doc-src/IsarRef/Thy/HOL_Specific.thy
author krauss
Mon Nov 23 15:06:37 2009 +0100 (2009-11-23)
changeset 33857 0cb5002c52db
parent 31998 2c7a24f74db9
child 33858 0c348f7997f7
permissions -rw-r--r--
clarified; checked
     1 theory HOL_Specific
     2 imports Main
     3 begin
     4 
     5 chapter {* Isabelle/HOL \label{ch:hol} *}
     6 
     7 section {* Primitive types \label{sec:hol-typedef} *}
     8 
     9 text {*
    10   \begin{matharray}{rcl}
    11     @{command_def (HOL) "typedecl"} & : & @{text "theory \<rightarrow> theory"} \\
    12     @{command_def (HOL) "typedef"} & : & @{text "theory \<rightarrow> proof(prove)"} \\
    13   \end{matharray}
    14 
    15   \begin{rail}
    16     'typedecl' typespec infix?
    17     ;
    18     'typedef' altname? abstype '=' repset
    19     ;
    20 
    21     altname: '(' (name | 'open' | 'open' name) ')'
    22     ;
    23     abstype: typespec infix?
    24     ;
    25     repset: term ('morphisms' name name)?
    26     ;
    27   \end{rail}
    28 
    29   \begin{description}
    30   
    31   \item @{command (HOL) "typedecl"}~@{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>n) t"} is similar
    32   to the original @{command "typedecl"} of Isabelle/Pure (see
    33   \secref{sec:types-pure}), but also declares type arity @{text "t ::
    34   (type, \<dots>, type) type"}, making @{text t} an actual HOL type
    35   constructor.  %FIXME check, update
    36   
    37   \item @{command (HOL) "typedef"}~@{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>n) t = A"} sets up
    38   a goal stating non-emptiness of the set @{text A}.  After finishing
    39   the proof, the theory will be augmented by a Gordon/HOL-style type
    40   definition, which establishes a bijection between the representing
    41   set @{text A} and the new type @{text t}.
    42   
    43   Technically, @{command (HOL) "typedef"} defines both a type @{text
    44   t} and a set (term constant) of the same name (an alternative base
    45   name may be given in parentheses).  The injection from type to set
    46   is called @{text Rep_t}, its inverse @{text Abs_t} (this may be
    47   changed via an explicit @{keyword (HOL) "morphisms"} declaration).
    48   
    49   Theorems @{text Rep_t}, @{text Rep_t_inverse}, and @{text
    50   Abs_t_inverse} provide the most basic characterization as a
    51   corresponding injection/surjection pair (in both directions).  Rules
    52   @{text Rep_t_inject} and @{text Abs_t_inject} provide a slightly
    53   more convenient view on the injectivity part, suitable for automated
    54   proof tools (e.g.\ in @{attribute simp} or @{attribute iff}
    55   declarations).  Rules @{text Rep_t_cases}/@{text Rep_t_induct}, and
    56   @{text Abs_t_cases}/@{text Abs_t_induct} provide alternative views
    57   on surjectivity; these are already declared as set or type rules for
    58   the generic @{method cases} and @{method induct} methods.
    59   
    60   An alternative name may be specified in parentheses; the default is
    61   to use @{text t} as indicated before.  The ``@{text "(open)"}''
    62   declaration suppresses a separate constant definition for the
    63   representing set.
    64 
    65   \end{description}
    66 
    67   Note that raw type declarations are rarely used in practice; the
    68   main application is with experimental (or even axiomatic!) theory
    69   fragments.  Instead of primitive HOL type definitions, user-level
    70   theories usually refer to higher-level packages such as @{command
    71   (HOL) "record"} (see \secref{sec:hol-record}) or @{command (HOL)
    72   "datatype"} (see \secref{sec:hol-datatype}).
    73 *}
    74 
    75 
    76 section {* Adhoc tuples *}
    77 
    78 text {*
    79   \begin{matharray}{rcl}
    80     @{attribute (HOL) split_format}@{text "\<^sup>*"} & : & @{text attribute} \\
    81   \end{matharray}
    82 
    83   \begin{rail}
    84     'split\_format' ((( name * ) + 'and') | ('(' 'complete' ')'))
    85     ;
    86   \end{rail}
    87 
    88   \begin{description}
    89   
    90   \item @{attribute (HOL) split_format}~@{text "p\<^sub>1 \<dots> p\<^sub>m \<AND> \<dots>
    91   \<AND> q\<^sub>1 \<dots> q\<^sub>n"} puts expressions of low-level tuple types into
    92   canonical form as specified by the arguments given; the @{text i}-th
    93   collection of arguments refers to occurrences in premise @{text i}
    94   of the rule.  The ``@{text "(complete)"}'' option causes \emph{all}
    95   arguments in function applications to be represented canonically
    96   according to their tuple type structure.
    97 
    98   Note that these operations tend to invent funny names for new local
    99   parameters to be introduced.
   100 
   101   \end{description}
   102 *}
   103 
   104 
   105 section {* Records \label{sec:hol-record} *}
   106 
   107 text {*
   108   In principle, records merely generalize the concept of tuples, where
   109   components may be addressed by labels instead of just position.  The
   110   logical infrastructure of records in Isabelle/HOL is slightly more
   111   advanced, though, supporting truly extensible record schemes.  This
   112   admits operations that are polymorphic with respect to record
   113   extension, yielding ``object-oriented'' effects like (single)
   114   inheritance.  See also \cite{NaraschewskiW-TPHOLs98} for more
   115   details on object-oriented verification and record subtyping in HOL.
   116 *}
   117 
   118 
   119 subsection {* Basic concepts *}
   120 
   121 text {*
   122   Isabelle/HOL supports both \emph{fixed} and \emph{schematic} records
   123   at the level of terms and types.  The notation is as follows:
   124 
   125   \begin{center}
   126   \begin{tabular}{l|l|l}
   127     & record terms & record types \\ \hline
   128     fixed & @{text "\<lparr>x = a, y = b\<rparr>"} & @{text "\<lparr>x :: A, y :: B\<rparr>"} \\
   129     schematic & @{text "\<lparr>x = a, y = b, \<dots> = m\<rparr>"} &
   130       @{text "\<lparr>x :: A, y :: B, \<dots> :: M\<rparr>"} \\
   131   \end{tabular}
   132   \end{center}
   133 
   134   \noindent The ASCII representation of @{text "\<lparr>x = a\<rparr>"} is @{text
   135   "(| x = a |)"}.
   136 
   137   A fixed record @{text "\<lparr>x = a, y = b\<rparr>"} has field @{text x} of value
   138   @{text a} and field @{text y} of value @{text b}.  The corresponding
   139   type is @{text "\<lparr>x :: A, y :: B\<rparr>"}, assuming that @{text "a :: A"}
   140   and @{text "b :: B"}.
   141 
   142   A record scheme like @{text "\<lparr>x = a, y = b, \<dots> = m\<rparr>"} contains fields
   143   @{text x} and @{text y} as before, but also possibly further fields
   144   as indicated by the ``@{text "\<dots>"}'' notation (which is actually part
   145   of the syntax).  The improper field ``@{text "\<dots>"}'' of a record
   146   scheme is called the \emph{more part}.  Logically it is just a free
   147   variable, which is occasionally referred to as ``row variable'' in
   148   the literature.  The more part of a record scheme may be
   149   instantiated by zero or more further components.  For example, the
   150   previous scheme may get instantiated to @{text "\<lparr>x = a, y = b, z =
   151   c, \<dots> = m'\<rparr>"}, where @{text m'} refers to a different more part.
   152   Fixed records are special instances of record schemes, where
   153   ``@{text "\<dots>"}'' is properly terminated by the @{text "() :: unit"}
   154   element.  In fact, @{text "\<lparr>x = a, y = b\<rparr>"} is just an abbreviation
   155   for @{text "\<lparr>x = a, y = b, \<dots> = ()\<rparr>"}.
   156   
   157   \medskip Two key observations make extensible records in a simply
   158   typed language like HOL work out:
   159 
   160   \begin{enumerate}
   161 
   162   \item the more part is internalized, as a free term or type
   163   variable,
   164 
   165   \item field names are externalized, they cannot be accessed within
   166   the logic as first-class values.
   167 
   168   \end{enumerate}
   169 
   170   \medskip In Isabelle/HOL record types have to be defined explicitly,
   171   fixing their field names and types, and their (optional) parent
   172   record.  Afterwards, records may be formed using above syntax, while
   173   obeying the canonical order of fields as given by their declaration.
   174   The record package provides several standard operations like
   175   selectors and updates.  The common setup for various generic proof
   176   tools enable succinct reasoning patterns.  See also the Isabelle/HOL
   177   tutorial \cite{isabelle-hol-book} for further instructions on using
   178   records in practice.
   179 *}
   180 
   181 
   182 subsection {* Record specifications *}
   183 
   184 text {*
   185   \begin{matharray}{rcl}
   186     @{command_def (HOL) "record"} & : & @{text "theory \<rightarrow> theory"} \\
   187   \end{matharray}
   188 
   189   \begin{rail}
   190     'record' typespec '=' (type '+')? (constdecl +)
   191     ;
   192   \end{rail}
   193 
   194   \begin{description}
   195 
   196   \item @{command (HOL) "record"}~@{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t = \<tau> + c\<^sub>1 :: \<sigma>\<^sub>1
   197   \<dots> c\<^sub>n :: \<sigma>\<^sub>n"} defines extensible record type @{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t"},
   198   derived from the optional parent record @{text "\<tau>"} by adding new
   199   field components @{text "c\<^sub>i :: \<sigma>\<^sub>i"} etc.
   200 
   201   The type variables of @{text "\<tau>"} and @{text "\<sigma>\<^sub>i"} need to be
   202   covered by the (distinct) parameters @{text "\<alpha>\<^sub>1, \<dots>,
   203   \<alpha>\<^sub>m"}.  Type constructor @{text t} has to be new, while @{text
   204   \<tau>} needs to specify an instance of an existing record type.  At
   205   least one new field @{text "c\<^sub>i"} has to be specified.
   206   Basically, field names need to belong to a unique record.  This is
   207   not a real restriction in practice, since fields are qualified by
   208   the record name internally.
   209 
   210   The parent record specification @{text \<tau>} is optional; if omitted
   211   @{text t} becomes a root record.  The hierarchy of all records
   212   declared within a theory context forms a forest structure, i.e.\ a
   213   set of trees starting with a root record each.  There is no way to
   214   merge multiple parent records!
   215 
   216   For convenience, @{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t"} is made a
   217   type abbreviation for the fixed record type @{text "\<lparr>c\<^sub>1 ::
   218   \<sigma>\<^sub>1, \<dots>, c\<^sub>n :: \<sigma>\<^sub>n\<rparr>"}, likewise is @{text
   219   "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m, \<zeta>) t_scheme"} made an abbreviation for
   220   @{text "\<lparr>c\<^sub>1 :: \<sigma>\<^sub>1, \<dots>, c\<^sub>n :: \<sigma>\<^sub>n, \<dots> ::
   221   \<zeta>\<rparr>"}.
   222 
   223   \end{description}
   224 *}
   225 
   226 
   227 subsection {* Record operations *}
   228 
   229 text {*
   230   Any record definition of the form presented above produces certain
   231   standard operations.  Selectors and updates are provided for any
   232   field, including the improper one ``@{text more}''.  There are also
   233   cumulative record constructor functions.  To simplify the
   234   presentation below, we assume for now that @{text "(\<alpha>\<^sub>1, \<dots>,
   235   \<alpha>\<^sub>m) t"} is a root record with fields @{text "c\<^sub>1 ::
   236   \<sigma>\<^sub>1, \<dots>, c\<^sub>n :: \<sigma>\<^sub>n"}.
   237 
   238   \medskip \textbf{Selectors} and \textbf{updates} are available for
   239   any field (including ``@{text more}''):
   240 
   241   \begin{matharray}{lll}
   242     @{text "c\<^sub>i"} & @{text "::"} & @{text "\<lparr>\<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr> \<Rightarrow> \<sigma>\<^sub>i"} \\
   243     @{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>"} \\
   244   \end{matharray}
   245 
   246   There is special syntax for application of updates: @{text "r\<lparr>x :=
   247   a\<rparr>"} abbreviates term @{text "x_update a r"}.  Further notation for
   248   repeated updates is also available: @{text "r\<lparr>x := a\<rparr>\<lparr>y := b\<rparr>\<lparr>z :=
   249   c\<rparr>"} may be written @{text "r\<lparr>x := a, y := b, z := c\<rparr>"}.  Note that
   250   because of postfix notation the order of fields shown here is
   251   reverse than in the actual term.  Since repeated updates are just
   252   function applications, fields may be freely permuted in @{text "\<lparr>x
   253   := a, y := b, z := c\<rparr>"}, as far as logical equality is concerned.
   254   Thus commutativity of independent updates can be proven within the
   255   logic for any two fields, but not as a general theorem.
   256 
   257   \medskip The \textbf{make} operation provides a cumulative record
   258   constructor function:
   259 
   260   \begin{matharray}{lll}
   261     @{text "t.make"} & @{text "::"} & @{text "\<sigma>\<^sub>1 \<Rightarrow> \<dots> \<sigma>\<^sub>n \<Rightarrow> \<lparr>\<^vec>c :: \<^vec>\<sigma>\<rparr>"} \\
   262   \end{matharray}
   263 
   264   \medskip We now reconsider the case of non-root records, which are
   265   derived of some parent.  In general, the latter may depend on
   266   another parent as well, resulting in a list of \emph{ancestor
   267   records}.  Appending the lists of fields of all ancestors results in
   268   a certain field prefix.  The record package automatically takes care
   269   of this by lifting operations over this context of ancestor fields.
   270   Assuming that @{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t"} has ancestor
   271   fields @{text "b\<^sub>1 :: \<rho>\<^sub>1, \<dots>, b\<^sub>k :: \<rho>\<^sub>k"},
   272   the above record operations will get the following types:
   273 
   274   \medskip
   275   \begin{tabular}{lll}
   276     @{text "c\<^sub>i"} & @{text "::"} & @{text "\<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr> \<Rightarrow> \<sigma>\<^sub>i"} \\
   277     @{text "c\<^sub>i_update"} & @{text "::"} & @{text "\<sigma>\<^sub>i \<Rightarrow> 
   278       \<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr> \<Rightarrow>
   279       \<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr>"} \\
   280     @{text "t.make"} & @{text "::"} & @{text "\<rho>\<^sub>1 \<Rightarrow> \<dots> \<rho>\<^sub>k \<Rightarrow> \<sigma>\<^sub>1 \<Rightarrow> \<dots> \<sigma>\<^sub>n \<Rightarrow>
   281       \<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>\<rparr>"} \\
   282   \end{tabular}
   283   \medskip
   284 
   285   \noindent Some further operations address the extension aspect of a
   286   derived record scheme specifically: @{text "t.fields"} produces a
   287   record fragment consisting of exactly the new fields introduced here
   288   (the result may serve as a more part elsewhere); @{text "t.extend"}
   289   takes a fixed record and adds a given more part; @{text
   290   "t.truncate"} restricts a record scheme to a fixed record.
   291 
   292   \medskip
   293   \begin{tabular}{lll}
   294     @{text "t.fields"} & @{text "::"} & @{text "\<sigma>\<^sub>1 \<Rightarrow> \<dots> \<sigma>\<^sub>n \<Rightarrow> \<lparr>\<^vec>c :: \<^vec>\<sigma>\<rparr>"} \\
   295     @{text "t.extend"} & @{text "::"} & @{text "\<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>\<rparr> \<Rightarrow>
   296       \<zeta> \<Rightarrow> \<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr>"} \\
   297     @{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>"} \\
   298   \end{tabular}
   299   \medskip
   300 
   301   \noindent Note that @{text "t.make"} and @{text "t.fields"} coincide
   302   for root records.
   303 *}
   304 
   305 
   306 subsection {* Derived rules and proof tools *}
   307 
   308 text {*
   309   The record package proves several results internally, declaring
   310   these facts to appropriate proof tools.  This enables users to
   311   reason about record structures quite conveniently.  Assume that
   312   @{text t} is a record type as specified above.
   313 
   314   \begin{enumerate}
   315   
   316   \item Standard conversions for selectors or updates applied to
   317   record constructor terms are made part of the default Simplifier
   318   context; thus proofs by reduction of basic operations merely require
   319   the @{method simp} method without further arguments.  These rules
   320   are available as @{text "t.simps"}, too.
   321   
   322   \item Selectors applied to updated records are automatically reduced
   323   by an internal simplification procedure, which is also part of the
   324   standard Simplifier setup.
   325 
   326   \item Inject equations of a form analogous to @{prop "(x, y) = (x',
   327   y') \<equiv> x = x' \<and> y = y'"} are declared to the Simplifier and Classical
   328   Reasoner as @{attribute iff} rules.  These rules are available as
   329   @{text "t.iffs"}.
   330 
   331   \item The introduction rule for record equality analogous to @{text
   332   "x r = x r' \<Longrightarrow> y r = y r' \<dots> \<Longrightarrow> r = r'"} is declared to the Simplifier,
   333   and as the basic rule context as ``@{attribute intro}@{text "?"}''.
   334   The rule is called @{text "t.equality"}.
   335 
   336   \item Representations of arbitrary record expressions as canonical
   337   constructor terms are provided both in @{method cases} and @{method
   338   induct} format (cf.\ the generic proof methods of the same name,
   339   \secref{sec:cases-induct}).  Several variations are available, for
   340   fixed records, record schemes, more parts etc.
   341   
   342   The generic proof methods are sufficiently smart to pick the most
   343   sensible rule according to the type of the indicated record
   344   expression: users just need to apply something like ``@{text "(cases
   345   r)"}'' to a certain proof problem.
   346 
   347   \item The derived record operations @{text "t.make"}, @{text
   348   "t.fields"}, @{text "t.extend"}, @{text "t.truncate"} are \emph{not}
   349   treated automatically, but usually need to be expanded by hand,
   350   using the collective fact @{text "t.defs"}.
   351 
   352   \end{enumerate}
   353 *}
   354 
   355 
   356 section {* Datatypes \label{sec:hol-datatype} *}
   357 
   358 text {*
   359   \begin{matharray}{rcl}
   360     @{command_def (HOL) "datatype"} & : & @{text "theory \<rightarrow> theory"} \\
   361   @{command_def (HOL) "rep_datatype"} & : & @{text "theory \<rightarrow> proof(prove)"} \\
   362   \end{matharray}
   363 
   364   \begin{rail}
   365     'datatype' (dtspec + 'and')
   366     ;
   367     'rep\_datatype' ('(' (name +) ')')? (term +)
   368     ;
   369 
   370     dtspec: parname? typespec infix? '=' (cons + '|')
   371     ;
   372     cons: name ( type * ) mixfix?
   373   \end{rail}
   374 
   375   \begin{description}
   376 
   377   \item @{command (HOL) "datatype"} defines inductive datatypes in
   378   HOL.
   379 
   380   \item @{command (HOL) "rep_datatype"} represents existing types as
   381   inductive ones, generating the standard infrastructure of derived
   382   concepts (primitive recursion etc.).
   383 
   384   \end{description}
   385 
   386   The induction and exhaustion theorems generated provide case names
   387   according to the constructors involved, while parameters are named
   388   after the types (see also \secref{sec:cases-induct}).
   389 
   390   See \cite{isabelle-HOL} for more details on datatypes, but beware of
   391   the old-style theory syntax being used there!  Apart from proper
   392   proof methods for case-analysis and induction, there are also
   393   emulations of ML tactics @{method (HOL) case_tac} and @{method (HOL)
   394   induct_tac} available, see \secref{sec:hol-induct-tac}; these admit
   395   to refer directly to the internal structure of subgoals (including
   396   internally bound parameters).
   397 *}
   398 
   399 
   400 section {* Recursive functions \label{sec:recursion} *}
   401 
   402 text {*
   403   \begin{matharray}{rcl}
   404     @{command_def (HOL) "primrec"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
   405     @{command_def (HOL) "fun"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
   406     @{command_def (HOL) "function"} & : & @{text "local_theory \<rightarrow> proof(prove)"} \\
   407     @{command_def (HOL) "termination"} & : & @{text "local_theory \<rightarrow> proof(prove)"} \\
   408   \end{matharray}
   409 
   410   \begin{rail}
   411     'primrec' target? fixes 'where' equations
   412     ;
   413     equations: (thmdecl? prop + '|')
   414     ;
   415     ('fun' | 'function') target? functionopts? fixes 'where' clauses
   416     ;
   417     clauses: (thmdecl? prop ('(' 'otherwise' ')')? + '|')
   418     ;
   419     functionopts: '(' (('sequential' | 'domintros' | 'tailrec' | 'default' term) + ',') ')'
   420     ;
   421     'termination' ( term )?
   422   \end{rail}
   423 
   424   \begin{description}
   425 
   426   \item @{command (HOL) "primrec"} defines primitive recursive
   427   functions over datatypes, see also \cite{isabelle-HOL}.
   428 
   429   \item @{command (HOL) "function"} defines functions by general
   430   wellfounded recursion. A detailed description with examples can be
   431   found in \cite{isabelle-function}. The function is specified by a
   432   set of (possibly conditional) recursive equations with arbitrary
   433   pattern matching. The command generates proof obligations for the
   434   completeness and the compatibility of patterns.
   435 
   436   The defined function is considered partial, and the resulting
   437   simplification rules (named @{text "f.psimps"}) and induction rule
   438   (named @{text "f.pinduct"}) are guarded by a generated domain
   439   predicate @{text "f_dom"}. The @{command (HOL) "termination"}
   440   command can then be used to establish that the function is total.
   441 
   442   \item @{command (HOL) "fun"} is a shorthand notation for ``@{command
   443   (HOL) "function"}~@{text "(sequential)"}, followed by automated
   444   proof attempts regarding pattern matching and termination.  See
   445   \cite{isabelle-function} for further details.
   446 
   447   \item @{command (HOL) "termination"}~@{text f} commences a
   448   termination proof for the previously defined function @{text f}.  If
   449   this is omitted, the command refers to the most recent function
   450   definition.  After the proof is closed, the recursive equations and
   451   the induction principle is established.
   452 
   453   \end{description}
   454 
   455   Recursive definitions introduced by the @{command (HOL) "function"}
   456   command accommodate
   457   reasoning by induction (cf.\ \secref{sec:cases-induct}): rule @{text
   458   "c.induct"} (where @{text c} is the name of the function definition)
   459   refers to a specific induction rule, with parameters named according
   460   to the user-specified equations. Cases are numbered (starting from 1).
   461 
   462   For @{command (HOL) "primrec"}, the induction principle coincides
   463   with structural recursion on the datatype the recursion is carried
   464   out.
   465 
   466   The equations provided by these packages may be referred later as
   467   theorem list @{text "f.simps"}, where @{text f} is the (collective)
   468   name of the functions defined.  Individual equations may be named
   469   explicitly as well.
   470 
   471   The @{command (HOL) "function"} command accepts the following
   472   options.
   473 
   474   \begin{description}
   475 
   476   \item @{text sequential} enables a preprocessor which disambiguates
   477   overlapping patterns by making them mutually disjoint.  Earlier
   478   equations take precedence over later ones.  This allows to give the
   479   specification in a format very similar to functional programming.
   480   Note that the resulting simplification and induction rules
   481   correspond to the transformed specification, not the one given
   482   originally. This usually means that each equation given by the user
   483   may result in several theroems.  Also note that this automatic
   484   transformation only works for ML-style datatype patterns.
   485 
   486   \item @{text domintros} enables the automated generation of
   487   introduction rules for the domain predicate. While mostly not
   488   needed, they can be helpful in some proofs about partial functions.
   489 
   490   \item @{text tailrec} generates the unconstrained recursive
   491   equations even without a termination proof, provided that the
   492   function is tail-recursive. This currently only works
   493 
   494   \item @{text "default d"} allows to specify a default value for a
   495   (partial) function, which will ensure that @{text "f x = d x"}
   496   whenever @{text "x \<notin> f_dom"}.
   497 
   498   \end{description}
   499 *}
   500 
   501 
   502 subsection {* Proof methods related to recursive definitions *}
   503 
   504 text {*
   505   \begin{matharray}{rcl}
   506     @{method_def (HOL) pat_completeness} & : & @{text method} \\
   507     @{method_def (HOL) relation} & : & @{text method} \\
   508     @{method_def (HOL) lexicographic_order} & : & @{text method} \\
   509   \end{matharray}
   510 
   511   \begin{rail}
   512     'relation' term
   513     ;
   514     'lexicographic\_order' ( clasimpmod * )
   515     ;
   516   \end{rail}
   517 
   518   \begin{description}
   519 
   520   \item @{method (HOL) pat_completeness} is a specialized method to
   521   solve goals regarding the completeness of pattern matching, as
   522   required by the @{command (HOL) "function"} package (cf.\
   523   \cite{isabelle-function}).
   524 
   525   \item @{method (HOL) relation}~@{text R} introduces a termination
   526   proof using the relation @{text R}.  The resulting proof state will
   527   contain goals expressing that @{text R} is wellfounded, and that the
   528   arguments of recursive calls decrease with respect to @{text R}.
   529   Usually, this method is used as the initial proof step of manual
   530   termination proofs.
   531 
   532   \item @{method (HOL) "lexicographic_order"} attempts a fully
   533   automated termination proof by searching for a lexicographic
   534   combination of size measures on the arguments of the function. The
   535   method accepts the same arguments as the @{method auto} method,
   536   which it uses internally to prove local descents.  The same context
   537   modifiers as for @{method auto} are accepted, see
   538   \secref{sec:clasimp}.
   539 
   540   In case of failure, extensive information is printed, which can help
   541   to analyse the situation (cf.\ \cite{isabelle-function}).
   542 
   543   \end{description}
   544 *}
   545 
   546 
   547 subsection {* Old-style recursive function definitions (TFL) *}
   548 
   549 text {*
   550   The old TFL commands @{command (HOL) "recdef"} and @{command (HOL)
   551   "recdef_tc"} for defining recursive are mostly obsolete; @{command
   552   (HOL) "function"} or @{command (HOL) "fun"} should be used instead.
   553 
   554   \begin{matharray}{rcl}
   555     @{command_def (HOL) "recdef"} & : & @{text "theory \<rightarrow> theory)"} \\
   556     @{command_def (HOL) "recdef_tc"}@{text "\<^sup>*"} & : & @{text "theory \<rightarrow> proof(prove)"} \\
   557   \end{matharray}
   558 
   559   \begin{rail}
   560     'recdef' ('(' 'permissive' ')')? \\ name term (prop +) hints?
   561     ;
   562     recdeftc thmdecl? tc
   563     ;
   564     hints: '(' 'hints' ( recdefmod * ) ')'
   565     ;
   566     recdefmod: (('recdef\_simp' | 'recdef\_cong' | 'recdef\_wf') (() | 'add' | 'del') ':' thmrefs) | clasimpmod
   567     ;
   568     tc: nameref ('(' nat ')')?
   569     ;
   570   \end{rail}
   571 
   572   \begin{description}
   573   
   574   \item @{command (HOL) "recdef"} defines general well-founded
   575   recursive functions (using the TFL package), see also
   576   \cite{isabelle-HOL}.  The ``@{text "(permissive)"}'' option tells
   577   TFL to recover from failed proof attempts, returning unfinished
   578   results.  The @{text recdef_simp}, @{text recdef_cong}, and @{text
   579   recdef_wf} hints refer to auxiliary rules to be used in the internal
   580   automated proof process of TFL.  Additional @{syntax clasimpmod}
   581   declarations (cf.\ \secref{sec:clasimp}) may be given to tune the
   582   context of the Simplifier (cf.\ \secref{sec:simplifier}) and
   583   Classical reasoner (cf.\ \secref{sec:classical}).
   584   
   585   \item @{command (HOL) "recdef_tc"}~@{text "c (i)"} recommences the
   586   proof for leftover termination condition number @{text i} (default
   587   1) as generated by a @{command (HOL) "recdef"} definition of
   588   constant @{text c}.
   589   
   590   Note that in most cases, @{command (HOL) "recdef"} is able to finish
   591   its internal proofs without manual intervention.
   592 
   593   \end{description}
   594 
   595   \medskip Hints for @{command (HOL) "recdef"} may be also declared
   596   globally, using the following attributes.
   597 
   598   \begin{matharray}{rcl}
   599     @{attribute_def (HOL) recdef_simp} & : & @{text attribute} \\
   600     @{attribute_def (HOL) recdef_cong} & : & @{text attribute} \\
   601     @{attribute_def (HOL) recdef_wf} & : & @{text attribute} \\
   602   \end{matharray}
   603 
   604   \begin{rail}
   605     ('recdef\_simp' | 'recdef\_cong' | 'recdef\_wf') (() | 'add' | 'del')
   606     ;
   607   \end{rail}
   608 *}
   609 
   610 
   611 section {* Inductive and coinductive definitions \label{sec:hol-inductive} *}
   612 
   613 text {*
   614   An \textbf{inductive definition} specifies the least predicate (or
   615   set) @{text R} closed under given rules: applying a rule to elements
   616   of @{text R} yields a result within @{text R}.  For example, a
   617   structural operational semantics is an inductive definition of an
   618   evaluation relation.
   619 
   620   Dually, a \textbf{coinductive definition} specifies the greatest
   621   predicate~/ set @{text R} that is consistent with given rules: every
   622   element of @{text R} can be seen as arising by applying a rule to
   623   elements of @{text R}.  An important example is using bisimulation
   624   relations to formalise equivalence of processes and infinite data
   625   structures.
   626 
   627   \medskip The HOL package is related to the ZF one, which is
   628   described in a separate paper,\footnote{It appeared in CADE
   629   \cite{paulson-CADE}; a longer version is distributed with Isabelle.}
   630   which you should refer to in case of difficulties.  The package is
   631   simpler than that of ZF thanks to implicit type-checking in HOL.
   632   The types of the (co)inductive predicates (or sets) determine the
   633   domain of the fixedpoint definition, and the package does not have
   634   to use inference rules for type-checking.
   635 
   636   \begin{matharray}{rcl}
   637     @{command_def (HOL) "inductive"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
   638     @{command_def (HOL) "inductive_set"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
   639     @{command_def (HOL) "coinductive"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
   640     @{command_def (HOL) "coinductive_set"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
   641     @{attribute_def (HOL) mono} & : & @{text attribute} \\
   642   \end{matharray}
   643 
   644   \begin{rail}
   645     ('inductive' | 'inductive\_set' | 'coinductive' | 'coinductive\_set') target? fixes ('for' fixes)? \\
   646     ('where' clauses)? ('monos' thmrefs)?
   647     ;
   648     clauses: (thmdecl? prop + '|')
   649     ;
   650     'mono' (() | 'add' | 'del')
   651     ;
   652   \end{rail}
   653 
   654   \begin{description}
   655 
   656   \item @{command (HOL) "inductive"} and @{command (HOL)
   657   "coinductive"} define (co)inductive predicates from the
   658   introduction rules given in the @{keyword "where"} part.  The
   659   optional @{keyword "for"} part contains a list of parameters of the
   660   (co)inductive predicates that remain fixed throughout the
   661   definition.  The optional @{keyword "monos"} section contains
   662   \emph{monotonicity theorems}, which are required for each operator
   663   applied to a recursive set in the introduction rules.  There
   664   \emph{must} be a theorem of the form @{text "A \<le> B \<Longrightarrow> M A \<le> M B"},
   665   for each premise @{text "M R\<^sub>i t"} in an introduction rule!
   666 
   667   \item @{command (HOL) "inductive_set"} and @{command (HOL)
   668   "coinductive_set"} are wrappers for to the previous commands,
   669   allowing the definition of (co)inductive sets.
   670 
   671   \item @{attribute (HOL) mono} declares monotonicity rules.  These
   672   rule are involved in the automated monotonicity proof of @{command
   673   (HOL) "inductive"}.
   674 
   675   \end{description}
   676 *}
   677 
   678 
   679 subsection {* Derived rules *}
   680 
   681 text {*
   682   Each (co)inductive definition @{text R} adds definitions to the
   683   theory and also proves some theorems:
   684 
   685   \begin{description}
   686 
   687   \item @{text R.intros} is the list of introduction rules as proven
   688   theorems, for the recursive predicates (or sets).  The rules are
   689   also available individually, using the names given them in the
   690   theory file;
   691 
   692   \item @{text R.cases} is the case analysis (or elimination) rule;
   693 
   694   \item @{text R.induct} or @{text R.coinduct} is the (co)induction
   695   rule.
   696 
   697   \end{description}
   698 
   699   When several predicates @{text "R\<^sub>1, \<dots>, R\<^sub>n"} are
   700   defined simultaneously, the list of introduction rules is called
   701   @{text "R\<^sub>1_\<dots>_R\<^sub>n.intros"}, the case analysis rules are
   702   called @{text "R\<^sub>1.cases, \<dots>, R\<^sub>n.cases"}, and the list
   703   of mutual induction rules is called @{text
   704   "R\<^sub>1_\<dots>_R\<^sub>n.inducts"}.
   705 *}
   706 
   707 
   708 subsection {* Monotonicity theorems *}
   709 
   710 text {*
   711   Each theory contains a default set of theorems that are used in
   712   monotonicity proofs.  New rules can be added to this set via the
   713   @{attribute (HOL) mono} attribute.  The HOL theory @{text Inductive}
   714   shows how this is done.  In general, the following monotonicity
   715   theorems may be added:
   716 
   717   \begin{itemize}
   718 
   719   \item Theorems of the form @{text "A \<le> B \<Longrightarrow> M A \<le> M B"}, for proving
   720   monotonicity of inductive definitions whose introduction rules have
   721   premises involving terms such as @{text "M R\<^sub>i t"}.
   722 
   723   \item Monotonicity theorems for logical operators, which are of the
   724   general form @{text "(\<dots> \<longrightarrow> \<dots>) \<Longrightarrow> \<dots> (\<dots> \<longrightarrow> \<dots>) \<Longrightarrow> \<dots> \<longrightarrow> \<dots>"}.  For example, in
   725   the case of the operator @{text "\<or>"}, the corresponding theorem is
   726   \[
   727   \infer{@{text "P\<^sub>1 \<or> P\<^sub>2 \<longrightarrow> Q\<^sub>1 \<or> Q\<^sub>2"}}{@{text "P\<^sub>1 \<longrightarrow> Q\<^sub>1"} & @{text "P\<^sub>2 \<longrightarrow> Q\<^sub>2"}}
   728   \]
   729 
   730   \item De Morgan style equations for reasoning about the ``polarity''
   731   of expressions, e.g.
   732   \[
   733   @{prop "\<not> \<not> P \<longleftrightarrow> P"} \qquad\qquad
   734   @{prop "\<not> (P \<and> Q) \<longleftrightarrow> \<not> P \<or> \<not> Q"}
   735   \]
   736 
   737   \item Equations for reducing complex operators to more primitive
   738   ones whose monotonicity can easily be proved, e.g.
   739   \[
   740   @{prop "(P \<longrightarrow> Q) \<longleftrightarrow> \<not> P \<or> Q"} \qquad\qquad
   741   @{prop "Ball A P \<equiv> \<forall>x. x \<in> A \<longrightarrow> P x"}
   742   \]
   743 
   744   \end{itemize}
   745 
   746   %FIXME: Example of an inductive definition
   747 *}
   748 
   749 
   750 section {* Arithmetic proof support *}
   751 
   752 text {*
   753   \begin{matharray}{rcl}
   754     @{method_def (HOL) arith} & : & @{text method} \\
   755     @{attribute_def (HOL) arith} & : & @{text attribute} \\
   756     @{attribute_def (HOL) arith_split} & : & @{text attribute} \\
   757   \end{matharray}
   758 
   759   The @{method (HOL) arith} method decides linear arithmetic problems
   760   (on types @{text nat}, @{text int}, @{text real}).  Any current
   761   facts are inserted into the goal before running the procedure.
   762 
   763   The @{attribute (HOL) arith} attribute declares facts that are
   764   always supplied to the arithmetic provers implicitly.
   765 
   766   The @{attribute (HOL) arith_split} attribute declares case split
   767   rules to be expanded before @{method (HOL) arith} is invoked.
   768 
   769   Note that a simpler (but faster) arithmetic prover is
   770   already invoked by the Simplifier.
   771 *}
   772 
   773 
   774 section {* Intuitionistic proof search *}
   775 
   776 text {*
   777   \begin{matharray}{rcl}
   778     @{method_def (HOL) iprover} & : & @{text method} \\
   779   \end{matharray}
   780 
   781   \begin{rail}
   782     'iprover' ('!' ?) ( rulemod * )
   783     ;
   784   \end{rail}
   785 
   786   The @{method (HOL) iprover} method performs intuitionistic proof
   787   search, depending on specifically declared rules from the context,
   788   or given as explicit arguments.  Chained facts are inserted into the
   789   goal before commencing proof search; ``@{method (HOL) iprover}@{text
   790   "!"}''  means to include the current @{fact prems} as well.
   791   
   792   Rules need to be classified as @{attribute (Pure) intro},
   793   @{attribute (Pure) elim}, or @{attribute (Pure) dest}; here the
   794   ``@{text "!"}'' indicator refers to ``safe'' rules, which may be
   795   applied aggressively (without considering back-tracking later).
   796   Rules declared with ``@{text "?"}'' are ignored in proof search (the
   797   single-step @{method rule} method still observes these).  An
   798   explicit weight annotation may be given as well; otherwise the
   799   number of rule premises will be taken into account here.
   800 *}
   801 
   802 
   803 section {* Coherent Logic *}
   804 
   805 text {*
   806   \begin{matharray}{rcl}
   807     @{method_def (HOL) "coherent"} & : & @{text method} \\
   808   \end{matharray}
   809 
   810   \begin{rail}
   811     'coherent' thmrefs?
   812     ;
   813   \end{rail}
   814 
   815   The @{method (HOL) coherent} method solves problems of
   816   \emph{Coherent Logic} \cite{Bezem-Coquand:2005}, which covers
   817   applications in confluence theory, lattice theory and projective
   818   geometry.  See @{"file" "~~/src/HOL/ex/Coherent.thy"} for some
   819   examples.
   820 *}
   821 
   822 
   823 section {* Checking and refuting propositions *}
   824 
   825 text {*
   826   Identifying incorrect propositions usually involves evaluation of
   827   particular assignments and systematic counter example search.  This
   828   is supported by the following commands.
   829 
   830   \begin{matharray}{rcl}
   831     @{command_def (HOL) "value"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
   832     @{command_def (HOL) "quickcheck"}@{text "\<^sup>*"} & : & @{text "proof \<rightarrow>"} \\
   833     @{command_def (HOL) "quickcheck_params"} & : & @{text "theory \<rightarrow> theory"}
   834   \end{matharray}
   835 
   836   \begin{rail}
   837     'value' ( ( '[' name ']' ) ? ) modes? term
   838     ;
   839 
   840     'quickcheck' ( ( '[' args ']' ) ? ) nat?
   841     ;
   842 
   843     'quickcheck_params' ( ( '[' args ']' ) ? )
   844     ;
   845 
   846     modes: '(' (name + ) ')'
   847     ;
   848 
   849     args: ( name '=' value + ',' )
   850     ;
   851   \end{rail}
   852 
   853   \begin{description}
   854 
   855   \item @{command (HOL) "value"}~@{text t} evaluates and prints a
   856     term; optionally @{text modes} can be specified, which are
   857     appended to the current print mode (see also \cite{isabelle-ref}).
   858     Internally, the evaluation is performed by registered evaluators,
   859     which are invoked sequentially until a result is returned.
   860     Alternatively a specific evaluator can be selected using square
   861     brackets; available evaluators include @{text nbe} for
   862     \emph{normalization by evaluation} and \emph{code} for code
   863     generation in SML.
   864 
   865   \item @{command (HOL) "quickcheck"} tests the current goal for
   866     counter examples using a series of arbitrary assignments for its
   867     free variables; by default the first subgoal is tested, an other
   868     can be selected explicitly using an optional goal index.
   869     A number of configuration options are supported for
   870     @{command (HOL) "quickcheck"}, notably:
   871 
   872     \begin{description}
   873 
   874       \item[size] specifies the maximum size of the search space for
   875         assignment values.
   876 
   877       \item[iterations] sets how many sets of assignments are
   878         generated for each particular size.
   879 
   880     \end{description}
   881 
   882     These option can be given within square brackets.
   883 
   884   \item @{command (HOL) "quickcheck_params"} changes quickcheck
   885     configuration options persitently.
   886 
   887   \end{description}
   888 *}
   889 
   890 
   891 section {* Invoking automated reasoning tools -- The Sledgehammer *}
   892 
   893 text {*
   894   Isabelle/HOL includes a generic \emph{ATP manager} that allows
   895   external automated reasoning tools to crunch a pending goal.
   896   Supported provers include E\footnote{\url{http://www.eprover.org}},
   897   SPASS\footnote{\url{http://www.spass-prover.org/}}, and Vampire.
   898   There is also a wrapper to invoke provers remotely via the
   899   SystemOnTPTP\footnote{\url{http://www.cs.miami.edu/~tptp/cgi-bin/SystemOnTPTP}}
   900   web service.
   901 
   902   The problem passed to external provers consists of the goal together
   903   with a smart selection of lemmas from the current theory context.
   904   The result of a successful proof search is some source text that
   905   usually reconstructs the proof within Isabelle, without requiring
   906   external provers again.  The Metis
   907   prover\footnote{\url{http://www.gilith.com/software/metis/}} that is
   908   integrated into Isabelle/HOL is being used here.
   909 
   910   In this mode of operation, heavy means of automated reasoning are
   911   used as a strong relevance filter, while the main proof checking
   912   works via explicit inferences going through the Isabelle kernel.
   913   Moreover, rechecking Isabelle proof texts with already specified
   914   auxiliary facts is much faster than performing fully automated
   915   search over and over again.
   916 
   917   \begin{matharray}{rcl}
   918     @{command_def (HOL) "sledgehammer"}@{text "\<^sup>*"} & : & @{text "proof \<rightarrow>"} \\
   919     @{command_def (HOL) "print_atps"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
   920     @{command_def (HOL) "atp_info"}@{text "\<^sup>*"} & : & @{text "any \<rightarrow>"} \\
   921     @{command_def (HOL) "atp_kill"}@{text "\<^sup>*"} & : & @{text "any \<rightarrow>"} \\
   922     @{command_def (HOL) "atp_messages"}@{text "\<^sup>*"} & : & @{text "any \<rightarrow>"} \\
   923     @{method_def (HOL) metis} & : & @{text method} \\
   924   \end{matharray}
   925 
   926   \begin{rail}
   927   'sledgehammer' ( nameref * )
   928   ;
   929   'atp\_messages' ('(' nat ')')?
   930   ;
   931 
   932   'metis' thmrefs
   933   ;
   934   \end{rail}
   935 
   936   \begin{description}
   937 
   938   \item @{command (HOL) sledgehammer}~@{text "prover\<^sub>1 \<dots> prover\<^sub>n"}
   939   invokes the specified automated theorem provers on the first
   940   subgoal.  Provers are run in parallel, the first successful result
   941   is displayed, and the other attempts are terminated.
   942 
   943   Provers are defined in the theory context, see also @{command (HOL)
   944   print_atps}.  If no provers are given as arguments to @{command
   945   (HOL) sledgehammer}, the system refers to the default defined as
   946   ``ATP provers'' preference by the user interface.
   947 
   948   There are additional preferences for timeout (default: 60 seconds),
   949   and the maximum number of independent prover processes (default: 5);
   950   excessive provers are automatically terminated.
   951 
   952   \item @{command (HOL) print_atps} prints the list of automated
   953   theorem provers available to the @{command (HOL) sledgehammer}
   954   command.
   955 
   956   \item @{command (HOL) atp_info} prints information about presently
   957   running provers, including elapsed runtime, and the remaining time
   958   until timeout.
   959 
   960   \item @{command (HOL) atp_kill} terminates all presently running
   961   provers.
   962 
   963   \item @{command (HOL) atp_messages} displays recent messages issued
   964   by automated theorem provers.  This allows to examine results that
   965   might have got lost due to the asynchronous nature of default
   966   @{command (HOL) sledgehammer} output.  An optional message limit may
   967   be specified (default 5).
   968 
   969   \item @{method (HOL) metis}~@{text "facts"} invokes the Metis prover
   970   with the given facts.  Metis is an automated proof tool of medium
   971   strength, but is fully integrated into Isabelle/HOL, with explicit
   972   inferences going through the kernel.  Thus its results are
   973   guaranteed to be ``correct by construction''.
   974 
   975   Note that all facts used with Metis need to be specified as explicit
   976   arguments.  There are no rule declarations as for other Isabelle
   977   provers, like @{method blast} or @{method fast}.
   978 
   979   \end{description}
   980 *}
   981 
   982 
   983 section {* Unstructured case analysis and induction \label{sec:hol-induct-tac} *}
   984 
   985 text {*
   986   The following tools of Isabelle/HOL support cases analysis and
   987   induction in unstructured tactic scripts; see also
   988   \secref{sec:cases-induct} for proper Isar versions of similar ideas.
   989 
   990   \begin{matharray}{rcl}
   991     @{method_def (HOL) case_tac}@{text "\<^sup>*"} & : & @{text method} \\
   992     @{method_def (HOL) induct_tac}@{text "\<^sup>*"} & : & @{text method} \\
   993     @{method_def (HOL) ind_cases}@{text "\<^sup>*"} & : & @{text method} \\
   994     @{command_def (HOL) "inductive_cases"}@{text "\<^sup>*"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
   995   \end{matharray}
   996 
   997   \begin{rail}
   998     'case\_tac' goalspec? term rule?
   999     ;
  1000     'induct\_tac' goalspec? (insts * 'and') rule?
  1001     ;
  1002     'ind\_cases' (prop +) ('for' (name +)) ?
  1003     ;
  1004     'inductive\_cases' (thmdecl? (prop +) + 'and')
  1005     ;
  1006 
  1007     rule: ('rule' ':' thmref)
  1008     ;
  1009   \end{rail}
  1010 
  1011   \begin{description}
  1012 
  1013   \item @{method (HOL) case_tac} and @{method (HOL) induct_tac} admit
  1014   to reason about inductive types.  Rules are selected according to
  1015   the declarations by the @{attribute cases} and @{attribute induct}
  1016   attributes, cf.\ \secref{sec:cases-induct}.  The @{command (HOL)
  1017   datatype} package already takes care of this.
  1018 
  1019   These unstructured tactics feature both goal addressing and dynamic
  1020   instantiation.  Note that named rule cases are \emph{not} provided
  1021   as would be by the proper @{method cases} and @{method induct} proof
  1022   methods (see \secref{sec:cases-induct}).  Unlike the @{method
  1023   induct} method, @{method induct_tac} does not handle structured rule
  1024   statements, only the compact object-logic conclusion of the subgoal
  1025   being addressed.
  1026   
  1027   \item @{method (HOL) ind_cases} and @{command (HOL)
  1028   "inductive_cases"} provide an interface to the internal @{ML_text
  1029   mk_cases} operation.  Rules are simplified in an unrestricted
  1030   forward manner.
  1031 
  1032   While @{method (HOL) ind_cases} is a proof method to apply the
  1033   result immediately as elimination rules, @{command (HOL)
  1034   "inductive_cases"} provides case split theorems at the theory level
  1035   for later use.  The @{keyword "for"} argument of the @{method (HOL)
  1036   ind_cases} method allows to specify a list of variables that should
  1037   be generalized before applying the resulting rule.
  1038 
  1039   \end{description}
  1040 *}
  1041 
  1042 
  1043 section {* Executable code *}
  1044 
  1045 text {*
  1046   Isabelle/Pure provides two generic frameworks to support code
  1047   generation from executable specifications.  Isabelle/HOL
  1048   instantiates these mechanisms in a way that is amenable to end-user
  1049   applications.
  1050 
  1051   One framework generates code from both functional and relational
  1052   programs to SML.  See \cite{isabelle-HOL} for further information
  1053   (this actually covers the new-style theory format as well).
  1054 
  1055   \begin{matharray}{rcl}
  1056     @{command_def (HOL) "code_module"} & : & @{text "theory \<rightarrow> theory"} \\
  1057     @{command_def (HOL) "code_library"} & : & @{text "theory \<rightarrow> theory"} \\
  1058     @{command_def (HOL) "consts_code"} & : & @{text "theory \<rightarrow> theory"} \\
  1059     @{command_def (HOL) "types_code"} & : & @{text "theory \<rightarrow> theory"} \\  
  1060     @{attribute_def (HOL) code} & : & @{text attribute} \\
  1061   \end{matharray}
  1062 
  1063   \begin{rail}
  1064   ( 'code\_module' | 'code\_library' ) modespec ? name ? \\
  1065     ( 'file' name ) ? ( 'imports' ( name + ) ) ? \\
  1066     'contains' ( ( name '=' term ) + | term + )
  1067   ;
  1068 
  1069   modespec: '(' ( name * ) ')'
  1070   ;
  1071 
  1072   'consts\_code' (codespec +)
  1073   ;
  1074 
  1075   codespec: const template attachment ?
  1076   ;
  1077 
  1078   'types\_code' (tycodespec +)
  1079   ;
  1080 
  1081   tycodespec: name template attachment ?
  1082   ;
  1083 
  1084   const: term
  1085   ;
  1086 
  1087   template: '(' string ')'
  1088   ;
  1089 
  1090   attachment: 'attach' modespec ? verblbrace text verbrbrace
  1091   ;
  1092 
  1093   'code' (name)?
  1094   ;
  1095   \end{rail}
  1096 
  1097   \medskip The other framework generates code from functional programs
  1098   (including overloading using type classes) to SML \cite{SML}, OCaml
  1099   \cite{OCaml} and Haskell \cite{haskell-revised-report}.
  1100   Conceptually, code generation is split up in three steps:
  1101   \emph{selection} of code theorems, \emph{translation} into an
  1102   abstract executable view and \emph{serialization} to a specific
  1103   \emph{target language}.  See \cite{isabelle-codegen} for an
  1104   introduction on how to use it.
  1105 
  1106   \begin{matharray}{rcl}
  1107     @{command_def (HOL) "export_code"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
  1108     @{command_def (HOL) "code_thms"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
  1109     @{command_def (HOL) "code_deps"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
  1110     @{command_def (HOL) "code_datatype"} & : & @{text "theory \<rightarrow> theory"} \\
  1111     @{command_def (HOL) "code_const"} & : & @{text "theory \<rightarrow> theory"} \\
  1112     @{command_def (HOL) "code_type"} & : & @{text "theory \<rightarrow> theory"} \\
  1113     @{command_def (HOL) "code_class"} & : & @{text "theory \<rightarrow> theory"} \\
  1114     @{command_def (HOL) "code_instance"} & : & @{text "theory \<rightarrow> theory"} \\
  1115     @{command_def (HOL) "code_monad"} & : & @{text "theory \<rightarrow> theory"} \\
  1116     @{command_def (HOL) "code_reserved"} & : & @{text "theory \<rightarrow> theory"} \\
  1117     @{command_def (HOL) "code_include"} & : & @{text "theory \<rightarrow> theory"} \\
  1118     @{command_def (HOL) "code_modulename"} & : & @{text "theory \<rightarrow> theory"} \\
  1119     @{command_def (HOL) "code_abort"} & : & @{text "theory \<rightarrow> theory"} \\
  1120     @{command_def (HOL) "print_codesetup"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
  1121     @{command_def (HOL) "print_codeproc"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
  1122     @{attribute_def (HOL) code} & : & @{text attribute} \\
  1123   \end{matharray}
  1124 
  1125   \begin{rail}
  1126     'export\_code' ( constexpr + ) ? \\
  1127       ( ( 'in' target ( 'module\_name' string ) ? \\
  1128         ( 'file' ( string | '-' ) ) ? ( '(' args ')' ) ?) + ) ?
  1129     ;
  1130 
  1131     'code\_thms' ( constexpr + ) ?
  1132     ;
  1133 
  1134     'code\_deps' ( constexpr + ) ?
  1135     ;
  1136 
  1137     const: term
  1138     ;
  1139 
  1140     constexpr: ( const | 'name.*' | '*' )
  1141     ;
  1142 
  1143     typeconstructor: nameref
  1144     ;
  1145 
  1146     class: nameref
  1147     ;
  1148 
  1149     target: 'OCaml' | 'SML' | 'Haskell'
  1150     ;
  1151 
  1152     'code\_datatype' const +
  1153     ;
  1154 
  1155     'code\_const' (const + 'and') \\
  1156       ( ( '(' target ( syntax ? + 'and' ) ')' ) + )
  1157     ;
  1158 
  1159     'code\_type' (typeconstructor + 'and') \\
  1160       ( ( '(' target ( syntax ? + 'and' ) ')' ) + )
  1161     ;
  1162 
  1163     'code\_class' (class + 'and') \\
  1164       ( ( '(' target \\ ( string ? + 'and' ) ')' ) + )
  1165     ;
  1166 
  1167     'code\_instance' (( typeconstructor '::' class ) + 'and') \\
  1168       ( ( '(' target ( '-' ? + 'and' ) ')' ) + )
  1169     ;
  1170 
  1171     'code\_monad' const const target
  1172     ;
  1173 
  1174     'code\_reserved' target ( string + )
  1175     ;
  1176 
  1177     'code\_include' target ( string ( string | '-') )
  1178     ;
  1179 
  1180     'code\_modulename' target ( ( string string ) + )
  1181     ;
  1182 
  1183     'code\_abort' ( const + )
  1184     ;
  1185 
  1186     syntax: string | ( 'infix' | 'infixl' | 'infixr' ) nat string
  1187     ;
  1188 
  1189     'code' ( 'del' ) ?
  1190     ;
  1191 
  1192     'code_unfold' ( 'del' ) ?
  1193     ;
  1194 
  1195     'code_post' ( 'del' ) ?
  1196     ;
  1197   \end{rail}
  1198 
  1199   \begin{description}
  1200 
  1201   \item @{command (HOL) "export_code"} is the canonical interface for
  1202   generating and serializing code: for a given list of constants, code
  1203   is generated for the specified target languages.  Abstract code is
  1204   cached incrementally.  If no constant is given, the currently cached
  1205   code is serialized.  If no serialization instruction is given, only
  1206   abstract code is cached.
  1207 
  1208   Constants may be specified by giving them literally, referring to
  1209   all executable contants within a certain theory by giving @{text
  1210   "name.*"}, or referring to \emph{all} executable constants currently
  1211   available by giving @{text "*"}.
  1212 
  1213   By default, for each involved theory one corresponding name space
  1214   module is generated.  Alternativly, a module name may be specified
  1215   after the @{keyword "module_name"} keyword; then \emph{all} code is
  1216   placed in this module.
  1217 
  1218   For \emph{SML} and \emph{OCaml}, the file specification refers to a
  1219   single file; for \emph{Haskell}, it refers to a whole directory,
  1220   where code is generated in multiple files reflecting the module
  1221   hierarchy.  The file specification ``@{text "-"}'' denotes standard
  1222   output.  For \emph{SML}, omitting the file specification compiles
  1223   code internally in the context of the current ML session.
  1224 
  1225   Serializers take an optional list of arguments in parentheses.  For
  1226   \emph{Haskell} a module name prefix may be given using the ``@{text
  1227   "root:"}'' argument; ``@{text string_classes}'' adds a ``@{verbatim
  1228   "deriving (Read, Show)"}'' clause to each appropriate datatype
  1229   declaration.
  1230 
  1231   \item @{command (HOL) "code_thms"} prints a list of theorems
  1232   representing the corresponding program containing all given
  1233   constants; if no constants are given, the currently cached code
  1234   theorems are printed.
  1235 
  1236   \item @{command (HOL) "code_deps"} visualizes dependencies of
  1237   theorems representing the corresponding program containing all given
  1238   constants; if no constants are given, the currently cached code
  1239   theorems are visualized.
  1240 
  1241   \item @{command (HOL) "code_datatype"} specifies a constructor set
  1242   for a logical type.
  1243 
  1244   \item @{command (HOL) "code_const"} associates a list of constants
  1245   with target-specific serializations; omitting a serialization
  1246   deletes an existing serialization.
  1247 
  1248   \item @{command (HOL) "code_type"} associates a list of type
  1249   constructors with target-specific serializations; omitting a
  1250   serialization deletes an existing serialization.
  1251 
  1252   \item @{command (HOL) "code_class"} associates a list of classes
  1253   with target-specific class names; omitting a serialization deletes
  1254   an existing serialization.  This applies only to \emph{Haskell}.
  1255 
  1256   \item @{command (HOL) "code_instance"} declares a list of type
  1257   constructor / class instance relations as ``already present'' for a
  1258   given target.  Omitting a ``@{text "-"}'' deletes an existing
  1259   ``already present'' declaration.  This applies only to
  1260   \emph{Haskell}.
  1261 
  1262   \item @{command (HOL) "code_monad"} provides an auxiliary mechanism
  1263   to generate monadic code for Haskell.
  1264 
  1265   \item @{command (HOL) "code_reserved"} declares a list of names as
  1266   reserved for a given target, preventing it to be shadowed by any
  1267   generated code.
  1268 
  1269   \item @{command (HOL) "code_include"} adds arbitrary named content
  1270   (``include'') to generated code.  A ``@{text "-"}'' as last argument
  1271   will remove an already added ``include''.
  1272 
  1273   \item @{command (HOL) "code_modulename"} declares aliasings from one
  1274   module name onto another.
  1275 
  1276   \item @{command (HOL) "code_abort"} declares constants which are not
  1277   required to have a definition by means of code equations; if
  1278   needed these are implemented by program abort instead.
  1279 
  1280   \item @{attribute (HOL) code} explicitly selects (or with option
  1281   ``@{text "del"}'' deselects) a code equation for code
  1282   generation.  Usually packages introducing code equations provide
  1283   a reasonable default setup for selection.
  1284 
  1285   \item @{attribute (HOL) code_inline} declares (or with
  1286   option ``@{text "del"}'' removes) inlining theorems which are
  1287   applied as rewrite rules to any code equation during
  1288   preprocessing.
  1289 
  1290   \item @{attribute (HOL) code_post} declares (or with
  1291   option ``@{text "del"}'' removes) theorems which are
  1292   applied as rewrite rules to any result of an evaluation.
  1293 
  1294   \item @{command (HOL) "print_codesetup"} gives an overview on
  1295   selected code equations and code generator datatypes.
  1296 
  1297   \item @{command (HOL) "print_codeproc"} prints the setup
  1298   of the code generator preprocessor.
  1299 
  1300   \end{description}
  1301 *}
  1302 
  1303 
  1304 section {* Definition by specification \label{sec:hol-specification} *}
  1305 
  1306 text {*
  1307   \begin{matharray}{rcl}
  1308     @{command_def (HOL) "specification"} & : & @{text "theory \<rightarrow> proof(prove)"} \\
  1309     @{command_def (HOL) "ax_specification"} & : & @{text "theory \<rightarrow> proof(prove)"} \\
  1310   \end{matharray}
  1311 
  1312   \begin{rail}
  1313   ('specification' | 'ax\_specification') '(' (decl +) ')' \\ (thmdecl? prop +)
  1314   ;
  1315   decl: ((name ':')? term '(' 'overloaded' ')'?)
  1316   \end{rail}
  1317 
  1318   \begin{description}
  1319 
  1320   \item @{command (HOL) "specification"}~@{text "decls \<phi>"} sets up a
  1321   goal stating the existence of terms with the properties specified to
  1322   hold for the constants given in @{text decls}.  After finishing the
  1323   proof, the theory will be augmented with definitions for the given
  1324   constants, as well as with theorems stating the properties for these
  1325   constants.
  1326 
  1327   \item @{command (HOL) "ax_specification"}~@{text "decls \<phi>"} sets up
  1328   a goal stating the existence of terms with the properties specified
  1329   to hold for the constants given in @{text decls}.  After finishing
  1330   the proof, the theory will be augmented with axioms expressing the
  1331   properties given in the first place.
  1332 
  1333   \item @{text decl} declares a constant to be defined by the
  1334   specification given.  The definition for the constant @{text c} is
  1335   bound to the name @{text c_def} unless a theorem name is given in
  1336   the declaration.  Overloaded constants should be declared as such.
  1337 
  1338   \end{description}
  1339 
  1340   Whether to use @{command (HOL) "specification"} or @{command (HOL)
  1341   "ax_specification"} is to some extent a matter of style.  @{command
  1342   (HOL) "specification"} introduces no new axioms, and so by
  1343   construction cannot introduce inconsistencies, whereas @{command
  1344   (HOL) "ax_specification"} does introduce axioms, but only after the
  1345   user has explicitly proven it to be safe.  A practical issue must be
  1346   considered, though: After introducing two constants with the same
  1347   properties using @{command (HOL) "specification"}, one can prove
  1348   that the two constants are, in fact, equal.  If this might be a
  1349   problem, one should use @{command (HOL) "ax_specification"}.
  1350 *}
  1351 
  1352 end