author  wenzelm 
Sun, 15 Oct 2000 19:50:35 +0200  
changeset 10220  2a726de6e124 
parent 9695  ec7d7f877712 
child 11052  1379e49c0ee9 
permissions  rwrr 
3201  1 

104  2 
%% $Id$ 
145  3 

104  4 
\chapter{Theories, Terms and Types} \label{theories} 
5 
\index{theories(}\index{signaturesbold} 

9695  6 
\index{reading!axiomssee{\texttt{assume_ax}}} Theories organize the syntax, 
7 
declarations and axioms of a mathematical development. They are built, 

8 
starting from the Pure or CPure theory, by extending and merging existing 

9 
theories. They have the \ML\ type \mltydx{theory}. Theory operations signal 

10 
errors by raising exception \xdx{THEORY}, returning a message and a list of 

11 
theories. 

104  12 

13 
Signatures, which contain information about sorts, types, constants and 

332  14 
syntax, have the \ML\ type~\mltydx{Sign.sg}. For identification, each 
864
d63b111b917a
quite a lot of minor and major revisions (inspecting theories, read_axm,
wenzelm
parents:
478
diff
changeset

15 
signature carries a unique list of \bfindex{stamps}, which are \ML\ 
324  16 
references to strings. The strings serve as humanreadable names; the 
104  17 
references serve as unique identifiers. Each primitive signature has a 
18 
single stamp. When two signatures are merged, their lists of stamps are 

19 
also merged. Every theory carries a unique signature. 

20 

21 
Terms and types are the underlying representation of logical syntax. Their 

275  22 
\ML\ definitions are irrelevant to naive Isabelle users. Programmers who 
23 
wish to extend Isabelle may need to know such details, say to code a tactic 

24 
that looks for subgoals of a particular form. Terms and types may be 

104  25 
`certified' to be wellformed with respect to a given signature. 
26 

324  27 

28 
\section{Defining theories}\label{sec:refdefiningtheories} 

104  29 

6571  30 
Theories are defined via theory files $name$\texttt{.thy} (there are also 
31 
\MLlevel interfaces which are only intended for people building advanced 

32 
theory definition packages). Appendix~\ref{app:TheorySyntax} presents the 

33 
concrete syntax for theory files; here follows an explanation of the 

34 
constituent parts. 

864
d63b111b917a
quite a lot of minor and major revisions (inspecting theories, read_axm,
wenzelm
parents:
478
diff
changeset

35 
\begin{description} 
6568  36 
\item[{\it theoryDef}] is the full definition. The new theory is called $id$. 
8136  37 
It is the union of the named \textbf{parent 
3108  38 
theories}\indexbold{theories!parent}, possibly extended with new 
6568  39 
components. \thydx{Pure} and \thydx{CPure} are the basic theories, which 
40 
contain only the metalogic. They differ just in their concrete syntax for 

41 
function applications. 

6571  42 

43 
The new theory begins as a merge of its parents. 

44 
\begin{ttbox} 

45 
Attempt to merge different versions of theories: "\(T@1\)", \(\ldots\), "\(T@n\)" 

46 
\end{ttbox} 

47 
This error may especially occur when a theory is redeclared  say to 

48 
change an inappropriate definition  and bindings to old versions persist. 

49 
Isabelle ensures that old and new theories of the same name are not involved 

50 
in a proof. 

324  51 

864
d63b111b917a
quite a lot of minor and major revisions (inspecting theories, read_axm,
wenzelm
parents:
478
diff
changeset

52 
\item[$classes$] 
d63b111b917a
quite a lot of minor and major revisions (inspecting theories, read_axm,
wenzelm
parents:
478
diff
changeset

53 
is a series of class declarations. Declaring {\tt$id$ < $id@1$ \dots\ 
324  54 
$id@n$} makes $id$ a subclass of the existing classes $id@1\dots 
55 
id@n$. This rules out cyclic class structures. Isabelle automatically 

56 
computes the transitive closure of subclass hierarchies; it is not 

6669  57 
necessary to declare \texttt{c < e} in addition to \texttt{c < d} and \texttt{d < 
324  58 
e}. 
59 

864
d63b111b917a
quite a lot of minor and major revisions (inspecting theories, read_axm,
wenzelm
parents:
478
diff
changeset

60 
\item[$default$] 
d63b111b917a
quite a lot of minor and major revisions (inspecting theories, read_axm,
wenzelm
parents:
478
diff
changeset

61 
introduces $sort$ as the new default sort for type variables. This applies 
d63b111b917a
quite a lot of minor and major revisions (inspecting theories, read_axm,
wenzelm
parents:
478
diff
changeset

62 
to unconstrained type variables in an input string but not to type 
d63b111b917a
quite a lot of minor and major revisions (inspecting theories, read_axm,
wenzelm
parents:
478
diff
changeset

63 
variables created internally. If omitted, the default sort is the listwise 
d63b111b917a
quite a lot of minor and major revisions (inspecting theories, read_axm,
wenzelm
parents:
478
diff
changeset

64 
union of the default sorts of the parent theories (i.e.\ their logical 
d63b111b917a
quite a lot of minor and major revisions (inspecting theories, read_axm,
wenzelm
parents:
478
diff
changeset

65 
intersection). 
3108  66 

7168  67 
\item[$sort$] is a finite set of classes. A single class $id$ abbreviates the 
68 
sort $\{id\}$. 

324  69 

864
d63b111b917a
quite a lot of minor and major revisions (inspecting theories, read_axm,
wenzelm
parents:
478
diff
changeset

70 
\item[$types$] 
324  71 
is a series of type declarations. Each declares a new type constructor 
72 
or type synonym. An $n$place type constructor is specified by 

73 
$(\alpha@1,\dots,\alpha@n)name$, where the type variables serve only to 

74 
indicate the number~$n$. 

75 

8136  76 
A \textbf{type synonym}\indexbold{type synonyms} is an abbreviation 
1387  77 
$(\alpha@1,\dots,\alpha@n)name = \tau$, where $name$ and $\tau$ can 
78 
be strings. 

864
d63b111b917a
quite a lot of minor and major revisions (inspecting theories, read_axm,
wenzelm
parents:
478
diff
changeset

79 

d63b111b917a
quite a lot of minor and major revisions (inspecting theories, read_axm,
wenzelm
parents:
478
diff
changeset

80 
\item[$infix$] 
8136  81 
declares a type or constant to be an infix operator having priority $nat$ 
82 
and associating to the left (\texttt{infixl}) or right (\texttt{infixr}). 

83 
Only 2place type constructors can have infix status; an example is {\tt 

3108  84 
('a,'b)~"*"~(infixr~20)}, which may express binary product types. 
324  85 

3108  86 
\item[$arities$] is a series of type arity declarations. Each assigns 
87 
arities to type constructors. The $name$ must be an existing type 

88 
constructor, which is given the additional arity $arity$. 

89 

5369  90 
\item[$nonterminals$]\index{*nonterminal symbols} declares purely 
91 
syntactic types to be used as nonterminal symbols of the context 

92 
free grammar. 

93 

3108  94 
\item[$consts$] is a series of constant declarations. Each new 
95 
constant $name$ is given the specified type. The optional $mixfix$ 

96 
annotations may attach concrete syntax to the constant. 

97 

98 
\item[$syntax$] \index{*syntax section}\index{print mode} is a variant 

99 
of $consts$ which adds just syntax without actually declaring 

100 
logical constants. This gives full control over a theory's context 

3485
f27a30a18a17
Now there are TWO spaces after each full stop, so that the Emacs sentence
paulson
parents:
3201
diff
changeset

101 
free grammar. The optional $mode$ specifies the print mode where the 
f27a30a18a17
Now there are TWO spaces after each full stop, so that the Emacs sentence
paulson
parents:
3201
diff
changeset

102 
mixfix productions should be added. If there is no \texttt{output} 
3108  103 
option given, all productions are also added to the input syntax 
104 
(regardless of the print mode). 

324  105 

106 
\item[$mixfix$] \index{mixfix declarations} 

107 
annotations can take three forms: 

273  108 
\begin{itemize} 
109 
\item A mixfix template given as a $string$ of the form 

110 
{\tt"}\dots{\tt\_}\dots{\tt\_}\dots{\tt"} where the $i$th underscore 

324  111 
indicates the position where the $i$th argument should go. The list 
112 
of numbers gives the priority of each argument. The final number gives 

113 
the priority of the whole construct. 

104  114 

324  115 
\item A constant $f$ of type $\tau@1\To(\tau@2\To\tau)$ can be given {\bf 
116 
infix} status. 

104  117 

324  118 
\item A constant $f$ of type $(\tau@1\To\tau@2)\To\tau$ can be given {\bf 
6669  119 
binder} status. The declaration \texttt{binder} $\cal Q$ $p$ causes 
286  120 
${\cal Q}\,x.F(x)$ to be treated 
121 
like $f(F)$, where $p$ is the priority. 

273  122 
\end{itemize} 
324  123 

864
d63b111b917a
quite a lot of minor and major revisions (inspecting theories, read_axm,
wenzelm
parents:
478
diff
changeset

124 
\item[$trans$] 
d63b111b917a
quite a lot of minor and major revisions (inspecting theories, read_axm,
wenzelm
parents:
478
diff
changeset

125 
specifies syntactic translation rules (macros). There are three forms: 
6669  126 
parse rules (\texttt{=>}), print rules (\texttt{<=}), and parse/print rules ({\tt 
864
d63b111b917a
quite a lot of minor and major revisions (inspecting theories, read_axm,
wenzelm
parents:
478
diff
changeset

127 
==}). 
324  128 

1650  129 
\item[$rules$] 
864
d63b111b917a
quite a lot of minor and major revisions (inspecting theories, read_axm,
wenzelm
parents:
478
diff
changeset

130 
is a series of rule declarations. Each has a name $id$ and the formula is 
d63b111b917a
quite a lot of minor and major revisions (inspecting theories, read_axm,
wenzelm
parents:
478
diff
changeset

131 
given by the $string$. Rule names must be distinct within any single 
3108  132 
theory. 
324  133 

1905  134 
\item[$defs$] is a series of definitions. They are just like $rules$, except 
135 
that every $string$ must be a definition (see below for details). 

1650  136 

137 
\item[$constdefs$] combines the declaration of constants and their 

3485
f27a30a18a17
Now there are TWO spaces after each full stop, so that the Emacs sentence
paulson
parents:
3201
diff
changeset

138 
definition. The first $string$ is the type, the second the definition. 
3108  139 

6625  140 
\item[$axclass$] \index{*axclass section} defines an \rmindex{axiomatic type 
141 
class} \cite{Wenzel:1997:TPHOL} as the intersection of existing classes, 

142 
with additional axioms holding. Class axioms may not contain more than one 

143 
type variable. The class axioms (with implicit sort constraints added) are 

144 
bound to the given names. Furthermore a class introduction rule is 

145 
generated, which is automatically employed by $instance$ to prove 

146 
instantiations of this class. 

3108  147 

148 
\item[$instance$] \index{*instance section} proves class inclusions or 

149 
type arities at the logical level and then transfers these to the 

3485
f27a30a18a17
Now there are TWO spaces after each full stop, so that the Emacs sentence
paulson
parents:
3201
diff
changeset

150 
type signature. The instantiation is proven and checked properly. 
3108  151 
The user has to supply sufficient witness information: theorems 
152 
($longident$), axioms ($string$), or even arbitrary \ML{} tactic 

153 
code $verbatim$. 

1650  154 

1846  155 
\item[$oracle$] links the theory to a trusted external reasoner. It is 
156 
allowed to create theorems, but each theorem carries a proof object 

157 
describing the oracle invocation. See \S\ref{sec:oracles} for details. 

4543  158 

5369  159 
\item[$local$, $global$] change the current name declaration mode. 
4543  160 
Initially, theories start in $local$ mode, causing all names of 
161 
types, constants, axioms etc.\ to be automatically qualified by the 

162 
theory name. Changing this to $global$ causes all names to be 

163 
declared as short base names only. 

164 

165 
The $local$ and $global$ declarations act like switches, affecting 

166 
all following theory sections until changed again explicitly. Also 

167 
note that the final state at the end of the theory will persist. In 

168 
particular, this determines how the names of theorems stored later 

169 
on are handled. 

5369  170 

171 
\item[$setup$]\index{*setup!theory} applies a list of ML functions to 

172 
the theory. The argument should denote a value of type 

173 
\texttt{(theory > theory) list}. Typically, ML packages are 

174 
initialized in this way. 

1846  175 

324  176 
\item[$ml$] \index{*ML section} 
864
d63b111b917a
quite a lot of minor and major revisions (inspecting theories, read_axm,
wenzelm
parents:
478
diff
changeset

177 
consists of \ML\ code, typically for parse and print translation functions. 
104  178 
\end{description} 
324  179 
% 
864
d63b111b917a
quite a lot of minor and major revisions (inspecting theories, read_axm,
wenzelm
parents:
478
diff
changeset

180 
Chapters~\ref{DefiningLogics} and \ref{chap:syntax} explain mixfix 
6669  181 
declarations, translation rules and the \texttt{ML} section in more detail. 
145  182 

183 

1905  184 
\subsection{Definitions}\indexbold{definitions} 
185 

8136  186 
\textbf{Definitions} are intended to express abbreviations. The simplest 
3108  187 
form of a definition is $f \equiv t$, where $f$ is a constant. 
188 
Isabelle also allows a derived forms where the arguments of~$f$ appear 

189 
on the left, abbreviating a string of $\lambda$abstractions. 

1905  190 

191 
Isabelle makes the following checks on definitions: 

192 
\begin{itemize} 

3108  193 
\item Arguments (on the lefthand side) must be distinct variables. 
1905  194 
\item All variables on the righthand side must also appear on the lefthand 
195 
side. 

3108  196 
\item All type variables on the righthand side must also appear on 
197 
the lefthand side; this prohibits definitions such as {\tt 

198 
(zero::nat) == length ([]::'a list)}. 

1905  199 
\item The definition must not be recursive. Most objectlogics provide 
200 
definitional principles that can be used to express recursion safely. 

201 
\end{itemize} 

202 
These checks are intended to catch the sort of errors that might be made 

203 
accidentally. Misspellings, for instance, might result in additional 

204 
variables appearing on the righthand side. More elaborate checks could be 

205 
made, but the cost might be overly strict rules on declaration order, etc. 

206 

207 

275  208 
\subsection{*Classes and arities} 
324  209 
\index{classes!context conditions}\index{arities!context conditions} 
145  210 

286  211 
In order to guarantee principal types~\cite{nipkowprehofer}, 
324  212 
arity declarations must obey two conditions: 
145  213 
\begin{itemize} 
3108  214 
\item There must not be any two declarations $ty :: (\vec{r})c$ and 
215 
$ty :: (\vec{s})c$ with $\vec{r} \neq \vec{s}$. For example, this 

216 
excludes the following: 

145  217 
\begin{ttbox} 
864
d63b111b917a
quite a lot of minor and major revisions (inspecting theories, read_axm,
wenzelm
parents:
478
diff
changeset

218 
arities 
8136  219 
foo :: (\{logic{\}}) logic 
220 
foo :: (\{{\}})logic 

145  221 
\end{ttbox} 
286  222 

145  223 
\item If there are two declarations $ty :: (s@1,\dots,s@n)c$ and $ty :: 
224 
(s@1',\dots,s@n')c'$ such that $c' < c$ then $s@i' \preceq s@i$ must hold 

225 
for $i=1,\dots,n$. The relationship $\preceq$, defined as 

226 
\[ s' \preceq s \iff \forall c\in s. \exists c'\in s'.~ c'\le c, \] 

3108  227 
expresses that the set of types represented by $s'$ is a subset of the 
228 
set of types represented by $s$. Assuming $term \preceq logic$, the 

229 
following is forbidden: 

145  230 
\begin{ttbox} 
864
d63b111b917a
quite a lot of minor and major revisions (inspecting theories, read_axm,
wenzelm
parents:
478
diff
changeset

231 
arities 
8136  232 
foo :: (\{logic{\}})logic 
233 
foo :: (\{{\}})term 

145  234 
\end{ttbox} 
286  235 

145  236 
\end{itemize} 
237 

104  238 

6568  239 
\section{The theory loader}\label{sec:moretheories} 
240 
\index{theories!reading}\index{files!reading} 

241 

242 
Isabelle's theory loader manages dependencies of the internal graph of theory 

243 
nodes (the \emph{theory database}) and the external view of the file system. 

244 
See \S\ref{sec:introtheories} for its most basic commands, such as 

245 
\texttt{use_thy}. There are a few more operations available. 

246 

864
d63b111b917a
quite a lot of minor and major revisions (inspecting theories, read_axm,
wenzelm
parents:
478
diff
changeset

247 
\begin{ttbox} 
6568  248 
use_thy_only : string > unit 
7136  249 
update_thy_only : string > unit 
6568  250 
touch_thy : string > unit 
6658  251 
remove_thy : string > unit 
8136  252 
delete_tmpfiles : bool ref \hfill\textbf{initially true} 
286  253 
\end{ttbox} 
254 

324  255 
\begin{ttdescription} 
6569  256 
\item[\ttindexbold{use_thy_only} "$name$";] is similar to \texttt{use_thy}, 
257 
but processes the actual theory file $name$\texttt{.thy} only, ignoring 

6568  258 
$name$\texttt{.ML}. This might be useful in replaying proof scripts by hand 
259 
from the very beginning, starting with the fresh theory. 

260 

7136  261 
\item[\ttindexbold{update_thy_only} "$name$";] is similar to 
262 
\texttt{update_thy}, but processes the actual theory file 

263 
$name$\texttt{.thy} only, ignoring $name$\texttt{.ML}. 

264 

6569  265 
\item[\ttindexbold{touch_thy} "$name$";] marks theory node $name$ of the 
6568  266 
internal graph as outdated. While the theory remains usable, subsequent 
267 
operations such as \texttt{use_thy} may cause a reload. 

268 

6658  269 
\item[\ttindexbold{remove_thy} "$name$";] deletes theory node $name$, 
270 
including \emph{all} of its descendants. Beware! This is a quick way to 

271 
dispose a large number of theories at once. Note that {\ML} bindings to 

272 
theorems etc.\ of removed theories may still persist. 

273 

6568  274 
\item[reset \ttindexbold{delete_tmpfiles};] processing theory files usually 
275 
involves temporary {\ML} files to be created. By default, these are deleted 

276 
afterwards. Resetting the \texttt{delete_tmpfiles} flag inhibits this, 

277 
leaving the generated code for debugging purposes. The basic location for 

278 
temporary files is determined by the \texttt{ISABELLE_TMP} environment 

6571  279 
variable (which is private to the running Isabelle process and may be 
6568  280 
retrieved by \ttindex{getenv} from {\ML}). 
324  281 
\end{ttdescription} 
138
9ba8bff1addc
added chapter "Defining Theories" and made changes for new Readthy functions
clasohm
parents:
104
diff
changeset

282 

6568  283 
\medskip Theory and {\ML} files are located by skimming through the 
284 
directories listed in Isabelle's internal load path, which merely contains the 

285 
current directory ``\texttt{.}'' by default. The load path may be accessed by 

286 
the following operations. 

332  287 

864
d63b111b917a
quite a lot of minor and major revisions (inspecting theories, read_axm,
wenzelm
parents:
478
diff
changeset

288 
\begin{ttbox} 
6568  289 
show_path: unit > string list 
290 
add_path: string > unit 

291 
del_path: string > unit 

292 
reset_path: unit > unit 

7440  293 
with_path: string > ('a > 'b) > 'a > 'b 
286  294 
\end{ttbox} 
295 

324  296 
\begin{ttdescription} 
6568  297 
\item[\ttindexbold{show_path}();] displays the load path components in 
6571  298 
canonical string representation (which is always according to Unix rules). 
6568  299 

6569  300 
\item[\ttindexbold{add_path} "$dir$";] adds component $dir$ to the beginning 
301 
of the load path. 

6568  302 

6569  303 
\item[\ttindexbold{del_path} "$dir$";] removes any occurrences of component 
6568  304 
$dir$ from the load path. 
305 

306 
\item[\ttindexbold{reset_path}();] resets the load path to ``\texttt{.}'' 

307 
(current directory) only. 

7440  308 

309 
\item[\ttindexbold{with_path} "$dir$" $f$ $x$;] temporarily adds component 

310 
$dir$ to the beginning of the load path before executing $(f~x)$. 

311 

324  312 
\end{ttdescription} 
286  313 

7440  314 
Furthermore, in operations referring indirectly to some file (e.g.\ 
315 
\texttt{use_dir}) the argument may be prefixed by a directory that will be 

316 
temporarily appended to the load path, too. 

104  317 

318 

6669  319 
\section{Locales} 
320 
\label{Locales} 

321 

322 
Locales \cite{kammuellerlocales} are a concept of local proof contexts. They 

323 
are introduced as named syntactic objects within theories and can be 

324 
opened in any descendant theory. 

325 

326 
\subsection{Declaring Locales} 

327 

328 
A locale is declared in a theory section that starts with the 

329 
keyword \texttt{locale}. It consists typically of three parts, the 

330 
\texttt{fixes} part, the \texttt{assumes} part, and the \texttt{defines} part. 

331 
Appendix \ref{app:TheorySyntax} presents the full syntax. 

332 

333 
\subsubsection{Parts of Locales} 

334 

335 
The subsection introduced by the keyword \texttt{fixes} declares the locale 

336 
constants in a way that closely resembles a global \texttt{consts} 

337 
declaration. In particular, there may be an optional pretty printing syntax 

338 
for the locale constants. 

339 

340 
The subsequent \texttt{assumes} part specifies the locale rules. They are 

341 
defined like \texttt{rules}: by an identifier followed by the rule 

342 
given as a string. Locale rules admit the statement of local assumptions 

343 
about the locale constants. The \texttt{assumes} part is optional. Nonfixed 

344 
variables in locale rules are automatically bound by the universal quantifier 

345 
\texttt{!!} of the metalogic. 

346 

347 
Finally, the \texttt{defines} part introduces the definitions that are 

348 
available in the locale. Locale constants declared in the \texttt{fixes} 

349 
section are defined using the metaequality \texttt{==}. If the 

350 
locale constant is a functiond then its definition can (as usual) have 

351 
variables on the lefthand side acting as formal parameters; they are 

352 
considered as schematic variables and are automatically generalized by 

353 
universal quantification of the metalogic. The right hand side of a 

354 
definition must not contain variables that are not already on the left hand 

355 
side. In so far locale definitions behave like theory level definitions. 

356 
However, the locale concept realizes \emph{dependent definitions}: any variable 

357 
that is fixed as a locale constant can occur on the right hand side of 

358 
definitions. For an illustration of these dependent definitions see the 

359 
occurrence of the locale constant \texttt{G} on the right hand side of the 

360 
definitions of the locale \texttt{group} below. Naturally, definitions can 

361 
already use the syntax of the locale constants in the \texttt{fixes} 

362 
subsection. The \texttt{defines} part is, as the \texttt{assumes} part, 

363 
optional. 

364 

365 
\subsubsection{Example for Definition} 

366 
The concrete syntax of locale definitions is demonstrated by example below. 

367 

368 
Locale \texttt{group} assumes the definition of groups in a theory 

369 
file\footnote{This and other examples are from \texttt{HOL/ex}.}. A locale 

370 
defining a convenient proof environment for group related proofs may be 

371 
added to the theory as follows: 

372 
\begin{ttbox} 

373 
locale group = 

374 
fixes 

375 
G :: "'a grouptype" 

376 
e :: "'a" 

377 
binop :: "'a => 'a => 'a" (infixr "#" 80) 

378 
inv :: "'a => 'a" ("i(_)" [90] 91) 

379 
assumes 

380 
Group_G "G: Group" 

381 
defines 

382 
e_def "e == unit G" 

383 
binop_def "x # y == bin_op G x y" 

384 
inv_def "i(x) == inverse G x" 

385 
\end{ttbox} 

386 

387 
\subsubsection{Polymorphism} 

388 

389 
In contrast to polymorphic definitions in theories, the use of the 

390 
same type variable for the declaration of different locale constants in the 

391 
fixes part means \emph{the same} type. In other words, the scope of the 

392 
polymorphic variables is extended over all constant declarations of a locale. 

393 
In the above example \texttt{'a} refers to the same type which is fixed inside 

394 
the locale. In an exported theorem (see \S\ref{sec:localeexport}) the 

395 
constructors of locale \texttt{group} are polymorphic, yet only simultaneously 

396 
instantiatable. 

397 

398 
\subsubsection{Nested Locales} 

399 

400 
A locale can be defined as the extension of a previously defined 

401 
locale. This operation of extension is optional and is syntactically 

402 
expressed as 

403 
\begin{ttbox} 

404 
locale foo = bar + ... 

405 
\end{ttbox} 

406 
The locale \texttt{foo} builds on the constants and syntax of the locale {\tt 

407 
bar}. That is, all contents of the locale \texttt{bar} can be used in 

408 
definitions and rules of the corresponding parts of the locale {\tt 

409 
foo}. Although locale \texttt{foo} assumes the \texttt{fixes} part of \texttt{bar} it 

410 
does not automatically subsume its rules and definitions. Normally, one 

411 
expects to use locale \texttt{foo} only if locale \texttt{bar} is already 

412 
active. These aspects of use and activation of locales are considered in the 

413 
subsequent section. 

414 

415 

416 
\subsection{Locale Scope} 

417 

418 
Locales are by default inactive, but they can be invoked. The list of 

419 
currently active locales is called \emph{scope}. The process of activating 

420 
them is called \emph{opening}; the reverse is \emph{closing}. 

421 

422 
\subsubsection{Scope} 

423 
The locale scope is part of each theory. It is a dynamic stack containing 

424 
all active locales at a certain point in an interactive session. 

425 
The scope lives until all locales are explicitly closed. At one time there 

426 
can be more than one locale open. The contents of these various active 

427 
locales are all visible in the scope. In case of nested locales for example, 

428 
the nesting is actually reflected to the scope, which contains the nested 

429 
locales as layers. To check the state of the scope during a development the 

430 
function \texttt{Print\_scope} may be used. It displays the names of all open 

431 
locales on the scope. The function \texttt{print\_locales} applied to a theory 

432 
displays all locales contained in that theory and in addition also the 

433 
current scope. 

434 

435 
The scope is manipulated by the commands for opening and closing of locales. 

436 

437 
\subsubsection{Opening} 

438 
Locales can be \emph{opened} at any point during a session where 

439 
we want to prove theorems concerning the locale. Opening a locale means 

440 
making its contents visible by pushing it onto the scope of the current 

441 
theory. Inside a scope of opened locales, theorems can use all definitions and 

442 
rules contained in the locales on the scope. The rules and definitions may 

443 
be accessed individually using the function \ttindex{thm}. This function is 

444 
applied to the names assigned to locale rules and definitions as 

445 
strings. The opening command is called \texttt{Open\_locale} and takes the 

446 
name of the locale to be opened as its argument. 

447 

448 
If one opens a locale \texttt{foo} that is defined by extension from locale 

449 
\texttt{bar}, the function \texttt{Open\_locale} checks if locale \texttt{bar} 

450 
is open. If so, then it just opens \texttt{foo}, if not, then it prints a 

451 
message and opens \texttt{bar} before opening \texttt{foo}. Naturally, this 

452 
carries on, if \texttt{bar} is again an extension. 

453 

454 
\subsubsection{Closing} 

455 

456 
\emph{Closing} means to cancel the last opened locale, pushing it out of the 

457 
scope. Theorems proved during the life cycle of this locale will be disabled, 

458 
unless they have been explicitly exported, as described below. However, when 

459 
the same locale is opened again these theorems may be used again as well, 

460 
provided that they were saved as theorems in the first place, using 

461 
\texttt{qed} or ML assignment. The command \texttt{Close\_locale} takes a 

462 
locale name as a string and checks if this locale is actually the topmost 

463 
locale on the scope. If this is the case, it removes this locale, otherwise 

464 
it prints a warning message and does not change the scope. 

465 

466 
\subsubsection{Export of Theorems} 

467 
\label{sec:localeexport} 

468 

469 
Export of theorems transports theorems out of the scope of locales. Locale 

470 
rules that have been used in the proof of an exported theorem inside the 

471 
locale are carried by the exported form of the theorem as its individual 

472 
metaassumptions. The locale constants are universally quantified variables 

473 
in these theorems, hence such theorems can be instantiated individually. 

474 
Definitions become unfolded; locale constants that were merely used for 

475 
definitions vanish. Logically, exporting corresponds to a combined 

476 
application of introduction rules for implication and universal 

477 
quantification. Exporting forms a kind of normalization of theorems in a 

478 
locale scope. 

479 

480 
According to the possibility of nested locales there are two different forms 

481 
of export. The first one is realized by the function \texttt{export} that 

482 
exports theorems through all layers of opened locales of the scope. Hence, 

483 
the application of export to a theorem yields a theorem of the global level, 

484 
that is, the current theory context without any local assumptions or 

485 
definitions. 

486 

487 
When locales are nested we might want to export a theorem, not to the global 

488 
level of the current theory but just to the previous level. The other export 

489 
function, \texttt{Export}, transports theorems one level up in the scope; the 

490 
theorem still uses locale constants, definitions and rules of the locales 

491 
underneath. 

492 

493 
\subsection{Functions for Locales} 

494 
\label{Syntax} 

495 
\index{locales!functions} 

496 

497 
Here is a quick reference list of locale functions. 

498 
\begin{ttbox} 

499 
Open_locale : xstring > unit 

500 
Close_locale : xstring > unit 

501 
export : thm > thm 

502 
Export : thm > thm 

503 
thm : xstring > thm 

504 
Print_scope : unit > unit 

505 
print_locales: theory > unit 

506 
\end{ttbox} 

507 
\begin{ttdescription} 

508 
\item[\ttindexbold{Open_locale} $xstring$] 

509 
opens the locale {\it xstring}, adding it to the scope of the theory of the 

510 
current context. If the opened locale is built by extension, the ancestors 

511 
are opened automatically. 

512 

513 
\item[\ttindexbold{Close_locale} $xstring$] eliminates the locale {\it 

514 
xstring} from the scope if it is the topmost item on it, otherwise it does 

515 
not change the scope and produces a warning. 

516 

517 
\item[\ttindexbold{export} $thm$] locale definitions become expanded in {\it 

518 
thm} and locale rules that were used in the proof of {\it thm} become part 

519 
of its individual assumptions. This normalization happens with respect to 

520 
\emph{all open locales} on the scope. 

521 

522 
\item[\ttindexbold{Export} $thm$] works like \texttt{export} but normalizes 

523 
theorems only up to the previous level of locales on the scope. 

524 

525 
\item[\ttindexbold{thm} $xstring$] applied to the name of a locale definition 

526 
or rule it returns the definition as a theorem. 

527 

528 
\item[\ttindexbold{Print_scope}()] prints the names of the locales in the 

529 
current scope of the current theory context. 

530 

531 
\item[\ttindexbold{print_locale} $theory$] prints all locales that are 

532 
contained in {\it theory} directly or indirectly. It also displays the 

533 
current scope similar to \texttt{Print\_scope}. 

534 
\end{ttdescription} 

535 

536 

866
2d3d020eef11
added documentation of bind_thm, qed, qed_goal, get_thm, thms_of
clasohm
parents:
864
diff
changeset

537 
\section{Basic operations on theories}\label{BasicOperationsOnTheories} 
4384  538 

539 
\subsection{*Theory inclusion} 

540 
\begin{ttbox} 

541 
subthy : theory * theory > bool 

542 
eq_thy : theory * theory > bool 

543 
transfer : theory > thm > thm 

544 
transfer_sg : Sign.sg > thm > thm 

545 
\end{ttbox} 

546 

547 
Inclusion and equality of theories is determined by unique 

548 
identification stamps that are created when declaring new components. 

549 
Theorems contain a reference to the theory (actually to its signature) 

550 
they have been derived in. Transferring theorems to super theories 

551 
has no logical significance, but may affect some operations in subtle 

552 
ways (e.g.\ implicit merges of signatures when applying rules, or 

553 
pretty printing of theorems). 

554 

555 
\begin{ttdescription} 

556 

557 
\item[\ttindexbold{subthy} ($thy@1$, $thy@2$)] determines if $thy@1$ 

558 
is included in $thy@2$ wrt.\ identification stamps. 

559 

560 
\item[\ttindexbold{eq_thy} ($thy@1$, $thy@2$)] determines if $thy@1$ 

561 
is exactly the same as $thy@2$. 

562 

563 
\item[\ttindexbold{transfer} $thy$ $thm$] transfers theorem $thm$ to 

564 
theory $thy$, provided the latter includes the theory of $thm$. 

565 

566 
\item[\ttindexbold{transfer_sg} $sign$ $thm$] is similar to 

567 
\texttt{transfer}, but identifies the super theory via its 

568 
signature. 

569 

570 
\end{ttdescription} 

571 

572 

6571  573 
\subsection{*Primitive theories} 
864
d63b111b917a
quite a lot of minor and major revisions (inspecting theories, read_axm,
wenzelm
parents:
478
diff
changeset

574 
\begin{ttbox} 
4317  575 
ProtoPure.thy : theory 
3108  576 
Pure.thy : theory 
577 
CPure.thy : theory 

286  578 
\end{ttbox} 
3108  579 
\begin{description} 
4317  580 
\item[\ttindexbold{ProtoPure.thy}, \ttindexbold{Pure.thy}, 
581 
\ttindexbold{CPure.thy}] contain the syntax and signature of the 

582 
metalogic. There are basically no axioms: metalevel inferences 

583 
are carried out by \ML\ functions. \texttt{Pure} and \texttt{CPure} 

584 
just differ in their concrete syntax of prefix function application: 

585 
$t(u@1, \ldots, u@n)$ in \texttt{Pure} vs.\ $t\,u@1,\ldots\,u@n$ in 

586 
\texttt{CPure}. \texttt{ProtoPure} is their common parent, 

587 
containing no syntax for printing prefix applications at all! 

6571  588 

864
d63b111b917a
quite a lot of minor and major revisions (inspecting theories, read_axm,
wenzelm
parents:
478
diff
changeset

589 
%% FIXME 
478  590 
%\item [\ttindexbold{extend_theory} $thy$ {\tt"}$T${\tt"} $\cdots$] extends 
591 
% the theory $thy$ with new types, constants, etc. $T$ identifies the theory 

592 
% internally. When a theory is redeclared, say to change an incorrect axiom, 

593 
% bindings to the old axiom may persist. Isabelle ensures that the old and 

594 
% new theories are not involved in the same proof. Attempting to combine 

595 
% different theories having the same name $T$ yields the fatal error 

596 
%extend_theory : theory > string > \(\cdots\) > theory 

864
d63b111b917a
quite a lot of minor and major revisions (inspecting theories, read_axm,
wenzelm
parents:
478
diff
changeset

597 
%\begin{ttbox} 
d63b111b917a
quite a lot of minor and major revisions (inspecting theories, read_axm,
wenzelm
parents:
478
diff
changeset

598 
%Attempt to merge different versions of theory: \(T\) 
d63b111b917a
quite a lot of minor and major revisions (inspecting theories, read_axm,
wenzelm
parents:
478
diff
changeset

599 
%\end{ttbox} 
3108  600 
\end{description} 
286  601 

864
d63b111b917a
quite a lot of minor and major revisions (inspecting theories, read_axm,
wenzelm
parents:
478
diff
changeset

602 
%% FIXME 
275  603 
%\item [\ttindexbold{extend_theory} $thy$ {\tt"}$T${\tt"} 
604 
% ($classes$, $default$, $types$, $arities$, $consts$, $sextopt$) $rules$] 

605 
%\hfill\break %%% include if line is just too short 

286  606 
%is the \ML{} equivalent of the following theory definition: 
275  607 
%\begin{ttbox} 
608 
%\(T\) = \(thy\) + 

609 
%classes \(c\) < \(c@1\),\(\dots\),\(c@m\) 

610 
% \dots 

611 
%default {\(d@1,\dots,d@r\)} 

612 
%types \(tycon@1\),\dots,\(tycon@i\) \(n\) 

613 
% \dots 

614 
%arities \(tycon@1'\),\dots,\(tycon@j'\) :: (\(s@1\),\dots,\(s@n\))\(c\) 

615 
% \dots 

616 
%consts \(b@1\),\dots,\(b@k\) :: \(\tau\) 

617 
% \dots 

618 
%rules \(name\) \(rule\) 

619 
% \dots 

620 
%end 

621 
%\end{ttbox} 

622 
%where 

623 
%\begin{tabular}[t]{l@{~=~}l} 

624 
%$classes$ & \tt[("$c$",["$c@1$",\dots,"$c@m$"]),\dots] \\ 

625 
%$default$ & \tt["$d@1$",\dots,"$d@r$"]\\ 

626 
%$types$ & \tt[([$tycon@1$,\dots,$tycon@i$], $n$),\dots] \\ 

627 
%$arities$ & \tt[([$tycon'@1$,\dots,$tycon'@j$], ([$s@1$,\dots,$s@n$],$c$)),\dots] 

628 
%\\ 

629 
%$consts$ & \tt[([$b@1$,\dots,$b@k$],$\tau$),\dots] \\ 

630 
%$rules$ & \tt[("$name$",$rule$),\dots] 

631 
%\end{tabular} 

104  632 

633 

864
d63b111b917a
quite a lot of minor and major revisions (inspecting theories, read_axm,
wenzelm
parents:
478
diff
changeset

634 
\subsection{Inspecting a theory}\label{sec:inspctthy} 
104  635 
\index{theories!inspectingbold} 
864
d63b111b917a
quite a lot of minor and major revisions (inspecting theories, read_axm,
wenzelm
parents:
478
diff
changeset

636 
\begin{ttbox} 
4317  637 
print_syntax : theory > unit 
638 
print_theory : theory > unit 

639 
parents_of : theory > theory list 

640 
ancestors_of : theory > theory list 

641 
sign_of : theory > Sign.sg 

642 
Sign.stamp_names_of : Sign.sg > string list 

104  643 
\end{ttbox} 
864
d63b111b917a
quite a lot of minor and major revisions (inspecting theories, read_axm,
wenzelm
parents:
478
diff
changeset

644 
These provide means of viewing a theory's components. 
324  645 
\begin{ttdescription} 
3108  646 
\item[\ttindexbold{print_syntax} $thy$] prints the syntax of $thy$ 
647 
(grammar, macros, translation functions etc., see 

648 
page~\pageref{pg:print_syn} for more details). 

649 

650 
\item[\ttindexbold{print_theory} $thy$] prints the logical parts of 

651 
$thy$, excluding the syntax. 

4317  652 

653 
\item[\ttindexbold{parents_of} $thy$] returns the direct ancestors 

654 
of~$thy$. 

655 

656 
\item[\ttindexbold{ancestors_of} $thy$] returns all ancestors of~$thy$ 

657 
(not including $thy$ itself). 

658 

659 
\item[\ttindexbold{sign_of} $thy$] returns the signature associated 

660 
with~$thy$. It is useful with functions like {\tt 

661 
read_instantiate_sg}, which take a signature as an argument. 

662 

663 
\item[\ttindexbold{Sign.stamp_names_of} $sg$]\index{signatures} 

664 
returns the names of the identification \rmindex{stamps} of ax 

665 
signature. These coincide with the names of its full ancestry 

666 
including that of $sg$ itself. 

104  667 

324  668 
\end{ttdescription} 
104  669 

1369  670 

104  671 
\section{Terms} 
672 
\index{termsbold} 

324  673 
Terms belong to the \ML\ type \mltydx{term}, which is a concrete datatype 
3108  674 
with six constructors: 
104  675 
\begin{ttbox} 
676 
type indexname = string * int; 

677 
infix 9 $; 

678 
datatype term = Const of string * typ 

679 
 Free of string * typ 

680 
 Var of indexname * typ 

681 
 Bound of int 

682 
 Abs of string * typ * term 

683 
 op $ of term * term; 

684 
\end{ttbox} 

324  685 
\begin{ttdescription} 
4317  686 
\item[\ttindexbold{Const} ($a$, $T$)] \index{constantsbold} 
8136  687 
is the \textbf{constant} with name~$a$ and type~$T$. Constants include 
286  688 
connectives like $\land$ and $\forall$ as well as constants like~0 
689 
and~$Suc$. Other constants may be required to define a logic's concrete 

864
d63b111b917a
quite a lot of minor and major revisions (inspecting theories, read_axm,
wenzelm
parents:
478
diff
changeset

690 
syntax. 
104  691 

4317  692 
\item[\ttindexbold{Free} ($a$, $T$)] \index{variables!freebold} 
8136  693 
is the \textbf{free variable} with name~$a$ and type~$T$. 
104  694 

4317  695 
\item[\ttindexbold{Var} ($v$, $T$)] \index{unknownsbold} 
8136  696 
is the \textbf{scheme variable} with indexname~$v$ and type~$T$. An 
324  697 
\mltydx{indexname} is a string paired with a nonnegative index, or 
698 
subscript; a term's scheme variables can be systematically renamed by 

699 
incrementing their subscripts. Scheme variables are essentially free 

700 
variables, but may be instantiated during unification. 

104  701 

324  702 
\item[\ttindexbold{Bound} $i$] \index{variables!boundbold} 
8136  703 
is the \textbf{bound variable} with de Bruijn index~$i$, which counts the 
324  704 
number of lambdas, starting from zero, between a variable's occurrence 
705 
and its binding. The representation prevents capture of variables. For 

706 
more information see de Bruijn \cite{debruijn72} or 

6592  707 
Paulson~\cite[page~376]{paulsonml2}. 
104  708 

4317  709 
\item[\ttindexbold{Abs} ($a$, $T$, $u$)] 
324  710 
\index{lambda abs@$\lambda$abstractionsbold} 
8136  711 
is the $\lambda$\textbf{abstraction} with body~$u$, and whose bound 
324  712 
variable has name~$a$ and type~$T$. The name is used only for parsing 
713 
and printing; it has no logical significance. 

104  714 

324  715 
\item[$t$ \$ $u$] \index{$@{\tt\$}bold} \index{function applicationsbold} 
8136  716 
is the \textbf{application} of~$t$ to~$u$. 
324  717 
\end{ttdescription} 
9695  718 
Application is written as an infix operator to aid readability. Here is an 
719 
\ML\ pattern to recognize FOL formulae of the form~$A\imp B$, binding the 

720 
subformulae to~$A$ and~$B$: 

864
d63b111b917a
quite a lot of minor and major revisions (inspecting theories, read_axm,
wenzelm
parents:
478
diff
changeset

721 
\begin{ttbox} 
104  722 
Const("Trueprop",_) $ (Const("op >",_) $ A $ B) 
723 
\end{ttbox} 

724 

725 

4317  726 
\section{*Variable binding} 
286  727 
\begin{ttbox} 
728 
loose_bnos : term > int list 

729 
incr_boundvars : int > term > term 

730 
abstract_over : term*term > term 

731 
variant_abs : string * typ * term > string * term 

8136  732 
aconv : term * term > bool\hfill\textbf{infix} 
286  733 
\end{ttbox} 
734 
These functions are all concerned with the de Bruijn representation of 

735 
bound variables. 

324  736 
\begin{ttdescription} 
864
d63b111b917a
quite a lot of minor and major revisions (inspecting theories, read_axm,
wenzelm
parents:
478
diff
changeset

737 
\item[\ttindexbold{loose_bnos} $t$] 
286  738 
returns the list of all dangling bound variable references. In 
6669  739 
particular, \texttt{Bound~0} is loose unless it is enclosed in an 
740 
abstraction. Similarly \texttt{Bound~1} is loose unless it is enclosed in 

286  741 
at least two abstractions; if enclosed in just one, the list will contain 
742 
the number 0. A wellformed term does not contain any loose variables. 

743 

864
d63b111b917a
quite a lot of minor and major revisions (inspecting theories, read_axm,
wenzelm
parents:
478
diff
changeset

744 
\item[\ttindexbold{incr_boundvars} $j$] 
332  745 
increases a term's dangling bound variables by the offset~$j$. This is 
286  746 
required when moving a subterm into a context where it is enclosed by a 
747 
different number of abstractions. Bound variables with a matching 

748 
abstraction are unaffected. 

749 

864
d63b111b917a
quite a lot of minor and major revisions (inspecting theories, read_axm,
wenzelm
parents:
478
diff
changeset

750 
\item[\ttindexbold{abstract_over} $(v,t)$] 
286  751 
forms the abstraction of~$t$ over~$v$, which may be any wellformed term. 
6669  752 
It replaces every occurrence of \(v\) by a \texttt{Bound} variable with the 
286  753 
correct index. 
754 

864
d63b111b917a
quite a lot of minor and major revisions (inspecting theories, read_axm,
wenzelm
parents:
478
diff
changeset

755 
\item[\ttindexbold{variant_abs} $(a,T,u)$] 
286  756 
substitutes into $u$, which should be the body of an abstraction. 
757 
It replaces each occurrence of the outermost bound variable by a free 

758 
variable. The free variable has type~$T$ and its name is a variant 

332  759 
of~$a$ chosen to be distinct from all constants and from all variables 
286  760 
free in~$u$. 
761 

864
d63b111b917a
quite a lot of minor and major revisions (inspecting theories, read_axm,
wenzelm
parents:
478
diff
changeset

762 
\item[$t$ \ttindexbold{aconv} $u$] 
286  763 
tests whether terms~$t$ and~$u$ are \(\alpha\)convertible: identical up 
764 
to renaming of bound variables. 

765 
\begin{itemize} 

766 
\item 

6669  767 
Two constants, \texttt{Free}s, or \texttt{Var}s are \(\alpha\)convertible 
286  768 
if their names and types are equal. 
769 
(Variables having the same name but different types are thus distinct. 

770 
This confusing situation should be avoided!) 

771 
\item 

772 
Two bound variables are \(\alpha\)convertible 

773 
if they have the same number. 

774 
\item 

775 
Two abstractions are \(\alpha\)convertible 

776 
if their bodies are, and their bound variables have the same type. 

777 
\item 

778 
Two applications are \(\alpha\)convertible 

779 
if the corresponding subterms are. 

780 
\end{itemize} 

781 

324  782 
\end{ttdescription} 
286  783 

864
d63b111b917a
quite a lot of minor and major revisions (inspecting theories, read_axm,
wenzelm
parents:
478
diff
changeset

784 
\section{Certified terms}\index{terms!certifiedbold}\index{signatures} 
8136  785 
A term $t$ can be \textbf{certified} under a signature to ensure that every type 
864
d63b111b917a
quite a lot of minor and major revisions (inspecting theories, read_axm,
wenzelm
parents:
478
diff
changeset

786 
in~$t$ is wellformed and every constant in~$t$ is a type instance of a 
d63b111b917a
quite a lot of minor and major revisions (inspecting theories, read_axm,
wenzelm
parents:
478
diff
changeset

787 
constant declared in the signature. The term must be welltyped and its use 
6669  788 
of bound variables must be wellformed. Metarules such as \texttt{forall_elim} 
864
d63b111b917a
quite a lot of minor and major revisions (inspecting theories, read_axm,
wenzelm
parents:
478
diff
changeset

789 
take certified terms as arguments. 
104  790 

324  791 
Certified terms belong to the abstract type \mltydx{cterm}. 
104  792 
Elements of the type can only be created through the certification process. 
793 
In case of error, Isabelle raises exception~\ttindex{TERM}\@. 

794 

795 
\subsection{Printing terms} 

324  796 
\index{terms!printing of} 
864
d63b111b917a
quite a lot of minor and major revisions (inspecting theories, read_axm,
wenzelm
parents:
478
diff
changeset

797 
\begin{ttbox} 
275  798 
string_of_cterm : cterm > string 
104  799 
Sign.string_of_term : Sign.sg > term > string 
800 
\end{ttbox} 

324  801 
\begin{ttdescription} 
864
d63b111b917a
quite a lot of minor and major revisions (inspecting theories, read_axm,
wenzelm
parents:
478
diff
changeset

802 
\item[\ttindexbold{string_of_cterm} $ct$] 
104  803 
displays $ct$ as a string. 
804 

864
d63b111b917a
quite a lot of minor and major revisions (inspecting theories, read_axm,
wenzelm
parents:
478
diff
changeset

805 
\item[\ttindexbold{Sign.string_of_term} $sign$ $t$] 
104  806 
displays $t$ as a string, using the syntax of~$sign$. 
324  807 
\end{ttdescription} 
104  808 

809 
\subsection{Making and inspecting certified terms} 

864
d63b111b917a
quite a lot of minor and major revisions (inspecting theories, read_axm,
wenzelm
parents:
478
diff
changeset

810 
\begin{ttbox} 
8136  811 
cterm_of : Sign.sg > term > cterm 
812 
read_cterm : Sign.sg > string * typ > cterm 

813 
cert_axm : Sign.sg > string * term > string * term 

814 
read_axm : Sign.sg > string * string > string * term 

815 
rep_cterm : cterm > \{T:typ, t:term, sign:Sign.sg, maxidx:int\} 

4543  816 
Sign.certify_term : Sign.sg > term > term * typ * int 
104  817 
\end{ttbox} 
324  818 
\begin{ttdescription} 
4543  819 

820 
\item[\ttindexbold{cterm_of} $sign$ $t$] \index{signatures} certifies 

821 
$t$ with respect to signature~$sign$. 

822 

823 
\item[\ttindexbold{read_cterm} $sign$ ($s$, $T$)] reads the string~$s$ 

824 
using the syntax of~$sign$, creating a certified term. The term is 

825 
checked to have type~$T$; this type also tells the parser what kind 

826 
of phrase to parse. 

827 

828 
\item[\ttindexbold{cert_axm} $sign$ ($name$, $t$)] certifies $t$ with 

829 
respect to $sign$ as a metaproposition and converts all exceptions 

830 
to an error, including the final message 

864
d63b111b917a
quite a lot of minor and major revisions (inspecting theories, read_axm,
wenzelm
parents:
478
diff
changeset

831 
\begin{ttbox} 
d63b111b917a
quite a lot of minor and major revisions (inspecting theories, read_axm,
wenzelm
parents:
478
diff
changeset

832 
The error(s) above occurred in axiom "\(name\)" 
d63b111b917a
quite a lot of minor and major revisions (inspecting theories, read_axm,
wenzelm
parents:
478
diff
changeset

833 
\end{ttbox} 
d63b111b917a
quite a lot of minor and major revisions (inspecting theories, read_axm,
wenzelm
parents:
478
diff
changeset

834 

4543  835 
\item[\ttindexbold{read_axm} $sign$ ($name$, $s$)] similar to {\tt 
836 
cert_axm}, but first reads the string $s$ using the syntax of 

837 
$sign$. 

838 

839 
\item[\ttindexbold{rep_cterm} $ct$] decomposes $ct$ as a record 

840 
containing its type, the term itself, its signature, and the maximum 

841 
subscript of its unknowns. The type and maximum subscript are 

842 
computed during certification. 

843 

844 
\item[\ttindexbold{Sign.certify_term}] is a more primitive version of 

845 
\texttt{cterm_of}, returning the internal representation instead of 

846 
an abstract \texttt{cterm}. 

864
d63b111b917a
quite a lot of minor and major revisions (inspecting theories, read_axm,
wenzelm
parents:
478
diff
changeset

847 

324  848 
\end{ttdescription} 
104  849 

850 

864
d63b111b917a
quite a lot of minor and major revisions (inspecting theories, read_axm,
wenzelm
parents:
478
diff
changeset

851 
\section{Types}\index{typesbold} 
d63b111b917a
quite a lot of minor and major revisions (inspecting theories, read_axm,
wenzelm
parents:
478
diff
changeset

852 
Types belong to the \ML\ type \mltydx{typ}, which is a concrete datatype with 
d63b111b917a
quite a lot of minor and major revisions (inspecting theories, read_axm,
wenzelm
parents:
478
diff
changeset

853 
three constructor functions. These correspond to type constructors, free 
d63b111b917a
quite a lot of minor and major revisions (inspecting theories, read_axm,
wenzelm
parents:
478
diff
changeset

854 
type variables and schematic type variables. Types are classified by sorts, 
d63b111b917a
quite a lot of minor and major revisions (inspecting theories, read_axm,
wenzelm
parents:
478
diff
changeset

855 
which are lists of classes (representing an intersection). A class is 
d63b111b917a
quite a lot of minor and major revisions (inspecting theories, read_axm,
wenzelm
parents:
478
diff
changeset

856 
represented by a string. 
104  857 
\begin{ttbox} 
858 
type class = string; 

859 
type sort = class list; 

860 

861 
datatype typ = Type of string * typ list 

862 
 TFree of string * sort 

863 
 TVar of indexname * sort; 

864 

865 
infixr 5 >; 

864
d63b111b917a
quite a lot of minor and major revisions (inspecting theories, read_axm,
wenzelm
parents:
478
diff
changeset

866 
fun S > T = Type ("fun", [S, T]); 
104  867 
\end{ttbox} 
324  868 
\begin{ttdescription} 
4317  869 
\item[\ttindexbold{Type} ($a$, $Ts$)] \index{type constructorsbold} 
8136  870 
applies the \textbf{type constructor} named~$a$ to the type operand list~$Ts$. 
324  871 
Type constructors include~\tydx{fun}, the binary function space 
872 
constructor, as well as nullary type constructors such as~\tydx{prop}. 

873 
Other type constructors may be introduced. In expressions, but not in 

874 
patterns, \hbox{\tt$S$>$T$} is a convenient shorthand for function 

875 
types. 

104  876 

4317  877 
\item[\ttindexbold{TFree} ($a$, $s$)] \index{type variablesbold} 
8136  878 
is the \textbf{type variable} with name~$a$ and sort~$s$. 
104  879 

4317  880 
\item[\ttindexbold{TVar} ($v$, $s$)] \index{type unknownsbold} 
8136  881 
is the \textbf{type unknown} with indexname~$v$ and sort~$s$. 
324  882 
Type unknowns are essentially free type variables, but may be 
883 
instantiated during unification. 

884 
\end{ttdescription} 

104  885 

886 

887 
\section{Certified types} 

888 
\index{types!certifiedbold} 

864
d63b111b917a
quite a lot of minor and major revisions (inspecting theories, read_axm,
wenzelm
parents:
478
diff
changeset

889 
Certified types, which are analogous to certified terms, have type 
275  890 
\ttindexbold{ctyp}. 
104  891 

892 
\subsection{Printing types} 

324  893 
\index{types!printing of} 
864
d63b111b917a
quite a lot of minor and major revisions (inspecting theories, read_axm,
wenzelm
parents:
478
diff
changeset

894 
\begin{ttbox} 
275  895 
string_of_ctyp : ctyp > string 
104  896 
Sign.string_of_typ : Sign.sg > typ > string 
897 
\end{ttbox} 

324  898 
\begin{ttdescription} 
864
d63b111b917a
quite a lot of minor and major revisions (inspecting theories, read_axm,
wenzelm
parents:
478
diff
changeset

899 
\item[\ttindexbold{string_of_ctyp} $cT$] 
104  900 
displays $cT$ as a string. 
901 

864
d63b111b917a
quite a lot of minor and major revisions (inspecting theories, read_axm,
wenzelm
parents:
478
diff
changeset

902 
\item[\ttindexbold{Sign.string_of_typ} $sign$ $T$] 
104  903 
displays $T$ as a string, using the syntax of~$sign$. 
324  904 
\end{ttdescription} 
104  905 

906 

907 
\subsection{Making and inspecting certified types} 

864
d63b111b917a
quite a lot of minor and major revisions (inspecting theories, read_axm,
wenzelm
parents:
478
diff
changeset

908 
\begin{ttbox} 
4543  909 
ctyp_of : Sign.sg > typ > ctyp 
8136  910 
rep_ctyp : ctyp > \{T: typ, sign: Sign.sg\} 
4543  911 
Sign.certify_typ : Sign.sg > typ > typ 
104  912 
\end{ttbox} 
324  913 
\begin{ttdescription} 
4543  914 

915 
\item[\ttindexbold{ctyp_of} $sign$ $T$] \index{signatures} certifies 

916 
$T$ with respect to signature~$sign$. 

917 

918 
\item[\ttindexbold{rep_ctyp} $cT$] decomposes $cT$ as a record 

919 
containing the type itself and its signature. 

920 

921 
\item[\ttindexbold{Sign.certify_typ}] is a more primitive version of 

922 
\texttt{ctyp_of}, returning the internal representation instead of 

923 
an abstract \texttt{ctyp}. 

104  924 

324  925 
\end{ttdescription} 
104  926 

1846  927 

4317  928 
\section{Oracles: calling trusted external reasoners} 
1846  929 
\label{sec:oracles} 
930 
\index{oracles(} 

931 

932 
Oracles allow Isabelle to take advantage of external reasoners such as 

933 
arithmetic decision procedures, model checkers, fast tautology checkers or 

934 
computer algebra systems. Invoked as an oracle, an external reasoner can 

935 
create arbitrary Isabelle theorems. It is your responsibility to ensure that 

936 
the external reasoner is as trustworthy as your application requires. 

937 
Isabelle's proof objects~(\S\ref{sec:proofObjects}) record how each theorem 

938 
depends upon oracle calls. 

939 

940 
\begin{ttbox} 

4317  941 
invoke_oracle : theory > xstring > Sign.sg * object > thm 
4597
a0bdee64194c
Fixed a lot of overfull and underfull lines (hboxes)
paulson
parents:
4543
diff
changeset

942 
Theory.add_oracle : bstring * (Sign.sg * object > term) > theory 
a0bdee64194c
Fixed a lot of overfull and underfull lines (hboxes)
paulson
parents:
4543
diff
changeset

943 
> theory 
1846  944 
\end{ttbox} 
945 
\begin{ttdescription} 

4317  946 
\item[\ttindexbold{invoke_oracle} $thy$ $name$ ($sign$, $data$)] 
947 
invokes the oracle $name$ of theory $thy$ passing the information 

948 
contained in the exception value $data$ and creating a theorem 

949 
having signature $sign$. Note that type \ttindex{object} is just an 

950 
abbreviation for \texttt{exn}. Errors arise if $thy$ does not have 

951 
an oracle called $name$, if the oracle rejects its arguments or if 

952 
its result is illtyped. 

953 

954 
\item[\ttindexbold{Theory.add_oracle} $name$ $fun$ $thy$] extends 

955 
$thy$ by oracle $fun$ called $name$. It is seldom called 

956 
explicitly, as there is concrete syntax for oracles in theory files. 

1846  957 
\end{ttdescription} 
958 

959 
A curious feature of {\ML} exceptions is that they are ordinary constructors. 

6669  960 
The {\ML} type \texttt{exn} is a datatype that can be extended at any time. (See 
1846  961 
my {\em {ML} for the Working Programmer}~\cite{paulsonml2}, especially 
962 
page~136.) The oracle mechanism takes advantage of this to allow an oracle to 

963 
take any information whatever. 

964 

965 
There must be some way of invoking the external reasoner from \ML, either 

966 
because it is coded in {\ML} or via an operating system interface. Isabelle 

967 
expects the {\ML} function to take two arguments: a signature and an 

4317  968 
exception object. 
1846  969 
\begin{itemize} 
970 
\item The signature will typically be that of a desendant of the theory 

971 
declaring the oracle. The oracle will use it to distinguish constants from 

972 
variables, etc., and it will be attached to the generated theorems. 

973 

974 
\item The exception is used to pass arbitrary information to the oracle. This 

975 
information must contain a full description of the problem to be solved by 

976 
the external reasoner, including any additional information that might be 

977 
required. The oracle may raise the exception to indicate that it cannot 

978 
solve the specified problem. 

979 
\end{itemize} 

980 

6669  981 
A trivial example is provided in theory \texttt{FOL/ex/IffOracle}. This 
4317  982 
oracle generates tautologies of the form $P\bimp\cdots\bimp P$, with 
983 
an even number of $P$s. 

1846  984 

4317  985 
The \texttt{ML} section of \texttt{IffOracle.thy} begins by declaring 
986 
a few auxiliary functions (suppressed below) for creating the 

987 
tautologies. Then it declares a new exception constructor for the 

988 
information required by the oracle: here, just an integer. It finally 

989 
defines the oracle function itself. 

1846  990 
\begin{ttbox} 
4317  991 
exception IffOracleExn of int;\medskip 
992 
fun mk_iff_oracle (sign, IffOracleExn n) = 

993 
if n > 0 andalso n mod 2 = 0 

6669  994 
then Trueprop \$ mk_iff n 
4317  995 
else raise IffOracleExn n; 
1846  996 
\end{ttbox} 
6669  997 
Observe the function's two arguments, the signature \texttt{sign} and the 
4317  998 
exception given as a pattern. The function checks its argument for 
999 
validity. If $n$ is positive and even then it creates a tautology 

1000 
containing $n$ occurrences of~$P$. Otherwise it signals error by 

1001 
raising its own exception (just by happy coincidence). Errors may be 

6669  1002 
signalled by other means, such as returning the theorem \texttt{True}. 
4317  1003 
Please ensure that the oracle's result is correctly typed; Isabelle 
1004 
will reject illtyped theorems by raising a cryptic exception at top 

1005 
level. 

1846  1006 

6669  1007 
The \texttt{oracle} section of \texttt{IffOracle.thy} installs above 
4317  1008 
\texttt{ML} function as follows: 
1846  1009 
\begin{ttbox} 
4317  1010 
IffOracle = FOL +\medskip 
1011 
oracle 

1012 
iff = mk_iff_oracle\medskip 

1846  1013 
end 
1014 
\end{ttbox} 

1015 

4317  1016 
Now in \texttt{IffOracle.ML} we first define a wrapper for invoking 
1017 
the oracle: 

1846  1018 
\begin{ttbox} 
4597
a0bdee64194c
Fixed a lot of overfull and underfull lines (hboxes)
paulson
parents:
4543
diff
changeset

1019 
fun iff_oracle n = invoke_oracle IffOracle.thy "iff" 
a0bdee64194c
Fixed a lot of overfull and underfull lines (hboxes)
paulson
parents:
4543
diff
changeset

1020 
(sign_of IffOracle.thy, IffOracleExn n); 
4317  1021 
\end{ttbox} 
1022 

1023 
Here are some example applications of the \texttt{iff} oracle. An 

1024 
argument of 10 is allowed, but one of 5 is forbidden: 

1025 
\begin{ttbox} 

1026 
iff_oracle 10; 

1846  1027 
{\out "P <> P <> P <> P <> P <> P <> P <> P <> P <> P" : thm} 
4317  1028 
iff_oracle 5; 
1846  1029 
{\out Exception IffOracleExn 5 raised} 
1030 
\end{ttbox} 

1031 

1032 
\index{oracles)} 

104  1033 
\index{theories)} 
5369  1034 

1035 

1036 
%%% Local Variables: 

1037 
%%% mode: latex 

1038 
%%% TeXmaster: "ref" 

1039 
%%% End: 