\documentclass[a4paper]{article}

\usepackage{a4wide}
\usepackage{pifont}
\usepackage{semantic}

\newcommand{\MATITA}{\ding{46}\textsf{\textbf{Matita}}}

\title{Extensible notation for \MATITA}
\author{Luca Padovani \qquad Stefano Zacchiroli \\
\small Department of Computer Science, University of Bologna \\
\small Mura Anteo Zamboni, 7 -- 40127 Bologna, ITALY \\
\small \{\texttt{lpadovan}, \texttt{zacchiro}\}\texttt{@cs.unibo.it}}

\newcommand{\BREAK}{\mathtt{break}}
\newcommand{\TVAR}[1]{#1:\mathtt{term}}
\newcommand{\NVAR}[1]{#1:\mathtt{number}}
\newcommand{\IVAR}[1]{#1:\mathtt{name}}
\newcommand{\FENCED}[1]{\texttt{\char'050}#1\texttt{\char'051}}
\newcommand{\IOT}[2]{|[#1|]_{\mathcal#2}^1}
\newcommand{\NAMES}{\mathit{names}}
\newcommand{\DOMAIN}{\mathit{domain}}

\begin{document}
  \maketitle

\section{Environment}

\[
\begin{array}{rcll}
  V & ::= & & \mbox{(\bf values)} \\
    &     & \verb+Term+~T & \mbox{(term)} \\
    &  |  & \verb+String+~s & \mbox{(string)} \\
    &  |  & \verb+Number+~n & \mbox{(number)} \\
    &  |  & \verb+None+ & \mbox{(optional value)} \\
    &  |  & \verb+Some+~V & \mbox{(optional value)} \\
    &  |  & [V_1,\dots,V_n] & \mbox{(list value)} \\[2ex]
\end{array}
\]

An environment is a map $\mathcal E : \mathit{Name} -> V$.

\section{Level 1: concrete syntax}

\begin{table}
\caption{\label{tab:l1c} Concrete syntax of level 1 patterns.\strut}
\hrule
\[
\begin{array}{rcll}
  P & ::= & & \mbox{(\bf patterns)} \\
    &     & S^{+} \\[2ex]
  S & ::= & & \mbox{(\bf simple patterns)} \\
    &     & l \\
    &  |  & S~\verb+\sub+~S\\
    &  |  & S~\verb+\sup+~S\\
    &  |  & S~\verb+\below+~S\\
    &  |  & S~\verb+\atop+~S\\
    &  |  & S~\verb+\over+~S\\
    &  |  & S~\verb+\atop+~S\\
    &  |  & \verb+\frac+~S~S \\
    &  |  & \verb+\sqrt+~S \\
    &  |  & \verb+\root+~S~\verb+\of+~S \\
    &  |  & \verb+(+~P~\verb+)+ \\
    &  |  & \verb+hbox (+~P~\verb+)+ \\
    &  |  & \verb+vbox (+~P~\verb+)+ \\
    &  |  & \verb+hvbox (+~P~\verb+)+ \\
    &  |  & \verb+hovbox (+~P~\verb+)+ \\
    &  |  & \verb+break+ \\
    &  |  & \verb+list0+~S~[\verb+sep+~l] \\
    &  |  & \verb+list1+~S~[\verb+sep+~l] \\
    &  |  & \verb+opt+~S \\
    &  |  & [\verb+term+]~x \\
    &  |  & \verb+number+~x \\
    &  |  & \verb+ident+~x \\
\end{array}
\]
\hrule
\end{table}

Rationale: while the layout schemata can occur in the concrete syntax
used by user, the box schemata and the magic patterns can only occur
when defining the notation. This is why the layout schemata are
``escaped'' with a backslash, so that they cannot be confused with
plain identifiers, wherease the others are not. Alternatively, they
could be defined as keywords, but this would prevent their names to be
used in different contexts.

\begin{table}
\caption{\label{tab:l1a} Abstract syntax of level 1 terms and patterns.\strut}
\hrule
\[
\begin{array}{@{}ll@{}}
\begin{array}[t]{rcll}
  T & ::= & & \mbox{(\bf terms)} \\
    &     & L_\kappa[T_1,\dots,T_n] & \mbox{(layout)} \\
    &  |  & B_\kappa^{ab}[T_1\cdots T_n] & \mbox{(box)} \\
    &  |  & \BREAK & \mbox{(breakpoint)} \\
    &  |  & \FENCED{T_1\cdots T_n} & \mbox{(fenced)} \\
    &  |  & l & \mbox{(literal)} \\[2ex]
  P & ::= & & \mbox{(\bf patterns)} \\
    &     & L_\kappa[P_1,\dots,P_n] & \mbox{(layout)} \\
    &  |  & B_\kappa^{ab}[P_1\cdots P_n] & \mbox{(box)} \\
    &  |  & \BREAK & \mbox{(breakpoint)} \\
    &  |  & \FENCED{P_1\cdots P_n} & \mbox{(fenced)} \\
    &  |  & M & \mbox{(magic)} \\
    &  |  & V & \mbox{(variable)} \\
    &  |  & l & \mbox{(literal)} \\
\end{array} &
\begin{array}[t]{rcll}
  V & ::= & & \mbox{(\bf variables)} \\
    &     & \TVAR{x} & \mbox{(term variable)} \\
    &  |  & \NVAR{x} & \mbox{(number variable)} \\
    &  |  & \IVAR{x} & \mbox{(name variable)} \\[2ex]
  M & ::= & & \mbox{(\bf magic patterns)} \\
    &     & \verb+list0+~P~l? & \mbox{(possibly empty list)} \\
    &  |  & \verb+list1+~P~l? & \mbox{(non-empty list)} \\
    &  |  & \verb+opt+~P & \mbox{(option)} \\[2ex]
\end{array}
\end{array}
\]
\hrule
\end{table}

\[
\IOT{\cdot}{{}} : P -> \mathit{Env} -> T
\]

\begin{table}
\caption{\label{tab:il1} Instantiation of level 1 patterns.\strut}
\hrule
\[
\begin{array}{rcll}
  \IOT{L_\kappa[P_1,\dots,P_n]}{E} & = & L_\kappa[\IOT{P_1}{E},\dots,\IOT{P_n}{E} ] \\
  \IOT{B_\kappa^{ab}[P_1\cdots P_n]}{E} & = & B_\kappa^{ab}[\IOT{P_1}{E}\cdots\IOT{P_n}{E}] \\
  \IOT{\BREAK}{E} & = & \BREAK \\
  \IOT{\FENCED{P_1\cdots P_n}}{E} & = & \FENCED{\IOT{P_1}{E}\cdots\IOT{P_n}{E}} \\
  \IOT{\TVAR{x}}{E} & = & t & \mathcal{E}(x) = \verb+Term+~t \\
  \IOT{\NVAR{x}}{E} & = & l & \mathcal{E}(x) = \verb+Number+~l \\
  \IOT{\IVAR{x}}{E} & = & l & \mathcal{E}(x) = \verb+String+~l \\
  \IOT{\mathtt{opt}~P}{E} & = & \varepsilon & \mathcal{E}(\NAMES(P)) = \{\mathtt{None}\} \\
  \IOT{\mathtt{opt}~P}{E} & = & \IOT{P}{E'} & \mathcal{E}(\NAMES(P)) = \{\mathtt{Some}~v_1,\dots,\mathtt{Some}~v_n\} \\
  & & & \mathcal{E}'(x)=\left\{
  \begin{array}{@{}ll}
    v, & \mathcal{E}(x) = \mathtt{Some}~v \\
    \mathcal{E}(x), & \mbox{otherwise}
  \end{array}
  \right. \\
  \IOT{\mathtt{list}k~P~l?}{E} & = & \IOT{P}{{E}_1}~{l?}\cdots {l?}~\IOT{P}{{E}_n} &
    \mathcal{E}(\NAMES(P)) = \{[v_{11},\dots,v_{1n}],\dots,[v_{m1},\dots,v_{mn}]\} \\
    & & & n\ge k \\
  & & & \mathcal{E}_i(x) = \left\{
  \begin{array}{@{}ll}
    v_i, & \mathcal{E}(x) = [v_1,\dots,v_n] \\
    \mathcal{E}(x), & \mbox{otherwise}
  \end{array}
  \right. \\
  \IOT{l}{E} & = & l \\

%%     &  |  & (P) & \mbox{(fenced)} \\
%%     &  |  & M & \mbox{(magic)} \\
%%     &  |  & V & \mbox{(variable)} \\
%%     &  |  & l & \mbox{(literal)} \\[2ex]
%%   V & ::= & & \mbox{(\bf variables)} \\
%%     &     & \TVAR{x} & \mbox{(term variable)} \\
%%     &  |  & \NVAR{x} & \mbox{(number variable)} \\
%%     &  |  & \IVAR{x} & \mbox{(name variable)} \\[2ex]
%%   M & ::= & & \mbox{(\bf magic patterns)} \\
%%     &     & \verb+list0+~S~l? & \mbox{(possibly empty list)} \\
%%     &  |  & \verb+list1+~S~l? & \mbox{(non-empty list)} \\
%%     &  |  & \verb+opt+~S & \mbox{(option)} \\[2ex]  
\end{array}
\]
\hrule
\end{table}

\begin{table}
\caption{\label{tab:wfl0} Well-formedness rules for level 1 patterns.\strut}
\hrule
\[
\renewcommand{\arraystretch}{3.5}
\begin{array}[t]{@{}c@{}}
  \inference[\sc layout]
    {P_i :: D_i & \forall i,j, i\ne j => \DOMAIN(D_i) \cap \DOMAIN(D_j) = \emptyset}
    {L_\kappa[P_1,\dots,P_n] :: D_1\oplus\cdots\oplus D_n}
  \\
  \inference[\sc box]
    {P_i :: D_i & \forall i,j, i\ne j => \DOMAIN(D_i) \cap \DOMAIN(D_j) = \emptyset}
    {B_\kappa^{ab}[P_1\cdots P_n] :: D_1\oplus\cdots\oplus D_n}
  \\
  \inference[\sc fenced]
    {P_i :: D_i & \forall i,j, i\ne j => \DOMAIN(D_i) \cap \DOMAIN(D_j) = \emptyset}
    {\FENCED{P_1\cdots P_n} :: D_1\oplus\cdots\oplus D_n}
  \\
  \inference[\sc breakpoint]
    {}
    {\BREAK :: \emptyset}
  \qquad
  \inference[\sc literal]
    {}
    {l}
  \qquad
  \inference[\sc tvar]
    {}
    {\TVAR{x} :: \TVAR{x}}
  \\
  \inference[\sc nvar]
    {}
    {\NVAR{x} :: \NVAR{x}}
  \qquad
  \inference[\sc ivar]
    {}
    {\IVAR{x} :: \IVAR{x}}
  \\
  \inference[\sc list0]
    {P :: D & \forall x\in\DOMAIN(D), D'(x) = D(x)~\mathtt{List}}
    {\mathtt{list0}~P~l? :: D'}
  \\
  \inference[\sc list1]
    {P :: D & \forall x\in\DOMAIN(D), D'(x) = D(x)~\mathtt{List}}
    {\mathtt{list1}~P~l? :: D'}
  \\
  \inference[\sc opt]
    {P :: D & \forall x\in\DOMAIN(D), D'(x) = D(x)~\mathtt{Option}}
    {\mathtt{opt}~P :: D'}
\end{array}
\]
\hrule
\end{table}

  \section{Level 2: abstract syntax}

\end{document}