--- /dev/null
+
+\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}
+
+\[
+\ITO{\cdot}{{}} : P -> \mathit{Env} -> T
+\]
+
+\begin{table}
+\caption{\label{tab:il1f2} Instantiation of level 1 patterns from level 2.\strut}
+\hrule
+\[
+\begin{array}{rcll}
+ \ITO{L_\kappa[P_1,\dots,P_n]}{E} & = & L_\kappa[\ITO{(P_1)}{E},\dots,\ITO{(P_n)}{E} ] \\
+ \ITO{B_\kappa^{ab}[P_1\cdots P_n]}{E} & = & B_\kappa^{ab}[\ITO{P_1}{E}\cdots\ITO{P_n}{E}] \\
+ \ITO{\BREAK}{E} & = & \BREAK \\
+ \ITO{(P)}{E} & = & \ITO{P}{E} \\
+ \ITO{(P_1\cdots P_n)}{E} & = & B_H^{00}[\ITO{P_1}{E}\cdots\ITO{P_n}{E}] \\
+ \ITO{\TVAR{x}}{E} & = & t & \mathcal{E}(x) = \verb+Term+~t \\
+ \ITO{\NVAR{x}}{E} & = & l & \mathcal{E}(x) = \verb+Number+~l \\
+ \ITO{\IVAR{x}}{E} & = & l & \mathcal{E}(x) = \verb+String+~l \\
+ \ITO{\mathtt{opt}~P}{E} & = & \varepsilon & \mathcal{E}(\NAMES(P)) = \{\mathtt{None}\} \\
+ \ITO{\mathtt{opt}~P}{E} & = & \ITO{P}{E'} & \mathcal{E}(\NAMES(P)) = \{\mathtt{Some}~v_1,\dots,\mathtt{Some}~v_n\} \\
+ & & & \mathcal{E}'(x)=\left\{
+ \begin{array}{@{}ll}
+ v, & \mathcal{E}(x) = \mathtt{Some}~v \\
+ \mathcal{E}(x), & \mbox{otherwise}
+ \end{array}
+ \right. \\
+ \ITO{\mathtt{list}k~P~l?}{E} & = & \ITO{P}{{E}_1}~{l?}\cdots {l?}~\ITO{P}{{E}_n} &
+ \mathcal{E}(\NAMES(P)) = \{[v_{11},\dots,v_{1n}],\dots,[v_{m1},\dots,v_{mn}]\} \\
+ & & & n\ge k \\
+ & & & \mathcal{E}_i(x) = \left\{
+ \begin{array}{@{}ll}
+ v_i, & \mathcal{E}(x) = [v_1,\dots,v_n] \\
+ \mathcal{E}(x), & \mbox{otherwise}
+ \end{array}
+ \right. \\
+ \ITO{l}{E} & = & l \\
+
+%% & | & (P) & \mbox{(fenced)} \\
+%% & | & M & \mbox{(magic)} \\
+%% & | & V & \mbox{(variable)} \\
+%% & | & l & \mbox{(literal)} \\[2ex]
+%% V & ::= & & \mbox{(\bf variables)} \\
+%% & & \TVAR{x} & \mbox{(term variable)} \\
+%% & | & \NVAR{x} & \mbox{(number variable)} \\
+%% & | & \IVAR{x} & \mbox{(name variable)} \\[2ex]
+%% M & ::= & & \mbox{(\bf magic patterns)} \\
+%% & & \verb+list0+~S~l? & \mbox{(possibly empty list)} \\
+%% & | & \verb+list1+~S~l? & \mbox{(non-empty list)} \\
+%% & | & \verb+opt+~S & \mbox{(option)} \\[2ex]
+\end{array}
+\]
+\hrule
+\end{table}
+
+\begin{table}
+\caption{\label{tab:wfl0} Well-formedness rules for level 1 patterns.\strut}
+\hrule
+\[
+\renewcommand{\arraystretch}{3.5}
+\begin{array}[t]{@{}c@{}}
+ \inference[\sc layout]
+ {P_i :: D_i & \forall i,j, i\ne j => \DOMAIN(D_i) \cap \DOMAIN(D_j) = \emptyset}
+ {L_\kappa[P_1,\dots,P_n] :: D_1\oplus\cdots\oplus D_n}
+ \\
+ \inference[\sc box]
+ {P_i :: D_i & \forall i,j, i\ne j => \DOMAIN(D_i) \cap \DOMAIN(D_j) = \emptyset}
+ {B_\kappa^{ab}[P_1\cdots P_n] :: D_1\oplus\cdots\oplus D_n}
+ \\
+ \inference[\sc fenced]
+ {P_i :: D_i & \forall i,j, i\ne j => \DOMAIN(D_i) \cap \DOMAIN(D_j) = \emptyset}
+ {\FENCED{P_1\cdots P_n} :: D_1\oplus\cdots\oplus D_n}
+ \\
+ \inference[\sc breakpoint]
+ {}
+ {\BREAK :: \emptyset}
+ \qquad
+ \inference[\sc literal]
+ {}
+ {l :: \emptyset}
+ \qquad
+ \inference[\sc tvar]
+ {}
+ {\TVAR{x} :: \TVAR{x}}
+ \\
+ \inference[\sc nvar]
+ {}
+ {\NVAR{x} :: \NVAR{x}}
+ \qquad
+ \inference[\sc ivar]
+ {}
+ {\IVAR{x} :: \IVAR{x}}
+ \\
+ \inference[\sc list0]
+ {P :: D & \forall x\in\DOMAIN(D), D'(x) = D(x)~\mathtt{List}}
+ {\mathtt{list0}~P~l? :: D'}
+ \\
+ \inference[\sc list1]
+ {P :: D & \forall x\in\DOMAIN(D), D'(x) = D(x)~\mathtt{List}}
+ {\mathtt{list1}~P~l? :: D'}
+ \\
+ \inference[\sc opt]
+ {P :: D & \forall x\in\DOMAIN(D), D'(x) = D(x)~\mathtt{Option}}
+ {\mathtt{opt}~P :: D'}
+\end{array}
+\]
+\hrule
+\end{table}
+
+\newcommand{\ATTRS}[1]{\langle#1\rangle}
+\newcommand{\ANNPOS}[2]{\mathit{pos}(#1)_{#2}}
+
+\begin{table}
+\caption{\label{tab:addparens} Can't read the AST and need parentheses? Here you go!.\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}
+\caption{\label{tab:synl2} Concrete syntax of level 2 patterns.\strut}
+\hrule
+\[
+\begin{array}{@{}rcll@{}}
+ \NT{term} & ::= & & \mbox{\bf terms} \\
+ & & x & \mbox{(identifier)} \\
+ & | & n & \mbox{(number)} \\
+ & | & s & \mbox{(symbol)} \\
+ & | & \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}) & \mbox{(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}
+\]
+\hrule
+\end{table}
+
+\begin{table}
+\caption{\label{tab:wfl2} Well-formedness rules for level 2 patterns.\strut}
+\hrule
+\[
+\renewcommand{\arraystretch}{3.5}
+\begin{array}{@{}c@{}}
+ \inference[\sc Constr]
+ {P_i :: D_i}
+ {\BLOB[P_1,\dots,P_n] :: D_i \oplus \cdots \oplus D_j} \\
+ \inference[\sc TermVar]
+ {}
+ {\mathtt{term}~x :: x : \mathtt{Term}}
+ \quad
+ \inference[\sc NumVar]
+ {}
+ {\mathtt{number}~x :: x : \mathtt{Number}}
+ \\
+ \inference[\sc IdentVar]
+ {}
+ {\mathtt{ident}~x :: x : \mathtt{String}}
+ \quad
+ \inference[\sc FreshVar]
+ {}
+ {\mathtt{fresh}~x :: x : \mathtt{String}}
+ \\
+ \inference[\sc Success]
+ {}
+ {\mathtt{anonymous} :: \emptyset}
+ \\
+ \inference[\sc Fold]
+ {P_1 :: D_1 & P_2 :: D_2 \oplus (x : \mathtt{Term}) & \DOMAIN(D_2)\ne\emptyset & \DOMAIN(D_1)\cap\DOMAIN(D_2)=\emptyset}
+ {\mathtt{fold}~P_1~\mathtt{rec}~x~P_2 :: D_1 \oplus D_2~\mathtt{List}}
+ \\
+ \inference[\sc Default]
+ {P_1 :: D \oplus D_1 & P_2 :: D & \DOMAIN(D_1) \ne \emptyset & \DOMAIN(D) \cap \DOMAIN(D_1) = \emptyset}
+ {\mathtt{default}~P_1~P_2 :: D \oplus D_1~\mathtt{Option}}
+ \\
+ \inference[\sc If]
+ {P_1 :: \emptyset & P_2 :: D & P_3 :: D }
+ {\mathtt{if}~P_1~\mathtt{then}~P_2~\mathtt{else}~P_3 :: D}
+ \qquad
+ \inference[\sc Fail]
+ {}
+ {\mathtt{fail} :: \emptyset}
+%% & | & \verb+if+~\NT{meta}~\verb+then+~\NT{meta}~\verb+else+~\NT{meta} \\
+%% & | & \verb+fail+
+\end{array}
+\]
+\hrule
+\end{table}
+
+\begin{table}
+ \caption{\label{tab:il2f1} Instantiation of level 2 patterns from level 1.
+ \strut}
+\hrule
+\[
+\begin{array}{rcll}
+
+\IOT{C[t_1,\dots,t_n]}{\mathcal{E}} & =
+& C[\IOT{t_1}{\mathcal{E}},\dots,\IOT{t_n}{\mathcal{E}}] \\
+
+\IOT{\mathtt{term}~x}{\mathcal{E}} & = & t & \mathcal{E}(x) = \mathtt{Term}~t \\
+
+\IOT{\mathtt{number}~x}{\mathcal{E}} & =
+& n & \mathcal{E}(x) = \mathtt{Number}~n \\
+
+\IOT{\mathtt{ident}~x}{\mathcal{E}} & =
+& y & \mathcal{E}(x) = \mathtt{String}~y \\
+
+\IOT{\mathtt{fresh}~x}{\mathcal{E}} & = & y & \mathcal{E}(x) = \mathtt{String}~y \\
+
+\IOT{\mathtt{default}~P_1~P_2}{\mathcal{E}} & =
+& \IOT{P_1}{\UPDATE{\mathcal{E}}{x_i|->v_i}}
+& \mathcal{E}(x_i)=\mathtt{Some}~v_i \\
+& & & \NAMES(P_1)\setminus\NAMES(P_2)=\{x_1,\dots,x_n\} \\
+
+\IOT{\mathtt{default}~P_1~P_2}{\mathcal{E}} & =
+& \IOT{P_2}{\UPDATE{\mathcal{E}}{x_i|->\bot}}
+& \mathcal{E}(x_i)=\mathtt{None} \\
+& & & \NAMES(P_1)\setminus\NAMES(P_2)=\{x_1,\dots,x_n\} \\
+
+\IOT{\mathtt{fold}~\mathtt{right}~P_1~\mathtt{rec}~x~P_2}{\mathcal{E}}
+& =
+& \IOT{P_1}{\mathcal{E}'}
+& \mathcal{E}(\NAMES(P_2)\setminus\{x\}) = \{[],\dots,[]\} \\
+& & \multicolumn{2}{l}{\mathcal{E}'=\UPDATE{\mathcal{E}}{\NAMES(P_2)\setminus\{x\}|->\bot}}
+\\
+
+\IOT{\mathtt{fold}~\mathtt{right}~P_1~\mathtt{rec}~x~P_2}{\mathcal{E}}
+& =
+& \IOT{P_2}{\mathcal{E}'}
+& \mathcal{E}(y_i) = [v_{i1},\dots,v_{in}] \\
+& & & \NAMES(P_2)\setminus\{x\}=\{y_1,\dots,y_m\} \\
+& & \multicolumn{2}{l}{\mathcal{E}'(y) =
+ \left\{
+ \begin{array}{@{}ll}
+ \IOT{\mathtt{fold}~\mathtt{right}~P_1~\mathtt{rec}~x~P_e}{\mathcal{E}''}
+ & y=x \\
+ v_{i1} & y=y_i \\
+ \mathcal{E}(y) & \mbox{otherwise} \\
+ \end{array}
+ \right.} \\
+& & \multicolumn{2}{l}{\mathcal{E}''(y) =
+ \left\{
+ \begin{array}{@{}ll}
+ [v_{i2};\dots;v_{in}] & y=y_i \\
+ \mathcal{E}(y) & \mbox{otherwise} \\
+ \end{array}
+ \right.} \\
+
+\IOT{\mathtt{fold}~\mathtt{left}~P_1~\mathtt{rec}~x~P_2}{\mathcal{E}}
+& =
+& \mathit{eval\_fold}(x,P_2,\mathcal{E}')
+& \\
+& & \multicolumn{2}{l}{\mathcal{E}' = \UPDATE{\mathcal{E}}{x|->
+\IOT{P_1}{\UPDATE{\mathcal{E}}{\NAMES(P_2)|->\bot}}}} \\
+
+\mathit{eval\_fold}(x,P,\mathcal{E})
+& =
+& \mathcal{E}(x)
+& \mathcal{E}(\NAMES(P)\setminus\{x\})=\{[],\dots,[]\} \\
+
+\mathit{eval\_fold}(x,P,\mathcal{E})
+& =
+& \mathit{eval\_fold}(x,P,\mathcal{E}')
+& \mathcal{E}(y_i) = [v_{i1},\dots,v_{in}] \\
+& & & \NAMES(P)\setminus{x}=\{y_1,\dots,y_m\} \\
+&
+& \multicolumn{2}{l}{\mathcal{E}' = \UPDATE{\mathcal{E}}{x|->\IOT{P}{\mathcal{E}''}; ~ y_i |-> [v_{i2};\dots;v_{in_i}]}}
+\\
+&
+& \multicolumn{2}{l}{\mathcal{E}''(y) =
+\left\{
+\begin{array}{ll}
+ v_1 & y\in \NAMES(P)\setminus\{x\} \\
+ \mathcal{E}(x) & y=x \\
+ \bot & \mathit{otherwise} \\
+\end{array}
+\right.
+}
+\\
+
+\end{array} \\
+\]
+\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 FoldRec]
+ {t \in P_2 ~> \mathcal E & \mathcal{E}(x) \in \mathtt{fold}~d~P_1~\mathtt{rec}~x~P_2 ~> \mathcal E'}
+ {t \in \mathtt{fold}~d~P_1~\mathtt{rec}~x~P_2 ~> \mathcal E''}
+ \\
+ \mbox{where}~\mathcal{E}''(y) = \left\{
+ \renewcommand{\arraystretch}{1}
+ \begin{array}{ll}
+ \mathcal{E}(y)::\mathcal{E}'(y) & y \in \NAMES(P_2) \setminus \{x\} \wedge d = \mathtt{right} \\
+ \mathcal{E}'(y)@[\mathcal{E}(y)] & y \in \NAMES(P_2) \setminus \{x\} \wedge d = \mathtt{left} \\
+ \mathcal{E}'(y) & \mbox{otherwise}
+ \end{array}
+ \right.
+ \\
+ \inference[\sc FoldBase]
+ {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}
+
+\begin{table}
+ \caption{\label{tab:synl3} Abstract syntax of level 3 terms and patterns.}
+ \hrule
+ \[
+ \begin{array}{@{}ll@{}}
+ \begin{array}[t]{rcll}
+ T & : := & & \mbox{(\bf terms)} \\
+ & & u & \mbox{(uri)} \\
+ & | & \lambda x.T & \mbox{($\lambda$-abstraction)} \\
+ & | & (T_1 \dots T_n) & \mbox{(application)} \\
+ & | & \dots \\[2ex]
+ \end{array} &
+ \begin{array}[t]{rcll}
+ P & : := & & \mbox{(\bf patterns)} \\
+ & & u & \mbox{(uri)} \\
+ & | & V & \mbox{(variable)} \\
+ & | & (P_1 \dots P_n) & \mbox{(application)} \\[2ex]
+ V & : := & & \mbox{(\bf variables)} \\
+ & & \TVAR{x} & \mbox{(term variable)} \\
+ & | & \IMPVAR & \mbox{(implicit variable)} \\
+ \end{array} \\
+ \end{array}
+ \]
+ \hrule
+\end{table}
+
+\begin{table}
+\caption{\label{tab:wfl3} Well-formedness rules for level 3 patterns.\strut}
+\hrule
+\[
+\renewcommand{\arraystretch}{3.5}
+\begin{array}{@{}c@{}}
+ \inference[\sc Uri] {} {u :: \emptyset} \quad
+ \inference[\sc ImpVar] {} {\TVAR{x} :: \emptyset} \quad
+ \inference[\sc TermVar] {} {\TVAR{x} :: x:\mathtt{Term}} \\
+ \inference[\sc Appl]
+ {P_i :: D_i
+ \quad \forall i,j,i\neq j=>\DOMAIN(D_i)\cap\DOMAIN(D_j)=\emptyset}
+ {P_1\cdots P_n :: D_1\oplus\cdots\oplus D_n} \\
+\end{array}
+\]
+\hrule
+\end{table}
+
+\begin{table}
+ \caption{\label{tab:synargp} Abstract syntax of applicative symbol patterns.}
+ \hrule
+ \[
+ \begin{array}{rcll}
+ P & : := & & \mbox{(\bf patterns)} \\
+ & & s ~ \{ \mathit{arg} \} & \mbox{(symbol pattern)} \\[2ex]
+ \mathit{arg} & : := & & \mbox{(\bf argument)} \\
+ & & \TVAR{x} & \mbox{(term variable)} \\
+ & | & \eta.\mathit{arg} & \mbox{($\eta$-abstraction)} \\
+ \end{array}
+ \]
+ \hrule
+\end{table}
+
+\begin{table}
+\caption{\label{tab:wfargp} Well-formedness rules for applicative symbol
+patterns.\strut}
+\hrule
+\[
+\renewcommand{\arraystretch}{3.5}
+\begin{array}{@{}c@{}}
+ \inference[\sc Pattern]
+ {\mathit{arg}_i :: D_i
+ \quad \forall i,j,i\neq j=>\DOMAIN(D_i)\cap\DOMAIN(D_j)=\emptyset}
+ {s~\mathit{arg}_1\cdots\mathit{arg}_n :: D_1\oplus\cdots\oplus D_n} \\
+ \inference[\sc TermVar]
+ {}
+ {\TVAR{x} :: x : \mathtt{Term}}
+ \quad
+ \inference[\sc EtaAbs]
+ {\mathit{arg} :: D}
+ {\eta.\mathit{arg} :: D}
+ \\
+\end{array}
+\]
+\hrule
+\end{table}
+
+\begin{table}
+\caption{\label{tab:l3match} Pattern matching of level 3 terms.\strut}
+\hrule
+\[
+\renewcommand{\arraystretch}{3.5}
+\begin{array}{@{}c@{}}
+ \inference[\sc Uri] {} {u\in u ~> []} \quad
+ \inference[\sc Appl] {t_i\in P_i ~> \mathcal{E}_i}
+ {(t_1\dots t_n)\in(P_1\dots P_n) ~>
+ \mathcal{E}_1\oplus\cdots\oplus\mathcal{E}_n} \\
+ \inference[\sc TermVar] {} {t\in \TVAR{x} ~> [x |-> \mathtt{Term}~t]} \quad
+ \inference[\sc ImpVar] {} {t\in \IMPVAR ~> []} \\
+\end{array}
+\]
+\hrule
+\end{table}
+
+\begin{table}
+\caption{\label{tab:iapf3} Instantiation of applicative symbol patterns (from
+level 3).\strut}
+\hrule
+\[
+\begin{array}{rcll}
+ \IAP{s~a_1\cdots a_n}{\mathcal{E}} & = &
+ (s~\IAPP{a_1}{\mathcal{E}}{0}\cdots\IAPP{a_n}{\mathcal{E}}{0}) & \\
+ \IAPP{\TVAR{x}}{\mathcal{E}}{0} & = & t & \mathcal{E}(x)=\mathtt{Term}~t \\
+ \IAPP{\TVAR{x}}{\mathcal{E}}{i+1} & = & \lambda y.\IAPP{t}{\mathcal{E}}{i}
+ & \mathcal{E}(x)=\mathtt{Term}~\lambda y.t \\
+ \IAPP{\TVAR{x}}{\mathcal{E}}{i+1} & =
+ & \lambda y_1.\cdots.\lambda y_{i+1}.t~y_1\cdots y_{i+1}
+ & \mathcal{E}(x)=\mathtt{Term}~t\wedge\forall y,t\neq\lambda y.t \\
+ \IAPP{\eta.a}{\mathcal{E}}{i} & = & \IAPP{a}{\mathcal{E}}{i+1} \\
+\end{array}
+\]
+\hrule
+\end{table}
+
+\section{Type checking}
+
+\subsection{Level 1 $<->$ Level 2}
+
+\newcommand{\GUARDED}{\mathit{guarded}}
+\newcommand{\TRUE}{\mathit{true}}
+\newcommand{\FALSE}{\mathit{false}}
+
+\newcommand{\TN}{\mathit{tn}}
+
+\begin{table}
+\caption{\label{tab:guarded} Guarded condition of level 2
+pattern. Note that the recursive case of the \texttt{fold} magic is
+not explicitly required to be guarded. The point is that it must
+contain at least two distinct names, and this guarantees that whatever
+is matched by the recursive pattern, the terms matched by those two
+names will be smaller than the whole matched term.\strut} \hrule
+\[
+\begin{array}{rcll}
+ \GUARDED(C(M(P))) & = & \GUARDED(P) \\
+ \GUARDED(C(t_1,\dots,t_n)) & = & \TRUE \\
+ \GUARDED(\mathtt{term}~x) & = & \FALSE \\
+ \GUARDED(\mathtt{number}~x) & = & \FALSE \\
+ \GUARDED(\mathtt{ident}~x) & = & \FALSE \\
+ \GUARDED(\mathtt{fresh}~x) & = & \FALSE \\
+ \GUARDED(\mathtt{anonymous}) & = & \TRUE \\
+ \GUARDED(\mathtt{default}~P_1~P_2) & = & \GUARDED(P_1) \wedge \GUARDED(P_2) \\
+ \GUARDED(\mathtt{if}~P_1~\mathtt{then}~P_2~\mathtt{else}~P_3) & = & \GUARDED(P_2) \wedge \GUARDED(P_3) \\
+ \GUARDED(\mathtt{fail}) & = & \TRUE \\
+ \GUARDED(\mathtt{fold}~d~P_1~\mathtt{rec}~x~P_2) & = & \GUARDED(P_1)
+\end{array}
+\]
+\hrule
+\end{table}
+
+%% 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 $P_1$ is a well-formed \emph{level 1 pattern} in some context $D$ and
+ $P_2$ is a well-formed \emph{level 2 pattern} in the very same context $D$,
+ that is $P_1 :: D$ and $P_2 :: D$;
+ \item the pattern $P_2$ is guarded, that is $\GUARDED(P_2)=\TRUE$;
+ \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).
+
+Relationship with grammatical frameworks?
+
+\subsection{Level 2 $<->$ Level 3}
+
+We say that an \emph{interpretation}
+\[
+ P_2 <=> P_3
+\]
+is well-formed if:
+\begin{itemize}
+ \item $P_2$ is a well-formed \emph{applicative symbol pattern} in some context
+ $D$ and $P_3$ is a well-formed \emph{level 3 pattern} in the very same
+ context $D$, that is $P_2 :: D$ and $P_3 :: D$.
+\end{itemize}
+
\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}
-
-\[
-\ITO{\cdot}{{}} : P -> \mathit{Env} -> T
-\]
-
-\begin{table}
-\caption{\label{tab:il1f2} Instantiation of level 1 patterns from level 2.\strut}
-\hrule
-\[
-\begin{array}{rcll}
- \ITO{L_\kappa[P_1,\dots,P_n]}{E} & = & L_\kappa[\ITO{(P_1)}{E},\dots,\ITO{(P_n)}{E} ] \\
- \ITO{B_\kappa^{ab}[P_1\cdots P_n]}{E} & = & B_\kappa^{ab}[\ITO{P_1}{E}\cdots\ITO{P_n}{E}] \\
- \ITO{\BREAK}{E} & = & \BREAK \\
- \ITO{(P)}{E} & = & \ITO{P}{E} \\
- \ITO{(P_1\cdots P_n)}{E} & = & B_H^{00}[\ITO{P_1}{E}\cdots\ITO{P_n}{E}] \\
- \ITO{\TVAR{x}}{E} & = & t & \mathcal{E}(x) = \verb+Term+~t \\
- \ITO{\NVAR{x}}{E} & = & l & \mathcal{E}(x) = \verb+Number+~l \\
- \ITO{\IVAR{x}}{E} & = & l & \mathcal{E}(x) = \verb+String+~l \\
- \ITO{\mathtt{opt}~P}{E} & = & \varepsilon & \mathcal{E}(\NAMES(P)) = \{\mathtt{None}\} \\
- \ITO{\mathtt{opt}~P}{E} & = & \ITO{P}{E'} & \mathcal{E}(\NAMES(P)) = \{\mathtt{Some}~v_1,\dots,\mathtt{Some}~v_n\} \\
- & & & \mathcal{E}'(x)=\left\{
- \begin{array}{@{}ll}
- v, & \mathcal{E}(x) = \mathtt{Some}~v \\
- \mathcal{E}(x), & \mbox{otherwise}
- \end{array}
- \right. \\
- \ITO{\mathtt{list}k~P~l?}{E} & = & \ITO{P}{{E}_1}~{l?}\cdots {l?}~\ITO{P}{{E}_n} &
- \mathcal{E}(\NAMES(P)) = \{[v_{11},\dots,v_{1n}],\dots,[v_{m1},\dots,v_{mn}]\} \\
- & & & n\ge k \\
- & & & \mathcal{E}_i(x) = \left\{
- \begin{array}{@{}ll}
- v_i, & \mathcal{E}(x) = [v_1,\dots,v_n] \\
- \mathcal{E}(x), & \mbox{otherwise}
- \end{array}
- \right. \\
- \ITO{l}{E} & = & l \\
-
-%% & | & (P) & \mbox{(fenced)} \\
-%% & | & M & \mbox{(magic)} \\
-%% & | & V & \mbox{(variable)} \\
-%% & | & l & \mbox{(literal)} \\[2ex]
-%% V & ::= & & \mbox{(\bf variables)} \\
-%% & & \TVAR{x} & \mbox{(term variable)} \\
-%% & | & \NVAR{x} & \mbox{(number variable)} \\
-%% & | & \IVAR{x} & \mbox{(name variable)} \\[2ex]
-%% M & ::= & & \mbox{(\bf magic patterns)} \\
-%% & & \verb+list0+~S~l? & \mbox{(possibly empty list)} \\
-%% & | & \verb+list1+~S~l? & \mbox{(non-empty list)} \\
-%% & | & \verb+opt+~S & \mbox{(option)} \\[2ex]
-\end{array}
-\]
-\hrule
-\end{table}
-
-\begin{table}
-\caption{\label{tab:wfl0} Well-formedness rules for level 1 patterns.\strut}
-\hrule
-\[
-\renewcommand{\arraystretch}{3.5}
-\begin{array}[t]{@{}c@{}}
- \inference[\sc layout]
- {P_i :: D_i & \forall i,j, i\ne j => \DOMAIN(D_i) \cap \DOMAIN(D_j) = \emptyset}
- {L_\kappa[P_1,\dots,P_n] :: D_1\oplus\cdots\oplus D_n}
- \\
- \inference[\sc box]
- {P_i :: D_i & \forall i,j, i\ne j => \DOMAIN(D_i) \cap \DOMAIN(D_j) = \emptyset}
- {B_\kappa^{ab}[P_1\cdots P_n] :: D_1\oplus\cdots\oplus D_n}
- \\
- \inference[\sc fenced]
- {P_i :: D_i & \forall i,j, i\ne j => \DOMAIN(D_i) \cap \DOMAIN(D_j) = \emptyset}
- {\FENCED{P_1\cdots P_n} :: D_1\oplus\cdots\oplus D_n}
- \\
- \inference[\sc breakpoint]
- {}
- {\BREAK :: \emptyset}
- \qquad
- \inference[\sc literal]
- {}
- {l :: \emptyset}
- \qquad
- \inference[\sc tvar]
- {}
- {\TVAR{x} :: \TVAR{x}}
- \\
- \inference[\sc nvar]
- {}
- {\NVAR{x} :: \NVAR{x}}
- \qquad
- \inference[\sc ivar]
- {}
- {\IVAR{x} :: \IVAR{x}}
- \\
- \inference[\sc list0]
- {P :: D & \forall x\in\DOMAIN(D), D'(x) = D(x)~\mathtt{List}}
- {\mathtt{list0}~P~l? :: D'}
- \\
- \inference[\sc list1]
- {P :: D & \forall x\in\DOMAIN(D), D'(x) = D(x)~\mathtt{List}}
- {\mathtt{list1}~P~l? :: D'}
- \\
- \inference[\sc opt]
- {P :: D & \forall x\in\DOMAIN(D), D'(x) = D(x)~\mathtt{Option}}
- {\mathtt{opt}~P :: D'}
-\end{array}
-\]
-\hrule
-\end{table}
-
-\newcommand{\ATTRS}[1]{\langle#1\rangle}
-\newcommand{\ANNPOS}[2]{\mathit{pos}(#1)_{#2}}
-
-\begin{table}
-\caption{\label{tab:addparens} Can't read the AST and need parentheses? Here you go!.\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}
-\caption{\label{tab:synl2} Concrete syntax of level 2 patterns.\strut}
-\hrule
-\[
-\begin{array}{@{}rcll@{}}
- \NT{term} & ::= & & \mbox{\bf terms} \\
- & & x & \mbox{(identifier)} \\
- & | & n & \mbox{(number)} \\
- & | & s & \mbox{(symbol)} \\
- & | & \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}) & \mbox{(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}
-\]
-\hrule
-\end{table}
-
-\begin{table}
-\caption{\label{tab:wfl2} Well-formedness rules for level 2 patterns.\strut}
-\hrule
-\[
-\renewcommand{\arraystretch}{3.5}
-\begin{array}{@{}c@{}}
- \inference[\sc Constr]
- {P_i :: D_i}
- {\BLOB[P_1,\dots,P_n] :: D_i \oplus \cdots \oplus D_j} \\
- \inference[\sc TermVar]
- {}
- {\mathtt{term}~x :: x : \mathtt{Term}}
- \quad
- \inference[\sc NumVar]
- {}
- {\mathtt{number}~x :: x : \mathtt{Number}}
- \\
- \inference[\sc IdentVar]
- {}
- {\mathtt{ident}~x :: x : \mathtt{String}}
- \quad
- \inference[\sc FreshVar]
- {}
- {\mathtt{fresh}~x :: x : \mathtt{String}}
- \\
- \inference[\sc Success]
- {}
- {\mathtt{anonymous} :: \emptyset}
- \\
- \inference[\sc Fold]
- {P_1 :: D_1 & P_2 :: D_2 \oplus (x : \mathtt{Term}) & \DOMAIN(D_2)\ne\emptyset & \DOMAIN(D_1)\cap\DOMAIN(D_2)=\emptyset}
- {\mathtt{fold}~P_1~\mathtt{rec}~x~P_2 :: D_1 \oplus D_2~\mathtt{List}}
- \\
- \inference[\sc Default]
- {P_1 :: D \oplus D_1 & P_2 :: D & \DOMAIN(D_1) \ne \emptyset & \DOMAIN(D) \cap \DOMAIN(D_1) = \emptyset}
- {\mathtt{default}~P_1~P_2 :: D \oplus D_1~\mathtt{Option}}
- \\
- \inference[\sc If]
- {P_1 :: \emptyset & P_2 :: D & P_3 :: D }
- {\mathtt{if}~P_1~\mathtt{then}~P_2~\mathtt{else}~P_3 :: D}
- \qquad
- \inference[\sc Fail]
- {}
- {\mathtt{fail} :: \emptyset}
-%% & | & \verb+if+~\NT{meta}~\verb+then+~\NT{meta}~\verb+else+~\NT{meta} \\
-%% & | & \verb+fail+
-\end{array}
-\]
-\hrule
-\end{table}
-
-\begin{table}
- \caption{\label{tab:il2f1} Instantiation of level 2 patterns from level 1.
- \strut}
-\hrule
-\[
-\begin{array}{rcll}
-
-\IOT{C[t_1,\dots,t_n]}{\mathcal{E}} & =
-& C[\IOT{t_1}{\mathcal{E}},\dots,\IOT{t_n}{\mathcal{E}}] \\
-
-\IOT{\mathtt{term}~x}{\mathcal{E}} & = & t & \mathcal{E}(x) = \mathtt{Term}~t \\
-
-\IOT{\mathtt{number}~x}{\mathcal{E}} & =
-& n & \mathcal{E}(x) = \mathtt{Number}~n \\
-
-\IOT{\mathtt{ident}~x}{\mathcal{E}} & =
-& y & \mathcal{E}(x) = \mathtt{String}~y \\
-
-\IOT{\mathtt{fresh}~x}{\mathcal{E}} & = & y & \mathcal{E}(x) = \mathtt{String}~y \\
-
-\IOT{\mathtt{default}~P_1~P_2}{\mathcal{E}} & =
-& \IOT{P_1}{\UPDATE{\mathcal{E}}{x_i|->v_i}}
-& \mathcal{E}(x_i)=\mathtt{Some}~v_i \\
-& & & \NAMES(P_1)\setminus\NAMES(P_2)=\{x_1,\dots,x_n\} \\
-
-\IOT{\mathtt{default}~P_1~P_2}{\mathcal{E}} & =
-& \IOT{P_2}{\UPDATE{\mathcal{E}}{x_i|->\bot}}
-& \mathcal{E}(x_i)=\mathtt{None} \\
-& & & \NAMES(P_1)\setminus\NAMES(P_2)=\{x_1,\dots,x_n\} \\
-
-\IOT{\mathtt{fold}~\mathtt{right}~P_1~\mathtt{rec}~x~P_2}{\mathcal{E}}
-& =
-& \IOT{P_1}{\mathcal{E}'}
-& \mathcal{E}(\NAMES(P_2)\setminus\{x\}) = \{[],\dots,[]\} \\
-& & \multicolumn{2}{l}{\mathcal{E}'=\UPDATE{\mathcal{E}}{\NAMES(P_2)\setminus\{x\}|->\bot}}
-\\
-
-\IOT{\mathtt{fold}~\mathtt{right}~P_1~\mathtt{rec}~x~P_2}{\mathcal{E}}
-& =
-& \IOT{P_2}{\mathcal{E}'}
-& \mathcal{E}(y_i) = [v_{i1},\dots,v_{in}] \\
-& & & \NAMES(P_2)\setminus\{x\}=\{y_1,\dots,y_m\} \\
-& & \multicolumn{2}{l}{\mathcal{E}'(y) =
- \left\{
- \begin{array}{@{}ll}
- \IOT{\mathtt{fold}~\mathtt{right}~P_1~\mathtt{rec}~x~P_e}{\mathcal{E}''}
- & y=x \\
- v_{i1} & y=y_i \\
- \mathcal{E}(y) & \mbox{otherwise} \\
- \end{array}
- \right.} \\
-& & \multicolumn{2}{l}{\mathcal{E}''(y) =
- \left\{
- \begin{array}{@{}ll}
- [v_{i2};\dots;v_{in}] & y=y_i \\
- \mathcal{E}(y) & \mbox{otherwise} \\
- \end{array}
- \right.} \\
-
-\IOT{\mathtt{fold}~\mathtt{left}~P_1~\mathtt{rec}~x~P_2}{\mathcal{E}}
-& =
-& \mathit{eval\_fold}(x,P_2,\mathcal{E}')
-& \\
-& & \multicolumn{2}{l}{\mathcal{E}' = \UPDATE{\mathcal{E}}{x|->
-\IOT{P_1}{\UPDATE{\mathcal{E}}{\NAMES(P_2)|->\bot}}}} \\
-
-\mathit{eval\_fold}(x,P,\mathcal{E})
-& =
-& \mathcal{E}(x)
-& \mathcal{E}(\NAMES(P)\setminus\{x\})=\{[],\dots,[]\} \\
-
-\mathit{eval\_fold}(x,P,\mathcal{E})
-& =
-& \mathit{eval\_fold}(x,P,\mathcal{E}')
-& \mathcal{E}(y_i) = [v_{i1},\dots,v_{in}] \\
-& & & \NAMES(P)\setminus{x}=\{y_1,\dots,y_m\} \\
-&
-& \multicolumn{2}{l}{\mathcal{E}' = \UPDATE{\mathcal{E}}{x|->\IOT{P}{\mathcal{E}''}; ~ y_i |-> [v_{i2};\dots;v_{in_i}]}}
-\\
-&
-& \multicolumn{2}{l}{\mathcal{E}''(y) =
-\left\{
-\begin{array}{ll}
- v_1 & y\in \NAMES(P)\setminus\{x\} \\
- \mathcal{E}(x) & y=x \\
- \bot & \mathit{otherwise} \\
-\end{array}
-\right.
-}
-\\
-
-\end{array} \\
-\]
-\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 FoldRec]
- {t \in P_2 ~> \mathcal E & \mathcal{E}(x) \in \mathtt{fold}~d~P_1~\mathtt{rec}~x~P_2 ~> \mathcal E'}
- {t \in \mathtt{fold}~d~P_1~\mathtt{rec}~x~P_2 ~> \mathcal E''}
- \\
- \mbox{where}~\mathcal{E}''(y) = \left\{
- \renewcommand{\arraystretch}{1}
- \begin{array}{ll}
- \mathcal{E}(y)::\mathcal{E}'(y) & y \in \NAMES(P_2) \setminus \{x\} \wedge d = \mathtt{right} \\
- \mathcal{E}'(y)@[\mathcal{E}(y)] & y \in \NAMES(P_2) \setminus \{x\} \wedge d = \mathtt{left} \\
- \mathcal{E}'(y) & \mbox{otherwise}
- \end{array}
- \right.
- \\
- \inference[\sc FoldBase]
- {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}
-
-\begin{table}
- \caption{\label{tab:synl3} Abstract syntax of level 3 terms and patterns.}
- \hrule
- \[
- \begin{array}{@{}ll@{}}
- \begin{array}[t]{rcll}
- T & : := & & \mbox{(\bf terms)} \\
- & & u & \mbox{(uri)} \\
- & | & \lambda x.T & \mbox{($\lambda$-abstraction)} \\
- & | & (T_1 \dots T_n) & \mbox{(application)} \\
- & | & \dots \\[2ex]
- \end{array} &
- \begin{array}[t]{rcll}
- P & : := & & \mbox{(\bf patterns)} \\
- & & u & \mbox{(uri)} \\
- & | & V & \mbox{(variable)} \\
- & | & (P_1 \dots P_n) & \mbox{(application)} \\[2ex]
- V & : := & & \mbox{(\bf variables)} \\
- & & \TVAR{x} & \mbox{(term variable)} \\
- & | & \IMPVAR & \mbox{(implicit variable)} \\
- \end{array} \\
- \end{array}
- \]
- \hrule
-\end{table}
-
-\begin{table}
-\caption{\label{tab:wfl3} Well-formedness rules for level 3 patterns.\strut}
-\hrule
-\[
-\renewcommand{\arraystretch}{3.5}
-\begin{array}{@{}c@{}}
- \inference[\sc Uri] {} {u :: \emptyset} \quad
- \inference[\sc ImpVar] {} {\TVAR{x} :: \emptyset} \quad
- \inference[\sc TermVar] {} {\TVAR{x} :: x:\mathtt{Term}} \\
- \inference[\sc Appl]
- {P_i :: D_i
- \quad \forall i,j,i\neq j=>\DOMAIN(D_i)\cap\DOMAIN(D_j)=\emptyset}
- {P_1\cdots P_n :: D_1\oplus\cdots\oplus D_n} \\
-\end{array}
-\]
-\hrule
-\end{table}
-
-\begin{table}
- \caption{\label{tab:synargp} Abstract syntax of applicative symbol patterns.}
- \hrule
- \[
- \begin{array}{rcll}
- P & : := & & \mbox{(\bf patterns)} \\
- & & s ~ \{ \mathit{arg} \} & \mbox{(symbol pattern)} \\[2ex]
- \mathit{arg} & : := & & \mbox{(\bf argument)} \\
- & & \TVAR{x} & \mbox{(term variable)} \\
- & | & \eta.\mathit{arg} & \mbox{($\eta$-abstraction)} \\
- \end{array}
- \]
- \hrule
-\end{table}
-
-\begin{table}
-\caption{\label{tab:wfargp} Well-formedness rules for applicative symbol
-patterns.\strut}
-\hrule
-\[
-\renewcommand{\arraystretch}{3.5}
-\begin{array}{@{}c@{}}
- \inference[\sc Pattern]
- {\mathit{arg}_i :: D_i
- \quad \forall i,j,i\neq j=>\DOMAIN(D_i)\cap\DOMAIN(D_j)=\emptyset}
- {s~\mathit{arg}_1\cdots\mathit{arg}_n :: D_1\oplus\cdots\oplus D_n} \\
- \inference[\sc TermVar]
- {}
- {\TVAR{x} :: x : \mathtt{Term}}
- \quad
- \inference[\sc EtaAbs]
- {\mathit{arg} :: D}
- {\eta.\mathit{arg} :: D}
- \\
-\end{array}
-\]
-\hrule
-\end{table}
-
-\begin{table}
-\caption{\label{tab:l3match} Pattern matching of level 3 terms.\strut}
-\hrule
-\[
-\renewcommand{\arraystretch}{3.5}
-\begin{array}{@{}c@{}}
- \inference[\sc Uri] {} {u\in u ~> []} \quad
- \inference[\sc Appl] {t_i\in P_i ~> \mathcal{E}_i}
- {(t_1\dots t_n)\in(P_1\dots P_n) ~>
- \mathcal{E}_1\oplus\cdots\oplus\mathcal{E}_n} \\
- \inference[\sc TermVar] {} {t\in \TVAR{x} ~> [x |-> \mathtt{Term}~t]} \quad
- \inference[\sc ImpVar] {} {t\in \IMPVAR ~> []} \\
-\end{array}
-\]
-\hrule
-\end{table}
-
-\begin{table}
-\caption{\label{tab:iapf3} Instantiation of applicative symbol patterns (from
-level 3).\strut}
-\hrule
-\[
-\begin{array}{rcll}
- \IAP{s~a_1\cdots a_n}{\mathcal{E}} & = &
- (s~\IAPP{a_1}{\mathcal{E}}{0}\cdots\IAPP{a_n}{\mathcal{E}}{0}) & \\
- \IAPP{\TVAR{x}}{\mathcal{E}}{0} & = & t & \mathcal{E}(x)=\mathtt{Term}~t \\
- \IAPP{\TVAR{x}}{\mathcal{E}}{i+1} & = & \lambda y.\IAPP{t}{\mathcal{E}}{i}
- & \mathcal{E}(x)=\mathtt{Term}~\lambda y.t \\
- \IAPP{\TVAR{x}}{\mathcal{E}}{i+1} & =
- & \lambda y_1.\cdots.\lambda y_{i+1}.t~y_1\cdots y_{i+1}
- & \mathcal{E}(x)=\mathtt{Term}~t\wedge\forall y,t\neq\lambda y.t \\
- \IAPP{\eta.a}{\mathcal{E}}{i} & = & \IAPP{a}{\mathcal{E}}{i+1} \\
-\end{array}
-\]
-\hrule
-\end{table}
-
-\section{Type checking}
-
-\subsection{Level 1 $<->$ Level 2}
-
-\newcommand{\GUARDED}{\mathit{guarded}}
-\newcommand{\TRUE}{\mathit{true}}
-\newcommand{\FALSE}{\mathit{false}}
-
-\newcommand{\TN}{\mathit{tn}}
-
-\begin{table}
-\caption{\label{tab:guarded} Guarded condition of level 2
-pattern. Note that the recursive case of the \texttt{fold} magic is
-not explicitly required to be guarded. The point is that it must
-contain at least two distinct names, and this guarantees that whatever
-is matched by the recursive pattern, the terms matched by those two
-names will be smaller than the whole matched term.\strut} \hrule
-\[
-\begin{array}{rcll}
- \GUARDED(C(M(P))) & = & \GUARDED(P) \\
- \GUARDED(C(t_1,\dots,t_n)) & = & \TRUE \\
- \GUARDED(\mathtt{term}~x) & = & \FALSE \\
- \GUARDED(\mathtt{number}~x) & = & \FALSE \\
- \GUARDED(\mathtt{ident}~x) & = & \FALSE \\
- \GUARDED(\mathtt{fresh}~x) & = & \FALSE \\
- \GUARDED(\mathtt{anonymous}) & = & \TRUE \\
- \GUARDED(\mathtt{default}~P_1~P_2) & = & \GUARDED(P_1) \wedge \GUARDED(P_2) \\
- \GUARDED(\mathtt{if}~P_1~\mathtt{then}~P_2~\mathtt{else}~P_3) & = & \GUARDED(P_2) \wedge \GUARDED(P_3) \\
- \GUARDED(\mathtt{fail}) & = & \TRUE \\
- \GUARDED(\mathtt{fold}~d~P_1~\mathtt{rec}~x~P_2) & = & \GUARDED(P_1)
-\end{array}
-\]
-\hrule
-\end{table}
-
-%% 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 $P_1$ is a well-formed \emph{level 1 pattern} in some context $D$ and
- $P_2$ is a well-formed \emph{level 2 pattern} in the very same context $D$,
- that is $P_1 :: D$ and $P_2 :: D$;
- \item the pattern $P_2$ is guarded, that is $\GUARDED(P_2)=\TRUE$;
- \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).
-
-Relationship with grammatical frameworks?
-
-\subsection{Level 2 $<->$ Level 3}
-
-We say that an \emph{interpretation}
-\[
- P_2 <=> P_3
-\]
-is well-formed if:
-\begin{itemize}
- \item $P_2$ is a well-formed \emph{applicative symbol pattern} in some context
- $D$ and $P_3$ is a well-formed \emph{level 3 pattern} in the very same
- context $D$, that is $P_2 :: D$ and $P_3 :: D$.
-\end{itemize}
+ \input{body}
\end{document}
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