doc-src/IsarRef/Thy/HOL_Specific.thy
 author wenzelm Thu May 08 23:02:23 2008 +0200 (2008-05-08) changeset 26860 7c749112261c parent 26852 a31203f58b20 child 26894 1120f6cc10b0 permissions -rw-r--r--
replaced some latex macros by antiquotations;
     1 (* $Id$ *)

     2

     3 theory HOL_Specific

     4 imports Main

     5 begin

     6

     7 chapter {* Isabelle/HOL \label{ch:hol} *}

     8

     9 section {* Primitive types \label{sec:hol-typedef} *}

    10

    11 text {*

    12   \begin{matharray}{rcl}

    13     @{command_def (HOL) "typedecl"} & : & \isartrans{theory}{theory} \\

    14     @{command_def (HOL) "typedef"} & : & \isartrans{theory}{proof(prove)} \\

    15   \end{matharray}

    16

    17   \begin{rail}

    18     'typedecl' typespec infix?

    19     ;

    20     'typedef' altname? abstype '=' repset

    21     ;

    22

    23     altname: '(' (name | 'open' | 'open' name) ')'

    24     ;

    25     abstype: typespec infix?

    26     ;

    27     repset: term ('morphisms' name name)?

    28     ;

    29   \end{rail}

    30

    31   \begin{descr}

    32

    33   \item [@{command (HOL) "typedecl"}~@{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>n)

    34   t"}] is similar to the original @{command "typedecl"} of

    35   Isabelle/Pure (see \secref{sec:types-pure}), but also declares type

    36   arity @{text "t :: (type, \<dots>, type) type"}, making @{text t} an

    37   actual HOL type constructor.   %FIXME check, update

    38

    39   \item [@{command (HOL) "typedef"}~@{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>n)

    40   t = A"}] sets up a goal stating non-emptiness of the set @{text A}.

    41   After finishing the proof, the theory will be augmented by a

    42   Gordon/HOL-style type definition, which establishes a bijection

    43   between the representing set @{text A} and the new type @{text t}.

    44

    45   Technically, @{command (HOL) "typedef"} defines both a type @{text

    46   t} and a set (term constant) of the same name (an alternative base

    47   name may be given in parentheses).  The injection from type to set

    48   is called @{text Rep_t}, its inverse @{text Abs_t} (this may be

    49   changed via an explicit @{keyword (HOL) "morphisms"} declaration).

    50

    51   Theorems @{text Rep_t}, @{text Rep_t_inverse}, and @{text

    52   Abs_t_inverse} provide the most basic characterization as a

    53   corresponding injection/surjection pair (in both directions).  Rules

    54   @{text Rep_t_inject} and @{text Abs_t_inject} provide a slightly

    55   more convenient view on the injectivity part, suitable for automated

    56   proof tools (e.g.\ in @{method simp} or @{method iff} declarations).

    57   Rules @{text Rep_t_cases}/@{text Rep_t_induct}, and @{text

    58   Abs_t_cases}/@{text Abs_t_induct} provide alternative views on

    59   surjectivity; these are already declared as set or type rules for

    60   the generic @{method cases} and @{method induct} methods.

    61

    62   An alternative name may be specified in parentheses; the default is

    63   to use @{text t} as indicated before.  The @{text "(open)"}''

    64   declaration suppresses a separate constant definition for the

    65   representing set.

    66

    67   \end{descr}

    68

    69   Note that raw type declarations are rarely used in practice; the

    70   main application is with experimental (or even axiomatic!) theory

    71   fragments.  Instead of primitive HOL type definitions, user-level

    72   theories usually refer to higher-level packages such as @{command

    73   (HOL) "record"} (see \secref{sec:hol-record}) or @{command (HOL)

    74   "datatype"} (see \secref{sec:hol-datatype}).

    75 *}

    76

    77

    78 section {* Adhoc tuples *}

    79

    80 text {*

    81   \begin{matharray}{rcl}

    82     @{attribute (HOL) split_format}@{text "\<^sup>*"} & : & \isaratt \\

    83   \end{matharray}

    84

    85   \begin{rail}

    86     'split\_format' (((name *) + 'and') | ('(' 'complete' ')'))

    87     ;

    88   \end{rail}

    89

    90   \begin{descr}

    91

    92   \item [@{method (HOL) split_format}~@{text "p\<^sub>1 \<dots> p\<^sub>m

    93   \<AND> \<dots> \<AND> q\<^sub>1 \<dots> q\<^sub>n"}] puts expressions of

    94   low-level tuple types into canonical form as specified by the

    95   arguments given; the @{text i}-th collection of arguments refers to

    96   occurrences in premise @{text i} of the rule.  The @{text

    97   "(complete)"}'' option causes \emph{all} arguments in function

    98   applications to be represented canonically according to their tuple

    99   type structure.

   100

   101   Note that these operations tend to invent funny names for new local

   102   parameters to be introduced.

   103

   104   \end{descr}

   105 *}

   106

   107

   108 section {* Records \label{sec:hol-record} *}

   109

   110 text {*

   111   In principle, records merely generalize the concept of tuples, where

   112   components may be addressed by labels instead of just position.  The

   113   logical infrastructure of records in Isabelle/HOL is slightly more

   114   advanced, though, supporting truly extensible record schemes.  This

   115   admits operations that are polymorphic with respect to record

   116   extension, yielding object-oriented'' effects like (single)

   117   inheritance.  See also \cite{NaraschewskiW-TPHOLs98} for more

   118   details on object-oriented verification and record subtyping in HOL.

   119 *}

   120

   121

   122 subsection {* Basic concepts *}

   123

   124 text {*

   125   Isabelle/HOL supports both \emph{fixed} and \emph{schematic} records

   126   at the level of terms and types.  The notation is as follows:

   127

   128   \begin{center}

   129   \begin{tabular}{l|l|l}

   130     & record terms & record types \\ \hline

   131     fixed & @{text "\<lparr>x = a, y = b\<rparr>"} & @{text "\<lparr>x :: A, y :: B\<rparr>"} \\

   132     schematic & @{text "\<lparr>x = a, y = b, \<dots> = m\<rparr>"} &

   133       @{text "\<lparr>x :: A, y :: B, \<dots> :: M\<rparr>"} \\

   134   \end{tabular}

   135   \end{center}

   136

   137   \noindent The ASCII representation of @{text "\<lparr>x = a\<rparr>"} is @{text

   138   "(| x = a |)"}.

   139

   140   A fixed record @{text "\<lparr>x = a, y = b\<rparr>"} has field @{text x} of value

   141   @{text a} and field @{text y} of value @{text b}.  The corresponding

   142   type is @{text "\<lparr>x :: A, y :: B\<rparr>"}, assuming that @{text "a :: A"}

   143   and @{text "b :: B"}.

   144

   145   A record scheme like @{text "\<lparr>x = a, y = b, \<dots> = m\<rparr>"} contains fields

   146   @{text x} and @{text y} as before, but also possibly further fields

   147   as indicated by the @{text "\<dots>"}'' notation (which is actually part

   148   of the syntax).  The improper field @{text "\<dots>"}'' of a record

   149   scheme is called the \emph{more part}.  Logically it is just a free

   150   variable, which is occasionally referred to as row variable'' in

   151   the literature.  The more part of a record scheme may be

   152   instantiated by zero or more further components.  For example, the

   153   previous scheme may get instantiated to @{text "\<lparr>x = a, y = b, z =

   154   c, \<dots> = m'\<rparr>"}, where @{text m'} refers to a different more part.

   155   Fixed records are special instances of record schemes, where

   156   @{text "\<dots>"}'' is properly terminated by the @{text "() :: unit"}

   157   element.  In fact, @{text "\<lparr>x = a, y = b\<rparr>"} is just an abbreviation

   158   for @{text "\<lparr>x = a, y = b, \<dots> = ()\<rparr>"}.

   159

   160   \medskip Two key observations make extensible records in a simply

   161   typed language like HOL work out:

   162

   163   \begin{enumerate}

   164

   165   \item the more part is internalized, as a free term or type

   166   variable,

   167

   168   \item field names are externalized, they cannot be accessed within

   169   the logic as first-class values.

   170

   171   \end{enumerate}

   172

   173   \medskip In Isabelle/HOL record types have to be defined explicitly,

   174   fixing their field names and types, and their (optional) parent

   175   record.  Afterwards, records may be formed using above syntax, while

   176   obeying the canonical order of fields as given by their declaration.

   177   The record package provides several standard operations like

   178   selectors and updates.  The common setup for various generic proof

   179   tools enable succinct reasoning patterns.  See also the Isabelle/HOL

   180   tutorial \cite{isabelle-hol-book} for further instructions on using

   181   records in practice.

   182 *}

   183

   184

   185 subsection {* Record specifications *}

   186

   187 text {*

   188   \begin{matharray}{rcl}

   189     @{command_def (HOL) "record"} & : & \isartrans{theory}{theory} \\

   190   \end{matharray}

   191

   192   \begin{rail}

   193     'record' typespec '=' (type '+')? (constdecl +)

   194     ;

   195   \end{rail}

   196

   197   \begin{descr}

   198

   199   \item [@{command (HOL) "record"}~@{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t

   200   = \<tau> + c\<^sub>1 :: \<sigma>\<^sub>1 \<dots> c\<^sub>n :: \<sigma>\<^sub>n"}] defines

   201   extensible record type @{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t"},

   202   derived from the optional parent record @{text "\<tau>"} by adding new

   203   field components @{text "c\<^sub>i :: \<sigma>\<^sub>i"} etc.

   204

   205   The type variables of @{text "\<tau>"} and @{text "\<sigma>\<^sub>i"} need to be

   206   covered by the (distinct) parameters @{text "\<alpha>\<^sub>1, \<dots>,

   207   \<alpha>\<^sub>m"}.  Type constructor @{text t} has to be new, while @{text

   208   \<tau>} needs to specify an instance of an existing record type.  At

   209   least one new field @{text "c\<^sub>i"} has to be specified.

   210   Basically, field names need to belong to a unique record.  This is

   211   not a real restriction in practice, since fields are qualified by

   212   the record name internally.

   213

   214   The parent record specification @{text \<tau>} is optional; if omitted

   215   @{text t} becomes a root record.  The hierarchy of all records

   216   declared within a theory context forms a forest structure, i.e.\ a

   217   set of trees starting with a root record each.  There is no way to

   218   merge multiple parent records!

   219

   220   For convenience, @{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t"} is made a

   221   type abbreviation for the fixed record type @{text "\<lparr>c\<^sub>1 ::

   222   \<sigma>\<^sub>1, \<dots>, c\<^sub>n :: \<sigma>\<^sub>n\<rparr>"}, likewise is @{text

   223   "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m, \<zeta>) t_scheme"} made an abbreviation for

   224   @{text "\<lparr>c\<^sub>1 :: \<sigma>\<^sub>1, \<dots>, c\<^sub>n :: \<sigma>\<^sub>n, \<dots> ::

   225   \<zeta>\<rparr>"}.

   226

   227   \end{descr}

   228 *}

   229

   230

   231 subsection {* Record operations *}

   232

   233 text {*

   234   Any record definition of the form presented above produces certain

   235   standard operations.  Selectors and updates are provided for any

   236   field, including the improper one @{text more}''.  There are also

   237   cumulative record constructor functions.  To simplify the

   238   presentation below, we assume for now that @{text "(\<alpha>\<^sub>1, \<dots>,

   239   \<alpha>\<^sub>m) t"} is a root record with fields @{text "c\<^sub>1 ::

   240   \<sigma>\<^sub>1, \<dots>, c\<^sub>n :: \<sigma>\<^sub>n"}.

   241

   242   \medskip \textbf{Selectors} and \textbf{updates} are available for

   243   any field (including @{text more}''):

   244

   245   \begin{matharray}{lll}

   246     @{text "c\<^sub>i"} & @{text "::"} & @{text "\<lparr>\<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr> \<Rightarrow> \<sigma>\<^sub>i"} \\

   247     @{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>"} \\

   248   \end{matharray}

   249

   250   There is special syntax for application of updates: @{text "r\<lparr>x :=

   251   a\<rparr>"} abbreviates term @{text "x_update a r"}.  Further notation for

   252   repeated updates is also available: @{text "r\<lparr>x := a\<rparr>\<lparr>y := b\<rparr>\<lparr>z :=

   253   c\<rparr>"} may be written @{text "r\<lparr>x := a, y := b, z := c\<rparr>"}.  Note that

   254   because of postfix notation the order of fields shown here is

   255   reverse than in the actual term.  Since repeated updates are just

   256   function applications, fields may be freely permuted in @{text "\<lparr>x

   257   := a, y := b, z := c\<rparr>"}, as far as logical equality is concerned.

   258   Thus commutativity of independent updates can be proven within the

   259   logic for any two fields, but not as a general theorem.

   260

   261   \medskip The \textbf{make} operation provides a cumulative record

   262   constructor function:

   263

   264   \begin{matharray}{lll}

   265     @{text "t.make"} & @{text "::"} & @{text "\<sigma>\<^sub>1 \<Rightarrow> \<dots> \<sigma>\<^sub>n \<Rightarrow> \<lparr>\<^vec>c :: \<^vec>\<sigma>\<rparr>"} \\

   266   \end{matharray}

   267

   268   \medskip We now reconsider the case of non-root records, which are

   269   derived of some parent.  In general, the latter may depend on

   270   another parent as well, resulting in a list of \emph{ancestor

   271   records}.  Appending the lists of fields of all ancestors results in

   272   a certain field prefix.  The record package automatically takes care

   273   of this by lifting operations over this context of ancestor fields.

   274   Assuming that @{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t"} has ancestor

   275   fields @{text "b\<^sub>1 :: \<rho>\<^sub>1, \<dots>, b\<^sub>k :: \<rho>\<^sub>k"},

   276   the above record operations will get the following types:

   277

   278   \medskip

   279   \begin{tabular}{lll}

   280     @{text "c\<^sub>i"} & @{text "::"} & @{text "\<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr> \<Rightarrow> \<sigma>\<^sub>i"} \\

   281     @{text "c\<^sub>i_update"} & @{text "::"} & @{text "\<sigma>\<^sub>i \<Rightarrow>

   282       \<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr> \<Rightarrow>

   283       \<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr>"} \\

   284     @{text "t.make"} & @{text "::"} & @{text "\<rho>\<^sub>1 \<Rightarrow> \<dots> \<rho>\<^sub>k \<Rightarrow> \<sigma>\<^sub>1 \<Rightarrow> \<dots> \<sigma>\<^sub>n \<Rightarrow>

   285       \<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>\<rparr>"} \\

   286   \end{tabular}

   287   \medskip

   288

   289   \noindent Some further operations address the extension aspect of a

   290   derived record scheme specifically: @{text "t.fields"} produces a

   291   record fragment consisting of exactly the new fields introduced here

   292   (the result may serve as a more part elsewhere); @{text "t.extend"}

   293   takes a fixed record and adds a given more part; @{text

   294   "t.truncate"} restricts a record scheme to a fixed record.

   295

   296   \medskip

   297   \begin{tabular}{lll}

   298     @{text "t.fields"} & @{text "::"} & @{text "\<sigma>\<^sub>1 \<Rightarrow> \<dots> \<sigma>\<^sub>n \<Rightarrow> \<lparr>\<^vec>c :: \<^vec>\<sigma>\<rparr>"} \\

   299     @{text "t.extend"} & @{text "::"} & @{text "\<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>\<rparr> \<Rightarrow>

   300       \<zeta> \<Rightarrow> \<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr>"} \\

   301     @{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>"} \\

   302   \end{tabular}

   303   \medskip

   304

   305   \noindent Note that @{text "t.make"} and @{text "t.fields"} coincide

   306   for root records.

   307 *}

   308

   309

   310 subsection {* Derived rules and proof tools *}

   311

   312 text {*

   313   The record package proves several results internally, declaring

   314   these facts to appropriate proof tools.  This enables users to

   315   reason about record structures quite conveniently.  Assume that

   316   @{text t} is a record type as specified above.

   317

   318   \begin{enumerate}

   319

   320   \item Standard conversions for selectors or updates applied to

   321   record constructor terms are made part of the default Simplifier

   322   context; thus proofs by reduction of basic operations merely require

   323   the @{method simp} method without further arguments.  These rules

   324   are available as @{text "t.simps"}, too.

   325

   326   \item Selectors applied to updated records are automatically reduced

   327   by an internal simplification procedure, which is also part of the

   328   standard Simplifier setup.

   329

   330   \item Inject equations of a form analogous to @{prop "(x, y) = (x',

   331   y') \<equiv> x = x' \<and> y = y'"} are declared to the Simplifier and Classical

   332   Reasoner as @{attribute iff} rules.  These rules are available as

   333   @{text "t.iffs"}.

   334

   335   \item The introduction rule for record equality analogous to @{text

   336   "x r = x r' \<Longrightarrow> y r = y r' \<dots> \<Longrightarrow> r = r'"} is declared to the Simplifier,

   337   and as the basic rule context as @{attribute intro}@{text "?"}''.

   338   The rule is called @{text "t.equality"}.

   339

   340   \item Representations of arbitrary record expressions as canonical

   341   constructor terms are provided both in @{method cases} and @{method

   342   induct} format (cf.\ the generic proof methods of the same name,

   343   \secref{sec:cases-induct}).  Several variations are available, for

   344   fixed records, record schemes, more parts etc.

   345

   346   The generic proof methods are sufficiently smart to pick the most

   347   sensible rule according to the type of the indicated record

   348   expression: users just need to apply something like @{text "(cases

   349   r)"}'' to a certain proof problem.

   350

   351   \item The derived record operations @{text "t.make"}, @{text

   352   "t.fields"}, @{text "t.extend"}, @{text "t.truncate"} are \emph{not}

   353   treated automatically, but usually need to be expanded by hand,

   354   using the collective fact @{text "t.defs"}.

   355

   356   \end{enumerate}

   357 *}

   358

   359

   360 section {* Datatypes \label{sec:hol-datatype} *}

   361

   362 text {*

   363   \begin{matharray}{rcl}

   364     @{command_def (HOL) "datatype"} & : & \isartrans{theory}{theory} \\

   365     @{command_def (HOL) "rep_datatype"} & : & \isartrans{theory}{theory} \\

   366   \end{matharray}

   367

   368   \begin{rail}

   369     'datatype' (dtspec + 'and')

   370     ;

   371     'rep\_datatype' (name *) dtrules

   372     ;

   373

   374     dtspec: parname? typespec infix? '=' (cons + '|')

   375     ;

   376     cons: name (type *) mixfix?

   377     ;

   378     dtrules: 'distinct' thmrefs 'inject' thmrefs 'induction' thmrefs

   379   \end{rail}

   380

   381   \begin{descr}

   382

   383   \item [@{command (HOL) "datatype"}] defines inductive datatypes in

   384   HOL.

   385

   386   \item [@{command (HOL) "rep_datatype"}] represents existing types as

   387   inductive ones, generating the standard infrastructure of derived

   388   concepts (primitive recursion etc.).

   389

   390   \end{descr}

   391

   392   The induction and exhaustion theorems generated provide case names

   393   according to the constructors involved, while parameters are named

   394   after the types (see also \secref{sec:cases-induct}).

   395

   396   See \cite{isabelle-HOL} for more details on datatypes, but beware of

   397   the old-style theory syntax being used there!  Apart from proper

   398   proof methods for case-analysis and induction, there are also

   399   emulations of ML tactics @{method (HOL) case_tac} and @{method (HOL)

   400   induct_tac} available, see \secref{sec:hol-induct-tac}; these admit

   401   to refer directly to the internal structure of subgoals (including

   402   internally bound parameters).

   403 *}

   404

   405

   406 section {* Recursive functions \label{sec:recursion} *}

   407

   408 text {*

   409   \begin{matharray}{rcl}

   410     @{command_def (HOL) "primrec"} & : & \isarkeep{local{\dsh}theory} \\

   411     @{command_def (HOL) "fun"} & : & \isarkeep{local{\dsh}theory} \\

   412     @{command_def (HOL) "function"} & : & \isartrans{local{\dsh}theory}{proof(prove)} \\

   413     @{command_def (HOL) "termination"} & : & \isartrans{local{\dsh}theory}{proof(prove)} \\

   414   \end{matharray}

   415

   416   \railalias{funopts}{function\_opts}  %FIXME ??

   417

   418   \begin{rail}

   419     'primrec' target? fixes 'where' equations

   420     ;

   421     equations: (thmdecl? prop + '|')

   422     ;

   423     ('fun' | 'function') (funopts)? fixes 'where' clauses

   424     ;

   425     clauses: (thmdecl? prop ('(' 'otherwise' ')')? + '|')

   426     ;

   427     funopts: '(' (('sequential' | 'in' name | 'domintros' | 'tailrec' |

   428     'default' term) + ',') ')'

   429     ;

   430     'termination' ( term )?

   431   \end{rail}

   432

   433   \begin{descr}

   434

   435   \item [@{command (HOL) "primrec"}] defines primitive recursive

   436   functions over datatypes, see also \cite{isabelle-HOL}.

   437

   438   \item [@{command (HOL) "function"}] defines functions by general

   439   wellfounded recursion. A detailed description with examples can be

   440   found in \cite{isabelle-function}. The function is specified by a

   441   set of (possibly conditional) recursive equations with arbitrary

   442   pattern matching. The command generates proof obligations for the

   443   completeness and the compatibility of patterns.

   444

   445   The defined function is considered partial, and the resulting

   446   simplification rules (named @{text "f.psimps"}) and induction rule

   447   (named @{text "f.pinduct"}) are guarded by a generated domain

   448   predicate @{text "f_dom"}. The @{command (HOL) "termination"}

   449   command can then be used to establish that the function is total.

   450

   451   \item [@{command (HOL) "fun"}] is a shorthand notation for

   452   @{command (HOL) "function"}~@{text "(sequential)"}, followed by

   453   automated proof attempts regarding pattern matching and termination.

   454   See \cite{isabelle-function} for further details.

   455

   456   \item [@{command (HOL) "termination"}~@{text f}] commences a

   457   termination proof for the previously defined function @{text f}.  If

   458   this is omitted, the command refers to the most recent function

   459   definition.  After the proof is closed, the recursive equations and

   460   the induction principle is established.

   461

   462   \end{descr}

   463

   464   %FIXME check

   465

   466   Recursive definitions introduced by both the @{command (HOL)

   467   "primrec"} and the @{command (HOL) "function"} command accommodate

   468   reasoning by induction (cf.\ \secref{sec:cases-induct}): rule @{text

   469   "c.induct"} (where @{text c} is the name of the function definition)

   470   refers to a specific induction rule, with parameters named according

   471   to the user-specified equations.  Case names of @{command (HOL)

   472   "primrec"} are that of the datatypes involved, while those of

   473   @{command (HOL) "function"} are numbered (starting from 1).

   474

   475   The equations provided by these packages may be referred later as

   476   theorem list @{text "f.simps"}, where @{text f} is the (collective)

   477   name of the functions defined.  Individual equations may be named

   478   explicitly as well.

   479

   480   The @{command (HOL) "function"} command accepts the following

   481   options.

   482

   483   \begin{descr}

   484

   485   \item [@{text sequential}] enables a preprocessor which

   486   disambiguates overlapping patterns by making them mutually disjoint.

   487   Earlier equations take precedence over later ones.  This allows to

   488   give the specification in a format very similar to functional

   489   programming.  Note that the resulting simplification and induction

   490   rules correspond to the transformed specification, not the one given

   491   originally. This usually means that each equation given by the user

   492   may result in several theroems.  Also note that this automatic

   493   transformation only works for ML-style datatype patterns.

   494

   495   \item [@{text "\<IN> name"}] gives the target for the definition.

   496   %FIXME ?!?

   497

   498   \item [@{text domintros}] enables the automated generation of

   499   introduction rules for the domain predicate. While mostly not

   500   needed, they can be helpful in some proofs about partial functions.

   501

   502   \item [@{text tailrec}] generates the unconstrained recursive

   503   equations even without a termination proof, provided that the

   504   function is tail-recursive. This currently only works

   505

   506   \item [@{text "default d"}] allows to specify a default value for a

   507   (partial) function, which will ensure that @{text "f x = d x"}

   508   whenever @{text "x \<notin> f_dom"}.

   509

   510   \end{descr}

   511 *}

   512

   513

   514 subsection {* Proof methods related to recursive definitions *}

   515

   516 text {*

   517   \begin{matharray}{rcl}

   518     @{method_def (HOL) pat_completeness} & : & \isarmeth \\

   519     @{method_def (HOL) relation} & : & \isarmeth \\

   520     @{method_def (HOL) lexicographic_order} & : & \isarmeth \\

   521   \end{matharray}

   522

   523   \begin{rail}

   524     'relation' term

   525     ;

   526     'lexicographic\_order' (clasimpmod *)

   527     ;

   528   \end{rail}

   529

   530   \begin{descr}

   531

   532   \item [@{method (HOL) pat_completeness}] is a specialized method to

   533   solve goals regarding the completeness of pattern matching, as

   534   required by the @{command (HOL) "function"} package (cf.\

   535   \cite{isabelle-function}).

   536

   537   \item [@{method (HOL) relation}~@{text R}] introduces a termination

   538   proof using the relation @{text R}.  The resulting proof state will

   539   contain goals expressing that @{text R} is wellfounded, and that the

   540   arguments of recursive calls decrease with respect to @{text R}.

   541   Usually, this method is used as the initial proof step of manual

   542   termination proofs.

   543

   544   \item [@{method (HOL) "lexicographic_order"}] attempts a fully

   545   automated termination proof by searching for a lexicographic

   546   combination of size measures on the arguments of the function. The

   547   method accepts the same arguments as the @{method auto} method,

   548   which it uses internally to prove local descents.  The same context

   549   modifiers as for @{method auto} are accepted, see

   550   \secref{sec:clasimp}.

   551

   552   In case of failure, extensive information is printed, which can help

   553   to analyse the situation (cf.\ \cite{isabelle-function}).

   554

   555   \end{descr}

   556 *}

   557

   558

   559 subsection {* Old-style recursive function definitions (TFL) *}

   560

   561 text {*

   562   The old TFL commands @{command (HOL) "recdef"} and @{command (HOL)

   563   "recdef_tc"} for defining recursive are mostly obsolete; @{command

   564   (HOL) "function"} or @{command (HOL) "fun"} should be used instead.

   565

   566   \begin{matharray}{rcl}

   567     @{command_def (HOL) "recdef"} & : & \isartrans{theory}{theory} \\

   568     @{command_def (HOL) "recdef_tc"}@{text "\<^sup>*"} & : & \isartrans{theory}{proof(prove)} \\

   569   \end{matharray}

   570

   571   \begin{rail}

   572     'recdef' ('(' 'permissive' ')')? \\ name term (prop +) hints?

   573     ;

   574     recdeftc thmdecl? tc

   575     ;

   576     hints: '(' 'hints' (recdefmod *) ')'

   577     ;

   578     recdefmod: (('recdef\_simp' | 'recdef\_cong' | 'recdef\_wf') (() | 'add' | 'del') ':' thmrefs) | clasimpmod

   579     ;

   580     tc: nameref ('(' nat ')')?

   581     ;

   582   \end{rail}

   583

   584   \begin{descr}

   585

   586   \item [@{command (HOL) "recdef"}] defines general well-founded

   587   recursive functions (using the TFL package), see also

   588   \cite{isabelle-HOL}.  The @{text "(permissive)"}'' option tells

   589   TFL to recover from failed proof attempts, returning unfinished

   590   results.  The @{text recdef_simp}, @{text recdef_cong}, and @{text

   591   recdef_wf} hints refer to auxiliary rules to be used in the internal

   592   automated proof process of TFL.  Additional @{syntax clasimpmod}

   593   declarations (cf.\ \secref{sec:clasimp}) may be given to tune the

   594   context of the Simplifier (cf.\ \secref{sec:simplifier}) and

   595   Classical reasoner (cf.\ \secref{sec:classical}).

   596

   597   \item [@{command (HOL) "recdef_tc"}~@{text "c (i)"}] recommences the

   598   proof for leftover termination condition number @{text i} (default

   599   1) as generated by a @{command (HOL) "recdef"} definition of

   600   constant @{text c}.

   601

   602   Note that in most cases, @{command (HOL) "recdef"} is able to finish

   603   its internal proofs without manual intervention.

   604

   605   \end{descr}

   606

   607   \medskip Hints for @{command (HOL) "recdef"} may be also declared

   608   globally, using the following attributes.

   609

   610   \begin{matharray}{rcl}

   611     @{attribute_def (HOL) recdef_simp} & : & \isaratt \\

   612     @{attribute_def (HOL) recdef_cong} & : & \isaratt \\

   613     @{attribute_def (HOL) recdef_wf} & : & \isaratt \\

   614   \end{matharray}

   615

   616   \begin{rail}

   617     ('recdef\_simp' | 'recdef\_cong' | 'recdef\_wf') (() | 'add' | 'del')

   618     ;

   619   \end{rail}

   620 *}

   621

   622

   623 section {* Definition by specification \label{sec:hol-specification} *}

   624

   625 text {*

   626   \begin{matharray}{rcl}

   627     @{command_def (HOL) "specification"} & : & \isartrans{theory}{proof(prove)} \\

   628     @{command_def (HOL) "ax_specification"} & : & \isartrans{theory}{proof(prove)} \\

   629   \end{matharray}

   630

   631   \begin{rail}

   632   ('specification' | 'ax\_specification') '(' (decl +) ')' \\ (thmdecl? prop +)

   633   ;

   634   decl: ((name ':')? term '(' 'overloaded' ')'?)

   635   \end{rail}

   636

   637   \begin{descr}

   638

   639   \item [@{command (HOL) "specification"}~@{text "decls \<phi>"}] sets up a

   640   goal stating the existence of terms with the properties specified to

   641   hold for the constants given in @{text decls}.  After finishing the

   642   proof, the theory will be augmented with definitions for the given

   643   constants, as well as with theorems stating the properties for these

   644   constants.

   645

   646   \item [@{command (HOL) "ax_specification"}~@{text "decls \<phi>"}] sets

   647   up a goal stating the existence of terms with the properties

   648   specified to hold for the constants given in @{text decls}.  After

   649   finishing the proof, the theory will be augmented with axioms

   650   expressing the properties given in the first place.

   651

   652   \item [@{text decl}] declares a constant to be defined by the

   653   specification given.  The definition for the constant @{text c} is

   654   bound to the name @{text c_def} unless a theorem name is given in

   655   the declaration.  Overloaded constants should be declared as such.

   656

   657   \end{descr}

   658

   659   Whether to use @{command (HOL) "specification"} or @{command (HOL)

   660   "ax_specification"} is to some extent a matter of style.  @{command

   661   (HOL) "specification"} introduces no new axioms, and so by

   662   construction cannot introduce inconsistencies, whereas @{command

   663   (HOL) "ax_specification"} does introduce axioms, but only after the

   664   user has explicitly proven it to be safe.  A practical issue must be

   665   considered, though: After introducing two constants with the same

   666   properties using @{command (HOL) "specification"}, one can prove

   667   that the two constants are, in fact, equal.  If this might be a

   668   problem, one should use @{command (HOL) "ax_specification"}.

   669 *}

   670

   671

   672 section {* Inductive and coinductive definitions \label{sec:hol-inductive} *}

   673

   674 text {*

   675   An \textbf{inductive definition} specifies the least predicate (or

   676   set) @{text R} closed under given rules: applying a rule to elements

   677   of @{text R} yields a result within @{text R}.  For example, a

   678   structural operational semantics is an inductive definition of an

   679   evaluation relation.

   680

   681   Dually, a \textbf{coinductive definition} specifies the greatest

   682   predicate~/ set @{text R} that is consistent with given rules: every

   683   element of @{text R} can be seen as arising by applying a rule to

   684   elements of @{text R}.  An important example is using bisimulation

   685   relations to formalise equivalence of processes and infinite data

   686   structures.

   687

   688   \medskip The HOL package is related to the ZF one, which is

   689   described in a separate paper,\footnote{It appeared in CADE

   690   \cite{paulson-CADE}; a longer version is distributed with Isabelle.}

   691   which you should refer to in case of difficulties.  The package is

   692   simpler than that of ZF thanks to implicit type-checking in HOL.

   693   The types of the (co)inductive predicates (or sets) determine the

   694   domain of the fixedpoint definition, and the package does not have

   695   to use inference rules for type-checking.

   696

   697   \begin{matharray}{rcl}

   698     @{command_def (HOL) "inductive"} & : & \isarkeep{local{\dsh}theory} \\

   699     @{command_def (HOL) "inductive_set"} & : & \isarkeep{local{\dsh}theory} \\

   700     @{command_def (HOL) "coinductive"} & : & \isarkeep{local{\dsh}theory} \\

   701     @{command_def (HOL) "coinductive_set"} & : & \isarkeep{local{\dsh}theory} \\

   702     @{attribute_def (HOL) mono} & : & \isaratt \\

   703   \end{matharray}

   704

   705   \begin{rail}

   706     ('inductive' | 'inductive\_set' | 'coinductive' | 'coinductive\_set') target? fixes ('for' fixes)? \\

   707     ('where' clauses)? ('monos' thmrefs)?

   708     ;

   709     clauses: (thmdecl? prop + '|')

   710     ;

   711     'mono' (() | 'add' | 'del')

   712     ;

   713   \end{rail}

   714

   715   \begin{descr}

   716

   717   \item [@{command (HOL) "inductive"} and @{command (HOL)

   718   "coinductive"}] define (co)inductive predicates from the

   719   introduction rules given in the @{keyword "where"} part.  The

   720   optional @{keyword "for"} part contains a list of parameters of the

   721   (co)inductive predicates that remain fixed throughout the

   722   definition.  The optional @{keyword "monos"} section contains

   723   \emph{monotonicity theorems}, which are required for each operator

   724   applied to a recursive set in the introduction rules.  There

   725   \emph{must} be a theorem of the form @{text "A \<le> B \<Longrightarrow> M A \<le> M B"},

   726   for each premise @{text "M R\<^sub>i t"} in an introduction rule!

   727

   728   \item [@{command (HOL) "inductive_set"} and @{command (HOL)

   729   "coinductive_set"}] are wrappers for to the previous commands,

   730   allowing the definition of (co)inductive sets.

   731

   732   \item [@{attribute (HOL) mono}] declares monotonicity rules.  These

   733   rule are involved in the automated monotonicity proof of @{command

   734   (HOL) "inductive"}.

   735

   736   \end{descr}

   737 *}

   738

   739

   740 subsection {* Derived rules *}

   741

   742 text {*

   743   Each (co)inductive definition @{text R} adds definitions to the

   744   theory and also proves some theorems:

   745

   746   \begin{description}

   747

   748   \item [@{text R.intros}] is the list of introduction rules as proven

   749   theorems, for the recursive predicates (or sets).  The rules are

   750   also available individually, using the names given them in the

   751   theory file;

   752

   753   \item [@{text R.cases}] is the case analysis (or elimination) rule;

   754

   755   \item [@{text R.induct} or @{text R.coinduct}] is the (co)induction

   756   rule.

   757

   758   \end{description}

   759

   760   When several predicates @{text "R\<^sub>1, \<dots>, R\<^sub>n"} are

   761   defined simultaneously, the list of introduction rules is called

   762   @{text "R\<^sub>1_\<dots>_R\<^sub>n.intros"}, the case analysis rules are

   763   called @{text "R\<^sub>1.cases, \<dots>, R\<^sub>n.cases"}, and the list

   764   of mutual induction rules is called @{text

   765   "R\<^sub>1_\<dots>_R\<^sub>n.inducts"}.

   766 *}

   767

   768

   769 subsection {* Monotonicity theorems *}

   770

   771 text {*

   772   Each theory contains a default set of theorems that are used in

   773   monotonicity proofs.  New rules can be added to this set via the

   774   @{attribute (HOL) mono} attribute.  The HOL theory @{text Inductive}

   775   shows how this is done.  In general, the following monotonicity

   776   theorems may be added:

   777

   778   \begin{itemize}

   779

   780   \item Theorems of the form @{text "A \<le> B \<Longrightarrow> M A \<le> M B"}, for proving

   781   monotonicity of inductive definitions whose introduction rules have

   782   premises involving terms such as @{text "M R\<^sub>i t"}.

   783

   784   \item Monotonicity theorems for logical operators, which are of the

   785   general form @{text "(\<dots> \<longrightarrow> \<dots>) \<Longrightarrow> \<dots> (\<dots> \<longrightarrow> \<dots>) \<Longrightarrow> \<dots> \<longrightarrow> \<dots>"}.  For example, in

   786   the case of the operator @{text "\<or>"}, the corresponding theorem is

   787   $  788 \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"}}   789$

   790

   791   \item De Morgan style equations for reasoning about the polarity''

   792   of expressions, e.g.

   793   $  794 @{prop "\<not> \<not> P \<longleftrightarrow> P"} \qquad\qquad   795 @{prop "\<not> (P \<and> Q) \<longleftrightarrow> \<not> P \<or> \<not> Q"}   796$

   797

   798   \item Equations for reducing complex operators to more primitive

   799   ones whose monotonicity can easily be proved, e.g.

   800   $  801 @{prop "(P \<longrightarrow> Q) \<longleftrightarrow> \<not> P \<or> Q"} \qquad\qquad   802 @{prop "Ball A P \<equiv> \<forall>x. x \<in> A \<longrightarrow> P x"}   803$

   804

   805   \end{itemize}

   806

   807   %FIXME: Example of an inductive definition

   808 *}

   809

   810

   811 section {* Arithmetic proof support *}

   812

   813 text {*

   814   \begin{matharray}{rcl}

   815     @{method_def (HOL) arith} & : & \isarmeth \\

   816     @{method_def (HOL) arith_split} & : & \isaratt \\

   817   \end{matharray}

   818

   819   The @{method (HOL) arith} method decides linear arithmetic problems

   820   (on types @{text nat}, @{text int}, @{text real}).  Any current

   821   facts are inserted into the goal before running the procedure.

   822

   823   The @{method (HOL) arith_split} attribute declares case split rules

   824   to be expanded before the arithmetic procedure is invoked.

   825

   826   Note that a simpler (but faster) version of arithmetic reasoning is

   827   already performed by the Simplifier.

   828 *}

   829

   830

   831 section {* Cases and induction: emulating tactic scripts \label{sec:hol-induct-tac} *}

   832

   833 text {*

   834   The following important tactical tools of Isabelle/HOL have been

   835   ported to Isar.  These should be never used in proper proof texts!

   836

   837   \begin{matharray}{rcl}

   838     @{method_def (HOL) case_tac}@{text "\<^sup>*"} & : & \isarmeth \\

   839     @{method_def (HOL) induct_tac}@{text "\<^sup>*"} & : & \isarmeth \\

   840     @{method_def (HOL) ind_cases}@{text "\<^sup>*"} & : & \isarmeth \\

   841     @{command_def (HOL) "inductive_cases"} & : & \isartrans{theory}{theory} \\

   842   \end{matharray}

   843

   844   \begin{rail}

   845     'case\_tac' goalspec? term rule?

   846     ;

   847     'induct\_tac' goalspec? (insts * 'and') rule?

   848     ;

   849     'ind\_cases' (prop +) ('for' (name +)) ?

   850     ;

   851     'inductive\_cases' (thmdecl? (prop +) + 'and')

   852     ;

   853

   854     rule: ('rule' ':' thmref)

   855     ;

   856   \end{rail}

   857

   858   \begin{descr}

   859

   860   \item [@{method (HOL) case_tac} and @{method (HOL) induct_tac}]

   861   admit to reason about inductive datatypes only (unless an

   862   alternative rule is given explicitly).  Furthermore, @{method (HOL)

   863   case_tac} does a classical case split on booleans; @{method (HOL)

   864   induct_tac} allows only variables to be given as instantiation.

   865   These tactic emulations feature both goal addressing and dynamic

   866   instantiation.  Note that named rule cases are \emph{not} provided

   867   as would be by the proper @{method induct} and @{method cases} proof

   868   methods (see \secref{sec:cases-induct}).

   869

   870   \item [@{method (HOL) ind_cases} and @{command (HOL)

   871   "inductive_cases"}] provide an interface to the internal @{ML_text

   872   mk_cases} operation.  Rules are simplified in an unrestricted

   873   forward manner.

   874

   875   While @{method (HOL) ind_cases} is a proof method to apply the

   876   result immediately as elimination rules, @{command (HOL)

   877   "inductive_cases"} provides case split theorems at the theory level

   878   for later use.  The @{keyword "for"} argument of the @{method (HOL)

   879   ind_cases} method allows to specify a list of variables that should

   880   be generalized before applying the resulting rule.

   881

   882   \end{descr}

   883 *}

   884

   885

   886 section {* Executable code *}

   887

   888 text {*

   889   Isabelle/Pure provides two generic frameworks to support code

   890   generation from executable specifications.  Isabelle/HOL

   891   instantiates these mechanisms in a way that is amenable to end-user

   892   applications.

   893

   894   One framework generates code from both functional and relational

   895   programs to SML.  See \cite{isabelle-HOL} for further information

   896   (this actually covers the new-style theory format as well).

   897

   898   \begin{matharray}{rcl}

   899     @{command_def (HOL) "value"}@{text "\<^sup>*"} & : & \isarkeep{theory~|~proof} \\

   900     @{command_def (HOL) "code_module"} & : & \isartrans{theory}{theory} \\

   901     @{command_def (HOL) "code_library"} & : & \isartrans{theory}{theory} \\

   902     @{command_def (HOL) "consts_code"} & : & \isartrans{theory}{theory} \\

   903     @{command_def (HOL) "types_code"} & : & \isartrans{theory}{theory} \\

   904     @{attribute_def (HOL) code} & : & \isaratt \\

   905   \end{matharray}

   906

   907   \begin{rail}

   908   'value' term

   909   ;

   910

   911   ( 'code\_module' | 'code\_library' ) modespec ? name ? \\

   912     ( 'file' name ) ? ( 'imports' ( name + ) ) ? \\

   913     'contains' ( ( name '=' term ) + | term + )

   914   ;

   915

   916   modespec: '(' ( name * ) ')'

   917   ;

   918

   919   'consts\_code' (codespec +)

   920   ;

   921

   922   codespec: const template attachment ?

   923   ;

   924

   925   'types\_code' (tycodespec +)

   926   ;

   927

   928   tycodespec: name template attachment ?

   929   ;

   930

   931   const: term

   932   ;

   933

   934   template: '(' string ')'

   935   ;

   936

   937   attachment: 'attach' modespec ? verblbrace text verbrbrace

   938   ;

   939

   940   'code' (name)?

   941   ;

   942   \end{rail}

   943

   944   \begin{descr}

   945

   946   \item [@{command (HOL) "value"}~@{text t}] evaluates and prints a

   947   term using the code generator.

   948

   949   \end{descr}

   950

   951   \medskip The other framework generates code from functional programs

   952   (including overloading using type classes) to SML \cite{SML}, OCaml

   953   \cite{OCaml} and Haskell \cite{haskell-revised-report}.

   954   Conceptually, code generation is split up in three steps:

   955   \emph{selection} of code theorems, \emph{translation} into an

   956   abstract executable view and \emph{serialization} to a specific

   957   \emph{target language}.  See \cite{isabelle-codegen} for an

   958   introduction on how to use it.

   959

   960   \begin{matharray}{rcl}

   961     @{command_def (HOL) "export_code"}@{text "\<^sup>*"} & : & \isarkeep{theory~|~proof} \\

   962     @{command_def (HOL) "code_thms"}@{text "\<^sup>*"} & : & \isarkeep{theory~|~proof} \\

   963     @{command_def (HOL) "code_deps"}@{text "\<^sup>*"} & : & \isarkeep{theory~|~proof} \\

   964     @{command_def (HOL) "code_datatype"} & : & \isartrans{theory}{theory} \\

   965     @{command_def (HOL) "code_const"} & : & \isartrans{theory}{theory} \\

   966     @{command_def (HOL) "code_type"} & : & \isartrans{theory}{theory} \\

   967     @{command_def (HOL) "code_class"} & : & \isartrans{theory}{theory} \\

   968     @{command_def (HOL) "code_instance"} & : & \isartrans{theory}{theory} \\

   969     @{command_def (HOL) "code_monad"} & : & \isartrans{theory}{theory} \\

   970     @{command_def (HOL) "code_reserved"} & : & \isartrans{theory}{theory} \\

   971     @{command_def (HOL) "code_include"} & : & \isartrans{theory}{theory} \\

   972     @{command_def (HOL) "code_modulename"} & : & \isartrans{theory}{theory} \\

   973     @{command_def (HOL) "code_exception"} & : & \isartrans{theory}{theory} \\

   974     @{command_def (HOL) "print_codesetup"}@{text "\<^sup>*"} & : & \isarkeep{theory~|~proof} \\

   975     @{attribute_def (HOL) code} & : & \isaratt \\

   976   \end{matharray}

   977

   978   \begin{rail}

   979     'export\_code' ( constexpr + ) ? \\

   980       ( ( 'in' target ( 'module\_name' string ) ? \\

   981         ( 'file' ( string | '-' ) ) ? ( '(' args ')' ) ?) + ) ?

   982     ;

   983

   984     'code\_thms' ( constexpr + ) ?

   985     ;

   986

   987     'code\_deps' ( constexpr + ) ?

   988     ;

   989

   990     const: term

   991     ;

   992

   993     constexpr: ( const | 'name.*' | '*' )

   994     ;

   995

   996     typeconstructor: nameref

   997     ;

   998

   999     class: nameref

  1000     ;

  1001

  1002     target: 'OCaml' | 'SML' | 'Haskell'

  1003     ;

  1004

  1005     'code\_datatype' const +

  1006     ;

  1007

  1008     'code\_const' (const + 'and') \\

  1009       ( ( '(' target ( syntax ? + 'and' ) ')' ) + )

  1010     ;

  1011

  1012     'code\_type' (typeconstructor + 'and') \\

  1013       ( ( '(' target ( syntax ? + 'and' ) ')' ) + )

  1014     ;

  1015

  1016     'code\_class' (class + 'and') \\

  1017       ( ( '(' target \\

  1018         ( ( string ('where' \\

  1019           ( const ( '==' | equiv ) string ) + ) ? ) ? + 'and' ) ')' ) + )

  1020     ;

  1021

  1022     'code\_instance' (( typeconstructor '::' class ) + 'and') \\

  1023       ( ( '(' target ( '-' ? + 'and' ) ')' ) + )

  1024     ;

  1025

  1026     'code\_monad' const const target

  1027     ;

  1028

  1029     'code\_reserved' target ( string + )

  1030     ;

  1031

  1032     'code\_include' target ( string ( string | '-') )

  1033     ;

  1034

  1035     'code\_modulename' target ( ( string string ) + )

  1036     ;

  1037

  1038     'code\_exception' ( const + )

  1039     ;

  1040

  1041     syntax: string | ( 'infix' | 'infixl' | 'infixr' ) nat string

  1042     ;

  1043

  1044     'code' ('func' | 'inline') ( 'del' )?

  1045     ;

  1046   \end{rail}

  1047

  1048   \begin{descr}

  1049

  1050   \item [@{command (HOL) "export_code"}] is the canonical interface

  1051   for generating and serializing code: for a given list of constants,

  1052   code is generated for the specified target languages.  Abstract code

  1053   is cached incrementally.  If no constant is given, the currently

  1054   cached code is serialized.  If no serialization instruction is

  1055   given, only abstract code is cached.

  1056

  1057   Constants may be specified by giving them literally, referring to

  1058   all executable contants within a certain theory by giving @{text

  1059   "name.*"}, or referring to \emph{all} executable constants currently

  1060   available by giving @{text "*"}.

  1061

  1062   By default, for each involved theory one corresponding name space

  1063   module is generated.  Alternativly, a module name may be specified

  1064   after the @{keyword "module_name"} keyword; then \emph{all} code is

  1065   placed in this module.

  1066

  1067   For \emph{SML} and \emph{OCaml}, the file specification refers to a

  1068   single file; for \emph{Haskell}, it refers to a whole directory,

  1069   where code is generated in multiple files reflecting the module

  1070   hierarchy.  The file specification @{text "-"}'' denotes standard

  1071   output.  For \emph{SML}, omitting the file specification compiles

  1072   code internally in the context of the current ML session.

  1073

  1074   Serializers take an optional list of arguments in parentheses.  For

  1075   \emph{Haskell} a module name prefix may be given using the @{text

  1076   "root:"}'' argument; @{text string_classes}'' adds a @{verbatim

  1077   "deriving (Read, Show)"}'' clause to each appropriate datatype

  1078   declaration.

  1079

  1080   \item [@{command (HOL) "code_thms"}] prints a list of theorems

  1081   representing the corresponding program containing all given

  1082   constants; if no constants are given, the currently cached code

  1083   theorems are printed.

  1084

  1085   \item [@{command (HOL) "code_deps"}] visualizes dependencies of

  1086   theorems representing the corresponding program containing all given

  1087   constants; if no constants are given, the currently cached code

  1088   theorems are visualized.

  1089

  1090   \item [@{command (HOL) "code_datatype"}] specifies a constructor set

  1091   for a logical type.

  1092

  1093   \item [@{command (HOL) "code_const"}] associates a list of constants

  1094   with target-specific serializations; omitting a serialization

  1095   deletes an existing serialization.

  1096

  1097   \item [@{command (HOL) "code_type"}] associates a list of type

  1098   constructors with target-specific serializations; omitting a

  1099   serialization deletes an existing serialization.

  1100

  1101   \item [@{command (HOL) "code_class"}] associates a list of classes

  1102   with target-specific class names; in addition, constants associated

  1103   with this class may be given target-specific names used for instance

  1104   declarations; omitting a serialization deletes an existing

  1105   serialization.  This applies only to \emph{Haskell}.

  1106

  1107   \item [@{command (HOL) "code_instance"}] declares a list of type

  1108   constructor / class instance relations as already present'' for a

  1109   given target.  Omitting a @{text "-"}'' deletes an existing

  1110   already present'' declaration.  This applies only to

  1111   \emph{Haskell}.

  1112

  1113   \item [@{command (HOL) "code_monad"}] provides an auxiliary

  1114   mechanism to generate monadic code.

  1115

  1116   \item [@{command (HOL) "code_reserved"}] declares a list of names as

  1117   reserved for a given target, preventing it to be shadowed by any

  1118   generated code.

  1119

  1120   \item [@{command (HOL) "code_include"}] adds arbitrary named content

  1121   (include'') to generated code.  A as last argument @{text "-"}''

  1122   will remove an already added include''.

  1123

  1124   \item [@{command (HOL) "code_modulename"}] declares aliasings from

  1125   one module name onto another.

  1126

  1127   \item [@{command (HOL) "code_exception"}] declares constants which

  1128   are not required to have a definition by a defining equations; these

  1129   are mapped on exceptions instead.

  1130

  1131   \item [@{attribute (HOL) code}~@{text func}] explicitly selects (or

  1132   with option @{text "del:"}'' deselects) a defining equation for

  1133   code generation.  Usually packages introducing defining equations

  1134   provide a resonable default setup for selection.

  1135

  1136   \item [@{attribute (HOL) code}@{text inline}] declares (or with

  1137   option @{text "del:"}'' removes) inlining theorems which are

  1138   applied as rewrite rules to any defining equation during

  1139   preprocessing.

  1140

  1141   \item [@{command (HOL) "print_codesetup"}] gives an overview on

  1142   selected defining equations, code generator datatypes and

  1143   preprocessor setup.

  1144

  1145   \end{descr}

  1146 *}

  1147

  1148 end

  1149