5 chapter {* Theory specifications *}
7 section {* Defining theories \label{sec:begin-thy} *}
10 \begin{matharray}{rcl}
11 @{command_def "theory"} & : & @{text "toplevel \<rightarrow> theory"} \\
12 @{command_def (global) "end"} & : & @{text "theory \<rightarrow> toplevel"} \\
15 Isabelle/Isar theories are defined via theory files, which may
16 contain both specifications and proofs; occasionally definitional
17 mechanisms also require some explicit proof. The theory body may be
18 sub-structured by means of \emph{local theory targets}, such as
19 @{command "locale"} and @{command "class"}.
21 The first proper command of a theory is @{command "theory"}, which
22 indicates imports of previous theories and optional dependencies on
23 other source files (usually in ML). Just preceding the initial
24 @{command "theory"} command there may be an optional @{command
25 "header"} declaration, which is only relevant to document
26 preparation: see also the other section markup commands in
29 A theory is concluded by a final @{command (global) "end"} command,
30 one that does not belong to a local theory target. No further
31 commands may follow such a global @{command (global) "end"},
32 although some user-interfaces might pretend that trailing input is
36 'theory' name 'imports' (name +) uses? 'begin'
39 uses: 'uses' ((name | parname) +);
44 \item @{command "theory"}~@{text "A \<IMPORTS> B\<^sub>1 \<dots> B\<^sub>n \<BEGIN>"}
45 starts a new theory @{text A} based on the merge of existing
46 theories @{text "B\<^sub>1 \<dots> B\<^sub>n"}.
48 Due to the possibility to import more than one ancestor, the
49 resulting theory structure of an Isabelle session forms a directed
50 acyclic graph (DAG). Isabelle's theory loader ensures that the
51 sources contributing to the development graph are always up-to-date:
52 changed files are automatically reloaded whenever a theory header
53 specification is processed.
55 The optional @{keyword_def "uses"} specification declares additional
56 dependencies on extra files (usually ML sources). Files will be
57 loaded immediately (as ML), unless the name is parenthesized. The
58 latter case records a dependency that needs to be resolved later in
59 the text, usually via explicit @{command_ref "use"} for ML files;
60 other file formats require specific load commands defined by the
61 corresponding tools or packages.
63 \item @{command (global) "end"} concludes the current theory
64 definition. Note that local theory targets involve a local
65 @{command (local) "end"}, which is clear from the nesting.
71 section {* Local theory targets \label{sec:target} *}
74 A local theory target is a context managed separately within the
75 enclosing theory. Contexts may introduce parameters (fixed
76 variables) and assumptions (hypotheses). Definitions and theorems
77 depending on the context may be added incrementally later on. Named
78 contexts refer to locales (cf.\ \secref{sec:locale}) or type classes
79 (cf.\ \secref{sec:class}); the name ``@{text "-"}'' signifies the
80 global theory context.
82 \begin{matharray}{rcll}
83 @{command_def "context"} & : & @{text "theory \<rightarrow> local_theory"} \\
84 @{command_def (local) "end"} & : & @{text "local_theory \<rightarrow> theory"} \\
87 \indexouternonterm{target}
89 'context' name 'begin'
92 target: '(' 'in' name ')'
98 \item @{command "context"}~@{text "c \<BEGIN>"} recommences an
99 existing locale or class context @{text c}. Note that locale and
100 class definitions allow to include the @{keyword "begin"} keyword as
101 well, in order to continue the local theory immediately after the
102 initial specification.
104 \item @{command (local) "end"} concludes the current local theory
105 and continues the enclosing global theory. Note that a global
106 @{command (global) "end"} has a different meaning: it concludes the
107 theory itself (\secref{sec:begin-thy}).
109 \item @{text "(\<IN> c)"} given after any local theory command
110 specifies an immediate target, e.g.\ ``@{command
111 "definition"}~@{text "(\<IN> c) \<dots>"}'' or ``@{command
112 "theorem"}~@{text "(\<IN> c) \<dots>"}''. This works both in a local or
113 global theory context; the current target context will be suspended
114 for this command only. Note that ``@{text "(\<IN> -)"}'' will
115 always produce a global result independently of the current target
120 The exact meaning of results produced within a local theory context
121 depends on the underlying target infrastructure (locale, type class
122 etc.). The general idea is as follows, considering a context named
123 @{text c} with parameter @{text x} and assumption @{text "A[x]"}.
125 Definitions are exported by introducing a global version with
126 additional arguments; a syntactic abbreviation links the long form
127 with the abstract version of the target context. For example,
128 @{text "a \<equiv> t[x]"} becomes @{text "c.a ?x \<equiv> t[?x]"} at the theory
129 level (for arbitrary @{text "?x"}), together with a local
130 abbreviation @{text "c \<equiv> c.a x"} in the target context (for the
131 fixed parameter @{text x}).
133 Theorems are exported by discharging the assumptions and
134 generalizing the parameters of the context. For example, @{text "a:
135 B[x]"} becomes @{text "c.a: A[?x] \<Longrightarrow> B[?x]"}, again for arbitrary
140 section {* Basic specification elements *}
143 \begin{matharray}{rcll}
144 @{command_def "axiomatization"} & : & @{text "theory \<rightarrow> theory"} & (axiomatic!)\\
145 @{command_def "definition"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
146 @{attribute_def "defn"} & : & @{text attribute} \\
147 @{command_def "abbreviation"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
148 @{command_def "print_abbrevs"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow> "} \\
151 These specification mechanisms provide a slightly more abstract view
152 than the underlying primitives of @{command "consts"}, @{command
153 "defs"} (see \secref{sec:consts}), and @{command "axioms"} (see
154 \secref{sec:axms-thms}). In particular, type-inference is commonly
155 available, and result names need not be given.
158 'axiomatization' target? fixes? ('where' specs)?
160 'definition' target? (decl 'where')? thmdecl? prop
162 'abbreviation' target? mode? (decl 'where')? prop
165 fixes: ((name ('::' type)? mixfix? | vars) + 'and')
167 specs: (thmdecl? props + 'and')
169 decl: name ('::' type)? mixfix?
175 \item @{command "axiomatization"}~@{text "c\<^sub>1 \<dots> c\<^sub>m \<WHERE> \<phi>\<^sub>1 \<dots> \<phi>\<^sub>n"}
176 introduces several constants simultaneously and states axiomatic
177 properties for these. The constants are marked as being specified
178 once and for all, which prevents additional specifications being
181 Note that axiomatic specifications are only appropriate when
182 declaring a new logical system; axiomatic specifications are
183 restricted to global theory contexts. Normal applications should
184 only use definitional mechanisms!
186 \item @{command "definition"}~@{text "c \<WHERE> eq"} produces an
187 internal definition @{text "c \<equiv> t"} according to the specification
188 given as @{text eq}, which is then turned into a proven fact. The
189 given proposition may deviate from internal meta-level equality
190 according to the rewrite rules declared as @{attribute defn} by the
191 object-logic. This usually covers object-level equality @{text "x =
192 y"} and equivalence @{text "A \<leftrightarrow> B"}. End-users normally need not
193 change the @{attribute defn} setup.
195 Definitions may be presented with explicit arguments on the LHS, as
196 well as additional conditions, e.g.\ @{text "f x y = t"} instead of
197 @{text "f \<equiv> \<lambda>x y. t"} and @{text "y \<noteq> 0 \<Longrightarrow> g x y = u"} instead of an
198 unrestricted @{text "g \<equiv> \<lambda>x y. u"}.
200 \item @{command "abbreviation"}~@{text "c \<WHERE> eq"} introduces a
201 syntactic constant which is associated with a certain term according
202 to the meta-level equality @{text eq}.
204 Abbreviations participate in the usual type-inference process, but
205 are expanded before the logic ever sees them. Pretty printing of
206 terms involves higher-order rewriting with rules stemming from
207 reverted abbreviations. This needs some care to avoid overlapping
208 or looping syntactic replacements!
210 The optional @{text mode} specification restricts output to a
211 particular print mode; using ``@{text input}'' here achieves the
212 effect of one-way abbreviations. The mode may also include an
213 ``@{keyword "output"}'' qualifier that affects the concrete syntax
214 declared for abbreviations, cf.\ @{command "syntax"} in
215 \secref{sec:syn-trans}.
217 \item @{command "print_abbrevs"} prints all constant abbreviations
218 of the current context.
224 section {* Generic declarations *}
227 Arbitrary operations on the background context may be wrapped-up as
228 generic declaration elements. Since the underlying concept of local
229 theories may be subject to later re-interpretation, there is an
230 additional dependency on a morphism that tells the difference of the
231 original declaration context wrt.\ the application context
232 encountered later on. A fact declaration is an important special
233 case: it consists of a theorem which is applied to the context by
234 means of an attribute.
236 \begin{matharray}{rcl}
237 @{command_def "declaration"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
238 @{command_def "declare"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
242 'declaration' target? text
244 'declare' target? (thmrefs + 'and')
250 \item @{command "declaration"}~@{text d} adds the declaration
251 function @{text d} of ML type @{ML_type declaration}, to the current
252 local theory under construction. In later application contexts, the
253 function is transformed according to the morphisms being involved in
254 the interpretation hierarchy.
256 \item @{command "declare"}~@{text thms} declares theorems to the
257 current local theory context. No theorem binding is involved here,
258 unlike @{command "theorems"} or @{command "lemmas"} (cf.\
259 \secref{sec:axms-thms}), so @{command "declare"} only has the effect
260 of applying attributes as included in the theorem specification.
266 section {* Locales \label{sec:locale} *}
269 Locales are named local contexts, consisting of a list of
270 declaration elements that are modeled after the Isar proof context
271 commands (cf.\ \secref{sec:proof-context}).
275 subsection {* Locale specifications *}
278 \begin{matharray}{rcl}
279 @{command_def "locale"} & : & @{text "theory \<rightarrow> local_theory"} \\
280 @{command_def "print_locale"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
281 @{command_def "print_locales"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
282 @{method_def intro_locales} & : & @{text method} \\
283 @{method_def unfold_locales} & : & @{text method} \\
286 \indexouternonterm{contextexpr}\indexouternonterm{contextelem}
287 \indexisarelem{fixes}\indexisarelem{constrains}\indexisarelem{assumes}
288 \indexisarelem{defines}\indexisarelem{notes}
290 'locale' name ('=' localeexpr)? 'begin'?
292 'print\_locale' '!'? localeexpr
294 localeexpr: ((contextexpr '+' (contextelem+)) | contextexpr | (contextelem+))
297 contextexpr: nameref | '(' contextexpr ')' |
298 (contextexpr (name mixfix? +)) | (contextexpr + '+')
300 contextelem: fixes | constrains | assumes | defines | notes
302 fixes: 'fixes' ((name ('::' type)? structmixfix? | vars) + 'and')
304 constrains: 'constrains' (name '::' type + 'and')
306 assumes: 'assumes' (thmdecl? props + 'and')
308 defines: 'defines' (thmdecl? prop proppat? + 'and')
310 notes: 'notes' (thmdef? thmrefs + 'and')
316 \item @{command "locale"}~@{text "loc = import + body"} defines a
317 new locale @{text loc} as a context consisting of a certain view of
318 existing locales (@{text import}) plus some additional elements
319 (@{text body}). Both @{text import} and @{text body} are optional;
320 the degenerate form @{command "locale"}~@{text loc} defines an empty
321 locale, which may still be useful to collect declarations of facts
322 later on. Type-inference on locale expressions automatically takes
323 care of the most general typing that the combined context elements
326 The @{text import} consists of a structured context expression,
327 consisting of references to existing locales, renamed contexts, or
328 merged contexts. Renaming uses positional notation: @{text "c
329 x\<^sub>1 \<dots> x\<^sub>n"} means that (a prefix of) the fixed
330 parameters of context @{text c} are named @{text "x\<^sub>1, \<dots>,
331 x\<^sub>n"}; a ``@{text _}'' (underscore) means to skip that
332 position. Renaming by default deletes concrete syntax, but new
333 syntax may by specified with a mixfix annotation. An exeption of
334 this rule is the special syntax declared with ``@{text
335 "(\<STRUCTURE>)"}'' (see below), which is neither deleted nor can it
336 be changed. Merging proceeds from left-to-right, suppressing any
337 duplicates stemming from different paths through the import
340 The @{text body} consists of basic context elements, further context
341 expressions may be included as well.
345 \item @{element "fixes"}~@{text "x :: \<tau> (mx)"} declares a local
346 parameter of type @{text \<tau>} and mixfix annotation @{text mx} (both
347 are optional). The special syntax declaration ``@{text
348 "(\<STRUCTURE>)"}'' means that @{text x} may be referenced
349 implicitly in this context.
351 \item @{element "constrains"}~@{text "x :: \<tau>"} introduces a type
352 constraint @{text \<tau>} on the local parameter @{text x}.
354 \item @{element "assumes"}~@{text "a: \<phi>\<^sub>1 \<dots> \<phi>\<^sub>n"}
355 introduces local premises, similar to @{command "assume"} within a
356 proof (cf.\ \secref{sec:proof-context}).
358 \item @{element "defines"}~@{text "a: x \<equiv> t"} defines a previously
359 declared parameter. This is similar to @{command "def"} within a
360 proof (cf.\ \secref{sec:proof-context}), but @{element "defines"}
361 takes an equational proposition instead of variable-term pair. The
362 left-hand side of the equation may have additional arguments, e.g.\
363 ``@{element "defines"}~@{text "f x\<^sub>1 \<dots> x\<^sub>n \<equiv> t"}''.
365 \item @{element "notes"}~@{text "a = b\<^sub>1 \<dots> b\<^sub>n"}
366 reconsiders facts within a local context. Most notably, this may
367 include arbitrary declarations in any attribute specifications
368 included here, e.g.\ a local @{attribute simp} rule.
370 The initial @{text import} specification of a locale expression
371 maintains a dynamic relation to the locales being referenced
372 (benefiting from any later fact declarations in the obvious manner).
376 Note that ``@{text "(\<IS> p\<^sub>1 \<dots> p\<^sub>n)"}'' patterns given
377 in the syntax of @{element "assumes"} and @{element "defines"} above
378 are illegal in locale definitions. In the long goal format of
379 \secref{sec:goals}, term bindings may be included as expected,
382 \medskip By default, locale specifications are ``closed up'' by
383 turning the given text into a predicate definition @{text
384 loc_axioms} and deriving the original assumptions as local lemmas
385 (modulo local definitions). The predicate statement covers only the
386 newly specified assumptions, omitting the content of included locale
387 expressions. The full cumulative view is only provided on export,
388 involving another predicate @{text loc} that refers to the complete
391 In any case, the predicate arguments are those locale parameters
392 that actually occur in the respective piece of text. Also note that
393 these predicates operate at the meta-level in theory, but the locale
394 packages attempts to internalize statements according to the
395 object-logic setup (e.g.\ replacing @{text \<And>} by @{text \<forall>}, and
396 @{text "\<Longrightarrow>"} by @{text "\<longrightarrow>"} in HOL; see also
397 \secref{sec:object-logic}). Separate introduction rules @{text
398 loc_axioms.intro} and @{text loc.intro} are provided as well.
400 \item @{command "print_locale"}~@{text "import + body"} prints the
401 specified locale expression in a flattened form. The notable
402 special case @{command "print_locale"}~@{text loc} just prints the
403 contents of the named locale, but keep in mind that type-inference
404 will normalize type variables according to the usual alphabetical
405 order. The command omits @{element "notes"} elements by default.
406 Use @{command "print_locale"}@{text "!"} to get them included.
408 \item @{command "print_locales"} prints the names of all locales
409 of the current theory.
411 \item @{method intro_locales} and @{method unfold_locales}
412 repeatedly expand all introduction rules of locale predicates of the
413 theory. While @{method intro_locales} only applies the @{text
414 loc.intro} introduction rules and therefore does not decend to
415 assumptions, @{method unfold_locales} is more aggressive and applies
416 @{text loc_axioms.intro} as well. Both methods are aware of locale
417 specifications entailed by the context, both from target statements,
418 and from interpretations (see below). New goals that are entailed
419 by the current context are discharged automatically.
425 subsection {* Interpretation of locales *}
428 Locale expressions (more precisely, \emph{context expressions}) may
429 be instantiated, and the instantiated facts added to the current
430 context. This requires a proof of the instantiated specification
431 and is called \emph{locale interpretation}. Interpretation is
432 possible in theories and locales (command @{command
433 "interpretation"}) and also within a proof body (command @{command
436 \begin{matharray}{rcl}
437 @{command_def "interpretation"} & : & @{text "theory \<rightarrow> proof(prove)"} \\
438 @{command_def "interpret"} & : & @{text "proof(state) | proof(chain \<rightarrow> proof(prove)"} \\
441 \indexouternonterm{interp}
443 'interpretation' (interp | name ('<' | subseteq) contextexpr)
447 instantiation: ('[' (inst+) ']')?
449 interp: (name ':')? \\ (contextexpr instantiation |
450 name instantiation 'where' (thmdecl? prop + 'and'))
456 \item @{command "interpretation"}~@{text "expr insts \<WHERE> eqns"}
458 The first form of @{command "interpretation"} interprets @{text
459 expr} in the theory. The instantiation is given as a list of terms
460 @{text insts} and is positional. All parameters must receive an
461 instantiation term --- with the exception of defined parameters.
462 These are, if omitted, derived from the defining equation and other
463 instantiations. Use ``@{text _}'' to omit an instantiation term.
465 The command generates proof obligations for the instantiated
466 specifications (assumes and defines elements). Once these are
467 discharged by the user, instantiated facts are added to the theory
468 in a post-processing phase.
470 Additional equations, which are unfolded in facts during
471 post-processing, may be given after the keyword @{keyword "where"}.
472 This is useful for interpreting concepts introduced through
473 definition specification elements. The equations must be proved.
474 Note that if equations are present, the context expression is
475 restricted to a locale name.
477 The command is aware of interpretations already active in the
478 theory, but does not simplify the goal automatically. In order to
479 simplify the proof obligations use methods @{method intro_locales}
480 or @{method unfold_locales}. Post-processing is not applied to
481 facts of interpretations that are already active. This avoids
482 duplication of interpreted facts, in particular. Note that, in the
483 case of a locale with import, parts of the interpretation may
484 already be active. The command will only process facts for new
487 The context expression may be preceded by a name, which takes effect
488 in the post-processing of facts. It is used to prefix fact names,
489 for example to avoid accidental hiding of other facts.
491 Adding facts to locales has the effect of adding interpreted facts
492 to the theory for all active interpretations also. That is,
493 interpretations dynamically participate in any facts added to
496 \item @{command "interpretation"}~@{text "name \<subseteq> expr"}
498 This form of the command interprets @{text expr} in the locale
499 @{text name}. It requires a proof that the specification of @{text
500 name} implies the specification of @{text expr}. As in the
501 localized version of the theorem command, the proof is in the
502 context of @{text name}. After the proof obligation has been
503 dischared, the facts of @{text expr} become part of locale @{text
504 name} as \emph{derived} context elements and are available when the
505 context @{text name} is subsequently entered. Note that, like
506 import, this is dynamic: facts added to a locale part of @{text
507 expr} after interpretation become also available in @{text name}.
508 Like facts of renamed context elements, facts obtained by
509 interpretation may be accessed by prefixing with the parameter
510 renaming (where the parameters are separated by ``@{text _}'').
512 Unlike interpretation in theories, instantiation is confined to the
513 renaming of parameters, which may be specified as part of the
514 context expression @{text expr}. Using defined parameters in @{text
515 name} one may achieve an effect similar to instantiation, though.
517 Only specification fragments of @{text expr} that are not already
518 part of @{text name} (be it imported, derived or a derived fragment
519 of the import) are considered by interpretation. This enables
520 circular interpretations.
522 If interpretations of @{text name} exist in the current theory, the
523 command adds interpretations for @{text expr} as well, with the same
524 prefix and attributes, although only for fragments of @{text expr}
525 that are not interpreted in the theory already.
527 \item @{command "interpret"}~@{text "expr insts \<WHERE> eqns"}
528 interprets @{text expr} in the proof context and is otherwise
529 similar to interpretation in theories.
534 Since attributes are applied to interpreted theorems,
535 interpretation may modify the context of common proof tools, e.g.\
536 the Simplifier or Classical Reasoner. Since the behavior of such
537 automated reasoning tools is \emph{not} stable under
538 interpretation morphisms, manual declarations might have to be
543 An interpretation in a theory may subsume previous
544 interpretations. This happens if the same specification fragment
545 is interpreted twice and the instantiation of the second
546 interpretation is more general than the interpretation of the
547 first. A warning is issued, since it is likely that these could
548 have been generalized in the first place. The locale package does
549 not attempt to remove subsumed interpretations.
554 section {* Classes \label{sec:class} *}
557 A class is a particular locale with \emph{exactly one} type variable
558 @{text \<alpha>}. Beyond the underlying locale, a corresponding type class
559 is established which is interpreted logically as axiomatic type
560 class \cite{Wenzel:1997:TPHOL} whose logical content are the
561 assumptions of the locale. Thus, classes provide the full
562 generality of locales combined with the commodity of type classes
563 (notably type-inference). See \cite{isabelle-classes} for a short
566 \begin{matharray}{rcl}
567 @{command_def "class"} & : & @{text "theory \<rightarrow> local_theory"} \\
568 @{command_def "instantiation"} & : & @{text "theory \<rightarrow> local_theory"} \\
569 @{command_def "instance"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
570 @{command_def "subclass"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
571 @{command_def "print_classes"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
572 @{method_def intro_classes} & : & @{text method} \\
576 'class' name '=' ((superclassexpr '+' (contextelem+)) | superclassexpr | (contextelem+)) \\
579 'instantiation' (nameref + 'and') '::' arity 'begin'
583 'subclass' target? nameref
588 superclassexpr: nameref | (nameref '+' superclassexpr)
594 \item @{command "class"}~@{text "c = superclasses + body"} defines
595 a new class @{text c}, inheriting from @{text superclasses}. This
596 introduces a locale @{text c} with import of all locales @{text
599 Any @{element "fixes"} in @{text body} are lifted to the global
600 theory level (\emph{class operations} @{text "f\<^sub>1, \<dots>,
601 f\<^sub>n"} of class @{text c}), mapping the local type parameter
602 @{text \<alpha>} to a schematic type variable @{text "?\<alpha> :: c"}.
604 Likewise, @{element "assumes"} in @{text body} are also lifted,
605 mapping each local parameter @{text "f :: \<tau>[\<alpha>]"} to its
606 corresponding global constant @{text "f :: \<tau>[?\<alpha> :: c]"}. The
607 corresponding introduction rule is provided as @{text
608 c_class_axioms.intro}. This rule should be rarely needed directly
609 --- the @{method intro_classes} method takes care of the details of
610 class membership proofs.
612 \item @{command "instantiation"}~@{text "t :: (s\<^sub>1, \<dots>, s\<^sub>n)s
613 \<BEGIN>"} opens a theory target (cf.\ \secref{sec:target}) which
614 allows to specify class operations @{text "f\<^sub>1, \<dots>, f\<^sub>n"} corresponding
615 to sort @{text s} at the particular type instance @{text "(\<alpha>\<^sub>1 :: s\<^sub>1,
616 \<dots>, \<alpha>\<^sub>n :: s\<^sub>n) t"}. A plain @{command "instance"} command in the
617 target body poses a goal stating these type arities. The target is
618 concluded by an @{command_ref (local) "end"} command.
620 Note that a list of simultaneous type constructors may be given;
621 this corresponds nicely to mutual recursive type definitions, e.g.\
624 \item @{command "instance"} in an instantiation target body sets
625 up a goal stating the type arities claimed at the opening @{command
626 "instantiation"}. The proof would usually proceed by @{method
627 intro_classes}, and then establish the characteristic theorems of
628 the type classes involved. After finishing the proof, the
629 background theory will be augmented by the proven type arities.
631 \item @{command "subclass"}~@{text c} in a class context for class
632 @{text d} sets up a goal stating that class @{text c} is logically
633 contained in class @{text d}. After finishing the proof, class
634 @{text d} is proven to be subclass @{text c} and the locale @{text
635 c} is interpreted into @{text d} simultaneously.
637 \item @{command "print_classes"} prints all classes in the current
640 \item @{method intro_classes} repeatedly expands all class
641 introduction rules of this theory. Note that this method usually
642 needs not be named explicitly, as it is already included in the
643 default proof step (e.g.\ of @{command "proof"}). In particular,
644 instantiation of trivial (syntactic) classes may be performed by a
645 single ``@{command ".."}'' proof step.
651 subsection {* The class target *}
656 A named context may refer to a locale (cf.\ \secref{sec:target}).
657 If this locale is also a class @{text c}, apart from the common
658 locale target behaviour the following happens.
662 \item Local constant declarations @{text "g[\<alpha>]"} referring to the
663 local type parameter @{text \<alpha>} and local parameters @{text "f[\<alpha>]"}
664 are accompanied by theory-level constants @{text "g[?\<alpha> :: c]"}
665 referring to theory-level class operations @{text "f[?\<alpha> :: c]"}.
667 \item Local theorem bindings are lifted as are assumptions.
669 \item Local syntax refers to local operations @{text "g[\<alpha>]"} and
670 global operations @{text "g[?\<alpha> :: c]"} uniformly. Type inference
671 resolves ambiguities. In rare cases, manual type annotations are
678 subsection {* Old-style axiomatic type classes \label{sec:axclass} *}
681 \begin{matharray}{rcl}
682 @{command_def "axclass"} & : & @{text "theory \<rightarrow> theory"} \\
683 @{command_def "instance"} & : & @{text "theory \<rightarrow> proof(prove)"} \\
686 Axiomatic type classes are Isabelle/Pure's primitive
687 \emph{definitional} interface to type classes. For practical
688 applications, you should consider using classes
689 (cf.~\secref{sec:classes}) which provide high level interface.
692 'axclass' classdecl (axmdecl prop +)
694 'instance' (nameref ('<' | subseteq) nameref | nameref '::' arity)
700 \item @{command "axclass"}~@{text "c \<subseteq> c\<^sub>1, \<dots>, c\<^sub>n axms"} defines an
701 axiomatic type class as the intersection of existing classes, with
702 additional axioms holding. Class axioms may not contain more than
703 one type variable. The class axioms (with implicit sort constraints
704 added) are bound to the given names. Furthermore a class
705 introduction rule is generated (being bound as @{text
706 c_class.intro}); this rule is employed by method @{method
707 intro_classes} to support instantiation proofs of this class.
709 The ``class axioms'' (which are derived from the internal class
710 definition) are stored as theorems according to the given name
711 specifications; the name space prefix @{text "c_class"} is added
712 here. The full collection of these facts is also stored as @{text
715 \item @{command "instance"}~@{text "c\<^sub>1 \<subseteq> c\<^sub>2"} and @{command
716 "instance"}~@{text "t :: (s\<^sub>1, \<dots>, s\<^sub>n)s"} setup a goal stating a class
717 relation or type arity. The proof would usually proceed by @{method
718 intro_classes}, and then establish the characteristic theorems of
719 the type classes involved. After finishing the proof, the theory
720 will be augmented by a type signature declaration corresponding to
721 the resulting theorem.
727 section {* Unrestricted overloading *}
730 Isabelle/Pure's definitional schemes support certain forms of
731 overloading (see \secref{sec:consts}). At most occassions
732 overloading will be used in a Haskell-like fashion together with
733 type classes by means of @{command "instantiation"} (see
734 \secref{sec:class}). Sometimes low-level overloading is desirable.
735 The @{command "overloading"} target provides a convenient view for
738 \begin{matharray}{rcl}
739 @{command_def "overloading"} & : & @{text "theory \<rightarrow> local_theory"} \\
744 ( string ( '==' | equiv ) term ( '(' 'unchecked' ')' )? + ) 'begin'
749 \item @{command "overloading"}~@{text "x\<^sub>1 \<equiv> c\<^sub>1 :: \<tau>\<^sub>1 \<AND> \<dots> x\<^sub>n \<equiv> c\<^sub>n :: \<tau>\<^sub>n \<BEGIN>"}
750 opens a theory target (cf.\ \secref{sec:target}) which allows to
751 specify constants with overloaded definitions. These are identified
752 by an explicitly given mapping from variable names @{text "x\<^sub>i"} to
753 constants @{text "c\<^sub>i"} at particular type instances. The
754 definitions themselves are established using common specification
755 tools, using the names @{text "x\<^sub>i"} as reference to the
756 corresponding constants. The target is concluded by @{command
759 A @{text "(unchecked)"} option disables global dependency checks for
760 the corresponding definition, which is occasionally useful for
761 exotic overloading. It is at the discretion of the user to avoid
762 malformed theory specifications!
768 section {* Incorporating ML code \label{sec:ML} *}
771 \begin{matharray}{rcl}
772 @{command_def "use"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
773 @{command_def "ML"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
774 @{command_def "ML_prf"} & : & @{text "proof \<rightarrow> proof"} \\
775 @{command_def "ML_val"} & : & @{text "any \<rightarrow>"} \\
776 @{command_def "ML_command"} & : & @{text "any \<rightarrow>"} \\
777 @{command_def "setup"} & : & @{text "theory \<rightarrow> theory"} \\
781 @{index_ML bind_thms: "string * thm list -> unit"} \\
782 @{index_ML bind_thm: "string * thm -> unit"} \\
788 ('ML' | 'ML\_prf' | 'ML\_val' | 'ML\_command' | 'setup') text
794 \item @{command "use"}~@{text "file"} reads and executes ML
795 commands from @{text "file"}. The current theory context is passed
796 down to the ML toplevel and may be modified, using @{ML [source=false]
797 "Context.>>"} or derived ML commands. The file name is checked with
798 the @{keyword_ref "uses"} dependency declaration given in the theory
799 header (see also \secref{sec:begin-thy}).
801 Top-level ML bindings are stored within the (global or local) theory
804 \item @{command "ML"}~@{text "text"} is similar to @{command "use"},
805 but executes ML commands directly from the given @{text "text"}.
806 Top-level ML bindings are stored within the (global or local) theory
809 \item @{command "ML_prf"} is analogous to @{command "ML"} but works
810 within a proof context.
812 Top-level ML bindings are stored within the proof context in a
813 purely sequential fashion, disregarding the nested proof structure.
814 ML bindings introduced by @{command "ML_prf"} are discarded at the
817 \item @{command "ML_val"} and @{command "ML_command"} are diagnostic
818 versions of @{command "ML"}, which means that the context may not be
819 updated. @{command "ML_val"} echos the bindings produced at the ML
820 toplevel, but @{command "ML_command"} is silent.
822 \item @{command "setup"}~@{text "text"} changes the current theory
823 context by applying @{text "text"}, which refers to an ML expression
824 of type @{ML_type [source=false] "theory -> theory"}. This enables
825 to initialize any object-logic specific tools and packages written
828 \item @{ML bind_thms}~@{text "(name, thms)"} stores a list of
829 theorems produced in ML both in the theory context and the ML
830 toplevel, associating it with the provided name. Theorems are put
831 into a global ``standard'' format before being stored.
833 \item @{ML bind_thm} is similar to @{ML bind_thms} but refers to a
840 section {* Primitive specification elements *}
842 subsection {* Type classes and sorts \label{sec:classes} *}
845 \begin{matharray}{rcll}
846 @{command_def "classes"} & : & @{text "theory \<rightarrow> theory"} \\
847 @{command_def "classrel"} & : & @{text "theory \<rightarrow> theory"} & (axiomatic!) \\
848 @{command_def "defaultsort"} & : & @{text "theory \<rightarrow> theory"} \\
849 @{command_def "class_deps"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
853 'classes' (classdecl +)
855 'classrel' (nameref ('<' | subseteq) nameref + 'and')
863 \item @{command "classes"}~@{text "c \<subseteq> c\<^sub>1, \<dots>, c\<^sub>n"} declares class
864 @{text c} to be a subclass of existing classes @{text "c\<^sub>1, \<dots>, c\<^sub>n"}.
865 Isabelle implicitly maintains the transitive closure of the class
866 hierarchy. Cyclic class structures are not permitted.
868 \item @{command "classrel"}~@{text "c\<^sub>1 \<subseteq> c\<^sub>2"} states subclass
869 relations between existing classes @{text "c\<^sub>1"} and @{text "c\<^sub>2"}.
870 This is done axiomatically! The @{command_ref "instance"} command
871 (see \secref{sec:axclass}) provides a way to introduce proven class
874 \item @{command "defaultsort"}~@{text s} makes sort @{text s} the
875 new default sort for any type variable that is given explicitly in
876 the text, but lacks a sort constraint (wrt.\ the current context).
877 Type variables generated by type inference are not affected.
879 Usually the default sort is only changed when defining a new
880 object-logic. For example, the default sort in Isabelle/HOL is
881 @{text type}, the class of all HOL types. %FIXME sort antiq?
883 When merging theories, the default sorts of the parents are
884 logically intersected, i.e.\ the representations as lists of classes
887 \item @{command "class_deps"} visualizes the subclass relation,
888 using Isabelle's graph browser tool (see also \cite{isabelle-sys}).
894 subsection {* Types and type abbreviations \label{sec:types-pure} *}
897 \begin{matharray}{rcll}
898 @{command_def "types"} & : & @{text "theory \<rightarrow> theory"} \\
899 @{command_def "typedecl"} & : & @{text "theory \<rightarrow> theory"} \\
900 @{command_def "arities"} & : & @{text "theory \<rightarrow> theory"} & (axiomatic!) \\
904 'types' (typespec '=' type infix? +)
906 'typedecl' typespec infix?
908 'arities' (nameref '::' arity +)
914 \item @{command "types"}~@{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>n) t = \<tau>"} introduces a
915 \emph{type synonym} @{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>n) t"} for the existing type
916 @{text "\<tau>"}. Unlike actual type definitions, as are available in
917 Isabelle/HOL for example, type synonyms are merely syntactic
918 abbreviations without any logical significance. Internally, type
919 synonyms are fully expanded.
921 \item @{command "typedecl"}~@{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>n) t"} declares a new
922 type constructor @{text t}. If the object-logic defines a base sort
923 @{text s}, then the constructor is declared to operate on that, via
924 the axiomatic specification @{command arities}~@{text "t :: (s, \<dots>,
927 \item @{command "arities"}~@{text "t :: (s\<^sub>1, \<dots>, s\<^sub>n)s"} augments
928 Isabelle's order-sorted signature of types by new type constructor
929 arities. This is done axiomatically! The @{command_ref "instance"}
930 command (see \secref{sec:axclass}) provides a way to introduce
937 subsection {* Co-regularity of type classes and arities *}
939 text {* The class relation together with the collection of
940 type-constructor arities must obey the principle of
941 \emph{co-regularity} as defined below.
943 \medskip For the subsequent formulation of co-regularity we assume
944 that the class relation is closed by transitivity and reflexivity.
945 Moreover the collection of arities @{text "t :: (\<^vec>s)c"} is
946 completed such that @{text "t :: (\<^vec>s)c"} and @{text "c \<subseteq> c'"}
947 implies @{text "t :: (\<^vec>s)c'"} for all such declarations.
949 Treating sorts as finite sets of classes (meaning the intersection),
950 the class relation @{text "c\<^sub>1 \<subseteq> c\<^sub>2"} is extended to sorts as
953 @{text "s\<^sub>1 \<subseteq> s\<^sub>2 \<equiv> \<forall>c\<^sub>2 \<in> s\<^sub>2. \<exists>c\<^sub>1 \<in> s\<^sub>1. c\<^sub>1 \<subseteq> c\<^sub>2"}
956 This relation on sorts is further extended to tuples of sorts (of
957 the same length) in the component-wise way.
959 \smallskip Co-regularity of the class relation together with the
960 arities relation means:
962 @{text "t :: (\<^vec>s\<^sub>1)c\<^sub>1 \<Longrightarrow> t :: (\<^vec>s\<^sub>2)c\<^sub>2 \<Longrightarrow> c\<^sub>1 \<subseteq> c\<^sub>2 \<Longrightarrow> \<^vec>s\<^sub>1 \<subseteq> \<^vec>s\<^sub>2"}
964 \noindent for all such arities. In other words, whenever the result
965 classes of some type-constructor arities are related, then the
966 argument sorts need to be related in the same way.
968 \medskip Co-regularity is a very fundamental property of the
969 order-sorted algebra of types. For example, it entails principle
970 types and most general unifiers, e.g.\ see \cite{nipkow-prehofer}.
974 subsection {* Constants and definitions \label{sec:consts} *}
977 Definitions essentially express abbreviations within the logic. The
978 simplest form of a definition is @{text "c :: \<sigma> \<equiv> t"}, where @{text
979 c} is a newly declared constant. Isabelle also allows derived forms
980 where the arguments of @{text c} appear on the left, abbreviating a
981 prefix of @{text \<lambda>}-abstractions, e.g.\ @{text "c \<equiv> \<lambda>x y. t"} may be
982 written more conveniently as @{text "c x y \<equiv> t"}. Moreover,
983 definitions may be weakened by adding arbitrary pre-conditions:
984 @{text "A \<Longrightarrow> c x y \<equiv> t"}.
986 \medskip The built-in well-formedness conditions for definitional
991 \item Arguments (on the left-hand side) must be distinct variables.
993 \item All variables on the right-hand side must also appear on the
996 \item All type variables on the right-hand side must also appear on
997 the left-hand side; this prohibits @{text "0 :: nat \<equiv> length ([] ::
998 \<alpha> list)"} for example.
1000 \item The definition must not be recursive. Most object-logics
1001 provide definitional principles that can be used to express
1006 Overloading means that a constant being declared as @{text "c :: \<alpha>
1007 decl"} may be defined separately on type instances @{text "c ::
1008 (\<beta>\<^sub>1, \<dots>, \<beta>\<^sub>n) t decl"} for each type constructor @{text
1009 t}. The right-hand side may mention overloaded constants
1010 recursively at type instances corresponding to the immediate
1011 argument types @{text "\<beta>\<^sub>1, \<dots>, \<beta>\<^sub>n"}. Incomplete
1012 specification patterns impose global constraints on all occurrences,
1013 e.g.\ @{text "d :: \<alpha> \<times> \<alpha>"} on the left-hand side means that all
1014 corresponding occurrences on some right-hand side need to be an
1015 instance of this, general @{text "d :: \<alpha> \<times> \<beta>"} will be disallowed.
1017 \begin{matharray}{rcl}
1018 @{command_def "consts"} & : & @{text "theory \<rightarrow> theory"} \\
1019 @{command_def "defs"} & : & @{text "theory \<rightarrow> theory"} \\
1020 @{command_def "constdefs"} & : & @{text "theory \<rightarrow> theory"} \\
1024 'consts' ((name '::' type mixfix?) +)
1026 'defs' ('(' 'unchecked'? 'overloaded'? ')')? \\ (axmdecl prop +)
1031 'constdefs' structs? (constdecl? constdef +)
1034 structs: '(' 'structure' (vars + 'and') ')'
1036 constdecl: ((name '::' type mixfix | name '::' type | name mixfix) 'where'?) | name 'where'
1038 constdef: thmdecl? prop
1044 \item @{command "consts"}~@{text "c :: \<sigma>"} declares constant @{text
1045 c} to have any instance of type scheme @{text \<sigma>}. The optional
1046 mixfix annotations may attach concrete syntax to the constants
1049 \item @{command "defs"}~@{text "name: eqn"} introduces @{text eqn}
1050 as a definitional axiom for some existing constant.
1052 The @{text "(unchecked)"} option disables global dependency checks
1053 for this definition, which is occasionally useful for exotic
1054 overloading. It is at the discretion of the user to avoid malformed
1055 theory specifications!
1057 The @{text "(overloaded)"} option declares definitions to be
1058 potentially overloaded. Unless this option is given, a warning
1059 message would be issued for any definitional equation with a more
1060 special type than that of the corresponding constant declaration.
1062 \item @{command "constdefs"} combines constant declarations and
1063 definitions, with type-inference taking care of the most general
1064 typing of the given specification (the optional type constraint may
1065 refer to type-inference dummies ``@{text _}'' as usual). The
1066 resulting type declaration needs to agree with that of the
1067 specification; overloading is \emph{not} supported here!
1069 The constant name may be omitted altogether, if neither type nor
1070 syntax declarations are given. The canonical name of the
1071 definitional axiom for constant @{text c} will be @{text c_def},
1072 unless specified otherwise. Also note that the given list of
1073 specifications is processed in a strictly sequential manner, with
1074 type-checking being performed independently.
1076 An optional initial context of @{text "(structure)"} declarations
1077 admits use of indexed syntax, using the special symbol @{verbatim
1078 "\<index>"} (printed as ``@{text "\<index>"}''). The latter concept is
1079 particularly useful with locales (see also \secref{sec:locale}).
1085 section {* Axioms and theorems \label{sec:axms-thms} *}
1088 \begin{matharray}{rcll}
1089 @{command_def "axioms"} & : & @{text "theory \<rightarrow> theory"} & (axiomatic!) \\
1090 @{command_def "lemmas"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
1091 @{command_def "theorems"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
1095 'axioms' (axmdecl prop +)
1097 ('lemmas' | 'theorems') target? (thmdef? thmrefs + 'and')
1103 \item @{command "axioms"}~@{text "a: \<phi>"} introduces arbitrary
1104 statements as axioms of the meta-logic. In fact, axioms are
1105 ``axiomatic theorems'', and may be referred later just as any other
1108 Axioms are usually only introduced when declaring new logical
1109 systems. Everyday work is typically done the hard way, with proper
1110 definitions and proven theorems.
1112 \item @{command "lemmas"}~@{text "a = b\<^sub>1 \<dots> b\<^sub>n"} retrieves and stores
1113 existing facts in the theory context, or the specified target
1114 context (see also \secref{sec:target}). Typical applications would
1115 also involve attributes, to declare Simplifier rules, for example.
1117 \item @{command "theorems"} is essentially the same as @{command
1118 "lemmas"}, but marks the result as a different kind of facts.
1124 section {* Oracles *}
1126 text {* Oracles allow Isabelle to take advantage of external reasoners
1127 such as arithmetic decision procedures, model checkers, fast
1128 tautology checkers or computer algebra systems. Invoked as an
1129 oracle, an external reasoner can create arbitrary Isabelle theorems.
1131 It is the responsibility of the user to ensure that the external
1132 reasoner is as trustworthy as the application requires. Another
1133 typical source of errors is the linkup between Isabelle and the
1134 external tool, not just its concrete implementation, but also the
1135 required translation between two different logical environments.
1137 Isabelle merely guarantees well-formedness of the propositions being
1138 asserted, and records within the internal derivation object how
1139 presumed theorems depend on unproven suppositions.
1141 \begin{matharray}{rcl}
1142 @{command_def "oracle"} & : & @{text "theory \<rightarrow> theory"} \\
1146 'oracle' name '=' text
1152 \item @{command "oracle"}~@{text "name = text"} turns the given ML
1153 expression @{text "text"} of type @{ML_text "'a -> cterm"} into an
1154 ML function of type @{ML_text "'a -> thm"}, which is bound to the
1155 global identifier @{ML_text name}. This acts like an infinitary
1156 specification of axioms! Invoking the oracle only works within the
1157 scope of the resulting theory.
1161 See @{"file" "~~/src/FOL/ex/IffOracle.thy"} for a worked example of
1162 defining a new primitive rule as oracle, and turning it into a proof
1167 section {* Name spaces *}
1170 \begin{matharray}{rcl}
1171 @{command_def "global"} & : & @{text "theory \<rightarrow> theory"} \\
1172 @{command_def "local"} & : & @{text "theory \<rightarrow> theory"} \\
1173 @{command_def "hide"} & : & @{text "theory \<rightarrow> theory"} \\
1177 'hide' ('(open)')? name (nameref + )
1181 Isabelle organizes any kind of name declarations (of types,
1182 constants, theorems etc.) by separate hierarchically structured name
1183 spaces. Normally the user does not have to control the behavior of
1184 name spaces by hand, yet the following commands provide some way to
1189 \item @{command "global"} and @{command "local"} change the current
1190 name declaration mode. Initially, theories start in @{command
1191 "local"} mode, causing all names to be automatically qualified by
1192 the theory name. Changing this to @{command "global"} causes all
1193 names to be declared without the theory prefix, until @{command
1194 "local"} is declared again.
1196 Note that global names are prone to get hidden accidently later,
1197 when qualified names of the same base name are introduced.
1199 \item @{command "hide"}~@{text "space names"} fully removes
1200 declarations from a given name space (which may be @{text "class"},
1201 @{text "type"}, @{text "const"}, or @{text "fact"}); with the @{text
1202 "(open)"} option, only the base name is hidden. Global
1203 (unqualified) names may never be hidden.
1205 Note that hiding name space accesses has no impact on logical
1206 declarations --- they remain valid internally. Entities that are no
1207 longer accessible to the user are printed with the special qualifier
1208 ``@{text "??"}'' prefixed to the full internal name.