doc-src/IsarRef/Thy/HOL_Specific.thy
author wenzelm
Thu Nov 13 21:34:23 2008 +0100 (2008-11-13)
changeset 28752 754f10154d73
parent 28687 150a8a1eae1a
child 28760 cbc435f7b16b
permissions -rw-r--r--
tuned;
     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 @{attribute simp} or @{attribute iff}
    57   declarations).  Rules @{text Rep_t_cases}/@{text Rep_t_induct}, and
    58   @{text Abs_t_cases}/@{text Abs_t_induct} provide alternative views
    59   on 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 [@{attribute (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}{proof} \\
   366   \end{matharray}
   367 
   368   \begin{rail}
   369     'datatype' (dtspec + 'and')
   370     ;
   371     'rep\_datatype' ('(' (name +) ')')? (term +)
   372     ;
   373 
   374     dtspec: parname? typespec infix? '=' (cons + '|')
   375     ;
   376     cons: name (type *) mixfix?
   377   \end{rail}
   378 
   379   \begin{descr}
   380 
   381   \item [@{command (HOL) "datatype"}] defines inductive datatypes in
   382   HOL.
   383 
   384   \item [@{command (HOL) "rep_datatype"}] represents existing types as
   385   inductive ones, generating the standard infrastructure of derived
   386   concepts (primitive recursion etc.).
   387 
   388   \end{descr}
   389 
   390   The induction and exhaustion theorems generated provide case names
   391   according to the constructors involved, while parameters are named
   392   after the types (see also \secref{sec:cases-induct}).
   393 
   394   See \cite{isabelle-HOL} for more details on datatypes, but beware of
   395   the old-style theory syntax being used there!  Apart from proper
   396   proof methods for case-analysis and induction, there are also
   397   emulations of ML tactics @{method (HOL) case_tac} and @{method (HOL)
   398   induct_tac} available, see \secref{sec:hol-induct-tac}; these admit
   399   to refer directly to the internal structure of subgoals (including
   400   internally bound parameters).
   401 *}
   402 
   403 
   404 section {* Recursive functions \label{sec:recursion} *}
   405 
   406 text {*
   407   \begin{matharray}{rcl}
   408     @{command_def (HOL) "primrec"} & : & \isarkeep{local{\dsh}theory} \\
   409     @{command_def (HOL) "fun"} & : & \isarkeep{local{\dsh}theory} \\
   410     @{command_def (HOL) "function"} & : & \isartrans{local{\dsh}theory}{proof(prove)} \\
   411     @{command_def (HOL) "termination"} & : & \isartrans{local{\dsh}theory}{proof(prove)} \\
   412   \end{matharray}
   413 
   414   \begin{rail}
   415     'primrec' target? fixes 'where' equations
   416     ;
   417     equations: (thmdecl? prop + '|')
   418     ;
   419     ('fun' | 'function') target? functionopts? fixes 'where' clauses
   420     ;
   421     clauses: (thmdecl? prop ('(' 'otherwise' ')')? + '|')
   422     ;
   423     functionopts: '(' (('sequential' | 'domintros' | 'tailrec' | 'default' term) + ',') ')'
   424     ;
   425     'termination' ( term )?
   426   \end{rail}
   427 
   428   \begin{descr}
   429 
   430   \item [@{command (HOL) "primrec"}] defines primitive recursive
   431   functions over datatypes, see also \cite{isabelle-HOL}.
   432 
   433   \item [@{command (HOL) "function"}] defines functions by general
   434   wellfounded recursion. A detailed description with examples can be
   435   found in \cite{isabelle-function}. The function is specified by a
   436   set of (possibly conditional) recursive equations with arbitrary
   437   pattern matching. The command generates proof obligations for the
   438   completeness and the compatibility of patterns.
   439 
   440   The defined function is considered partial, and the resulting
   441   simplification rules (named @{text "f.psimps"}) and induction rule
   442   (named @{text "f.pinduct"}) are guarded by a generated domain
   443   predicate @{text "f_dom"}. The @{command (HOL) "termination"}
   444   command can then be used to establish that the function is total.
   445 
   446   \item [@{command (HOL) "fun"}] is a shorthand notation for
   447   ``@{command (HOL) "function"}~@{text "(sequential)"}, followed by
   448   automated proof attempts regarding pattern matching and termination.
   449   See \cite{isabelle-function} for further details.
   450 
   451   \item [@{command (HOL) "termination"}~@{text f}] commences a
   452   termination proof for the previously defined function @{text f}.  If
   453   this is omitted, the command refers to the most recent function
   454   definition.  After the proof is closed, the recursive equations and
   455   the induction principle is established.
   456 
   457   \end{descr}
   458 
   459   %FIXME check
   460 
   461   Recursive definitions introduced by the @{command (HOL) "function"}
   462   command accommodate
   463   reasoning by induction (cf.\ \secref{sec:cases-induct}): rule @{text
   464   "c.induct"} (where @{text c} is the name of the function definition)
   465   refers to a specific induction rule, with parameters named according
   466   to the user-specified equations.
   467   For the @{command (HOL) "primrec"} the induction principle coincides
   468   with structural recursion on the datatype the recursion is carried
   469   out.
   470   Case names of @{command (HOL)
   471   "primrec"} are that of the datatypes involved, while those of
   472   @{command (HOL) "function"} are numbered (starting from 1).
   473 
   474   The equations provided by these packages may be referred later as
   475   theorem list @{text "f.simps"}, where @{text f} is the (collective)
   476   name of the functions defined.  Individual equations may be named
   477   explicitly as well.
   478 
   479   The @{command (HOL) "function"} command accepts the following
   480   options.
   481 
   482   \begin{descr}
   483 
   484   \item [@{text sequential}] enables a preprocessor which
   485   disambiguates overlapping patterns by making them mutually disjoint.
   486   Earlier equations take precedence over later ones.  This allows to
   487   give the specification in a format very similar to functional
   488   programming.  Note that the resulting simplification and induction
   489   rules correspond to the transformed specification, not the one given
   490   originally. This usually means that each equation given by the user
   491   may result in several theroems.  Also note that this automatic
   492   transformation only works for ML-style datatype patterns.
   493 
   494   \item [@{text domintros}] enables the automated generation of
   495   introduction rules for the domain predicate. While mostly not
   496   needed, they can be helpful in some proofs about partial functions.
   497 
   498   \item [@{text tailrec}] generates the unconstrained recursive
   499   equations even without a termination proof, provided that the
   500   function is tail-recursive. This currently only works
   501 
   502   \item [@{text "default d"}] allows to specify a default value for a
   503   (partial) function, which will ensure that @{text "f x = d x"}
   504   whenever @{text "x \<notin> f_dom"}.
   505 
   506   \end{descr}
   507 *}
   508 
   509 
   510 subsection {* Proof methods related to recursive definitions *}
   511 
   512 text {*
   513   \begin{matharray}{rcl}
   514     @{method_def (HOL) pat_completeness} & : & \isarmeth \\
   515     @{method_def (HOL) relation} & : & \isarmeth \\
   516     @{method_def (HOL) lexicographic_order} & : & \isarmeth \\
   517   \end{matharray}
   518 
   519   \begin{rail}
   520     'relation' term
   521     ;
   522     'lexicographic\_order' (clasimpmod *)
   523     ;
   524   \end{rail}
   525 
   526   \begin{descr}
   527 
   528   \item [@{method (HOL) pat_completeness}] is a specialized method to
   529   solve goals regarding the completeness of pattern matching, as
   530   required by the @{command (HOL) "function"} package (cf.\
   531   \cite{isabelle-function}).
   532 
   533   \item [@{method (HOL) relation}~@{text R}] introduces a termination
   534   proof using the relation @{text R}.  The resulting proof state will
   535   contain goals expressing that @{text R} is wellfounded, and that the
   536   arguments of recursive calls decrease with respect to @{text R}.
   537   Usually, this method is used as the initial proof step of manual
   538   termination proofs.
   539 
   540   \item [@{method (HOL) "lexicographic_order"}] attempts a fully
   541   automated termination proof by searching for a lexicographic
   542   combination of size measures on the arguments of the function. The
   543   method accepts the same arguments as the @{method auto} method,
   544   which it uses internally to prove local descents.  The same context
   545   modifiers as for @{method auto} are accepted, see
   546   \secref{sec:clasimp}.
   547 
   548   In case of failure, extensive information is printed, which can help
   549   to analyse the situation (cf.\ \cite{isabelle-function}).
   550 
   551   \end{descr}
   552 *}
   553 
   554 
   555 subsection {* Old-style recursive function definitions (TFL) *}
   556 
   557 text {*
   558   The old TFL commands @{command (HOL) "recdef"} and @{command (HOL)
   559   "recdef_tc"} for defining recursive are mostly obsolete; @{command
   560   (HOL) "function"} or @{command (HOL) "fun"} should be used instead.
   561 
   562   \begin{matharray}{rcl}
   563     @{command_def (HOL) "recdef"} & : & \isartrans{theory}{theory} \\
   564     @{command_def (HOL) "recdef_tc"}@{text "\<^sup>*"} & : & \isartrans{theory}{proof(prove)} \\
   565   \end{matharray}
   566 
   567   \begin{rail}
   568     'recdef' ('(' 'permissive' ')')? \\ name term (prop +) hints?
   569     ;
   570     recdeftc thmdecl? tc
   571     ;
   572     hints: '(' 'hints' (recdefmod *) ')'
   573     ;
   574     recdefmod: (('recdef\_simp' | 'recdef\_cong' | 'recdef\_wf') (() | 'add' | 'del') ':' thmrefs) | clasimpmod
   575     ;
   576     tc: nameref ('(' nat ')')?
   577     ;
   578   \end{rail}
   579 
   580   \begin{descr}
   581   
   582   \item [@{command (HOL) "recdef"}] defines general well-founded
   583   recursive functions (using the TFL package), see also
   584   \cite{isabelle-HOL}.  The ``@{text "(permissive)"}'' option tells
   585   TFL to recover from failed proof attempts, returning unfinished
   586   results.  The @{text recdef_simp}, @{text recdef_cong}, and @{text
   587   recdef_wf} hints refer to auxiliary rules to be used in the internal
   588   automated proof process of TFL.  Additional @{syntax clasimpmod}
   589   declarations (cf.\ \secref{sec:clasimp}) may be given to tune the
   590   context of the Simplifier (cf.\ \secref{sec:simplifier}) and
   591   Classical reasoner (cf.\ \secref{sec:classical}).
   592   
   593   \item [@{command (HOL) "recdef_tc"}~@{text "c (i)"}] recommences the
   594   proof for leftover termination condition number @{text i} (default
   595   1) as generated by a @{command (HOL) "recdef"} definition of
   596   constant @{text c}.
   597   
   598   Note that in most cases, @{command (HOL) "recdef"} is able to finish
   599   its internal proofs without manual intervention.
   600 
   601   \end{descr}
   602 
   603   \medskip Hints for @{command (HOL) "recdef"} may be also declared
   604   globally, using the following attributes.
   605 
   606   \begin{matharray}{rcl}
   607     @{attribute_def (HOL) recdef_simp} & : & \isaratt \\
   608     @{attribute_def (HOL) recdef_cong} & : & \isaratt \\
   609     @{attribute_def (HOL) recdef_wf} & : & \isaratt \\
   610   \end{matharray}
   611 
   612   \begin{rail}
   613     ('recdef\_simp' | 'recdef\_cong' | 'recdef\_wf') (() | 'add' | 'del')
   614     ;
   615   \end{rail}
   616 *}
   617 
   618 
   619 section {* Inductive and coinductive definitions \label{sec:hol-inductive} *}
   620 
   621 text {*
   622   An \textbf{inductive definition} specifies the least predicate (or
   623   set) @{text R} closed under given rules: applying a rule to elements
   624   of @{text R} yields a result within @{text R}.  For example, a
   625   structural operational semantics is an inductive definition of an
   626   evaluation relation.
   627 
   628   Dually, a \textbf{coinductive definition} specifies the greatest
   629   predicate~/ set @{text R} that is consistent with given rules: every
   630   element of @{text R} can be seen as arising by applying a rule to
   631   elements of @{text R}.  An important example is using bisimulation
   632   relations to formalise equivalence of processes and infinite data
   633   structures.
   634 
   635   \medskip The HOL package is related to the ZF one, which is
   636   described in a separate paper,\footnote{It appeared in CADE
   637   \cite{paulson-CADE}; a longer version is distributed with Isabelle.}
   638   which you should refer to in case of difficulties.  The package is
   639   simpler than that of ZF thanks to implicit type-checking in HOL.
   640   The types of the (co)inductive predicates (or sets) determine the
   641   domain of the fixedpoint definition, and the package does not have
   642   to use inference rules for type-checking.
   643 
   644   \begin{matharray}{rcl}
   645     @{command_def (HOL) "inductive"} & : & \isarkeep{local{\dsh}theory} \\
   646     @{command_def (HOL) "inductive_set"} & : & \isarkeep{local{\dsh}theory} \\
   647     @{command_def (HOL) "coinductive"} & : & \isarkeep{local{\dsh}theory} \\
   648     @{command_def (HOL) "coinductive_set"} & : & \isarkeep{local{\dsh}theory} \\
   649     @{attribute_def (HOL) mono} & : & \isaratt \\
   650   \end{matharray}
   651 
   652   \begin{rail}
   653     ('inductive' | 'inductive\_set' | 'coinductive' | 'coinductive\_set') target? fixes ('for' fixes)? \\
   654     ('where' clauses)? ('monos' thmrefs)?
   655     ;
   656     clauses: (thmdecl? prop + '|')
   657     ;
   658     'mono' (() | 'add' | 'del')
   659     ;
   660   \end{rail}
   661 
   662   \begin{descr}
   663 
   664   \item [@{command (HOL) "inductive"} and @{command (HOL)
   665   "coinductive"}] define (co)inductive predicates from the
   666   introduction rules given in the @{keyword "where"} part.  The
   667   optional @{keyword "for"} part contains a list of parameters of the
   668   (co)inductive predicates that remain fixed throughout the
   669   definition.  The optional @{keyword "monos"} section contains
   670   \emph{monotonicity theorems}, which are required for each operator
   671   applied to a recursive set in the introduction rules.  There
   672   \emph{must} be a theorem of the form @{text "A \<le> B \<Longrightarrow> M A \<le> M B"},
   673   for each premise @{text "M R\<^sub>i t"} in an introduction rule!
   674 
   675   \item [@{command (HOL) "inductive_set"} and @{command (HOL)
   676   "coinductive_set"}] are wrappers for to the previous commands,
   677   allowing the definition of (co)inductive sets.
   678 
   679   \item [@{attribute (HOL) mono}] declares monotonicity rules.  These
   680   rule are involved in the automated monotonicity proof of @{command
   681   (HOL) "inductive"}.
   682 
   683   \end{descr}
   684 *}
   685 
   686 
   687 subsection {* Derived rules *}
   688 
   689 text {*
   690   Each (co)inductive definition @{text R} adds definitions to the
   691   theory and also proves some theorems:
   692 
   693   \begin{description}
   694 
   695   \item [@{text R.intros}] is the list of introduction rules as proven
   696   theorems, for the recursive predicates (or sets).  The rules are
   697   also available individually, using the names given them in the
   698   theory file;
   699 
   700   \item [@{text R.cases}] is the case analysis (or elimination) rule;
   701 
   702   \item [@{text R.induct} or @{text R.coinduct}] is the (co)induction
   703   rule.
   704 
   705   \end{description}
   706 
   707   When several predicates @{text "R\<^sub>1, \<dots>, R\<^sub>n"} are
   708   defined simultaneously, the list of introduction rules is called
   709   @{text "R\<^sub>1_\<dots>_R\<^sub>n.intros"}, the case analysis rules are
   710   called @{text "R\<^sub>1.cases, \<dots>, R\<^sub>n.cases"}, and the list
   711   of mutual induction rules is called @{text
   712   "R\<^sub>1_\<dots>_R\<^sub>n.inducts"}.
   713 *}
   714 
   715 
   716 subsection {* Monotonicity theorems *}
   717 
   718 text {*
   719   Each theory contains a default set of theorems that are used in
   720   monotonicity proofs.  New rules can be added to this set via the
   721   @{attribute (HOL) mono} attribute.  The HOL theory @{text Inductive}
   722   shows how this is done.  In general, the following monotonicity
   723   theorems may be added:
   724 
   725   \begin{itemize}
   726 
   727   \item Theorems of the form @{text "A \<le> B \<Longrightarrow> M A \<le> M B"}, for proving
   728   monotonicity of inductive definitions whose introduction rules have
   729   premises involving terms such as @{text "M R\<^sub>i t"}.
   730 
   731   \item Monotonicity theorems for logical operators, which are of the
   732   general form @{text "(\<dots> \<longrightarrow> \<dots>) \<Longrightarrow> \<dots> (\<dots> \<longrightarrow> \<dots>) \<Longrightarrow> \<dots> \<longrightarrow> \<dots>"}.  For example, in
   733   the case of the operator @{text "\<or>"}, the corresponding theorem is
   734   \[
   735   \infer{@{text "P\<^sub>1 \<or> P\<^sub>2 \<longrightarrow> Q\<^sub>1 \<or> Q\<^sub>2"}}{@{text "P\<^sub>1 \<longrightarrow> Q\<^sub>1"} & @{text "P\<^sub>2 \<longrightarrow> Q\<^sub>2"}}
   736   \]
   737 
   738   \item De Morgan style equations for reasoning about the ``polarity''
   739   of expressions, e.g.
   740   \[
   741   @{prop "\<not> \<not> P \<longleftrightarrow> P"} \qquad\qquad
   742   @{prop "\<not> (P \<and> Q) \<longleftrightarrow> \<not> P \<or> \<not> Q"}
   743   \]
   744 
   745   \item Equations for reducing complex operators to more primitive
   746   ones whose monotonicity can easily be proved, e.g.
   747   \[
   748   @{prop "(P \<longrightarrow> Q) \<longleftrightarrow> \<not> P \<or> Q"} \qquad\qquad
   749   @{prop "Ball A P \<equiv> \<forall>x. x \<in> A \<longrightarrow> P x"}
   750   \]
   751 
   752   \end{itemize}
   753 
   754   %FIXME: Example of an inductive definition
   755 *}
   756 
   757 
   758 section {* Arithmetic proof support *}
   759 
   760 text {*
   761   \begin{matharray}{rcl}
   762     @{method_def (HOL) arith} & : & \isarmeth \\
   763     @{attribute_def (HOL) arith_split} & : & \isaratt \\
   764   \end{matharray}
   765 
   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.
   769 
   770   The @{attribute (HOL) arith_split} attribute declares case split
   771   rules to be expanded before the arithmetic procedure is invoked.
   772 
   773   Note that a simpler (but faster) version of arithmetic reasoning is
   774   already performed by the Simplifier.
   775 *}
   776 
   777 
   778 section {* Invoking automated reasoning tools -- The Sledgehammer *}
   779 
   780 text {*
   781   Isabelle/HOL includes a generic \emph{ATP manager} that allows
   782   external automated reasoning tools to crunch a pending goal.
   783   Supported provers include E\footnote{\url{http://www.eprover.org}},
   784   SPASS\footnote{\url{http://www.spass-prover.org/}}, and Vampire.
   785   There is also a wrapper to invoke provers remotely via the
   786   SystemOnTPTP\footnote{\url{http://www.cs.miami.edu/~tptp/cgi-bin/SystemOnTPTP}}
   787   web service.
   788 
   789   The problem passed to external provers consists of the goal together
   790   with a smart selection of lemmas from the current theory context.
   791   The result of a successful proof search is some source text that
   792   usually reconstructs the proof within Isabelle, without requiring
   793   external provers again.  The Metis
   794   prover\footnote{\url{http://www.gilith.com/software/metis/}} that is
   795   integrated into Isabelle/HOL is being used here.
   796 
   797   In this mode of operation, heavy means of automated reasoning are
   798   used as a strong relevance filter, while the main proof checking
   799   works via explicit inferences going through the Isabelle kernel.
   800   Moreover, rechecking Isabelle proof texts with already specified
   801   auxiliary facts is much faster than performing fully automated
   802   search over and over again.
   803 
   804   \begin{matharray}{rcl}
   805     @{command_def (HOL) "sledgehammer"}@{text "\<^sup>*"} & : & \isarkeep{proof} \\
   806     @{command_def (HOL) "print_atps"}@{text "\<^sup>*"} & : & \isarkeep{theory~|~proof} \\
   807     @{command_def (HOL) "atp_info"}@{text "\<^sup>*"} & : & \isarkeep{\cdot} \\
   808     @{command_def (HOL) "atp_kill"}@{text "\<^sup>*"} & : & \isarkeep{\cdot} \\
   809     @{method_def (HOL) metis} & : & \isarmeth \\
   810   \end{matharray}
   811 
   812   \begin{rail}
   813   'sledgehammer' (nameref *)
   814   ;
   815 
   816   'metis' thmrefs
   817   ;
   818   \end{rail}
   819 
   820   \begin{descr}
   821 
   822   \item [@{command (HOL) sledgehammer}~@{text "prover\<^sub>1 \<dots>
   823   prover\<^sub>n"}] invokes the specified automated theorem provers on
   824   the first subgoal.  Provers are run in parallel, the first
   825   successful result is displayed, and the other attempts are
   826   terminated.
   827 
   828   Provers are defined in the theory context, see also @{command (HOL)
   829   print_atps}.  If no provers are given as arguments to @{command
   830   (HOL) sledgehammer}, the system refers to the default defined as
   831   ``ATP provers'' preference by the user interface.
   832 
   833   There are additional preferences for timeout (default: 60 seconds),
   834   and the maximum number of independent prover processes (default: 5);
   835   excessive provers are automatically terminated.
   836 
   837   \item [@{command (HOL) print_atps}] prints the list of automated
   838   theorem provers available to the @{command (HOL) sledgehammer}
   839   command.
   840 
   841   \item [@{command (HOL) atp_info}] prints information about presently
   842   running provers, including elapsed runtime, and the remaining time
   843   until timeout.
   844 
   845   \item [@{command (HOL) atp_kill}] terminates all presently running
   846   provers.
   847 
   848   \item [@{method (HOL) metis}~@{text "facts"}] invokes the Metis
   849   prover with the given facts.  Metis is an automated proof tool of
   850   medium strength, but is fully integrated into Isabelle/HOL, with
   851   explicit inferences going through the kernel.  Thus its results are
   852   guaranteed to be ``correct by construction''.
   853 
   854   Note that all facts used with Metis need to be specified as explicit
   855   arguments.  There are no rule declarations as for other Isabelle
   856   provers, like @{method blast} or @{method fast}.
   857 
   858   \end{descr}
   859 *}
   860 
   861 
   862 section {* Unstructured case analysis and induction \label{sec:hol-induct-tac} *}
   863 
   864 text {*
   865   The following tools of Isabelle/HOL support cases analysis and
   866   induction in unstructured tactic scripts; see also
   867   \secref{sec:cases-induct} for proper Isar versions of similar ideas.
   868 
   869   \begin{matharray}{rcl}
   870     @{method_def (HOL) case_tac}@{text "\<^sup>*"} & : & \isarmeth \\
   871     @{method_def (HOL) induct_tac}@{text "\<^sup>*"} & : & \isarmeth \\
   872     @{method_def (HOL) ind_cases}@{text "\<^sup>*"} & : & \isarmeth \\
   873     @{command_def (HOL) "inductive_cases"}@{text "\<^sup>*"} & : & \isartrans{theory}{theory} \\
   874   \end{matharray}
   875 
   876   \begin{rail}
   877     'case\_tac' goalspec? term rule?
   878     ;
   879     'induct\_tac' goalspec? (insts * 'and') rule?
   880     ;
   881     'ind\_cases' (prop +) ('for' (name +)) ?
   882     ;
   883     'inductive\_cases' (thmdecl? (prop +) + 'and')
   884     ;
   885 
   886     rule: ('rule' ':' thmref)
   887     ;
   888   \end{rail}
   889 
   890   \begin{descr}
   891 
   892   \item [@{method (HOL) case_tac} and @{method (HOL) induct_tac}]
   893   admit to reason about inductive types.  Rules are selected according
   894   to the declarations by the @{attribute cases} and @{attribute
   895   induct} attributes, cf.\ \secref{sec:cases-induct}.  The @{command
   896   (HOL) datatype} package already takes care of this.
   897 
   898   These unstructured tactics feature both goal addressing and dynamic
   899   instantiation.  Note that named rule cases are \emph{not} provided
   900   as would be by the proper @{method cases} and @{method induct} proof
   901   methods (see \secref{sec:cases-induct}).  Unlike the @{method
   902   induct} method, @{method induct_tac} does not handle structured rule
   903   statements, only the compact object-logic conclusion of the subgoal
   904   being addressed.
   905   
   906   \item [@{method (HOL) ind_cases} and @{command (HOL)
   907   "inductive_cases"}] provide an interface to the internal @{ML_text
   908   mk_cases} operation.  Rules are simplified in an unrestricted
   909   forward manner.
   910 
   911   While @{method (HOL) ind_cases} is a proof method to apply the
   912   result immediately as elimination rules, @{command (HOL)
   913   "inductive_cases"} provides case split theorems at the theory level
   914   for later use.  The @{keyword "for"} argument of the @{method (HOL)
   915   ind_cases} method allows to specify a list of variables that should
   916   be generalized before applying the resulting rule.
   917 
   918   \end{descr}
   919 *}
   920 
   921 
   922 section {* Executable code *}
   923 
   924 text {*
   925   Isabelle/Pure provides two generic frameworks to support code
   926   generation from executable specifications.  Isabelle/HOL
   927   instantiates these mechanisms in a way that is amenable to end-user
   928   applications.
   929 
   930   One framework generates code from both functional and relational
   931   programs to SML.  See \cite{isabelle-HOL} for further information
   932   (this actually covers the new-style theory format as well).
   933 
   934   \begin{matharray}{rcl}
   935     @{command_def (HOL) "value"}@{text "\<^sup>*"} & : & \isarkeep{theory~|~proof} \\
   936     @{command_def (HOL) "code_module"} & : & \isartrans{theory}{theory} \\
   937     @{command_def (HOL) "code_library"} & : & \isartrans{theory}{theory} \\
   938     @{command_def (HOL) "consts_code"} & : & \isartrans{theory}{theory} \\
   939     @{command_def (HOL) "types_code"} & : & \isartrans{theory}{theory} \\  
   940     @{attribute_def (HOL) code} & : & \isaratt \\
   941   \end{matharray}
   942 
   943   \begin{rail}
   944   'value' term
   945   ;
   946 
   947   ( 'code\_module' | 'code\_library' ) modespec ? name ? \\
   948     ( 'file' name ) ? ( 'imports' ( name + ) ) ? \\
   949     'contains' ( ( name '=' term ) + | term + )
   950   ;
   951 
   952   modespec: '(' ( name * ) ')'
   953   ;
   954 
   955   'consts\_code' (codespec +)
   956   ;
   957 
   958   codespec: const template attachment ?
   959   ;
   960 
   961   'types\_code' (tycodespec +)
   962   ;
   963 
   964   tycodespec: name template attachment ?
   965   ;
   966 
   967   const: term
   968   ;
   969 
   970   template: '(' string ')'
   971   ;
   972 
   973   attachment: 'attach' modespec ? verblbrace text verbrbrace
   974   ;
   975 
   976   'code' (name)?
   977   ;
   978   \end{rail}
   979 
   980   \begin{descr}
   981 
   982   \item [@{command (HOL) "value"}~@{text t}] evaluates and prints a
   983   term using the code generator.
   984 
   985   \end{descr}
   986 
   987   \medskip The other framework generates code from functional programs
   988   (including overloading using type classes) to SML \cite{SML}, OCaml
   989   \cite{OCaml} and Haskell \cite{haskell-revised-report}.
   990   Conceptually, code generation is split up in three steps:
   991   \emph{selection} of code theorems, \emph{translation} into an
   992   abstract executable view and \emph{serialization} to a specific
   993   \emph{target language}.  See \cite{isabelle-codegen} for an
   994   introduction on how to use it.
   995 
   996   \begin{matharray}{rcl}
   997     @{command_def (HOL) "export_code"}@{text "\<^sup>*"} & : & \isarkeep{theory~|~proof} \\
   998     @{command_def (HOL) "code_thms"}@{text "\<^sup>*"} & : & \isarkeep{theory~|~proof} \\
   999     @{command_def (HOL) "code_deps"}@{text "\<^sup>*"} & : & \isarkeep{theory~|~proof} \\
  1000     @{command_def (HOL) "code_datatype"} & : & \isartrans{theory}{theory} \\
  1001     @{command_def (HOL) "code_const"} & : & \isartrans{theory}{theory} \\
  1002     @{command_def (HOL) "code_type"} & : & \isartrans{theory}{theory} \\
  1003     @{command_def (HOL) "code_class"} & : & \isartrans{theory}{theory} \\
  1004     @{command_def (HOL) "code_instance"} & : & \isartrans{theory}{theory} \\
  1005     @{command_def (HOL) "code_monad"} & : & \isartrans{theory}{theory} \\
  1006     @{command_def (HOL) "code_reserved"} & : & \isartrans{theory}{theory} \\
  1007     @{command_def (HOL) "code_include"} & : & \isartrans{theory}{theory} \\
  1008     @{command_def (HOL) "code_modulename"} & : & \isartrans{theory}{theory} \\
  1009     @{command_def (HOL) "code_abort"} & : & \isartrans{theory}{theory} \\
  1010     @{command_def (HOL) "print_codesetup"}@{text "\<^sup>*"} & : & \isarkeep{theory~|~proof} \\
  1011     @{attribute_def (HOL) code} & : & \isaratt \\
  1012   \end{matharray}
  1013 
  1014   \begin{rail}
  1015     'export\_code' ( constexpr + ) ? \\
  1016       ( ( 'in' target ( 'module\_name' string ) ? \\
  1017         ( 'file' ( string | '-' ) ) ? ( '(' args ')' ) ?) + ) ?
  1018     ;
  1019 
  1020     'code\_thms' ( constexpr + ) ?
  1021     ;
  1022 
  1023     'code\_deps' ( constexpr + ) ?
  1024     ;
  1025 
  1026     const: term
  1027     ;
  1028 
  1029     constexpr: ( const | 'name.*' | '*' )
  1030     ;
  1031 
  1032     typeconstructor: nameref
  1033     ;
  1034 
  1035     class: nameref
  1036     ;
  1037 
  1038     target: 'OCaml' | 'SML' | 'Haskell'
  1039     ;
  1040 
  1041     'code\_datatype' const +
  1042     ;
  1043 
  1044     'code\_const' (const + 'and') \\
  1045       ( ( '(' target ( syntax ? + 'and' ) ')' ) + )
  1046     ;
  1047 
  1048     'code\_type' (typeconstructor + 'and') \\
  1049       ( ( '(' target ( syntax ? + 'and' ) ')' ) + )
  1050     ;
  1051 
  1052     'code\_class' (class + 'and') \\
  1053       ( ( '(' target \\ ( string ? + 'and' ) ')' ) + )
  1054     ;
  1055 
  1056     'code\_instance' (( typeconstructor '::' class ) + 'and') \\
  1057       ( ( '(' target ( '-' ? + 'and' ) ')' ) + )
  1058     ;
  1059 
  1060     'code\_monad' const const target
  1061     ;
  1062 
  1063     'code\_reserved' target ( string + )
  1064     ;
  1065 
  1066     'code\_include' target ( string ( string | '-') )
  1067     ;
  1068 
  1069     'code\_modulename' target ( ( string string ) + )
  1070     ;
  1071 
  1072     'code\_abort' ( const + )
  1073     ;
  1074 
  1075     syntax: string | ( 'infix' | 'infixl' | 'infixr' ) nat string
  1076     ;
  1077 
  1078     'code' ( 'inline' ) ? ( 'del' ) ?
  1079     ;
  1080   \end{rail}
  1081 
  1082   \begin{descr}
  1083 
  1084   \item [@{command (HOL) "export_code"}] is the canonical interface
  1085   for generating and serializing code: for a given list of constants,
  1086   code is generated for the specified target languages.  Abstract code
  1087   is cached incrementally.  If no constant is given, the currently
  1088   cached code is serialized.  If no serialization instruction is
  1089   given, only abstract code is cached.
  1090 
  1091   Constants may be specified by giving them literally, referring to
  1092   all executable contants within a certain theory by giving @{text
  1093   "name.*"}, or referring to \emph{all} executable constants currently
  1094   available by giving @{text "*"}.
  1095 
  1096   By default, for each involved theory one corresponding name space
  1097   module is generated.  Alternativly, a module name may be specified
  1098   after the @{keyword "module_name"} keyword; then \emph{all} code is
  1099   placed in this module.
  1100 
  1101   For \emph{SML} and \emph{OCaml}, the file specification refers to a
  1102   single file; for \emph{Haskell}, it refers to a whole directory,
  1103   where code is generated in multiple files reflecting the module
  1104   hierarchy.  The file specification ``@{text "-"}'' denotes standard
  1105   output.  For \emph{SML}, omitting the file specification compiles
  1106   code internally in the context of the current ML session.
  1107 
  1108   Serializers take an optional list of arguments in parentheses.  For
  1109   \emph{Haskell} a module name prefix may be given using the ``@{text
  1110   "root:"}'' argument; ``@{text string_classes}'' adds a ``@{verbatim
  1111   "deriving (Read, Show)"}'' clause to each appropriate datatype
  1112   declaration.
  1113 
  1114   \item [@{command (HOL) "code_thms"}] prints a list of theorems
  1115   representing the corresponding program containing all given
  1116   constants; if no constants are given, the currently cached code
  1117   theorems are printed.
  1118 
  1119   \item [@{command (HOL) "code_deps"}] visualizes dependencies of
  1120   theorems representing the corresponding program containing all given
  1121   constants; if no constants are given, the currently cached code
  1122   theorems are visualized.
  1123 
  1124   \item [@{command (HOL) "code_datatype"}] specifies a constructor set
  1125   for a logical type.
  1126 
  1127   \item [@{command (HOL) "code_const"}] associates a list of constants
  1128   with target-specific serializations; omitting a serialization
  1129   deletes an existing serialization.
  1130 
  1131   \item [@{command (HOL) "code_type"}] associates a list of type
  1132   constructors with target-specific serializations; omitting a
  1133   serialization deletes an existing serialization.
  1134 
  1135   \item [@{command (HOL) "code_class"}] associates a list of classes
  1136   with target-specific class names; omitting a
  1137   serialization deletes an existing serialization.
  1138   This applies only to \emph{Haskell}.
  1139 
  1140   \item [@{command (HOL) "code_instance"}] declares a list of type
  1141   constructor / class instance relations as ``already present'' for a
  1142   given target.  Omitting a ``@{text "-"}'' deletes an existing
  1143   ``already present'' declaration.  This applies only to
  1144   \emph{Haskell}.
  1145 
  1146   \item [@{command (HOL) "code_monad"}] provides an auxiliary
  1147   mechanism to generate monadic code for Haskell.
  1148 
  1149   \item [@{command (HOL) "code_reserved"}] declares a list of names as
  1150   reserved for a given target, preventing it to be shadowed by any
  1151   generated code.
  1152 
  1153   \item [@{command (HOL) "code_include"}] adds arbitrary named content
  1154   (``include'') to generated code.  A ``@{text "-"}'' as last argument
  1155   will remove an already added ``include''.
  1156 
  1157   \item [@{command (HOL) "code_modulename"}] declares aliasings from
  1158   one module name onto another.
  1159 
  1160   \item [@{command (HOL) "code_abort"}] declares constants which
  1161   are not required to have a definition by means of defining equations;
  1162   if needed these are implemented by program abort instead.
  1163 
  1164   \item [@{attribute (HOL) code}] explicitly selects (or
  1165   with option ``@{text "del"}'' deselects) a defining equation for
  1166   code generation.  Usually packages introducing defining equations
  1167   provide a reasonable default setup for selection.
  1168 
  1169   \item [@{attribute (HOL) code}@{text inline}] declares (or with
  1170   option ``@{text "del"}'' removes) inlining theorems which are
  1171   applied as rewrite rules to any defining equation during
  1172   preprocessing.
  1173 
  1174   \item [@{command (HOL) "print_codesetup"}] gives an overview on
  1175   selected defining equations, code generator datatypes and
  1176   preprocessor setup.
  1177 
  1178   \end{descr}
  1179 *}
  1180 
  1181 
  1182 section {* Definition by specification \label{sec:hol-specification} *}
  1183 
  1184 text {*
  1185   \begin{matharray}{rcl}
  1186     @{command_def (HOL) "specification"} & : & \isartrans{theory}{proof(prove)} \\
  1187     @{command_def (HOL) "ax_specification"} & : & \isartrans{theory}{proof(prove)} \\
  1188   \end{matharray}
  1189 
  1190   \begin{rail}
  1191   ('specification' | 'ax\_specification') '(' (decl +) ')' \\ (thmdecl? prop +)
  1192   ;
  1193   decl: ((name ':')? term '(' 'overloaded' ')'?)
  1194   \end{rail}
  1195 
  1196   \begin{descr}
  1197 
  1198   \item [@{command (HOL) "specification"}~@{text "decls \<phi>"}] sets up a
  1199   goal stating the existence of terms with the properties specified to
  1200   hold for the constants given in @{text decls}.  After finishing the
  1201   proof, the theory will be augmented with definitions for the given
  1202   constants, as well as with theorems stating the properties for these
  1203   constants.
  1204 
  1205   \item [@{command (HOL) "ax_specification"}~@{text "decls \<phi>"}] sets
  1206   up a goal stating the existence of terms with the properties
  1207   specified to hold for the constants given in @{text decls}.  After
  1208   finishing the proof, the theory will be augmented with axioms
  1209   expressing the properties given in the first place.
  1210 
  1211   \item [@{text decl}] declares a constant to be defined by the
  1212   specification given.  The definition for the constant @{text c} is
  1213   bound to the name @{text c_def} unless a theorem name is given in
  1214   the declaration.  Overloaded constants should be declared as such.
  1215 
  1216   \end{descr}
  1217 
  1218   Whether to use @{command (HOL) "specification"} or @{command (HOL)
  1219   "ax_specification"} is to some extent a matter of style.  @{command
  1220   (HOL) "specification"} introduces no new axioms, and so by
  1221   construction cannot introduce inconsistencies, whereas @{command
  1222   (HOL) "ax_specification"} does introduce axioms, but only after the
  1223   user has explicitly proven it to be safe.  A practical issue must be
  1224   considered, though: After introducing two constants with the same
  1225   properties using @{command (HOL) "specification"}, one can prove
  1226   that the two constants are, in fact, equal.  If this might be a
  1227   problem, one should use @{command (HOL) "ax_specification"}.
  1228 *}
  1229 
  1230 end