5 chapter {* Isabelle/HOL \label{ch:hol} *}
7 section {* Typedef axiomatization \label{sec:hol-typedef} *}
10 \begin{matharray}{rcl}
11 @{command_def (HOL) "typedef"} & : & @{text "local_theory \<rightarrow> proof(prove)"} \\
15 'typedef' altname? abstype '=' repset
18 altname: '(' (name | 'open' | 'open' name) ')'
20 abstype: typespecsorts mixfix?
22 repset: term ('morphisms' name name)?
28 \item @{command (HOL) "typedef"}~@{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>n) t = A"}
29 axiomatizes a Gordon/HOL-style type definition in the background
30 theory of the current context, depending on a non-emptiness result
31 of the set @{text A} (which needs to be proven interactively).
33 The raw type may not depend on parameters or assumptions of the
34 context --- this is logically impossible in Isabelle/HOL --- but the
35 non-emptiness property can be local, potentially resulting in
36 multiple interpretations in target contexts. Thus the established
37 bijection between the representing set @{text A} and the new type
38 @{text t} may semantically depend on local assumptions.
40 By default, @{command (HOL) "typedef"} defines both a type @{text t}
41 and a set (term constant) of the same name, unless an alternative
42 base name is given in parentheses, or the ``@{text "(open)"}''
43 declaration is used to suppress a separate constant definition
44 altogether. The injection from type to set is called @{text Rep_t},
45 its inverse @{text Abs_t} --- this may be changed via an explicit
46 @{keyword (HOL) "morphisms"} declaration.
48 Theorems @{text Rep_t}, @{text Rep_t_inverse}, and @{text
49 Abs_t_inverse} provide the most basic characterization as a
50 corresponding injection/surjection pair (in both directions). Rules
51 @{text Rep_t_inject} and @{text Abs_t_inject} provide a slightly
52 more convenient view on the injectivity part, suitable for automated
53 proof tools (e.g.\ in @{attribute simp} or @{attribute iff}
54 declarations). Rules @{text Rep_t_cases}/@{text Rep_t_induct}, and
55 @{text Abs_t_cases}/@{text Abs_t_induct} provide alternative views
56 on surjectivity; these are already declared as set or type rules for
57 the generic @{method cases} and @{method induct} methods.
59 An alternative name for the set definition (and other derived
60 entities) may be specified in parentheses; the default is to use
61 @{text t} as indicated before.
67 section {* Adhoc tuples *}
70 \begin{matharray}{rcl}
71 @{attribute (HOL) split_format}@{text "\<^sup>*"} & : & @{text attribute} \\
75 'split_format' '(' 'complete' ')'
81 \item @{attribute (HOL) split_format}\ @{text "(complete)"} causes
82 arguments in function applications to be represented canonically
83 according to their tuple type structure.
85 Note that this operation tends to invent funny names for new local
86 parameters introduced.
92 section {* Records \label{sec:hol-record} *}
95 In principle, records merely generalize the concept of tuples, where
96 components may be addressed by labels instead of just position. The
97 logical infrastructure of records in Isabelle/HOL is slightly more
98 advanced, though, supporting truly extensible record schemes. This
99 admits operations that are polymorphic with respect to record
100 extension, yielding ``object-oriented'' effects like (single)
101 inheritance. See also \cite{NaraschewskiW-TPHOLs98} for more
102 details on object-oriented verification and record subtyping in HOL.
106 subsection {* Basic concepts *}
109 Isabelle/HOL supports both \emph{fixed} and \emph{schematic} records
110 at the level of terms and types. The notation is as follows:
113 \begin{tabular}{l|l|l}
114 & record terms & record types \\ \hline
115 fixed & @{text "\<lparr>x = a, y = b\<rparr>"} & @{text "\<lparr>x :: A, y :: B\<rparr>"} \\
116 schematic & @{text "\<lparr>x = a, y = b, \<dots> = m\<rparr>"} &
117 @{text "\<lparr>x :: A, y :: B, \<dots> :: M\<rparr>"} \\
121 \noindent The ASCII representation of @{text "\<lparr>x = a\<rparr>"} is @{text
124 A fixed record @{text "\<lparr>x = a, y = b\<rparr>"} has field @{text x} of value
125 @{text a} and field @{text y} of value @{text b}. The corresponding
126 type is @{text "\<lparr>x :: A, y :: B\<rparr>"}, assuming that @{text "a :: A"}
127 and @{text "b :: B"}.
129 A record scheme like @{text "\<lparr>x = a, y = b, \<dots> = m\<rparr>"} contains fields
130 @{text x} and @{text y} as before, but also possibly further fields
131 as indicated by the ``@{text "\<dots>"}'' notation (which is actually part
132 of the syntax). The improper field ``@{text "\<dots>"}'' of a record
133 scheme is called the \emph{more part}. Logically it is just a free
134 variable, which is occasionally referred to as ``row variable'' in
135 the literature. The more part of a record scheme may be
136 instantiated by zero or more further components. For example, the
137 previous scheme may get instantiated to @{text "\<lparr>x = a, y = b, z =
138 c, \<dots> = m'\<rparr>"}, where @{text m'} refers to a different more part.
139 Fixed records are special instances of record schemes, where
140 ``@{text "\<dots>"}'' is properly terminated by the @{text "() :: unit"}
141 element. In fact, @{text "\<lparr>x = a, y = b\<rparr>"} is just an abbreviation
142 for @{text "\<lparr>x = a, y = b, \<dots> = ()\<rparr>"}.
144 \medskip Two key observations make extensible records in a simply
145 typed language like HOL work out:
149 \item the more part is internalized, as a free term or type
152 \item field names are externalized, they cannot be accessed within
153 the logic as first-class values.
157 \medskip In Isabelle/HOL record types have to be defined explicitly,
158 fixing their field names and types, and their (optional) parent
159 record. Afterwards, records may be formed using above syntax, while
160 obeying the canonical order of fields as given by their declaration.
161 The record package provides several standard operations like
162 selectors and updates. The common setup for various generic proof
163 tools enable succinct reasoning patterns. See also the Isabelle/HOL
164 tutorial \cite{isabelle-hol-book} for further instructions on using
169 subsection {* Record specifications *}
172 \begin{matharray}{rcl}
173 @{command_def (HOL) "record"} & : & @{text "theory \<rightarrow> theory"} \\
177 'record' typespecsorts '=' (type '+')? (constdecl +)
183 \item @{command (HOL) "record"}~@{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t = \<tau> + c\<^sub>1 :: \<sigma>\<^sub>1
184 \<dots> c\<^sub>n :: \<sigma>\<^sub>n"} defines extensible record type @{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t"},
185 derived from the optional parent record @{text "\<tau>"} by adding new
186 field components @{text "c\<^sub>i :: \<sigma>\<^sub>i"} etc.
188 The type variables of @{text "\<tau>"} and @{text "\<sigma>\<^sub>i"} need to be
189 covered by the (distinct) parameters @{text "\<alpha>\<^sub>1, \<dots>,
190 \<alpha>\<^sub>m"}. Type constructor @{text t} has to be new, while @{text
191 \<tau>} needs to specify an instance of an existing record type. At
192 least one new field @{text "c\<^sub>i"} has to be specified.
193 Basically, field names need to belong to a unique record. This is
194 not a real restriction in practice, since fields are qualified by
195 the record name internally.
197 The parent record specification @{text \<tau>} is optional; if omitted
198 @{text t} becomes a root record. The hierarchy of all records
199 declared within a theory context forms a forest structure, i.e.\ a
200 set of trees starting with a root record each. There is no way to
201 merge multiple parent records!
203 For convenience, @{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t"} is made a
204 type abbreviation for the fixed record type @{text "\<lparr>c\<^sub>1 ::
205 \<sigma>\<^sub>1, \<dots>, c\<^sub>n :: \<sigma>\<^sub>n\<rparr>"}, likewise is @{text
206 "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m, \<zeta>) t_scheme"} made an abbreviation for
207 @{text "\<lparr>c\<^sub>1 :: \<sigma>\<^sub>1, \<dots>, c\<^sub>n :: \<sigma>\<^sub>n, \<dots> ::
214 subsection {* Record operations *}
217 Any record definition of the form presented above produces certain
218 standard operations. Selectors and updates are provided for any
219 field, including the improper one ``@{text more}''. There are also
220 cumulative record constructor functions. To simplify the
221 presentation below, we assume for now that @{text "(\<alpha>\<^sub>1, \<dots>,
222 \<alpha>\<^sub>m) t"} is a root record with fields @{text "c\<^sub>1 ::
223 \<sigma>\<^sub>1, \<dots>, c\<^sub>n :: \<sigma>\<^sub>n"}.
225 \medskip \textbf{Selectors} and \textbf{updates} are available for
226 any field (including ``@{text more}''):
228 \begin{matharray}{lll}
229 @{text "c\<^sub>i"} & @{text "::"} & @{text "\<lparr>\<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr> \<Rightarrow> \<sigma>\<^sub>i"} \\
230 @{text "c\<^sub>i_update"} & @{text "::"} & @{text "\<sigma>\<^sub>i \<Rightarrow> \<lparr>\<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr> \<Rightarrow> \<lparr>\<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr>"} \\
233 There is special syntax for application of updates: @{text "r\<lparr>x :=
234 a\<rparr>"} abbreviates term @{text "x_update a r"}. Further notation for
235 repeated updates is also available: @{text "r\<lparr>x := a\<rparr>\<lparr>y := b\<rparr>\<lparr>z :=
236 c\<rparr>"} may be written @{text "r\<lparr>x := a, y := b, z := c\<rparr>"}. Note that
237 because of postfix notation the order of fields shown here is
238 reverse than in the actual term. Since repeated updates are just
239 function applications, fields may be freely permuted in @{text "\<lparr>x
240 := a, y := b, z := c\<rparr>"}, as far as logical equality is concerned.
241 Thus commutativity of independent updates can be proven within the
242 logic for any two fields, but not as a general theorem.
244 \medskip The \textbf{make} operation provides a cumulative record
245 constructor function:
247 \begin{matharray}{lll}
248 @{text "t.make"} & @{text "::"} & @{text "\<sigma>\<^sub>1 \<Rightarrow> \<dots> \<sigma>\<^sub>n \<Rightarrow> \<lparr>\<^vec>c :: \<^vec>\<sigma>\<rparr>"} \\
251 \medskip We now reconsider the case of non-root records, which are
252 derived of some parent. In general, the latter may depend on
253 another parent as well, resulting in a list of \emph{ancestor
254 records}. Appending the lists of fields of all ancestors results in
255 a certain field prefix. The record package automatically takes care
256 of this by lifting operations over this context of ancestor fields.
257 Assuming that @{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t"} has ancestor
258 fields @{text "b\<^sub>1 :: \<rho>\<^sub>1, \<dots>, b\<^sub>k :: \<rho>\<^sub>k"},
259 the above record operations will get the following types:
263 @{text "c\<^sub>i"} & @{text "::"} & @{text "\<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr> \<Rightarrow> \<sigma>\<^sub>i"} \\
264 @{text "c\<^sub>i_update"} & @{text "::"} & @{text "\<sigma>\<^sub>i \<Rightarrow>
265 \<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr> \<Rightarrow>
266 \<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr>"} \\
267 @{text "t.make"} & @{text "::"} & @{text "\<rho>\<^sub>1 \<Rightarrow> \<dots> \<rho>\<^sub>k \<Rightarrow> \<sigma>\<^sub>1 \<Rightarrow> \<dots> \<sigma>\<^sub>n \<Rightarrow>
268 \<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>\<rparr>"} \\
272 \noindent Some further operations address the extension aspect of a
273 derived record scheme specifically: @{text "t.fields"} produces a
274 record fragment consisting of exactly the new fields introduced here
275 (the result may serve as a more part elsewhere); @{text "t.extend"}
276 takes a fixed record and adds a given more part; @{text
277 "t.truncate"} restricts a record scheme to a fixed record.
281 @{text "t.fields"} & @{text "::"} & @{text "\<sigma>\<^sub>1 \<Rightarrow> \<dots> \<sigma>\<^sub>n \<Rightarrow> \<lparr>\<^vec>c :: \<^vec>\<sigma>\<rparr>"} \\
282 @{text "t.extend"} & @{text "::"} & @{text "\<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>\<rparr> \<Rightarrow>
283 \<zeta> \<Rightarrow> \<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr>"} \\
284 @{text "t.truncate"} & @{text "::"} & @{text "\<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr> \<Rightarrow> \<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>\<rparr>"} \\
288 \noindent Note that @{text "t.make"} and @{text "t.fields"} coincide
293 subsection {* Derived rules and proof tools *}
296 The record package proves several results internally, declaring
297 these facts to appropriate proof tools. This enables users to
298 reason about record structures quite conveniently. Assume that
299 @{text t} is a record type as specified above.
303 \item Standard conversions for selectors or updates applied to
304 record constructor terms are made part of the default Simplifier
305 context; thus proofs by reduction of basic operations merely require
306 the @{method simp} method without further arguments. These rules
307 are available as @{text "t.simps"}, too.
309 \item Selectors applied to updated records are automatically reduced
310 by an internal simplification procedure, which is also part of the
311 standard Simplifier setup.
313 \item Inject equations of a form analogous to @{prop "(x, y) = (x',
314 y') \<equiv> x = x' \<and> y = y'"} are declared to the Simplifier and Classical
315 Reasoner as @{attribute iff} rules. These rules are available as
318 \item The introduction rule for record equality analogous to @{text
319 "x r = x r' \<Longrightarrow> y r = y r' \<dots> \<Longrightarrow> r = r'"} is declared to the Simplifier,
320 and as the basic rule context as ``@{attribute intro}@{text "?"}''.
321 The rule is called @{text "t.equality"}.
323 \item Representations of arbitrary record expressions as canonical
324 constructor terms are provided both in @{method cases} and @{method
325 induct} format (cf.\ the generic proof methods of the same name,
326 \secref{sec:cases-induct}). Several variations are available, for
327 fixed records, record schemes, more parts etc.
329 The generic proof methods are sufficiently smart to pick the most
330 sensible rule according to the type of the indicated record
331 expression: users just need to apply something like ``@{text "(cases
332 r)"}'' to a certain proof problem.
334 \item The derived record operations @{text "t.make"}, @{text
335 "t.fields"}, @{text "t.extend"}, @{text "t.truncate"} are \emph{not}
336 treated automatically, but usually need to be expanded by hand,
337 using the collective fact @{text "t.defs"}.
343 section {* Datatypes \label{sec:hol-datatype} *}
346 \begin{matharray}{rcl}
347 @{command_def (HOL) "datatype"} & : & @{text "theory \<rightarrow> theory"} \\
348 @{command_def (HOL) "rep_datatype"} & : & @{text "theory \<rightarrow> proof(prove)"} \\
352 'datatype' (dtspec + 'and')
354 'rep_datatype' ('(' (name +) ')')? (term +)
357 dtspec: parname? typespec mixfix? '=' (cons + '|')
359 cons: name ( type * ) mixfix?
364 \item @{command (HOL) "datatype"} defines inductive datatypes in
367 \item @{command (HOL) "rep_datatype"} represents existing types as
368 inductive ones, generating the standard infrastructure of derived
369 concepts (primitive recursion etc.).
373 The induction and exhaustion theorems generated provide case names
374 according to the constructors involved, while parameters are named
375 after the types (see also \secref{sec:cases-induct}).
377 See \cite{isabelle-HOL} for more details on datatypes, but beware of
378 the old-style theory syntax being used there! Apart from proper
379 proof methods for case-analysis and induction, there are also
380 emulations of ML tactics @{method (HOL) case_tac} and @{method (HOL)
381 induct_tac} available, see \secref{sec:hol-induct-tac}; these admit
382 to refer directly to the internal structure of subgoals (including
383 internally bound parameters).
387 section {* Functorial structure of types *}
390 \begin{matharray}{rcl}
391 @{command_def (HOL) "enriched_type"} & : & @{text "local_theory \<rightarrow> proof(prove)"}
395 'enriched_type' (prefix ':')? term
401 \item @{command (HOL) "enriched_type"} allows to prove and register
402 properties about the functorial structure of type constructors;
403 these properties then can be used by other packages to
404 deal with those type constructors in certain type constructions.
405 Characteristic theorems are noted in the current local theory; by
406 default, they are prefixed with the base name of the type constructor,
407 an explicit prefix can be given alternatively.
409 The given term @{text "m"} is considered as \emph{mapper} for the
410 corresponding type constructor and must conform to the following
413 \begin{matharray}{lll}
414 @{text "m"} & @{text "::"} &
415 @{text "\<sigma>\<^isub>1 \<Rightarrow> \<dots> \<sigma>\<^isub>k \<Rightarrow> (\<^vec>\<alpha>\<^isub>n) t \<Rightarrow> (\<^vec>\<beta>\<^isub>n) t"} \\
418 \noindent where @{text t} is the type constructor, @{text
419 "\<^vec>\<alpha>\<^isub>n"} and @{text "\<^vec>\<beta>\<^isub>n"} are distinct
420 type variables free in the local theory and @{text "\<sigma>\<^isub>1"},
421 \ldots, @{text "\<sigma>\<^isub>k"} is a subsequence of @{text "\<alpha>\<^isub>1 \<Rightarrow>
422 \<beta>\<^isub>1"}, @{text "\<beta>\<^isub>1 \<Rightarrow> \<alpha>\<^isub>1"}, \ldots,
423 @{text "\<alpha>\<^isub>n \<Rightarrow> \<beta>\<^isub>n"}, @{text "\<beta>\<^isub>n \<Rightarrow>
430 section {* Recursive functions \label{sec:recursion} *}
433 \begin{matharray}{rcl}
434 @{command_def (HOL) "primrec"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
435 @{command_def (HOL) "fun"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
436 @{command_def (HOL) "function"} & : & @{text "local_theory \<rightarrow> proof(prove)"} \\
437 @{command_def (HOL) "termination"} & : & @{text "local_theory \<rightarrow> proof(prove)"} \\
441 'primrec' target? fixes 'where' equations
443 ('fun' | 'function') target? functionopts? fixes \\ 'where' equations
445 equations: (thmdecl? prop + '|')
447 functionopts: '(' (('sequential' | 'domintros') + ',') ')'
449 'termination' ( term )?
454 \item @{command (HOL) "primrec"} defines primitive recursive
455 functions over datatypes, see also \cite{isabelle-HOL}.
457 \item @{command (HOL) "function"} defines functions by general
458 wellfounded recursion. A detailed description with examples can be
459 found in \cite{isabelle-function}. The function is specified by a
460 set of (possibly conditional) recursive equations with arbitrary
461 pattern matching. The command generates proof obligations for the
462 completeness and the compatibility of patterns.
464 The defined function is considered partial, and the resulting
465 simplification rules (named @{text "f.psimps"}) and induction rule
466 (named @{text "f.pinduct"}) are guarded by a generated domain
467 predicate @{text "f_dom"}. The @{command (HOL) "termination"}
468 command can then be used to establish that the function is total.
470 \item @{command (HOL) "fun"} is a shorthand notation for ``@{command
471 (HOL) "function"}~@{text "(sequential)"}, followed by automated
472 proof attempts regarding pattern matching and termination. See
473 \cite{isabelle-function} for further details.
475 \item @{command (HOL) "termination"}~@{text f} commences a
476 termination proof for the previously defined function @{text f}. If
477 this is omitted, the command refers to the most recent function
478 definition. After the proof is closed, the recursive equations and
479 the induction principle is established.
483 Recursive definitions introduced by the @{command (HOL) "function"}
485 reasoning by induction (cf.\ \secref{sec:cases-induct}): rule @{text
486 "c.induct"} (where @{text c} is the name of the function definition)
487 refers to a specific induction rule, with parameters named according
488 to the user-specified equations. Cases are numbered (starting from 1).
490 For @{command (HOL) "primrec"}, the induction principle coincides
491 with structural recursion on the datatype the recursion is carried
494 The equations provided by these packages may be referred later as
495 theorem list @{text "f.simps"}, where @{text f} is the (collective)
496 name of the functions defined. Individual equations may be named
499 The @{command (HOL) "function"} command accepts the following
504 \item @{text sequential} enables a preprocessor which disambiguates
505 overlapping patterns by making them mutually disjoint. Earlier
506 equations take precedence over later ones. This allows to give the
507 specification in a format very similar to functional programming.
508 Note that the resulting simplification and induction rules
509 correspond to the transformed specification, not the one given
510 originally. This usually means that each equation given by the user
511 may result in several theorems. Also note that this automatic
512 transformation only works for ML-style datatype patterns.
514 \item @{text domintros} enables the automated generation of
515 introduction rules for the domain predicate. While mostly not
516 needed, they can be helpful in some proofs about partial functions.
522 subsection {* Proof methods related to recursive definitions *}
525 \begin{matharray}{rcl}
526 @{method_def (HOL) pat_completeness} & : & @{text method} \\
527 @{method_def (HOL) relation} & : & @{text method} \\
528 @{method_def (HOL) lexicographic_order} & : & @{text method} \\
529 @{method_def (HOL) size_change} & : & @{text method} \\
535 'lexicographic_order' ( clasimpmod * )
537 'size_change' ( orders ( clasimpmod * ) )
539 orders: ( 'max' | 'min' | 'ms' ) *
544 \item @{method (HOL) pat_completeness} is a specialized method to
545 solve goals regarding the completeness of pattern matching, as
546 required by the @{command (HOL) "function"} package (cf.\
547 \cite{isabelle-function}).
549 \item @{method (HOL) relation}~@{text R} introduces a termination
550 proof using the relation @{text R}. The resulting proof state will
551 contain goals expressing that @{text R} is wellfounded, and that the
552 arguments of recursive calls decrease with respect to @{text R}.
553 Usually, this method is used as the initial proof step of manual
556 \item @{method (HOL) "lexicographic_order"} attempts a fully
557 automated termination proof by searching for a lexicographic
558 combination of size measures on the arguments of the function. The
559 method accepts the same arguments as the @{method auto} method,
560 which it uses internally to prove local descents. The same context
561 modifiers as for @{method auto} are accepted, see
562 \secref{sec:clasimp}.
564 In case of failure, extensive information is printed, which can help
565 to analyse the situation (cf.\ \cite{isabelle-function}).
567 \item @{method (HOL) "size_change"} also works on termination goals,
568 using a variation of the size-change principle, together with a
569 graph decomposition technique (see \cite{krauss_phd} for details).
570 Three kinds of orders are used internally: @{text max}, @{text min},
571 and @{text ms} (multiset), which is only available when the theory
572 @{text Multiset} is loaded. When no order kinds are given, they are
573 tried in order. The search for a termination proof uses SAT solving
576 For local descent proofs, the same context modifiers as for @{method
577 auto} are accepted, see \secref{sec:clasimp}.
582 subsection {* Functions with explicit partiality *}
585 \begin{matharray}{rcl}
586 @{command_def (HOL) "partial_function"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
587 @{attribute_def (HOL) "partial_function_mono"} & : & @{text attribute} \\
591 'partial_function' target? '(' mode ')' fixes \\ 'where' thmdecl? prop
596 \item @{command (HOL) "partial_function"} defines recursive
597 functions based on fixpoints in complete partial orders. No
598 termination proof is required from the user or constructed
599 internally. Instead, the possibility of non-termination is modelled
600 explicitly in the result type, which contains an explicit bottom
603 Pattern matching and mutual recursion are currently not supported.
604 Thus, the specification consists of a single function described by a
605 single recursive equation.
607 There are no fixed syntactic restrictions on the body of the
608 function, but the induced functional must be provably monotonic
609 wrt.\ the underlying order. The monotonicitity proof is performed
610 internally, and the definition is rejected when it fails. The proof
611 can be influenced by declaring hints using the
612 @{attribute (HOL) partial_function_mono} attribute.
614 The mandatory @{text mode} argument specifies the mode of operation
615 of the command, which directly corresponds to a complete partial
616 order on the result type. By default, the following modes are
620 \item @{text option} defines functions that map into the @{type
621 option} type. Here, the value @{term None} is used to model a
622 non-terminating computation. Monotonicity requires that if @{term
623 None} is returned by a recursive call, then the overall result
624 must also be @{term None}. This is best achieved through the use of
625 the monadic operator @{const "Option.bind"}.
627 \item @{text tailrec} defines functions with an arbitrary result
628 type and uses the slightly degenerated partial order where @{term
629 "undefined"} is the bottom element. Now, monotonicity requires that
630 if @{term undefined} is returned by a recursive call, then the
631 overall result must also be @{term undefined}. In practice, this is
632 only satisfied when each recursive call is a tail call, whose result
633 is directly returned. Thus, this mode of operation allows the
634 definition of arbitrary tail-recursive functions.
637 Experienced users may define new modes by instantiating the locale
638 @{const "partial_function_definitions"} appropriately.
640 \item @{attribute (HOL) partial_function_mono} declares rules for
641 use in the internal monononicity proofs of partial function
648 subsection {* Old-style recursive function definitions (TFL) *}
651 The old TFL commands @{command (HOL) "recdef"} and @{command (HOL)
652 "recdef_tc"} for defining recursive are mostly obsolete; @{command
653 (HOL) "function"} or @{command (HOL) "fun"} should be used instead.
655 \begin{matharray}{rcl}
656 @{command_def (HOL) "recdef"} & : & @{text "theory \<rightarrow> theory)"} \\
657 @{command_def (HOL) "recdef_tc"}@{text "\<^sup>*"} & : & @{text "theory \<rightarrow> proof(prove)"} \\
661 'recdef' ('(' 'permissive' ')')? \\ name term (prop +) hints?
665 hints: '(' 'hints' ( recdefmod * ) ')'
667 recdefmod: (('recdef_simp' | 'recdef_cong' | 'recdef_wf') (() | 'add' | 'del') ':' thmrefs) | clasimpmod
669 tc: nameref ('(' nat ')')?
675 \item @{command (HOL) "recdef"} defines general well-founded
676 recursive functions (using the TFL package), see also
677 \cite{isabelle-HOL}. The ``@{text "(permissive)"}'' option tells
678 TFL to recover from failed proof attempts, returning unfinished
679 results. The @{text recdef_simp}, @{text recdef_cong}, and @{text
680 recdef_wf} hints refer to auxiliary rules to be used in the internal
681 automated proof process of TFL. Additional @{syntax clasimpmod}
682 declarations (cf.\ \secref{sec:clasimp}) may be given to tune the
683 context of the Simplifier (cf.\ \secref{sec:simplifier}) and
684 Classical reasoner (cf.\ \secref{sec:classical}).
686 \item @{command (HOL) "recdef_tc"}~@{text "c (i)"} recommences the
687 proof for leftover termination condition number @{text i} (default
688 1) as generated by a @{command (HOL) "recdef"} definition of
691 Note that in most cases, @{command (HOL) "recdef"} is able to finish
692 its internal proofs without manual intervention.
696 \medskip Hints for @{command (HOL) "recdef"} may be also declared
697 globally, using the following attributes.
699 \begin{matharray}{rcl}
700 @{attribute_def (HOL) recdef_simp} & : & @{text attribute} \\
701 @{attribute_def (HOL) recdef_cong} & : & @{text attribute} \\
702 @{attribute_def (HOL) recdef_wf} & : & @{text attribute} \\
706 ('recdef_simp' | 'recdef_cong' | 'recdef_wf') (() | 'add' | 'del')
712 section {* Inductive and coinductive definitions \label{sec:hol-inductive} *}
715 An \textbf{inductive definition} specifies the least predicate (or
716 set) @{text R} closed under given rules: applying a rule to elements
717 of @{text R} yields a result within @{text R}. For example, a
718 structural operational semantics is an inductive definition of an
721 Dually, a \textbf{coinductive definition} specifies the greatest
722 predicate~/ set @{text R} that is consistent with given rules: every
723 element of @{text R} can be seen as arising by applying a rule to
724 elements of @{text R}. An important example is using bisimulation
725 relations to formalise equivalence of processes and infinite data
728 \medskip The HOL package is related to the ZF one, which is
729 described in a separate paper,\footnote{It appeared in CADE
730 \cite{paulson-CADE}; a longer version is distributed with Isabelle.}
731 which you should refer to in case of difficulties. The package is
732 simpler than that of ZF thanks to implicit type-checking in HOL.
733 The types of the (co)inductive predicates (or sets) determine the
734 domain of the fixedpoint definition, and the package does not have
735 to use inference rules for type-checking.
737 \begin{matharray}{rcl}
738 @{command_def (HOL) "inductive"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
739 @{command_def (HOL) "inductive_set"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
740 @{command_def (HOL) "coinductive"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
741 @{command_def (HOL) "coinductive_set"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
742 @{attribute_def (HOL) mono} & : & @{text attribute} \\
746 ('inductive' | 'inductive_set' | 'coinductive' | 'coinductive_set') target? fixes ('for' fixes)? \\
747 ('where' clauses)? ('monos' thmrefs)?
749 clauses: (thmdecl? prop + '|')
751 'mono' (() | 'add' | 'del')
757 \item @{command (HOL) "inductive"} and @{command (HOL)
758 "coinductive"} define (co)inductive predicates from the
759 introduction rules given in the @{keyword "where"} part. The
760 optional @{keyword "for"} part contains a list of parameters of the
761 (co)inductive predicates that remain fixed throughout the
762 definition. The optional @{keyword "monos"} section contains
763 \emph{monotonicity theorems}, which are required for each operator
764 applied to a recursive set in the introduction rules. There
765 \emph{must} be a theorem of the form @{text "A \<le> B \<Longrightarrow> M A \<le> M B"},
766 for each premise @{text "M R\<^sub>i t"} in an introduction rule!
768 \item @{command (HOL) "inductive_set"} and @{command (HOL)
769 "coinductive_set"} are wrappers for to the previous commands,
770 allowing the definition of (co)inductive sets.
772 \item @{attribute (HOL) mono} declares monotonicity rules. These
773 rule are involved in the automated monotonicity proof of @{command
780 subsection {* Derived rules *}
783 Each (co)inductive definition @{text R} adds definitions to the
784 theory and also proves some theorems:
788 \item @{text R.intros} is the list of introduction rules as proven
789 theorems, for the recursive predicates (or sets). The rules are
790 also available individually, using the names given them in the
793 \item @{text R.cases} is the case analysis (or elimination) rule;
795 \item @{text R.induct} or @{text R.coinduct} is the (co)induction
800 When several predicates @{text "R\<^sub>1, \<dots>, R\<^sub>n"} are
801 defined simultaneously, the list of introduction rules is called
802 @{text "R\<^sub>1_\<dots>_R\<^sub>n.intros"}, the case analysis rules are
803 called @{text "R\<^sub>1.cases, \<dots>, R\<^sub>n.cases"}, and the list
804 of mutual induction rules is called @{text
805 "R\<^sub>1_\<dots>_R\<^sub>n.inducts"}.
809 subsection {* Monotonicity theorems *}
812 Each theory contains a default set of theorems that are used in
813 monotonicity proofs. New rules can be added to this set via the
814 @{attribute (HOL) mono} attribute. The HOL theory @{text Inductive}
815 shows how this is done. In general, the following monotonicity
816 theorems may be added:
820 \item Theorems of the form @{text "A \<le> B \<Longrightarrow> M A \<le> M B"}, for proving
821 monotonicity of inductive definitions whose introduction rules have
822 premises involving terms such as @{text "M R\<^sub>i t"}.
824 \item Monotonicity theorems for logical operators, which are of the
825 general form @{text "(\<dots> \<longrightarrow> \<dots>) \<Longrightarrow> \<dots> (\<dots> \<longrightarrow> \<dots>) \<Longrightarrow> \<dots> \<longrightarrow> \<dots>"}. For example, in
826 the case of the operator @{text "\<or>"}, the corresponding theorem is
828 \infer{@{text "P\<^sub>1 \<or> P\<^sub>2 \<longrightarrow> Q\<^sub>1 \<or> Q\<^sub>2"}}{@{text "P\<^sub>1 \<longrightarrow> Q\<^sub>1"} & @{text "P\<^sub>2 \<longrightarrow> Q\<^sub>2"}}
831 \item De Morgan style equations for reasoning about the ``polarity''
834 @{prop "\<not> \<not> P \<longleftrightarrow> P"} \qquad\qquad
835 @{prop "\<not> (P \<and> Q) \<longleftrightarrow> \<not> P \<or> \<not> Q"}
838 \item Equations for reducing complex operators to more primitive
839 ones whose monotonicity can easily be proved, e.g.
841 @{prop "(P \<longrightarrow> Q) \<longleftrightarrow> \<not> P \<or> Q"} \qquad\qquad
842 @{prop "Ball A P \<equiv> \<forall>x. x \<in> A \<longrightarrow> P x"}
847 %FIXME: Example of an inductive definition
851 section {* Arithmetic proof support *}
854 \begin{matharray}{rcl}
855 @{method_def (HOL) arith} & : & @{text method} \\
856 @{attribute_def (HOL) arith} & : & @{text attribute} \\
857 @{attribute_def (HOL) arith_split} & : & @{text attribute} \\
860 The @{method (HOL) arith} method decides linear arithmetic problems
861 (on types @{text nat}, @{text int}, @{text real}). Any current
862 facts are inserted into the goal before running the procedure.
864 The @{attribute (HOL) arith} attribute declares facts that are
865 always supplied to the arithmetic provers implicitly.
867 The @{attribute (HOL) arith_split} attribute declares case split
868 rules to be expanded before @{method (HOL) arith} is invoked.
870 Note that a simpler (but faster) arithmetic prover is
871 already invoked by the Simplifier.
875 section {* Intuitionistic proof search *}
878 \begin{matharray}{rcl}
879 @{method_def (HOL) iprover} & : & @{text method} \\
883 'iprover' ( rulemod * )
887 The @{method (HOL) iprover} method performs intuitionistic proof
888 search, depending on specifically declared rules from the context,
889 or given as explicit arguments. Chained facts are inserted into the
890 goal before commencing proof search.
892 Rules need to be classified as @{attribute (Pure) intro},
893 @{attribute (Pure) elim}, or @{attribute (Pure) dest}; here the
894 ``@{text "!"}'' indicator refers to ``safe'' rules, which may be
895 applied aggressively (without considering back-tracking later).
896 Rules declared with ``@{text "?"}'' are ignored in proof search (the
897 single-step @{method rule} method still observes these). An
898 explicit weight annotation may be given as well; otherwise the
899 number of rule premises will be taken into account here.
903 section {* Coherent Logic *}
906 \begin{matharray}{rcl}
907 @{method_def (HOL) "coherent"} & : & @{text method} \\
915 The @{method (HOL) coherent} method solves problems of
916 \emph{Coherent Logic} \cite{Bezem-Coquand:2005}, which covers
917 applications in confluence theory, lattice theory and projective
918 geometry. See @{file "~~/src/HOL/ex/Coherent.thy"} for some
923 section {* Checking and refuting propositions *}
926 Identifying incorrect propositions usually involves evaluation of
927 particular assignments and systematic counter example search. This
928 is supported by the following commands.
930 \begin{matharray}{rcl}
931 @{command_def (HOL) "value"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
932 @{command_def (HOL) "quickcheck"}@{text "\<^sup>*"} & : & @{text "proof \<rightarrow>"} \\
933 @{command_def (HOL) "quickcheck_params"} & : & @{text "theory \<rightarrow> theory"}
937 'value' ( ( '[' name ']' ) ? ) modes? term
940 'quickcheck' ( ( '[' args ']' ) ? ) nat?
943 'quickcheck_params' ( ( '[' args ']' ) ? )
946 modes: '(' (name + ) ')'
949 args: ( name '=' value + ',' )
955 \item @{command (HOL) "value"}~@{text t} evaluates and prints a
956 term; optionally @{text modes} can be specified, which are
957 appended to the current print mode (see also \cite{isabelle-ref}).
958 Internally, the evaluation is performed by registered evaluators,
959 which are invoked sequentially until a result is returned.
960 Alternatively a specific evaluator can be selected using square
961 brackets; typical evaluators use the current set of code equations
962 to normalize and include @{text simp} for fully symbolic evaluation
963 using the simplifier, @{text nbe} for \emph{normalization by evaluation}
964 and \emph{code} for code generation in SML.
966 \item @{command (HOL) "quickcheck"} tests the current goal for
967 counter examples using a series of assignments for its
968 free variables; by default the first subgoal is tested, an other
969 can be selected explicitly using an optional goal index.
970 Assignments can be chosen exhausting the search space upto a given
971 size or using a fixed number of random assignments in the search space.
972 By default, quickcheck uses exhaustive testing.
973 A number of configuration options are supported for
974 @{command (HOL) "quickcheck"}, notably:
978 \item[@{text tester}] specifies how to explore the search space
979 (e.g. exhaustive or random).
980 An unknown configuration option is treated as an argument to tester,
981 making @{text "tester ="} optional.
982 \item[@{text size}] specifies the maximum size of the search space
983 for assignment values.
985 \item[@{text eval}] takes a term or a list of terms and evaluates
986 these terms under the variable assignment found by quickcheck.
988 \item[@{text iterations}] sets how many sets of assignments are
989 generated for each particular size.
991 \item[@{text no_assms}] specifies whether assumptions in
992 structured proofs should be ignored.
994 \item[@{text timeout}] sets the time limit in seconds.
996 \item[@{text default_type}] sets the type(s) generally used to
997 instantiate type variables.
999 \item[@{text report}] if set quickcheck reports how many tests
1000 fulfilled the preconditions.
1002 \item[@{text quiet}] if not set quickcheck informs about the
1003 current size for assignment values.
1005 \item[@{text expect}] can be used to check if the user's
1006 expectation was met (@{text no_expectation}, @{text
1007 no_counterexample}, or @{text counterexample}).
1011 These option can be given within square brackets.
1013 \item @{command (HOL) "quickcheck_params"} changes quickcheck
1014 configuration options persitently.
1020 section {* Unstructured case analysis and induction \label{sec:hol-induct-tac} *}
1023 The following tools of Isabelle/HOL support cases analysis and
1024 induction in unstructured tactic scripts; see also
1025 \secref{sec:cases-induct} for proper Isar versions of similar ideas.
1027 \begin{matharray}{rcl}
1028 @{method_def (HOL) case_tac}@{text "\<^sup>*"} & : & @{text method} \\
1029 @{method_def (HOL) induct_tac}@{text "\<^sup>*"} & : & @{text method} \\
1030 @{method_def (HOL) ind_cases}@{text "\<^sup>*"} & : & @{text method} \\
1031 @{command_def (HOL) "inductive_cases"}@{text "\<^sup>*"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
1035 'case_tac' goalspec? term rule?
1037 'induct_tac' goalspec? (insts * 'and') rule?
1039 'ind_cases' (prop +) ('for' (name +)) ?
1041 'inductive_cases' (thmdecl? (prop +) + 'and')
1044 rule: ('rule' ':' thmref)
1050 \item @{method (HOL) case_tac} and @{method (HOL) induct_tac} admit
1051 to reason about inductive types. Rules are selected according to
1052 the declarations by the @{attribute cases} and @{attribute induct}
1053 attributes, cf.\ \secref{sec:cases-induct}. The @{command (HOL)
1054 datatype} package already takes care of this.
1056 These unstructured tactics feature both goal addressing and dynamic
1057 instantiation. Note that named rule cases are \emph{not} provided
1058 as would be by the proper @{method cases} and @{method induct} proof
1059 methods (see \secref{sec:cases-induct}). Unlike the @{method
1060 induct} method, @{method induct_tac} does not handle structured rule
1061 statements, only the compact object-logic conclusion of the subgoal
1064 \item @{method (HOL) ind_cases} and @{command (HOL)
1065 "inductive_cases"} provide an interface to the internal @{ML_text
1066 mk_cases} operation. Rules are simplified in an unrestricted
1069 While @{method (HOL) ind_cases} is a proof method to apply the
1070 result immediately as elimination rules, @{command (HOL)
1071 "inductive_cases"} provides case split theorems at the theory level
1072 for later use. The @{keyword "for"} argument of the @{method (HOL)
1073 ind_cases} method allows to specify a list of variables that should
1074 be generalized before applying the resulting rule.
1080 section {* Executable code *}
1083 Isabelle/Pure provides two generic frameworks to support code
1084 generation from executable specifications. Isabelle/HOL
1085 instantiates these mechanisms in a way that is amenable to end-user
1088 \medskip One framework generates code from functional programs
1089 (including overloading using type classes) to SML \cite{SML}, OCaml
1090 \cite{OCaml}, Haskell \cite{haskell-revised-report} and Scala
1091 \cite{scala-overview-tech-report}.
1092 Conceptually, code generation is split up in three steps:
1093 \emph{selection} of code theorems, \emph{translation} into an
1094 abstract executable view and \emph{serialization} to a specific
1095 \emph{target language}. Inductive specifications can be executed
1096 using the predicate compiler which operates within HOL.
1097 See \cite{isabelle-codegen} for an introduction.
1099 \begin{matharray}{rcl}
1100 @{command_def (HOL) "export_code"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
1101 @{attribute_def (HOL) code} & : & @{text attribute} \\
1102 @{command_def (HOL) "code_abort"} & : & @{text "theory \<rightarrow> theory"} \\
1103 @{command_def (HOL) "code_datatype"} & : & @{text "theory \<rightarrow> theory"} \\
1104 @{command_def (HOL) "print_codesetup"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
1105 @{attribute_def (HOL) code_inline} & : & @{text attribute} \\
1106 @{attribute_def (HOL) code_post} & : & @{text attribute} \\
1107 @{command_def (HOL) "print_codeproc"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
1108 @{command_def (HOL) "code_thms"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
1109 @{command_def (HOL) "code_deps"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
1110 @{command_def (HOL) "code_const"} & : & @{text "theory \<rightarrow> theory"} \\
1111 @{command_def (HOL) "code_type"} & : & @{text "theory \<rightarrow> theory"} \\
1112 @{command_def (HOL) "code_class"} & : & @{text "theory \<rightarrow> theory"} \\
1113 @{command_def (HOL) "code_instance"} & : & @{text "theory \<rightarrow> theory"} \\
1114 @{command_def (HOL) "code_reserved"} & : & @{text "theory \<rightarrow> theory"} \\
1115 @{command_def (HOL) "code_monad"} & : & @{text "theory \<rightarrow> theory"} \\
1116 @{command_def (HOL) "code_include"} & : & @{text "theory \<rightarrow> theory"} \\
1117 @{command_def (HOL) "code_modulename"} & : & @{text "theory \<rightarrow> theory"} \\
1118 @{command_def (HOL) "code_reflect"} & : & @{text "theory \<rightarrow> theory"}
1122 'export_code' ( constexpr + ) \\
1123 ( ( 'in' target ( 'module_name' string ) ? \\
1124 ( 'file' ( string | '-' ) ) ? ( '(' args ')' ) ?) + ) ?
1130 constexpr: ( const | 'name._' | '_' )
1133 typeconstructor: nameref
1139 target: 'SML' | 'OCaml' | 'Haskell' | 'Scala'
1142 'code' ( 'del' | 'abstype' | 'abstract' ) ?
1145 'code_abort' ( const + )
1148 'code_datatype' ( const + )
1151 'code_inline' ( 'del' ) ?
1154 'code_post' ( 'del' ) ?
1157 'code_thms' ( constexpr + ) ?
1160 'code_deps' ( constexpr + ) ?
1163 'code_const' (const + 'and') \\
1164 ( ( '(' target ( syntax ? + 'and' ) ')' ) + )
1167 'code_type' (typeconstructor + 'and') \\
1168 ( ( '(' target ( syntax ? + 'and' ) ')' ) + )
1171 'code_class' (class + 'and') \\
1172 ( ( '(' target \\ ( string ? + 'and' ) ')' ) + )
1175 'code_instance' (( typeconstructor '::' class ) + 'and') \\
1176 ( ( '(' target ( '-' ? + 'and' ) ')' ) + )
1179 'code_reserved' target ( string + )
1182 'code_monad' const const target
1185 'code_include' target ( string ( string | '-') )
1188 'code_modulename' target ( ( string string ) + )
1191 'code_reflect' string \\
1192 ( 'datatypes' ( string '=' ( '_' | ( string + '|' ) + 'and' ) ) ) ? \\
1193 ( 'functions' ( string + ) ) ? ( 'file' string ) ?
1196 syntax: string | ( 'infix' | 'infixl' | 'infixr' ) nat string
1203 \item @{command (HOL) "export_code"} generates code for a given list
1204 of constants in the specified target language(s). If no
1205 serialization instruction is given, only abstract code is generated
1208 Constants may be specified by giving them literally, referring to
1209 all executable contants within a certain theory by giving @{text
1210 "name.*"}, or referring to \emph{all} executable constants currently
1211 available by giving @{text "*"}.
1213 By default, for each involved theory one corresponding name space
1214 module is generated. Alternativly, a module name may be specified
1215 after the @{keyword "module_name"} keyword; then \emph{all} code is
1216 placed in this module.
1218 For \emph{SML}, \emph{OCaml} and \emph{Scala} the file specification
1219 refers to a single file; for \emph{Haskell}, it refers to a whole
1220 directory, where code is generated in multiple files reflecting the
1221 module hierarchy. Omitting the file specification denotes standard
1224 Serializers take an optional list of arguments in parentheses. For
1225 \emph{SML} and \emph{OCaml}, ``@{text no_signatures}`` omits
1226 explicit module signatures.
1228 For \emph{Haskell} a module name prefix may be given using the
1229 ``@{text "root:"}'' argument; ``@{text string_classes}'' adds a
1230 ``@{verbatim "deriving (Read, Show)"}'' clause to each appropriate
1231 datatype declaration.
1233 \item @{attribute (HOL) code} explicitly selects (or with option
1234 ``@{text "del"}'' deselects) a code equation for code generation.
1235 Usually packages introducing code equations provide a reasonable
1236 default setup for selection. Variants @{text "code abstype"} and
1237 @{text "code abstract"} declare abstract datatype certificates or
1238 code equations on abstract datatype representations respectively.
1240 \item @{command (HOL) "code_abort"} declares constants which are not
1241 required to have a definition by means of code equations; if needed
1242 these are implemented by program abort instead.
1244 \item @{command (HOL) "code_datatype"} specifies a constructor set
1247 \item @{command (HOL) "print_codesetup"} gives an overview on
1248 selected code equations and code generator datatypes.
1250 \item @{attribute (HOL) code_inline} declares (or with option
1251 ``@{text "del"}'' removes) inlining theorems which are applied as
1252 rewrite rules to any code equation during preprocessing.
1254 \item @{attribute (HOL) code_post} declares (or with option ``@{text
1255 "del"}'' removes) theorems which are applied as rewrite rules to any
1256 result of an evaluation.
1258 \item @{command (HOL) "print_codeproc"} prints the setup of the code
1259 generator preprocessor.
1261 \item @{command (HOL) "code_thms"} prints a list of theorems
1262 representing the corresponding program containing all given
1263 constants after preprocessing.
1265 \item @{command (HOL) "code_deps"} visualizes dependencies of
1266 theorems representing the corresponding program containing all given
1267 constants after preprocessing.
1269 \item @{command (HOL) "code_const"} associates a list of constants
1270 with target-specific serializations; omitting a serialization
1271 deletes an existing serialization.
1273 \item @{command (HOL) "code_type"} associates a list of type
1274 constructors with target-specific serializations; omitting a
1275 serialization deletes an existing serialization.
1277 \item @{command (HOL) "code_class"} associates a list of classes
1278 with target-specific class names; omitting a serialization deletes
1279 an existing serialization. This applies only to \emph{Haskell}.
1281 \item @{command (HOL) "code_instance"} declares a list of type
1282 constructor / class instance relations as ``already present'' for a
1283 given target. Omitting a ``@{text "-"}'' deletes an existing
1284 ``already present'' declaration. This applies only to
1287 \item @{command (HOL) "code_reserved"} declares a list of names as
1288 reserved for a given target, preventing it to be shadowed by any
1291 \item @{command (HOL) "code_monad"} provides an auxiliary mechanism
1292 to generate monadic code for Haskell.
1294 \item @{command (HOL) "code_include"} adds arbitrary named content
1295 (``include'') to generated code. A ``@{text "-"}'' as last argument
1296 will remove an already added ``include''.
1298 \item @{command (HOL) "code_modulename"} declares aliasings from one
1299 module name onto another.
1301 \item @{command (HOL) "code_reflect"} without a ``@{text "file"}''
1302 argument compiles code into the system runtime environment and
1303 modifies the code generator setup that future invocations of system
1304 runtime code generation referring to one of the ``@{text
1305 "datatypes"}'' or ``@{text "functions"}'' entities use these precompiled
1306 entities. With a ``@{text "file"}'' argument, the corresponding code
1307 is generated into that specified file without modifying the code
1312 The other framework generates code from both functional and
1313 relational programs to SML. See \cite{isabelle-HOL} for further
1314 information (this actually covers the new-style theory format as
1317 \begin{matharray}{rcl}
1318 @{command_def (HOL) "code_module"} & : & @{text "theory \<rightarrow> theory"} \\
1319 @{command_def (HOL) "code_library"} & : & @{text "theory \<rightarrow> theory"} \\
1320 @{command_def (HOL) "consts_code"} & : & @{text "theory \<rightarrow> theory"} \\
1321 @{command_def (HOL) "types_code"} & : & @{text "theory \<rightarrow> theory"} \\
1322 @{attribute_def (HOL) code} & : & @{text attribute} \\
1326 ( 'code_module' | 'code_library' ) modespec ? name ? \\
1327 ( 'file' name ) ? ( 'imports' ( name + ) ) ? \\
1328 'contains' ( ( name '=' term ) + | term + )
1331 modespec: '(' ( name * ) ')'
1334 'consts_code' (codespec +)
1337 codespec: const template attachment ?
1340 'types_code' (tycodespec +)
1343 tycodespec: name template attachment ?
1349 template: '(' string ')'
1352 attachment: 'attach' modespec ? verblbrace text verbrbrace
1362 section {* Definition by specification \label{sec:hol-specification} *}
1365 \begin{matharray}{rcl}
1366 @{command_def (HOL) "specification"} & : & @{text "theory \<rightarrow> proof(prove)"} \\
1367 @{command_def (HOL) "ax_specification"} & : & @{text "theory \<rightarrow> proof(prove)"} \\
1371 ('specification' | 'ax_specification') '(' (decl +) ')' \\ (thmdecl? prop +)
1373 decl: ((name ':')? term '(' 'overloaded' ')'?)
1378 \item @{command (HOL) "specification"}~@{text "decls \<phi>"} sets up a
1379 goal stating the existence of terms with the properties specified to
1380 hold for the constants given in @{text decls}. After finishing the
1381 proof, the theory will be augmented with definitions for the given
1382 constants, as well as with theorems stating the properties for these
1385 \item @{command (HOL) "ax_specification"}~@{text "decls \<phi>"} sets up
1386 a goal stating the existence of terms with the properties specified
1387 to hold for the constants given in @{text decls}. After finishing
1388 the proof, the theory will be augmented with axioms expressing the
1389 properties given in the first place.
1391 \item @{text decl} declares a constant to be defined by the
1392 specification given. The definition for the constant @{text c} is
1393 bound to the name @{text c_def} unless a theorem name is given in
1394 the declaration. Overloaded constants should be declared as such.
1398 Whether to use @{command (HOL) "specification"} or @{command (HOL)
1399 "ax_specification"} is to some extent a matter of style. @{command
1400 (HOL) "specification"} introduces no new axioms, and so by
1401 construction cannot introduce inconsistencies, whereas @{command
1402 (HOL) "ax_specification"} does introduce axioms, but only after the
1403 user has explicitly proven it to be safe. A practical issue must be
1404 considered, though: After introducing two constants with the same
1405 properties using @{command (HOL) "specification"}, one can prove
1406 that the two constants are, in fact, equal. If this might be a
1407 problem, one should use @{command (HOL) "ax_specification"}.