doc-src/IsarRef/Thy/HOL_Specific.thy
author wenzelm
Thu Nov 13 21:45:40 2008 +0100 (2008-11-13)
changeset 28761 9ec4482c9201
parent 28760 cbc435f7b16b
child 29112 f2b45eea6dac
permissions -rw-r--r--
updated/refined types of Isar language elements, removed special LaTeX macros;
     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"} & : & @{text "theory \<rightarrow> theory"} \\
    14     @{command_def (HOL) "typedef"} & : & @{text "theory \<rightarrow> 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{description}
    32   
    33   \item @{command (HOL) "typedecl"}~@{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>n) t"} is similar
    34   to the original @{command "typedecl"} of Isabelle/Pure (see
    35   \secref{sec:types-pure}), but also declares type arity @{text "t ::
    36   (type, \<dots>, type) type"}, making @{text t} an actual HOL type
    37   constructor.  %FIXME check, update
    38   
    39   \item @{command (HOL) "typedef"}~@{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>n) t = A"} sets up
    40   a goal stating non-emptiness of the set @{text A}.  After finishing
    41   the proof, the theory will be augmented by a Gordon/HOL-style type
    42   definition, which establishes a bijection between the representing
    43   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{description}
    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>*"} & : & @{text attribute} \\
    83   \end{matharray}
    84 
    85   \begin{rail}
    86     'split\_format' (((name *) + 'and') | ('(' 'complete' ')'))
    87     ;
    88   \end{rail}
    89 
    90   \begin{description}
    91   
    92   \item @{attribute (HOL) split_format}~@{text "p\<^sub>1 \<dots> p\<^sub>m \<AND> \<dots>
    93   \<AND> q\<^sub>1 \<dots> q\<^sub>n"} puts expressions of low-level tuple types into
    94   canonical form as specified by the arguments given; the @{text i}-th
    95   collection of arguments refers to occurrences in premise @{text i}
    96   of the rule.  The ``@{text "(complete)"}'' option causes \emph{all}
    97   arguments in function applications to be represented canonically
    98   according to their tuple type structure.
    99 
   100   Note that these operations tend to invent funny names for new local
   101   parameters to be introduced.
   102 
   103   \end{description}
   104 *}
   105 
   106 
   107 section {* Records \label{sec:hol-record} *}
   108 
   109 text {*
   110   In principle, records merely generalize the concept of tuples, where
   111   components may be addressed by labels instead of just position.  The
   112   logical infrastructure of records in Isabelle/HOL is slightly more
   113   advanced, though, supporting truly extensible record schemes.  This
   114   admits operations that are polymorphic with respect to record
   115   extension, yielding ``object-oriented'' effects like (single)
   116   inheritance.  See also \cite{NaraschewskiW-TPHOLs98} for more
   117   details on object-oriented verification and record subtyping in HOL.
   118 *}
   119 
   120 
   121 subsection {* Basic concepts *}
   122 
   123 text {*
   124   Isabelle/HOL supports both \emph{fixed} and \emph{schematic} records
   125   at the level of terms and types.  The notation is as follows:
   126 
   127   \begin{center}
   128   \begin{tabular}{l|l|l}
   129     & record terms & record types \\ \hline
   130     fixed & @{text "\<lparr>x = a, y = b\<rparr>"} & @{text "\<lparr>x :: A, y :: B\<rparr>"} \\
   131     schematic & @{text "\<lparr>x = a, y = b, \<dots> = m\<rparr>"} &
   132       @{text "\<lparr>x :: A, y :: B, \<dots> :: M\<rparr>"} \\
   133   \end{tabular}
   134   \end{center}
   135 
   136   \noindent The ASCII representation of @{text "\<lparr>x = a\<rparr>"} is @{text
   137   "(| x = a |)"}.
   138 
   139   A fixed record @{text "\<lparr>x = a, y = b\<rparr>"} has field @{text x} of value
   140   @{text a} and field @{text y} of value @{text b}.  The corresponding
   141   type is @{text "\<lparr>x :: A, y :: B\<rparr>"}, assuming that @{text "a :: A"}
   142   and @{text "b :: B"}.
   143 
   144   A record scheme like @{text "\<lparr>x = a, y = b, \<dots> = m\<rparr>"} contains fields
   145   @{text x} and @{text y} as before, but also possibly further fields
   146   as indicated by the ``@{text "\<dots>"}'' notation (which is actually part
   147   of the syntax).  The improper field ``@{text "\<dots>"}'' of a record
   148   scheme is called the \emph{more part}.  Logically it is just a free
   149   variable, which is occasionally referred to as ``row variable'' in
   150   the literature.  The more part of a record scheme may be
   151   instantiated by zero or more further components.  For example, the
   152   previous scheme may get instantiated to @{text "\<lparr>x = a, y = b, z =
   153   c, \<dots> = m'\<rparr>"}, where @{text m'} refers to a different more part.
   154   Fixed records are special instances of record schemes, where
   155   ``@{text "\<dots>"}'' is properly terminated by the @{text "() :: unit"}
   156   element.  In fact, @{text "\<lparr>x = a, y = b\<rparr>"} is just an abbreviation
   157   for @{text "\<lparr>x = a, y = b, \<dots> = ()\<rparr>"}.
   158   
   159   \medskip Two key observations make extensible records in a simply
   160   typed language like HOL work out:
   161 
   162   \begin{enumerate}
   163 
   164   \item the more part is internalized, as a free term or type
   165   variable,
   166 
   167   \item field names are externalized, they cannot be accessed within
   168   the logic as first-class values.
   169 
   170   \end{enumerate}
   171 
   172   \medskip In Isabelle/HOL record types have to be defined explicitly,
   173   fixing their field names and types, and their (optional) parent
   174   record.  Afterwards, records may be formed using above syntax, while
   175   obeying the canonical order of fields as given by their declaration.
   176   The record package provides several standard operations like
   177   selectors and updates.  The common setup for various generic proof
   178   tools enable succinct reasoning patterns.  See also the Isabelle/HOL
   179   tutorial \cite{isabelle-hol-book} for further instructions on using
   180   records in practice.
   181 *}
   182 
   183 
   184 subsection {* Record specifications *}
   185 
   186 text {*
   187   \begin{matharray}{rcl}
   188     @{command_def (HOL) "record"} & : & @{text "theory \<rightarrow> theory"} \\
   189   \end{matharray}
   190 
   191   \begin{rail}
   192     'record' typespec '=' (type '+')? (constdecl +)
   193     ;
   194   \end{rail}
   195 
   196   \begin{description}
   197 
   198   \item @{command (HOL) "record"}~@{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t = \<tau> + c\<^sub>1 :: \<sigma>\<^sub>1
   199   \<dots> c\<^sub>n :: \<sigma>\<^sub>n"} defines extensible record type @{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t"},
   200   derived from the optional parent record @{text "\<tau>"} by adding new
   201   field components @{text "c\<^sub>i :: \<sigma>\<^sub>i"} etc.
   202 
   203   The type variables of @{text "\<tau>"} and @{text "\<sigma>\<^sub>i"} need to be
   204   covered by the (distinct) parameters @{text "\<alpha>\<^sub>1, \<dots>,
   205   \<alpha>\<^sub>m"}.  Type constructor @{text t} has to be new, while @{text
   206   \<tau>} needs to specify an instance of an existing record type.  At
   207   least one new field @{text "c\<^sub>i"} has to be specified.
   208   Basically, field names need to belong to a unique record.  This is
   209   not a real restriction in practice, since fields are qualified by
   210   the record name internally.
   211 
   212   The parent record specification @{text \<tau>} is optional; if omitted
   213   @{text t} becomes a root record.  The hierarchy of all records
   214   declared within a theory context forms a forest structure, i.e.\ a
   215   set of trees starting with a root record each.  There is no way to
   216   merge multiple parent records!
   217 
   218   For convenience, @{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t"} is made a
   219   type abbreviation for the fixed record type @{text "\<lparr>c\<^sub>1 ::
   220   \<sigma>\<^sub>1, \<dots>, c\<^sub>n :: \<sigma>\<^sub>n\<rparr>"}, likewise is @{text
   221   "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m, \<zeta>) t_scheme"} made an abbreviation for
   222   @{text "\<lparr>c\<^sub>1 :: \<sigma>\<^sub>1, \<dots>, c\<^sub>n :: \<sigma>\<^sub>n, \<dots> ::
   223   \<zeta>\<rparr>"}.
   224 
   225   \end{description}
   226 *}
   227 
   228 
   229 subsection {* Record operations *}
   230 
   231 text {*
   232   Any record definition of the form presented above produces certain
   233   standard operations.  Selectors and updates are provided for any
   234   field, including the improper one ``@{text more}''.  There are also
   235   cumulative record constructor functions.  To simplify the
   236   presentation below, we assume for now that @{text "(\<alpha>\<^sub>1, \<dots>,
   237   \<alpha>\<^sub>m) t"} is a root record with fields @{text "c\<^sub>1 ::
   238   \<sigma>\<^sub>1, \<dots>, c\<^sub>n :: \<sigma>\<^sub>n"}.
   239 
   240   \medskip \textbf{Selectors} and \textbf{updates} are available for
   241   any field (including ``@{text more}''):
   242 
   243   \begin{matharray}{lll}
   244     @{text "c\<^sub>i"} & @{text "::"} & @{text "\<lparr>\<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr> \<Rightarrow> \<sigma>\<^sub>i"} \\
   245     @{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>"} \\
   246   \end{matharray}
   247 
   248   There is special syntax for application of updates: @{text "r\<lparr>x :=
   249   a\<rparr>"} abbreviates term @{text "x_update a r"}.  Further notation for
   250   repeated updates is also available: @{text "r\<lparr>x := a\<rparr>\<lparr>y := b\<rparr>\<lparr>z :=
   251   c\<rparr>"} may be written @{text "r\<lparr>x := a, y := b, z := c\<rparr>"}.  Note that
   252   because of postfix notation the order of fields shown here is
   253   reverse than in the actual term.  Since repeated updates are just
   254   function applications, fields may be freely permuted in @{text "\<lparr>x
   255   := a, y := b, z := c\<rparr>"}, as far as logical equality is concerned.
   256   Thus commutativity of independent updates can be proven within the
   257   logic for any two fields, but not as a general theorem.
   258 
   259   \medskip The \textbf{make} operation provides a cumulative record
   260   constructor function:
   261 
   262   \begin{matharray}{lll}
   263     @{text "t.make"} & @{text "::"} & @{text "\<sigma>\<^sub>1 \<Rightarrow> \<dots> \<sigma>\<^sub>n \<Rightarrow> \<lparr>\<^vec>c :: \<^vec>\<sigma>\<rparr>"} \\
   264   \end{matharray}
   265 
   266   \medskip We now reconsider the case of non-root records, which are
   267   derived of some parent.  In general, the latter may depend on
   268   another parent as well, resulting in a list of \emph{ancestor
   269   records}.  Appending the lists of fields of all ancestors results in
   270   a certain field prefix.  The record package automatically takes care
   271   of this by lifting operations over this context of ancestor fields.
   272   Assuming that @{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t"} has ancestor
   273   fields @{text "b\<^sub>1 :: \<rho>\<^sub>1, \<dots>, b\<^sub>k :: \<rho>\<^sub>k"},
   274   the above record operations will get the following types:
   275 
   276   \medskip
   277   \begin{tabular}{lll}
   278     @{text "c\<^sub>i"} & @{text "::"} & @{text "\<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr> \<Rightarrow> \<sigma>\<^sub>i"} \\
   279     @{text "c\<^sub>i_update"} & @{text "::"} & @{text "\<sigma>\<^sub>i \<Rightarrow> 
   280       \<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr> \<Rightarrow>
   281       \<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr>"} \\
   282     @{text "t.make"} & @{text "::"} & @{text "\<rho>\<^sub>1 \<Rightarrow> \<dots> \<rho>\<^sub>k \<Rightarrow> \<sigma>\<^sub>1 \<Rightarrow> \<dots> \<sigma>\<^sub>n \<Rightarrow>
   283       \<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>\<rparr>"} \\
   284   \end{tabular}
   285   \medskip
   286 
   287   \noindent Some further operations address the extension aspect of a
   288   derived record scheme specifically: @{text "t.fields"} produces a
   289   record fragment consisting of exactly the new fields introduced here
   290   (the result may serve as a more part elsewhere); @{text "t.extend"}
   291   takes a fixed record and adds a given more part; @{text
   292   "t.truncate"} restricts a record scheme to a fixed record.
   293 
   294   \medskip
   295   \begin{tabular}{lll}
   296     @{text "t.fields"} & @{text "::"} & @{text "\<sigma>\<^sub>1 \<Rightarrow> \<dots> \<sigma>\<^sub>n \<Rightarrow> \<lparr>\<^vec>c :: \<^vec>\<sigma>\<rparr>"} \\
   297     @{text "t.extend"} & @{text "::"} & @{text "\<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>\<rparr> \<Rightarrow>
   298       \<zeta> \<Rightarrow> \<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr>"} \\
   299     @{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>"} \\
   300   \end{tabular}
   301   \medskip
   302 
   303   \noindent Note that @{text "t.make"} and @{text "t.fields"} coincide
   304   for root records.
   305 *}
   306 
   307 
   308 subsection {* Derived rules and proof tools *}
   309 
   310 text {*
   311   The record package proves several results internally, declaring
   312   these facts to appropriate proof tools.  This enables users to
   313   reason about record structures quite conveniently.  Assume that
   314   @{text t} is a record type as specified above.
   315 
   316   \begin{enumerate}
   317   
   318   \item Standard conversions for selectors or updates applied to
   319   record constructor terms are made part of the default Simplifier
   320   context; thus proofs by reduction of basic operations merely require
   321   the @{method simp} method without further arguments.  These rules
   322   are available as @{text "t.simps"}, too.
   323   
   324   \item Selectors applied to updated records are automatically reduced
   325   by an internal simplification procedure, which is also part of the
   326   standard Simplifier setup.
   327 
   328   \item Inject equations of a form analogous to @{prop "(x, y) = (x',
   329   y') \<equiv> x = x' \<and> y = y'"} are declared to the Simplifier and Classical
   330   Reasoner as @{attribute iff} rules.  These rules are available as
   331   @{text "t.iffs"}.
   332 
   333   \item The introduction rule for record equality analogous to @{text
   334   "x r = x r' \<Longrightarrow> y r = y r' \<dots> \<Longrightarrow> r = r'"} is declared to the Simplifier,
   335   and as the basic rule context as ``@{attribute intro}@{text "?"}''.
   336   The rule is called @{text "t.equality"}.
   337 
   338   \item Representations of arbitrary record expressions as canonical
   339   constructor terms are provided both in @{method cases} and @{method
   340   induct} format (cf.\ the generic proof methods of the same name,
   341   \secref{sec:cases-induct}).  Several variations are available, for
   342   fixed records, record schemes, more parts etc.
   343   
   344   The generic proof methods are sufficiently smart to pick the most
   345   sensible rule according to the type of the indicated record
   346   expression: users just need to apply something like ``@{text "(cases
   347   r)"}'' to a certain proof problem.
   348 
   349   \item The derived record operations @{text "t.make"}, @{text
   350   "t.fields"}, @{text "t.extend"}, @{text "t.truncate"} are \emph{not}
   351   treated automatically, but usually need to be expanded by hand,
   352   using the collective fact @{text "t.defs"}.
   353 
   354   \end{enumerate}
   355 *}
   356 
   357 
   358 section {* Datatypes \label{sec:hol-datatype} *}
   359 
   360 text {*
   361   \begin{matharray}{rcl}
   362     @{command_def (HOL) "datatype"} & : & @{text "theory \<rightarrow> theory"} \\
   363   @{command_def (HOL) "rep_datatype"} & : & @{text "theory \<rightarrow> proof(prove)"} \\
   364   \end{matharray}
   365 
   366   \begin{rail}
   367     'datatype' (dtspec + 'and')
   368     ;
   369     'rep\_datatype' ('(' (name +) ')')? (term +)
   370     ;
   371 
   372     dtspec: parname? typespec infix? '=' (cons + '|')
   373     ;
   374     cons: name (type *) mixfix?
   375   \end{rail}
   376 
   377   \begin{description}
   378 
   379   \item @{command (HOL) "datatype"} defines inductive datatypes in
   380   HOL.
   381 
   382   \item @{command (HOL) "rep_datatype"} represents existing types as
   383   inductive ones, generating the standard infrastructure of derived
   384   concepts (primitive recursion etc.).
   385 
   386   \end{description}
   387 
   388   The induction and exhaustion theorems generated provide case names
   389   according to the constructors involved, while parameters are named
   390   after the types (see also \secref{sec:cases-induct}).
   391 
   392   See \cite{isabelle-HOL} for more details on datatypes, but beware of
   393   the old-style theory syntax being used there!  Apart from proper
   394   proof methods for case-analysis and induction, there are also
   395   emulations of ML tactics @{method (HOL) case_tac} and @{method (HOL)
   396   induct_tac} available, see \secref{sec:hol-induct-tac}; these admit
   397   to refer directly to the internal structure of subgoals (including
   398   internally bound parameters).
   399 *}
   400 
   401 
   402 section {* Recursive functions \label{sec:recursion} *}
   403 
   404 text {*
   405   \begin{matharray}{rcl}
   406     @{command_def (HOL) "primrec"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
   407     @{command_def (HOL) "fun"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
   408     @{command_def (HOL) "function"} & : & @{text "local_theory \<rightarrow> proof(prove)"} \\
   409     @{command_def (HOL) "termination"} & : & @{text "local_theory \<rightarrow> proof(prove)"} \\
   410   \end{matharray}
   411 
   412   \begin{rail}
   413     'primrec' target? fixes 'where' equations
   414     ;
   415     equations: (thmdecl? prop + '|')
   416     ;
   417     ('fun' | 'function') target? functionopts? fixes 'where' clauses
   418     ;
   419     clauses: (thmdecl? prop ('(' 'otherwise' ')')? + '|')
   420     ;
   421     functionopts: '(' (('sequential' | 'domintros' | 'tailrec' | 'default' term) + ',') ')'
   422     ;
   423     'termination' ( term )?
   424   \end{rail}
   425 
   426   \begin{description}
   427 
   428   \item @{command (HOL) "primrec"} defines primitive recursive
   429   functions over datatypes, see also \cite{isabelle-HOL}.
   430 
   431   \item @{command (HOL) "function"} defines functions by general
   432   wellfounded recursion. A detailed description with examples can be
   433   found in \cite{isabelle-function}. The function is specified by a
   434   set of (possibly conditional) recursive equations with arbitrary
   435   pattern matching. The command generates proof obligations for the
   436   completeness and the compatibility of patterns.
   437 
   438   The defined function is considered partial, and the resulting
   439   simplification rules (named @{text "f.psimps"}) and induction rule
   440   (named @{text "f.pinduct"}) are guarded by a generated domain
   441   predicate @{text "f_dom"}. The @{command (HOL) "termination"}
   442   command can then be used to establish that the function is total.
   443 
   444   \item @{command (HOL) "fun"} is a shorthand notation for ``@{command
   445   (HOL) "function"}~@{text "(sequential)"}, followed by automated
   446   proof attempts regarding pattern matching and termination.  See
   447   \cite{isabelle-function} for further details.
   448 
   449   \item @{command (HOL) "termination"}~@{text f} commences a
   450   termination proof for the previously defined function @{text f}.  If
   451   this is omitted, the command refers to the most recent function
   452   definition.  After the proof is closed, the recursive equations and
   453   the induction principle is established.
   454 
   455   \end{description}
   456 
   457   %FIXME check
   458 
   459   Recursive definitions introduced by the @{command (HOL) "function"}
   460   command accommodate
   461   reasoning by induction (cf.\ \secref{sec:cases-induct}): rule @{text
   462   "c.induct"} (where @{text c} is the name of the function definition)
   463   refers to a specific induction rule, with parameters named according
   464   to the user-specified equations.
   465   For the @{command (HOL) "primrec"} the induction principle coincides
   466   with structural recursion on the datatype the recursion is carried
   467   out.
   468   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{description}
   481 
   482   \item @{text sequential} enables a preprocessor which disambiguates
   483   overlapping patterns by making them mutually disjoint.  Earlier
   484   equations take precedence over later ones.  This allows to give the
   485   specification in a format very similar to functional programming.
   486   Note that the resulting simplification and induction rules
   487   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{description}
   505 *}
   506 
   507 
   508 subsection {* Proof methods related to recursive definitions *}
   509 
   510 text {*
   511   \begin{matharray}{rcl}
   512     @{method_def (HOL) pat_completeness} & : & @{text method} \\
   513     @{method_def (HOL) relation} & : & @{text method} \\
   514     @{method_def (HOL) lexicographic_order} & : & @{text method} \\
   515   \end{matharray}
   516 
   517   \begin{rail}
   518     'relation' term
   519     ;
   520     'lexicographic\_order' (clasimpmod *)
   521     ;
   522   \end{rail}
   523 
   524   \begin{description}
   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{description}
   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"} & : & @{text "theory \<rightarrow> theory)"} \\
   562     @{command_def (HOL) "recdef_tc"}@{text "\<^sup>*"} & : & @{text "theory \<rightarrow> 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{description}
   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{description}
   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} & : & @{text attribute} \\
   606     @{attribute_def (HOL) recdef_cong} & : & @{text attribute} \\
   607     @{attribute_def (HOL) recdef_wf} & : & @{text attribute} \\
   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"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
   644     @{command_def (HOL) "inductive_set"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
   645     @{command_def (HOL) "coinductive"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
   646     @{command_def (HOL) "coinductive_set"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
   647     @{attribute_def (HOL) mono} & : & @{text attribute} \\
   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{description}
   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{description}
   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} & : & @{text method} \\
   761     @{attribute_def (HOL) arith_split} & : & @{text attribute} \\
   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 {* Invoking automated reasoning tools -- The Sledgehammer *}
   777 
   778 text {*
   779   Isabelle/HOL includes a generic \emph{ATP manager} that allows
   780   external automated reasoning tools to crunch a pending goal.
   781   Supported provers include E\footnote{\url{http://www.eprover.org}},
   782   SPASS\footnote{\url{http://www.spass-prover.org/}}, and Vampire.
   783   There is also a wrapper to invoke provers remotely via the
   784   SystemOnTPTP\footnote{\url{http://www.cs.miami.edu/~tptp/cgi-bin/SystemOnTPTP}}
   785   web service.
   786 
   787   The problem passed to external provers consists of the goal together
   788   with a smart selection of lemmas from the current theory context.
   789   The result of a successful proof search is some source text that
   790   usually reconstructs the proof within Isabelle, without requiring
   791   external provers again.  The Metis
   792   prover\footnote{\url{http://www.gilith.com/software/metis/}} that is
   793   integrated into Isabelle/HOL is being used here.
   794 
   795   In this mode of operation, heavy means of automated reasoning are
   796   used as a strong relevance filter, while the main proof checking
   797   works via explicit inferences going through the Isabelle kernel.
   798   Moreover, rechecking Isabelle proof texts with already specified
   799   auxiliary facts is much faster than performing fully automated
   800   search over and over again.
   801 
   802   \begin{matharray}{rcl}
   803     @{command_def (HOL) "sledgehammer"}@{text "\<^sup>*"} & : & @{text "proof \<rightarrow>"} \\
   804     @{command_def (HOL) "print_atps"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
   805     @{command_def (HOL) "atp_info"}@{text "\<^sup>*"} & : & @{text "any \<rightarrow>"} \\
   806     @{command_def (HOL) "atp_kill"}@{text "\<^sup>*"} & : & @{text "any \<rightarrow>"} \\
   807     @{method_def (HOL) metis} & : & @{text method} \\
   808   \end{matharray}
   809 
   810   \begin{rail}
   811   'sledgehammer' (nameref *)
   812   ;
   813 
   814   'metis' thmrefs
   815   ;
   816   \end{rail}
   817 
   818   \begin{description}
   819 
   820   \item @{command (HOL) sledgehammer}~@{text "prover\<^sub>1 \<dots> prover\<^sub>n"}
   821   invokes the specified automated theorem provers on the first
   822   subgoal.  Provers are run in parallel, the first successful result
   823   is displayed, and the other attempts are terminated.
   824 
   825   Provers are defined in the theory context, see also @{command (HOL)
   826   print_atps}.  If no provers are given as arguments to @{command
   827   (HOL) sledgehammer}, the system refers to the default defined as
   828   ``ATP provers'' preference by the user interface.
   829 
   830   There are additional preferences for timeout (default: 60 seconds),
   831   and the maximum number of independent prover processes (default: 5);
   832   excessive provers are automatically terminated.
   833 
   834   \item @{command (HOL) print_atps} prints the list of automated
   835   theorem provers available to the @{command (HOL) sledgehammer}
   836   command.
   837 
   838   \item @{command (HOL) atp_info} prints information about presently
   839   running provers, including elapsed runtime, and the remaining time
   840   until timeout.
   841 
   842   \item @{command (HOL) atp_kill} terminates all presently running
   843   provers.
   844 
   845   \item @{method (HOL) metis}~@{text "facts"} invokes the Metis prover
   846   with the given facts.  Metis is an automated proof tool of medium
   847   strength, but is fully integrated into Isabelle/HOL, with explicit
   848   inferences going through the kernel.  Thus its results are
   849   guaranteed to be ``correct by construction''.
   850 
   851   Note that all facts used with Metis need to be specified as explicit
   852   arguments.  There are no rule declarations as for other Isabelle
   853   provers, like @{method blast} or @{method fast}.
   854 
   855   \end{description}
   856 *}
   857 
   858 
   859 section {* Unstructured case analysis and induction \label{sec:hol-induct-tac} *}
   860 
   861 text {*
   862   The following tools of Isabelle/HOL support cases analysis and
   863   induction in unstructured tactic scripts; see also
   864   \secref{sec:cases-induct} for proper Isar versions of similar ideas.
   865 
   866   \begin{matharray}{rcl}
   867     @{method_def (HOL) case_tac}@{text "\<^sup>*"} & : & @{text method} \\
   868     @{method_def (HOL) induct_tac}@{text "\<^sup>*"} & : & @{text method} \\
   869     @{method_def (HOL) ind_cases}@{text "\<^sup>*"} & : & @{text method} \\
   870     @{command_def (HOL) "inductive_cases"}@{text "\<^sup>*"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
   871   \end{matharray}
   872 
   873   \begin{rail}
   874     'case\_tac' goalspec? term rule?
   875     ;
   876     'induct\_tac' goalspec? (insts * 'and') rule?
   877     ;
   878     'ind\_cases' (prop +) ('for' (name +)) ?
   879     ;
   880     'inductive\_cases' (thmdecl? (prop +) + 'and')
   881     ;
   882 
   883     rule: ('rule' ':' thmref)
   884     ;
   885   \end{rail}
   886 
   887   \begin{description}
   888 
   889   \item @{method (HOL) case_tac} and @{method (HOL) induct_tac} admit
   890   to reason about inductive types.  Rules are selected according to
   891   the declarations by the @{attribute cases} and @{attribute induct}
   892   attributes, cf.\ \secref{sec:cases-induct}.  The @{command (HOL)
   893   datatype} package already takes care of this.
   894 
   895   These unstructured tactics feature both goal addressing and dynamic
   896   instantiation.  Note that named rule cases are \emph{not} provided
   897   as would be by the proper @{method cases} and @{method induct} proof
   898   methods (see \secref{sec:cases-induct}).  Unlike the @{method
   899   induct} method, @{method induct_tac} does not handle structured rule
   900   statements, only the compact object-logic conclusion of the subgoal
   901   being addressed.
   902   
   903   \item @{method (HOL) ind_cases} and @{command (HOL)
   904   "inductive_cases"} provide an interface to the internal @{ML_text
   905   mk_cases} operation.  Rules are simplified in an unrestricted
   906   forward manner.
   907 
   908   While @{method (HOL) ind_cases} is a proof method to apply the
   909   result immediately as elimination rules, @{command (HOL)
   910   "inductive_cases"} provides case split theorems at the theory level
   911   for later use.  The @{keyword "for"} argument of the @{method (HOL)
   912   ind_cases} method allows to specify a list of variables that should
   913   be generalized before applying the resulting rule.
   914 
   915   \end{description}
   916 *}
   917 
   918 
   919 section {* Executable code *}
   920 
   921 text {*
   922   Isabelle/Pure provides two generic frameworks to support code
   923   generation from executable specifications.  Isabelle/HOL
   924   instantiates these mechanisms in a way that is amenable to end-user
   925   applications.
   926 
   927   One framework generates code from both functional and relational
   928   programs to SML.  See \cite{isabelle-HOL} for further information
   929   (this actually covers the new-style theory format as well).
   930 
   931   \begin{matharray}{rcl}
   932     @{command_def (HOL) "value"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
   933     @{command_def (HOL) "code_module"} & : & @{text "theory \<rightarrow> theory"} \\
   934     @{command_def (HOL) "code_library"} & : & @{text "theory \<rightarrow> theory"} \\
   935     @{command_def (HOL) "consts_code"} & : & @{text "theory \<rightarrow> theory"} \\
   936     @{command_def (HOL) "types_code"} & : & @{text "theory \<rightarrow> theory"} \\  
   937     @{attribute_def (HOL) code} & : & @{text attribute} \\
   938   \end{matharray}
   939 
   940   \begin{rail}
   941   'value' term
   942   ;
   943 
   944   ( 'code\_module' | 'code\_library' ) modespec ? name ? \\
   945     ( 'file' name ) ? ( 'imports' ( name + ) ) ? \\
   946     'contains' ( ( name '=' term ) + | term + )
   947   ;
   948 
   949   modespec: '(' ( name * ) ')'
   950   ;
   951 
   952   'consts\_code' (codespec +)
   953   ;
   954 
   955   codespec: const template attachment ?
   956   ;
   957 
   958   'types\_code' (tycodespec +)
   959   ;
   960 
   961   tycodespec: name template attachment ?
   962   ;
   963 
   964   const: term
   965   ;
   966 
   967   template: '(' string ')'
   968   ;
   969 
   970   attachment: 'attach' modespec ? verblbrace text verbrbrace
   971   ;
   972 
   973   'code' (name)?
   974   ;
   975   \end{rail}
   976 
   977   \begin{description}
   978 
   979   \item @{command (HOL) "value"}~@{text t} evaluates and prints a term
   980   using the code generator.
   981 
   982   \end{description}
   983 
   984   \medskip The other framework generates code from functional programs
   985   (including overloading using type classes) to SML \cite{SML}, OCaml
   986   \cite{OCaml} and Haskell \cite{haskell-revised-report}.
   987   Conceptually, code generation is split up in three steps:
   988   \emph{selection} of code theorems, \emph{translation} into an
   989   abstract executable view and \emph{serialization} to a specific
   990   \emph{target language}.  See \cite{isabelle-codegen} for an
   991   introduction on how to use it.
   992 
   993   \begin{matharray}{rcl}
   994     @{command_def (HOL) "export_code"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
   995     @{command_def (HOL) "code_thms"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
   996     @{command_def (HOL) "code_deps"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
   997     @{command_def (HOL) "code_datatype"} & : & @{text "theory \<rightarrow> theory"} \\
   998     @{command_def (HOL) "code_const"} & : & @{text "theory \<rightarrow> theory"} \\
   999     @{command_def (HOL) "code_type"} & : & @{text "theory \<rightarrow> theory"} \\
  1000     @{command_def (HOL) "code_class"} & : & @{text "theory \<rightarrow> theory"} \\
  1001     @{command_def (HOL) "code_instance"} & : & @{text "theory \<rightarrow> theory"} \\
  1002     @{command_def (HOL) "code_monad"} & : & @{text "theory \<rightarrow> theory"} \\
  1003     @{command_def (HOL) "code_reserved"} & : & @{text "theory \<rightarrow> theory"} \\
  1004     @{command_def (HOL) "code_include"} & : & @{text "theory \<rightarrow> theory"} \\
  1005     @{command_def (HOL) "code_modulename"} & : & @{text "theory \<rightarrow> theory"} \\
  1006     @{command_def (HOL) "code_abort"} & : & @{text "theory \<rightarrow> theory"} \\
  1007     @{command_def (HOL) "print_codesetup"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
  1008     @{attribute_def (HOL) code} & : & @{text attribute} \\
  1009   \end{matharray}
  1010 
  1011   \begin{rail}
  1012     'export\_code' ( constexpr + ) ? \\
  1013       ( ( 'in' target ( 'module\_name' string ) ? \\
  1014         ( 'file' ( string | '-' ) ) ? ( '(' args ')' ) ?) + ) ?
  1015     ;
  1016 
  1017     'code\_thms' ( constexpr + ) ?
  1018     ;
  1019 
  1020     'code\_deps' ( constexpr + ) ?
  1021     ;
  1022 
  1023     const: term
  1024     ;
  1025 
  1026     constexpr: ( const | 'name.*' | '*' )
  1027     ;
  1028 
  1029     typeconstructor: nameref
  1030     ;
  1031 
  1032     class: nameref
  1033     ;
  1034 
  1035     target: 'OCaml' | 'SML' | 'Haskell'
  1036     ;
  1037 
  1038     'code\_datatype' const +
  1039     ;
  1040 
  1041     'code\_const' (const + 'and') \\
  1042       ( ( '(' target ( syntax ? + 'and' ) ')' ) + )
  1043     ;
  1044 
  1045     'code\_type' (typeconstructor + 'and') \\
  1046       ( ( '(' target ( syntax ? + 'and' ) ')' ) + )
  1047     ;
  1048 
  1049     'code\_class' (class + 'and') \\
  1050       ( ( '(' target \\ ( string ? + 'and' ) ')' ) + )
  1051     ;
  1052 
  1053     'code\_instance' (( typeconstructor '::' class ) + 'and') \\
  1054       ( ( '(' target ( '-' ? + 'and' ) ')' ) + )
  1055     ;
  1056 
  1057     'code\_monad' const const target
  1058     ;
  1059 
  1060     'code\_reserved' target ( string + )
  1061     ;
  1062 
  1063     'code\_include' target ( string ( string | '-') )
  1064     ;
  1065 
  1066     'code\_modulename' target ( ( string string ) + )
  1067     ;
  1068 
  1069     'code\_abort' ( const + )
  1070     ;
  1071 
  1072     syntax: string | ( 'infix' | 'infixl' | 'infixr' ) nat string
  1073     ;
  1074 
  1075     'code' ( 'inline' ) ? ( 'del' ) ?
  1076     ;
  1077   \end{rail}
  1078 
  1079   \begin{description}
  1080 
  1081   \item @{command (HOL) "export_code"} is the canonical interface for
  1082   generating and serializing code: for a given list of constants, code
  1083   is generated for the specified target languages.  Abstract code is
  1084   cached incrementally.  If no constant is given, the currently cached
  1085   code is serialized.  If no serialization instruction is given, only
  1086   abstract code is cached.
  1087 
  1088   Constants may be specified by giving them literally, referring to
  1089   all executable contants within a certain theory by giving @{text
  1090   "name.*"}, or referring to \emph{all} executable constants currently
  1091   available by giving @{text "*"}.
  1092 
  1093   By default, for each involved theory one corresponding name space
  1094   module is generated.  Alternativly, a module name may be specified
  1095   after the @{keyword "module_name"} keyword; then \emph{all} code is
  1096   placed in this module.
  1097 
  1098   For \emph{SML} and \emph{OCaml}, the file specification refers to a
  1099   single file; for \emph{Haskell}, it refers to a whole directory,
  1100   where code is generated in multiple files reflecting the module
  1101   hierarchy.  The file specification ``@{text "-"}'' denotes standard
  1102   output.  For \emph{SML}, omitting the file specification compiles
  1103   code internally in the context of the current ML session.
  1104 
  1105   Serializers take an optional list of arguments in parentheses.  For
  1106   \emph{Haskell} a module name prefix may be given using the ``@{text
  1107   "root:"}'' argument; ``@{text string_classes}'' adds a ``@{verbatim
  1108   "deriving (Read, Show)"}'' clause to each appropriate datatype
  1109   declaration.
  1110 
  1111   \item @{command (HOL) "code_thms"} prints a list of theorems
  1112   representing the corresponding program containing all given
  1113   constants; if no constants are given, the currently cached code
  1114   theorems are printed.
  1115 
  1116   \item @{command (HOL) "code_deps"} visualizes dependencies of
  1117   theorems representing the corresponding program containing all given
  1118   constants; if no constants are given, the currently cached code
  1119   theorems are visualized.
  1120 
  1121   \item @{command (HOL) "code_datatype"} specifies a constructor set
  1122   for a logical type.
  1123 
  1124   \item @{command (HOL) "code_const"} associates a list of constants
  1125   with target-specific serializations; omitting a serialization
  1126   deletes an existing serialization.
  1127 
  1128   \item @{command (HOL) "code_type"} associates a list of type
  1129   constructors with target-specific serializations; omitting a
  1130   serialization deletes an existing serialization.
  1131 
  1132   \item @{command (HOL) "code_class"} associates a list of classes
  1133   with target-specific class names; omitting a serialization deletes
  1134   an existing serialization.  This applies only to \emph{Haskell}.
  1135 
  1136   \item @{command (HOL) "code_instance"} declares a list of type
  1137   constructor / class instance relations as ``already present'' for a
  1138   given target.  Omitting a ``@{text "-"}'' deletes an existing
  1139   ``already present'' declaration.  This applies only to
  1140   \emph{Haskell}.
  1141 
  1142   \item @{command (HOL) "code_monad"} provides an auxiliary mechanism
  1143   to generate monadic code for Haskell.
  1144 
  1145   \item @{command (HOL) "code_reserved"} declares a list of names as
  1146   reserved for a given target, preventing it to be shadowed by any
  1147   generated code.
  1148 
  1149   \item @{command (HOL) "code_include"} adds arbitrary named content
  1150   (``include'') to generated code.  A ``@{text "-"}'' as last argument
  1151   will remove an already added ``include''.
  1152 
  1153   \item @{command (HOL) "code_modulename"} declares aliasings from one
  1154   module name onto another.
  1155 
  1156   \item @{command (HOL) "code_abort"} declares constants which are not
  1157   required to have a definition by means of defining equations; if
  1158   needed these are implemented by program abort instead.
  1159 
  1160   \item @{attribute (HOL) code} explicitly selects (or with option
  1161   ``@{text "del"}'' deselects) a defining equation for code
  1162   generation.  Usually packages introducing defining equations provide
  1163   a reasonable default setup for selection.
  1164 
  1165   \item @{attribute (HOL) code}~@{text inline} declares (or with
  1166   option ``@{text "del"}'' removes) inlining theorems which are
  1167   applied as rewrite rules to any defining equation during
  1168   preprocessing.
  1169 
  1170   \item @{command (HOL) "print_codesetup"} gives an overview on
  1171   selected defining equations, code generator datatypes and
  1172   preprocessor setup.
  1173 
  1174   \end{description}
  1175 *}
  1176 
  1177 
  1178 section {* Definition by specification \label{sec:hol-specification} *}
  1179 
  1180 text {*
  1181   \begin{matharray}{rcl}
  1182     @{command_def (HOL) "specification"} & : & @{text "theory \<rightarrow> proof(prove)"} \\
  1183     @{command_def (HOL) "ax_specification"} & : & @{text "theory \<rightarrow> proof(prove)"} \\
  1184   \end{matharray}
  1185 
  1186   \begin{rail}
  1187   ('specification' | 'ax\_specification') '(' (decl +) ')' \\ (thmdecl? prop +)
  1188   ;
  1189   decl: ((name ':')? term '(' 'overloaded' ')'?)
  1190   \end{rail}
  1191 
  1192   \begin{description}
  1193 
  1194   \item @{command (HOL) "specification"}~@{text "decls \<phi>"} sets up a
  1195   goal stating the existence of terms with the properties specified to
  1196   hold for the constants given in @{text decls}.  After finishing the
  1197   proof, the theory will be augmented with definitions for the given
  1198   constants, as well as with theorems stating the properties for these
  1199   constants.
  1200 
  1201   \item @{command (HOL) "ax_specification"}~@{text "decls \<phi>"} sets up
  1202   a goal stating the existence of terms with the properties specified
  1203   to hold for the constants given in @{text decls}.  After finishing
  1204   the proof, the theory will be augmented with axioms expressing the
  1205   properties given in the first place.
  1206 
  1207   \item @{text decl} declares a constant to be defined by the
  1208   specification given.  The definition for the constant @{text c} is
  1209   bound to the name @{text c_def} unless a theorem name is given in
  1210   the declaration.  Overloaded constants should be declared as such.
  1211 
  1212   \end{description}
  1213 
  1214   Whether to use @{command (HOL) "specification"} or @{command (HOL)
  1215   "ax_specification"} is to some extent a matter of style.  @{command
  1216   (HOL) "specification"} introduces no new axioms, and so by
  1217   construction cannot introduce inconsistencies, whereas @{command
  1218   (HOL) "ax_specification"} does introduce axioms, but only after the
  1219   user has explicitly proven it to be safe.  A practical issue must be
  1220   considered, though: After introducing two constants with the same
  1221   properties using @{command (HOL) "specification"}, one can prove
  1222   that the two constants are, in fact, equal.  If this might be a
  1223   problem, one should use @{command (HOL) "ax_specification"}.
  1224 *}
  1225 
  1226 end