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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 

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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 & \\ 

34 
$\downarrow$ & parser \\ 

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{} & \\ 

46 
$\downarrow$ & print \AST{} translation, printer \\ 

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))}. 

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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 

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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 

96 
\section{Transforming parse trees to \AST{}s}\label{sec:astofpt} 

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 

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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}, 

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\ndx{var}, \ndx{tid}, \ndx{tvar}, \ndx{xnum} or \ndx{xstr} token, and $s$ 
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its associated string. Note that for {\tt xstr} this does not include the 
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quotes. 
323  119 

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\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 

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\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 

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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 

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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)) \\ 

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

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

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\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) \\ 

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

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

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

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

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"['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} 

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

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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 

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\section{Transforming \AST{}s to terms}\label{sec:termofast} 

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\index{terms!made from ASTs} 

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\newcommand\termofast[1]{\lbrakk#1\rbrakk} 

187 

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

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

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

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

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

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correct types or rejects the input. 

194 

195 
Another set of translation functions, namely parse 

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

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

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

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

200 

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

202 
\begin{itemize} 

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

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\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 

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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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\[ \astofterm{\ttfct{Abs} (x, \tau, t)} = 
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\Appl{\Constant \mtt{"_abs"}, 
323  262 
constrain(\Variable x', \tau), \astofterm{t'}}. 
263 
\] 

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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 

307 
If an application $(\mtt"c\mtt"~ x@1 \ldots x@m)$ has more arguments than the 

308 
corresponding production, it is first split into $((\mtt"c\mtt"~ x@1 \ldots 

309 
x@n) ~ x@{n+1} \ldots x@m)$. Applications with too few arguments or with 

310 
nonconstant head or without a corresponding production are printed as 

311 
$f(x@1, \ldots, x@l)$ or $(\alpha@1, \ldots, \alpha@l) ty$. An occurrence of 

312 
$\Variable x$ is simply printed as~$x$. 

313 

314 
Blanks are {\em not\/} inserted automatically. If blanks are required to 

315 
separate tokens, specify them in the mixfix declaration, possibly preceded 

316 
by a slash~({\tt/}) to allow a line break. 

317 
\index{ASTs)} 

318 

319 

320 

321 
\section{Macros: Syntactic rewriting} \label{sec:macros} 

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\index{macros(}\index{rewriting!syntactic(} 
323  323 

324 
Mixfix declarations alone can handle situations where there is a direct 

325 
connection between the concrete syntax and the underlying term. Sometimes 

326 
we require a more elaborate concrete syntax, such as quantifiers and list 

327 
notation. Isabelle's {\bf macros} and {\bf translation functions} can 

328 
perform translations such as 

329 
\begin{center}\tt 

330 
\begin{tabular}{r@{$\quad\protect\rightleftharpoons\quad$}l} 

331 
ALL x:A.P & Ball(A, \%x.P) \\ \relax 

332 
[x, y, z] & Cons(x, Cons(y, Cons(z, Nil))) 

333 
\end{tabular} 

334 
\end{center} 

335 
Translation functions (see \S\ref{sec:tr_funs}) must be coded in ML; they 

336 
are the most powerful translation mechanism but are difficult to read or 

337 
write. Macros are specified by firstorder rewriting systems that operate 

338 
on abstract syntax trees. They are usually easy to read and write, and can 

339 
express all but the most obscure translations. 

340 

341 
Figure~\ref{fig:set_trans} defines a fragment of firstorder logic and set 

342 
theory.\footnote{This and the following theories are complete working 

343 
examples, though they specify only syntax, no axioms. The file {\tt 

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ZF/ZF.thy} presents a full set theory definition, including many 
323  345 
macro rules.} Theory {\tt SET} defines constants for set comprehension 
346 
({\tt Collect}), replacement ({\tt Replace}) and bounded universal 

347 
quantification ({\tt Ball}). Each of these binds some variables. Without 

332  348 
additional syntax we should have to write $\forall x \in A. P$ as {\tt 
323  349 
Ball(A,\%x.P)}, and similarly for the others. 
350 

351 
\begin{figure} 

352 
\begin{ttbox} 

353 
SET = Pure + 

354 
types 

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i o 
323  356 
arities 
357 
i, o :: logic 

358 
consts 

359 
Trueprop :: "o => prop" ("_" 5) 

360 
Collect :: "[i, i => o] => i" 

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Replace :: "[i, [i, i] => o] => i" 
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Ball :: "[i, i => o] => o" 
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syntax 
323  364 
"{\at}Collect" :: "[idt, i, o] => i" ("(1{\ttlbrace}_:_./ _{\ttrbrace})") 
365 
"{\at}Replace" :: "[idt, idt, i, o] => i" ("(1{\ttlbrace}_./ _:_, _{\ttrbrace})") 

366 
"{\at}Ball" :: "[idt, i, o] => o" ("(3ALL _:_./ _)" 10) 

367 
translations 

368 
"{\ttlbrace}x:A. P{\ttrbrace}" == "Collect(A, \%x. P)" 

369 
"{\ttlbrace}y. x:A, Q{\ttrbrace}" == "Replace(A, \%x y. Q)" 

370 
"ALL x:A. P" == "Ball(A, \%x. P)" 

371 
end 

372 
\end{ttbox} 

373 
\caption{Macro example: set theory}\label{fig:set_trans} 

374 
\end{figure} 

375 

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The theory specifies a variablebinding syntax through additional productions 
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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  382 

383 
The translations cause the replacement of external forms by internal forms 

384 
after parsing, and vice versa before printing of terms. As a specification 

385 
of the set theory notation, they should be largely selfexplanatory. The 

386 
syntactic constants, {\tt\at Collect}, {\tt\at Replace} and {\tt\at Ball}, 

387 
appear implicitly in the macro rules via their mixfix forms. 

388 

389 
Macros can define variablebinding syntax because they operate on \AST{}s, 

390 
which have no inbuilt notion of bound variable. The macro variables {\tt 

391 
x} and~{\tt y} have type~{\tt idt} and therefore range over identifiers, 

392 
in this case bound variables. The macro variables {\tt P} and~{\tt Q} 

393 
range over formulae containing bound variable occurrences. 

394 

395 
Other applications of the macro system can be less straightforward, and 

396 
there are peculiarities. The rest of this section will describe in detail 

397 
how Isabelle macros are preprocessed and applied. 

398 

399 

400 
\subsection{Specifying macros} 

401 
Macros are basically rewrite rules on \AST{}s. But unlike other macro 

402 
systems found in programming languages, Isabelle's macros work in both 

403 
directions. Therefore a syntax contains two lists of rewrites: one for 

404 
parsing and one for printing. 

405 

406 
\index{*translations section} 

407 
The {\tt translations} section specifies macros. The syntax for a macro is 

408 
\[ (root)\; string \quad 

409 
\left\{\begin{array}[c]{c} \mtt{=>} \\ \mtt{<=} \\ \mtt{==} \end{array} 

410 
\right\} \quad 

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411 
(root)\; string 
323  412 
\] 
413 
% 

414 
This specifies a parse rule ({\tt =>}), a print rule ({\tt <=}), or both 

415 
({\tt ==}). The two strings specify the left and righthand sides of the 

416 
macro rule. The $(root)$ specification is optional; it specifies the 

417 
nonterminal for parsing the $string$ and if omitted defaults to {\tt 

418 
logic}. \AST{} rewrite rules $(l, r)$ must obey certain conditions: 

419 
\begin{itemize} 

420 
\item Rules must be left linear: $l$ must not contain repeated variables. 

421 

422 
\item Rules must have constant heads, namely $l = \mtt"c\mtt"$ or $l = 

423 
(\mtt"c\mtt" ~ x@1 \ldots x@n)$. 

424 

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

426 
\end{itemize} 

427 

428 
Macro rules may refer to any syntax from the parent theories. They may 

499  429 
also refer to anything defined before the {\tt .thy} file's {\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 

443 
The result of this preprocessing is two lists of macro rules, each stored 

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444 
as a pair of \AST{}s. They can be viewed using {\tt print_syntax} 
323  445 
(sections \ttindex{parse_rules} and \ttindex{print_rules}). For 
446 
theory~{\tt SET} of Fig.~\ref{fig:set_trans} these are 

447 
\begin{ttbox} 

448 
parse_rules: 

449 
("{\at}Collect" x A P) > ("Collect" A ("_abs" x P)) 

450 
("{\at}Replace" y x A Q) > ("Replace" A ("_abs" x ("_abs" y Q))) 

451 
("{\at}Ball" x A P) > ("Ball" A ("_abs" x P)) 

452 
print_rules: 

453 
("Collect" A ("_abs" x P)) > ("{\at}Collect" x A P) 

454 
("Replace" A ("_abs" x ("_abs" y Q))) > ("{\at}Replace" y x A Q) 

455 
("Ball" A ("_abs" x P)) > ("{\at}Ball" x A P) 

456 
\end{ttbox} 

457 

458 
\begin{warn} 

459 
Avoid choosing variable names that have previously been used as 

460 
constants, types or type classes; the {\tt consts} section in the output 

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461 
of {\tt print_syntax} lists all such names. If a macro rule works 
323  462 
incorrectly, inspect its internal form as shown above, recalling that 
463 
constants appear as quoted strings and variables without quotes. 

464 
\end{warn} 

465 

466 
\begin{warn} 

467 
If \ttindex{eta_contract} is set to {\tt true}, terms will be 

468 
$\eta$contracted {\em before\/} the \AST{} rewriter sees them. Thus some 

469 
abstraction nodes needed for print rules to match may vanish. For example, 

332  470 
\verbBall(A, %x. P(x)) contracts to {\tt Ball(A, P)}; the print rule does 
323  471 
not apply and the output will be {\tt Ball(A, P)}. This problem would not 
472 
occur if \ML{} translation functions were used instead of macros (as is 

473 
done for binder declarations). 

474 
\end{warn} 

475 

476 

477 
\begin{warn} 

478 
Another trap concerns type constraints. If \ttindex{show_types} is set to 

479 
{\tt true}, bound variables will be decorated by their meta types at the 

480 
binding place (but not at occurrences in the body). Matching with 

481 
\verbCollect(A, %x. P) binds {\tt x} to something like {\tt ("_constrain" y 

482 
"i")} rather than only {\tt y}. \AST{} rewriting will cause the constraint to 

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483 
appear in the external form, say \verb{y::i:A::i. P::o}. 
323  484 

485 
To allow such constraints to be reread, your syntax should specify bound 

486 
variables using the nonterminal~\ndx{idt}. This is the case in our 

487 
example. Choosing {\tt id} instead of {\tt idt} is a common error, 

488 
especially since it appears in former versions of most of Isabelle's 

489 
objectlogics. 

490 
\end{warn} 

491 

492 

493 

494 
\subsection{Applying rules} 

495 
As a term is being parsed or printed, an \AST{} is generated as an 

496 
intermediate form (recall Fig.\ts\ref{fig:parse_print}). The \AST{} is 

332  497 
normalised by applying macro rules in the manner of a traditional term 
323  498 
rewriting system. We first examine how a single rule is applied. 
499 

332  500 
Let $t$ be the abstract syntax tree to be normalised and $(l, r)$ some 
323  501 
translation rule. A subtree~$u$ of $t$ is a {\bf redex} if it is an 
502 
instance of~$l$; in this case $l$ is said to {\bf match}~$u$. A redex 

503 
matched by $l$ may be replaced by the corresponding instance of~$r$, thus 

504 
{\bf rewriting} the \AST~$t$. Matching requires some notion of {\bf 

505 
placeholders} that may occur in rule patterns but not in ordinary 

506 
\AST{}s; {\tt Variable} atoms serve this purpose. 

507 

508 
The matching of the object~$u$ by the pattern~$l$ is performed as follows: 

509 
\begin{itemize} 

510 
\item Every constant matches itself. 

511 

512 
\item $\Variable x$ in the object matches $\Constant x$ in the pattern. 

513 
This point is discussed further below. 

514 

515 
\item Every \AST{} in the object matches $\Variable x$ in the pattern, 

516 
binding~$x$ to~$u$. 

517 

518 
\item One application matches another if they have the same number of 

519 
subtrees and corresponding subtrees match. 

520 

521 
\item In every other case, matching fails. In particular, {\tt 

522 
Constant}~$x$ can only match itself. 

523 
\end{itemize} 

524 
A successful match yields a substitution that is applied to~$r$, generating 

525 
the instance that replaces~$u$. 

526 

527 
The second case above may look odd. This is where {\tt Variable}s of 

528 
nonrule \AST{}s behave like {\tt Constant}s. Recall that \AST{}s are not 

529 
far removed from parse trees; at this level it is not yet known which 

530 
identifiers will become constants, bounds, frees, types or classes. As 

531 
\S\ref{sec:asts} describes, former parse tree heads appear in \AST{}s as 

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532 
{\tt Constant}s, while the name tokens \ndx{id}, \ndx{var}, \ndx{tid}, 
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533 
\ndx{tvar}, \ndx{xnum} and \ndx{xstr} become {\tt Variable}s. On the other 
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534 
hand, when \AST{}s generated from terms for printing, all constants and type 
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535 
constructors become {\tt Constant}s; see \S\ref{sec:asts}. Thus \AST{}s may 
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536 
contain a messy mixture of {\tt Variable}s and {\tt Constant}s. This is 
323  537 
insignificant at macro level because matching treats them alike. 
538 

539 
Because of this behaviour, different kinds of atoms with the same name are 

540 
indistinguishable, which may make some rules prone to misbehaviour. Example: 

541 
\begin{ttbox} 

542 
types 

543 
Nil 

544 
consts 

545 
Nil :: "'a list" 

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546 
syntax 
323  547 
"[]" :: "'a list" ("[]") 
548 
translations 

549 
"[]" == "Nil" 

550 
\end{ttbox} 

551 
The term {\tt Nil} will be printed as {\tt []}, just as expected. 

552 
The term \verb%Nil.t will be printed as \verb%[].t, which might not be 

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553 
expected! Guess how type~{\tt Nil} is printed? 
323  554 

555 
Normalizing an \AST{} involves repeatedly applying macro rules until none 

332  556 
are applicable. Macro rules are chosen in the order that they appear in the 
323  557 
{\tt translations} section. You can watch the normalization of \AST{}s 
558 
during parsing and printing by setting \ttindex{Syntax.trace_norm_ast} to 

559 
{\tt true}.\index{tracing!of macros} Alternatively, use 

560 
\ttindex{Syntax.test_read}. The information displayed when tracing 

561 
includes the \AST{} before normalization ({\tt pre}), redexes with results 

562 
({\tt rewrote}), the normal form finally reached ({\tt post}) and some 

563 
statistics ({\tt normalize}). If tracing is off, 

564 
\ttindex{Syntax.stat_norm_ast} can be set to {\tt true} in order to enable 

565 
printing of the normal form and statistics only. 

566 

567 

568 
\subsection{Example: the syntax of finite sets} 

569 
\index{examples!of macros} 

570 

571 
This example demonstrates the use of recursive macros to implement a 

572 
convenient notation for finite sets. 

573 
\index{*empty constant}\index{*insert constant}\index{{}@\verb'{}' symbol} 

574 
\index{"@Enum@{\tt\at Enum} constant} 

575 
\index{"@Finset@{\tt\at Finset} constant} 

576 
\begin{ttbox} 

577 
FINSET = SET + 

578 
types 

579 
is 

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580 
syntax 
323  581 
"" :: "i => is" ("_") 
582 
"{\at}Enum" :: "[i, is] => is" ("_,/ _") 

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583 
consts 
323  584 
empty :: "i" ("{\ttlbrace}{\ttrbrace}") 
585 
insert :: "[i, i] => i" 

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586 
syntax 
323  587 
"{\at}Finset" :: "is => i" ("{\ttlbrace}(_){\ttrbrace}") 
588 
translations 

589 
"{\ttlbrace}x, xs{\ttrbrace}" == "insert(x, {\ttlbrace}xs{\ttrbrace})" 

590 
"{\ttlbrace}x{\ttrbrace}" == "insert(x, {\ttlbrace}{\ttrbrace})" 

591 
end 

592 
\end{ttbox} 

593 
Finite sets are internally built up by {\tt empty} and {\tt insert}. The 

594 
declarations above specify \verb{x, y, z} as the external representation 

595 
of 

596 
\begin{ttbox} 

597 
insert(x, insert(y, insert(z, empty))) 

598 
\end{ttbox} 

599 
The nonterminal symbol~\ndx{is} stands for one or more objects of type~{\tt 

600 
i} separated by commas. The mixfix declaration \hbox{\verb"_,/ _"} 

601 
allows a line break after the comma for \rmindex{pretty printing}; if no 

602 
line break is required then a space is printed instead. 

603 

604 
The nonterminal is declared as the type~{\tt is}, but with no {\tt arities} 

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605 
declaration. Hence {\tt is} is not a logical type and may be used safely as 
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606 
a new nonterminal for custom syntax. The nonterminal~{\tt is} can later be 
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607 
reused for other enumerations of type~{\tt i} like lists or tuples. If we 
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608 
had needed polymorphic enumerations, we could have used the predefined 
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609 
nonterminal symbol \ndx{args} and skipped this part altogether. 
323  610 

611 
\index{"@Finset@{\tt\at Finset} constant} 

612 
Next follows {\tt empty}, which is already equipped with its syntax 

613 
\verb{}, and {\tt insert} without concrete syntax. The syntactic 

614 
constant {\tt\at Finset} provides concrete syntax for enumerations of~{\tt 

615 
i} enclosed in curly braces. Remember that a pair of parentheses, as in 

616 
\verb"{(_)}", specifies a block of indentation for pretty printing. 

617 

618 
The translations may look strange at first. Macro rules are best 

619 
understood in their internal forms: 

620 
\begin{ttbox} 

621 
parse_rules: 

622 
("{\at}Finset" ("{\at}Enum" x xs)) > ("insert" x ("{\at}Finset" xs)) 

623 
("{\at}Finset" x) > ("insert" x "empty") 

624 
print_rules: 

625 
("insert" x ("{\at}Finset" xs)) > ("{\at}Finset" ("{\at}Enum" x xs)) 

626 
("insert" x "empty") > ("{\at}Finset" x) 

627 
\end{ttbox} 

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628 
This shows that \verb{x,xs} indeed matches any set enumeration of at least 
323  629 
two elements, binding the first to {\tt x} and the rest to {\tt xs}. 
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630 
Likewise, \verb{xs} and \verb{x} represent any set enumeration. 
323  631 
The parse rules only work in the order given. 
632 

633 
\begin{warn} 

332  634 
The \AST{} rewriter cannot distinguish constants from variables and looks 
323  635 
only for names of atoms. Thus the names of {\tt Constant}s occurring in 
636 
the (internal) lefthand side of translation rules should be regarded as 

637 
\rmindex{reserved words}. Choose nonidentifiers like {\tt\at Finset} or 

638 
sufficiently long and strange names. If a bound variable's name gets 

639 
rewritten, the result will be incorrect; for example, the term 

640 
\begin{ttbox} 

641 
\%empty insert. insert(x, empty) 

642 
\end{ttbox} 

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643 
\par\noindent is incorrectly printed as \verb%empty insert. {x}. 
323  644 
\end{warn} 
645 

646 

647 
\subsection{Example: a parse macro for dependent types}\label{prod_trans} 

648 
\index{examples!of macros} 

649 

650 
As stated earlier, a macro rule may not introduce new {\tt Variable}s on 

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651 
the righthand side. Something like \verb"K(B)" => "%x.B" is illegal; 
323  652 
if allowed, it could cause variable capture. In such cases you usually 
653 
must fall back on translation functions. But a trick can make things 

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654 
readable in some cases: {\em calling\/} translation functions by parse 
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655 
macros: 
323  656 
\begin{ttbox} 
657 
PROD = FINSET + 

658 
consts 

659 
Pi :: "[i, i => i] => i" 

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660 
syntax 
323  661 
"{\at}PROD" :: "[idt, i, i] => i" ("(3PROD _:_./ _)" 10) 
662 
"{\at}>" :: "[i, i] => i" ("(_ >/ _)" [51, 50] 50) 

663 
\ttbreak 

664 
translations 

665 
"PROD x:A. B" => "Pi(A, \%x. B)" 

666 
"A > B" => "Pi(A, _K(B))" 

667 
end 

668 
ML 

669 
val print_translation = [("Pi", dependent_tr' ("{\at}PROD", "{\at}>"))]; 

670 
\end{ttbox} 

671 

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672 
Here {\tt Pi} is a logical constant for constructing general products. 
323  673 
Two external forms exist: the general case {\tt PROD x:A.B} and the 
674 
function space {\tt A > B}, which abbreviates \verbPi(A, %x.B) when {\tt B} 

675 
does not depend on~{\tt x}. 

676 

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677 
The second parse macro introduces {\tt _K(B)}, which later becomes 
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678 
\verb%x.B due to a parse translation associated with \cdx{_K}. 
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679 
Unfortunately there is no such trick for printing, so we have to add a {\tt 
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680 
ML} section for the print translation \ttindex{dependent_tr'}. 
323  681 

682 
Recall that identifiers with a leading {\tt _} are allowed in translation 

683 
rules, but not in ordinary terms. Thus we can create \AST{}s containing 

684 
names that are not directly expressible. 

685 

686 
The parse translation for {\tt _K} is already installed in Pure, and {\tt 

687 
dependent_tr'} is exported by the syntax module for public use. See 

688 
\S\ref{sec:tr_funs} below for more of the arcane lore of translation functions. 

689 
\index{macros)}\index{rewriting!syntactic)} 

690 

691 

692 
\section{Translation functions} \label{sec:tr_funs} 

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693 
\index{translations(} 
323  694 
% 
695 
This section describes the translation function mechanism. By writing 

696 
\ML{} functions, you can do almost everything with terms or \AST{}s during 

697 
parsing and printing. The logic \LK\ is a good example of sophisticated 

332  698 
transformations between internal and external representations of sequents; 
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699 
here, macros would be useless. 
323  700 

701 
A full understanding of translations requires some familiarity 

702 
with Isabelle's internals, especially the datatypes {\tt term}, {\tt typ}, 

703 
{\tt Syntax.ast} and the encodings of types and terms as such at the various 

704 
stages of the parsing or printing process. Most users should never need to 

705 
use translation functions. 

706 

707 
\subsection{Declaring translation functions} 

708 
There are four kinds of translation functions. Each such function is 

709 
associated with a name, which triggers calls to it. Such names can be 

710 
constants (logical or syntactic) or type constructors. 

711 

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712 
{\tt print_syntax} displays the sets of names associated with the 
323  713 
translation functions of a {\tt Syntax.syntax} under 
714 
\ttindex{parse_ast_translation}, \ttindex{parse_translation}, 

715 
\ttindex{print_translation} and \ttindex{print_ast_translation}. You can 

716 
add new ones via the {\tt ML} section\index{*ML section} of 

717 
a {\tt .thy} file. There may never be more than one function of the same 

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718 
kind per name. Even though the {\tt ML} section is the very last part of a 
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719 
{\tt .thy} file, newly installed translation functions are effective when 
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720 
processing the preceding section. 
323  721 

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722 
The {\tt ML} section is copied verbatim near the beginning of the \ML\ file 
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723 
generated from a {\tt .thy} file. Definitions made here are accessible as 
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724 
components of an \ML\ structure; to make some definitions private, use an 
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725 
\ML{} {\tt local} declaration. The {\tt ML} section may install translation 
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726 
functions by declaring any of the following identifiers: 
323  727 
\begin{ttbox} 
728 
val parse_ast_translation : (string * (ast list > ast)) list 

729 
val print_ast_translation : (string * (ast list > ast)) list 

730 
val parse_translation : (string * (term list > term)) list 

731 
val print_translation : (string * (term list > term)) list 

732 
\end{ttbox} 

733 

734 
\subsection{The translation strategy} 

735 
All four kinds of translation functions are treated similarly. They are 

736 
called during the transformations between parse trees, \AST{}s and terms 

737 
(recall Fig.\ts\ref{fig:parse_print}). Whenever a combination of the form 

738 
$(\mtt"c\mtt"~x@1 \ldots x@n)$ is encountered, and a translation function 

739 
$f$ of appropriate kind exists for $c$, the result is computed by the \ML{} 

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function call $f \mtt[ x@1, \ldots, x@n \mtt]$. 

741 

742 
For \AST{} translations, the arguments $x@1, \ldots, x@n$ are \AST{}s. A 

743 
combination has the form $\Constant c$ or $\Appl{\Constant c, x@1, \ldots, 

744 
x@n}$. For term translations, the arguments are terms and a combination 

745 
has the form $\ttfct{Const} (c, \tau)$ or $\ttfct{Const} (c, \tau) \ttapp 

746 
x@1 \ttapp \ldots \ttapp x@n$. Terms allow more sophisticated 

747 
transformations than \AST{}s do, typically involving abstractions and bound 

748 
variables. 

749 

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Regardless of whether they act on terms or \AST{}s, parse translations differ 
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from print translations in their overall behaviour fundamentally: 
323  752 
\begin{description} 
753 
\item[Parse translations] are applied bottomup. The arguments are already 

754 
in translated form. The translations must not fail; exceptions trigger 

755 
an error message. 

756 

757 
\item[Print translations] are applied topdown. They are supplied with 

758 
arguments that are partly still in internal form. The result again 

759 
undergoes translation; therefore a print translation should not introduce 

760 
as head the very constant that invoked it. The function may raise 

761 
exception \xdx{Match} to indicate failure; in this event it has no 

762 
effect. 

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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 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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fun dependent_tr' (q, r) (A :: Abs (x, T, B) :: ts) = 
323  806 
if 0 mem (loose_bnos B) then 
807 
let val (x', B') = variant_abs (x, dummyT, B); 

808 
in list_comb (Const (q, dummyT) $ Free (x', T) $ A $ B', ts) 

809 
end 

810 
else list_comb (Const (r, dummyT) $ A $ B, ts) 

811 
 dependent_tr' _ _ = raise Match; 

812 
\end{ttbox} 

332  813 
The argument {\tt (q,r)} is supplied to the curried function {\tt 
814 
dependent_tr'} by a partial application during its installation. We 

815 
can set up print translations for both {\tt Pi} and {\tt Sigma} by 

816 
including 

323  817 
\begin{ttbox}\index{*ML section} 
818 
val print_translation = 

819 
[("Pi", dependent_tr' ("{\at}PROD", "{\at}>")), 

820 
("Sigma", dependent_tr' ("{\at}SUM", "{\at}*"))]; 

821 
\end{ttbox} 

822 
within the {\tt ML} section. The first of these transforms ${\tt Pi}(A, 

823 
\mtt{Abs}(x, T, B))$ into $\hbox{\tt{\at}PROD}(x', A, B')$ or 

332  824 
$\hbox{\tt{\at}>}(A, B)$, choosing the latter form if $B$ does not depend 
323  825 
on~$x$. It checks this using \ttindex{loose_bnos}, yet another function 
826 
from {\tt Pure/term.ML}. Note that $x'$ is a version of $x$ renamed away 

332  827 
from all names in $B$, and $B'$ is the body $B$ with {\tt Bound} nodes 
828 
referring to the {\tt Abs} node replaced by $\ttfct{Free} (x', 

323  829 
\mtt{dummyT})$. 
830 

831 
We must be careful with types here. While types of {\tt Const}s are 

832 
ignored, type constraints may be printed for some {\tt Free}s and 

833 
{\tt Var}s if \ttindex{show_types} is set to {\tt true}. Variables of type 

834 
\ttindex{dummyT} are never printed with constraint, though. The line 

835 
\begin{ttbox} 

836 
let val (x', B') = variant_abs (x, dummyT, B); 

837 
\end{ttbox}\index{*variant_abs} 

838 
replaces bound variable occurrences in~$B$ by the free variable $x'$ with 

839 
type {\tt dummyT}. Only the binding occurrence of~$x'$ is given the 

840 
correct type~{\tt T}, so this is the only place where a type 

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841 
constraint might appear. 
323  842 
\index{translations)} 
843 
\index{syntax!transformations)} 