doc-src/IsarRef/Thy/HOL_Specific.thy
author wenzelm
Mon Jun 02 23:11:51 2008 +0200 (2008-06-02)
changeset 27045 4e7ecec1b685
parent 27041 22dcf2fc0aa2
child 27103 d8549f4d900b
permissions -rw-r--r--
moved (ax_)specification to end;
     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}{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   \begin{rail}
   417     'primrec' target? fixes 'where' equations
   418     ;
   419     equations: (thmdecl? prop + '|')
   420     ;
   421     ('fun' | 'function') target? functionopts? fixes 'where' clauses
   422     ;
   423     clauses: (thmdecl? prop ('(' 'otherwise' ')')? + '|')
   424     ;
   425     functionopts: '(' (('sequential' | 'domintros' | 'tailrec' | 'default' term) + ',') ')'
   426     ;
   427     'termination' ( term )?
   428   \end{rail}
   429 
   430   \begin{descr}
   431 
   432   \item [@{command (HOL) "primrec"}] defines primitive recursive
   433   functions over datatypes, see also \cite{isabelle-HOL}.
   434 
   435   \item [@{command (HOL) "function"}] defines functions by general
   436   wellfounded recursion. A detailed description with examples can be
   437   found in \cite{isabelle-function}. The function is specified by a
   438   set of (possibly conditional) recursive equations with arbitrary
   439   pattern matching. The command generates proof obligations for the
   440   completeness and the compatibility of patterns.
   441 
   442   The defined function is considered partial, and the resulting
   443   simplification rules (named @{text "f.psimps"}) and induction rule
   444   (named @{text "f.pinduct"}) are guarded by a generated domain
   445   predicate @{text "f_dom"}. The @{command (HOL) "termination"}
   446   command can then be used to establish that the function is total.
   447 
   448   \item [@{command (HOL) "fun"}] is a shorthand notation for
   449   ``@{command (HOL) "function"}~@{text "(sequential)"}, followed by
   450   automated proof attempts regarding pattern matching and termination.
   451   See \cite{isabelle-function} for further details.
   452 
   453   \item [@{command (HOL) "termination"}~@{text f}] commences a
   454   termination proof for the previously defined function @{text f}.  If
   455   this is omitted, the command refers to the most recent function
   456   definition.  After the proof is closed, the recursive equations and
   457   the induction principle is established.
   458 
   459   \end{descr}
   460 
   461   %FIXME check
   462 
   463   Recursive definitions introduced by both the @{command (HOL)
   464   "primrec"} and the @{command (HOL) "function"} command accommodate
   465   reasoning by induction (cf.\ \secref{sec:cases-induct}): rule @{text
   466   "c.induct"} (where @{text c} is the name of the function definition)
   467   refers to a specific induction rule, with parameters named according
   468   to the user-specified equations.  Case names of @{command (HOL)
   469   "primrec"} are that of the datatypes involved, while those of
   470   @{command (HOL) "function"} are numbered (starting from 1).
   471 
   472   The equations provided by these packages may be referred later as
   473   theorem list @{text "f.simps"}, where @{text f} is the (collective)
   474   name of the functions defined.  Individual equations may be named
   475   explicitly as well.
   476 
   477   The @{command (HOL) "function"} command accepts the following
   478   options.
   479 
   480   \begin{descr}
   481 
   482   \item [@{text sequential}] enables a preprocessor which
   483   disambiguates overlapping patterns by making them mutually disjoint.
   484   Earlier equations take precedence over later ones.  This allows to
   485   give the specification in a format very similar to functional
   486   programming.  Note that the resulting simplification and induction
   487   rules correspond to the transformed specification, not the one given
   488   originally. This usually means that each equation given by the user
   489   may result in several theroems.  Also note that this automatic
   490   transformation only works for ML-style datatype patterns.
   491 
   492   \item [@{text domintros}] enables the automated generation of
   493   introduction rules for the domain predicate. While mostly not
   494   needed, they can be helpful in some proofs about partial functions.
   495 
   496   \item [@{text tailrec}] generates the unconstrained recursive
   497   equations even without a termination proof, provided that the
   498   function is tail-recursive. This currently only works
   499 
   500   \item [@{text "default d"}] allows to specify a default value for a
   501   (partial) function, which will ensure that @{text "f x = d x"}
   502   whenever @{text "x \<notin> f_dom"}.
   503 
   504   \end{descr}
   505 *}
   506 
   507 
   508 subsection {* Proof methods related to recursive definitions *}
   509 
   510 text {*
   511   \begin{matharray}{rcl}
   512     @{method_def (HOL) pat_completeness} & : & \isarmeth \\
   513     @{method_def (HOL) relation} & : & \isarmeth \\
   514     @{method_def (HOL) lexicographic_order} & : & \isarmeth \\
   515   \end{matharray}
   516 
   517   \begin{rail}
   518     'relation' term
   519     ;
   520     'lexicographic\_order' (clasimpmod *)
   521     ;
   522   \end{rail}
   523 
   524   \begin{descr}
   525 
   526   \item [@{method (HOL) pat_completeness}] is a specialized method to
   527   solve goals regarding the completeness of pattern matching, as
   528   required by the @{command (HOL) "function"} package (cf.\
   529   \cite{isabelle-function}).
   530 
   531   \item [@{method (HOL) relation}~@{text R}] introduces a termination
   532   proof using the relation @{text R}.  The resulting proof state will
   533   contain goals expressing that @{text R} is wellfounded, and that the
   534   arguments of recursive calls decrease with respect to @{text R}.
   535   Usually, this method is used as the initial proof step of manual
   536   termination proofs.
   537 
   538   \item [@{method (HOL) "lexicographic_order"}] attempts a fully
   539   automated termination proof by searching for a lexicographic
   540   combination of size measures on the arguments of the function. The
   541   method accepts the same arguments as the @{method auto} method,
   542   which it uses internally to prove local descents.  The same context
   543   modifiers as for @{method auto} are accepted, see
   544   \secref{sec:clasimp}.
   545 
   546   In case of failure, extensive information is printed, which can help
   547   to analyse the situation (cf.\ \cite{isabelle-function}).
   548 
   549   \end{descr}
   550 *}
   551 
   552 
   553 subsection {* Old-style recursive function definitions (TFL) *}
   554 
   555 text {*
   556   The old TFL commands @{command (HOL) "recdef"} and @{command (HOL)
   557   "recdef_tc"} for defining recursive are mostly obsolete; @{command
   558   (HOL) "function"} or @{command (HOL) "fun"} should be used instead.
   559 
   560   \begin{matharray}{rcl}
   561     @{command_def (HOL) "recdef"} & : & \isartrans{theory}{theory} \\
   562     @{command_def (HOL) "recdef_tc"}@{text "\<^sup>*"} & : & \isartrans{theory}{proof(prove)} \\
   563   \end{matharray}
   564 
   565   \begin{rail}
   566     'recdef' ('(' 'permissive' ')')? \\ name term (prop +) hints?
   567     ;
   568     recdeftc thmdecl? tc
   569     ;
   570     hints: '(' 'hints' (recdefmod *) ')'
   571     ;
   572     recdefmod: (('recdef\_simp' | 'recdef\_cong' | 'recdef\_wf') (() | 'add' | 'del') ':' thmrefs) | clasimpmod
   573     ;
   574     tc: nameref ('(' nat ')')?
   575     ;
   576   \end{rail}
   577 
   578   \begin{descr}
   579   
   580   \item [@{command (HOL) "recdef"}] defines general well-founded
   581   recursive functions (using the TFL package), see also
   582   \cite{isabelle-HOL}.  The ``@{text "(permissive)"}'' option tells
   583   TFL to recover from failed proof attempts, returning unfinished
   584   results.  The @{text recdef_simp}, @{text recdef_cong}, and @{text
   585   recdef_wf} hints refer to auxiliary rules to be used in the internal
   586   automated proof process of TFL.  Additional @{syntax clasimpmod}
   587   declarations (cf.\ \secref{sec:clasimp}) may be given to tune the
   588   context of the Simplifier (cf.\ \secref{sec:simplifier}) and
   589   Classical reasoner (cf.\ \secref{sec:classical}).
   590   
   591   \item [@{command (HOL) "recdef_tc"}~@{text "c (i)"}] recommences the
   592   proof for leftover termination condition number @{text i} (default
   593   1) as generated by a @{command (HOL) "recdef"} definition of
   594   constant @{text c}.
   595   
   596   Note that in most cases, @{command (HOL) "recdef"} is able to finish
   597   its internal proofs without manual intervention.
   598 
   599   \end{descr}
   600 
   601   \medskip Hints for @{command (HOL) "recdef"} may be also declared
   602   globally, using the following attributes.
   603 
   604   \begin{matharray}{rcl}
   605     @{attribute_def (HOL) recdef_simp} & : & \isaratt \\
   606     @{attribute_def (HOL) recdef_cong} & : & \isaratt \\
   607     @{attribute_def (HOL) recdef_wf} & : & \isaratt \\
   608   \end{matharray}
   609 
   610   \begin{rail}
   611     ('recdef\_simp' | 'recdef\_cong' | 'recdef\_wf') (() | 'add' | 'del')
   612     ;
   613   \end{rail}
   614 *}
   615 
   616 
   617 section {* Inductive and coinductive definitions \label{sec:hol-inductive} *}
   618 
   619 text {*
   620   An \textbf{inductive definition} specifies the least predicate (or
   621   set) @{text R} closed under given rules: applying a rule to elements
   622   of @{text R} yields a result within @{text R}.  For example, a
   623   structural operational semantics is an inductive definition of an
   624   evaluation relation.
   625 
   626   Dually, a \textbf{coinductive definition} specifies the greatest
   627   predicate~/ set @{text R} that is consistent with given rules: every
   628   element of @{text R} can be seen as arising by applying a rule to
   629   elements of @{text R}.  An important example is using bisimulation
   630   relations to formalise equivalence of processes and infinite data
   631   structures.
   632 
   633   \medskip The HOL package is related to the ZF one, which is
   634   described in a separate paper,\footnote{It appeared in CADE
   635   \cite{paulson-CADE}; a longer version is distributed with Isabelle.}
   636   which you should refer to in case of difficulties.  The package is
   637   simpler than that of ZF thanks to implicit type-checking in HOL.
   638   The types of the (co)inductive predicates (or sets) determine the
   639   domain of the fixedpoint definition, and the package does not have
   640   to use inference rules for type-checking.
   641 
   642   \begin{matharray}{rcl}
   643     @{command_def (HOL) "inductive"} & : & \isarkeep{local{\dsh}theory} \\
   644     @{command_def (HOL) "inductive_set"} & : & \isarkeep{local{\dsh}theory} \\
   645     @{command_def (HOL) "coinductive"} & : & \isarkeep{local{\dsh}theory} \\
   646     @{command_def (HOL) "coinductive_set"} & : & \isarkeep{local{\dsh}theory} \\
   647     @{attribute_def (HOL) mono} & : & \isaratt \\
   648   \end{matharray}
   649 
   650   \begin{rail}
   651     ('inductive' | 'inductive\_set' | 'coinductive' | 'coinductive\_set') target? fixes ('for' fixes)? \\
   652     ('where' clauses)? ('monos' thmrefs)?
   653     ;
   654     clauses: (thmdecl? prop + '|')
   655     ;
   656     'mono' (() | 'add' | 'del')
   657     ;
   658   \end{rail}
   659 
   660   \begin{descr}
   661 
   662   \item [@{command (HOL) "inductive"} and @{command (HOL)
   663   "coinductive"}] define (co)inductive predicates from the
   664   introduction rules given in the @{keyword "where"} part.  The
   665   optional @{keyword "for"} part contains a list of parameters of the
   666   (co)inductive predicates that remain fixed throughout the
   667   definition.  The optional @{keyword "monos"} section contains
   668   \emph{monotonicity theorems}, which are required for each operator
   669   applied to a recursive set in the introduction rules.  There
   670   \emph{must} be a theorem of the form @{text "A \<le> B \<Longrightarrow> M A \<le> M B"},
   671   for each premise @{text "M R\<^sub>i t"} in an introduction rule!
   672 
   673   \item [@{command (HOL) "inductive_set"} and @{command (HOL)
   674   "coinductive_set"}] are wrappers for to the previous commands,
   675   allowing the definition of (co)inductive sets.
   676 
   677   \item [@{attribute (HOL) mono}] declares monotonicity rules.  These
   678   rule are involved in the automated monotonicity proof of @{command
   679   (HOL) "inductive"}.
   680 
   681   \end{descr}
   682 *}
   683 
   684 
   685 subsection {* Derived rules *}
   686 
   687 text {*
   688   Each (co)inductive definition @{text R} adds definitions to the
   689   theory and also proves some theorems:
   690 
   691   \begin{description}
   692 
   693   \item [@{text R.intros}] is the list of introduction rules as proven
   694   theorems, for the recursive predicates (or sets).  The rules are
   695   also available individually, using the names given them in the
   696   theory file;
   697 
   698   \item [@{text R.cases}] is the case analysis (or elimination) rule;
   699 
   700   \item [@{text R.induct} or @{text R.coinduct}] is the (co)induction
   701   rule.
   702 
   703   \end{description}
   704 
   705   When several predicates @{text "R\<^sub>1, \<dots>, R\<^sub>n"} are
   706   defined simultaneously, the list of introduction rules is called
   707   @{text "R\<^sub>1_\<dots>_R\<^sub>n.intros"}, the case analysis rules are
   708   called @{text "R\<^sub>1.cases, \<dots>, R\<^sub>n.cases"}, and the list
   709   of mutual induction rules is called @{text
   710   "R\<^sub>1_\<dots>_R\<^sub>n.inducts"}.
   711 *}
   712 
   713 
   714 subsection {* Monotonicity theorems *}
   715 
   716 text {*
   717   Each theory contains a default set of theorems that are used in
   718   monotonicity proofs.  New rules can be added to this set via the
   719   @{attribute (HOL) mono} attribute.  The HOL theory @{text Inductive}
   720   shows how this is done.  In general, the following monotonicity
   721   theorems may be added:
   722 
   723   \begin{itemize}
   724 
   725   \item Theorems of the form @{text "A \<le> B \<Longrightarrow> M A \<le> M B"}, for proving
   726   monotonicity of inductive definitions whose introduction rules have
   727   premises involving terms such as @{text "M R\<^sub>i t"}.
   728 
   729   \item Monotonicity theorems for logical operators, which are of the
   730   general form @{text "(\<dots> \<longrightarrow> \<dots>) \<Longrightarrow> \<dots> (\<dots> \<longrightarrow> \<dots>) \<Longrightarrow> \<dots> \<longrightarrow> \<dots>"}.  For example, in
   731   the case of the operator @{text "\<or>"}, the corresponding theorem is
   732   \[
   733   \infer{@{text "P\<^sub>1 \<or> P\<^sub>2 \<longrightarrow> Q\<^sub>1 \<or> Q\<^sub>2"}}{@{text "P\<^sub>1 \<longrightarrow> Q\<^sub>1"} & @{text "P\<^sub>2 \<longrightarrow> Q\<^sub>2"}}
   734   \]
   735 
   736   \item De Morgan style equations for reasoning about the ``polarity''
   737   of expressions, e.g.
   738   \[
   739   @{prop "\<not> \<not> P \<longleftrightarrow> P"} \qquad\qquad
   740   @{prop "\<not> (P \<and> Q) \<longleftrightarrow> \<not> P \<or> \<not> Q"}
   741   \]
   742 
   743   \item Equations for reducing complex operators to more primitive
   744   ones whose monotonicity can easily be proved, e.g.
   745   \[
   746   @{prop "(P \<longrightarrow> Q) \<longleftrightarrow> \<not> P \<or> Q"} \qquad\qquad
   747   @{prop "Ball A P \<equiv> \<forall>x. x \<in> A \<longrightarrow> P x"}
   748   \]
   749 
   750   \end{itemize}
   751 
   752   %FIXME: Example of an inductive definition
   753 *}
   754 
   755 
   756 section {* Arithmetic proof support *}
   757 
   758 text {*
   759   \begin{matharray}{rcl}
   760     @{method_def (HOL) arith} & : & \isarmeth \\
   761     @{attribute_def (HOL) arith_split} & : & \isaratt \\
   762   \end{matharray}
   763 
   764   The @{method (HOL) arith} method decides linear arithmetic problems
   765   (on types @{text nat}, @{text int}, @{text real}).  Any current
   766   facts are inserted into the goal before running the procedure.
   767 
   768   The @{attribute (HOL) arith_split} attribute declares case split
   769   rules to be expanded before the arithmetic procedure is invoked.
   770 
   771   Note that a simpler (but faster) version of arithmetic reasoning is
   772   already performed by the Simplifier.
   773 *}
   774 
   775 
   776 section {* Cases and induction: emulating tactic scripts \label{sec:hol-induct-tac} *}
   777 
   778 text {*
   779   The following important tactical tools of Isabelle/HOL have been
   780   ported to Isar.  These should be never used in proper proof texts!
   781 
   782   \begin{matharray}{rcl}
   783     @{method_def (HOL) case_tac}@{text "\<^sup>*"} & : & \isarmeth \\
   784     @{method_def (HOL) induct_tac}@{text "\<^sup>*"} & : & \isarmeth \\
   785     @{method_def (HOL) ind_cases}@{text "\<^sup>*"} & : & \isarmeth \\
   786     @{command_def (HOL) "inductive_cases"} & : & \isartrans{theory}{theory} \\
   787   \end{matharray}
   788 
   789   \begin{rail}
   790     'case\_tac' goalspec? term rule?
   791     ;
   792     'induct\_tac' goalspec? (insts * 'and') rule?
   793     ;
   794     'ind\_cases' (prop +) ('for' (name +)) ?
   795     ;
   796     'inductive\_cases' (thmdecl? (prop +) + 'and')
   797     ;
   798 
   799     rule: ('rule' ':' thmref)
   800     ;
   801   \end{rail}
   802 
   803   \begin{descr}
   804 
   805   \item [@{method (HOL) case_tac} and @{method (HOL) induct_tac}]
   806   admit to reason about inductive datatypes only (unless an
   807   alternative rule is given explicitly).  Furthermore, @{method (HOL)
   808   case_tac} does a classical case split on booleans; @{method (HOL)
   809   induct_tac} allows only variables to be given as instantiation.
   810   These tactic emulations feature both goal addressing and dynamic
   811   instantiation.  Note that named rule cases are \emph{not} provided
   812   as would be by the proper @{method induct} and @{method cases} proof
   813   methods (see \secref{sec:cases-induct}).
   814   
   815   \item [@{method (HOL) ind_cases} and @{command (HOL)
   816   "inductive_cases"}] provide an interface to the internal @{ML_text
   817   mk_cases} operation.  Rules are simplified in an unrestricted
   818   forward manner.
   819 
   820   While @{method (HOL) ind_cases} is a proof method to apply the
   821   result immediately as elimination rules, @{command (HOL)
   822   "inductive_cases"} provides case split theorems at the theory level
   823   for later use.  The @{keyword "for"} argument of the @{method (HOL)
   824   ind_cases} method allows to specify a list of variables that should
   825   be generalized before applying the resulting rule.
   826 
   827   \end{descr}
   828 *}
   829 
   830 
   831 section {* Executable code *}
   832 
   833 text {*
   834   Isabelle/Pure provides two generic frameworks to support code
   835   generation from executable specifications.  Isabelle/HOL
   836   instantiates these mechanisms in a way that is amenable to end-user
   837   applications.
   838 
   839   One framework generates code from both functional and relational
   840   programs to SML.  See \cite{isabelle-HOL} for further information
   841   (this actually covers the new-style theory format as well).
   842 
   843   \begin{matharray}{rcl}
   844     @{command_def (HOL) "value"}@{text "\<^sup>*"} & : & \isarkeep{theory~|~proof} \\
   845     @{command_def (HOL) "code_module"} & : & \isartrans{theory}{theory} \\
   846     @{command_def (HOL) "code_library"} & : & \isartrans{theory}{theory} \\
   847     @{command_def (HOL) "consts_code"} & : & \isartrans{theory}{theory} \\
   848     @{command_def (HOL) "types_code"} & : & \isartrans{theory}{theory} \\  
   849     @{attribute_def (HOL) code} & : & \isaratt \\
   850   \end{matharray}
   851 
   852   \begin{rail}
   853   'value' term
   854   ;
   855 
   856   ( 'code\_module' | 'code\_library' ) modespec ? name ? \\
   857     ( 'file' name ) ? ( 'imports' ( name + ) ) ? \\
   858     'contains' ( ( name '=' term ) + | term + )
   859   ;
   860 
   861   modespec: '(' ( name * ) ')'
   862   ;
   863 
   864   'consts\_code' (codespec +)
   865   ;
   866 
   867   codespec: const template attachment ?
   868   ;
   869 
   870   'types\_code' (tycodespec +)
   871   ;
   872 
   873   tycodespec: name template attachment ?
   874   ;
   875 
   876   const: term
   877   ;
   878 
   879   template: '(' string ')'
   880   ;
   881 
   882   attachment: 'attach' modespec ? verblbrace text verbrbrace
   883   ;
   884 
   885   'code' (name)?
   886   ;
   887   \end{rail}
   888 
   889   \begin{descr}
   890 
   891   \item [@{command (HOL) "value"}~@{text t}] evaluates and prints a
   892   term using the code generator.
   893 
   894   \end{descr}
   895 
   896   \medskip The other framework generates code from functional programs
   897   (including overloading using type classes) to SML \cite{SML}, OCaml
   898   \cite{OCaml} and Haskell \cite{haskell-revised-report}.
   899   Conceptually, code generation is split up in three steps:
   900   \emph{selection} of code theorems, \emph{translation} into an
   901   abstract executable view and \emph{serialization} to a specific
   902   \emph{target language}.  See \cite{isabelle-codegen} for an
   903   introduction on how to use it.
   904 
   905   \begin{matharray}{rcl}
   906     @{command_def (HOL) "export_code"}@{text "\<^sup>*"} & : & \isarkeep{theory~|~proof} \\
   907     @{command_def (HOL) "code_thms"}@{text "\<^sup>*"} & : & \isarkeep{theory~|~proof} \\
   908     @{command_def (HOL) "code_deps"}@{text "\<^sup>*"} & : & \isarkeep{theory~|~proof} \\
   909     @{command_def (HOL) "code_datatype"} & : & \isartrans{theory}{theory} \\
   910     @{command_def (HOL) "code_const"} & : & \isartrans{theory}{theory} \\
   911     @{command_def (HOL) "code_type"} & : & \isartrans{theory}{theory} \\
   912     @{command_def (HOL) "code_class"} & : & \isartrans{theory}{theory} \\
   913     @{command_def (HOL) "code_instance"} & : & \isartrans{theory}{theory} \\
   914     @{command_def (HOL) "code_monad"} & : & \isartrans{theory}{theory} \\
   915     @{command_def (HOL) "code_reserved"} & : & \isartrans{theory}{theory} \\
   916     @{command_def (HOL) "code_include"} & : & \isartrans{theory}{theory} \\
   917     @{command_def (HOL) "code_modulename"} & : & \isartrans{theory}{theory} \\
   918     @{command_def (HOL) "code_exception"} & : & \isartrans{theory}{theory} \\
   919     @{command_def (HOL) "print_codesetup"}@{text "\<^sup>*"} & : & \isarkeep{theory~|~proof} \\
   920     @{attribute_def (HOL) code} & : & \isaratt \\
   921   \end{matharray}
   922 
   923   \begin{rail}
   924     'export\_code' ( constexpr + ) ? \\
   925       ( ( 'in' target ( 'module\_name' string ) ? \\
   926         ( 'file' ( string | '-' ) ) ? ( '(' args ')' ) ?) + ) ?
   927     ;
   928 
   929     'code\_thms' ( constexpr + ) ?
   930     ;
   931 
   932     'code\_deps' ( constexpr + ) ?
   933     ;
   934 
   935     const: term
   936     ;
   937 
   938     constexpr: ( const | 'name.*' | '*' )
   939     ;
   940 
   941     typeconstructor: nameref
   942     ;
   943 
   944     class: nameref
   945     ;
   946 
   947     target: 'OCaml' | 'SML' | 'Haskell'
   948     ;
   949 
   950     'code\_datatype' const +
   951     ;
   952 
   953     'code\_const' (const + 'and') \\
   954       ( ( '(' target ( syntax ? + 'and' ) ')' ) + )
   955     ;
   956 
   957     'code\_type' (typeconstructor + 'and') \\
   958       ( ( '(' target ( syntax ? + 'and' ) ')' ) + )
   959     ;
   960 
   961     'code\_class' (class + 'and') \\
   962       ( ( '(' target \\
   963         ( ( string ('where' \\
   964           ( const ( '==' | equiv ) string ) + ) ? ) ? + 'and' ) ')' ) + )
   965     ;
   966 
   967     'code\_instance' (( typeconstructor '::' class ) + 'and') \\
   968       ( ( '(' target ( '-' ? + 'and' ) ')' ) + )
   969     ;
   970 
   971     'code\_monad' const const target
   972     ;
   973 
   974     'code\_reserved' target ( string + )
   975     ;
   976 
   977     'code\_include' target ( string ( string | '-') )
   978     ;
   979 
   980     'code\_modulename' target ( ( string string ) + )
   981     ;
   982 
   983     'code\_exception' ( const + )
   984     ;
   985 
   986     syntax: string | ( 'infix' | 'infixl' | 'infixr' ) nat string
   987     ;
   988 
   989     'code' ('func' | 'inline') ( 'del' )?
   990     ;
   991   \end{rail}
   992 
   993   \begin{descr}
   994 
   995   \item [@{command (HOL) "export_code"}] is the canonical interface
   996   for generating and serializing code: for a given list of constants,
   997   code is generated for the specified target languages.  Abstract code
   998   is cached incrementally.  If no constant is given, the currently
   999   cached code is serialized.  If no serialization instruction is
  1000   given, only abstract code is cached.
  1001 
  1002   Constants may be specified by giving them literally, referring to
  1003   all executable contants within a certain theory by giving @{text
  1004   "name.*"}, or referring to \emph{all} executable constants currently
  1005   available by giving @{text "*"}.
  1006 
  1007   By default, for each involved theory one corresponding name space
  1008   module is generated.  Alternativly, a module name may be specified
  1009   after the @{keyword "module_name"} keyword; then \emph{all} code is
  1010   placed in this module.
  1011 
  1012   For \emph{SML} and \emph{OCaml}, the file specification refers to a
  1013   single file; for \emph{Haskell}, it refers to a whole directory,
  1014   where code is generated in multiple files reflecting the module
  1015   hierarchy.  The file specification ``@{text "-"}'' denotes standard
  1016   output.  For \emph{SML}, omitting the file specification compiles
  1017   code internally in the context of the current ML session.
  1018 
  1019   Serializers take an optional list of arguments in parentheses.  For
  1020   \emph{Haskell} a module name prefix may be given using the ``@{text
  1021   "root:"}'' argument; ``@{text string_classes}'' adds a ``@{verbatim
  1022   "deriving (Read, Show)"}'' clause to each appropriate datatype
  1023   declaration.
  1024 
  1025   \item [@{command (HOL) "code_thms"}] prints a list of theorems
  1026   representing the corresponding program containing all given
  1027   constants; if no constants are given, the currently cached code
  1028   theorems are printed.
  1029 
  1030   \item [@{command (HOL) "code_deps"}] visualizes dependencies of
  1031   theorems representing the corresponding program containing all given
  1032   constants; if no constants are given, the currently cached code
  1033   theorems are visualized.
  1034 
  1035   \item [@{command (HOL) "code_datatype"}] specifies a constructor set
  1036   for a logical type.
  1037 
  1038   \item [@{command (HOL) "code_const"}] associates a list of constants
  1039   with target-specific serializations; omitting a serialization
  1040   deletes an existing serialization.
  1041 
  1042   \item [@{command (HOL) "code_type"}] associates a list of type
  1043   constructors with target-specific serializations; omitting a
  1044   serialization deletes an existing serialization.
  1045 
  1046   \item [@{command (HOL) "code_class"}] associates a list of classes
  1047   with target-specific class names; in addition, constants associated
  1048   with this class may be given target-specific names used for instance
  1049   declarations; omitting a serialization deletes an existing
  1050   serialization.  This applies only to \emph{Haskell}.
  1051 
  1052   \item [@{command (HOL) "code_instance"}] declares a list of type
  1053   constructor / class instance relations as ``already present'' for a
  1054   given target.  Omitting a ``@{text "-"}'' deletes an existing
  1055   ``already present'' declaration.  This applies only to
  1056   \emph{Haskell}.
  1057 
  1058   \item [@{command (HOL) "code_monad"}] provides an auxiliary
  1059   mechanism to generate monadic code.
  1060 
  1061   \item [@{command (HOL) "code_reserved"}] declares a list of names as
  1062   reserved for a given target, preventing it to be shadowed by any
  1063   generated code.
  1064 
  1065   \item [@{command (HOL) "code_include"}] adds arbitrary named content
  1066   (``include'') to generated code.  A as last argument ``@{text "-"}''
  1067   will remove an already added ``include''.
  1068 
  1069   \item [@{command (HOL) "code_modulename"}] declares aliasings from
  1070   one module name onto another.
  1071 
  1072   \item [@{command (HOL) "code_exception"}] declares constants which
  1073   are not required to have a definition by a defining equations; these
  1074   are mapped on exceptions instead.
  1075 
  1076   \item [@{attribute (HOL) code}~@{text func}] explicitly selects (or
  1077   with option ``@{text "del:"}'' deselects) a defining equation for
  1078   code generation.  Usually packages introducing defining equations
  1079   provide a resonable default setup for selection.
  1080 
  1081   \item [@{attribute (HOL) code}@{text inline}] declares (or with
  1082   option ``@{text "del:"}'' removes) inlining theorems which are
  1083   applied as rewrite rules to any defining equation during
  1084   preprocessing.
  1085 
  1086   \item [@{command (HOL) "print_codesetup"}] gives an overview on
  1087   selected defining equations, code generator datatypes and
  1088   preprocessor setup.
  1089 
  1090   \end{descr}
  1091 *}
  1092 
  1093 
  1094 section {* Definition by specification \label{sec:hol-specification} *}
  1095 
  1096 text {*
  1097   \begin{matharray}{rcl}
  1098     @{command_def (HOL) "specification"} & : & \isartrans{theory}{proof(prove)} \\
  1099     @{command_def (HOL) "ax_specification"} & : & \isartrans{theory}{proof(prove)} \\
  1100   \end{matharray}
  1101 
  1102   \begin{rail}
  1103   ('specification' | 'ax\_specification') '(' (decl +) ')' \\ (thmdecl? prop +)
  1104   ;
  1105   decl: ((name ':')? term '(' 'overloaded' ')'?)
  1106   \end{rail}
  1107 
  1108   \begin{descr}
  1109 
  1110   \item [@{command (HOL) "specification"}~@{text "decls \<phi>"}] sets up a
  1111   goal stating the existence of terms with the properties specified to
  1112   hold for the constants given in @{text decls}.  After finishing the
  1113   proof, the theory will be augmented with definitions for the given
  1114   constants, as well as with theorems stating the properties for these
  1115   constants.
  1116 
  1117   \item [@{command (HOL) "ax_specification"}~@{text "decls \<phi>"}] sets
  1118   up a goal stating the existence of terms with the properties
  1119   specified to hold for the constants given in @{text decls}.  After
  1120   finishing the proof, the theory will be augmented with axioms
  1121   expressing the properties given in the first place.
  1122 
  1123   \item [@{text decl}] declares a constant to be defined by the
  1124   specification given.  The definition for the constant @{text c} is
  1125   bound to the name @{text c_def} unless a theorem name is given in
  1126   the declaration.  Overloaded constants should be declared as such.
  1127 
  1128   \end{descr}
  1129 
  1130   Whether to use @{command (HOL) "specification"} or @{command (HOL)
  1131   "ax_specification"} is to some extent a matter of style.  @{command
  1132   (HOL) "specification"} introduces no new axioms, and so by
  1133   construction cannot introduce inconsistencies, whereas @{command
  1134   (HOL) "ax_specification"} does introduce axioms, but only after the
  1135   user has explicitly proven it to be safe.  A practical issue must be
  1136   considered, though: After introducing two constants with the same
  1137   properties using @{command (HOL) "specification"}, one can prove
  1138   that the two constants are, in fact, equal.  If this might be a
  1139   problem, one should use @{command (HOL) "ax_specification"}.
  1140 *}
  1141 
  1142 end