X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Focaml%2Fcic_notation%2Fdoc%2Fmain.tex;h=f73dc21681561fe95b051bced72e9f0bec16eb62;hb=4496bf2f1647f61657441b8249c05dab979da091;hp=16b29bf8a30796a42f96de356ff036292e803a99;hpb=2d9b039d9b2f1f20fae18e577306ead3b5d2090d;p=helm.git diff --git a/helm/ocaml/cic_notation/doc/main.tex b/helm/ocaml/cic_notation/doc/main.tex index 16b29bf8a..f73dc2168 100644 --- a/helm/ocaml/cic_notation/doc/main.tex +++ b/helm/ocaml/cic_notation/doc/main.tex @@ -1,10 +1,14 @@ \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 \\ @@ -16,6 +20,14 @@ \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 @@ -30,53 +42,128 @@ & | & \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 \[ \begin{array}{rcll} - \IOT{S_1\cdots S_n}{E} & = & \IOT{S_1}{E}\cdots\IOT{S_n}{E} \\ - \IOT{L_\kappa[S_1,\dots,S_n]}{E} & = & L_\kappa[\IOT{S_1}{E},\dots,\IOT{S_n}{E} ] \\ - \IOT{B_\kappa^{ab}[P]}{E} & = & B_\kappa^{ab}[\IOT{P}{E}] \\ + \IOT{L_\kappa[P_1,\dots,P_n]}{E} & = & L_\kappa[\IOT{(P_1)}{E},\dots,\IOT{(P_n)}{E} ] \\ + \IOT{B_\kappa^{ab}[P_1\cdots P_n]}{E} & = & B_\kappa^{ab}[\IOT{P_1}{E}\cdots\IOT{P_n}{E}] \\ \IOT{\BREAK}{E} & = & \BREAK \\ - \IOT{(P)}{E} & = & (\IOT{P}{E}) \\ + \IOT{(P)}{E} & = & \IOT{P}{E} \\ + \IOT{(P_1\cdots P_n)}{E} & = & B_H^{00}[\IOT{P_1}{E}\cdots\IOT{P_n}{E}] \\ \IOT{\TVAR{x}}{E} & = & t & \mathcal{E}(x) = \verb+Term+~t \\ \IOT{\NVAR{x}}{E} & = & l & \mathcal{E}(x) = \verb+Number+~l \\ \IOT{\IVAR{x}}{E} & = & l & \mathcal{E}(x) = \verb+String+~l \\ - \IOT{\mathtt{opt}~S}{E} & = & \varepsilon & \forall x \in \NAMES(S), \mathcal{E}(x) = \mathtt{None} \\ - \IOT{\mathtt{opt}~S}{E} & = & \varepsilon & \forall x \in \NAMES(S), \mathcal{E}(x) = \mathtt{None} \\ + \IOT{\mathtt{opt}~P}{E} & = & \varepsilon & \mathcal{E}(\NAMES(P)) = \{\mathtt{None}\} \\ + \IOT{\mathtt{opt}~P}{E} & = & \IOT{P}{E'} & \mathcal{E}(\NAMES(P)) = \{\mathtt{Some}~v_1,\dots,\mathtt{Some}~v_n\} \\ + & & & \mathcal{E}'(x)=\left\{ + \begin{array}{@{}ll} + v, & \mathcal{E}(x) = \mathtt{Some}~v \\ + \mathcal{E}(x), & \mbox{otherwise} + \end{array} + \right. \\ + \IOT{\mathtt{list}k~P~l?}{E} & = & \IOT{P}{{E}_1}~{l?}\cdots {l?}~\IOT{P}{{E}_n} & + \mathcal{E}(\NAMES(P)) = \{[v_{11},\dots,v_{1n}],\dots,[v_{m1},\dots,v_{mn}]\} \\ + & & & n\ge k \\ + & & & \mathcal{E}_i(x) = \left\{ + \begin{array}{@{}ll} + v_i, & \mathcal{E}(x) = [v_1,\dots,v_n] \\ + \mathcal{E}(x), & \mbox{otherwise} + \end{array} + \right. \\ + \IOT{l}{E} & = & l \\ %% & | & (P) & \mbox{(fenced)} \\ %% & | & M & \mbox{(magic)} \\ @@ -92,7 +179,379 @@ An environment is a map $\mathcal E : \mathit{Name} -> V$. %% & | & \verb+opt+~S & \mbox{(option)} \\[2ex] \end{array} \] +\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} \ No newline at end of file