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323  1 
%% $Id$ 
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\chapter{Syntax Transformations} \label{chap:syntax} 

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\newcommand\ttapp{\mathrel{\hbox{\tt\$}}} 

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\newcommand\mtt[1]{\mbox{\tt #1}} 

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\newcommand\ttfct[1]{\mathop{\mtt{#1}}\nolimits} 

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\newcommand\Constant{\ttfct{Constant}} 

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\newcommand\Variable{\ttfct{Variable}} 

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\newcommand\Appl[1]{\ttfct{Appl}\,[#1]} 

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\index{syntax!transformations(} 

10 

11 
This chapter is intended for experienced Isabelle users who need to define 

12 
macros or code their own translation functions. It describes the 

13 
transformations between parse trees, abstract syntax trees and terms. 

14 

15 

16 
\section{Abstract syntax trees} \label{sec:asts} 

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\index{ASTs(} 
323  18 

19 
The parser, given a token list from the lexer, applies productions to yield 

20 
a parse tree\index{parse trees}. By applying some internal transformations 

21 
the parse tree becomes an abstract syntax tree, or \AST{}. Macro 

22 
expansion, further translations and finally type inference yields a 

23 
welltyped term. The printing process is the reverse, except for some 

24 
subtleties to be discussed later. 

25 

26 
Figure~\ref{fig:parse_print} outlines the parsing and printing process. 

27 
Much of the complexity is due to the macro mechanism. Using macros, you 

28 
can specify most forms of concrete syntax without writing any \ML{} code. 

29 

30 
\begin{figure} 

31 
\begin{center} 

32 
\begin{tabular}{cl} 

33 
string & \\ 

3108  34 
$\downarrow$ & lexer, parser \\ 
323  35 
parse tree & \\ 
36 
$\downarrow$ & parse \AST{} translation \\ 

37 
\AST{} & \\ 

38 
$\downarrow$ & \AST{} rewriting (macros) \\ 

39 
\AST{} & \\ 

40 
$\downarrow$ & parse translation, type inference \\ 

41 
 welltyped term  & \\ 

42 
$\downarrow$ & print translation \\ 

43 
\AST{} & \\ 

44 
$\downarrow$ & \AST{} rewriting (macros) \\ 

45 
\AST{} & \\ 

3108  46 
$\downarrow$ & print \AST{} translation, token translation \\ 
323  47 
string & 
48 
\end{tabular} 

49 

50 
\end{center} 

51 
\caption{Parsing and printing}\label{fig:parse_print} 

52 
\end{figure} 

53 

54 
Abstract syntax trees are an intermediate form between the raw parse trees 

55 
and the typed $\lambda$terms. An \AST{} is either an atom (constant or 

56 
variable) or a list of {\em at least two\/} subtrees. Internally, they 

57 
have type \mltydx{Syntax.ast}: \index{*Constant} \index{*Variable} 

58 
\index{*Appl} 

59 
\begin{ttbox} 

60 
datatype ast = Constant of string 

61 
 Variable of string 

62 
 Appl of ast list 

63 
\end{ttbox} 

64 
% 

65 
Isabelle uses an Sexpression syntax for abstract syntax trees. Constant 

66 
atoms are shown as quoted strings, variable atoms as nonquoted strings and 

332  67 
applications as a parenthesised list of subtrees. For example, the \AST 
323  68 
\begin{ttbox} 
69 
Appl [Constant "_constrain", 

70 
Appl [Constant "_abs", Variable "x", Variable "t"], 

71 
Appl [Constant "fun", Variable "'a", Variable "'b"]] 

72 
\end{ttbox} 

73 
is shown as {\tt ("_constrain" ("_abs" x t) ("fun" 'a 'b))}. 

74 
Both {\tt ()} and {\tt (f)} are illegal because they have too few 

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subtrees. 
323  76 

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The resemblance to Lisp's Sexpressions is intentional, but there are two 

78 
kinds of atomic symbols: $\Constant x$ and $\Variable x$. Do not take the 

79 
names {\tt Constant} and {\tt Variable} too literally; in the later 

80 
translation to terms, $\Variable x$ may become a constant, free or bound 

81 
variable, even a type constructor or class name; the actual outcome depends 

82 
on the context. 

83 

84 
Similarly, you can think of ${\tt (} f~x@1~\ldots~x@n{\tt )}$ as the 

85 
application of~$f$ to the arguments $x@1, \ldots, x@n$. But the kind of 

86 
application is determined later by context; it could be a type constructor 

87 
applied to types. 

88 

89 
Forms like {\tt (("_abs" x $t$) $u$)} are legal, but \AST{}s are 

90 
firstorder: the {\tt "_abs"} does not bind the {\tt x} in any way. Later 

91 
at the term level, {\tt ("_abs" x $t$)} will become an {\tt Abs} node and 

92 
occurrences of {\tt x} in $t$ will be replaced by bound variables (the term 

93 
constructor \ttindex{Bound}). 

94 

95 

6618  96 
\section{Transforming parse trees to ASTs}\label{sec:astofpt} 
323  97 
\index{ASTs!made from parse trees} 
98 
\newcommand\astofpt[1]{\lbrakk#1\rbrakk} 

99 

100 
The parse tree is the raw output of the parser. Translation functions, 

101 
called {\bf parse AST translations}\indexbold{translations!parse AST}, 

102 
transform the parse tree into an abstract syntax tree. 

103 

104 
The parse tree is constructed by nesting the righthand sides of the 

105 
productions used to recognize the input. Such parse trees are simply lists 

106 
of tokens and constituent parse trees, the latter representing the 

107 
nonterminals of the productions. Let us refer to the actual productions in 

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the form displayed by {\tt print_syntax} (see \S\ref{sec:inspctthy} for an 
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example). 
323  110 

111 
Ignoring parse \AST{} translations, parse trees are transformed to \AST{}s 

112 
by stripping out delimiters and copy productions. More precisely, the 

113 
mapping $\astofpt{}$ is derived from the productions as follows: 

114 
\begin{itemize} 

115 
\item Name tokens: $\astofpt{t} = \Variable s$, where $t$ is an \ndx{id}, 

14947  116 
\ndx{var}, \ndx{tid}, \ndx{tvar}, \ndx{num}, \ndx{xnum} or \ndx{xstr} token, 
117 
and $s$ its associated string. Note that for {\tt xstr} this does not 

118 
include the quotes. 

323  119 

120 
\item Copy productions:\index{productions!copy} 

121 
$\astofpt{\ldots P \ldots} = \astofpt{P}$. Here $\ldots$ stands for 

122 
strings of delimiters, which are discarded. $P$ stands for the single 

123 
constituent that is not a delimiter; it is either a nonterminal symbol or 

124 
a name token. 

125 

126 
\item 0ary productions: $\astofpt{\ldots \mtt{=>} c} = \Constant c$. 

127 
Here there are no constituents other than delimiters, which are 

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discarded. 
323  129 

130 
\item $n$ary productions, where $n \ge 1$: delimiters are discarded and 

131 
the remaining constituents $P@1$, \ldots, $P@n$ are built into an 

132 
application whose head constant is~$c$: 

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\[ \astofpt{\ldots P@1 \ldots P@n \ldots \mtt{=>} c} = 
323  134 
\Appl{\Constant c, \astofpt{P@1}, \ldots, \astofpt{P@n}} 
135 
\] 

136 
\end{itemize} 

137 
Figure~\ref{fig:parse_ast} presents some simple examples, where {\tt ==}, 

138 
{\tt _appl}, {\tt _args}, and so forth name productions of the Pure syntax. 

139 
These examples illustrate the need for further translations to make \AST{}s 

140 
closer to the typed $\lambda$calculus. The Pure syntax provides 

141 
predefined parse \AST{} translations\index{translations!parse AST} for 

142 
ordinary applications, type applications, nested abstractions, meta 

143 
implications and function types. Figure~\ref{fig:parse_ast_tr} shows their 

144 
effect on some representative input strings. 

145 

146 

147 
\begin{figure} 

148 
\begin{center} 

149 
\tt\begin{tabular}{ll} 

150 
\rm input string & \rm \AST \\\hline 

151 
"f" & f \\ 

152 
"'a" & 'a \\ 

153 
"t == u" & ("==" t u) \\ 

154 
"f(x)" & ("_appl" f x) \\ 

155 
"f(x, y)" & ("_appl" f ("_args" x y)) \\ 

156 
"f(x, y, z)" & ("_appl" f ("_args" x ("_args" y z))) \\ 

157 
"\%x y.\ t" & ("_lambda" ("_idts" x y) t) \\ 

158 
\end{tabular} 

159 
\end{center} 

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\caption{Parsing examples using the Pure syntax}\label{fig:parse_ast} 
323  161 
\end{figure} 
162 

163 
\begin{figure} 

164 
\begin{center} 

165 
\tt\begin{tabular}{ll} 

166 
\rm input string & \rm \AST{} \\\hline 

167 
"f(x, y, z)" & (f x y z) \\ 

168 
"'a ty" & (ty 'a) \\ 

169 
"('a, 'b) ty" & (ty 'a 'b) \\ 

170 
"\%x y z.\ t" & ("_abs" x ("_abs" y ("_abs" z t))) \\ 

171 
"\%x ::\ 'a.\ t" & ("_abs" ("_constrain" x 'a) t) \\ 

172 
"[ P; Q; R ] => S" & ("==>" P ("==>" Q ("==>" R S))) \\ 

173 
"['a, 'b, 'c] => 'd" & ("fun" 'a ("fun" 'b ("fun" 'c 'd))) 

174 
\end{tabular} 

175 
\end{center} 

176 
\caption{Builtin parse \AST{} translations}\label{fig:parse_ast_tr} 

177 
\end{figure} 

178 

179 
The names of constant heads in the \AST{} control the translation process. 

180 
The list of constants invoking parse \AST{} translations appears in the 

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output of {\tt print_syntax} under {\tt parse_ast_translation}. 
323  182 

183 

6618  184 
\section{Transforming ASTs to terms}\label{sec:termofast} 
323  185 
\index{terms!made from ASTs} 
186 
\newcommand\termofast[1]{\lbrakk#1\rbrakk} 

187 

188 
The \AST{}, after application of macros (see \S\ref{sec:macros}), is 

189 
transformed into a term. This term is probably illtyped since type 

190 
inference has not occurred yet. The term may contain type constraints 

191 
consisting of applications with head {\tt "_constrain"}; the second 

192 
argument is a type encoded as a term. Type inference later introduces 

193 
correct types or rejects the input. 

194 

195 
Another set of translation functions, namely parse 

196 
translations\index{translations!parse}, may affect this process. If we 

197 
ignore parse translations for the time being, then \AST{}s are transformed 

198 
to terms by mapping \AST{} constants to constants, \AST{} variables to 

199 
schematic or free variables and \AST{} applications to applications. 

200 

201 
More precisely, the mapping $\termofast{}$ is defined by 

202 
\begin{itemize} 

203 
\item Constants: $\termofast{\Constant x} = \ttfct{Const} (x, 

204 
\mtt{dummyT})$. 

205 

206 
\item Schematic variables: $\termofast{\Variable \mtt{"?}xi\mtt"} = 

207 
\ttfct{Var} ((x, i), \mtt{dummyT})$, where $x$ is the base name and $i$ 

208 
the index extracted from~$xi$. 

209 

210 
\item Free variables: $\termofast{\Variable x} = \ttfct{Free} (x, 

211 
\mtt{dummyT})$. 

212 

213 
\item Function applications with $n$ arguments: 

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\[ \termofast{\Appl{f, x@1, \ldots, x@n}} = 
323  215 
\termofast{f} \ttapp 
216 
\termofast{x@1} \ttapp \ldots \ttapp \termofast{x@n} 

217 
\] 

218 
\end{itemize} 

219 
Here \ttindex{Const}, \ttindex{Var}, \ttindex{Free} and 

220 
\verb$\index{$@{\tt\$}} are constructors of the datatype \mltydx{term}, 

221 
while \ttindex{dummyT} stands for some dummy type that is ignored during 

222 
type inference. 

223 

224 
So far the outcome is still a firstorder term. Abstractions and bound 

225 
variables (constructors \ttindex{Abs} and \ttindex{Bound}) are introduced 

226 
by parse translations. Such translations are attached to {\tt "_abs"}, 

227 
{\tt "!!"} and userdefined binders. 

228 

229 

230 
\section{Printing of terms} 

231 
\newcommand\astofterm[1]{\lbrakk#1\rbrakk}\index{ASTs!made from terms} 

232 

233 
The output phase is essentially the inverse of the input phase. Terms are 

234 
translated via abstract syntax trees into strings. Finally the strings are 

235 
pretty printed. 

236 

237 
Print translations (\S\ref{sec:tr_funs}) may affect the transformation of 

238 
terms into \AST{}s. Ignoring those, the transformation maps 

239 
term constants, variables and applications to the corresponding constructs 

240 
on \AST{}s. Abstractions are mapped to applications of the special 

241 
constant {\tt _abs}. 

242 

243 
More precisely, the mapping $\astofterm{}$ is defined as follows: 

244 
\begin{itemize} 

245 
\item $\astofterm{\ttfct{Const} (x, \tau)} = \Constant x$. 

246 

247 
\item $\astofterm{\ttfct{Free} (x, \tau)} = constrain (\Variable x, 

248 
\tau)$. 

249 

250 
\item $\astofterm{\ttfct{Var} ((x, i), \tau)} = constrain (\Variable 

251 
\mtt{"?}xi\mtt", \tau)$, where $\mtt?xi$ is the string representation of 

252 
the {\tt indexname} $(x, i)$. 

253 

254 
\item For the abstraction $\lambda x::\tau.t$, let $x'$ be a variant 

255 
of~$x$ renamed to differ from all names occurring in~$t$, and let $t'$ 

256 
be obtained from~$t$ by replacing all bound occurrences of~$x$ by 

257 
the free variable $x'$. This replaces corresponding occurrences of the 

258 
constructor \ttindex{Bound} by the term $\ttfct{Free} (x', 

259 
\mtt{dummyT})$: 

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\[ \astofterm{\ttfct{Abs} (x, \tau, t)} = 
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\Appl{\Constant \mtt{"_abs"}, 
8136  262 
constrain(\Variable x', \tau), \astofterm{t'}} 
323  263 
\] 
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\item $\astofterm{\ttfct{Bound} i} = \Variable \mtt{"B.}i\mtt"$. 
323  266 
The occurrence of constructor \ttindex{Bound} should never happen 
267 
when printing welltyped terms; it indicates a de Bruijn index with no 

268 
matching abstraction. 

269 

270 
\item Where $f$ is not an application, 

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\[ \astofterm{f \ttapp x@1 \ttapp \ldots \ttapp x@n} = 
323  272 
\Appl{\astofterm{f}, \astofterm{x@1}, \ldots,\astofterm{x@n}} 
273 
\] 

274 
\end{itemize} 

275 
% 

332  276 
Type constraints\index{type constraints} are inserted to allow the printing 
277 
of types. This is governed by the boolean variable \ttindex{show_types}: 

323  278 
\begin{itemize} 
279 
\item $constrain(x, \tau) = x$ \ if $\tau = \mtt{dummyT}$ \index{*dummyT} or 

332  280 
\ttindex{show_types} is set to {\tt false}. 
323  281 

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\item $constrain(x, \tau) = \Appl{\Constant \mtt{"_constrain"}, x, 
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\astofterm{\tau}}$ \ otherwise. 
323  284 

285 
Here, $\astofterm{\tau}$ is the \AST{} encoding of $\tau$: type 

286 
constructors go to {\tt Constant}s; type identifiers go to {\tt 

287 
Variable}s; type applications go to {\tt Appl}s with the type 

288 
constructor as the first element. If \ttindex{show_sorts} is set to 

289 
{\tt true}, some type variables are decorated with an \AST{} encoding 

290 
of their sort. 

291 
\end{itemize} 

292 
% 

293 
The \AST{}, after application of macros (see \S\ref{sec:macros}), is 

294 
transformed into the final output string. The builtin {\bf print AST 

332  295 
translations}\indexbold{translations!print AST} reverse the 
323  296 
parse \AST{} translations of Fig.\ts\ref{fig:parse_ast_tr}. 
297 

298 
For the actual printing process, the names attached to productions 

299 
of the form $\ldots A^{(p@1)}@1 \ldots A^{(p@n)}@n \ldots \mtt{=>} c$ play 

300 
a vital role. Each \AST{} with constant head $c$, namely $\mtt"c\mtt"$ or 

301 
$(\mtt"c\mtt"~ x@1 \ldots x@n)$, is printed according to the production 

302 
for~$c$. Each argument~$x@i$ is converted to a string, and put in 

303 
parentheses if its priority~$(p@i)$ requires this. The resulting strings 

304 
and their syntactic sugar (denoted by \dots{} above) are joined to make a 

305 
single string. 

306 

3108  307 
If an application $(\mtt"c\mtt"~ x@1 \ldots x@m)$ has more arguments 
308 
than the corresponding production, it is first split into 

309 
$((\mtt"c\mtt"~ x@1 \ldots x@n) ~ x@{n+1} \ldots x@m)$. Applications 

310 
with too few arguments or with nonconstant head or without a 

311 
corresponding production are printed as $f(x@1, \ldots, x@l)$ or 

312 
$(\alpha@1, \ldots, \alpha@l) ty$. Multiple productions associated 

313 
with some name $c$ are tried in order of appearance. An occurrence of 

323  314 
$\Variable x$ is simply printed as~$x$. 
315 

316 
Blanks are {\em not\/} inserted automatically. If blanks are required to 

317 
separate tokens, specify them in the mixfix declaration, possibly preceded 

318 
by a slash~({\tt/}) to allow a line break. 

319 
\index{ASTs)} 

320 

321 

322 

3108  323 
\section{Macros: syntactic rewriting} \label{sec:macros} 
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\index{macros(}\index{rewriting!syntactic(} 
323  325 

326 
Mixfix declarations alone can handle situations where there is a direct 

327 
connection between the concrete syntax and the underlying term. Sometimes 

328 
we require a more elaborate concrete syntax, such as quantifiers and list 

329 
notation. Isabelle's {\bf macros} and {\bf translation functions} can 

330 
perform translations such as 

331 
\begin{center}\tt 

332 
\begin{tabular}{r@{$\quad\protect\rightleftharpoons\quad$}l} 

333 
ALL x:A.P & Ball(A, \%x.P) \\ \relax 

334 
[x, y, z] & Cons(x, Cons(y, Cons(z, Nil))) 

335 
\end{tabular} 

336 
\end{center} 

337 
Translation functions (see \S\ref{sec:tr_funs}) must be coded in ML; they 

338 
are the most powerful translation mechanism but are difficult to read or 

339 
write. Macros are specified by firstorder rewriting systems that operate 

340 
on abstract syntax trees. They are usually easy to read and write, and can 

341 
express all but the most obscure translations. 

342 

3108  343 
Figure~\ref{fig:set_trans} defines a fragment of firstorder logic and 
344 
set theory.\footnote{This and the following theories are complete 

345 
working examples, though they specify only syntax, no axioms. The 

346 
file {\tt ZF/ZF.thy} presents a full set theory definition, 

347 
including many macro rules.} Theory {\tt SetSyntax} declares 

348 
constants for set comprehension ({\tt Collect}), replacement ({\tt 

349 
Replace}) and bounded universal quantification ({\tt Ball}). Each 

350 
of these binds some variables. Without additional syntax we should 

351 
have to write $\forall x \in A. P$ as {\tt Ball(A,\%x.P)}, and 

352 
similarly for the others. 

323  353 

354 
\begin{figure} 

355 
\begin{ttbox} 

3108  356 
SetSyntax = Pure + 
323  357 
types 
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i o 
323  359 
arities 
360 
i, o :: logic 

361 
consts 

1387  362 
Trueprop :: o => prop ("_" 5) 
363 
Collect :: [i, i => o] => i 

364 
Replace :: [i, [i, i] => o] => i 

365 
Ball :: [i, i => o] => o 

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syntax 
1387  367 
"{\at}Collect" :: [idt, i, o] => i ("(1{\ttlbrace}_:_./ _{\ttrbrace})") 
368 
"{\at}Replace" :: [idt, idt, i, o] => i ("(1{\ttlbrace}_./ _:_, _{\ttrbrace})") 

369 
"{\at}Ball" :: [idt, i, o] => o ("(3ALL _:_./ _)" 10) 

323  370 
translations 
371 
"{\ttlbrace}x:A. P{\ttrbrace}" == "Collect(A, \%x. P)" 

372 
"{\ttlbrace}y. x:A, Q{\ttrbrace}" == "Replace(A, \%x y. Q)" 

373 
"ALL x:A. P" == "Ball(A, \%x. P)" 

374 
end 

375 
\end{ttbox} 

376 
\caption{Macro example: set theory}\label{fig:set_trans} 

377 
\end{figure} 

378 

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The theory specifies a variablebinding syntax through additional productions 
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380 
that have mixfix declarations. Each noncopy production must specify some 
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constant, which is used for building \AST{}s. \index{constants!syntactic} The 
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additional constants are decorated with {\tt\at} to stress their purely 
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syntactic purpose; they may not occur within the final welltyped terms, 
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being declared as {\tt syntax} rather than {\tt consts}. 
323  385 

386 
The translations cause the replacement of external forms by internal forms 

387 
after parsing, and vice versa before printing of terms. As a specification 

388 
of the set theory notation, they should be largely selfexplanatory. The 

389 
syntactic constants, {\tt\at Collect}, {\tt\at Replace} and {\tt\at Ball}, 

390 
appear implicitly in the macro rules via their mixfix forms. 

391 

392 
Macros can define variablebinding syntax because they operate on \AST{}s, 

393 
which have no inbuilt notion of bound variable. The macro variables {\tt 

394 
x} and~{\tt y} have type~{\tt idt} and therefore range over identifiers, 

395 
in this case bound variables. The macro variables {\tt P} and~{\tt Q} 

396 
range over formulae containing bound variable occurrences. 

397 

398 
Other applications of the macro system can be less straightforward, and 

399 
there are peculiarities. The rest of this section will describe in detail 

400 
how Isabelle macros are preprocessed and applied. 

401 

402 

403 
\subsection{Specifying macros} 

404 
Macros are basically rewrite rules on \AST{}s. But unlike other macro 

405 
systems found in programming languages, Isabelle's macros work in both 

406 
directions. Therefore a syntax contains two lists of rewrites: one for 

407 
parsing and one for printing. 

408 

409 
\index{*translations section} 

410 
The {\tt translations} section specifies macros. The syntax for a macro is 

411 
\[ (root)\; string \quad 

412 
\left\{\begin{array}[c]{c} \mtt{=>} \\ \mtt{<=} \\ \mtt{==} \end{array} 

413 
\right\} \quad 

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414 
(root)\; string 
323  415 
\] 
416 
% 

417 
This specifies a parse rule ({\tt =>}), a print rule ({\tt <=}), or both 

418 
({\tt ==}). The two strings specify the left and righthand sides of the 

419 
macro rule. The $(root)$ specification is optional; it specifies the 

420 
nonterminal for parsing the $string$ and if omitted defaults to {\tt 

421 
logic}. \AST{} rewrite rules $(l, r)$ must obey certain conditions: 

422 
\begin{itemize} 

423 
\item Rules must be left linear: $l$ must not contain repeated variables. 

424 

425 
\item Every variable in~$r$ must also occur in~$l$. 

426 
\end{itemize} 

427 

3108  428 
Macro rules may refer to any syntax from the parent theories. They 
429 
may also refer to anything defined before the current {\tt 

323  430 
translations} section  including any mixfix declarations. 
431 

432 
Upon declaration, both sides of the macro rule undergo parsing and parse 

433 
\AST{} translations (see \S\ref{sec:asts}), but do not themselves undergo 

434 
macro expansion. The lexer runs in a different mode that additionally 

435 
accepts identifiers of the form $\_~letter~quasiletter^*$ (like {\tt _idt}, 

436 
{\tt _K}). Thus, a constant whose name starts with an underscore can 

437 
appear in macro rules but not in ordinary terms. 

438 

439 
Some atoms of the macro rule's \AST{} are designated as constants for 

440 
matching. These are all names that have been declared as classes, types or 

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441 
constants (logical and syntactic). 
323  442 

3108  443 
The result of this preprocessing is two lists of macro rules, each 
444 
stored as a pair of \AST{}s. They can be viewed using {\tt 

445 
print_syntax} (sections \ttindex{parse_rules} and 

446 
\ttindex{print_rules}). For theory~{\tt SetSyntax} of 

447 
Fig.~\ref{fig:set_trans} these are 

323  448 
\begin{ttbox} 
449 
parse_rules: 

450 
("{\at}Collect" x A P) > ("Collect" A ("_abs" x P)) 

451 
("{\at}Replace" y x A Q) > ("Replace" A ("_abs" x ("_abs" y Q))) 

452 
("{\at}Ball" x A P) > ("Ball" A ("_abs" x P)) 

453 
print_rules: 

454 
("Collect" A ("_abs" x P)) > ("{\at}Collect" x A P) 

455 
("Replace" A ("_abs" x ("_abs" y Q))) > ("{\at}Replace" y x A Q) 

456 
("Ball" A ("_abs" x P)) > ("{\at}Ball" x A P) 

457 
\end{ttbox} 

458 

459 
\begin{warn} 

460 
Avoid choosing variable names that have previously been used as 

461 
constants, types or type classes; the {\tt consts} section in the output 

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462 
of {\tt print_syntax} lists all such names. If a macro rule works 
323  463 
incorrectly, inspect its internal form as shown above, recalling that 
464 
constants appear as quoted strings and variables without quotes. 

465 
\end{warn} 

466 

467 
\begin{warn} 

468 
If \ttindex{eta_contract} is set to {\tt true}, terms will be 

469 
$\eta$contracted {\em before\/} the \AST{} rewriter sees them. Thus some 

470 
abstraction nodes needed for print rules to match may vanish. For example, 

332  471 
\verbBall(A, %x. P(x)) contracts to {\tt Ball(A, P)}; the print rule does 
323  472 
not apply and the output will be {\tt Ball(A, P)}. This problem would not 
473 
occur if \ML{} translation functions were used instead of macros (as is 

474 
done for binder declarations). 

475 
\end{warn} 

476 

477 

478 
\begin{warn} 

479 
Another trap concerns type constraints. If \ttindex{show_types} is set to 

480 
{\tt true}, bound variables will be decorated by their meta types at the 

481 
binding place (but not at occurrences in the body). Matching with 

482 
\verbCollect(A, %x. P) binds {\tt x} to something like {\tt ("_constrain" y 

483 
"i")} rather than only {\tt y}. \AST{} rewriting will cause the constraint to 

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484 
appear in the external form, say \verb{y::i:A::i. P::o}. 
323  485 

486 
To allow such constraints to be reread, your syntax should specify bound 

487 
variables using the nonterminal~\ndx{idt}. This is the case in our 

3108  488 
example. Choosing {\tt id} instead of {\tt idt} is a common error. 
323  489 
\end{warn} 
490 

491 

492 

493 
\subsection{Applying rules} 

494 
As a term is being parsed or printed, an \AST{} is generated as an 

495 
intermediate form (recall Fig.\ts\ref{fig:parse_print}). The \AST{} is 

332  496 
normalised by applying macro rules in the manner of a traditional term 
323  497 
rewriting system. We first examine how a single rule is applied. 
498 

332  499 
Let $t$ be the abstract syntax tree to be normalised and $(l, r)$ some 
323  500 
translation rule. A subtree~$u$ of $t$ is a {\bf redex} if it is an 
501 
instance of~$l$; in this case $l$ is said to {\bf match}~$u$. A redex 

502 
matched by $l$ may be replaced by the corresponding instance of~$r$, thus 

503 
{\bf rewriting} the \AST~$t$. Matching requires some notion of {\bf 

504 
placeholders} that may occur in rule patterns but not in ordinary 

505 
\AST{}s; {\tt Variable} atoms serve this purpose. 

506 

507 
The matching of the object~$u$ by the pattern~$l$ is performed as follows: 

508 
\begin{itemize} 

509 
\item Every constant matches itself. 

510 

511 
\item $\Variable x$ in the object matches $\Constant x$ in the pattern. 

512 
This point is discussed further below. 

513 

514 
\item Every \AST{} in the object matches $\Variable x$ in the pattern, 

515 
binding~$x$ to~$u$. 

516 

517 
\item One application matches another if they have the same number of 

518 
subtrees and corresponding subtrees match. 

519 

520 
\item In every other case, matching fails. In particular, {\tt 

521 
Constant}~$x$ can only match itself. 

522 
\end{itemize} 

523 
A successful match yields a substitution that is applied to~$r$, generating 

524 
the instance that replaces~$u$. 

525 

526 
The second case above may look odd. This is where {\tt Variable}s of 

527 
nonrule \AST{}s behave like {\tt Constant}s. Recall that \AST{}s are not 

528 
far removed from parse trees; at this level it is not yet known which 

529 
identifiers will become constants, bounds, frees, types or classes. As 

530 
\S\ref{sec:asts} describes, former parse tree heads appear in \AST{}s as 

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531 
{\tt Constant}s, while the name tokens \ndx{id}, \ndx{var}, \ndx{tid}, 
14947  532 
\ndx{tvar}, \ndx{num}, \ndx{xnum} and \ndx{xstr} become {\tt Variable}s. On the other 
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533 
hand, when \AST{}s generated from terms for printing, all constants and type 
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534 
constructors become {\tt Constant}s; see \S\ref{sec:asts}. Thus \AST{}s may 
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535 
contain a messy mixture of {\tt Variable}s and {\tt Constant}s. This is 
323  536 
insignificant at macro level because matching treats them alike. 
537 

538 
Because of this behaviour, different kinds of atoms with the same name are 

539 
indistinguishable, which may make some rules prone to misbehaviour. Example: 

540 
\begin{ttbox} 

541 
types 

542 
Nil 

543 
consts 

1387  544 
Nil :: 'a list 
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545 
syntax 
1387  546 
"[]" :: 'a list ("[]") 
323  547 
translations 
548 
"[]" == "Nil" 

549 
\end{ttbox} 

550 
The term {\tt Nil} will be printed as {\tt []}, just as expected. 

551 
The term \verb%Nil.t will be printed as \verb%[].t, which might not be 

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552 
expected! Guess how type~{\tt Nil} is printed? 
323  553 

14893  554 
Normalizing an \AST{} involves repeatedly applying macro rules until none are 
555 
applicable. Macro rules are chosen in order of appearance in the theory 

556 
definitions. You can watch the normalization of \AST{}s during parsing and 

557 
printing by setting \ttindex{Syntax.trace_ast} to {\tt true}.\index{tracing!of 

558 
macros} The information displayed when tracing includes the \AST{} before 

559 
normalization ({\tt pre}), redexes with results ({\tt rewrote}), the normal 

560 
form finally reached ({\tt post}) and some statistics ({\tt normalize}). 

323  561 

562 
\subsection{Example: the syntax of finite sets} 

563 
\index{examples!of macros} 

564 

565 
This example demonstrates the use of recursive macros to implement a 

566 
convenient notation for finite sets. 

567 
\index{*empty constant}\index{*insert constant}\index{{}@\verb'{}' symbol} 

568 
\index{"@Enum@{\tt\at Enum} constant} 

569 
\index{"@Finset@{\tt\at Finset} constant} 

570 
\begin{ttbox} 

3108  571 
FinSyntax = SetSyntax + 
323  572 
types 
573 
is 

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574 
syntax 
1387  575 
"" :: i => is ("_") 
576 
"{\at}Enum" :: [i, is] => is ("_,/ _") 

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577 
consts 
1387  578 
empty :: i ("{\ttlbrace}{\ttrbrace}") 
579 
insert :: [i, i] => i 

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580 
syntax 
1387  581 
"{\at}Finset" :: is => i ("{\ttlbrace}(_){\ttrbrace}") 
323  582 
translations 
583 
"{\ttlbrace}x, xs{\ttrbrace}" == "insert(x, {\ttlbrace}xs{\ttrbrace})" 

584 
"{\ttlbrace}x{\ttrbrace}" == "insert(x, {\ttlbrace}{\ttrbrace})" 

585 
end 

586 
\end{ttbox} 

587 
Finite sets are internally built up by {\tt empty} and {\tt insert}. The 

588 
declarations above specify \verb{x, y, z} as the external representation 

589 
of 

590 
\begin{ttbox} 

591 
insert(x, insert(y, insert(z, empty))) 

592 
\end{ttbox} 

593 
The nonterminal symbol~\ndx{is} stands for one or more objects of type~{\tt 

594 
i} separated by commas. The mixfix declaration \hbox{\verb"_,/ _"} 

595 
allows a line break after the comma for \rmindex{pretty printing}; if no 

596 
line break is required then a space is printed instead. 

597 

598 
The nonterminal is declared as the type~{\tt is}, but with no {\tt arities} 

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599 
declaration. Hence {\tt is} is not a logical type and may be used safely as 
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600 
a new nonterminal for custom syntax. The nonterminal~{\tt is} can later be 
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601 
reused for other enumerations of type~{\tt i} like lists or tuples. If we 
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602 
had needed polymorphic enumerations, we could have used the predefined 
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603 
nonterminal symbol \ndx{args} and skipped this part altogether. 
323  604 

605 
\index{"@Finset@{\tt\at Finset} constant} 

606 
Next follows {\tt empty}, which is already equipped with its syntax 

607 
\verb{}, and {\tt insert} without concrete syntax. The syntactic 

608 
constant {\tt\at Finset} provides concrete syntax for enumerations of~{\tt 

609 
i} enclosed in curly braces. Remember that a pair of parentheses, as in 

610 
\verb"{(_)}", specifies a block of indentation for pretty printing. 

611 

612 
The translations may look strange at first. Macro rules are best 

613 
understood in their internal forms: 

614 
\begin{ttbox} 

615 
parse_rules: 

616 
("{\at}Finset" ("{\at}Enum" x xs)) > ("insert" x ("{\at}Finset" xs)) 

617 
("{\at}Finset" x) > ("insert" x "empty") 

618 
print_rules: 

619 
("insert" x ("{\at}Finset" xs)) > ("{\at}Finset" ("{\at}Enum" x xs)) 

620 
("insert" x "empty") > ("{\at}Finset" x) 

621 
\end{ttbox} 

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622 
This shows that \verb{x,xs} indeed matches any set enumeration of at least 
323  623 
two elements, binding the first to {\tt x} and the rest to {\tt xs}. 
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624 
Likewise, \verb{xs} and \verb{x} represent any set enumeration. 
323  625 
The parse rules only work in the order given. 
626 

627 
\begin{warn} 

332  628 
The \AST{} rewriter cannot distinguish constants from variables and looks 
323  629 
only for names of atoms. Thus the names of {\tt Constant}s occurring in 
630 
the (internal) lefthand side of translation rules should be regarded as 

631 
\rmindex{reserved words}. Choose nonidentifiers like {\tt\at Finset} or 

632 
sufficiently long and strange names. If a bound variable's name gets 

633 
rewritten, the result will be incorrect; for example, the term 

634 
\begin{ttbox} 

635 
\%empty insert. insert(x, empty) 

636 
\end{ttbox} 

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637 
\par\noindent is incorrectly printed as \verb%empty insert. {x}. 
323  638 
\end{warn} 
639 

640 

641 
\subsection{Example: a parse macro for dependent types}\label{prod_trans} 

642 
\index{examples!of macros} 

643 

644 
As stated earlier, a macro rule may not introduce new {\tt Variable}s on 

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645 
the righthand side. Something like \verb"K(B)" => "%x.B" is illegal; 
323  646 
if allowed, it could cause variable capture. In such cases you usually 
647 
must fall back on translation functions. But a trick can make things 

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648 
readable in some cases: {\em calling\/} translation functions by parse 
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649 
macros: 
323  650 
\begin{ttbox} 
3135  651 
ProdSyntax = SetSyntax + 
323  652 
consts 
1387  653 
Pi :: [i, i => i] => i 
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654 
syntax 
1387  655 
"{\at}PROD" :: [idt, i, i] => i ("(3PROD _:_./ _)" 10) 
656 
"{\at}>" :: [i, i] => i ("(_ >/ _)" [51, 50] 50) 

323  657 
\ttbreak 
658 
translations 

659 
"PROD x:A. B" => "Pi(A, \%x. B)" 

660 
"A > B" => "Pi(A, _K(B))" 

661 
end 

662 
ML 

663 
val print_translation = [("Pi", dependent_tr' ("{\at}PROD", "{\at}>"))]; 

664 
\end{ttbox} 

665 

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666 
Here {\tt Pi} is a logical constant for constructing general products. 
323  667 
Two external forms exist: the general case {\tt PROD x:A.B} and the 
668 
function space {\tt A > B}, which abbreviates \verbPi(A, %x.B) when {\tt B} 

669 
does not depend on~{\tt x}. 

670 

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671 
The second parse macro introduces {\tt _K(B)}, which later becomes 
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672 
\verb%x.B due to a parse translation associated with \cdx{_K}. 
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673 
Unfortunately there is no such trick for printing, so we have to add a {\tt 
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674 
ML} section for the print translation \ttindex{dependent_tr'}. 
323  675 

676 
Recall that identifiers with a leading {\tt _} are allowed in translation 

677 
rules, but not in ordinary terms. Thus we can create \AST{}s containing 

678 
names that are not directly expressible. 

679 

8136  680 
The parse translation for {\tt _K} is already installed in Pure, and the 
681 
function {\tt dependent_tr'} is exported by the syntax module for public use. 

682 
See \S\ref{sec:tr_funs} below for more of the arcane lore of translation 

683 
functions. \index{macros)}\index{rewriting!syntactic)} 

323  684 

685 

686 
\section{Translation functions} \label{sec:tr_funs} 

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687 
\index{translations(} 
323  688 
% 
9695  689 
This section describes the translation function mechanism. By writing \ML{} 
690 
functions, you can do almost everything to terms or \AST{}s during parsing and 

691 
printing. The logic LK is a good example of sophisticated transformations 

692 
between internal and external representations of sequents; here, macros would 

693 
be useless. 

323  694 

695 
A full understanding of translations requires some familiarity 

696 
with Isabelle's internals, especially the datatypes {\tt term}, {\tt typ}, 

697 
{\tt Syntax.ast} and the encodings of types and terms as such at the various 

698 
stages of the parsing or printing process. Most users should never need to 

699 
use translation functions. 

700 

701 
\subsection{Declaring translation functions} 

3108  702 
There are four kinds of translation functions, with one of these 
703 
coming in two variants. Each such function is associated with a name, 

704 
which triggers calls to it. Such names can be constants (logical or 

705 
syntactic) or type constructors. 

323  706 

6343  707 
Function {\tt print_syntax} displays the sets of names associated with the 
708 
translation functions of a theory under \texttt{parse_ast_translation}, etc. 

709 
You can add new ones via the {\tt ML} section\index{*ML section} of a theory 

710 
definition file. Even though the {\tt ML} section is the very last part of 

711 
the file, newly installed translation functions are already effective when 

712 
processing all of the preceding sections. 

323  713 

3108  714 
The {\tt ML} section's contents are simply copied verbatim near the 
715 
beginning of the \ML\ file generated from a theory definition file. 

716 
Definitions made here are accessible as components of an \ML\ 

717 
structure; to make some parts private, use an \ML{} {\tt local} 

718 
declaration. The {\ML} code may install translation functions by 

719 
declaring any of the following identifiers: 

323  720 
\begin{ttbox} 
3108  721 
val parse_ast_translation : (string * (ast list > ast)) list 
722 
val print_ast_translation : (string * (ast list > ast)) list 

723 
val parse_translation : (string * (term list > term)) list 

724 
val print_translation : (string * (term list > term)) list 

4375  725 
val typed_print_translation : 
726 
(string * (bool > typ > term list > term)) list 

323  727 
\end{ttbox} 
728 

729 
\subsection{The translation strategy} 

3108  730 
The different kinds of translation functions are called during the 
731 
transformations between parse trees, \AST{}s and terms (recall 

732 
Fig.\ts\ref{fig:parse_print}). Whenever a combination of the form 

733 
$(\mtt"c\mtt"~x@1 \ldots x@n)$ is encountered, and a translation 

734 
function $f$ of appropriate kind exists for $c$, the result is 

735 
computed by the \ML{} function call $f \mtt[ x@1, \ldots, x@n \mtt]$. 

323  736 

3108  737 
For \AST{} translations, the arguments $x@1, \ldots, x@n$ are \AST{}s. 
738 
A combination has the form $\Constant c$ or $\Appl{\Constant c, x@1, 

739 
\ldots, x@n}$. For term translations, the arguments are terms and a 

740 
combination has the form $\ttfct{Const} (c, \tau)$ or $\ttfct{Const} 

741 
(c, \tau) \ttapp x@1 \ttapp \ldots \ttapp x@n$. Terms allow more 

742 
sophisticated transformations than \AST{}s do, typically involving 

743 
abstractions and bound variables. {\em Typed} print translations may 

4375  744 
even peek at the type $\tau$ of the constant they are invoked on; they 
745 
are also passed the current value of the \ttindex{show_sorts} flag. 

323  746 

3108  747 
Regardless of whether they act on terms or \AST{}s, translation 
748 
functions called during the parsing process differ from those for 

749 
printing more fundamentally in their overall behaviour: 

323  750 
\begin{description} 
6343  751 
\item[Parse translations] are applied bottomup. The arguments are already in 
752 
translated form. The translations must not fail; exceptions trigger an 

753 
error message. There may never be more than one function associated with 

754 
any syntactic name. 

755 

323  756 
\item[Print translations] are applied topdown. They are supplied with 
757 
arguments that are partly still in internal form. The result again 

6343  758 
undergoes translation; therefore a print translation should not introduce as 
759 
head the very constant that invoked it. The function may raise exception 

760 
\xdx{Match} to indicate failure; in this event it has no effect. Multiple 

761 
functions associated with some syntactic name are tried in an unspecified 

762 
order. 

323  763 
\end{description} 
764 

765 
Only constant atoms  constructor \ttindex{Constant} for \AST{}s and 

766 
\ttindex{Const} for terms  can invoke translation functions. This 

767 
causes another difference between parsing and printing. 

768 

769 
Parsing starts with a string and the constants are not yet identified. 

770 
Only parse tree heads create {\tt Constant}s in the resulting \AST, as 

771 
described in \S\ref{sec:astofpt}. Macros and parse \AST{} translations may 

772 
introduce further {\tt Constant}s. When the final \AST{} is converted to a 

773 
term, all {\tt Constant}s become {\tt Const}s, as described in 

774 
\S\ref{sec:termofast}. 

775 

776 
Printing starts with a welltyped term and all the constants are known. So 

777 
all logical constants and type constructors may invoke print translations. 

778 
These, and macros, may introduce further constants. 

779 

780 

781 
\subsection{Example: a print translation for dependent types} 

782 
\index{examples!of translations}\indexbold{*dependent_tr'} 

783 

784 
Let us continue the dependent type example (page~\pageref{prod_trans}) by 

785 
examining the parse translation for~\cdx{_K} and the print translation 

786 
{\tt dependent_tr'}, which are both builtin. By convention, parse 

787 
translations have names ending with {\tt _tr} and print translations have 

788 
names ending with {\tt _tr'}. Search for such names in the Isabelle 

789 
sources to locate more examples. 

790 

791 
Here is the parse translation for {\tt _K}: 

792 
\begin{ttbox} 

793 
fun k_tr [t] = Abs ("x", dummyT, incr_boundvars 1 t) 

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794 
 k_tr ts = raise TERM ("k_tr", ts); 
323  795 
\end{ttbox} 
796 
If {\tt k_tr} is called with exactly one argument~$t$, it creates a new 

797 
{\tt Abs} node with a body derived from $t$. Since terms given to parse 

798 
translations are not yet typed, the type of the bound variable in the new 

799 
{\tt Abs} is simply {\tt dummyT}. The function increments all {\tt Bound} 

800 
nodes referring to outer abstractions by calling \ttindex{incr_boundvars}, 

801 
a basic term manipulation function defined in {\tt Pure/term.ML}. 

802 

803 
Here is the print translation for dependent types: 

804 
\begin{ttbox} 

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805 
fun dependent_tr' (q, r) (A :: Abs (x, T, B) :: ts) = 
323  806 
if 0 mem (loose_bnos B) then 
3108  807 
let val (x', B') = Syntax.variant_abs' (x, dummyT, B) in 
808 
list_comb 

8136  809 
(Const (q,dummyT) $ 
810 
Syntax.mark_boundT (x',{\thinspace}T) $ A $ B', ts) 

323  811 
end 
812 
else list_comb (Const (r, dummyT) $ A $ B, ts) 

813 
 dependent_tr' _ _ = raise Match; 

814 
\end{ttbox} 

3135  815 
The argument {\tt (q,{\thinspace}r)} is supplied to the curried function {\tt 
3108  816 
dependent_tr'} by a partial application during its installation. 
817 
For example, we could set up print translations for both {\tt Pi} and 

818 
{\tt Sigma} by including 

323  819 
\begin{ttbox}\index{*ML section} 
820 
val print_translation = 

821 
[("Pi", dependent_tr' ("{\at}PROD", "{\at}>")), 

822 
("Sigma", dependent_tr' ("{\at}SUM", "{\at}*"))]; 

823 
\end{ttbox} 

3108  824 
within the {\tt ML} section. The first of these transforms ${\tt 
825 
Pi}(A, \mtt{Abs}(x, T, B))$ into $\hbox{\tt{\at}PROD}(x', A, B')$ or 

826 
$\hbox{\tt{\at}>}(A, B)$, choosing the latter form if $B$ does not 

827 
depend on~$x$. It checks this using \ttindex{loose_bnos}, yet another 

828 
function from {\tt Pure/term.ML}. Note that $x'$ is a version of $x$ 

829 
renamed away from all names in $B$, and $B'$ is the body $B$ with {\tt 

830 
Bound} nodes referring to the {\tt Abs} node replaced by 

831 
$\ttfct{Free} (x', \mtt{dummyT})$ (but marked as representing a bound 

832 
variable). 

323  833 

834 
We must be careful with types here. While types of {\tt Const}s are 

835 
ignored, type constraints may be printed for some {\tt Free}s and 

836 
{\tt Var}s if \ttindex{show_types} is set to {\tt true}. Variables of type 

837 
\ttindex{dummyT} are never printed with constraint, though. The line 

838 
\begin{ttbox} 

3108  839 
let val (x', B') = Syntax.variant_abs' (x, dummyT, B); 
840 
\end{ttbox}\index{*Syntax.variant_abs'} 

323  841 
replaces bound variable occurrences in~$B$ by the free variable $x'$ with 
842 
type {\tt dummyT}. Only the binding occurrence of~$x'$ is given the 

843 
correct type~{\tt T}, so this is the only place where a type 

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844 
constraint might appear. 
3108  845 

846 
Also note that we are responsible to mark free identifiers that 

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847 
actually represent bound variables. This is achieved by 
3108  848 
\ttindex{Syntax.variant_abs'} and \ttindex{Syntax.mark_boundT} above. 
849 
Failing to do so may cause these names to be printed in the wrong 

850 
style. \index{translations)} \index{syntax!transformations)} 

851 

852 

853 
\section{Token translations} \label{sec:tok_tr} 

854 
\index{token translations(} 

855 
% 

856 
Isabelle's metalogic features quite a lot of different kinds of 

857 
identifiers, namely {\em class}, {\em tfree}, {\em tvar}, {\em free}, 

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858 
{\em bound}, {\em var}. One might want to have these printed in 
3108  859 
different styles, e.g.\ in bold or italic, or even transcribed into 
860 
something more readable like $\alpha, \alpha', \beta$ instead of {\tt 

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861 
'a}, {\tt 'aa}, {\tt 'b} for type variables. Token translations 
3108  862 
provide a means to such ends, enabling the user to install certain 
863 
\ML{} functions associated with any logical \rmindex{token class} and 

864 
depending on some \rmindex{print mode}. 

865 

866 
The logical class of identifiers can not necessarily be determined by 

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867 
its syntactic category, though. For example, consider free vs.\ bound 
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868 
variables. So Isabelle's pretty printing mechanism, starting from 
3108  869 
fully typed terms, has to be careful to preserve this additional 
870 
information\footnote{This is done by marking atoms in abstract syntax 

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871 
trees appropriately. The marks are actually visible by print 
3108  872 
translation functions  they are just special constants applied to 
873 
atomic asts, for example \texttt{("_bound" x)}.}. In particular, 

874 
usersupplied print translation functions operating on terms have to 

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875 
be wellbehaved in this respect. Free identifiers introduced to 
3108  876 
represent bound variables have to be marked appropriately (cf.\ the 
877 
example at the end of \S\ref{sec:tr_funs}). 

878 

879 
\medskip Token translations may be installed by declaring the 

6343  880 
\ttindex{token_translation} value within the \texttt{ML} section of a theory 
881 
definition file: 

3108  882 
\begin{ttbox} 
8136  883 
val token_translation: 
884 
(string * string * (string > string * real)) list 

8701  885 
\end{ttbox} 
6343  886 
The elements of this list are of the form $(m, c, f)$, where $m$ is a print 
887 
mode identifier, $c$ a token class, and $f\colon string \to string \times 

888 
real$ the actual translation function. Assuming that $x$ is of identifier 

889 
class $c$, and print mode $m$ is the first (active) mode providing some 

890 
translation for $c$, then $x$ is output according to $f(x) = (x', len)$. 

891 
Thereby $x'$ is the modified identifier name and $len$ its visual length in 

892 
terms of characters (e.g.\ length 1.0 would correspond to $1/2$\,em in 

893 
\LaTeX). Thus $x'$ may include nonprinting parts like control sequences or 

894 
markup information for typesetting systems. 

3108  895 

896 

897 
\index{token translations)} 

5371  898 

899 

900 
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901 
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902 
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903 
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