%%% perfectcut.sty  documentation
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%
%%% Author: Guillaume Munch-Maccagnoni
%%% http://www.pps.univ-paris-diderot.fr/~munch/
%%%
%%% 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. Refer to the README file.
%%%
\documentclass[12pt,a4paper,british]{scrartcl}
\usepackage[T1]{fontenc}
\usepackage[latin9]{inputenc}
\setcounter{secnumdepth}{1}
\setcounter{tocdepth}{1}
\usepackage{babel}
\usepackage{array}
\usepackage{booktabs}
\usepackage{calc}
\usepackage{amsmath}
\usepackage[unicode=true]{hyperref}
\usepackage{perfectcut}
\usepackage{stix}
\renewcommand*\ttdefault{txtt}
\usepackage[oldstyle,lining,scale=0.97]{sourcesanspro}
\usepackage[protrusion=true,expansion=true,tracking=false,kerning=true,spacing=true]{microtype}

\begin{document}
\global\long\def\cut#1#2{\perfectcut{#1}{#2}}
\global\long\def\mt{\bar{\mu}}


\title{\texttt{perfectcut.sty} documentation}


\author{\noindent {\large{}Guillaume Munch-Maccagnoni}%
\thanks{\protect\href{http://www.pps.univ-paris-diderot.fr/~munch/}{http://www.pps.univ-paris-diderot.fr/$\sim$munch/}%
}}


\date{2nd November 2014}

\maketitle


\tableofcontents{}


\section{Use}


\paragraph{Original use}

This package supplies the following commands:

\begin{center}
\begin{tabular}{ll}
\toprule 
Command & Produces\tabularnewline
\midrule
\texttt{\textbackslash{}perfectcut\{\#1\}\{\#2\}} & $\perfectcut{\#1}{\#2}$\tabularnewline
\texttt{\textbackslash{}perfectbra\{\#1\}} & $\perfectbra{\#1}$\tabularnewline
\texttt{\textbackslash{}perfectket\{\#1\}} & $\perfectket{\#1}$\tabularnewline
\bottomrule
\end{tabular}\quad{}%
\begin{tabular}{ll}
\toprule 
Command & Produces\tabularnewline
\midrule
\texttt{\textbackslash{}perfectcase\{\#1\}} & $\perfectcase{\#1}{\#2}$\tabularnewline
\texttt{\textbackslash{}perfectbrackets\{\#1\}} & $\perfectbrackets{\#1}$\tabularnewline
\texttt{\textbackslash{}perfectparens\{\#1\}} & $\perfectparens{\#1}$\tabularnewline
\bottomrule
\end{tabular}
\par\end{center}

The effect of these commands is to let the delimiters grow according
to the number of nested \texttt{\textbackslash{}perfect}\texttt{\emph{command}}s
(regardless of the size of contents). The package is originally intended
for solving a notational issue regarding the representation of abstract-machine-like
calculi in proof theory and computer science.


\paragraph{General use}

The package also defines \texttt{\textbackslash{}perfectunary} and
\texttt{\textbackslash{}perfectbinary} for defining custom delimiters
that behave similarly to the above ones. These commands should be considered when
facing the need of nested delimiters that consistently grow regardless
of the contents. (See ``Advanced Use''.)\medskip{}


If the package causes incompatibility errors, refer to the option
\texttt{nomathstyle} below.


\section{Example}




\subsection{Using \texttt{perfectcut.sty}}

\global\long\def\cut#1#2{\cutprimitive{#1}{#2}}


\noindent \texttt{\footnotesize{}\textbackslash{}usepackage\{perfectcut\}}~\\
\texttt{\footnotesize{}\textbackslash{}let\textbackslash{}cut\textbackslash{}perfectcut}{\footnotesize \par}

\begin{center}
\fbox{\begin{minipage}[t]{0.8\columnwidth}%
The following states the idempotency of an adjunction: 
\[
\cut t{\mt x.\cut{\mu\alpha.\cut ue}{e'}}=\cut{\mu\alpha.\cut t{\mt x.\cut ue}}{e'}
\]


The following states the commutativity of a strong monad:
\[
\cut t{\mt x.\cut u{\mt y.\cut ve}}=\cut u{\mt y.\cut t{\mt x.\cut ve}}
\]
Using \texttt{\textbackslash{}underline} to mark redexes:
\begin{align*}
 & \delta(V,x.y,x.y)\\
 & =\mu{\star}.\cut V{\perfectcase{\mt x.\underline{\cut y{\star}}}{\mt x.\underline{\cut y{\star}}}}\\
 & =\mu{\star}.\cut V{\underline{\perfectcase{\mt x.\cut{\iota_{1}(x)}{\mt z.\cut y{\star}}}{\mt x.\cut{\iota_{2}(x)}{\mt z.\cut y{\star}}}}}\\
 & =\mu{\star}.\cut V{\underline{\mt z.}\cut y{\star}}\\
 & =\mu{\star}.\cut y{\star}=y
\end{align*}
 %
\end{minipage}}
\par\end{center}


\subsection{Using \texttt{\textbackslash{}left}, \texttt{\textbackslash{}middle}
and \texttt{\textbackslash{}right} instead}

\global\long\def\fakecut#1#2{\left\langle #1\middle|\mkern-2mu\middle|#2\right\rangle }


\noindent \texttt{\footnotesize{}\textbackslash{}renewcommand\{\textbackslash{}cut\}{[}2{]}\{\textbackslash{}left\textbackslash{}langle
\#1\textbackslash{}middle|\textbackslash{}mkern-2mu\textbackslash{}middle|\#2\textbackslash{}right\textbackslash{}rangle\}}{\footnotesize \par}

\begin{center}
\fbox{\parbox[t]{0.8\columnwidth}{%
The following states the idempotency of an adjunction: 
\[
\fakecut t{\mt x.\fakecut{\mu\alpha.\fakecut ue}{e'}}=\fakecut{\mu\alpha.\fakecut t{\mt x.\fakecut ue}}{e'}
\]


The following states the commutativity of a strong monad:
\[
\fakecut t{\mt x.\fakecut u{\mt y.\fakecut ve}}=\fakecut u{\mt y.\fakecut t{\mt x.\fakecut ve}}
\]


Using \texttt{\textbackslash{}underline} to mark redexes:
\begin{align*}
 & \delta(V,x.y,x.y)\\
 & =\mu{\star}.\fakecut V{\left[\mt x.\underline{\fakecut y{\star}}\middle|\mt x.\underline{\fakecut y{\star}}\right]}\\
 & =\mu{\star}.\fakecut V{\underline{\left[\mt x.\fakecut{\iota_{1}(x)}{\mt z.\fakecut y{\star}}\middle|\mt x.\fakecut{\iota_{2}(x)}{\mt z.\fakecut y{\star}}\right]}}\\
 & =\mu{\star}.\fakecut V{\underline{\mt z.}\fakecut y{\star}}\\
 & =\mu{\star}.\fakecut y{\star}=y
\end{align*}
%
}}
\par\end{center}

As we can see, the legibility of the above rendering is hampered by
multiple issues: the delimiters grow inconsistently, vertical bars
have the wrong size, accents or underlines uselessly make the delimiters
grow, and the spacing could be improved. The package is designed to
fix these issues.


\section{Advanced use}

The package lets you define your own growing delimiters. Let us first
stress that the size of these delimiters is entirely determined by
the number of nestings and is insensitive to the size of the contents.
If you need the size of the contents to be taken into account then
it is probably sufficient to use \texttt{\textbackslash{}left} and
\texttt{\textbackslash{}right} while tweaking \texttt{\textbackslash{}delimitershortfall}
and \texttt{\textbackslash{}delimiterfactor}.


\subsection{Example}

The following displays a set $\{\#1\mid\#2\}$ with delimiters appropriately
sized if there are other \texttt{\textbackslash{}perfectcommands}
inside \texttt{\#1} and \texttt{\#2}.\medskip{}


\texttt{\footnotesize{}\textbackslash{}def\textbackslash{}Set\#1\#2\{\textbackslash{}perfectbinary\{IncreaseHeight\}\textbackslash{}\{|\textbackslash{}\}\{\#1\textbackslash{}mathrel\{\}\}\{\textbackslash{}mathrel\{\}\#2\}\}}{\footnotesize \par}

\texttt{\footnotesize{}\textbackslash{}{[}\textbackslash{}Set\{\textbackslash{}perfectparens\{a\}\}\{\textbackslash{}perfectparens\{b\}\}\textbackslash{}{]}}{\footnotesize \par}

\def\Set#1#2{\perfectbinary{IncreaseHeight}\{|\}{#1\mathrel{}}{\mathrel{}#2}}

\[
\fbox{\ensuremath{\Set{\perfectparens{a}}{\perfectparens{b}}}}
\]



\subsection{Custom delimiters}
\begin{description}
\item [{\texttt{\textbackslash{}perfectunary\#1\#2\#3\#4}}] Displays \texttt{\#2}
\texttt{\#4} \texttt{\#3} where \texttt{\#2} and \texttt{\#3} are
delimiters. The delimiters grow according to \texttt{\#1} which must
be one of \texttt{IncreaseHeight}, \texttt{CurrentHeight}, or \texttt{CurrentHeightPlusOne}.
\item [{\texttt{\textbackslash{}perfectbinary\#1\#2\#3\#4\#5\#6}}] Displays
\texttt{\#2} \texttt{\#5} \texttt{\#3} \texttt{\#6} \texttt{\#4} where
\texttt{\#2}, \texttt{\#3} and \texttt{\#4} are delimiters. The delimiters
grow according to \texttt{\#1} which must be one of \texttt{IncreaseHeight},
\texttt{CurrentHeight}, or \texttt{CurrentHeightPlusOne}.
\end{description}

\subsection{Stock delimiters}

The stock commands behave as follow:

\begin{center}
\begin{tabular}{>{\raggedright}p{11em}lll}
\toprule 
Command & Produces & Growth & Inserts skips\tabularnewline
\midrule
\texttt{\textbackslash{}perfectcut\{\#1\}\{\#2\}} & $\perfectcut{\#1}{\#2}$ & \texttt{IncreaseHeight} & Yes\tabularnewline
\texttt{\textbackslash{}perfectbra\{\#1\}} & $\perfectbra{\#1}$ & \texttt{IncreaseHeight} & Yes\tabularnewline
\texttt{\textbackslash{}perfectket\{\#1\}} & $\perfectket{\#1}$ & \texttt{IncreaseHeight} & Yes\tabularnewline
\texttt{\textbackslash{}perfectcase\{\#1\}} & $\perfectcase{\#1}{\#2}$ & \texttt{CurrentHeightPlusOne} & Yes\tabularnewline
\texttt{\textbackslash{}perfectbrackets\{\#1\}} & $\perfectbrackets{\#1}$ & \texttt{CurrentHeightPlusOne} & Only inside\tabularnewline
\texttt{\textbackslash{}perfectparens\{\#1\}} & $\perfectparens{\#1}$ & \texttt{CurrentHeight} & Only inside\tabularnewline
\texttt{\textbackslash{}perfectunary\{\#1\}}~\\
\texttt{~\{\#2\}\{\#3\}\{\#4\}} & $\#2\,\#4\,\#3$ & \#1 & No\tabularnewline
\texttt{\textbackslash{}perfectbinary\{\#1\}}~\\
\texttt{~\{\#2\}\{\#3\}\{\#4\}\{\#5\}\{\#6\}} & $\#2\,\#5\,\#3\,\#6\,\#4$ & \#1 & No\tabularnewline
\bottomrule
\end{tabular}
\par\end{center}


\section{Options}


\subsection{Option \texttt{nomathstyle}}

The use of \texttt{\textbackslash{}currentmathstyle} from the package
\texttt{mathstyle} prevents the exponential blowup in recursions that
would occur if we used \texttt{\textbackslash{}mathpalette} instead.
To record the \texttt{\textbackslash{}currentmath\-style}, \texttt{mathstyle}
redefines many  primitives and is therefore a source of incompatibilities.
If you run into such issues, please refer to the \texttt{mathstyle}
manual.

If you want to  disable the loading of \texttt{mathstyle}, use the
\texttt{nomathstyle} option.  With the \texttt{nomathstyle} option,
the style is set to \texttt{\textbackslash{}cutstyle}, which you must
 define by hand to be \texttt{\textbackslash{}displaystyle}, \texttt{\textbackslash{}textstyle
}(default), \texttt{\textbackslash{}scriptstyle},  \texttt{\textbackslash{}scriptscriptstyle}.
This means that cuts in subscripts and superscripts do  not have the
proper size unless \texttt{\textbackslash{}cutstyle }is redefined.


\subsection{Option \texttt{realVert}}

With the option \texttt{realVert}, the double bars are obtained with
the \texttt{\textbackslash{}Vert} command. Without it, two \texttt{\textbackslash{}vert}
symbols are used and their spacing is controlled with  \texttt{\textbackslash{}cutinterbarskip}.
But, if \texttt{realVert} is not activated, then a penalty (\texttt{binoppenalty})
is added, such that $\cut{\mu\alpha.\cut ab}{\mt x.\cut cd}$ splits
across lines.


\subsection{Option \texttt{fixxits}}

For some reason that the author was unable to identify, the vertical
alignment is wrong with the Opentype XITS math font with XeTeX. The
option \texttt{fixxits} fixes this behaviour.


\subsection{Customisation}

The following mu-skips can be redefined in your preamble:

\begin{center}
\begin{tabular}{ll}
\toprule 
Command & Defines the spacing...\tabularnewline
\midrule
\texttt{\textbackslash{}cutbarskip=5.0mu plus 8mu minus 2.0mu} & around bars\tabularnewline
\texttt{\textbackslash{}cutangleskip=0.0mu plus 8mu minus 1.0mu} & around delimiters (inside)\tabularnewline
\texttt{\textbackslash{}cutangleouterskip=0.0mu plus 8mu minus 0mu} & around delimiters (outside)\tabularnewline
\texttt{\textbackslash{}cutinterbarskip=0.8mu plus 0mu minus 0mu} & between bars (excl. \texttt{realVert})\tabularnewline
\bottomrule
\end{tabular}
\par\end{center}

\noindent (1 mu equals $1/18$ of an em in the current math font.)


\section{Reimplementation of fixed-size delimiters}

In addition, I provide the following corrections and generalisations
of  \texttt{\textbackslash{}big},\texttt{\textbackslash{}bigg}, etc.
Why not using the latter? Because both the plain \TeX{} and the \texttt{amsmath}
versions can be incorrect when changing the math font, the font size,
the math style or \texttt{\textbackslash{}delimitershortfall}. Moreover,
Opentype math fonts sometimes offer more than five sizes. For this
package we need a robust solution.

\begin{center}
\begin{tabular}{lll}
\toprule 
Command & Example & \tabularnewline
\midrule 
\multicolumn{3}{l}{\emph{\#1-th size of delimiter \#2}}\tabularnewline
\texttt{\textbackslash{}nthleft\{\#1\}\{\#2\} } & \texttt{\textbackslash{}nthleft\{2\}(} & $\nthleft{2}($\tabularnewline\addlinespace[0.1em]
\texttt{\textbackslash{}nthmiddle\{\#1\}\{\#2\}} & \texttt{\textbackslash{}nthmiddle\{2\}\textbackslash{}Vert} & $\nthmiddle{2}\Vert$\tabularnewline\addlinespace[0.1em]
\texttt{\textbackslash{}nthright\{\#1\}\{\#2\}} & \texttt{\textbackslash{}nthright\{2\})} & $\nthright{2})$\tabularnewline\addlinespace[0.1em]
\multicolumn{1}{l}{\emph{delimiter \#2 of height at least \#1}} &  & \tabularnewline\addlinespace[0.1em]
\texttt{\textbackslash{}lenleft\{\#1\}\{\#2\}}  & \texttt{\textbackslash{}lenleft\{3.2mm\}{[}} & $\lenleft{3mm}[$\tabularnewline\addlinespace[0.1em]
\texttt{\textbackslash{}lenmiddle\{\#1\}\{\#2\}} & \texttt{\textbackslash{}lenmiddle\{3.2mm\}|} & $\lenmiddle{3mm}|$\tabularnewline\addlinespace[0.1em]
\texttt{\textbackslash{}lenright\{\#1\}\{\#2\}} & \texttt{\textbackslash{}lenright\{3.2mm\}{]}} & $\lenright{3mm}]$\tabularnewline\addlinespace[0.1em]
\multicolumn{3}{l}{\emph{delimiter \#2 of height exactly \#1 obtained by scaling the
above one}}\tabularnewline\addlinespace[0.1em]
\texttt{\textbackslash{}reallenleft\{\#1\}\{\#2\}}  & \texttt{\textbackslash{}reallenleft\{3.2mm\}{[}} & $\reallenleft{3mm}[$\tabularnewline\addlinespace[0.1em]
\texttt{\textbackslash{}reallenmiddle\{\#1\}\{\#2\}} & \texttt{\textbackslash{}reallenmiddle\{3.2mm\}|} & $\reallenmiddle{3mm}|$\tabularnewline\addlinespace[0.1em]
\texttt{\textbackslash{}reallenright\{\#1\}\{\#2\}} & \texttt{\textbackslash{}reallenright\{3.2mm\}{]}} & $\reallenright{3mm}]$\tabularnewline
\bottomrule
\end{tabular}  
\par\end{center}


\subsection{Example with \texttt{\textbackslash{}nthleft}}

\texttt{\footnotesize{}\textbackslash{}nrthleft0{[}\textbackslash{}nthleft1{[}\textbackslash{}nthleft2{[}\textbackslash{}nthleft3{[}\textbackslash{}nthleft4{[}\textbackslash{}nthleft5{[}\textbackslash{}nthleft6{[}}{\footnotesize \par}

\[
\nthleft0(\nthleft1(\nthleft2(\nthleft3(\nthleft4(\nthleft5(\nthleft6(
\]



\subsection{Example with \texttt{\textbackslash{}big},\texttt{\textbackslash{}Big},\texttt{\textbackslash{}bigg},\texttt{\textbackslash{}Bigg}}

\texttt{\footnotesize{}{[}\textbackslash{}big{[}\textbackslash{}Big{[}\textbackslash{}bigg{[}\textbackslash{}Bigg{[}}{\footnotesize \par}

\[
(\big(\Big(\bigg(\Bigg(
\]
The above uses the \texttt{\textbackslash{}big} commands from the
\texttt{amsmath} package. The \texttt{amsmath} package corrects issues
with the original \TeX{} commands, but I could still notice inconsistencies,
such as \texttt{\textbackslash{}big} starting at size 2, under some
font combinations. \texttt{\textbackslash{}nthleft}, \texttt{\textbackslash{}nthright}
and \texttt{\textbackslash{}nthmiddle} are implemented in a more robust
way.


\section{License}

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. Refer to the \texttt{README}
file.
\end{document}
