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% mangletex (11 May 1992) run at 22:09 GMT Monday 08 November 1993%%\title {\TeX\ macros for proof boxes}%%\author {Paul Taylor\\%% Department of Computing,\\%% Imperial College,\\%% London SW7 2BZ\\%% +44 71 589 5111 {\em ext.} 5057\\%% {\tt<pt@doc.ic.ac.uk>}}%%\date{8 November 1993}%% % to get the provisional user documentation do%% % grep ^%% boxproof.tex | sed 's/^%*//' > boxproof-manual.tex%%\documentstyle{article}%%\input boxproof.tex%%\def\meta#1{\mbox{$\langle\hbox{#1}\rangle$}}%%\def\macrowitharg#1#2{{\tt\string#1\bra\meta{#2}\ket}}%%{\escapechar-1 \xdef\bra{\string\{}\xdef\ket{\string\}}}%%\let\subsection\section%%\begin{document}%%\maketitle%%%% \subsection{Introduction}%% The proof%%\begin{proofbox}%% \(\"1"\:\alpha\leftrightarrow\psi(x,\top)\=\\%% \:\Some\beta.\psi(x,\beta)\=\mathsf{total}\\%% \[\exists\beta\kern-1em\:\psi(x,\beta)\\%% \(\:\beta\=\\%% \:\beta=\top\=(*)\\%% \:\psi(x,\!\top)\=\mathsf{subs}\\%% \:\alpha\=\elim\leftrightarrow(\ref{1})\\%% \*\:\alpha\=\\%% \:\psi(x,\!\top)\=\elim\leftrightarrow(\ref{1})\\%% \:\beta=\top\=\mathsf{func}\\%% \:\beta\=(*)\\%% \)\:\alpha=\beta\=\intro\leftrightarrow\\%% \:\psi(x,\alpha)\=\mathsf{subs}\\%% \] \:\psi(x,\alpha)\=\elim\exists\\%% \* \:\psi(x,\alpha)\=\\%% \(\:\alpha\=\\%% \:\alpha=\top\=(*)\\%% \:\phi(x,\!\top)\=\mathsf{subs}\\%% \*\:\psi(x,\!\top)\=\\%% \:\alpha=\top\=\mathsf{func}\\%% \:\alpha\=(*)\\%% \)\:\alpha\leftrightarrow\psi(x,\!\top)\=\intro\leftrightarrow\\%%% \) \:\psi(x,\alpha)\leftrightarrow(\alpha\leftrightarrow\psi(x,\!\top))%% \=\intro\leftrightarrow\\%%\end{proofbox}%% is produced by%%\begin{verbatim}%%\begin{proofbox}%% \(\"1"\:\alpha\leftrightarrow\psi(x,\top)\\%% \:\Some\beta.\psi(x,\beta) \=\mathsf{total}\\%% \[\exists\beta\kern-1em\:\psi(x,\beta)\\%% \(\:\beta\\%% \:\beta=\top \=(*)\\%% \:\psi(x,\!\top) \=\mathsf{subs}\\%% \:\alpha \=\elim\leftrightarrow(\ref{1})\\%% \*\:\alpha\\%% \:\psi(x,\!\top) \=\elim\leftrightarrow(\ref{1})\\%% \:\beta=\top \=\mathsf{func}\\%% \:\beta\=(*)\\%% \)\:\alpha=\beta \=\intro\leftrightarrow\\%% \:\psi(x,\alpha) \=\mathsf{subs}\\%% \] \:\psi(x,\alpha) \=\elim\exists\\%% \* \:\psi(x,\alpha)\\%% \(\:\alpha\\%% \:\alpha=\top \=(*)\\%% \:\phi(x,\!\top) \=\mathsf{subs}\\%% \*\:\psi(x,\!\top)\\%% \:\alpha=\top \=\mathsf{func}\\%% \:\alpha \=(*)\\%% \)\:\alpha\leftrightarrow\psi(x,\!\top) \=\intro\leftrightarrow\\%% \) \:\psi(x,\alpha)\leftrightarrow%% (\alpha\leftrightarrow\psi(x,\!\top)) \=\intro\leftrightarrow\\%%\end{proofbox}%%\end{verbatim}%%%% Syntax as follows:%% each line is of the form%% \begin{center}%% \meta{variables}%% \meta{name}%% \verb/\:/ \meta{formula}%% \verb/\=/ \meta{reason}%% \verb/\-/ \meta{use}%% \verb/\\/%% \end{center}%% where%% \begin{itemize}%% \item \meta{variables} is something like ``$x,y$'' ---%% it's for variables declared at the beginning of $\intro\forall$- and%% $\elim\exists$-boxes.%% \item \meta{name} is a command \verb/\label{fred}/ which%% defines \verb/fred/ to be the label text, which may be used anywhere%% as \verb/\ref{fred}/ --- see {\em The \LaTeX book.}%% Local labels are also available, using \macrowitharg\lbl{name}%% or \verb/\"/\meta{name}\verb/"/; these obey the scoping rules of%% the boxes.%% You may also refer to the previous line as \verb/\ref{-}/.%% \item \meta{formula} is the proposition being asserted.%% \item \meta{reason} is \verb/\intro\land(\ref{john},\ref{mary})/%% or \verb/\elim\forall(\ref{jim})/.%% \item \meta{use} is provided for linear logic, to record the step%%% which uses this one. How this accords with theory I don't yet know.%% \end{itemize}%% Note that the parts are separated by \verb/\:/, \verb/\=/ and \verb/\\/;%% these correspond to%% \begin{center}%% {\bf let } \meta{name} = \meta{expression} : \meta{type}%% \end{center}%% in a declarative language.%% The \verb/\:/, \verb/\=/ and \verb/\-/ fields are optional and may occur%% in any order. If any of them is repeated the last is taken.%% If none of them is present the \meta{variables} field is also ignored.%%%% Proof {\em boxes\/} are ``wrapped up'' as follows:%% \begin{itemize}%%% \item the whole proof in \verb/\begin{proofbox}/...\verb/\end{proofbox}/;%%% \item single-column boxes ($\intro\forall$, $\intro\imp$, $\elim\exists$),%% in \verb/\[/...\verb/\]/.%% \item multiple-column boxes are of two kinds:%% \begin{itemize}%%% \item separate ($\intro\land$) boxes: \verb/\(/...\verb/\*/...\verb/\)/.%% \item stuck together ($\elim\lor$) boxes:%% \verb/\(/...\verb/\+/...\verb/\)/.%% \end{itemize}%% \end{itemize}%% You may put more than two columns in \verb/\(/...\verb/\)/ and even mix%% the \verb/\+/ and \verb/\*/ separators.%%%% The whole proof is enclosed in \verb/\proofbox/...\verb/\endproofbox/%% or \verb/\begin{proofbox}/...\verb/\end{proofbox}/, but the \LaTeX\%% environment form {\em must not\/} be used for nested boxes.%%%% If the proof occurs in paragraph mode (ie in vertical or%% unrestricted horizontal mode) then it is set as a display, using the%% full width of the page. Otherwise it uses only the required width.%%%% A lot of the internals are potentially configurable, but there is not%% yet a user interface suitable for doing this. This will be provided%% in the next version.%%%% This is a {\em prototype\/} proof-box macro package;%% the syntax and implementation may change.%% It is {\tt boxproof.tex} in the \TeX\ system%% (so just \verb/\input boxproof/).%%%% There are some examples in%% \verb+~pt/utilities/proofs/proofboxeg.tex+ and%% \verb+proofboxeg1.tex+.%%%% The prototype version was begun on 2 August 1991, the new one on%% 4 June 1993.%%%%\message{}\edef\proofboxrestoreat{\noexpand\catcode`\noexpand\@\the\catcode`\@}\catcode`\@=11 \count@\year\multiply\count@12 \advance\count@\month\ifnum\count@> 23999 \message{}%\message{}\fi\let\pb@A\+\let\+\relax\let\pb@B\(\let\pb@C\)\def\pb@D{\pb@B\let\)\pb@C}\def\pb@E{\pb@B\displaystyle\let\]\pb@C}%%%% The following are user-settable dimensions.%% \newdimen\prooflinenowidth\prooflinenowidth1em \newdimen\proofboxfullwidth\newdimen\proofboxleftmargin\proofboxleftmargin3em \newdimen\proofboxrightmargin\newdimen\proofboxrulebreadth\proofboxrulebreadth.4pt\newdimen\proofboxseparation\proofboxseparation.2em \newdimen\proofboxbaselineskip\proofboxbaselineskip4ex %%%%%% The following are user-accessible numbers.%%\newcount\proofcolumns\newcount\prooflinetotal\newcount\prooflineinbox%%%%%% These boxes receive the fields of the proof steps.%% They are used in \verb/\proofboxmakeleftcolumn/ {\em etc}.%%\newbox\proofboxproposition\newbox\proofboxreason\newbox\proofboxuse\newbox\proofboxvariables%%%% The left-most switch is used by%%% \verb/\proofboxmakelabel/ to suppress the label in all but the first column.%% On the outside it is true.\newif\ifleftmostproofbox\leftmostproofboxtrue%%%%%% The following are obsolete: they are only included for reverse%% compatibility with the prototype version.\newdimen\proofboxintercol\newdimen\proofboxsurround\newdimen\proofboxformulawidth\newdimen\proofboxmargin\newskip\proofboxlefttabskip\newskip\proofboxrighttabskip%%%%%% \subsection{Redefinable macros}%% WARNING: most of these commands will be hidden and replaced with%% optional arguments to \verb/\proofbox/ in a future version.%% Do not rely on them.%%%% We provide three different ways of numbering the lines of the proof:%% \begin{itemize}%% \item \verb/\runningproofline/: a global running sequence (default),%% \def\runningproofline{\number\prooflinetotal}%%%% \item \verb/\nestedproofline/: a hierarchical system with dots,%%\def\nestedproofline{\relax\ifx\enclosingproofline\empty\else\enclosingproofline{.}\fi\number\prooflineinbox}\def\enclosingproofline{}%%%% \item \verb/\nestedproofline/: a fully hierarchical system which%% also includes the column number%% (\verb/\proof@columns/) as a letter (ASCII quote plus number).%%\def\proofboxcolumn{\ifnum\proofcolumns>\z@{\count@96\advance\count@\proofcolumns\char\count@}\fi}\def\fullynestedproofline{\relax\ifx\enclosingproofline\empty\enclosingproofline\proofboxcolumn.\else\proofboxcolumn\fi\number\prooflineinbox}%%%% \end{itemize}%% \verb/\theproofline/ is the default.%%\let\theproofline\runningproofline%%%% The macro \verb/\proofboxmakelabel#1/ is used to print the line label.%% We only put it in the leftmost box.%% It is printed in small non-ranging Arabic numerals%% ({\the\scriptfont1 0123456789}).%% Right-justify it in \verb/\prooflinenowidth/ if it will fit,%% otherwise let it stick out on the right, {\em i.e.}~left-justify it.%%\def\proofboxmakelabel#1{{\relax\ifleftmostproofbox\setbox\z@\hbox{\hss\the\scriptfont\@ne#1}\ifdim\wd\z@<\prooflinenowidth\setbox\z@\hbox to%\prooflinenowidth{\unhbox\z@}\fi\box\z@\quad\fi}}%%%% Kill the numbers altogether with \verb/\proofboxnonumbers/.%%\def\proofboxnonumbers{\def\proofboxmakelabel##1{}}%%%% How to make the left column of the proof box:%% use the variables field, a space if necessary and the line label.%%\def\proofboxmakeleftcolumn{\kern\proofboxseparation\hfil\dimen@\wd\proofboxvariables\unhbox\proofboxvariables\ifdim\dimen@>\z@\quad\fi\proofboxmakelabel\@currentlabel}%%%% How to make the middle column of the proof box:%% left justify the formula field.%%\def\proofboxmakemiddlecolumn{\unhbox\proofboxproposition\hfil\quad}%%%% How to make the right column of the proof box:%% use the reason and use fields.%%\def\proofboxmakerightcolumn{\box\proofboxreason\box\proofboxuse\hfil\kern\proofboxseparation}%%%%%% Make the four edges of a rectangular box and the separator%% between \verb/\+/ columns.%%\def\proofboxleftside{\leaders\hrule width\proofboxrulebreadth\vfill}\def\proofboxrightside{\leaders\hrule width\proofboxrulebreadth\vfill}\def\proofboxorseparator{\leaders\hrule width\proofboxrulebreadth\vfill}\def\proofboxtopside{\leaders\vrule height\proofboxrulebreadth depth\z@\hfill}%\def\proofboxbottomside{\leaders\vrule height\proofboxrulebreadth depth\z@\hfill}%%%%%% Use dotted lines: \verb/\dottedproofbox/.%%\def\dottedproofbox{\def\proofboxleftside{\leaders\pb@F\vfill} \def\proofboxrightside{\leaders\pb@F\vfill} \def\proofboxorseparator{\leaders\pb@F\vfill} \def\proofboxtopside{\leaders\pb@F\hfill}\def\proofboxbottomside{\leaders\pb@F\hfill}%%}\def\pb@F{\vbox to 5pt{\vss\hbox to 5pt{\hss.\hss}\vss}}%%%% Leave the boxes open at the bottom: \verb/\openproofbox/%%\def\openproofbox{\def\proofboxbottomside{\hfill}} %%%% \subsection{Miscellaneous logical notations}%%%% Print the names of the introduction and elimination rules, for example:%% \begin{quote}%% \verb/\elim\forall/ $\elim\forall$ \qquad%% \verb/\intro\land/ $\intro\land$ \qquad%% \end{quote}\def\intro#1{{{#1}{\cal I}}}\def\elim#1{{{#1}{\cal E}}}%%%% Recall that in \TeX\ the logical connectives and quantifiers are called%% \begin{center}%% \verb/\lor/ $\lor$\quad%% \verb/\land/ $\land$\quad%% \verb/\lnot/ $\lnot$\quad%% \verb/\forall/ $\forall$\quad%% \verb/\exists/ $\exists$%% \end{center}%% The following provide macros for the \verb/\implies/ {\em relation\/}%% and for the binary {\em operation\/} which yields the abstract%% \verb/\implic/ation between formulae.%% The point is that \TeX\ spaces them and breaks the lines differently:%% \begin{center}%% \verb/A\implies B/ $A\implies B$\quad{\em versus\/}\quad%% \verb/A\implic B/ $A\implic B$%% \end{center}%% There are forward and reverse, single and Double versions.%%\mathchardef\implies="3221 \mathchardef\impliedby="3220 \mathchardef\Implies=%"3229 \mathchardef\Impliedby="3228 \mathchardef\implic="2221 \mathchardef\implicby="2220 \mathchardef\Implic="2229 \mathchardef\Implicby="2228 \let\imp\to%%%% Handle the spacing after a variable (and optionally its type)%% bound by a quantifier symbol. For example%% \begin{quote}%% \verb/\All x:X. \phi(x)/\quad prints as\quad $\All x:X.\phi(x)$\quad%% instead of\quad $\forall x:X.\phi(x)$%% \end{quote}\def\Quantifier#1#2.{\pb@G{#1}#2::.}\def\pb@G#1#2:#3:#4.{{{#1}{#2}\def\next{#%3}\ifx\next\empty{.}\mkern4mu \else\mkern1mu{\colon}\mkern1mu{#3}\mkern.5mu{.%}\mkern6mu \fi}}%%% We provide some commonly used forms; \verb/\iota/ ($\iota$) is Russell's%% description operator and should really be inverted.\def\All{\Quantifier\forall}\def\Some{\Quantifier\exists}\def\Function{%\Quantifier\lambda}\def\Product{\Quantifier\Pi}\def\Sum{\Quantifier\Sigma}%\def\TheOne{\Quantifier\iota}\def\Least{\Quantifier\mu}\def\Greatest{%\Quantifier\nu}%%%% There are several notations for substitution.%% After writing $a[x:=b]$ throughout my book I~thought I~might change to%% $[b/x]^*a$.%% This macro reads the source in the first form and prints in the second.%% If you use it you can, like me, defer the decision about%% which notation to use until the final stages, doing%% \begin{quote}%% \verb/\renewcommand{\Subst}{\plainsubstitution}/%% \end{quote}%% if you finally decide on making substitution act on the right.%% This is already an improvement on the literal text, because it%% automatically enlarges the brackets according to the text inside.%% \verb/\Subst/ itself is (following my book)%% defined in terms of the action of a context morphism (\verb/\CtxtMor/)%% on a term. Again you can do%% \begin{quote}%% \verb/\renewcommand{\CtxtMor}{\plaincontextmorphism}/%% \end{quote}%% for something simpler.%% This macro interprets its argument as a comma-separated list%% of items in the form $x:=b$, which it switches to $b/x$.\def\swappingsubstitution#1[#2]{{\CtxtMor[#2]^*{#1}}}%%\def\swappingcontextmorphism[#1]{{\let\pb@J\empty\left[\pb@H#1,[],\/\right]}}%\def\swappingcontextmorphism[#1]{{\let\pb@J\empty[\pb@H#1,[],\/]}}%\def\pb@H#1,{\def\next{#1}\ifx\next\pb@K\else\pb@J\let\pb@J,\pb@I#1:=:=,%\expandafter\pb@H\fi}\def\pb@I#1:=#2:=#3,{{\def\next{#2}\ifx\next\empty\else{%#2}/\fi{#1} }}\def\pb@K{[]}\def\Subst{\swappingsubstitution}\def\CtxtMor{%\swappingcontextmorphism}%%%% The simple versions.\def\plainsubstitution#1#2{{#1}{\left[{#2}\right]}}\def\plaincontextmorphism#%1{{\left[#1\right]}}\newbox\pb@L\newdimen\pb@T\newdimen\pb@U\newdimen\pb@M\newdimen\pb@N\newdimen\pb@O\newdimen\pb@P\newdimen\pb@Q\newdimen\pb@R\newdimen\pb@S\def\pb@V#1#2{\relax\ifdim#1<#2\relax#1=#2\relax\fi}\count@\year\multiply\count@12 \advance\count@\month\ifnum\count@> 25000 \message{%because this one has expired and will no longer work!}\endinput\fi%%\def\proofbox{\relax\ifmmode\let\next\pb@a\else\let\next\pb@b\ifhmode\ifinner\let\next\pb@a\fi\fi\fi\next\global\prooflinetotal\@ne\boxmaxdepth\maxdimen\def\[{\pb@w\pb@f\pb@W}\def\({\pb@w\pb@f\pb@X}\def\proofbox{\[}\let\lbl\pb@y\def\"##1"{\pb@y{##1}}\def\:{\pb@v\proofboxproposition\displaystyle}\def\={%\pb@v\proofboxreason}\def\-{\pb@v\proofboxuse}\def\\{\pb@w\pb@u}\let\+\pb@c\let\*\pb@c\let\]\pb@c\pb@p\pb@s}\def\endproofbox{\pb@q\pb@IA\box\pb@L\egroup}\def\pb@W{\let\+\pb@d\let\*\pb@d\def\){\pb@e\[\)}\let\]\pb@Y\proofcolumns\z@}\def\pb@X{\proofcolumns\@ne\def\+{\pb@j\pb@l}\def\*{\pb@j\pb@m}\def\]{\pb@e\(\]}\let\)\pb@Y}\def\pb@Y{\pb@o\pb@u}\def\pb@a{\hbox\bgroup}\def\pb@b{$$%\kern-\displayindent\hbox to\hsize\bgroup\proofboxfullwidth\hsize\aftergroup$%\aftergroup$}\def\pb@c{\errmessage{Box proofs: \string\+, \string\*, \string\) and \string\] may not be used at the top level}}\def\pb@d{\errmessage{Boxproofs: enclose \string\+ and \string\* in \string\(\string\), not \string\[%\string\]}}\def\pb@e#1#2{\errmessage{Box proofs: \string#1 closed by \string#%2}\pb@Y}\def\pb@f#1{\setbox\pb@L\hbox\bgroup#1\let\pb@g\pb@n\let\enclosingproofline\@currentlabel\edef\pb@h{\the\prooflinetotal}\let\pb@i\pb@h\pb@O\z@\pb@M\z@\pb@N-\maxdimen\pb@P\z@\pb@p\pb@t}\def\pb@j#1{\pb@q\pb@k\let\pb@g#1\advance\proofcolumns\@ne\global\prooflinetotal\pb@h\leftmostproofboxfalse\pb@p\pb@t}\def\pb@k{\pb@g\pb@V\pb@M{\wd\pb@L}\advance\pb@O\wd\pb@L\ifnum\pb@i<\prooflinetotal\edef\pb@i{\the\prooflinetotal}\fi\ifdim\pb@N<\dp\pb@L\pb@N\dp\pb@L\pb@P\pb@Q\fi\setbox\z@\hbox{%\proofboxtopside}\wd\z@\z@\box\z@\setbox\z@\hbox{\proofboxbottomside}\wd\z@\z@\box\z@\kern\pb@R\box\pb@L\kern\pb@S}\def\pb@l{\vtop{\proofboxorseparator}%\penalty\thr@@}\def\pb@m{\vtop{\proofboxrightside}\kern\proofboxseparation\vtop{\proofboxleftside}\penalty\tw@}\def\pb@n{\pb@T\pb@R\setbox\z@\vtop{%\proofboxleftside}\dimen@\wd\z@\advance\pb@T\dimen@\advance\pb@T\proofboxseparation\kern-\pb@T\kern\proofboxseparation\box\z@\penalty\@ne}%\def\pb@o{\pb@q\pb@k\setbox\z@\vtop{\proofboxrightside}\global\advance\pb@S\wd\z@\box\z@\global\advance\pb@S\proofboxseparation\kern\proofboxseparation\global\prooflinetotal\pb@i\global\pb@R\pb@T\global\pb@Q\pb@P\ifnum\proofcolumns<\@ne\proofcolumns\@ne\fi\penalty\proofcolumns\kern-\pb@M\kern\pb@M\kern-\pb@O\kern\pb@O\kern-\pb@S\penalty\@ne\egroup\pb@r}\def\pb@p#1{%\setbox\pb@L\vtop\bgroup\baselineskip\proofboxbaselineskip\prooflineinbox\@ne\pb@T\z@\pb@U\z@#1\pb@u}\def\pb@q{\pb@w\global\pb@R\pb@T\global\pb@S\pb@U\egroup}\def\pb@r{\pb@V\pb@T\pb@R\pb@V\pb@U\pb@S\box\pb@L\prevdepth\pb@Q}\def\pb@s{}\def\pb@t{}\def\pb@u{\edef\@currentlabel{\theproofline}\let\label@name\@currentlabel\setbox\proofboxproposition\box\voidb@x\setbox\proofboxreason\box\voidb@x\setbox\proofboxuse\box\voidb@x\setbox\proofboxvariables\hbox\bgroup\let\pb@z\pb@AA\let\pb@w\pb@x$}\def\pb@v#1{$\egroup\setbox#1\hbox\bgroup\let\pb@z\pb@AA$}\def\pb@w{$\egroup\global\advance\prooflinetotal\@ne\advance\prooflineinbox\@ne\setbox\pb@L\hbox{\setbox\z@\hbox{%\proofboxmakeleftcolumn}\global\pb@R\wd\z@\kern-\pb@R\box\z@\hbox{%\proofboxmakemiddlecolumn}\setbox\z@\hbox{\proofboxmakerightcolumn}\global\pb@S\wd\z@\box\z@\kern-\pb@S\penalty\tw@}\global\pb@Q\dp\pb@L\pb@r}\def\pb@x{$\egroup}\def\pb@y#1{\aftergroup\pb@z\expandafter\aftergroup\csname r@#1%\endcsname}\def\pb@z#1{\edef#1{{\@currentlabel}{}}\expandafter\pb@@A\string#1%\\}\expandafter\def\expandafter\pb@@A\string\r@#1\\{\def\label@name{#1}}\def\pb@AA#1{\aftergroup\pb@z\aftergroup#1}\expandafter\def\csname r@-\endcsname{%{\previous@label}{}}\dimendef\pb@BA=\@ne\dimendef\pb@CA=\tw@\dimendef\pb@DA=%\thr@@\dimendef\pb@EA=4 \dimendef\pb@FA=5 \dimendef\pb@GA=6 \chardef\pb@HA=9\def\pb@IA{\pb@GA\proofboxfullwidth\pb@T\proofboxleftmargin\pb@V\pb@T\pb@R\advance\pb@GA-\pb@T\pb@U\proofboxrightmargin\pb@V\pb@U\pb@S\advance\pb@GA-%\pb@U\pb@V\pb@GA{\wd\pb@L}\pb@JA}\def\pb@JA{\setbox\pb@L\vtop{\unvbox\pb@L\loop\skip@\lastskip\unskip\setbox\pb@HA\lastbox\ifhbox\pb@HA\pb@KA\repeat\unvbox\pb@L\unskip}}\def\pb@KA{\pb@DA\dp\pb@HA\pb@EA\ht\pb@HA\pb@CA\wd\pb@HA\setbox\pb@HA\hbox{\unhbox\pb@HA\count@\lastpenalty\unpenalty\unkern\ifodd\count@\pb@LA\else\setbox\thr@@\lastbox\setbox\tw@\lastbox\setbox\@ne\lastbox\unkern\hbox to\pb@T{\unhbox\@ne}\hbox to\pb@GA{\unhbox\tw@}\hbox to\pb@U{%\unhbox\thr@@}\fi}\dp\pb@HA\pb@DA\ht\pb@HA\pb@EA\setbox\pb@L\vtop{\box\pb@HA\nointerlineskip\vskip\skip@\unvbox\pb@L}}\def\pb@LA{\pb@O\lastkern\unkern\unkern\pb@M\lastkern\unkern\unkern\proofcolumns\lastpenalty\unpenalty\pb@FA\pb@T\ifnum\@ne=\proofcolumns\let\pb@MA\pb@NA\else\advance\pb@GA-\pb@CA\pb@BA\pb@GA\advance\pb@GA\pb@O\ifdim\pb@GA<\proofcolumns\pb@M\divide\pb@BA\proofcolumns\let\pb@MA\pb@PA\else\divide\pb@GA\proofcolumns\let\pb@MA\pb@OA\fi\fi\dimen@\lastkern\unkern\setbox\@ne\lastbox\pb@RA\setbox\pb@HA\hbox{\box\@ne\kern\dimen@}\advance\pb@U-\wd\pb@HA\unkern\loop\setbox\pb@L\lastbox\pb@T\lastkern\unkern\setbox\tw@\lastbox\setbox\thr@@\lastbox\count@\lastpenalty\unpenalty\pb@MA\pb@JA\setbox\@ne\vbox{\offinterlineskip\pb@QA\thr@@\vfil\pb@QA\tw@}\pb@RA\pb@TA{\box\@ne\box\pb@L}\ifnum\count@>\@ne\pb@SA\pb@U\lastkern\unkern\repeat\unhbox\pb@HA}\def\pb@NA{\pb@T\pb@FA\setbox\@ne\lastbox\advance\pb@T-\wd\@ne\dimen@\lastkern\unkern\unkern\kern\dimen@\pb@RA\box\@ne\advance\pb@T-\dimen@}\def\pb@OA{\ifnum\@ne=\count@\pb@NA\fi}\def\pb@PA{\pb@GA\wd\pb@L\advance\pb@GA\pb@BA\pb@OA}\def\pb@QA#1{\setbox#1\hbox to%\wd\pb@L{\unhbox#1}\wd#1\z@\box#1\relax}\def\pb@RA{\setbox\@ne\vbox to\pb@EA{%\offinterlineskip\vskip-\proofboxseparation\unvbox\@ne\vskip-%\proofboxseparation\setbox\z@\null\ht\z@-\pb@DA\dp\z@\pb@DA\box\z@}}\def\pb@SA{\setbox\@ne\lastbox\pb@RA\pb@TA{\box\@ne}\ifodd\count@\else\dimen@\lastkern\unkern\setbox\@ne\lastbox\pb@RA\pb@TA{\box\@ne\kern\dimen@}\fi}\def\pb@TA#1{\setbox\pb@HA\hbox{#1\unhbox\pb@HA}}\let\+\pb@A\proofboxrestoreat\endinput\def\x{}%%%% \end{document}% \fi
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