1 \documentclass[a4paper]{article}
7 \usepackage{stmaryrd,latexsym}
9 \newcommand{\BLOB}{\raisebox{0ex}{\small\manstar}}
11 \newcommand{\MATITA}{\ding{46}\textsf{\textbf{Matita}}}
13 \title{Extensible notation for \MATITA}
14 \author{Luca Padovani \qquad Stefano Zacchiroli \\
15 \small Department of Computer Science, University of Bologna \\
16 \small Mura Anteo Zamboni, 7 -- 40127 Bologna, ITALY \\
17 \small \{\texttt{lpadovan}, \texttt{zacchiro}\}\texttt{@cs.unibo.it}}
19 \newcommand{\BREAK}{\mathtt{break}}
20 \newcommand{\TVAR}[1]{#1:\mathtt{term}}
21 \newcommand{\NVAR}[1]{#1:\mathtt{number}}
22 \newcommand{\IVAR}[1]{#1:\mathtt{name}}
23 \newcommand{\FENCED}[1]{\texttt{\char'050}#1\texttt{\char'051}}
24 \newcommand{\IOT}[2]{|[#1|]_{\mathcal#2}^1}
25 \newcommand{\ADDPARENS}[2]{\llparenthesis#1\rrparenthesis^{#2}}
26 \newcommand{\NAMES}{\mathit{names}}
27 \newcommand{\DOMAIN}{\mathit{domain}}
29 \mathlig{~>}{\leadsto}
30 \mathlig{|->}{\mapsto}
39 V & ::= & & \mbox{(\bf values)} \\
40 & & \verb+Term+~T & \mbox{(term)} \\
41 & | & \verb+String+~s & \mbox{(string)} \\
42 & | & \verb+Number+~n & \mbox{(number)} \\
43 & | & \verb+None+ & \mbox{(optional value)} \\
44 & | & \verb+Some+~V & \mbox{(optional value)} \\
45 & | & [V_1,\dots,V_n] & \mbox{(list value)} \\[2ex]
49 An environment is a map $\mathcal E : \mathit{Name} -> V$.
51 \section{Level 1: concrete syntax}
54 \caption{\label{tab:l1c} Concrete syntax of level 1 patterns.\strut}
58 P & ::= & & \mbox{(\bf patterns)} \\
60 S & ::= & & \mbox{(\bf simple patterns)} \\
62 & | & S~\verb+\sub+~S\\
63 & | & S~\verb+\sup+~S\\
64 & | & S~\verb+\below+~S\\
65 & | & S~\verb+\atop+~S\\
66 & | & S~\verb+\over+~S\\
67 & | & S~\verb+\atop+~S\\
68 & | & \verb+\frac+~S~S \\
69 & | & \verb+\sqrt+~S \\
70 & | & \verb+\root+~S~\verb+\of+~S \\
71 & | & \verb+(+~P~\verb+)+ \\
72 & | & \verb+hbox (+~P~\verb+)+ \\
73 & | & \verb+vbox (+~P~\verb+)+ \\
74 & | & \verb+hvbox (+~P~\verb+)+ \\
75 & | & \verb+hovbox (+~P~\verb+)+ \\
77 & | & \verb+list0+~S~[\verb+sep+~l] \\
78 & | & \verb+list1+~S~[\verb+sep+~l] \\
80 & | & [\verb+term+]~x \\
81 & | & \verb+number+~x \\
82 & | & \verb+ident+~x \\
88 Rationale: while the layout schemata can occur in the concrete syntax
89 used by user, the box schemata and the magic patterns can only occur
90 when defining the notation. This is why the layout schemata are
91 ``escaped'' with a backslash, so that they cannot be confused with
92 plain identifiers, wherease the others are not. Alternatively, they
93 could be defined as keywords, but this would prevent their names to be
94 used in different contexts.
97 \caption{\label{tab:l1a} Abstract syntax of level 1 terms and patterns.\strut}
100 \begin{array}{@{}ll@{}}
101 \begin{array}[t]{rcll}
102 T & ::= & & \mbox{(\bf terms)} \\
103 & & L_\kappa[T_1,\dots,T_n] & \mbox{(layout)} \\
104 & | & B_\kappa^{ab}[T_1\cdots T_n] & \mbox{(box)} \\
105 & | & \BREAK & \mbox{(breakpoint)} \\
106 & | & \FENCED{T_1\cdots T_n} & \mbox{(fenced)} \\
107 & | & l & \mbox{(literal)} \\[2ex]
108 P & ::= & & \mbox{(\bf patterns)} \\
109 & & L_\kappa[P_1,\dots,P_n] & \mbox{(layout)} \\
110 & | & B_\kappa^{ab}[P_1\cdots P_n] & \mbox{(box)} \\
111 & | & \BREAK & \mbox{(breakpoint)} \\
112 & | & \FENCED{P_1\cdots P_n} & \mbox{(fenced)} \\
113 & | & M & \mbox{(magic)} \\
114 & | & V & \mbox{(variable)} \\
115 & | & l & \mbox{(literal)} \\
117 \begin{array}[t]{rcll}
118 V & ::= & & \mbox{(\bf variables)} \\
119 & & \TVAR{x} & \mbox{(term variable)} \\
120 & | & \NVAR{x} & \mbox{(number variable)} \\
121 & | & \IVAR{x} & \mbox{(name variable)} \\[2ex]
122 M & ::= & & \mbox{(\bf magic patterns)} \\
123 & & \verb+list0+~P~l? & \mbox{(possibly empty list)} \\
124 & | & \verb+list1+~P~l? & \mbox{(non-empty list)} \\
125 & | & \verb+opt+~P & \mbox{(option)} \\[2ex]
133 \IOT{\cdot}{{}} : P -> \mathit{Env} -> T
137 \caption{\label{tab:il1} Instantiation of level 1 patterns.\strut}
141 \IOT{L_\kappa[P_1,\dots,P_n]}{E} & = & L_\kappa[\IOT{(P_1)}{E},\dots,\IOT{(P_n)}{E} ] \\
142 \IOT{B_\kappa^{ab}[P_1\cdots P_n]}{E} & = & B_\kappa^{ab}[\IOT{P_1}{E}\cdots\IOT{P_n}{E}] \\
143 \IOT{\BREAK}{E} & = & \BREAK \\
144 \IOT{(P)}{E} & = & \IOT{P}{E} \\
145 \IOT{(P_1\cdots P_n)}{E} & = & B_H^{00}[\IOT{P_1}{E}\cdots\IOT{P_n}{E}] \\
146 \IOT{\TVAR{x}}{E} & = & t & \mathcal{E}(x) = \verb+Term+~t \\
147 \IOT{\NVAR{x}}{E} & = & l & \mathcal{E}(x) = \verb+Number+~l \\
148 \IOT{\IVAR{x}}{E} & = & l & \mathcal{E}(x) = \verb+String+~l \\
149 \IOT{\mathtt{opt}~P}{E} & = & \varepsilon & \mathcal{E}(\NAMES(P)) = \{\mathtt{None}\} \\
150 \IOT{\mathtt{opt}~P}{E} & = & \IOT{P}{E'} & \mathcal{E}(\NAMES(P)) = \{\mathtt{Some}~v_1,\dots,\mathtt{Some}~v_n\} \\
151 & & & \mathcal{E}'(x)=\left\{
153 v, & \mathcal{E}(x) = \mathtt{Some}~v \\
154 \mathcal{E}(x), & \mbox{otherwise}
157 \IOT{\mathtt{list}k~P~l?}{E} & = & \IOT{P}{{E}_1}~{l?}\cdots {l?}~\IOT{P}{{E}_n} &
158 \mathcal{E}(\NAMES(P)) = \{[v_{11},\dots,v_{1n}],\dots,[v_{m1},\dots,v_{mn}]\} \\
160 & & & \mathcal{E}_i(x) = \left\{
162 v_i, & \mathcal{E}(x) = [v_1,\dots,v_n] \\
163 \mathcal{E}(x), & \mbox{otherwise}
166 \IOT{l}{E} & = & l \\
168 %% & | & (P) & \mbox{(fenced)} \\
169 %% & | & M & \mbox{(magic)} \\
170 %% & | & V & \mbox{(variable)} \\
171 %% & | & l & \mbox{(literal)} \\[2ex]
172 %% V & ::= & & \mbox{(\bf variables)} \\
173 %% & & \TVAR{x} & \mbox{(term variable)} \\
174 %% & | & \NVAR{x} & \mbox{(number variable)} \\
175 %% & | & \IVAR{x} & \mbox{(name variable)} \\[2ex]
176 %% M & ::= & & \mbox{(\bf magic patterns)} \\
177 %% & & \verb+list0+~S~l? & \mbox{(possibly empty list)} \\
178 %% & | & \verb+list1+~S~l? & \mbox{(non-empty list)} \\
179 %% & | & \verb+opt+~S & \mbox{(option)} \\[2ex]
186 \caption{\label{tab:wfl0} Well-formedness rules for level 1 patterns.\strut}
189 \renewcommand{\arraystretch}{3.5}
190 \begin{array}[t]{@{}c@{}}
191 \inference[\sc layout]
192 {P_i :: D_i & \forall i,j, i\ne j => \DOMAIN(D_i) \cap \DOMAIN(D_j) = \emptyset}
193 {L_\kappa[P_1,\dots,P_n] :: D_1\oplus\cdots\oplus D_n}
196 {P_i :: D_i & \forall i,j, i\ne j => \DOMAIN(D_i) \cap \DOMAIN(D_j) = \emptyset}
197 {B_\kappa^{ab}[P_1\cdots P_n] :: D_1\oplus\cdots\oplus D_n}
199 \inference[\sc fenced]
200 {P_i :: D_i & \forall i,j, i\ne j => \DOMAIN(D_i) \cap \DOMAIN(D_j) = \emptyset}
201 {\FENCED{P_1\cdots P_n} :: D_1\oplus\cdots\oplus D_n}
203 \inference[\sc breakpoint]
205 {\BREAK :: \emptyset}
207 \inference[\sc literal]
213 {\TVAR{x} :: \TVAR{x}}
217 {\NVAR{x} :: \NVAR{x}}
221 {\IVAR{x} :: \IVAR{x}}
223 \inference[\sc list0]
224 {P :: D & \forall x\in\DOMAIN(D), D'(x) = D(x)~\mathtt{List}}
225 {\mathtt{list0}~P~l? :: D'}
227 \inference[\sc list1]
228 {P :: D & \forall x\in\DOMAIN(D), D'(x) = D(x)~\mathtt{List}}
229 {\mathtt{list1}~P~l? :: D'}
232 {P :: D & \forall x\in\DOMAIN(D), D'(x) = D(x)~\mathtt{Option}}
233 {\mathtt{opt}~P :: D'}
239 \newcommand{\ATTRS}[1]{\langle#1\rangle}
240 \newcommand{\ANNPOS}[2]{\mathit{pos}(#1)_{#2}}
243 \caption{\label{tab:addparens} Where are parentheses added? Look here.\strut}
247 \ADDPARENS{l}{n} & = & l \\
248 \ADDPARENS{\BREAK}{n} & = & \BREAK \\
249 \ADDPARENS{\ATTRS{\mathit{prec}=m}T}{n} & = & \ADDPARENS{T}{m} & n < m \\
250 \ADDPARENS{\ATTRS{\mathit{prec}=m}T}{n} & = & \FENCED{\ADDPARENS{T}{\bot}} & n > m \\
251 \ADDPARENS{\ATTRS{\mathit{prec}=n,\mathit{assoc}=L,\mathit{pos}=R}T}{n} & = & \FENCED{\ADDPARENS{T}{\bot}} \\
252 \ADDPARENS{\ATTRS{\mathit{prec}=n,\mathit{assoc}=R,\mathit{pos}=L}T}{n} & = & \FENCED{\ADDPARENS{T}{\bot}} \\
253 \ADDPARENS{\ATTRS{\cdots}T}{n} & = & \ADDPARENS{T}{n} \\
254 \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}] \\
255 \ADDPARENS{B_\kappa^{ab}[T_1,\dots,T_m]}{n} & = & B_\kappa^{ab}[\ADDPARENS{T_1}{n},\dots,\ADDPARENS{T_m}{n}]
262 \caption{\label{tab:annpos} Annotation of level 1 meta variable with position information.\strut}
266 \ANNPOS{l}{p,q} & = & l \\
267 \ANNPOS{\BREAK}{p,q} & = & \BREAK \\
268 \ANNPOS{x}{1,0} & = & \ATTRS{\mathit{pos}=L}{x} \\
269 \ANNPOS{x}{0,1} & = & \ATTRS{\mathit{pos}=R}{x} \\
270 \ANNPOS{x}{p,q} & = & \ATTRS{\mathit{pos}=I}{x} \\
271 \ANNPOS{B_\kappa^{ab}[P]}{p,q} & = & B_\kappa^{ab}[\ANNPOS{P}{p,q}] \\
272 \ANNPOS{B_\kappa^{ab}[\{\BREAK\} P_1\cdots P_n\{\BREAK\}]}{p,q} & = & B_\kappa^{ab}[\begin{array}[t]{@{}l}
273 \{\BREAK\} \ANNPOS{P_1}{p,0} \\
274 \ANNPOS{P_2}{0,0}\cdots\ANNPOS{P_{n-1}}{0,0} \\
275 \ANNPOS{P_n}{0,q}\{\BREAK\}]
278 %% & & L_\kappa[P_1,\dots,P_n] & \mbox{(layout)} \\
279 %% & | & \BREAK & \mbox{(breakpoint)} \\
280 %% & | & \FENCED{P_1\cdots P_n} & \mbox{(fenced)} \\
281 %% V & ::= & & \mbox{(\bf variables)} \\
282 %% & & \TVAR{x} & \mbox{(term variable)} \\
283 %% & | & \NVAR{x} & \mbox{(number variable)} \\
284 %% & | & \IVAR{x} & \mbox{(name variable)} \\[2ex]
285 %% M & ::= & & \mbox{(\bf magic patterns)} \\
286 %% & & \verb+list0+~P~l? & \mbox{(possibly empty list)} \\
287 %% & | & \verb+list1+~P~l? & \mbox{(non-empty list)} \\
288 %% & | & \verb+opt+~P & \mbox{(option)} \\[2ex]
294 \section{Level 2: abstract syntax}
296 \newcommand{\NT}[1]{\langle\mathit{#1}\rangle}
300 \begin{array}{@{}rcll@{}}
301 \NT{term} & ::= & & \mbox{\bf terms} \\
302 & & x & \mbox{(identifier)} \\
303 & | & n & \mbox{(number)} \\
304 & | & \mathrm{URI} & \mbox{(URI)} \\
305 & | & \verb+?+ & \mbox{(implicit)} \\
306 & | & \verb+%+ & \mbox{(placeholder)} \\
307 & | & \verb+?+n~[\verb+[+~\{\NT{subst}\}~\verb+]+] & \mbox{(meta)} \\
308 & | & \verb+let+~\NT{ptname}~\verb+\def+~\NT{term}~\verb+in+~\NT{term} \\
309 & | & \verb+let+~\NT{kind}~\NT{defs}~\verb+in+~\NT{term} \\
310 & | & \NT{binder}~\{\NT{ptnames}\}^{+}~\verb+.+~\NT{term} \\
311 & | & \NT{term}~\NT{term} & \mbox{(application)} \\
312 & | & \verb+Prop+ \mid \verb+Set+ \mid \verb+Type+ \mid \verb+CProp+ & \mbox{(sort)} \\
313 & | & [\verb+[+~\NT{term}~\verb+]+]~\verb+match+~\NT{term}~\verb+with [+~[\NT{rule}~\{\verb+|+~\NT{rule}\}]~\verb+]+ & \mbox{(pattern match)} \\
314 & | & \verb+(+~\NT{term}~\verb+:+~\NT{term}~\verb+)+ & \mbox{(cast)} \\
315 & | & \verb+(+~\NT{term}~\verb+)+ \\
316 & | & \BLOB(\NT{meta},\dots,\NT{meta}) & \mbox{(meta blob)} \\
317 \NT{defs} & ::= & & \mbox{\bf mutual definitions} \\
318 & & \NT{fun}~\{\verb+and+~\NT{fun}\} \\
319 \NT{fun} & ::= & & \mbox{\bf functions} \\
320 & & \NT{arg}~\{\NT{ptnames}\}^{+}~[\verb+on+~x]~\verb+\def+~\NT{term} \\
321 \NT{binder} & ::= & & \mbox{\bf binders} \\
322 & & \verb+\Pi+ \mid \verb+\exists+ \mid \verb+\forall+ \mid \verb+\lambda+ \\
323 \NT{arg} & ::= & & \mbox{\bf single argument} \\
324 & & \verb+_+ \mid x \mid \BLOB(\NT{meta},\dots,\NT{meta}) \\
325 \NT{ptname} & ::= & & \mbox{\bf possibly typed name} \\
327 & | & \verb+(+~\NT{arg}~\verb+:+~\NT{term}~\verb+)+ \\
328 \NT{ptnames} & ::= & & \mbox{\bf bound variables} \\
330 & | & \verb+(+~\NT{arg}~\{\verb+,+~\NT{arg}\}~[\verb+:+~\NT{term}]~\verb+)+ \\
331 \NT{kind} & ::= & & \mbox{\bf induction kind} \\
332 & & \verb+rec+ \mid \verb+corec+ \\
333 \NT{rule} & ::= & & \mbox{\bf rules} \\
334 & & x~\{\NT{ptname}\}~\verb+\Rightarrow+~\NT{term} \\[10ex]
336 \NT{meta} & ::= & & \mbox{\bf meta} \\
337 & & \BLOB(\NT{term},\dots,\NT{term}) & (term blob) \\
338 & | & [\verb+term+]~x \\
339 & | & \verb+number+~x \\
340 & | & \verb+ident+~x \\
341 & | & \verb+fresh+~x \\
342 & | & \verb+anonymous+ \\
343 & | & \verb+fold+~[\verb+left+\mid\verb+right+]~\NT{meta}~\verb+rec+~x~\NT{meta} \\
344 & | & \verb+default+~\NT{meta}~\NT{meta} \\
345 & | & \verb+if+~\NT{meta}~\verb+then+~\NT{meta}~\verb+else+~\NT{meta} \\
352 \caption{\label{tab:wfl2} Well-formedness rules for level 2 patterns.\strut}
355 \renewcommand{\arraystretch}{3.5}
356 \begin{array}{@{}c@{}}
357 \inference[\sc Constr]
358 {P_i :: D_i & \forall i,j, i\ne j => \DOMAIN(D_i) \cap \DOMAIN(D_j) = \emptyset}
359 {\BLOB[P_1,\dots,P_n] :: D_i \oplus \cdots \oplus D_j} \\
360 \inference[\sc TermVar]
362 {[\mathtt{term}]~x :: x : \mathtt{Term}}
364 \inference[\sc NumVar]
366 {\mathtt{number}~x :: x : \mathtt{Number}}
368 \inference[\sc IdentVar]
370 {\mathtt{ident}~x :: x : \mathtt{String}}
372 \inference[\sc FreshVar]
374 {\mathtt{fresh}~x :: x : \mathtt{String}}
376 \inference[\sc Success]
378 {\mathtt{anonymous} :: \emptyset}
381 {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}
382 {\mathtt{fold}~P_1~\mathtt{rec}~x~P_2 :: D_1 \oplus D_2~\mathtt{List}}
384 \inference[\sc Default]
385 {P_1 :: D \oplus D_1 & P_2 :: D & \DOMAIN(D_1) \ne \emptyset & \DOMAIN(D) \cap \DOMAIN(D_1) = \emptyset}
386 {\mathtt{default}~P_1~P_2 :: D \oplus D_1~\mathtt{Option}}
389 {P_1 :: \emptyset & P_2 :: D & P_3 :: D }
390 {\mathtt{if}~P_1~\mathtt{then}~P_2~\mathtt{else}~P_3 :: D}
394 {\mathtt{fail} : \emptyset}
395 %% & | & \verb+if+~\NT{meta}~\verb+then+~\NT{meta}~\verb+else+~\NT{meta} \\
403 \caption{\label{tab:l2match} Pattern matching of level 2 terms.\strut}
406 \renewcommand{\arraystretch}{3.5}
407 \begin{array}{@{}c@{}}
408 \inference[\sc Constr]
409 {t_i \in P_i ~> \mathcal E_i & i\ne j => \DOMAIN(\mathcal E_i)\cap\DOMAIN(\mathcal E_j)=\emptyset}
410 {C[t_1,\dots,t_n] \in C[P_1,\dots,P_n] ~> \mathcal E_1 \oplus \cdots \oplus \mathcal E_n}
412 \inference[\sc TermVar]
414 {t \in [\mathtt{term}]~x ~> [x |-> \mathtt{Term}~t]}
416 \inference[\sc NumVar]
418 {n \in \mathtt{number}~x ~> [x |-> \mathtt{Number}~n]}
420 \inference[\sc IdentVar]
422 {x \in \mathtt{ident}~x ~> [x |-> \mathtt{String}~x]}
424 \inference[\sc FreshVar]
426 {x \in \mathtt{fresh}~x ~> [x |-> \mathtt{String}~x]}
428 \inference[\sc Success]
430 {t \in \mathtt{anonymous} ~> \emptyset}
432 \inference[\sc DefaultT]
433 {t \in P_1 ~> \mathcal E}
434 {t \in \mathtt{default}~P_1~P_2 ~> \mathcal E'}
436 \mathcal E'(x) = \left\{
437 \renewcommand{\arraystretch}{1}
439 \mathtt{Some}~\mathcal{E}(x) & x \in \NAMES(P_1) \setminus \NAMES(P_2) \\
440 \mathcal{E}(x) & \mbox{otherwise}
444 \inference[\sc DefaultF]
445 {t \not\in P_1 & t \in P_2 ~> \mathcal E}
446 {t \in \mathtt{default}~P_1~P_2 ~> \mathcal E'}
448 \mathcal E'(x) = \left\{
449 \renewcommand{\arraystretch}{1}
451 \mathtt{None} & x \in \NAMES(P_1) \setminus \NAMES(P_2) \\
452 \mathcal{E}(x) & \mbox{otherwise}
457 {t \in P_1 ~> \mathcal E' & t \in P_2 ~> \mathcal E}
458 {t \in \mathtt{if}~P_1~\mathtt{then}~P_2~\mathtt{else}~P_3 ~> \mathcal E}
461 {t \not\in P_1 & t \in P_3 ~> \mathcal E}
462 {t \in \mathtt{if}~P_1~\mathtt{then}~P_2~\mathtt{else}~P_3 ~> \mathcal E}
464 \inference[\sc FoldR]
465 {t \in P_2 ~> \mathcal E & \mathcal{E}(x) \in \mathtt{fold}~P_1~\mathtt{rec}~x~P_2 ~> \mathcal E'}
466 {t \in \mathtt{fold}~P_1~\mathtt{rec}~x~P_2 ~> \mathcal E''}
468 \mathcal E''(y) = \left\{
469 \renewcommand{\arraystretch}{1}
471 \mathcal{E}(y)::\mathcal{E}'(y) & y \in \NAMES(P_2) \setminus \{x\} \\
472 \mathcal{E}'(y) & \mbox{otherwise}
476 \inference[\sc FoldB]
477 {t \not\in P_2 & t \in P_1 ~> \mathcal E}
478 {t \in \mathtt{fold}~P_1~\mathtt{rec}~x~P_2 ~> \mathcal E'}
480 \mathcal E'(y) = \left\{
481 \renewcommand{\arraystretch}{1}
483 [] & y \in \NAMES(P_2) \setminus \{x\} \\
484 \mathcal{E}(y) & \mbox{otherwise}