/
amstex.sty
2025 lines (2015 loc) · 68.1 KB
/
amstex.sty
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%
% \iffalse meta-comment
%
% Copyright (C) 1995, 1999 American Mathematical Society.
% Copyright (C) 2016-2021 LaTeX Project and American Mathematical Society.
%
% This work may be distributed and/or modified under the
% conditions of the LaTeX Project Public License, either version 1.3c
% of this license or (at your option) any later version.
% The latest version of this license is in
% https://www.latex-project.org/lppl.txt
% and version 1.3c or later is part of all distributions of LaTeX
% version 2005/12/01 or later.
%
% This work has the LPPL maintenance status `maintained'.
%
% The Current Maintainer of this work is the LaTeX Project.
%
% \fi
%
% This file was generated from version 1.2beta of amstex.dtx and then
% underwent additional hand-editing.
\if@compatibility \else \catcode`\%=9 \fi
%\PackageError{amstex}{%
%Package `amstex' is obsolete, substituting `amsmath'}{You
%should use \string\usepackage{amsmath} instead. If
%that does not give satisfactory results, rename your previous copy
%of amstex.sty to amstex12beta.sty and call it with
%\string\usepackage{amstex12beta}.}
%\catcode`\%=14 \relax \input{amsmath.sty}\endinput
%
\ProvidesPackage{amstex}[1999/11/15 v1.2f AMS extra math features]
\let\@amsfonts=T
\DeclareOption{noamsfonts}{\let\@amsfonts=F}
\DeclareOption{psamsfonts}{\let\@amsfonts=P}
\DeclareOption{intlimits}{\let\ilimits@\displaylimits}
\DeclareOption{nointlimits}{\let\ilimits@\nolimits}
\DeclareOption{sumlimits}{\let\slimits@\displaylimits}
\DeclareOption{nosumlimits}{\let\slimits@\nolimits}
\DeclareOption{namelimits}{\let\nlimits@\displaylimits}
\DeclareOption{nonamelimits}{\let\nlimits@\nolimits}
\newif\ifctagsplit@
\newif\iftagsleft@
\DeclareOption{leqno}{\tagsleft@true}
\DeclareOption{reqno}{\tagsleft@false}
\DeclareOption{centertags}{\ctagsplit@true}
\DeclareOption{tbtags}{\ctagsplit@false}
\newif\if@fleqn
\newskip\@mathmargin
\@mathmargin\@centering
\DeclareOption{fleqn}{%
\@fleqntrue
\@mathmargin\m@ne sp
\AtBeginDocument{%
\ifdim\@mathmargin=\m@ne sp
\@mathmargin\leftmargini
\fi
}%
}
\DeclareOption{righttag}{\tagsleft@false}
\DeclareOption{ctagsplt}{\ctagsplit@true}
\DeclareOption{intlim}{\let\ilimits@\displaylimits}
\DeclareOption{nosumlim}{\let\slimits@\nolimits}
\DeclareOption{nonamelm}{\let\nlimits@\nolimits}
\ExecuteOptions{nointlimits,sumlimits,namelimits,centertags}
\ProcessOptions
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% In this section \RequirePackage statements were replaced with the
%% contents of the referenced files.
%%% filename = "amsgen.dtx",
%%% version = "1.2beta",
%%% date = "1994/12/13",
\providecommand{\@saveprimitive}[2]{\begingroup\escapechar`\\\relax
\edef\@tempa{\string#1}\edef\@tempb{\meaning#1}%
\ifx\@tempa\@tempb \global\let#2#1%
\else
\edef\@tempb{\meaning#2}%
\ifx\@tempa\@tempb
\else
\@latex@error{Unable to properly define \string#2; primitive
\noexpand#1no longer primitive}\@eha
\fi
\fi
\endgroup}
\let\@xp=\expandafter
\let\@nx=\noexpand
\newtoks\@emptytoks
\def\@oparg#1[#2]{\@ifnextchar[{#1}{#1[#2]}}
\def\@ifempty#1{\@xifempty#1@@..\@nil}
\long\def\@xifempty#1#2@#3#4#5\@nil{%
\ifx#3#4\@xp\@firstoftwo\else\@xp\@secondoftwo\fi}
\long\def\@ifnotempty#1{\@ifempty{#1}{}}
\def\FN@{\futurelet\@let@token}
\def\DN@{\def\next@}
\def\RIfM@{\relax\ifmmode}
\def\setboxz@h{\setbox\z@\hbox}
\def\wdz@{\wd\z@}
\def\boxz@{\box\z@}
\def\relaxnext@{\let\@let@token\relax}
\def\new@ifnextchar#1#2#3{%
\let\@tempe #1\def\@tempa{#2}\def\@tempb{#3}\futurelet
\@tempc\new@ifnch}
\def\new@ifnch{\ifx\@tempc \@tempe \let\@tempd\@tempa
\else\let\@tempd\@tempb\fi\@tempd}
\def\new@ifstar#1#2{\new@ifnextchar *{\def\@tempa*{#1}\@tempa}{#2}}
\@ifundefined{every@math@size}{%
\let\every@math@size=\every@size
\def\glb@settings{%
\expandafter\ifx\csname S@\f@size\endcsname\relax
\calculate@math@sizes
\fi
\csname S@\f@size\endcsname
\ifmath@fonts
\begingroup
\escapechar\m@ne
\csname mv@\math@version \endcsname
\globaldefs\@ne
\let \glb@currsize \f@size
\math@fonts
\endgroup
\the\every@math@size
\else
\fi
}
\def\set@fontsize#1#2#3{%
\@defaultunits\@tempdimb#2pt\relax\@nnil
\edef\f@size{\strip@pt\@tempdimb}%
\@defaultunits\@tempskipa#3pt\relax\@nnil
\edef\f@baselineskip{\the\@tempskipa}%
\edef\f@linespread{#1}%
\let\baselinestretch\f@linespread
\def\size@update{%
\baselineskip\f@baselineskip\relax
\baselineskip\f@linespread\baselineskip
\normalbaselineskip\baselineskip
\setbox\strutbox\hbox{%
\vrule\@height.7\baselineskip
\@depth.3\baselineskip
\@width\z@}%
%%% \the\every@size
\let\size@update\relax}%
}
}{}% end \@ifundefined test
\newdimen\ex@
\addto@hook\every@math@size{\compute@ex@}
\def\compute@ex@{%
\begingroup
\dimen@-\f@size\p@
\ifdim\dimen@<-20\p@
\global\ex@ 1.5\p@
\else
\advance\dimen@10\p@ \multiply\dimen@\tw@
\edef\@tempa{\ifdim\dimen@>\z@ -\fi}%
\dimen@ \ifdim\dimen@<\z@ -\fi \dimen@
\advance\dimen@-\@m sp % fudge factor
\vfuzz\p@
\def\do{\ifdim\dimen@>\z@
\vfuzz=.97\vfuzz
\advance\dimen@ -\p@
\@xp\do \fi}%
\do
\dimen@\p@ \advance\dimen@-\vfuzz
\global\ex@\p@
\global\advance\ex@ \@tempa\dimen@
\fi
\endgroup
}
\def\@addpunct#1{\ifnum\spacefactor>\@m \else#1\fi}
\def\frenchspacing{\sfcode`\.1006\sfcode`\?1005\sfcode`\!1004%
\sfcode`\:1003\sfcode`\;1002\sfcode`\,1001 }
\def\@mathmeasure#1#2#3{\setbox#1\hbox{\frozen@everymath\@emptytoks
\m@th$#2#3$}}
\def\nomath@env{\PackageError{amstex}{%
\string\begin{\@currenvir} allowed only in paragraph mode%
}\@ehb% "You've lost some text"
}
\def\Invalid@@{Invalid use of \string}
\def\nonmatherr@#1{\PackageError{amsfonts}%
{\noexpand#1allowed only in math mode}\@ehd}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% filename = "amstext.dtx",
%%% version = "1.2beta",
%%% date = "1994/12/13",
\DeclareRobustCommand{\text}{%
\ifmmode\expandafter\text@\else\expandafter\mbox\fi}
\let\nfss@text\text
\def\text@#1{\mathchoice
{\textdef@\displaystyle\f@size{#1}}%
{\textdef@\textstyle\tf@size{\firstchoice@false #1}}%
{\textdef@\textstyle\sf@size{\firstchoice@false #1}}%
{\textdef@\textstyle \ssf@size{\firstchoice@false #1}}%
\check@mathfonts
}
\def\textdef@#1#2#3{\hbox{{%
\everymath{#1}%
\let\f@size#2\selectfont
#3}}}
\newif\iffirstchoice@
\firstchoice@true
\def\stepcounter#1{%
\iffirstchoice@
\global\expandafter\advance\csname c@#1\endcsname \@ne
\begingroup
\let\@elt\@stpelt
\csname cl@#1\endcsname
\endgroup
\fi}
\toks@{\iffirstchoice@\else\@xp\@gobblefour\fi\protect}
\edef\GenericInfo{\the\toks@
\@xp\@nx\csname GenericInfo \endcsname}
\edef\GenericWarning{\the\toks@
\@xp\@nx\csname GenericWarning \endcsname}
\toks@{\iffirstchoice@\else
\@xp\@xp\@xp\@gobblefour
\@xp\@gobbletwo\fi\protect}
\edef\GenericError{\the\toks@
\@xp\@nx\csname GenericError \endcsname}
\def\label#1{\@bsphack\iffirstchoice@\if@filesw {\let\thepage\relax
\def\protect{\noexpand\noexpand\noexpand}%
\xdef\@gtempa{\write\@auxout{\string
\newlabel{#1}{{\@currentlabel}{\thepage}}}}}\@gtempa
\if@nobreak \ifvmode\nobreak\fi\fi\fi\fi\@esphack}
\begingroup \catcode`\"=12
\gdef\mathhexbox#1#2#3{\text{$\m@th\mathchar"#1#2#3$}}
\endgroup
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% filename = "amsfonts.dtx",
%%% version = "1.2beta",
%%% date = "1994/12/13",
\newbox\Mathstrutbox@
\setbox\Mathstrutbox@=\hbox{}
\def\Mathstrut@{\copy\Mathstrutbox@}
\addto@hook\every@math@size{\setbox\z@\hbox{\normalfont(}%
\ht\Mathstrutbox@\ht\z@ \dp\Mathstrutbox@\dp\z@}
\newbox\strutbox@
\def\strut@{\copy\strutbox@}
\addto@hook\every@math@size{%
\global\setbox\strutbox@\hbox{\lower.5\normallineskiplimit
\vbox{\kern-\normallineskiplimit\copy\strutbox}}}
\def\big{\bBigg@\@ne}
\def\Big{\bBigg@{1.5}}
\def\bigg{\bBigg@\tw@}
\def\Bigg{\bBigg@{2.5}}
\def\bBigg@#1#2{%
{%
\hbox{$\left#2\vcenter to#1\big@size{}\right.%
\n@space
$}}}
\addto@hook\every@math@size{%
\global\big@size 1.2\ht\Mathstrutbox@
\global\advance\big@size 1.2\dp\Mathstrutbox@ }
\newdimen\big@size
%% msam, msbm, eufm fonts dealt with at end of file
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% filename = "amsbsy.dtx",
%%% version = "1.2beta",
%%% date = "1994/10/27",
\DeclareRobustCommand{\boldsymbol}[1]{%
\begingroup
\let\@nomath\@gobble \mathversion{bold}%
\math@atom{#1}{%
\mathchoice%
{\hbox{$\m@th\displaystyle#1$}}%
{\hbox{$\m@th\textstyle#1$}}%
{\hbox{$\m@th\scriptstyle#1$}}%
{\hbox{$\m@th\scriptscriptstyle#1$}}}%
\endgroup}
\def\math@atom#1#2{%
\binrel@{#1}\binrel@@{#2}}
\DeclareRobustCommand{\pmb}{%
\ifmmode\else \expandafter\pmb@@\fi\mathpalette\pmb@}
\def\pmb@@#1#2#3{\leavevmode\setboxz@h{#3}%
\dimen@-\wdz@
\kern-.5\ex@\copy\z@
\kern\dimen@\kern.25\ex@\raise.4\ex@\copy\z@
\kern\dimen@\kern.25\ex@\box\z@
}
\newdimen\pmbraise@
\def\pmb@#1#2{\setbox8\hbox{$\m@th#1{#2}$}%
\setboxz@h{$\m@th#1\mkern.5mu$}\pmbraise@\wdz@
\binrel@{#2}%
\dimen@-\wd8 %
\binrel@@{%
\mkern-.8mu\copy8 %
\kern\dimen@\mkern.4mu\raise\pmbraise@\copy8 %
\kern\dimen@\mkern.4mu\box8 }%
}
\def\binrel@#1{\begingroup
\setboxz@h{\thinmuskip0mu
\medmuskip\m@ne mu\thickmuskip\@ne mu
\setbox\tw@\hbox{$#1\m@th$}\kern-\wd\tw@
${}#1{}\m@th$}%
\edef\@tempa{\endgroup\let\noexpand\binrel@@
\ifdim\wdz@<\z@ \mathbin
\else\ifdim\wdz@>\z@ \mathrel
\else \relax\fi\fi}%
\@tempa
}
\let\binrel@@\relax
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\AmSTeX{\protect\AmS-\protect\TeX{}}
\def\AmS{{\protect\AmSfont
A\kern-.1667em\lower.5ex\hbox{M}\kern-.125emS}}
\def\AmSfont{%
\usefont{OMS}{cmsy}{\if\@xp\@car\f@series\@nil bb\else m\fi}{n}}
\def\prim@s{\prime\futurelet\@let@token\pr@m@s}
\def\pr@m@s{%
\ifx\@let@token'\DN@##1{\prim@s}\else\let\next@\egroup\fi\next@}
\let\@prime=\prime
\renewcommand{\prime}{{\kern\z@\@prime}}
\DeclareRobustCommand{\mspace}[3]{%
\ifmmode\mskip#1#2\else\kern#1#3\fi\relax}
\renewcommand{\,}{\mspace+\thinmuskip{.1667em}}
\let\thinspace\,
\renewcommand{\!}{\mspace-\thinmuskip{.1667em}}
\let\negthinspace\!
\renewcommand{\:}{\mspace+\medmuskip{.2222em}}
\let\medspace\:
\newcommand{\negmedspace}{\mspace-\medmuskip{.2222em}}
\renewcommand{\;}{\mspace+\thickmuskip{.2777em}}
\let\thickspace\;
\newcommand{\negthickspace}{\mspace-\thickmuskip{.2777em}}
\def\implies{\DOTSB\;\Longrightarrow\;}
\def\impliedby{\DOTSB\;\Longleftarrow\;}
\begingroup \catcode`\"=12 % in case activated by a preceding package
\gdef\And{\DOTSB\;\mathchar"3026 \;}
\gdef\lvert{\delimiter"426A30C }
\gdef\rvert{\delimiter"526A30C }
\gdef\lVert{\delimiter"426B30D }
\gdef\rVert{\delimiter"526B30D }
\endgroup % restore "
\DeclareRobustCommand{\frac}{\new@ifnextchar[{\frfrac{}}{\frfrac{}[]}}
\def\frfrac#1[#2]#3#4{\def\next@{#2}%
\ifx\next@\@empty \def\next@{#1{#3\over#4}}%
\else \def\next@{{#1{#3\above#2\relax#4}}}%
\fi
\next@}
\def\dfrac{\protect\frfrac\displaystyle[]}
\def\tfrac{\protect\frfrac\textstyle[]}
\DeclareRobustCommand{\binom}{\frbinom{}}
\def\frbinom#1#2#3{{#1{#2\atopwithdelims()#3}}}
\def\dbinom{\protect\frbinom\displaystyle}
\def\tbinom{\protect\frbinom\textstyle}
\def\colon{\nobreak\mskip2mu\mathpunct{}\nonscript
\mkern-\thinmuskip{:}\mskip6muplus1mu\relax}
\begingroup \catcode`\"=12
\DeclareMathSymbol{\coprod@}{\mathop}{largesymbols}{"60}
\gdef\coprod{\DOTSB\coprod@\slimits@}
\DeclareMathSymbol{\bigvee@}{\mathop}{largesymbols}{"57}
\gdef\bigvee{\DOTSB\bigvee@\slimits@}
\DeclareMathSymbol{\bigwedge@}{\mathop}{largesymbols}{"56}
\gdef\bigwedge{\DOTSB\bigwedge@\slimits@}
\DeclareMathSymbol{\biguplus@}{\mathop}{largesymbols}{"55}
\gdef\biguplus{\DOTSB\biguplus@\slimits@}
\DeclareMathSymbol{\bigcap@}{\mathop}{largesymbols}{"54}
\gdef\bigcap{\DOTSB\bigcap@\slimits@}
\DeclareMathSymbol{\bigcup@}{\mathop}{largesymbols}{"53}
\gdef\bigcup{\DOTSB\bigcup@\slimits@}
\DeclareMathSymbol{\prod@}{\mathop}{largesymbols}{"51}
\gdef\prod{\DOTSB\prod@\slimits@}
\DeclareMathSymbol{\sum@}{\mathop}{largesymbols}{"50}
\gdef\sum{\DOTSB\sum@\slimits@}
\DeclareMathSymbol{\bigotimes@}{\mathop}{largesymbols}{"4E}
\gdef\bigotimes{\DOTSB\bigotimes@\slimits@}
\DeclareMathSymbol{\bigoplus@}{\mathop}{largesymbols}{"4C}
\gdef\bigoplus{\DOTSB\bigoplus@\slimits@}
\DeclareMathSymbol{\bigodot@}{\mathop}{largesymbols}{"4A}
\gdef\bigodot{\DOTSB\bigodot@\slimits@}
\DeclareMathSymbol{\bigsqcup@}{\mathop}{largesymbols}{"46}
\gdef\bigsqcup{\DOTSB\bigsqcup@\slimits@}
\endgroup
\def\dddot#1{{\mathop{#1}\limits^{\vbox to-1.4\ex@{\kern-\tw@\ex@
\hbox{\normalfont ...}\vss}}}}
\def\ddddot#1{{\mathop{#1}\limits^{\vbox to-1.4\ex@{\kern-\tw@\ex@
\hbox{\normalfont....}\vss}}}}
\def~{\nobreakspace}
\DeclareRobustCommand{\nobreakspace}{\unskip\nobreak\ \ignorespaces}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newif\if@display
\everydisplay\expandafter{\the\everydisplay \@displaytrue}
\def\int{\DOTSI\intop\ilimits@}
\def\oint{\DOTSI\ointop\ilimits@}
\def\intkern@{\mkern-6mu\mathchoice{\mkern-3mu}{}{}{}}
\def\intdots@{\mathchoice{\@cdots}%
{{\cdotp}\mkern1.5mu{\cdotp}\mkern1.5mu{\cdotp}}%
{{\cdotp}\mkern1mu{\cdotp}\mkern1mu{\cdotp}}%
{{\cdotp}\mkern1mu{\cdotp}\mkern1mu{\cdotp}}}
\def\iint{\DOTSI\protect\ints@\tw@}
\def\iiint{\DOTSI\protect\ints@\thr@@}
\def\iiiint{\DOTSI\protect\ints@{4}}
\def\idotsint{\DOTSI\protect\ints@\z@}
\def\ints@#1{%
\mkern-7mu\mathchoice{\mkern-2mu}{}{}{}%
\mathop{\mkern7mu\mathchoice{\mkern2mu}{}{}{}%
\intop\ifnum#1=\z@\intdots@
\else\intkern@\fi
\ifnum#1>\tw@\intop\intkern@\fi
\ifnum#1>\thr@@\intop\intkern@\fi
\intop
}\ilimits@
}
\def\nolimits@{%
\DN@{\nolimits\ifx\@let@token\limits\expandafter\@gobble\fi}%
\FN@\next@}
\begingroup \catcode`\"=12
\gdef\newmcodes@{\mathcode`\'39\mathcode`\*42\mathcode`\."613A%
\mathcode`\-45\mathcode`\/47\mathcode`\:"603A\relax}
\endgroup
\DeclareRobustCommand{\operatorname}{%
\@ifstar{\qopname\newmcodes@ n}%
{\qopname\newmcodes@{no}}}%
\DeclareRobustCommand{\qopname}[3]{\mathop{#1\kern\z@\operator@font#3}%
\csname #2limits@\endcsname}
\def\arccos{\qopname\relax{no}{arccos}}
\def\arcsin{\qopname\relax{no}{arcsin}}
\def\arctan{\qopname\relax{no}{arctan}}
\def\arg{\qopname\relax{no}{arg}}
\def\cos{\qopname\relax{no}{cos}}
\def\cosh{\qopname\relax{no}{cosh}}
\def\cot{\qopname\relax{no}{cot}}
\def\coth{\qopname\relax{no}{coth}}
\def\csc{\qopname\relax{no}{csc}}
\def\deg{\qopname\relax{no}{deg}}
\def\det{\qopname\relax n{det}}
\def\dim{\qopname\relax{no}{dim}}
\def\exp{\qopname\relax{no}{exp}}
\def\gcd{\qopname\relax n{gcd}}
\def\hom{\qopname\relax{no}{hom}}
\def\inf{\qopname\relax n{inf}}
\def\injlim{\qopname\relax n{inj\,lim}}
\def\ker{\qopname\relax{no}{ker}}
\def\lg{\qopname\relax{no}{lg}}
\def\lim{\qopname\relax n{lim}}
\def\liminf{\qopname\relax n{lim\,inf}}
\def\limsup{\qopname\relax n{lim\,sup}}
\def\ln{\qopname\relax{no}{ln}}
\def\log{\qopname\relax{no}{log}}
\def\max{\qopname\relax n{max}}
\def\min{\qopname\relax n{min}}
\def\Pr{\qopname\relax n{Pr}}
\def\projlim{\qopname\relax n{proj\,lim}}
\def\sec{\qopname\relax{no}{sec}}
\def\sin{\qopname\relax{no}{sin}}
\def\sinh{\qopname\relax{no}{sinh}}
\def\sup{\qopname\relax n{sup}}
\def\tan{\qopname\relax{no}{tan}}
\def\tanh{\qopname\relax{no}{tanh}}
\def\operator@font{\mathgroup\symoperators}
\def\operatornamewithlimits{\operatorname*}
\def\varlim@#1#2{\mathop{\vtop{\ialign{##\crcr
\hfil$#1\m@th\operator@font lim$\hfil\crcr
\noalign{\nointerlineskip\kern\ex@}#2#1\crcr
\noalign{\nointerlineskip\kern-\ex@}\crcr}}}}
\def\varinjlim{\mathpalette\varlim@\rightarrowfill@}
\def\varprojlim{\mathpalette\varlim@\leftarrowfill@}
\def\varliminf{\mathpalette\varliminf@{}}
\def\varliminf@#1{\mathop{\@@underline{\vrule\@depth.2\ex@\@width\z@
\hbox{$#1\m@th\operator@font lim$}}}}
\def\varlimsup{\mathpalette\varlimsup@{}}
\def\varlimsup@#1{\mathop{\@@overline
{\hbox{$#1\m@th\operator@font lim$}}}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\numberwithin#1#2{\@ifundefined{c@#1}{\@nocounterr{#1}}{%
\@ifundefined{c@#2}{\@nocnterr{#2}}{%
\@addtoreset{#1}{#2}%
\toks@\expandafter\expandafter\expandafter{\csname the#1\endcsname}%
\expandafter\xdef\csname the#1\endcsname
{\expandafter\noexpand\csname the#2\endcsname
.\the\toks@}}}}
\def\eqref#1{\textup{\tagform@{\ref{#1}}}}
\def\bmod{\mskip-\medmuskip\mkern5mu\mathbin
{\operator@font mod}\penalty900
\mkern5mu\mskip-\medmuskip}
\def\pod#1{\allowbreak\if@display\mkern18mu\else\mkern8mu\fi(#1)}
\def\pmod#1{\pod{{\operator@font mod}\mkern6mu#1}}
\def\mod#1{\allowbreak\if@display\mkern18mu
\else\mkern12mu\fi{\operator@font mod}\,\,#1}
\newcommand{\cfrac}[3][c]{{\displaystyle\frac{%
\strut\ifx r#1\hfill\fi#2\ifx l#1\hfill\fi}{#3}}%
\kern-\nulldelimiterspace}
\def\lcfrac{\cfrac[l]}
\def\rcfrac{\cfrac[r]}
\def\overset#1#2{\binrel@{#2}%
\binrel@@{\mathop{\kern\z@#2}\limits^{#1}}}
\def\underset#1#2{\binrel@{#2}%
\binrel@@{\mathop{\kern\z@#2}\limits_{#1}}}
\def\sideset#1#2#3{%
\setbox\z@\hbox{$\displaystyle{\vphantom{#3}}#1{#3}\m@th$}%
\setbox\tw@\hbox{$\displaystyle{#3}#2\m@th$}%
\hskip\wd\z@\hskip-\wd\tw@\mathop{\hskip\wd\tw@\hskip-\wd\z@
{\vphantom{#3}}#1{#3}#2}}
\renewcommand{\smash}[2][tb]{%
\def\smash@{#1}%
\ifmmode\expandafter\mathpalette\expandafter\mathsm@sh\else
\expandafter\makesm@sh\fi{#2}}
\def\finsm@sh{\def\mb@t{\ht\z@\z@}\def\mb@b{\dp\z@\z@}%
\def\mb@tb{\mb@t\mb@b}%
{\csname mb@\smash@\endcsname}%
\leavevmode\boxz@}
\def\rightarrowfill@#1{\m@th\setboxz@h{$#1\relbar$}\ht\z@\z@
$#1\copy\z@\mkern-6mu\cleaders
\hbox{$#1\mkern-2mu\box\z@\mkern-2mu$}\hfill
\mkern-6mu\mathord\rightarrow$}
\def\leftarrowfill@#1{\m@th\setboxz@h{$#1\relbar$}\ht\z@\z@
$#1\mathord\leftarrow\mkern-6mu\cleaders
\hbox{$#1\mkern-2mu\copy\z@\mkern-2mu$}\hfill
\mkern-6mu\box\z@$}
\def\leftrightarrowfill@#1{\m@th\setboxz@h{$#1\relbar$}\ht\z@\z@
$#1\mathord\leftarrow\mkern-6mu\cleaders
\hbox{$#1\mkern-2mu\box\z@\mkern-2mu$}\hfill
\mkern-6mu\mathord\rightarrow$}
\def\overarrow@#1#2#3{\vbox{\ialign{##\crcr#1#2\crcr
\noalign{\kern-\ex@\nointerlineskip}$\m@th\hfil#2#3\hfil$\crcr}}}
\def\overrightarrow{\mathpalette{\overarrow@\rightarrowfill@}}
\def\overleftarrow{\mathpalette{\overarrow@\leftarrowfill@}}
\def\overleftrightarrow{\mathpalette{\overarrow@\leftrightarrowfill@}}
\def\underarrow@#1#2#3{%
\vtop{\ialign{##\crcr$\m@th\hfil#2#3\hfil$\crcr
\noalign{\nointerlineskip\kern-.5\ex@}#1#2\crcr}}}
\def\underrightarrow{\mathpalette{\underarrow@\rightarrowfill@}}
\def\underleftarrow{\mathpalette{\underarrow@\leftarrowfill@}}
\def\underleftrightarrow{\mathpalette{\underarrow@\leftrightarrowfill@}}
\let\ifgtest@\iffalse % initial value
\def\gtest@true{\global\let\ifgtest@\iftrue}
\def\gtest@false{\global\let\ifgtest@\iffalse}
\let\DOTSI\relax
\let\DOTSB\relax
\let\DOTSX\relax
{\uccode`7=`\\ \uccode`8=`m \uccode`9=`a \uccode`0=`t \uccode`!=`h
\uppercase{%
\gdef\math@#1#2#3#4#5#6\math@{\gtest@false\ifx 7#1\ifx 8#2%
\ifx 9#3\ifx 0#4\ifx !#5\xdef\meaning@{#6}\gtest@true
\fi\fi\fi\fi\fi}}}
{\uccode`7=`c \uccode`8=`h \uccode`9=`\"
\uppercase{\gdef\mathch@#1#2#3#4#5#6\mathch@{\gtest@false
\ifx 7#1\ifx 8#2\ifx 9#5\gtest@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}
{\uccode`4=`b \uccode`5=`i \uccode`6=`n
\uppercase{\gdef\mathbin@#1#2#3{\relaxnext@
\def\nextii@##1\mathbin@{\ifx\@sptoken\@let@token\gtest@true\fi}%
\gtest@false\DN@##1\mathbin@{}%
\ifx 4#1\ifx 5#2\ifx 6#3\DN@{\FN@\nextii@}\fi\fi\fi\next@}}}
{\uccode`4=`r \uccode`5=`e \uccode`6=`l
\uppercase{\gdef\mathrel@#1#2#3{\relaxnext@
\def\nextii@##1\mathrel@{\ifx\@sptoken\@let@token\gtest@true\fi}%
\gtest@false\DN@##1\mathrel@{}%
\ifx 4#1\ifx 5#2\ifx 6#3\DN@{\FN@\nextii@}\fi\fi\fi\next@}}}
{\uccode`5=`m \uccode`6=`a \uccode`7=`c
\uppercase{\gdef\macro@#1#2#3#4\macro@{\gtest@false
\ifx 5#1\ifx 6#2\ifx 7#3\gtest@true
\xdef\meaning@{\macro@@#4\macro@@}\fi\fi\fi}}}
\def\macro@@#1->#2\macro@@{#2}
\newcount\DOTSCASE@
{\uccode`6=`\\ \uccode`7=`D \uccode`8=`O \uccode`9=`T \uccode`0=`S
\uppercase{\gdef\DOTS@#1#2#3#4#5{\gtest@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@
\def\nextii@##1\DOTS@{\ifx\@sptoken\@let@token\gtest@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@}}}
{\uccode`5=`\\ \uccode`6=`n \uccode`7=`o \uccode`8=`t
\uppercase{\gdef\not@#1#2#3#4{\relaxnext@
\def\nextii@##1\not@{\ifx\@sptoken\@let@token\gtest@true\fi}%
\gtest@false\DN@##1\not@{}%
\ifx 5#1\ifx 6#2\ifx 7#3\ifx 8#4\DN@{\FN@\nextii@}\fi\fi\fi
\fi\next@}}}
\def\keybin@{\gtest@true
\ifx\@let@token+\else\ifx\@let@token=\else\ifx\@let@token<%
\else\ifx\@let@token>\else\ifx\@let@token-\else\ifx\@let@token*%
\else\ifx\@let@token:\else\gtest@false\fi\fi\fi\fi\fi\fi\fi}
\@ifundefined{@ldots}{\def\@ldots{\mathellipsis}}{}
\DeclareRobustCommand{\dots}{\relax
\csname\ifmmode m\else t\fi dots@\endcsname}
\def\tdots@{\leavevmode\unskip\relaxnext@
\DN@{$\m@th\@ldots\,
\ifx\@let@token,\,$\else\ifx\@let@token.\,$\else\ifx\@let@token;\,$%
\else\ifx\@let@token:\,$\else\ifx\@let@token?\,$%
\else\ifx\@let@token!\,$\else$ \fi\fi\fi\fi\fi\fi}%
\ \FN@\next@}
\def\mdots@{\FN@\mdots@@}
\def\mdots@@{\gdef\thedots@{\dotso@}%
\ifx\@let@token\boldsymbol\gdef\thedots@\boldsymbol{\boldsymboldots@}%
\else\ifx,\@let@token\gdef\thedots@{\dotsc}%
\else\ifx\not\@let@token\gdef\thedots@{\dotsb@}%
\else\keybin@
\ifgtest@\gdef\thedots@{\dotsb@}%
\else\xdef\meaning@{\meaning\@let@token..........}%
\xdef\meaning@@{\meaning@}%
\expandafter\math@\meaning@\math@
\ifgtest@
\expandafter\mathch@\meaning@\mathch@
\ifgtest@\expandafter\getmathch@\meaning@\getmathch@\fi
\else\expandafter\macro@\meaning@@\macro@
\ifgtest@
\expandafter\not@\meaning@\not@\ifgtest@\gdef\thedots@{\dotsb@}%
\else\expandafter\DOTS@\meaning@\DOTS@
\ifgtest@
\ifcase\number\DOTSCASE@\gdef\thedots@{\dotsb@}%
\or\gdef\thedots@{\dotsi}\else\fi
\else\expandafter\math@\meaning@\math@
\ifgtest@\expandafter\mathbin@\meaning@\mathbin@
\ifgtest@\gdef\thedots@{\dotsb@}%
\else\expandafter\mathrel@\meaning@\mathrel@
\ifgtest@\gdef\thedots@{\dotsb@}%
\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi
\thedots@}
\def\boldsymboldots@#1{%
\bold@true\let\@let@token=#1\let\delayed@=#1\mdots@@
\boldsymbol#1\bold@false}
\def\@cdots{\mathinner{\cdotp\cdotp\cdotp}}
\def\dotsi{\!\@cdots}
\let\dotsb@\@cdots
\def\rightdelim@{\gtest@true
\ifx\@let@token)\else
\ifx\@let@token]\else
\ifx\@let@token\rbrack\else
\ifx\@let@token\}\else
\ifx\@let@token\rbrace\else
\ifx\@let@token\rangle\else
\ifx\@let@token\rceil\else
\ifx\@let@token\rfloor\else
\ifx\@let@token\rgroup\else
\ifx\@let@token\rmoustache\else
\ifx\@let@token\right\else
\ifx\@let@token\bigr\else
\ifx\@let@token\biggr\else
\ifx\@let@token\Bigr\else
\ifx\@let@token\Biggr\else\gtest@false
\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi}
\def\extra@{%
\rightdelim@\ifgtest@
\else\ifx\@let@token$\gtest@true
\else\xdef\meaning@{\meaning\@let@token..........}%
\expandafter\macro@\meaning@\macro@\ifgtest@
\expandafter\DOTS@\meaning@\DOTS@
\ifgtest@
\ifnum\DOTSCASE@=\tw@\gtest@true\else\gtest@false
\fi\fi\fi\fi\fi}
\newif\ifbold@
\def\dotso@{\relaxnext@
\ifbold@
\let\@let@token\delayed@
\def\nextii@{\extra@\@ldots\ifgtest@\,\fi}%
\else
\def\nextii@{\DN@{\extra@\@ldots\ifgtest@\,\fi}\FN@\next@}%
\fi
\nextii@}
\def\extrap@#1{%
\DN@{#1\,}%
\ifx\@let@token,\else
\ifx\@let@token;\else
\ifx\@let@token.\else\extra@
\ifgtest@\else
\let\next@#1\fi\fi\fi\fi\next@}
\DeclareRobustCommand{\ldots}{\relax
\ifmmode \DN@{\extrap@\@ldots}%
\else \let\next@\tdots@\fi
\FN@\next@}
\DeclareRobustCommand{\cdots}{\DN@{\extrap@\@cdots}\FN@\next@}
\let\dotso\ldots
\let\dotsb\cdots
\let\dotsm\dotsb
\DeclareRobustCommand{\dotsc}{%
\DN@{\ifx\@let@token;\@ldots\,%
\else \ifx\@let@token.\@ldots\,%
\else \extra@\@ldots \ifgtest@\,\fi
\fi\fi}%
\FN@\next@}
\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\;}
\begingroup \catcode`\"=12
\mathchardef\@tempa="7000
\ifx\@tempa\Gamma
\DeclareMathSymbol{\Gamma}{\mathord}{operators}{"00}
\DeclareMathSymbol{\Delta}{\mathord}{operators}{"01}
\DeclareMathSymbol{\Theta}{\mathord}{operators}{"02}
\DeclareMathSymbol{\Lambda}{\mathord}{operators}{"03}
\DeclareMathSymbol{\Xi}{\mathord}{operators}{"04}
\DeclareMathSymbol{\Pi}{\mathord}{operators}{"05}
\DeclareMathSymbol{\Sigma}{\mathord}{operators}{"06}
\DeclareMathSymbol{\Upsilon}{\mathord}{operators}{"07}
\DeclareMathSymbol{\Phi}{\mathord}{operators}{"08}
\DeclareMathSymbol{\Psi}{\mathord}{operators}{"09}
\DeclareMathSymbol{\Omega}{\mathord}{operators}{"0A}
\DeclareMathSymbol{\varGamma}{\mathord}{letters}{"00}
\DeclareMathSymbol{\varDelta}{\mathord}{letters}{"01}
\DeclareMathSymbol{\varTheta}{\mathord}{letters}{"02}
\DeclareMathSymbol{\varLambda}{\mathord}{letters}{"03}
\DeclareMathSymbol{\varXi}{\mathord}{letters}{"04}
\DeclareMathSymbol{\varPi}{\mathord}{letters}{"05}
\DeclareMathSymbol{\varSigma}{\mathord}{letters}{"06}
\DeclareMathSymbol{\varUpsilon}{\mathord}{letters}{"07}
\DeclareMathSymbol{\varPhi}{\mathord}{letters}{"08}
\DeclareMathSymbol{\varPsi}{\mathord}{letters}{"09}
\DeclareMathSymbol{\varOmega}{\mathord}{letters}{"0A}
\fi
\endgroup
\def\accentclass@{7}
\def\noaccents@{\def\accentclass@{0}}
\DeclareFontEncoding{OML}{}{\noaccents@}
\DeclareFontEncoding{OMS}{}{\noaccents@}
\edef\@tempa#1#2{\def#1{\mathaccent\string"\noexpand\accentclass@#2 }}
\@tempa\hat{05E}\@tempa\check{014}\@tempa\tilde{07E}\@tempa\acute{013}
\@tempa\grave{012}\@tempa\dot{05F}\@tempa\ddot{07F}\@tempa\breve{015}
\@tempa\bar{016}
\newcount\skewcharcount@
\newcount\familycount@
\def\theskewchar@{\familycount@\@ne
\global\skewcharcount@\the\skewchar\textfont\@ne
\ifnum\mathgroup>\m@ne\ifnum\mathgroup<16
\global\familycount@\the\mathgroup\relax
\global\skewcharcount@\the\skewchar\textfont\the\mathgroup\relax\fi\fi
\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}
\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\mathaccent@#1#2{\ifnum\mathgroup=\m@ne\xdef\thefam@{1}\else
\xdef\thefam@{\the\mathgroup}\fi
\accentdimen@\z@
\setboxz@h{\unbracefonts@$\m@th\mathgroup\thefam@\relax#2$}%
\ifdim\accentdimen@=\z@\DN@{\mathaccent#1{#2}}%
\setbox\@ne\hbox{\unbracefonts@
$\m@th\mathgroup\thefam@\relax#2\theskewchar@$}
\setbox\tw@\hbox{$\m@th\ifnum\skewcharcount@=\m@ne\else
\mathchar\skewcharcount@\fi$}%
\global\accentdimen@\wd\@ne\global\advance\accentdimen@-\wdz@
\global\advance\accentdimen@-\wd\tw@
\global\multiply\accentdimen@\tw@
\dimentomu@\global\advance\accentmu@\@ne
\else\DN@{{\mathaccent#1{#2\mkern\accentmu@ mu}%
\mkern-\accentmu@ mu}{}}\fi
\next@}
\def\unbracefonts@{\let\math@bgroup\@empty\let\math@egroup\@empty}
\begingroup \catcode`\"=12
\def\@tempa#1#2{\gdef#1{\RIfM@\DN@{\mathaccent@{"\accentclass@#2 }}%
\else\DN@{\nonmatherr@{#1}}\fi\next@}}
\@tempa\Hat{05E}\@tempa\Check{014}\@tempa\Tilde{07E}\@tempa\Acute{013}
\@tempa\Grave{012}\@tempa\Dot{05F}\@tempa\Ddot{07F}\@tempa\Breve{015}
\@tempa\Bar{016}
\gdef\Vec{\RIfM@\DN@{\mathaccent@{"017E }}\else
\DN@{\nonmatherr@\Vec}\fi\next@}
\endgroup
\begingroup \catcode`\"=12
\gdef\@@sqrt#1{\radical"270370 {#1}}
\endgroup
\@saveprimitive\overline\@@overline
\def\overline#1{\@@overline{#1}}
\def\boxed#1{\fbox{\m@th$\displaystyle#1$}}
\def\setb@ck{\iffirstchoice@\global\advance\c@equation\m@ne\fi}
\newcount\dspbrk@lvl
\dspbrk@lvl=-1
\interdisplaylinepenalty\@M
\def\allowdisplaybreaks{%
\new@ifnextchar[\allowdspbrks@{\allowdspbrks@[4]}}
\def\allowdspbrks@[#1]{%
\interdisplaylinepenalty\getdsp@pen{#1}}
\def\getdsp@pen#1{%
\ifcase #1\relax \@M
\or 9999
\or 6999
\or 2999
\or \z@\fi}
\def\displaybreak{\PackageError{amstex}{\Invalid@@\displaybreak}\@eha}
\def\displaybreak@{%
\def\displaybreak{\new@ifnextchar[\dspbrk@{\dspbrk@[4]}}}
\def\dspbrk@[#1]{\global\dspbrk@lvl #1\relax}
\def\math@cr{{\ifnum0=`}\fi
\new@ifstar{\global\@eqpen\@M\math@cr@}%
{\global\@eqpen
\ifnum\dspbrk@lvl <\z@ \interdisplaylinepenalty
\else -\@getpen\dspbrk@lvl \fi
\math@cr@}}
\def\math@cr@{\new@ifnextchar[\math@cr@@{\math@cr@@[\z@]}}
\def\math@cr@@[#1]{\ifnum0=`{\fi}\math@cr@@@
\noalign{\vskip#1\relax}}
\def\Let@{\let\\\math@cr}
\def\restore@math@cr{\def\math@cr@@@{\cr}}
\restore@math@cr
\def\intertext{\PackageError{amstex}{\Invalid@@\intertext}\@eha}
\def\intertext@{\def\intertext##1{\noalign{%
\penalty\postdisplaypenalty\vskip\belowdisplayskip
\vbox{\normalbaselines\noindent##1}%
\penalty\predisplaypenalty\vskip\abovedisplayskip}}}
\newhelp\tag@help
{tag cannot be used at this point.\space
If you don't understand why^^Jyou should consult
the documentation.^^JBut don't worry: just continue, and I'll
forget what happened.}
\def\gobble@tag{\new@ifstar\@gobble\@gobble}
\def\invalid@tag#1{\PackageError{amstex}{#1}{\the\tag@help}\gobble@tag}
\def\dft@tag{\invalid@tag{\string\tag\space not allowed here}}
\def\default@tag{\let\tag\dft@tag}
\default@tag
\def\maketag@@{\new@ifstar\maketag@@@\tagform@}
\def\maketag@@@#1{\hbox{\m@th\normalfont#1}}
\def\tagform@#1{\maketag@@@{(\ignorespaces#1\unskip)}}
\def\thetag{\leavevmode\tagform@}
\def\make@df@tag{\new@ifstar\make@df@tag@@\make@df@tag@@@}
\def\make@df@tag@@#1{%
\def\@currentlabel{#1}%
\gdef\df@tag{\maketag@@@{#1}}}
\def\make@df@tag@@@#1{\@seteqlabel{#1}\gdef\df@tag{\tagform@{#1}}}
\def\@seteqlabel#1{\toks@\@xp{\p@equation{#1}}%
\edef\@currentlabel{\the\toks@}}
\def\notag{\nonumber}
\iftagsleft@
\def\@eqnnum{\hbox to1sp{}\rlap{\normalfont
\hskip -\displaywidth\tagform@\theequation}}
\fi
\let\@adjusttag\@empty
\def\adjusttag#1{\skip@#1\relax\gdef\@adjusttag{#1}}
\def\@movetag{%
\ifx\@adjusttag\@empty \else
\vskip\@adjusttag\relax
\global\let\@adjusttag\@empty
\fi
}
\def\multilimits@{\bgroup
\Let@
\restore@math@cr
\default@tag
\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\Sp{^\multilimits@}
\def\endSb{\crcr\egroup\egroup\egroup}
\let\endSp=\endSb
\def\smallmatrix{\null\,\vcenter\bgroup
\Let@\restore@math@cr\default@tag
\baselineskip6\ex@ \lineskip1.5\ex@ \lineskiplimit\lineskip
\ialign\bgroup\hfil$\m@th\scriptstyle{##}$\hfil&&\thickspace\hfil
$\m@th\scriptstyle{##}$\hfil\crcr}
\def\endsmallmatrix{\crcr\egroup\egroup\,}
\newcount\c@MaxMatrixCols
\c@MaxMatrixCols=10
\def\matrix{\hskip -\arraycolsep\array{*\c@MaxMatrixCols c}}
\def\endmatrix{\endarray \hskip -\arraycolsep}
\@ifundefined{prepnext@tok}{%
\def\@array[#1]#2{%
\setbox\@arstrutbox\hbox{\vrule\@height\arraystretch\ht\strutbox
\@depth\arraystretch\dp\strutbox \@width\z@}%
\@mkpream{#2}\edef\@preamble{\ialign \noexpand\@halignto
\bgroup \tabskip\z@ \@arstrut \@preamble \tabskip\z@ \cr}%
\let\@startpbox\@@startpbox \let\@endpbox\@@endpbox
\if #1t\vtop \else \if#1b\vbox \else \vcenter \fi\fi
\bgroup \let\par\@empty
\let\@sharp##\let\protect\relax
\lineskip\z@\baselineskip\z@\@preamble}%
\begingroup \lccode`\~=`\@ \catcode`\~=\active % just to be sure!
\lowercase{%
\gdef\@xexpast#1*#2#3#4\@@{\begingroup
\def~{@}\edef\reserved@a{#1}\@tempcnta#2\relax
\ifnum\@tempcnta >\z@
\@whilenum\@tempcnta >\z@\do
{\edef\reserved@a{\reserved@a#3}\advance\@tempcnta \m@ne}%
\let\reserved@b\@xexpast
\else \let\reserved@b\@xexnoop\fi
\expandafter\reserved@b \reserved@a #4\@@
\expandafter\endgroup
\expandafter\def\expandafter\reserved@a\expandafter{\reserved@a}%
\let\@tempa\reserved@a
}% End of \@xexpast
}% End of \lowercase
\endgroup
}{}% End of \@ifundefined test.
\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\|}
\let\hdots\@ldots
\def\hdotsfor#1{\if\noexpand#1[\expandafter\shdots@for\else
\hdots@for\@ne{#1}\fi}
\newmuskip\dotsspace@
\def\shdots@for#1]{\hdots@for{#1}}
\def\hdots@for#1#2{\multicolumn{#2}c%
{\m@th\dotsspace@1.5mu\mkern-#1\dotsspace@
\xleaders\hbox{$\m@th\mkern#1\dotsspace@.\mkern#1\dotsspace@$}%
\hfill
\mkern-#1\dotsspace@}%
}
\def\cases{\left\{\def\arraystretch{1.2}\hskip-\arraycolsep
\array{l@{\quad}l}}
\def\endcases{\endarray\hskip-\arraycolsep\right.}
\newif\ifinany@
\newif\ifinalign@
\newif\ifingather@
\newcount\and@
\newif\iftag@
\newhelp\andhelp@
{An extra & here is so disastrous that you should probably exit^^J
and fix things up.}
\newif\ifst@rred
\newtoks\@envbody
\def\addto@envbody#1{\@envbody\expandafter{\the\@envbody#1}}
\def\collect@body#1{%
\@envbody{}%
\def\process@envbody{%
\expandafter#1\expandafter{\the\@envbody}}%
\expandafter\let\csname\@currenvir\endcsname\collect@@body
\csname\@currenvir\endcsname}
\def\collect@@body#1\end#2{%
\def\@tempa{#2}%
\ifx\@tempa\@currenvir
\addto@envbody{#1}%
\expandafter\edef\csname\@currenvir\endcsname
{\noexpand\process@envbody\noexpand\end{\@tempa}}%
\else
\addto@envbody{#1\end{#2}}%
\fi
\csname\@currenvir\endcsname}
\newenvironment{aligned}{%
\relax\ifmmode\else\nonmatherr@{\begin{aligned}}\fi
\null\,\new@ifnextchar[{\aligned@}{\aligned@[c]}%
}{%
\crcr\egroup\egroup
}
\def\aligned@[#1]{%
\if #1t\vtop \else \if#1b\vbox \else \vcenter \fi\fi
\bgroup
\Let@\restore@math@cr
\default@tag
\ifinany@\else\openup\jot\fi
\ialign\bgroup\hfil\strut@$\m@th\displaystyle{##}$&%
$\m@th\displaystyle{{}##}$\hfil\crcr}
\newtoks\hashtoks@
\hashtoks@{#}
\newcount\atcount@
\def\doat@#1{\toks@{\hfil\strut@$\m@th
\displaystyle{\the\hashtoks@}$&$\m@th\displaystyle
{{}\the\hashtoks@}$\hfil}%
\atcount@#1\relax\advance\atcount@\m@ne
\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
\xdef\preamble@{\the\toks@}\xdef\preamble@@{\preamble@}}
\newenvironment{alignedat}[2][c]{%
\null\,%
\if #1t\vtop \else \if#1b\vbox \else \vcenter \fi\fi
\bgroup\doat@{#2}\Let@ \restore@math@cr\default@tag
\ifinany@\else\openup\jot\fi\ialign\bgroup\span\preamble@@\crcr
}{%
\endaligned
}
\newenvironment{gathered}[1][c]{%
\relax\ifmmode\else\nonmatherr@{\begin{gathered}}\fi
\null\,%
\if #1t\vtop \else \if#1b\vbox \else \vcenter \fi\fi
\bgroup\Let@\restore@math@cr
\ifinany@\else\openup\jot\fi\ialign
\bgroup\hfil\strut@$\m@th\displaystyle##$\hfil\crcr
}{%
\endaligned
}
\def\tag@in@align{\relax