From e0c6872cf40896c7be36b11dcc744620f10adf1d Mon Sep 17 00:00:00 2001
From: Norbert Preining <norbert@preining.info>
Date: Mon, 2 Sep 2019 13:46:59 +0900
Subject: Initial commit

---
 macros/latex/contrib/breqn/breqn.dtx | 7357 ++++++++++++++++++++++++++++++++++
 1 file changed, 7357 insertions(+)
 create mode 100644 macros/latex/contrib/breqn/breqn.dtx

(limited to 'macros/latex/contrib/breqn/breqn.dtx')

diff --git a/macros/latex/contrib/breqn/breqn.dtx b/macros/latex/contrib/breqn/breqn.dtx
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--- /dev/null
+++ b/macros/latex/contrib/breqn/breqn.dtx
@@ -0,0 +1,7357 @@
+% \iffalse meta-comment
+%
+% Copyright (C) 1997-2003 by Michael J. Downes
+% Copyright (C) 2007-2008 by Morten Hoegholm
+% Copyright (C) 2007-2014 by Lars Madsen
+% Copyright (C) 2007-2014 by Will Robertson
+% Copyright (C) 2010-2017 by Joseph Wright
+%
+% This work may be distributed and/or modified under the
+% conditions of the LaTeX Project Public License, either
+% version 1.3 of this license or (at your option) any later
+% version. The latest version of this license is in
+%    http://www.latex-project.org/lppl.txt
+% and version 1.3 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 Will Robertson.
+%
+% This work consists of the main source file breqn.dtx
+% and the derived files
+%    breqn.sty, breqn.pdf, breqn.ins.
+%
+% Distribution:
+%    CTAN:macros/latex/contrib/mh/breqn.dtx
+%    CTAN:macros/latex/contrib/mh/breqn.pdf
+%
+% Unpacking:
+%    (a) If breqn.ins is present:
+%           tex breqn.ins
+%    (b) Without breqn.ins:
+%           tex breqn.dtx
+%    (c) If you insist on using LaTeX
+%           latex \let\install=y\input{breqn.dtx}
+%        (quote the arguments according to the demands of your shell)
+%
+% Documentation:
+%    The class ltxdoc loads the configuration file ltxdoc.cfg
+%    if available. Here you can specify further options, e.g.
+%    use A4 as paper format:
+%       \PassOptionsToClass{a4paper}{article}
+%
+%    Programm calls to get the documentation (example):
+%       pdflatex breqn.dtx
+%       makeindex -s gind.ist breqn.idx
+%       pdflatex breqn.dtx
+%       makeindex -s gind.ist breqn.idx
+%       pdflatex breqn.dtx
+%
+% Installation:
+%    TDS:tex/latex/breqn/breqn.sty
+%    TDS:doc/latex/breqn/breqn.pdf
+%    TDS:source/latex/breqn/breqn.dtx
+%
+%<*ignore>
+\begingroup
+  \def\x{LaTeX2e}
+\expandafter\endgroup
+\ifcase 0\ifx\install y1\fi\expandafter
+         \ifx\csname processbatchFile\endcsname\relax\else1\fi
+         \ifx\fmtname\x\else 1\fi\relax
+\else\csname fi\endcsname
+%</ignore>
+%<*install>
+\input docstrip.tex
+\Msg{************************************************************************}
+\Msg{* Installation: breqn}
+\Msg{************************************************************************}
+
+\keepsilent
+\askforoverwritefalse
+
+\preamble
+
+This is a generated file.
+
+Copyright (C) 1997-2003 by Michael J. Downes
+Copyright (C) 2007-2008 by Morten Hoegholm
+Copyright (C) 2007-2014 by Lars Madsen
+Copyright (C) 2007-2018 by Will Robertson
+Copyright (C) 2010-2017 by Joseph Wright
+
+This work may be distributed and/or modified under the
+conditions of the LaTeX Project Public License, either
+version 1.3 of this license or (at your option) any later
+version. The latest version of this license is in
+   http://www.latex-project.org/lppl.txt
+and version 1.3 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 Will Robertson.
+
+This work consists of the main source file breqn.dtx
+and the derived files
+   breqn.sty, breqn.pdf, breqn.ins.
+
+\endpreamble
+
+\generate{%
+  \file{breqn.ins}{\from{breqn.dtx}{install}}%
+  \usedir{tex/latex/breqn}%
+  \file{breqn.sty}{\from{breqn.dtx}{package}}%
+}
+
+\obeyspaces
+\Msg{************************************************************************}
+\Msg{*}
+\Msg{* To finish the installation you have to move the following}
+\Msg{* file into a directory searched by TeX:}
+\Msg{*}
+\Msg{*     breqn.sty}
+\Msg{*}
+\Msg{* Happy TeXing!}
+\Msg{*}
+\Msg{************************************************************************}
+
+\endbatchfile
+%</install>
+%<*ignore>
+\fi
+%</ignore>
+%<*driver>
+\NeedsTeXFormat{LaTeX2e}
+\ProvidesFile{breqn.drv}[2018/09/14 0.98f breqn]
+\documentclass{ltxdoc}
+\CodelineIndex
+\EnableCrossrefs
+\setcounter{IndexColumns}{2}
+\usepackage{color,verbatim,xspace,varioref,listings,amsmath,trace}
+
+\usepackage{geometry}
+
+\AtBeginDocument{\addtocontents{toc}{\protect\begin{multicols}{2}}}
+\AtEndDocument{\addtocontents{toc}{\protect\end{multicols}}}
+
+\def\partname{Part}
+
+\definecolor{hilite}{rgb}{0.2,0.4,0.7}
+\def\theCodelineNo{\textcolor{hilite}{\sffamily\tiny\arabic{CodelineNo}}}
+
+\lstloadlanguages{[AlLaTeX]TeX}
+
+\lstnewenvironment{literalcode}
+  {\lstset{gobble=2,columns=fullflexible,basicstyle=\color{hilite}\ttfamily}}
+  {}
+\makeatletter
+
+{\catcode`\%=12
+ \long\gdef\@gobble@percent@space#1{\ifx
+   #1%\expandafter\@gobble\else\expandafter#1\fi}}
+
+
+\AtBeginDocument{\def\verbatim@processline{\expandafter\check@percent
+  \the\verbatim@line\par}}
+\newwrite\tmp@out
+\newcounter{xio}
+\newenvironment{xio}{% example input and output
+  \par\addvspace\bigskipamount
+  \hbox{\itshape
+    \refstepcounter{xio}\kern-\p@ Example \thexio}\@nobreaktrue
+  \immediate\openout\tmp@out\jobname.tmp \relax
+  \begingroup
+  \let\do\@makeother\dospecials\catcode`\^^M\active
+  \def\verbatim@processline{
+    \immediate\write\tmp@out{\expandafter\@gobble@percent@space
+      \the\verbatim@line}}%
+  \verbatim@start
+}{%
+  \immediate\closeout\tmp@out
+  \@verbatim\frenchspacing\@vobeyspaces
+  \@@input \jobname.tmp \relax
+  \endgroup
+  \par\medskip
+  \noindent\ignorespaces
+  \@@input \jobname.tmp \relax
+  \par\medskip
+}
+
+
+
+
+\providecommand*\pkg[1]{\textsf{#1}}
+\providecommand*\cls[1]{\textsf{#1}}
+\providecommand*\opt[1]{\texttt{#1}}
+\providecommand*\env[1]{\texttt{#1}}
+\providecommand*\fn[1]{\texttt{#1}}
+
+
+\providecommand*\cn[1]{\cs{#1}}
+\providecommand*\csarg[1]{\texttt{\char`\{#1\char`\}}}
+
+\providecommand*\tex{\TeX\xspace}
+\providecommand*\latex{\LaTeX\xspace}
+\providecommand*\dbldollars{\texttt{\detokenize{$$}}}%$$
+\providecommand*\arg{}
+\edef\arg{\expandafter\@gobble\string\#}
+
+\newenvironment{aside}{\begin{quote}\bfseries}{\end{quote}}
+\newenvironment{dn}{\begin{quote}\bfseries}{\end{quote}}
+
+\newcommand\dash{\textemdash}
+\newcommand\dbslash[1]{\texttt{\string\\}}
+\newcommand\thepkg{the \pkg{breqn} package\xspace}
+
+\providecommand*\texbook{\textsl{The \protect\TeX{}book}\xspace}
+
+\providecommand*\dotsc{\ldots}
+\providecommand*\eqref[1]{(\ref{#1})}
+
+\providecommand*\qq[1]{\textquotedblleft#1\textquotedblright}
+\providecommand*\quoted[1]{\textquoteleft#1\textquoteright}
+\providecommand*\dquoted[1]{\textquotedblleft#1\textquotedblright}
+
+\providecommand*\ie{i.e.,\xspace}
+\providecommand*\eg{e.g.,\xspace}
+\providecommand*\etc{etc.\xspace}
+\providecommand*\cf{cf.\xspace}
+
+\providecommand*\ndash{\unskip\textendash\ignorespaces}
+\providecommand*\mdash{\unskip\textemdash\ignorespaces}
+
+\makeatother
+
+\usepackage{breqn}
+
+\begin{document}
+  \DocInput{breqn.dtx}
+\end{document}
+%</driver>
+% \fi
+%
+%
+% \GetFileInfo{breqn.drv}
+% \let\pkgdate\filedate\let\pkgversion\fileversion
+% \title{The \pkg{breqn} package}
+% \date{\pkg{breqn} bundle: \pkgdate\space\pkgversion}
+% \author{Authors: Michael J. Downes, Morten H\o gholm\\ Maintained by Morten H\o gholm, Will Robertson\\ Feedback: \texttt{https://github.com/wspr/breqn/issues}}
+%
+% \maketitle
+% \begin{abstract}
+%   The \pkg{breqn} package facilitates automatic line-breaking of
+%   displayed math expressions.
+% \end{abstract}
+%
+%
+% \tableofcontents
+%
+%
+% \part{User's guide}
+%
+% \section{A bit of history}
+%
+% Originally \pkg{breqn}, \pkg{flexisym}, and \pkg{mathstyle} were
+% created by Michael J.~Downes from the American Mathematical Society
+% during the 1990's up to late 2002. Sadly---and much to the shock of
+% the \TeX\ world---Michael passed away in early 2003 at the age of
+% only~44.
+%
+% The American Mathematical Society kindly allowed Morten H\o gholm to
+% assume maintainership of this part of his work and we wish to
+% express our gratitude to them and to Barbara Beeton in particular for
+% providing the files needed.
+%
+% MH brought Michael's work to a wider audience, thereby allowing users to create
+% more \emph{masterpieces of the publishing art} as we think he would
+% have wanted.
+%
+% Following the July 2008 breqn release, \pkg{breqn} was left in the hands
+% of a maintenance team, while MH moved on with other projects.
+%
+% \section{Package loading}
+%
+%
+% The recommended way of loading the \pkg{breqn} package is to load it
+% \emph{after} other packages dealing with math, \ie, after
+% \pkg{amsmath}, \pkg{amssymb}, or packages such as \pkg{mathpazo} or
+% \pkg{mathptmx}.
+%
+% The \pkg{flexisym} package (described in section \vref{flexisym}) is
+% required by \pkg{breqn} and ensures the math symbols are set up
+% correctly. By default \pkg{breqn} loads it with support for Computer
+% Modern but if you use a different math package requiring slightly
+% different definitions, it must be loaded before \pkg{breqn}. Below
+% is an example of how you enable \pkg{breqn} to work with the widely
+% used \pkg{mathpazo} package.
+% \begin{verbatim}
+%\usepackage{mathpazo}
+%\usepackage[mathpazo]{flexisym}
+%\usepackage{breqn}
+% \end{verbatim}
+% Currently, the packages \pkg{mathpazo} and \pkg{mathptmx} are
+% supported. Despair not: Chances are that the package will work using
+% the default settings. If you find that a particular math font
+% package doesn't work then please see implementation in
+% \fn{flexisym.dtx} for how to create a support file---it is easier
+% than one might think. Contributions welcome.
+%
+% The documentation for the package was formerly found in
+% \fn{breqndoc}. It has now been added to this implementation
+% file. Below follows the contents of the original \pkg{breqn}
+% documentation. Not all details hold anymore but I have prioritized
+% fixing the package.
+%
+% \section{Introduction}
+%
+% The \pkg{breqn} package for \LaTeX\ provides solutions to a number of
+% common difficulties in writing displayed equations and getting
+% high-quality output. For example, it is a well-known inconvenience that
+% if an equation must be broken into more than one line, \cs{left} \dots\
+% \cs{right} constructs cannot span lines. The \pkg{breqn} package makes
+% them work as one would expect whether or not there is an intervening
+% line break.
+%
+% The single most ambitious goal of the \pkg{breqn} package, however, is
+% to support automatic linebreaking of displayed equations. Such
+% linebreaking cannot be done without substantial changes under the hood
+% in the way math formulas are processed. For this reason, especially in
+% the alpha release, users should proceed with care and keep an eye out
+% for unexpected glitches or side effects.
+%
+% \section{Principal features}
+% The principal features of the \pkg{breqn} package are:
+% \begin{description}
+%
+% \item[semantically oriented structure] The way in which compound
+% displayed formulas are subdivided matches the logical structure more
+% closely than, say, the standard \env{eqnarray} environment. Separate
+% equations in a group of equations are written as separate environments
+% instead of being bounded merely by \dbslash/ commands. Among other
+% things, this clears up a common problem of wrong math symbol spacing at
+% the beginning of continuation lines. It also makes it possible to
+% specify different vertical space values for the space between lines of a
+% long, broken equation and the space between separate equations in a
+% group of equations.
+%
+% \item[automatic line breaking] Overlong equations will be broken
+% automatically to the prevailing column width, and continuation lines
+% will be indented following standard conventions.
+%
+% \item[line breaks within delimiters] Line breaks within \cs{left} \dots\
+% \cs{right} delimiters work in a natural way. Line breaks can be
+% forbidden below a given depth of delimiter nesting through a package
+% option.
+%
+% \item[mixed math and text] Display equations that contain mixed
+% math and text, or even text only, are handled naturally by means of a
+% \env{dseries} environment that starts out in text mode instead of math
+% mode.
+%
+% \item[ending punctuation] The punctuation at the end of a displayed
+% equation can be handled in a natural way that makes it easier to promote
+% or demote formulas from\slash to inline math, and to apply special
+% effects such as adding space before the punctuation.
+%
+% \item[flexible numbering] Equation numbering is handled in a natural
+% way, with all the flexibility of the \pkg{amsmath} package and with no
+% need for a special \cs{nonumber} command.
+%
+% \item[special effects] It is easy to apply special effects to individual
+% displays, e.g., changing the type size or adding a frame.
+%
+% \item[using available space] Horizontal shrink is made use of
+% whenever feasible. With most other equation macros it is frozen when it
+% occurs between \cs{left} \dots\ \cs{right} delimiters, or in any sort of
+% multiline structure, so that some expressions require two lines that would
+% otherwise fit on one.
+%
+% \item[high-quality spacing] The \cs{abovedisplayshortskip} is used when
+% applicable (other equation macros fail to apply it in equations of more
+% than one line).
+%
+% \item[abbreviations] Unlike the \pkg{amsmath} equation environments, the
+% \pkg{breqn} environments can be called through user-defined abbreviations
+% such as \cs{beq} \dots\ \cs{eeq}.
+%
+% \end{description}
+%
+% \section{Shortcomings of the package}
+% The principal known deficiencies of the \pkg{breqn} package are:
+%
+% \subsection{Incompatibilities} As it pushes the envelope
+% of what is possible within the context of \LaTeXe, \thepkg will tend
+% to break other packages when used in combination with them, or to fail
+% itself, when there are any areas of internal overlap; successful use may
+% in some cases depend on package loading order.
+%
+%
+%
+% \subsection{Indention of delimited fragments} When line breaks within
+% delimiters are involved, the automatic indention of continuation lines
+% is likely to be unsatisfactory and need manual adjustment. I don't see
+% any easy way to provide a general solution for this, though I have some
+% ideas on how to attain partial improvements.
+%
+% \subsection{Math symbol subversion}
+% In order for automatic line breaking to work, the operation of all the
+% math symbols of class 2, 3, 4, and 5 must be altered (relations, binary
+% operators, opening delimiters, closing delimiters). This is done by an
+% auxiliary package \pkg{flexisym}. As long as you stick to the advertised
+% \LaTeX\ interface for defining math symbols (\cs{DeclareMathSymbol}),
+% things should work OK most of the time. Any more complex math symbol
+% setup is quite likely to quarrel with the \pkg{flexisym} package.
+% See Section~\vref{flexisym} for further information.
+%
+% \subsection{Subscripts and superscripts}
+%
+% Because of the changes to math symbols of class 2--5, writing certain
+% combinations such as \verb'^+' or \verb'_\pm' or \verb'^\geq' without
+% braces would lead to error messages; (The problem described here
+% already exists in standard \LaTeX\ to a lesser extent, as you may know
+% if you ever tried \verb'^\neq' or \verb'^\cong'; and indeed there are
+% no examples in the \LaTeX\ book to indicate any sanction for omitting
+% braces around a subscript or superscript.)
+%
+% The \pkg{flexisym} package therefore calls, as of version 0.92, another
+% package called \pkg{mathstyle} which turns \verb'^' and \verb'_' into
+% active characters. This is something that I believe is desirable in any
+% case, in the long run, because having a proper mathstyle variable
+% eliminates some enormous burdens that affect almost any
+% nontrivial math macros, as well as many other things where the
+% connection is not immediately obvious, e.g., the \LaTeX\ facilities for
+% loading fonts on demand.
+%
+% Not that this doesn't introduce new and interesting problems of its
+% own---for example, you don't want to put usepackage statements
+% after flexisym for any package that refers to, e.g., \verb'^^J' or
+% \verb'^^M'
+% internally (too bad that the \LaTeX\ package loading code does not
+% include automatic defenses to ensure normal catcodes in the interior of
+% a package; but it only handles the \verb'@' character).
+%
+% But I took a random AMS journal article, with normal end-user kind of
+% \LaTeX\ writing, did some straightforward substitutions to change all
+% the equations into dmath environments, and ran it with active math
+% sub/sup: everything worked OK. This suggests to me that it can work in
+% the real world, without an impossible amount of compatibility work.
+%
+% \section{Incomplete}
+% In addition, in the \textbf{alpha release [1997/10/30]} the following
+% gaps remain to be filled in:
+% \begin{description}
+% \item[documentation]
+% The documentation could use amplification, especially more
+% illustrations, and I have undoubtedly overlooked more than a few errors.
+%
+% \item[group alignment] The algorithm for doing alignment
+% of mathrel symbols across equations in a \env{dgroup} environment
+% needs work. Currently the standard and \opt{noalign} alternatives
+% produce the same output.
+%
+% \item[single group number] When a \env{dgroup} has a group number and
+% the individual equations are unnumbered, the handling and placement of
+% the group number aren't right.
+%
+% \item[group frame] Framing a group doesn't work, you might be able to
+% get frames on the individual equations at best.
+%
+% \item[group brace] The \opt{brace} option for \env{dgroup} is intended
+% to produce a large brace encompassing the whole group. This hasn't been
+% implemented yet.
+%
+% \item[darray environment] The \env{darray} environment is unfinished.
+%
+% \item[dseries environment] The syntax and usage for the \env{dseries}
+% environment are in doubt and may change.
+%
+% \item[failure arrangements] When none of the line-breaking passes for a
+% \env{dmath} environment succeeds\dash i.e., at least one line is
+% overfull\dash the final arrangement is usually rather poor. A better
+% fall-back arrangement in the failure case is needed.
+%
+% \end{description}
+%
+% \section{Package options}
+%
+% Many of the package options for \thepkg are the same as options of the
+% \env{dmath} or \env{dgroup} environments, and some of them require an
+% argument, which is something that cannot be done through the normal
+% package option mechanism. Therefore most of the \pkg{breqn} package
+% options are designed to be set with a \cs{setkeys} command after the
+% package is loaded. For example, to load the package and set the
+% maximum delimiter nesting depth for line breaks to~1:
+% \begin{verbatim}
+% \usepackage{breqn}
+% \setkeys{breqn}{breakdepth={1}}
+% \end{verbatim}
+%
+% See the discussion of environment options, Section~\vref{envopts}, for
+% more information.
+%
+% Debugging information is no longer available as a package
+% option. Instead, the tracing information has been added in a fashion
+% so that it can be enabled as a docstrip option:
+% \begin{verbatim}
+% \generate{\file{breqn.sty}{\from{breqn.dtx}{package,trace}}}
+% \end{verbatim}
+%
+%
+% \section{Environments and commands}
+% \subsection{Environments}
+% All of the following environments take an optional argument for
+% applying local effects such as changing the typesize or adding a
+% frame to an individual equation.
+%
+% \begin{description}
+% \item[\env{dmath}] Like \env{equation} but supports line breaking and variant
+% numbers.
+%
+% \item[\env{dmath*}] Unnumbered; like \env{displaymath} but supports line
+% breaking
+%
+% \item[\env{dseries}] Like \env{equation} but starts out in text mode;
+% intended for series of mathematical expressions of the form `A, B, and
+% C'. As a special feature, if you use
+% \begin{verbatim}
+% \begin{math} ... \end{math}
+% \end{verbatim}
+% for each expression in the series, a suitable amount of inter-expression
+% space will be automatically added. This is a small step in the direction of
+% facilitating conversion of display math to inline math, and vice versa: If
+% you write a display as
+% \begin{verbatim}
+% \begin{dseries}
+% \begin{math}A\end{math},
+% \begin{math}B\end{math},
+% and
+% \begin{math}C\end{math}.
+% \end{dseries}
+% \end{verbatim}
+% then conversion to inline form is simply a matter of removing the
+% \verb'\begin{dseries}' and \verb'\end{dseries}' lines; the contents of the
+% display need no alterations.
+%
+% It would be nice to provide the same feature for \verb'$' notation but
+% there is no easy way to do that because the \verb'$' function has no
+% entry point to allow changing what happens before math mode is entered.
+% Making it work would therefore require turning \verb'$' into an active
+% character, something that I hesitate to do in a \LaTeXe\ context.
+%
+% \item[\env{dseries*}] Unnumbered variant of \env{dseries}
+%
+% \item[\env{dgroup}] Like the \env{align} environment of \pkg{amsmath},
+% but with each constituent equation wrapped in a \env{dmath},
+% \env{dmath*}, \env{dseries}, or \env{dseries*} environment instead of being
+% separated by \dbslash/. The equations are numbered with a group number.
+% When the constituent environments are the numbered forms (\env{dmath} or
+% \env{dseries}) they automatically switch to `subequations'-style
+% numbering, i.e., something like (3a), (3b), (3c), \dots, depending on
+% the current form of non-grouped equation numbers. See also
+% \env{dgroup*}.
+%
+% \item[\env{dgroup*}] Unnumbered variant of \env{dgroup}. If the
+% constituent environments are the numbered forms, they get normal
+% individual equation numbers, i.e., something like (3), (4), (5), \dots~.
+%
+% \item[\env{darray}] Similar to \env{eqnarray} but with an argument like
+% \env{array} for giving column specs. Automatic line breaking is not
+% done here.
+%
+% \item[\env{darray*}] Unnumbered variant of \env{darray}, rather like
+% \env{array} except in using \cs{displaystyle} for all column
+% entries.
+%
+% \item[\env{dsuspend}] Suspend the current display in order to print
+%   some text, without loss of the alignment. There is also a command
+%   form of the same thing, \cs{intertext}.
+% \end{description}
+%
+% \subsection{Commands}
+%
+% The commands provided by \thepkg are:
+% \begin{description}
+% \item[\cs{condition}] This command is used for
+% a part of a display which functions as a condition on the main
+% assertion. For example:
+% \begin{verbatim}
+% \begin{dmath}
+% f(x)=\frac{1}{x} \condition{for $x\neq 0$}
+% \end{dmath}.
+% \end{verbatim}
+% \begin{dmath}
+% f(x)=\frac{1}{x} \condition{for $x\neq 0$}
+% \end{dmath}.
+% The \cs{condition} command automatically switches to text mode (so that
+% interword spaces function the way they should), puts in a comma, and
+% adds an appropriate amount of space. To facilitate promotion\slash
+% demotion of formulas, \cs{condition} \qq{does the right thing} if used
+% outside of display math.
+%
+% To substitute a different punctuation mark instead of the default comma,
+% supply it as an optional argument for the \cs{condition} command:
+% \begin{verbatim}
+% \condition[;]{...}
+% \end{verbatim}
+% (Thus, to get no punctuation: \verb'\condition[]{...}'.)
+%
+% For conditions that contain no text, you can use the starred form of the
+% command, which means to stay in math mode:
+% \begin{verbatim}
+% \begin{dmath}
+% f(x)=\frac{1}{x} \condition*{x\neq 0}
+% \end{dmath}.
+%
+% If your material contains a lot of conditions like these, you might like
+% to define shorter abbreviations, e.g.,
+% \begin{verbatim}
+% \newcommand{\mc}{\condition*}% math condition
+% \newcommand{\tc}{\condition}%  text condition
+% \end{verbatim}
+% But \thepkg refrains from predefining such abbreviations in order that
+% they may be left to the individual author's taste.
+%
+% \item[\cs{hiderel}] In a compound equation it is sometimes desired to
+% use a later relation symbol as the alignment point, rather than the
+% first one. To do this, mark all the relation symbols up to the desired
+% one with \cs{hiderel}:
+% \begin{verbatim}
+% T(n) \hiderel{\leq} T(2^n) \leq c(3^n - 2^n) ...
+% \end{verbatim}
+% \end{description}
+%
+% \section{Various environment options}\label{envopts}
+%
+% The following options are recognized for the \env{dmath}, \env{dgroup},
+% \env{darray}, and \env{dseries} environments; some of the options do not
+% make sense for all of the environments, but if an option is used where
+% not applicable it is silently ignored rather than treated as an error.
+% \changes{v0.98b}{2010/08/27}{replaced relindent with indentstep}
+%
+% \begin{verbatim}
+% \begin{dmath}[style={\small}]
+% \begin{dmath}[number={BV}]
+% \begin{dmath}[labelprefix={eq:}]
+% \begin{dmath}[label={xyz}]
+% \begin{dmath}[indentstep={2em}]
+% \begin{dmath}[compact]
+% \begin{dmath}[spread={1pt}]
+% \begin{dmath}[frame]
+% \begin{dmath}[frame={1pt},framesep={2pt}]
+% \begin{dmath}[background={red}]
+% \begin{dmath}[color={purple}]
+% \begin{dmath}[breakdepth={0}]
+% \end{verbatim}
+%
+% Use the \opt{style} option to change the type size of an individual
+% equation. This option can also serve as a catch-all option for
+% altering the equation style in other ways; the contents are simply
+% executed directly within the context of the equation.
+%
+% Use the \opt{number} option if you want the number for a particular
+% equation to fall outside of the usual sequence. If this option is used
+% the equation counter is not incremented. If for some reason you need to
+% increment the counter and change the number at the same time, use the
+% \opt{style} option in addition to the \opt{number} option:
+% \begin{verbatim}
+% style={\refstepcounter{equation}}
+% \end{verbatim}
+%
+% Use of the normal \cs{label} command instead of the \opt{label} option
+% works, I think, most of the time (untested). \opt{labelprefix} prepends
+% its argument to the label (only useful as a global option, really),
+% and must be called before \opt{label}.
+%
+% \changes{v0.98b}{2010/08/27}{replaced relindent with indentstep}
+% Use the \opt{indentstep} option to specify something other than the
+% default amount for the indention of relation symbols. The default is
+% 8pt.
+%
+% Use the \opt{compact} option in compound equations to inhibit line
+% breaks at relation symbols. By default a line break will be taken before
+% each relation symbol except the first one. With the \opt{compact} option
+% \LaTeX\ will try to fit as much material as possible on each line, but
+% breaks at relation symbols will still be preferred over breaks at binary
+% operator symbols.
+%
+% Use the \opt{spread} option to increase (or decrease) the amount of
+% interline space in an equation. See the example given above.
+%
+% Use the \opt{frame} option to produce a frame around the body of the
+% equation. The thickness of the frame can optionally be specified by
+% giving it as an argument of the option. The default thickness is
+% \cs{fboxrule}.
+%
+% Use the \opt{framesep} option to change the amount of space separating
+% the frame from what it encloses. The default space is \cs{fboxsep}.
+%
+% Use the \opt{background} option to produce a colored background for the
+% equation body. The \pkg{breqn} package doesn't automatically load the
+% \pkg{color} package, so this option won't work unless you remember
+% to load the \pkg{color} package yourself.
+%
+% Use the \opt{color} option to specify a different color for the contents
+% of the equation. Like the \opt{background} option, this doesn't work if
+% you forgot to load the \pkg{color} package.
+%
+% Use the \opt{breakdepth} option to change the level of delimiter nesting
+% to which line breaks are allowed. To prohibit line breaks within
+% delimiters, set this to 0:
+% \begin{verbatim}
+% \begin{dmath}[breakdepth={0}]
+% \end{verbatim}
+% The default value for breakdepth is 2. Even when breaks are allowed
+% inside delimiters, they are marked as less desirable than breaks outside
+% delimiters. Most of the time a break will not be taken within delimiters
+% until the alternatives have been exhausted.
+%
+% Options for the \env{dgroup} environment: all of the above, and also
+% \begin{verbatim}
+% \begin{dgroup}[noalign]
+% \begin{dgroup}[brace]
+% \end{verbatim}
+%
+% By default the equations in a \env{dgroup} are mutually aligned on their
+% relation symbols ($=$, $<$, $\geq$, and the like). With the
+% \opt{noalign} option each equation is placed individually without
+% reference to the others.
+%
+% The \opt{brace} option means to place a large brace encompassing the
+% whole group on the same side as the equation number.
+%
+% Options for the \env{darray} environment: all of the above (where
+% sensible), and also
+% \begin{verbatim}
+% \begin{darray}[cols={lcr@{\hspace{2em}}lcr}]
+% \end{verbatim}
+% The value of the \opt{cols} option for the darray environment should be
+% a series of column specs as for the \env{array} environment, with the
+% following differences:
+% \begin{itemize}
+% \item For l, c, and r what you get is not text, but math, and
+% displaystyle math at that. To get text you must use a 'p' column
+% specifier, or put an \cs{mbox} in each of the individual cells.
+%
+% \item Vertical rules don't connect across lines.
+% \end{itemize}
+%
+% \section{The \pkg{flexisym} package}\label{flexisym}
+%
+% The \pkg{flexisym} package does some radical changes in the setup for
+% math symbols to allow their definitions to change dynamically throughout
+% a document. The \pkg{breqn} package uses this to make symbols of classes
+% 2, 3, 4, 5 run special functions inside an environment such as
+% \env{dmath} that provide the necessary support for automatic line
+% breaking.
+%
+% The method used to effect these changes is to change the definitions of
+% \cs{DeclareMathSymbol} and \cs{DeclareMathDelimiter}, and then
+% re-execute the standard set of \LaTeX\ math symbol definitions.
+% Consequently, additional mathrel and mathbin symbols defined by other
+% packages will get proper line-breaking behavior if the other package is
+% loaded after the \pkg{flexisym} package and the symbols are defined
+% through the standard interface.
+%
+%
+%
+% \section{Caution! Warning!}
+% Things to keep in mind when writing documents with \thepkg:
+% \begin{itemize}
+%
+% \item The notation $:=$ must be written with the command \cs{coloneq}.
+%   Otherwise the $:$ and the $=$ will be treated as two separate relation
+%   symbols with an \qq{empty RHS} between them, and they will be printed
+%   on separate lines.
+%
+% \item Watch out for constructions like \verb'^+' where a single binary
+% operator or binary relation symbol is subscripted or superscripted. When
+% the \pkg{breqn} or \pkg{flexisym} package is used, braces are mandatory
+% in such constructions: \verb'^{+}'. This applies for both display and
+% in-line math.
+%
+% \item If you want \LaTeX\ to make intelligent decisions about line
+% breaks when vert bars are involved, use proper pairing versions of the
+% vert-bar symbols according to context: \verb'\lvert n\rvert' instead of
+% \verb'|n|'. With the nondirectional \verb'|' there is no way for \LaTeX\
+% to reliably deduce which potential breakpoints are inside delimiters
+% (more highly discouraged) and which are not.
+%
+% \item If you use the \pkg{german} package or some other package that
+% turns double quote \verb'"' into a special character, you may encounter
+% some problems with named math symbols of type mathbin, mathrel,
+% mathopen, or mathclose in moving arguments. For example, \cs{leq} in a
+% section title will be written to the \fn{.aux} file as something like
+% \verb'\mathchar "3214'. This situation probably ought to be improved,
+% but for now use \cs{protect}.
+%
+% \item Watch out for the \texttt{[} character at the beginning of a
+% \env{dmath} or similar environment, if it is supposed to be interpreted
+% as mathematical content rather than the start of the environment's
+% optional argument.
+%
+% This is OK:
+% \begin{verbatim}
+% \begin{dmath}
+% [\lambda,1]...
+% \end{dmath}
+% \end{verbatim}
+% This will not work as expected:
+% \begin{verbatim}
+% \begin{dmath}[\lambda,1]...\end{dmath}
+% \end{verbatim}
+%
+% \item Watch out for unpaired delimiter symbols (in display math only):
+% \begin{verbatim}
+% ( ) [ ] \langle \rangle \{ \} \lvert \rvert ...
+% \end{verbatim}
+% If an open delimiter is used without a close delimiter, or vice versa,
+% it is normally harmless but may adversely affect line breaking. This is only
+% for symbols that have a natural left or right directionality. Unpaired
+% \cs{vert} and so on are fine.
+%
+% When a null delimiter is used as the other member of the pair
+% (\verb'\left.' or \verb'\right.') this warning doesn't apply.
+%
+% \item If you inadvertently apply \cs{left} or \cs{right} to something
+% that is not a delimiter, the error messages are likely to be a bit
+% more confusing than usual. The normal \LaTeX\ response to an error such
+% as
+% \begin{verbatim}
+% \left +
+% \end{verbatim}
+% is an immediate message
+% \begin{verbatim}
+% ! Missing delimiter (. inserted).
+% \end{verbatim}
+% When \thepkg is in use, \LaTeX\ will fail to realize anything is wrong
+% until it hits the end of the math formula, or a closing delimiter
+% without a matching opening delimiter, and then the first message is an
+% apparently pointless
+% \begin{verbatim}
+% ! Missing \endgroup inserted.
+% \end{verbatim}
+%
+% \end{itemize}
+%
+% \section{Examples}
+%
+% \renewcommand\theequation{\thesection.\arabic{equation}}
+% % Knuth, SNA p74
+% \begin{xio}
+% Replace $j$ by $h-j$ and by $k-j$ in these sums to get [cf.~(26)]
+% \begin{dmath}[label={sna74}]
+% \frac{1}{6} \left(\sigma(k,h,0) +\frac{3(h-1)}{h}\right)
+%   +\frac{1}{6} \left(\sigma(h,k,0) +\frac{3(k-1)}{k}\right)
+% =\frac{1}{6} \left(\frac{h}{k} +\frac{k}{h} +\frac{1}{hk}\right)
+%   +\frac{1}{2} -\frac{1}{2h} -\frac{1}{2k},
+% \end{dmath}
+% which is equivalent to the desired result.
+% \end{xio}
+%
+% % Knuth, SNA 4.6.2, p387
+% \begin{xio}
+% \newcommand\mx[1]{\begin{math}#1\end{math}}% math expression
+% %
+% Now every column which has no circled entry is completely zero;
+% so when $k=6$ and $k=7$ the algorithm outputs two more vectors,
+% namely
+% \begin{dseries}[frame]
+% \mx{v^{[2]} =(0,5,5,0,9,5,1,0)},
+% \mx{v^{[3]} =(0,9,11,9,10,12,0,1)}.
+% \end{dseries}
+% From the form of the matrix $A$ after $k=5$, it is evident that
+% these vectors satisfy the equation $vA =(0,\dotsc,0)$.
+% \end{xio}
+%
+% \begin{xio}
+% \begin{dmath*}
+% T(n) \hiderel{\leq} T(2^{\lceil\lg n\rceil})
+%   \leq c(3^{\lceil\lg n\rceil}
+%     -2^{\lceil\lg n\rceil})
+%   <3c\cdot3^{\lg n}
+%   =3c\,n^{\lg3}
+% \end{dmath*}.
+% \end{xio}
+%
+% \begin{xio}
+% The reduced minimal Gr\"obner basis for $I^q_3$ consists of
+% \begin{dgroup*}
+% \begin{dmath*}
+% H_1^3 = x_1 + x_2 + x_3
+% \end{dmath*},
+% \begin{dmath*}
+% H_2^2 = x_1^2 + x_1 x_2 + x_2^2 - q_1 - q_2
+% \end{dmath*},
+% \begin{dsuspend}
+% and
+% \end{dsuspend}
+% \begin{dmath*}
+% H_3^1 = x_1^3 - 2x_1 q_1 - x_2 q_1
+% \end{dmath*}.
+% \end{dgroup*}
+% \end{xio}
+%
+%
+%
+%\section{Technical notes on tag placement}
+%
+%The method used by the breqn package to place the equation number is
+%rather more complicated than you might think, and the whole reason is
+%to allow the number to stay properly centered on the total height even
+%when the height fluctuates due to stretching or shrinking of the page.
+%
+%
+%Consider the following equation:
+%\begin{dmath}[number={3.15}]
+%  N_{0} \simeq \left( \frac{\nu}{\lVert u\rVert_{H^{i}}} \right)
+%  \lvert I\rvert^{-1/2}
+%\end{dmath}
+%It will have only one line, if the column width is not too narrow.
+%
+%Scrutinizing the vertical list will shed light on some of the basic properties
+%shared by all breqn equations. After that we will look at what would happen if
+%two or more lines were needed. The numbers added on the left in the following
+%\cs{showlists} output mark the points of interest.
+%\begin{verbatim}
+%[1] \penalty 10000
+%    \glue(\abovedisplayskip) 0.0
+%    \penalty 10000
+%    \glue(\belowdisplayskip) 0.0
+%[2] \glue 4.0 plus 4.0
+%    \glue(\lineskip) 1.0
+%[3] \vbox(16.53902+0.0)x0.0, glue set 16.53902fil
+%    .\glue 0.0 plus 1.0fil minus 1.0fil
+%    \penalty 10000
+%[4] \glue -8.51945
+%[5] \hbox(7.5+2.5)x25.55563
+%    .\OT1/cmr/m/n/10 (
+%    .\OT1/cmr/m/n/10 3
+%    .\OT1/cmr/m/n/10 .
+%    .\OT1/cmr/m/n/10 1
+%    .\OT1/cmr/m/n/10 5
+%    .\kern 0.0
+%    .\OT1/cmr/m/n/10 )
+%    \penalty 10000
+%[6] \glue(\parskip) -18.01956
+%[7] \hbox(16.53902+9.50012)x360.0, glue set 1.78647
+%\end{verbatim}
+%\begin{enumerate}
+%\item These four lines are a hidden display structure from \TeX's
+%  primitive \texttt{\$\$} mechanism. It is used only to get the value
+%  of \cs{predisplaysize} so that we can later calculate by hand
+%  whether to use the short display skips or the regular ones. (The
+%  reason that we have to do it by hand traces back to the fact that
+%  \TeX\ 3.x does not allow unhboxing in math mode.) The penalties come
+%  from \cs{predisplaypenalty} and \cs{postdisplaypenalty}, which were
+%  locally set to 10000 to ensure there would be no unintended page
+%  breaks at these glue nodes.
+%
+%\item These two glue nodes are the ones that would normally have been
+%  produced at the top of a display; the first one is the above-display
+%  skip node (though we had to put it in by hand with \cs{vskip}) and
+%  the second one is the usual baselineskip/lineskip node.
+%
+%\item This is a dummy copy of the equation's first line, which is
+%  thrown in here to get the proper value of baselineskip (or lineskip
+%  in this case). Why do we need this? Because this ensures that we get
+%  the top spacing right before we fiddle with the glue nodes
+%  surrounding the equation number. And if the equation has a frame,
+%  this box is a good place to add it from.
+%
+%\item This is a special glue node that brings us to the right vertical
+%  position for adding the equation number. Its value is calculated
+%  from the variables that you would expect, given the presence of the
+%  dummy first line above the num- ber: starting position of the
+%  equation, height of first line, total height of equation body. If
+%  the equation body had more than one line, with stretchable glue
+%  between the lines, half of the stretch would be added in this glue
+%  node.
+%
+%\item The hbox containing the equation number.
+%
+%\item Backspace to bring the equation body to the right starting point. We use
+%  \cs{parskip} to put this glue in place because we're going to get a
+%  parskip node here in any case when we add the equation body with (in
+%  essence). If we didn't do this we'd get two glue nodes instead of
+%  one, to no purpose.
+%
+%\cs{\noindent} \cs{unhbox}\cs{EQ@box}.
+%
+%
+%\item And lastly we see here the first line of the equation body,
+%  which appears to have height 16.5pt and depth 9.5pt.
+%\end{enumerate}
+%
+%For comparison, the vertical list produced from the above equation in
+%standard \LaTeX\ would look like this, if the same values of
+%columnwidth and abovedisplayskip are used:
+%\begin{verbatim}
+%[1] \penalty 10000
+%[2] \glue(\abovedisplayskip) 4.0 plus 4.0
+%    \glue(\lineskip) 1.0
+%    \hbox(16.53902+9.50012)x232.94844
+%[3] .\hbox(7.5+2.5)x25.55563
+%    ..\hbox(7.5+2.5)x25.55563
+%    ...\OT1/cmr/m/n/10 (
+%    ...\OT1/cmr/m/n/10 3
+%    ...\OT1/cmr/m/n/10 .
+%    ...\OT1/cmr/m/n/10 1
+%    ...\OT1/cmr/m/n/10 5
+%    ...\kern 0.0
+%    ...\OT1/cmr/m/n/10 )
+%    .\kern101.49591
+%[4] .\hbox(16.53902+9.50012)x105.8969
+%    ...
+%[5] \penalty 0
+%[6] \glue(\belowdisplayskip) 4.0 plus 4.0
+%    \glue(\lineskip) 1.0
+%    \hbox(6.94444+1.94444)x345.0, glue set 62.1106fil
+%\end{verbatim}
+%
+%\begin{enumerate}
+%  \item \cs{predisplaypenalty}
+%  \item \cs{abovedisplayskip}
+%  \item  equation number box
+%  \item equation body
+%  \item \cs{postdisplaypenaltly}
+%  \item \cs{belowdisplayskip}
+%\end{enumerate}
+%
+%
+% \section{Technical notes on Equation Layouts}
+% \textbf{MJD [1998/12/28]}
+%
+% \providecommand{\qq}[1]{\textquotedblleft#1\textquotedblright}
+% \providecommand{\mdash}{\textemdash}
+% \providecommand{\ndash}{\textendash}
+%
+% \newcommand{\ititle}[1]{\textit{#1}}
+%
+% \newcommand{\LR}[2][.4]{%
+%   \framebox[#1\displaywidth]{$\displaystyle{}#2$}%
+% }
+%
+% \newcommand{\LHS}[1]{\LR[\relifactor]{#1}}
+%
+% \newdimen\relindent \newdimen\rhswd
+%
+% \newcommand{\dwline}{%
+%   \hbox to\curdw{\vrule height1ex
+%     \leaders\hrule height.55ex depth-.45ex\hfil
+%     \tiny \space display width
+%     \leaders\hrule height.55ex depth-.45ex\hfil
+%     \vrule height1ex}%
+% }
+%
+% \newenvironment{layout}[1][.15]{%
+%   \noindent
+%   $$\edef\curdw{\the\displaywidth}%
+%     \def\relifactor{#1}%
+%     \gdef\layoutcr{\cr}\def\\{\layoutcr}%
+%     \binoppenalty 10000 \relpenalty 10000
+%   \setbox8\vbox\bgroup
+%     \advance\baselineskip .35\baselineskip
+%     \advance\lineskip .35\baselineskip \lineskiplimit\lineskip
+%     \relindent=#1\displaywidth
+%     \rhswd=\displaywidth \advance\rhswd-\relindent
+%     \global\row 0 \gdef\rhsskew{}%
+%     \halign\bgroup \global\advance\row 1 $\hfil\displaystyle{}##$&%
+%       \ifnum\row>1 \rhsskew \fi $\displaystyle{}##\hfil$\cr
+% }{%
+%   \crcr\egroup\egroup
+%   \vcenter{\halign{\hfil##\hfil\cr
+%     \hbox{\hss\dwline\hss}\cr\noalign{\vskip.6\baselineskip}\box8 \cr}}%
+%   $$\relax
+%   \ignorespacesafterend
+% }
+%
+% \newcommand{\stagger}{\omit\span\gdef\layoutcr{\cr\omit\span}}
+%
+% \newcount\row
+%
+% \newcommand{\rhsskew}{}
+% \newcommand{\skewleft}[1]{\gdef\rhsskew{\kern-#1\relax}}
+%
+%
+% \subsection{Misc examples}
+%
+% Let us consider which of these have 50\% or more of wasted whitespace
+% \emph{within the bounding box of the visible material}.
+% \begin{layout}[.4]
+% \LHS{L}&=\LR[.35]{R_{1}}\\
+% &=\LR[.25]{R_{1}}
+% \end{layout}
+%
+% \subsection{Ladder and step layouts}
+%
+% \subsubsection{Straight ladder layout}
+% This is distinguished by a relatively short LHS and one or more RHS's of
+% any length.
+% \begin{layout}
+% \LHS{L} &= \LR[.5]{R_{1}}\\
+% &=\LR[.3]{R_{2}}\\
+% &=\LR[.25]{R_{3}}\\
+% &\qquad\ldots
+% \end{layout}
+% The simplest kind of equation that fits on one line and has only one RHS
+% may be viewed as a trivial subcase of the straight ladder layout:
+% \begin{layout}
+% \LHS{L} &= \LR[.5]{R}
+% \end{layout}
+% If some of the RHS's are too wide to fit on a single line they may be
+% broken at binary operator symbols such as plus or minus. This is still
+% classified as a straight ladder layout if none of the fragments intrude
+% into the LHS column, because the underlying parshape is the same.
+% \begin{layout}
+% \LHS{L} &= \LR[.4]{R_{1a}}\\
+% &\quad +\LR[.6]{R_{1b}}\\
+% &=\LR[.3]{R_{2}}\\
+% &=\LR[.25]{R_{3a}}\\
+% &\quad +\LR[.45]{R_{3b}}\\
+% &\quad +\LR[.54]{R_{3c}}\\
+% &\qquad\ldots
+% \end{layout}
+%
+% \subsubsection{Skew ladder layout}
+% \begin{layout}[.5]
+% \skewleft{.35\displaywidth}
+% \LHS{L}&= \LR[.3]{R_{1}}\\
+% &=\LR[.6]{R_{2}}\\
+% &=\LR[.25]{R_{3}}\\
+% &\qquad\ldots
+% \end{layout}
+% In a skew ladder layout, the combined LHS width plus width of $R_{1}$
+% does not exceed the available width, but one of the other RHS's is so
+% wide that aligning its relation symbol with the others cannot be done
+% without making it run over the right margin: $\mbox{width}(L) +
+% \mbox{width}_{\mathrm{max}}(R_{i})>\mbox{width}_{\mathrm{avail}}$. In
+% that case we next try aligning all but the first relation symbol,
+% allowing all the $R_{i}$ after $R_1$ to shift leftward.
+%
+% \subsubsection{Drop ladder layout}
+% \begin{layout}[.6]
+% \makebox[.15\displaywidth][l]{\LHS{L}}\\
+% &= \LR[.6]{R_{1}}\\
+% &=\LR[.3]{R_{2}}\\
+% &=\LR[.25]{R_{3}}\\
+% &\qquad\ldots
+% \end{layout}
+% The drop ladder layout is similar to the skew ladder layout but with the
+% width of $R_1$ too large for it to fit on the same line as the LHS. Then
+% we move $R_1$ down to a separate line and try again to align all the
+% relation symbols. Note that this layout consumes more vertical space
+% than the skew ladder layout.
+%
+% \subsubsection{Step layout}
+% \begin{layout}[.6]
+% \stagger
+% \LHS{R_{a}}\\
+% \qquad + \LR[.7]{R_{b}}\\
+% \qquad\qquad + \LR[.6]{R_{c}}\\
+% \qquad\qquad\qquad + \LR[.45]{R_{d}}\\
+% \qquad\qquad\qquad\qquad\ldots
+% \end{layout}
+% The chief characteristic of the step layout is that there is no relation
+% symbol, so that the available line breaks are (usually) all at binary
+% operator symbols. Let $w_1$ and $w_l$ be the widths of the first and
+% last fragments. We postulate that the ideal presentation is as follows:
+% Choose a small stairstep indent $I$ (let's say 1 or 2 em). We want the
+% last fragment to be offset at least $I$ from the start of the first
+% fragment, and to end at least $I$ past the end of the first fragment. If
+% there are only two lines these requirements determine a target width
+% $w_T=\max(w_1+I,w_l+I)$. If there are more than two lines ($l>2$) then
+% use $w_T = \max(w_1 + (l-1)I, w_l+I, w_{\mathrm{avail}}$ and reset $I$
+% to $w_T/(l-1)$ if $w_T = w_{\mathrm{avail}}$.
+%
+% Furthermore, we would like the material to be distributed as evenly as
+% possible over all the lines rather than leave the last line exceedingly
+% short. If the total width is $1.1(\mbox{width}_{\mathrm{avail}})$, we
+% don't want to have .9 of that on line 1 and .2 of it on line 2:
+% \begin{layout}[.9]
+% \stagger
+% \LHS{R_{a}\hfil+\hfil R_{b}\hfil+\hfil R_{c}}\\
+% \qquad + \LR[.1]{R_{d}}
+% \end{layout}
+% Better to split it as evenly as possible, if the available breakpoints
+% permit.
+% \begin{layout}[.5]
+% \stagger
+% \LHS{R_{a}\hfil+\hfil R_{b}}\\
+% \qquad + \LR[.5]{R_{c}\hfil+\hfil R_d}
+% \end{layout}
+% A degenerate step layout may arise if an unbreakable fragment of
+% the equation is so wide that indenting it to its appointed starting
+% point would cause it to run over the right margin. In that case, we want
+% to shift the fragment leftward just enough to bring it within the right
+% margin:
+% \begin{layout}[.6]
+% \stagger
+% \LHS{L_{a}}\\
+% \quad + \LR[.8]{L_{b}}\\
+% \qquad + \LR[.7]{L_{c}}\\
+% \; + \LR[.87]{L_{d}}\\
+% \qquad\ldots
+% \end{layout}
+% And then we may want to regularize the indents as in the drop ladder
+% layout. Let's call this a dropped step layout:
+% \begin{layout}[.6]
+% \stagger
+% \LHS{L_{a}}\\
+% \quad + \LR[.8]{L_{b}}\\
+% \quad + \LR[.7]{L_{c}}\\
+% \quad + \LR[.87]{L_{d}}\\
+% \qquad\ldots
+% \end{layout}
+%
+% \subsection{Strategy}
+%
+% Here is the basic procedure for deciding which equation layout to use,
+% before complications like equation numbers and delimiter clearance come
+% into the picture. Let $A$ be the available width, $w_{\mathrm{total}}$
+% the total width of the equation contents, $w(L)$ the width of the
+% left-hand side, $w_{\max}(R)$ the max width of the right-hand sides, $I$
+% the standard indent for step layout, and $O$ the standard offset for
+% binary operators if a break occurs in the middle of an RHS. Also let
+% $t_L$ and $t_R$ represent certain thresholds for the width of the LHS or
+% the RHS at which a layout decision may change, as explained below.
+%
+% \begin{small}
+% \begin{enumerate}
+% \renewcommand{\labelenumi}{(\theenumi)}
+% \item \ititle{Does everything fit on one line?}\label{i:LR}
+%   $w_{\mathrm{total}}\leq A$?
+%
+% Yes: print the equation on a single line (done).
+%
+% No: Check whether the equation has both LHS and RHS (\ref{i:lhs-check}).
+%
+% \item \ititle{Is there a left-hand side?}\label{i:lhs-check}
+% Are there any relation symbols in the equation?
+%
+% Yes: Try a ladder layout (\ref{i:ladder}).
+%
+% No: Try a step layout (\ref{i:step}).
+%
+% \item\ititle{Does the LHS leave room to fit the widest RHS?}\label{i:ladder}
+%   $w(L)+w_{\max}(R)<A$?
+%
+% Yes: Use a straight ladder layout (\ref{i:straight-ladder}).
+%
+% No: Check the width of the LHS (\ref{i:check-lhs}).
+%
+% \item\ititle{Is the LHS relatively short?}\label{i:check-lhs}
+% $w(L)\leq t_L$? (where $t_L$ is typically $0.4A$).
+%
+% Yes: Subdividing one or more of the RHS's may permit us to use a
+% straight ladder layout (\ref{i:straight-ladder}).
+%
+% No: The straight ladder layout is unlikely to work.
+% Try a skew or drop ladder layout (\ref{i:skew-drop}).
+%
+% \item\ititle{Straight ladder layout}\label{i:straight-ladder}
+% Set up a straight ladder parshape [0pt $A$ $w(L)$ $A-w(L)$] and run
+% a trial break. If the combined width of the LHS plus the longest RHS is
+% no greater than $A$ then we should get a satisfactory layout with all
+% line breaks occurring at major division points (relation symbols).
+% Otherwise, we hope, some additional line breaks at minor division points
+% will allow everything to fit within the text column.
+%
+% \ititle{Line breaks OK?}
+%
+% \begingroup \hbadness=9999
+% Yes: The straight ladder layout succeeded
+%   (done).\par\endgroup
+%
+% No: Try a skew or drop ladder layout (\ref{i:skew-drop}).
+%
+% \item\ititle{Do the LHS and the first RHS fit on one
+%     line?}\label{i:skew-drop} $w(L)+w(R_1) \leq A$?
+%
+% Yes: Try a skew ladder layout (\ref{i:skew}).
+%
+% No: Try a drop ladder layout (\ref{i:drop}).
+%
+% \item\ititle{Skew ladder layout}\label{i:skew}
+% Set up a parshape [0pt $A$ $I$ $A-I$] and run a trial break.
+%
+% \ititle{Line breaks OK?}
+%
+% Yes: Skew ladder layout succeeded (done).
+%
+% No: One of the unbreakable fragments of the $R_i$ ($i>1$) is wider than
+% $A-I$; try an almost-columnar layout (\ref{i:almost-columnar}).
+%
+% \item\ititle{Drop ladder layout}\label{i:drop}
+% Set up a parshape [0pt $w(L)$ $I$ $A-I$] and run a trial break.
+% This is the same parshape as for a skew ladder layout except that the
+% width of the first line is limited to the LHS width, so that the RHS is
+% forced to drop down to the next line.
+%
+% \ititle{Line breaks OK?}
+%
+% Yes: Drop ladder layout succeeded (done).
+%
+% No: One of the unbreakable fragments of the $R_i$ ($i>1$) is wider than
+% $A-I$; try an almost-columnar layout (\ref{i:almost-columnar}).
+%
+% \item\ititle{Almost-columnar layout}\label{i:almost-columnar}
+% This presupposes a trial break that yielded a series of expressions or
+% fragments, one per line. Let $w(F)$ denote the width of the first
+% fragment and $w(R_i)$ the widths of the remaining fragments.
+% Set up a parshape [0pt $w(F)$ $A-w_{\max}(R_i)$ $w_{\max}(R_i)$]: in other
+% words, set the first line flush left and the longest line flush right
+% and all other lines indented to the same position as the longest line.
+% But as a matter of fact there is one other refinement for extreme cases:
+% if $w_{\max}(R_i)>A$ then the parshape can be simplified without loss to
+% [0pt $w(F)$ 0pt $A$]\mdash for that is the net effect of substituting
+% $\min(A,w_{\max})$ in stead of $w_{\max}$.
+% (Done.)
+%
+% \item\ititle{Step layout}\label{i:step} Set target width $w_T$ to $A -
+%   2I$.  Set parshape to [0pt $w_T$ $I$ $w_T -I$ $2I$ $w_T -2I$ \ldots\
+%   $(l-1)I$ $w_T - (l-1)I$], where $l=\lceil w_{\mathrm{total}}/A\rceil$
+%   is the expected number of lines that will be required.  Trial break
+%   with that parshape in order to find out the width of the last line.
+%
+% \ititle{Indents OK?}
+%
+% Yes: Step layout succeeded (done).
+%
+% No: One of the fragments is too wide to fit in
+% the allotted line width, after subtracting the indent specified by the
+% parshape. Try a dropped step layout (\ref{i:drop-step})
+%
+% \item\ititle{Dropped step layout}\label{i:drop-step} Set up a parshape
+%   [0pt $A$ $I$ $A-I$] and run a trial break.  Note that this is actually
+%   the same parshape as for a skew ladder layout.
+%
+% \ititle{Line breaks OK?}
+%
+% Yes: Dropped step layout succeeded (done).
+%
+% No: One of the unbreakable fragments of the $R_i$ ($i>1$) is wider than
+% $A-I$; as a last resort try an almost-columnar layout (\ref{i:almost-columnar}).
+%
+% \end{enumerate}
+% \par\end{small}
+%
+% \section{To do}
+%
+% \begin{itemize}
+% \item Handling of QED
+% \item Space between \verb'\end{dmath}' and following punctuation will
+% prevent the punctuation from being drawn into the equation.
+% \item Overriding the equation layout
+% \item Overriding the placement of the equation number
+% \item \qq{alignid} option for more widely separated equations where
+%   shared alignment is desired (requires two passes)
+% \item Or maybe provide an \qq{alignwidths} option where you give
+%   lhs/rhs width in terms of ems? And get feedback later on discrepancies
+%   with the actual measured contents?
+% \item \cs{intertext} not needed within dgroup! But currently there are
+%   limitations on floating objects within dgroup.
+% \item \verb'align={1}' or 2, 3, 4 expressing various levels of demand
+%   for group-wide alignment. Level 4 means force alignment even if some
+%   lines then have to run over the right margin! Level 1, the default,
+%   means first break LHS-RHS equations as if it occurred by itself, then
+%   move them left or right within the current line width to align them if
+%   possible. Levels 2 and 3 mean try harder to align but give up if
+%   overfull lines result.
+% \item Need an \cs{hshift} command to help with alignment of
+%   lines broken at a discretionary times sign. Also useful for adjusting
+%   inside-delimiter breaks.
+% \end{itemize}
+%
+% \StopEventually{}
+% \clearpage
+% \newgeometry{left=4cm}
+% \part{Implementation}
+%
+%
+% The package version here is Michael's v0.90 updated by Bruce
+% Miller. Michael's changes between v0.90 and his last v0.94 will be
+% incorporated where applicable.
+%
+%
+% The original sources of \pkg{breqn} and related files exist in a
+% non-dtx format devised by Michael Downes himself.
+% Lars Madsen has kindly written a Perl script for transforming the
+% original source files into near-perfect dtx state, requiring only
+% very little hand tuning. Without his help it would have been nigh
+% impossible to incorporate the original sources with Michael's
+% comments. A big, big thank you to him.
+%
+%
+%
+% \section{Introduction}
+% The \pkg{breqn} package provides environments
+% \env{dmath}, \env{dseries}, and \env{dgroup} for
+% displayed equations with \emph{automatic line breaking},
+% including automatic indention of relation symbols and binary operator
+% symbols at the beginning of broken lines.    These environments
+% automatically pull in following punctuation so that it can be written in
+% a natural way.    The \pkg{breqn} package also provides a
+% \env{darray} environment similar to the \env{array}
+% environment but using \cs{displaystyle} for all the array cells and
+% providing better interline spacing (because the vertical ruling
+% feature of \env{array} is dropped).
+% These are all autonumbered environments like \env{equation}
+% and have starred forms that don't add a number.    For a more
+% comprehensive and detailed description of the features and intended
+% usage of the \pkg{breqn} package see \fn{breqndoc.tex}.
+%
+%
+%
+%
+% \section{Strategy}
+% Features of particular note are the ability to have
+% linebreaks even within a \cs{left} \ndash  \cs{right} pair of
+% delimiters, and the automatic alignment on relations and binary
+% operators of a split equation.    To make \env{dmath} handle
+% all this, we begin by setting the body of the equation in a special
+% paragraph form with strategic line breaks whose purpose is not to
+% produce line breaks in the final printed output but rather to mark
+% significant points in the equation and give us entry points for
+% unpacking \cn{left} \ndash  \cn{right} boxes.
+% After the initial typesetting, we take the resulting stack of line
+% fragments and, working backward, splice them into a new, single-line
+% paragraph; this will eventually be poured into a custom parshape, after
+% we do some measuring to calculate what that parshape should be.
+% This streamlined horizontal list may contain embedded material
+% from user commands intended to alter line breaks, horizontal alignment,
+% and interline spacing; such material requires special handling.
+%
+% To make the `shortskip' possibility work even for
+% multiline equations, we must plug in a dummy \tex  display to give us
+% the value of \cs{predisplaysize}, and calculate for ourselves when
+% to apply the short skips.
+%
+% In order to measure the equation body and do various
+% enervating calculations on whether the equation number will fit and so
+% on, we have to set it in a box.    Among other things, this means
+% that we can't unhbox it inside \dbldollars  \dots  \dbldollars , or
+% even \verb"$" \dots  \verb"$": \tex  doesn't allow you to
+% \cs{unhbox} in math mode.    But we do want to unhbox it rather
+% than just call \cs{box}, otherwise we can't take advantage of
+% available shrink from \cs{medmuskip} to make equations shrink to
+% fit in the available width.    So even for simple one-line equations
+% we are forced to fake a whole display without going through \tex 's
+% primitive display mechanism (except for using it to get
+% \cs{predisplaysize} as mentioned above).
+%
+%
+% In the case of a framed equation body, the current implementation is
+% to set the frame in a separate box, of width zero and height zero,
+% pinned to the upper left corner of the equation body, and then print the
+% equation body on top of it.
+% For attaching an equation number it would be much simpler to wrap
+% the equation body in the frame and from then on treat the body as a
+% single box instead of multiple line boxes.
+% But I had a notion that it might be possible some day to support
+% vertical stretching of the frame.
+%
+%
+%
+%
+% \section{Prelim}
+%
+%    \begin{macrocode}
+%<*package>
+\NeedsTeXFormat{LaTeX2e}
+%    \end{macrocode}
+%
+% Declare package name and date.
+%    \begin{macrocode}
+\RequirePackage{expl3}
+\ProvidesExplPackage{breqn}{2018/09/14}{0.98f}{Breaking equations}
+%    \end{macrocode}
+%   Regrettably, \pkg{breqn} is internally a mess, so we have to take
+%   some odd steps.
+%    \begin{macrocode}
+\ExplSyntaxOff
+%    \end{macrocode}
+%
+%
+% \section{Package options}
+%
+% Most options are set with the \cs{options} command (which
+% calls \cs{setkeys}) because the standard package option
+% mechanism doesn't provide support for key-value syntax.
+%
+% First we need to get the catcodes sorted out.
+%    \begin{macrocode}
+\edef\breqnpopcats{%
+  \catcode\number`\"=\number\catcode`\"
+  \relax}
+\AtEndOfPackage{\breqnpopcats}%
+\catcode`\^=7 \catcode`\_=8 \catcode`\"=12 \relax
+\DeclareOption{mathstyleoff}{%
+  \PassOptionsToPackage{mathstyleoff}{flexisym}%
+}
+%    \end{macrocode}
+% Process options.
+%    \begin{macrocode}
+\ProcessOptions\relax
+%    \end{macrocode}
+%
+%
+%
+%
+% \section{Required packages}
+% The \pkg{flexisym} package makes it possible to attach
+% extra actions to math symbols, in particular mathbin, mathrel, mathopen,
+% and mathclose symbols.
+% Normally it would suffice to call \cs{RequirePackage} without
+% any extra testing, but the nature of the package is such that it is
+% likely to be called earlier with different (no) options.
+% Then is it really helpful to be always warning the user about
+% \quoted{Incompatible Package Options!}?
+% I don't think so.
+%    \begin{macrocode}
+\@ifpackageloaded{flexisym}{}{%
+  \RequirePackage{flexisym}[2009/08/07]
+    \edef\breqnpopcats{\breqnpopcats
+    \catcode\number`\^=\number\catcode`\^
+    \catcode\number`\_=\number\catcode`\_
+  }%
+  \catcode`\^=7 \catcode`\_=8 \catcode`\"=12 \relax
+}
+%    \end{macrocode}
+% The \pkg{keyval} package for handling equation options and
+% \pkg{calc} to ease writing computations.
+%    \begin{macrocode}
+\RequirePackage{keyval,calc}\relax
+%    \end{macrocode}
+%
+%
+% And add an \cs{options} cmd for processing package
+% options that require an argument.    Maybe this will get added to
+% the \pkg{keyval} package eventually.
+%    \begin{macrocode}
+\@ifundefined{options}{%
+%    \end{macrocode}
+%
+%
+% \begin{macro}{\options}
+% Get the package options and run setkeys on them.
+%    \begin{macrocode}
+\newcommand{\options}[2]{%
+  \expandafter\options@a\csname opt@#1.sty\endcsname{#2}%
+  \setkeys{#1}{#2}%
+}
+%    \end{macrocode}
+% \end{macro}
+%
+%
+% \begin{macro}{\options@a}
+% \begin{macro}{\options@b}
+% \begin{macro}{\options@c}
+% \begin{macro}{\options@d}
+% Redefine \cs{opt@pkgname.sty} as we go along to take out
+% the options that are handled and leave the ones that are not.
+%    \begin{macrocode}
+\def\options@a#1#2{%
+  \edef\@tempa{\options@b#2,\@empty\@nil}%
+  \ifx#1\relax \let#1\@empty\fi
+  \xdef#1{#1\ifx#1\@empty\@xp\@gobble\@tempa\@empty\else\@tempa \fi}%
+}
+%    \end{macrocode}
+% Add the next option, and recurse if there remain more
+% options.
+%    \begin{macrocode}
+\def\options@b#1,#2#3\@nil{%
+  \options@c#1 \@nil
+  \ifx#2\@empty \else\options@b#2#3\@nil\fi
+}
+%    \end{macrocode}
+% Discard everything after the first space.
+%    \begin{macrocode}
+\def\options@c#1 #2\@nil{\options@d#1=\@nil}
+%    \end{macrocode}
+% Discard everything after the first = sign; add a comma only if the
+% remainder is not empty.
+%    \begin{macrocode}
+\def\options@d#1=#2\@nil{\ifx\@empty #1\@empty\else,\fi#1}
+%    \end{macrocode}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+%
+%
+%
+% The tail of the \cs{@ifundefined} test.
+%    \begin{macrocode}
+}{}% end @ifundefined test
+%    \end{macrocode}
+%
+%
+%
+%
+% \section{Some useful tools}
+%
+% \begin{macro}{\@nx}
+% \begin{macro}{\@xp}
+% The comparative brevity of \cs{@nx} and \cs{@xp} is
+% valuable not so much for typing convenience as for reducing visual
+% clutter in code sections that require a lot of expansion control.
+%    \begin{macrocode}
+\let\@nx\noexpand
+\let\@xp\expandafter
+%    \end{macrocode}
+% \end{macro}
+% \end{macro}
+%
+%
+% \begin{macro}{\@emptytoks}
+% Constant empty token register, analogous to \cs{@empty}.
+%    \begin{macrocode}
+\@ifundefined{@emptytoks}{\newtoks\@emptytoks}{}
+%    \end{macrocode}
+% \end{macro}
+%
+%
+% \begin{macro}{\f@ur}
+% Constants 0\ndash 3 are provided in plain \tex , but not 4.
+%    \begin{macrocode}
+\chardef\f@ur=4
+%    \end{macrocode}
+% \end{macro}
+%
+%
+% \begin{macro}{\inf@bad}
+% \cs{inf@bad} is for testing box badness.
+%    \begin{macrocode}
+\newcount\inf@bad \inf@bad=1000000
+%    \end{macrocode}
+% \end{macro}
+%
+%
+% \begin{macro}{\maxint}
+%
+% We want to use \cs{maxint} rather than coerced
+% \cs{maxdimen} for \cs{linepenalty} in one place.
+%    \begin{macrocode}
+\newcount\maxint \maxint=2147483647
+%    \end{macrocode}
+% \end{macro}
+%
+%
+% \begin{macro}{\int@a}
+% \begin{macro}{\int@b}
+% \begin{macro}{\int@b}
+%
+% Provide some shorter aliases for various scratch registers.
+%    \begin{macrocode}
+\let\int@a=\@tempcnta
+\let\int@b=\@tempcntb
+\let\int@c=\count@
+%    \end{macrocode}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+%
+%
+% \begin{macro}{\dim@a}
+% \begin{macro}{\dim@b}
+% \begin{macro}{\dim@c}
+% \begin{macro}{\dim@d}
+% \begin{macro}{\dim@e}
+% \begin{macro}{\dim@A}
+%
+% Same for dimen registers.
+%    \begin{macrocode}
+\let\dim@a\@tempdima
+\let\dim@b\@tempdimb
+\let\dim@c\@tempdimc
+\let\dim@d\dimen@
+\let\dim@e\dimen@ii
+\let\dim@A\dimen@i
+%    \end{macrocode}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+%
+%
+% \begin{macro}{\skip@a}
+% \begin{macro}{\skip@b}
+% \begin{macro}{\skip@c}
+%
+% Same for skip registers.
+%    \begin{macrocode}
+\let\skip@a\@tempskipa
+\let\skip@b\@tempskipb
+\let\skip@c\skip@
+%    \end{macrocode}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+%
+%
+% \begin{macro}{\toks@a}
+% \begin{macro}{\toks@b}
+% \begin{macro}{\toks@c}
+% \begin{macro}{\toks@d}
+% \begin{macro}{\toks@e}
+% \begin{macro}{\toks@f}
+%
+% Same for token registers.
+%    \begin{macrocode}
+\let\toks@a\@temptokena
+\let\toks@b\toks@
+\toksdef\toks@c=2
+\toksdef\toks@d=4
+\toksdef\toks@e=6
+\toksdef\toks@f=8
+%    \end{macrocode}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+%
+%
+% \begin{macro}{\abs@num}
+% We need an absolute value function for comparing
+% penalties.
+%    \begin{macrocode}
+\def\abs@num#1{\ifnum#1<\z@-\fi#1}
+%    \end{macrocode}
+% \end{macro}
+%
+%
+% \begin{macro}{\@ifnext}
+% \begin{macro}{\@ifnexta}
+% The \cs{@ifnext} function is a variation of
+% \cs{@ifnextchar} that doesn't skip over intervening whitespace.
+% We use it for the optional arg of \dbslash  inside
+% \env{dmath} \etc  because we don't want
+% unwary users to be tripped up by an unexpected attempt on \latex 's part
+% to interpret a bit of math as an optional arg:
+% \begin{literalcode}
+% \begin{equation}
+% ...\\
+% [z,w]...
+% \end{equation}
+% \end{literalcode}
+% .
+%    \begin{macrocode}
+\def\@ifnext#1#2#3{%
+  \let\@tempd= #1\def\@tempa{#2}\def\@tempb{#3}%
+  \futurelet\@tempc\@ifnexta
+}
+%    \end{macrocode}
+% Switch to \cs{@tempa} iff the next token matches.
+%    \begin{macrocode}
+\def\@ifnexta{\ifx\@tempc\@tempd \let\@tempb\@tempa \fi \@tempb}
+%    \end{macrocode}
+% \end{macro}
+% \end{macro}
+%
+%
+% \begin{macro}{\@ifstar}
+% Similarly let's remove space-skipping from \cs{@ifstar}
+% because in some rare case of \dbslash  inside an equation, followed by
+% a space and a \verb"*" where the \verb"*" is intended as the math
+% binary operator, it would be a disservice to gobble the star as an
+% option of the \dbslash  command.    In all other contexts the chance
+% of having a space \emph{before} the star is extremely small: either
+% the command is a control word which will get no space token after it in
+% any case because of \tex 's tokenization rules; or it is a control
+% symbol such as \dbslash  \verb"*" which is exceedingly unlikely to be
+% written as \dbslash  \verb"*" by any one who really wants the
+% \verb"*" to act as a modifier for the \dbslash  command.
+%    \begin{macrocode}
+\def\@ifstar#1#2{%
+  \let\@tempd*\def\@tempa*{#1}\def\@tempb{#2}%
+  \futurelet\@tempc\@ifnexta
+}
+%    \end{macrocode}
+% \end{macro}
+%
+%
+% \begin{macro}{\@optarg}
+% Utility function for reading an optional arg
+% \emph{without} skipping over any intervening spaces.
+%    \begin{macrocode}
+\def\@optarg#1#2{\@ifnext[{#1}{#1[#2]}}
+%    \end{macrocode}
+% \end{macro}
+%
+%
+% \begin{macro}{\@True}
+% \begin{macro}{\@False}
+% \begin{macro}{\@Not}
+% \begin{macro}{\@And}
+% After \verb"\let\foo\@True" the test
+% \begin{literalcode}
+% \if\foo
+% \end{literalcode}
+% evaluates to true.    Would rather avoid \cs{newif} because it
+% uses three csnames per Boolean variable; this uses only one.
+%    \begin{macrocode}
+\def\@True{00}
+\def\@False{01}
+\def\@Not#1{0\ifcase#11 \or\@xp 1\else \@xp 0\fi}
+\def\@And#1#2{0\ifcase#1#2 \@xp 0\else \@xp 1\fi}
+\def\@Or#1#2{0\ifnum#1#2<101 \@xp 0\else \@xp 1\fi}
+%    \end{macrocode}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+%
+%
+%
+%    \begin{macrocode}
+\def\theb@@le#1{\if#1 True\else False\fi}
+%    \end{macrocode}
+% \begin{macro}{\freeze@glue}
+%
+% Remove the stretch and shrink from a glue register.
+%    \begin{macrocode}
+\def\freeze@glue#1{#11#1\relax}
+%    \end{macrocode}
+% \end{macro}
+% \begin{macro}{\z@rule}
+% \begin{macro}{\keep@glue}
+% Note well
+% the intentional absence of \cs{relax} at the end of the replacement
+% text of \cs{z@rule}; use it with care.
+%    \begin{macrocode}
+\def\z@rule{\vrule\@width\z@}% no \relax ! use with care
+%    \end{macrocode}
+% Different ways to keep a bit of glue from disappearing at the
+% beginning of a line after line breaking:
+% \begin{itemize}
+% \item Zero-thickness rule
+% \item Null character
+% \item \cs{vadjust}\verb"{}" (\texbook , Exercise ??)
+% \end{itemize}
+% The null character idea would be nice except it
+% creates a mathord which then screws up math spacing for \eg  a following
+% unary minus sign.    (the vrule \emph{is} transparent to
+% the math spacing).    The vadjust is the cheapest in terms of box
+% memory\mdash it vanishes after the pass through \tex 's
+% paragrapher.
+% It is what I would have used, except that the equation contents get
+% run through two paragraphing passes, once for breaking up LR boxes and
+% once for the real typesetting.
+% If \cs{keep@glue} were done with an empty vadjust, it would
+% disappear after the first pass and\mdash in particular\mdash the
+% pre-bin-op adjustment for relation symbols would disappear at a line break.
+%    \begin{macrocode}
+\def\keep@glue{\z@rule\relax}
+%    \end{macrocode}
+% \end{macro}
+% \end{macro}
+%
+%
+% \begin{macro}{\replicate}
+%
+% This is a fully expandable way of making N copies of a token
+% list.
+% Based on a post of David Kastrup to comp.text.tex circa January
+% 1999.
+% The extra application of \cs{number} is needed for maximal
+% robustness in case the repeat count N is given in some weird \tex  form
+% such as \verb|"E9| or \verb|\count9|.
+%    \begin{macrocode}
+% usage: \message{H\replicate{5}{i h}ow de doo dee!}
+\begingroup \catcode`\&=11
+\gdef\replicate#1{%
+  \csname &\expandafter\replicate@a\romannumeral\number\number#1 000q\endcsname
+}
+\endgroup
+%    \end{macrocode}
+% \end{macro}
+%
+%
+% \begin{macro}{\replicate@a}
+%    \begin{macrocode}
+\long\def\replicate@a#1#2\endcsname#3{#1\endcsname{#3}#2}
+%    \end{macrocode}
+% \end{macro}
+%
+%
+% \begin{macro}{\8m}% fix
+%    \begin{macrocode}
+\begingroup \catcode`\&=11
+\long\gdef\&m#1#2{#1\csname &#2\endcsname{#1}}
+\endgroup
+%    \end{macrocode}
+% \end{macro}
+%
+%
+% \begin{macro}{\8q}% fix
+%    \begin{macrocode}
+\@xp\let\csname\string &q\endcsname\@gobble
+%    \end{macrocode}
+% \end{macro}
+%
+% \begin{macro}{\mathchars@reset}
+%
+% Need to patch up this function from flexisym a little, to better
+% handle certain constructed symbols like \cs{neq}.
+%    \begin{macrocode}
+\ExplSyntaxOn
+\g@addto@macro\mathchars@reset{%
+  %\let\@symRel\@secondoftwo \let\@symBin\@secondoftwo
+  %\let\@symDeL\@secondoftwo \let\@symDeR\@secondoftwo
+  %\let\@symDeB\@secondoftwo
+  \cs_set_eq:NN \math_csym_Rel:Nn \use_ii:nn
+  \cs_set_eq:NN \math_csym_Bin:Nn \use_ii:nn
+  \cs_set_eq:NN \math_csym_DeL:Nn \use_ii:nn
+  \cs_set_eq:NN \math_csym_DeR:Nn \use_ii:nn
+  \cs_set_eq:NN \math_csym_DeB:Nn \use_ii:nn
+}
+\ExplSyntaxOff
+%    \end{macrocode}
+% \end{macro}
+%
+%
+% \begin{macro}{\eq@cons}
+%
+% \latex 's \cs{@cons} appends to the end of a list, but we need
+% a function that adds material at the beginning.
+%    \begin{macrocode}
+\def\eq@cons#1#2{%
+  \begingroup \let\@elt\relax \xdef#1{\@elt{#2}#1}\endgroup
+}
+%    \end{macrocode}
+% \end{macro}
+%
+% \begin{macro}{\@saveprimitive}
+% If some preceding package redefined one of the
+% primitives that we must change, we had better do some checking to make
+% sure that we are able to save the primitive meaning for internal use.
+% This is handled by the \cs{@saveprimitive} function.    We
+% follow the example of \cs{@@input} where the primitive meaning is
+% stored in an internal control sequence with a \verb"@@" prefix.
+% Primitive control sequences can be distinguished by the fact that
+% \cs{string} and \cs{meaning} return the same information.
+% Well, not quite all: \cs{nullfont} and \cs{topmark}
+% and the other \cs{...mark} primitives being the exceptions.
+%    \begin{macrocode}
+\providecommand{\@saveprimitive}[2]{%
+  \begingroup
+  \edef\@tempa{\string#1}\edef\@tempb{\meaning#1}%
+  \ifx\@tempa\@tempb \global\let#2#1%
+  \else
+%    \end{macrocode}%
+% If [arg1] is no longer primitive, then we are in trouble unless
+% [arg2] was already given the desired primitive meaning somewhere
+% else.
+%    \begin{macrocode}
+    \edef\@tempb{\meaning#2}%
+    \ifx\@tempa\@tempb
+    \else \@saveprimitive@a#1#2%
+    \fi
+  \fi
+  \endgroup
+}
+%    \end{macrocode}
+% Aux function, check for the special cases.
+% Most of the time this branch will be skipped so we can
+% stuff a lot of work into it without worrying about speed costs.
+%    \begin{macrocode}
+\providecommand\@saveprimitive@a[2]{%
+  \begingroup
+  \def\@tempb##1#1##2{\edef\@tempb{##2}\@car{}}%
+  \@tempb\nullfont{select font nullfont}%
+    \topmark{\string\topmark:}%
+    \firstmark{\string\firstmark:}%
+    \botmark{\string\botmark:}%
+    \splitfirstmark{\string\splitfirstmark:}%
+    \splitbotmark{\string\splitbotmark:}%
+    #1{\string#1}%
+    \@nil % for the \@car
+  \edef\@tempa{\expandafter\strip@prefix\meaning\@tempb}%
+  \edef\@tempb{\meaning#1}%
+  \ifx\@tempa\@tempb \global\let#2#1%
+  \else
+    \PackageError{breqn}%
+      {Unable to properly define \string#2; primitive
+      \noexpand#1no longer primitive}\@eha
+    \fi
+  \fi
+  \endgroup
+}
+%    \end{macrocode}
+% \end{macro}
+%
+%
+% \begin{macro}{\@@math}
+% \begin{macro}{\@@endmath}
+% \begin{macro}{\@@display}
+% \begin{macro}{\@@enddisplay}
+% Move the math-start and math-end functions into control
+% sequences.    If I were redesigning \tex  I guess I'd put these
+% functions into primitive control words instead of linking them to a
+% catcode.    That way \tex  would not have to do the special
+% lookahead at a \verb"$" to see if there's another one coming up.
+% Of course that's related to the question of how to provide user
+% shorthand for common constructions: \tex , or an editing interface of
+% some sort.
+%    \begin{macrocode}
+\begingroup \catcode`\$=\thr@@ % just to make sure
+  \global\let\@@math=$ \gdef\@@display{$$}% $$$
+\endgroup
+\let\@@endmath=\@@math
+\let\@@enddisplay=\@@display
+%    \end{macrocode}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+%
+%
+% \begin{macro}{\@@insert}
+% \begin{macro}{\@@mark}
+% \begin{macro}{\@@vadjust}
+% Save the primitives \cs{vadjust}, \cs{insert},
+% \cs{mark} because we will want to change them locally during
+% equation measuring to keep them from getting in the way of our vertical
+% decomposition procedures.    We follow the example of
+% \cs{@@input}, \cs{@@end}, \cs{@@par} where the primitive
+% meaning is stored in an internal control sequence with a \verb"@@"
+% prefix.
+%    \begin{macrocode}
+\@saveprimitive\vadjust\@@vadjust
+\@saveprimitive\insert\@@insert
+\@saveprimitive\mark\@@mark
+%    \end{macrocode}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+%
+%
+%
+%
+% \section{Debugging}
+% Debugging help.
+%    \begin{macrocode}
+%<*trace>
+\errorcontextlines=2000\relax
+\typeout{BREQN DEBUGGING MODE ACTIVE}
+%    \end{macrocode}
+%
+% \begin{macro}{\breqn@debugmsg}
+% Print a debugging message.
+%    \begin{macrocode}
+\long\def\breqn@debugmsg#1{\GenericWarning{||}{||=\space#1}}
+%    \end{macrocode}
+% \end{macro}
+%
+% \begin{macro}{\debugwr}
+% Sometimes the newline behavior of \cs{message} is
+% unsatisfactory; this provides an alternative.
+%    \begin{macrocode}
+\def\debugwr#1{\immediate\write\sixt@@n{||= #1}}
+%    \end{macrocode}
+% \end{macro}
+%
+%
+% \begin{macro}{\debug@box}
+% Record the contents of a box in the log file, without stopping.
+%    \begin{macrocode}
+\def\debug@box#1{%
+  \batchmode{\showboxbreadth\maxdimen\showboxdepth99\showbox#1}%
+  \errorstopmode
+}
+%    \end{macrocode}
+% \end{macro}
+%
+%
+% \begin{macro}{\eqinfo}
+% Show lots of info about the material before launching into the
+% trials.
+%    \begin{macrocode}
+\def\eqinfo{%
+  \debug@box\EQ@copy
+  \wlog{!! EQ@copy: \the\wd\EQ@copy\space x
+    \the\ht\EQ@copy+\the\dp\EQ@copy
+  }%
+}
+%    \end{macrocode}
+% \end{macro}
+%
+%
+% \begin{macro}{\debug@para}
+% Check params that affect line breaking.
+%    \begin{macrocode}
+\def\debug@para{%
+  \debugwr{\hsize\the\hsize, \parfillskip\the\parfillskip}%
+  \breqn@debugmsg{\leftskip\the\leftskip, \rightskip\the\rightskip}%
+  \breqn@debugmsg{\linepenalty\the\linepenalty, \adjdemerits\the\adjdemerits}%
+  \breqn@debugmsg{\pretolerance\the\pretolerance, \tolerance\the\tolerance,
+    \parindent\the\parindent}%
+}
+%    \end{macrocode}
+% \end{macro}
+%
+%
+%    \begin{macrocode}
+%</trace>
+%    \end{macrocode}
+%
+%
+%
+%
+% \section{The \cs{listwidth} variable}
+% The dimen variable \cs{listwidth} is \cs{linewidth}
+% plus \cs{leftmargin} plus \cs{rightmargin}, which is typically
+% less than \cs{hsize} if the list depth is greater than one.
+% In case a future package will provide this variable, define it only
+% if not yet defined.
+%    \begin{macrocode}
+\@ifundefined{listwidth}{\newdimen\listwidth}{}
+\listwidth=\z@
+%    \end{macrocode}
+%
+%
+%
+%
+% \section{Parameters}
+%
+% Here follows a list of parameters needed.
+%
+% \begin{macro}{\eqfontsize}
+% \begin{macro}{\eqcolor}
+% \begin{macro}{\eqmargin}
+% \begin{macro}{\eqindent}
+% \begin{macro}{\eqbinoffset}
+% \begin{macro}{\eqnumside}
+% \begin{macro}{\eqnumplace}
+% \begin{macro}{\eqnumsep}
+% \begin{macro}{\eqnumfont}
+% \begin{macro}{\eqnumform}
+% \begin{macro}{\eqnumsize}
+% \begin{macro}{\eqnumcolor}
+% \begin{macro}{\eqlinespacing}
+% \begin{macro}{\eqlineskip}
+% \begin{macro}{\eqlineskiplimit}
+% \begin{macro}{\eqstyle}
+% \begin{macro}{\eqinterlinepenalty}
+% \begin{macro}{\intereqpenalty}
+% \begin{macro}{\intereqskip}
+%
+% Note: avoid M, m, P, p because they look like they might be the
+% start of a keyword \quoted{minus} or \quoted{plus}.    Then
+% \tex  looks further to see if the next letter is i or l.    And if
+% the next thing is an undefined macro, the attempt to expand the macro
+% results in an error message.
+%    \begin{macrocode}
+\def\eqfontsize{}         % Inherit from context    [NOT USED?]
+\def\eqcolor{black}       % Default to black        [NOT USED?]
+\newdimen\eqnumsep \eqnumsep=10pt        % Min space between equ number and body
+\newdimen\eqmargin \eqmargin=8pt         % For `multline' gap emulation
+%    \end{macrocode}
+% The \cs{eqindent} and \cs{eqnumside} variables need to
+% have their values initialized from context, actually.    But
+% that takes a bit of work, which is postponed till later.
+%    \begin{macrocode}
+\def\eqindent{C}%         % C or I, centered or indented
+\def\eqnumside{R}%        % R or L, right or left
+\def\eqnumplace{M}%       % M or T or B, middle top or bottom
+%    \end{macrocode}
+% Typesetting the equation number is done thus:
+% \begin{literalcode}
+% {\eqnumcolor \eqnumsize \eqnumfont{\eqnumform{\eq@number}}}
+% \end{literalcode}
+% .
+%    \begin{macrocode}
+%d\eqnumfont{\upshape}% % Upright even when surrounding text is slanted
+\def\eqnumfont{}%         % Null for easier debugging [mjd,1997/09/26]
+\def\eqnumform#1{(#1\@@italiccorr)} % Add parens
+\def\eqnumsize{}          % Allow numbers to have different typesize ...
+%    \end{macrocode}
+% Tricky questions on \cs{eqnumsize}.    Should the default
+% be \cs{normalsize}?    Then the user can scale down the
+% equation body with \cs{small} and not affect the equation
+% number.    Or should the default be empty?    Then in large
+% sections of smaller text, like the dangerous bend stuff in
+% \emph{\TeX book}, the equation number size will keep in synch
+% with the context.
+% Maybe need an \cs{eqbodysize} param as well to allow separating
+% the two cases.
+%    \begin{macrocode}
+\def\eqnumcolor{}         % ... or color than eq body e.g. \color{blue}
+\newlength\eqlinespacing \eqlinespacing=14pt plus2pt % Base-to-base space between lines
+\newlength\eqlineskip \eqlineskip=3pt plus2pt % Min space if eqlinespacing too small
+\newdimen\eqlineskiplimit \eqlineskiplimit=2pt  % Threshold for switching to eqlineskip
+%    \end{macrocode}
+% The value of \cs{eqbinoffset} should include a negative shrink
+% component that cancels the shrink component of medmuskip, otherwise
+% there can be a noticeable variation in the indent of adjacent lines if
+% one is shrunken a lot and the other isn't.
+%    \begin{macrocode}
+\newmuskip \eqbinoffset \eqbinoffset=15mu minus-3mu % Offset from mathrel alignment pt for mathbins
+\newmuskip\eqdelimoffset \eqdelimoffset=2mu    % Additional offset for break inside delims
+\newdimen\eqindentstep \eqindentstep=8pt     % Indent used when LHS wd is n/a or too large
+\newtoks\eqstyle           % Customization hook
+\newcount\eqbreakdepth \eqbreakdepth=2       % Allow breaks within delimiters to this depth
+\newcount \eqinterlinepenalty \eqinterlinepenalty=10000 % No page breaks between equation lines
+\newcount \intereqpenalty \intereqpenalty=1000   % Pagebreak penalty between equations [BRM: Was \@M]
+\newlength \intereqskip \intereqskip=3pt plus2pt % Additional vert space between equations
+\newcount\prerelpenalty \prerelpenalty=-\@M   % Linebreak penalty before mathrel symbols
+\newcount\prebinoppenalty \prebinoppenalty=888  % Linebreak penalty before mathbins
+%    \end{macrocode}
+% When breaking equations we never right-justify, so a stretch
+% component of the muskip is never helpful and sometimes it is definitely
+% undesirable.    Note that thick\slash medmuskips frozen inside a
+% fraction or radical may turn out noticeably larger than neighboring
+% unfrozen ones.    Nonetheless I think this way is the best
+% compromise short of a new \tex  that can make those built-up objects
+% shrink horizontally in proportion; the alternative is to pretty much
+% eliminate the shrink possibility completely in displays.
+%    \begin{macrocode}
+\newmuskip \Dmedmuskip \Dmedmuskip=4mu minus 3mu % medmuskip in displays
+\newmuskip \Dthickmuskip \Dthickmuskip=5mu minus 2mu % thickmuskip in displays
+%    \end{macrocode}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+%
+% And now some internal variables.    1997/10/22: some of
+% these are dead branches that need to be pruned.
+%
+% MH: Started cleaning up a bit. No more funny loops.
+%    \begin{macrocode}
+\def\eq@number{}          % Internal variable
+\newlength\eqleftskip \eqleftskip=\@centering  % Space on the left  [NOT USED?]
+\newlength\eqrightskip \eqrightskip=\@centering % Space on the right [NOT USED?]
+\newlength\eq@vspan \eq@vspan=\z@skip     % Glue used to vcenter the eq number
+\newmuskip\eq@binoffset \eq@binoffset=\eqbinoffset % Roughly, \eqbinoffset + \eqdelimoffset
+\newsavebox\EQ@box               % Storage for equation body
+\newsavebox\EQ@copy              % For eq body sans vadjust/insert/mark material
+\newsavebox\EQ@numbox            % For equation number
+\newdimen\eq@wdNum         % width of number + separation [NEW]
+\newsavebox\GRP@numbox            % For group number [NEW]
+\newdimen\grp@wdNum         % width of number + separation [NEW]
+%%B\EQ@vimbox            % Vadjust, insert, or mark material
+%%B\EQ@vimcopy           % Spare copy of same
+%%B\eq@impinging         % Temporary box for measuring number placement
+\newcount \eq@lines          % Internal counter, actual number of lines
+\newcount \eq@curline       % Loop counter
+\newcount \eq@badness      % Used in testing for overfull lines
+\newcount \EQ@vims          % For bookkeeping
+\def\@eq@numbertrue{\let\eq@hasNumber\@True}%
+\def\@eq@numberfalse{\let\eq@hasNumber\@False}%
+\let\eq@hasNumber\@False
+%    \end{macrocode}
+% Here for the dimens, it would be advisable to do some more careful
+% management to conserve dimen registers.    First of all, most of the
+% dimen registers are needed in the measuring phase, which is a tightly
+% contained step that happens after the contents of the equation have been
+% typeset into a box and before any external functions have a chance to
+% regain control\mdash  \eg , the output routine.
+% Therefore it is possible to make use of the the dimen registers 0--9,
+% reserved by convention for scratch use, without fear of conflict with
+% other macros.    But I don't want to use them directly with the
+% available names:
+% \begin{literalcode}
+% \dimen@ \dimen@i \dimen@ii \dimen3 \dimen4 ... \dimen9
+% \end{literalcode}
+% .    It would be much more useful to have names for these registers
+% indicative of way they are used.
+%
+% Another source whence dimen registers could be borrowed is the
+% \pkg{amsmath} package, which allocates six registers for
+% equation-measuring purposes.    We can reuse them under different
+% names since the \pkg{amsmath} functions and our functions will
+% never be used simultaneously.
+% \begin{literalcode}
+% \eqnshift@ \alignsep@ \tagshift@ \tagwidth@ \totwidth@ \lineht@
+% \end{literalcode}
+%    \begin{macrocode}
+\newdimen\eq@dp         % Depth of last line
+\newdimen\eq@wdL        % Width of the left-hand-side
+\newdimen\eq@wdT        % Total width for framing
+\newdimen\eq@wdMin      % Width of narrowest line in equation
+\newdimen\grp@wdL       % Max width of LHS's in a group
+\newdimen\grp@wdR       % Max RHS of all equations in a group
+\newdimen\grp@wdT
+\newdimen\eq@wdRmax
+\newdimen\eq@firstht    % Height of first line
+%    \end{macrocode}
+% BRM: measure the condition too.
+%    \begin{macrocode}
+\newdimen\eq@wdCond
+\newdimen\eq@indentstep % Indent amount when LHS is not present
+\newdimen\eq@linewidth  % Width actually used for display
+\newdimen\grp@linewidth % Max eq@linewidth over a group
+%    \end{macrocode}
+% Maybe \cs{eq@hshift} could share the same register as
+% \cs{mathindent} [mjd,1997/10/22].
+%    \begin{macrocode}
+\newdimen\eq@hshift
+\let\eq@isIntertext\@False
+%    \end{macrocode}
+% Init \cs{eq@indentstep} to a nonzero value so that we can
+% detect and refrain from clobbering a user setting of zero.
+% And \cs{eq@sidespace} to \cs{maxdimen} because
+% that is the right init before computing a min.
+%    \begin{macrocode}
+\eq@indentstep=\maxdimen
+\newdimen\eq@given@sidespace
+%    \end{macrocode}
+%
+% \begin{macro}{\eq@overrun}
+%   MH: Appears to be unused.
+%
+%   Not a dimen register; don't need to advance it.
+%    \begin{macrocode}
+\def\eq@overrun{0pt}
+%    \end{macrocode}
+% \end{macro}
+%
+%
+% To initialize \cs{eqnumside} and \cs{eqindent} properly,
+% we may need to grub around a bit in \cs{@filelist}.    However,
+% if the \pkg{amsmath} package was used, we can use its option
+% data.    More trouble: if a documentclass sends an option of
+% \opt{leqno} to \pkg{amsmath} by default, and it gets
+% overridden by the user with a \opt{reqno} documentclass option,
+% then \pkg{amsmath} believes itself to have received
+% \emph{both} options.
+%    \begin{macrocode}
+\@ifpackagewith{amsmath}{leqno}{%
+  \@ifpackagewith{amsmath}{reqno}{}{\def\eqnumside{L}}%
+}{%
+%    \end{macrocode}
+% If the \pkg{amsmath} package was not used, the next
+% method for testing the \opt{leqno} option is to see if
+% \fn{leqno.clo} is present in \cs{@filelist}.
+%    \begin{macrocode}
+  \def\@tempa#1,leqno.clo,#2#3\@nil{%
+    \ifx @#2\relax\else \def\eqnumside{L}\fi
+  }%
+  \@xp\@tempa\@filelist,leqno.clo,@\@nil
+%    \end{macrocode}
+% Even that test may fail in the case of \cls{amsart} if it does
+% not load \pkg{amsmath}.    Then we have to look whether
+% \cs{iftagsleft@} is defined, and if so whether it is true.
+% This is tricky if you want to be careful about conditional nesting
+% and don't want to put anything in the hash table unnecessarily.
+%    \begin{macrocode}
+  \if L\eqnumside
+  \else
+    \@ifundefined{iftagsleft@}{}{%
+      \edef\eqnumside{%
+        \if TT\csname fi\endcsname\csname iftagsleft@\endcsname
+          L\else R\fi
+      }%
+    }
+  \fi
+}
+%    \end{macrocode}
+% A similar sequence of tests handles the \quoted{fleqn or not fleqn}
+% question for the \cls{article} and \cls{amsart}
+% documentclasses.
+%    \begin{macrocode}
+\@ifpackagewith{amsmath}{fleqn}{%
+  \def\eqindent{I}%
+}{%
+  \def\@tempa#1,fleqn.clo,#2#3\@nil{%
+    \ifx @#2\relax\else \def\eqindent{I}\fi
+  }%
+  \@xp\@tempa\@filelist,fleqn.clo,@\@nil
+  \if I\eqindent
+  \else
+    \@ifundefined{if@fleqn}{}{%
+      \edef\eqindent{%
+        \if TT\csname fi\endcsname\csname if@fleqn\endcsname
+          I\else C\fi
+      }%
+    }%
+  \fi
+}
+%    \end{macrocode}
+% BRM: This conditional implies we must use ALL indented or ALL centered?
+%    \begin{macrocode}
+%\if I\eqindent
+  \@ifundefined{mathindent}{%
+    \newdimen\mathindent
+  }{%
+    \@ifundefined{@mathmargin}{}{%
+      \mathindent\@mathmargin
+    }%
+  }
+%\fi
+%    \end{macrocode}
+%
+%
+%
+%
+% \section{Measuring equation components}
+% Measure the left-hand side of an equation.    This
+% function is called by mathrel symbols.    For the first mathrel we
+% want to discourage a line break more than for following mathrels; so
+% \cs{mark@lhs} gobbles the following \cs{rel@break} and
+% substitutes a higher penalty.
+% \begin{aside}
+% Maybe the LHS should be kept in a separate box.
+% \end{aside}
+%
+%
+%
+% \begin{macro}{\EQ@hasLHS}
+%
+% Boolean: does this equation have a \dquoted{left-hand side}?
+%    \begin{macrocode}
+\let\EQ@hasLHS=\@False
+%    \end{macrocode}
+% \end{macro}
+%
+%
+% \begin{macro}{\EQ@QED}
+%
+% If nonempty: the qed material that should be incorporated into this
+% equation after the final punctuation.
+%    \begin{macrocode}
+\let\EQ@QED=\@empty
+%    \end{macrocode}
+% \end{macro}
+%
+%
+% \begin{macro}{\mark@lhs}
+%
+%    \begin{macrocode}
+\def\mark@lhs#1{%
+  \ifnum\lr@level<\@ne
+    \let\mark@lhs\relax
+    \global\let\EQ@hasLHS=\@True
+    \global\let\EQ@prebin@space\EQ@prebin@space@a
+    \mark@lhs@a
+%    \end{macrocode}
+% But the penalty for the first mathrel should still be lower than a
+% binoppenalty.    If not, when the LHS contains a binop, the split
+% will occur inside the LHS rather than at the mathrel.
+% On the other hand if we end up with a multline sort of equation
+% layout where the RHS is very short, the break before the relation symbol
+% should be made \emph{less} desirable than the breakpoints inside
+% the LHS.
+% Since a lower penalty takes precedence over a higher one, we start
+% by putting in the highest relpenalty; during subsequent measuring if we
+% find that that RHS is not excessively short then we put in an extra
+% \dquoted{normal} relpenalty when rejoining the LHS and RHS.
+%    \begin{macrocode}
+    \penalty9999 % instead of normal \rel@break
+  % else no penalty = forbid break
+  \fi
+}
+%    \end{macrocode}
+% \end{macro}
+%
+%
+% \begin{macro}{\mark@lhs@a}
+%
+% Temporarily add an extra thickmuskip to the LHS; it will be removed
+% later.    This is necessary to compensate for the disappearance of
+% the thickmuskip glue preceding a mathrel if a line break is taken at
+% that point.    Otherwise we would have to make our definition of
+% mathrel symbols more complicated, like the one for mathbins.    The
+% penalty of $2$ put in with vadjust is a flag for
+% \cs{eq@repack} to suggest that the box containing this line should
+% be measured to find the value of \cs{eq@wdL}.    The
+% second vadjust ensures that the normal prerelpenalty and thickmuskip
+% will not get lost at the line break during this preliminary pass.
+%
+% BRM: I originally thought the \verb"\mskip\thickmuskip" was messing
+% up summation limits in LHS.  But I may have fixed that problem by
+% fixing other things\ldots
+%    \begin{macrocode}
+\def\mark@lhs@a{%
+  \mskip\thickmuskip \@@vadjust{\penalty\tw@}\penalty-\@Mi\@@vadjust{}%
+}
+%    \end{macrocode}
+% \end{macro}
+%
+%
+% \begin{macro}{\hiderel}
+% If you want the LHS to extend past the first mathrel symbol to a
+% following one, mark the first one with \cs{hiderel}:
+% \begin{literalcode}
+% a \hiderel{=} b = c...
+% \end{literalcode}
+% .
+% \begin{aside}
+% I'm not sure now why I didn't use \cs{begingroup}
+% \cs{endgroup} here \begin{dn}
+% mjd,1999/01/21
+% \end{dn}
+% .
+% \end{aside}
+%
+%    \begin{macrocode}
+\newcommand\hiderel[1]{\mathrel{\advance\lr@level\@ne#1}}
+%    \end{macrocode}
+% \end{macro}
+%
+%
+% \begin{macro}{\m@@Bin}
+% \begin{macro}{\m@@Rel}
+% \begin{macro}{\bin@break}
+% \begin{macro}{\rel@break}
+% \begin{macro}{\bin@mark}
+% \begin{macro}{\rel@mark}
+% \begin{macro}{\d@@Bin}
+% \begin{macro}{\d@@Rel}
+%
+% \cf  \pkg{flexisym} handling of mathbins and mathrels.    These
+% are alternate definitions of \cs{m@Bin} and \cs{m@Rel},
+% activated by \cs{display@setup}.
+%    \begin{macrocode}
+%%%%\let\m@@Bin\m@Bin
+%%%%%\let\m@@Rel\m@Rel
+\let\EQ@prebin@space\relax
+\def\EQ@prebin@space@a{\mskip-\eq@binoffset \keep@glue \mskip\eq@binoffset}
+\def\bin@break{\ifnum\lastpenalty=\z@\penalty\prebinoppenalty\fi
+  \EQ@prebin@space}
+\def\rel@break{%
+  \ifnum\abs@num\lastpenalty <\abs@num\prerelpenalty
+    \penalty\prerelpenalty
+  \fi
+}
+\ExplSyntaxOn
+%%%\def\d@@Bin{\bin@break \m@@Bin}
+%%%%\def\d@@Rel{\mark@lhs \rel@break \m@@Rel}
+\cs_set:Npn \math_dsym_Bin:Nn {\bin@break\math_bsym_Bin:Nn}
+\cs_set:Npn \math_dsym_Rel:Nn {\mark@lhs \rel@break \math_bsym_Rel:Nn }
+\ExplSyntaxOff
+%    \end{macrocode}
+% The difficulty of dealing properly with the subscripts and
+% superscripts sometimes appended to mathbins and mathrels is one of the
+% reasons that we do not attempt to handle the mathrels as a separate
+% \quoted{column} a la \env{eqnarray}.
+%
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+%
+%
+% \begin{macro}{\m@@symRel}
+% \begin{macro}{\d@@symRel}
+% \begin{macro}{\m@@symBin}
+% \begin{macro}{\d@@symBin}
+% \begin{macro}{\m@@symDel}
+% \begin{macro}{\d@@symDel}
+% \begin{macro}{\m@@symDeR}
+% \begin{macro}{\d@@symDeR}
+% \begin{macro}{\m@@symDeB}
+% \begin{macro}{\d@@symDeB}
+% \begin{macro}{\m@@symDeA}
+% \begin{macro}{\d@@symDeA}
+%
+% More of the same.
+%    \begin{macrocode}
+\ExplSyntaxOn
+%%\let\m@@symRel\@symRel
+%%%\def\d@@symRel{\mark@lhs \rel@break \m@@symRel}
+
+\cs_set_protected:Npn \math_dcsym_Bin:Nn {\bin@break \math_bcsym_Bin:Nn}
+\cs_set_protected:Npn \math_dcsym_Rel:Nn { \mark@lhs \rel@break \math_bcsym_Rel:Nn}
+
+
+%%\let\m@@symBin\@symBin \def\d@@symBin{\bin@break \m@@symBin}
+%%\let\m@@symDel\@symDel
+%%\let\m@@symDeR\@symDeR
+%%\let\m@@symDeB\@symDeB
+%%\let\m@@symDeA\@symDeA
+
+%    \end{macrocode}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+%
+%
+% \begin{macro}{\display@setup}
+% \begin{macro}{\everydisplay}
+% Setup.    Note that \latex  reserves the primitive
+% \cs{everydisplay} under the name \cs{frozen@everydisplay}.
+% BRM: Disable this! It also affects non-breqn math!!!!
+%    \begin{macrocode}
+%\global\everydisplay\expandafter{\the\everydisplay \display@setup}
+%    \end{macrocode}
+% Change some math symbol function calls.
+%    \begin{macrocode}
+\def\display@setup{%
+  \medmuskip\Dmedmuskip \thickmuskip\Dthickmuskip
+   \math_setup_display_symbols:
+  %%\let\m@Bin\d@@Bin \let\m@Rel\d@@Rel
+  %%\let\@symRel\d@@symRel \let\@symBin\d@@symBin
+  %%\let\m@DeL\d@@DeL \let\m@DeR\d@@DeR \let\m@DeB\d@@DeB
+  %%\let\m@DeA\d@@DeA
+  %%\let\@symDeL\d@@symDeL \let\@symDeR\d@@symDeR
+  %%\let\@symDeB\d@@symDeB \let\@symDeA\d@@symDeA
+  \let\left\eq@left \let\right\eq@right \global\lr@level\z@
+  \global\eq@wdCond\z@          %BRM: new
+%    \end{macrocode}
+% If we have an embedded array environment (for example), we
+% don't want to have each math cell within the array resetting
+% \cs{lr@level} globally to 0\mdash not good!
+% And in general I think it is safe to say that whenever we have a
+% subordinate level of boxing we want to revert to a normal math setup.
+%    \begin{macrocode}
+  \everyhbox{\everyhbox\@emptytoks
+    \let\display@setup\relax \textmath@setup \let\textmath@setup\relax
+  }%
+  \everyvbox{\everyvbox\@emptytoks
+    \let\display@setup\relax \textmath@setup \let\textmath@setup\relax
+  }%
+}
+%    \end{macrocode}
+% The \cs{textmath@setup} function is needed for embedded inline
+% math inside text inside a display.
+%
+% BRM: DS Experiment: Variant of \cs{display@setup} for use within
+% dseries environmnents
+%    \begin{macrocode}
+\def\dseries@display@setup{%
+  \medmuskip\Dmedmuskip \thickmuskip\Dthickmuskip
+  \math_setup_display_symbols:
+%%%%  \let\m@Bin\d@@Bin
+%%%\let\m@Rel\d@@Rel
+%%%  \let\@symRel\d@@symRel
+%%% \let\@symBin\d@@symBin
+%%%  \let\m@DeL\d@@DeL \let\m@DeR\d@@DeR \let\m@DeB\d@@DeB
+%%%  \let\m@DeA\d@@DeA
+%%%  \let\@symDeL\d@@symDeL \let\@symDeR\d@@symDeR
+%%%  \let\@symDeB\d@@symDeB \let\@symDeA\d@@symDeA
+  \let\left\eq@left \let\right\eq@right \global\lr@level\z@
+  \everyhbox{\everyhbox\@emptytoks
+    \let\display@setup\relax \textmath@setup \let\textmath@setup\relax
+  }%
+  \everyvbox{\everyvbox\@emptytoks
+    \let\display@setup\relax \textmath@setup \let\textmath@setup\relax
+  }%
+ \displaystyle
+}
+%    \end{macrocode}
+%
+%    \begin{macrocode}
+\def\textmath@setup{%
+   \math_setup_inline_symbols:
+%%%%  \let\m@Bin\m@@Bin \let\m@Rel\m@@Rel
+%%%%  \let\@symRel\m@@symRel \let\@symBin\m@@symBin
+%%%%  \let\m@DeL\m@@DeL \let\m@DeR\m@@DeR \let\m@DeB\m@@DeB
+%%%%  \let\m@DeA\m@@DeA
+%%%%  \let\@symDeL\m@@symDeL \let\@symDeR\m@@symDeR
+%%%%  \let\@symDeB\m@@symDeB \let\@symDeA\m@@symDeA
+  \let\left\@@left \let\right\@@right
+}
+
+\ExplSyntaxOff
+%    \end{macrocode}
+% \end{macro}
+% \end{macro}
+%
+% \begin{macro}{\if@display}
+% \begin{macro}{\everydisplay}
+% The test \cs{ifinner} is unreliable for distinguishing
+% whether we are in a displayed formula or an inline formula: any display
+% more complex than a simple one-line equation typically involves the use
+% of \verb"$" \cs{displaystyle} \dots  \verb"$" instead of
+% \dbldollars  \dots  \dbldollars .    So we provide a more reliable
+% test.    But it might have been provided already by the
+% \pkg{amsmath} package.
+%    \begin{macrocode}
+\@ifundefined{@displaytrue}{%
+  \@xp\newif\csname if@display\endcsname
+  \everydisplay\@xp{\the\everydisplay \@displaytrue}%
+}{}
+%    \end{macrocode}
+%
+% \begin{aside}
+% Is there any reason to maintain separate
+% \cs{everydisplay} and \cn{eqstyle}?
+%
+% \end{aside}
+%
+%
+% \end{macro}
+% \end{macro}
+%
+%
+%
+%
+% \section{The \env{dmath} and \env{dmath*} environments}
+%
+% Options for the \env{dmath} and \env{dmath*}
+% environments.
+% \begin{literalcode}
+% \begin{dmath}[label={eq:xyz}]
+% \begin{dmath}[labelprefix={eq:},label={xyz}]
+% \end{literalcode}
+% WSPR: added the option for a label prefix, designed to be used in the preamble like so:
+% \begin{literalcode}
+% \setkeys{breqn}{labelprefix={eq:}}
+% \end{literalcode}
+%    \begin{macrocode}
+\define@key{breqn}{label}{%
+  \edef\next@label{\noexpand\label{\next@label@pre#1}}%
+  \let\next@label@pre\@empty}
+\define@key{breqn}{labelprefix}{\def\next@label@pre{#1}}
+\global\let\next@label\@empty
+\global\let\next@label@pre\@empty
+%    \end{macrocode}
+% Allow a variant number.
+% \begin{literalcode}
+% \begin{dmath}[number={\nref{foo}\textprime}]
+% \end{literalcode}
+%    \begin{macrocode}
+\define@key{breqn}{number}{\def\eq@number{#1}%
+  \let\@currentlabel\eq@number
+}
+%    \end{macrocode}
+% \begin{literalcode}
+% \begin{dmath}[shiftnumber]
+% \begin{dmath}[holdnumber]
+% \end{literalcode}
+% Holding or shifting the number.
+%    \begin{macrocode}
+\define@key{breqn}{shiftnumber}{\let\eq@shiftnumber\@True}
+\define@key{breqn}{holdnumber}{\let\eq@holdnumber\@True}
+%    \end{macrocode}
+% \begin{literalcode}
+% \begin{dmath}[density={.5}]
+% \end{literalcode}
+%    \begin{macrocode}
+\define@key{breqn}{density}{\def\eq@density@factor{#1}}
+%    \end{macrocode}
+% \begin{literalcode}
+% \begin{dmath}[indentstep={1em}]
+% \end{literalcode}
+% To change the amount of indent for post-initial lines.    Note:
+% for lines that begin with a mathbin symbol there is a fixed amount of
+% indent already built in (\cs{eqbinoffset}) and it cannot be
+% reduced through this option.    The indentstep amount is the indent
+% used for lines that begin with a mathrel symbol.
+%    \begin{macrocode}
+\define@key{breqn}{indentstep}{\eqindentstep#1\relax}
+%    \end{macrocode}
+% \begin{literalcode}
+% \begin{dmath}[compact]
+% \begin{dmath}[compact=-2000]
+% \end{literalcode}
+% To make mathrels stay inline to the extent possible, use the compact
+% option.
+% Can give a numeric value in the range $-10000 \dots  10000$
+% to adjust the behavior.
+% $-10000$: always break at a rel symbol; $10000$: never
+% break at a rel symbol.
+%    \begin{macrocode}
+\define@key{breqn}{compact}[-99]{\prerelpenalty=#1\relax}
+%    \end{macrocode}
+% \begin{literalcode}
+% \begin{dmath}[layout={S}]%
+% \end{literalcode}
+% Specify a particular layout.
+% We take care to ensure that \cs{eq@layout} ends up containing
+% one and only one letter.
+%    \begin{macrocode}
+\define@key{breqn}{layout}[?]{%
+  \edef\eq@layout{\@car#1?\@nil}%
+}
+%    \end{macrocode}
+% \begin{literalcode}
+% \begin{dmath}[spread={1pt}]
+% \end{literalcode}
+% To change the interline spacing in a particular equation.
+%    \begin{macrocode}
+\define@key{breqn}{spread}{%
+  \addtolength\eqlinespacing{#1}%
+  \addtolength\eqlineskip{#1}%
+  \eqlineskiplimit\eqlineskip
+}
+%    \end{macrocode}
+% To change the amount of space on the side for \dquoted{multline} layout.
+%    \begin{macrocode}
+\define@key{breqn}{sidespace}{%
+  \setlength\eq@given@sidespace{#1}%
+}
+%    \end{macrocode}
+% \begin{literalcode}
+% \begin{dmath}[style={\small}]
+% \end{literalcode}
+% The \opt{style} option is mainly intended for changing the
+% type size of an equation but as a matter of fact you could put arbitrary
+% \latex  code here \mdash  thus the option name is \quoted{style} rather
+% than just \quoted{typesize}.    In order for this option to work when
+% setting options globally, we need to put the code in
+% \cs{eqstyle} rather than execute it directly.
+%    \begin{macrocode}
+\define@key{breqn}{style}{\eqstyle\@xp{\the\eqstyle #1}}
+%    \end{macrocode}
+% \begin{literalcode}
+% \begin{dmath}[shortskiplimit={1em}]
+% \end{literalcode}
+% If the line immediately preceeding a display has length $l$, the
+% first line of the display is indented $i$, and a shortskip limit $s$
+% is set, then the spacing above the display is equal to
+% \cs{abovedisplayshortskip} if $l+s < i $ and \cs{abovedisplayskip}
+% otherwise. The default shortskip limit is 2\,em which is what \TeX\
+% hardcodes but this parameter overrides that.
+%    \begin{macrocode}
+\define@key{breqn}{shortskiplimit}{\def\eq@shortskiplimit{#1}}
+\def\eq@shortskiplimit{2em}
+%    \end{macrocode}
+%
+% \begin{literalcode}
+% \begin{dmath}[frame]
+% \end{literalcode}
+% The \opt{frame} option merely puts a framebox around the body
+% of the equation.    To change the thickness of the frame, give the
+% thickness as the argument of the option.    For greater control, you
+% can change the appearance of the frame by redefining
+% \cs{eqframe}.    It must be a command taking two arguments, the
+% width and height of the equation body.    The top left corner of the
+% box produced by \cs{eqframe} will be pinned to the top-left corner
+% of the equation body.
+%    \begin{macrocode}
+\define@key{breqn}{frame}[\fboxrule]{\def\eq@frame{T}%
+  \dim@a#1\relax\edef\eq@framewd{\the\dim@a}%
+%    \end{macrocode}
+% Until such time as we provide a frame implementation that allows the
+% frame to stretch and shrink, we'd better remove any stretch/shrink from
+% the interline glue in this case.
+%    \begin{macrocode}
+  \freeze@glue\eqlinespacing \freeze@glue\eqlineskip
+}
+\define@key{breqn}{fullframe}[]{\def\eq@frame{U}%
+  \freeze@glue\eqlinespacing \freeze@glue\eqlineskip
+}
+\def\eq@frame{F} % no frame
+\def\eq@framewd{\fboxrule}
+%    \end{macrocode}
+% Wishful thinking?
+% \begin{literalcode}
+% \begin{dmath}[frame={width={2pt},color={blue},sep={2pt}}]
+% \end{literalcode}
+% To change the space between the frame and the equation there is a
+% framesep option.
+%    \begin{macrocode}
+\define@key{breqn}{framesep}[\fboxsep]{%
+  \if\eq@frame F\def\eq@frame{T}\fi
+  \dim@a#1\relax \edef\eq@framesep{\the\dim@a}%
+  \freeze@glue\eqlinespacing \freeze@glue\eqlineskip
+}
+\def\eq@framesep{\fboxsep}
+%    \end{macrocode}
+% \begin{literalcode}
+% \begin{dmath}[background={red}]
+% \end{literalcode}
+% Foreground and background colors for the equation.    By default
+% the background area that is colored is the size of the equation, plus
+% fboxsep.    If you need anything fancier for the background, you'd
+% better do it by defining \cs{eqframe} in terms of
+% \cs{colorbox} or \cs{fcolorbox}.
+%    \begin{macrocode}
+\define@key{breqn}{background}{\def\eq@background{#1}%
+  \freeze@glue\eqlinespacing \freeze@glue\eqlineskip
+}
+%   \end{macrocode}
+% \begin{literalcode}
+% \begin{dmath}[color={purple}]
+% \end{literalcode}
+%    \begin{macrocode}
+\define@key{breqn}{color}{\def\eq@foreground{#1}}
+%    \end{macrocode}
+% \begin{literalcode}
+% \begin{dmath}[center]
+% \begin{dmath}[nocenter]
+% \end{literalcode}
+% The \opt{center} option means add leftskip stretch to make the
+% individual lines be centered; this is the default for
+% \env{dseries}.
+%    \begin{macrocode}
+\define@key{breqn}{center}[]{\let\eq@centerlines\@True}
+\define@key{breqn}{nocenter}[]{\let\eq@centerlines\@False}
+\let\eq@centerlines\@False
+%    \end{macrocode}
+% \begin{literalcode}
+% \begin{dgroup}[noalign]
+% \end{literalcode}
+% Equation groups normally have alignment of the primary relation
+% symbols across the whole group.    The \opt{noalign} option
+% switches that behavior.
+%    \begin{macrocode}
+\define@key{breqn}{noalign}[]{\let\grp@aligned\@False}
+\let\grp@aligned\@True % default
+%    \end{macrocode}
+% \begin{literalcode}
+% \begin{dgroup}[breakdepth={2}]
+% \end{literalcode}
+% Break depth of 2 means that breaks are allowed at mathbin symbols
+% inside two pairs of  delimiters, but not three.
+%    \begin{macrocode}
+\define@key{breqn}{breakdepth}{\eqbreakdepth#1\relax}
+%    \end{macrocode}
+% \begin{literalcode}
+% \begin{darray}[cols={lcrlcr}]
+% \end{literalcode}
+% The \opt{cols} option only makes sense for the
+% \env{darray} environment but we liberally allow all the options to
+% be used with all the environments and just ignore any unsensible ones
+% that happen to come along.
+%    \begin{macrocode}
+\define@key{breqn}{cols}{\global\let\@preamble\@empty
+  \darray@mkpream#1\@percentchar
+}
+%    \end{macrocode}
+%
+% FORMAT STATUS%
+% \begin{verbatim}
+% \def\eq@frame{T}%
+% CLM works tolerably
+%  \def\eqindent{C}\def\eqnumside{L}\def\eqnumplace{M}
+% CLT works tolerably
+%  \def\eqindent{C}\def\eqnumside{L}\def\eqnumplace{T}
+% ILM
+%  \def\eqindent{I}\def\eqnumside{L}\def\eqnumplace{M}\mathindent40\p@
+% ILT
+%  \def\eqindent{I}\def\eqnumside{L}\def\eqnumplace{T}\mathindent40\p@
+% Indended w/left number
+%    work ok if mathindent is larger than number width,
+%    but then equations must fit into smaller space.
+%    Is shiftnumber allowed to put eqn at left, instead of indent?
+% CRM
+%  \def\eqindent{C}\def\eqnumside{R}\def\eqnumplace{M}
+% CRB
+%  \def\eqindent{C}\def\eqnumside{R}\def\eqnumplace{B}
+% IRM
+%  \def\eqindent{I}\def\eqnumside{R}\def\eqnumplace{M}\mathindent10\p@
+% IRB
+%  \def\eqindent{I}\def\eqnumside{R}\def\eqnumplace{B}\mathindent10\p@
+% \end{verbatim}
+%
+% The main environments.
+%
+%BRM: The following incorporates several changes:
+%  1) modifications supplied by MJD to fix the eaten \cs{paragraph} problem.
+%  2) Added \cs{display@setup} here, rather than globally.
+%
+% \begin{macro}{\@dmath@start@hook}
+% \begin{macro}{\@dgroup@start@hook}
+%    \begin{macrocode}
+\let\@dmath@start@hook\@empty
+\let\@dgroup@start@hook\@empty
+%    \end{macrocode}
+% \end{macro}
+% \end{macro}
+%
+% \begin{macro}{\dmath}
+% \begin{macro}{\enddmath}
+% For the \env{dmath} environment we don't want the standard
+% optional arg processing because of the way it skips over whitespace,
+% including newline, while looking for the \verb"[" char; which is not good
+% for math material.    So we call \cs{@optarg} instead.
+%    \begin{macrocode}
+\newenvironment{dmath}{%
+ \@dmath@start@hook
+ \let\eq@hasNumber\@True \@optarg\@dmath{}}{}
+\def\@dmath[#1]{%
+%<trace>  \breqn@debugmsg{=== DMATH ==================================================}%
+  \everydisplay\expandafter{\the\everydisplay \display@setup}%
+  \if@noskipsec \leavevmode \fi
+  \if@inlabel \leavevmode \global\@inlabelfalse \fi
+  \if\eq@group\else\eq@prelim\fi
+  \setkeys{breqn}{#1}%
+  \the\eqstyle
+%    \end{macrocode}
+% The equation number might have been overridden in \verb|#1|.
+%    \begin{macrocode}
+  \eq@setnumber
+%    \end{macrocode}
+% Start up the displayed equation by reading the contents into a
+% box register.    Enclose this phase in an extra group so that
+% modified \cs{hsize} and other params will be auto-restored
+% afterwards.
+%    \begin{macrocode}
+  \begingroup
+  \eq@setup@a
+  \eq@startup
+}
+%    \end{macrocode}
+% Before it finishes off the box holding the equation body,
+% \cs{enddmath} needs to look ahead for punctuation (and
+% \cs{qed}?).
+%    \begin{macrocode}
+\def\enddmath#1{%
+  \check@punct@or@qed
+}
+\def\end@dmath{%
+  \gdef\EQ@setwdL{}% Occasionally undefined ???
+  \eq@capture
+  \endgroup
+  \EQ@setwdL
+%    \end{macrocode}
+% Measure (a copy of) the equation body to find the minimum width
+% required to get acceptable line breaks, how many lines will be required
+% at that width, and whether the equation number needs to be shifted to
+% avoid overlapping.    This information will then be used by
+% \cs{eq@finish} to do the typesetting of the real equation body.
+%    \begin{macrocode}
+  \eq@measure
+%    \end{macrocode}
+% Piece together the equation from its constituents, recognizing
+% current constraints.    If we are in an equation group, this might
+% just save the material on a stack for later processing.
+%    \begin{macrocode}
+  \if\eq@group \grp@push \else \eq@finish\fi
+}
+%    \end{macrocode}
+% \end{macro}
+% \end{macro}
+%
+% \begin{macro}{\dmath*}
+% \begin{macro}{\enddmath*}
+% Ah yes, now the lovely \env{dmath*} environment.
+%    \begin{macrocode}
+\newenvironment{dmath*}{%
+  \@dmath@start@hook
+  \let\eq@hasNumber\@False \@optarg\@dmath{}%
+}{}
+\@namedef{end@dmath*}{\end@dmath}
+\@namedef{enddmath*}#1{\check@punct@or@qed}
+%    \end{macrocode}
+% \end{macro}
+% \end{macro}
+%
+%
+% \begin{macro}{\eq@prelim}
+% If \cs{everypar} has a non-null value, it's probably
+% some code from \cs{@afterheading} that sets \cs{clubpenalty}
+% and\slash or removes the parindent box.    Both of those actions
+% are irrelevant and interfering for our purposes and need to be deflected
+% for the time being.
+% If an equation appears at the very beginning of a list item
+% (possibly from a trivlist such as \env{proof}), we need to
+% trigger the item label.
+%    \begin{macrocode}
+\def\eq@prelim{%
+  \if@inlabel \indent \par \fi
+  \if@nobreak \global\@nobreakfalse \predisplaypenalty\@M \fi
+  \everypar\@emptytoks
+%    \end{macrocode}
+% If for some reason \env{dmath} is called between paragraphs,
+% \cn{noindent} is better than \cn{leavevmode}, which would produce
+% an indent box and an empty line to hold it.    If we are in a list
+% environment, \cn{par} is defined as \verb"{\@@par}" to preserve
+% \cs{parshape}.
+%    \begin{macrocode}
+  \noindent
+  \eq@nulldisplay
+  \par %% \eq@saveparinfo %% needs work
+  \let\intertext\breqn@intertext
+}
+%    \end{macrocode}
+% \end{macro}
+% \begin{macro}{\breqn@parshape@warning}
+% Warning message extracted to a separate function to streamline the
+% calling function.
+%    \begin{macrocode}
+\def\breqn@parshape@warning{%
+  \PackageWarning{breqn}{%
+    Complex paragraph shape cannot be followed by this equation}%
+}
+%    \end{macrocode}
+% \end{macro}
+%
+%
+% \begin{macro}{\eq@prevshape}
+% Storage; see \cs{eq@saveparinfo}.
+%    \begin{macrocode}
+\let\eq@prevshape\@empty
+%    \end{macrocode}
+% \end{macro}
+%
+%
+% \begin{macro}{\eq@saveparinfo}
+% Save the number of lines and parshape info for the text preceding
+% the equation.
+%    \begin{macrocode}
+\def\eq@saveparinfo{%
+  \count@\prevgraf \advance\count@-\thr@@ % for the null display
+  \edef\eq@prevshape{\prevgraf\the\count@\space}%
+  \ifcase\parshape
+    % case 0: no action required
+  \or \edef\eq@prevshape{\eq@prevshape
+        \parshape\@ne\displayindent\displaywidth\relax
+      }%
+%    \end{macrocode}
+% Maybe best to set \cs{eq@prevshape} the same in the else case
+% also.    Better than nothing.
+%    \begin{macrocode}
+  \else
+    \breqn@parshape@warning
+  \fi
+}
+%    \end{macrocode}
+% \end{macro}
+%
+%
+% \begin{macro}{\eq@setnumber}
+% If the current equation number is not explicitly given, then
+% use an auto-generated number, unless the no-number switch has been
+% thrown (\env{dmath*}).
+% \cs{theequation} is the number form to be used for all equations,
+% \cs{eq@number} is the actual value for the current equation
+% (might be an exception to the usual sequence).
+%    \begin{macrocode}
+\def\eq@setnumber{%
+  \eq@wdNum\z@
+  \if\eq@hasNumber
+    \ifx\eq@number\@empty
+      \stepcounter{equation}\let\eq@number\theequation
+    \fi
+%  \fi
+%    \end{macrocode}
+ % This sets up numbox, etc, even if unnumbered?????
+%    \begin{macrocode}
+    \ifx\eq@number\@empty
+    \else
+%    \end{macrocode}
+% Put the number in a box so we can use its measurements in our
+% number-placement calculations.    The extra braces around
+% \cs{eqnumform} make it possible for \cs{eqnumfont} to have
+% either an \cs{itshape} (recommended) or a \cs{textit}
+% value.
+%    \begin{macrocode}
+%<trace>      \breqn@debugmsg{Number \eq@number}%
+      \set@label{equation}\eq@number
+      \global\sbox\EQ@numbox{%
+        \next@label \global\let\next@label\@empty
+        \eqnumcolor\eqnumsize\eqnumfont{\eqnumform{\eq@number}}%
+      }%
+      \global\eq@wdNum\wd\EQ@numbox\global\advance\eq@wdNum\eqnumsep
+%    \let\eq@hasNumber\@True % locally true
+    \fi
+  \fi
+}
+%    \end{macrocode}
+% \end{macro}
+%
+% \begin{macro}{\eq@finish}
+% The information available at this point from preliminary
+% measuring includes the number of lines required, the width of the
+% equation number, the total height of the equation body, and (most
+% important) the parshape spec that was used in determining height and
+% number of lines.
+%
+% Invoke the equation formatter for the requested centering/indentation
+% having worked out the best parshape.
+% BRM: This portion is extensively refactored to get common operations
+% together (so corrections get consistently applied).
+%
+% MH: I've destroyed Bruce's nice refactoring a bit to get the
+% abovedisplayskips correct for both groups of equations and single
+% \env{dmath} environments.  I will have to redo that later.
+%    \begin{macrocode}
+\newcount\eq@final@linecount
+\let\eq@GRP@first@dmath\@True
+\def\eq@finish{%
+  \begingroup
+%<trace>    \breqn@debugmsg{Formatting equation}%
+%<trace>    \debug@showmeasurements
+    \if F\eq@frame\else
+      \freeze@glue\eqlinespacing \freeze@glue\eqlineskip
+    \fi
+%    \eq@topspace{\vskip\parskip}% Set top spacing
+    \csname eq@\eqindent @setsides\endcsname % Compute \leftskip,\rightskip
+    \adjust@parshape\eq@parshape% Final adjustment of parshape for left|right skips
+%    \end{macrocode}
+% If we are in an a group of equations we don't want to calculate the
+% top space for the first one as that will be delayed until later when
+% the space for the group is calculated. However, we do need to store
+% the leftskip used here as that will be used later on for calculating
+% the top space.
+%    \begin{macrocode}
+    \if\eq@group
+      \if\eq@GRP@first@dmath
+        \global\let\eq@GRP@first@dmath\@False
+        \xdef\dmath@first@leftskip{\leftskip=\the\leftskip\relax}%
+%<trace> \breqn@debugmsg{Stored\space\dmath@first@leftskip}
+      \else
+        \eq@topspace{\vskip\parskip}% Set top spacing
+      \fi
+    \else
+      \eq@topspace{\vskip\parskip}% Set top spacing
+    \fi
+%<trace>    \debug@showformat
+%    \end{macrocode}
+% We now know the final line count of the display. If it is a
+% single-line display, we want to know as that greatly simplifies the
+% equation tag placement (until such a time where this algorithm has
+% been straightened out).
+%    \begin{macrocode}
+    \afterassignment\remove@to@nnil
+    \eq@final@linecount=\expandafter\@gobble\eq@parshape\@nnil
+%    \end{macrocode}
+% Now, invoke the appropriate typesetter according to number placement
+%    \begin{macrocode}
+    \if\eq@hasNumber
+      \if\eq@shiftnumber
+        \csname eq@typeset@\eqnumside Shifted\endcsname
+      \else
+%    \end{macrocode}
+% If there is only one line and the tag doesn't have to be shifted, we
+% call a special procedure to put the tag correctly.
+%    \begin{macrocode}
+        \ifnum\eq@final@linecount=\@ne
+          \csname eq@typeset@\eqnumside @single\endcsname
+        \else
+          \csname eq@typeset@\eqnumside\eqnumplace\endcsname
+        \fi
+      \fi
+    \else
+      \eq@typeset@Unnumbered
+    \fi
+  \endgroup
+  \eq@botspace
+}
+%    \end{macrocode}
+% \end{macro}
+%
+%
+% These are temporary until the tag position algorithm gets
+% rewritten. At least the tag is positioned correctly for single-line
+% displays. The horizontal frame position is not correct but the
+% problem lies elsewhere.
+%    \begin{macrocode}
+\def\eq@typeset@L@single{%
+  \nobreak
+  \eq@params\eq@parshape
+  \nointerlineskip\noindent
+  \add@grp@label
+  \rlap{\kern-\leftskip\box\EQ@numbox}%
+  \if F\eq@frame
+  \else
+    \rlap{\raise\eq@firstht\hbox to\z@{\eq@addframe\hss}}%
+  \fi
+  \eq@dump@box\unhbox\EQ@box \@@par
+}
+\def\eq@typeset@R@single{%
+  \nobreak
+  \eq@params\eq@parshape
+  \nointerlineskip\noindent
+  \add@grp@label
+  \if F\eq@frame
+  \else
+    \rlap{\raise\eq@firstht\hbox to\z@{\eq@addframe\hss}}%
+  \fi
+  \rlap{\kern-\leftskip\kern\linewidth\kern-\wd\EQ@numbox\copy\EQ@numbox}%
+  \eq@dump@box\unhbox\EQ@box
+  \@@par
+}
+%    \end{macrocode}
+%
+%
+%
+% \section{Special processing for end-of-equation}
+%
+% At the end of a displayed equation environment we need to peek ahead
+% for two things: following punction such as period or command that
+% should be pulled in for inclusion at the end of the equation; and
+% possibly also an \verb"\end{proof}" with an implied \dquoted{qed}
+% symbol that is traditionally included at the end of the display rather
+% than typeset on a separate line.
+% We could require that the users type \cs{qed} explicitly at the
+% end of the display when they want to have the display take notice of it.
+% But the reason for doing that would only be to save work for the
+% programmer; the most natural document markup would allow an inline
+% equation and a displayed equation at the end of a proof to differ only
+% in the environment name:
+% \begin{literalcode}
+% ... \begin{math} ... \end{math}.
+% \end{proof}
+% \end{literalcode}
+% versus
+% \begin{literalcode}
+% ...
+% \begin{dmath}
+%  ...
+% \end{dmath}.
+% \end{proof}
+% \end{literalcode}
+% .
+% The technical difficulties involved in supporting this markup within
+% \latex2e  are, admittedly, nontrivial.
+% Nonetheless, let's see how far we can go.
+%
+%
+% The variations that we will support are only the most
+% straightforward ones:
+% \begin{literalcode}
+% \end{dmath}.
+% \end{proof}
+% \end{literalcode}
+% or
+% \begin{literalcode}
+% \end{dmath}.
+% Perhaps a comment
+% \end{proof}
+% \end{literalcode}
+% .
+% If there is anything more complicated than a space after the
+% period we will not attempt to scan any further for a possible
+% \verb"\end{proof}".
+% This includes material such as:
+% \begin{literalcode}
+% \begin{figure}...\end{figure}%
+% \footnote{...}
+% \renewcommand{\foo}{...}
+% \par
+% \end{literalcode}
+% or even a blank line\mdash because in \latex  a blank line is
+% equivalent to \cs{par} and the meaning of \cs{par} is
+% \dquoted{end-paragraph}; in my opinion if explicit end-of-paragraph
+% markup is given before the end of an element, it has to be respected,
+% and the preceding paragraph has to be fully finished off before
+% proceeding further, even inside an element like \dquoted{proof} whose
+% end-element formatting requires integration with the end of the
+% paragraph text.
+% And \tex nically speaking, a \cs{par} token that comes from a
+% blank line and one that comes from the sequence of characters
+% \verb"\" \verb"p" \verb"a" \verb"r" are equally explicit.
+% I hope to add support for \cs{footnote} in the future, as it
+% seems to be a legitimate markup possibility in that context from a
+% purely logical point of view, but there are additional technical
+% complications if one wants to handle it in full generality
+% \begin{dn}
+% mjd,1999/02/08
+% \end{dn}
+% .
+%
+%
+% \begin{macro}{\peek@branch}
+% This is a generalized \dquoted{look at next token and choose some action
+% based on it} function.
+%    \begin{macrocode}
+\def\peek@branch#1#2{%
+  \let\peek@b#1\let\peek@space#2\futurelet\@let@token\peek@a
+}
+\def\peek@skipping@spaces#1{\peek@branch#1\peek@skip@space}
+\def\peek@a{%
+  \ifx\@let@token\@sptoken \expandafter\peek@space
+  \else \expandafter\peek@b\fi
+}
+\lowercase{\def\peek@skip@space} {\futurelet\@let@token\peek@a}%
+%    \end{macrocode}
+% \end{macro}
+%
+%
+% \begin{macro}{\check@punct}
+%   \changes{v0.96a}{2007/12/17}{Insert \cs{finish@end} if no special
+%     case is found.}
+% For this one we need to recognize and grab for inclusion any of the
+% following tokens: \verb",;.!?", both catcode 12 (standard \latex
+% value) and catcode 13 (as might hold when the Babel package is
+% being used).
+% We do not support a space preceding the punctuation since that would
+% be considered simply invalid markup if a display-math environment were
+% demoted to in-line math; and we want to keep their markup as parallel as
+% possible.
+% If punctuation does not follow, then the \cs{check@qed} branch
+% is not applicable.
+%    \begin{macrocode}
+\def\check@punct{\futurelet\@let@token\check@punct@a}
+\def\check@punct@a{%
+  \edef\@tempa{%
+    \ifx\@let@token\@sptoken\@nx\finish@end
+    \else\ifx\@let@token ,\@nx\check@qed
+    \else\ifx\@let@token .\@nx\check@qed
+    \else\check@punct@b % check the less common possibilities
+    \fi\fi\fi
+  }%
+  \@tempa
+}
+\begingroup
+\toks@a{%
+  \ifx\@let@token ;\@nx\check@qed
+  \else\ifx\@let@token ?\@nx\check@qed
+  \else\ifx\@let@token !\@nx\check@qed
+}
+\toks@c{\fi\fi\fi}% matching with \toks@a
+\catcode`\.=\active \catcode`\,=\active \catcode`\;=\active
+\catcode`\?=\active \catcode`\!=\active
+\toks@b{%
+  \else\ifx\@let@token ,\@nx\check@qed
+  \else\ifx\@let@token .\@nx\check@qed
+  \else\ifx\@let@token ;\@nx\check@qed
+  \else\ifx\@let@token ?\@nx\check@qed
+  \else\ifx\@let@token !\@nx\check@qed
+  \else\@nx\finish@end
+  \fi\fi\fi\fi\fi
+}
+\xdef\check@punct@b{%
+  \the\toks@a\the\toks@b\the\toks@c
+}
+\endgroup
+%    \end{macrocode}
+%
+%    \begin{macrocode}
+\let\found@punct\@empty
+\def\check@qed#1{%
+  \gdef\found@punct{#1}%
+  \peek@skipping@spaces\check@qed@a
+}
+\def\check@qed@a{%
+  \ifx\end\@let@token \@xp\check@qed@b
+  \else \@xp\finish@end
+  \fi
+}
+%    \end{macrocode}
+% For each environment ENV that takes an implied qed at the end, the
+% control sequence ENVqed must be defined; and it must include suitable
+% code to yield the desired results in a displayed equation.
+%    \begin{macrocode}
+\def\check@qed@b#1#2{%
+  \@ifundefined{#2qed}{}{%
+    \toks@\@xp{\found@punct\csname#2qed\endcsname}%
+    \xdef\found@punct{\the\toks@}%
+  }%
+  \finish@end
+  \end{#2}%
+}
+%    \end{macrocode}
+% \end{macro}
+%
+%
+% \begin{macro}{\latex@end}
+% \begin{macro}{\finish@end}
+% The lookahead for punctuation following a display requires
+% mucking about with the normal operation of \cn{end}.    Although
+% this is not exactly something to be done lightly, on the other hand this
+% whole package is so over-the-top anyway, what's a little more
+% going to hurt?    And rationalizing this aspect of
+% equation markup is a worthy cause.    Here is the usual
+% definition of \cs{end}.
+% \begin{literalcode}
+% \def\end#1{
+%   \csname end#1\endcsname \@checkend{#1}%
+%   \expandafter\endgroup\if@endpe\@doendpe\fi
+%   \if@ignore \global\@ignorefalse \ignorespaces \fi
+% }
+% \end{literalcode}
+% We can improve the chances of this code surviving through future
+% minor changes in the fundamental definition of \cs{end} by taking a
+% little care in saving the original meaning.
+%    \begin{macrocode}
+\def\@tempa#1\endcsname#2\@nil{\def\latex@end##1{#2}}
+\expandafter\@tempa\end{#1}\@nil
+\def\end#1{\csname end#1\endcsname \latex@end{#1}}%
+%    \end{macrocode}
+% Why don't we call \cs{CheckCommand} here?    Because that
+% doesn't help end users much; it works better to use it during package
+% testing by the maintainer.
+%
+%
+% If a particular environment needs to call a different end action, the
+% end command of the environment should be defined to gobble two args and
+% then call a function like \cs{check@punct@or@qed}.
+%    \begin{macrocode}
+\def\check@punct@or@qed#1{%
+  \xdef\found@punct{\@empty}% BRM: punctuation was being remembered past this eqn.
+  % WSPR: err, why isn't that just \global\let\found@punct\@empty ?
+  \def\finish@end{\csname end@#1\endcsname\latex@end{#1}}%
+  \check@punct
+}
+%    \end{macrocode}
+% \end{macro}
+% \end{macro}
+%
+% \begin{macro}{\eqpunct}
+% User-settable function for handling
+% the punctuation at the end of an equation.    You could, for example,
+% define it to just discard the punctuation.
+%    \begin{macrocode}
+\newcommand\eqpunct[1]{\thinspace#1}
+%    \end{macrocode}
+% \end{macro}
+%
+% \begin{macro}{\set@label}
+% \cs{set@label} just sets \cs{@currentlabel} but it
+% takes the counter as an argument, in the hope that \latex  will some
+% day provide an improved labeling system that includes type info on the
+% labels.
+%    \begin{macrocode}
+\providecommand\set@label[2]{\protected@edef\@currentlabel{#2}}
+%    \end{macrocode}
+% \end{macro}
+%
+% \begin{macro}{\eq@topspace}
+% \begin{macro}{\eq@botspace}
+% The action of \cs{eq@topspace} is complicated by the
+% need to test whether the \quoted{short} versions of the display skips
+% should be used.    This can be done only after the final parshape
+% and indent have been determined, so the calls of this function are
+% buried relatively deeply in the code by comparison to the calls of
+% \cs{eq@botspace}.    This also allows us to optimize
+% slightly by setting the above-skip with \cs{parskip} instead of
+% \cs{vskip}.    \verb|#1|  is either \cs{noindent} or
+% \verb"\vskip\parskip".
+%
+% BRM: Hmm; we need to do *@setspace BEFORE this for small skips to work!
+%    \begin{macrocode}
+\def\eq@topspace#1{%
+  \begingroup
+    \global\let\EQ@shortskips\@False
+%    \end{macrocode}
+% If we are in \env{dgroup} or \env{dgroup*} and not before the top
+% one, we just insert \cs{intereqskip}. Otherwise we must check for
+% shortskip.
+%    \begin{macrocode}
+    \if\@And{\eq@group}{\@Not\eq@GRP@first@dmath}%
+%<trace>\breqn@debugmsg{Between lines}%
+        \parskip\intereqskip \penalty\intereqpenalty
+%<trace>\breqn@debugmsg{parskip=\the\parskip}%
+    \else
+      \eq@check@shortskip
+      \if\EQ@shortskips
+        \parskip\abovedisplayshortskip
+        \aftergroup\belowdisplayskip\aftergroup\belowdisplayshortskip
+%    \end{macrocode}
+% BRM: Not exactly \TeX's approach, but seems right\ldots
+%    \begin{macrocode}
+        \ifdim\predisplaysize>\z@\nointerlineskip\fi
+      \else
+        \parskip\abovedisplayskip
+      \fi
+    \fi
+    \if F\eq@frame
+    \else
+      \addtolength\parskip{\eq@framesep+\eq@framewd}%
+    \fi
+%<*trace>
+    \breqn@debugmsg{Topspace: \theb@@le\EQ@shortskips, \parskip=\the\parskip,
+      \predisplaysize=\the\predisplaysize}%
+%</trace>
+    #1%
+  \endgroup
+}
+%    \end{macrocode}
+% \begin{macro}{\eq@check@shortskip}
+%   \changes{v0.96a}{2007/12/17}{Insert \cs{finish@end} if no special
+%     case is found.}
+%   \changes{v0.97a}{2007/12/22}{Use design parameter and fix
+%   shortskips properly.}
+%    \begin{macrocode}
+\def\eq@check@shortskip {%
+  \global\let\EQ@shortskips\@False
+  \setlength\dim@a{\abovedisplayskip+\ht\EQ@numbox}%
+%    \end{macrocode}
+% Here we work around the hardwired standard TeX value and use the
+% designer parameter instead.
+%    \begin{macrocode}
+  \ifdim\leftskip<\predisplaysize
+  \else
+%    \end{macrocode}
+% If the display was preceeded by a blank line, \cs{predisplaysize} is
+% $-\cs{maxdimen}$ and so we should insert a fairly large skip to
+% separate paragraphs, i.e., no short skip. Perhaps this should be a
+% third parameter \cs{abovedisplayparskip}.
+%    \begin{macrocode}
+    \ifdim -\maxdimen=\predisplaysize
+    \else
+      \if R\eqnumside
+        \global\let\EQ@shortskips\@True
+      \else
+        \if\eq@shiftnumber
+        \else
+          \if T\eqnumplace
+            \ifdim\dim@a<\eq@firstht
+              \global\let\EQ@shortskips\@True
+            \fi
+          \else
+            \setlength\dim@b{\eq@vspan/2}%
+            \ifdim\dim@a<\dim@b
+              \global\let\EQ@shortskips\@True
+            \fi
+          \fi
+        \fi
+      \fi
+    \fi
+  \fi
+}
+%    \end{macrocode}
+% \end{macro}
+%
+% At the end of an equation, need to put in a pagebreak penalty
+% and some vertical space.    Also set some flags to remove parindent
+% and extra word space if the current paragraph text continues without an
+% intervening \cs{par}.
+%    \begin{macrocode}
+\def\eq@botspace{%
+  \penalty\postdisplaypenalty
+%    \end{macrocode}
+% Earlier calculations will have set \cs{belowdisplayskip} locally
+% to \cs{belowdisplayshortskip} if applicable.    So we can just use
+% it here.
+%    \begin{macrocode}
+  \if F\eq@frame
+  \else
+    \addtolength\belowdisplayskip{\eq@framesep+\eq@framewd}%
+  \fi
+  \vskip\belowdisplayskip
+  \@endpetrue % kill parindent if current paragraph continues
+  \global\@ignoretrue % ignore following spaces
+  \eq@resume@parshape
+}
+%    \end{macrocode}
+% \end{macro}
+% \end{macro}
+%
+% \begin{macro}{\eq@resume@parshape}
+% This should calculate the total height of the equation,
+% including space above and below, and set prevgraf to the number it would
+% be if that height were taken up by normally-spaced normal-height
+% lines.    We also need to restore parshape if it had a non-null
+% value before the equation.    Not implemented yet.
+%    \begin{macrocode}
+\def\eq@resume@parshape{}
+%    \end{macrocode}
+% \end{macro}
+%
+% \section{Preprocessing the equation body}
+% \begin{macro}{\eq@startup}
+% Here is the function that initially collects the equation
+% material in a box.
+%
+%    \begin{macrocode}
+\def\eq@startup{%
+  \global\let\EQ@hasLHS\@False
+  \setbox\z@\vbox\bgroup
+    \noindent \@@math \displaystyle
+    \penalty-\@Mi
+}
+%    \end{macrocode}
+%
+% This setup defines the environment for the first typesetting
+% pass, note the \cs{hsize} value for example.
+%    \begin{macrocode}
+\def\eq@setup@a{%
+  \everymath\everydisplay
+  %\let\@newline\eq@newline % future possibility?
+  \let\\\eq@newline
+  \let\insert\eq@insert \let\mark\eq@mark \let\vadjust\eq@vadjust
+  \hsize\maxdimen \pretolerance\@M
+%    \end{macrocode}
+% Here it is better not to use \cs{@flushglue} (0pt
+% plus1fil) for \cs{rightskip}, or else a negative penalty
+% (such as $-99$ for \cs{prerelpenalty}) will tempt
+% \tex  to use more line breaks than necessary in the first typesetting
+% pass.    Ideal values for \cs{rightskip} and
+% \cs{linepenalty} are unclear to me, but they are rather sensitively
+% interdependent.    Choice of 10000 pt for rightskip is derived by
+% saying, let's use a value smaller than 1 fil and smaller than
+% \cs{hsize}, but more than half of \cs{hsize} so that if a line
+% is nearly empty, the glue stretch factor will always be less than 2.0
+% and so the badness will be less than 100 and so \tex  will not issue
+% badness warnings.
+%    \begin{macrocode}
+  \linepenalty\@m
+  \rightskip\z@\@plus\@M\p@ \leftskip\z@skip \parfillskip\z@skip
+  \clubpenalty\@ne \widowpenalty\z@ \interlinepenalty\z@
+%    \end{macrocode}
+% After a relation symbol is discovered, binop symbols should start
+% including a special offset space.
+% But until then \cs{EQ@prebin@space} is a no-op.
+%    \begin{macrocode}
+  \global\let\EQ@prebin@space\relax
+%    \end{macrocode}
+% Set binoppenalty and relpenalty high to prohibit line breaks
+% after mathbins and mathrels.    As a matter of fact, the penalties are
+% then omitted by \tex , since bare glue without a penalty is
+% \emph{not} a valid breakpoint if it occurs within
+% mathon\ndash mathoff items.
+%    \begin{macrocode}
+  \binoppenalty\@M \relpenalty\@M
+}
+%    \end{macrocode}
+% \end{macro}
+%
+%
+% \begin{figure}
+%   \centering
+% The contents of an equation after the initial typesetting pass,
+% as shown by \cs{showlists}.    This is the material on which the
+% \cs{eq@repack} function operates.    The equation was
+% \begin{literalcode}
+% a=b +\left(\frac{c\sp 2}{2} -d\right) +(e -f) +g
+% \end{literalcode}
+% .    The contents are shown in four parts in this figure and the next
+% three.    The first part contains two line boxes, one for the mathon
+% node and one for the LHS.
+% \begin{literalcode}
+% \hbox(0.0+0.0)x16383.99998, glue set 1.6384
+% .\mathon
+% .\penalty -10000
+% .\glue(\rightskip) 0.0 plus 10000.0
+% \penalty 1
+% \glue(\baselineskip) 7.69446
+% \hbox(4.30554+0.0)x16383.99998, glue set 1.63759
+% .\OML/cmm/m/it/10 a
+% .\glue 2.77771 minus 1.11108
+% .\penalty -10001
+% .\glue(\rightskip) 0.0 plus 10000.0
+% \penalty 2
+% \glue(\lineskip) 1.0
+% ...
+% \end{literalcode}
+%   \caption{Preliminary equation contents, part 1}
+% \end{figure}
+% \begin{figure}\centering
+% This is the first part of the RHS, up to the
+% \cs{right}, where a line break has been forced so that we can break
+% open the left-right box.
+% \begin{literalcode}
+% ...
+% \penalty 2
+% \glue(\lineskip) 1.0
+% \hbox(14.9051+9.50012)x16383.99998, glue set 1.63107
+% .\penalty -99
+% .\glue(\thickmuskip) 2.77771 minus 1.11108
+% .\OT1/cmr/m/n/10 =
+% .\glue(\thickmuskip) 2.77771 minus 1.11108
+% .\OML/cmm/m/it/10 b
+% .\penalty 888
+% .\glue -10.5553
+% .\rule(*+*)x0.0
+% .\penalty 10000
+% .\glue 10.5553
+% .\glue(\medmuskip) 2.22217 minus 1.66663
+% .\OT1/cmr/m/n/10 +
+% .\glue(\medmuskip) 2.22217 minus 1.66663
+% .\hbox(14.9051+9.50012)x43.36298
+% ..\hbox(0.39998+23.60025)x7.36115, shifted -14.10013
+% ...\OMX/cmex/m/n/5 \hat \hat R
+% ..\hbox(14.9051+6.85951)x11.21368
+% ...\hbox(14.9051+6.85951)x11.21368
+%   ... [fraction contents, elided]
+% ..\penalty 5332
+% ..\glue -10.5553
+% ..\rule(*+*)x0.0
+% ..\penalty 10000
+% ..\glue 10.5553
+% ..\glue(\medmuskip) 2.22217 minus 1.66663
+% ..\OMS/cmsy/m/n/10 \hat \hat @
+% ..\glue(\medmuskip) 2.22217 minus 1.66663
+% ..\OML/cmm/m/it/10 d
+% ..\hbox(0.39998+23.60025)x7.36115, shifted -14.10013
+% ...\OMX/cmex/m/n/5 \hat \hat S
+% .\penalty -10000
+% .\glue(\rightskip) 0.0 plus 10000.0
+% \penalty 3
+% \glue(\lineskip) 1.0
+% ...
+% \end{literalcode}
+% \caption{Preliminary equation contents, part 2}
+% \end{figure}
+%
+% \begin{figure}
+%   \centering
+% This is the remainder of the RHS after the post-\cs{right}
+% split.
+% \begin{literalcode}
+% ...
+% \penalty 3
+% \glue(\lineskip) 1.0
+% \hbox(7.5+2.5)x16383.99998, glue set 1.63239
+% .\penalty 888
+% .\glue -10.5553
+% .\rule(*+*)x0.0
+% .\penalty 10000
+% .\glue 10.5553
+% .\glue(\medmuskip) 2.22217 minus 1.66663
+% .\OT1/cmr/m/n/10 +
+% .\glue(\medmuskip) 2.22217 minus 1.66663
+% .\OT1/cmr/m/n/10 (
+% .\OML/cmm/m/it/10 e
+% .\penalty 5332
+% .\glue -10.5553
+% .\rule(*+*)x0.0
+% .\penalty 10000
+% .\glue 10.5553
+% .\glue(\medmuskip) 2.22217 minus 1.66663
+% .\OMS/cmsy/m/n/10 \hat \hat @
+% .\glue(\medmuskip) 2.22217 minus 1.66663
+% .\OML/cmm/m/it/10 f
+% .\kern1.0764
+% .\OT1/cmr/m/n/10 )
+% .\penalty 888
+% .\glue -10.5553
+% .\rule(*+*)x0.0
+% .\penalty 10000
+% .\glue 10.5553
+% .\glue(\medmuskip) 2.22217 minus 1.66663
+% .\OT1/cmr/m/n/10 +
+% .\glue(\medmuskip) 2.22217 minus 1.66663
+% .\OML/cmm/m/it/10 g
+% .\kern0.35878
+% .\penalty -10000
+% .\glue(\rightskip) 0.0 plus 10000.0
+% \glue(\baselineskip) 9.5
+% ...
+% \end{literalcode}
+% \caption{Preliminary equation contents, part 3}
+% \end{figure}
+%
+% \begin{figure}
+%   \centering
+% This is the mathoff fragment.
+% \begin{literalcode}
+% ...
+% \glue(\baselineskip) 9.5
+% \hbox(0.0+0.0)x16383.99998, glue set 1.6384
+% .\mathoff
+% .\penalty 10000
+% .\glue(\parfillskip) 0.0
+% .\glue(\rightskip) 0.0 plus 10000.0
+% \end{literalcode}
+% \caption{Preliminary equation contents, part 4}
+% \end{figure}
+%
+% \begin{macro}{\eq@capture}
+% \begin{macro}{\eq@punct}
+% If an equation ends with a \cs{right} delim, the last thing
+% on the math list will be a force-break penalty.    Then don't
+% redundantly add another forcing penalty.    (question: when does a
+% penalty after a linebreak not disappear?    Answer: when you have
+% two forced break penalties in a row).    Ending punctuation, if
+% any, goes into the last box with the mathoff kern.    If the math list
+% ends with a slanted letter, then there will be an italic correction
+% added after it by \tex .    Should we remove it?    I guess
+% so.
+%
+%
+% \subsection{Capturing the equation}
+%
+% BRM: There's a problem here (or with \cs{ss@scan}).  If the LHS has
+% \cs{left} \cs{right} pairs, \ss@scan gets involved.  It seems to produce
+% a separate box marked w/\cs{penalty} 3.  But it appears that \cs{eq@repack}
+% is only expecting a single box for the LHS; when it measures that
+% box it's missing the (typically larger) bracketted section,
+% so the LHS is measured => 0pt (or very small).
+%  I'm not entirely clear what Michael had in mind for this case;
+% whether it's an oversight, or whether I've introduced some other bug.
+% At any rate, my solution is to measure the RHS (accumulated in \cs{EQ@box}),
+% at the time of the relation, and subtract that from the total size.
+%    \begin{macrocode}
+\newdimen\eq@wdR\eq@wdR\z@%BRM
+\def\eq@capture{%
+  \ifnum\lastpenalty>-\@M \penalty-\@Mi \fi
+%    \end{macrocode}
+% We want to keep the mathoff kern from vanishing at the line break,
+% so that we can reuse it later.
+%    \begin{macrocode}
+  \keep@glue\@@endmath
+  \eq@addpunct
+  \@@par
+  \eq@wdL\z@
+%    \end{macrocode}
+% First snip the last box, which contains the mathoff node, and put it
+% into \cs{EQ@box}.    Then when we call \cs{eq@repack} it
+% will recurse properly.
+%    \begin{macrocode}
+  \setbox\tw@\lastbox
+  \global\setbox\EQ@box\hbox{\unhbox\tw@\unskip\unskip\unpenalty}%
+  \unskip\unpenalty
+  \global\setbox\EQ@copy\copy\EQ@box
+%%  \global\setbox\EQ@vimcopy\copy\EQ@vimbox
+  \clubpenalty\z@
+%\batchmode\showboxbreadth\maxdimen\showboxdepth99\showlists\errorstopmode
+  \eq@wdR\z@%BRM: eq@wdL patch
+  \eq@repack % recursive
+%    \end{macrocode}
+% Finally, add the mathon item to \cs{EQ@box} and \cs{EQ@copy}.
+%    \begin{macrocode}
+  \setbox\tw@\lastbox
+  \global\setbox\EQ@box\hbox{\unhcopy\tw@\unskip\unpenalty \unhbox\EQ@box}%
+  \global\setbox\EQ@copy\hbox{\unhbox\tw@\unskip\unpenalty \unhbox\EQ@copy}%
+%\batchmode\showbox\EQ@copy \showthe\eq@wdL\errorstopmode
+  \ifdim\eq@wdR>\z@% BRM:  eq@wdL patch
+    \setlength\dim@a{\wd\EQ@box-\eq@wdR
+    % Apparently missing a \thickmuskip = 5mu = 5/18em=0.27777777777.. ?
+       + 0.2777777777777em}% FUDGE??!?!?!
+    \ifdim\dim@a>\eq@wdL
+%<*trace>
+      \breqn@debugmsg{Correcting LHS from \the\eq@wdL\space to
+                \the\dim@a = \the\wd\EQ@box - \the\eq@wdR}%
+%</trace>
+      \eq@wdL\dim@a
+      \xdef\EQ@setwdL{\eq@wdL\the\eq@wdL\relax}%
+    \fi
+  \fi
+%<*trace>
+  \breqn@debugmsg{Capture: total length=\the\wd\EQ@box \MessageBreak
+           ==== has LHS=\theb@@le\EQ@hasLHS, \eq@wdL=\the\eq@wdL, \eq@wdR=\the\eq@wdR,
+           \MessageBreak
+           ==== \eq@wdCond=\the\eq@wdCond}%
+%</trace>
+  \egroup % end vbox started earlier
+%<*trace>
+%\debugwr{EQ@box}\debug@box\EQ@box
+%\debugwr{EQ@copy}\debug@box\EQ@copy
+%</trace>
+}
+%    \end{macrocode}
+% Now we have two copies of the equation, one in \cs{EQ@box},
+% and one in \cs{EQ@copy} with inconvenient stuff like inserts and
+% marks omitted.
+%
+% \cs{eq@addpunct} is for tacking on text punctuation at the end
+% of a display, if any was captured by the \quoted{gp} lookahead.
+%    \begin{macrocode}
+\def\eq@addpunct{%
+  \ifx\found@punct\@empty
+  \else \eqpunct{\found@punct}%
+  \fi
+  % BRM: Added; the punctuation kept  getting carried to following environs
+  \xdef\found@punct{\@empty}%
+  \EQ@afterspace
+}
+%    \end{macrocode}
+% Needed for the \env{dseries} environment, among other things.
+%    \begin{macrocode}
+\global\let\EQ@afterspace\@empty
+%    \end{macrocode}
+% \end{macro}
+% \end{macro}
+%
+% \begin{macro}{\eq@repack}
+% The \cs{eq@repack} function looks at the information at hand
+% and proceeds accordingly.
+%
+% TeX Note: this scans BACKWARDS from the end of the math.
+%    \begin{macrocode}
+\def\eq@repack{%
+% A previous penalty of 3 on the vertical list means that we need
+% to break open a left-right box.
+%    \begin{macrocode}
+  \ifcase\lastpenalty
+     % case 0: normal case
+    \setbox\tw@\lastbox
+    \eq@repacka\EQ@copy \eq@repacka\EQ@box
+    \unskip
+  \or % case 1: finished recursing
+%    \end{macrocode}
+% Grab the mathon object since we need it to inhibit line breaking at
+% bare glue nodes later.
+%    \begin{macrocode}
+    \unpenalty
+    \setbox\tw@\lastbox
+    \eq@repacka\EQ@copy \eq@repacka\EQ@box
+    \@xp\@gobble
+  \or % case 2: save box width = LHS width
+%    \end{macrocode}
+% Don't need to set \cs{EQ@hasLHS} here because it was set earlier
+% if applicable.
+%    \begin{macrocode}
+    \unpenalty
+    \setbox\tw@\lastbox
+    \setbox\z@\copy\tw@ \setbox\z@\hbox{\unhbox\z@\unskip\unpenalty}%
+    \addtolength\eq@wdL{\wd\z@}
+    \setlength\eq@wdR{\wd\EQ@box}% BRM:  eq@wdL patch
+    \xdef\EQ@setwdL{\eq@wdL\the\eq@wdL\relax}%
+%    \end{macrocode}
+% At this point, box 2 typically ends with
+% \begin{literalcode}
+% .\mi10 a
+% .\glue 2.77771 plus 2.77771
+% .\penalty -10001
+% .\glue(\rightskip) 0.0 plus 10000.0
+% \end{literalcode}
+% and we want to ensure that the thickmuskip glue gets removed.
+% And we now arrange for \cs{EQ@copy} and \cs{EQ@box} to
+% keep the LHS in a separate subbox; this is so that we can introduce a
+% different penalty before the first relation symbol if necessary,
+% depending on the layout decisions that are made later.
+%    \begin{macrocode}
+    \global\setbox\EQ@copy\hbox{%
+      \hbox{\unhcopy\tw@\unskip\unpenalty\unskip}%
+      \box\EQ@copy
+    }%
+    \global\setbox\EQ@box\hbox{%
+      \hbox{\unhbox\tw@\unskip\unpenalty\unskip}%
+      \box\EQ@box
+    }%
+    \unskip
+  \or % case 3: unpack left-right box
+    \unpenalty
+    \eq@lrunpack
+  \else
+    \breqn@repack@err
+  \fi
+  \eq@repack % RECURSE
+}
+%    \end{macrocode}
+% Error message extracted to streamline calling function.
+%    \begin{macrocode}
+\def\breqn@repack@err{%
+  \PackageError{breqn}{eq@repack penalty neq 0,1,2,3}\relax
+}
+%    \end{macrocode}
+% \end{macro}
+%
+%
+% \begin{macro}{\eq@repacka}
+% We need to transfer each line into two separate boxes, one
+% containing everything and one that omits stuff like \cs{insert}s
+% that would interfere with measuring.
+%    \begin{macrocode}
+\def\eq@repacka#1{%
+  \global\setbox#1\hbox{\unhcopy\tw@ \unskip
+    \count@-\lastpenalty
+    \ifnum\count@<\@M \else \advance\count@-\@M \fi
+    \unpenalty
+%    \end{macrocode}
+% If creating the measure copy, ignore all cases above case 3 by
+% folding them into case 1.
+%    \begin{macrocode}
+    \ifx\EQ@copy#1\ifnum\count@>\thr@@ \count@\@ne\fi\fi
+    \ifcase\count@
+        % case 0, normal line break
+      \penalty-\@M % put back the linebreak penalty
+    \or % case 1, do nothing (end of equation)
+      \relax
+    \or % case 2, no-op (obsolete case)
+    \or % case 3, transfer vspace and/or penalty
+      \ifx#1\EQ@box \eq@revspace \else \eq@revspaceb \fi
+    \or % case 4, put back an insert
+      \eq@reinsert
+    \or % case 5, put back a mark
+      \eq@remark
+    \or % case 6, put back a vadjust
+      \eq@readjust
+    \else % some other break penalty
+      \penalty-\count@
+    \fi
+    \unhbox#1}%
+}
+%    \end{macrocode}
+% \end{macro}
+%
+%
+% \begin{macro}{\eq@nulldisplay}
+% Throw in a null display in order to get predisplaysize \etc .
+% My original approach here was to start the null display, then measure
+% the equation, and set a phantom of the equation's first line before
+% ending the null display.    That would allow finding out if \tex  used
+% the short displayskips instead of the normal ones.    But because of
+% some complications with grouping and the desirability of omitting
+% unnecessary invisible material on the vertical list, it seems better to
+% just collect information about the display (getting \cs{prevdepth}
+% requires \cs{halign}) and manually perform our own version of
+% \TeX's shortskip calculations.    This approach also gives greater
+% control, \eg , the threshold amount of horizontal space between
+% predisplaysize and the equation's left edge that determines when the
+% short skips kick in becomes a designer-settable parameter rather than
+% hardwired into \TeX .
+%    \begin{macrocode}
+\def\eq@nulldisplay{%
+  \begingroup \frozen@everydisplay\@emptytoks
+  \@@display
+  \predisplaypenalty\@M \postdisplaypenalty\@M
+  \abovedisplayskip\z@skip \abovedisplayshortskip\z@skip
+  \belowdisplayskip\z@skip \belowdisplayshortskip\z@skip
+  \xdef\EQ@displayinfo{%
+    \prevgraf\the\prevgraf \predisplaysize\the\predisplaysize
+    \displaywidth\the\displaywidth \displayindent\the\displayindent
+    \listwidth\the\linewidth
+%    \end{macrocode}
+% Not sure how best to test whether leftmargin should be
+% added.    Let's do this for now [mjd,1997/10/08].
+%    \begin{macrocode}
+    \ifdim\displayindent>\z@
+      \advance\listwidth\the\leftmargin
+      \advance\listwidth\the\rightmargin
+    \fi
+    \relax}%
+%    \end{macrocode}
+% An \cs{halign} containing only one \cs{cr} (for the
+% preamble) puts no box on the vertical list, which means that no
+% \cs{baselineskip} will be added (so we didn't need to set it to
+% zero) and the previous value of prevdepth carries through.    Those
+% properties do not hold for an empty simple equation without
+% \cs{halign}.
+%    \begin{macrocode}
+  \halign{##\cr}%
+  \@@enddisplay
+  \par
+  \endgroup
+  \EQ@displayinfo
+}
+%    \end{macrocode}
+% \end{macro}
+%
+%
+% \begin{macro}{\eq@newline}
+% \begin{macro}{\eq@newlinea}
+% \begin{macro}{\eq@newlineb}
+% Here we use \cs{@ifnext} so that in a sequence like
+% \begin{literalcode}
+% ...\\
+% [a,b]
+% \end{literalcode}
+% \latex  does not attempt to interpret the \verb"[a,b]" as a
+% vertical space amount.    We would have used \cs{eq@break} in the
+% definition of \cs{eq@newlineb} except that it puts in a
+% \cs{keep@glue} object which is not such a good idea if a mathbin
+% symbol follows \mdash  the indent of the mathbin will be wrong because
+% the leading negative glue will not disappear as it should at the line
+% break.
+%    \begin{macrocode}
+\def\eq@newline{%
+  \@ifstar{\eq@newlinea\@M}{\eq@newlinea\eqinterlinepenalty}}
+\def\eq@newlinea#1{%
+  \@ifnext[{\eq@newlineb{#1}}{\eq@newlineb{#1}[\maxdimen]}}
+\def\eq@newlineb#1[#2]{\penalty-\@M}
+%    \end{macrocode}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+%
+%
+% \begin{macro}{\eq@revspace}
+% \begin{macro}{\eq@revspaceb}
+% When \cs{eq@revspace} (re-vspace) is called, we are the
+% end of an equation line; we need to remove the existing penalty of
+% $-10002$ in order to put a vadjust object in front of it, then put
+% back the penalty so that the line break will still take place in the
+% final result.
+%    \begin{macrocode}
+\def\eq@revspace{%
+  \global\setbox\EQ@vimbox\vbox{\unvbox\EQ@vimbox
+    \unpenalty
+    \global\setbox\@ne\lastbox}%
+  \@@vadjust{\unvbox\@ne}%
+  \penalty-\@M
+}
+%    \end{macrocode}
+% The b version is used for the \cs{EQ@copy} box.
+%    \begin{macrocode}
+\def\eq@revspaceb{%
+  \global\setbox\EQ@vimcopy\vbox{\unvbox\EQ@vimcopy
+    \unpenalty
+    \global\setbox\@ne\lastbox}%
+  \@@vadjust{\unvbox\@ne}%
+  \penalty-\@M
+}
+%    \end{macrocode}
+% \end{macro}
+% \end{macro}
+%
+%
+% \begin{macro}{\eq@break}
+% The function \cs{eq@break} does a preliminary linebreak with
+% a flag penalty.
+%    \begin{macrocode}
+\def\eq@break#1{\penalty-1000#1 \keep@glue}
+%    \end{macrocode}
+% \end{macro}
+%
+%
+%
+%
+% \section{Choosing optimal line breaks}
+% The question of what line width to use when breaking an
+% equation into several lines is best examined in the light of an extreme
+% example.    Suppose we have a two-column layout and a displayed
+% equation falls inside a second-level list with nonzero leftmargin and
+% rightmargin.    Then we want to try in succession a number of
+% different possibilities.    In each case if the next possibility is
+% no wider than the previous one, skip ahead to the one after.
+% \begin{enumerate}
+% \item First try linewidth(2), the linewidth for the current
+% level-2 list.
+%
+%
+% \item If we cannot find adequate linebreaks at that width, next try
+% listwidth(2), the sum of leftmargin, linewidth, and rightmargin for
+% the current list.
+%
+%
+% \item If we cannot find linebreaks at that width, next try linewidth
+% (1) (skipping this step if it is no larger then
+% listwidth(2)).
+%
+%
+% \item If we cannot find linebreaks at that width, next try
+% listwidth(1).
+%
+%
+% \item If we cannot find linebreaks at that width, next try column
+% width.
+%
+%
+% \item If we cannot find linebreaks at that width, next try text
+% width.
+%
+%
+% \item If we cannot find linebreaks at that width, next try equation
+% width, if it exceeds text width (\ie , if the style allows equations
+% to extend into the margins).
+%
+%
+% \end{enumerate}
+%
+%
+% \begin{figure}
+%   \centering
+%   needs work
+% \caption{first-approximation parshape for equations}\label{f:parshape-1}
+% \end{figure}
+%
+% At any given line width, we run through a series of parshape
+% trials and, essentially, use the first one that gives decent line
+% breaks.
+% But the process is a bit more complicated in fact.
+% In order to do a really good job of setting up the parshapes, we
+% need to know how many lines the equation will require.
+% And of course the number of lines needed depends on the parshape!
+% So as our very first trial we run a simple first-approximation
+% parshape (Figure~\vref{f:parshape-1}) whose
+% main purpose is to get an estimate on the number of lines that will be
+% needed; it chooses a uniform indent for all lines after the first one
+% and does not take any account of the equation number.
+% A substantial majority of equations only require one line anyway,
+% and for them this first trial will succeed.
+% In the one-line case if there is an equation number and it doesn't
+% fit on the same line as the equation body, we don't go on to other
+% trials because breaking up the equation body will not gain us
+% anything\mdash we know that we'll have to use two lines in any case
+% \mdash  so we might as well keep the equation body together on one line
+% and shift the number to a separate line.
+%
+% If we learn from the first trial that the equation body
+% requires more than one line, the next parshape trial involves adjusting
+% the previous parshape to leave room for the equation number, if
+% present.    If no number is present, again no further trials are
+% needed.
+%
+% Some remarks about parshape handling.    The \tex
+% primitive doesn't store the line specs anywhere, \verb"\the\parshape"
+% only returns the number of line specs.    This makes it well nigh
+% impossible for different packages that use \cs{parshape} to work
+% together.    Not that it would be terribly easy for the package
+% authors to make inter-package collaboration work, if it were
+% possible.    If we optimistically conjecture that
+% someone some day may take on such a task, then the thing to do,
+% obviously, is provide a parshape interface that includes a record of all
+% the line specs.    For that we designate a macro \cs{@parshape}
+% which includes not only the line specs, but also the line count and even
+% the leading \cs{parshape} token.
+% This allows it to be directly executed without an auxiliary if-empty
+% test.    It should include a trailing \cs{relax} when it has a
+% nonempty value.
+%    \begin{macrocode}
+\let\@parshape\@empty
+%    \end{macrocode}
+%
+%
+% The function \cs{eq@measure} runs line-breaking trials
+% on the copy of the equation body that is stored in the box register
+% \cs{EQ@copy}, trying various possible layouts in order of
+% preference until we get successful line breaks, where \quoted{successful}
+% means there were no overfull lines.    The result of the trials is,
+% first, a parshape spec that can be used for typesetting the real
+% equation body in \cs{EQ@box}, and second, some information that
+% depends on the line breaks such as the depth of the last line, the
+% height of the first line, and positioning information for the equation
+% number.   The two main variables in the equation layout are the line
+% width and the placement of the equation number, if one is present.
+%
+%
+% \begin{macro}{\eq@measure}
+% Run linebreak trials on the equation contents and measure the
+% results.
+%    \begin{macrocode}
+\def\eq@measure{%
+%    \end{macrocode}
+% If an override value is given for indentstep in the env options, use
+% it.
+%    \begin{macrocode}
+  \ifdim\eq@indentstep=\maxdimen \eq@indentstep\eqindentstep \fi
+%    \end{macrocode}
+% If \cs{eq@linewidth} is nonzero at this point, it means that
+% the user specified a particular target width for this equation.
+% In that case we override the normal list of trial widths.
+%    \begin{macrocode}
+  \ifdim\eq@linewidth=\z@ \else \edef\eq@linewidths{{\the\eq@linewidth}}\fi
+  \begingroup \eq@params
+  \leftskip\z@skip
+%    \end{macrocode}
+% Even if \cs{hfuzz} is greater than zero a box whose contents
+% exceed the target width by less then hfuzz still has a reported badness
+% value of 1000000 (infinitely bad).    Because we use inf-bad
+% to test whether a particular trial succeeds or fails, we want to make
+% such boxes return a smaller badness.    To this end we include an
+% \cs{hfuzz} allowance in \cs{rightskip}.    In fact,
+% \cs{eq@params} ensures that \cs{hfuzz} for equations is at
+% least 1pt.
+%    \begin{macrocode}
+  \rightskip\z@\@plus\columnwidth\@minus\hfuzz
+%  \eqinfo
+  \global\EQ@continue{\eq@trial}%
+  \eq@trial % uses \eq@linewidths
+  \eq@failout % will be a no-op if the trial succeeded
+  \endgroup
+%    \end{macrocode}
+% \quoted{local} parameter settings are passed outside the endgroup through
+% \cs{EQ@trial}.
+%    \begin{macrocode}
+  \EQ@trial
+}
+%    \end{macrocode}
+% \end{macro}
+%    \begin{macrocode}
+%<*trace>
+\def\debug@showmeasurements{%
+  \breqn@debugmsg{=> \number\eq@lines\space lines}%
+  \begingroup
+  \def\@elt##1X##2{\MessageBreak==== \space\space##1/##2}%
+  \let\@endelt\@empty
+  \breqn@debugmsg{=> trial info:\eq@measurements}%
+  \breqn@debugmsg{=> bounding box: \the\eq@wdT x\the\eq@vspan, badness=\the\eq@badness}%
+  \let\@elt\relax \let\@endelt\relax
+  \endgroup
+}
+\def\debug@showmeasurements{%
+  \begingroup
+  \def\@elt##1X##2{\MessageBreak====   ##1/##2}%
+  \let\@endelt\@empty
+  \breqn@debugmsg{===> Measurements: \number\eq@lines\space lines
+           \eq@measurements
+           \MessageBreak
+           ==== bounding box: \the\eq@wdT x\the\eq@vspan, badness=\the\eq@badness
+           \MessageBreak
+           ==== \leftskip=\the\leftskip, \rightskip=\the\rightskip}%
+ \endgroup
+}
+%</trace>
+%    \end{macrocode}
+%
+% Layout Trials Driver
+% Basically, trying different sequences of parshapes.
+%
+%
+% \begin{macro}{\EQ@trial}
+% Init.
+%    \begin{macrocode}
+\let\EQ@trial\@empty
+%    \end{macrocode}
+% \end{macro}
+%
+%
+% \begin{macro}{\EQ@continue}
+% This is a token register used to carry trial info past a
+% group boundary with only one global assignment.
+%    \begin{macrocode}
+\newtoks\EQ@continue
+%    \end{macrocode}
+% \end{macro}
+%
+%
+% \begin{macro}{\EQ@widths}
+% This is used for storing the actual line-width info of the equation
+% contents after breaking.
+%    \begin{macrocode}
+\let\EQ@widths\@empty
+%    \end{macrocode}
+% \end{macro}
+% \begin{macro}{\EQ@fallback}
+%    \begin{macrocode}
+\let\EQ@fallback\@empty
+%    \end{macrocode}
+% \end{macro}
+% \begin{macro}{\eq@linewidths}
+% This is the list of target widths for line breaking.
+%
+%========================================
+% BRM: Odd; I don't think I've seen this use anything but \cs{displaywidth}...
+%    \begin{macrocode}
+\def\eq@linewidths{\displaywidth\linewidth\columnwidth}
+%    \end{macrocode}
+% \end{macro}
+%
+%
+% \begin{macro}{\eq@trial}
+% The \cs{eq@trial} function tries each candidate
+% line width in \cs{eq@linewidths} until an equation layout is found
+% that yields satisfactory line breaks.
+%    \begin{macrocode}
+\def\eq@trial{%
+  \ifx\@empty\eq@linewidths
+    \global\EQ@continue{}%
+  \else
+    \iffalse{\fi \@xp\eq@trial@a \eq@linewidths}%
+  \fi
+  \the\EQ@continue
+}
+%    \end{macrocode}
+% \end{macro}
+%
+%
+% \begin{macro}{\eq@trial@a}
+% The \cs{eq@trial@a} function reads the leading line
+% width from \cs{eq@linewidths}; if the new line width is greater
+% than the previous one, start running trials with it; otherwise do
+% nothing with it.
+% Finally, run a peculiar \cs{edef} that leaves
+% \cs{eq@linewidths} redefined to be the tail of the list.
+% If we succeed in finding satisfactory line breaks
+% for the equation, we will reset \cs{EQ@continue} in such a
+% way that it will terminate the current trials.
+% An obvious branch here would be to check whether the width of
+% \cs{EQ@copy} is less than \cs{eq@linewidth} and go immediately
+% to the one-line case if so.
+% However, if the equation contains more than one RHS, by
+% default each additional RHS starts on a new line\mdash \ie , we want
+% the ladder layout anyway.
+% So we choose the initial trial on an assumption of multiple lines
+% and leave the one-line case to fall out naturally at a later point.
+%    \begin{macrocode}
+\def\eq@trial@a#1{%
+  \dim@c#1\relax
+  \if T\eq@frame \eq@frame@adjust\dim@c \fi
+  \ifdim\dim@c>\eq@linewidth
+    \eq@linewidth\dim@c
+%<trace>    \breqn@debugmsg{Choose Shape for width(#1)=\the\eq@linewidth}%
+    \let\eq@trial@b\eq@trial@d
+    \csname eq@try@layout@\eq@layout\endcsname
+%<trace>  \else
+%<trace>    \breqn@debugmsg{Next width (#1) is shorter; skip it}%
+  \fi
+  \edef\eq@linewidths{\iffalse}\fi
+}
+\def\eq@frame@adjust#1{%
+  %\addtolength#1{-2\eq@framewd-2\eq@framesep}%
+  \dim@a\eq@framewd \advance\dim@a\eq@framesep
+  \advance#1-2\dim@a
+}
+%    \end{macrocode}
+% \end{macro}
+%========================================
+% Note curious control structure.
+% Try to understand interaction of \cs{EQ@fallback}, \cs{EQ@continue},
+% \cs{eq@failout}
+%    \begin{macrocode}
+\def\eq@trial@succeed{%
+  \aftergroup\@gobbletwo % cancel the \EQ@fallback code; see \eq@trial@c (?)
+  \global\EQ@continue{\eq@trial@done}%
+}
+%    \end{macrocode}
+% \begin{macro}{\eq@trial@done}
+% Success.
+%    \begin{macrocode}
+\def\eq@trial@done{%
+%<trace>  \breqn@debugmsg{End trial: Success!}%
+  \let\eq@failout\relax
+}
+%    \end{macrocode}
+% \end{macro}
+%
+% \begin{macro}{\eq@trial@init}
+% This is called from \cs{eq@trial@b} to initialize or
+% re-initialize certain variables as needed when running one or more
+% trials at a given line width.
+% By default assume success, skip the fallback code.
+%    \begin{macrocode}
+\def\eq@trial@init{\global\let\EQ@fallback\eq@nextlayout}
+%    \end{macrocode}
+% \end{macro}
+% \begin{macro}{\eq@nextlayout}
+%
+% In the fallback case cancel the current group to avoid unnecessary
+% group nesting (with associated save-stack cost, \etc ).
+%    \begin{macrocode}
+\def\eq@nextlayout#1{%
+  \endgroup
+%<trace>  \breqn@debugmsg{Nope ... that ain't gonna work.}%
+  \begingroup #1%
+}
+%    \end{macrocode}
+% \end{macro}
+% \begin{macro}{\eq@failout}
+%
+% .
+%    \begin{macrocode}
+\def\eq@failout{%
+%<trace>\breqn@debugmsg{End trial: failout}%
+  \global\let\EQ@trial\EQ@last@trial
+}
+%    \end{macrocode}
+% \end{macro}
+% \begin{macro}{\eq@trial@save}
+%
+% Save the parameters of the current trial.
+%    \begin{macrocode}
+\def\eq@trial@save#1{%
+%<*trace>
+%  \begingroup \def\@elt##1X##2{\MessageBreak==== \space\space##1/##2}\let\@endelt\@empty\breqn@debugmsg{=> trial info:\eq@measurements}%
+%         \breqn@debugmsg{=> bounding box: \the\eq@wdT x\the\eq@vspan, badness=\the\eq@badness\MessageBreak}%
+%         \let\@elt\relax \let\@endelt\relax
+%  \endgroup
+%</trace>
+  \xdef#1{%
+    \eq@linewidth\the\eq@linewidth
+    % save info about the fit
+    \eq@lines\the\eq@lines \eq@badness\the\eq@badness \def\@nx\eq@badline{\eq@badline}%
+    % save size info
+    \eq@wdT\the\eq@wdT \eq@wdMin\the\eq@wdMin
+    \eq@vspan\the\eq@vspan \eq@dp\the\eq@dp \eq@firstht\the\eq@firstht
+    % save info about the LHS
+    \eq@wdL\the\eq@wdL \def\@nx\EQ@hasLHS{\EQ@hasLHS}%
+    % save info about the numbering
+    \def\@nx\eq@hasNumber{\eq@hasNumber}%
+    % save info about the chosen layout
+    \def\@nx\eq@layout{\eq@layout}%
+    \def\@nx\eq@parshape{\@parshape}%
+    \def\@nx\eq@measurements{\eq@measurements}%
+    \def\@nx\adjust@rel@penalty{\adjust@rel@penalty}%
+    \def\@nx\eq@shiftnumber{\eq@shiftnumber}%
+    \def\@nx\eq@isIntertext{\@False}%
+  }%
+}
+%    \end{macrocode}
+% \end{macro}
+% \begin{macro}{\eq@trial@b}
+%
+% By default this just runs \cs{eq@trial@c}; \cf
+% \cs{eq@trial@d}.
+%    \begin{macrocode}
+\def\eq@trial@b{\eq@trial@c}
+%    \end{macrocode}
+% \end{macro}
+%
+%
+% \begin{macro}{\eq@trial@c}
+%
+% Run the equation contents through the current parshape.
+%    \begin{macrocode}
+\def\eq@trial@c#1#2{%
+%<trace>  \breqn@debugmsg{Trying layout "#1" with\MessageBreak==== parshape\space\@xp\@gobble\@parshape}%
+  \begingroup
+  \eq@trial@init
+  \def\eq@layout{#1}%
+  \setbox\z@\vbox{%
+    \hfuzz\maxdimen
+    \eq@trial@p % run the given parshape
+    \if\@Not{\eq@badline}%
+      \eq@trial@save\EQ@trial
+%    \end{macrocode}
+% If there is a number, try the same parshape again with adjustments
+% to make room for the number.
+%
+% This is an awkward place for this:
+% It only allows trying to fit the number w/the SAME layout shape!
+%    \begin{macrocode}
+      \if\eq@hasNumber\eq@retry@with@number\fi
+      \if L\eq@layout \eq@check@density
+      \else
+        \if\@Not{\eq@badline}%
+           \eq@trial@succeed
+        \fi
+      \fi
+    \else
+      \eq@trial@save\EQ@last@trial
+    \fi
+  }%
+  \EQ@fallback{#2}%
+  \endgroup
+}
+%    \end{macrocode}
+% \end{macro}
+% \begin{macro}{\eq@trial@d}
+%    \begin{macrocode}
+\def\eq@trial@d#1#2{\eq@trial@c{#1}{}}
+%    \end{macrocode}
+% \end{macro}
+%
+% \begin{macro}{\eq@check@density}
+%    \begin{macrocode}
+\def\eq@check@density{%
+%<trace>  \breqn@debugmsg{Checking density for layout L}%
+  \if\@Or{\@Not\EQ@hasLHS}{\eq@shortLHS}%
+%<trace>    \breqn@debugmsg{Density check: No LHS, or is short; OK}%
+    \eq@trial@succeed
+  \else\if\eq@dense@enough
+    \eq@trial@succeed
+  \fi\fi
+}
+%    \end{macrocode}
+% \end{macro}
+%
+% \begin{macro}{\eq@shortLHS}
+% Test to see if we need to apply the \cs{eq@dense@enough} test.
+%    \begin{macrocode}
+\def\eq@shortLHS{\ifdim\eq@wdL>.44\eq@wdT 1\else 0\fi 0}
+%    \end{macrocode}
+% \end{macro}
+%
+%\verb|\def\eq@shortLHS{\@False}|
+%========================================
+% \begin{macro}{\eq@trial@p}
+% Run a trial with the current \cs{@parshape} and measure it.
+%    \begin{macrocode}
+\def\eq@trial@p{%
+  \@parshape %
+  \eq@dump@box\unhcopy\EQ@copy
+  {\@@par}% leave \parshape readable
+  \eq@lines\prevgraf
+  \eq@fix@lastline
+  \let\eq@badline\@False
+  \if i\eq@layout \ifnum\eq@lines>\@ne \let\eq@badline\@True \fi\fi
+  \eq@curline\eq@lines % loop counter for eq@measure@lines
+  \let\eq@measurements\@empty
+  \eq@ml@record@indents
+  \eq@measure@lines
+  \eq@recalc
+%<trace>  \debug@showmeasurements
+}
+%    \end{macrocode}
+% \end{macro}
+%
+%
+% \begin{macro}{\adjust@rel@penalty}
+% Normally this is a no-op.
+%    \begin{macrocode}
+\let\adjust@rel@penalty\@empty
+%    \end{macrocode}
+% \end{macro}
+%
+% \begin{macro}{\eq@fix@lastline}
+%  Remove parfillskip from the last line box.
+%    \begin{macrocode}
+\def\eq@fix@lastline{%
+  \setbox\tw@\lastbox \dim@b\wd\tw@
+  \eq@dp\dp\tw@
+%    \end{macrocode}
+% Remove \cs{parfillskip} but retain \cs{rightskip}.
+% Need to keep the original line width for later shrink testing.
+%    \begin{macrocode}
+  \nointerlineskip\hbox to\dim@b{\unhbox\tw@
+    \skip@c\lastskip \unskip\unskip\hskip\skip@c
+  }%
+}
+%    \end{macrocode}
+% \end{macro}
+%
+% \begin{macro}{\eq@recalc}
+% Calculate \cs{eq@wdT} et cetera.
+%    \begin{macrocode}
+\def\eq@recalc{%
+  \eq@wdT\z@ \eq@wdMin\maxdimen \eq@vspan\z@skip \eq@badness\z@
+  \let\@elt\eq@recalc@a \eq@measurements \let\@elt\relax
+}
+%    \end{macrocode}
+% \end{macro}
+%
+%
+% \begin{macro}{\eq@recalc@a}
+%    \begin{macrocode}
+\def\eq@recalc@a#1x#2+#3\@endelt{%
+  \eq@firstht#2\relax
+  \let\@elt\eq@recalc@b
+  \@elt#1x#2+#3\@endelt
+}
+%    \end{macrocode}
+% \end{macro}
+%
+%
+% \begin{macro}{\eq@recalc@b}
+%    \begin{macrocode}
+\def\eq@recalc@b#1X#2,#3x#4+#5@#6\@endelt{%
+  \setlength\dim@a{#2+#3}%
+  \ifdim\dim@a>\eq@wdT \eq@wdT\dim@a \fi
+  \ifdim\dim@a<\eq@wdMin \eq@wdMin\dim@a \fi
+  \eq@dp#5\relax
+  \addtolength\eq@vspan{#1+#4+\eq@dp}%
+%    \end{macrocode}
+% Record the max badness of all the lines in \cs{eq@badness}.
+%    \begin{macrocode}
+  \ifnum#6>\eq@badness \eq@badness#6\relax\fi
+}
+%    \end{macrocode}
+% \end{macro}
+%
+% \begin{macro}{\eq@layout}
+% A value of \verb"?" for \cs{eq@layout} means that we should
+% deduce which layout to use by looking at the size of the components.
+% Any other value means we have a user-specified override on the
+% layout.
+%
+% Layout Definitions.
+% Based on initial equation measurements, we can choose a sequence of
+% candidate parshapes that the equation might fit into.
+% We accept the first shape that `works', else fall to next one.
+% [The sequence is hardcoded in the \cs{eq@try@layout@}<shape>
+%  Would it be useful be more flexible? (eg. try layouts LDA, in order...)]
+%    \begin{macrocode}
+\def\eq@layout{?}
+%    \end{macrocode}
+% \end{macro}
+%
+%
+% \begin{macro}{\eq@try@layout@?}
+% This is a branching function used to choose a suitable layout if
+% the user didn't specify one in particular.
+%
+% Default layout:
+%  Try Single line layout first, else try Multiline layouts
+%    \begin{macrocode}
+\@namedef{eq@try@layout@?}{%
+  \let\eq@trial@b\eq@trial@c
+  \edef\@parshape{\parshape 1 0pt \the\eq@linewidth\relax}%
+%  \eq@trial@b{i}{\eq@try@layout@multi}%
+  \setlength\dim@a{\wd\EQ@copy-2em}% Fudge; can't shrink more than this?
+  % if we're in a numbered group, try hard to fit within the numbers
+  \dim@b\eq@linewidth
+  \if\eq@shiftnumber\else\if\eq@group
+    \if\eq@hasNumber\addtolength\dim@b{-\wd\EQ@numbox-\eqnumsep}%
+    \else\if\grp@hasNumber\addtolength\dim@b{-\wd\GRP@numbox-\eqnumsep}%
+  \fi\fi\fi\fi
+  \ifdim\dim@a<\dim@b% Do we even have a chance of fitting to one line?
+%<trace>    \breqn@debugmsg{Choose Shape: (\the\wd\EQ@copy) may fit in \the\dim@b}%
+%    \end{macrocode}
+% BRM: assuming it might fit, don't push too hard
+%    \begin{macrocode}
+    \setlength\dim@b{\columnwidth-\dim@a+\eq@wdCond}%
+    \rightskip\z@\@plus\dim@b\@minus\hfuzz
+    \eq@trial@b{i}{\eq@try@layout@multi}%
+  \else
+%<*trace>
+  \breqn@debugmsg{Choose Shape: Too long (\the\wd\EQ@copy) for one line
+            (free width=\the\dim@b)}%
+%</trace>
+   \eq@try@layout@multi
+  \fi
+}
+%    \end{macrocode}
+% Layout Multiline layout:
+%  If no LHS, try Stepped(S) layout
+%  Else try Stepped(S), Ladder(L), Drop-ladder(D) or Stepladder(l), depending on LHS length.
+%    \begin{macrocode}
+\def\eq@try@layout@multi{%
+  \if\EQ@hasLHS
+    \ifdim\eq@wdL>\eq@linewidth
+%<trace>       \breqn@debugmsg{Choose Shape: LHS \the\eq@wdL > linewidth}%
+%    \end{macrocode}
+% Find the total width of the RHS.
+% If it is relatively short, a step layout is the thing to try.
+%    \begin{macrocode}
+       \setlength\dim@a{\wd\EQ@copy-\eq@wdL}%
+       \ifdim\dim@a<.25\eq@linewidth \eq@try@layout@S
+       \else \eq@try@layout@l
+       \fi
+    % BRM: Originally .7: Extreme for L since rhs has to wrap within the remaining 30+%!
+    \else\ifdim\eq@wdL>.50\eq@linewidth
+%<*trace>
+      \breqn@debugmsg{Choose Shape: LHS (\the\eq@wdL) > .50 linewidth (linewidth=\the\eq@linewidth)}%
+%</trace>
+      \eq@try@layout@D
+    \else
+%<trace>      \breqn@debugmsg{Choose Shape: LHS (\the\eq@wdL) not extraordinarily wide}%
+      \eq@try@layout@L
+    \fi\fi
+  \else
+%<trace>    \breqn@debugmsg{Choose Shape: No LHS here}%
+%    \end{macrocode}
+% Try one-line layout first, then step layout.
+%    \begin{macrocode}
+    \eq@try@layout@S % (already checked case i)
+  \fi
+}
+%    \end{macrocode}
+% \end{macro}
+%
+% \begin{macro}{\eq@try@layout@D}
+% Change the penalty before the first mathrel symbol to encourage a
+% break there.
+%
+% Layout D=Drop-Ladder Layout, for wide LHS.
+% \begin{literalcode}
+%   LOOOOOOOONG LHS
+%    = RHS
+%    = ...
+% \end{literalcode}
+% If fails, try Almost-Columnar layout
+%    \begin{macrocode}
+\def\eq@try@layout@D{%
+  \setlength\dim@a{\eq@linewidth -\eq@indentstep}%
+  \edef\@parshape{\parshape 2
+    0pt \the\eq@wdL\space \the\eq@indentstep\space \the\dim@a\relax
+  }%
+  \def\adjust@rel@penalty{\penalty-99 }%
+  \eq@trial@b{D}{\eq@try@layout@A}%
+}
+%    \end{macrocode}
+% \end{macro}
+% \begin{macro}{\eq@try@layout@L}
+% Try a straight ladder layout.
+% Preliminary filtering ensures that \cs{eq@wdL} is less than 70%
+% of the current line width.
+% \begin{literalcode}
+% Layout L=Ladder layout
+%  LHS = RHS
+%      = RHS
+%      ...
+% \end{literalcode}
+% If fails, try Drop-ladder layout.
+% NOTE: This is great for some cases (multi relations?), but
+% tends to break really badly when it fails....
+%    \begin{macrocode}
+\def\eq@try@layout@L{%
+  \setlength\dim@b{\eq@linewidth-\eq@wdL}%
+  \edef\@parshape{\parshape 2 0pt \the\eq@linewidth\space
+    \the\eq@wdL\space \the\dim@b\relax
+  }%
+  \eq@trial@b{L}{\eq@try@layout@D}%
+}
+%    \end{macrocode}
+% \end{macro}
+%
+% \begin{macro}{\eq@try@layout@S}
+% In the \dquoted{stepped} layout there is no LHS, or LHS
+% is greater than the line width and RHS is small.
+% Then we want to split up the equation into lines of roughly equal
+% width and stagger them downwards to the right, leaving a small amount of
+% whitespace on both sides.
+% But also, if there is an equation number, we want to try first a
+% layout that leaves room for the number.
+% Otherwise it would nearly always be the case that the number would
+% get thrown on a separate line.
+%
+% Layout S=Stepped layout, typically no LHS or very long, variations on
+% \begin{literalcode}
+%  STUFF ....
+%    + MORE STUFF ...
+%      + MORE STUFF ...
+% \end{literalcode}
+% If fails, try Almost-Columnar layout
+%    \begin{macrocode}
+\def\eq@try@layout@S{%
+  \setlength\dim@b{\eq@linewidth-2\eqmargin}% \advance\dim@b-1em%
+%    \end{macrocode}
+% About how many lines will we need if dim@b is the line width?
+%    \begin{macrocode}
+  \int@a\wd\EQ@copy \divide\int@a\dim@b
+%    \end{macrocode}
+% Adjust the target width by number of lines times indentstep.
+% We don't need to decrement \cs{int@a} because \tex
+% division is integer division with truncation.
+%    \begin{macrocode}
+  \addtolength\dim@b{-\int@a\eq@indentstep}%
+%    \end{macrocode}
+% Adjust for equation number.
+% But try not to leave too little room for the equation body.
+%    \begin{macrocode}
+  \if\eq@hasNumber
+    \ifdim\dim@b>15em%
+%      \advance\dim@b-\eqnumsep \advance\dim@b-\wd\EQ@numbox
+      \addtolength\dim@b{-\eq@wdNum}%
+    \fi
+  \fi
+%    \end{macrocode}
+% Now some hand-waving to set up the parshape.
+%    \begin{macrocode}
+  \int@b\z@
+  \def\@tempa{\dim}%
+  \edef\@parshape{\parshape 2 0pt \the\dim@b\space
+    \the\eqmargin\space\the\dim@b\relax}%
+  \eq@trial@b{S}{\eq@try@layout@A}%
+}
+%    \end{macrocode}
+% \end{macro}
+%
+% \begin{macro}{\eq@try@layout@l}
+% This is the \dquoted{step-ladder} layout: similar to the drop-ladder
+% layout but the LHS is too wide and needs to be broken up.
+%
+% Layout l = Stepladder
+% Similar to Drop-Ladder, but LHS is long and needs to be broken up.
+% If fails, try Almost-Columnar layout
+%    \begin{macrocode}
+\def\eq@try@layout@l{%
+  \setlength\dim@a{\eq@linewidth -\eq@indentstep}%
+  \int@a\eq@wdL \divide\int@a\dim@a
+  \advance\int@a\tw@
+  \edef\@parshape{\parshape \number\int@a\space
+    0pt \the\eq@linewidth
+  }%
+  \advance\int@a-\tw@
+  \setlength\dim@b{2\eq@indentstep}%
+  \setlength\dim@c{\eq@linewidth -\dim@b}%
+  \edef\@parshape{\@parshape
+    \replicate{\int@a}{\space\the\eq@indentstep\space\the\dim@a}%
+    \space\the\dim@b\space\the\dim@c\relax
+  }%
+  \eq@trial@b{l}{\eq@try@layout@A}%
+}
+%    \end{macrocode}
+% \end{macro}
+%
+% \begin{macro}{\eq@try@layout@A}
+% In the \dquoted{almost-columnar} layout, which is the layout of last
+% resort, we let all lines run to the full width and leave the adjusting
+% of the indents to later.
+%
+% Layout A = Almost-Columnar layout.
+% Pretty much straight full width, more of a last-resort.
+% If fails, give up.
+%    \begin{macrocode}
+\def\eq@try@layout@A{%
+  \edef\@parshape{\parshape 1 0pt \the\eq@linewidth\relax}%
+  \if\EQ@hasLHS \def\adjust@rel@penalty{\penalty-99 }\fi
+  \eq@trial@b{A}{}%
+}
+%    \end{macrocode}
+% \end{macro}
+%
+% \begin{macro}{\eq@shiftnumber}
+%  MH: Should be moved to a section where all keys are set to defaults.
+%    \begin{macrocode}
+\let\eq@shiftnumber\@False
+%    \end{macrocode}
+% \end{macro}
+%
+% \begin{macro}{\eq@retry@with@number@a}
+% Number placement adjustments
+%    \begin{macrocode}
+\def\eq@retry@with@number{%
+ \if\eq@shiftnumber
+%<trace>   \breqn@debugmsg{Place number: Shifted number requested}%
+ \else
+%    \end{macrocode}
+% Condition and right numbers? We're just going to have to shift.
+%    \begin{macrocode}
+    \ifdim\eq@wdCond>\z@\if R\eqnumside
+%<trace>      \breqn@debugmsg{Place number: Condition w/Right number => Shift number}%
+      \let\eq@shiftnumber\@True
+    \fi\fi
+%    \end{macrocode}
+% Compute free space.
+%    \begin{macrocode}
+%    \dim@b\eqnumsep\advance\dim@b\wd\EQ@numbox
+    \dim@b\eq@wdNum
+    \if L\eqnumside
+      \ifdim\@totalleftmargin>\dim@b\dim@b\@totalleftmargin\fi
+    \else
+      \addtolength\dim@b{\@totalleftmargin}%
+    \fi
+    \setlength\dim@a{\eq@linewidth-\dim@b}%\advance\dim@a1em\relax% Allowance for shrink?
+%    \end{macrocode}
+% Set up test against 1-line case only if not in a group
+%    \begin{macrocode}
+    \int@a\@ne\if\eq@group\int@a\maxint\fi
+%    \end{macrocode}
+% Now check for cases.
+%    \begin{macrocode}
+    \if\eq@shiftnumber               % Already know we need to shift
+    \else\ifdim\eq@wdT<\dim@a % Fits!
+%    \end{macrocode}
+% left \& right skips will be done later, and parshape adjusted if
+% needed.
+%    \begin{macrocode}
+%<trace>      \breqn@debugmsg{Place number: eqn and number fit together}%
+%    \else\ifnum\eq@lines=\int@a %  Shift, if single line, unless inside a dgroup.
+%    \end{macrocode}
+% NOTE: this is too strong for dgroup!
+%    \begin{macrocode}
+%<*trace>
+%      \breqn@debugmsg{Place number: single line too long with number => Shift number \the\int@a}%
+%</trace>
+%      \let\eq@shiftnumber\@True
+    \else
+%    \end{macrocode}
+%      % Retry: use leftskip for space for number(for now; whether
+%      % right/left) \& adjust parshape
+%    \begin{macrocode}
+%      \leftskip\wd\EQ@numbox\advance\leftskip\eqnumsep
+      \setlength\leftskip{\eq@wdNum}%
+      \setlength\rightskip{\z@\@plus\dim@a}%
+      \adjust@parshape\@parshape
+%<*trace>
+      \breqn@debugmsg{Place number: Try with \leftskip=\the\leftskip, \rightskip=\the\rightskip,
+                \MessageBreak==== parshape\space\@xp\@gobble\@parshape}%
+%</trace>
+      \nointerlineskip
+      \edef\eq@prev@lines{\the\eq@lines}%
+      \edef\eq@prev@badness{\the\eq@badness}% BRM
+      \eq@trial@p
+      \int@a\eq@prev@badness\relax\advance\int@a 50\relax%?
+      \int@b\eq@prev@lines  \if\eq@group\advance\int@b\@ne\fi% Allow 1 extra line in group
+      \ifnum\eq@lines>\int@b % \eq@prev@lines
+%<trace>        \breqn@debugmsg{Adjustment causes more breaks => Shift number}%
+        \let\eq@shiftnumber\@True
+      \else\if\eq@badline
+%<trace>        \breqn@debugmsg{Adjustment causes bad lines (\the\eq@badness) => Shift}%
+        \let\eq@shiftnumber\@True
+      \else\ifnum\eq@badness>\int@a % BRM: New case
+%<*trace>
+        \breqn@debugmsg{Adjustment is badder than previous
+                  (\the\eq@badness >> \eq@prev@badness) => Shift}%
+%</trace>
+        \let\eq@shiftnumber\@True
+      \else
+%<trace>        \breqn@debugmsg{Adjustment succeeded}%
+      \fi\fi%\fi
+   \fi\fi\fi
+%    \end{macrocode}
+%       If we got shifted, restore parshape, etc,
+%    \begin{macrocode}
+   \if\eq@shiftnumber
+     \EQ@trial% Restore parshape & other params,
+     \leftskip\z@\let\eq@shiftnumber\@True % But set shift & leftskip
+     \edef\@parshape{\eq@parshape}% And copy saved parshape back to `working copy' !?!?
+   \fi
+   \eq@trial@save\EQ@trial      % Either way, save the trial state.
+ \fi
+}
+%    \end{macrocode}
+% \end{macro}
+%
+% \begin{macro}{\adjust@parshape}
+% Varies depending on the layout.
+%
+% Adjust a parshape variable for a given set of left\textbar right skips.
+% Note that the fixed part of the left\textbar right skips effectively
+% comes out of the parshape widths (NOT in addition to it).
+% We also must trim the widths so that the sum of skips, indents
+% and widths add up to no more than the \cs{eq@linewidth}.
+%    \begin{macrocode}
+\def\adjust@parshape#1{%
+  \@xp\adjust@parshape@a#1\relax
+  \edef#1{\temp@a}%
+}
+%    \end{macrocode}
+% \end{macro}
+%
+%
+% \begin{macro}{\adjust@parshape@a}
+% \begin{macro}{\adjust@parshape@b}
+%    \begin{macrocode}
+\def\adjust@parshape@a#1 #2\relax{%
+  \setlength\dim@a{\leftskip+\rightskip}%
+  \edef\temp@a{#1}%
+  \adjust@parshape@b#2 @ @ \relax
+}
+\def\adjust@parshape@b#1 #2 {%
+  \ifx @#1\edef\temp@a{\temp@a\relax}%
+    \@xp\@gobble
+  \else
+    \dim@b#1\relax
+    \dim@c#2\relax
+    \addtolength\dim@c{\dim@a+\dim@b}%
+    \ifdim\dim@c>\eq@linewidth\setlength\dim@c{\eq@linewidth}\fi
+     \addtolength\dim@c{-\dim@b}%
+    \edef\temp@a{\temp@a\space\the\dim@b\space\the\dim@c}%
+  \fi
+  \adjust@parshape@b
+}
+%    \end{macrocode}
+% \end{macro}
+% \end{macro}
+%
+% \begin{macro}{\eq@ml@record@indents}
+% Plunk the parshape's indent values into an array for easy access
+% when constructing \cs{eq@measurements}.
+%    \begin{macrocode}
+\def\eq@ml@record@indents{%
+  \int@a\z@
+  \def\@tempa{%
+    \advance\int@a\@ne
+    \@xp\edef\csname eq@i\number\int@a\endcsname{\the\dim@a}%
+    \ifnum\int@a<\int@b \afterassignment\@tempb \fi
+    \dim@a
+  }%
+  \def\@tempb{\afterassignment\@tempa \dim@a}%
+  \def\@tempc##1##2 {\int@b##2\afterassignment\@tempa\dim@a}%
+  \@xp\@tempc\@parshape
+}
+%    \end{macrocode}
+% \end{macro}
+%
+% \begin{macro}{\@endelt}
+% This is a scan marker.
+% It should get a non-expandable definition.
+% It could be \cs{relax}, but let's try a chardef instead.
+%    \begin{macrocode}
+\chardef\@endelt=`\?
+%    \end{macrocode}
+% \end{macro}
+%
+%
+% \begin{macro}{\eq@measurements}
+% This is similar to a parshape spec but for each line we record more
+% info: space above, indent, width x height + dp, and badness.
+%    \begin{macrocode}
+\def\eq@measurements{%
+  \@elt 4.5pt/5.0pt,66.0ptx6.8pt+2.4pt@27\@endelt
+  ...
+}
+%    \end{macrocode}
+% \end{macro}
+%
+%
+% \begin{macro}{\eq@measure@lines}
+% Loop through the list of boxes to measure things like total
+% height (including interline stretch), \etc .    We check the
+% actual width of the current line against the natural width \mdash
+% after removing rightskip \mdash  in case the former is
+% \emph{less} than the latter because of shrinkage.    In that
+% case we do not want to use the natural width for RHS-max-width because
+% it might unnecessarily exceed the right margin.
+%    \begin{macrocode}
+\def\eq@measure@lines{%
+  \let\eq@ml@continue\eq@measure@lines
+  \setbox\tw@\lastbox \dim@b\wd\tw@ % find target width of line
+  \setbox\z@\hbox to\dim@b{\unhbox\tw@}% check for overfull
+  \eq@badness\badness
+  \ifnum\eq@badness<\inf@bad \else \let\eq@badline\@True \fi
+  \eq@ml@a \eq@ml@continue
+}
+%    \end{macrocode}
+% \end{macro}
+%
+%
+% \begin{macro}{\eq@ml@a}
+%
+%    \begin{macrocode}
+\def\eq@ml@a{%
+  \setbox\tw@\hbox{\unhbox\z@ \unskip}% find natural width
+%<*trace>
+  \ifnum\eq@badness<\inf@bad\else\breqn@debugmsg{!?! Overfull: \the\wd\tw@ >\the\dim@b}\fi
+%</trace>
+%    \end{macrocode}
+% Is actual width less than natural width?
+%    \begin{macrocode}
+  \ifdim\dim@b<\wd\tw@ \setlength\dim@a{\dim@b}% shrunken line
+  \else                \setlength\dim@a{\wd\tw@}% OK to use natural width
+  \fi
+  \addtolength\dim@a{-\leftskip}% BRM: Deduct the skip if we're retrying w/number
+%    \end{macrocode}
+% If there's no aboveskip, assume we've reached the top of the
+% equation.
+%    \begin{macrocode}
+  \skip@a\lastskip \unskip \unpenalty
+  \ifdim\skip@a=\z@
+    \let\eq@ml@continue\relax % end the recursion
+  \else
+    % Sum repeated vskips if present
+    \def\@tempa{%
+      \ifdim \lastskip=\z@
+      \else \addtolength\skip@a{\lastskip}\unskip\unpenalty \@xp\@tempa
+      \fi
+    }%
+  \fi
+  \edef\eq@measurements{\@elt
+    \the\skip@a\space X% extra space to facilitate extracting only the
+                        % dimen part later
+    \csname eq@i%
+      \ifnum\eq@curline<\parshape \number\eq@curline
+      \else\number\parshape
+      \fi
+    \endcsname,\the\dim@a x\the\ht\tw@+\the\dp\tw@ @\the\eq@badness\@endelt
+    \eq@measurements
+  }%
+  \advance\eq@curline\m@ne
+  \ifnum\eq@curline=\z@ \let\eq@ml@continue\relax\fi
+}
+%    \end{macrocode}
+% \end{macro}
+%
+%
+% \begin{macro}{\eq@ml@vspace}
+%
+% Handle an embedded vspace.
+%    \begin{macrocode}
+\def\eq@ml@vspace{%
+  \global\advance\eq@vspan\lastskip \unskip\unpenalty
+  \ifdim\lastskip=\z@ \else \@xp\eq@ml@vspace \fi
+}
+%    \end{macrocode}
+% \end{macro}
+%
+%
+% \begin{macro}{\eq@dense@enough}
+%
+%    \begin{macrocode}
+\def\eq@dense@enough{%
+  \ifnum\eq@lines<\thr@@
+%<trace>    \breqn@debugmsg{Density check: less than 3 lines; OK}%
+    \@True
+  \else
+    \ifdim\eq@wdL >.7\eq@wdT
+%<trace>     \breqn@debugmsg{Density check: LHS too long; NOT OK}%
+      \@False
+    \else \@xp\@xp\@xp\eq@dense@enough@a
+    \fi
+  \fi
+}
+%    \end{macrocode}
+% \end{macro}
+%
+%
+% \begin{macro}{\true@true@true}
+% \begin{macro}{\true@false@true}
+% \begin{macro}{\false@true@false}
+% \begin{macro}{\false@false@false}
+%    \begin{macrocode}
+\def\true@true@true   {\fi\fi\iftrue \iftrue \iftrue }
+\def\true@false@true  {\fi\fi\iftrue \iffalse\iftrue }
+\def\false@true@false {\fi\fi\iffalse\iftrue \iffalse}
+\def\false@false@false{\fi\fi\iffalse\iffalse\iffalse}
+%    \end{macrocode}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+%
+%
+% \begin{macro}{\eq@density@factor}
+%
+% This number specifies, for the ladder layout, how much of the
+% equation's bounding box should contain visible material rather than
+% whitespace.
+% If the amount of visible material drops below this value, then we
+% switch to the drop-ladder layout.
+% The optimality of this factor is highly dependent on the equation
+% contents; .475 was chosen as the default just because it worked well
+% with the sample equation, designed to be as average as possible, that I
+% used for testing.
+%    \begin{macrocode}
+\def\eq@density@factor{.475}
+%    \end{macrocode}
+% \end{macro}
+%
+%
+% \begin{macro}{\eq@dense@enough@a}
+%
+% Calculate whether there is more
+% visible material than whitespace within the equation's bounding box.
+% Sum up the actual line widths and compare to the total
+% \dquoted{area} of the bounding box.
+% But if we have an extremely large number of lines, fall back to an
+% approximate calculation that is more conservative about the danger of
+% exceeding \cs{maxdimen}.
+%    \begin{macrocode}
+\def\eq@dense@enough@a{%
+  \@True \fi
+  \ifnum\eq@lines>\sixt@@n
+    \eq@dense@enough@b
+  \else
+    \dim@b\z@ \let\@elt\eq@delt \eq@measurements
+    \dim@c\eq@density@factor\eq@wdT \multiply\dim@c\eq@lines
+%<trace>    \breqn@debugmsg{Density check: black \the\dim@b/\eq@density@factor total \the\dim@c}%
+    \ifdim\dim@b>\dim@c \true@false@true \else \false@false@false \fi
+  \fi
+}
+%    \end{macrocode}
+% \end{macro}
+%
+%
+% \begin{macro}{\eq@delt}
+% Args are space-above, indent, width, height, depth, badness.
+%    \begin{macrocode}
+\def\eq@delt#1X#2,#3x#4+#5@#6\@endelt{\addtolength\dim@b{#3}}%
+%    \end{macrocode}
+% \end{macro}
+%
+%
+% \begin{macro}{\eq@dense@enough@b}
+% This is an approximate calculation used to keep from going over
+% \cs{maxdimen} if the number of lines in our trial break is large
+% enough to make that a threat.
+% If l, t, n represent left-side-width, total-width, and number of
+% lines, the formula is
+% \begin{literalcode}
+% l/t < .4n/(.9n-1)
+% \end{literalcode}
+% or equivalently, since rational arithmetic is awkward in \tex :
+% b
+% \begin{literalcode}
+% l/t < 4n/(9n-10)
+% \end{literalcode}
+% .
+%    \begin{macrocode}
+\def\eq@dense@enough@b{%
+  \int@b\eq@wdT \divide\int@b\p@
+  \dim@b\eq@wdL \divide\dim@b\int@b
+  \dim@c\eq@lines\p@ \multiply\dim@c\f@ur
+  \int@b\eq@lines \multiply\int@b 9 \advance\int@b -10%
+  \divide\dim@c\int@b
+%<trace>  \breqn@debugmsg{Density check: l/t \the\dim@b\space< \the\dim@c\space 4n/(9n-10)?}%
+  \ifdim\dim@b<\dim@c \true@true@true \else \false@true@false \fi
+}
+%    \end{macrocode}
+% \end{macro}
+%
+% \begin{macro}{\eq@parshape}
+%    \begin{macrocode}
+\let\eq@parshape\@empty
+%    \end{macrocode}
+% \end{macro}
+%
+%
+% \begin{macro}{\eq@params}
+% The interline spacing and penalties in \cs{eq@params}
+% are used during both preliminary line breaking and final typesetting.
+%    \begin{macrocode}
+\def\eq@params{%
+  \baselineskip\eqlinespacing
+  \lineskip\eqlineskip \lineskiplimit\eqlineskiplimit
+%    \end{macrocode}
+% Forbid absolutely a pagebreak that separates the first line or last
+% line of a multiline equation from the rest of it.    Or in other
+% words: no equation of three lines or less will be broken at the bottom
+% of a page; instead it will be moved whole to the top of the next
+% page.    If you really really need a page break that splits the
+% first or last line from the rest of the equation, you can always fall
+% back to\cs{pagebreak}, I suppose.
+%
+%    \begin{macrocode}
+  \clubpenalty\@M \widowpenalty\@M \interlinepenalty\eqinterlinepenalty
+  \linepenalty199 \exhyphenpenalty5000 % was 9999: make breaks at, eg. \* a bit easier.
+%    \end{macrocode}
+% For equations, hfuzz should be at least 1pt.
+% But we have to fake it a little because we are running the equation
+% through \tex 's paragrapher.
+% In our trials we use minus 1pt in the rightskip rather than hfuzz;
+% and we must do the same during final breaking of the equation, otherwise
+% in borderline cases \tex  will use two lines instead of one when our
+% trial indicated that one line would be enough.
+%    \begin{macrocode}
+  \ifdim\hfuzz<\p@ \hfuzz\p@ \fi
+%\hfuzz=2pt
+%  \ifdim\hfuzz<2pt\relax \hfuzz2pt \fi
+  \parfillskip\z@skip
+%  \hfuzz\z@
+%    \end{macrocode}
+% Make sure we skip \tex 's preliminary line-breaking pass to save
+% processing time.
+%    \begin{macrocode}
+  \tolerance9999 \pretolerance\m@ne
+}
+%    \end{macrocode}
+% \end{macro}
+%
+%
+%
+%
+% \section{Equation layout options}
+% Using the notation C centered, I indented (applied to
+% the equation body), T top, B bottom, M
+% middle, L left, R right (applied to the equation number),
+% the commonly used equation types are C, CRM, CRB, CLM, CLT,
+% I, IRM, IRB, ILM, ILT.    In other words, CLM stands for Centered equation
+% body with Left-hand Middle-placed equation number, and IRB stands for
+% Indented equation with Right-hand Bottom-placed equation number.
+%
+% Here are some general thoughts on how to place an equation
+% tag. Currently it does not work as desired: the L option positions
+% the tag app. 10 lines below the math expression, the RM doesn't
+% position the tag on the baseline for single-line math
+% expressions. Therefore I am going to first write what I think is
+% supposed to happen and then implement it.
+%
+% Below is a small list where especially the two three specifications
+% should be quite obvious, I just don't want to forget anything and it
+% is important to the implementation.
+% \begin{description}
+% \item[Definition 1] If a display consists of exactly one line, the
+%   tag should always be placed on the same baseline as the math
+%   expression.
+% \end{description}
+% The remaining comments refer to multi-line displays.
+% \begin{description}
+% \item[Definition 2] If a tag is to be positioned at the top (T), it
+%   should be placed such that the baseline of the tag aligns with the
+%   baseline of the top line of the display.
+%
+% \item[Definition 3] If a tag is to be positioned at the bottom (B),
+%   it should be placed such that the baseline of the tag aligns with
+%   the baseline of the bottom line of the display.
+%
+% \item[Definition 4] If a tag is to be positioned vertically centered
+%   (M), it should be placed such that the baseline of the tag is
+%   positioned exactly halfway between the baseline of the top line of
+%   the display and the baseline of the bottom line of the display.
+% \end{description}
+%
+% Definitions 1--3 are almost axiomatic in their
+% simplicity. Definition~4 is different because I saw at least two
+% possibilities for which area to span:
+% \begin{itemize}
+% \item Calculate distance from top of top line to the bottom of the
+%   bottom line, position the vertical center of the tag exactly
+%   halfway between those two extremes.
+%
+% \item Calculate the distance from the baseline of the top line to
+%   the baseline of the bottom line, position the baseline of the tag
+%   exactly halfway between these two extremes.
+% \end{itemize}
+% Additional combinations of these methods are possible but make
+% little sense in my opinion. I have two reasons for choosing the
+% latter of these possibilities: Firstly, two expressions looking
+% completely identical with the exception of a superscript in the
+% first line or a subscript in the last line will have the tag
+% positioned identically. Secondly, then M means halfway between T and
+% B positions which makes good sense and then also automatically
+% fulfills Definition~1.
+%
+% From an implementation perspective, these definitions should also
+% make it possible to fix a deficiency in the current implementation,
+% namely that the tag does not influence the height of a display, even
+% if the display is a single line. This means that two single-line
+% expressions in a \env{dgroup} can be closer together than
+% \cs{intereqskip} if the math expressions are (vertically) smaller
+% than the tag.
+%
+% \section{Centered Right-Number Equations}
+%
+% \begin{macro}{\eq@dump@box}
+% \arg1  might be \cs{unhbox} or \cs{unhcopy}; \arg2  is
+% the box name.
+%    \begin{macrocode}
+\def\eq@dump@box#1#2{%
+%\debug@box#1%
+  \noindent #1#2\setbox\f@ur\lastbox \setbox\tw@\lastbox
+%    \end{macrocode}
+% If the LHS contains shrinkable glue, in an L layout the alignment
+% could be thrown off if the first line is shrunk noticeably.
+% For the time being, disable shrinking on the left-hand side.
+% The proper solution requires more work \begin{dn}
+% mjd,1999/03/17
+% \end{dn}
+% .
+%    \begin{macrocode}
+  \if L\eq@layout \box\tw@ \else\unhbox\tw@\fi
+  \adjust@rel@penalty \unhbox\f@ur
+}
+%    \end{macrocode}
+% \end{macro}
+%
+% Various typesetting bits, invoked from \cs{eq@finish}
+% BRM: This has been extensively refactored from the original breqn,
+% initially to get left\textbar right skips and parshape used consistently,
+% ultimately to get most things handled the same way, in the same order.
+%
+% Given that left and right skips have been set,
+% typeset the frame, number and equation with the
+% given number side and placement
+%
+%    \begin{macrocode}
+\def\eq@typeset@Unnumbered{%
+  \eq@typeset@frame
+  \eq@typeset@equation
+}
+\def\eq@typeset@LM{%
+  \setlength\dim@a{(\eq@vspan+\ht\EQ@numbox-\dp\EQ@numbox)/2}%
+  \eq@typeset@leftnumber
+  \eq@typeset@frame
+  \eq@typeset@equation
+}
+%    \end{macrocode}
+% Typeset equation and left-top number (and shifted)
+%    \begin{macrocode}
+\def\eq@typeset@LT{%
+  \dim@a\eq@firstht
+  \eq@typeset@leftnumber
+  \eq@typeset@frame
+  \eq@typeset@equation
+}
+%    \end{macrocode}
+% Typeset equation and left shifted number
+%    \begin{macrocode}
+\def\eq@typeset@LShifted{%
+  % place number
+  \copy\EQ@numbox \penalty\@M
+  \dim@a\eqlineskip
+  \if F\eq@frame\else
+     \setlength\dim@a{\eq@framesep+\eq@framewd}%
+  \fi
+  \kern\dim@a
+  \eq@typeset@frame
+  \eq@typeset@equation
+}
+%    \end{macrocode}
+% Typeset equation and right middle number
+%    \begin{macrocode}
+\def\eq@typeset@RM{%
+  \setlength\dim@a{(\eq@vspan+\ht\EQ@numbox-\dp\EQ@numbox)/2}%
+  \eq@typeset@rightnumber
+  \eq@typeset@frame
+  \eq@typeset@equation
+}
+%    \end{macrocode}
+% Typeset equation and right bottom number
+%    \begin{macrocode}
+\def\eq@typeset@RB{%
+  % NOTE: is \eq@dp useful here
+  \setlength\dim@a{\eq@vspan-\ht\EQ@numbox-\dp\EQ@numbox}%
+  \eq@typeset@rightnumber
+  \eq@typeset@frame
+  \eq@typeset@equation
+}
+%    \end{macrocode}
+% Typeset equation and right shifted number
+%    \begin{macrocode}
+\def\eq@typeset@RShifted{%
+  % place number
+  \eq@typeset@frame
+  \eq@typeset@equation
+  \penalty\@M
+  \dim@a\eqlineskip
+  \if F\eq@frame\else
+    \addtolength\dim@a{\eq@framesep+\eq@framewd}%
+  \fi
+  \parskip\dim@a
+  \hbox to\hsize{\hfil\copy\EQ@numbox}\@@par%
+}
+%    \end{macrocode}
+%
+% Debugging aid to show all relevant formatting info for a given eqn.
+%    \begin{macrocode}
+%<*trace>
+\def\debug@showformat{%
+  \breqn@debugmsg{Formatting Layout:\eq@layout\space Center/indent: \eqindent\space
+    Number placement \eqnumside\eqnumplace:
+    \MessageBreak==== \eq@linewidth=\the\eq@linewidth, \@totalleftmargin=\the\@totalleftmargin,
+    \MessageBreak==== Centered Lines=\theb@@le\eq@centerlines, Shift Number=\theb@@le\eq@shiftnumber,
+    \MessageBreak==== \eq@wdT=\the\eq@wdT, \eq@wdMin=\the\eq@wdMin,
+    \MessageBreak==== LHS=\theb@@le\EQ@hasLHS: \eq@wdL=\the\eq@wdL,
+    \MessageBreak==== \eq@firstht=\the\eq@firstht, \eq@vspan=\the\eq@vspan
+    \MessageBreak==== \eq@wdNum=\the\eq@wdNum
+    \MessageBreak==== \eq@wdCond=\the\eq@wdCond, \conditionsep=\the\conditionsep,
+    \MessageBreak==== \leftskip=\the\leftskip, \rightskip=\the\rightskip,
+    \MessageBreak==== \abovedisplayskip=\the\abovedisplayskip,
+    \MessageBreak==== \belowdisplayskip=\the\belowdisplayskip
+    \MessageBreak==== parshape=\eq@parshape}%
+}
+%</trace>
+%    \end{macrocode}
+%
+% Set left \& right skips for centered equations,
+% making allowances for numbers (if any, right, left) and constraint.
+%
+% Amazingly, I've managed to collect all the positioning logic for
+% centered equations in one place, so it's more manageable.
+% Unfortunately, by the time it does all it needs to do,
+% it has evolved I'm (re)using so many temp variables, it's becoming
+% unmanageble!
+%
+%    \begin{macrocode}
+\def\eq@C@setsides{%
+  % \dim@c = space for number, if any, and not shifted.
+  \dim@c\z@
+  \if\eq@hasNumber\if\eq@shiftnumber\else
+    \dim@c\eq@wdNum
+  \fi\fi
+  % \dim@e = space for condition(on right), if any and formula is only a single line.(to center nicely)
+  % but only count it as being right-aligned if we're not framing, since the frame must enclose it.
+  \dim@e\z@
+  \if F\eq@frame
+    \ifnum\eq@lines=\@ne\ifdim\eq@wdCond>\z@
+      \setlength\dim@e{\eq@wdCond+\conditionsep}%
+  \fi\fi\fi
+  % \dim@b = minimum needed on left max(totalleftmargin, left number space)
+  \dim@b\z@
+  \if L\eqnumside\ifdim\dim@b<\dim@c
+    \dim@b\dim@c
+  \fi\fi
+  \ifdim\dim@b<\@totalleftmargin
+    \dim@b\z@
+  \else
+    \addtolength\dim@b{-\@totalleftmargin}%
+  \fi
+  % \dim@d = minimum needed on right max(condition, right number space)
+  \dim@d\dim@e
+  \if R\eqnumside\ifdim\dim@d<\dim@c
+    \dim@d\dim@c
+  \fi\fi
+  % \dim@a = left margin; initially half available space
+  % \dim@c = right margin;  ditto
+  \setlength\dim@a{(\eq@linewidth-\eq@wdT+\dim@e+\@totalleftmargin)/2}%
+  \dim@c=\dim@a
+  % If too far to the left
+  \ifdim\dim@a<\dim@b
+     \addtolength\dim@c{\dim@a-\dim@b}%
+     \ifdim\dim@c<\z@\dim@c=\z@\fi
+     \dim@a=\dim@b
+  % Or if too far to the right
+  \else\ifdim\dim@c<\dim@d
+     \addtolength\dim@a{\dim@c-\dim@d}%
+     \ifdim\dim@a<\z@\dim@a=\z@\fi
+     \dim@c=\dim@d
+  \fi\fi
+  % Now, \dim@d,\dim@e is the left & right glue to center each line for centerlines
+  \setlength\dim@e{\eq@wdT-\eq@wdMin}\dim@d=\z@
+%    \end{macrocode}
+% NOTE: Need some work here centering when there's a condition
+%    \begin{macrocode}
+%  \advance\dim@e-\eq@wdT\multiply\dim@e-1\relax
+%  \if\eq@wdMin<\dim@e\dim@e\eq@wdMin\fi
+%  \multiply\dim@e-1\relax\advance\dim@e\eq@wdT
+  \dim@d\z@
+  \if\eq@centerlines
+    \divide\dim@e2\relax
+    \dim@d=\dim@e
+  \fi
+  \setlength\leftskip{\dim@a\@plus\dim@d}%
+  \addtolength\dim@e{\dim@c}%
+  \setlength\rightskip{\z@\@plus\dim@e}%\@minus5\p@
+  % Special case: if framing, reduce the stretchiness of the formula (eg. condition)
+  % Or if we have a right number, FORCE space for it
+  \dim@b\z@
+  \if F\eq@frame\else
+    \dim@b\dim@c
+  \fi
+  \if\@And{\eq@hasNumber}{\@Not{\eq@shiftnumber}}%
+    \if R\eqnumside
+      \dim@c\eq@wdNum
+      \ifdim\dim@c>\dim@b
+        \dim@b\dim@c
+      \fi
+    \fi
+  \fi
+  % If either of those cases requires hard rightskip, move that part from glue.
+  \ifdim\dim@b>\z@
+    \addtolength\dim@e{-\dim@c}%
+    \rightskip\dim@b\@plus\dim@e%\@minus5\p@
+  \fi
+  % And peculiar further special case: in indented environs, width isn't where it would seem
+  \ifdim\eq@wdCond>\z@
+    \addtolength\rightskip{-\@totalleftmargin}%
+  \fi
+  \parfillskip\z@skip
+}
+%    \end{macrocode}
+%
+% Set the left and right side spacing for indented equations
+% Some things handled by eq@C@setsides that probably apply here????
+% \begin{itemize}
+% \item centerlines
+% \item \cs{@totalleftmargin}: SHOULD we move farther right?
+% \end{itemize}
+% Leftskip is normally just the requested indentation
+%    \begin{macrocode}
+\def\eq@I@setsides{%
+  \leftskip\mathindent
+%    \end{macrocode}
+% But move left, if shifted number presumably because of clashed w/ number?
+%    \begin{macrocode}
+  \if\eq@shiftnumber
+    \setlength\dim@a{\eq@linewidth-\eq@wdT-\mathindent}%
+    \ifdim\dim@a<\z@
+      \leftskip=\z@ % Or something minimal?
+    \fi
+  \fi
+%    \end{macrocode}
+% Push gently from right.
+%    \begin{macrocode}
+  \dim@a=\z@
+  \setlength\dim@b{\eq@linewidth-\leftskip-\eq@wdMin}%
+%    \end{macrocode}
+% Special case: if framing be much more rigid(?)
+%    \begin{macrocode}
+  \if F\eq@frame
+  \else
+    \setlength\dim@a{\eq@linewidth-\leftskip-\eq@wdT}
+    \addtolength\dim@b{-\dim@a}%
+  \fi
+  % Or force the space for right number, if needed
+%    \begin{macrocode}
+  \if\@And{\eq@hasNumber}{\@Not{\eq@shiftnumber}}%
+    \if R\eqnumside
+      \dim@c=\eq@wdNum
+      \if\dim@c>\dim@a
+        \addtolength\dim@b{-\dim@c}%
+        \dim@a=\dim@c
+      \fi
+    \fi
+  \fi
+  \setlength\rightskip{\dim@a\@plus\dim@b \@minus\hfuzz }%\hfuzz\z@
+  \parfillskip\z@skip
+}
+%    \end{macrocode}
+% \paragraph{Typesetting pieces: frame, equation and number (if any)}
+% \cs{dim@a} should contain the downward displacement of number's baseline
+%    \begin{macrocode}
+\def\eq@typeset@leftnumber{%
+  \setlength\skip@c{\dim@a-\ht\EQ@numbox}%
+  \vglue\skip@c% NON discardable
+  \copy\EQ@numbox \penalty\@M
+  \kern-\dim@a
+}
+\def\eq@typeset@rightnumber{%
+  \setlength\skip@c{\dim@a-\ht\EQ@numbox}%
+  \vglue\skip@c% NON discardable
+  \hbox to \hsize{\hfil\copy\EQ@numbox}\penalty\@M
+  \kern-\dim@a
+}
+\def\eq@typeset@equation{%
+  \nobreak
+  \eq@params\eq@parshape
+  \nointerlineskip\noindent
+  \add@grp@label
+  \eq@dump@box\unhbox\EQ@box\@@par
+}
+%    \end{macrocode}
+%
+% \section{Framing an equation}
+% \begin{macro}{\eqframe}
+% The \cs{eqframe} function is called in vertical mode
+% with the reference point at the top left corner of the equation, including
+% any allowance for \cs{fboxsep}.    Its arguments are the width
+% and height of the equation body, plus fboxsep.
+% \changes{v0.95}{2007/12/03}{Made \cs{eqframe} obey the key settings
+%   for frame and framesep.}
+%    \begin{macrocode}
+\newcommand\eqframe[2]{%
+  \begingroup
+  \fboxrule=\eq@framewd\relax\fboxsep=\eq@framesep\relax
+  \framebox{\z@rule\@height#2\kern#1}%
+  \endgroup
+}
+%    \end{macrocode}
+% \end{macro}
+% The frame is not typeset at the correct horizontal position. Will
+% fix later.
+%    \begin{macrocode}
+\def\eq@addframe{%
+  \hbox to\z@{%
+    \setlength\dim@a{\eq@framesep+\eq@framewd}%
+    \kern-\dim@a
+    \vbox to\z@{\kern-\dim@a
+      \hbox{\eqframe{\eq@wdT}{\eq@vspan}}%
+      \vss
+    }%
+    \hss
+  }%
+}
+\def\eq@typeset@frame{%
+  \if F\eq@frame\else
+   % Tricky: put before \noindent, so it's not affected by glue in \leftskip
+   \nobreak\nointerlineskip
+   \vbox to\eq@firstht{\moveright\leftskip\hbox to\z@{\eq@addframe\hss}\vss}%
+   \kern-\eq@firstht
+  \fi
+}
+%    \end{macrocode}
+%
+% \section{Delimiter handling}
+% The special handling of delimiters is rather complex, but
+% everything is driven by two motives: to mark line breaks inside
+% delimiters as less desirable than line breaks elsewhere, and to make it
+% possible to break open left-right boxes so that line breaks between
+% \cs{left} and \cs{right} delimiters are not absolutely
+% prohibited.    To control the extent to which line breaks will be
+% allowed inside delimiters, set \cs{eqbreakdepth} to the maximum
+% nesting depth.    Depth 0 means never break inside delimiters.
+%
+% Note: \cs{eqbreakdepth} is not implemented as a \latex
+% counter because changes done by \cs{setcounter} \etc  are always
+% global.
+%
+% It would be natural to use grouping in the implementation
+% \mdash  at an open delimiter, start a group and increase mathbin
+% penalties; at a close delimiter, close the group.    But this gives us
+% trouble in situations like the \env{array} environment, where a
+% close delimiter might fall in a different cell of the \cs{halign}
+% than the open delimiter.
+% Ok then, here's what we want the various possibilities to
+% expand to.    Note that \cs{right} and \cs{biggr} are
+% being unnaturally applied to a naturally open-type delimiter.
+% \begin{literalcode}
+% ( -> \delimiter"4... \after@open
+% \left( ->
+%   \@@left \delimiter"4... \after@open
+% \right( ->
+%   \@@right \delimiter"4... \after@close
+% \biggl( ->
+%   \mathopen{\@@left \delimiter... \vrule...\@@right.}
+%   \after@open
+% \biggr( ->
+%   \mathclose{\@@left \delimiter... \vrule...\@@right.}
+%   \after@close
+% \bigg\vert ->
+%   \mathord{\@@left \delimiter... \vrule...\@@right.}
+% \biggm\vert ->
+%   \mathrel{\@@left \delimiter... \vrule...\@@right.}
+% \end{literalcode}
+%
+% First save the primitive meanings of \cs{left} and
+% \cs{right}.
+%    \begin{macrocode}
+\@saveprimitive\left\@@left
+\@saveprimitive\right\@@right
+%    \end{macrocode}
+%
+% The variable \cs{lr@level} is used by the first mathrel in
+% an equation to tell whether it is at top level: yes? break and measure
+% the LHS, no? keep going.
+%    \begin{macrocode}
+\newcount\lr@level
+%    \end{macrocode}
+%
+% It would be nice to have better error checking here if the
+% argument is not a delimiter symbol at all.
+%
+% Ah, a small problem when renaming commands. In the original version,
+% |\delimiter| is hijacked in order to remove the |\after@bidir| (or
+% open or close) instruction following the delimiter declaration.
+%    \begin{macrocode}
+\ExplSyntaxOn
+\def\eq@left{%
+  \@ifnext .{\eq@nullleft}{\begingroup \let\math_delimiter:NNnNn \eq@left@a}%
+}
+\def\eq@right{%
+  \@ifnext .{\eq@nullright}{\begingroup \let \math_delimiter:NNnNn \eq@right@a}%
+}
+%    \end{macrocode}
+% The arguments are: \arg1  delim symbol, \arg2 .
+%    \begin{macrocode}
+%\def\eq@left@a#1 #2{\endgroup\@@left\delimiter#1\space \after@open}
+\def\eq@left@a#1#2#3#4#5#6{\endgroup
+  \@@left \math_delimiter:NNnNn #1#2{#3}#4{#5}\after@open}
+\def\eq@right@a#1#2#3#4#5#6{\endgroup
+  \@@right \math_delimiter:NNnNn #1#2{#3}#4{#5}\after@close \ss@scan{#1#2{#3}#4{#5}}%
+}
+\ExplSyntaxOff
+%    \end{macrocode}
+% The null versions.
+%    \begin{macrocode}
+\def\eq@nullleft#1{\@@left#1\after@open}
+\def\eq@nullright#1{\@@right#1\after@close}
+%    \end{macrocode}
+%
+% Here is the normal operation of \cs{biggl}, for example.
+% \begin{literalcode}
+% \biggl ->\mathopen \bigg
+% {\mathopen}
+%
+% \bigg #1->{\hbox {$\left #1\vbox to14.5\p@ {}\right .\n@space $}}
+% #1<-(
+% \end{literalcode}
+% ^^AFor paren matching: )
+% Like \cs{left}, \cs{biggl} coerces its delimiter to be of
+% mathopen type even if its natural inclination is towards closing.
+%
+% The function \cs{delim@reset} makes delimiter characters
+% work just about the same as they would in normal \latex .
+%    \begin{macrocode}
+\def\delim@reset{%
+  \let\after@open\relax \let\after@close\relax
+  \let\left\@@left \let\right\@@right
+}
+%    \end{macrocode}
+% If the \pkg{amsmath} or \pkg{exscale} package is loaded, it
+% will have defined \cs{bBigg@}; if not, the macros \cs{big} and
+% variants will have hard-coded point sizes as inherited through the ages
+% from \fn{plain.tex}.    In this case we can kluge a little by
+% setting \cs{big@size} to \cs{p@}, so that our definition of
+% \cs{bBigg@} will work equally well with the different multipliers.
+%    \begin{macrocode}
+\@ifundefined{bBigg@}{% not defined
+  \let\big@size\p@
+  \def\big{\bBigg@{8.5}}\def\Big{\bBigg@{11.5}}%
+  \def\bigg{\bBigg@{14.5}}\def\Bigg{\bBigg@{17.5}}%
+  \def\biggg{\bBigg@{20.5}}\def\Biggg{\bBigg@{23.5}}%
+}{}
+\def\bBigg@#1#2{%
+  {\delim@reset
+   \left#2%
+   \vrule\@height#1\big@size\@width-\nulldelimiterspace
+   \right.
+  }%
+}
+%    \end{macrocode}
+% .
+%    \begin{macrocode}
+\def\bigl#1{\mathopen\big{#1}\after@open}
+\def\Bigl#1{\mathopen\Big{#1}\after@open}
+\def\biggl#1{\mathopen\bigg{#1}\after@open}
+\def\Biggl#1{\mathopen\Bigg{#1}\after@open}
+\def\bigggl#1{\mathopen\biggg{#1}\after@open}
+\def\Bigggl#1{\mathopen\Biggg{#1}\after@open}
+
+\def\bigr#1{\mathclose\big{#1}\after@close}
+\def\Bigr#1{\mathclose\Big{#1}\after@close}
+\def\biggr#1{\mathclose\bigg{#1}\after@close}
+\def\Biggr#1{\mathclose\Bigg{#1}\after@close}
+\def\bigggr#1{\mathclose\biggg{#1}\after@close}
+\def\Bigggr#1{\mathclose\Biggg{#1}\after@close}
+
+%% No change needed, I think. [mjd,1998/12/04]
+%%\def\bigm{\mathrel\big}
+%%\def\Bigm{\mathrel\Big}
+%%\def\biggm{\mathrel\bigg}
+%%\def\Biggm{\mathrel\Bigg}
+%%\def\bigggm{\mathrel\biggg}
+%%\def\Bigggm{\mathrel\Biggg}
+%    \end{macrocode}
+%
+%
+% \begin{macro}{\m@@DeL} \begin{macro}{\d@@DeL}
+% \begin{macro}{\m@@DeR} \begin{macro}{\d@@DeR}
+% \begin{macro}{\m@@DeB} \begin{macro}{\d@@DeB}
+% Original definition of \cs{m@DeL} from
+% \pkg{flexisym} is as follows.    \cs{m@DeR} and
+% \cs{m@DeB} are the same except for the math class number.
+% \begin{literalcode}
+% \def\m@DeL#1#2#3{%
+%   \delimiter"4\@xp\delim@a\csname sd@#1#2#3\endcsname #1#2#3 }
+% \end{literalcode}
+%
+% Save the existing meanings of \cs{m@De[LRB]}.
+%
+% Define display variants of DeL, DeR, DeB
+%    \begin{macrocode}
+\ExplSyntaxOn
+\cs_set:Npn \math_dsym_DeL:Nn #1#2{\math_bsym_DeL:Nn #1{#2}\after@open}
+\cs_set:Npn \math_dsym_DeR:Nn #1#2{\math_bsym_DeR:Nn #1{#2}\after@close}
+\cs_set:Npn \math_dsym_DeB:Nn #1#2{\math_bsym_DeB:Nn #1{#2}\after@bidir}
+
+%%%%%
+%%%%%\let\m@@DeL\m@DeL \let\m@@DeR\m@DeR \let\m@@DeB\m@DeB
+%%%%%\def\d@@DeL#1#2#3{%
+%%%%%  \delimiter"4\@xp\delim@a\csname sd@#1#2#3\endcsname #1#2#3 \after@open
+%%%%%}
+%%%%%\def\d@@DeR#1#2#3{%
+%%%%%  \delimiter"5\@xp\delim@a\csname sd@#1#2#3\endcsname #1#2#3 \after@close
+%%%%%}
+%%%%%\def\d@@DeB#1#2#3{%
+%%%%%  \delimiter"0\@xp\delim@a\csname sd@#1#2#3\endcsname #1#2#3 \after@bidir
+%%%%%}
+%    \end{macrocode}
+%%BRM: These weren't defined, but apparently should be.
+% Are these the right values???
+%    \begin{macrocode}
+%%%%%%\let\m@@DeA\m@DeA\let\d@@DeA\m@DeA%
+%    \end{macrocode}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+%
+% \begin{macro}{\after@open}
+% \begin{macro}{\after@close}
+% \begin{macro}{\after@bidir}
+% \begin{macro}{\zero@bop}
+% \begin{macro}{\bop@incr}
+% \cs{after@open} and \cs{after@close} are carefully
+% written to avoid the use of grouping and to run as fast as possible.
+% \cs{zero@bop} is the value used for \cs{prebinoppenalty} at
+% delimiter level 0, while \cs{bop@incr} is added for each level of
+% nesting.    The standard values provide that breaks will be prohibited
+% within delimiters below nesting level 2.
+%    \begin{macrocode}
+\let\after@bidir\@empty
+\mathchardef\zero@bop=888 \relax
+\mathchardef\bop@incr=4444 \relax
+\def\after@open{%
+  \global\advance\lr@level\@ne
+  \prebinoppenalty\bop@incr \multiply\prebinoppenalty\lr@level
+  \advance\prebinoppenalty\zero@bop
+  \ifnum\eqbreakdepth<\lr@level
+     \cs_set_eq:NN \math_sym_Bin:Nn \math_isym_Bin:Nn %%%%%%\let\m@Bin\m@@Bin
+%    \end{macrocode}
+% Inside delimiters, add some fillglue before binops so that a broken off
+% portion will get thrown flush right.    Also shift it slightly
+% further to the right to ensure that it clears the opening delimiter.
+%    \begin{macrocode}
+  \else
+    \eq@binoffset=\eqbinoffset
+    \advance\eq@binoffset\lr@level\eqdelimoffset plus1fill\relax
+    \def\dt@fill@cancel{\hskip\z@ minus1fill\relax}%
+  \fi
+  \penalty\@M % BRM: discourage break after an open fence?
+}
+\def\after@close{%
+  \global\advance\lr@level\m@ne
+  \prebinoppenalty\bop@incr \multiply\prebinoppenalty\lr@level
+  \advance\prebinoppenalty\zero@bop
+  \ifnum\eqbreakdepth<\lr@level
+  \else \cs_set_eq:NN \math_sym_Bin:Nn \math_dsym_Bin:Nn %%%%%%\let\m@Bin\d@@Bin
+  \fi
+%    \end{macrocode}
+% When we get back to level 0, no delimiters, remove the stretch
+% component of \cs{eqbinoffset}.
+%    \begin{macrocode}
+  \ifnum\lr@level<\@ne \eq@binoffset=\eqbinoffset\relax \fi
+}
+
+\ExplSyntaxOff
+
+%    \end{macrocode}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+% \end{macro}
+%
+%
+% \begin{macro}{\subsup@flag}
+% \begin{macro}{\ss@scan}
+% \cs{ss@scan} is called after a \cs{right} delimiter and
+% looks ahead for sub and superscript tokens.
+% If sub and/or superscripts are present, we adjust the line-ending
+% penalty to distinguish the various cases (sub, sup, or both).
+% This facilitates the later work of excising the sub/sup box and
+% reattaching it with proper shifting.
+%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% Sub/Superscript measurement
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% BRM: There's possibly a problem here.
+% When \cs{ss@scan} gets invoked after a \cs{left}...\cs{right} pair in the LHS
+% during \cs{eq@measure}, it produces an extra box (marked with \cs{penalty} 3);
+% Apparently \cs{eq@repack} expects only one for the LHS.
+% The end result is \cs{eq@wdL} => 0.0pt !!! (or at least very small)
+%    \begin{macrocode}
+\let\subsup@flag=\count@
+\def\ss@delim@a@new#1#2#3#4#5{\xdef\right@delim@code{\number"#4#5}}
+%    \end{macrocode}
+% The argument of \cs{ss@scan} is an expanded form of a
+% right-delimiter macro.
+% We want to use the last three digits in the expansion
+% to define \cs{right@delim@code}.
+% The assignment to a temp register is just a way to scan away the
+% leading digits that we don't care about.
+%    \begin{macrocode}
+\def\ss@scan#1{%
+%    \end{macrocode}
+% This part of the code.
+%    \begin{macrocode}
+  \begingroup
+    \ss@delim@a@new #1%
+  \endgroup
+  \subsup@flag\@M \afterassignment\ss@scan@a \let\@let@token=}
+\def\ss@scan@a{%
+  \let\breqn@next\ss@scan@b
+  \ifx\@let@token\sb \advance\subsup@flag\@ne\else
+  \ifx\@let@token\@@subscript \advance\subsup@flag\@ne\else
+  \ifx\@let@token\@@subscript@other \advance\subsup@flag\@ne\else
+  \ifx\@let@token\sp \advance\subsup@flag\tw@\else
+  \ifx\@let@token\@@superscript \advance\subsup@flag\tw@\else
+  \ifx\@let@token\@@superscript@other \advance\subsup@flag\tw@\else
+    \ss@finish
+    \let\breqn@next\relax
+  \fi\fi\fi\fi\fi\fi
+  \breqn@next\@let@token
+}
+%    \end{macrocode}
+%
+%    \begin{macrocode}
+\ExplSyntaxOn
+\def\ss@scan@b#1#2{#1{%
+%    \end{macrocode}
+% hack! coff!
+%    \begin{macrocode}
+  %%%%%\let\m@Bin\m@@Bin  \let\m@Rel\m@@Rel
+  \cs_set_eq:NN \math_sym_Bin:Nn \math_isym_Bin:Nn
+  \cs_set_eq:NN \math_sym_Rel:Nn \math_isym_Rel:Nn
+  #2}\afterassignment\ss@scan@a \let\@let@token=}%
+\ExplSyntaxOff
+%    \end{macrocode}
+% We need to keep following glue from disappearing
+% \mdash  \eg , a thickmuskip or medmuskip from a following mathrel or
+% mathbin symbol.
+%    \begin{macrocode}
+\def\ss@finish{%
+  \@@vadjust{\penalty\thr@@}%
+  \penalty\right@delim@code \penalty-\subsup@flag \keep@glue
+}
+%    \end{macrocode}
+% \end{macro}
+% \end{macro}
+%
+%
+% \begin{macro}{\eq@lrunpack}
+% For \cs{eq@lrunpack} we need to break open a left-right box and
+% reset it just in case it contains any more special breaks.    After
+% it is unpacked the recursion of \cs{eq@repack} will continue,
+% acting on the newly created lines.
+%    \begin{macrocode}
+\def\eq@lrunpack{\setbox\z@\lastbox
+%    \end{macrocode}
+% We remove the preceding glue item and deactivate
+% baselineskip for the next line, otherwise we would end up with
+% three items of glue (counting parskip) at this point instead of
+% the single one expected by our recursive repacking
+% procedure.
+%    \begin{macrocode}
+  \unskip \nointerlineskip
+%    \end{macrocode}
+% Then we open box 0, take the left-right box at the right end of
+% it, and break that open.    If the line-ending penalty is greater than
+% 10000, it means a sub and/or superscript is present on the right
+% delimiter and the box containing them must be taken off first.
+%    \begin{macrocode}
+  \noindent\unhbox\z@ \unskip
+  \subsup@flag-\lastpenalty \unpenalty
+  \xdef\right@delim@code{\number\lastpenalty}%
+  \unpenalty
+  \ifnum\subsup@flag>\@M
+    \advance\subsup@flag-\@M
+    \setbox\tw@\lastbox
+  \else \setbox\tw@\box\voidb@x
+  \fi
+  \setbox\z@\lastbox
+  \ifvoid\tw@ \unhbox\z@
+  \else \lrss@reattach % uses \subsup@flag, box\z@, box\tw@
+  \fi
+%    \end{macrocode}
+% The reason for adding a null last line here is that the last
+% line will contain parfillskip in addition to rightskip, and a final
+% penalty of $10000$ instead of $-1000N$
+% ($1\leq N\leq 9$), which would interfere with the usual
+% processing.    Setting a null last line and discarding it dodges
+% this complication.    The penalty value $-10001$ is a no-op case
+% in the case statement of \cs{eq@repacka}.
+%    \begin{macrocode}
+  \penalty-\@Mi\z@rule\@@par
+  \setbox\z@\lastbox \unskip\unpenalty
+%%{\showboxbreadth\maxdimen\showboxdepth99\showlists}%
+}
+%    \end{macrocode}
+% \end{macro}
+%
+%
+% \begin{macro}{\lrss@reattach}
+%
+% Well, for a small self-contained computation, carefully
+% hand-allocated dimens should be safe enough.    But let the
+% maintainer beware!    This code cannot be arbitrarily transplanted
+% or shaken up without regard to grouping and interaction with other
+% hand-allocated dimens.
+%    \begin{macrocode}
+\dimendef\sub@depth=8 \dimendef\sup@base=6
+\dimendef\prelim@sub@depth=4 \dimendef\prelim@sup@base=2
+\def\sym@xheight{\fontdimen5\textfont\tw@}
+\def\sup@base@one{\fontdimen13\textfont\tw@}
+\def\sub@base@one{\fontdimen16\textfont\tw@}
+\def\sub@base@two{\fontdimen17\textfont\tw@}
+%    \end{macrocode}
+% Note that only \cs{sup@drop} and \cs{sub@drop} come from
+% the next smaller math style.
+%    \begin{macrocode}
+\def\sup@drop{\fontdimen18\scriptfont\tw@}
+\def\sub@drop{\fontdimen19\scriptfont\tw@}
+%    \end{macrocode}
+% Provide a mnemonic name for the math axis fontdimen, if it's not
+% already defined.
+%    \begin{macrocode}
+\providecommand{\mathaxis}{\fontdimen22\textfont\tw@}
+%    \end{macrocode}
+%
+% Assumes box 2 contains the sub/sup and box 0 contains the left-right
+% box.    This is just a repeat of the algorithm in \fn{tex.web},
+% with some modest simplifications from knowing that this is only going to
+% be called at top level in a displayed equation, thus always mathstyle =
+% uncramped displaystyle.
+%    \begin{macrocode}
+\def\lrss@reattach{%
+  \begingroup
+  % "The TeXbook" Appendix G step 18:
+  \setlength\prelim@sup@base{\ht\z@-\sup@drop}%
+  \setlength\prelim@sub@depth{\dp\z@ +\sub@drop}%
+  \unhbox\z@
+  \ifcase\subsup@flag      % case 0: this can't happen
+  \or \lr@subscript   % case 1: subscript only
+  \or \lr@superscript % case 2: superscript only
+  \else \lr@subsup    % case 3: sub and superscript both
+  \fi
+  \endgroup
+}
+%    \end{macrocode}
+%    \begin{macrocode}
+\def\lr@subscript{%
+  \sub@depth\sub@base@one
+  \ifdim\prelim@sub@depth>\sub@depth \sub@depth\prelim@sub@depth\fi
+  \setlength\dim@a{\ht\tw@  -.8\sym@xheight}%
+  \ifdim\dim@a>\sub@depth \sub@depth=\dim@a \fi
+  \twang@adjust\sub@depth
+  \lower\sub@depth\box\tw@
+}
+%    \end{macrocode}
+%
+%    \begin{macrocode}
+\def\lr@superscript{%
+  \sup@base\sup@base@one
+  \ifdim\prelim@sup@base>\sup@base \sup@base\prelim@sup@base\fi
+  \setlength\dim@a{\dp\tw@ -.25\sym@xheight}%
+  \ifdim\dim@a>\sup@base \sup@base\dim@a \fi
+  \twang@adjust\sup@base
+  \raise\sup@base\box\tw@
+}
+%    \end{macrocode}
+%
+%    \begin{macrocode}
+\def\lr@subsup{%
+  \sub@depth\sub@base@two
+  \ifdim\prelim@sub@depth>\sub@depth \sub@depth\prelim@sub@depth \fi
+  \twang@adjust\sub@depth
+  \lower\sub@depth\box\tw@
+}
+%    \end{macrocode}
+%
+% For delimiters that curve top and bottom, the twang factor allows
+% horizontal shifting of the sub and superscripts so they don't
+% fall too far away (or too close for that matter).    This is
+% accomplished by arranging for (\eg ) \verb"\right\rangle" to leave
+% a penalty $N$ in the math list before the subsup penalty that triggers
+% \cs{lrss@reattach}, where $N$ is the mathcode of
+% \cs{rangle} (ignoring \dquoted{small} variant).
+%    \begin{macrocode}
+\def\twang@adjust#1{%
+  \begingroup
+    \@ifundefined{twang@\right@delim@code}{}{%
+      \setlength\dim@d{#1-\mathaxis}%
+      % put an upper limit on the adjustment
+      \ifdim\dim@d>1em \dim@d 1em \fi
+      \kern\csname twang@\right@delim@code\endcsname\dim@d
+    }%
+  \endgroup
+}
+%    \end{macrocode}
+% The method used to apply a \dquoted{twang} adjustment is just an
+% approximate solution to a complicated problem.
+% We make the following assumptions that hold true, approximately,
+% for the most common kinds of delimiters:
+% \begin{enumerate}
+% \item
+% The right delimiter is symmetrical top to bottom.
+%
+%
+% \item There is an upper limit on the size of the adjustment.
+%
+%
+% \item When we have a superscript, the amount of left-skew that we
+% want to apply is linearly proportional to the distance of the bottom
+% left corner of the superscript from the math axis, with the ratio
+% depending on the shape of the delimiter symbol.
+%
+%
+% \end{enumerate}
+% .
+% By symmetry, Assumption 3 is true also for subscripts (upper left
+% corner).
+% Assumption 2 is more obviously true for parens and braces, where the
+% largest super-extended versions consist of truly vertical parts with
+% slight bending on the ends, than it is for a \cs{rangle}.
+% But suppose for the sake of expediency that it is
+% approximately true for rangle symbols also.
+%
+%
+% Here are some passable twang factors for the most common types of
+% delimiters in \fn{cmex10}, as determined by rough measurements from
+% magnified printouts.
+% \begin{literalcode}
+%   vert bar, double vert:  0
+%          square bracket: -.1
+%             curly brace: -.25
+%             parenthesis: -.33
+%                  rangle: -.4
+% \end{literalcode}
+% Let's provide a non-private command for changing the twang factor of
+% a given symbol.
+%    \begin{macrocode}
+\newcommand{\DeclareTwang}[2]{%
+  \ifcat.\@nx#1\begingroup
+    \lccode`\~=`#1\lowercase{\endgroup \DeclareTwang{~}}{#2}%
+  \else
+    \@xp\decl@twang#1?\@nil{#2}%
+  \fi
+}
+%    \end{macrocode}
+% Note that this is dependent on a fixed interpretation of the
+% mathgroup number \arg4 .
+%    \begin{macrocode}
+\def\decl@twang#1#2#3#4#5#6#7\@nil#8{%
+  \@namedef{twang@\number"#4#5#6}{#8}%
+}
+\DeclareTwang{\rangle}{-.4}
+\DeclareTwang{)}{-.33}
+\DeclareTwang{\rbrace}{-.25}
+%    \end{macrocode}
+% \end{macro}
+%
+%
+%
+% \section{Series of expressions}
+% The \env{dseries} environment is for a display
+% containing a series of expressions of the form \quoted{A, B} or \quoted{A and
+% B} or \quoted{A, B, and C} and so on.    Typically the expressions
+% are separated by a double quad of space.    If the expressions in a
+% series don't all fit in a single line, they are continued onto extra
+% lines in a ragged-center format.
+%    \begin{macrocode}
+\newenvironment{dseries}{\let\eq@hasNumber\@True \@optarg\@dseries{}}{}%
+\def\enddseries#1{\check@punct@or@qed}%
+%    \end{macrocode}
+%
+% And the unnumbered version of same.
+%    \begin{macrocode}
+\newenvironment{dseries*}{\let\eq@hasNumber\@False \@optarg\@dseries{}}{}%
+\@namedef{enddseries*}#1{\check@punct@or@qed}%
+\@namedef{end@dseries*}{\end@dseries}%
+\def\@dseries[#1]{%
+%    \end{macrocode}
+% Turn off the special breaking behavior of mathrels \etc  for math
+% formulas embedded in a \env{dseries} environment.
+%
+%BRM: DS Expermient: Use alternative display setup.
+%    \begin{macrocode}
+%  \def\display@setup{\displaystyle}%
+  \let\display@setup\dseries@display@setup
+  % Question: should this be the default for dseries???
+%  \let\eq@centerlines\@True
+  \global\eq@wdCond\z@
+%    \end{macrocode}
+% BRM: use special layout for dseries
+%    \begin{macrocode}
+%  \@dmath[#1]%
+  \@dmath[layout={M},#1]%
+  \mathsurround\z@\@@math \penalty\@Mi
+  \let\endmath\ends@math
+  \def\premath{%
+%    \end{macrocode}
+% BRM: Tricky to cleanup space OR add space ONLY BETWEEN math!
+%    \begin{macrocode}
+    \ifdim\lastskip<.3em \unskip
+    \else\ifnum\lastpenalty<\@M \dquad\fi\fi
+}%
+%    \end{macrocode}
+%BRM: Tricky; if a subformula breaks, we'd like to start the next on new line!
+%    \begin{macrocode}
+  \def\postmath{\unpenalty\eq@addpunct \penalty\intermath@penalty \dquad \@ignoretrue}%
+\ignorespaces
+}
+\def\end@dseries{%
+  \unskip\unpenalty
+  \@@endmath \mathsurround\z@ \end@dmath
+}
+%    \end{macrocode}
+% BRM: Try this layout for dseries: Essentially layout i, but w/o
+% limit to 1 line.  And no fallback!
+%    \begin{macrocode}
+\def\eq@try@layout@M{%
+  \edef\@parshape{\parshape 1 0pt \the\eq@linewidth\relax}%
+  \eq@trial@b{M}{}%
+}
+%    \end{macrocode}
+% BRM: Tricky to get right value here.
+% Prefer breaks between formula if we've got to break at all.
+%    \begin{macrocode}
+%\def\intermath@penalty{-201}%
+\def\intermath@penalty{-221}%
+%    \end{macrocode}
+% BRM: A bit tighter than it was ( 1em minus.25em )
+%    \begin{macrocode}
+%\newcommand\dquad{\hskip0.4em}
+\newcommand\dquad{\hskip0.6em minus.3em}
+\newcommand\premath{}\newcommand\postmath{}
+%    \end{macrocode}
+%
+% Change the \env{math} environment to add
+% \cs{premath} and \cs{postmath}.    They are no-ops except
+% inside a \env{dseries} environment.
+%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% Redefinition of math environment to take advantage of dseries env.
+%    \begin{macrocode}
+\renewenvironment{math}{%
+  \leavevmode \premath
+  \ifmmode\@badmath\else\@@math\fi
+}{%
+  \ifmmode\@@endmath\else\@badmath\fi
+}
+\def\ends@math#1{\check@punct@or@qed}
+\def\end@math{%
+  \ifmmode\@@endmath\else\@badmath\fi
+  \postmath
+}
+%    \end{macrocode}
+%
+%
+%
+%
+% \section{Equation groups}
+% For many equation groups the strategy is easy: just center each
+% equation individually following the normal rules for a single
+% equation.    In some groups, each equation gets its own number; in
+% others, a single number applies to the whole group (and may need to be
+% vertically centered on the height of the group).    In still other
+% groups, the equations share a parent number but get individual equation
+% numbers consisting of parent number plus a letter.
+%
+% If the main relation symbols in a group of equations are to be
+% aligned, then the final alignment computations cannot be done until the
+% end of the group \mdash  \ie , the horizontal positioning of the first
+% $n - 1$ equations cannot be done immediately.    Yet because of
+% the automatic line breaking, we cannot calculate an initial value of
+% RHS-max over the whole group unless we do a trial run on each equation
+% first to find an RHS-max for that equation.    Once we know RHS-group-max
+% and LHS-group-max we must redo the trial set of each equation because
+% they may affect the line breaks.    If the second trial for an
+% equation fails (one of its lines exceeds the available width), but
+% the first one succeeded, fall back to the first trial, \ie  let that
+% equation fall out of alignment with the rest of the group.
+%
+%
+% All right then, here is the general idea of the whole algorithm for
+% group alignment.
+% To start with, ignore the possibility of equation numbers so that
+% our equation group has the form:
+% \begin{literalcode}
+% LHS[1] RHS[1,1] RHS[1,2] ... RHS[1,n[1]]
+% LHS[2] RHS[2,1] RHS[2,2] ... RHS[2,n[2]]
+%   ...
+% LHS[3] RHS[3,1] RHS[3,2] ... RHS[3,n[3]]
+% \end{literalcode}
+% The number of RHS's might not be the same for all of the
+% equations.
+% First, accumulate all of the equation contents in a queue, checking
+% along the way to find the maximum width of all the LHS's and the maximum
+% width of all the RHS's.
+% Call these widths maxwd\_L and maxwd\_R.
+% Clearly if maxwd\_L + maxwd\_R is less than or equal to the available
+% equation width then aligning all of the equations is going to be simple.
+%
+%
+% Otherwise we are going to have to break at least one of the RHS's
+% and/or at least one of the LHS's.
+% The first thing to try is using maxwd\_L for the LHS's and breaking
+% all the RHS's as needed to fit in the remaining space.
+% However, this might be a really dumb strategy if one or more of the
+% LHS's is extraordinarily wide.
+% So before trying that we check whether maxwd\_L exceeds some
+% threshold width beyond which it would be unsensible not to break the LHS.
+% Such as, max(one-third of the available width; six ems), or
+% something like that.
+% Or how about this?
+% Compare the average LHS width and RHS width and divide up the available
+% width in the same ratio for line breaking purposes.
+%
+%
+% BRM: Fairly broad changes; it mostly didn't work before (for me).
+%
+% \begin{description}
+% \item[\cs{begin}\csarg{dgroup} produces a `numbered' group]
+%   The number is the next equation number.
+%   There are 2 cases:
+% \begin{itemize}
+% \item If ANY contained equations are numbered (|\begin{dmath}|),
+%       then they will be subnumbered: eg 1.1a
+%       and the group number is not otherwise displayed.
+% \item If ALL contained equations are unnumbered (|\begin{dmath*}|)
+%       then the group, as a whole, gets a number displayed,
+%       using the same number placement as for equations.
+% \end{itemize}
+% \item[\cs{begin}\csarg{dgroup*} produces an unnumbered group.]
+%    Contained equations are numbered, or not, as normal.
+%    But note that in the mixed case, it's too late to
+%    force the unnumbered eqns to \cs{retry@with@number}
+%    We'll just do a simple check of dimensions, after the fact,
+%    and force a shiftnumber if we're stuck.
+%
+% NOTE: Does this work for dseries, as well? (alignment?)
+%
+% NOTE: Does \cs{label} attach to the expected thing?
+%
+% \item[For number placement] We use shiftnumber placement on ALL equations
+%    if ANY equations need it, or if an unnumbered equation is too
+%    wide to be aligned, given that the group or other eqns are numbered.
+%    [does this latter case interract with the chosen alignment?]
+%
+% \item[For Alignment]
+%   As currently coded, it tries to align on relations, by default.
+%   If LHS's are not all present, or too long, it switches to left-justify.
+%   Maybe there are other cases that should switch?
+%   Should there be a case for centered?
+%
+% NOTE: Should there be some options to choose alignment?
+% \end{description}
+%
+% \begin{macro}{\eq@group}
+% \begin{macro}{\GRP@top}
+%
+%    \begin{macrocode}
+\let\eq@group\@False
+\let\grp@shiftnumber\@False
+\let\grp@hasNumber\@False
+\let\grp@eqs@numbered\@False
+\let\grp@aligned\@True
+%    \end{macrocode}
+% \end{macro}
+% \end{macro}
+%
+%
+% Definition of the \env{dgroup} environment.
+%    \begin{macrocode}
+\newenvironment{dgroup}{%
+  \@dgroup@start@hook
+  \let\grp@hasNumber\@True\@optarg\@dgroup{}%
+}{%
+  \end@dgroup
+}
+%    \end{macrocode}
+% And the.
+%    \begin{macrocode}
+\newtoks\GRP@queue
+\newenvironment{dgroup*}{%
+  \let\grp@hasNumber\@False\@optarg\@dgroup{}%
+}{%
+  \end@dgroup
+}
+\def\@dgroup[#1]{%
+%<trace>  \breqn@debugmsg{=== DGROUP ==================================================}%
+  \let\eq@group\@True \global\let\eq@GRP@first@dmath\@True
+  \global\GRP@queue\@emptytoks \global\setbox\GRP@box\box\voidb@x
+  \global\let\GRP@label\@empty
+  \global\grp@wdL\z@\global\grp@wdR\z@\global\grp@wdT\z@
+  \global\grp@linewidth\z@\global\grp@wdNum\z@
+  \global\let\grp@eqs@numbered\@False
+  \global\let\grp@aligned\@True
+  \global\let\grp@shiftnumber\@False
+  \eq@prelim
+  \setkeys{breqn}{#1}%
+  \if\grp@hasNumber \grp@setnumber \fi
+}
+\def\end@dgroup{%
+  \EQ@displayinfo \grp@finish
+  \if\grp@hasNumber\grp@resetnumber\fi
+}
+%    \end{macrocode}
+% If the \pkg{amsmath} package is not loaded the parentequation
+% counter will not be defined.
+%    \begin{macrocode}
+\@ifundefined{c@parentequation}{\newcounter{parentequation}}{}
+%    \end{macrocode}
+% Init.
+%    \begin{macrocode}
+\global\let\GRP@label\@empty
+\def\add@grp@label{%
+  \ifx\@empty\GRP@label
+  \else \GRP@label \global\let\GRP@label\@empty
+  \fi
+}
+%    \end{macrocode}
+% Before sending down the `equation' counter to the subordinate level,
+% set the current number in \cs{EQ@numbox}.    The
+% \cs{eq@setnumber} function does everything we need here.    If
+% the child equations are unnumbered, \cs{EQ@numbox} will retain the
+% group number at the end of the group.
+%    \begin{macrocode}
+\def\grp@setnumber{%
+  \global\let\GRP@label\next@label \global\let\next@label\@empty
+  % Trick \eq@setnumber to doing our work for us.
+  \let\eq@hasNumber\@True
+  \eq@setnumber
+%    \end{macrocode}
+% Define \cn{theparentequation} equivalent to current
+% \cn{theequation}.    \cn{edef} is necessary to expand the
+% current value of the equation counter.    This might in rare cases
+% cause something to blow up, in which case the user needs to add
+% \cn{protect}.
+%    \begin{macrocode}
+  \global\sbox\GRP@numbox{\unhbox\EQ@numbox}%
+  \grp@wdNum\eq@wdNum
+  \let\eq@hasNumber\@False
+  \let\eq@number\@empty
+  \eq@wdNum\z@
+%
+  \protected@edef\theparentequation{\theequation}%
+  \setcounter{parentequation}{\value{equation}}%
+%    \end{macrocode}
+% And set the equation counter to 0, so that the normal incrementing
+% processes will produce the desired results if the child equations are
+% numbered.
+%    \begin{macrocode}
+  \setcounter{equation}{0}%
+  \def\theequation{\theparentequation\alph{equation}}%
+%<trace>  \breqn@debugmsg{Group Number \theequation}%
+}
+%    \end{macrocode}
+% At the end of a group, need to reset the equation counter.
+%    \begin{macrocode}
+\def\grp@resetnumber{%
+  \setcounter{equation}{\value{parentequation}}%
+}
+\newbox\GRP@box
+\newbox\GRP@wholebox
+%    \end{macrocode}
+% Save data for this equation in the group
+% \begin{itemize}
+% \item push the trial data onto end of \cs{GRP@queue}.
+% \item push an hbox onto the front of \cs{GRP@box} containing:
+%   \cs{EQ@box}, \cs{EQ@copy}, \cs{penalty} 1 and \cs{EQ@numbox}.
+% \end{itemize}
+% \begin{macro}{\grp@push}
+%
+% For putting the equation on a queue.
+%    \begin{macrocode}
+\def\grp@push{%
+  \global\GRP@queue\@xp\@xp\@xp{\@xp\the\@xp\GRP@queue
+    \@xp\@elt\@xp{\EQ@trial}%
+  }%
+  \global\setbox\GRP@box\vbox{%
+    \hbox{\box\EQ@box\box\EQ@copy\penalty\@ne\copy\EQ@numbox}%
+    \unvbox\GRP@box
+  }%
+  \EQ@trial
+  \if\eq@isIntertext\else
+    \ifdim\eq@wdL>\grp@wdL \global\grp@wdL\eq@wdL \fi
+    \ifdim\eq@wdT>\grp@wdT \global\grp@wdT\eq@wdT \fi
+    \setlength\dim@a{\eq@wdT-\eq@wdL}%
+    \ifdim\dim@a>\grp@wdR \global\grp@wdR\dim@a \fi
+    \ifdim\eq@linewidth>\grp@linewidth \global\grp@linewidth\eq@linewidth\fi
+    \if\eq@hasNumber
+       \global\let\grp@eqs@numbered\@True
+       \ifdim\eq@wdNum>\grp@wdNum\global\grp@wdNum\eq@wdNum\fi
+    \fi
+    \if\EQ@hasLHS\else\global\let\grp@aligned\@False\fi
+    \if D\eq@layout \global\let\grp@aligned\@False\fi % Layout D (usually) puts rel on 2nd line.
+    \if\eq@shiftnumber\global\let\grp@shiftnumber\@True\fi % One eq shifted forces all.
+  \fi
+}
+%    \end{macrocode}
+% \end{macro}
+% \begin{macro}{\grp@finish}
+%
+% Set accumulated equations from a \env{dgroup} environment.
+%
+% BRM: Questionable patch!!
+% When processing the \cs{GRP@queue}, put it into a \cs{vbox}, then \cs{unvbox} it.
+% This since there's a bizarre problem when the \cs{output} routine
+% gets invoked at an inopportune moment: All the not-yet-processed
+% \cs{GRP@queue} ends up in the \cs{@freelist} and bad name clashes happen.
+% Of course, it could be due to some other problem entirely!!!
+%    \begin{macrocode}
+\def\grp@finish{%
+%  \debug@box\GRP@box
+%  \breqn@debugmsg{\GRP@queue: \the\GRP@queue}%
+%    \end{macrocode}
+% == Now that we know the collective measurements, make final decision
+% about alignment \& shifting.  Check if alignment is still possible
+%    \begin{macrocode}
+  \setlength\dim@a{\grp@wdL+\grp@wdR-4em}% Allowance for shrink?
+  \if\grp@aligned
+    \ifdim\dim@a>\grp@linewidth
+      \global\let\grp@aligned\@False
+    \fi
+  \fi
+%    \end{macrocode}
+% If we're adding an unshifted group number that equations didn't know
+% about, re-check shifting
+%    \begin{macrocode}
+  \addtolength\dim@a{\grp@wdNum }% Effective length
+  \if\grp@shiftnumber
+  \else
+    \if\@And{\grp@hasNumber}{\@Not\grp@eqs@numbered}
+      \ifdim\dim@a>\grp@linewidth
+        \global\let\grp@shiftnumber\@True
+      \fi
+    \fi
+  \fi
+%    \end{macrocode}
+% If we can still align, total width is sum of maximum LHS \& RHS
+%    \begin{macrocode}
+  \if\grp@aligned
+     \global\grp@wdT\grp@wdL
+     \global\advance\grp@wdT\grp@wdR
+  \fi
+%<*trace>
+  \breqn@debugmsg{======= DGROUP Formatting
+   \MessageBreak==== \grp@wdL=\the\grp@wdL, \grp@wdR=\the\grp@wdR
+   \MessageBreak==== Shift Number=\theb@@le\grp@shiftnumber, Eqns. numbered=\theb@@le\grp@eqs@numbered
+   \MessageBreak==== Aligned=\theb@@le\grp@aligned
+   \MessageBreak==== \grp@wdNum=\the\grp@wdNum}%
+%</trace>
+%    \end{macrocode}
+% BRM: Originally this stuff was dumped directly, without capturing it
+% in a \cs{vbox}
+%    \begin{macrocode}
+  \setbox\GRP@wholebox\vbox{%
+    \let\@elt\eqgrp@elt
+    \the\GRP@queue
+  }%
+%    \end{macrocode}
+% If we're placing a group number (not individual eqn numbers)
+% NOTE: For now, just code up LM number
+% NOTE: Come back and handle other cases.
+% NOTE: Vertical spacing is off, perhaps because of inter eqn. glue
+%
+% A bit of a hack to get the top spacing correct. Fix this logic
+% properly some day.  Also, we do the calculation in a group for
+% maximum safety.
+%    \begin{macrocode}
+  \global\let\eq@GRP@first@dmath\@True
+  \begingroup
+  \dmath@first@leftskip
+  \eq@topspace{\vskip\parskip}%
+  \endgroup
+  \if\@And{\grp@hasNumber}{\@Not{\grp@eqs@numbered}}%
+%    \eq@topspace{\vskip\parskip}%
+    \if\grp@shiftnumber
+      \copy\GRP@numbox \penalty\@M
+      \kern\eqlineskip
+    \else
+      \setlength\dim@a{%
+        (\ht\GRP@wholebox+\dp\GRP@wholebox+\ht\GRP@numbox-\dp\GRP@numbox)/2}%
+      \setlength\skip@c{\dim@a-\ht\GRP@numbox}%
+      \vglue\skip@c% NON discardable
+      \copy\GRP@numbox \penalty\@M
+%<*trace>
+\breqn@debugmsg{GROUP NUMBER: preskip:\the\skip@c,  postkern:\the\dim@a, height:\the\ht\GRP@wholebox,
+  \MessageBreak==== box height:\the\ht\GRP@numbox, box depth:\the\dp\GRP@numbox}%
+%</trace>
+      \kern-\dim@a
+      \kern-\abovedisplayskip % To cancel the topspace above the first eqn.
+    \fi
+  \fi
+%<*trace>
+%\debug@box\GRP@wholebox
+%</trace>
+  \unvbox\GRP@wholebox
+  \let\@elt\relax
+%    \end{macrocode}
+% We'd need to handle shifted, right number here, too!!!
+%    \begin{macrocode}
+  \eq@botspace % not needed unless bottom number?
+}
+%    \end{macrocode}
+% \end{macro}
+%
+% \begin{macro}{\eqgrp@elt}
+%
+% Mission is to typeset the next equation from the group queue.
+%
+% The arg is an \cs{EQ@trial}
+%    \begin{macrocode}
+\def\eqgrp@elt#1{%
+  \global\setbox\GRP@box\vbox{%
+    \unvbox\GRP@box
+    \setbox\z@\lastbox
+    \setbox\tw@\hbox{\unhbox\z@
+      \ifnum\lastpenalty=\@ne
+      \else
+        \global\setbox\EQ@numbox\lastbox
+      \fi
+      \unpenalty
+      \global\setbox\EQ@copy\lastbox
+      \global\setbox\EQ@box\lastbox
+    }%
+  }%
+  \begingroup \let\eq@botspace\relax
+  #1%
+  \if\eq@isIntertext
+    \vskip\belowdisplayskip
+    \unvbox\EQ@copy
+  \else
+    \grp@override
+    \eq@finish
+  \fi
+  \endgroup
+}
+%    \end{macrocode}
+% \end{macro}
+% Override the \cs{eq@trial} data as needed for this equation in this group
+% NOTE: w/ numbering variations (see above), we may need to tell
+%  \cs{eq@finish} to allocate space for a number, but not actually have one
+%    \begin{macrocode}
+\def\grp@override{%
+%    \end{macrocode}
+% For aligned (possibly becomes an option?)
+% For now ASSUMING we started out as CLM!!!
+%    \begin{macrocode}
+  \def\eqindent{I}%
+%    \end{macrocode}
+% compute nominal left for centering the group
+%    \begin{macrocode}
+  \setlength\dim@a{(\grp@linewidth-\grp@wdT)/2}%
+%    \end{macrocode}
+% Make sure L+R not too wide; should already have unset alignment
+%    \begin{macrocode}
+  \ifdim\dim@a<\z@\dim@a=\z@\fi
+  \dim@b\if L\eqnumside\grp@wdNum\else\z@\fi
+%    \end{macrocode}
+% make sure room for number on left, if needed.
+%    \begin{macrocode}
+  \if\grp@shiftnumber\else
+    \ifdim\dim@b>\dim@a\dim@a\dim@b\fi
+  \fi
+  \if\grp@aligned
+    \addtolength\dim@a{\grp@wdL-\eq@wdL}%
+  \fi
+  \mathindent\dim@a
+  \ifdim\dim@b>\dim@a
+    \let\eq@shiftnumber\@True
+  \fi
+%    \end{macrocode}
+% Could set |\def\eqnumplace{T}| (or even (m) if indentation is enough).
+%
+% NOTE: Work out how this should interact with the various formats!!!
+% NOTE: should recognize the case where the LHS's are a bit Wild,
+%  and then do simple left align (not on relation)
+%    \begin{macrocode}
+}
+%    \end{macrocode}
+%
+%
+%
+% \section{The \env{darray} environment}
+% There are two potential applications for darray.    One
+% is like eqnarray where the natural structure of the material crosses the
+% table cell boundaries, and math operator spacing needs to be preserved
+% across cell boundaries.    And there is also the feature of
+% attaching an equation number to each row.    The other application
+% is like a regular array but with automatic displaystyle math in each
+% cell and better interline spacing to accommodate outsize cell
+% contents.    In this case it is difficult to keep the vert ruling
+% capabilities of the standard \env{array} environment without
+% redoing the implementation along the lines of Arseneau's
+% \pkg{tabls} package.    Because the vert ruling feature is at
+% cross purposes with the feature of allowing interline stretch and page
+% breaks within a multiline array of equations, the \env{darray}
+% environment is targeted primarily as an alternative to
+% \env{eqnarray}, and does not support vertical ruling.
+%
+% Overall strategy for \env{darray} is to use
+% \cs{halign} for the body.    In the case of a group, use a
+% single halign for the whole group!
+% \begin{aside}
+% What about intertext?
+% \end{aside}
+%
+% That's the most reliable way
+% to get accurate column widths.    Don't spread the halign to the
+% column width, just use the natural width.    Then, if we repack the
+% contents of the halign into \cs{EQ@box} and \cs{EQ@copy}, as
+% done for dmath, and twiddle a bit with the widths of the first and last
+% cell in each row, we can use the same algorithms for centering and
+% equation number placement as dmath!    As well as handling footnotes
+% and vadjust objects the same way.
+%
+% We can't just use \cs{arraycolsep} for \env{darray}, if
+% we want to be able to change it without screwing up interior arrays.
+% So let's make a new colsep variable.    The initial value is
+% \quoted{2em, but let it shrink if necessary}.
+%    \begin{macrocode}
+\newskip\darraycolsep \darraycolsep 20pt plus1fil minus12pt
+%    \end{macrocode}
+% Let's make a nice big default setup with eighteen columns, split up
+% into six sets of lcr like \env{eqnarray}.
+%    \begin{macrocode}
+\newcount\cur@row \newcount\cur@col
+\def\@tempa#1#2#3{%
+  \cur@col#1 \hfil
+  \setbox\z@\hbox{$\displaystyle####\m@th$}\@nx\col@box
+  \tabskip\z@skip
+  &\cur@col#2 \hfil
+  \setbox\z@\hbox{$\displaystyle\mathord{}####\mathord{}\m@th$}\@nx\col@box
+  \hfil
+  &\cur@col#3 \setbox\z@\hbox{$\displaystyle####\m@th$}\@nx\col@box
+  \hfil\tabskip\darraycolsep
+}
+\xdef\darray@preamble{%
+  \@tempa 123&\@tempa 456&\@tempa 789%
+  &\@tempa{10}{11}{12}&\@tempa{13}{14}{15}&\@tempa{16}{17}{18}%
+  \cr
+}
+\@ifundefined{Mathstrut@}{\let\Mathstrut@\strut}{}
+\def\darray@cr{\Mathstrut@\cr}
+\def\col@box{%
+%<*trace>
+%\breqn@debugmsg{Col \number\cur@row,\number\cur@col: \the\wd\z@\space x \the\ht\z@+\the\dp\z@}%
+%</trace>
+  \unhbox\z@
+}
+\newenvironment{darray}{\@optarg\@darray{}}{}
+\def\@darray[#1]{%
+%<trace>  \breqn@debugmsg{=== DARRAY ==================================================}%
+  \if\eq@group\else\eq@prelim\fi
+%    \end{macrocode}
+% Init the halign preamble to empty, then unless the \quoted{cols} key is
+% used to provide a non-null preamble just use the
+% default darray preamble which is a multiple lcr.
+%    \begin{macrocode}
+  \global\let\@preamble\@empty
+  \setkeys{breqn}{#1}%
+  \the\eqstyle \eq@setnumber
+  \ifx\@preamble\@empty \global\let\@preamble\darray@preamble \fi
+  \check@mathfonts
+  % \let\check@mathfonts\relax % tempting, but too risky
+  \@xp\let\csname\string\ \endcsname\darray@cr
+  \setbox\z@\vbox\bgroup
+  \everycr{\noalign{\global\advance\cur@row\@ne}}%
+  \tabskip\z@skip \cur@col\z@
+  \global\cur@row\z@
+  \penalty\@ne % flag for \dar@repack
+  \halign\@xp\bgroup\@preamble
+}
+%    \end{macrocode}
+% Assimilate following punctuation.
+%    \begin{macrocode}
+\def\enddarray#1{\check@punct@or@qed}
+\def\end@darray{%
+  \ifvmode\else \eq@addpunct \Mathstrut@\fi\crcr \egroup
+  \dar@capture
+  \egroup
+}
+%    \end{macrocode}
+%
+% The \cs{dar@capture} function steps back through the
+% list of row boxes and grinds them up in the best possible way.
+%    \begin{macrocode}
+\def\dar@capture{%
+%% \showboxbreadth\maxdimen\showboxdepth99\showlists
+  \eq@wdL\z@ \eq@wdRmax\z@
+  \dar@repack
+}
+%    \end{macrocode}
+%
+% The \cs{dar@repack} function is a variation of
+% \cs{eq@repack}.
+%    \begin{macrocode}
+\def\dar@repack{%
+  \unpenalty
+  \setbox\tw@\lastbox
+%\batchmode{\showboxbreadth\maxdimen\showboxdepth99\showbox\tw@}\errorstopmode
+  \global\setbox\EQ@box\hbox{%
+    \hbox{\unhcopy\tw@\unskip}\penalty-\@M \unhbox\EQ@box}%
+  \global\setbox\EQ@copy\hbox{%
+    \hbox{\unhbox\tw@\unskip}\penalty-\@M \unhbox\EQ@copy}%
+  \unskip
+  \ifcase\lastpenalty \else\@xp\@gobble\fi
+  \dar@repack
+}
+%    \end{macrocode}
+%
+%
+%
+%
+% \section{Miscellaneous}
+% The \cs{condition} command.    With
+% the star form, set the argument in math mode instead of text mode.
+% In a series of conditions, use less space between members of the
+% series than between the conditions and the main equation body.
+%
+% WSPR: tidied/fixed things up as it made sense to me but might have
+% broken something else!
+%    \begin{macrocode}
+\newskip\conditionsep \conditionsep=10pt minus5pt%
+\newcommand{\conditionpunct}{,}
+%    \end{macrocode}
+% \begin{macro}{\condition}
+%    \begin{macrocode}
+\newcommand\condition{%
+  \begingroup\@tempswatrue
+    \@ifstar{\@tempswafalse \condition@a}{\condition@a}}
+%    \end{macrocode}
+% \end{macro}
+% \begin{macro}{\condition@a}
+%    \begin{macrocode}
+\newcommand\condition@a[2][\conditionpunct]{%
+  \unpenalty\unskip\unpenalty\unskip % BRM Added
+  \hbox{#1}%
+  \penalty -201\relax\hbox{}% Penalty to allow breaks here.
+  \hskip\conditionsep
+  \setbox\z@\if@tempswa\hbox{#2}\else\hbox{$\textmath@setup #2$}\fi
+%    \end{macrocode}
+% BRM's layout is achieved with this line commented out but it has the nasty side-effect of shifting the equation number to the next line:
+%    \begin{macrocode}
+%  \global\eq@wdCond\wd\z@
+  \usebox\z@
+  \endgroup}
+%    \end{macrocode}
+% \end{macro}
+%
+% The \env{dsuspend} environment.    First the old one that didn't work.
+%    \begin{macrocode}
+\newenvironment{XXXXdsuspend}{%
+  \global\setbox\EQ@box\vbox\bgroup \@parboxrestore
+%    \end{macrocode}
+% If we are inside a list environment, \cs{displayindent} and
+% \cs{displaywidth} give us \cs{@totalleftmargin} and
+% \cs{linewidth}.
+%    \begin{macrocode}
+    \parshape 1 \displayindent \displaywidth\relax
+    \hsize=\columnwidth \noindent\ignorespaces
+}{%
+  \par\egroup
+%    \end{macrocode}
+% Let's try giving \cs{EQ@box} the correct height for the first
+% line and \cs{EQ@copy} the depth of the last line.
+%    \begin{macrocode}
+  \global\setbox\GRP@box\vbox{%
+    \vbox{\copy\EQ@box\vtop{\unvbox\EQ@box}}%
+    \unvbox\GRP@box
+  }%
+%    \end{macrocode}
+% Need to add a dummy element to \cs{GRP@queue}.
+%    \begin{macrocode}
+  \global\GRP@queue\@xp{\the\GRP@queue
+    \@elt{\gdef\EQ@trial{}}%
+  }%
+}
+%    \end{macrocode}
+% And then the one that does work.
+%    \begin{macrocode}
+\newenvironment{dsuspend}{%
+  \global\setbox\EQ@box\vbox\bgroup \@parboxrestore
+    \parshape 1 \displayindent \displaywidth\relax
+    \hsize=\columnwidth \noindent\ignorespaces
+}{%
+  \par\egroup
+  \global\setbox\GRP@box\vbox{%
+    \hbox{\copy\EQ@box\vtop{\unvbox\EQ@box}}%
+    \unvbox\GRP@box
+  }%
+  \global\GRP@queue\@xp{\the\GRP@queue
+%    \@elt{\gdef\EQ@trial{\let\eq@isIntertext\@True}}%
+     \@elt{\let\eq@isIntertext\@True}%
+  }%
+}
+%    \end{macrocode}
+% Allow \cn{intertext} as a short form of the \env{dsuspend}
+% environment; it's more convenient to write, but it doesn't support
+% embedded verbatim because it reads the material as a macro argument.
+% To support simultaneous use of \pkg{amsmath} and
+% \pkg{breqn}, the user command \cs{intertext} is left alone
+% until we enter a \pkg{breqn} environment.
+%    \begin{macrocode}
+\newcommand\breqn@intertext[1]{\dsuspend#1\enddsuspend}
+%    \end{macrocode}
+%
+%
+% \begin{macro}{\*}
+% \begin{macro}{\discretionarytimes}
+% Discretionary times sign.    Standard \latex  definition
+% serves only for inline math.    Should the thin space be
+% included?    Not sure.
+%    \begin{macrocode}
+\renewcommand{\*}{%
+  \if@display
+%    \end{macrocode}
+% Since \cs{eq@binoffset} is mu-glue, we can't use it directly
+% with \cs{kern} but have to measure it separately in a box.
+%    \begin{macrocode}
+    \setbox\z@\hbox{\mathsurround\z@$\mkern\eq@binoffset$}%
+    \discretionary{}{%
+      \kern\the\wd\z@ \textchar\discretionarytimes
+    }{}%
+    \thinspace
+  \else
+    \discretionary{\thinspace\textchar\discretionarytimes}{}{}%
+  \fi
+}
+%    \end{macrocode}
+% This is only the symbol; it can be changed to some other symbol if
+% desired.
+%    \begin{macrocode}
+\newcommand{\discretionarytimes}{\times}
+%    \end{macrocode}
+% \end{macro}
+% \end{macro}
+%
+%
+% \begin{macro}{\nref}
+%
+% This is like \cs{ref} but doesn't apply font changes or other
+% guff if the reference is undefined.
+% And it is fully expandable for use as a label value.
+% \begin{aside}
+%
+% Can break with Babel if author uses active characters in label key;
+% need to address that \begin{dn}
+% mjd,1999/01/21
+% \end{dn}
+% .
+% \end{aside}
+%
+%    \begin{macrocode}
+\def\nref#1{\@xp\@nref\csname r@#1\endcsname}
+\def\@nref#1#2{\ifx\relax#1??\else \@xp\@firstoftwo#1\fi}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%    \end{macrocode}
+% \end{macro}
+%
+%
+% \section{Compatibility}
+%
+% \paragraph{lineno} (or at the very least, allow documents to compile!)
+%    \begin{macrocode}
+\AtBeginDocument{%
+  \@ifpackageloaded{lineno}{%
+    \g@addto@macro\@dmath@start@hook{\nolinenumbers}%
+    \g@addto@macro\@dgroup@start@hook{\nolinenumbers}%
+  }{}%
+}
+%    \end{macrocode}
+%
+%
+%
+% \section{Wrap-up}
+% The usual endinput.
+%    \begin{macrocode}
+%</package>
+%    \end{macrocode}
+%
+%
+%
+%
+% \section{To do}
+% \begin{enumerate}
+% \item Alignment for equation groups.
+%
+%
+% \item
+% Use dpc's code for package options in keyval form.
+%
+% \item
+% Encapsulate \dquoted{break math} into a subroutine taking suitable
+% arguments.
+%
+% \item
+% Need a density check for layout S when linewidth is very small.
+%
+% \item
+% Make \verb":=" trigger a warning about using \cs{coloneq}
+% instead.
+%
+% \item Ill-centered multiline equation (three-line case) in
+% test008.
+%
+% \item Attaching a single group number.
+%
+%
+% \item
+% Make sure to dump out box registers after done using them.
+%
+% \item Do the implementation for \cs{eq@resume@parshape}.
+%
+%
+% \item Check on stackrel and buildrel and relbar and ???.
+%
+%
+% \item Test math symbols at the beginning of array cells.
+%
+% \item Test \dbslash  cmd in and out of delims.
+%
+% \item Framing the equation body: the parshape and number placement
+% need adjusting when a frame is present.
+%
+%
+% \item Cascading line widths in list env.
+%
+%
+% \item Noalign option for dmath = multline arrangement?
+%
+%
+% \item Nocompact option, suggested 1998/05/19 by Andrew
+% Swann.
+%
+%
+% \item \cs{delbreak} cmd to add discretionary space at a break
+% within delimiters.
+%
+%
+% \item Reduce above/below skip when the number is shifted.
+%
+%
+% \item Need a \cs{middelim} command for marking a delimiter symbol
+% as nondirectional if it has an innate directionality \verb"()[]" \etc .
+%
+%
+% \item
+% \cs{xrightarrow} from amsmath won't participate in line
+% breaking unless something extra is done.
+% Make \cs{BreakingRel} and \cs{BreakingBin} functions?
+%
+% \item Placement of number in an indented quotation or
+% abstract.
+%
+% \item If $LHSwd > 2em$, it might be a good idea to try with
+% eq@indentstep = 2em before shifting the number.    Currently this
+% doesn't happen if the first trial pass (without the number)
+% succeeds with $indentstep = LHSwd > 2em$.
+%
+%
+% \item Read past \verb"\end{enumerate}" when checking
+% for \verb"\end{proof}"?
+%
+% \item
+% Look into using a \dquoted{qed-list} of environment names instead of
+% checking the existence of \cs{proofqed}.
+%
+% \item Pick up the vadjust\slash footnote\slash mark handling.
+%
+%
+% \item Forcing\slash prohibiting page breaks after\slash before
+% an equation.
+%
+%
+% \item Adding a spanner brace on the left and individual numbers on
+% the right (indy-numbered cases).
+%
+%
+% \item Provide \cs{shiftnumber}, \cs{holdnumber} to
+% override the decision.
+%
+% \item  Provide a mechanism for adjusting the vertical position of
+% the number.    Here a version-specific selection macro would be
+% useful.
+% \begin{literalcode}
+% \begin{dmath}[
+%   style={\foredition{1}{\raisenumber{13pt}}}
+% ]
+% \end{literalcode}
+%
+%
+% \item
+% Add an alignleft option for an equation group to mean, break and
+% align to a ladder layout as usual within the equations, but for the
+% group alignment used the leftmost point (for equations that don't
+% have an LHS, this makes no difference).
+%
+% \item
+% Test with Arseneau's wrapfig for parshape\slash everypar
+% interaction.
+%
+%
+% \item Fix up the macro/def elements.
+%
+% \item Convert the literal examples in section \quoted{Equation types and
+% forms} to typeset form.
+%
+%
+% \item Compile comparison-examples: \eg , a standard equation
+% env with big left-right objects that don't shrink, versus how shrinking
+% can allow it to fit.
+%
+%
+% \item Frame the \dquoted{figures} since they are mostly
+% text.
+%
+% \end{enumerate}
+%
+%
+%
+% Possible enhancements:
+% \begin{enumerate}
+% \item Provide a \opt{pull} option meaning to pull the first
+% and last lines out to the margin, like the \env{multline}
+% environment of the \pkg{amsmath} package.    Maybe this should
+% get an optional argument, actually, to specify the amount of space left
+% at the margin.
+%
+% \item With the draft option, one would like to see the equation
+% labels in the left margin.    Need to check with the
+% \pkg{showkeys} package.
+%
+%
+% \item Options for break preferences: if there's not enough room, do
+% we first shift the number, or first try to break up the equation
+% body?.    In an aligned group, does sticking to the group alignment
+% take precedence over minimizing the number of line breaks needed for
+% individual equations?.    And the general preferences probably need
+% to be overridable for individual instances.
+%
+% \item Extend suppress-breaks-inside-delimiters support to inline
+% math (suggestion of Michael Doob).
+%
+% \item Use belowdisplayshortskip above a dsuspend fragment if the
+% fragment is only one line and short enough compared to the equation line
+% above it.
+%
+%
+% \item Add \cs{eqfuzz} distinct from \cs{hfuzz}.
+% Make use of it in the measuring phase.
+%
+%
+% \item Provision for putting in a \quoted{continued} note.
+%
+% \item Conserve box mem: modify frac, sub, sup, overline, underline,
+% sqrt, to turn off \cs{bin@break} and (less urgently)
+% \cs{rel@break}.
+%
+%
+% \item More explicit support for Russian typesetting conventions (cf
+% Grinchuk article).
+%
+%
+% \item With package option \opt{refnumbers},
+% leave unnumbered all uncited equations, even if they are not done with
+% the star form (Bertolazzi's easyeqn idea).
+%
+% \item In an equation group, use a vertical bracket with the
+% equation number to mark the lines contained in that equation.
+%
+%
+% \item For a two-line multline thingamabob, try to
+% make sure that the lines overlap in the middle by 2 em or whatever
+% (settable design variable).
+%
+% \item Provide a separate vertical column for the principal mathrel
+% symbols and center them within the column if they aren't all the same
+% width.    Maybe an option for \env{dmath}: relwidth=x, so that two
+% passes are not required to get the max width of all the mathrels.
+% Or, no, just require it to be an halign or provide a macro to be
+% applied to all the shorter rels:
+% \begin{literalcode}
+% lhs \widerel{19pt}{=} ...
+%     \xrightarrow{foo} ...
+% \end{literalcode}
+%
+%
+% \item try to use vadjust for keepglue
+%
+% \end{enumerate}
+%
+% \PrintIndex
+%
+% \Finale
+
-- 
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