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+The prftree package provides a package to write proof trees for
+natural deduction calculi, sequent-like calculi, and similar.
+
+The package is composed by the filed
+- prftree.sty: containing the LaTeX engine to typeset proof trees;
+- prftreedoc.pdf: the human readable documentation for the package;
+- prftreedoc.tex: the source code for the documentation.
+
+The package is distributed under the GNU General Public License.
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+\documentclass{amsart}
+\usepackage[ND,SEQ]{prftree}
+\usepackage{url}
+
+\setlength{\fboxsep}{0pt}
+
+\begin{document}
+\title{Proof Trees in \LaTeX}
+\date{}
+\author{Marco Benini}
+\address{Dipartimento di Scienza e Alta Tecnologia\\
+  Universit\`a degli Studi dell'Insubria\\
+  via Valleggio 11, I-22100 Como, Italy}
+\email{marco.benini@uninsubria.it}
+\urladdr{http://marcobenini.wordpress.com}
+\maketitle
+
+% --------------------------
+
+\section{Introduction}\label{sec:introduction}
+Writing proofs in natural deduction or in similar, tree-like calculi,
+is always a challenge: from the typographical point of view, these
+proofs are complex objects that cannot be simply typeset using the
+standard \LaTeX{} commands. Thus, many packages have been developed:
+Sam Buss's \texttt{bussproofs.sty},
+\url{http://math.ucsd.edu/~sbuss/ResearchWeb/bussproofs/}; Makoto
+Tatsuta's \texttt{proof.sty},
+\url{http://research.nii.ac.jp/~tatsuta/proof-sty.html}; and
+\texttt{prooftree.sty} by Paul Taylor,
+\url{http://mirror.ctan.org/macros/generic/proofs/taylor}.
+
+All these packages have their merits and weaknesses. For example,
+Buss's package is extremely flexible but inference rules with more
+than five assumptions cannot be directly typeset. On the other hand,
+Tatsuta's package provides a very simple set of commands doing a
+fine job, but customisation is very limited. Taylor's package provides
+a natural syntax for writing proofs, but customisation is limited, and
+the package has an expire date.
+
+The package presented in the following provides most of the features
+which are already present in Buss's package, coupled with some new
+ones. This package uses a syntax which is closer to Tatsuta's one, but
+almost all the typesetting process is parametric, so that each bit of
+a proof can be customised at will.
+
+The graphical appearance of a proof is similar to the one obtained
+using Taylor's package, but the additional features allow to set up
+the graphical output to follow the style of some of the standard
+textbooks, e.g., A.S.~Troelstra and H.~Schwichtenberg, \textit{Basic
+  Proof Theory}, Cambridge University Press (2000).
+
+% --------------------------
+\clearpage
+\section{Basic Commands}\label{sec:basic_commands}
+The package is invoked by putting \verb|\usepackage{prfree.sty}| in
+the preamble of the document, and installation reduces to put the file
+\texttt{prftree.sty} somewhere in the \LaTeX{} search
+path.\vspace{2ex} 
+
+A proof tree constructs a box with the following internal structure:
+\begin{center}
+  {\setlength{\unitlength}{1em}
+  \begin{picture}(31,6)
+    \put(7,4){\framebox(17,2){$\mbox{assumption}_1 \cdots
+        \mbox{assumption}_n$}} 
+    \put(6,3){\line(1,0){19}}
+    \put(26,2){\framebox(5,2){rule name}}
+    \put(0,2){\framebox(5,2){label}}
+    \put(10,0){\framebox(11,2){conclusion}}
+  \end{picture}}
+\end{center}
+In turn, each assumption is typeset as a box which has usually the
+shape of another proof tree, while the rule name and the label are
+typeset in a text box, and the conclusion in a math box. The aspect of
+the proof line is controlled by suitable options, as is the presence
+of the rule name and of the label. Options cover other aspects of the
+graphical rendering of a proof tree, as it will be explained
+later. The basic command to build a proof tree is \verb|\prftree|.
+
+For example, the proof of $A \supset \neg\neg A$ in natural deduction
+is:
+\begin{displaymath}
+  \prftree[r]{$\supset$I}
+    {\prftree[r]{$\supset$I}
+      {\prftree[r]{$\supset$E}
+        {\prfboundedassumption{A}}
+        {\prfboundedassumption{\neg A}}
+        {\bot}}
+      {\neg\neg A}}
+    {A \supset \neg\neg A}
+\end{displaymath}
+This proof is generated by the following \LaTeX{} code:
+\begin{verbatim}
+  \begin{displaymath}
+    \prftree[r]{$\supset$I}
+      {\prftree[r]{$\supset$I}
+        {\prftree[r]{$\supset$E}
+          {\prfboundedassumption{A}}
+          {\prfboundedassumption{\neg A}}
+          {\bot}}
+        {\neg\neg A}}
+      {A \supset \neg\neg A}
+  \end{displaymath}
+\end{verbatim}
+
+In general, the syntax of the \verb|\prftree| command is:
+\begin{displaymath}
+  \verb|\prftree|[\mbox{options}] \cdots
+  [\mbox{options}]\{\mbox{assumption}_1\} \cdots
+  \{\mbox{assumption}_n\}\{\mbox{conclusion}\}
+\end{displaymath}
+Assumptions are optional and there may be any number of them. Each
+assumption may contain a proof tree, which is typeset
+independently. The conclusion is mandatory, and it is supposed to be a
+formula. Assumptions and the conclusion are typeset in a display style
+math environment. Options control the way the proof is generated: in
+the example, the \verb|r| option has been used to signal that the
+first argument of \verb|\prftree| is the name of the inference rule.
+
+The available options are:
+\begin{itemize}
+\item\ [\textbf{r}], [\textbf{rule}], [\textbf{by rule}],
+  [\textbf{by}], [\textbf{right}]: the first argument after the
+  options is the rule name, which is typeset in text mode;
+\item\ [\textbf{l}], [\textbf{left}], [\textbf{label}]: the first
+  argument after the options is the label of the rule, which is
+  typeset in text mode. If a rule name is present, the first argument
+  is the rule name, and the second one is the label;
+\item\ [\textbf{straight}], [\textbf{straight line}],
+  [\textbf{straightline}]: makes the proof line solid;
+\item\ [\textbf{dotted}], [\textbf{dotted line}],
+  [\textbf{dottedline}]: makes the proof line  dotted;
+\item\ [\textbf{dashed}], [\textbf{dashed line}],
+  [\textbf{dashedline}]: makes the proof line dashed;
+\item\ [\textbf{f}], [\textbf{fancy}], [\textbf{fancy line}],
+  [\textbf{fancyline}]: the proof line will be fancy;
+\item\ [\textbf{s}], [\textbf{single}], [\textbf{single line}],
+  [\textbf{singleline}]: makes the proof line single;
+\item\ [\textbf{d}], [\textbf{double}], [\textbf{double line}],
+  [\textbf{doubleline}]: makes the proof line double;
+\item\ [\textbf{noline}]: suppresses the proof line (prevails over all
+  other line options);
+\item\ [\textbf{summary}]: renders the proof line as the summary
+  symbol (prevails over all other line options except \textbf{noline}).
+\end{itemize}
+By default the proof line is straight and single.  Options may be
+written in sequence, as in \verb|[r,f,d]|, which means that the proof
+tree will have a rule name, and the proof line will be fancy and
+double, or separately, as in \verb|[r][f][d]|, or even as a
+combination, like \verb|[r][f,d]|. Options are evaluated
+left-to-right, so \verb|[d,s]| is the same as \verb|[s]|, while
+\verb|[noline,straight,d]| is the same as \verb|[noline]|.
+
+The conjunction introduction rule illustrates the various line
+options\footnote{The reader is invited to look at the source code of
+  the documentation to see how these examples have been implemented.}:
+\begin{displaymath}
+  \begin{array}{lcc@{\qquad}l}
+    \mbox{default (single straight)} &
+    \prftree{A}{B}{A \wedge B} &
+    \prftree[r]{$\wedge$I}{A}{B}{A \wedge B} &
+    \texttt{[straight]} \\
+    \mbox{double straight} &
+    \prftree[d]{A}{B}{A \wedge B} &
+    \prftree[d,r]{$\wedge$I}{A}{B}{A \wedge B} &
+    \texttt{[double,straight]} \\
+    \mbox{single dotted} &
+    \prftree[dotted]{A}{B}{A \wedge B} &
+    \prftree[dotted,r]{$\wedge$I}{A}{B}{A \wedge B} &
+    \texttt{[dotted]} \\
+    \mbox{double dotted} &
+    \prftree[dotted,d]{A}{B}{A \wedge B} &
+    \prftree[dotted,d,r]{$\wedge$I}{A}{B}{A \wedge B} &
+    \texttt{[double,dotted]} \\
+    \mbox{single dashed} &
+    \prftree[dashed]{A}{B}{A \wedge B} &
+    \prftree[dashed,r]{$\wedge$I}{A}{B}{A \wedge B} &
+    \texttt{[dashed]} \\
+    \mbox{double dashed} &
+    \prftree[dashed,d]{A}{B}{A \wedge B} &
+    \prftree[dashed,d,r]{$\wedge$I}{A}{B}{A \wedge B} &
+    \texttt{[double,dashed]} \\
+    \mbox{single fancy} &
+    \prftree[f]{A}{B}{A \wedge B} &
+    \prftree[f,r]{$\wedge$I}{A}{B}{A \wedge B} &
+    \texttt{[fancy]} \\
+    \mbox{double fancy} &
+    \prftree[f,d]{A}{B}{A \wedge B} &
+    \prftree[f,d,r]{$\wedge$I}{A}{B}{A \wedge B} &
+    \texttt{[double,fancy]} \\
+    \mbox{noline} &
+    \prftree[noline]{A}{B}{A \wedge B} &
+    \prftree[noline,r]{$\wedge$I}{A}{B}{A \wedge B} &
+    \texttt{[noline]}
+  \end{array}
+\end{displaymath}\vspace{1ex}
+
+An assumption is a special proof tree, built by the command:
+\begin{displaymath}
+  \verb|\prfassumption|\{\text{formula}\}
+\end{displaymath}
+Similarly, a bounded assumption is produced by the command:
+\begin{displaymath}
+  \verb|\prfboundedassumption|\{\text{formula}\}
+\end{displaymath}
+as in the previous example.
+
+Although it is possible to type assumptions directly as argument of
+\verb|\prftree|, it is better to use the commands above: as explained
+later, since a proof tree is a box with an internal structure, the
+assumption commands take care of building this structure
+appropriately, while the direct typing does not, which may produce
+unexpected results.\vspace{2ex}
+
+Similarly, axioms are produced by the commands
+\begin{displaymath}
+  \verb|\prfaxiom|\{\mbox{axiom}\}
+\end{displaymath}
+and
+\begin{displaymath}
+  \verb|\prfbyaxiom|\{\mbox{name}\}\{\mbox{axiom}\}
+\end{displaymath}
+For example, the axiom stating that equality is reflexive, is 
+\begin{displaymath}
+  \begin{array}{cc}
+    \prfaxiom{\forall x\, x = x} &
+    \prfbyaxiom{refl}{\forall x\, x = x}
+  \end{array}
+\end{displaymath}
+and they are generated by the \LaTeX{} code
+\begin{displaymath}
+  \begin{array}{cc}
+    \verb|\prfaxiom{\forall x\, x = x}|& 
+    \verb|\prfbyaxiom{refl}{\forall x\, x = x}|
+  \end{array}
+\end{displaymath}\vspace{-.2ex}
+
+Finally, a proof summary is used to summarise a proof. The
+corresponding command is:
+\begin{displaymath}
+  \verb|\prfsummary|[\mbox{name}]\{\mbox{assumption}_1\} \cdots
+  \{\mbox{assumption}_n\}\{\mbox{conclusion}\}
+\end{displaymath}
+The name of the proof is optional, while the assumptions and the
+conclusion are treated as in \verb|\prftree|. When present, the proof
+name is typeset in text mode.
+
+For example, \verb|\prfsummary{\forall x\, x = x}| produces
+\begin{displaymath}
+  \prfsummary{\forall x\, x = x}
+\end{displaymath}
+while \verb|\prfsummary[name]{A(x)}{B(y)}{B(y) \wedge A(x)}| gives
+\begin{displaymath}
+  \prfsummary[name]{A(x)}{B(y)}{B(y) \wedge A(x)}
+\end{displaymath}\vspace{-.2ex}
+
+In general, a proof tree is a \TeX{} box containing all the pieces of
+the tree, with strict bounds: for example,
+\begin{displaymath}
+  \fbox{\prfsummary[name]{A(x)}{B(y)}{B(y) \wedge A(x)}}
+\end{displaymath}
+
+% --------------------------
+\clearpage
+\section{Parameters}\label{sec:parameters}
+A number of parameters may be used to control the typesetting of proof
+trees. They may be changed globally or locally, following the usual
+scoping rules of \TeX{}. In this respect, remember that each
+assumption is typeset independently, so parameters may be changed on a
+sub-proof basis, as will be done in most examples.\vspace{2ex}
+
+There are various \TeX{} dimensions that influence how proofs are
+constructed:
+\begin{itemize}
+\item\ \verb|\prflinepad| (default 0.3ex): the space between the
+  bottom line of assumptions and the proof line, and also the space
+  between the proof line and the top of the conclusion;
+\item\ \verb|\prflineextra| (default 0.3em): the length which extends
+  on the left and on the right the proof line so that it is slightly
+  longer than the largest between the conclusion and the list of
+  (direct) assumptions;
+\item\ \verb|\prflinethickness| (default 0.2pt): the thickness of the
+  proof line;
+\item\ \verb|\prfemptylinethickness| (default 4 times the line
+  thickness): in the rare case when the line is empty, but there are
+  assumptions, this is the distance between the assumptions and the
+  conclusion;
+\item\ \verb|\prfrulenameskip| (default 0.2em): the space between the
+  proof line and the rule name; 
+\item\ \verb|\prflabelskip| (default 0.2em): the space between the
+  proof label and the proof line; 
+\item\ \verb|\prfinterspace| (default .6em): the space between two
+  subsequent assumptions in the assumption list;
+\item\ \verb|\prfdoublelineinterspace| (default 1.2pt): the space
+  between the two lines of a double line.
+\end{itemize}
+
+For example, 
+\begin{displaymath}
+  \prflinepad=.7ex
+  \prftree[r]{$\supset$I}
+  {\prftree[r]{$\supset$I}
+    {\prftree[r]{$\supset$E}
+      {\prfboundedassumption{A}}
+      {\prfboundedassumption{\neg A}}
+      {\bot}}
+    {\neg\neg A}}
+  {A \supset \neg\neg A}
+\end{displaymath}
+is typeset by
+\begin{verbatim}
+  \prflinepad=.7ex
+  \prftree[r]{$\supset$I}
+    {\prftree[r]{$\supset$I}
+      {\prftree[r]{$\supset$E}
+        {\prfboundedassumption{A}}
+        {\prfboundedassumption{\neg A}}
+        {\bot}}
+      {\neg\neg A}}
+    {A \supset \neg\neg A}
+\end{verbatim}
+
+Similarly, \verb|\prflineextra=-.4em| and \verb|\prfrulenameskip=.8em|
+produce: 
+\begin{displaymath}
+  {\prflineextra=-.4em
+    \prfrulenameskip=.8em
+  \prftree[r]{$\supset$I}
+    {\prftree[r]{$\supset$I}
+      {\prftree[r]{$\supset$E}
+        {\prfboundedassumption{A}}
+        {\prfboundedassumption{\neg A}}
+        {\bot}}
+      {\neg\neg A}}
+    {A \supset \neg\neg A}}
+\end{displaymath}
+
+Also, \verb|\prflinethickness=3pt| and
+\verb|\prfdoublelineinterspace=2pt| in the upper sub-proof generate:
+\begin{displaymath}
+  \prftree[r]{$\supset$I}
+    {\prftree[r]{$\supset$I}
+      {\prflinethickness=3pt
+        \prfdoublelineinterspace=2pt
+        \prftree[r,d]{$\supset$E}
+        {\prfboundedassumption{A}}
+        {\prfboundedassumption{\neg A}}
+        {\bot}}
+      {\neg\neg A}}
+    {A \supset \neg\neg A}
+\end{displaymath}
+The corresponding code is
+\begin{verbatim}
+  \prftree[r]{$\supset$I}
+    {\prftree[r]{$\supset$I}
+      {\prflinethickness=3pt
+        \prfdoublelineinterspace=2pt
+        \prftree[r,d]{$\supset$E}
+        {\prfboundedassumption{A}}
+        {\prfboundedassumption{\neg A}}
+        {\bot}}
+      {\neg\neg A}}
+    {A \supset \neg\neg A}
+\end{verbatim}
+
+Line thickness does not affect dashed, dotted, and fancy lines, but
+interline space does: in the example,
+\verb|\prfdoublelineinterspace=4pt| on a fancy line produces
+\begin{displaymath}
+  \prftree[r]{$\supset$I}
+    {\prftree[r]{$\supset$I}
+      {\prfdoublelineinterspace=4pt
+        \prftree[r,d,f]{$\supset$E}
+        {\prfboundedassumption{A}}
+        {\prfboundedassumption{\neg A}}
+        {\bot}}
+      {\neg\neg A}}
+    {A \supset \neg\neg A}
+\end{displaymath}\vspace{.2ex}
+
+Fancy lines are drawn by the \verb|\prffancyline| command. This can be
+redefined: as a guideline, the package defines it as
+\begin{verbatim}
+  \def\prffancyline{\cleaders\hbox to .63em%
+    {\hss\raisebox{-.5ex}[.2ex][0pt]{$\sim$}\hss}\hfill}
+\end{verbatim}\vspace{2ex}
+
+Label spacing works exactly as rule name spacing. Actually, it is
+possible to have a proof with both a label and a rule name:
+\begin{displaymath}
+  \prftree[r]{$\supset$I}
+    {\prflabelskip=.7em
+      \prftree[r,l]{$\supset$I}
+              {[\textsl{$\bot\mathrm{E}$ will not work here!}]}
+      {\prftree[r]{$\supset$E}
+        {\prfboundedassumption{A}}
+        {\prfboundedassumption{\neg A}}
+        {\bot}}
+      {\neg\neg A}}
+    {A \supset \neg\neg A}
+\end{displaymath}
+which has been typeset by
+\begin{verbatim}
+  \prftree[r]{$\supset$I}
+    {\prflabelskip=.7em
+      \prftree[r,l]{$\supset$I}
+              {[\textsl{$\bot\mathrm{E}$ will not work here!}]} 
+      {\prftree[r]{$\supset$E}
+        {\prfboundedassumption{A}}
+        {\prfboundedassumption{\neg A}}
+        {\bot}}
+      {\neg\neg A}}
+    {A \supset \neg\neg A}
+\end{verbatim}\vspace{2ex}
+
+The \verb|\prfinterspace| controls the distance between
+assumptions. Specifically, this is the space between the \emph{boxes}
+containing two assumptions. 
+
+Consider the following example
+\begin{displaymath}
+  \prftree
+  {\prftree
+    {\prftree
+      {\prftree
+        {\prftree
+          {\prfboundedassumption{A \rightarrow (B \rightarrow C)}}
+          {\prfboundedassumption{A}}
+          {B \rightarrow C}}
+        {\prftree
+          {\prfboundedassumption{A \rightarrow B}}
+          {\prfboundedassumption{A}}
+          {B}}
+        {C}}
+      {A \rightarrow C}}
+    {(A \rightarrow B) \rightarrow (A \rightarrow C)}}
+  {(A \rightarrow (B \rightarrow C)) \rightarrow ((A \rightarrow B) 
+    \rightarrow (A \rightarrow C))} 
+\end{displaymath}
+Although the assumptions in the top line are well spaced, the two
+sub-proofs on the top are too close. This can be corrected in two
+different ways: by putting an explicit space, via \verb|\hspace|, in
+front of the second sub-proof, or after the first
+sub-proof---remember, they are just boxes
+\begin{displaymath}
+  \prftree
+  {\prftree
+    {\prftree
+      {\prftree
+        {\prftree
+          {\prfboundedassumption{A \rightarrow (B \rightarrow C)}}
+          {\prfboundedassumption{A}}
+          {B \rightarrow C}\hspace{1.5em}}
+        {\prftree
+          {\prfboundedassumption{A \rightarrow B}}
+          {\prfboundedassumption{A}}
+          {B}}
+        {C}}
+      {A \rightarrow C}}
+    {(A \rightarrow B) \rightarrow (A \rightarrow C)}}
+  {(A \rightarrow (B \rightarrow C)) \rightarrow ((A \rightarrow B) 
+    \rightarrow (A \rightarrow C))} 
+\end{displaymath}
+otherwise, putting $\verb|\prfinterspace|=1.5\mathrm{em}$ before the
+sub-proof whose conclusion is $C$, one obtains the more pleasant
+\begin{displaymath}
+  \prftree
+  {\prftree
+    {\prftree
+      {\prfinterspace=1.5em
+        \prftree
+        {\prftree
+          {\prfboundedassumption{A \rightarrow (B \rightarrow C)}}
+          {\prfboundedassumption{A}}
+          {B \rightarrow C}}
+        {\prftree
+          {\prfboundedassumption{A \rightarrow B}}
+          {\prfboundedassumption{A}}
+          {B}}
+        {C}}
+      {A \rightarrow C}}
+    {(A \rightarrow B) \rightarrow (A \rightarrow C)}}
+  {(A \rightarrow (B \rightarrow C)) \rightarrow ((A \rightarrow B) 
+    \rightarrow (A \rightarrow C))} 
+\end{displaymath}\vspace{.2ex}
+
+The rendering of bounded assumptions is modified by
+\verb|\prfboundedstyle|. When $\verb|\prfboundedstyle| = 0$, the
+format of the assumption is $[\mbox{formula}]$, which is the default
+behaviour; with $\verb|\prfboundedstyle| = 1$, the formula is
+cancelled by a horizontal line; with $\verb|\prfboundedstyle| > 1$,
+the custom \verb|\prfdiscargedassumption| command is invoked:
+\begin{displaymath}
+  \begin{array}{c@{\qquad}c@{\qquad}c}
+    \prfboundedassumption{A(x)} &
+    {\prfboundedstyle=1\prfboundedassumption{A(x)}} &
+    {\prfboundedstyle=2\prfboundedassumption{A(x)}}
+  \end{array}
+\end{displaymath}
+
+The \verb|\prfdiscargedassumption| can be freely redefined. The
+package provides a reference implementation:
+\begin{verbatim}
+  \def\prfdiscargedassumption#1{\left\langle{#1}\right\rangle}
+\end{verbatim}\vspace{2ex}
+
+Proof summaries are drawn according to \verb|\prfsummarystyle|.  The
+default value is $0$, which produces a vertical dotted line. Setting
+$\verb|\prfsummarystyle| = 1$ produces a huge $\Pi$, while
+$\verb|\prfsummarystyle| = 2$ produces a $\prod$. The value $3$ uses a
+$\mathcal{D}$ as the derivation symbol. Values greater than $3$ force
+the summary to be rendered by the \verb|\prffancysummarybox| command.
+\begin{displaymath}
+  \begin{array}{@{}c@{\quad}c@{\qquad}c@{\qquad}c@{}}
+    \verb|\prfsummarystyle| = 0 &
+    {\prfsummary{\forall x.\, x = x}} &
+    {\prfsummary{B(x)}{A(x)}} &
+    {\prfsummary[name]{A(y)}{D(x)}{B(x) \wedge C(x)}} \\[2ex]
+    \verb|\prfsummarystyle| = 1 &
+    {\prfsummarystyle1\prfsummary{\forall x.\, x = x}} &
+    {\prfsummarystyle1\prfsummary{B(x)}{A(x)}} &
+    {\prfsummarystyle1\prfsummary[name]{A(y)}{D(x)}{B(x) \wedge
+        C(x)}} \\[1ex]
+    \verb|\prfsummarystyle| = 2 &
+    {\prfsummarystyle2\prfsummary{\forall x.\, x = x}} &
+    {\prfsummarystyle2\prfsummary{B(x)}{A(x)}} &
+    {\prfsummarystyle2\prfsummary[name]{A(y)}{D(x)}{B(x) \wedge
+        C(x)}} \\[1ex]
+    \verb|\prfsummarystyle| = 3 &
+    {\prfsummarystyle3\prfsummary{\forall x.\, x = x}} &
+    {\prfsummarystyle3\prfsummary{B(x)}{A(x)}} &
+    {\prfsummarystyle3\prfsummary[name]{A(y)}{D(x)}{B(x) \wedge C(x)}}
+    \\[1ex]
+    \verb|\prfsummarystyle| = 4 &
+    {\prfsummarystyle4\prfsummary{\forall x.\, x = x}} &
+    {\prfsummarystyle4\prfsummary{B(x)}{A(x)}} &
+    {\prfsummarystyle4\prfsummary[name]{A(y)}{D(x)}{B(x) \wedge C(x)}}
+  \end{array}
+\end{displaymath}
+
+The fancy summary box is composed by the \verb|\prffancysummarybox|
+command. This can be modified at will. The package defines it as
+\begin{verbatim}
+  \newbox\prf@@fancysummarybox\newdimen\prf@@fancysymmarylen
+  \def\prffancysummarybox{%
+    \sbox{\prf@@fancysummarybox}{\Huge$\bigtriangledown$}%
+    \prf@@fancysymmarylen\ht\prf@@fancysummarybox%
+    \advance\prf@@fancysymmarylen\dp\prf@@fancysummarybox%
+    \sbox{\prf@@fancysummarybox}{%
+      \raisebox{.25\prf@@fancysymmarylen}[.8\prf@@fancysymmarylen]%
+      [0pt]{\usebox{\prf@@fancysummarybox}}}%
+    \prf@@fancysymmarylen\wd\prf@summary@label%
+    \ifdim\prf@@fancysymmarylen>\z@\relax%
+      \prf@@fancysymmarylen\wd\prf@@fancysummarybox%
+      \wd\prf@summary@label.4em%
+      \hbox to\prf@@fancysymmarylen{%
+        \usebox\prf@@fancysummarybox}\kern-.4em%
+        \box\prf@summary@label%
+    \else\usebox\prf@@fancysummarybox\fi}
+\end{verbatim}\vspace{2ex}
+
+The assumptions, conclusions, labels, and rule names are drawn using
+the following commands, which may be redefined:
+\begin{verbatim}
+  \def\prfConclusionBox#1{\hbox%
+    {$\displaystyle\begingroup#1\endgroup\mathstrut$}}
+  \def\prfAssumptionBox#1{\hbox%
+    {$\displaystyle\begingroup#1\endgroup\mathstrut$}}
+  \def\prfRuleNameBox#1{\hbox{\begingroup#1\endgroup\strut}}
+  \def\prfLabelBox#1{\hbox{\begingroup#1\endgroup\strut}}
+\end{verbatim}
+It is not advisable to change these commands in a radical way, unless
+one understands how the graphical engine works.
+
+% -------------------------------------
+\clearpage
+\section{Labels and References}\label{sec:references}
+As discharged assumptions are often hard to track in a proof, the
+package provides a mechanism to label them and to reference them
+inside a proof tree. A reference is made up of three pieces: the
+\emph{label}, which is the name to denote the reference inside the
+text, the \emph{reference value}, which is the value denoted by the
+label, and the \emph{anchor}, which is the graphical rendering of the
+value aside the labelled point of the proof.
+
+For example,
+\begin{displaymath}
+  \begin{prooftree}
+    \prftree[r]{$\supset\mathrm{I}_{\prfref<assum:A>}$}
+    {\prftree[r]{$\supset\mathrm{I}_{\prfref<assum:not_A>}$}
+      {\prftree[r]{$\supset$E}
+        {\prfboundedassumption<assum:A>{A}}
+        {\prfboundedassumption<assum:not_A>{\neg A}}
+        {\bot}}
+      {\neg\neg A}}
+    {A \supset \neg\neg A}
+  \end{prooftree}
+\end{displaymath}
+is generated by the following code
+\begin{verbatim}
+  \begin{prooftree}
+    \prftree[r]{$\supset\mathrm{I}_{\prfref<assum:A>}$}
+    {\prftree[r]{$\supset\mathrm{I}_{\prfref<assum:not_A>}$}
+      {\prftree[r]{$\supset$E}
+        {\prfboundedassumption<assum:A>{A}}
+        {\prfboundedassumption<assum:not_A>{\neg A}}
+        {\bot}}
+      {\neg\neg A}}
+    {A \supset \neg\neg A}
+  \end{prooftree}
+\end{verbatim}
+The labels are \verb|assum:A| and \verb|assum:not_A|, the reference
+values are $1$ and $2$, respectively, and the anchors are these values
+on the discharged assumptions on the top of the proof. The references
+to these labels are the values in the rule names.\vspace{2ex}
+
+The \verb|prooftree| environment delimits the scope of labels: the
+\verb|\end{prooftree}| declaration makes the labels still available
+for reference, but numbering of new labels will restart from
+$1$. Enclosing a proof tree in a \verb|prooftree| environment is not
+mandatory: in such case, labels will be global to the
+document.\vspace{2ex}
+
+Sometimes, labels require two compilation steps to be correctly
+generated: in fact, as \LaTeX{} labels, forward references may be
+undefined in the first compilation step. The package issues a warning
+in this case, and display a \verb|??| for the invalid reference. Also,
+notice how the assumption reference mechanism is analogous to \LaTeX{}
+labels, but it is independent from it.\vspace{2ex}
+
+A reference to a label is made by the
+$\verb|\prfref|\langle\mathrm{label}\rangle$ command: its argument is
+a label, i.e., a string of text following the same rules as the
+argument of the \LaTeX{} \verb|\label| command. As in the \verb|\ref|
+command, the resulting value has no formatting.\vspace{2ex}
+
+A labelled assumption is generated by the following commands:
+\begin{displaymath}
+  \begin{array}{l}
+    \verb|\prfassumption|\langle[\mathrm{option}]\mathrm{label}\rangle
+    \{\mathrm{assumption}\}
+    \\ 
+    \verb|\prfboundedassumption|\langle[\mathrm{option}]
+    \mathrm{label}\rangle\{\mathrm{assumption}\} 
+  \end{array}
+\end{displaymath}
+The first one acts as \verb|\prfassumption| but also declares the
+assumption label and decorates the assumption text with the
+anchor. The second one does the same on bounded assumptions. 
+
+The generation of labels is controlled by the option value:
+\begin{itemize}
+\item \textbf{n}, \textbf{number}, \textbf{arabic}: generates a number
+  (default);
+\item \textbf{r}, \textbf{roman}: generates a lowercase roman number;
+\item \textbf{R}, \textbf{Roman}: generates an uppercase
+  roman number;
+\item \textbf{a}, \textbf{alph}, \textbf{alpha}, \textbf{alphabetic}:
+  produces a lowercase letter;
+\item \textbf{A}, \textbf{Alph}, \textbf{Alpha}, \textbf{Alphabetic}:
+  produces an uppercase letter;
+\item \textbf{f}, \textbf{s}, \textbf{function}, \textbf{symbol},
+  \textbf{function symbol}: produces a footnote symbol, as in
+  Section~C.8.4 of Lamport's, \textit{\LaTeX: A document preparation
+    system};
+\item \textbf{l}, \textbf{label}: tells that the label has not to be
+  defined. This is used to generate a labelled assumption sharing the
+  label with another one, which declares the value and the format.
+\end{itemize}
+Except for \textbf{l} and \textbf{label}, all the options are used to
+format the anchor following the standard \LaTeX{} way available for
+counters. No multiple options are allowed.
+
+For example, the disjunction elimination rule is a perfect way to
+illustrate the reason behind the \textbf{label} option, i.e., the need
+to discharge a pair of assumptions: 
+\begin{displaymath}
+  \begin{prooftree}
+    \prfinterspace=1.2em
+    \prftree[r]{$\vee\mathrm{E}_{\prfref<assum:orE>}$}
+    {\prfsummary{\Gamma}{A \vee B}}
+    {\prfsummary{\Gamma, 
+        \prfboundedassumption<assum:orE>{A}}{C}}
+    {\prfsummary{\Gamma, 
+        \prfboundedassumption<[l]assum:orE>{B}}{C}}{C}
+  \end{prooftree}
+\end{displaymath}
+\begin{verbatim}
+    \prftree[r]{$\vee\mathrm{E}_{\prfref<assum:orE>}$}
+    {\prfsummary{\Gamma}{A \vee B}}
+    {\prfsummary{\Gamma, 
+        \prfboundedassumption<assum:orE>{A}}{C}}
+    {\prfsummary{\Gamma, 
+        \prfboundedassumption<[l]assum:orE>{B}}{C}}{C}
+\end{verbatim}
+
+If a label is declared more than once, a warning is issued when the
+\textbf{label} option is not used: although this is not a mistake, it
+may indicate that a label is reused when it should not.
+
+The same example can be used to show how the other options work:
+\begin{displaymath}
+  \begin{array}{ccc}
+    \begin{prooftree}
+      \prfsummarystyle=2
+      \prftree[r]{$\vee\mathrm{E}_{\prfref<assum:orEn>}$}
+      {\prfsummary{\Gamma}{A \vee B}}
+      {\prfsummary{\Gamma, 
+          \prfboundedassumption<[n]assum:orEn>{A}}{C}}
+      {\prfsummary{\Gamma, 
+          \prfboundedassumption<[l]assum:orEn>{B}}{C}}
+      {C}
+    \end{prooftree} &
+    \begin{prooftree}
+      \prfsummarystyle=2
+      \prftree[r]{$\vee\mathrm{E}_{\prfref<assum:orEr>}$}
+      {\prfsummary{\Gamma}{A \vee B}}
+      {\prfsummary{\Gamma, 
+          \prfboundedassumption<[r]assum:orEr>{A}}{C}}
+      {\prfsummary{\Gamma, 
+          \prfboundedassumption<[l]assum:orEr>{B}}{C}}
+      {C}
+    \end{prooftree} &
+    \begin{prooftree}
+      \prfsummarystyle=2
+      \prftree[r]{$\vee\mathrm{E}_{\prfref<assum:orER>}$}
+      {\prfsummary{\Gamma}{A \vee B}}
+      {\prfsummary{\Gamma, 
+          \prfboundedassumption<[R]assum:orER>{A}}{C}}
+      {\prfsummary{\Gamma, 
+          \prfboundedassumption<[l]assum:orER>{B}}{C}}
+      {C}
+    \end{prooftree} \\
+    \begin{prooftree}
+      \prfsummarystyle=2
+      \prftree[r]{$\vee\mathrm{E}_{\prfref<assum:orEa>}$}
+      {\prfsummary{\Gamma}{A \vee B}}
+      {\prfsummary{\Gamma, 
+          \prfboundedassumption<[a]assum:orEa>{A}}{C}}
+      {\prfsummary{\Gamma, 
+          \prfboundedassumption<[l]assum:orEa>{B}}{C}}
+      {C}
+    \end{prooftree} &
+    \begin{prooftree}
+      \prfsummarystyle=2
+      \prftree[r]{$\vee\mathrm{E}_{\prfref<assum:orEA>}$}
+      {\prfsummary{\Gamma}{A \vee B}}
+      {\prfsummary{\Gamma, 
+          \prfboundedassumption<[A]assum:orEA>{A}}{C}}
+      {\prfsummary{\Gamma, 
+          \prfboundedassumption<[l]assum:orEA>{B}}{C}}
+      {C}
+    \end{prooftree} &
+    \begin{prooftree}
+      \prfsummarystyle=2
+      \prftree[r]{$\vee\mathrm{E}_{\prfref<assum:orEf>}$}
+      {\prfsummary{\Gamma}{A \vee B}}
+      {\prfsummary{\Gamma, 
+          \prfboundedassumption<[f]assum:orEf>{A}}{C}}
+      {\prfsummary{\Gamma, 
+          \prfboundedassumption<[l]assum:orEf>{B}}{C}}
+      {C}
+    \end{prooftree}
+  \end{array}
+\end{displaymath}
+
+Also, as the \verb|\prfboundedstyle| varies, the resulting proof trees
+are: 
+\begin{displaymath}
+  \begin{array}{ccc}
+    \begin{prooftree}
+      \prfboundedstyle=0
+      \prfsummarystyle=4
+      \prftree[r]{$\vee\mathrm{E}_{\prfref<assum:AorE>}$}
+      {\prfsummary{\Gamma}{A \vee B}}
+      {\prfsummary{\Gamma, 
+          \prfboundedassumption<assum:AorE>{A}}{C}}
+      {\prfsummary{\Gamma, 
+          \prfboundedassumption<[l]assum:AorE>{B}}{C}}
+      {C}
+    \end{prooftree} &
+    \begin{prooftree}
+      \prfboundedstyle=1
+      \prfsummarystyle=4
+      \prftree[r]{$\vee\mathrm{E}_{\prfref<assum:BorE>}$}
+      {\prfsummary{\Gamma}{A \vee B}}
+      {\prfsummary{\Gamma, 
+          \prfboundedassumption<assum:BorE>{A}}{C}}
+      {\prfsummary{\Gamma, 
+          \prfboundedassumption<[l]assum:BorE>{B}}{C}}
+      {C}
+    \end{prooftree} &
+    \begin{prooftree}
+      \prfboundedstyle=2
+      \prfsummarystyle=4
+      \prftree[r]{$\vee\mathrm{E}_{\prfref<assum:CorE>}$}
+      {\prfsummary{\Gamma}{A \vee B}}
+      {\prfsummary{\Gamma, 
+          \prfboundedassumption<assum:CorE>{A}}{C}}
+      {\prfsummary{\Gamma, 
+          \prfboundedassumption<[l]assum:CorE>{B}}{C}}
+      {C}
+    \end{prooftree}
+  \end{array}
+\end{displaymath}\vspace{.2ex}
+
+The \verb|prfassumptioncounter| is the \LaTeX{} counter used to
+generate the assumption values. It contains the last used value, and
+initially, it is set to $0$. By modifying its value, e.g., to
+\verb|\setcounter{prfassumptioncounter}{1}|,
+\begin{displaymath}
+  \begin{prooftree}
+    \setcounter{prfassumptioncounter}{1}
+    \prfsummarystyle=2
+    \prftree[r]{$\vee\mathrm{E}_{\prfref<assum:orEff>}$}
+    {\prfsummary{\Gamma}{A \vee B}}
+    {\prfsummary{\Gamma, 
+        \prfboundedassumption<[f]assum:orEff>{A}}{C}}
+    {\prfsummary{\Gamma, 
+        \prfboundedassumption<[l]assum:orEff>{B}}{C}}
+    {C}
+  \end{prooftree}
+\end{displaymath}\vspace{.2ex}
+
+A labelled assumption box is graphically constructed by the package
+command \verb|\prflabelledassumptionbox| which can be redefined if
+needed. It takes two arguments: the assumption and the anchor. Its
+standard definition is
+\begin{verbatim}
+  \def\prflabelledassumptionbox#1#2{%
+    \setbox\prf@fancybox\hbox{${#1}$}%
+    \prf@tmp\wd\prf@fancybox%
+    \setbox\prf@fancybox\hbox{$\box\prf@fancybox^{#2}$}%
+    \wd\prf@fancybox\prf@tmp%
+    \prf@assumption{\box\prf@fancybox}}
+\end{verbatim}
+
+Moreover, also a labelled and bounded assumption is graphically
+rendered by the same command. There is just one exception: when
+$\verb|\prfboundedstyle| > 1$. In fact, since that style is
+controlled by a command that can be redefined, the same must hold for
+references in that style. The command which is called in this case is
+\verb|\prflabelleddiscargedassumption| which can be redefined if
+needed; its standard definition in the package is
+\begin{verbatim}
+  \def\prflabelleddiscargedassumption#1#2{%
+    \prflabelledassumptionbox{\left\langle{#1}\right\rangle}{#2}}
+\end{verbatim}\vspace{2ex}
+
+Also proof summaries can be labelled and referenced. The syntax
+extends the \verb|\prfsummary| command:
+\begin{displaymath}
+  \verb|\prfsummary|\langle[\mathrm{option}]\mathrm{label}\rangle
+  [\mathrm{name}]\{\mathrm{assumption}1\} \cdots
+  \{\mathrm{assumption}_n\}\{\mathrm{conclusion}\} 
+\end{displaymath}
+The reference argument works in the same way as the corresponding one
+for assumptions, and the options are the same.
+
+\begin{displaymath}
+  \setcounter{prfsummarycounter}{0}
+  \begin{array}{c@{\qquad}c@{\qquad}c@{\qquad}c@{\qquad}c}
+    {\prfsummarystyle=0
+      \prfsummary<proof:a0>{A}{B}{A \wedge B}} &
+    {\prfsummarystyle=1
+      \prfsummary<proof:a1>{A}{B}{A \wedge B}} &
+    {\prfsummarystyle=2
+      \prfsummary<proof:a2>{A}{B}{A \wedge B}} &
+    {\prfsummarystyle=3
+      \prfsummary<proof:a3>{A}{B}{A \wedge B}} &
+    {\prfsummarystyle=4
+      \prfsummary<proof:a4>{A}{B}{A \wedge B}} 
+  \end{array}
+\end{displaymath}
+
+These examples have been generated by the following code snippet:
+\begin{verbatim}
+  {\prfsummarystyle=X
+   \prfsummary<proof:aX>{A}{B}{A \wedge B}}
+\end{verbatim}
+
+The \verb|[option]| part of the label specification is optional, and
+it works exactly as the option field of labelled assumptions. This is
+best illustrated by an example:
+\begin{displaymath}
+  \setcounter{prfsummarycounter}{0}
+  \begin{array}{cccc}
+    {\prfsummarystyle=1
+    \prfsummary<[r]proof:b1>{A}{B}{A \wedge B}} &
+    {\prfsummarystyle=1
+    \prfsummary<[R]proof:b2>{A}{B}{A \wedge B}} &
+    {\prfsummarystyle=1
+    \prfsummary<[f]proof:b3>{A}{B}{A \wedge B}} &
+    {\prfsummarystyle=1
+    \prfsummary<[a]proof:b4>{A}{B}{A \wedge B}} \\ &
+    {\prfsummarystyle=1
+    \prfsummary<[A]proof:b5>{A}{B}{A \wedge B}} &
+    {\prfsummarystyle=1
+    \prfsummary<[l]proof:b3>{A}{B}{A \wedge B}} &
+  \end{array}
+\end{displaymath}
+
+These examples have been generated by the following code snippet:
+\begin{verbatim}
+  {\prfsummarystyle=1
+   \prfsummary<[r]proof:bX>{A}{B}{A \wedge B}}
+\end{verbatim}
+and the last line uses the \verb|label| option.\vspace{2ex}
+
+The value of the summary labelling is controlled by the
+\verb|prfsummarycounter| counter, which is initially $0$ and contains
+the last used value.
+
+% -------------------------------------
+\clearpage
+\section{Simplified Commands}\label{sec:simplified_commands}
+The basic commands illustrated so far allow to control proof trees in
+all aspects, but they tend to be verbose in practise. Thus, a number
+of abbreviations are provided to make handier the writing of proofs.
+
+Since they may collide with other packages, these macros are activated
+by suitable options. By loading the package as
+\verb|\usepackage[ND]{prftree.sty}|, the following abbreviations are
+available, which correspond to the inference rule of natural deduction
+calculi:
+\begin{itemize}
+\item \verb|\NDA|: assumption;
+\item \verb|\NDAL|: labelled assumption;
+\item \verb|\NDD|: bounded assumption;
+\item \verb|\NDDL|: labelled bounded assumption;
+\item \verb|\NDP|: generic proof tree;
+\item \verb|\NDANDI|: conjunction introduction;
+\item \verb|\NDANDER|: conjunction elimination, right;
+\item \verb|\NDANDEL|: conjunction elimination, left;
+\item \verb|\NDANDE|: conjunction elimination, unspecified;
+\item \verb|\NDIMPI|: implication introduction;
+\item \verb|\NDIMPIL|: implication introduction with the label of the
+  discharged assumption;
+\item \verb|\NDIMPE|: implication elimination;
+\item \verb|\NDORIR|: disjunction introduction, right;
+\item \verb|\NDORIL|: disjunction introduction, left;
+\item \verb|\NDORI|: disjunction introduction, unspecified;
+\item \verb|\NDORE|: disjunction elimination;
+\item \verb|\NDOREL|: disjunction elimination with the label of the
+  discharged assumptions;
+\item \verb|\NDALLI|: universal quantifier introduction;
+\item \verb|\NDALLE|: universal quantifier elimination;
+\item \verb|\NDEXI|: existential quantifier introduction;
+\item \verb|\NDEXE|: existential quantifier elimination;
+\item \verb|\NDEXE|: existential quantifier elimination with the label
+  of the discharged assumption;
+\item \verb|\NDTI|: truth introduction;
+\item \verb|\NDFE|: falsity elimination;
+\item \verb|\NDLEM|: Law of Excluded Middle.
+\end{itemize}
+
+For example, the proof
+\begin{displaymath}
+  \begin{prooftree}
+    \NDOREL{simp:notA}{\NDLEM{A \vee \neg A}}
+    {\NDIMPI{\NDDL{[l]simp:notA}{A}}{\neg\neg A \supset A}}
+    {\NDIMPIL{simp:notnotA}
+      {\NDFE{\NDIMPE{\NDDL{simp:notnotA}{\neg\neg A}}
+          {\NDDL{simp:notA}{\neg A}}{\bot}}{A}} 
+      {\neg\neg A \supset A}}
+    {\neg\neg A \supset A}
+  \end{prooftree}
+\end{displaymath}
+is typeset in abbreviated form by the following code
+\begin{verbatim}
+    \NDOREL{simp:notA}{\NDLEM{A \vee \neg A}}
+    {\NDIMPI{\NDDL{[l]simp:notA}{A}}{\neg\neg A \supset A}}
+    {\NDIMPIL{simp:notnotA}
+      {\NDFE{\NDIMPE{\NDDL{simp:notnotA}{\neg\neg A}}
+          {\NDDL{simp:notA}{\neg A}}{\bot}}{A}} 
+      {\neg\neg A \supset A}}
+    {\neg\neg A \supset A}
+\end{verbatim}\vspace{2ex}
+
+Similarly, by loading the package as
+\verb|\usepackage[SEQ]{prooftree.sty}|, the following abbreviations
+are available, which roughly correspond to the inference rule of
+sequent calculi:
+\begin{itemize}
+\item \verb|\SEQA|: assumption;
+\item \verb|\SEQD|: bounded assumption;
+\item \verb|\SEQP|: generic proof;
+\item \verb|\SEQAX|: axiom rule;
+\item \verb|\SEQLF|: left falsity;
+\item \verb|\SEQLW|: left weakening;
+\item \verb|\SEQRW|: right weakening;
+\item \verb|\SEQLC|: left contraction;
+\item \verb|\SEQRC|: right contraction;
+\item \verb|\SEQLAND|: left conjunction;
+\item \verb|\SEQRAND|: right conjunction;
+\item \verb|\SEQLOR|: left disjunction;
+\item \verb|\SEQROR|: right disjunction;
+\item \verb|\SEQLIMP|: left implication;
+\item \verb|\SEQRIMP|: right implication;
+\item \verb|\SEQLALL|: left universal quantification;
+\item \verb|\SEQRALL|: right universal quantification;
+\item \verb|\SEQLEX|:  left existential quantification;
+\item \verb|\SEQREX|: right existential quantification;
+\item \verb|\SEQCUT|: cut rule.
+\end{itemize}
+
+One can load the package with both options at the same
+time.\vspace{2ex}
+
+Since the implication symbol is usually represented either as
+$\rightarrow$ or as $\supset$, the package allows to choose which
+representation to use. By default, implication is $\rightarrow$, but
+loading the package with the \verb|[IMP]| option switches to
+$\supset$. The same effect is obtained by the commands
+\verb|\prfIMPOptiontrue| (implication is $\supset$) and
+\verb|prfIMPOptionfalse| (implication is $\rightarrow$).
+
+Of course, the reader is encouraged to develop her own abbreviations
+starting from the provided ones.
+
+% -------------------------------------
+\clearpage
+\section{Hints and Tricks}\label{sec:hints_and_tricks}
+This section shows a few hints and tricks to use the package at its
+best.\vspace{2ex}
+
+Consider the proof:
+\begin{displaymath}
+  \begin{prooftree}
+    \NDOREL{a:notA}{\NDLEM{A \vee \neg A}}
+    {\NDIMPI{\NDDL{a:notA}{A}}{\neg\neg A \supset A}}
+    {\NDIMPIL{a:notnotA}
+      {\NDFE{\NDIMPE{\NDDL{a:notnotA}{\neg\neg A}}
+          {\NDDL{[l]a:notA}{\neg A}}{\bot}}{A}} 
+      {\neg\neg A \supset A}}
+    {\neg\neg A \supset A}
+  \end{prooftree}
+\end{displaymath}
+the space between the axiom and the sub-proof of the positive case is
+visually much less than the space between the positive and the
+negative cases. Looking at boxes, the space is exactly the same, but
+the perception is that spacing is wrong.
+
+We can correct this perception in two distinct ways: by adding space
+between the axiom and the positive case; or, conversely, by moving the
+negative case closer to the positive one.
+
+The first strategy yields:
+\begin{displaymath}
+  \begin{prooftree}
+    \NDOREL{a:notA}{\NDLEM{A \vee \neg A}\hspace{.8em}}
+    {\NDIMPI{\NDDL{[l]a:notA}{A}}{\neg\neg A \supset A}}
+    {\NDIMPIL{a:notnotA}
+      {\NDFE{\NDIMPE{\NDDL{[l]a:notnotA}{\neg\neg A}}
+          {\NDDL{[l]a:notA}{\neg A}}{\bot}}{A}} 
+      {\neg\neg A \supset A}}
+    {\neg\neg A \supset A}
+  \end{prooftree}
+\end{displaymath}
+and this effect is given by adding an appropriate \verb|\hspace| after
+the axiom, as in
+\begin{verbatim}
+  \NDOREL{a:notA}{\NDLEM{A \vee \neg A}\hspace{.4em}}
+  {\NDIMPI{\NDDL{[l]a:notA}{A}}{\neg\neg A \supset A}}
+  {\NDIMPIL{a:notnotA}
+    {\NDFE{\NDIMPE{\NDDL{[l]a:notnotA}{\neg\neg A}}
+        {\NDDL{[l]a:notA}{\neg A}}{\bot}}{A}} 
+    {\neg\neg A \supset A}}
+  {\neg\neg A \supset A}
+\end{verbatim}
+
+Adding the same space in front of the positive case is equivalent.
+
+The second strategy yields:
+\begin{displaymath}
+  \begin{prooftree}
+    \NDOREL{a:notA}{\NDLEM{A \vee \neg A}}
+    {\NDIMPI{\NDDL{[l]a:notA}{A}}{\neg\neg A \supset A}}
+    {\hspace{-.4em}\NDIMPIL{a:notnotA}
+      {\NDFE{\NDIMPE{\NDDL{[l]a:notnotA}{\neg\neg A}}
+          {\NDDL{[l]a:notA}{\neg A}}{\bot}}{A}} 
+      {\neg\neg A \supset A}}
+    {\neg\neg A \supset A}
+  \end{prooftree}
+\end{displaymath}
+Again, this is obtained by adding a negative \verb|hspace| after the
+positive case, or, equivalently, before the negative one:
+\begin{verbatim}
+  \NDOREL{a:notA}{\NDLEM{A \vee \neg A}}
+  {\NDIMPI{\NDDL{[l]a:notA}{A}}{\neg\neg A \supset A}}
+  {\hspace{-.8em}\NDIMPIL{a:notnotA}
+    {\NDFE{\NDIMPE{\NDDL{[l]a:notnotA}{\neg\neg A}}
+        {\NDDL{[l]a:notA}{\neg A}}{\bot}}{A}} 
+    {\neg\neg A \supset A}}
+  {\neg\neg A \supset A}
+\end{verbatim}
+
+In general, to make a wide proof \emph{compact}, one can appropriately
+add negative spaces in front of sub-proofs so to make them closer and
+letting them to overlap as boxes, but not visually, thus \emph{tiling}
+the space.\vspace{2ex}
+
+Since proof trees are boxes, it is easy to align them on need. For
+example the following proof tree, with the bounding box put in
+evidence
+\begin{displaymath}
+  \fbox{\prfsummarystyle=1
+    \prfsummary{A}{B}{A \wedge B}}
+\end{displaymath}
+can be used wherever a box may appear. In the flow of text, it will
+look like \fbox{\prfsummarystyle=1\prfsummary{A}{B}{A \wedge B}}, so
+that the conclusion is aligned with the baseline. This makes easier to
+align proof trees, as in
+\begin{center}
+  \fbox{\prfsummarystyle=1
+    \prfsummary{f}{g}{f \wedge g}}\qquad
+  \fbox{$\begin{prooftree}
+      \NDOREL{a:notA}{\NDLEM{A \vee \neg A}}
+      {\NDIMPI{\NDDL{[l]a:notA}{A}}{\neg\neg A \supset A}}
+      {\hspace{-.4em}\NDIMPIL{a:notnotA}
+        {\NDFE{\NDIMPE{\NDDL{[l]a:notnotA}{\neg\neg A}}
+            {\NDDL{[l]a:notA}{\neg A}}{\bot}}{A}} 
+        {\neg\neg A \supset A}}
+      {\neg\neg A \supset A}
+    \end{prooftree}$}
+\end{center}
+since this is the natural way to put proofs side by side:
+\begin{verbatim}
+  \fbox{\prfsummarystyle=1
+    \prfsummary{f}{g}{f \wedge g}}\qquad
+  \fbox{$
+  \NDOREL{a:notA}{\NDLEM{A \vee \neg A}}
+  {\NDIMPI{\NDDL{[l]a:notA}{A}}{\neg\neg A \supset A}}
+  {\hspace{-.4em}\NDIMPIL{a:notnotA}
+    {\NDFE{\NDIMPE{\NDDL{[l]a:notnotA}{\neg\neg A}}
+        {\NDDL{[l]a:notA}{\neg A}}{\bot}}{A}} 
+    {\neg\neg A \supset A}}
+  {\neg\neg A \supset A}$}
+\end{verbatim}
+
+But, if really one has to include a proof tree in the flow of text, it
+is slightly better to vertically centre the box, as in
+\fbox{$\vcenter{\prfsummary{A}{B}{A \wedge B}}$}. This is obtained by
+\begin{verbatim}
+  $\vcenter{\prfsummary{A}{B}{A \wedge B}}$
+\end{verbatim}
+
+Of course, the result is not pleasant, because rows are far apart,
+which is unavoidable because of the height of the proof tree.  The
+same principle applies also to arrays of proof trees:
+\begin{displaymath}
+  \begin{array}{lcccc}
+    \text{some text} &
+    \setcounter{prfsummarycounter}{0}
+    \setcounter{prfassumptioncounter}{0}
+    {\prfsummarystyle=1
+    \prfsummary<[l]proof:b1>{A}{B}{A \wedge B}} &
+    {\prfsummarystyle=1
+    \prfsummary<[l]proof:b2>{A}{B}{A \wedge B}} &
+    {\prfsummarystyle=1
+    \prfsummary<[l]proof:b3>{A}{B}{A \wedge B}} &
+    {\prfsummarystyle=1
+    \prfsummary<[l]proof:b4>{A}{B}{A \wedge B}}
+  \end{array}
+\end{displaymath}
+\begin{verbatim}
+  \begin{array}{lcccc}
+    \text{some text} &
+    {\prfsummarystyle=1
+    \prfsummary<[l]proof:b1>{A}{B}{A \wedge B}} &
+    {\prfsummarystyle=1
+    \prfsummary<[l]proof:b2>{A}{B}{A \wedge B}} &
+    {\prfsummarystyle=1
+    \prfsummary<[l]proof:b3>{A}{B}{A \wedge B}} &
+    {\prfsummarystyle=1
+    \prfsummary<[l]proof:b4>{A}{B}{A \wedge B}}
+  \end{array}
+\end{verbatim}
+vertically aligns the cells to their baselines.
+
+On the contrary
+\begin{displaymath}
+  \begin{array}{lcccc}
+    \text{some text} &
+    \setcounter{prfsummarycounter}{0}
+    \setcounter{prfassumptioncounter}{0}
+    \vcenter{\prfsummarystyle=1
+    \prfsummary<[l]proof:b1>{A}{B}{A \wedge B}} &
+    \vcenter{\prfsummarystyle=1
+    \prfsummary<[l]proof:b2>{A}{B}{A \wedge B}} &
+    \vcenter{\prfsummarystyle=1
+    \prfsummary<[l]proof:b3>{A}{B}{A \wedge B}} &
+    \vcenter{\prfsummarystyle=1
+    \prfsummary<[l]proof:b4>{A}{B}{A \wedge B}}
+  \end{array}
+\end{displaymath}
+is much better, and it is obtained by
+\begin{verbatim}
+  \begin{array}{lcccc}
+    \text{some text} &
+    \vcenter{\prfsummarystyle=1
+    \prfsummary<[l]proof:b1>{A}{B}{A \wedge B}} &
+    \vcenter{\prfsummarystyle=1
+    \prfsummary<[l]proof:b2>{A}{B}{A \wedge B}} &
+    \vcenter{\prfsummarystyle=1
+    \prfsummary<[l]proof:b3>{A}{B}{A \wedge B}} &
+    \vcenter{\prfsummarystyle=1
+    \prfsummary<[l]proof:b4>{A}{B}{A \wedge B}}
+  \end{array}
+\end{verbatim}\vspace{2ex}
+
+The labelling of proof summaries is useful when a proof is very large
+and there is the need to split it. The strategy is to select some
+sub-proofs and to show them as summaries: instead of writing
+\begin{displaymath}
+  \setcounter{prfsummarycounter}{0}
+  \setcounter{prfassumptioncounter}{0}
+  \NDOREL{a:notA}{\NDLEM{A \vee \neg A}}
+  {\NDIMPI{\NDDL{[l]a:notA}{A}}{\neg\neg A \supset A}}
+  {\NDIMPIL{a:notnotA}
+    {\NDFE{\NDIMPE{\NDDL{[l]a:notnotA}{\neg\neg A}}
+        {\NDDL{[l]a:notA}{\neg A}}{\bot}}{A}} 
+    {\neg\neg A \supset A}}
+  {\neg\neg A \supset A}
+\end{displaymath}
+we may consider to define
+\begin{displaymath}
+  \setcounter{prfsummarycounter}{0}
+  \setcounter{prfassumptioncounter}{0}
+  \mbox{Let }
+  \vcenter{\vbox{\prfsummary<s:abbrev>
+      {\NDDL{s:notnotA}{\neg\neg A}}
+      {\NDAL{s:notA}{\neg A}}
+      {\neg\neg A \supset A}}}
+  \equiv
+  \vcenter{\hbox{$\NDIMPIL{s:notnotA}
+      {\NDFE{\NDIMPE{\NDDL{[l]s:notnotA}{\neg\neg A}}
+          {\NDAL{[l]s:notA}{\neg A}}{\bot}}{A}} 
+      {\neg\neg A \supset A}$}}
+\end{displaymath}
+allowing to abbreviate the whole proof as
+\begin{displaymath}
+  \NDOREL{s:notA}{\NDLEM{A \vee \neg A}}
+  {\NDIMPI{\NDDL{[l]s:notA}{A}}{\neg\neg A \supset A}}
+  {\prfsummary<s:abbrev>
+    {\NDDL{[l]s:notnotA}{\neg\neg A}}
+    {\NDDL{[l]s:notA}{\neg A}}
+      {\neg\neg A \supset A}}
+  {\neg\neg A \supset A}
+\end{displaymath}
+
+The corresponding \LaTeX{} code is
+\begin{verbatim}
+  \setcounter{prfsummarycounter}{0}
+  \setcounter{prfassumptioncounter}{0}
+  \mbox{Let }
+  \vcenter{\vbox{\prfsummary<s:abbrev>
+      {\NDDL{s:notnotA}{\neg\neg A}}
+      {\NDAL{s:notA}{\neg A}}
+      {\neg\neg A \supset A}}}
+  \equiv
+  \vcenter{\hbox{$\NDIMPIL{s:notnotA}
+      {\NDFE{\NDIMPE{\NDDL{[l]s:notnotA}{\neg\neg A}}
+          {\NDAL{[l]s:notA}{\neg A}}{\bot}}{A}} 
+      {\neg\neg A \supset A}$}}
+\end{verbatim}
+for the definition of the proof summary, and
+\begin{verbatim}
+  \NDOREL{s:notA}{\NDLEM{A \vee \neg A}}
+  {\NDIMPI{\NDDL{[l]s:notA}{A}}{\neg\neg A \supset A}}
+  {\prfsummary<s:abbrev>
+    {\NDDL{[l]s:notnotA}{\neg\neg A}}
+    {\NDDL{[l]s:notA}{\neg A}}
+      {\neg\neg A \supset A}}
+  {\neg\neg A \supset A}
+\end{verbatim}
+for its use.
+
+% -------------------------------------
+\clearpage
+\section{More Examples}\label{sec:examples}
+This section shows a number of examples illustrating the package. See
+the previous sections for the description of the features.\vspace{2ex}
+
+The disjunction elimination rule, with various line options:
+\begin{displaymath}
+  \begin{array}{@{}ccc@{}}
+    {\prfsummarystyle=1
+      \prftree{\prfsummary{\Gamma}{A \vee B}}
+      {\prfsummary{\Gamma, \prfboundedassumption{A}}{C}}
+      {\prfsummary{\Gamma, \prfboundedassumption{B}}{C}}
+      {C}} &
+    {\prfsummarystyle=1
+      \prftree[r]{$\vee$E}{\prfsummary{\Gamma}{A \vee B}}
+      {\prfsummary{\Gamma, \prfboundedassumption{A}}{C}}
+      {\prfsummary{\Gamma, \prfboundedassumption{B}}{C}}
+      {C}} &
+    {\prfsummarystyle=1
+      \prftree[l]{$\vee$E}{\prfsummary{\Gamma}{A \vee B}}
+      {\prfsummary{\Gamma, \prfboundedassumption{A}}{C}}
+      {\prfsummary{\Gamma, \prfboundedassumption{B}}{C}}
+      {C}} \\
+    {\prfsummarystyle=1
+      \prftree[d]{\prfsummary{\Gamma}{A \vee B}}
+      {\prfsummary{\Gamma, \prfboundedassumption{A}}{C}}
+      {\prfsummary{\Gamma, \prfboundedassumption{B}}{C}}
+      {C}} &
+    {\prfsummarystyle=1
+      \prftree[r][d]{$\vee$E}{\prfsummary{\Gamma}{A \vee B}}
+      {\prfsummary{\Gamma, \prfboundedassumption{A}}{C}}
+      {\prfsummary{\Gamma, \prfboundedassumption{B}}{C}}
+      {C}} &
+    {\prfsummarystyle=1
+      \prftree[l][d]{$\vee$E}{\prfsummary{\Gamma}{A \vee B}}
+      {\prfsummary{\Gamma, \prfboundedassumption{A}}{C}}
+      {\prfsummary{\Gamma, \prfboundedassumption{B}}{C}}
+      {C}} \\
+    {\prfsummarystyle=1
+      \prftree[dotted]{\prfsummary{\Gamma}{A \vee B}}
+      {\prfsummary{\Gamma, \prfboundedassumption{A}}{C}}
+      {\prfsummary{\Gamma, \prfboundedassumption{B}}{C}}
+      {C}} &
+    {\prfsummarystyle=1
+      \prftree[r,dotted]{$\vee$E}{\prfsummary{\Gamma}{A \vee B}}
+      {\prfsummary{\Gamma, \prfboundedassumption{A}}{C}}
+      {\prfsummary{\Gamma, \prfboundedassumption{B}}{C}}
+      {C}} &
+    {\prfsummarystyle=1
+      \prftree[l,dotted]{$\vee$E}{\prfsummary{\Gamma}{A \vee B}}
+      {\prfsummary{\Gamma, \prfboundedassumption{A}}{C}}
+      {\prfsummary{\Gamma, \prfboundedassumption{B}}{C}}
+      {C}} \\
+    {\prfsummarystyle=1
+      \prftree[d,dotted]{\prfsummary{\Gamma}{A \vee B}}
+      {\prfsummary{\Gamma, \prfboundedassumption{A}}{C}}
+      {\prfsummary{\Gamma, \prfboundedassumption{B}}{C}}
+      {C}} &
+    {\prfsummarystyle=1
+      \prftree[r,d,dotted]{$\vee$E}{\prfsummary{\Gamma}{A \vee B}}
+      {\prfsummary{\Gamma, \prfboundedassumption{A}}{C}}
+      {\prfsummary{\Gamma, \prfboundedassumption{B}}{C}}
+      {C}} &
+    {\prfsummarystyle=1
+      \prftree[l,d,dotted]{$\vee$E}{\prfsummary{\Gamma}{A \vee B}}
+      {\prfsummary{\Gamma, \prfboundedassumption{A}}{C}}
+      {\prfsummary{\Gamma, \prfboundedassumption{B}}{C}}
+      {C}} \\
+    {\prfsummarystyle=1
+      \prftree[dashed]{\prfsummary{\Gamma}{A \vee B}}
+      {\prfsummary{\Gamma, \prfboundedassumption{A}}{C}}
+      {\prfsummary{\Gamma, \prfboundedassumption{B}}{C}}
+      {C}} &
+    {\prfsummarystyle=1
+      \prftree[r,dashed]{$\vee$E}{\prfsummary{\Gamma}{A \vee B}}
+      {\prfsummary{\Gamma, \prfboundedassumption{A}}{C}}
+      {\prfsummary{\Gamma, \prfboundedassumption{B}}{C}}
+      {C}} &
+    {\prfsummarystyle=1
+      \prftree[l,dashed]{$\vee$E}{\prfsummary{\Gamma}{A \vee B}}
+      {\prfsummary{\Gamma, \prfboundedassumption{A}}{C}}
+      {\prfsummary{\Gamma, \prfboundedassumption{B}}{C}}
+      {C}} \\
+    {\prfsummarystyle=1
+      \prftree[d,dashed]{\prfsummary{\Gamma}{A \vee B}}
+      {\prfsummary{\Gamma, \prfboundedassumption{A}}{C}}
+      {\prfsummary{\Gamma, \prfboundedassumption{B}}{C}}
+      {C}} &
+    {\prfsummarystyle=1
+      \prftree[d,r,dashed]{$\vee$E}{\prfsummary{\Gamma}{A \vee B}}
+      {\prfsummary{\Gamma, \prfboundedassumption{A}}{C}}
+      {\prfsummary{\Gamma, \prfboundedassumption{B}}{C}}
+      {C}} &
+    {\prfsummarystyle=1
+      \prftree[d,l,dashed]{$\vee$E}{\prfsummary{\Gamma}{A \vee B}}
+      {\prfsummary{\Gamma, \prfboundedassumption{A}}{C}}
+      {\prfsummary{\Gamma, \prfboundedassumption{B}}{C}}
+      {C}} \\
+    {\prfsummarystyle=1
+      \prftree[f]{\prfsummary{\Gamma}{A \vee B}}
+      {\prfsummary{\Gamma, \prfboundedassumption{A}}{C}}
+      {\prfsummary{\Gamma, \prfboundedassumption{B}}{C}}
+      {C}} &
+    {\prfsummarystyle=1
+      \prftree[r,f]{$\vee$E}{\prfsummary{\Gamma}{A \vee B}}
+      {\prfsummary{\Gamma, \prfboundedassumption{A}}{C}}
+      {\prfsummary{\Gamma, \prfboundedassumption{B}}{C}}
+      {C}} &
+    {\prfsummarystyle=1
+      \prftree[l,f]{$\vee$E}{\prfsummary{\Gamma}{A \vee B}}
+      {\prfsummary{\Gamma, \prfboundedassumption{A}}{C}}
+      {\prfsummary{\Gamma, \prfboundedassumption{B}}{C}}
+      {C}} \\
+    {\prfsummarystyle=1
+      \prftree[noline]{\prfsummary{\Gamma}{A \vee B}}
+      {\prfsummary{\Gamma, \prfboundedassumption{A}}{C}}
+      {\prfsummary{\Gamma, \prfboundedassumption{B}}{C}}
+      {C}} &
+    {\prfsummarystyle=1
+      \prftree[noline][r]{$\vee$E}{\prfsummary{\Gamma}{A \vee B}}
+      {\prfsummary{\Gamma, \prfboundedassumption{A}}{C}}
+      {\prfsummary{\Gamma, \prfboundedassumption{B}}{C}}
+      {C}} &
+    {\prfsummarystyle=1
+      \prftree[noline][l]{$\vee$E}{\prfsummary{\Gamma}{A \vee B}}
+      {\prfsummary{\Gamma, \prfboundedassumption{A}}{C}}
+      {\prfsummary{\Gamma, \prfboundedassumption{B}}{C}}
+      {C}}
+  \end{array}
+\end{displaymath}
+
+Proof that the Law of Excluded middle implies $\neg\neg A \supset A$:
+\begin{displaymath}
+  \prftree[r]{$\vee$E}
+  {\prfbyaxiom{LEM}
+    {A \vee \neg A}\hspace{.4em}}
+  {\prftree[r]{$\supset$I}
+    {\prfboundedassumption{A}}
+    {\neg\neg A \supset A}}
+  {\prftree[r]{$\supset$I}
+    {\prftree[r]{$\bot$E}
+      {\prftree[r]{$\supset$E}
+        {\prfboundedassumption{\neg\neg A}}
+        {\prfboundedassumption{\neg A}}
+        {\bot}}
+      {A}}
+    {\neg\neg A \supset A}}
+  {\neg\neg A \supset A}
+\end{displaymath}
+
+Proof that the Law of Excluded middle implies $\neg\neg A \supset A$
+with labels instead of rule names, except on axioms:
+\begin{displaymath}
+  \prftree[l]{$\vee$E}
+  {\prfbyaxiom{LEM}
+    {A \vee \neg A}\hspace{.6em}}
+  {\prftree[l]{$\supset$I}
+    {\prfboundedassumption{A}}
+    {\neg\neg A \supset A}}
+  {\prftree[l]{$\supset$I}
+    {\prftree[l]{$\bot$E}
+      {\prftree[l]{$\supset$E}
+        {\prfboundedassumption{\neg\neg A}}
+        {\prfboundedassumption{\neg A}}
+        {\bot}}
+      {A}}
+    {\neg\neg A \supset A}}
+  {\neg\neg A \supset A}
+\end{displaymath}
+
+Another simple proof in natural deduction:
+\begin{displaymath}
+  \prftree
+  {\prftree
+    {\prftree
+      {\prftree
+        {\prftree
+          {\prfboundedassumption{A \rightarrow (B \rightarrow C)}}
+          {\prfboundedassumption{A}}
+          {B \rightarrow C}\hspace{2em}}
+        {\prftree
+          {\prfboundedassumption{A \rightarrow B}}
+          {\prfboundedassumption{A}}
+          {B}}
+        {C}}
+      {A \rightarrow C}}
+    {(A \rightarrow B) \rightarrow (A \rightarrow C)}}
+  {(A \rightarrow (B \rightarrow C)) \rightarrow ((A \rightarrow B) 
+    \rightarrow (A \rightarrow C))} 
+\end{displaymath}
+
+The same proof, under the proposition-as-types interpretation:
+\begin{displaymath}
+  \prftree
+  {\prftree
+    {\prftree
+      {\prftree
+        {\prftree
+          {\prfassumption{u\colon A \rightarrow (B \rightarrow C)}}
+          {\prfassumption{w\colon A}}
+          {u w\colon B \rightarrow C}\hspace{2em}}
+        {\prftree
+          {\prfassumption{v\colon A \rightarrow B}}
+          {\prfassumption{w\colon A}}
+          {v w\colon B}}
+        {u w(v w)\colon C}}
+      {\lambda w.\, u w(v w)\colon A \rightarrow C}}
+    {\lambda v w.\, u w(v w)\colon (A \rightarrow B) \rightarrow (A
+      \rightarrow C)}}
+  {\lambda u v w.\, u w(v w)\colon (A \rightarrow (B \rightarrow C))
+    \rightarrow ((A \rightarrow B) \rightarrow (A \rightarrow C))}
+\end{displaymath}
+
+A deduction in a sequent calculus:
+\begin{displaymath}
+  \prfinterspace=1.2em
+  \prftree
+  {\prftree
+    {\prftree
+      {\prftree
+        {\prfassumption{A \Rightarrow A}}
+        {\prftree
+          {\prfassumption{A \Rightarrow A}}
+          {\prftree
+            {B \Rightarrow B}
+            {C \Rightarrow C}
+            {B, B \rightarrow C \Rightarrow C}}
+          {A, A \rightarrow B, B \rightarrow C \Rightarrow C}}
+        {A, A \rightarrow B, A \rightarrow (B \rightarrow C)
+          \Rightarrow C}}
+      {A \rightarrow B, A \rightarrow (B \rightarrow C) \Rightarrow A 
+        \rightarrow C}}
+    {A \rightarrow (B \rightarrow C) \Rightarrow (A \rightarrow B) 
+      \rightarrow (A \rightarrow C)}}
+  {\Rightarrow (A \rightarrow (B \rightarrow C)) \rightarrow ((A
+    \rightarrow B) \rightarrow (A \rightarrow C))}
+\end{displaymath}
+
+% -------------------------------------
+\clearpage
+\section{Internals}\label{sec:internals}
+A proof tree is typeset as a \TeX{} box in horizontal mode. This means
+that wherever a character can stay, so does a proof: in principle,
+there is no need to put the proof in a math environment. Also, the
+width of a proof is exactly the width of the box; the height of the
+proof is the height of the conclusion plus the total height of all the
+matter above it; the depth of the proof is the depth of the
+conclusion. The proof is aligned so that the current baseline is the
+baseline of the conclusion.
+
+For example, the proof of $g \supset \neg\neg g$ in natural deduction
+is:
+\begin{displaymath}
+  \mbox{proof} \equiv 
+  \fbox{\prftree[r]{$\supset$I}
+    {\prftree[r]{$\supset$I}
+      {\prftree[r]{$\supset$E}
+        {\prfboundedassumption{g}}
+        {\prfboundedassumption{\neg g}}
+        {\bot}}
+      {\neg\neg g}}
+    {g \supset \neg\neg g}}
+\end{displaymath}
+The proof has been surrounded by a framebox to make evident its
+bounds. Also, since the letter $g$ has a depth, the example shows how
+depth in the conclusion influences the alignment of the proof with
+respect to the preceding text.\vspace{2ex}
+
+Actually, the fundamental command in the package is \verb|\prftree|:
+the commands to construct assumptions (\verb|\prfassumption| and
+\verb|\prfboundedassumption|), those to generate axioms
+(\verb|\prfaxiom| and \verb|\prfbyaxiom|), and \verb|\prfsummary| are
+just appropriate instances.\vspace{2ex}
+
+The \verb|\prftree| command is composed by a parser, which takes care
+of reading the various options and parameters, and by a graphical
+engine, \verb|\prf@draw|, which calculates and draw the box containing
+the proof tree.
+
+It may be useful to understand how the graphical engine works. In the
+first place, each proof tree is a box with a structure:
+\begin{center}
+  {\setlength{\unitlength}{1em}
+  \begin{picture}(27,6)
+    \put(0.8,0){\framebox(26.2,6){}}
+    \put(5,4){\framebox(18,1.8){$\cdots$}}
+    \put(5.2,4.2){\framebox(6,1.4){$\mbox{assumption}_1$}} 
+    \put(16.8,4.2){\framebox(6,1.4){$\mbox{assumption}_n$}} 
+    \put(7,3){\line(1,0){14}}
+    \put(22,2.3){\framebox(4.8,1.4){rule name}}
+    \put(1,2.3){\framebox(4.8,1.4){label}}
+    \put(8.5,0.2){\framebox(11,1.8){conclusion}}
+  \end{picture}}
+\end{center}
+
+The conclusion, the proof line, and the \emph{assumption line} are
+centred. The assumption line is the line whose first element is the
+conclusion of the first assumption, and whose last element is the
+conclusion of the last assumption, properly spaced so that all the
+assumptions fit in between. The width of the proof line is calculated
+as the maximum of the width of the assumption line and the conclusion,
+with the rule name and the label, if present, hanging on the right and
+the left, respectively.
+
+To calculate the assumption line, the engine keeps track of the
+position of the conclusion within a proof tree, which reduces to
+remember how far is the conclusion from the left margin
+(\verb|Lassum|), and how far it is from the right margin
+(\verb|Rassum|). So, the assumption line starts from the value of
+\verb|Lassum| of the first assumption, and finishes at \verb|Rassum|
+of the last assumption. 
+
+Thus, with these values it is not difficult to figure out the
+mathematics to place the various boxes around, so to combine them into
+a proof tree. This is exactly what the graphical engine does.
+
+Unfortunately, when one writes assumptions as simple formulae, without
+the \verb|\prfassumption| command, the corresponding \verb|Lassum| and
+\verb|Rassum| are not set to $0$, which is the right value. In fact,
+the recursive expansion of the \verb|\prf@draw| macro follows the
+\emph{natural} order in the construction of the proof box, which is
+extremely useful because it allows to locally modify parameters in
+sub-proofs; but this order conflicts with proper rendering of
+assumptions which are not proof trees.
+
+Also, the hints on how to put space between assumptions, see
+Section~\ref{sec:hints_and_tricks}, may have strange effects: if space
+is added in front of the first assumption or behind the last one, this
+space makes invalid the values of \verb|Lassum| and \verb|Rassum|,
+respectively, yielding hard to predict results.
+
+It is worth remarking that the mathematics of the graphical engine is
+sound, which means that zero or negative values for the various
+dimensions specified as parameters, or using \emph{bizarre} boxes in
+the fancy commands, yields the expected results, as far as boxes do
+not have parts which extends beyond the bounds.\vspace{2ex}
+
+The implementation of references mimics the implementation of
+\verb|\label| and \verb|\ref| in \LaTeX. Whenever a reference is
+defined, through a command with the $\langle \mathrm{label}\rangle$ as
+the first argument, the reference value is created according to the
+options, and it gets stored in the \texttt{.aux} file, by writing
+$\verb|\prfauxvalue|\{\mathrm{label}\}\{\mathrm{value}\}$ in the
+file. Then, when the source code will be recompiled, and the
+\texttt{.aux} file read, this command will be executed before any
+occurrence of a reference, which can be resolved.
+
+Most difficulties in the implementation of references lie in the way
+to construct the boxes to be used in the proof tree. But, the tricky
+part is the interaction with the \LaTeX{} and \TeX{} kernel for error
+reporting. Actually, it is in this part that the bugs signalled in the
+next section have their origin. 
+
+% -------------------------------------
+\clearpage
+\section{Future Features and Bugs}\label{sec:future_features}
+Essentially, all the features of Buss's package have been implemented
+but one: alignment of proofs according to the $\vdash$ (or equivalent)
+sign. While this feature is occasionally useful in the writing of
+sequent proofs, it requires some trickery in the graphical engine, so
+it has been postponed for the moment.\vspace{2ex}
+
+Moreover, automatic compact proofs have been analysed, but not
+implemented. A compact proof minimises the amount of space between
+subsequent assumptions, eventually making the upper trees to overlap
+as boxes, but not as typed text. 
+
+The algorithm to obtain this result is not immediate: one should keep
+track of the left and right \emph{skylines} of a proof. Comparing the
+left skyline of an assumption with the right skyline of the next one,
+one can calculate what is the distance between the boxes so that the
+distance between the closest points in the skylines is exactly
+\verb|\prfinterspace|. 
+
+It is not simple to code such an algorithm in \TeX{}, but the real
+difficulty is how to represent skylines and how to store them, since
+\TeX{} provides no abstract data structures. Hence, the implementation
+of this feature has been postponed to a remote future, or to the will
+of a real \TeX{} magician.\vspace{2ex}
+
+There are three bugs in the packages. 
+
+The first one is that \verb|\mathrm| and similar may break a proof
+tree when used in the rule name. I have not been able to track down
+why this happens. The effect is that the proof tree is correctly
+constructed but it cannot be used as a box, e.g., it cannot be put
+inside a \verb|\fbox| or used in normal text. Although disappointing
+this bug can be easily circumvented by typesetting the proof tree in a
+math environment, e.g., by putting it into a math display or by
+enclosing it in a pair of dollar signs.\vspace{2ex}
+
+The second bug is minimal and in a future version it could be
+solved. If one considers the following proof:
+\begin{displaymath}
+  \begin{prooftree}
+    \fbox{%
+      \prflineextra=0pt
+      \prftree
+      {\prftree
+        {\prftree
+          {\prfboundedassumption{\neg\neg A}}
+          {\prfboundedassumption<bug:1>{\neg A}}
+          {\bot}}
+        {A}}
+      {\neg\neg A \supset A}}
+  \end{prooftree}
+\end{displaymath}
+the anchor of assumption (\prfref<bug:1>) is out of the bounding
+box. Usually, this is not a problem and, in case, it can be manually
+corrected
+\begin{displaymath}
+  \begin{prooftree}
+    \fbox{%
+      \prflineextra=0pt
+      \hbox{\prftree
+      {\prftree
+        {\prftree
+          {\prfboundedassumption{\neg\neg A}}
+          {\prfboundedassumption<[l]bug:1>{\neg A}}
+          {\bot}}
+        {A}}
+      {\neg\neg A \supset A}\hspace{.34em}}}
+  \end{prooftree}
+\end{displaymath}
+as in the following code:
+\begin{verbatim}
+  \prfassumption{\prftree{\prftree{\prftree
+        {\prfboundedassumption{\neg\neg A}}
+        {\prfboundedassumption<bug:1>{\neg A}}{\bot}}
+      {A}}{\neg\neg A \supset A}\hspace{.34em}}}
+\end{verbatim}\vspace{2ex}
+
+The third bug happens the first time a reference is created: if it is
+referred by \verb|\prfref| in the rule name, a strange ``immediate''
+follows it. This is not a problem, since the code has to be recompiled
+anyway to complete the definition of references, and this is enough to
+make the problem to disappear. Since it is a transient problem, I have
+not investigated any further.  
+
+\vfill
+
+\end{document}
diff --git a/Master/texmf-dist/tex/latex/prftree/prftree.sty b/Master/texmf-dist/tex/latex/prftree/prftree.sty
new file mode 100644
index 00000000000..8396b7b9cf5
--- /dev/null
+++ b/Master/texmf-dist/tex/latex/prftree/prftree.sty
@@ -0,0 +1,1046 @@
+%
+% prftree.sty
+% by Marco Benini - 30th November 2014
+%
+% A package to typeset natural deduction proofs, or sequent proofs, or
+% tableau proofs
+%
+% This package is distributed under the GNU General Public License
+%
+
+\NeedsTeXFormat{LaTeX2e}
+\ProvidesPackage{prftree}[2014/11/30 Natural Deduction Proofs]
+
+% Package options
+\newif\ifprf@NDOption\prf@NDOptionfalse
+\newif\ifprf@SEQOption\prf@SEQOptionfalse
+\newif\ifprfIMPOption\prfIMPOptionfalse
+
+\DeclareOption{ND}{\prf@NDOptiontrue}
+\DeclareOption{SEQ}{\prf@SEQOptiontrue}
+\DeclareOption{IMP}{\prf@IMPOptiontrue}
+\ProcessOptions\relax
+
+%--------------------------------------------------------------------
+% A proof tree is composed by a generic cell:
+%  A1 A2 A3
+% ---------- R
+%      C
+% We call A1, A2, A3. ... assumptions, C the conclusion and R the rule
+% name. The line dividing the assumptions from the conclusion is
+% called the proof line. A proof tree with no assumptions is an axiom.
+% An assumption which is not a proof tree is a pure assumption.
+%
+% A proof tree is typeset by
+% \prftree[<options>]...[<options>]
+%         {<assumption1>}...{<assumptionN>}
+%         {<conclusion>}
+% The possible options are:
+%   r             : the first argument after the options is the rule
+%                   name, typeset in text mode
+%   rule          : synonym for 'r'
+%   by rule       : synonym for 'r'
+%   by            : synonym for 'r'
+%   right         : synonym for 'r'
+%   l             : the first argument after the options is the rule
+%                   label, typeset in text mode; if there is also a
+%                   rule name, the label is the second argument
+%   label         : synonym for 'l'
+%   left          : synonym for 'l'
+%   straight      : the proof line will be a solid line (default) 
+%   straight line : synonym for 'l'
+%   straightline  : synonym for 'l'
+%   dotted        : the proof line will be a dotted line
+%   dotted line   : synonym for 'dotted'
+%   dottedline    : synonym for 'dotted'
+%   dashed        : the proof line will be a dashed line
+%   dashed line   : synonym for 'dashed'
+%   dashedline    : synonym for 'dashed'
+%   f             : the proof line will be fancy
+%   fancy         : synonym for 'f'
+%   fancy line    : synonym for 'f'
+%   fancyline     : synonym for 'f'
+%   s             : the proof line will be single, not double (default)
+%   single        : synonym for 's'
+%   single line   : synonym for 's'
+%   singleline    : synonym for 's'
+%   d             : the proof line will be double, not single
+%   double        : synonym for 'd'
+%   double line   : synonym for 'd'
+%   doubleline    : synonym for 'd'
+%   noline        : suppresses the proof line (prevails over all other
+%                   line options)
+%   summary       : renders the proof line as the summary symbol
+%                   (prevails over all other line options except
+%                   noline) 
+% 
+% The format of options is [<opt1>,...,<optM>]; it is correct to have
+% empty options. Options are parsed left-to-right: in case of
+% conflicting options, the latest prevails.
+%
+% If present, the rule name is typeset in text style, while the
+% conclusion and the assumptions are typeset in display math mode.
+% The result is a box with the correct height and width, whose
+% conclusion is aligned with the current baseline.
+%
+% Notice that the conclusion must be present, while the assumptions
+% may be absent.
+%
+% If there are no assumptions, i.e., if the proof tree represents an
+% axiom, you may use the following commands:
+%  \prfaxiom{<axiom>}
+%  \prfbyaxiom{<label>}{<axiom>}
+% the latter includes also the axiom name.
+%
+% A pure assumption is just a proof tree composed by the conclusion
+% only. It uses an option to suppress the drawing of the proof line.
+% The commands are:
+%   \prfassumption<reference>{<formula>}
+%   \prfboundedassumption<reference>{<formula>}
+% the second command discharges the assumption.
+%
+% A proof summary is just a proof tree whose proof line is bizarre. It
+% uses an option to specify the type of summary.
+% The command is:
+%   \prfsummary<reference>[<label>]
+%              {<assumption1>}...{<assumptionN>} 
+%              {<conclusion>}
+% The label is optional. The shape of the summary is controlled by the
+% global option \prfsummarystyle.
+%
+% The <reference> field above is used for labelling and
+% referencing, as explained in the documentation.
+%--------------------------------------------------------------------
+
+%--------------------------------------------------------------------
+% Parameters
+% These values can be safely used to configure the behaviour of the
+% macros in the package. Beware to modify them only outside a proof
+% tree as the order in which a proof tree is typeset may vary in
+% future version of the package.
+%--------------------------------------------------------------------
+
+% \prflinepad                default 0.3ex
+% the space between the baseline of assumptions and the proof line and
+% also the space between the proof line and the top of the conclusion
+\newdimen\prflinepad\prflinepad.3ex
+
+% \prflineextra              default 0.3em 
+% the length which extends on the left and on the right the proof line
+% so that it is slightly longer than the largest between the
+% conclusion and the list of (direct) assumptions
+\newdimen\prflineextra\prflineextra.3em
+
+% \prflinethickness          default 0.2pt
+% the thickness of the proof line
+\newdimen\prflinethickness\prflinethickness.2pt
+
+% \prfemptylinethickness     default 4 times the linethickness
+% the thickness of the proof line which has to be drawn but it is empty
+\newdimen\prfemptylinethickness\prfemptylinethickness4\prflinethickness
+
+% \prfrulenameskip           default 0.2em
+% the space between the proof line and the rule name
+\newdimen\prfrulenameskip\prfrulenameskip.2em
+
+% \prflabelskip           default 0.2em
+% the space between the label and the proof line
+\newdimen\prflabelskip\prflabelskip.2em
+
+% \prfinterspace             default .6em
+% the space between two subsequent assumptions
+\newdimen\prfinterspace\prfinterspace.6em
+
+% \prfdoublelineinterspace   default 1.2pt
+% the space between a double line
+\newdimen\prfdoublelineinterspace\prfdoublelineinterspace1.2pt
+
+% \prfboundedstyle           default 0
+% defines the style of bounded (discharged) assumptions:
+% 0 = [<formula>]
+% 1 = <formula> cancelled by a horizontal line
+% >1 = uses the custom \prfdiscargedassumption command
+\newcount\prfboundedstyle\prfboundedstyle0
+
+% \prfsummarystyle           default 0
+% defines the style for the summary box. The argument takes the
+% following values
+% 0 = a vertical dotted line (default)
+% 1 = a huge \Pi
+% 2 = \prod
+% 3 = a calligraphic D
+% >3 = uses the custom \prffancysummarybox command
+\newcount\prfsummarystyle\prfsummarystyle0
+
+% \prffancyline
+% the command to draw "fancy" lines
+\def\prffancyline{\cleaders\hbox to .63em%
+  {\hss\raisebox{-.5ex}[.2ex][0pt]{$\sim$}\hss}\hfill}
+
+% \prfConclusionBox
+% the command to draw the conclusion box
+\def\prfConclusionBox#1{%
+  \hbox{$\displaystyle\begingroup#1\endgroup\mathstrut$}}
+
+% \prfAssumptionBox
+% the command to draw the assumption box
+\def\prfAssumptionBox#1{%
+  \hbox{$\displaystyle\begingroup#1\endgroup\mathstrut$}}
+
+% \prfRuleNameBox
+% the command to draw the rule name box.
+\def\prfRuleNameBox#1{\hbox{\begingroup#1\endgroup\strut}}
+
+% \prfLabelBox
+% the command to draw the label box.
+\def\prfLabelBox#1{\hbox{\begingroup#1\endgroup\strut}}
+
+% \prfdiscargedassumption
+% the command to draw a discharged assumption in a custom way.
+% Users may consider the cancel.sty package
+\def\prfdiscargedassumption#1{\left\langle{#1}\right\rangle}
+
+% \prffancysummarybox
+% the command to draw the body of a custom summary
+\newbox\prf@@fancysummarybox
+\newdimen\prf@@fancysymmarylen
+\def\prffancysummarybox{%
+  \sbox{\prf@@fancysummarybox}{\Huge$\bigtriangledown$}%
+  \prf@@fancysymmarylen\ht\prf@@fancysummarybox%
+  \advance\prf@@fancysymmarylen\dp\prf@@fancysummarybox%
+  \sbox{\prf@@fancysummarybox}{%
+    \raisebox{.25\prf@@fancysymmarylen}[.8\prf@@fancysymmarylen]%
+    [0pt]{\usebox{\prf@@fancysummarybox}}}%
+  \prf@@fancysymmarylen\wd\prf@summary@label%
+  \ifdim\prf@@fancysymmarylen>\z@\relax%
+    \prf@@fancysymmarylen\wd\prf@@fancysummarybox%
+    \wd\prf@summary@label.4em%
+    \hbox to\prf@@fancysymmarylen{%
+      \usebox\prf@@fancysummarybox}\kern-.4em%
+      \box\prf@summary@label%
+  \else\usebox\prf@@fancysummarybox\fi}
+
+%====================================================================
+
+%--------------------------------------------------------------------
+% What follows is intended for internal use of the package.
+% It shouldn't be used in documents and may vary in future versions.
+%
+% Notice that we use temporary registers shared with TeX & LaTeX, and
+% that some registers are shared also inside the drawing procedure!
+%--------------------------------------------------------------------
+
+% Lengths
+\newskip\prf@Lassum%     space between the left border of the
+                   %     assumption box and the left border of the
+                   %     conclusion of the first assumption;
+\newskip\prf@Rassum%     space between the right border of the
+                   %     assumption box and the right border of the
+                   %     conclusion of the last assumption 
+\skipdef\prf@assumline1%
+\skipdef\prf@concline3%
+\skipdef\prf@linewd5%
+\skipdef\prf@lineht3%
+\skipdef\prf@linedp1%
+\skipdef\prf@temp7%
+\skipdef\prf@tmp7%
+\skipdef\prf@Aval9%
+
+% Boxes
+\newbox\prf@assumptionsbox%   Assumptions
+\newbox\prf@conclusionbox%    Conclusion
+\newbox\prf@rulenamebox%      Rule name
+\newbox\prf@labelbox%         Label
+\newbox\prf@linebox%          Proof line
+\newbox\prf@summarybox%       Summary
+\newbox\prf@fancybox%         To compute the fancy line
+
+%--------------------------------------------------------------------
+% Stack implementation
+%--------------------------------------------------------------------
+%% End-of-stack, wrapper and stack declaration 
+\def\prf@eos{\noexpand\prf@eos}
+\def\prf@cons{\noexpand\prf@cons}
+\let\prf@stack\prf@eos
+
+%% Pushing a value puts the (decorated) value in front
+%% of the stack list
+\def\prf@push#1{\expandafter\edef\expandafter\prf@stack%
+  {\prf@cons{#1}\prf@stack}}
+
+%% Top retrieves the topmost value in the stack
+\def\prf@@top\prf@cons#1#2\prf@eos{#1}
+\def\prf@top{\expandafter\prf@@top\prf@stack}
+
+%% Pop discard the topmost value in the stack
+\def\prf@@pop\prf@cons#1#2\prf@eos{%
+  \expandafter\edef\expandafter\prf@stack{#2\prf@eos}}
+\def\prf@pop{\expandafter\prf@@pop\prf@stack}
+
+%--------------------------------------------------------------------
+% Checks whether two strings are equal
+% \prf@streq{A}{B} if A = B then the boolean value \ifprf@same is 
+% true otherwise false 
+%
+% There is also \prf@strlisteq{A}{L} which checks whether A is equal
+% to any element in the comma-separated list L of strings
+%--------------------------------------------------------------------
+\newif\ifprf@same
+
+\newcommand{\prf@streq}[2]{%
+  \global\prf@samefalse\begingroup\def\1{#1}\def\2{#2}%
+  \ifx\1\2\global\prf@sametrue\fi\endgroup}
+
+\def\prf@strlisteq#1#2{\prf@@strlisteq{#1}#2,@}
+\def\prf@@strlisteq#1#2,{\prf@streq{#1}{#2}%
+  \ifprf@same\expandafter\prf@@strlistignore%
+  \else\expandafter\prf@@@strlisteq\fi{#1}}
+\def\prf@@strlistignore#1@{\relax}
+\def\prf@@@strlisteq#1#2{%
+  \begingroup\def\1{@}\def\2{#2}%
+  \ifx\1\2\prf@@@strlistignore\else\prf@hop\fi\endgroup{#1}#2}
+\def\prf@hop\fi\endgroup#1#2{\fi\endgroup\prf@@strlisteq{#1}#2}
+\def\prf@@@strlistignore\else\prf@hop\fi\endgroup#1@{\fi\endgroup}
+
+% -------------------------------------------------------------------
+% Labelling and referencing
+% The following commands implement the reference mechanism to allow to
+% label and then refer to assumptions and summaries of proofs
+% -------------------------------------------------------------------
+
+% The counter to keep track of assumptions
+\newcounter{prfassumptioncounter}
+\setcounter{prfassumptioncounter}{0}
+
+% The counter to keep track of summaries
+\newcounter{prfsummarycounter}
+\setcounter{prfsummarycounter}{0}
+
+% To keep track of the options
+\newif\ifprf@lbl@arabic\prf@lbl@arabictrue
+\newif\ifprf@lbl@roman\prf@lbl@romanfalse
+\newif\ifprf@lbl@Roman\prf@lbl@Romanfalse
+\newif\ifprf@lbl@alph\prf@lbl@alphfalse
+\newif\ifprf@lbl@Alph\prf@lbl@Alphfalse
+\newif\ifprf@lbl@fnsymbol\prf@lbl@fnsymbolfalse
+\newif\ifprf@lbl@label\prf@lbl@labelfalse
+
+% The prooftree environment is used to define the scope of the
+% assumption references
+\newenvironment{prooftree}%
+  {\setcounter{prfassumptioncounter}{0}}%
+  {\setcounter{prfassumptioncounter}{0}}
+
+% Test whether a reference is defined
+\def\ifprf@undefref<#1>{%
+  \expandafter\ifx\csname prf@ref@@#1\endcsname\relax}
+
+% Test whether a reference has been written in the .aux file
+\def\ifprf@unwrittenref<#1>{%
+  \expandafter\ifx\csname prf@w@ref@#1\endcsname\relax}
+
+% Write a reference
+\def\prfref<#1>{%
+  \ifprf@undefref<#1>{??}%
+    \PackageWarningNoLine{prooftree}{Undefined reference <#1>}%
+  \else\csname prf@ref@@#1\endcsname\relax\fi} 
+
+% Create the anchor from what stored in the .aux file
+\def\prfauxlabel#1#2{%
+  \ifprf@undefref<#1>%
+    \expandafter\gdef\csname prf@ref@@#1\endcsname{{#2}}\fi}
+
+% Write the newly created label in the .aux file
+\def\prf@auxwrite<#1>#2{%
+  \ifprf@unwrittenref<#1>%
+    \protected@write\@auxout{}{\string\prfauxlabel{#1}{#2}}%
+    \expandafter\gdef\csname prf@w@ref@#1\endcsname{0}\fi}
+
+% Generate an assumption label
+\def\prf@assum@lbl<#1>{\prf@generic@lbl<#1>{prfassumptioncounter}}
+
+% Generate a summary label 
+\def\prf@summary@lbl<#1>{\prf@generic@lbl<#1>{prfsummarycounter}}
+
+% Generate a label on a given counter
+\def\prf@generic@lbl<#1>#2{%
+  \prf@warningtrue\stepcounter{#2}%
+  \ifprf@lbl@arabic\prf@warningfalse%
+    \prf@auxwrite<#1>{\arabic{#2}}\fi%
+  \ifprf@lbl@roman\prf@warningfalse%
+    \prf@auxwrite<#1>{\roman{#2}}\fi%
+  \ifprf@lbl@Roman\prf@warningfalse%
+    \prf@auxwrite<#1>{\Roman{#2}}\fi%
+  \ifprf@lbl@alph\prf@warningfalse%
+    \prf@auxwrite<#1>{\alph{#2}}\fi%
+  \ifprf@lbl@Alph\prf@warningfalse%
+    \prf@auxwrite<#1>{\Alph{#2}}\fi%
+  \ifprf@lbl@fnsymbol\prf@warningfalse%
+    \prf@auxwrite<#1>{\fnsymbol{#2}}\fi%
+  \ifprf@warning\prf@auxwrite<#1>{\arabic{#2}}\fi}
+
+% Parsing of the labelled assumption options
+\def\prf@lblopt#1{%
+  \prf@warningtrue%
+  \global\prf@lbl@arabicfalse% 
+  \global\prf@lbl@romanfalse% 
+  \global\prf@lbl@Romanfalse%
+  \global\prf@lbl@alphfalse%
+  \global\prf@lbl@Alphfalse%
+  \global\prf@lbl@fnsymbolfalse%
+  \global\prf@lbl@labelfalse%
+  \prf@strlisteq{#1}{n,number,arabic}%
+  \ifprf@same\prf@warningfalse\global\prf@lbl@arabictrue\fi%
+  \prf@strlisteq{#1}{r,roman}%
+  \ifprf@same\prf@warningfalse\global\prf@lbl@romantrue\fi%
+  \prf@strlisteq{#1}{R,Roman}%
+  \ifprf@same\prf@warningfalse\global\prf@lbl@Romantrue\fi%
+  \prf@strlisteq{#1}{a,alph,alpha,alphabetic}%
+  \ifprf@same\prf@warningfalse\global\prf@lbl@alphtrue\fi%
+  \prf@strlisteq{#1}{A,Alph,Alpha,Alphabetic}%
+  \ifprf@same\prf@warningfalse\global\prf@lbl@Alphtrue\fi
+  \prf@strlisteq{#1}{f,s,function,symbol,function symbol}%
+  \ifprf@same\prf@warningfalse\global\prf@lbl@fnsymboltrue\fi%
+  \prf@strlisteq{#1}{l,label}%
+  \ifprf@same\prf@warningfalse\global\prf@lbl@labeltrue\fi%
+  \ifprf@warning%
+    \PackageWarning{prftree}{Unrecognised option #1}\fi}
+
+% How a labelled assumption is graphically rendered
+\def\prflabelledassumptionbox#1#2{%
+  \setbox\prf@fancybox\hbox{${#1}$}%
+  \prf@tmp\wd\prf@fancybox%
+  \setbox\prf@fancybox\hbox{$\box\prf@fancybox^{#2}$}%
+  \wd\prf@fancybox\prf@tmp%
+  \prf@assumption{\box\prf@fancybox}}
+
+% How a discharged fancy labelled assumption is graphically rendered
+\def\prflabelleddiscargedassumption#1#2{%
+  \prflabelledassumptionbox{\left\langle{#1}\right\rangle}{#2}}
+
+%--------------------------------------------------------------------
+% \prfassumption{<formula>}
+% \prfboundedassumption{<formula>}
+% 
+% Commands to draw a pure assumption: the former draws a pure
+% assumption, while the latter draws a pure assumption which is
+% discharged. The way to mark discharging is controlled by the global
+% option \prfboundedstyle
+%--------------------------------------------------------------------
+\def\prfassumption{\@ifnextchar<{\prf@assum@one}{\prf@assumption}}
+\def\prf@assum@one<#1>{\global\prf@lbl@labelfalse\prf@assum@two#1>}
+\def\prf@assum@two{\@ifnextchar[{\prf@assum@three}{\prf@assum@four}}
+\def\prf@assum@three[#1]{\prf@lblopt{#1}\prf@assum@four}
+\def\prf@assum@four#1>#2{%
+  \ifprf@unwrittenref<#1>\relax%
+  \else\ifprf@lbl@label\relax\else%
+    \global\prf@lbl@labeltrue%
+    \PackageWarningNoLine{prftree}{Label <#1> is [label]!}\fi\fi%
+  \ifprf@lbl@label\relax\else\prf@assum@lbl<#1>\fi%
+  \prflabelledassumptionbox{#2}{\prfref<#1>}}
+\def\prf@assumption#1{\prftree[noline]{#1}\relax}
+
+\def\prfboundedassumption{%
+  \@ifnextchar<{\prf@bassum@one}{\prf@boundedassumption}}
+\def\prf@bassum@one<#1>{\global\prf@lbl@labelfalse\prf@bassum@two#1>}
+\def\prf@bassum@two{\@ifnextchar[{\prf@bassum@three}{\prf@bassum@four}}
+\def\prf@bassum@three[#1]{\prf@lblopt{#1}\prf@bassum@four}
+\def\prf@bassum@four#1>#2{%
+  \ifprf@unwrittenref<#1>\relax%
+  \else\ifprf@lbl@label\relax\else%
+    \global\prf@lbl@labeltrue%
+    \PackageWarningNoLine{prftree}{Label <#1> is [label]!}\fi\fi%
+  \ifprf@lbl@label\relax\else\prf@assum@lbl<#1>\fi%
+  \prftree[noline]{\prf@labelledbounded{#2}{#1}}\relax}
+\def\prf@boundedassumption#1{\prftree[noline]{\prf@bounded{#1}}\relax}
+\def\prf@bounded#1{%
+  \ifnum\prfboundedstyle=0\left[{#1}\right]%
+  \else\ifnum\prfboundedstyle=1%
+      \setbox\prf@conclusionbox\prfConclusionBox{#1}%
+      \prf@tmp\wd\prf@conclusionbox%
+      \prf@linedp.5\ht\prf@conclusionbox%
+      \advance\prf@linedp-.5\dp\prf@conclusionbox%
+      \prf@lineht\prf@linedp\advance\prf@lineht\prflinethickness%
+      \edef\prf@cancel{\hskip-\prf@tmp%
+        \vrule width\prf@tmp height\prf@lineht depth-\prf@linedp}%
+      {#1}\prf@cancel%
+    \else\prfdiscargedassumption{#1}\fi\fi}  
+\def\prf@labelledbounded#1#2{%
+  \ifnum\prfboundedstyle=0%
+    \prflabelledassumptionbox{\left[{#1}\right]}{\prfref<#2>}%
+  \else\ifnum\prfboundedstyle=1%
+      \setbox\prf@conclusionbox\prfConclusionBox{#1}%
+      \prf@tmp\wd\prf@conclusionbox%
+      \prf@linedp.5\ht\prf@conclusionbox%
+      \advance\prf@linedp-.5\dp\prf@conclusionbox%
+      \prf@lineht\prf@linedp\advance\prf@lineht\prflinethickness%
+      \edef\prf@cancel{\hskip-\prf@tmp%
+        \vrule width\prf@tmp height\prf@lineht depth-\prf@linedp}%
+      \prflabelledassumptionbox{{#1}\prf@cancel}{\prfref<#2>}%
+    \else\prflabelleddiscargedassumption{#1}{\prfref<#2>}\fi\fi}  
+
+%--------------------------------------------------------------------
+% \prfaxiom{<axiom>}
+% \prfbyaxiom{<label>}{<axiom>}
+%
+% Commands to draw an axiom: the former draws the axiom, while the
+% latter includes also a rule name for the axiom.
+%--------------------------------------------------------------------
+\def\prfaxiom#1{\prftree[single,straight]{#1}\relax}
+\def\prfbyaxiom#1#2{\prftree[single,straight,rule]{#1}{#2}\relax}
+
+%--------------------------------------------------------------------
+% \prfsummary[<label>]
+%            {<assumption1>}...{<assumptionN>} 
+%            {<conclusion>}
+%
+% Command to draw a proof summary, i.e., a structure with assumptions
+% 1 to N and a conclusion which stands for a complex proof not
+% explicited. The label is optional.
+% The shape of the summary is controlled by the global option
+% \prfsummarystyle 
+%--------------------------------------------------------------------
+\def\prfsummary{\@ifnextchar<{\prf@summary@one}{\prf@summary}}
+\def\prf@summary@one<#1>{%
+  \global\prf@lbl@labelfalse\prf@summary@four#1>}
+\def\prf@summary@four{%
+  \@ifnextchar[{\prf@summary@five}{\prf@summary@six}}
+\def\prf@summary@five[#1]#2>{%
+  \prf@lblopt{#1}%
+  \ifprf@unwrittenref<#2>\relax%
+  \else\ifprf@lbl@label\relax\else%
+    \global\prf@lbl@labeltrue%
+    \PackageWarningNoLine{prftree}{Label <#2> is [label]!}\fi\fi%
+  \ifprf@lbl@label\relax\else\prf@summary@lbl<#2>\fi%
+  \setbox\prf@summary@label\hbox{\scriptsize\prfref<#2>}%
+  \prf@summary}
+\def\prf@summary@six#1>{%
+  \ifprf@lbl@label\relax\else\prf@summary@lbl<#1>\fi%
+  \setbox\prf@summary@label\hbox{\scriptsize\prfref<#1>}%
+  \prf@summary}
+\def\prf@summary{\@ifnextchar[{\prf@summary@two}{\prf@summary@three}}
+\def\prf@summary@two[#1]{\prftree[summary,rule]{#1}}
+\def\prf@summary@three{\prftree[summary]}
+
+\newbox\prf@summary@label
+\def\prf@@summarystyle{%
+  \ifnum3=\prfsummarystyle\relax%
+    \setbox\prf@summarybox\hbox{$\mathcal{D}$}%
+    \prf@tmp\wd\prf@summarybox%
+    \setbox\prf@summarybox\hbox{$\mathcal{D}%
+      \hbox{\box\prf@summary@label}\mathstrut$}%
+    \wd\prf@summarybox\prf@tmp%
+  \else\ifnum2=\prfsummarystyle\relax%
+    \setbox\prf@summarybox\hbox{$\prod$}%
+    \prf@tmp\wd\prf@summarybox%
+    \setbox\prf@summarybox\hbox{$\prod\hbox{\box\prf@summary@label}$}% 
+    \wd\prf@summarybox\prf@tmp%
+  \else\ifnum1=\prfsummarystyle\relax%
+    \setbox\prf@summarybox\hbox{\hbox{\Huge$\Pi$}}%
+    \prf@tmp\wd\prf@summarybox%
+    \setbox\prf@summarybox\hbox{\hbox{\Huge$\Pi$}%
+      \hbox{\box\prf@summary@label}}%
+    \wd\prf@summarybox\prf@tmp%
+  \else\ifnum0=\prfsummarystyle\relax%
+    \setbox\prf@summarybox\hbox{$\cdot$}%
+    \prf@tmp\wd\prf@summarybox%
+    \setbox\prf@summarybox%
+      \vbox to5.2ex{\cleaders\hbox{$\cdot$}\vfill}%
+    \setbox\prf@summarybox\hbox{\usebox{\prf@summarybox}%
+      \hbox{\ \box\prf@summary@label}}%
+    \wd\prf@summarybox\prf@tmp%
+  \else\setbox\prf@summarybox\hbox{\prffancysummarybox}\fi\fi\fi\fi% 
+  \prf@tmp.5\ht\prf@summarybox%
+  \advance\prf@tmp-.5\dp\prf@summarybox%
+  \setbox\prf@summarybox%
+    \hbox{\lower\prf@tmp\hbox{\box\prf@summarybox}}}%
+
+%--------------------------------------------------------------------
+% \prftree[<options>]...[<options>]
+%         {<assumption1>}...{<assumptionN>}
+%         {<conclusion>}
+% 
+% The format of options is [<opt1>,...,<optM>]; it is correct to have
+% empty options. Options are parsed left-to-right: in case of
+% conflicting options, the latest prevails. See the documentation for
+% their description.
+%
+% If present, the rule name is typesetted in text style, while the
+% conclusion and the assumptions are typesetted in math mode.
+% The result is a box with the correct height and width, whose
+% conclusion is aligned with the current baseline.
+%
+% Notice that the conclusion must be present, while the assumptions
+% may be absent.
+%
+% Options and the way things are drawn can be modified via the global
+% settings of the package (see above). Notice that changing these
+% settings inside a proof tree may lead to impredictable results, as
+% the typesetting may proceed in an arbitrary order.
+%--------------------------------------------------------------------
+\newif\ifprf@rule%
+\newif\ifprf@label%
+\newif\ifprf@noline%
+\newif\ifprf@doubleline%
+\newif\ifprf@straightline%
+\newif\ifprf@dottedline%
+\newif\ifprf@dashedline%
+\newif\ifprf@fancyline%
+\newif\ifprf@summary%
+\newif\ifprf@first%
+\newif\ifprf@warning%
+\newif\ifprf@assum
+
+% The real parsing command is \prf@Ntree; the external command sets
+% the right values for the various switches controlling the options
+\def\prftree{\prf@init\prf@Ntree}
+\def\prf@init{%
+  \global\prf@rulefalse%
+  \global\prf@labelfalse%
+  \global\prf@nolinefalse%
+  \global\prf@doublelinefalse%
+  \global\prf@straightlinetrue%
+  \global\prf@dottedlinefalse%
+  \global\prf@dashedlinefalse%
+  \global\prf@fancylinefalse%
+  \global\prf@summaryfalse}
+
+% If there are options then parse them, eventually setting the
+% booleans to their appropriate values. Then, expand \prf@@tree
+\def\prf@Ntree{\@ifnextchar[{\prf@options}{\prf@@tree}}
+
+% Parses a block [<opt>,...,<opt>] of options one by one, left to
+% right 
+\def\prf@options[#1]{\prf@opts#1,]\prf@Ntree}
+\def\prf@opts#1,{\prf@opt#1,\@ifnextchar]{\prf@optsskip}{\prf@opts}} 
+\def\prf@optsskip]{\relax} % to consume the final ]
+
+% Parses a single option eventually setting the appropriate flag 
+\def\prf@opt#1,{\prf@warningtrue%
+  \prf@strlisteq{#1}{r,by rule,rule,by,right}%
+  \ifprf@same\prf@warningfalse\global\prf@ruletrue\fi%
+  \prf@strlisteq{#1}{l,label,left}%
+  \ifprf@same\prf@warningfalse\global\prf@labeltrue\fi%
+  \prf@strlisteq{#1}{s,single line,single,singleline}%
+  \ifprf@same\prf@warningfalse\global\prf@doublelinefalse\fi%
+  \prf@strlisteq{#1}{d,double line,double,doubleline}%
+  \ifprf@same\prf@warningfalse\global\prf@doublelinetrue\fi%
+  \prf@strlisteq{#1}{straight,straight line,straightline}%
+  \ifprf@same\prf@warningfalse%
+  \global\prf@straightlinetrue\global\prf@dottedlinefalse%
+  \global\prf@dashedlinefalse\global\prf@fancylinefalse\fi%
+  \prf@strlisteq{#1}{dotted,dotted line,dottedline}%
+  \ifprf@same\prf@warningfalse%
+  \global\prf@straightlinefalse\global\prf@dottedlinetrue%
+  \global\prf@dashedlinefalse\global\prf@fancylinefalse\fi%
+  \prf@strlisteq{#1}{dashed,dashed line,dashedline}%
+  \ifprf@same\prf@warningfalse%
+  \global\prf@straightlinefalse\global\prf@dottedlinefalse%
+  \global\prf@dashedlinetrue\global\prf@fancylinefalse\fi%
+  \prf@strlisteq{#1}{f,fancy line,fancy,fancyline}%
+  \ifprf@same\prf@warningfalse%
+  \global\prf@straightlinefalse\global\prf@dottedlinefalse%
+  \global\prf@dashedlinefalse\global\prf@fancylinetrue\fi%
+  \prf@strlisteq{#1}{noline}%
+  \ifprf@same\prf@warningfalse\global\prf@nolinetrue\fi%
+  \prf@strlisteq{#1}{summary}%
+  \ifprf@same\prf@warningfalse\global\prf@summarytrue\fi%
+  \ifprf@warning\PackageWarning{prftree}%
+                               {Unrecognised option #1}\fi}%
+
+% At this point, options have been parsed. So, we process them
+% preparing the right parameters for the drawing procedure. 
+% We start from the proof line options:
+%   0 = single straight line
+%   1 = double straight line 
+%   2 = single dotted line
+%   3 = double dotted line
+%   4 = single dashed line
+%   5 = double dashed line
+%   6 = single fancy line
+%   7 = double fancy line
+%   8 = no line
+%   9 = summary
+\def\prf@@tree{%
+  \countdef\prf@linetype1\prf@linetype0%
+  \ifprf@dottedline\prf@linetype2\fi%
+  \ifprf@dashedline\prf@linetype4\fi%
+  \ifprf@fancyline\prf@linetype6\fi%
+  \ifprf@doubleline\advance\prf@linetype1\fi%
+  \ifprf@noline\prf@linetype8\fi%
+  \ifprf@summary\prf@linetype9\fi\relax%
+  \expandafter\prf@@Ntree\the\prf@linetype}
+
+% If there is no rule name, we put an empty one in place 
+% #1 the line options
+\def\prf@@Ntree#1{%
+  \edef\prf@choice{\ifprf@rule{#1}\else{#1}{}\fi}%
+  \expandafter\prf@@NNtree\prf@choice}
+
+% If there is no label, we put an empty one in place 
+% #1 the line options
+% #2 the rule name
+\def\prf@@NNtree#1#2{%
+  \edef\prf@choice{\ifprf@label{#1}{#2}\else{#1}{#2}{}\fi}%
+  \expandafter\prf@tree\prf@choice}
+
+% We collect the first assumption, if any, or the conclusion otherwise
+% and we look if there is another argument. If not, we draw the proof.
+% #1 the line options' value
+% #2 the rule name
+% #3 the label
+% #4 the conclusion or the first assumption 
+\def\prf@tree#1#2#3#4{\@ifnextchar\bgroup%
+  {\prf@assumptions{#1}{#2}{#3}{\prf@preprf{#4}}}%
+  {\prf@draw{#1}{#2}{#3}{#4}{\prf@preprf}{}}}
+
+% We examine assumptions from left to right. If we hit the last
+% argument, it is the conclusion and the proof is ready to be drawn. 
+% #1 the line options' value
+% #2 the rule name
+% #3 the label
+% #4 the first assumptions
+% #5 the current argument
+\def\prf@assumptions#1#2#3#4#5{\@ifnextchar\bgroup%
+  {\prf@assumenext{#1}{#2}{#3}{#4}{\prf@interprf{#5}}}%
+  {\prf@draw{#1}{#2}{#3}{#5}{#4}{}}}
+
+% An intermediate assumption is added to the assumptions' box, until
+% the last assumption has been collected. Then the proof is drawn.
+% #1 the line options' value
+% #2 the rule name
+% #3 the label
+% #4 the first assumption
+% #5 the complete list of assumptions
+% #6 the new argument
+\def\prf@assumenext#1#2#3#4#5#6{\@ifnextchar\bgroup%
+  {\prf@assumenext{#1}{#2}{#3}{#4}{{#5}\prf@interprf{#6}}}%
+  {\prf@draw{#1}{#2}{#3}{#6}{#4}{#5}}}
+
+% We need to reset some internal variables during the generation of
+% the assumptions
+\def\prf@preprf{\global\prf@Lassum\z@\global\prf@Rassum\z@}
+
+% We need to put some space between assumptions when generating them 
+\def\prf@interprf{\hskip\prfinterspace}
+
+%--------------------------------------------------------------------
+% The drawing procedure, i.e., the "core" of the package
+% #1 the line options' value
+% #2 the rule name
+% #3 the label 
+% #4 the conclusion
+% #5 the first assumption
+% #6 the list of assumptions
+%--------------------------------------------------------------------
+
+\def\prf@draw#1#2#3#4#5#6{%
+  % We construct the boxes which will constitute the proof
+  \ifprf@first\prf@push{1}\else\prf@push{0}\fi%
+  \global\prf@firsttrue%
+  \setbox\prf@assumptionsbox\prfAssumptionBox{#5}%
+  \prf@push{\the\prf@Lassum}%
+  \global\prf@firstfalse%
+  \ifdim\wd\prf@assumptionsbox>\z@\relax\prf@assumtrue%
+    \setbox\prf@assumptionsbox%
+      \hbox{\box\prf@assumptionsbox\prfAssumptionBox{#6}}%
+  \else\prf@assumfalse%
+    \setbox\prf@assumptionsbox\hbox{}\fi%
+  \prf@push{\the\prf@Rassum}%
+  \setbox\prf@conclusionbox\prfConclusionBox{#4}%
+  \setbox\prf@rulenamebox\prfRuleNameBox{#2}%
+  \setbox\prf@labelbox\prfLabelBox{#3}%
+  % We set up the flags needed to format the proof tree box
+  \prf@nolinefalse\prf@doublelinefalse\prf@straightlinetrue%
+  \prf@dottedlinefalse\prf@dashedlinefalse\prf@fancylinefalse%
+  \prf@summaryfalse\prf@rulefalse\prf@labelfalse%
+  \ifcase#1\relax\or%
+  \prf@doublelinetrue\or%
+  \prf@dottedlinetrue\prf@straightlinefalse\or%
+  \prf@doublelinetrue\prf@dottedlinetrue\prf@straightlinefalse\or%
+  \prf@dashedlinetrue\prf@straightlinefalse\or%
+  \prf@doublelinetrue\prf@dashedlinetrue\prf@straightlinefalse\or%
+  \prf@fancylinetrue\prf@straightlinefalse\or%
+  \prf@doublelinetrue\prf@fancylinetrue\prf@straightlinefalse\or%
+  \prf@nolinetrue\prf@straightlinefalse\or%
+  \prf@summarytrue\prf@straightlinefalse\fi%
+  \ifdim\wd\prf@rulenamebox>\z@\relax\prf@ruletrue%
+  \else\prf@rulefalse\fi%
+  \ifdim\wd\prf@labelbox>\z@\relax\prf@labeltrue%
+  \else\prf@labelfalse\fi%
+  \ifdim\wd\prf@assumptionsbox>\z@\relax\prf@assumtrue%
+  \else\prf@assumfalse\fi%
+  % Now, we retrieve the right values for Lassum and Rassum
+  % since box construction may have overwritten them. Also, we need to
+  % know if this proof is the first of a list of assumptions
+  \edef\prf@tok{\prf@top}\prf@pop\global\prf@Rassum\prf@tok%
+  \edef\prf@tok{\prf@top}\prf@pop\global\prf@Lassum\prf@tok%
+  \edef\prf@tok{\prf@top}\prf@pop\global\prf@firstfalse%
+  \ifnum\prf@tok=1\global\prf@firsttrue\fi%
+  % The assumptions' box has the following shape
+  %  --------------------------------
+  % |<---------------------wd------->|
+  % | -----------       ------------ |
+  % ||<-->       |     |      <---->||
+  % || Lassum    | ... |   Rassum   ||
+  % ||   |       |     |      |     ||
+  % ||   V       |     |      V     ||
+  % | -----------       ------------ |
+  %  --------------------------------
+  % The assumptions' line is composed by the conclusions of the
+  % proofs which are the antecedents of the application of the
+  % inference rule
+  % The final box will be something like
+  %  --------------------------------
+  % |<---------------------wd------->|
+  % | -----------       ------------ |
+  % ||<-->       |     |      <---->||
+  % || Lassum    | ... |   Rassum   ||
+  % ||   |       |     |      |     ||
+  % ||   V       |     |      V     ||
+  % | -----------       ------------ |
+  %  --------------------------------
+  %     ========================   rule name
+  %             conclusion
+  % <--Lassum-->          <-----Rassum----->
+  % 
+  % We calculate the length of the assumptions' line:
+  %   assumline = assumptionsbox - Lassum - Rassum
+  % if there are no assumption, assumline = 0
+  % (for efficiency reasons, we actually calculate 1/2assumline)
+  \ifprf@assum\prf@assumline.5\wd\prf@assumptionsbox%
+    \advance\prf@assumline-.5\prf@Lassum%
+    \advance\prf@assumline-.5\prf@Rassum%
+    \else\prf@assumline\z@\fi%
+  % The length of the conclusionbox
+  % (for efficiency reasons, we actually calculate 1/2concline)
+  \prf@concline.5\wd\prf@conclusionbox%
+  % The length of the proof line is 
+  %   linewd = if summary then width of summary box else
+  %               max(concline,assumline)
+  %   if there is a line, we add 2extra
+  % (for efficiency reasons, we actually calculate 1/2linewd)
+  \ifprf@summary\prf@@summarystyle\prf@linewd.5\wd\prf@summarybox%
+  \else\prf@linewd\prf@assumline%
+    \ifdim\prf@assumline<\prf@concline\prf@linewd\prf@concline\fi% 
+    \ifprf@noline\relax\else\advance\prf@linewd\prflineextra\fi\fi%
+  % We calculate the proofline without label and rule name.
+  \ifprf@noline\setbox\prf@fancybox\hbox{\hskip2\prf@linewd}%
+  \else\ifprf@summary\setbox\prf@fancybox\copy\prf@summarybox%
+  \else\ifprf@fancyline%
+      \setbox\prf@fancybox\hbox to2\prf@linewd{\prffancyline}\else%
+      \ifprf@dashedline%
+      \setbox\prf@fancybox%
+      \hbox to2\prf@linewd{\cleaders\hbox to.5em{\hss\_\hss}\hfill}\else%
+      \ifprf@dottedline%
+      \setbox\prf@fancybox%
+      \hbox to2\prf@linewd{\cleaders\hbox to.33em{\hss.\hss}\hfill}\else% 
+   % it must be a straight line!
+      \setbox\prf@fancybox\hbox{\vrule width2\prf@linewd%
+        height\prflinethickness}\fi\fi\fi%
+  % If the line is double, we draw it twice with enough
+  % (doublelineinterspace) space between the two copies. 
+    \ifprf@doubleline%
+      \setbox\prf@fancybox\hbox{\vbox{\copy\prf@fancybox%
+         \nointerlineskip\vskip\prfdoublelineinterspace\nointerlineskip%
+         \box\prf@fancybox}}\fi\fi\fi%
+  \prf@tmp\dp\prf@fancybox%
+  \setbox\prf@fancybox\hbox{\raise\prf@tmp\box\prf@fancybox}%
+  % Let A = max(1/2assumline + Lassum, 1/2conclusionbox, 1/2linewd +
+  % delta), where delta is 0 if there is no label, and it is the width
+  % of the label box plus prflabelskip otherwise.
+  % Then
+  %   Assumptions shift         = A - (1/2assumline + Lassum)
+  %   Rule line shift           = A - (1/2linewd + delta)
+  %   Lassum = Conclusion shift = A - 1/2conclusionbox
+  % And, so, we can prepare the assumptions box and the conclusion box 
+  \prf@Aval\prf@assumline\advance\prf@Aval\prf@Lassum%
+  \prf@tmp\prf@linewd\relax%
+  \ifprf@label\advance\prf@tmp\wd\prf@labelbox%
+    \advance\prf@tmp\prflabelskip\fi%
+  \ifdim\prf@Aval<\prf@tmp\prf@Aval\prf@tmp\fi%
+  \ifdim\prf@Aval<\prf@concline\prf@Aval\prf@concline\fi%
+  \prf@temp\prf@Aval\advance\prf@temp-\prf@assumline%
+  \advance\prf@temp-\prf@Lassum%
+  \setbox\prf@assumptionsbox%
+    \hbox{\hskip\prf@temp\box\prf@assumptionsbox}%
+  \prf@temp\prf@Aval\advance\prf@temp-\prf@concline%
+  \setbox\prf@conclusionbox%
+    \hbox{\hskip\prf@temp\box\prf@conclusionbox}%
+  \global\prf@Lassum\prf@temp%
+  % Rassum will be the width of the composition of all boxes (but we
+  % still have to cope with heigths...) minus (concline + conclusion
+  % shift) 
+  \global\prf@Rassum-2\prf@concline%
+  \global\advance\prf@Rassum-\prf@temp%
+  % Then, if there is a rule name, the line is raised to align with
+  % the centre of the rule name and the rule box gets attached to the
+  % proof line box.
+  \prf@linedp\z@%
+  \ifprf@noline\setbox\prf@fancybox\hbox to2\prf@linewd{}\fi%
+  \ifprf@rule\advance\prf@linedp.5\ht\prf@fancybox%
+    \advance\prf@linedp-.5\ht\prf@rulenamebox%
+    \advance\prf@linedp.5\dp\prf@rulenamebox%
+    \setbox\prf@linebox%
+      \hbox{\copy\prf@fancybox\hskip\prfrulenameskip%
+        \raise\prf@linedp\box\prf@rulenamebox}%
+  \else\setbox\prf@linebox\copy\prf@fancybox\fi%
+  \setbox\prf@linebox\hbox{\raise\dp\prf@linebox\box\prf@linebox}%
+  % Then, if there is a label, the line is raised to align with
+  % the centre of the label and the label box gets attached to the
+  % proof line box.
+  \prf@linedp\z@%
+  \prf@temp\prf@Aval\advance\prf@temp-\prf@linewd%
+  \ifprf@label\advance\prf@linedp.5\ht\prf@linebox%
+    \advance\prf@linedp-.5\ht\prf@labelbox%
+    \advance\prf@linedp.5\dp\prf@labelbox%
+    \advance\prf@temp-\prflabelskip%
+    \advance\prf@temp-\wd\prf@labelbox%
+    \setbox\prf@linebox%
+      \hbox{\raise\prf@linedp\box\prf@labelbox%
+        \hskip\prflabelskip\box\prf@linebox}\fi%
+  \setbox\prf@linebox%
+    \hbox{\hskip\prf@temp\raise\dp\prf@linebox\box\prf@linebox}%
+  % We need to calculate the distance between the assumptions and the
+  % proofline: if there are no assumptions, it is zero; if there are
+  % assumptions but no rule name, it is \prflinepad; otherwise it is
+  % \linepad plus half of the "real" line's height
+  \prf@tmp\z@%
+  \ifprf@assum%
+    \ifprf@rule\prf@tmp-.5\ht\prf@linebox%
+      \advance\prf@tmp.5\ht\prf@fancybox%
+      \advance\prf@tmp.5\dp\prf@fancybox\fi\relax%
+    \ifprf@label\ifprf@rule\relax\else%
+      \prf@tmp-.5\ht\prf@linebox%
+      \advance\prf@tmp.5\ht\prf@fancybox%
+      \advance\prf@tmp.5\dp\prf@fancybox\fi\fi%
+    \advance\prf@tmp\prflinepad%
+    \setbox\prf@assumptionsbox%
+      \vbox{\box\prf@assumptionsbox\nointerlineskip\vskip\prf@tmp}\fi%
+  % We need to calculate the distance between the proofline and the
+  % conclusion: it is slightly complex to explain... try to draw the
+  % whole thing to make it clear!! 
+  \prf@temp\prflinepad%
+  \ifprf@noline%
+    \ifprf@assum%
+      \advance\prf@temp\prfemptylinethickness\fi%
+    \advance\prf@temp-\prflinepad\fi%
+  \ifprf@rule%
+    \advance\prf@temp-.5\ht\prf@linebox%
+    \advance\prf@temp.5\ht\prf@fancybox%
+    \advance\prf@temp.5\dp\prf@fancybox\fi%
+  \ifprf@label\ifprf@rule\relax\else%
+    \advance\prf@temp-.5\ht\prf@linebox%
+    \advance\prf@temp.5\ht\prf@fancybox%
+    \advance\prf@temp.5\dp\prf@fancybox\fi\fi%
+  \setbox\prf@conclusionbox%
+    \vbox{\vskip\prf@temp\box\prf@conclusionbox}%
+  % And, finally, at the very end, we can prepare the proof box and we
+  % can calculate the right value from Rassum. At the very end, we can
+  % output the result (!!)
+  \setbox\prf@rulenamebox%
+    \vbox{\box\prf@assumptionsbox\nointerlineskip%
+      \box\prf@linebox\nointerlineskip%
+      \box\prf@conclusionbox\nointerlineskip}%
+  \global\advance\prf@Rassum\wd\prf@rulenamebox%
+  \hbox{$\box\prf@rulenamebox$}}
+
+% -------------------------------------------------------------------
+% The following macros are used to simplify the writing of proofs in
+% natural deduction, and they roughly follow the format of proofs as
+% in Troelstra, Schwichtenberg "Basic Proof Theory".
+% 
+% Since they may conflict with other packages, they are controlled by
+% options in the package. 
+% -------------------------------------------------------------------
+
+% -------------------------------------------------------------------
+% Natural deduction systems
+%
+% Package option [ND]
+% -------------------------------------------------------------------
+
+\newbox\prf@charbox
+\def\prfscript#1{\setbox\prf@charbox%
+  \hbox{$\scriptstyle\mathrm{#1}$}\copy\prf@charbox}
+\ifprf@NDOption%
+  \def\NDA{\prfassumption}
+  \def\NDD{\prfboundedassumption}
+  \def\NDAL#1{\prfassumption<#1>}
+  \def\NDDL#1{\prfboundedassumption<#1>}
+  \def\NDP{\prftree}
+  \def\NDANDI{\prftree[by]{$\scriptstyle\wedge\prfscript{I}$}}
+  \def\NDANDER{\prftree[by]{%
+      $\scriptstyle\wedge\prfscript{E}_{\prfscript{R}}$}} 
+  \def\NDANDEL{\prftree[by]{%
+      $\scriptstyle\wedge\prfscript{E}_{\prfscript{L}}$}}
+  \def\NDANDE{\prftree[by]{$\scriptstyle\wedge\prfscript{E}$}}
+  \ifprfIMPOption
+    \def\NDIMPI{\prftree[by]{$\scriptstyle\supset\prfscript{I}$}}
+    \def\NDIMPE{\prftree[by]{$\scriptstyle\supset\prfscript{E}$}}
+    \def\NDIMPIL#1{\prftree[by]{$\scriptstyle\supset%
+        \prfscript{I}_{\prfref<#1>}$}}
+  \else
+    \def\NDIMPI{\prftree[by]{$\scriptstyle\rightarrow%
+        \prfscript{I}$}}
+    \def\NDIMPE{\prftree[by]{$\scriptstyle\rightarrow%
+        \prfscript{E}$}}
+    \def\NDIMPIL#1{%
+      \prftree[by]{$\scriptstyle\rightarrow%
+        \prfscript{I}_{\prfref<#1>}$}}\fi
+  \def\NDORIR{\prftree[by]{$\scriptstyle\vee%
+      \prfscript{I}_{\prfscript{R}}$}}
+  \def\NDORIL{\prftree[by]{$\scriptstyle\vee%
+      \prfscript{I}_{\prfscript{L}}$}}
+  \def\NDORI{\prftree[by]{$\scriptstyle\vee\prfscript{I}$}}
+  \def\NDORE{\prftree[by]{$\scriptstyle\vee\prfscript{E}$}}
+  \def\NDOREL#1{\prftree[by]{$\scriptstyle\vee%
+      \prfscript{I}_{\prfref<#1>}$}}
+  \def\NDALLI{\prftree[by]{$\scriptstyle\forall\prfscript{I}$}}
+  \def\NDALLE{\prftree[by]{$\scriptstyle\forall\prfscript{E}$}}
+  \def\NDEXI{\prftree[by]{$\scriptstyle\exists\prfscript{I}$}}
+  \def\NDEXE{\prftree[by]{$\scriptstyle\exists\prfscript{E}$}}
+  \def\NDEXEL#1{%
+    \prftree[by]{$\scriptstyle\exists\prfscript{E}_{\prfref<#1>}$}}
+  \def\NDTI{\prftree[by]{$\scriptstyle\top\prfscript{I}$}}
+  \def\NDFE{\prftree[by]{$\scriptstyle\bot\prfscript{E}$}}
+  \def\NDLEM{\prftree[by]{$\scriptstyle\prfscript{lem}$}}
+\fi
+
+% -------------------------------------------------------------------
+% Sequent systems
+%
+% Package option [SEQ]
+% -------------------------------------------------------------------
+
+\ifprf@SEQOption%
+  \def\SEQA{\prfassumption}
+  \def\SEQD{\prfboundedassumption}
+  \def\SEQP{\prftree}
+  \def\SEQAX{\prftree[by]{$\scriptstyle\prfscript{Ax}$}}
+  \def\SEQLF{\prftree[by]{$\scriptstyle\prfscript{L}\bot$}}
+  \def\SEQLW{\prftree[by]{$\scriptstyle\prfscript{LW}$}}
+  \def\SEQRW{\prftree[by]{$\scriptstyle\prfscript{RW}$}}
+  \def\SEQLC{\prftree[by]{$\scriptstyle\prfscript{LC}$}}
+  \def\SEQRC{\prftree[by]{$\scriptstyle\prfscript{RC}$}}
+  \def\SEQLAND{\prftree[by]{$\scriptstyle\prfscript{L}\wedge$}}
+  \def\SEQRAND{\prftree[by]{$\scriptstyle\prfscript{R}\wedge$}}
+  \def\SEQLOR{\prftree[by]{$\scriptstyle\prfscript{L}\vee$}}
+  \def\SEQROR{\prftree[by]{$\scriptstyle\prfscript{R}\vee$}}
+  \ifprfIMPOption
+    \def\SEQLIMP{\prftree[by]{$\scriptstyle\prfscript{L}\supset$}}
+    \def\SEQRIMP{\prftree[by]{$\scriptstyle\prfscript{R}\supset$}}
+  \else
+    \def\SEQLIMP{\prftree[by]{$\scriptstyle\prfscript{L}%
+        \rightarrow$}}
+    \def\SEQRIMP{\prftree[by]{$\scriptstyle\prfscript{R}%
+        \rightarrow$}}
+  \fi
+  \def\SEQLALL{\prftree[by]{$\scriptstyle\prfscript{L}\forall$}}
+  \def\SEQRALL{\prftree[by]{$\scriptstyle\prfscript{R}\forall$}}
+  \def\SEQLEX{\prftree[by]{$\scriptstyle\prfscript{L}\exists$}}
+  \def\SEQREX{\prftree[by]{$\scriptstyle\prfscript{R}\exists$}}
+  \def\SEQCUT{\prftree[by]{$\scriptstyle\prfscript{Cut}$}}
+\fi
+
+% -------------------------------------------------------------------
diff --git a/Master/tlpkg/bin/tlpkg-ctan-check b/Master/tlpkg/bin/tlpkg-ctan-check
index 9a9eb72c952..d3b9855e0e8 100755
--- a/Master/tlpkg/bin/tlpkg-ctan-check
+++ b/Master/tlpkg/bin/tlpkg-ctan-check
@@ -394,7 +394,7 @@ my @TLP_working = qw(
     polyglossia polynom polynomial
     polytable postcards poster-mac powerdot powerdot-FUBerlin
     ppr-prv pracjourn preprint prerex present presentations presentations-en
-    pressrelease prettyref preview printlen proba probsoln procIAGssymp
+    pressrelease prettyref preview prftree printlen proba probsoln procIAGssymp
     prodint productbox program
     progress progressbar proposal properties
     prosper protex protocol przechlewski-book
diff --git a/Master/tlpkg/tlpsrc/collection-mathextra.tlpsrc b/Master/tlpkg/tlpsrc/collection-mathextra.tlpsrc
index 4608c02b8ee..c2670198b42 100644
--- a/Master/tlpkg/tlpsrc/collection-mathextra.tlpsrc
+++ b/Master/tlpkg/tlpsrc/collection-mathextra.tlpsrc
@@ -37,6 +37,7 @@ depend nath
 depend ot-tableau
 depend oubraces
 depend perfectcut
+depend prftree
 depend proba
 depend rec-thy
 depend ribbonproofs
diff --git a/Master/tlpkg/tlpsrc/prftree.tlpsrc b/Master/tlpkg/tlpsrc/prftree.tlpsrc
new file mode 100644
index 00000000000..e69de29bb2d
--- /dev/null
+++ b/Master/tlpkg/tlpsrc/prftree.tlpsrc