%%% ====================================================================
%%%  @LaTeX-file{
%%%     filename        = "amsintsm.dtx",
%%%     version         = "1.2beta",
%%%     date            = "1994/10/28",
%%%     time            = "12:16:44 EDT",
%%%     author          = "American Mathematical Society",
%%%     copyright       = "Copyright (C) 1994 American Mathematical Society,
%%%                        all rights reserved.  Copying of this file is
%%%                        authorized only if either:
%%%                        (1) you make absolutely no changes to your copy,
%%%                        including name; OR
%%%                        (2) if you do make changes, you first rename it
%%%                        to some other name.",
%%%     address         = "American Mathematical Society,
%%%                        Technical Support,
%%%                        Electronic Products and Services,
%%%                        P. O. Box 6248,
%%%                        Providence, RI 02940,
%%%                        USA",
%%%     telephone       = "401-455-4080 or (in the USA and Canada)
%%%                        800-321-4AMS (321-4267)",
%%%     FAX             = "401-331-3842",
%%%     checksum        = "51467 355 1811 14705",
%%%     email           = "tech-support@math.ams.org (Internet)",
%%%     codetable       = "ISO/ASCII",
%%%     keywords        = "latex, amslatex, ams-latex, integral, limits",
%%%     supported       = "yes",
%%%     abstract        = "This is part of the AMS-\LaTeX{} distribution.
%%%                        It provides some commands for printing
%%%                        integrals, sums, and derivatives.",
%%%     docstring       = "The checksum field above contains a CRC-16
%%%                        checksum as the first value, followed by the
%%%                        equivalent of the standard UNIX wc (word
%%%                        count) utility output of lines, words, and
%%%                        characters.  This is produced by Robert
%%%                        Solovay's checksum utility.",
%%%  }
%%% ====================================================================
% \iffalse
%<*driver>
\NeedsTeXFormat{LaTeX2e}
\documentclass{amsdtx}
\title{The \pkg{amsintsm} package}
\author{American Mathematical Society\\Michael Downes}
\date{Version \fileversion, \filedate}
\begin{document}
\hDocInput{amsintsm.dtx}
\end{document}
%</driver>
% \fi
%
% \maketitle
%
% \MakeShortVerb\|
%^^A LaTeX's tabular works poorly when cell contents have unusual/varying
%^^A heights. Let's define a substitute.
% \newenvironment{tab1}{\par\addvspace\medskipamount
%   \tabskip0pt plus\textwidth \def\\{\cr}\openup2pt
%   \def\hline{\noalign{\nobreak\vskip2pt\hrule\nobreak\vskip3pt}}%
%   \halign to\columnwidth\bgroup\tabskip1em minus.6em
%     \tbox{.3}\xverb##\strut\par\egroup\hfil&
%     $\displaystyle##$\hfil&
%     \tbox{.3}\xverb##\strut\par\egroup\hfil
%       \tabskip0pt plus\textwidth\cr
% }{%
%   \crcr\egroup \addvspace\medskipamount
% }
%
%^^A Column head for tab1:
% \newcommand\ch[1]{\omit\textbf{#1}\hfil}
%
% \newcommand{\tbox}[1]{%
%   \vtop\bgroup\hsize#1\columnwidth\raggedright\def\\{\cr}%
%   \noindent\hangindent1em \strut}
%
%^^A Define a pseudo-\verb that allows friendlier line breaking
% \def\xverb"{\verb"\catcode`\\\active \catcode`\ =10
%   \hyphenchar\font=`\}\relax}
% \begingroup \lccode`\~=`\\\lowercase{\endgroup
%   \def~}{\penalty100 \char92 }
%
%^^A Emulate Sb so we don't need to load amstex for this documentation.
% \newenvironment{Sb}{_\bgroup\renewcommand{\arraystretch}{0.8}%
%   \everymath{\scriptstyle}\array{@{}c@{}}%
% }{
%   \endarray\egroup
% }
%
% \section{Introduction}
%
%    In a new package \pkg{amsintsm} (`amsintsum' shortened to 8
%    letters) I attempt to provide a more rational syntax for the
%    \cn{int} and \cn{sum} commands, with a view to supporting
%    information processing possibilities that require better markup
%    (e.g., indicating the end of the sum or integral body for a
%    vision-impaired `reader' hearing the formula through a
%    text-to-speech converter; or changing the font of the differential
%    $d$ according to different conventions).
%
%    The basic goal is to make the simplest form of the \cn{int} command
%    cover most of the common variations for integral notation in a way
%    that clearly marks the variable of integration, the lower and upper
%    limits if present, and the boundaries of the integral body---and to
%    do this in a way that is convenient for authors to write. Because
%    the notation can vary widely, we won't attempt to make the primary
%    \cn{int} command handle all possibilities, but instead provide some
%    alternate versions of the \cn{int} command to handle special cases.
%    Here is a table showing various integral possibilities, typical
%    markup under current conventions, and proposed `rational' markup.
% \begin{tab1}
%    \ch{Old markup}& \ch{Printed result}& \ch{Proposed markup}\\
%    \hline
%    "\int f(x)"& \int f(x)&
%      "\int{f(x)}"\\
%    "\int f(x)\,dx"& \int f(x)\,dx&
%      "\int[x]{f(x)}"\\
%    "\int_0^1 f(x)\,dx"& \int_0^1 f(x)\,dx&
%      "\int[x,0,1]{f(x)}"\\
%    "\int_S f(x)\,dx"& \int_S f(x)\,dx&
%      "\int[x,S]{f(x)}"\\
%    "\int_0^1 \frac{dx}{f(x)}"& \int_0^1 \frac{dx}{f(x)}&
%      "\int[,0,1] {\frac{\diffl{x}}{f(x)}"\\
%    "\int_A^B\!\!\int_{S'} f(x,y)\,dS\,dx"&
%      \int_A^B\!\!\int_{S'} f(x,y)\,dS\,dx&
%        "\int[x,A,B] {\int[S,S'] {f(x,y)}}"\\
%    "\int\!\!\int f(x,y)\,dy\,dx"& \int\!\!\int f(x,y)\,dy\,dx&
%      "\ints[2,y,x] {f(x,y)}"\\
%    "\int\!\!\int\!\!\int f(x,y,z)\,dz\,dy\,dx"&
%      \int\!\!\int\!\!\int f(x,y,z)\,dz\,dy\,dx&
%        "\ints[3,z,y,x] {f(x,y,z)}"
% \end{tab1}
%
%    Thus the \cn{int} command takes one mandatory argument (the
%    integral body) and three optional arguments in the form
%    \verb'[a,b,c]', where trailing arguments in the comma-separated
%    list can be dropped if not applicable. The \cn{ints} command takes
%    the body as a mandatory argument, and the number of optional
%    arguments varies according to the multiplicity of the integral
%    symbol, which is given as the first optional argument. For a first
%    argument $n=1,2,\ldots$, the next $n$ arguments are taken to be the
%    variables of integration and printed in reverse order at the end.
%    That is, \verb'\ints[2,x,y,S]' means a double integral with
%    variables $x$ and $y$, over the area labeled $S$.
%
%    Commands \cn{dx}, \cn{dy}, etc., are not provided, but rather a
%    differential command \cn{diffl} taking one argument. If you need to
%    use \cn{diffl} frequently in a document with particular variables,
%    then you would add \cn{newcommand} statements in the document
%    preamble to define abbreviations for and only for the ones you
%    need:
% \begin{verbatim}
% \newcommand{\dx}{\diffl{x}}
% \newcommand{\dt}{\diffl{t}}
% \end{verbatim}
%    Naturally, the syntax of \cn{sum}, \cn{prod}, and other commands
%    would follow the general lines of the proposed \cn{int} syntax.
% \begin{tab1}
%    \ch{Old markup}& \ch{Printed result}& \ch{Proposed markup}\\
%    \hline
%    "\sum a_i"& \sum a_i& "\sum{a_i}"\\
%    "\sum_{a_i=\theta_i r_i} f(a)"&
%      \sum_{a_i=\theta_i r_i} f(a)&
%        "\sum[i,a_i=\theta_i r_i] {f(a)}"\\
%    "\bigcup_{0<t\leq h} \prod_{i=1}^n"
%      \xverb"\left( \frac{a_i} {t^i}, \frac{b_i} {t^i} \right)"&
%      \bigcup_{0<t\leq h} \prod_{i=1}^n \left( \frac{a_i} {t^i},
%        \frac{b_i} {t^i} \right)&
%      "\bigcup[t,0<t\leq h] {\prod[i,1,n]"
%        \xverb"{\left( \frac{a_i} {t^i}, \frac{b_i} {t^i} \right) }}"
% \end{tab1}
%
%    Here are some problem cases: How to handle overrides of the limit
%    placement; a compound differential for a single integral; a comma
%    inside one of the limits; multiline subscripts.
% \begin{tab1}
%    \ch{Old markup}& \ch{Printed result}& \ch{Proposed markup}\\
%    \hline
%    "\int\limits_{r\leq\Delta P/2} f(r)\,dr"&
%      \int\limits_{r\leq\Delta P/2} f(r)\,dr&
%        "\int*[r, r\leq\Delta P/2] {f(r)}"\\
%    "\int_\Omega f\,dx_1,\ldots,dx_n"& \int_\Omega f\,dx_1,\ldots,dx_n&
%      "\int[,\Omega] {f\dx_1,\ldots,\dx_n}"\\
%    "\sum_{i,j=1}^n f(x_i,x_j)"&
%      \sum_{i,j=1}^n f(x_i,x_j)&
%        "\sum[,{i,j=1},n] {f(x_i,x_j)}"\\
%    "\sum\begin{Sb}a_i<b_i\\ g(b_i)<k\end{Sb} f(a_i,b_i)"&
%      \sum\begin{Sb}a_i<b_i\\ g(b_i)<k\end{Sb} f(a_i,b_i)&
%        "\sum[i,a_i<b_i\\ g(b_i)<k] {f(a_i,b_i)}"
% \end{tab1}
%
%    The \verb'*' option meaning `place limits above and below' has no
%    counterpart that means `place limits to the side'. I can't think of
%    any convenient syntax, so I guess that means the user is left to
%    write
% \begin{verbatim}
% \textstyle\sum ...
% \end{verbatim}
%    But I think in practice this might not be a major problem; in the most
%    likely cases where you'd want to force the limits to the side, \TeX{}
%    will already make that happen by its style-shifting practices, e.g. inside
%    fractions or radicals.
%
% \subsection{Open questions}
% \begin{itemize}
%    \item Should the proposed syntax be provided under different
%    command names, leaving the original \cn{sum}, \cn{int}, etc.,
%    unchanged?
%
%    \item Should the variable of summation be given a slot in the
%    optional arguments list or just omitted? You could think of writing
% \begin{displaymath}
%    \sum_{i=1}^\infty a_i
% \end{displaymath}
%    as |\sum[i,1,\infty]{a_i}|, making a nice parallel with the
%    \cn{int} syntax, and having the default action be to combine args 1
%    and 2 with an intervening |=|. But then what syntax do you use for
%    variant lower limits like $i,j>0$ that often occur? Or you might
%    give the variable of summation but for printing purposes just
%    ignore it: |\sum[i,i=1,\infty]{a_i}|, that is, always give the
%    limits in full. Or you might just leave out that slot and make the
%    first optional argument the lower limit of summation.
%
%    \item On a similar note, for the purpose of accurately describing a
%    given integral it would be better to retain the variable of
%    integration in the argument list even when it is not printed in the
%    normal place. That is, instead of writing
% \begin{verbatim}
% \int[,0,1] {\frac{\diffl{x}}{f(x)}
% \end{verbatim}
%    when the $dx$ goes into the numerator of a fraction, we would like
%    some other syntax that retains the `logical' argument list
%    |[x,0,1]| but somehow indicates that the $dx$ should be printed in
%    a different spot. Perhaps a different command name \cn{fracint}
%    instead of \cn{int} that has two arguments instead of one for the
%    body (numerator, denominator) and places the $dx$ at the end of the
%    numerator?
%
%    \item What is the natural order for the arguments in a multiple
%    integral such as
% \begin{verbatim}
% \ints[3,z,y,x] {f(x,y,z)}
% \end{verbatim}
%    That is, should it be |z,y,x| or |x,y,z|?
%
%    \item What should be provided in the way of a generalized integral
%    or sum command that can handle more problematic variations? For
%    example, one might want to have something like
% \begin{verbatim}
% \gint{variable}{lower limit}{upper limit}{body}{differential}
% \end{verbatim}
%    where the first argument is ignored (just included for
%    informational purposes), any of the arguments is allowed to be
%    empty, and nothing is added to any of them except the obvious
%    sub\slash superscripting operations and horizontal spacing.
%
% \end{itemize}
%
% \subsection{Note}
%    The current implementation [1994/10/28] redefines \cn{int} and
%    \cn{sum}, defines \cn{diffl}, \cn{pdiffl} for differential
%    notation, and defines \cn{deriv}, \cn{pderiv} for the most common
%    kinds of derivative fractions. Remaining to be done: \cn{ints};
%    \cn{prod}, \cn{bigcup}, and the other large operator symbols that
%    should behave like \cn{sum}; the * option of \cn{int} and \cn{sum};
%    etc.
%
% \StopEventually{}
%
% \section{Implementation}
%
%    Standard package info.
%    \begin{macrocode}
\NeedsTeXFormat{LaTeX2e}
\ProvidesPackage{amsintsm}[1994/10/12 v1.2beta]
%    \end{macrocode}
%
%    Need \cs{new@ifnextchar}, \cs{@ifnotempty} from the \pkg{amsgen}
%    package.
%    \begin{macrocode}
\RequirePackage{amsgen}
%    \end{macrocode}
%
%    The \pkg{amstex} package defines \cs{DOTSB}, \cs{slimits@},
%    \cs{ilimits@}.
%    \begin{macrocode}
\@ifundefined{DOTSB}{%
  \let\DOTSB\relax
  \let\slimits@\displaylimits
  \let\ilimits@\nolimits
%    \end{macrocode}
%    Revert the meaning of \cn{sum} in preparation for saving it as
%    \cn{sumsymbol}.
%    If the \pkg{amstex} package is loaded, \cn{sum} is no longer a
%    simple mathchardef, we need to get \cs{sum@} instead.
%    \begin{macrocode}
  \let\sum=\sum@
}{}%
%    \end{macrocode}
%
%    Save the previous definitions for the sum and integral symbols.
%    Assumption: \cn{sum} and \cn{intop} are suitable mathchardefs for
%    the math fonts currently in use.
%    \begin{macrocode}
\def\intsymbol{\intop\ilimits@}
\let\sumsymbol=\sum
%    \end{macrocode}
%
%    \begin{macrocode}
\DeclareRobustCommand{\int}{\DOTSB
  \new@ifnextchar[{\@integral}{\@integral[]}}
%    \end{macrocode}
%
%    \begin{macrocode}
\def\@integral[#1]{\@integrala#1,,,\@nil}
\def\@integrala#1,#2,#3,#4\@nil#5{%
 \intop\ilimits@
   \@ifnotempty{#2}{_{#2}}%
   \@ifnotempty{#3}{^{#3}}%
   #5\@ifnotempty{#1}{\diffl{#1}}%
}
%    \end{macrocode}
%
%    Differential d should be roman according to some conventions,
%    instead of italic. If the \cn{diffl} function is used the output
%    can be changed according to the prevailing convention. I guess this
%    should be made into a package option. For the moment you get the
%    default which is italic.
%    \begin{macrocode}
\DeclareRobustCommand{\diffl}[1]{\mathinner{d#1}}
\DeclareRobustCommand{\pdiffl}[1]{\mathinner{\partial#1}}
%    \end{macrocode}
%
%    A couple of things for derivatives:
%    \begin{macrocode}
\DeclareRobustCommand{\deriv}[2]{\frac{\diffl{#1}}{\diffl{#2}}}
\DeclareRobustCommand{\pderiv}[2]{\frac{\pdiffl{#1}}{\pdiffl{#2}}}
%    \end{macrocode}
%
%    \begin{macrocode}
\DeclareRobustCommand{\sum}{\DOTSB
  \new@ifnextchar[{\@summation}{\@summation[]}}
\def\@summation[#1]{\@summationa#1,,\@nil}
\def\@summationa#1,#2,#3\@nil{\sumsymbol\slimits@
  \@ifnotempty{#1}{_{#1}}\@ifnotempty{#2}{^{#2}}\mathinner{}}
%    \end{macrocode}
%
%    The usual \cs{endinput} to ensure that random garbage at the end of
%    the file doesn't get copied by \fn{docstrip}.
%    \begin{macrocode}
\endinput
%    \end{macrocode}
%
% \CheckSum{78}
% \Finale