doc-src/IsarRef/Thy/HOL_Specific.thy
 author hoelzl Wed Apr 14 19:46:36 2010 +0200 (2010-04-14) changeset 36139 0c2538afe8e8 parent 35841 94f901e4969a child 36158 d2ad76e374d3 permissions -rw-r--r--
Spelling error: theroems -> theorems
1 theory HOL_Specific
2 imports Main
3 begin
5 chapter {* Isabelle/HOL \label{ch:hol} *}
7 section {* Typedef axiomatization \label{sec:hol-typedef} *}
9 text {*
10   \begin{matharray}{rcl}
11     @{command_def (HOL) "typedef"} & : & @{text "local_theory \<rightarrow> proof(prove)"} \\
12   \end{matharray}
14   \begin{rail}
15     'typedef' altname? abstype '=' repset
16     ;
18     altname: '(' (name | 'open' | 'open' name) ')'
19     ;
20     abstype: typespecsorts mixfix?
21     ;
22     repset: term ('morphisms' name name)?
23     ;
24   \end{rail}
26   \begin{description}
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.
63   \end{description}
64 *}
67 section {* Adhoc tuples *}
69 text {*
70   \begin{matharray}{rcl}
71     @{attribute (HOL) split_format}@{text "\<^sup>*"} & : & @{text attribute} \\
72   \end{matharray}
74   \begin{rail}
75     'split\_format' ((( name * ) + 'and') | ('(' 'complete' ')'))
76     ;
77   \end{rail}
79   \begin{description}
81   \item @{attribute (HOL) split_format}~@{text "p\<^sub>1 \<dots> p\<^sub>m \<AND> \<dots>
82   \<AND> q\<^sub>1 \<dots> q\<^sub>n"} puts expressions of low-level tuple types into
83   canonical form as specified by the arguments given; the @{text i}-th
84   collection of arguments refers to occurrences in premise @{text i}
85   of the rule.  The @{text "(complete)"}'' option causes \emph{all}
86   arguments in function applications to be represented canonically
87   according to their tuple type structure.
89   Note that these operations tend to invent funny names for new local
90   parameters to be introduced.
92   \end{description}
93 *}
96 section {* Records \label{sec:hol-record} *}
98 text {*
99   In principle, records merely generalize the concept of tuples, where
100   components may be addressed by labels instead of just position.  The
101   logical infrastructure of records in Isabelle/HOL is slightly more
102   advanced, though, supporting truly extensible record schemes.  This
103   admits operations that are polymorphic with respect to record
104   extension, yielding object-oriented'' effects like (single)
106   details on object-oriented verification and record subtyping in HOL.
107 *}
110 subsection {* Basic concepts *}
112 text {*
113   Isabelle/HOL supports both \emph{fixed} and \emph{schematic} records
114   at the level of terms and types.  The notation is as follows:
116   \begin{center}
117   \begin{tabular}{l|l|l}
118     & record terms & record types \\ \hline
119     fixed & @{text "\<lparr>x = a, y = b\<rparr>"} & @{text "\<lparr>x :: A, y :: B\<rparr>"} \\
120     schematic & @{text "\<lparr>x = a, y = b, \<dots> = m\<rparr>"} &
121       @{text "\<lparr>x :: A, y :: B, \<dots> :: M\<rparr>"} \\
122   \end{tabular}
123   \end{center}
125   \noindent The ASCII representation of @{text "\<lparr>x = a\<rparr>"} is @{text
126   "(| x = a |)"}.
128   A fixed record @{text "\<lparr>x = a, y = b\<rparr>"} has field @{text x} of value
129   @{text a} and field @{text y} of value @{text b}.  The corresponding
130   type is @{text "\<lparr>x :: A, y :: B\<rparr>"}, assuming that @{text "a :: A"}
131   and @{text "b :: B"}.
133   A record scheme like @{text "\<lparr>x = a, y = b, \<dots> = m\<rparr>"} contains fields
134   @{text x} and @{text y} as before, but also possibly further fields
135   as indicated by the @{text "\<dots>"}'' notation (which is actually part
136   of the syntax).  The improper field @{text "\<dots>"}'' of a record
137   scheme is called the \emph{more part}.  Logically it is just a free
138   variable, which is occasionally referred to as row variable'' in
139   the literature.  The more part of a record scheme may be
140   instantiated by zero or more further components.  For example, the
141   previous scheme may get instantiated to @{text "\<lparr>x = a, y = b, z =
142   c, \<dots> = m'\<rparr>"}, where @{text m'} refers to a different more part.
143   Fixed records are special instances of record schemes, where
144   @{text "\<dots>"}'' is properly terminated by the @{text "() :: unit"}
145   element.  In fact, @{text "\<lparr>x = a, y = b\<rparr>"} is just an abbreviation
146   for @{text "\<lparr>x = a, y = b, \<dots> = ()\<rparr>"}.
148   \medskip Two key observations make extensible records in a simply
149   typed language like HOL work out:
151   \begin{enumerate}
153   \item the more part is internalized, as a free term or type
154   variable,
156   \item field names are externalized, they cannot be accessed within
157   the logic as first-class values.
159   \end{enumerate}
161   \medskip In Isabelle/HOL record types have to be defined explicitly,
162   fixing their field names and types, and their (optional) parent
163   record.  Afterwards, records may be formed using above syntax, while
164   obeying the canonical order of fields as given by their declaration.
165   The record package provides several standard operations like
166   selectors and updates.  The common setup for various generic proof
168   tutorial \cite{isabelle-hol-book} for further instructions on using
169   records in practice.
170 *}
173 subsection {* Record specifications *}
175 text {*
176   \begin{matharray}{rcl}
177     @{command_def (HOL) "record"} & : & @{text "theory \<rightarrow> theory"} \\
178   \end{matharray}
180   \begin{rail}
181     'record' typespec '=' (type '+')? (constdecl +)
182     ;
183   \end{rail}
185   \begin{description}
187   \item @{command (HOL) "record"}~@{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t = \<tau> + c\<^sub>1 :: \<sigma>\<^sub>1
188   \<dots> c\<^sub>n :: \<sigma>\<^sub>n"} defines extensible record type @{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t"},
189   derived from the optional parent record @{text "\<tau>"} by adding new
190   field components @{text "c\<^sub>i :: \<sigma>\<^sub>i"} etc.
192   The type variables of @{text "\<tau>"} and @{text "\<sigma>\<^sub>i"} need to be
193   covered by the (distinct) parameters @{text "\<alpha>\<^sub>1, \<dots>,
194   \<alpha>\<^sub>m"}.  Type constructor @{text t} has to be new, while @{text
195   \<tau>} needs to specify an instance of an existing record type.  At
196   least one new field @{text "c\<^sub>i"} has to be specified.
197   Basically, field names need to belong to a unique record.  This is
198   not a real restriction in practice, since fields are qualified by
199   the record name internally.
201   The parent record specification @{text \<tau>} is optional; if omitted
202   @{text t} becomes a root record.  The hierarchy of all records
203   declared within a theory context forms a forest structure, i.e.\ a
204   set of trees starting with a root record each.  There is no way to
205   merge multiple parent records!
207   For convenience, @{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t"} is made a
208   type abbreviation for the fixed record type @{text "\<lparr>c\<^sub>1 ::
209   \<sigma>\<^sub>1, \<dots>, c\<^sub>n :: \<sigma>\<^sub>n\<rparr>"}, likewise is @{text
210   "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m, \<zeta>) t_scheme"} made an abbreviation for
211   @{text "\<lparr>c\<^sub>1 :: \<sigma>\<^sub>1, \<dots>, c\<^sub>n :: \<sigma>\<^sub>n, \<dots> ::
212   \<zeta>\<rparr>"}.
214   \end{description}
215 *}
218 subsection {* Record operations *}
220 text {*
221   Any record definition of the form presented above produces certain
222   standard operations.  Selectors and updates are provided for any
223   field, including the improper one @{text more}''.  There are also
224   cumulative record constructor functions.  To simplify the
225   presentation below, we assume for now that @{text "(\<alpha>\<^sub>1, \<dots>,
226   \<alpha>\<^sub>m) t"} is a root record with fields @{text "c\<^sub>1 ::
227   \<sigma>\<^sub>1, \<dots>, c\<^sub>n :: \<sigma>\<^sub>n"}.
229   \medskip \textbf{Selectors} and \textbf{updates} are available for
230   any field (including @{text more}''):
232   \begin{matharray}{lll}
233     @{text "c\<^sub>i"} & @{text "::"} & @{text "\<lparr>\<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr> \<Rightarrow> \<sigma>\<^sub>i"} \\
234     @{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>"} \\
235   \end{matharray}
237   There is special syntax for application of updates: @{text "r\<lparr>x :=
238   a\<rparr>"} abbreviates term @{text "x_update a r"}.  Further notation for
239   repeated updates is also available: @{text "r\<lparr>x := a\<rparr>\<lparr>y := b\<rparr>\<lparr>z :=
240   c\<rparr>"} may be written @{text "r\<lparr>x := a, y := b, z := c\<rparr>"}.  Note that
241   because of postfix notation the order of fields shown here is
242   reverse than in the actual term.  Since repeated updates are just
243   function applications, fields may be freely permuted in @{text "\<lparr>x
244   := a, y := b, z := c\<rparr>"}, as far as logical equality is concerned.
245   Thus commutativity of independent updates can be proven within the
246   logic for any two fields, but not as a general theorem.
248   \medskip The \textbf{make} operation provides a cumulative record
249   constructor function:
251   \begin{matharray}{lll}
252     @{text "t.make"} & @{text "::"} & @{text "\<sigma>\<^sub>1 \<Rightarrow> \<dots> \<sigma>\<^sub>n \<Rightarrow> \<lparr>\<^vec>c :: \<^vec>\<sigma>\<rparr>"} \\
253   \end{matharray}
255   \medskip We now reconsider the case of non-root records, which are
256   derived of some parent.  In general, the latter may depend on
257   another parent as well, resulting in a list of \emph{ancestor
258   records}.  Appending the lists of fields of all ancestors results in
259   a certain field prefix.  The record package automatically takes care
260   of this by lifting operations over this context of ancestor fields.
261   Assuming that @{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t"} has ancestor
262   fields @{text "b\<^sub>1 :: \<rho>\<^sub>1, \<dots>, b\<^sub>k :: \<rho>\<^sub>k"},
263   the above record operations will get the following types:
265   \medskip
266   \begin{tabular}{lll}
267     @{text "c\<^sub>i"} & @{text "::"} & @{text "\<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr> \<Rightarrow> \<sigma>\<^sub>i"} \\
268     @{text "c\<^sub>i_update"} & @{text "::"} & @{text "\<sigma>\<^sub>i \<Rightarrow>
269       \<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr> \<Rightarrow>
270       \<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr>"} \\
271     @{text "t.make"} & @{text "::"} & @{text "\<rho>\<^sub>1 \<Rightarrow> \<dots> \<rho>\<^sub>k \<Rightarrow> \<sigma>\<^sub>1 \<Rightarrow> \<dots> \<sigma>\<^sub>n \<Rightarrow>
272       \<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>\<rparr>"} \\
273   \end{tabular}
274   \medskip
276   \noindent Some further operations address the extension aspect of a
277   derived record scheme specifically: @{text "t.fields"} produces a
278   record fragment consisting of exactly the new fields introduced here
279   (the result may serve as a more part elsewhere); @{text "t.extend"}
280   takes a fixed record and adds a given more part; @{text
281   "t.truncate"} restricts a record scheme to a fixed record.
283   \medskip
284   \begin{tabular}{lll}
285     @{text "t.fields"} & @{text "::"} & @{text "\<sigma>\<^sub>1 \<Rightarrow> \<dots> \<sigma>\<^sub>n \<Rightarrow> \<lparr>\<^vec>c :: \<^vec>\<sigma>\<rparr>"} \\
286     @{text "t.extend"} & @{text "::"} & @{text "\<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>\<rparr> \<Rightarrow>
287       \<zeta> \<Rightarrow> \<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr>"} \\
288     @{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>"} \\
289   \end{tabular}
290   \medskip
292   \noindent Note that @{text "t.make"} and @{text "t.fields"} coincide
293   for root records.
294 *}
297 subsection {* Derived rules and proof tools *}
299 text {*
300   The record package proves several results internally, declaring
301   these facts to appropriate proof tools.  This enables users to
302   reason about record structures quite conveniently.  Assume that
303   @{text t} is a record type as specified above.
305   \begin{enumerate}
307   \item Standard conversions for selectors or updates applied to
308   record constructor terms are made part of the default Simplifier
309   context; thus proofs by reduction of basic operations merely require
310   the @{method simp} method without further arguments.  These rules
311   are available as @{text "t.simps"}, too.
313   \item Selectors applied to updated records are automatically reduced
314   by an internal simplification procedure, which is also part of the
315   standard Simplifier setup.
317   \item Inject equations of a form analogous to @{prop "(x, y) = (x',
318   y') \<equiv> x = x' \<and> y = y'"} are declared to the Simplifier and Classical
319   Reasoner as @{attribute iff} rules.  These rules are available as
320   @{text "t.iffs"}.
322   \item The introduction rule for record equality analogous to @{text
323   "x r = x r' \<Longrightarrow> y r = y r' \<dots> \<Longrightarrow> r = r'"} is declared to the Simplifier,
324   and as the basic rule context as @{attribute intro}@{text "?"}''.
325   The rule is called @{text "t.equality"}.
327   \item Representations of arbitrary record expressions as canonical
328   constructor terms are provided both in @{method cases} and @{method
329   induct} format (cf.\ the generic proof methods of the same name,
330   \secref{sec:cases-induct}).  Several variations are available, for
331   fixed records, record schemes, more parts etc.
333   The generic proof methods are sufficiently smart to pick the most
334   sensible rule according to the type of the indicated record
335   expression: users just need to apply something like @{text "(cases
336   r)"}'' to a certain proof problem.
338   \item The derived record operations @{text "t.make"}, @{text
339   "t.fields"}, @{text "t.extend"}, @{text "t.truncate"} are \emph{not}
340   treated automatically, but usually need to be expanded by hand,
341   using the collective fact @{text "t.defs"}.
343   \end{enumerate}
344 *}
347 section {* Datatypes \label{sec:hol-datatype} *}
349 text {*
350   \begin{matharray}{rcl}
351     @{command_def (HOL) "datatype"} & : & @{text "theory \<rightarrow> theory"} \\
352   @{command_def (HOL) "rep_datatype"} & : & @{text "theory \<rightarrow> proof(prove)"} \\
353   \end{matharray}
355   \begin{rail}
356     'datatype' (dtspec + 'and')
357     ;
358     'rep\_datatype' ('(' (name +) ')')? (term +)
359     ;
361     dtspec: parname? typespec mixfix? '=' (cons + '|')
362     ;
363     cons: name ( type * ) mixfix?
364   \end{rail}
366   \begin{description}
368   \item @{command (HOL) "datatype"} defines inductive datatypes in
369   HOL.
371   \item @{command (HOL) "rep_datatype"} represents existing types as
372   inductive ones, generating the standard infrastructure of derived
373   concepts (primitive recursion etc.).
375   \end{description}
377   The induction and exhaustion theorems generated provide case names
378   according to the constructors involved, while parameters are named
381   See \cite{isabelle-HOL} for more details on datatypes, but beware of
382   the old-style theory syntax being used there!  Apart from proper
383   proof methods for case-analysis and induction, there are also
384   emulations of ML tactics @{method (HOL) case_tac} and @{method (HOL)
385   induct_tac} available, see \secref{sec:hol-induct-tac}; these admit
386   to refer directly to the internal structure of subgoals (including
387   internally bound parameters).
388 *}
391 section {* Recursive functions \label{sec:recursion} *}
393 text {*
394   \begin{matharray}{rcl}
395     @{command_def (HOL) "primrec"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
396     @{command_def (HOL) "fun"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
397     @{command_def (HOL) "function"} & : & @{text "local_theory \<rightarrow> proof(prove)"} \\
398     @{command_def (HOL) "termination"} & : & @{text "local_theory \<rightarrow> proof(prove)"} \\
399   \end{matharray}
401   \begin{rail}
402     'primrec' target? fixes 'where' equations
403     ;
404     equations: (thmdecl? prop + '|')
405     ;
406     ('fun' | 'function') target? functionopts? fixes 'where' clauses
407     ;
408     clauses: (thmdecl? prop ('(' 'otherwise' ')')? + '|')
409     ;
410     functionopts: '(' (('sequential' | 'domintros' | 'tailrec' | 'default' term) + ',') ')'
411     ;
412     'termination' ( term )?
413   \end{rail}
415   \begin{description}
417   \item @{command (HOL) "primrec"} defines primitive recursive
420   \item @{command (HOL) "function"} defines functions by general
421   wellfounded recursion. A detailed description with examples can be
422   found in \cite{isabelle-function}. The function is specified by a
423   set of (possibly conditional) recursive equations with arbitrary
424   pattern matching. The command generates proof obligations for the
425   completeness and the compatibility of patterns.
427   The defined function is considered partial, and the resulting
428   simplification rules (named @{text "f.psimps"}) and induction rule
429   (named @{text "f.pinduct"}) are guarded by a generated domain
430   predicate @{text "f_dom"}. The @{command (HOL) "termination"}
431   command can then be used to establish that the function is total.
433   \item @{command (HOL) "fun"} is a shorthand notation for @{command
434   (HOL) "function"}~@{text "(sequential)"}, followed by automated
435   proof attempts regarding pattern matching and termination.  See
436   \cite{isabelle-function} for further details.
438   \item @{command (HOL) "termination"}~@{text f} commences a
439   termination proof for the previously defined function @{text f}.  If
440   this is omitted, the command refers to the most recent function
441   definition.  After the proof is closed, the recursive equations and
442   the induction principle is established.
444   \end{description}
446   Recursive definitions introduced by the @{command (HOL) "function"}
447   command accommodate
448   reasoning by induction (cf.\ \secref{sec:cases-induct}): rule @{text
449   "c.induct"} (where @{text c} is the name of the function definition)
450   refers to a specific induction rule, with parameters named according
451   to the user-specified equations. Cases are numbered (starting from 1).
453   For @{command (HOL) "primrec"}, the induction principle coincides
454   with structural recursion on the datatype the recursion is carried
455   out.
457   The equations provided by these packages may be referred later as
458   theorem list @{text "f.simps"}, where @{text f} is the (collective)
459   name of the functions defined.  Individual equations may be named
460   explicitly as well.
462   The @{command (HOL) "function"} command accepts the following
463   options.
465   \begin{description}
467   \item @{text sequential} enables a preprocessor which disambiguates
468   overlapping patterns by making them mutually disjoint.  Earlier
469   equations take precedence over later ones.  This allows to give the
470   specification in a format very similar to functional programming.
471   Note that the resulting simplification and induction rules
472   correspond to the transformed specification, not the one given
473   originally. This usually means that each equation given by the user
474   may result in several theorems.  Also note that this automatic
475   transformation only works for ML-style datatype patterns.
477   \item @{text domintros} enables the automated generation of
478   introduction rules for the domain predicate. While mostly not
479   needed, they can be helpful in some proofs about partial functions.
481   \item @{text tailrec} generates the unconstrained recursive
482   equations even without a termination proof, provided that the
483   function is tail-recursive. This currently only works
485   \item @{text "default d"} allows to specify a default value for a
486   (partial) function, which will ensure that @{text "f x = d x"}
487   whenever @{text "x \<notin> f_dom"}.
489   \end{description}
490 *}
493 subsection {* Proof methods related to recursive definitions *}
495 text {*
496   \begin{matharray}{rcl}
497     @{method_def (HOL) pat_completeness} & : & @{text method} \\
498     @{method_def (HOL) relation} & : & @{text method} \\
499     @{method_def (HOL) lexicographic_order} & : & @{text method} \\
500     @{method_def (HOL) size_change} & : & @{text method} \\
501   \end{matharray}
503   \begin{rail}
504     'relation' term
505     ;
506     'lexicographic\_order' ( clasimpmod * )
507     ;
508     'size\_change' ( orders ( clasimpmod * ) )
509     ;
510     orders: ( 'max' | 'min' | 'ms' ) *
511   \end{rail}
513   \begin{description}
515   \item @{method (HOL) pat_completeness} is a specialized method to
516   solve goals regarding the completeness of pattern matching, as
517   required by the @{command (HOL) "function"} package (cf.\
518   \cite{isabelle-function}).
520   \item @{method (HOL) relation}~@{text R} introduces a termination
521   proof using the relation @{text R}.  The resulting proof state will
522   contain goals expressing that @{text R} is wellfounded, and that the
523   arguments of recursive calls decrease with respect to @{text R}.
524   Usually, this method is used as the initial proof step of manual
525   termination proofs.
527   \item @{method (HOL) "lexicographic_order"} attempts a fully
528   automated termination proof by searching for a lexicographic
529   combination of size measures on the arguments of the function. The
530   method accepts the same arguments as the @{method auto} method,
531   which it uses internally to prove local descents.  The same context
532   modifiers as for @{method auto} are accepted, see
533   \secref{sec:clasimp}.
535   In case of failure, extensive information is printed, which can help
536   to analyse the situation (cf.\ \cite{isabelle-function}).
538   \item @{method (HOL) "size_change"} also works on termination goals,
539   using a variation of the size-change principle, together with a
540   graph decomposition technique (see \cite{krauss_phd} for details).
541   Three kinds of orders are used internally: @{text max}, @{text min},
542   and @{text ms} (multiset), which is only available when the theory
543   @{text Multiset} is loaded. When no order kinds are given, they are
544   tried in order. The search for a termination proof uses SAT solving
545   internally.
547  For local descent proofs, the same context modifiers as for @{method
548   auto} are accepted, see \secref{sec:clasimp}.
550   \end{description}
551 *}
554 subsection {* Old-style recursive function definitions (TFL) *}
556 text {*
557   The old TFL commands @{command (HOL) "recdef"} and @{command (HOL)
558   "recdef_tc"} for defining recursive are mostly obsolete; @{command
559   (HOL) "function"} or @{command (HOL) "fun"} should be used instead.
561   \begin{matharray}{rcl}
562     @{command_def (HOL) "recdef"} & : & @{text "theory \<rightarrow> theory)"} \\
563     @{command_def (HOL) "recdef_tc"}@{text "\<^sup>*"} & : & @{text "theory \<rightarrow> proof(prove)"} \\
564   \end{matharray}
566   \begin{rail}
567     'recdef' ('(' 'permissive' ')')? \\ name term (prop +) hints?
568     ;
569     recdeftc thmdecl? tc
570     ;
571     hints: '(' 'hints' ( recdefmod * ) ')'
572     ;
573     recdefmod: (('recdef\_simp' | 'recdef\_cong' | 'recdef\_wf') (() | 'add' | 'del') ':' thmrefs) | clasimpmod
574     ;
575     tc: nameref ('(' nat ')')?
576     ;
577   \end{rail}
579   \begin{description}
581   \item @{command (HOL) "recdef"} defines general well-founded
583   \cite{isabelle-HOL}.  The @{text "(permissive)"}'' option tells
584   TFL to recover from failed proof attempts, returning unfinished
585   results.  The @{text recdef_simp}, @{text recdef_cong}, and @{text
586   recdef_wf} hints refer to auxiliary rules to be used in the internal
587   automated proof process of TFL.  Additional @{syntax clasimpmod}
588   declarations (cf.\ \secref{sec:clasimp}) may be given to tune the
589   context of the Simplifier (cf.\ \secref{sec:simplifier}) and
590   Classical reasoner (cf.\ \secref{sec:classical}).
592   \item @{command (HOL) "recdef_tc"}~@{text "c (i)"} recommences the
593   proof for leftover termination condition number @{text i} (default
594   1) as generated by a @{command (HOL) "recdef"} definition of
595   constant @{text c}.
597   Note that in most cases, @{command (HOL) "recdef"} is able to finish
598   its internal proofs without manual intervention.
600   \end{description}
602   \medskip Hints for @{command (HOL) "recdef"} may be also declared
603   globally, using the following attributes.
605   \begin{matharray}{rcl}
606     @{attribute_def (HOL) recdef_simp} & : & @{text attribute} \\
607     @{attribute_def (HOL) recdef_cong} & : & @{text attribute} \\
608     @{attribute_def (HOL) recdef_wf} & : & @{text attribute} \\
609   \end{matharray}
611   \begin{rail}
612     ('recdef\_simp' | 'recdef\_cong' | 'recdef\_wf') (() | 'add' | 'del')
613     ;
614   \end{rail}
615 *}
618 section {* Inductive and coinductive definitions \label{sec:hol-inductive} *}
620 text {*
621   An \textbf{inductive definition} specifies the least predicate (or
622   set) @{text R} closed under given rules: applying a rule to elements
623   of @{text R} yields a result within @{text R}.  For example, a
624   structural operational semantics is an inductive definition of an
625   evaluation relation.
627   Dually, a \textbf{coinductive definition} specifies the greatest
628   predicate~/ set @{text R} that is consistent with given rules: every
629   element of @{text R} can be seen as arising by applying a rule to
630   elements of @{text R}.  An important example is using bisimulation
631   relations to formalise equivalence of processes and infinite data
632   structures.
634   \medskip The HOL package is related to the ZF one, which is
635   described in a separate paper,\footnote{It appeared in CADE
636   \cite{paulson-CADE}; a longer version is distributed with Isabelle.}
637   which you should refer to in case of difficulties.  The package is
638   simpler than that of ZF thanks to implicit type-checking in HOL.
639   The types of the (co)inductive predicates (or sets) determine the
640   domain of the fixedpoint definition, and the package does not have
641   to use inference rules for type-checking.
643   \begin{matharray}{rcl}
644     @{command_def (HOL) "inductive"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
645     @{command_def (HOL) "inductive_set"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
646     @{command_def (HOL) "coinductive"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
647     @{command_def (HOL) "coinductive_set"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
648     @{attribute_def (HOL) mono} & : & @{text attribute} \\
649   \end{matharray}
651   \begin{rail}
652     ('inductive' | 'inductive\_set' | 'coinductive' | 'coinductive\_set') target? fixes ('for' fixes)? \\
653     ('where' clauses)? ('monos' thmrefs)?
654     ;
655     clauses: (thmdecl? prop + '|')
656     ;
657     'mono' (() | 'add' | 'del')
658     ;
659   \end{rail}
661   \begin{description}
663   \item @{command (HOL) "inductive"} and @{command (HOL)
664   "coinductive"} define (co)inductive predicates from the
665   introduction rules given in the @{keyword "where"} part.  The
666   optional @{keyword "for"} part contains a list of parameters of the
667   (co)inductive predicates that remain fixed throughout the
668   definition.  The optional @{keyword "monos"} section contains
669   \emph{monotonicity theorems}, which are required for each operator
670   applied to a recursive set in the introduction rules.  There
671   \emph{must} be a theorem of the form @{text "A \<le> B \<Longrightarrow> M A \<le> M B"},
672   for each premise @{text "M R\<^sub>i t"} in an introduction rule!
674   \item @{command (HOL) "inductive_set"} and @{command (HOL)
675   "coinductive_set"} are wrappers for to the previous commands,
676   allowing the definition of (co)inductive sets.
678   \item @{attribute (HOL) mono} declares monotonicity rules.  These
679   rule are involved in the automated monotonicity proof of @{command
680   (HOL) "inductive"}.
682   \end{description}
683 *}
686 subsection {* Derived rules *}
688 text {*
689   Each (co)inductive definition @{text R} adds definitions to the
690   theory and also proves some theorems:
692   \begin{description}
694   \item @{text R.intros} is the list of introduction rules as proven
695   theorems, for the recursive predicates (or sets).  The rules are
696   also available individually, using the names given them in the
697   theory file;
699   \item @{text R.cases} is the case analysis (or elimination) rule;
701   \item @{text R.induct} or @{text R.coinduct} is the (co)induction
702   rule.
704   \end{description}
706   When several predicates @{text "R\<^sub>1, \<dots>, R\<^sub>n"} are
707   defined simultaneously, the list of introduction rules is called
708   @{text "R\<^sub>1_\<dots>_R\<^sub>n.intros"}, the case analysis rules are
709   called @{text "R\<^sub>1.cases, \<dots>, R\<^sub>n.cases"}, and the list
710   of mutual induction rules is called @{text
711   "R\<^sub>1_\<dots>_R\<^sub>n.inducts"}.
712 *}
715 subsection {* Monotonicity theorems *}
717 text {*
718   Each theory contains a default set of theorems that are used in
719   monotonicity proofs.  New rules can be added to this set via the
720   @{attribute (HOL) mono} attribute.  The HOL theory @{text Inductive}
721   shows how this is done.  In general, the following monotonicity
724   \begin{itemize}
726   \item Theorems of the form @{text "A \<le> B \<Longrightarrow> M A \<le> M B"}, for proving
727   monotonicity of inductive definitions whose introduction rules have
728   premises involving terms such as @{text "M R\<^sub>i t"}.
730   \item Monotonicity theorems for logical operators, which are of the
731   general form @{text "(\<dots> \<longrightarrow> \<dots>) \<Longrightarrow> \<dots> (\<dots> \<longrightarrow> \<dots>) \<Longrightarrow> \<dots> \<longrightarrow> \<dots>"}.  For example, in
732   the case of the operator @{text "\<or>"}, the corresponding theorem is
733   $734 \infer{@{text "P\<^sub>1 \<or> P\<^sub>2 \<longrightarrow> Q\<^sub>1 \<or> Q\<^sub>2"}}{@{text "P\<^sub>1 \<longrightarrow> Q\<^sub>1"} & @{text "P\<^sub>2 \<longrightarrow> Q\<^sub>2"}} 735$
737   \item De Morgan style equations for reasoning about the polarity''
738   of expressions, e.g.
739   $740 @{prop "\<not> \<not> P \<longleftrightarrow> P"} \qquad\qquad 741 @{prop "\<not> (P \<and> Q) \<longleftrightarrow> \<not> P \<or> \<not> Q"} 742$
744   \item Equations for reducing complex operators to more primitive
745   ones whose monotonicity can easily be proved, e.g.
746   $747 @{prop "(P \<longrightarrow> Q) \<longleftrightarrow> \<not> P \<or> Q"} \qquad\qquad 748 @{prop "Ball A P \<equiv> \<forall>x. x \<in> A \<longrightarrow> P x"} 749$
751   \end{itemize}
753   %FIXME: Example of an inductive definition
754 *}
757 section {* Arithmetic proof support *}
759 text {*
760   \begin{matharray}{rcl}
761     @{method_def (HOL) arith} & : & @{text method} \\
762     @{attribute_def (HOL) arith} & : & @{text attribute} \\
763     @{attribute_def (HOL) arith_split} & : & @{text attribute} \\
764   \end{matharray}
766   The @{method (HOL) arith} method decides linear arithmetic problems
767   (on types @{text nat}, @{text int}, @{text real}).  Any current
768   facts are inserted into the goal before running the procedure.
770   The @{attribute (HOL) arith} attribute declares facts that are
771   always supplied to the arithmetic provers implicitly.
773   The @{attribute (HOL) arith_split} attribute declares case split
774   rules to be expanded before @{method (HOL) arith} is invoked.
776   Note that a simpler (but faster) arithmetic prover is
777   already invoked by the Simplifier.
778 *}
781 section {* Intuitionistic proof search *}
783 text {*
784   \begin{matharray}{rcl}
785     @{method_def (HOL) iprover} & : & @{text method} \\
786   \end{matharray}
788   \begin{rail}
789     'iprover' ( rulemod * )
790     ;
791   \end{rail}
793   The @{method (HOL) iprover} method performs intuitionistic proof
794   search, depending on specifically declared rules from the context,
795   or given as explicit arguments.  Chained facts are inserted into the
796   goal before commencing proof search.
798   Rules need to be classified as @{attribute (Pure) intro},
799   @{attribute (Pure) elim}, or @{attribute (Pure) dest}; here the
800   @{text "!"}'' indicator refers to safe'' rules, which may be
801   applied aggressively (without considering back-tracking later).
802   Rules declared with @{text "?"}'' are ignored in proof search (the
803   single-step @{method rule} method still observes these).  An
804   explicit weight annotation may be given as well; otherwise the
805   number of rule premises will be taken into account here.
806 *}
809 section {* Coherent Logic *}
811 text {*
812   \begin{matharray}{rcl}
813     @{method_def (HOL) "coherent"} & : & @{text method} \\
814   \end{matharray}
816   \begin{rail}
817     'coherent' thmrefs?
818     ;
819   \end{rail}
821   The @{method (HOL) coherent} method solves problems of
822   \emph{Coherent Logic} \cite{Bezem-Coquand:2005}, which covers
823   applications in confluence theory, lattice theory and projective
824   geometry.  See @{"file" "~~/src/HOL/ex/Coherent.thy"} for some
825   examples.
826 *}
829 section {* Checking and refuting propositions *}
831 text {*
832   Identifying incorrect propositions usually involves evaluation of
833   particular assignments and systematic counter example search.  This
834   is supported by the following commands.
836   \begin{matharray}{rcl}
837     @{command_def (HOL) "value"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
838     @{command_def (HOL) "quickcheck"}@{text "\<^sup>*"} & : & @{text "proof \<rightarrow>"} \\
839     @{command_def (HOL) "quickcheck_params"} & : & @{text "theory \<rightarrow> theory"}
840   \end{matharray}
842   \begin{rail}
843     'value' ( ( '[' name ']' ) ? ) modes? term
844     ;
846     'quickcheck' ( ( '[' args ']' ) ? ) nat?
847     ;
849     'quickcheck_params' ( ( '[' args ']' ) ? )
850     ;
852     modes: '(' (name + ) ')'
853     ;
855     args: ( name '=' value + ',' )
856     ;
857   \end{rail}
859   \begin{description}
861   \item @{command (HOL) "value"}~@{text t} evaluates and prints a
862     term; optionally @{text modes} can be specified, which are
864     Internally, the evaluation is performed by registered evaluators,
865     which are invoked sequentially until a result is returned.
866     Alternatively a specific evaluator can be selected using square
867     brackets; available evaluators include @{text nbe} for
868     \emph{normalization by evaluation} and \emph{code} for code
869     generation in SML.
871   \item @{command (HOL) "quickcheck"} tests the current goal for
872     counter examples using a series of arbitrary assignments for its
873     free variables; by default the first subgoal is tested, an other
874     can be selected explicitly using an optional goal index.
875     A number of configuration options are supported for
876     @{command (HOL) "quickcheck"}, notably:
878     \begin{description}
880       \item[size] specifies the maximum size of the search space for
881         assignment values.
883       \item[iterations] sets how many sets of assignments are
884         generated for each particular size.
886       \item[no\_assms] specifies whether assumptions in
887         structured proofs should be ignored.
889     \end{description}
891     These option can be given within square brackets.
893   \item @{command (HOL) "quickcheck_params"} changes quickcheck
894     configuration options persitently.
896   \end{description}
897 *}
900 section {* Invoking automated reasoning tools --- The Sledgehammer *}
902 text {*
903   Isabelle/HOL includes a generic \emph{ATP manager} that allows
904   external automated reasoning tools to crunch a pending goal.
905   Supported provers include E\footnote{\url{http://www.eprover.org}},
906   SPASS\footnote{\url{http://www.spass-prover.org/}}, and Vampire.
907   There is also a wrapper to invoke provers remotely via the
908   SystemOnTPTP\footnote{\url{http://www.cs.miami.edu/~tptp/cgi-bin/SystemOnTPTP}}
909   web service.
911   The problem passed to external provers consists of the goal together
912   with a smart selection of lemmas from the current theory context.
913   The result of a successful proof search is some source text that
914   usually reconstructs the proof within Isabelle, without requiring
915   external provers again.  The Metis
916   prover\footnote{\url{http://www.gilith.com/software/metis/}} that is
917   integrated into Isabelle/HOL is being used here.
919   In this mode of operation, heavy means of automated reasoning are
920   used as a strong relevance filter, while the main proof checking
921   works via explicit inferences going through the Isabelle kernel.
922   Moreover, rechecking Isabelle proof texts with already specified
923   auxiliary facts is much faster than performing fully automated
924   search over and over again.
926   \begin{matharray}{rcl}
927     @{command_def (HOL) "sledgehammer"}@{text "\<^sup>*"} & : & @{text "proof \<rightarrow>"} \\
928     @{command_def (HOL) "print_atps"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
929     @{command_def (HOL) "atp_info"}@{text "\<^sup>*"} & : & @{text "any \<rightarrow>"} \\
930     @{command_def (HOL) "atp_kill"}@{text "\<^sup>*"} & : & @{text "any \<rightarrow>"} \\
931     @{command_def (HOL) "atp_messages"}@{text "\<^sup>*"} & : & @{text "any \<rightarrow>"} \\
932     @{method_def (HOL) metis} & : & @{text method} \\
933   \end{matharray}
935   \begin{rail}
936   'sledgehammer' ( nameref * )
937   ;
938   'atp\_messages' ('(' nat ')')?
939   ;
941   'metis' thmrefs
942   ;
943   \end{rail}
945   \begin{description}
947   \item @{command (HOL) sledgehammer}~@{text "prover\<^sub>1 \<dots> prover\<^sub>n"}
948   invokes the specified automated theorem provers on the first
949   subgoal.  Provers are run in parallel, the first successful result
950   is displayed, and the other attempts are terminated.
952   Provers are defined in the theory context, see also @{command (HOL)
953   print_atps}.  If no provers are given as arguments to @{command
954   (HOL) sledgehammer}, the system refers to the default defined as
955   ATP provers'' preference by the user interface.
957   There are additional preferences for timeout (default: 60 seconds),
958   and the maximum number of independent prover processes (default: 5);
959   excessive provers are automatically terminated.
961   \item @{command (HOL) print_atps} prints the list of automated
962   theorem provers available to the @{command (HOL) sledgehammer}
963   command.
965   \item @{command (HOL) atp_info} prints information about presently
966   running provers, including elapsed runtime, and the remaining time
967   until timeout.
969   \item @{command (HOL) atp_kill} terminates all presently running
970   provers.
972   \item @{command (HOL) atp_messages} displays recent messages issued
973   by automated theorem provers.  This allows to examine results that
974   might have got lost due to the asynchronous nature of default
975   @{command (HOL) sledgehammer} output.  An optional message limit may
976   be specified (default 5).
978   \item @{method (HOL) metis}~@{text "facts"} invokes the Metis prover
979   with the given facts.  Metis is an automated proof tool of medium
980   strength, but is fully integrated into Isabelle/HOL, with explicit
981   inferences going through the kernel.  Thus its results are
982   guaranteed to be correct by construction''.
984   Note that all facts used with Metis need to be specified as explicit
985   arguments.  There are no rule declarations as for other Isabelle
986   provers, like @{method blast} or @{method fast}.
988   \end{description}
989 *}
992 section {* Unstructured case analysis and induction \label{sec:hol-induct-tac} *}
994 text {*
995   The following tools of Isabelle/HOL support cases analysis and
997   \secref{sec:cases-induct} for proper Isar versions of similar ideas.
999   \begin{matharray}{rcl}
1000     @{method_def (HOL) case_tac}@{text "\<^sup>*"} & : & @{text method} \\
1001     @{method_def (HOL) induct_tac}@{text "\<^sup>*"} & : & @{text method} \\
1002     @{method_def (HOL) ind_cases}@{text "\<^sup>*"} & : & @{text method} \\
1003     @{command_def (HOL) "inductive_cases"}@{text "\<^sup>*"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
1004   \end{matharray}
1006   \begin{rail}
1007     'case\_tac' goalspec? term rule?
1008     ;
1009     'induct\_tac' goalspec? (insts * 'and') rule?
1010     ;
1011     'ind\_cases' (prop +) ('for' (name +)) ?
1012     ;
1013     'inductive\_cases' (thmdecl? (prop +) + 'and')
1014     ;
1016     rule: ('rule' ':' thmref)
1017     ;
1018   \end{rail}
1020   \begin{description}
1022   \item @{method (HOL) case_tac} and @{method (HOL) induct_tac} admit
1023   to reason about inductive types.  Rules are selected according to
1024   the declarations by the @{attribute cases} and @{attribute induct}
1025   attributes, cf.\ \secref{sec:cases-induct}.  The @{command (HOL)
1026   datatype} package already takes care of this.
1028   These unstructured tactics feature both goal addressing and dynamic
1029   instantiation.  Note that named rule cases are \emph{not} provided
1030   as would be by the proper @{method cases} and @{method induct} proof
1031   methods (see \secref{sec:cases-induct}).  Unlike the @{method
1032   induct} method, @{method induct_tac} does not handle structured rule
1033   statements, only the compact object-logic conclusion of the subgoal
1036   \item @{method (HOL) ind_cases} and @{command (HOL)
1037   "inductive_cases"} provide an interface to the internal @{ML_text
1038   mk_cases} operation.  Rules are simplified in an unrestricted
1039   forward manner.
1041   While @{method (HOL) ind_cases} is a proof method to apply the
1042   result immediately as elimination rules, @{command (HOL)
1043   "inductive_cases"} provides case split theorems at the theory level
1044   for later use.  The @{keyword "for"} argument of the @{method (HOL)
1045   ind_cases} method allows to specify a list of variables that should
1046   be generalized before applying the resulting rule.
1048   \end{description}
1049 *}
1052 section {* Executable code *}
1054 text {*
1055   Isabelle/Pure provides two generic frameworks to support code
1056   generation from executable specifications.  Isabelle/HOL
1057   instantiates these mechanisms in a way that is amenable to end-user
1058   applications.
1060   One framework generates code from both functional and relational
1061   programs to SML.  See \cite{isabelle-HOL} for further information
1062   (this actually covers the new-style theory format as well).
1064   \begin{matharray}{rcl}
1065     @{command_def (HOL) "code_module"} & : & @{text "theory \<rightarrow> theory"} \\
1066     @{command_def (HOL) "code_library"} & : & @{text "theory \<rightarrow> theory"} \\
1067     @{command_def (HOL) "consts_code"} & : & @{text "theory \<rightarrow> theory"} \\
1068     @{command_def (HOL) "types_code"} & : & @{text "theory \<rightarrow> theory"} \\
1069     @{attribute_def (HOL) code} & : & @{text attribute} \\
1070   \end{matharray}
1072   \begin{rail}
1073   ( 'code\_module' | 'code\_library' ) modespec ? name ? \\
1074     ( 'file' name ) ? ( 'imports' ( name + ) ) ? \\
1075     'contains' ( ( name '=' term ) + | term + )
1076   ;
1078   modespec: '(' ( name * ) ')'
1079   ;
1081   'consts\_code' (codespec +)
1082   ;
1084   codespec: const template attachment ?
1085   ;
1087   'types\_code' (tycodespec +)
1088   ;
1090   tycodespec: name template attachment ?
1091   ;
1093   const: term
1094   ;
1096   template: '(' string ')'
1097   ;
1099   attachment: 'attach' modespec ? verblbrace text verbrbrace
1100   ;
1102   'code' (name)?
1103   ;
1104   \end{rail}
1106   \medskip The other framework generates code from functional programs
1109   Conceptually, code generation is split up in three steps:
1110   \emph{selection} of code theorems, \emph{translation} into an
1111   abstract executable view and \emph{serialization} to a specific
1112   \emph{target language}.  See \cite{isabelle-codegen} for an
1113   introduction on how to use it.
1115   \begin{matharray}{rcl}
1116     @{command_def (HOL) "export_code"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
1117     @{command_def (HOL) "code_thms"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
1118     @{command_def (HOL) "code_deps"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
1119     @{command_def (HOL) "code_datatype"} & : & @{text "theory \<rightarrow> theory"} \\
1120     @{command_def (HOL) "code_const"} & : & @{text "theory \<rightarrow> theory"} \\
1121     @{command_def (HOL) "code_type"} & : & @{text "theory \<rightarrow> theory"} \\
1122     @{command_def (HOL) "code_class"} & : & @{text "theory \<rightarrow> theory"} \\
1123     @{command_def (HOL) "code_instance"} & : & @{text "theory \<rightarrow> theory"} \\
1124     @{command_def (HOL) "code_monad"} & : & @{text "theory \<rightarrow> theory"} \\
1125     @{command_def (HOL) "code_reserved"} & : & @{text "theory \<rightarrow> theory"} \\
1126     @{command_def (HOL) "code_include"} & : & @{text "theory \<rightarrow> theory"} \\
1127     @{command_def (HOL) "code_modulename"} & : & @{text "theory \<rightarrow> theory"} \\
1128     @{command_def (HOL) "code_abort"} & : & @{text "theory \<rightarrow> theory"} \\
1129     @{command_def (HOL) "print_codesetup"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
1130     @{command_def (HOL) "print_codeproc"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
1131     @{attribute_def (HOL) code} & : & @{text attribute} \\
1132   \end{matharray}
1134   \begin{rail}
1135     'export\_code' ( constexpr + ) \\
1136       ( ( 'in' target ( 'module\_name' string ) ? \\
1137         ( 'file' ( string | '-' ) ) ? ( '(' args ')' ) ?) + ) ?
1138     ;
1140     'code\_thms' ( constexpr + ) ?
1141     ;
1143     'code\_deps' ( constexpr + ) ?
1144     ;
1146     const: term
1147     ;
1149     constexpr: ( const | 'name.*' | '*' )
1150     ;
1152     typeconstructor: nameref
1153     ;
1155     class: nameref
1156     ;
1158     target: 'OCaml' | 'SML' | 'Haskell'
1159     ;
1161     'code\_datatype' const +
1162     ;
1164     'code\_const' (const + 'and') \\
1165       ( ( '(' target ( syntax ? + 'and' ) ')' ) + )
1166     ;
1168     'code\_type' (typeconstructor + 'and') \\
1169       ( ( '(' target ( syntax ? + 'and' ) ')' ) + )
1170     ;
1172     'code\_class' (class + 'and') \\
1173       ( ( '(' target \\ ( string ? + 'and' ) ')' ) + )
1174     ;
1176     'code\_instance' (( typeconstructor '::' class ) + 'and') \\
1177       ( ( '(' target ( '-' ? + 'and' ) ')' ) + )
1178     ;
1181     ;
1183     'code\_reserved' target ( string + )
1184     ;
1186     'code\_include' target ( string ( string | '-') )
1187     ;
1189     'code\_modulename' target ( ( string string ) + )
1190     ;
1192     'code\_abort' ( const + )
1193     ;
1195     syntax: string | ( 'infix' | 'infixl' | 'infixr' ) nat string
1196     ;
1198     'code' ( 'del' ) ?
1199     ;
1201     'code_unfold' ( 'del' ) ?
1202     ;
1204     'code_post' ( 'del' ) ?
1205     ;
1206   \end{rail}
1208   \begin{description}
1210   \item @{command (HOL) "export_code"} is the canonical interface for
1211   generating and serializing code: for a given list of constants, code
1212   is generated for the specified target languages.  If no serialization
1213   instruction is given, only abstract code is generated internally.
1215   Constants may be specified by giving them literally, referring to
1216   all executable contants within a certain theory by giving @{text
1217   "name.*"}, or referring to \emph{all} executable constants currently
1218   available by giving @{text "*"}.
1220   By default, for each involved theory one corresponding name space
1221   module is generated.  Alternativly, a module name may be specified
1222   after the @{keyword "module_name"} keyword; then \emph{all} code is
1223   placed in this module.
1225   For \emph{SML} and \emph{OCaml}, the file specification refers to a
1226   single file; for \emph{Haskell}, it refers to a whole directory,
1227   where code is generated in multiple files reflecting the module
1228   hierarchy.  The file specification @{text "-"}'' denotes standard
1229   output.  For \emph{SML}, omitting the file specification compiles
1230   code internally in the context of the current ML session.
1232   Serializers take an optional list of arguments in parentheses.  For
1233   \emph{SML} and \emph{OCaml}, @{text no_signatures} omits
1234   explicit module signatures.
1236   For \emph{Haskell} a module name prefix may be given using the @{text
1237   "root:"}'' argument; @{text string_classes}'' adds a @{verbatim
1238   "deriving (Read, Show)"}'' clause to each appropriate datatype
1239   declaration.
1241   \item @{command (HOL) "code_thms"} prints a list of theorems
1242   representing the corresponding program containing all given
1243   constants.
1245   \item @{command (HOL) "code_deps"} visualizes dependencies of
1246   theorems representing the corresponding program containing all given
1247   constants.
1249   \item @{command (HOL) "code_datatype"} specifies a constructor set
1250   for a logical type.
1252   \item @{command (HOL) "code_const"} associates a list of constants
1253   with target-specific serializations; omitting a serialization
1254   deletes an existing serialization.
1256   \item @{command (HOL) "code_type"} associates a list of type
1257   constructors with target-specific serializations; omitting a
1258   serialization deletes an existing serialization.
1260   \item @{command (HOL) "code_class"} associates a list of classes
1261   with target-specific class names; omitting a serialization deletes
1262   an existing serialization.  This applies only to \emph{Haskell}.
1264   \item @{command (HOL) "code_instance"} declares a list of type
1265   constructor / class instance relations as already present'' for a
1266   given target.  Omitting a @{text "-"}'' deletes an existing
1267   already present'' declaration.  This applies only to
1270   \item @{command (HOL) "code_monad"} provides an auxiliary mechanism
1273   \item @{command (HOL) "code_reserved"} declares a list of names as
1274   reserved for a given target, preventing it to be shadowed by any
1275   generated code.
1277   \item @{command (HOL) "code_include"} adds arbitrary named content
1278   (include'') to generated code.  A @{text "-"}'' as last argument
1279   will remove an already added include''.
1281   \item @{command (HOL) "code_modulename"} declares aliasings from one
1282   module name onto another.
1284   \item @{command (HOL) "code_abort"} declares constants which are not
1285   required to have a definition by means of code equations; if
1286   needed these are implemented by program abort instead.
1288   \item @{attribute (HOL) code} explicitly selects (or with option
1289   @{text "del"}'' deselects) a code equation for code
1290   generation.  Usually packages introducing code equations provide
1291   a reasonable default setup for selection.
1293   \item @{attribute (HOL) code_inline} declares (or with
1294   option @{text "del"}'' removes) inlining theorems which are
1295   applied as rewrite rules to any code equation during
1296   preprocessing.
1298   \item @{attribute (HOL) code_post} declares (or with
1299   option @{text "del"}'' removes) theorems which are
1300   applied as rewrite rules to any result of an evaluation.
1302   \item @{command (HOL) "print_codesetup"} gives an overview on
1303   selected code equations and code generator datatypes.
1305   \item @{command (HOL) "print_codeproc"} prints the setup
1306   of the code generator preprocessor.
1308   \end{description}
1309 *}
1312 section {* Definition by specification \label{sec:hol-specification} *}
1314 text {*
1315   \begin{matharray}{rcl}
1316     @{command_def (HOL) "specification"} & : & @{text "theory \<rightarrow> proof(prove)"} \\
1317     @{command_def (HOL) "ax_specification"} & : & @{text "theory \<rightarrow> proof(prove)"} \\
1318   \end{matharray}
1320   \begin{rail}
1321   ('specification' | 'ax\_specification') '(' (decl +) ')' \\ (thmdecl? prop +)
1322   ;
1323   decl: ((name ':')? term '(' 'overloaded' ')'?)
1324   \end{rail}
1326   \begin{description}
1328   \item @{command (HOL) "specification"}~@{text "decls \<phi>"} sets up a
1329   goal stating the existence of terms with the properties specified to
1330   hold for the constants given in @{text decls}.  After finishing the
1331   proof, the theory will be augmented with definitions for the given
1332   constants, as well as with theorems stating the properties for these
1333   constants.
1335   \item @{command (HOL) "ax_specification"}~@{text "decls \<phi>"} sets up
1336   a goal stating the existence of terms with the properties specified
1337   to hold for the constants given in @{text decls}.  After finishing
1338   the proof, the theory will be augmented with axioms expressing the
1339   properties given in the first place.
1341   \item @{text decl} declares a constant to be defined by the
1342   specification given.  The definition for the constant @{text c} is
1343   bound to the name @{text c_def} unless a theorem name is given in
1344   the declaration.  Overloaded constants should be declared as such.
1346   \end{description}
1348   Whether to use @{command (HOL) "specification"} or @{command (HOL)
1349   "ax_specification"} is to some extent a matter of style.  @{command
1350   (HOL) "specification"} introduces no new axioms, and so by
1351   construction cannot introduce inconsistencies, whereas @{command
1352   (HOL) "ax_specification"} does introduce axioms, but only after the
1353   user has explicitly proven it to be safe.  A practical issue must be
1354   considered, though: After introducing two constants with the same
1355   properties using @{command (HOL) "specification"}, one can prove
1356   that the two constants are, in fact, equal.  If this might be a
1357   problem, one should use @{command (HOL) "ax_specification"}.
1358 *}
1360 end