removed FIXME: "try_methods" doesn't take a clasimpmod, because it needs to generate Isar proof text in case of success and hence needs total control over its arguments
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 @@{command (HOL) typedef} altname? abstype '=' repset
18 altname: '(' (@{syntax name} | @'open' | @'open' @{syntax name}) ')'
20 abstype: @{syntax typespec_sorts} @{syntax mixfix}?
22 repset: @{syntax term} (@'morphisms' @{syntax name} @{syntax name})?
27 \item @{command (HOL) "typedef"}~@{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>n) t = A"}
28 axiomatizes a Gordon/HOL-style type definition in the background
29 theory of the current context, depending on a non-emptiness result
30 of the set @{text A} (which needs to be proven interactively).
32 The raw type may not depend on parameters or assumptions of the
33 context --- this is logically impossible in Isabelle/HOL --- but the
34 non-emptiness property can be local, potentially resulting in
35 multiple interpretations in target contexts. Thus the established
36 bijection between the representing set @{text A} and the new type
37 @{text t} may semantically depend on local assumptions.
39 By default, @{command (HOL) "typedef"} defines both a type @{text t}
40 and a set (term constant) of the same name, unless an alternative
41 base name is given in parentheses, or the ``@{text "(open)"}''
42 declaration is used to suppress a separate constant definition
43 altogether. The injection from type to set is called @{text Rep_t},
44 its inverse @{text Abs_t} --- this may be changed via an explicit
45 @{keyword (HOL) "morphisms"} declaration.
47 Theorems @{text Rep_t}, @{text Rep_t_inverse}, and @{text
48 Abs_t_inverse} provide the most basic characterization as a
49 corresponding injection/surjection pair (in both directions). Rules
50 @{text Rep_t_inject} and @{text Abs_t_inject} provide a slightly
51 more convenient view on the injectivity part, suitable for automated
52 proof tools (e.g.\ in @{attribute simp} or @{attribute iff}
53 declarations). Rules @{text Rep_t_cases}/@{text Rep_t_induct}, and
54 @{text Abs_t_cases}/@{text Abs_t_induct} provide alternative views
55 on surjectivity; these are already declared as set or type rules for
56 the generic @{method cases} and @{method induct} methods.
58 An alternative name for the set definition (and other derived
59 entities) may be specified in parentheses; the default is to use
60 @{text t} as indicated before.
66 section {* Adhoc tuples *}
69 \begin{matharray}{rcl}
70 @{attribute_def (HOL) split_format}@{text "\<^sup>*"} & : & @{text attribute} \\
74 @@{attribute (HOL) split_format} ('(' 'complete' ')')?
79 \item @{attribute (HOL) split_format}\ @{text "(complete)"} causes
80 arguments in function applications to be represented canonically
81 according to their tuple type structure.
83 Note that this operation tends to invent funny names for new local
84 parameters introduced.
90 section {* Records \label{sec:hol-record} *}
93 In principle, records merely generalize the concept of tuples, where
94 components may be addressed by labels instead of just position. The
95 logical infrastructure of records in Isabelle/HOL is slightly more
96 advanced, though, supporting truly extensible record schemes. This
97 admits operations that are polymorphic with respect to record
98 extension, yielding ``object-oriented'' effects like (single)
99 inheritance. See also \cite{NaraschewskiW-TPHOLs98} for more
100 details on object-oriented verification and record subtyping in HOL.
104 subsection {* Basic concepts *}
107 Isabelle/HOL supports both \emph{fixed} and \emph{schematic} records
108 at the level of terms and types. The notation is as follows:
111 \begin{tabular}{l|l|l}
112 & record terms & record types \\ \hline
113 fixed & @{text "\<lparr>x = a, y = b\<rparr>"} & @{text "\<lparr>x :: A, y :: B\<rparr>"} \\
114 schematic & @{text "\<lparr>x = a, y = b, \<dots> = m\<rparr>"} &
115 @{text "\<lparr>x :: A, y :: B, \<dots> :: M\<rparr>"} \\
119 \noindent The ASCII representation of @{text "\<lparr>x = a\<rparr>"} is @{text
122 A fixed record @{text "\<lparr>x = a, y = b\<rparr>"} has field @{text x} of value
123 @{text a} and field @{text y} of value @{text b}. The corresponding
124 type is @{text "\<lparr>x :: A, y :: B\<rparr>"}, assuming that @{text "a :: A"}
125 and @{text "b :: B"}.
127 A record scheme like @{text "\<lparr>x = a, y = b, \<dots> = m\<rparr>"} contains fields
128 @{text x} and @{text y} as before, but also possibly further fields
129 as indicated by the ``@{text "\<dots>"}'' notation (which is actually part
130 of the syntax). The improper field ``@{text "\<dots>"}'' of a record
131 scheme is called the \emph{more part}. Logically it is just a free
132 variable, which is occasionally referred to as ``row variable'' in
133 the literature. The more part of a record scheme may be
134 instantiated by zero or more further components. For example, the
135 previous scheme may get instantiated to @{text "\<lparr>x = a, y = b, z =
136 c, \<dots> = m'\<rparr>"}, where @{text m'} refers to a different more part.
137 Fixed records are special instances of record schemes, where
138 ``@{text "\<dots>"}'' is properly terminated by the @{text "() :: unit"}
139 element. In fact, @{text "\<lparr>x = a, y = b\<rparr>"} is just an abbreviation
140 for @{text "\<lparr>x = a, y = b, \<dots> = ()\<rparr>"}.
142 \medskip Two key observations make extensible records in a simply
143 typed language like HOL work out:
147 \item the more part is internalized, as a free term or type
150 \item field names are externalized, they cannot be accessed within
151 the logic as first-class values.
155 \medskip In Isabelle/HOL record types have to be defined explicitly,
156 fixing their field names and types, and their (optional) parent
157 record. Afterwards, records may be formed using above syntax, while
158 obeying the canonical order of fields as given by their declaration.
159 The record package provides several standard operations like
160 selectors and updates. The common setup for various generic proof
161 tools enable succinct reasoning patterns. See also the Isabelle/HOL
162 tutorial \cite{isabelle-hol-book} for further instructions on using
167 subsection {* Record specifications *}
170 \begin{matharray}{rcl}
171 @{command_def (HOL) "record"} & : & @{text "theory \<rightarrow> theory"} \\
175 @@{command (HOL) record} @{syntax typespec_sorts} '=' \\
176 (@{syntax type} '+')? (@{syntax constdecl} +)
181 \item @{command (HOL) "record"}~@{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t = \<tau> + c\<^sub>1 :: \<sigma>\<^sub>1
182 \<dots> c\<^sub>n :: \<sigma>\<^sub>n"} defines extensible record type @{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t"},
183 derived from the optional parent record @{text "\<tau>"} by adding new
184 field components @{text "c\<^sub>i :: \<sigma>\<^sub>i"} etc.
186 The type variables of @{text "\<tau>"} and @{text "\<sigma>\<^sub>i"} need to be
187 covered by the (distinct) parameters @{text "\<alpha>\<^sub>1, \<dots>,
188 \<alpha>\<^sub>m"}. Type constructor @{text t} has to be new, while @{text
189 \<tau>} needs to specify an instance of an existing record type. At
190 least one new field @{text "c\<^sub>i"} has to be specified.
191 Basically, field names need to belong to a unique record. This is
192 not a real restriction in practice, since fields are qualified by
193 the record name internally.
195 The parent record specification @{text \<tau>} is optional; if omitted
196 @{text t} becomes a root record. The hierarchy of all records
197 declared within a theory context forms a forest structure, i.e.\ a
198 set of trees starting with a root record each. There is no way to
199 merge multiple parent records!
201 For convenience, @{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t"} is made a
202 type abbreviation for the fixed record type @{text "\<lparr>c\<^sub>1 ::
203 \<sigma>\<^sub>1, \<dots>, c\<^sub>n :: \<sigma>\<^sub>n\<rparr>"}, likewise is @{text
204 "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m, \<zeta>) t_scheme"} made an abbreviation for
205 @{text "\<lparr>c\<^sub>1 :: \<sigma>\<^sub>1, \<dots>, c\<^sub>n :: \<sigma>\<^sub>n, \<dots> ::
212 subsection {* Record operations *}
215 Any record definition of the form presented above produces certain
216 standard operations. Selectors and updates are provided for any
217 field, including the improper one ``@{text more}''. There are also
218 cumulative record constructor functions. To simplify the
219 presentation below, we assume for now that @{text "(\<alpha>\<^sub>1, \<dots>,
220 \<alpha>\<^sub>m) t"} is a root record with fields @{text "c\<^sub>1 ::
221 \<sigma>\<^sub>1, \<dots>, c\<^sub>n :: \<sigma>\<^sub>n"}.
223 \medskip \textbf{Selectors} and \textbf{updates} are available for
224 any field (including ``@{text more}''):
226 \begin{matharray}{lll}
227 @{text "c\<^sub>i"} & @{text "::"} & @{text "\<lparr>\<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr> \<Rightarrow> \<sigma>\<^sub>i"} \\
228 @{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>"} \\
231 There is special syntax for application of updates: @{text "r\<lparr>x :=
232 a\<rparr>"} abbreviates term @{text "x_update a r"}. Further notation for
233 repeated updates is also available: @{text "r\<lparr>x := a\<rparr>\<lparr>y := b\<rparr>\<lparr>z :=
234 c\<rparr>"} may be written @{text "r\<lparr>x := a, y := b, z := c\<rparr>"}. Note that
235 because of postfix notation the order of fields shown here is
236 reverse than in the actual term. Since repeated updates are just
237 function applications, fields may be freely permuted in @{text "\<lparr>x
238 := a, y := b, z := c\<rparr>"}, as far as logical equality is concerned.
239 Thus commutativity of independent updates can be proven within the
240 logic for any two fields, but not as a general theorem.
242 \medskip The \textbf{make} operation provides a cumulative record
243 constructor function:
245 \begin{matharray}{lll}
246 @{text "t.make"} & @{text "::"} & @{text "\<sigma>\<^sub>1 \<Rightarrow> \<dots> \<sigma>\<^sub>n \<Rightarrow> \<lparr>\<^vec>c :: \<^vec>\<sigma>\<rparr>"} \\
249 \medskip We now reconsider the case of non-root records, which are
250 derived of some parent. In general, the latter may depend on
251 another parent as well, resulting in a list of \emph{ancestor
252 records}. Appending the lists of fields of all ancestors results in
253 a certain field prefix. The record package automatically takes care
254 of this by lifting operations over this context of ancestor fields.
255 Assuming that @{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t"} has ancestor
256 fields @{text "b\<^sub>1 :: \<rho>\<^sub>1, \<dots>, b\<^sub>k :: \<rho>\<^sub>k"},
257 the above record operations will get the following types:
261 @{text "c\<^sub>i"} & @{text "::"} & @{text "\<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr> \<Rightarrow> \<sigma>\<^sub>i"} \\
262 @{text "c\<^sub>i_update"} & @{text "::"} & @{text "\<sigma>\<^sub>i \<Rightarrow>
263 \<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr> \<Rightarrow>
264 \<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr>"} \\
265 @{text "t.make"} & @{text "::"} & @{text "\<rho>\<^sub>1 \<Rightarrow> \<dots> \<rho>\<^sub>k \<Rightarrow> \<sigma>\<^sub>1 \<Rightarrow> \<dots> \<sigma>\<^sub>n \<Rightarrow>
266 \<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>\<rparr>"} \\
270 \noindent Some further operations address the extension aspect of a
271 derived record scheme specifically: @{text "t.fields"} produces a
272 record fragment consisting of exactly the new fields introduced here
273 (the result may serve as a more part elsewhere); @{text "t.extend"}
274 takes a fixed record and adds a given more part; @{text
275 "t.truncate"} restricts a record scheme to a fixed record.
279 @{text "t.fields"} & @{text "::"} & @{text "\<sigma>\<^sub>1 \<Rightarrow> \<dots> \<sigma>\<^sub>n \<Rightarrow> \<lparr>\<^vec>c :: \<^vec>\<sigma>\<rparr>"} \\
280 @{text "t.extend"} & @{text "::"} & @{text "\<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>\<rparr> \<Rightarrow>
281 \<zeta> \<Rightarrow> \<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr>"} \\
282 @{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>"} \\
286 \noindent Note that @{text "t.make"} and @{text "t.fields"} coincide
291 subsection {* Derived rules and proof tools *}
294 The record package proves several results internally, declaring
295 these facts to appropriate proof tools. This enables users to
296 reason about record structures quite conveniently. Assume that
297 @{text t} is a record type as specified above.
301 \item Standard conversions for selectors or updates applied to
302 record constructor terms are made part of the default Simplifier
303 context; thus proofs by reduction of basic operations merely require
304 the @{method simp} method without further arguments. These rules
305 are available as @{text "t.simps"}, too.
307 \item Selectors applied to updated records are automatically reduced
308 by an internal simplification procedure, which is also part of the
309 standard Simplifier setup.
311 \item Inject equations of a form analogous to @{prop "(x, y) = (x',
312 y') \<equiv> x = x' \<and> y = y'"} are declared to the Simplifier and Classical
313 Reasoner as @{attribute iff} rules. These rules are available as
316 \item The introduction rule for record equality analogous to @{text
317 "x r = x r' \<Longrightarrow> y r = y r' \<dots> \<Longrightarrow> r = r'"} is declared to the Simplifier,
318 and as the basic rule context as ``@{attribute intro}@{text "?"}''.
319 The rule is called @{text "t.equality"}.
321 \item Representations of arbitrary record expressions as canonical
322 constructor terms are provided both in @{method cases} and @{method
323 induct} format (cf.\ the generic proof methods of the same name,
324 \secref{sec:cases-induct}). Several variations are available, for
325 fixed records, record schemes, more parts etc.
327 The generic proof methods are sufficiently smart to pick the most
328 sensible rule according to the type of the indicated record
329 expression: users just need to apply something like ``@{text "(cases
330 r)"}'' to a certain proof problem.
332 \item The derived record operations @{text "t.make"}, @{text
333 "t.fields"}, @{text "t.extend"}, @{text "t.truncate"} are \emph{not}
334 treated automatically, but usually need to be expanded by hand,
335 using the collective fact @{text "t.defs"}.
341 section {* Datatypes \label{sec:hol-datatype} *}
344 \begin{matharray}{rcl}
345 @{command_def (HOL) "datatype"} & : & @{text "theory \<rightarrow> theory"} \\
346 @{command_def (HOL) "rep_datatype"} & : & @{text "theory \<rightarrow> proof(prove)"} \\
350 @@{command (HOL) datatype} (spec + @'and')
352 @@{command (HOL) rep_datatype} ('(' (@{syntax name} +) ')')? (@{syntax term} +)
355 spec: @{syntax parname}? @{syntax typespec} @{syntax mixfix}? '=' (cons + '|')
357 cons: @{syntax name} (@{syntax type} * ) @{syntax mixfix}?
362 \item @{command (HOL) "datatype"} defines inductive datatypes in
365 \item @{command (HOL) "rep_datatype"} represents existing types as
366 inductive ones, generating the standard infrastructure of derived
367 concepts (primitive recursion etc.).
371 The induction and exhaustion theorems generated provide case names
372 according to the constructors involved, while parameters are named
373 after the types (see also \secref{sec:cases-induct}).
375 See \cite{isabelle-HOL} for more details on datatypes, but beware of
376 the old-style theory syntax being used there! Apart from proper
377 proof methods for case-analysis and induction, there are also
378 emulations of ML tactics @{method (HOL) case_tac} and @{method (HOL)
379 induct_tac} available, see \secref{sec:hol-induct-tac}; these admit
380 to refer directly to the internal structure of subgoals (including
381 internally bound parameters).
385 section {* Functorial structure of types *}
388 \begin{matharray}{rcl}
389 @{command_def (HOL) "enriched_type"} & : & @{text "local_theory \<rightarrow> proof(prove)"}
393 @@{command (HOL) enriched_type} (@{syntax name} ':')? @{syntax term}
399 \item @{command (HOL) "enriched_type"}~@{text "prefix: m"} allows to
400 prove and register properties about the functorial structure of type
401 constructors. These properties then can be used by other packages
402 to deal with those type constructors in certain type constructions.
403 Characteristic theorems are noted in the current local theory. By
404 default, they are prefixed with the base name of the type
405 constructor, an explicit prefix can be given alternatively.
407 The given term @{text "m"} is considered as \emph{mapper} for the
408 corresponding type constructor and must conform to the following
411 \begin{matharray}{lll}
412 @{text "m"} & @{text "::"} &
413 @{text "\<sigma>\<^isub>1 \<Rightarrow> \<dots> \<sigma>\<^isub>k \<Rightarrow> (\<^vec>\<alpha>\<^isub>n) t \<Rightarrow> (\<^vec>\<beta>\<^isub>n) t"} \\
416 \noindent where @{text t} is the type constructor, @{text
417 "\<^vec>\<alpha>\<^isub>n"} and @{text "\<^vec>\<beta>\<^isub>n"} are distinct
418 type variables free in the local theory and @{text "\<sigma>\<^isub>1"},
419 \ldots, @{text "\<sigma>\<^isub>k"} is a subsequence of @{text "\<alpha>\<^isub>1 \<Rightarrow>
420 \<beta>\<^isub>1"}, @{text "\<beta>\<^isub>1 \<Rightarrow> \<alpha>\<^isub>1"}, \ldots,
421 @{text "\<alpha>\<^isub>n \<Rightarrow> \<beta>\<^isub>n"}, @{text "\<beta>\<^isub>n \<Rightarrow>
428 section {* Recursive functions \label{sec:recursion} *}
431 \begin{matharray}{rcl}
432 @{command_def (HOL) "primrec"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
433 @{command_def (HOL) "fun"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
434 @{command_def (HOL) "function"} & : & @{text "local_theory \<rightarrow> proof(prove)"} \\
435 @{command_def (HOL) "termination"} & : & @{text "local_theory \<rightarrow> proof(prove)"} \\
439 @@{command (HOL) primrec} @{syntax target}? @{syntax \"fixes\"} @'where' equations
441 (@@{command (HOL) fun} | @@{command (HOL) function}) @{syntax target}? functionopts?
442 @{syntax \"fixes\"} \\ @'where' equations
445 equations: (@{syntax thmdecl}? @{syntax prop} + '|')
447 functionopts: '(' (('sequential' | 'domintros') + ',') ')'
449 @@{command (HOL) termination} @{syntax 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} \\
533 @@{method (HOL) relation} @{syntax term}
535 @@{method (HOL) lexicographic_order} (@{syntax clasimpmod} * )
537 @@{method (HOL) size_change} ( orders (@{syntax 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 @@{command (HOL) partial_function} @{syntax target}?
592 '(' @{syntax nameref} ')' @{syntax \"fixes\"} \\
593 @'where' @{syntax thmdecl}? @{syntax prop}
598 \item @{command (HOL) "partial_function"}~@{text "(mode)"} defines
599 recursive functions based on fixpoints in complete partial
600 orders. No termination proof is required from the user or
601 constructed internally. Instead, the possibility of non-termination
602 is modelled explicitly in the result type, which contains an
603 explicit bottom element.
605 Pattern matching and mutual recursion are currently not supported.
606 Thus, the specification consists of a single function described by a
607 single recursive equation.
609 There are no fixed syntactic restrictions on the body of the
610 function, but the induced functional must be provably monotonic
611 wrt.\ the underlying order. The monotonicitity proof is performed
612 internally, and the definition is rejected when it fails. The proof
613 can be influenced by declaring hints using the
614 @{attribute (HOL) partial_function_mono} attribute.
616 The mandatory @{text mode} argument specifies the mode of operation
617 of the command, which directly corresponds to a complete partial
618 order on the result type. By default, the following modes are
622 \item @{text option} defines functions that map into the @{type
623 option} type. Here, the value @{term None} is used to model a
624 non-terminating computation. Monotonicity requires that if @{term
625 None} is returned by a recursive call, then the overall result
626 must also be @{term None}. This is best achieved through the use of
627 the monadic operator @{const "Option.bind"}.
629 \item @{text tailrec} defines functions with an arbitrary result
630 type and uses the slightly degenerated partial order where @{term
631 "undefined"} is the bottom element. Now, monotonicity requires that
632 if @{term undefined} is returned by a recursive call, then the
633 overall result must also be @{term undefined}. In practice, this is
634 only satisfied when each recursive call is a tail call, whose result
635 is directly returned. Thus, this mode of operation allows the
636 definition of arbitrary tail-recursive functions.
639 Experienced users may define new modes by instantiating the locale
640 @{const "partial_function_definitions"} appropriately.
642 \item @{attribute (HOL) partial_function_mono} declares rules for
643 use in the internal monononicity proofs of partial function
650 subsection {* Old-style recursive function definitions (TFL) *}
653 The old TFL commands @{command (HOL) "recdef"} and @{command (HOL)
654 "recdef_tc"} for defining recursive are mostly obsolete; @{command
655 (HOL) "function"} or @{command (HOL) "fun"} should be used instead.
657 \begin{matharray}{rcl}
658 @{command_def (HOL) "recdef"} & : & @{text "theory \<rightarrow> theory)"} \\
659 @{command_def (HOL) "recdef_tc"}@{text "\<^sup>*"} & : & @{text "theory \<rightarrow> proof(prove)"} \\
663 @@{command (HOL) recdef} ('(' @'permissive' ')')? \\
664 @{syntax name} @{syntax term} (@{syntax prop} +) hints?
666 recdeftc @{syntax thmdecl}? tc
668 hints: '(' @'hints' ( recdefmod * ) ')'
670 recdefmod: (('recdef_simp' | 'recdef_cong' | 'recdef_wf')
671 (() | 'add' | 'del') ':' @{syntax thmrefs}) | @{syntax clasimpmod}
673 tc: @{syntax nameref} ('(' @{syntax nat} ')')?
678 \item @{command (HOL) "recdef"} defines general well-founded
679 recursive functions (using the TFL package), see also
680 \cite{isabelle-HOL}. The ``@{text "(permissive)"}'' option tells
681 TFL to recover from failed proof attempts, returning unfinished
682 results. The @{text recdef_simp}, @{text recdef_cong}, and @{text
683 recdef_wf} hints refer to auxiliary rules to be used in the internal
684 automated proof process of TFL. Additional @{syntax clasimpmod}
685 declarations (cf.\ \secref{sec:clasimp}) may be given to tune the
686 context of the Simplifier (cf.\ \secref{sec:simplifier}) and
687 Classical reasoner (cf.\ \secref{sec:classical}).
689 \item @{command (HOL) "recdef_tc"}~@{text "c (i)"} recommences the
690 proof for leftover termination condition number @{text i} (default
691 1) as generated by a @{command (HOL) "recdef"} definition of
694 Note that in most cases, @{command (HOL) "recdef"} is able to finish
695 its internal proofs without manual intervention.
699 \medskip Hints for @{command (HOL) "recdef"} may be also declared
700 globally, using the following attributes.
702 \begin{matharray}{rcl}
703 @{attribute_def (HOL) recdef_simp} & : & @{text attribute} \\
704 @{attribute_def (HOL) recdef_cong} & : & @{text attribute} \\
705 @{attribute_def (HOL) recdef_wf} & : & @{text attribute} \\
709 (@@{attribute (HOL) recdef_simp} | @@{attribute (HOL) recdef_cong} |
710 @@{attribute (HOL) recdef_wf}) (() | 'add' | 'del')
715 section {* Inductive and coinductive definitions \label{sec:hol-inductive} *}
718 An \textbf{inductive definition} specifies the least predicate (or
719 set) @{text R} closed under given rules: applying a rule to elements
720 of @{text R} yields a result within @{text R}. For example, a
721 structural operational semantics is an inductive definition of an
724 Dually, a \textbf{coinductive definition} specifies the greatest
725 predicate~/ set @{text R} that is consistent with given rules: every
726 element of @{text R} can be seen as arising by applying a rule to
727 elements of @{text R}. An important example is using bisimulation
728 relations to formalise equivalence of processes and infinite data
731 \medskip The HOL package is related to the ZF one, which is
732 described in a separate paper,\footnote{It appeared in CADE
733 \cite{paulson-CADE}; a longer version is distributed with Isabelle.}
734 which you should refer to in case of difficulties. The package is
735 simpler than that of ZF thanks to implicit type-checking in HOL.
736 The types of the (co)inductive predicates (or sets) determine the
737 domain of the fixedpoint definition, and the package does not have
738 to use inference rules for type-checking.
740 \begin{matharray}{rcl}
741 @{command_def (HOL) "inductive"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
742 @{command_def (HOL) "inductive_set"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
743 @{command_def (HOL) "coinductive"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
744 @{command_def (HOL) "coinductive_set"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
745 @{attribute_def (HOL) mono} & : & @{text attribute} \\
749 (@@{command (HOL) inductive} | @@{command (HOL) inductive_set} |
750 @@{command (HOL) coinductive} | @@{command (HOL) coinductive_set})
751 @{syntax target}? @{syntax \"fixes\"} (@'for' @{syntax \"fixes\"})? \\
752 (@'where' clauses)? (@'monos' @{syntax thmrefs})?
754 clauses: (@{syntax thmdecl}? @{syntax prop} + '|')
756 @@{attribute (HOL) mono} (() | 'add' | 'del')
761 \item @{command (HOL) "inductive"} and @{command (HOL)
762 "coinductive"} define (co)inductive predicates from the
763 introduction rules given in the @{keyword "where"} part. The
764 optional @{keyword "for"} part contains a list of parameters of the
765 (co)inductive predicates that remain fixed throughout the
766 definition. The optional @{keyword "monos"} section contains
767 \emph{monotonicity theorems}, which are required for each operator
768 applied to a recursive set in the introduction rules. There
769 \emph{must} be a theorem of the form @{text "A \<le> B \<Longrightarrow> M A \<le> M B"},
770 for each premise @{text "M R\<^sub>i t"} in an introduction rule!
772 \item @{command (HOL) "inductive_set"} and @{command (HOL)
773 "coinductive_set"} are wrappers for to the previous commands,
774 allowing the definition of (co)inductive sets.
776 \item @{attribute (HOL) mono} declares monotonicity rules. These
777 rule are involved in the automated monotonicity proof of @{command
784 subsection {* Derived rules *}
787 Each (co)inductive definition @{text R} adds definitions to the
788 theory and also proves some theorems:
792 \item @{text R.intros} is the list of introduction rules as proven
793 theorems, for the recursive predicates (or sets). The rules are
794 also available individually, using the names given them in the
797 \item @{text R.cases} is the case analysis (or elimination) rule;
799 \item @{text R.induct} or @{text R.coinduct} is the (co)induction
804 When several predicates @{text "R\<^sub>1, \<dots>, R\<^sub>n"} are
805 defined simultaneously, the list of introduction rules is called
806 @{text "R\<^sub>1_\<dots>_R\<^sub>n.intros"}, the case analysis rules are
807 called @{text "R\<^sub>1.cases, \<dots>, R\<^sub>n.cases"}, and the list
808 of mutual induction rules is called @{text
809 "R\<^sub>1_\<dots>_R\<^sub>n.inducts"}.
813 subsection {* Monotonicity theorems *}
816 Each theory contains a default set of theorems that are used in
817 monotonicity proofs. New rules can be added to this set via the
818 @{attribute (HOL) mono} attribute. The HOL theory @{text Inductive}
819 shows how this is done. In general, the following monotonicity
820 theorems may be added:
824 \item Theorems of the form @{text "A \<le> B \<Longrightarrow> M A \<le> M B"}, for proving
825 monotonicity of inductive definitions whose introduction rules have
826 premises involving terms such as @{text "M R\<^sub>i t"}.
828 \item Monotonicity theorems for logical operators, which are of the
829 general form @{text "(\<dots> \<longrightarrow> \<dots>) \<Longrightarrow> \<dots> (\<dots> \<longrightarrow> \<dots>) \<Longrightarrow> \<dots> \<longrightarrow> \<dots>"}. For example, in
830 the case of the operator @{text "\<or>"}, the corresponding theorem is
832 \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"}}
835 \item De Morgan style equations for reasoning about the ``polarity''
838 @{prop "\<not> \<not> P \<longleftrightarrow> P"} \qquad\qquad
839 @{prop "\<not> (P \<and> Q) \<longleftrightarrow> \<not> P \<or> \<not> Q"}
842 \item Equations for reducing complex operators to more primitive
843 ones whose monotonicity can easily be proved, e.g.
845 @{prop "(P \<longrightarrow> Q) \<longleftrightarrow> \<not> P \<or> Q"} \qquad\qquad
846 @{prop "Ball A P \<equiv> \<forall>x. x \<in> A \<longrightarrow> P x"}
851 %FIXME: Example of an inductive definition
855 section {* Arithmetic proof support *}
858 \begin{matharray}{rcl}
859 @{method_def (HOL) arith} & : & @{text method} \\
860 @{attribute_def (HOL) arith} & : & @{text attribute} \\
861 @{attribute_def (HOL) arith_split} & : & @{text attribute} \\
864 The @{method (HOL) arith} method decides linear arithmetic problems
865 (on types @{text nat}, @{text int}, @{text real}). Any current
866 facts are inserted into the goal before running the procedure.
868 The @{attribute (HOL) arith} attribute declares facts that are
869 always supplied to the arithmetic provers implicitly.
871 The @{attribute (HOL) arith_split} attribute declares case split
872 rules to be expanded before @{method (HOL) arith} is invoked.
874 Note that a simpler (but faster) arithmetic prover is
875 already invoked by the Simplifier.
879 section {* Intuitionistic proof search *}
882 \begin{matharray}{rcl}
883 @{method_def (HOL) iprover} & : & @{text method} \\
887 @@{method (HOL) iprover} ( @{syntax rulemod} * )
890 The @{method (HOL) iprover} method performs intuitionistic proof
891 search, depending on specifically declared rules from the context,
892 or given as explicit arguments. Chained facts are inserted into the
893 goal before commencing proof search.
895 Rules need to be classified as @{attribute (Pure) intro},
896 @{attribute (Pure) elim}, or @{attribute (Pure) dest}; here the
897 ``@{text "!"}'' indicator refers to ``safe'' rules, which may be
898 applied aggressively (without considering back-tracking later).
899 Rules declared with ``@{text "?"}'' are ignored in proof search (the
900 single-step @{method (Pure) rule} method still observes these). An
901 explicit weight annotation may be given as well; otherwise the
902 number of rule premises will be taken into account here.
906 section {* Coherent Logic *}
909 \begin{matharray}{rcl}
910 @{method_def (HOL) "coherent"} & : & @{text method} \\
914 @@{method (HOL) coherent} @{syntax thmrefs}?
917 The @{method (HOL) coherent} method solves problems of
918 \emph{Coherent Logic} \cite{Bezem-Coquand:2005}, which covers
919 applications in confluence theory, lattice theory and projective
920 geometry. See @{file "~~/src/HOL/ex/Coherent.thy"} for some
925 section {* Proving propositions *}
928 In addition to the standard proof methods, a number of diagnosis
929 tools search for proofs and provide an Isar proof snippet on success.
930 These tools are available via the following commands.
932 \begin{matharray}{rcl}
933 @{command_def (HOL) "solve_direct"}@{text "\<^sup>*"} & : & @{text "proof \<rightarrow>"} \\
934 @{command_def (HOL) "try_methods"}@{text "\<^sup>*"} & : & @{text "proof \<rightarrow>"} \\
935 @{command_def (HOL) "sledgehammer"}@{text "\<^sup>*"} & : & @{text "proof \<rightarrow>"} \\
936 @{command_def (HOL) "sledgehammer_params"} & : & @{text "theory \<rightarrow> theory"}
940 @@{command (HOL) try_methods} ( ( ( 'simp' | 'intro' | 'elim' | 'dest' ) ':' @{syntax thmrefs} ) + ) ?
943 @@{command (HOL) sledgehammer} ( '[' args ']' )? facts? @{syntax nat}?
946 @@{command (HOL) sledgehammer_params} ( ( '[' args ']' ) ? )
949 args: ( @{syntax name} '=' value + ',' )
952 facts: '(' ( ( ( ( 'add' | 'del' ) ':' ) ? @{syntax thmrefs} ) + ) ? ')'
954 "} % FIXME check args "value"
958 \item @{command (HOL) "solve_direct"} checks whether the current subgoals can
959 be solved directly by an existing theorem. Duplicate lemmas can be detected
962 \item @{command (HOL) "try_methods"} attempts to prove a subgoal using a combination
963 of standard proof methods (@{text auto}, @{text simp}, @{text blast}, etc.).
964 Additional facts supplied via @{text "simp:"}, @{text "intro:"},
965 @{text "elim:"}, and @{text "dest:"} are passed to the appropriate proof
968 \item @{command (HOL) "sledgehammer"} attempts to prove a subgoal using external
969 automatic provers (resolution provers and SMT solvers). See the Sledgehammer
970 manual \cite{isabelle-sledgehammer} for details.
972 \item @{command (HOL) "sledgehammer_params"} changes
973 @{command (HOL) "sledgehammer"} configuration options persistently.
979 section {* Checking and refuting propositions *}
982 Identifying incorrect propositions usually involves evaluation of
983 particular assignments and systematic counterexample search. This
984 is supported by the following commands.
986 \begin{matharray}{rcl}
987 @{command_def (HOL) "value"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
988 @{command_def (HOL) "quickcheck"}@{text "\<^sup>*"} & : & @{text "proof \<rightarrow>"} \\
989 @{command_def (HOL) "refute"}@{text "\<^sup>*"} & : & @{text "proof \<rightarrow>"} \\
990 @{command_def (HOL) "nitpick"}@{text "\<^sup>*"} & : & @{text "proof \<rightarrow>"} \\
991 @{command_def (HOL) "quickcheck_params"} & : & @{text "theory \<rightarrow> theory"} \\
992 @{command_def (HOL) "refute_params"} & : & @{text "theory \<rightarrow> theory"} \\
993 @{command_def (HOL) "nitpick_params"} & : & @{text "theory \<rightarrow> theory"}
997 @@{command (HOL) value} ( '[' name ']' )? modes? @{syntax term}
1000 (@@{command (HOL) quickcheck} | @@{command (HOL) refute} | @@{command (HOL) nitpick})
1001 ( '[' args ']' )? @{syntax nat}?
1004 (@@{command (HOL) quickcheck_params} | @@{command (HOL) refute_params} |
1005 @@{command (HOL) nitpick_params}) ( '[' args ']' )?
1008 modes: '(' (@{syntax name} +) ')'
1011 args: ( @{syntax name} '=' value + ',' )
1013 "} % FIXME check "value"
1017 \item @{command (HOL) "value"}~@{text t} evaluates and prints a
1018 term; optionally @{text modes} can be specified, which are
1019 appended to the current print mode (see also \cite{isabelle-ref}).
1020 Internally, the evaluation is performed by registered evaluators,
1021 which are invoked sequentially until a result is returned.
1022 Alternatively a specific evaluator can be selected using square
1023 brackets; typical evaluators use the current set of code equations
1024 to normalize and include @{text simp} for fully symbolic evaluation
1025 using the simplifier, @{text nbe} for \emph{normalization by evaluation}
1026 and \emph{code} for code generation in SML.
1028 \item @{command (HOL) "quickcheck"} tests the current goal for
1029 counterexamples using a series of assignments for its
1030 free variables; by default the first subgoal is tested, an other
1031 can be selected explicitly using an optional goal index.
1032 Assignments can be chosen exhausting the search space upto a given
1033 size or using a fixed number of random assignments in the search space.
1034 By default, quickcheck uses exhaustive testing.
1035 A number of configuration options are supported for
1036 @{command (HOL) "quickcheck"}, notably:
1040 \item[@{text tester}] specifies how to explore the search space
1041 (e.g. exhaustive or random).
1042 An unknown configuration option is treated as an argument to tester,
1043 making @{text "tester ="} optional.
1044 \item[@{text size}] specifies the maximum size of the search space
1045 for assignment values.
1047 \item[@{text eval}] takes a term or a list of terms and evaluates
1048 these terms under the variable assignment found by quickcheck.
1050 \item[@{text iterations}] sets how many sets of assignments are
1051 generated for each particular size.
1053 \item[@{text no_assms}] specifies whether assumptions in
1054 structured proofs should be ignored.
1056 \item[@{text timeout}] sets the time limit in seconds.
1058 \item[@{text default_type}] sets the type(s) generally used to
1059 instantiate type variables.
1061 \item[@{text report}] if set quickcheck reports how many tests
1062 fulfilled the preconditions.
1064 \item[@{text quiet}] if not set quickcheck informs about the
1065 current size for assignment values.
1067 \item[@{text expect}] can be used to check if the user's
1068 expectation was met (@{text no_expectation}, @{text
1069 no_counterexample}, or @{text counterexample}).
1073 These option can be given within square brackets.
1075 \item @{command (HOL) "quickcheck_params"} changes
1076 @{command (HOL) "quickcheck"} configuration options persistently.
1078 \item @{command (HOL) "refute"} tests the current goal for
1079 counterexamples using a reduction to SAT. The following configuration
1080 options are supported:
1084 \item[@{text minsize}] specifies the minimum size (cardinality) of the
1085 models to search for.
1087 \item[@{text maxsize}] specifies the maximum size (cardinality) of the
1088 models to search for. Nonpositive values mean $\infty$.
1090 \item[@{text maxvars}] specifies the maximum number of Boolean variables
1091 to use when transforming the term into a propositional formula.
1092 Nonpositive values mean $\infty$.
1094 \item[@{text satsolver}] specifies the SAT solver to use.
1096 \item[@{text no_assms}] specifies whether assumptions in
1097 structured proofs should be ignored.
1099 \item[@{text maxtime}] sets the time limit in seconds.
1101 \item[@{text expect}] can be used to check if the user's
1102 expectation was met (@{text genuine}, @{text potential},
1103 @{text none}, or @{text unknown}).
1107 These option can be given within square brackets.
1109 \item @{command (HOL) "refute_params"} changes
1110 @{command (HOL) "refute"} configuration options persistently.
1112 \item @{command (HOL) "nitpick"} tests the current goal for counterexamples
1113 using a reduction to first-order relational logic. See the Nitpick manual
1114 \cite{isabelle-nitpick} for details.
1116 \item @{command (HOL) "nitpick_params"} changes
1117 @{command (HOL) "nitpick"} configuration options persistently.
1123 section {* Unstructured case analysis and induction \label{sec:hol-induct-tac} *}
1126 The following tools of Isabelle/HOL support cases analysis and
1127 induction in unstructured tactic scripts; see also
1128 \secref{sec:cases-induct} for proper Isar versions of similar ideas.
1130 \begin{matharray}{rcl}
1131 @{method_def (HOL) case_tac}@{text "\<^sup>*"} & : & @{text method} \\
1132 @{method_def (HOL) induct_tac}@{text "\<^sup>*"} & : & @{text method} \\
1133 @{method_def (HOL) ind_cases}@{text "\<^sup>*"} & : & @{text method} \\
1134 @{command_def (HOL) "inductive_cases"}@{text "\<^sup>*"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
1138 @@{method (HOL) case_tac} @{syntax goal_spec}? @{syntax term} rule?
1140 @@{method (HOL) induct_tac} @{syntax goal_spec}? (@{syntax insts} * @'and') rule?
1142 @@{method (HOL) ind_cases} (@{syntax prop}+) (@'for' (@{syntax name}+))?
1144 @@{command (HOL) inductive_cases} (@{syntax thmdecl}? (@{syntax prop}+) + @'and')
1147 rule: 'rule' ':' @{syntax thmref}
1152 \item @{method (HOL) case_tac} and @{method (HOL) induct_tac} admit
1153 to reason about inductive types. Rules are selected according to
1154 the declarations by the @{attribute cases} and @{attribute induct}
1155 attributes, cf.\ \secref{sec:cases-induct}. The @{command (HOL)
1156 datatype} package already takes care of this.
1158 These unstructured tactics feature both goal addressing and dynamic
1159 instantiation. Note that named rule cases are \emph{not} provided
1160 as would be by the proper @{method cases} and @{method induct} proof
1161 methods (see \secref{sec:cases-induct}). Unlike the @{method
1162 induct} method, @{method induct_tac} does not handle structured rule
1163 statements, only the compact object-logic conclusion of the subgoal
1166 \item @{method (HOL) ind_cases} and @{command (HOL)
1167 "inductive_cases"} provide an interface to the internal @{ML_text
1168 mk_cases} operation. Rules are simplified in an unrestricted
1171 While @{method (HOL) ind_cases} is a proof method to apply the
1172 result immediately as elimination rules, @{command (HOL)
1173 "inductive_cases"} provides case split theorems at the theory level
1174 for later use. The @{keyword "for"} argument of the @{method (HOL)
1175 ind_cases} method allows to specify a list of variables that should
1176 be generalized before applying the resulting rule.
1182 section {* Executable code *}
1184 text {* For validation purposes, it is often useful to \emph{execute}
1185 specifications. In principle, execution could be simulated by
1186 Isabelle's inference kernel, i.e. by a combination of resolution and
1187 simplification. Unfortunately, this approach is rather inefficient.
1188 A more efficient way of executing specifications is to translate
1189 them into a functional programming language such as ML.
1191 Isabelle provides two generic frameworks to support code generation
1192 from executable specifications. Isabelle/HOL instantiates these
1193 mechanisms in a way that is amenable to end-user applications.
1197 subsection {* The new code generator (F. Haftmann) *}
1199 text {* This framework generates code from functional programs
1200 (including overloading using type classes) to SML \cite{SML}, OCaml
1201 \cite{OCaml}, Haskell \cite{haskell-revised-report} and Scala
1202 \cite{scala-overview-tech-report}. Conceptually, code generation is
1203 split up in three steps: \emph{selection} of code theorems,
1204 \emph{translation} into an abstract executable view and
1205 \emph{serialization} to a specific \emph{target language}.
1206 Inductive specifications can be executed using the predicate
1207 compiler which operates within HOL. See \cite{isabelle-codegen} for
1210 \begin{matharray}{rcl}
1211 @{command_def (HOL) "export_code"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
1212 @{attribute_def (HOL) code} & : & @{text attribute} \\
1213 @{command_def (HOL) "code_abort"} & : & @{text "theory \<rightarrow> theory"} \\
1214 @{command_def (HOL) "code_datatype"} & : & @{text "theory \<rightarrow> theory"} \\
1215 @{command_def (HOL) "print_codesetup"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
1216 @{attribute_def (HOL) code_inline} & : & @{text attribute} \\
1217 @{attribute_def (HOL) code_post} & : & @{text attribute} \\
1218 @{command_def (HOL) "print_codeproc"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
1219 @{command_def (HOL) "code_thms"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
1220 @{command_def (HOL) "code_deps"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
1221 @{command_def (HOL) "code_const"} & : & @{text "theory \<rightarrow> theory"} \\
1222 @{command_def (HOL) "code_type"} & : & @{text "theory \<rightarrow> theory"} \\
1223 @{command_def (HOL) "code_class"} & : & @{text "theory \<rightarrow> theory"} \\
1224 @{command_def (HOL) "code_instance"} & : & @{text "theory \<rightarrow> theory"} \\
1225 @{command_def (HOL) "code_reserved"} & : & @{text "theory \<rightarrow> theory"} \\
1226 @{command_def (HOL) "code_monad"} & : & @{text "theory \<rightarrow> theory"} \\
1227 @{command_def (HOL) "code_include"} & : & @{text "theory \<rightarrow> theory"} \\
1228 @{command_def (HOL) "code_modulename"} & : & @{text "theory \<rightarrow> theory"} \\
1229 @{command_def (HOL) "code_reflect"} & : & @{text "theory \<rightarrow> theory"}
1233 @@{command (HOL) export_code} ( constexpr + ) \\
1234 ( ( @'in' target ( @'module_name' @{syntax string} ) ? \\
1235 ( @'file' ( @{syntax string} | '-' ) ) ? ( '(' args ')' ) ?) + ) ?
1238 const: @{syntax term}
1241 constexpr: ( const | 'name._' | '_' )
1244 typeconstructor: @{syntax nameref}
1247 class: @{syntax nameref}
1250 target: 'SML' | 'OCaml' | 'Haskell' | 'Scala'
1253 @@{attribute (HOL) code} ( 'del' | 'abstype' | 'abstract' )?
1256 @@{command (HOL) code_abort} ( const + )
1259 @@{command (HOL) code_datatype} ( const + )
1262 @@{attribute (HOL) code_inline} ( 'del' ) ?
1265 @@{attribute (HOL) code_post} ( 'del' ) ?
1268 @@{command (HOL) code_thms} ( constexpr + ) ?
1271 @@{command (HOL) code_deps} ( constexpr + ) ?
1274 @@{command (HOL) code_const} (const + @'and') \\
1275 ( ( '(' target ( syntax ? + @'and' ) ')' ) + )
1278 @@{command (HOL) code_type} (typeconstructor + @'and') \\
1279 ( ( '(' target ( syntax ? + @'and' ) ')' ) + )
1282 @@{command (HOL) code_class} (class + @'and') \\
1283 ( ( '(' target \\ ( @{syntax string} ? + @'and' ) ')' ) + )
1286 @@{command (HOL) code_instance} (( typeconstructor '::' class ) + @'and') \\
1287 ( ( '(' target ( '-' ? + @'and' ) ')' ) + )
1290 @@{command (HOL) code_reserved} target ( @{syntax string} + )
1293 @@{command (HOL) code_monad} const const target
1296 @@{command (HOL) code_include} target ( @{syntax string} ( @{syntax string} | '-') )
1299 @@{command (HOL) code_modulename} target ( ( @{syntax string} @{syntax string} ) + )
1302 @@{command (HOL) code_reflect} @{syntax string} \\
1303 ( @'datatypes' ( @{syntax string} '=' ( '_' | ( @{syntax string} + '|' ) + @'and' ) ) ) ? \\
1304 ( @'functions' ( @{syntax string} + ) ) ? ( @'file' @{syntax string} ) ?
1307 syntax: @{syntax string} | ( @'infix' | @'infixl' | @'infixr' ) @{syntax nat} @{syntax string}
1312 \item @{command (HOL) "export_code"} generates code for a given list
1313 of constants in the specified target language(s). If no
1314 serialization instruction is given, only abstract code is generated
1317 Constants may be specified by giving them literally, referring to
1318 all executable contants within a certain theory by giving @{text
1319 "name.*"}, or referring to \emph{all} executable constants currently
1320 available by giving @{text "*"}.
1322 By default, for each involved theory one corresponding name space
1323 module is generated. Alternativly, a module name may be specified
1324 after the @{keyword "module_name"} keyword; then \emph{all} code is
1325 placed in this module.
1327 For \emph{SML}, \emph{OCaml} and \emph{Scala} the file specification
1328 refers to a single file; for \emph{Haskell}, it refers to a whole
1329 directory, where code is generated in multiple files reflecting the
1330 module hierarchy. Omitting the file specification denotes standard
1333 Serializers take an optional list of arguments in parentheses. For
1334 \emph{SML} and \emph{OCaml}, ``@{text no_signatures}`` omits
1335 explicit module signatures.
1337 For \emph{Haskell} a module name prefix may be given using the
1338 ``@{text "root:"}'' argument; ``@{text string_classes}'' adds a
1339 ``@{verbatim "deriving (Read, Show)"}'' clause to each appropriate
1340 datatype declaration.
1342 \item @{attribute (HOL) code} explicitly selects (or with option
1343 ``@{text "del"}'' deselects) a code equation for code generation.
1344 Usually packages introducing code equations provide a reasonable
1345 default setup for selection. Variants @{text "code abstype"} and
1346 @{text "code abstract"} declare abstract datatype certificates or
1347 code equations on abstract datatype representations respectively.
1349 \item @{command (HOL) "code_abort"} declares constants which are not
1350 required to have a definition by means of code equations; if needed
1351 these are implemented by program abort instead.
1353 \item @{command (HOL) "code_datatype"} specifies a constructor set
1356 \item @{command (HOL) "print_codesetup"} gives an overview on
1357 selected code equations and code generator datatypes.
1359 \item @{attribute (HOL) code_inline} declares (or with option
1360 ``@{text "del"}'' removes) inlining theorems which are applied as
1361 rewrite rules to any code equation during preprocessing.
1363 \item @{attribute (HOL) code_post} declares (or with option ``@{text
1364 "del"}'' removes) theorems which are applied as rewrite rules to any
1365 result of an evaluation.
1367 \item @{command (HOL) "print_codeproc"} prints the setup of the code
1368 generator preprocessor.
1370 \item @{command (HOL) "code_thms"} prints a list of theorems
1371 representing the corresponding program containing all given
1372 constants after preprocessing.
1374 \item @{command (HOL) "code_deps"} visualizes dependencies of
1375 theorems representing the corresponding program containing all given
1376 constants after preprocessing.
1378 \item @{command (HOL) "code_const"} associates a list of constants
1379 with target-specific serializations; omitting a serialization
1380 deletes an existing serialization.
1382 \item @{command (HOL) "code_type"} associates a list of type
1383 constructors with target-specific serializations; omitting a
1384 serialization deletes an existing serialization.
1386 \item @{command (HOL) "code_class"} associates a list of classes
1387 with target-specific class names; omitting a serialization deletes
1388 an existing serialization. This applies only to \emph{Haskell}.
1390 \item @{command (HOL) "code_instance"} declares a list of type
1391 constructor / class instance relations as ``already present'' for a
1392 given target. Omitting a ``@{text "-"}'' deletes an existing
1393 ``already present'' declaration. This applies only to
1396 \item @{command (HOL) "code_reserved"} declares a list of names as
1397 reserved for a given target, preventing it to be shadowed by any
1400 \item @{command (HOL) "code_monad"} provides an auxiliary mechanism
1401 to generate monadic code for Haskell.
1403 \item @{command (HOL) "code_include"} adds arbitrary named content
1404 (``include'') to generated code. A ``@{text "-"}'' as last argument
1405 will remove an already added ``include''.
1407 \item @{command (HOL) "code_modulename"} declares aliasings from one
1408 module name onto another.
1410 \item @{command (HOL) "code_reflect"} without a ``@{text "file"}''
1411 argument compiles code into the system runtime environment and
1412 modifies the code generator setup that future invocations of system
1413 runtime code generation referring to one of the ``@{text
1414 "datatypes"}'' or ``@{text "functions"}'' entities use these precompiled
1415 entities. With a ``@{text "file"}'' argument, the corresponding code
1416 is generated into that specified file without modifying the code
1423 subsection {* The old code generator (S. Berghofer) *}
1425 text {* This framework generates code from both functional and
1426 relational programs to SML, as explained below.
1428 \begin{matharray}{rcl}
1429 @{command_def "code_module"} & : & @{text "theory \<rightarrow> theory"} \\
1430 @{command_def "code_library"} & : & @{text "theory \<rightarrow> theory"} \\
1431 @{command_def "consts_code"} & : & @{text "theory \<rightarrow> theory"} \\
1432 @{command_def "types_code"} & : & @{text "theory \<rightarrow> theory"} \\
1433 @{attribute_def code} & : & @{text attribute} \\
1437 ( @@{command code_module} | @@{command code_library} ) modespec? @{syntax name}? \\
1438 ( @'file' name ) ? ( @'imports' ( @{syntax name} + ) ) ? \\
1439 @'contains' ( ( @{syntax name} '=' @{syntax term} ) + | @{syntax term} + )
1442 modespec: '(' ( @{syntax name} * ) ')'
1445 @@{command (HOL) consts_code} (codespec +)
1448 codespec: const template attachment ?
1451 @@{command (HOL) types_code} (tycodespec +)
1454 tycodespec: @{syntax name} template attachment ?
1457 const: @{syntax term}
1460 template: '(' @{syntax string} ')'
1463 attachment: 'attach' modespec? '{' @{syntax text} '}'
1466 @@{attribute code} name?
1471 subsubsection {* Invoking the code generator *}
1473 text {* The code generator is invoked via the @{command code_module}
1474 and @{command code_library} commands, which correspond to
1475 \emph{incremental} and \emph{modular} code generation, respectively.
1479 \item [Modular] For each theory, an ML structure is generated,
1480 containing the code generated from the constants defined in this
1483 \item [Incremental] All the generated code is emitted into the same
1484 structure. This structure may import code from previously generated
1485 structures, which can be specified via @{keyword "imports"}.
1486 Moreover, the generated structure may also be referred to in later
1487 invocations of the code generator.
1491 After the @{command code_module} and @{command code_library}
1492 keywords, the user may specify an optional list of ``modes'' in
1493 parentheses. These can be used to instruct the code generator to
1494 emit additional code for special purposes, e.g.\ functions for
1495 converting elements of generated datatypes to Isabelle terms, or
1496 test data generators. The list of modes is followed by a module
1497 name. The module name is optional for modular code generation, but
1498 must be specified for incremental code generation.
1500 The code can either be written to a file, in which case a file name
1501 has to be specified after the @{keyword "file"} keyword, or be loaded
1502 directly into Isabelle's ML environment. In the latter case, the
1503 @{command ML} theory command can be used to inspect the results
1504 interactively, for example.
1506 The terms from which to generate code can be specified after the
1507 @{keyword "contains"} keyword, either as a list of bindings, or just
1508 as a list of terms. In the latter case, the code generator just
1509 produces code for all constants and types occuring in the term, but
1510 does not bind the compiled terms to ML identifiers.
1516 contains test = "foldl op + (0 :: int) [1, 2, 3, 4, 5]"
1518 text {* \noindent This binds the result of compiling the given term to
1519 the ML identifier @{ML Test.test}. *}
1521 ML {* @{assert} (Test.test = 15) *}
1524 subsubsection {* Configuring the code generator *}
1526 text {* When generating code for a complex term, the code generator
1527 recursively calls itself for all subterms. When it arrives at a
1528 constant, the default strategy of the code generator is to look up
1529 its definition and try to generate code for it. Constants which
1530 have no definitions that are immediately executable, may be
1531 associated with a piece of ML code manually using the @{command_ref
1532 consts_code} command. It takes a list whose elements consist of a
1533 constant (given in usual term syntax -- an explicit type constraint
1534 accounts for overloading), and a mixfix template describing the ML
1535 code. The latter is very much the same as the mixfix templates used
1536 when declaring new constants. The most notable difference is that
1537 terms may be included in the ML template using antiquotation
1538 brackets @{verbatim "{"}@{verbatim "*"}~@{text "..."}~@{verbatim
1539 "*"}@{verbatim "}"}.
1541 A similar mechanism is available for types: @{command_ref
1542 types_code} associates type constructors with specific ML code.
1544 For example, the following declarations copied from @{file
1545 "~~/src/HOL/Product_Type.thy"} describe how the product type of
1546 Isabelle/HOL should be compiled to ML. *}
1548 typedecl ('a, 'b) prod
1549 consts Pair :: "'a \<Rightarrow> 'b \<Rightarrow> ('a, 'b) prod"
1551 types_code prod ("(_ */ _)")
1552 consts_code Pair ("(_,/ _)")
1554 text {* Sometimes, the code associated with a constant or type may
1555 need to refer to auxiliary functions, which have to be emitted when
1556 the constant is used. Code for such auxiliary functions can be
1557 declared using @{keyword "attach"}. For example, the @{const wfrec}
1558 function can be implemented as follows:
1561 consts_code wfrec ("\<module>wfrec?") (* FIXME !? *)
1562 attach {* fun wfrec f x = f (wfrec f) x *}
1564 text {* If the code containing a call to @{const wfrec} resides in an
1565 ML structure different from the one containing the function
1566 definition attached to @{const wfrec}, the name of the ML structure
1567 (followed by a ``@{text "."}'') is inserted in place of ``@{text
1568 "\<module>"}'' in the above template. The ``@{text "?"}'' means that
1569 the code generator should ignore the first argument of @{const
1570 wfrec}, i.e.\ the termination relation, which is usually not
1573 \medskip Another possibility of configuring the code generator is to
1574 register theorems to be used for code generation. Theorems can be
1575 registered via the @{attribute code} attribute. It takes an optional
1576 name as an argument, which indicates the format of the
1577 theorem. Currently supported formats are equations (this is the
1578 default when no name is specified) and horn clauses (this is
1579 indicated by the name \texttt{ind}). The left-hand sides of
1580 equations may only contain constructors and distinct variables,
1581 whereas horn clauses must have the same format as introduction rules
1582 of inductive definitions.
1584 The following example specifies three equations from which to
1585 generate code for @{term "op <"} on natural numbers (see also
1586 @{"file" "~~/src/HOL/Nat.thy"}). *}
1588 lemma [code]: "(Suc m < Suc n) = (m < n)"
1589 and [code]: "((n::nat) < 0) = False"
1590 and [code]: "(0 < Suc n) = True" by simp_all
1593 subsubsection {* Specific HOL code generators *}
1595 text {* The basic code generator framework offered by Isabelle/Pure
1596 has already been extended with additional code generators for
1597 specific HOL constructs. These include datatypes, recursive
1598 functions and inductive relations. The code generator for inductive
1599 relations can handle expressions of the form @{text "(t\<^sub>1, \<dots>, t\<^sub>n) \<in>
1600 r"}, where @{text "r"} is an inductively defined relation. If at
1601 least one of the @{text "t\<^sub>i"} is a dummy pattern ``@{text "_"}'',
1602 the above expression evaluates to a sequence of possible answers. If
1603 all of the @{text "t\<^sub>i"} are proper terms, the expression evaluates
1606 The following example demonstrates this for beta-reduction on lambda
1607 terms (see also @{"file" "~~/src/HOL/Proofs/Lambda/Lambda.thy"}).
1612 | App dB dB (infixl "\<degree>" 200)
1615 primrec lift :: "dB \<Rightarrow> nat \<Rightarrow> dB"
1617 "lift (Var i) k = (if i < k then Var i else Var (i + 1))"
1618 | "lift (s \<degree> t) k = lift s k \<degree> lift t k"
1619 | "lift (Abs s) k = Abs (lift s (k + 1))"
1621 primrec subst :: "dB \<Rightarrow> dB \<Rightarrow> nat \<Rightarrow> dB" ("_[_'/_]" [300, 0, 0] 300)
1624 (if k < i then Var (i - 1) else if i = k then s else Var i)"
1625 | "(t \<degree> u)[s/k] = t[s/k] \<degree> u[s/k]"
1626 | "(Abs t)[s/k] = Abs (t[lift s 0 / k+1])"
1628 inductive beta :: "dB \<Rightarrow> dB \<Rightarrow> bool" (infixl "\<rightarrow>\<^sub>\<beta>" 50)
1630 beta: "Abs s \<degree> t \<rightarrow>\<^sub>\<beta> s[t/0]"
1631 | appL: "s \<rightarrow>\<^sub>\<beta> t \<Longrightarrow> s \<degree> u \<rightarrow>\<^sub>\<beta> t \<degree> u"
1632 | appR: "s \<rightarrow>\<^sub>\<beta> t \<Longrightarrow> u \<degree> s \<rightarrow>\<^sub>\<beta> u \<degree> t"
1633 | abs: "s \<rightarrow>\<^sub>\<beta> t \<Longrightarrow> Abs s \<rightarrow>\<^sub>\<beta> Abs t"
1637 test1 = "Abs (Var 0) \<degree> Var 0 \<rightarrow>\<^sub>\<beta> Var 0"
1638 test2 = "Abs (Abs (Var 0 \<degree> Var 0) \<degree> (Abs (Var 0) \<degree> Var 0)) \<rightarrow>\<^sub>\<beta> _"
1641 In the above example, @{ML Test.test1} evaluates to a boolean,
1642 whereas @{ML Test.test2} is a lazy sequence whose elements can be
1643 inspected separately.
1646 ML {* @{assert} Test.test1 *}
1647 ML {* val results = DSeq.list_of Test.test2 *}
1648 ML {* @{assert} (length results = 2) *}
1651 \medskip The theory underlying the HOL code generator is described
1652 more detailed in \cite{Berghofer-Nipkow:2002}. More examples that
1653 illustrate the usage of the code generator can be found e.g.\ in
1654 @{"file" "~~/src/HOL/MicroJava/J/JListExample.thy"} and @{"file"
1655 "~~/src/HOL/MicroJava/JVM/JVMListExample.thy"}.
1659 section {* Definition by specification \label{sec:hol-specification} *}
1662 \begin{matharray}{rcl}
1663 @{command_def (HOL) "specification"} & : & @{text "theory \<rightarrow> proof(prove)"} \\
1664 @{command_def (HOL) "ax_specification"} & : & @{text "theory \<rightarrow> proof(prove)"} \\
1668 (@@{command (HOL) specification} | @@{command (HOL) ax_specification})
1669 '(' (decl +) ')' \\ (@{syntax thmdecl}? @{syntax prop} +)
1671 decl: ((@{syntax name} ':')? @{syntax term} '(' @'overloaded' ')'?)
1676 \item @{command (HOL) "specification"}~@{text "decls \<phi>"} sets up a
1677 goal stating the existence of terms with the properties specified to
1678 hold for the constants given in @{text decls}. After finishing the
1679 proof, the theory will be augmented with definitions for the given
1680 constants, as well as with theorems stating the properties for these
1683 \item @{command (HOL) "ax_specification"}~@{text "decls \<phi>"} sets up
1684 a goal stating the existence of terms with the properties specified
1685 to hold for the constants given in @{text decls}. After finishing
1686 the proof, the theory will be augmented with axioms expressing the
1687 properties given in the first place.
1689 \item @{text decl} declares a constant to be defined by the
1690 specification given. The definition for the constant @{text c} is
1691 bound to the name @{text c_def} unless a theorem name is given in
1692 the declaration. Overloaded constants should be declared as such.
1696 Whether to use @{command (HOL) "specification"} or @{command (HOL)
1697 "ax_specification"} is to some extent a matter of style. @{command
1698 (HOL) "specification"} introduces no new axioms, and so by
1699 construction cannot introduce inconsistencies, whereas @{command
1700 (HOL) "ax_specification"} does introduce axioms, but only after the
1701 user has explicitly proven it to be safe. A practical issue must be
1702 considered, though: After introducing two constants with the same
1703 properties using @{command (HOL) "specification"}, one can prove
1704 that the two constants are, in fact, equal. If this might be a
1705 problem, one should use @{command (HOL) "ax_specification"}.