\documentclass[a4paper]{article}
+\usepackage{manfnt}
+\usepackage{a4wide}
\usepackage{pifont}
\usepackage{semantic}
+\usepackage{stmaryrd,latexsym}
-\newcommand{\MATITA}{\ding{46}\textsf{Matita}}
+\newcommand{\BLOB}{\raisebox{0ex}{\small\manstar}}
+\newcommand{\MATITA}{\ding{46}\textsf{\textbf{Matita}}}
\title{Extensible notation for \MATITA}
\author{Luca Padovani \qquad Stefano Zacchiroli \\
\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{\ADDPARENS}[2]{\llparenthesis#1\rrparenthesis^{#2}}
+\newcommand{\NAMES}{\mathit{names}}
+\newcommand{\DOMAIN}{\mathit{domain}}
+
+\mathlig{~>}{\leadsto}
+\mathlig{|->}{\mapsto}
\begin{document}
\maketitle
& | & \verb+Number+~n & \mbox{(number)} \\
& | & \verb+None+ & \mbox{(optional value)} \\
& | & \verb+Some+~V & \mbox{(optional value)} \\
- & | & [ V^{*} ] & \mbox{(list value)} \\[2ex]
+ & | & [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 patterns}
+\section{Level 1: concrete syntax}
+\begin{table}
+\caption{\label{tab:l1c} Concrete syntax of level 1 patterns.\strut}
+\hrule
\[
\begin{array}{rcll}
- P & ::= & & \mbox{(patterns)} \\
+ P & ::= & & \mbox{(\bf patterns)} \\
& & S^{+} \\[2ex]
S & ::= & & \mbox{(\bf simple patterns)} \\
- & & L_\kappa[S_1,\dots,S_n] & \mbox{(layout)} \\
- & | & B_\kappa^{ab}[P] & \mbox{(box)} \\
+ & & l \\
+ & | & S~\verb+\sub+~S\\
+ & | & S~\verb+\sup+~S\\
+ & | & S~\verb+\below+~S\\
+ & | & S~\verb+\atop+~S\\
+ & | & S~\verb+\over+~S\\
+ & | & S~\verb+\atop+~S\\
+ & | & \verb+\frac+~S~S \\
+ & | & \verb+\sqrt+~S \\
+ & | & \verb+\root+~S~\verb+\of+~S \\
+ & | & \verb+(+~P~\verb+)+ \\
+ & | & \verb+hbox (+~P~\verb+)+ \\
+ & | & \verb+vbox (+~P~\verb+)+ \\
+ & | & \verb+hvbox (+~P~\verb+)+ \\
+ & | & \verb+hovbox (+~P~\verb+)+ \\
+ & | & \verb+break+ \\
+ & | & \verb+list0+~S~[\verb+sep+~l] \\
+ & | & \verb+list1+~S~[\verb+sep+~l] \\
+ & | & \verb+opt+~S \\
+ & | & [\verb+term+]~x \\
+ & | & \verb+number+~x \\
+ & | & \verb+ident+~x \\
+\end{array}
+\]
+\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
+\[
+\begin{array}{@{}ll@{}}
+\begin{array}[t]{rcll}
+ T & ::= & & \mbox{(\bf terms)} \\
+ & & L_\kappa[T_1,\dots,T_n] & \mbox{(layout)} \\
+ & | & B_\kappa^{ab}[T_1\cdots T_n] & \mbox{(box)} \\
+ & | & \BREAK & \mbox{(breakpoint)} \\
+ & | & \FENCED{T_1\cdots T_n} & \mbox{(fenced)} \\
+ & | & l & \mbox{(literal)} \\[2ex]
+ P & ::= & & \mbox{(\bf patterns)} \\
+ & & L_\kappa[P_1,\dots,P_n] & \mbox{(layout)} \\
+ & | & B_\kappa^{ab}[P_1\cdots P_n] & \mbox{(box)} \\
& | & \BREAK & \mbox{(breakpoint)} \\
- & | & (P) & \mbox{(fenced)} \\
+ & | & \FENCED{P_1\cdots P_n} & \mbox{(fenced)} \\
& | & M & \mbox{(magic)} \\
& | & V & \mbox{(variable)} \\
- & | & l & \mbox{(literal)} \\[2ex]
+ & | & l & \mbox{(literal)} \\
+\end{array} &
+\begin{array}[t]{rcll}
V & ::= & & \mbox{(\bf variables)} \\
& & \TVAR{x} & \mbox{(term variable)} \\
& | & \NVAR{x} & \mbox{(number variable)} \\
& | & \IVAR{x} & \mbox{(name variable)} \\[2ex]
M & ::= & & \mbox{(\bf magic patterns)} \\
- & & \verb+list0+~S~l? & \mbox{(possibly empty list)} \\
- & | & \verb+list1+~S~l? & \mbox{(non-empty list)} \\
- & | & \verb+opt+~S & \mbox{(option)} \\[2ex]
+ & & \verb+list0+~P~l? & \mbox{(possibly empty list)} \\
+ & | & \verb+list1+~P~l? & \mbox{(non-empty list)} \\
+ & | & \verb+opt+~P & \mbox{(option)} \\[2ex]
+\end{array}
\end{array}
\]
+\hrule
+\end{table}
-% IOT = Instantiate Two to One
-\newcommand{\IOT}[2]{|[#1|]_{\mathcal{#2}}}
-\newcommand{\NAMES}{\mathit{names}}
+\[
+\IOT{\cdot}{{}} : P -> \mathit{Env} -> T
+\]
+\begin{table}
+\caption{\label{tab:il1} Instantiation of level 1 patterns.\strut}
+\hrule
\[
\begin{array}{rcll}
- \IOT{S_1\cdots S_n}{E} & = & \IOT{S_1}{E}\cdots\IOT{S_n}{E} \\
- \IOT{L_\kappa[S_1,\dots,S_n]}{E} & = & L_\kappa[\IOT{S_1}{E},\dots,\IOT{S_n}{E} ] \\
- \IOT{B_\kappa^{ab}[P]}{E} & = & B_\kappa^{ab}[\IOT{P}{E}] \\
+ \IOT{L_\kappa[P_1,\dots,P_n]}{E} & = & L_\kappa[\IOT{(P_1)}{E},\dots,\IOT{(P_n)}{E} ] \\
+ \IOT{B_\kappa^{ab}[P_1\cdots P_n]}{E} & = & B_\kappa^{ab}[\IOT{P_1}{E}\cdots\IOT{P_n}{E}] \\
\IOT{\BREAK}{E} & = & \BREAK \\
- \IOT{(P)}{E} & = & (\IOT{P}{E}) \\
+ \IOT{(P)}{E} & = & \IOT{P}{E} \\
+ \IOT{(P_1\cdots P_n)}{E} & = & B_H^{00}[\IOT{P_1}{E}\cdots\IOT{P_n}{E}] \\
\IOT{\TVAR{x}}{E} & = & t & \mathcal{E}(x) = \verb+Term+~t \\
\IOT{\NVAR{x}}{E} & = & l & \mathcal{E}(x) = \verb+Number+~l \\
\IOT{\IVAR{x}}{E} & = & l & \mathcal{E}(x) = \verb+String+~l \\
- \IOT{\mathtt{opt}~S}{E} & = & \varepsilon & \forall x \in \NAMES(S), \mathcal{E}(x) = \mathtt{None} \\
- \IOT{\mathtt{opt}~S}{E} & = & \varepsilon & \forall x \in \NAMES(S), \mathcal{E}(x) = \mathtt{None} \\
+ \IOT{\mathtt{opt}~P}{E} & = & \varepsilon & \mathcal{E}(\NAMES(P)) = \{\mathtt{None}\} \\
+ \IOT{\mathtt{opt}~P}{E} & = & \IOT{P}{E'} & \mathcal{E}(\NAMES(P)) = \{\mathtt{Some}~v_1,\dots,\mathtt{Some}~v_n\} \\
+ & & & \mathcal{E}'(x)=\left\{
+ \begin{array}{@{}ll}
+ v, & \mathcal{E}(x) = \mathtt{Some}~v \\
+ \mathcal{E}(x), & \mbox{otherwise}
+ \end{array}
+ \right. \\
+ \IOT{\mathtt{list}k~P~l?}{E} & = & \IOT{P}{{E}_1}~{l?}\cdots {l?}~\IOT{P}{{E}_n} &
+ \mathcal{E}(\NAMES(P)) = \{[v_{11},\dots,v_{1n}],\dots,[v_{m1},\dots,v_{mn}]\} \\
+ & & & n\ge k \\
+ & & & \mathcal{E}_i(x) = \left\{
+ \begin{array}{@{}ll}
+ v_i, & \mathcal{E}(x) = [v_1,\dots,v_n] \\
+ \mathcal{E}(x), & \mbox{otherwise}
+ \end{array}
+ \right. \\
+ \IOT{l}{E} & = & l \\
%% & | & (P) & \mbox{(fenced)} \\
%% & | & M & \mbox{(magic)} \\
%% & | & \verb+opt+~S & \mbox{(option)} \\[2ex]
\end{array}
\]
+\hrule
+\end{table}
+
+\begin{table}
+\caption{\label{tab:wfl0} Well-formedness rules for level 1 patterns.\strut}
+\hrule
+\[
+\renewcommand{\arraystretch}{3.5}
+\begin{array}[t]{@{}c@{}}
+ \inference[\sc layout]
+ {P_i :: D_i & \forall i,j, i\ne j => \DOMAIN(D_i) \cap \DOMAIN(D_j) = \emptyset}
+ {L_\kappa[P_1,\dots,P_n] :: D_1\oplus\cdots\oplus D_n}
+ \\
+ \inference[\sc box]
+ {P_i :: D_i & \forall i,j, i\ne j => \DOMAIN(D_i) \cap \DOMAIN(D_j) = \emptyset}
+ {B_\kappa^{ab}[P_1\cdots P_n] :: D_1\oplus\cdots\oplus D_n}
+ \\
+ \inference[\sc fenced]
+ {P_i :: D_i & \forall i,j, i\ne j => \DOMAIN(D_i) \cap \DOMAIN(D_j) = \emptyset}
+ {\FENCED{P_1\cdots P_n} :: D_1\oplus\cdots\oplus D_n}
+ \\
+ \inference[\sc breakpoint]
+ {}
+ {\BREAK :: \emptyset}
+ \qquad
+ \inference[\sc literal]
+ {}
+ {l :: \emptyset}
+ \qquad
+ \inference[\sc tvar]
+ {}
+ {\TVAR{x} :: \TVAR{x}}
+ \\
+ \inference[\sc nvar]
+ {}
+ {\NVAR{x} :: \NVAR{x}}
+ \qquad
+ \inference[\sc ivar]
+ {}
+ {\IVAR{x} :: \IVAR{x}}
+ \\
+ \inference[\sc list0]
+ {P :: D & \forall x\in\DOMAIN(D), D'(x) = D(x)~\mathtt{List}}
+ {\mathtt{list0}~P~l? :: D'}
+ \\
+ \inference[\sc list1]
+ {P :: D & \forall x\in\DOMAIN(D), D'(x) = D(x)~\mathtt{List}}
+ {\mathtt{list1}~P~l? :: D'}
+ \\
+ \inference[\sc opt]
+ {P :: D & \forall x\in\DOMAIN(D), D'(x) = D(x)~\mathtt{Option}}
+ {\mathtt{opt}~P :: D'}
+\end{array}
+\]
+\hrule
+\end{table}
+
+\newcommand{\ATTRS}[1]{\langle#1\rangle}
+\newcommand{\ANNPOS}[2]{\mathit{pos}(#1)_{#2}}
+
+\begin{table}
+\caption{\label{tab:addparens} Where are parentheses added? Look here.\strut}
+\hrule
+\[
+\begin{array}{rcll}
+ \ADDPARENS{l}{n} & = & l \\
+ \ADDPARENS{\BREAK}{n} & = & \BREAK \\
+ \ADDPARENS{\ATTRS{\mathit{prec}=m}T}{n} & = & \ADDPARENS{T}{m} & n < m \\
+ \ADDPARENS{\ATTRS{\mathit{prec}=m}T}{n} & = & \FENCED{\ADDPARENS{T}{\bot}} & n > m \\
+ \ADDPARENS{\ATTRS{\mathit{prec}=n,\mathit{assoc}=L,\mathit{pos}=R}T}{n} & = & \FENCED{\ADDPARENS{T}{\bot}} \\
+ \ADDPARENS{\ATTRS{\mathit{prec}=n,\mathit{assoc}=R,\mathit{pos}=L}T}{n} & = & \FENCED{\ADDPARENS{T}{\bot}} \\
+ \ADDPARENS{\ATTRS{\cdots}T}{n} & = & \ADDPARENS{T}{n} \\
+ \ADDPARENS{L_\kappa[T_1,\dots,\underline{T_k},\dots,T_m]}{n} & = & L_\kappa[\ADDPARENS{T_1}{n},\dots,\ADDPARENS{T_k}{\bot},\dots,\ADDPARENS{T_m}{n}] \\
+ \ADDPARENS{B_\kappa^{ab}[T_1,\dots,T_m]}{n} & = & B_\kappa^{ab}[\ADDPARENS{T_1}{n},\dots,\ADDPARENS{T_m}{n}]
+\end{array}
+\]
+\hrule
+\end{table}
+
+\begin{table}
+\caption{\label{tab:annpos} Annotation of level 1 meta variable with position information.\strut}
+\hrule
+\[
+\begin{array}{rcll}
+ \ANNPOS{l}{p,q} & = & l \\
+ \ANNPOS{\BREAK}{p,q} & = & \BREAK \\
+ \ANNPOS{x}{1,0} & = & \ATTRS{\mathit{pos}=L}{x} \\
+ \ANNPOS{x}{0,1} & = & \ATTRS{\mathit{pos}=R}{x} \\
+ \ANNPOS{x}{p,q} & = & \ATTRS{\mathit{pos}=I}{x} \\
+ \ANNPOS{B_\kappa^{ab}[P]}{p,q} & = & B_\kappa^{ab}[\ANNPOS{P}{p,q}] \\
+ \ANNPOS{B_\kappa^{ab}[\{\BREAK\} P_1\cdots P_n\{\BREAK\}]}{p,q} & = & B_\kappa^{ab}[\begin{array}[t]{@{}l}
+ \{\BREAK\} \ANNPOS{P_1}{p,0} \\
+ \ANNPOS{P_2}{0,0}\cdots\ANNPOS{P_{n-1}}{0,0} \\
+ \ANNPOS{P_n}{0,q}\{\BREAK\}]
+ \end{array}
+
+%% & & L_\kappa[P_1,\dots,P_n] & \mbox{(layout)} \\
+%% & | & \BREAK & \mbox{(breakpoint)} \\
+%% & | & \FENCED{P_1\cdots P_n} & \mbox{(fenced)} \\
+%% V & ::= & & \mbox{(\bf variables)} \\
+%% & & \TVAR{x} & \mbox{(term variable)} \\
+%% & | & \NVAR{x} & \mbox{(number variable)} \\
+%% & | & \IVAR{x} & \mbox{(name variable)} \\[2ex]
+%% M & ::= & & \mbox{(\bf magic patterns)} \\
+%% & & \verb+list0+~P~l? & \mbox{(possibly empty list)} \\
+%% & | & \verb+list1+~P~l? & \mbox{(non-empty list)} \\
+%% & | & \verb+opt+~P & \mbox{(option)} \\[2ex]
+\end{array}
+\]
+\hrule
+\end{table}
+
+\section{Level 2: abstract syntax}
+
+\newcommand{\NT}[1]{\langle\mathit{#1}\rangle}
+
+\begin{table}
+\[
+\begin{array}{@{}rcll@{}}
+ \NT{term} & ::= & & \mbox{\bf terms} \\
+ & & x & \mbox{(identifier)} \\
+ & | & n & \mbox{(number)} \\
+ & | & \mathrm{URI} & \mbox{(URI)} \\
+ & | & \verb+?+ & \mbox{(implicit)} \\
+ & | & \verb+%+ & \mbox{(placeholder)} \\
+ & | & \verb+?+n~[\verb+[+~\{\NT{subst}\}~\verb+]+] & \mbox{(meta)} \\
+ & | & \verb+let+~\NT{ptname}~\verb+\def+~\NT{term}~\verb+in+~\NT{term} \\
+ & | & \verb+let+~\NT{kind}~\NT{defs}~\verb+in+~\NT{term} \\
+ & | & \NT{binder}~\{\NT{ptnames}\}^{+}~\verb+.+~\NT{term} \\
+ & | & \NT{term}~\NT{term} & \mbox{(application)} \\
+ & | & \verb+Prop+ \mid \verb+Set+ \mid \verb+Type+ \mid \verb+CProp+ & \mbox{(sort)} \\
+ & | & [\verb+[+~\NT{term}~\verb+]+]~\verb+match+~\NT{term}~\verb+with [+~[\NT{rule}~\{\verb+|+~\NT{rule}\}]~\verb+]+ & \mbox{(pattern match)} \\
+ & | & \verb+(+~\NT{term}~\verb+:+~\NT{term}~\verb+)+ & \mbox{(cast)} \\
+ & | & \verb+(+~\NT{term}~\verb+)+ \\
+ & | & \BLOB(\NT{meta},\dots,\NT{meta}) & \mbox{(meta blob)} \\
+ \NT{defs} & ::= & & \mbox{\bf mutual definitions} \\
+ & & \NT{fun}~\{\verb+and+~\NT{fun}\} \\
+ \NT{fun} & ::= & & \mbox{\bf functions} \\
+ & & \NT{arg}~\{\NT{ptnames}\}^{+}~[\verb+on+~x]~\verb+\def+~\NT{term} \\
+ \NT{binder} & ::= & & \mbox{\bf binders} \\
+ & & \verb+\Pi+ \mid \verb+\exists+ \mid \verb+\forall+ \mid \verb+\lambda+ \\
+ \NT{arg} & ::= & & \mbox{\bf single argument} \\
+ & & \verb+_+ \mid x \mid \BLOB(\NT{meta},\dots,\NT{meta}) \\
+ \NT{ptname} & ::= & & \mbox{\bf possibly typed name} \\
+ & & \NT{arg} \\
+ & | & \verb+(+~\NT{arg}~\verb+:+~\NT{term}~\verb+)+ \\
+ \NT{ptnames} & ::= & & \mbox{\bf bound variables} \\
+ & & \NT{arg} \\
+ & | & \verb+(+~\NT{arg}~\{\verb+,+~\NT{arg}\}~[\verb+:+~\NT{term}]~\verb+)+ \\
+ \NT{kind} & ::= & & \mbox{\bf induction kind} \\
+ & & \verb+rec+ \mid \verb+corec+ \\
+ \NT{rule} & ::= & & \mbox{\bf rules} \\
+ & & x~\{\NT{ptname}\}~\verb+\Rightarrow+~\NT{term} \\[10ex]
+
+ \NT{meta} & ::= & & \mbox{\bf meta} \\
+ & & \BLOB(\NT{term},\dots,\NT{term}) & (term blob) \\
+ & | & [\verb+term+]~x \\
+ & | & \verb+number+~x \\
+ & | & \verb+ident+~x \\
+ & | & \verb+fresh+~x \\
+ & | & \verb+anonymous+ \\
+ & | & \verb+fold+~[\verb+left+\mid\verb+right+]~\NT{meta}~\verb+rec+~x~\NT{meta} \\
+ & | & \verb+default+~\NT{meta}~\NT{meta} \\
+ & | & \verb+if+~\NT{meta}~\verb+then+~\NT{meta}~\verb+else+~\NT{meta} \\
+ & | & \verb+fail+
+\end{array}
+\]
+\end{table}
+
+\begin{table}
+\caption{\label{tab:wfl2} Well-formedness rules for level 2 patterns.\strut}
+\hrule
+\[
+\renewcommand{\arraystretch}{3.5}
+\begin{array}{@{}c@{}}
+ \inference[\sc Constr]
+ {P_i :: D_i & \forall i,j, i\ne j => \DOMAIN(D_i) \cap \DOMAIN(D_j) = \emptyset}
+ {\BLOB[P_1,\dots,P_n] :: D_i \oplus \cdots \oplus D_j} \\
+ \inference[\sc TermVar]
+ {}
+ {[\mathtt{term}]~x :: x : \mathtt{Term}}
+ \quad
+ \inference[\sc NumVar]
+ {}
+ {\mathtt{number}~x :: x : \mathtt{Number}}
+ \\
+ \inference[\sc IdentVar]
+ {}
+ {\mathtt{ident}~x :: x : \mathtt{String}}
+ \quad
+ \inference[\sc FreshVar]
+ {}
+ {\mathtt{fresh}~x :: x : \mathtt{String}}
+ \\
+ \inference[\sc Success]
+ {}
+ {\mathtt{anonymous} :: \emptyset}
+ \\
+ \inference[\sc Fold]
+ {P_1 :: D_1 & P_2 :: D_2 \oplus (x : \mathtt{Term}) & \DOMAIN(D_2)\ne\emptyset & \DOMAIN(D_1)\cap\DOMAIN(D_2)=\emptyset}
+ {\mathtt{fold}~P_1~\mathtt{rec}~x~P_2 :: D_1 \oplus D_2~\mathtt{List}}
+ \\
+ \inference[\sc Default]
+ {P_1 :: D \oplus D_1 & P_2 :: D & \DOMAIN(D_1) \ne \emptyset & \DOMAIN(D) \cap \DOMAIN(D_1) = \emptyset}
+ {\mathtt{default}~P_1~P_2 :: D \oplus D_1~\mathtt{Option}}
+ \\
+ \inference[\sc If]
+ {P_1 :: \emptyset & P_2 :: D & P_3 :: D }
+ {\mathtt{if}~P_1~\mathtt{then}~P_2~\mathtt{else}~P_3 :: D}
+ \qquad
+ \inference[\sc Fail]
+ {}
+ {\mathtt{fail} : \emptyset}
+%% & | & \verb+if+~\NT{meta}~\verb+then+~\NT{meta}~\verb+else+~\NT{meta} \\
+%% & | & \verb+fail+
+\end{array}
+\]
+\hrule
+\end{table}
+
+\begin{table}
+\caption{\label{tab:l2match} Pattern matching of level 2 terms.\strut}
+\hrule
+\[
+\renewcommand{\arraystretch}{3.5}
+\begin{array}{@{}c@{}}
+ \inference[\sc Constr]
+ {t_i \in P_i ~> \mathcal E_i & i\ne j => \DOMAIN(\mathcal E_i)\cap\DOMAIN(\mathcal E_j)=\emptyset}
+ {C[t_1,\dots,t_n] \in C[P_1,\dots,P_n] ~> \mathcal E_1 \oplus \cdots \oplus \mathcal E_n}
+ \\
+ \inference[\sc TermVar]
+ {}
+ {t \in [\mathtt{term}]~x ~> [x |-> \mathtt{Term}~t]}
+ \quad
+ \inference[\sc NumVar]
+ {}
+ {n \in \mathtt{number}~x ~> [x |-> \mathtt{Number}~n]}
+ \\
+ \inference[\sc IdentVar]
+ {}
+ {x \in \mathtt{ident}~x ~> [x |-> \mathtt{String}~x]}
+ \quad
+ \inference[\sc FreshVar]
+ {}
+ {x \in \mathtt{fresh}~x ~> [x |-> \mathtt{String}~x]}
+ \\
+ \inference[\sc Success]
+ {}
+ {t \in \mathtt{anonymous} ~> \emptyset}
+ \\
+ \inference[\sc DefaultT]
+ {t \in P_1 ~> \mathcal E}
+ {t \in \mathtt{default}~P_1~P_2 ~> \mathcal E'}
+ \quad
+ \mathcal E'(x) = \left\{
+ \renewcommand{\arraystretch}{1}
+ \begin{array}{ll}
+ \mathtt{Some}~\mathcal{E}(x) & x \in \NAMES(P_1) \setminus \NAMES(P_2) \\
+ \mathcal{E}(x) & \mbox{otherwise}
+ \end{array}
+ \right.
+ \\
+ \inference[\sc DefaultF]
+ {t \not\in P_1 & t \in P_2 ~> \mathcal E}
+ {t \in \mathtt{default}~P_1~P_2 ~> \mathcal E'}
+ \quad
+ \mathcal E'(x) = \left\{
+ \renewcommand{\arraystretch}{1}
+ \begin{array}{ll}
+ \mathtt{None} & x \in \NAMES(P_1) \setminus \NAMES(P_2) \\
+ \mathcal{E}(x) & \mbox{otherwise}
+ \end{array}
+ \right.
+ \\
+ \inference[\sc IfT]
+ {t \in P_1 ~> \mathcal E' & t \in P_2 ~> \mathcal E}
+ {t \in \mathtt{if}~P_1~\mathtt{then}~P_2~\mathtt{else}~P_3 ~> \mathcal E}
+ \quad
+ \inference[\sc IfF]
+ {t \not\in P_1 & t \in P_3 ~> \mathcal E}
+ {t \in \mathtt{if}~P_1~\mathtt{then}~P_2~\mathtt{else}~P_3 ~> \mathcal E}
+ \\
+ \inference[\sc FoldR]
+ {t \in P_2 ~> \mathcal E & \mathcal{E}(x) \in \mathtt{fold}~P_1~\mathtt{rec}~x~P_2 ~> \mathcal E'}
+ {t \in \mathtt{fold}~P_1~\mathtt{rec}~x~P_2 ~> \mathcal E''}
+ \quad
+ \mathcal E''(y) = \left\{
+ \renewcommand{\arraystretch}{1}
+ \begin{array}{ll}
+ \mathcal{E}(y)::\mathcal{E}'(y) & y \in \NAMES(P_2) \setminus \{x\} \\
+ \mathcal{E}'(y) & \mbox{otherwise}
+ \end{array}
+ \right.
+ \\
+ \inference[\sc FoldB]
+ {t \not\in P_2 & t \in P_1 ~> \mathcal E}
+ {t \in \mathtt{fold}~P_1~\mathtt{rec}~x~P_2 ~> \mathcal E'}
+ \quad
+ \mathcal E'(y) = \left\{
+ \renewcommand{\arraystretch}{1}
+ \begin{array}{ll}
+ [] & y \in \NAMES(P_2) \setminus \{x\} \\
+ \mathcal{E}(y) & \mbox{otherwise}
+ \end{array}
+ \right.
+\end{array}
+\]
+\hrule
+\end{table}
+
+\section{Type checking}
+
+\newcommand{\TN}{\mathit{tn}}
+
+Assume that we have two corresponding patterns $P_1$ (level 1) and
+$P_2$ (level 2) and that we have to check whether they are
+``correct''. First we define the notion of \emph{top-level names} of
+$P_1$ and $P_2$, as follows:
+\[
+\begin{array}{rcl}
+ \TN(C_1[P'_1,\dots,P'_2]) & = & \TN(P'_1) \cup \cdots \cup \TN(P'_2) \\
+ \TN(\TVAR{x}) & = & \{x\} \\
+ \TN(\NVAR{x}) & = & \{x\} \\
+ \TN(\IVAR{x}) & = & \{x\} \\
+ \TN(\mathtt{list0}~P'~l?) & = & \emptyset \\
+ \TN(\mathtt{list1}~P'~l?) & = & \emptyset \\
+ \TN(\mathtt{opt}~P') & = & \emptyset \\[3ex]
+
+ \TN(\BLOB(P''_1,\dots,P''_2)) & = & \TN(P''_1) \cup \cdots \cup \TN(P''_2) \\
+ \TN(\mathtt{term}~x) & = & \{x\} \\
+ \TN(\mathtt{number}~x) & = & \{x\} \\
+ \TN(\mathtt{ident}~x) & = & \{x\} \\
+ \TN(\mathtt{fresh}~x) & = & \{x\} \\
+ \TN(\mathtt{anonymous}) & = & \emptyset \\
+ \TN(\mathtt{fold}~P''_1~\mathtt{rec}~x~P''_2) & = & \TN(P''_1) \\
+ \TN(\mathtt{default}~P''_1~P''_2) & = & \TN(P''_1) \cap \TN(P''_2) \\
+ \TN(\mathtt{if}~P''_1~\mathtt{then}~P''_2~\mathtt{else}~P''_3) & = & \TN(P''_2) \\
+ \TN(\mathtt{fail}) & = & \emptyset
+\end{array}
+\]
+
+We say that a \emph{bidirectional transformation}
+\[
+ P_1 <=> P_2
+\]
+is well-formed if:
+\begin{itemize}
+ \item the two patterns are well-formed in the same context $D$, that
+ is $P_1 :: D$ and $P_2 :: D$;
+ \item for any direct sub-pattern $\mathtt{opt}~P'_1$ of $P_1$ such
+ that $\mathtt{opt}~P'_1 :: X$ there is a sub-pattern
+ $\mathtt{default}~P'_2~P''_2$ of $P_2$ such that
+ $\mathtt{default}~P'_2~P''_2 :: X \oplus Y$ for some context $Y$;
+ \item for any direct sub-pattern $\mathtt{list}~P'_1~l?$ of $P_1$
+ such that $\mathtt{list}~P'_1~l? :: X$ there is a sub-pattern
+ $\mathtt{fold}~P'_2~\mathtt{rec}~x~P''_2$ of $P_2$ such that
+ $\mathtt{fold}~P'_2~\mathtt{rec}~x~P''_2 :: X \oplus Y$ for some
+ context $Y$.
+\end{itemize}
+
+A \emph{left-to-right transformation}
+\[
+ P_1 => P_2
+\]
+is well-formed if $P_2$ does not contain \texttt{if}, \texttt{fail},
+or \texttt{anonymous} meta patterns.
+
+Note that the transformations are in a sense asymmetric. Moving from
+the concrete syntax (level 1) to the abstract syntax (level 2) we
+forget about syntactic details. Moving from the abstract syntax to the
+concrete syntax we may want to forget about redundant structure
+(types).
- \section{Level 2: abstract syntax}
+Relationship with grammatical frameworks?
\end{document}
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