%% fitchdoc.tex
%% Macros for Fitch-style natural deduction (documentation)
%% Copyright 2002-2023 Peter Selinger
%
% This work may be distributed and/or modified under the
% conditions of the LaTeX Project Public License, either version 1.3
% of this license or (at your option) any later version.
% The latest version of this license is in
%   https://www.latex-project.org/lppl.txt
% and version 1.3c or later is part of all distributions of LaTeX
% version 2008 or later.
%
% This work has the LPPL maintenance status `maintained'.
% 
% The Current Maintainer of this work is Richard Zach <rzach@ucalgary.ca>
%
% This work consists of the files fitch.sty and fitchdoc.tex.

% Original Author: Peter Selinger, Dalhousie University
% Created: Jan 14, 2002
% Modified: Dec 17, 2023
% Version: 1.0
% Copyright: (C) 2002-2023 Peter Selinger, Richard Zach
% Filename: fitch.sty
% Documentation: fitchdoc.tex
% https://github.com/OpenLogicProject/fitch/

\documentclass{ltxdoc}

\usepackage{fitch,graphicx,showexpl}

\lstset{%
  basicstyle=\ttfamily\small,
  commentstyle=\itshape\ttfamily\small,
  showspaces=false,
  showstringspaces=false,
  breaklines=true,
  breakautoindent=true,
  captionpos=t
}

\addtolength{\oddsidemargin}{-1cm}
\addtolength{\textwidth}{2cm}

\newcommand\NewIn[1]{\leavevmode
  \marginpar{\hfill\fbox{\fbox{New in #1}}\hspace*{1em}}\ignorespaces}

\title{\texttt{fitch.sty}: Fitch-style natural deduction macros}

\author{Peter Selinger\\Dalhousie University \and Richard
Zach\\University of Calgary\thanks{The current maintainer of this
package is \href{https://richardzach.org}{Richard Zach}. The package
repository is at \url{https://github.com/OpenLogicProject/fitch/},
where you can also report any
\href{https://github.com/OpenLogicProject/fitch/issues}{issues} with
it.}}

\date{Version 1.0\\ December 17, 2023}

\begin{document}
\maketitle

\section{Overview}

This document describes how to use the {\tt fitch.sty} macros for
typesetting Fitch-style natural deduction derivations. To load the
macros, simply put |\usepackage{fitch}| into the preamble of your
{\LaTeX} file. 

Here is a natural deduction derivation, together with the code that
produced it:

\begin{LTXexample}
$\begin{nd}
  \hypo {1}  {P\vee Q}
  \hypo {2}  {\neg Q}
  \open
  \hypo {3} {P}
  \have {4} {P}        \r{3}
  \close
  \open
  \hypo {aa} {Q}
  \have {6} {\neg Q}   \r{2}
  \have {7} {\bot}     \ne{aa,6}
  \have {8} {P}        \be{7}
  \close
  \have {9} {P}        \oe{1,3-4,aa-8}
\end{nd}$
\end{LTXexample}

A derivation consists of \emph{lines}, and each line contains a {\em
line label} and a \emph{formula}. Some lines also carry a {\em
justification}. Moreover, each line is either a \emph{hypothesis} or a
\emph{derived formula}. Usually, derived formulas carry a
justification, whereas hypotheses do not; however, the macros do not
enforce this.

Proofs set using |fitch.sty| will have lines numbered automatically in
the output. It is possible to override the automatic numbering (see
Section~\ref{subsec-customline}). You can refer to line numbers in the
text using |\ndref| (see Section~\ref{subsec-ndref}). Various
dimensions, formatting of justifications and line references, and the
shorthand macros used to produce rule names can be customized (see
Section~\ref{sec-customization}).

\NewIn{1.0} Several new commands and customization options have been
introduced in v1.0. It is mostly backwards compatible with earlier
versions, but see section~\ref{compat}. In particular, if you
redefined any internal |fitch| commands, you will have to change |nd*|
to |nd@|.

\section{Usage}\label{sec-usage}

\DescribeEnv{nd}\DescribeEnv{nd} Derivations are typeset
inside the |nd| environment. By default, the standard |array|
environment is used to do this, so the |nd| environment must must be
used in math mode, i.e., it should be surrounded by |$...$| or
|\[...\]|.

\NewIn{1.0} The environment |fitchproof| typesets the proof on its own
in text, not math mode. Proofs will be set flush left, with the
default |\partopsep| spacing surrounding lists added. You do not have
to insert |$| to switch to math mode for |fitchproof|---by default.
However, it only works if you generate proofs using the |tabular|
environment, e.g., by loading |fitch| with the |arrayenv=tabular|
option.

\DescribeMacro{\hypo}\DescribeMacro{\have}
The commands |\hypo| and |\have| are used
to typeset one line of the derivation; |\hypo| is used for
hypotheses, and |\have| for derived formulas. Both commands take
a label and a formula as an argument. Note that the labels used to
identify lines in the derivation need not be actual line numbers; for
instance, in the above example, we used the label $aa$ instead of $5$.
In the output, lines are automatically numbered consecutively. Labels
may not contain any punctuation characters or spaces.

\DescribeMacro{\open}\DescribeMacro{\close}
Subderivations are opened and closed with the commands |\open| and
|\close|. Finally, the following commands are provided for
annotating lines with justifications:
\begin{center}
\begin{tabular}[t]{@{}ll@{}}
  |\r|  & reiteration \\
  |\ii| & implication introduction \\
  |\ie| & implication elimination \\
  |\ai| & and introduction \\
  |\ae| & and elimination \\
  |\oi| & or introduction \\
  |\oe| & or elimination 
\end{tabular} 
\qquad
\begin{tabular}[t]{ll}
  |\ni| & not introduction \\
  |\ne| & not elimination \\
  |\be| & bottom elimination \\
  |\nne| & double negation elimination \\
  |\Ai| & forall introduction \\
  |\Ae| & forall elimination \\
  |\Ei| & exists introduction \\
  |\Ee| & exists elimination \\
\end{tabular} 
\end{center}
\NewIn{1.0} These commands and what they produce can be customized,
see Section~\ref{sec-customization}.

Each such command takes a \emph{reference list} as an argument. A
reference list is a string made from labels, commas, and hyphens, for
instance |1,3a-3b,4a-4d|.

\section{Details}

\subsection{Guards}

Some natural deduction derivations with quantifiers use \emph{guards}, as in
the following example:

\begin{LTXexample}
$
\begin{nd}
  \hypo {1} {\exists x\forall y.P(x,y)}
  \open[v]
  \open[u]
  \hypo {2} {\forall y.P(u,y)}
  \have {3} {P(u,v)}                     \Ae{2}
  \have {4} {\exists x.P(x,v)}           \Ei{3}
  \close
  \have {5} {\exists x.P(x,v)}           \Ee{1,2-5}
  \close
  \have {6} {\forall y\exists x.P(x,y)}  \Ai{2-5}
\end{nd}
$
\end{LTXexample}

The guards $v$ and $u$ in line 2 were typeset by giving optional
arguments to the |\open| commands of the respective
subderivations. 

\DescribeMacro{\guard}
For most purposes, the above way of specifying guards is sufficient.
However, there is another method, which allows a more flexible
placement of guards: before any line, you can give the command
|\guard{u}| to add a guard $u$ to the top-level subderivation at
that line, or |\guard[n]{u}| to add a guard to the $n$th
enclosing subderivation at that line. Thus, the above example could
have also been typeset by inserting the two commands |\guard{u}| and
|\guard[2]{v}| just after the second |\open| statement.

% For backward compatibility, there is a third way of specifying guards
% by giving an optional second argument to the |\hypo| and
% |\have| commands. The use of this feature is discouraged.

\subsection{Label and reference list details}\label{subsec-ndref}

Labels for lines given to the |\have| and |\hypo| commands need not be
numeric, although the package will \emph{output} them as consecutive
numbers (see Section~\ref{subsec-customline} for how to adjust the
numbering). Labels, however, may not contain commas, periods,
semicolons, hyphens, parentheses, or spaces. In a reference list,
spaces are ignored (even within a label!), whereas commas, periods,
semicolons, parentheses, and hyphens are copied to the output. All
other characters are interpreted as part of a label. Attempting to
reference a label which has not been previously defined by any |\hypo|
or |\have| command produces a \NewIn{0.6} warning message of the form:
\begin{verbatim}
  ! Package fitch Warning: Undefined line reference: lab17.
\end{verbatim}
(In earlier versions, this resulted in an error, not a warning.)

\DescribeMacro{\ndref}
Labels defined in an |nd| environment can be referenced in the
text with the |\ndref| command. This command takes a reference
list as an argument, and produces the corresponding output. However,
it is only possible to reference labels \emph{after} the corresponding
derivation has been typeset. There is currently no convenient way of
defining forward references. Also, if a label is used more than once,
|\ndref| will always refer to the most recent time it was used.

\subsection{Generic justifications}

\DescribeMacro{\by}
Non-standard justifications can be created with the |\by|
command. This command takes two arguments: a name and a reference
list. For instance, the command |\by{De Morgan}{lab3,lab4}| might
produce the output ``\mbox{De Morgan, 3, 4}''. Note that the justification
is typeset in text mode.

In the default justification format, a comma is automatically inserted
between the name and the reference list, unless the reference list is
empty. The formatting of justifications can be changed, see
Section~\ref{sec-customization}. If the second argument (the reference
list) is empty, only the first argument (without formatting or
punctuation) is printed. \NewIn{1.0} If the first argument is empty,
only the reference list is printed.

Since the |\by| command outputs its first argument without additional
formatting when the second argument is empty, you can use |\by{...}{}|
to produce arbitrary text in the justification. You can use the
|\ndref| command here.
\begin{LTXexample}
$
\begin{nd}
  \hypo {a} {A \Rightarrow B} 
    \by{Premise}{}
  \hypo {b} {A} \by{Premise}{}
  \have {c} {B}
    \by{\ndref{a,b}
      (but \emph{how?})}{}
  \have {d} {A \wedge B} \by{}{b,c}
\end{nd}
$
\end{LTXexample}

\subsection{Scope}

The commands |\hypo|, |\have|, |\open|, |\close|, |\by|, |\r|, |\ii|, and so
forth are only available inside an |nd| environment. These commands
may have a different meaning elsewhere in the same document. The only
commands provided by the |fitch.sty|  package which are visible
outside an |nd| environment are the command |\ndref| described in
Section~\ref{subsec-ndref}, the commands |\ndrules|, |\ndjustformat|,
|\ndrefformat|, and |\nddim|, and the dimension |\ndindent| described
in Section~\ref{sec-customization}.

\subsection{Breaking it across pages}\label{subsec-break}

\DescribeEnv{ndresume}
\DescribeEnv{fitchproof*}
The |nd| environment is derived from the {\LaTeX} |array|
environment, and thus it does not break across pages automatically. 
However, if a derivation is too long to fit on a single page, it is
possible to split it manually into physically independent, but
logically consecutive subparts. For this purpose, the |ndresume|
environment is provided to continue a previously interrupted
derivation. 
\NewIn{1.0} The environment |fitchproof*| works the same way, except typesets the
proof just like the |fitchproof| environment (no need for math mode,
flush left, spacing before and after). However, like |fitchproof|, it
requires the |arrayenv=tabular| option. Here is an example:

\begin{LTXexample}
\begin{fitchproof}[arrayenv=tabular]
  \hypo {1}  {P\vee Q}
  \hypo {2}  {\neg Q}
  \open
  \hypo {3} {P}
  \have {4} {P}      \r{3}
  \close
  \open
  \hypo {aa} {Q}
  \have {6} {\neg Q} \r{2}
\end{fitchproof}
Derivations can be interrupted and 
resumed at any point.
\begin{fitchproof*}[arrayenv=tabular]
  \have {7} {\bot}  \ne{aa,6}
  \have {8} {P}     \be{7}
  \close
  \have {9} {P}     \oe{1,3-4,aa-8}
\end{fitchproof*}
\end{LTXexample}

\NewIn{1.0} You can also have derivations break across pages
automatically. In order to do this, you have to load the
\href{https://ctan.org/pkg/longtable}{|longtable|} package, and load
|fitch| with the |arrayenv=longtable| option. Since the |longtable|
environment works in text (not math) mode, you should then only use
|fitchproof|, or the |nd| environment but \emph{not} in text mode.
Note that the |longtable| package does not work in 2-column mode or
inside a |multicolumn| environment. You can always produce a proof
inside a |multicolumn| environment by passing |arrayenv=tabular| as an
option to the |nd| or |fitchproof| environment.


\subsection{Custom line numbers}\label{subsec-customline}

One often needs to write derivation schemas, rather than derivations.
This often requires the use of symbolic constants such as $n$, $n+1$,
etc, instead of actual line numbers. The |\have| and |\hypo|
commands have an optional first argument which is a symbolic constant.
For instance, |\have[n]| will cause the current line to be
numbered with the symbolic constant $n$. Subsequent lines are
automatically numbered $n+1$ etc. An initial offset can be given as a
second optional argument, as in |\have[n][-1]|, which will cause
the current line to be numbered $n-1$, the following line $n$, etc. In
an explicit offset is given, the symbolic constant can also be absent:
for instance, the command |\have[][7]| resets the current line
number to $7$. The following example illustrates this behavior:

\begin{LTXexample}
$
\begin{nd}[justsep=1em]
  \hypo          {1} {P\vee Q}
  \open
  \hypo          {2} {P}
  \have [\vdots] {3} {\vdots}
  \have [n][-1]  {4} {A\wedge B}
  \close
  \open
  \hypo          {5} {Q}
  \have [\vdots] {6} {\vdots}
  \have [m]      {7} {A\wedge B}
  \close
  \have          {8} {A\wedge B}
      \oe{1,2-(4),5-7}
  \have [\vdots] {9} {\vdots}
  \have [][100] {10} {A} \ae{8}
\end{nd}
$
\end{LTXexample}

Note that in the justification for line $m+1$, parentheses had to be
put around the label $4$. There is currently no way of doing this
automatically. 

{\bf Exercise.} How does one typeset an empty line number?

{\bf Solution.} Since |\have[]| has a special meaning as explained
above, we need to use |\have[~]| instead.

\subsection{Continuation lines}\label{subsec-continuation}

Sometimes one has to typeset a very long formula that does not fit on
a single line. As of version 0.5 of the {\tt fitch.sty} macros, it is
possible to break a formula into several lines using |\\| as a
line separator. Continuation lines are automatically indented, as
shown in the following example.

\begin{LTXexample}
$
\begin{nd}
  \hypo{1}  {A\wedge B}
  \hypo{2}  {A\wedge B\Rightarrow{} \\
             C\wedge D}
  \have{3}  {C\wedge D}  \ie{1,2}
  \have{4}  {A\wedge B\wedge{} \\
             C\wedge D}  \ai{1,3}
\end{nd}
$
\end{LTXexample}

\DescribeMacro{\hypocont}
\DescribeMacro{\havecont}
Alternatively, the |\havecont| and |\hypocont|  commands can
be used to specify each continuation line separately, as the following
example illustrates.

\begin{LTXexample}
$
\begin{nd}
  \hypo{1}  {A\wedge B}
  \hypo{2}  {A\wedge B\Rightarrow{}}
  \hypocont {C\wedge D}
  \have{3}  {C\wedge D}         \ie{1,2}
  \have{4}  {A\wedge B\wedge{}} \ai{1,3}
  \havecont {C\wedge D}
\end{nd}
$
\end{LTXexample}

This latter style gives slightly more flexibility in the placement of
justifications, since each line and continuation line can have its own
justification and its own guard (via the |\guard| command).  It
also allows a derivation to be interrupted between a line and its
continuation, as discussed in Section~\ref{subsec-break}.

\section{Customization}\label{sec-customization}

The relative sizes of the various elements of a natural deduction
proof are preset to reasonable values depending on the size of the
currently selected font. However, it will sometimes be necessary to
customize these dimensions. The customizable dimensions are given in
the following diagram.
\begin{center}
  \includegraphics{fitchdoc-dimen}
\end{center}
In addition, \meta{linethickness} determines the thickness of scope
lines, and \meta{cindent} the extra indentation of continuation lines
(as discussed in Section~\ref{subsec-continuation}). The default
dimensions are:
\begin{center}
\begin{tabular}{ll@{\qquad}ll}
  \meta{height} & 4.5ex &
  \meta{topheight} & 3.5ex\\
  \meta{depth} & 1.5ex &
  \meta{labelsep} & 1em\\
  \meta{indent} & 1.6em &
  \meta{hsep} & .5em\\
  \meta{justsep} & 2.5em &
  \meta{linethickness} & .2mm\\
  \meta{cindent} & 1em
\end{tabular}
\end{center}

\NewIn{1.0} To change these default dimensions, pass a list of
key-value pairs as package options, as optional arguments to the
|nd| or |fitchproof| environment, or use the |\setkeys|
command:
\begin{verbatim}
  \usepackage[justsep=1em]{fitch}
  \begin{nd}[rules=myrules]
  \begin{fitchproof}[linethickness=1pt]
  \setkeys{fitch}{hsep=1em,indent=1em}\end{verbatim}

In addition, the macros used to generate the table containing the
proof, to format justifications, format line number references, and to
initialize macros to produce justifications in the proof can be
customized:
\begin{center}
  \begin{tabular}{ll}
  option & default\\\hline
  |rules=|\meta{macroname} & |ndrules|\\
  |justformat=|\meta{macroname} & |ndjustformat|\\
  |refformat=|\meta{macroname} & |ndrefformat|\\
  |arrayenv=|\meta{envname} & |array|
  \end{tabular}
\end{center}
For compatibility with earlier versions of |fitch.sty|, the default
for \meta{arrayenv} is |array|, i.e., the table containing the proof
is generated using an |array| environment, and must therefore occur
inside math mode. The |fitchproof| and |fitchproof*| environments
assume that you use a text table command instead, such as |tabular|.
Hence, you should \emph{not} use |fitchproof| in math mode, and you
must use the option |arrayenv=tabular| (when loading |fitch|, as an
optional argument to |fitchproof|, or using
|\setkeys{fitch}{arrayenv=tabular}|. Any other table environment that
takes the same table format argument as |array| and |tabular| can be
used here, e.g., the |longtable| environment from the |longtable|
package (which must be loaded separately).

The package defines the macros
\DescribeMacro{\ndrules}\cmd{\ndrules}, which defines the rule
macros given at the end of Section~\ref{sec-usage}, using
\begin{verbatim}
  \def\ndrules{%
  \def\ii{\by{$\Rightarrow$I}}%
  \def\ie{\by{$\Rightarrow$E}}%
  \def\Ai{\by{$\forall$I}}%
  \def\Ae{\by{$\forall$E}}%
  \def\Ei{\by{$\exists$I}}%
  \def\Ee{\by{$\exists$E}}%
  \def\ai{\by{$\wedge$I}}%
  \def\ae{\by{$\wedge$E}}%
  \def\ai{\by{$\wedge$I}}%
  \def\ae{\by{$\wedge$E}}%
  \def\oi{\by{$\vee$I}}%
  \def\oe{\by{$\vee$E}}%
  \def\ni{\by{$\neg$I}}%
  \def\ne{\by{$\neg$E}}%
  \def\be{\by{$\bot$E}}%
  \def\nne{\by{$\neg\neg$E}}%
  \def\r{\by{R}}}
\end{verbatim}
\DescribeMacro{\ndjustformat}
The macro \cmd{\ndjustformat} is defined as
\begin{verbatim}
  \newcommand{\ndjustformat}[2]{#1, #2}
\end{verbatim}
The first argument takes the rule name, the second the reference list.
It is used to typeset the justification.

\DescribeMacro{\ndrefformat}
The macro \cmd{\ndrefformat} is defined as
\begin{verbatim}
  \newcommand{\ndrefformat}[1]{#1}
\end{verbatim}
It is used to typeset the line numbers in justifications.

\cmd{\ndrules}, \cmd{\ndjustformat}, and \cmd{\ndrefformat} can
be redefined using \cmd{\renewcommand}, or you can define your own
commands to provide the rule names, the justification format, and line
number format, and pass the names (without initial |\|) as an option
to the \cmd{\usepackage} or individual \cmd{\nd} or \cmd{\fitchproof}
commands.
\begin{LTXexample}
\newcommand{\myjust}[2]
  {#2 by \textsf{#1}}
\newcommand{\myrules}{
  \ndrules % include standard rules
  \def\ds{\by{DS}}}
\renewcommand{\ndrefformat}[1]{(#1)}
$
\begin{nd}[rules=myrules,
  justformat=myjust,
  indent=1.5cm,
  linethickness=1.5pt,
  justsep=1cm]
  \hypo {1} {A\vee B}
  \hypo {2} {\neg B}
  \open
  \hypo {a} {B}
  \have {3} {A}          \ds{1,2}
  \have {b} {A \wedge B} \ai{a,3}
\end{nd}
$
\end{LTXexample}

The boolean option |outerline| can be set to false to suppress the
leftmost scope line. This may be useful when printing inference rules,
e.g.,
\begin{LTXexample}
$
\begin{nd}[outerline=false,
  labelsep=0pt]
  \open
  \hypo [n]{1} {A}
  \have [~]{2} {\raisebox{-1ex}{\vdots}}
  \have [m]{3} {B}
  \close
  \have [~]{b} {A \Rightarrow B} \ii{1-3}
\end{nd}
$
\end{LTXexample}

\section{Obsolete commands and backwards compatibility}\label{compat}

\DescribeMacro{\nddim}
The dimensions can also be changed with the
|\nddim| command. The syntax of the command is as follows:
\begin{center}
  \cmd{\nddim}\marg{height}\marg{topheight}\marg{depth}\marg{labelsep}
  \marg{indent}\marg{hsep}\marg{justsep}\marg{linethickness},
\end{center}
where each of the eight parameters is a dimension. This still works,
but using the key-value pair options is the preferred method.

\NewIn{1.0}\DescribeMacro{\ndindent} In versions before v1.0, the recommended way
to change the extra indentation used on continuation lines was to
change dimension |\ndindent| directly using |\setlength|. As of v1.0,
you should use the |cindent| option instead.

The original code ``hid'' the internal macros by naming
them |\nd*...|. In v1.0 this has been changed to the standard
|\nd@...|. Any low-level redefinition of |fitch| internals that uses
|*| will break in v1.0.

\section{Other comments}

The goal was to design a flexible package which would not impose any
constraints on the form of derivations, while making typesetting easy.
With this package, it is in fact possible to typeset incomplete,
ill-formed, or invalid derivations. Sometimes it is pedagogically necessary
to do so.

There are no arbitrary limits on the size or nesting depth of a derivation,
except for the obvious requirement of fitting horizontally on the
printed page.

\section{Copyright and license}

This document and the accompanying |fitch.sty| macros are {\copyright}
2002--2023 by Peter Selinger and Richard Zach and distributed under
the terms of the \href{https://www.latex-project.org/lppl/}{LPPL}.

\end{document}