doc-src/IsarRef/Thy/HOL_Specific.thy
author blanchet
Tue Feb 23 14:11:36 2010 +0100 (2010-02-23)
changeset 35331 450ab945c451
parent 34172 4301e9ea1c54
child 35351 7425aece4ee3
permissions -rw-r--r--
document Quickcheck's "no_assms" option
     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     @{method_def (HOL) size_change} & : & @{text method} \\
   510   \end{matharray}
   511 
   512   \begin{rail}
   513     'relation' term
   514     ;
   515     'lexicographic\_order' ( clasimpmod * )
   516     ;
   517     'size\_change' ( orders ( clasimpmod * ) )
   518     ;
   519     orders: ( 'max' | 'min' | 'ms' ) *
   520   \end{rail}
   521 
   522   \begin{description}
   523 
   524   \item @{method (HOL) pat_completeness} is a specialized method to
   525   solve goals regarding the completeness of pattern matching, as
   526   required by the @{command (HOL) "function"} package (cf.\
   527   \cite{isabelle-function}).
   528 
   529   \item @{method (HOL) relation}~@{text R} introduces a termination
   530   proof using the relation @{text R}.  The resulting proof state will
   531   contain goals expressing that @{text R} is wellfounded, and that the
   532   arguments of recursive calls decrease with respect to @{text R}.
   533   Usually, this method is used as the initial proof step of manual
   534   termination proofs.
   535 
   536   \item @{method (HOL) "lexicographic_order"} attempts a fully
   537   automated termination proof by searching for a lexicographic
   538   combination of size measures on the arguments of the function. The
   539   method accepts the same arguments as the @{method auto} method,
   540   which it uses internally to prove local descents.  The same context
   541   modifiers as for @{method auto} are accepted, see
   542   \secref{sec:clasimp}.
   543 
   544   In case of failure, extensive information is printed, which can help
   545   to analyse the situation (cf.\ \cite{isabelle-function}).
   546 
   547   \item @{method (HOL) "size_change"} also works on termination goals,
   548   using a variation of the size-change principle, together with a
   549   graph decomposition technique (see \cite{krauss_phd} for details).
   550   Three kinds of orders are used internally: @{text max}, @{text min},
   551   and @{text ms} (multiset), which is only available when the theory
   552   @{text Multiset} is loaded. When no order kinds are given, they are
   553   tried in order. The search for a termination proof uses SAT solving
   554   internally.
   555 
   556  For local descent proofs, the same context modifiers as for @{method
   557   auto} are accepted, see \secref{sec:clasimp}.
   558 
   559   \end{description}
   560 *}
   561 
   562 
   563 subsection {* Old-style recursive function definitions (TFL) *}
   564 
   565 text {*
   566   The old TFL commands @{command (HOL) "recdef"} and @{command (HOL)
   567   "recdef_tc"} for defining recursive are mostly obsolete; @{command
   568   (HOL) "function"} or @{command (HOL) "fun"} should be used instead.
   569 
   570   \begin{matharray}{rcl}
   571     @{command_def (HOL) "recdef"} & : & @{text "theory \<rightarrow> theory)"} \\
   572     @{command_def (HOL) "recdef_tc"}@{text "\<^sup>*"} & : & @{text "theory \<rightarrow> proof(prove)"} \\
   573   \end{matharray}
   574 
   575   \begin{rail}
   576     'recdef' ('(' 'permissive' ')')? \\ name term (prop +) hints?
   577     ;
   578     recdeftc thmdecl? tc
   579     ;
   580     hints: '(' 'hints' ( recdefmod * ) ')'
   581     ;
   582     recdefmod: (('recdef\_simp' | 'recdef\_cong' | 'recdef\_wf') (() | 'add' | 'del') ':' thmrefs) | clasimpmod
   583     ;
   584     tc: nameref ('(' nat ')')?
   585     ;
   586   \end{rail}
   587 
   588   \begin{description}
   589   
   590   \item @{command (HOL) "recdef"} defines general well-founded
   591   recursive functions (using the TFL package), see also
   592   \cite{isabelle-HOL}.  The ``@{text "(permissive)"}'' option tells
   593   TFL to recover from failed proof attempts, returning unfinished
   594   results.  The @{text recdef_simp}, @{text recdef_cong}, and @{text
   595   recdef_wf} hints refer to auxiliary rules to be used in the internal
   596   automated proof process of TFL.  Additional @{syntax clasimpmod}
   597   declarations (cf.\ \secref{sec:clasimp}) may be given to tune the
   598   context of the Simplifier (cf.\ \secref{sec:simplifier}) and
   599   Classical reasoner (cf.\ \secref{sec:classical}).
   600   
   601   \item @{command (HOL) "recdef_tc"}~@{text "c (i)"} recommences the
   602   proof for leftover termination condition number @{text i} (default
   603   1) as generated by a @{command (HOL) "recdef"} definition of
   604   constant @{text c}.
   605   
   606   Note that in most cases, @{command (HOL) "recdef"} is able to finish
   607   its internal proofs without manual intervention.
   608 
   609   \end{description}
   610 
   611   \medskip Hints for @{command (HOL) "recdef"} may be also declared
   612   globally, using the following attributes.
   613 
   614   \begin{matharray}{rcl}
   615     @{attribute_def (HOL) recdef_simp} & : & @{text attribute} \\
   616     @{attribute_def (HOL) recdef_cong} & : & @{text attribute} \\
   617     @{attribute_def (HOL) recdef_wf} & : & @{text attribute} \\
   618   \end{matharray}
   619 
   620   \begin{rail}
   621     ('recdef\_simp' | 'recdef\_cong' | 'recdef\_wf') (() | 'add' | 'del')
   622     ;
   623   \end{rail}
   624 *}
   625 
   626 
   627 section {* Inductive and coinductive definitions \label{sec:hol-inductive} *}
   628 
   629 text {*
   630   An \textbf{inductive definition} specifies the least predicate (or
   631   set) @{text R} closed under given rules: applying a rule to elements
   632   of @{text R} yields a result within @{text R}.  For example, a
   633   structural operational semantics is an inductive definition of an
   634   evaluation relation.
   635 
   636   Dually, a \textbf{coinductive definition} specifies the greatest
   637   predicate~/ set @{text R} that is consistent with given rules: every
   638   element of @{text R} can be seen as arising by applying a rule to
   639   elements of @{text R}.  An important example is using bisimulation
   640   relations to formalise equivalence of processes and infinite data
   641   structures.
   642 
   643   \medskip The HOL package is related to the ZF one, which is
   644   described in a separate paper,\footnote{It appeared in CADE
   645   \cite{paulson-CADE}; a longer version is distributed with Isabelle.}
   646   which you should refer to in case of difficulties.  The package is
   647   simpler than that of ZF thanks to implicit type-checking in HOL.
   648   The types of the (co)inductive predicates (or sets) determine the
   649   domain of the fixedpoint definition, and the package does not have
   650   to use inference rules for type-checking.
   651 
   652   \begin{matharray}{rcl}
   653     @{command_def (HOL) "inductive"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
   654     @{command_def (HOL) "inductive_set"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
   655     @{command_def (HOL) "coinductive"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
   656     @{command_def (HOL) "coinductive_set"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
   657     @{attribute_def (HOL) mono} & : & @{text attribute} \\
   658   \end{matharray}
   659 
   660   \begin{rail}
   661     ('inductive' | 'inductive\_set' | 'coinductive' | 'coinductive\_set') target? fixes ('for' fixes)? \\
   662     ('where' clauses)? ('monos' thmrefs)?
   663     ;
   664     clauses: (thmdecl? prop + '|')
   665     ;
   666     'mono' (() | 'add' | 'del')
   667     ;
   668   \end{rail}
   669 
   670   \begin{description}
   671 
   672   \item @{command (HOL) "inductive"} and @{command (HOL)
   673   "coinductive"} define (co)inductive predicates from the
   674   introduction rules given in the @{keyword "where"} part.  The
   675   optional @{keyword "for"} part contains a list of parameters of the
   676   (co)inductive predicates that remain fixed throughout the
   677   definition.  The optional @{keyword "monos"} section contains
   678   \emph{monotonicity theorems}, which are required for each operator
   679   applied to a recursive set in the introduction rules.  There
   680   \emph{must} be a theorem of the form @{text "A \<le> B \<Longrightarrow> M A \<le> M B"},
   681   for each premise @{text "M R\<^sub>i t"} in an introduction rule!
   682 
   683   \item @{command (HOL) "inductive_set"} and @{command (HOL)
   684   "coinductive_set"} are wrappers for to the previous commands,
   685   allowing the definition of (co)inductive sets.
   686 
   687   \item @{attribute (HOL) mono} declares monotonicity rules.  These
   688   rule are involved in the automated monotonicity proof of @{command
   689   (HOL) "inductive"}.
   690 
   691   \end{description}
   692 *}
   693 
   694 
   695 subsection {* Derived rules *}
   696 
   697 text {*
   698   Each (co)inductive definition @{text R} adds definitions to the
   699   theory and also proves some theorems:
   700 
   701   \begin{description}
   702 
   703   \item @{text R.intros} is the list of introduction rules as proven
   704   theorems, for the recursive predicates (or sets).  The rules are
   705   also available individually, using the names given them in the
   706   theory file;
   707 
   708   \item @{text R.cases} is the case analysis (or elimination) rule;
   709 
   710   \item @{text R.induct} or @{text R.coinduct} is the (co)induction
   711   rule.
   712 
   713   \end{description}
   714 
   715   When several predicates @{text "R\<^sub>1, \<dots>, R\<^sub>n"} are
   716   defined simultaneously, the list of introduction rules is called
   717   @{text "R\<^sub>1_\<dots>_R\<^sub>n.intros"}, the case analysis rules are
   718   called @{text "R\<^sub>1.cases, \<dots>, R\<^sub>n.cases"}, and the list
   719   of mutual induction rules is called @{text
   720   "R\<^sub>1_\<dots>_R\<^sub>n.inducts"}.
   721 *}
   722 
   723 
   724 subsection {* Monotonicity theorems *}
   725 
   726 text {*
   727   Each theory contains a default set of theorems that are used in
   728   monotonicity proofs.  New rules can be added to this set via the
   729   @{attribute (HOL) mono} attribute.  The HOL theory @{text Inductive}
   730   shows how this is done.  In general, the following monotonicity
   731   theorems may be added:
   732 
   733   \begin{itemize}
   734 
   735   \item Theorems of the form @{text "A \<le> B \<Longrightarrow> M A \<le> M B"}, for proving
   736   monotonicity of inductive definitions whose introduction rules have
   737   premises involving terms such as @{text "M R\<^sub>i t"}.
   738 
   739   \item Monotonicity theorems for logical operators, which are of the
   740   general form @{text "(\<dots> \<longrightarrow> \<dots>) \<Longrightarrow> \<dots> (\<dots> \<longrightarrow> \<dots>) \<Longrightarrow> \<dots> \<longrightarrow> \<dots>"}.  For example, in
   741   the case of the operator @{text "\<or>"}, the corresponding theorem is
   742   \[
   743   \infer{@{text "P\<^sub>1 \<or> P\<^sub>2 \<longrightarrow> Q\<^sub>1 \<or> Q\<^sub>2"}}{@{text "P\<^sub>1 \<longrightarrow> Q\<^sub>1"} & @{text "P\<^sub>2 \<longrightarrow> Q\<^sub>2"}}
   744   \]
   745 
   746   \item De Morgan style equations for reasoning about the ``polarity''
   747   of expressions, e.g.
   748   \[
   749   @{prop "\<not> \<not> P \<longleftrightarrow> P"} \qquad\qquad
   750   @{prop "\<not> (P \<and> Q) \<longleftrightarrow> \<not> P \<or> \<not> Q"}
   751   \]
   752 
   753   \item Equations for reducing complex operators to more primitive
   754   ones whose monotonicity can easily be proved, e.g.
   755   \[
   756   @{prop "(P \<longrightarrow> Q) \<longleftrightarrow> \<not> P \<or> Q"} \qquad\qquad
   757   @{prop "Ball A P \<equiv> \<forall>x. x \<in> A \<longrightarrow> P x"}
   758   \]
   759 
   760   \end{itemize}
   761 
   762   %FIXME: Example of an inductive definition
   763 *}
   764 
   765 
   766 section {* Arithmetic proof support *}
   767 
   768 text {*
   769   \begin{matharray}{rcl}
   770     @{method_def (HOL) arith} & : & @{text method} \\
   771     @{attribute_def (HOL) arith} & : & @{text attribute} \\
   772     @{attribute_def (HOL) arith_split} & : & @{text attribute} \\
   773   \end{matharray}
   774 
   775   The @{method (HOL) arith} method decides linear arithmetic problems
   776   (on types @{text nat}, @{text int}, @{text real}).  Any current
   777   facts are inserted into the goal before running the procedure.
   778 
   779   The @{attribute (HOL) arith} attribute declares facts that are
   780   always supplied to the arithmetic provers implicitly.
   781 
   782   The @{attribute (HOL) arith_split} attribute declares case split
   783   rules to be expanded before @{method (HOL) arith} is invoked.
   784 
   785   Note that a simpler (but faster) arithmetic prover is
   786   already invoked by the Simplifier.
   787 *}
   788 
   789 
   790 section {* Intuitionistic proof search *}
   791 
   792 text {*
   793   \begin{matharray}{rcl}
   794     @{method_def (HOL) iprover} & : & @{text method} \\
   795   \end{matharray}
   796 
   797   \begin{rail}
   798     'iprover' ('!' ?) ( rulemod * )
   799     ;
   800   \end{rail}
   801 
   802   The @{method (HOL) iprover} method performs intuitionistic proof
   803   search, depending on specifically declared rules from the context,
   804   or given as explicit arguments.  Chained facts are inserted into the
   805   goal before commencing proof search; ``@{method (HOL) iprover}@{text
   806   "!"}''  means to include the current @{fact prems} as well.
   807   
   808   Rules need to be classified as @{attribute (Pure) intro},
   809   @{attribute (Pure) elim}, or @{attribute (Pure) dest}; here the
   810   ``@{text "!"}'' indicator refers to ``safe'' rules, which may be
   811   applied aggressively (without considering back-tracking later).
   812   Rules declared with ``@{text "?"}'' are ignored in proof search (the
   813   single-step @{method rule} method still observes these).  An
   814   explicit weight annotation may be given as well; otherwise the
   815   number of rule premises will be taken into account here.
   816 *}
   817 
   818 
   819 section {* Coherent Logic *}
   820 
   821 text {*
   822   \begin{matharray}{rcl}
   823     @{method_def (HOL) "coherent"} & : & @{text method} \\
   824   \end{matharray}
   825 
   826   \begin{rail}
   827     'coherent' thmrefs?
   828     ;
   829   \end{rail}
   830 
   831   The @{method (HOL) coherent} method solves problems of
   832   \emph{Coherent Logic} \cite{Bezem-Coquand:2005}, which covers
   833   applications in confluence theory, lattice theory and projective
   834   geometry.  See @{"file" "~~/src/HOL/ex/Coherent.thy"} for some
   835   examples.
   836 *}
   837 
   838 
   839 section {* Checking and refuting propositions *}
   840 
   841 text {*
   842   Identifying incorrect propositions usually involves evaluation of
   843   particular assignments and systematic counter example search.  This
   844   is supported by the following commands.
   845 
   846   \begin{matharray}{rcl}
   847     @{command_def (HOL) "value"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
   848     @{command_def (HOL) "quickcheck"}@{text "\<^sup>*"} & : & @{text "proof \<rightarrow>"} \\
   849     @{command_def (HOL) "quickcheck_params"} & : & @{text "theory \<rightarrow> theory"}
   850   \end{matharray}
   851 
   852   \begin{rail}
   853     'value' ( ( '[' name ']' ) ? ) modes? term
   854     ;
   855 
   856     'quickcheck' ( ( '[' args ']' ) ? ) nat?
   857     ;
   858 
   859     'quickcheck_params' ( ( '[' args ']' ) ? )
   860     ;
   861 
   862     modes: '(' (name + ) ')'
   863     ;
   864 
   865     args: ( name '=' value + ',' )
   866     ;
   867   \end{rail}
   868 
   869   \begin{description}
   870 
   871   \item @{command (HOL) "value"}~@{text t} evaluates and prints a
   872     term; optionally @{text modes} can be specified, which are
   873     appended to the current print mode (see also \cite{isabelle-ref}).
   874     Internally, the evaluation is performed by registered evaluators,
   875     which are invoked sequentially until a result is returned.
   876     Alternatively a specific evaluator can be selected using square
   877     brackets; available evaluators include @{text nbe} for
   878     \emph{normalization by evaluation} and \emph{code} for code
   879     generation in SML.
   880 
   881   \item @{command (HOL) "quickcheck"} tests the current goal for
   882     counter examples using a series of arbitrary assignments for its
   883     free variables; by default the first subgoal is tested, an other
   884     can be selected explicitly using an optional goal index.
   885     A number of configuration options are supported for
   886     @{command (HOL) "quickcheck"}, notably:
   887 
   888     \begin{description}
   889 
   890       \item[size] specifies the maximum size of the search space for
   891         assignment values.
   892 
   893       \item[iterations] sets how many sets of assignments are
   894         generated for each particular size.
   895 
   896       \item[no\_assms] specifies whether assumptions in
   897         structured proofs should be ignored.
   898 
   899     \end{description}
   900 
   901     These option can be given within square brackets.
   902 
   903   \item @{command (HOL) "quickcheck_params"} changes quickcheck
   904     configuration options persitently.
   905 
   906   \end{description}
   907 *}
   908 
   909 
   910 section {* Invoking automated reasoning tools -- The Sledgehammer *}
   911 
   912 text {*
   913   Isabelle/HOL includes a generic \emph{ATP manager} that allows
   914   external automated reasoning tools to crunch a pending goal.
   915   Supported provers include E\footnote{\url{http://www.eprover.org}},
   916   SPASS\footnote{\url{http://www.spass-prover.org/}}, and Vampire.
   917   There is also a wrapper to invoke provers remotely via the
   918   SystemOnTPTP\footnote{\url{http://www.cs.miami.edu/~tptp/cgi-bin/SystemOnTPTP}}
   919   web service.
   920 
   921   The problem passed to external provers consists of the goal together
   922   with a smart selection of lemmas from the current theory context.
   923   The result of a successful proof search is some source text that
   924   usually reconstructs the proof within Isabelle, without requiring
   925   external provers again.  The Metis
   926   prover\footnote{\url{http://www.gilith.com/software/metis/}} that is
   927   integrated into Isabelle/HOL is being used here.
   928 
   929   In this mode of operation, heavy means of automated reasoning are
   930   used as a strong relevance filter, while the main proof checking
   931   works via explicit inferences going through the Isabelle kernel.
   932   Moreover, rechecking Isabelle proof texts with already specified
   933   auxiliary facts is much faster than performing fully automated
   934   search over and over again.
   935 
   936   \begin{matharray}{rcl}
   937     @{command_def (HOL) "sledgehammer"}@{text "\<^sup>*"} & : & @{text "proof \<rightarrow>"} \\
   938     @{command_def (HOL) "print_atps"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
   939     @{command_def (HOL) "atp_info"}@{text "\<^sup>*"} & : & @{text "any \<rightarrow>"} \\
   940     @{command_def (HOL) "atp_kill"}@{text "\<^sup>*"} & : & @{text "any \<rightarrow>"} \\
   941     @{command_def (HOL) "atp_messages"}@{text "\<^sup>*"} & : & @{text "any \<rightarrow>"} \\
   942     @{method_def (HOL) metis} & : & @{text method} \\
   943   \end{matharray}
   944 
   945   \begin{rail}
   946   'sledgehammer' ( nameref * )
   947   ;
   948   'atp\_messages' ('(' nat ')')?
   949   ;
   950 
   951   'metis' thmrefs
   952   ;
   953   \end{rail}
   954 
   955   \begin{description}
   956 
   957   \item @{command (HOL) sledgehammer}~@{text "prover\<^sub>1 \<dots> prover\<^sub>n"}
   958   invokes the specified automated theorem provers on the first
   959   subgoal.  Provers are run in parallel, the first successful result
   960   is displayed, and the other attempts are terminated.
   961 
   962   Provers are defined in the theory context, see also @{command (HOL)
   963   print_atps}.  If no provers are given as arguments to @{command
   964   (HOL) sledgehammer}, the system refers to the default defined as
   965   ``ATP provers'' preference by the user interface.
   966 
   967   There are additional preferences for timeout (default: 60 seconds),
   968   and the maximum number of independent prover processes (default: 5);
   969   excessive provers are automatically terminated.
   970 
   971   \item @{command (HOL) print_atps} prints the list of automated
   972   theorem provers available to the @{command (HOL) sledgehammer}
   973   command.
   974 
   975   \item @{command (HOL) atp_info} prints information about presently
   976   running provers, including elapsed runtime, and the remaining time
   977   until timeout.
   978 
   979   \item @{command (HOL) atp_kill} terminates all presently running
   980   provers.
   981 
   982   \item @{command (HOL) atp_messages} displays recent messages issued
   983   by automated theorem provers.  This allows to examine results that
   984   might have got lost due to the asynchronous nature of default
   985   @{command (HOL) sledgehammer} output.  An optional message limit may
   986   be specified (default 5).
   987 
   988   \item @{method (HOL) metis}~@{text "facts"} invokes the Metis prover
   989   with the given facts.  Metis is an automated proof tool of medium
   990   strength, but is fully integrated into Isabelle/HOL, with explicit
   991   inferences going through the kernel.  Thus its results are
   992   guaranteed to be ``correct by construction''.
   993 
   994   Note that all facts used with Metis need to be specified as explicit
   995   arguments.  There are no rule declarations as for other Isabelle
   996   provers, like @{method blast} or @{method fast}.
   997 
   998   \end{description}
   999 *}
  1000 
  1001 
  1002 section {* Unstructured case analysis and induction \label{sec:hol-induct-tac} *}
  1003 
  1004 text {*
  1005   The following tools of Isabelle/HOL support cases analysis and
  1006   induction in unstructured tactic scripts; see also
  1007   \secref{sec:cases-induct} for proper Isar versions of similar ideas.
  1008 
  1009   \begin{matharray}{rcl}
  1010     @{method_def (HOL) case_tac}@{text "\<^sup>*"} & : & @{text method} \\
  1011     @{method_def (HOL) induct_tac}@{text "\<^sup>*"} & : & @{text method} \\
  1012     @{method_def (HOL) ind_cases}@{text "\<^sup>*"} & : & @{text method} \\
  1013     @{command_def (HOL) "inductive_cases"}@{text "\<^sup>*"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
  1014   \end{matharray}
  1015 
  1016   \begin{rail}
  1017     'case\_tac' goalspec? term rule?
  1018     ;
  1019     'induct\_tac' goalspec? (insts * 'and') rule?
  1020     ;
  1021     'ind\_cases' (prop +) ('for' (name +)) ?
  1022     ;
  1023     'inductive\_cases' (thmdecl? (prop +) + 'and')
  1024     ;
  1025 
  1026     rule: ('rule' ':' thmref)
  1027     ;
  1028   \end{rail}
  1029 
  1030   \begin{description}
  1031 
  1032   \item @{method (HOL) case_tac} and @{method (HOL) induct_tac} admit
  1033   to reason about inductive types.  Rules are selected according to
  1034   the declarations by the @{attribute cases} and @{attribute induct}
  1035   attributes, cf.\ \secref{sec:cases-induct}.  The @{command (HOL)
  1036   datatype} package already takes care of this.
  1037 
  1038   These unstructured tactics feature both goal addressing and dynamic
  1039   instantiation.  Note that named rule cases are \emph{not} provided
  1040   as would be by the proper @{method cases} and @{method induct} proof
  1041   methods (see \secref{sec:cases-induct}).  Unlike the @{method
  1042   induct} method, @{method induct_tac} does not handle structured rule
  1043   statements, only the compact object-logic conclusion of the subgoal
  1044   being addressed.
  1045   
  1046   \item @{method (HOL) ind_cases} and @{command (HOL)
  1047   "inductive_cases"} provide an interface to the internal @{ML_text
  1048   mk_cases} operation.  Rules are simplified in an unrestricted
  1049   forward manner.
  1050 
  1051   While @{method (HOL) ind_cases} is a proof method to apply the
  1052   result immediately as elimination rules, @{command (HOL)
  1053   "inductive_cases"} provides case split theorems at the theory level
  1054   for later use.  The @{keyword "for"} argument of the @{method (HOL)
  1055   ind_cases} method allows to specify a list of variables that should
  1056   be generalized before applying the resulting rule.
  1057 
  1058   \end{description}
  1059 *}
  1060 
  1061 
  1062 section {* Executable code *}
  1063 
  1064 text {*
  1065   Isabelle/Pure provides two generic frameworks to support code
  1066   generation from executable specifications.  Isabelle/HOL
  1067   instantiates these mechanisms in a way that is amenable to end-user
  1068   applications.
  1069 
  1070   One framework generates code from both functional and relational
  1071   programs to SML.  See \cite{isabelle-HOL} for further information
  1072   (this actually covers the new-style theory format as well).
  1073 
  1074   \begin{matharray}{rcl}
  1075     @{command_def (HOL) "code_module"} & : & @{text "theory \<rightarrow> theory"} \\
  1076     @{command_def (HOL) "code_library"} & : & @{text "theory \<rightarrow> theory"} \\
  1077     @{command_def (HOL) "consts_code"} & : & @{text "theory \<rightarrow> theory"} \\
  1078     @{command_def (HOL) "types_code"} & : & @{text "theory \<rightarrow> theory"} \\  
  1079     @{attribute_def (HOL) code} & : & @{text attribute} \\
  1080   \end{matharray}
  1081 
  1082   \begin{rail}
  1083   ( 'code\_module' | 'code\_library' ) modespec ? name ? \\
  1084     ( 'file' name ) ? ( 'imports' ( name + ) ) ? \\
  1085     'contains' ( ( name '=' term ) + | term + )
  1086   ;
  1087 
  1088   modespec: '(' ( name * ) ')'
  1089   ;
  1090 
  1091   'consts\_code' (codespec +)
  1092   ;
  1093 
  1094   codespec: const template attachment ?
  1095   ;
  1096 
  1097   'types\_code' (tycodespec +)
  1098   ;
  1099 
  1100   tycodespec: name template attachment ?
  1101   ;
  1102 
  1103   const: term
  1104   ;
  1105 
  1106   template: '(' string ')'
  1107   ;
  1108 
  1109   attachment: 'attach' modespec ? verblbrace text verbrbrace
  1110   ;
  1111 
  1112   'code' (name)?
  1113   ;
  1114   \end{rail}
  1115 
  1116   \medskip The other framework generates code from functional programs
  1117   (including overloading using type classes) to SML \cite{SML}, OCaml
  1118   \cite{OCaml} and Haskell \cite{haskell-revised-report}.
  1119   Conceptually, code generation is split up in three steps:
  1120   \emph{selection} of code theorems, \emph{translation} into an
  1121   abstract executable view and \emph{serialization} to a specific
  1122   \emph{target language}.  See \cite{isabelle-codegen} for an
  1123   introduction on how to use it.
  1124 
  1125   \begin{matharray}{rcl}
  1126     @{command_def (HOL) "export_code"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
  1127     @{command_def (HOL) "code_thms"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
  1128     @{command_def (HOL) "code_deps"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
  1129     @{command_def (HOL) "code_datatype"} & : & @{text "theory \<rightarrow> theory"} \\
  1130     @{command_def (HOL) "code_const"} & : & @{text "theory \<rightarrow> theory"} \\
  1131     @{command_def (HOL) "code_type"} & : & @{text "theory \<rightarrow> theory"} \\
  1132     @{command_def (HOL) "code_class"} & : & @{text "theory \<rightarrow> theory"} \\
  1133     @{command_def (HOL) "code_instance"} & : & @{text "theory \<rightarrow> theory"} \\
  1134     @{command_def (HOL) "code_monad"} & : & @{text "theory \<rightarrow> theory"} \\
  1135     @{command_def (HOL) "code_reserved"} & : & @{text "theory \<rightarrow> theory"} \\
  1136     @{command_def (HOL) "code_include"} & : & @{text "theory \<rightarrow> theory"} \\
  1137     @{command_def (HOL) "code_modulename"} & : & @{text "theory \<rightarrow> theory"} \\
  1138     @{command_def (HOL) "code_abort"} & : & @{text "theory \<rightarrow> theory"} \\
  1139     @{command_def (HOL) "print_codesetup"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
  1140     @{command_def (HOL) "print_codeproc"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
  1141     @{attribute_def (HOL) code} & : & @{text attribute} \\
  1142   \end{matharray}
  1143 
  1144   \begin{rail}
  1145     'export\_code' ( constexpr + ) \\
  1146       ( ( 'in' target ( 'module\_name' string ) ? \\
  1147         ( 'file' ( string | '-' ) ) ? ( '(' args ')' ) ?) + ) ?
  1148     ;
  1149 
  1150     'code\_thms' ( constexpr + ) ?
  1151     ;
  1152 
  1153     'code\_deps' ( constexpr + ) ?
  1154     ;
  1155 
  1156     const: term
  1157     ;
  1158 
  1159     constexpr: ( const | 'name.*' | '*' )
  1160     ;
  1161 
  1162     typeconstructor: nameref
  1163     ;
  1164 
  1165     class: nameref
  1166     ;
  1167 
  1168     target: 'OCaml' | 'SML' | 'Haskell'
  1169     ;
  1170 
  1171     'code\_datatype' const +
  1172     ;
  1173 
  1174     'code\_const' (const + 'and') \\
  1175       ( ( '(' target ( syntax ? + 'and' ) ')' ) + )
  1176     ;
  1177 
  1178     'code\_type' (typeconstructor + 'and') \\
  1179       ( ( '(' target ( syntax ? + 'and' ) ')' ) + )
  1180     ;
  1181 
  1182     'code\_class' (class + 'and') \\
  1183       ( ( '(' target \\ ( string ? + 'and' ) ')' ) + )
  1184     ;
  1185 
  1186     'code\_instance' (( typeconstructor '::' class ) + 'and') \\
  1187       ( ( '(' target ( '-' ? + 'and' ) ')' ) + )
  1188     ;
  1189 
  1190     'code\_monad' const const target
  1191     ;
  1192 
  1193     'code\_reserved' target ( string + )
  1194     ;
  1195 
  1196     'code\_include' target ( string ( string | '-') )
  1197     ;
  1198 
  1199     'code\_modulename' target ( ( string string ) + )
  1200     ;
  1201 
  1202     'code\_abort' ( const + )
  1203     ;
  1204 
  1205     syntax: string | ( 'infix' | 'infixl' | 'infixr' ) nat string
  1206     ;
  1207 
  1208     'code' ( 'del' ) ?
  1209     ;
  1210 
  1211     'code_unfold' ( 'del' ) ?
  1212     ;
  1213 
  1214     'code_post' ( 'del' ) ?
  1215     ;
  1216   \end{rail}
  1217 
  1218   \begin{description}
  1219 
  1220   \item @{command (HOL) "export_code"} is the canonical interface for
  1221   generating and serializing code: for a given list of constants, code
  1222   is generated for the specified target languages.  If no serialization
  1223   instruction is given, only abstract code is generated internally.
  1224 
  1225   Constants may be specified by giving them literally, referring to
  1226   all executable contants within a certain theory by giving @{text
  1227   "name.*"}, or referring to \emph{all} executable constants currently
  1228   available by giving @{text "*"}.
  1229 
  1230   By default, for each involved theory one corresponding name space
  1231   module is generated.  Alternativly, a module name may be specified
  1232   after the @{keyword "module_name"} keyword; then \emph{all} code is
  1233   placed in this module.
  1234 
  1235   For \emph{SML} and \emph{OCaml}, the file specification refers to a
  1236   single file; for \emph{Haskell}, it refers to a whole directory,
  1237   where code is generated in multiple files reflecting the module
  1238   hierarchy.  The file specification ``@{text "-"}'' denotes standard
  1239   output.  For \emph{SML}, omitting the file specification compiles
  1240   code internally in the context of the current ML session.
  1241 
  1242   Serializers take an optional list of arguments in parentheses.  For
  1243   \emph{SML} and \emph{OCaml}, ``@{text no_signatures}`` omits
  1244   explicit module signatures.
  1245   
  1246   For \emph{Haskell} a module name prefix may be given using the ``@{text
  1247   "root:"}'' argument; ``@{text string_classes}'' adds a ``@{verbatim
  1248   "deriving (Read, Show)"}'' clause to each appropriate datatype
  1249   declaration.
  1250 
  1251   \item @{command (HOL) "code_thms"} prints a list of theorems
  1252   representing the corresponding program containing all given
  1253   constants.
  1254 
  1255   \item @{command (HOL) "code_deps"} visualizes dependencies of
  1256   theorems representing the corresponding program containing all given
  1257   constants.
  1258 
  1259   \item @{command (HOL) "code_datatype"} specifies a constructor set
  1260   for a logical type.
  1261 
  1262   \item @{command (HOL) "code_const"} associates a list of constants
  1263   with target-specific serializations; omitting a serialization
  1264   deletes an existing serialization.
  1265 
  1266   \item @{command (HOL) "code_type"} associates a list of type
  1267   constructors with target-specific serializations; omitting a
  1268   serialization deletes an existing serialization.
  1269 
  1270   \item @{command (HOL) "code_class"} associates a list of classes
  1271   with target-specific class names; omitting a serialization deletes
  1272   an existing serialization.  This applies only to \emph{Haskell}.
  1273 
  1274   \item @{command (HOL) "code_instance"} declares a list of type
  1275   constructor / class instance relations as ``already present'' for a
  1276   given target.  Omitting a ``@{text "-"}'' deletes an existing
  1277   ``already present'' declaration.  This applies only to
  1278   \emph{Haskell}.
  1279 
  1280   \item @{command (HOL) "code_monad"} provides an auxiliary mechanism
  1281   to generate monadic code for Haskell.
  1282 
  1283   \item @{command (HOL) "code_reserved"} declares a list of names as
  1284   reserved for a given target, preventing it to be shadowed by any
  1285   generated code.
  1286 
  1287   \item @{command (HOL) "code_include"} adds arbitrary named content
  1288   (``include'') to generated code.  A ``@{text "-"}'' as last argument
  1289   will remove an already added ``include''.
  1290 
  1291   \item @{command (HOL) "code_modulename"} declares aliasings from one
  1292   module name onto another.
  1293 
  1294   \item @{command (HOL) "code_abort"} declares constants which are not
  1295   required to have a definition by means of code equations; if
  1296   needed these are implemented by program abort instead.
  1297 
  1298   \item @{attribute (HOL) code} explicitly selects (or with option
  1299   ``@{text "del"}'' deselects) a code equation for code
  1300   generation.  Usually packages introducing code equations provide
  1301   a reasonable default setup for selection.
  1302 
  1303   \item @{attribute (HOL) code_inline} declares (or with
  1304   option ``@{text "del"}'' removes) inlining theorems which are
  1305   applied as rewrite rules to any code equation during
  1306   preprocessing.
  1307 
  1308   \item @{attribute (HOL) code_post} declares (or with
  1309   option ``@{text "del"}'' removes) theorems which are
  1310   applied as rewrite rules to any result of an evaluation.
  1311 
  1312   \item @{command (HOL) "print_codesetup"} gives an overview on
  1313   selected code equations and code generator datatypes.
  1314 
  1315   \item @{command (HOL) "print_codeproc"} prints the setup
  1316   of the code generator preprocessor.
  1317 
  1318   \end{description}
  1319 *}
  1320 
  1321 
  1322 section {* Definition by specification \label{sec:hol-specification} *}
  1323 
  1324 text {*
  1325   \begin{matharray}{rcl}
  1326     @{command_def (HOL) "specification"} & : & @{text "theory \<rightarrow> proof(prove)"} \\
  1327     @{command_def (HOL) "ax_specification"} & : & @{text "theory \<rightarrow> proof(prove)"} \\
  1328   \end{matharray}
  1329 
  1330   \begin{rail}
  1331   ('specification' | 'ax\_specification') '(' (decl +) ')' \\ (thmdecl? prop +)
  1332   ;
  1333   decl: ((name ':')? term '(' 'overloaded' ')'?)
  1334   \end{rail}
  1335 
  1336   \begin{description}
  1337 
  1338   \item @{command (HOL) "specification"}~@{text "decls \<phi>"} sets up a
  1339   goal stating the existence of terms with the properties specified to
  1340   hold for the constants given in @{text decls}.  After finishing the
  1341   proof, the theory will be augmented with definitions for the given
  1342   constants, as well as with theorems stating the properties for these
  1343   constants.
  1344 
  1345   \item @{command (HOL) "ax_specification"}~@{text "decls \<phi>"} sets up
  1346   a goal stating the existence of terms with the properties specified
  1347   to hold for the constants given in @{text decls}.  After finishing
  1348   the proof, the theory will be augmented with axioms expressing the
  1349   properties given in the first place.
  1350 
  1351   \item @{text decl} declares a constant to be defined by the
  1352   specification given.  The definition for the constant @{text c} is
  1353   bound to the name @{text c_def} unless a theorem name is given in
  1354   the declaration.  Overloaded constants should be declared as such.
  1355 
  1356   \end{description}
  1357 
  1358   Whether to use @{command (HOL) "specification"} or @{command (HOL)
  1359   "ax_specification"} is to some extent a matter of style.  @{command
  1360   (HOL) "specification"} introduces no new axioms, and so by
  1361   construction cannot introduce inconsistencies, whereas @{command
  1362   (HOL) "ax_specification"} does introduce axioms, but only after the
  1363   user has explicitly proven it to be safe.  A practical issue must be
  1364   considered, though: After introducing two constants with the same
  1365   properties using @{command (HOL) "specification"}, one can prove
  1366   that the two constants are, in fact, equal.  If this might be a
  1367   problem, one should use @{command (HOL) "ax_specification"}.
  1368 *}
  1369 
  1370 end