$
\ub{iff} (the equivalence in $\circledast_3$) 
$g^{-1}_{\alpha_2}(\zeta_2) < g^{-1}_{\alpha_2}(\zeta_1)$ \ub{iff} by
the choice of $\zeta_1,g^{-1}_{\alpha_2}(\zeta_1) =
\alpha_0),g^{-1}_{\alpha_2}(\zeta_2) < \alpha_0$ \ub{iff} (as 
$c'_5\{\alpha_0,\alpha_2\} = c'_5\{\beta_0,\beta_1\}$ and $\zeta_2 <
\zeta_1 = g_{\alpha_1}(\beta_0)),
g^{-1}_{\beta_1}(\zeta_2) < \beta_0$ \ub{iff} 
(as $\beta_0 = g^{-1}_{\beta_1}(\zeta_1)),g^{-1}_{\beta_1}(\zeta_2) <
g^{-1}_{\beta_1}(\zeta_1)$, clear contradiction. \nl
So we have proved $\boxdot_5$.
\enddemo
\bn
We can now sum up, i.e.:
\demo{Proof of \scite{t.6}(2) from $\boxdot_1 - \boxdot_5$}   We are given
$c_0:[\mu^+]^{< \aleph_0} \rightarrow \mu$.  First we apply $\boxdot_2$
for $c=c_0$ and get $c_2:[\mu^+]^2 \rightarrow \mu$ as there. \nl
Second, we apply $\boxdot_5$ for $c=c_2$ and get $c_5$ as there.  Let
us check that $c_5$ is as required on $c^*$ in \scite{t.6}(2).  So assume
$(*)_0 + (*)_1$ below and (as the case $n=2$ is trivial)
assume $n \ge 3$ where
\mr
\item "{$(*)_0$}"  $\{\alpha_0,\dotsc,\alpha_{n-1}\} \in [\mu^+]^n$
and $\{\beta_0,\dotsc,\beta_{n-1}\} \in [\mu^+]^n$ and
\sn
\item "{$(*)_1$}"  $\ell < k < n \Rightarrow
c_5\{\alpha_\ell,\alpha_k\} = c_5\{\beta_\ell,\beta_k\}$. 
\ermn
Without loss of generality (by renaming)
\mr
\item "{$(*)_2$}"   $\alpha_0 < \ldots < \alpha_{n-1}$. 
\ermn
and it is enough to prove that $c_0\{\alpha_0,\dotsc,\alpha_{n-1}\} =
c_0\{\beta_0,\dotsc,\beta_{n-1}\}$. 
By clause (a) of $\boxdot_5$ we have
\mr
\item "{$(*)_3$}"   $\ell < k < n \Rightarrow
c_2\{\alpha_\ell,\alpha_k\} = c_2\{\beta_\ell,\beta_k\}$. 
\ermn
By clause (c) of $\boxdot_5$ we have
\mr
\item "{$(*)_4$}"  $\ell < k < n \Rightarrow
c_4\{\alpha_\ell,\alpha_k\} = c_4\{\beta_\ell,\beta_k\}$. 
\ermn
Hence by $\boxdot_4$ we have
\mr
\item "{$(*)_5$}"   if $\ell < k < n$ and $\ell < n-2$ then
$\beta_\ell < \beta_k$.
\ermn
[Why?  Apply $\boxdot_4$  to $\alpha_\ell,\alpha_{\ell
+1},\alpha_k;\beta_\ell,\beta_{\ell +1},\beta_k$ if $\ell +1 < k$, and
apply $\boxdot_4$ to \nl
$\alpha_\ell,\alpha_{\ell +1},\alpha_{\ell
+2};\beta_\ell,\beta_{\ell +1},\beta_{\ell +2}$ if $\ell+1=k$.] \nl
So
\mr
\item "{$(*)_6(i)$}"  $\beta_0 < \beta_1 < \ldots < \beta_{n-3} <
\beta_{n-2} < \beta_{n-1}$ or
\sn
\item "{${{}}(ii)$}"  $\beta_0 < \beta_1 < \ldots < \beta_{n-3} <
\beta_{n-1} < \beta_{n-2}$.
\ermn
So clause $(\beta)$ of \scite{t.6} holds.

If $(i)$ of $(*)_6$ holds, then the choice of $c_2$, i.e., by $\boxdot_2$
and $(*)_3$ above we get $c_0\{\alpha_0,\dotsc,\alpha_{n-1}\} =
c_0\{\beta_0,\dotsc,\beta_{n-1}\}$ so we are done.  Otherwise we have
$(ii)$ of $(*)_6$ so by clause (b) of $\boxdot_5$ we have
\mr
\item "{$(*)_7$}"  if $\ell < n-2$ then $f_{\alpha_{n-1}}(\alpha_\ell)
= f_{\beta_{n-2}}(\beta_\ell)$.
\ermn
[Why?  Apply $\boxdot_5(b)$ to $\alpha_\ell,\alpha_{n-2},\alpha_{n-1};
\beta_\ell,\beta_{n-2},\beta_{n-1}$.] \nl
So by $\boxdot_3$ we get $c_0\{\alpha_0,\dotsc,\alpha_{n-1}\} = 
c_0\{\beta_0,\dotsc,\beta_{n-1}\}$ finishing.
\hfill$\square_{\scite{t.6}}$\margincite{t.6}
\enddemo
\bigskip

\proclaim{\stag{t.7} Claim}  Defining {\rm ID}$(\lambda,\mu)$, we
can restrict ourselves to $c:[\lambda]^{< \aleph_0} \rightarrow \mu$
such that $c \restriction [\lambda]^1$ is constant \ub{if} 
{\rm cf}$(\lambda) > \mu$.
\endproclaim
\bigskip

\proclaim{\stag{t.7A} Claim}  1) Assume $\mu = \mu^{< \mu}$ and $n \in
[1,\omega)$.  The identities of {\rm ID}$(\mu^{+n},\mu)$ are $(n+1)$-simple
(and also {\rm OID}$(\mu^+,\mu))$.
\endproclaim
\bigskip

\demo{Proof}  As in \scite{t.6}, only easier in the additional cases.
\hfill$\square_{\scite{t.8}}$\margincite{t.8}
\enddemo
\newpage

\head {\S2 Successor of strong limit above supercompact: 2-identities} \endhead  \resetall \sectno=2
 \spuriousreset
\bigskip

So we know that if $\mu$ is strong limit singular and there is a compact
cardinal in (cf$(\mu),\mu$) then ID$_2(\mu^+,\mu) 
\ne \text{ ID}_2(\aleph_1,\aleph_0)$.  It seems desirable to find
explicitly such 2-identity.
\bn
The proof of the following does much more.
\proclaim{\stag{t.8} Claim}  Assume
\mr
\item "{$(a)$}"  $\bold s_k = (k + \binom k2,e_{{\bold s}_k})$ where the
non-singleton $e_{\bold s_k}$-equivalence classes are the set sets
here $\binom 12 = 0$) \nl

$\{\{\ell_0,\ell_2\}:\ell_0 < k$ and for some $\ell_1 \in 
\{\ell_0+1,\dotsc,k-1\}$ we have 
$\ell_2 = k + {\binom{\ell_1}2} + \ell_0\}$ and \nl

$\{\{\ell_1,\ell_2\}:\ell_1 < k$ and for some $\ell_0 < \ell_1$ we
have $\ell_2 = k + \binom{\ell_1}2 + \ell_0\}$
\sn
\item "{$(b)$}"  $\mu$ is strong limit, $\theta$ a compact cardinal and
\, ${\text{\rm cf\/}}(\mu) < \theta < \mu$.
\ermn
1) $\bold s_k \in { \text{\rm ID\/}}_2(\mu^+,\mu)$, moreover $\bold s_k \in
{ \text{\rm OID\/}}_2(\mu^+,\mu)$. \nl
2) $\bold s_k \notin { \text{\rm ID\/}}_2(\aleph_1,\aleph_0)$ for $k \ge 3$ so
for $k=3$ we have 
$\bold s_k = (6,e_{\bold s})$ and the non-singleton equivalence
classes, after permuting $\{3,5\}$ are $\{\{1,3\},\{0,4\},\{0,5\}\}$ and
$\{\{1,5\},\{2,3\},\{2,4\}\}$. 
\endproclaim
\bigskip

\demo{Proof}  Part (1) 
follows from subclaim \scite{t.8A}(3) below and part (2) follows from
\scite{t.8B} below.  \hfill$\square_{\scite{t.8}}$\margincite{t.8}
\enddemo
\bigskip

\proclaim{\stag{t.8A} Claim}  Assume  
\mr
\item "{$(a)$}"  $\mu$ is strong limit,
\sn
\item "{$(b)$}"   $\theta$ is compact and \, ${\text{\rm cf\/}}(\mu) 
< \theta < \mu$
\sn
\item "{$(c)$}"  $\kappa = \,{\text{\rm cf\/}}(\mu),
\langle \lambda_i:i < \kappa
\rangle$ is increasing with limit $\mu$
\sn
\item "{$(d)$}"  $c:[\mu^+]^2 \rightarrow \mu$
\sn
\item "{$(e)$}"  $d\{\alpha,\beta\} = \,{\text{\rm Min\/}}\{i:c\{\alpha,\beta\}
< \lambda_i\}$.
\ermn
1) We can find $i(*),A,f$ such that
\mr
\item "{$(*)(i)$}"  $i(*) < \kappa,A \in [\mu^+]^{\mu^+}$ and $f:A \rightarrow
\lambda_{i(*)}$ 
\sn
\item "{$(ii)$}"  for every set $B \subseteq A$ of cardinality $<
\theta$ there are $\mu^+$ ordinals $\gamma \in A$ satisfying $(\forall
\alpha \in B)[d\{\alpha,\gamma\} = f(\alpha)]$.
\ermn
2) In part (1) we also have: if $A_1 \subseteq A,|A_1| \ge
\beth_n(\lambda)^+$ and $\lambda_{i(*)} \le \lambda < \mu$, \ub{then} for
some $\langle \gamma_\ell:\ell < n \rangle \in {}^n(\lambda_{i(*)})$ 
and $B \in [A_1]^\lambda$ for every $\alpha_0 <
\ldots < \alpha_{n-1}$ from $B$ for arbitrarily large $\beta <
\lambda$ we have $\ell < n \Rightarrow c\{\alpha_\ell,\beta\} =
\gamma_\ell$. \nl
3) $\bold s_k \in { \text{\rm ID\/}}_2(c)$ 
where $\bold s_k$ is from clause (a) of \scite{t.8}.
\endproclaim
\bigskip

\demo{Proof}  1) Let $D$ be a uniform $\theta$-complete ultrafilter on
$\mu^+$.

Define $f:\mu^+ \rightarrow \kappa$ by $f(\alpha) = i \Leftrightarrow
\{\gamma < \mu^+:d\{\alpha,\gamma\} = i\} \in D$, note that the
function $f$ is well defined as $D$
is a $\theta$-complete ultrafilter on $\mu^+$ and 
$\theta > \kappa$.  So for some $i(*)$, the
set $A =: \{\alpha < \mu^+:f(\alpha) = i(*)\}$ belongs to $D$ and 
check that $(*)$ holds, that is (i) + (ii) hold. \nl
2) Define $c^*:[A]^n \rightarrow {}^n(\lambda_{i(*)})$ such that
\mr
\item "{$\circledast$}"  if $\alpha_0 < \ldots < \alpha_{n-1}$ are
from $A$ then for $\mu^+$ ordinals $\beta < \mu^+$ we have $\langle
c\{\alpha_\ell,\beta\}:\ell < n\}\rangle =
c^*\{\alpha_0,\dotsc,\alpha_{n-1}\}$.
\ermn
So Rang$(c^*)$ has cardinality $\le (\lambda_{i(*)})^n =
\lambda_{i(*)}$ hence by the Erd\"os-Rado theorem there is $B \subseteq A_1$
infinite (even of any pregiven cardinality $< \lambda$) such that $c^*
\restriction [B]^n$ is constant.
 \nl
3) Straight: in part (2) use $n=2,A_1=A$ and get $B$ and $\langle
\gamma_0,\gamma_1 \rangle \in {}^2(\lambda_{i(*)})$ as there and choose
$\alpha_0 < \ldots < \alpha_{k-1}$ from $B$.  
Next choose $\alpha_\ell$ for $\ell = 0,1,\dotsc,\binom k2 -1$,
choosing $\beta_\ell$ by induction on $\ell$.  If 
$\ell = \binom{\ell_1}2 + \ell_0$ and $\ell_0 < \ell_1 < k$
choose $\beta_\ell \in A$ satisfying $\beta_\ell > \alpha_{k-1}$ and 
$\beta_\ell > \beta_m$ for $m < \ell$ such that
$c\{\alpha_{\ell_0},\beta_\ell\} =
\gamma_0,c\{\alpha_{\ell_1},\beta_\ell\} = \gamma_1$. \nl
Now let $\alpha_{k + \ell} = \beta_\ell$ for $\ell < \binom k2$, and
clearly $\langle \alpha_\ell:\ell < k + \binom k2 \rangle$ realize the
identity $\bold s_k$.  \nl
${{}}$  \hfill$\square_{\scite{t.8A}}$\margincite{t.8A}
\enddemo
\bigskip

\proclaim{\stag{t.8B} Subclaim}  1) If $\bold s \in 
{ \text{\rm ID\/}}_2(\aleph_1,\aleph_0)$, \ub{then} we can find a function
$h:[{\text{\rm Dom\/}}_{\bold s}]^2/\bold s \rightarrow \omega$
respecting $e_{\bold s}$ (i.e. $\{\ell_1,\ell_2\} e_{\bold
s}\{\ell_3,\ell_4\} \Rightarrow h\{\ell_1,\ell_2\} =
h\{\ell_3,\ell_4\}$) and there is a linear order $<$ of 
{\rm Dom}$_{\bold s}$ satisfying
\mr
\item "{$\circledast$}"  for any equivalence class $\bold a$ of $e$
there are $a_0,a_1$ such that
{\roster
\itemitem{ $(i)$ }  $a_0,a_1$ are disjoint finite subsets of
{\rm Dom}$_{\bold s}$
\sn
\itemitem{ $(ii)$ }  if $\{\ell_0,\ell_1\} \in \bold a$ and $\ell_0 <
\ell_1$ then $\ell_0 \in a_0 \and \ell_1 \in a_1$
\sn
\itemitem{ $(iii)$ }  if $\ell_0 \ne \ell_1$ are from $a_0 \cup a_1$
and $\{\ell_0,\ell_1\} \notin \bold a$ \ub{then} $h(\{\ell_0,\ell_1\})
> h(\bold a)$.
\endroster}
\ermn
2) We can add in $\circledast$
\mr
\item "{$(iv)$}"   if $\bold a_0,\bold a_1$ are distinct $\bold
e_{\bold s}$-equivalence classes \ub{then} for some $m \in \{0,1\}$ we have
$[\cup \bold a_m]^2 \backslash \bold a_m$ is disjoint to $\bold
a_{1-m}$
\sn
\item "{$(v)$}"  in $\circledast$ above $a_0,a_1$ can be defined as
$\{\ell_0:\{\ell_0,\ell_1\} \in \bold a,\ell_0 <
\ell_1\},\{\ell_1:\{\ell_0,\ell_1\} \in \bold a,\ell_0 < \ell_1\}$
respectively. 
\ermn
3) If $k \ge 3,\bold s_k$ from \scite{t.8} clause (a) \ub{then} $\bold
s_k$ does not belong to {\rm ID}$_2(\aleph_1,\aleph_0)$.
\endproclaim
\bigskip

\demo{Proof}  1) Remember that by \scite{0.5A} 
we can deal with OID$(\aleph_1,\aleph_0)$.
By \cite{Sh:74} we know what is
ID$(\aleph_1,\aleph_0)$, i.e., the  family of identities in
OID$(\aleph_1,\aleph_0)$ is generated by two operations; one is called
duplication and the other of restriction (see below) from the trivial identity
(i.e. $|\text{dom}_{\bold s}| = 1$) and 
we prove $\circledast$ by induction on $n$, the number of times we
need to apply the operations.
\nl
Recall that $(a,e)$ is gotten by duplication if 
we can find sets $a_0,a_1,a_2$ and a function $g$ such that
\mr
\item "{$\circledast_1(a)$}"  $a_0 < a_1 
< a_2$ (i.e. $\ell_0 \in a_0,\ell_1 \in
a_1,\ell_2 \in a_2 \Rightarrow \ell_0 < \ell_1 < \ell_2$)
\sn
\item "{$(b)$}"  $a = a_0 \cup a_1 \cup a_2$
\sn
\item "{$(c)$}"  $g$ a one-to-one order preserving function from $a_0
\cup a_1$ onto $a_0 \cup a_1$ (so $g \restriction a_0 = 
\text{ id}_{a_0}$; let $g_1 = g,g_2 = g^{-1}$
\sn
\item "{$(d)$}"  for $\ell_0 \ne \ell_1 \in (a_0 \cup a_1)$ we have
$\{\ell_0,\ell_1\} e \{g(\ell_0),g(\ell_1)\}$
\sn
\item "{$(e)$}"  if $\ell_1 \in a_1,\ell_2 \in a_2$ then
$\{\ell_1,\ell_2\}/e$ is a singleton
\sn
\item "{$(f)$}"  $\bold s_\ell = (a_0 \cup a_\ell,e \restriction [a_0
\cup a_\ell]^2)$ is from a lower level (up to isomorphism).
\ermn
Recall that $(a,e)$ is gotten by restriction from $(a',e')$ if $a
\subseteq a',e = e' \restriction [a]^2$.

Now we prove the existence of $h$ as required by induction on the
level.  If $|\text{Dom}_{\bold s}| =1$ this is trivial.  If $\bold s$
is gotten by restriction it is trivial too, (as if $\bold s = (a,e),s'
= (a',e'),a' \subseteq a,e' = e \restriction a'$ and $h:[a]^2/e$ is as
guaranteed then we let $h'(\{\ell_0,\ell_1\}/e') = h(\{\ell_0,\ell_1\}/e)$ for
$\ell_0 < \ell_1$.  Easily $h'$ is as required).  So assume $\bold s =
(a,e)$ is gotten by duplication, so let $a_0,a_1,a_2,g_1,g_2$ be as in
$\circledast_1$ and let $h_1$ be as required for $\bold s_1 = (a_0 \cup a_1,e
\restriction [a_0 \cup a_1)^2)$ and similarly define $h_2$ by 
$h_2\{\alpha,\beta\} =
h_1\{g_2(\alpha),g_2(\beta)\}$.  Let $n^* = \text{ max Rang}(h_1)$ and
define $h:[a_0 \cup a_1 \cup a_2]^2 \Rightarrow \omega$ by $h
\supseteq h_1,h \supseteq h_2$ and if $k \in a_1,\ell \in a_2$ then we
let $h\{k,\ell\} = n^*+1$.  Now check.
\nl
2) By symmetry, \wilog \, $h(\bold a_0) < h(\bold a_1)$ and now $m=1$
satisfies the requirement by applying $\circledast_1$ to the equivalence
class $\bold a = \bold a_1$. \nl
3) It is enough to deal with $\bold s_3$.   
By direct checking the criterion in part (2) fails. \nl
${{}}$  \hfill$\square_{\scite{t.8B}}$\margincite{t.8B}
\enddemo
\bn
The following is like \scite{t.8} with $\mu$ just limit (not
necessarily a strong limit cardinal) so
\proclaim{\stag{t.9} Claim}   Assume
\mr
\item "{$(a)$}"  $\bold s'_n \in { \text{\rm OID\/}}_2$ is $(2n+n^2,
e_{{\bold s}'_n})$ where the non-singleton $e_{{\bold s}'_n}$-equivalence
classes are \nl
$\{\{\ell_0,2n+n \ell_0 + \ell_1\}:\ell_0,\ell_1 < n\}$ and \nl
$\{\{n+ \ell_1,2n+n \ell_0 + \ell_1\}:\ell_0,\ell_1 < n\}$
\sn
\item "{$(b)$}"  $\mu$ is a limit cardinal, 
$\mu > \theta > { \text{\rm cf\/}}(\mu)$ and $\theta$ is a compact cardinal
\sn
\item "{$(c)$}"  $s''_n \in { \text{\rm OID\/}}_n$ is $(2^n +
2^{2n},e_{{\bold s}''_s})$ where the non-singleton 
$e_{{\bold s}''_n}$-equivalence classes are: for 
$m<n,\eta \in {}^m 2,i=0,1$ let
$\bold a^i_\eta = \{\{\ell_i,2^n + \binom{2^n}{\ell_0} +
\ell_1\}:\ell_0,\ell_1 < 2^n$ and for some $\nu_0,\nu_1 \in {}^n 2$ we
have $\eta \char 94 \langle 0 \rangle \trianglelefteq \nu_0,\eta \char
94 \langle 1 \rangle \trianglelefteq \nu_1$ and $\ell_0 = \Sigma\{\nu_0(j)
2^j:j < n\}$ and $\ell_1 = \Sigma\{\nu_1(j)2^j:j <n\}\}$.
\ermn
1) $\bold s'_n \in { \text{\rm ID\/}}_2(\mu^+,\mu)$, moreover $\bold s'_n \in
{ \text{\rm OID\/}}_2(\mu^+,\mu)$ similarly for $\bold s'_n$. \nl
2) $\bold s'_n \notin { \text{\rm ID\/}}_2(\aleph_1,\aleph_0)$ for $n
\ge 2$, similarly for $\bold s''_n$.
\endproclaim
\bigskip

\demo{Proof}  1) Like the proof of \scite{t.8A} using \cite{Sh:49} (or
just \cite[\S5]{Sh:604}) instead of the Erd\"os-Rado theorem. \nl
2) Otherwise there is $(a,e) \in \text{ ID}_2(\aleph_1,\aleph_0)$ and
an embedding $h$ of $\bold s'_n$ into $(a,e)$ and by \scite{0.5A}
without loss of generality $h$ is order preserving and $(a,e) \in
\text{ OID}_2(\aleph_1,\aleph_0)$.  Now
\mr
\item "{$(*)_1$}"    if $\ell_0 < n,\ell_1 < n$ and $\ell = 2n + n
\ell_0 + \ell_1$ \ub{then} $h(\ell_0) < h(\ell)$. \nl
[Why?  Choose $\ell'_1 < n,\ell'_1 \ne \ell_1$ and $\ell' = 2n + n
\ell_0 + \ell'_1$, so $\ell \ne \ell'$ and 
$\{\ell_0,\ell\}e_{{\bold s}'_n} \{\ell_0,n + \ell'\}$ hence
$\{h(\ell_0),h(\ell)\},\{h(\ell_0),h(\ell')\}$ are
$e$-equivalent and $h(\ell) \ne h(\ell')$.  But on $(a,e)$
we know that if $\{m_0,m_1\} e \{m_0,m_2\}$ then $m_2 < m_1 < m_0$ and
$m_2 < m_0 < m_1$ are impossible (see \scite{t.10}(2) below) so we are done.]
\sn
\item "{$(*)_2$}"  if $\ell_0 < n,\ell_1 < n$ and $\ell = 2n +
n\ell_0 + \ell_1$ then $h(\ell_1) < h(\ell)$. \nl
[Why?  Like $(*)_1$.]
\ermn
Now we apply \scite{t.8B}(1) + (2) above so $\bold s'_n \notin \text{
ID}_2(\aleph_2,\aleph_1)$.  The conclusion about $\bold s''_n$
follows.  \hfill$\square_{\scite{t.9}}$\margincite{t.9}
\enddemo
\bigskip

\demo{\stag{t.10} Observation}  1) If $k \ge 2,\bold s = 
(n,e) \in \text{ OID}_2(\mu^+,\mu)$ \ub{then} we can find 
$\bold s' = (n',e')$ in fact $n' = 2n-1$ such that:
\mr
\widestnumber\item{$(iii)$}
\item "{$(i)$}"  $e' \restriction [n]^2 = e$
\sn
\item "{$(ii)$}"  $\bold s' \in \text{ ID}(\mu^+,\mu)$
\sn
\item "{$(iii)$}"   for every $c:[\mu^+]^{< \aleph_0} \rightarrow \mu$
there is $c':[\mu^+]^{< \aleph_0} \rightarrow \mu$ refining $c$
(i.e. $c'(u_1) = c'(u_2) \Rightarrow c(u_1) = c(u_2))$ such that: if
$h:\{0,\dotsc,2n-2\} \rightarrow \mu^+$ is one to one and
satisfies $u_1 e' u_2 \Rightarrow c'(h''(u_1)) 
= c'(h''(u_2))$ then $h \restriction \{0,\dotsc,n-1\}$ is increasing.
\ermn
2) There is $c:[\mu^+]^2 \rightarrow \mu$ such that:

if $\alpha,\beta,\gamma$ are distinct and $c\{\alpha,\beta\} =
c\{\alpha,\gamma\}$ then $\alpha < \beta \and \alpha < \gamma$. \nl
3) We can replace in (1), $(\mu^+,\mu)$ by $(\lambda,\mu)$ if there is
$\bold s = (n,e) \in \text{ ID}(\lambda,\mu)$ such that for some
$c:[\lambda]^{< \aleph_0} \rightarrow \mu$ such that
\mr
\item "{$\circledast$}"   if $h:n \rightarrow \lambda$ induces 
$e_{\bold s}$ then $h(0) < h(1)$.
\endroster
\enddemo
\bigskip

\demo{Proof}  1) Define $e':u_1 e' u_2$ \ub{iff} $u_1
e u_2 \vee u_1 = u_2 \vee \dsize \bigvee_{\ell < n-1}
(u_1 = \{\ell,n+ \ell+1\} \and u_2 e \{\ell,\ell+1\}) \vee \dsize
\bigvee_{\ell <n} (u_2 = \{\ell,n + \ell +1\} \and u_1 e \{\ell,\ell +1\})$.  
Now use (2). 
\nl
2) Let $f_\alpha:\alpha \rightarrow \mu$ be one to one and let $<^*$ a dense
linear order on $\mu^+$ with $\{\alpha:\alpha < \mu\}$ a dense
subset.  Now choose $c_1:[\mu^+]^2 \rightarrow \mu$ such that $\alpha < \beta
\Rightarrow \alpha \le^* c_1\{\alpha,\beta\} <^* \beta$ and
$c:[\mu^+]^2 \rightarrow \mu$ be $\alpha < \beta \Rightarrow
c\{\alpha,\beta\} = \text{ pr}(f_\beta(\alpha),c_1\{\alpha,\beta\})$
for some pairing function pr. \nl
3) Similar to part (1) only $|\text{Dom}_{\bold s'}|$ is larger.
\hfill$\square_{\scite{t.10}}$\margincite{t.10}
\enddemo
\newpage


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

\nocite{ignore-this-bibtex-warning} 
%% you may want to move the following lines up a bit
\newpage
    
REFERENCES.  
\bibliographystyle{lit-plain}
\bibliography{lista,listb,listx,listf,liste}

\enddocument %%

   



824.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}{ShVa 790}
\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[Fu65]{Fu65}E.~G. Furkhen.
\newblock {Languages with added quantifier ``there exist at least
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\newblock In J.~V. Addison, L.~A. Henkin, and A.~Tarski, editors, {\em {The
  Theory of Models}}, pages 121--131. North-Holland Publishing Company, 1965.

\bibitem[GcSh 491]{GcSh:491}Martin Gilchrist and Saharon Shelah.
\newblock {Identities on cardinals less than $\aleph_\omega$}.
\newblock {\em {Journal of Symbolic Logic}}, {\bf 61}:780--787, 1996.
\newblock math.LO/9505215.

\bibitem[GcSh 583]{GcSh:583}Martin Gilchrist and Saharon Shelah.
\newblock {The Consistency of ${\rm ZFC}+2^{\aleph_{0}}>\aleph_{\omega}+ {\cal
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\newblock {\em {Journal of Symbolic Logic}}, {\bf 62}:1151--1160, 1997.
\newblock math.LO/9603219.

\bibitem[Ke70]{Ke70}Jerome~H. Keisler.
\newblock {Logic with the quantifier "there exist uncountably many"}.
\newblock {\em Annals of Mathematical Logic}, {\bf 1}:1--93, 1970.

\bibitem[M1]{M1}J.~Donald Monk.
\newblock {\em {Cardinal Invariants of Boolean Algebras}}.
\newblock Lectures in Mathematics. ETH Zurich, Birkhauser Verlag, Basel Boston
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\bibitem[MoVa62]{MoVa62}M.~D. Morley and R.~L. Vaught.
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\bibitem[Sh 604]{Sh:604}Saharon Shelah.
\newblock {The pair $(\aleph_n,\aleph_0)$ may fail $\aleph_0$--compactness}.
\newblock In M.~Baaz, S.~Friedman, and J.~Krajicek, editors, {\em {Proceedings
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\bibitem[Sh:E17]{Sh:E17}Saharon Shelah.
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\bibitem[Sh 8]{Sh:8}Saharon Shelah.
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\bibitem[Sh 49]{Sh:49}Saharon Shelah.
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\bibitem[Sh 74]{Sh:74}Saharon Shelah.
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\newblock {\em {Annals of Mathematical Logic}}, {\bf 14}:223--226, 1978.

\bibitem[ShVa 790]{ShVa:790}Saharon Shelah and Jouko V\"a\"an\"anen.
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\newblock {\em Preprint}.

\end{thebibliography}
