% \iffalse meta-comment
%
% Copyright (C) 2005 by Kevin W. Hamlen
% -------------------------------------
%
% This file may be distributed and/or modified under the
% conditions of the LaTeX Project Public License, either
% version 1.3a 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 1.3a or later is part of all distributions of
% LaTeX version 2004/10/01 or later.
%
% \fi
%
% \iffalse
%<*driver>
\ProvidesFile{trfrac.dtx}
%</driver>
%<package>\NeedsTeXFormat{LaTeX2e}
%<package>\ProvidesPackage{trfrac}
%<*package>
  [2006/05/01 v2.0 Tree Fraction Package]
%</package>
%
%<*driver>
\documentclass{ltxdoc}
\usepackage{amsmath}
\usepackage{hyperref}
\usepackage{trfrac}
\newcount\seclevel%
\newcommand{\getseclevel}{%
  \global\seclevel=3\relax%
  \ifnum\arabic{subsection}=0\global\seclevel=2\relax\fi%
}
{\catcode`/=0 \catcode`\\=11
  /gdef/DescMacro#1{%
    /getseclevel%
    /pdfbookmark[/the/seclevel]{\\#1}{macdef-#1}%
    /expandafter/DescribeMacro/expandafter{/csname #1/endcsname}%
    /hyperdef{macro}{#1}{}/kern0pt%
  }
}
\newcommand{\DescEnv}[1]{%
  \getseclevel%
  \pdfbookmark[\the\seclevel]{#1}{envdef-#1}%
  \DescribeEnv{#1}%
  \hyperdef{env}{#1}{}\kern0pt%
}
\newcommand{\mac}[1]{{\tt\hyperlink{macro.#1}{\char92 #1}}}
\newcommand{\env}[1]{{\tt\hyperlink{env.#1}{#1}}}
\EnableCrossrefs
\CodelineIndex
\RecordChanges
\begin{document}
  \DocInput{trfrac.dtx}
\end{document}
%</driver>
% \fi
%
% \CheckSum{443}
%
% \CharacterTable
%  {Upper-case    \A\B\C\D\E\F\G\H\I\J\K\L\M\N\O\P\Q\R\S\T\U\V\W\X\Y\Z
%   Lower-case    \a\b\c\d\e\f\g\h\i\j\k\l\m\n\o\p\q\r\s\t\u\v\w\x\y\z
%   Digits        \0\1\2\3\4\5\6\7\8\9
%   Exclamation   \!     Double quote  \"     Hash (number) \#
%   Dollar        \$     Percent       \%     Ampersand     \&
%   Acute accent  \'     Left paren    \(     Right paren   \)
%   Asterisk      \*     Plus          \+     Comma         \,
%   Minus         \-     Point         \.     Solidus       \/
%   Colon         \:     Semicolon     \;     Less than     \<
%   Equals        \=     Greater than  \>     Question mark \?
%   Commercial at \@     Left bracket  \[     Backslash     \\
%   Right bracket \]     Circumflex    \^     Underscore    \_
%   Grave accent  \`     Left brace    \{     Vertical bar  \|
%   Right brace   \}     Tilde         \~}
%
% \changes{v1.0}{2005/09/05}{Initial version}
% \changes{v2.0}{2006/05/01}{Renamed to TrFrac package}
%
% \GetFileInfo{trfrac.dtx}
%
% \title{The \textsf{trfrac} package\thanks{This document corresponds to
%    \textsf{trfrac}~\fileversion, dated~\filedate.}}
% \author{Kevin W.~Hamlen}
%
% \maketitle
%
% \begin{abstract}
% The |trfrac| package provides convenient mechanisms for typesetting
% derivation trees in \LaTeX.
% Derivation trees look like nested fractions; thus, the primary contribution
% of this software is a macro |\trfrac| that functions much like \LaTeX's
% |\frac| for fractions, but with a number of important differences that make
% it better suited to derivation trees.
% \end{abstract}
%
% \section{Introduction}
%
% Typing rules and typing derivations in formal typing systems are usually
% written like fractions in which the numerator consists of one or more
% premise judgments or derivations, and the denominator consists of a
% single conclusion judgment.
% For example, the typing rule for application in the simply-typed lambda
% calculus is typically written
% \[ \trfrac[\small(\emph{app})]
%           {\Gamma \vdash e_1:\tau\rightarrow\tau' \qquad
%            \Gamma \vdash e_2:\tau}
%           {\Gamma \vdash e_1 e_2 : \tau'} \]
% where $\Gamma \vdash e_1:\tau\rightarrow\tau'$ and $\Gamma \vdash e_2:\tau$
% are the two premise judgments of the rule, and
% $\Gamma \vdash e_1 e_2 : \tau'$ is the conclusion judgment.
%
% Simple rules like the above are easy to typeset with \LaTeX's built-in
% |\frac| macro, or with fraction macros like |\dfrac| provided by the
% |amslatex| package; but when derivations consist of a tree of nested rules,
% |\frac| and |\dfrac| often prove inadequate because they yield spacing
% and alignment that is unsuited to derivation trees.
% For example, consider the derivation
% \[ \trfrac[\small(\emph{app})]
%           {\Gamma \vdash e_1:\tau\rightarrow\tau' \qquad
%            \trfrac[\small(\emph{app})]
%                   {\Gamma \vdash e_2:\tau''\rightarrow\tau \qquad
%                    \Gamma \vdash e_3:\tau''}
%                   {\Gamma \vdash e_2 e_3:\tau}}
%           {\Gamma \vdash e_1 (e_2 e_3):\tau'} \]
% which contains a nested subderivation of one of its premise judgments.
% This was easily typeset using the |\trfrac| macro provided by
% the |trfrac| package.
% The complete \LaTeX\ code is as follows:
% \begin{verbatim}
% \[ \trfrac[\small(\emph{app})]
%           {\Gamma \vdash e_1:\tau\rightarrow\tau' \qquad
%             \trfrac[\small(\emph{app})]
%                    {\Gamma \vdash e_2:\tau''\rightarrow\tau \qquad
%                     \Gamma \vdash e_3:\tau''}
%                    {\Gamma \vdash e_2 e_3:\tau}}
%           {\Gamma \vdash e_1 (e_2 e_3):\tau'} \]
% \end{verbatim}
% In contrast, if one writes analogous code using |\dfrac| instead of
% |\trfrac|, the result is
% \[ \dfrac{\Gamma \vdash e_1:\tau\rightarrow\tau' \qquad
%           \dfrac{\Gamma \vdash e_2:\tau''\rightarrow\tau \qquad
%                  \Gamma \vdash e_3:\tau''}
%                 {\Gamma \vdash e_2 e_3:\tau}\text{\small(\emph{app})}}
%          {\Gamma \vdash e_1 (e_2 e_3):\tau'}\text{\small(\emph{app})} \]
% which is unsuitable for two reasons:
% First, the leftmost premise is floating up above the fraction bar when
% it should be sitting on the bar.
% Second, the bottom fraction bar is longer than it should be; it has been
% unnecessarily extended rightward to span the entire ``numerator'' of the
% ``fraction''.
% This can become problematic when derivations are large and horizontal
% space is at a premium.
%
% In addition to providing |\trfrac| as an alternative to fraction macros,
% the |trfrac| package also provides two environments that can be helpful
% for squeezing numerous and/or lengthy premise judgments into a smaller
% amount of horizontal space.
% Derivations can sometimes contain long judgments that must be wrapped, or
% numerous premise judgments that must be placed above one another for lack
% of space.
% The |trfrac| package provides math environments |trgather| and |tralign|
% for stacking sets of judgments vertically within a derivation tree.
% The |trgather| environment horizontally centers each judgment in the stack,
% whereas the |tralign| environment aligns each judgment at a specified point.
% For example, the derivation
% \[ \trfrac[\small(\emph{case})]
%           {\begin{trgather}
%              \Gamma \vdash e:\tau_1+\tau_2 \\
%              \Gamma[x_1\mapsto\tau_1] \vdash e_1:\tau \qquad
%              \Gamma[x_2\mapsto\tau_2] \vdash e_2:\tau
%            \end{trgather}}
%           {\begin{tralign}
%              \Gamma \vdash \mathbf{case}~e~\mathbf{of}~
%                  &\mathit{in}_1(x_1)\Rightarrow e_1 \\
%              \mid\,&\mathit{in}_2(x_2)\Rightarrow e_2 : \tau
%            \end{tralign}} \]
% can be produced with
% \begin{verbatim}
% \[ \trfrac[\small(\emph{case})]
%           {\begin{trgather}
%              \Gamma \vdash e:\tau_1+\tau_2 \\
%              \Gamma[x_1\mapsto\tau_1] \vdash e_1:\tau \qquad
%              \Gamma[x_2\mapsto\tau_2] \vdash e_2:\tau
%            \end{trgather}}
%           {\begin{tralign}
%              \Gamma \vdash \mathbf{case}~e~\mathbf{of}~
%                  &\mathit{in}_1(x_1)\Rightarrow e_1 \\
%              \mid\,&\mathit{in}_2(x_2)\Rightarrow e_2 : \tau
%            \end{tralign}} \]
% \end{verbatim}
%
% The |split| and |aligned| environments provided by the |amsmath| package
% provide facilities for vertically stacking sub-equations in similar ways,
% but like |\dfrac|, these often prove inadequate for derivation trees
% because they vertically center the material they typeset, causing any
% adjacent material to float above the fraction bar.
%
% \section{Usage}
%
% To start using the macros and environments provided by the |trfrac| package,
% place the line
%
% \medskip{\narrower|\usepackage{trfrac}|}\medskip
%
% \noindent
% somewhere near the top of your \TeX\ file.
%
% \DescMacro{trfrac}
% Subsequently, within any math environment you can use
%
% \medskip{\narrower
%   |\trfrac[|\meta{name}|]{|\meta{premises}|}{|\meta{consequent}|}|
% }\medskip
%
% \noindent
% to typeset a derivation rule, where
% \meta{name} is optional material processed in horizontal text mode
% that is to sit to the right of the fraction bar,
% \meta{premises} is math material that is to sit above the fraction bar, and
% \meta{consequent} is math material that is to sit below the fraction bar.
% The \meta{consequent} and \meta{name} may not contain additional |\trfrac|
% macros, but the \meta{premises} may.
%
% The outermost |\trfrac| in a derivation will be vertically centered on its
% fraction bar, but inner |\trfrac|'s in a derivation will have baselines at
% the bottom of their consequents.
% This makes it possible to typeset something like
% \[ {\cal D} =
%    \trfrac[\small(\emph{app})]
%           {\Gamma \vdash e_1:\tau\rightarrow\tau' \qquad
%            \trfrac[\small(\emph{app})]
%                   {\Gamma \vdash e_2:\tau''\rightarrow\tau \qquad
%                    \Gamma \vdash e_3:\tau''}
%                   {\Gamma \vdash e_2 e_3:\tau}}
%           {\Gamma \vdash e_1 (e_2 e_3):\tau'} \]
% in which the material ``${\cal D} =$'' should be vertically centered with
% respect to the bottom fraction bar, but the left premise should not be
% centered with respect to the right subderivation's fraction bar.
%
% \DescEnv{trgather}
% The environment
%
% \medskip{\narrower\parindent=0pt
%   |\begin{trgather}|\par
%   \hskip1em \meta{eqn$_1$} |\\[|\meta{len$_1$}|]|\par
%   \hskip1em \meta{eqn$_2$} |\\[|\meta{len$_2$}|]|\par
%   \hskip2em $\vdots$\par
%   \hskip1em \meta{eqn$_n$} |\\[|\meta{len$_n$}|]|\par
%   |\end{trgather}|\par
% }\medskip
%
% \noindent
% can be used in math mode to stack a set of equations, each of which is to
% be centered horizontally over one another.
% Here, \meta{eqn$_i$} is material processed in math mode and
% \meta{len$_i$} is an optional length that modifies the vertical spacing
% between the current equation and the next.
% The baseline of the resulting stack of equations will be the baseline of
% the bottom equation in the stack.
%
% \DescEnv{tralign}
% The environment
%
% \medskip{\narrower\parindent=0pt
%   |\begin{tralign}|\par
%   \hskip1em \meta{left$_1$}|&|\meta{right$_1$} |\\[|\meta{len$_1$}|]|\par
%   \hskip1em \meta{left$_2$}|&|\meta{right$_2$} |\\[|\meta{len$_2$}|]|\par
%   \hskip2em $\vdots$\par
%   \hskip1em \meta{left$_n$}|&|\meta{right$_n$} |\\[|\meta{len$_n$}|]|\par
%   |\end{tralign}|\par
% }\medskip
%
% \noindent
% can be used in math mode to stack a set of equations, each of which is to
% be aligned horizontally by the position of the |&| in the equation.
% Here, \meta{left$_i$} and \meta{right$_i$} is the math material to be
% placed to the left and right, respectively, of the vertical center axis of
% the resulting stack of equations;
% and \meta{len$_i$} is an optional length that modifies the vertical
% spacing between the current equation and the next.
% The baseline of the resulting stack of equations will be the baseline of
% the bottom equation in the stack.
%
% \StopEventually{\PrintIndex}
%
% \section{Implementation}
%
% The following is a verbatim listing of the |trfrac| package implementation
% with comments explaining how it works.
%
% \begin{macro}{\TRF@adjust}
% The following dimen register will hold half the total vertical length
% (height plus depth) of a box passed to |\TRF@squeeze|.
%    \begin{macrocode}
\newdimen\TRF@adjust
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\TRF@squeeze}
% Take a box and output it so that it is vertically centered across the
% baseline, it consumes no horizontal space (like |\rlap|), and it consumes
% no vertical space.
% This is used for adding equation numbers to derivations.
%    \begin{macrocode}
\newcommand\TRF@squeeze[1]{%
  \TRF@adjust0.5\ht#1%
  \advance\TRF@adjust0.5\dp#1%
  \vbox to\z@{%
    \kern-\TRF@adjust%
    \hb@xt@\z@{\box#1\hss}%
    \kern-\TRF@adjust%
  }%
}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\TRF@startp}
% \begin{macro}{\TRF@endp}
% When processing a numerator, the |\TRF@trfrac| code must be able to determine
% if (a)~the current derivation is the first thing being added to the current
% horizontal list, and if (b)~the current horizontal list ends in a derivation.
% To accomplish this, two penalties of ``magic'' values are added to
% horizontal lists: |\TRF@startp| is added at the beginning of a horizontal
% list comprising a numerator, and |\TRF@endp| is added just after every
% derivation.
% Then |\lastpenalty| can be used to inquire whether the current horizontal
% list ends in either ``magic'' value.
% (Since derivations encased within numerators cannot undergo line-breaking,
% these extra penalties will have no effect on \LaTeX's processing of the
% actual output.)
%    \begin{macrocode}
\newcount\TRF@startp\TRF@startp31415926
\newcount\TRF@endp\TRF@endp27182818
%    \end{macrocode}
% \end{macro}
% \end{macro}
%
% \begin{macro}{\TRF@lastpenalty}
% In accordance with the above, the result of |\lastpenalty| will be saved in
% the following register before the current horizontal list is modified.
%    \begin{macrocode}
\newcount\TRF@lastpenalty
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\TRF@level}
% The outermost derivation behaves slightly differently than subderivations
% within a derivation tree.
% (Its baseline is its fraction bar instead of its denominator's baseline,
% and it should not include the spurious |\TRF@endp| penalty, since
% line-breaking might occur.)
% The |\TRF@level| counter therefore tracks the current nesting level of
% derivations.
%    \begin{macrocode}
\newcount\TRF@level\TRF@level0
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\ifTRF@derivok}
% Subderivations are not permitted to appear within denominators or within
% rule names.
% The |\ifTRF@derivok| conditional is set to |false| when processing such
% material.
%    \begin{macrocode}
\newif\ifTRF@derivok\TRF@derivoktrue
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\TRF@lhang}
% \begin{macro}{\TRF@rhang}
% After processing a horizontal list that might contain one or more
% derivations, |\TRF@lhang| will hold the horizontal distance by which the
% leftmost derivation's numerator overhangs its fraction bar to the left
% (or |0pt| if the leftmost list item is not a derivation),
% and |\TRF@rhang| will hold the amount by which the rightmost derivation's
% numerator overhangs its fraction bar to the right
% (or |0pt| if the rightmost list item is not a derivation).
%    \begin{macrocode}
\newdimen\TRF@lhang\TRF@lhang\z@
\newdimen\TRF@rhang\TRF@rhang\z@
%    \end{macrocode}
% \end{macro}
% \end{macro}
%
% \begin{macro}{\TRF@xtrarhang}
% While creating a derivation tree, |\TRF@xtrarhang| holds the amount by which
% the current derivation rule's name extends beyond the |\TRF@rhang| of its
% numerator.
%    \begin{macrocode}
\newdimen\TRF@xtrarhang
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\TRF@lsave}
% Because derivations are recursive structures, the various count and dimen
% registers used in their construction must conform to some delicate scoping
% rules.
% Most are local, |\TRF@rhang| is global, but |\TRF@lhang| must be treated as
% either local or global depending on where the derivation appears.
% To accomplish this, we use |\TRF@lsave| as a local variable that saves the
% current global |\TRF@lhang| value on entrance to the derivation code.
% Upon exit, this value is either restored or not depending on whether
% |\TRF@lhang| is being treated as local or global for this particular
% derivation.
%    \begin{macrocode}
\newdimen\TRF@lsave
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\TRF@linewd}
% The following dimen register holds the width of the current derivation's
% fraction bar.
%    \begin{macrocode}
\newdimen\TRF@linewd
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\TRF@totalwd}
% The following dimen register holds the overall width of the current
% derivation.
% That is, it is the sum of |\TRF@lhang|, |\TRF@linewd|, |\TRF@rhang|, and
% |\TRF@xtrarhang|.
%    \begin{macrocode}
\newdimen\TRF@totalwd
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\TRF@numwd}
% \begin{macro}{\TRF@denomwd}
% The following dimen registers hold the width of the current derivation's
% numerator and denominator.
%    \begin{macrocode}
\newdimen\TRF@numwd
\newdimen\TRF@denomwd
%    \end{macrocode}
% \end{macro}
% \end{macro}
%
% \begin{macro}{\TRF@numbox}
% \begin{macro}{\TRF@denombox}
% \begin{macro}{\TRF@eqbox}
% While constructing a derivation, the following three boxes hold the
% derivation's numerator, denominator, and rule name, respectively.
%    \begin{macrocode}
\newbox\TRF@numbox
\newbox\TRF@denombox
\newbox\TRF@eqbox
%    \end{macrocode}
% \end{macro}
% \end{macro}
% \end{macro}
%
% \begin{macro}{\TRF@abovebar}
% \begin{macro}{\TRF@belowbar}
% The following two dimensions hold the computed distances of the numerator
% above the barline and the denominator below the barline, respectively.
%    \begin{macrocode}
\newdimen\TRF@abovebar
\newdimen\TRF@belowbar
%    \end{macrocode}
% \end{macro}
% \end{macro}
%
% \begin{macro}{\TRF@trfrac}
% Here is the real meat of the code.
% The following macro produces a single derivation rule in a derivation tree.
% Its arguments are
% (1)~the rule name (processed in horizontal mode),
% (2)~the numerator (processed in math mode), and
% (3)~the denominator (processed in math mode).
%    \begin{macrocode}
\newcommand\TRF@trfrac[3]{\begingroup%
  \TRF@lastpenalty\the\lastpenalty%
  \TRF@lsave\TRF@lhang%
  \setbox\TRF@eqbox\hbox{\TRF@derivokfalse#1}%
  \ifdim\wd\TRF@eqbox>\z@%
    \setbox\TRF@eqbox\hbox{\kern\p@\unhbox\TRF@eqbox}%
  \fi%
  \TRF@lhang\z@%
  \advance\TRF@level\@ne%
  \setbox\TRF@numbox\hbox{\m@th$%
    \strut%
    \penalty\TRF@startp%
    #2%
    \ifnum\the\lastpenalty=\TRF@endp\else%
      \global\TRF@rhang\z@%
    \fi%
  $}%
  \advance\TRF@level\m@ne%
  \setbox\TRF@denombox\hbox{\m@th$%
    \strut%
    \TRF@derivokfalse%
    #3%
  $}%
  \TRF@numwd\wd\TRF@numbox%
  \TRF@denomwd\wd\TRF@denombox%
  \ifdim\TRF@numwd>\z@\else%
    \setbox\TRF@numbox\hbox{\vbox to0.5ex{\vfil}}%
  \fi%
  \ifdim\wd\TRF@numbox>\TRF@denomwd%
    \TRF@totalwd\wd\TRF@numbox%
  \else%
    \TRF@totalwd\TRF@denomwd%
  \fi%
  \TRF@linewd\wd\TRF@numbox%
  \advance\TRF@linewd-\TRF@lhang%
  \advance\TRF@linewd-\TRF@rhang%
  \ifdim\TRF@denomwd>\TRF@linewd%
    \advance\TRF@linewd-\TRF@denomwd%
    \advance\TRF@lhang0.5\TRF@linewd%
    \global\advance\TRF@rhang0.5\TRF@linewd%
    \ifdim\TRF@lhang<\z@\TRF@lhang\z@\fi%
    \ifdim\TRF@rhang<\z@\global\TRF@rhang\z@\fi%
    \TRF@linewd\TRF@denomwd%
  \fi%
  \ifdim\wd\TRF@eqbox>\TRF@rhang%
    \TRF@xtrarhang\wd\TRF@eqbox%
    \advance\TRF@xtrarhang-\TRF@rhang%
  \else%
    \TRF@xtrarhang\z@%
  \fi%
  \advance\TRF@totalwd\TRF@xtrarhang%
  \TRF@abovebar\fontdimen8\textfont\tw@%
  \advance\TRF@abovebar-\fontdimen22\textfont\tw@%
  \advance\TRF@abovebar-0.5\fontdimen8\textfont\thr@@%
  \advance\TRF@abovebar-\dp\TRF@numbox%
  \ifdim\TRF@abovebar<3\fontdimen8\textfont\thr@@%
    \TRF@abovebar3\fontdimen8\textfont\thr@@%
  \fi%
  \TRF@belowbar\fontdimen11\textfont\tw@%
  \advance\TRF@belowbar\fontdimen22\textfont\tw@%
  \advance\TRF@belowbar-0.5\fontdimen8\textfont\thr@@%
  \advance\TRF@belowbar-\ht\TRF@denombox%
  \ifdim\TRF@belowbar<3\fontdimen8\textfont\thr@@%
    \TRF@belowbar3\fontdimen8\textfont\thr@@%
  \fi%
  \mathinner{%
    \ifnum\TRF@level=0 %
      \raise\fontdimen22\textfont\tw@\vtop%
    \else%
      \vbox%
    \fi%
    {%
      \offinterlineskip%
      \hbox{\vbox{%
        \hb@xt@\TRF@totalwd{%
          \hfil%
          \unhbox\TRF@numbox%
          \hfil%
          \kern\TRF@xtrarhang%
        }%
        \kern\TRF@abovebar%
        \hb@xt@\TRF@totalwd{%
          \kern\TRF@lhang%
          \vrule\@width\TRF@linewd%
                \@height0.5\fontdimen8\textfont\thr@@%
                \@depth0.5\fontdimen8\textfont\thr@@%
          \TRF@squeeze{\TRF@eqbox}%
          \kern\TRF@rhang%
          \kern\TRF@xtrarhang%
        }%
      }}%
      \kern\TRF@belowbar%
      \hb@xt@\TRF@totalwd{%
        \kern\TRF@lhang%
        \hfil%
        \box\TRF@denombox%
        \hfil%
        \kern\TRF@rhang%
        \kern\TRF@xtrarhang%
      }%
    }%
  }%
  \ifnum\TRF@level>0 %
    \penalty\TRF@endp%
  \fi%
  \global\advance\TRF@rhang\TRF@xtrarhang%
  \ifdim\TRF@denomwd<\TRF@linewd%
    \advance\TRF@denomwd-\TRF@linewd%
    \advance\TRF@lhang-0.5\TRF@denomwd%
    \global\advance\TRF@rhang-0.5\TRF@denomwd%
  \fi%
  \ifnum\TRF@lastpenalty=\TRF@startp%
    \global\TRF@lhang\TRF@lhang%
  \else%
    \global\TRF@lhang\TRF@lsave%
  \fi%
\endgroup}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\trfrac}
% The following is the entrypoint to the derivation code.
% The |\trfrac| macro takes one optional argument which, if present, is the
% name of the rule (processed in horizontal mode),
% followed by two mandatory arguments---the rule's numerator and denominator.
% Derivations can only appear in math mode, and cannot appear within the
% denominator of another derivation or within its rule name.
%    \begin{macrocode}
\newcommand\trfrac[3][]{%
  \ifTRF@derivok%
    \ifmmode%
      \TRF@trfrac{#1}{#2}{#3}%
    \else%
      \PackageError{trfrac}{\protect\trfrac\space%
         only allowed in math mode}%
        {I encountered a \protect\trfrac\space%
         macro without first encountering a begin-math token.}%
    \fi%
  \else%
    \PackageError{trfrac}{\protect\trfrac\space%
       not allowed in the consequent of another \protect\trfrac}%
      {\protect\trfrac\space can only appear in the premise (numerator%
       ) of another \protect\trfrac, not in the consequent (denominator%
       ) of a \protect\trfrac.}%
  \fi%
}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\TRF@next}
% The |\TRF@next| token is here reserved so that it can be used in conjunction
% with |\futurelet| (when processing optional arguments).
%    \begin{macrocode}
\newcommand\TRF@next{}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\TRF@sep}
% In a |trgather| or |tralign| environment, the |\\| macro is used to delimit
% lines.
% It is temporarily set equal to |\TRF@sep| (below), which is just like |\cr|
% in an |\halign| except that it takes an optional length argument that
% has the effect of inserting a vertical space between the lines.
%    \begin{macrocode}
\newcommand\TRF@sep{\futurelet\TRF@next\TRF@@sep}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\TRF@@sep}
% The following processes the optional argument to |\TRF@sep|.
%    \begin{macrocode}
\newcommand\TRF@@sep{%
  \ifx[\TRF@next%
    \let\TRF@next\TRF@separg%
  \else%
    \let\TRF@next\TRF@sepnoarg%
  \fi%
  \TRF@next%
}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\TRF@separg}
% The following is the code for a |\TRF@sep| that has been provided the
% optional argument.
%    \begin{macrocode}
\newcommand\TRF@separg{}
\def\TRF@separg[#1]{\cr\noalign{\kern#1}}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\TRF@sepnoarg}
% The following is the code for a |\TRF@sep| that has not been provided the
% optional argument.
% You might think that it would have been simpler to use |\let| instead of
% |\newcommand| in the line below, or even to just use |\cr| within some of
% the code above instead of defining this new macro at all,
% but |\cr| is a strange beast; it can only be expanded onto the token
% stream when we're sure of everything that is to follow it.
%    \begin{macrocode}
\newcommand\TRF@sepnoarg{\cr}
%    \end{macrocode}
% \end{macro}
%
% \begin{environment}{trgather}
% A |trgather| environment is just like |amsmath|'s |gather| environment except
% that its baseline is at the bottom instead of in the middle.
% Also, it can appear nested within other math environments, whereas |gather|
% can only be used at the outermost level.
%    \begin{macrocode}
\newenvironment{trgather}{%
  \vbox\bgroup%
    \let\\\TRF@sep%
    \ialign\bgroup\hfil{\m@th$##$}\hfil\cr%
}{%
  \crcr\egroup\egroup%
}
%    \end{macrocode}
% \end{environment}
%
% \begin{environment}{tralign}
% A |tralign| environment is like |amsmath|'s |aligned| environment except that
% its baseline is at the bottom instead of in the middle, and it's restricted
% to two columns (for simplicity).
%    \begin{macrocode}
\newenvironment{tralign}{%
  \vbox\bgroup%
    \let\\\TRF@sep%
    \ialign\bgroup\hfil{\m@th$##\null$}&{\m@th$\null##$}\hfil\cr%
}{%
  \crcr\egroup\egroup%
}
%    \end{macrocode}
% \end{environment}
%
% \Finale
\endinput