doc-src/IsarRef/Thy/HOL_Specific.thy
author bulwahn
Fri Oct 29 08:44:43 2010 +0200 (2010-10-29)
changeset 40245 59f011c1877a
parent 40171 1fa547166a1d
child 40254 6d1ebaa7a4ba
permissions -rw-r--r--
updating documentation on quickcheck in the Isar reference
     1 theory HOL_Specific
     2 imports Main
     3 begin
     4 
     5 chapter {* Isabelle/HOL \label{ch:hol} *}
     6 
     7 section {* Typedef axiomatization \label{sec:hol-typedef} *}
     8 
     9 text {*
    10   \begin{matharray}{rcl}
    11     @{command_def (HOL) "typedef"} & : & @{text "local_theory \<rightarrow> proof(prove)"} \\
    12   \end{matharray}
    13 
    14   \begin{rail}
    15     'typedef' altname? abstype '=' repset
    16     ;
    17 
    18     altname: '(' (name | 'open' | 'open' name) ')'
    19     ;
    20     abstype: typespecsorts mixfix?
    21     ;
    22     repset: term ('morphisms' name name)?
    23     ;
    24   \end{rail}
    25 
    26   \begin{description}
    27   
    28   \item @{command (HOL) "typedef"}~@{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>n) t = A"}
    29   axiomatizes a Gordon/HOL-style type definition in the background
    30   theory of the current context, depending on a non-emptiness result
    31   of the set @{text A} (which needs to be proven interactively).
    32 
    33   The raw type may not depend on parameters or assumptions of the
    34   context --- this is logically impossible in Isabelle/HOL --- but the
    35   non-emptiness property can be local, potentially resulting in
    36   multiple interpretations in target contexts.  Thus the established
    37   bijection between the representing set @{text A} and the new type
    38   @{text t} may semantically depend on local assumptions.
    39   
    40   By default, @{command (HOL) "typedef"} defines both a type @{text t}
    41   and a set (term constant) of the same name, unless an alternative
    42   base name is given in parentheses, or the ``@{text "(open)"}''
    43   declaration is used to suppress a separate constant definition
    44   altogether.  The injection from type to set is called @{text Rep_t},
    45   its inverse @{text Abs_t} --- this may be changed via an explicit
    46   @{keyword (HOL) "morphisms"} declaration.
    47   
    48   Theorems @{text Rep_t}, @{text Rep_t_inverse}, and @{text
    49   Abs_t_inverse} provide the most basic characterization as a
    50   corresponding injection/surjection pair (in both directions).  Rules
    51   @{text Rep_t_inject} and @{text Abs_t_inject} provide a slightly
    52   more convenient view on the injectivity part, suitable for automated
    53   proof tools (e.g.\ in @{attribute simp} or @{attribute iff}
    54   declarations).  Rules @{text Rep_t_cases}/@{text Rep_t_induct}, and
    55   @{text Abs_t_cases}/@{text Abs_t_induct} provide alternative views
    56   on surjectivity; these are already declared as set or type rules for
    57   the generic @{method cases} and @{method induct} methods.
    58   
    59   An alternative name for the set definition (and other derived
    60   entities) may be specified in parentheses; the default is to use
    61   @{text t} as indicated before.
    62 
    63   \end{description}
    64 *}
    65 
    66 
    67 section {* Adhoc tuples *}
    68 
    69 text {*
    70   \begin{matharray}{rcl}
    71     @{attribute (HOL) split_format}@{text "\<^sup>*"} & : & @{text attribute} \\
    72   \end{matharray}
    73 
    74   \begin{rail}
    75     'split\_format' ((( name * ) + 'and') | ('(' 'complete' ')'))
    76     ;
    77   \end{rail}
    78 
    79   \begin{description}
    80   
    81   \item @{attribute (HOL) split_format}~@{text "p\<^sub>1 \<dots> p\<^sub>m \<AND> \<dots>
    82   \<AND> q\<^sub>1 \<dots> q\<^sub>n"} puts expressions of low-level tuple types into
    83   canonical form as specified by the arguments given; the @{text i}-th
    84   collection of arguments refers to occurrences in premise @{text i}
    85   of the rule.  The ``@{text "(complete)"}'' option causes \emph{all}
    86   arguments in function applications to be represented canonically
    87   according to their tuple type structure.
    88 
    89   Note that these operations tend to invent funny names for new local
    90   parameters to be introduced.
    91 
    92   \end{description}
    93 *}
    94 
    95 
    96 section {* Records \label{sec:hol-record} *}
    97 
    98 text {*
    99   In principle, records merely generalize the concept of tuples, where
   100   components may be addressed by labels instead of just position.  The
   101   logical infrastructure of records in Isabelle/HOL is slightly more
   102   advanced, though, supporting truly extensible record schemes.  This
   103   admits operations that are polymorphic with respect to record
   104   extension, yielding ``object-oriented'' effects like (single)
   105   inheritance.  See also \cite{NaraschewskiW-TPHOLs98} for more
   106   details on object-oriented verification and record subtyping in HOL.
   107 *}
   108 
   109 
   110 subsection {* Basic concepts *}
   111 
   112 text {*
   113   Isabelle/HOL supports both \emph{fixed} and \emph{schematic} records
   114   at the level of terms and types.  The notation is as follows:
   115 
   116   \begin{center}
   117   \begin{tabular}{l|l|l}
   118     & record terms & record types \\ \hline
   119     fixed & @{text "\<lparr>x = a, y = b\<rparr>"} & @{text "\<lparr>x :: A, y :: B\<rparr>"} \\
   120     schematic & @{text "\<lparr>x = a, y = b, \<dots> = m\<rparr>"} &
   121       @{text "\<lparr>x :: A, y :: B, \<dots> :: M\<rparr>"} \\
   122   \end{tabular}
   123   \end{center}
   124 
   125   \noindent The ASCII representation of @{text "\<lparr>x = a\<rparr>"} is @{text
   126   "(| x = a |)"}.
   127 
   128   A fixed record @{text "\<lparr>x = a, y = b\<rparr>"} has field @{text x} of value
   129   @{text a} and field @{text y} of value @{text b}.  The corresponding
   130   type is @{text "\<lparr>x :: A, y :: B\<rparr>"}, assuming that @{text "a :: A"}
   131   and @{text "b :: B"}.
   132 
   133   A record scheme like @{text "\<lparr>x = a, y = b, \<dots> = m\<rparr>"} contains fields
   134   @{text x} and @{text y} as before, but also possibly further fields
   135   as indicated by the ``@{text "\<dots>"}'' notation (which is actually part
   136   of the syntax).  The improper field ``@{text "\<dots>"}'' of a record
   137   scheme is called the \emph{more part}.  Logically it is just a free
   138   variable, which is occasionally referred to as ``row variable'' in
   139   the literature.  The more part of a record scheme may be
   140   instantiated by zero or more further components.  For example, the
   141   previous scheme may get instantiated to @{text "\<lparr>x = a, y = b, z =
   142   c, \<dots> = m'\<rparr>"}, where @{text m'} refers to a different more part.
   143   Fixed records are special instances of record schemes, where
   144   ``@{text "\<dots>"}'' is properly terminated by the @{text "() :: unit"}
   145   element.  In fact, @{text "\<lparr>x = a, y = b\<rparr>"} is just an abbreviation
   146   for @{text "\<lparr>x = a, y = b, \<dots> = ()\<rparr>"}.
   147   
   148   \medskip Two key observations make extensible records in a simply
   149   typed language like HOL work out:
   150 
   151   \begin{enumerate}
   152 
   153   \item the more part is internalized, as a free term or type
   154   variable,
   155 
   156   \item field names are externalized, they cannot be accessed within
   157   the logic as first-class values.
   158 
   159   \end{enumerate}
   160 
   161   \medskip In Isabelle/HOL record types have to be defined explicitly,
   162   fixing their field names and types, and their (optional) parent
   163   record.  Afterwards, records may be formed using above syntax, while
   164   obeying the canonical order of fields as given by their declaration.
   165   The record package provides several standard operations like
   166   selectors and updates.  The common setup for various generic proof
   167   tools enable succinct reasoning patterns.  See also the Isabelle/HOL
   168   tutorial \cite{isabelle-hol-book} for further instructions on using
   169   records in practice.
   170 *}
   171 
   172 
   173 subsection {* Record specifications *}
   174 
   175 text {*
   176   \begin{matharray}{rcl}
   177     @{command_def (HOL) "record"} & : & @{text "theory \<rightarrow> theory"} \\
   178   \end{matharray}
   179 
   180   \begin{rail}
   181     'record' typespecsorts '=' (type '+')? (constdecl +)
   182     ;
   183   \end{rail}
   184 
   185   \begin{description}
   186 
   187   \item @{command (HOL) "record"}~@{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t = \<tau> + c\<^sub>1 :: \<sigma>\<^sub>1
   188   \<dots> c\<^sub>n :: \<sigma>\<^sub>n"} defines extensible record type @{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t"},
   189   derived from the optional parent record @{text "\<tau>"} by adding new
   190   field components @{text "c\<^sub>i :: \<sigma>\<^sub>i"} etc.
   191 
   192   The type variables of @{text "\<tau>"} and @{text "\<sigma>\<^sub>i"} need to be
   193   covered by the (distinct) parameters @{text "\<alpha>\<^sub>1, \<dots>,
   194   \<alpha>\<^sub>m"}.  Type constructor @{text t} has to be new, while @{text
   195   \<tau>} needs to specify an instance of an existing record type.  At
   196   least one new field @{text "c\<^sub>i"} has to be specified.
   197   Basically, field names need to belong to a unique record.  This is
   198   not a real restriction in practice, since fields are qualified by
   199   the record name internally.
   200 
   201   The parent record specification @{text \<tau>} is optional; if omitted
   202   @{text t} becomes a root record.  The hierarchy of all records
   203   declared within a theory context forms a forest structure, i.e.\ a
   204   set of trees starting with a root record each.  There is no way to
   205   merge multiple parent records!
   206 
   207   For convenience, @{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t"} is made a
   208   type abbreviation for the fixed record type @{text "\<lparr>c\<^sub>1 ::
   209   \<sigma>\<^sub>1, \<dots>, c\<^sub>n :: \<sigma>\<^sub>n\<rparr>"}, likewise is @{text
   210   "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m, \<zeta>) t_scheme"} made an abbreviation for
   211   @{text "\<lparr>c\<^sub>1 :: \<sigma>\<^sub>1, \<dots>, c\<^sub>n :: \<sigma>\<^sub>n, \<dots> ::
   212   \<zeta>\<rparr>"}.
   213 
   214   \end{description}
   215 *}
   216 
   217 
   218 subsection {* Record operations *}
   219 
   220 text {*
   221   Any record definition of the form presented above produces certain
   222   standard operations.  Selectors and updates are provided for any
   223   field, including the improper one ``@{text more}''.  There are also
   224   cumulative record constructor functions.  To simplify the
   225   presentation below, we assume for now that @{text "(\<alpha>\<^sub>1, \<dots>,
   226   \<alpha>\<^sub>m) t"} is a root record with fields @{text "c\<^sub>1 ::
   227   \<sigma>\<^sub>1, \<dots>, c\<^sub>n :: \<sigma>\<^sub>n"}.
   228 
   229   \medskip \textbf{Selectors} and \textbf{updates} are available for
   230   any field (including ``@{text more}''):
   231 
   232   \begin{matharray}{lll}
   233     @{text "c\<^sub>i"} & @{text "::"} & @{text "\<lparr>\<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr> \<Rightarrow> \<sigma>\<^sub>i"} \\
   234     @{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>"} \\
   235   \end{matharray}
   236 
   237   There is special syntax for application of updates: @{text "r\<lparr>x :=
   238   a\<rparr>"} abbreviates term @{text "x_update a r"}.  Further notation for
   239   repeated updates is also available: @{text "r\<lparr>x := a\<rparr>\<lparr>y := b\<rparr>\<lparr>z :=
   240   c\<rparr>"} may be written @{text "r\<lparr>x := a, y := b, z := c\<rparr>"}.  Note that
   241   because of postfix notation the order of fields shown here is
   242   reverse than in the actual term.  Since repeated updates are just
   243   function applications, fields may be freely permuted in @{text "\<lparr>x
   244   := a, y := b, z := c\<rparr>"}, as far as logical equality is concerned.
   245   Thus commutativity of independent updates can be proven within the
   246   logic for any two fields, but not as a general theorem.
   247 
   248   \medskip The \textbf{make} operation provides a cumulative record
   249   constructor function:
   250 
   251   \begin{matharray}{lll}
   252     @{text "t.make"} & @{text "::"} & @{text "\<sigma>\<^sub>1 \<Rightarrow> \<dots> \<sigma>\<^sub>n \<Rightarrow> \<lparr>\<^vec>c :: \<^vec>\<sigma>\<rparr>"} \\
   253   \end{matharray}
   254 
   255   \medskip We now reconsider the case of non-root records, which are
   256   derived of some parent.  In general, the latter may depend on
   257   another parent as well, resulting in a list of \emph{ancestor
   258   records}.  Appending the lists of fields of all ancestors results in
   259   a certain field prefix.  The record package automatically takes care
   260   of this by lifting operations over this context of ancestor fields.
   261   Assuming that @{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t"} has ancestor
   262   fields @{text "b\<^sub>1 :: \<rho>\<^sub>1, \<dots>, b\<^sub>k :: \<rho>\<^sub>k"},
   263   the above record operations will get the following types:
   264 
   265   \medskip
   266   \begin{tabular}{lll}
   267     @{text "c\<^sub>i"} & @{text "::"} & @{text "\<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr> \<Rightarrow> \<sigma>\<^sub>i"} \\
   268     @{text "c\<^sub>i_update"} & @{text "::"} & @{text "\<sigma>\<^sub>i \<Rightarrow> 
   269       \<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr> \<Rightarrow>
   270       \<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr>"} \\
   271     @{text "t.make"} & @{text "::"} & @{text "\<rho>\<^sub>1 \<Rightarrow> \<dots> \<rho>\<^sub>k \<Rightarrow> \<sigma>\<^sub>1 \<Rightarrow> \<dots> \<sigma>\<^sub>n \<Rightarrow>
   272       \<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>\<rparr>"} \\
   273   \end{tabular}
   274   \medskip
   275 
   276   \noindent Some further operations address the extension aspect of a
   277   derived record scheme specifically: @{text "t.fields"} produces a
   278   record fragment consisting of exactly the new fields introduced here
   279   (the result may serve as a more part elsewhere); @{text "t.extend"}
   280   takes a fixed record and adds a given more part; @{text
   281   "t.truncate"} restricts a record scheme to a fixed record.
   282 
   283   \medskip
   284   \begin{tabular}{lll}
   285     @{text "t.fields"} & @{text "::"} & @{text "\<sigma>\<^sub>1 \<Rightarrow> \<dots> \<sigma>\<^sub>n \<Rightarrow> \<lparr>\<^vec>c :: \<^vec>\<sigma>\<rparr>"} \\
   286     @{text "t.extend"} & @{text "::"} & @{text "\<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>\<rparr> \<Rightarrow>
   287       \<zeta> \<Rightarrow> \<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr>"} \\
   288     @{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>"} \\
   289   \end{tabular}
   290   \medskip
   291 
   292   \noindent Note that @{text "t.make"} and @{text "t.fields"} coincide
   293   for root records.
   294 *}
   295 
   296 
   297 subsection {* Derived rules and proof tools *}
   298 
   299 text {*
   300   The record package proves several results internally, declaring
   301   these facts to appropriate proof tools.  This enables users to
   302   reason about record structures quite conveniently.  Assume that
   303   @{text t} is a record type as specified above.
   304 
   305   \begin{enumerate}
   306   
   307   \item Standard conversions for selectors or updates applied to
   308   record constructor terms are made part of the default Simplifier
   309   context; thus proofs by reduction of basic operations merely require
   310   the @{method simp} method without further arguments.  These rules
   311   are available as @{text "t.simps"}, too.
   312   
   313   \item Selectors applied to updated records are automatically reduced
   314   by an internal simplification procedure, which is also part of the
   315   standard Simplifier setup.
   316 
   317   \item Inject equations of a form analogous to @{prop "(x, y) = (x',
   318   y') \<equiv> x = x' \<and> y = y'"} are declared to the Simplifier and Classical
   319   Reasoner as @{attribute iff} rules.  These rules are available as
   320   @{text "t.iffs"}.
   321 
   322   \item The introduction rule for record equality analogous to @{text
   323   "x r = x r' \<Longrightarrow> y r = y r' \<dots> \<Longrightarrow> r = r'"} is declared to the Simplifier,
   324   and as the basic rule context as ``@{attribute intro}@{text "?"}''.
   325   The rule is called @{text "t.equality"}.
   326 
   327   \item Representations of arbitrary record expressions as canonical
   328   constructor terms are provided both in @{method cases} and @{method
   329   induct} format (cf.\ the generic proof methods of the same name,
   330   \secref{sec:cases-induct}).  Several variations are available, for
   331   fixed records, record schemes, more parts etc.
   332   
   333   The generic proof methods are sufficiently smart to pick the most
   334   sensible rule according to the type of the indicated record
   335   expression: users just need to apply something like ``@{text "(cases
   336   r)"}'' to a certain proof problem.
   337 
   338   \item The derived record operations @{text "t.make"}, @{text
   339   "t.fields"}, @{text "t.extend"}, @{text "t.truncate"} are \emph{not}
   340   treated automatically, but usually need to be expanded by hand,
   341   using the collective fact @{text "t.defs"}.
   342 
   343   \end{enumerate}
   344 *}
   345 
   346 
   347 section {* Datatypes \label{sec:hol-datatype} *}
   348 
   349 text {*
   350   \begin{matharray}{rcl}
   351     @{command_def (HOL) "datatype"} & : & @{text "theory \<rightarrow> theory"} \\
   352   @{command_def (HOL) "rep_datatype"} & : & @{text "theory \<rightarrow> proof(prove)"} \\
   353   \end{matharray}
   354 
   355   \begin{rail}
   356     'datatype' (dtspec + 'and')
   357     ;
   358     'rep\_datatype' ('(' (name +) ')')? (term +)
   359     ;
   360 
   361     dtspec: parname? typespec mixfix? '=' (cons + '|')
   362     ;
   363     cons: name ( type * ) mixfix?
   364   \end{rail}
   365 
   366   \begin{description}
   367 
   368   \item @{command (HOL) "datatype"} defines inductive datatypes in
   369   HOL.
   370 
   371   \item @{command (HOL) "rep_datatype"} represents existing types as
   372   inductive ones, generating the standard infrastructure of derived
   373   concepts (primitive recursion etc.).
   374 
   375   \end{description}
   376 
   377   The induction and exhaustion theorems generated provide case names
   378   according to the constructors involved, while parameters are named
   379   after the types (see also \secref{sec:cases-induct}).
   380 
   381   See \cite{isabelle-HOL} for more details on datatypes, but beware of
   382   the old-style theory syntax being used there!  Apart from proper
   383   proof methods for case-analysis and induction, there are also
   384   emulations of ML tactics @{method (HOL) case_tac} and @{method (HOL)
   385   induct_tac} available, see \secref{sec:hol-induct-tac}; these admit
   386   to refer directly to the internal structure of subgoals (including
   387   internally bound parameters).
   388 *}
   389 
   390 
   391 section {* Recursive functions \label{sec:recursion} *}
   392 
   393 text {*
   394   \begin{matharray}{rcl}
   395     @{command_def (HOL) "primrec"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
   396     @{command_def (HOL) "fun"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
   397     @{command_def (HOL) "function"} & : & @{text "local_theory \<rightarrow> proof(prove)"} \\
   398     @{command_def (HOL) "termination"} & : & @{text "local_theory \<rightarrow> proof(prove)"} \\
   399   \end{matharray}
   400 
   401   \begin{rail}
   402     'primrec' target? fixes 'where' equations
   403     ;
   404     ('fun' | 'function') target? functionopts? fixes \\ 'where' equations
   405     ;
   406     equations: (thmdecl? prop + '|')
   407     ;
   408     functionopts: '(' (('sequential' | 'domintros' | 'tailrec' | 'default' term) + ',') ')'
   409     ;
   410     'termination' ( term )?
   411   \end{rail}
   412 
   413   \begin{description}
   414 
   415   \item @{command (HOL) "primrec"} defines primitive recursive
   416   functions over datatypes, see also \cite{isabelle-HOL}.
   417 
   418   \item @{command (HOL) "function"} defines functions by general
   419   wellfounded recursion. A detailed description with examples can be
   420   found in \cite{isabelle-function}. The function is specified by a
   421   set of (possibly conditional) recursive equations with arbitrary
   422   pattern matching. The command generates proof obligations for the
   423   completeness and the compatibility of patterns.
   424 
   425   The defined function is considered partial, and the resulting
   426   simplification rules (named @{text "f.psimps"}) and induction rule
   427   (named @{text "f.pinduct"}) are guarded by a generated domain
   428   predicate @{text "f_dom"}. The @{command (HOL) "termination"}
   429   command can then be used to establish that the function is total.
   430 
   431   \item @{command (HOL) "fun"} is a shorthand notation for ``@{command
   432   (HOL) "function"}~@{text "(sequential)"}, followed by automated
   433   proof attempts regarding pattern matching and termination.  See
   434   \cite{isabelle-function} for further details.
   435 
   436   \item @{command (HOL) "termination"}~@{text f} commences a
   437   termination proof for the previously defined function @{text f}.  If
   438   this is omitted, the command refers to the most recent function
   439   definition.  After the proof is closed, the recursive equations and
   440   the induction principle is established.
   441 
   442   \end{description}
   443 
   444   Recursive definitions introduced by the @{command (HOL) "function"}
   445   command accommodate
   446   reasoning by induction (cf.\ \secref{sec:cases-induct}): rule @{text
   447   "c.induct"} (where @{text c} is the name of the function definition)
   448   refers to a specific induction rule, with parameters named according
   449   to the user-specified equations. Cases are numbered (starting from 1).
   450 
   451   For @{command (HOL) "primrec"}, the induction principle coincides
   452   with structural recursion on the datatype the recursion is carried
   453   out.
   454 
   455   The equations provided by these packages may be referred later as
   456   theorem list @{text "f.simps"}, where @{text f} is the (collective)
   457   name of the functions defined.  Individual equations may be named
   458   explicitly as well.
   459 
   460   The @{command (HOL) "function"} command accepts the following
   461   options.
   462 
   463   \begin{description}
   464 
   465   \item @{text sequential} enables a preprocessor which disambiguates
   466   overlapping patterns by making them mutually disjoint.  Earlier
   467   equations take precedence over later ones.  This allows to give the
   468   specification in a format very similar to functional programming.
   469   Note that the resulting simplification and induction rules
   470   correspond to the transformed specification, not the one given
   471   originally. This usually means that each equation given by the user
   472   may result in several theorems.  Also note that this automatic
   473   transformation only works for ML-style datatype patterns.
   474 
   475   \item @{text domintros} enables the automated generation of
   476   introduction rules for the domain predicate. While mostly not
   477   needed, they can be helpful in some proofs about partial functions.
   478 
   479   \item @{text tailrec} generates the unconstrained recursive
   480   equations even without a termination proof, provided that the
   481   function is tail-recursive. This currently only works
   482 
   483   \item @{text "default d"} allows to specify a default value for a
   484   (partial) function, which will ensure that @{text "f x = d x"}
   485   whenever @{text "x \<notin> f_dom"}.
   486 
   487   \end{description}
   488 *}
   489 
   490 
   491 subsection {* Proof methods related to recursive definitions *}
   492 
   493 text {*
   494   \begin{matharray}{rcl}
   495     @{method_def (HOL) pat_completeness} & : & @{text method} \\
   496     @{method_def (HOL) relation} & : & @{text method} \\
   497     @{method_def (HOL) lexicographic_order} & : & @{text method} \\
   498     @{method_def (HOL) size_change} & : & @{text method} \\
   499   \end{matharray}
   500 
   501   \begin{rail}
   502     'relation' term
   503     ;
   504     'lexicographic\_order' ( clasimpmod * )
   505     ;
   506     'size\_change' ( orders ( clasimpmod * ) )
   507     ;
   508     orders: ( 'max' | 'min' | 'ms' ) *
   509   \end{rail}
   510 
   511   \begin{description}
   512 
   513   \item @{method (HOL) pat_completeness} is a specialized method to
   514   solve goals regarding the completeness of pattern matching, as
   515   required by the @{command (HOL) "function"} package (cf.\
   516   \cite{isabelle-function}).
   517 
   518   \item @{method (HOL) relation}~@{text R} introduces a termination
   519   proof using the relation @{text R}.  The resulting proof state will
   520   contain goals expressing that @{text R} is wellfounded, and that the
   521   arguments of recursive calls decrease with respect to @{text R}.
   522   Usually, this method is used as the initial proof step of manual
   523   termination proofs.
   524 
   525   \item @{method (HOL) "lexicographic_order"} attempts a fully
   526   automated termination proof by searching for a lexicographic
   527   combination of size measures on the arguments of the function. The
   528   method accepts the same arguments as the @{method auto} method,
   529   which it uses internally to prove local descents.  The same context
   530   modifiers as for @{method auto} are accepted, see
   531   \secref{sec:clasimp}.
   532 
   533   In case of failure, extensive information is printed, which can help
   534   to analyse the situation (cf.\ \cite{isabelle-function}).
   535 
   536   \item @{method (HOL) "size_change"} also works on termination goals,
   537   using a variation of the size-change principle, together with a
   538   graph decomposition technique (see \cite{krauss_phd} for details).
   539   Three kinds of orders are used internally: @{text max}, @{text min},
   540   and @{text ms} (multiset), which is only available when the theory
   541   @{text Multiset} is loaded. When no order kinds are given, they are
   542   tried in order. The search for a termination proof uses SAT solving
   543   internally.
   544 
   545  For local descent proofs, the same context modifiers as for @{method
   546   auto} are accepted, see \secref{sec:clasimp}.
   547 
   548   \end{description}
   549 *}
   550 
   551 subsection {* Functions with explicit partiality *}
   552 
   553 text {*
   554   \begin{matharray}{rcl}
   555     @{command_def (HOL) "partial_function"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
   556     @{attribute_def (HOL) "partial_function_mono"} & : & @{text attribute} \\
   557   \end{matharray}
   558 
   559   \begin{rail}
   560     'partial_function' target? '(' mode ')' fixes \\ 'where' thmdecl? prop
   561   \end{rail}
   562 
   563   \begin{description}
   564 
   565   \item @{command (HOL) "partial_function"} defines recursive
   566   functions based on fixpoints in complete partial orders. No
   567   termination proof is required from the user or constructed
   568   internally. Instead, the possibility of non-termination is modelled
   569   explicitly in the result type, which contains an explicit bottom
   570   element.
   571 
   572   Pattern matching and mutual recursion are currently not supported.
   573   Thus, the specification consists of a single function described by a
   574   single recursive equation.
   575 
   576   There are no fixed syntactic restrictions on the body of the
   577   function, but the induced functional must be provably monotonic
   578   wrt.\ the underlying order.  The monotonicitity proof is performed
   579   internally, and the definition is rejected when it fails. The proof
   580   can be influenced by declaring hints using the
   581   @{attribute (HOL) partial_function_mono} attribute.
   582 
   583   The mandatory @{text mode} argument specifies the mode of operation
   584   of the command, which directly corresponds to a complete partial
   585   order on the result type. By default, the following modes are
   586   defined: 
   587 
   588   \begin{description}
   589   \item @{text option} defines functions that map into the @{type
   590   option} type. Here, the value @{term None} is used to model a
   591   non-terminating computation. Monotonicity requires that if @{term
   592   None} is returned by a recursive call, then the overall result
   593   must also be @{term None}. This is best achieved through the use of
   594   the monadic operator @{const "Option.bind"}.
   595   
   596   \item @{text tailrec} defines functions with an arbitrary result
   597   type and uses the slightly degenerated partial order where @{term
   598   "undefined"} is the bottom element.  Now, monotonicity requires that
   599   if @{term undefined} is returned by a recursive call, then the
   600   overall result must also be @{term undefined}. In practice, this is
   601   only satisfied when each recursive call is a tail call, whose result
   602   is directly returned. Thus, this mode of operation allows the
   603   definition of arbitrary tail-recursive functions.
   604   \end{description}
   605 
   606   Experienced users may define new modes by instantiating the locale
   607   @{const "partial_function_definitions"} appropriately.
   608 
   609   \item @{attribute (HOL) partial_function_mono} declares rules for
   610   use in the internal monononicity proofs of partial function
   611   definitions.
   612 
   613   \end{description}
   614 
   615 *}
   616 
   617 subsection {* Old-style recursive function definitions (TFL) *}
   618 
   619 text {*
   620   The old TFL commands @{command (HOL) "recdef"} and @{command (HOL)
   621   "recdef_tc"} for defining recursive are mostly obsolete; @{command
   622   (HOL) "function"} or @{command (HOL) "fun"} should be used instead.
   623 
   624   \begin{matharray}{rcl}
   625     @{command_def (HOL) "recdef"} & : & @{text "theory \<rightarrow> theory)"} \\
   626     @{command_def (HOL) "recdef_tc"}@{text "\<^sup>*"} & : & @{text "theory \<rightarrow> proof(prove)"} \\
   627   \end{matharray}
   628 
   629   \begin{rail}
   630     'recdef' ('(' 'permissive' ')')? \\ name term (prop +) hints?
   631     ;
   632     recdeftc thmdecl? tc
   633     ;
   634     hints: '(' 'hints' ( recdefmod * ) ')'
   635     ;
   636     recdefmod: (('recdef\_simp' | 'recdef\_cong' | 'recdef\_wf') (() | 'add' | 'del') ':' thmrefs) | clasimpmod
   637     ;
   638     tc: nameref ('(' nat ')')?
   639     ;
   640   \end{rail}
   641 
   642   \begin{description}
   643   
   644   \item @{command (HOL) "recdef"} defines general well-founded
   645   recursive functions (using the TFL package), see also
   646   \cite{isabelle-HOL}.  The ``@{text "(permissive)"}'' option tells
   647   TFL to recover from failed proof attempts, returning unfinished
   648   results.  The @{text recdef_simp}, @{text recdef_cong}, and @{text
   649   recdef_wf} hints refer to auxiliary rules to be used in the internal
   650   automated proof process of TFL.  Additional @{syntax clasimpmod}
   651   declarations (cf.\ \secref{sec:clasimp}) may be given to tune the
   652   context of the Simplifier (cf.\ \secref{sec:simplifier}) and
   653   Classical reasoner (cf.\ \secref{sec:classical}).
   654   
   655   \item @{command (HOL) "recdef_tc"}~@{text "c (i)"} recommences the
   656   proof for leftover termination condition number @{text i} (default
   657   1) as generated by a @{command (HOL) "recdef"} definition of
   658   constant @{text c}.
   659   
   660   Note that in most cases, @{command (HOL) "recdef"} is able to finish
   661   its internal proofs without manual intervention.
   662 
   663   \end{description}
   664 
   665   \medskip Hints for @{command (HOL) "recdef"} may be also declared
   666   globally, using the following attributes.
   667 
   668   \begin{matharray}{rcl}
   669     @{attribute_def (HOL) recdef_simp} & : & @{text attribute} \\
   670     @{attribute_def (HOL) recdef_cong} & : & @{text attribute} \\
   671     @{attribute_def (HOL) recdef_wf} & : & @{text attribute} \\
   672   \end{matharray}
   673 
   674   \begin{rail}
   675     ('recdef\_simp' | 'recdef\_cong' | 'recdef\_wf') (() | 'add' | 'del')
   676     ;
   677   \end{rail}
   678 *}
   679 
   680 
   681 section {* Inductive and coinductive definitions \label{sec:hol-inductive} *}
   682 
   683 text {*
   684   An \textbf{inductive definition} specifies the least predicate (or
   685   set) @{text R} closed under given rules: applying a rule to elements
   686   of @{text R} yields a result within @{text R}.  For example, a
   687   structural operational semantics is an inductive definition of an
   688   evaluation relation.
   689 
   690   Dually, a \textbf{coinductive definition} specifies the greatest
   691   predicate~/ set @{text R} that is consistent with given rules: every
   692   element of @{text R} can be seen as arising by applying a rule to
   693   elements of @{text R}.  An important example is using bisimulation
   694   relations to formalise equivalence of processes and infinite data
   695   structures.
   696 
   697   \medskip The HOL package is related to the ZF one, which is
   698   described in a separate paper,\footnote{It appeared in CADE
   699   \cite{paulson-CADE}; a longer version is distributed with Isabelle.}
   700   which you should refer to in case of difficulties.  The package is
   701   simpler than that of ZF thanks to implicit type-checking in HOL.
   702   The types of the (co)inductive predicates (or sets) determine the
   703   domain of the fixedpoint definition, and the package does not have
   704   to use inference rules for type-checking.
   705 
   706   \begin{matharray}{rcl}
   707     @{command_def (HOL) "inductive"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
   708     @{command_def (HOL) "inductive_set"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
   709     @{command_def (HOL) "coinductive"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
   710     @{command_def (HOL) "coinductive_set"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
   711     @{attribute_def (HOL) mono} & : & @{text attribute} \\
   712   \end{matharray}
   713 
   714   \begin{rail}
   715     ('inductive' | 'inductive\_set' | 'coinductive' | 'coinductive\_set') target? fixes ('for' fixes)? \\
   716     ('where' clauses)? ('monos' thmrefs)?
   717     ;
   718     clauses: (thmdecl? prop + '|')
   719     ;
   720     'mono' (() | 'add' | 'del')
   721     ;
   722   \end{rail}
   723 
   724   \begin{description}
   725 
   726   \item @{command (HOL) "inductive"} and @{command (HOL)
   727   "coinductive"} define (co)inductive predicates from the
   728   introduction rules given in the @{keyword "where"} part.  The
   729   optional @{keyword "for"} part contains a list of parameters of the
   730   (co)inductive predicates that remain fixed throughout the
   731   definition.  The optional @{keyword "monos"} section contains
   732   \emph{monotonicity theorems}, which are required for each operator
   733   applied to a recursive set in the introduction rules.  There
   734   \emph{must} be a theorem of the form @{text "A \<le> B \<Longrightarrow> M A \<le> M B"},
   735   for each premise @{text "M R\<^sub>i t"} in an introduction rule!
   736 
   737   \item @{command (HOL) "inductive_set"} and @{command (HOL)
   738   "coinductive_set"} are wrappers for to the previous commands,
   739   allowing the definition of (co)inductive sets.
   740 
   741   \item @{attribute (HOL) mono} declares monotonicity rules.  These
   742   rule are involved in the automated monotonicity proof of @{command
   743   (HOL) "inductive"}.
   744 
   745   \end{description}
   746 *}
   747 
   748 
   749 subsection {* Derived rules *}
   750 
   751 text {*
   752   Each (co)inductive definition @{text R} adds definitions to the
   753   theory and also proves some theorems:
   754 
   755   \begin{description}
   756 
   757   \item @{text R.intros} is the list of introduction rules as proven
   758   theorems, for the recursive predicates (or sets).  The rules are
   759   also available individually, using the names given them in the
   760   theory file;
   761 
   762   \item @{text R.cases} is the case analysis (or elimination) rule;
   763 
   764   \item @{text R.induct} or @{text R.coinduct} is the (co)induction
   765   rule.
   766 
   767   \end{description}
   768 
   769   When several predicates @{text "R\<^sub>1, \<dots>, R\<^sub>n"} are
   770   defined simultaneously, the list of introduction rules is called
   771   @{text "R\<^sub>1_\<dots>_R\<^sub>n.intros"}, the case analysis rules are
   772   called @{text "R\<^sub>1.cases, \<dots>, R\<^sub>n.cases"}, and the list
   773   of mutual induction rules is called @{text
   774   "R\<^sub>1_\<dots>_R\<^sub>n.inducts"}.
   775 *}
   776 
   777 
   778 subsection {* Monotonicity theorems *}
   779 
   780 text {*
   781   Each theory contains a default set of theorems that are used in
   782   monotonicity proofs.  New rules can be added to this set via the
   783   @{attribute (HOL) mono} attribute.  The HOL theory @{text Inductive}
   784   shows how this is done.  In general, the following monotonicity
   785   theorems may be added:
   786 
   787   \begin{itemize}
   788 
   789   \item Theorems of the form @{text "A \<le> B \<Longrightarrow> M A \<le> M B"}, for proving
   790   monotonicity of inductive definitions whose introduction rules have
   791   premises involving terms such as @{text "M R\<^sub>i t"}.
   792 
   793   \item Monotonicity theorems for logical operators, which are of the
   794   general form @{text "(\<dots> \<longrightarrow> \<dots>) \<Longrightarrow> \<dots> (\<dots> \<longrightarrow> \<dots>) \<Longrightarrow> \<dots> \<longrightarrow> \<dots>"}.  For example, in
   795   the case of the operator @{text "\<or>"}, the corresponding theorem is
   796   \[
   797   \infer{@{text "P\<^sub>1 \<or> P\<^sub>2 \<longrightarrow> Q\<^sub>1 \<or> Q\<^sub>2"}}{@{text "P\<^sub>1 \<longrightarrow> Q\<^sub>1"} & @{text "P\<^sub>2 \<longrightarrow> Q\<^sub>2"}}
   798   \]
   799 
   800   \item De Morgan style equations for reasoning about the ``polarity''
   801   of expressions, e.g.
   802   \[
   803   @{prop "\<not> \<not> P \<longleftrightarrow> P"} \qquad\qquad
   804   @{prop "\<not> (P \<and> Q) \<longleftrightarrow> \<not> P \<or> \<not> Q"}
   805   \]
   806 
   807   \item Equations for reducing complex operators to more primitive
   808   ones whose monotonicity can easily be proved, e.g.
   809   \[
   810   @{prop "(P \<longrightarrow> Q) \<longleftrightarrow> \<not> P \<or> Q"} \qquad\qquad
   811   @{prop "Ball A P \<equiv> \<forall>x. x \<in> A \<longrightarrow> P x"}
   812   \]
   813 
   814   \end{itemize}
   815 
   816   %FIXME: Example of an inductive definition
   817 *}
   818 
   819 
   820 section {* Arithmetic proof support *}
   821 
   822 text {*
   823   \begin{matharray}{rcl}
   824     @{method_def (HOL) arith} & : & @{text method} \\
   825     @{attribute_def (HOL) arith} & : & @{text attribute} \\
   826     @{attribute_def (HOL) arith_split} & : & @{text attribute} \\
   827   \end{matharray}
   828 
   829   The @{method (HOL) arith} method decides linear arithmetic problems
   830   (on types @{text nat}, @{text int}, @{text real}).  Any current
   831   facts are inserted into the goal before running the procedure.
   832 
   833   The @{attribute (HOL) arith} attribute declares facts that are
   834   always supplied to the arithmetic provers implicitly.
   835 
   836   The @{attribute (HOL) arith_split} attribute declares case split
   837   rules to be expanded before @{method (HOL) arith} is invoked.
   838 
   839   Note that a simpler (but faster) arithmetic prover is
   840   already invoked by the Simplifier.
   841 *}
   842 
   843 
   844 section {* Intuitionistic proof search *}
   845 
   846 text {*
   847   \begin{matharray}{rcl}
   848     @{method_def (HOL) iprover} & : & @{text method} \\
   849   \end{matharray}
   850 
   851   \begin{rail}
   852     'iprover' ( rulemod * )
   853     ;
   854   \end{rail}
   855 
   856   The @{method (HOL) iprover} method performs intuitionistic proof
   857   search, depending on specifically declared rules from the context,
   858   or given as explicit arguments.  Chained facts are inserted into the
   859   goal before commencing proof search.
   860 
   861   Rules need to be classified as @{attribute (Pure) intro},
   862   @{attribute (Pure) elim}, or @{attribute (Pure) dest}; here the
   863   ``@{text "!"}'' indicator refers to ``safe'' rules, which may be
   864   applied aggressively (without considering back-tracking later).
   865   Rules declared with ``@{text "?"}'' are ignored in proof search (the
   866   single-step @{method rule} method still observes these).  An
   867   explicit weight annotation may be given as well; otherwise the
   868   number of rule premises will be taken into account here.
   869 *}
   870 
   871 
   872 section {* Coherent Logic *}
   873 
   874 text {*
   875   \begin{matharray}{rcl}
   876     @{method_def (HOL) "coherent"} & : & @{text method} \\
   877   \end{matharray}
   878 
   879   \begin{rail}
   880     'coherent' thmrefs?
   881     ;
   882   \end{rail}
   883 
   884   The @{method (HOL) coherent} method solves problems of
   885   \emph{Coherent Logic} \cite{Bezem-Coquand:2005}, which covers
   886   applications in confluence theory, lattice theory and projective
   887   geometry.  See @{"file" "~~/src/HOL/ex/Coherent.thy"} for some
   888   examples.
   889 *}
   890 
   891 
   892 section {* Checking and refuting propositions *}
   893 
   894 text {*
   895   Identifying incorrect propositions usually involves evaluation of
   896   particular assignments and systematic counter example search.  This
   897   is supported by the following commands.
   898 
   899   \begin{matharray}{rcl}
   900     @{command_def (HOL) "value"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
   901     @{command_def (HOL) "quickcheck"}@{text "\<^sup>*"} & : & @{text "proof \<rightarrow>"} \\
   902     @{command_def (HOL) "quickcheck_params"} & : & @{text "theory \<rightarrow> theory"}
   903   \end{matharray}
   904 
   905   \begin{rail}
   906     'value' ( ( '[' name ']' ) ? ) modes? term
   907     ;
   908 
   909     'quickcheck' ( ( '[' args ']' ) ? ) nat?
   910     ;
   911 
   912     'quickcheck_params' ( ( '[' args ']' ) ? )
   913     ;
   914 
   915     modes: '(' (name + ) ')'
   916     ;
   917 
   918     args: ( name '=' value + ',' )
   919     ;
   920   \end{rail}
   921 
   922   \begin{description}
   923 
   924   \item @{command (HOL) "value"}~@{text t} evaluates and prints a
   925     term; optionally @{text modes} can be specified, which are
   926     appended to the current print mode (see also \cite{isabelle-ref}).
   927     Internally, the evaluation is performed by registered evaluators,
   928     which are invoked sequentially until a result is returned.
   929     Alternatively a specific evaluator can be selected using square
   930     brackets; typical evaluators use the current set of code equations
   931     to normalize and include @{text simp} for fully symbolic evaluation
   932     using the simplifier, @{text nbe} for \emph{normalization by evaluation}
   933     and \emph{code} for code generation in SML.
   934 
   935   \item @{command (HOL) "quickcheck"} tests the current goal for
   936     counter examples using a series of arbitrary assignments for its
   937     free variables; by default the first subgoal is tested, an other
   938     can be selected explicitly using an optional goal index.
   939     A number of configuration options are supported for
   940     @{command (HOL) "quickcheck"}, notably:
   941 
   942     \begin{description}
   943 
   944       \item[size] specifies the maximum size of the search space for
   945         assignment values.
   946 
   947       \item[iterations] sets how many sets of assignments are
   948         generated for each particular size.
   949 
   950       \item[no\_assms] specifies whether assumptions in
   951         structured proofs should be ignored.
   952 
   953       \item[timeout] sets the time limit in seconds.
   954 
   955       \item[default\_type] sets the type(s) generally used to instantiate
   956         type variables.
   957 
   958       \item[report] if set quickcheck reports how many tests fulfilled
   959         the preconditions.
   960 
   961       \item[quiet] if not set quickcheck informs about the current size
   962         for assignment values.
   963 
   964       \item[expect] can be used to check if the user's expectation was met
   965         (no\_expectation, no\_counterexample, or counterexample)
   966 
   967     \end{description}
   968 
   969     These option can be given within square brackets.
   970 
   971   \item @{command (HOL) "quickcheck_params"} changes quickcheck
   972     configuration options persitently.
   973 
   974   \end{description}
   975 *}
   976 
   977 
   978 section {* Unstructured case analysis and induction \label{sec:hol-induct-tac} *}
   979 
   980 text {*
   981   The following tools of Isabelle/HOL support cases analysis and
   982   induction in unstructured tactic scripts; see also
   983   \secref{sec:cases-induct} for proper Isar versions of similar ideas.
   984 
   985   \begin{matharray}{rcl}
   986     @{method_def (HOL) case_tac}@{text "\<^sup>*"} & : & @{text method} \\
   987     @{method_def (HOL) induct_tac}@{text "\<^sup>*"} & : & @{text method} \\
   988     @{method_def (HOL) ind_cases}@{text "\<^sup>*"} & : & @{text method} \\
   989     @{command_def (HOL) "inductive_cases"}@{text "\<^sup>*"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
   990   \end{matharray}
   991 
   992   \begin{rail}
   993     'case\_tac' goalspec? term rule?
   994     ;
   995     'induct\_tac' goalspec? (insts * 'and') rule?
   996     ;
   997     'ind\_cases' (prop +) ('for' (name +)) ?
   998     ;
   999     'inductive\_cases' (thmdecl? (prop +) + 'and')
  1000     ;
  1001 
  1002     rule: ('rule' ':' thmref)
  1003     ;
  1004   \end{rail}
  1005 
  1006   \begin{description}
  1007 
  1008   \item @{method (HOL) case_tac} and @{method (HOL) induct_tac} admit
  1009   to reason about inductive types.  Rules are selected according to
  1010   the declarations by the @{attribute cases} and @{attribute induct}
  1011   attributes, cf.\ \secref{sec:cases-induct}.  The @{command (HOL)
  1012   datatype} package already takes care of this.
  1013 
  1014   These unstructured tactics feature both goal addressing and dynamic
  1015   instantiation.  Note that named rule cases are \emph{not} provided
  1016   as would be by the proper @{method cases} and @{method induct} proof
  1017   methods (see \secref{sec:cases-induct}).  Unlike the @{method
  1018   induct} method, @{method induct_tac} does not handle structured rule
  1019   statements, only the compact object-logic conclusion of the subgoal
  1020   being addressed.
  1021   
  1022   \item @{method (HOL) ind_cases} and @{command (HOL)
  1023   "inductive_cases"} provide an interface to the internal @{ML_text
  1024   mk_cases} operation.  Rules are simplified in an unrestricted
  1025   forward manner.
  1026 
  1027   While @{method (HOL) ind_cases} is a proof method to apply the
  1028   result immediately as elimination rules, @{command (HOL)
  1029   "inductive_cases"} provides case split theorems at the theory level
  1030   for later use.  The @{keyword "for"} argument of the @{method (HOL)
  1031   ind_cases} method allows to specify a list of variables that should
  1032   be generalized before applying the resulting rule.
  1033 
  1034   \end{description}
  1035 *}
  1036 
  1037 
  1038 section {* Executable code *}
  1039 
  1040 text {*
  1041   Isabelle/Pure provides two generic frameworks to support code
  1042   generation from executable specifications.  Isabelle/HOL
  1043   instantiates these mechanisms in a way that is amenable to end-user
  1044   applications.
  1045 
  1046   \medskip One framework generates code from functional programs
  1047   (including overloading using type classes) to SML \cite{SML}, OCaml
  1048   \cite{OCaml}, Haskell \cite{haskell-revised-report} and Scala
  1049   \cite{scala-overview-tech-report}.
  1050   Conceptually, code generation is split up in three steps:
  1051   \emph{selection} of code theorems, \emph{translation} into an
  1052   abstract executable view and \emph{serialization} to a specific
  1053   \emph{target language}.  Inductive specifications can be executed
  1054   using the predicate compiler which operates within HOL.
  1055   See \cite{isabelle-codegen} for an introduction.
  1056 
  1057   \begin{matharray}{rcl}
  1058     @{command_def (HOL) "export_code"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
  1059     @{attribute_def (HOL) code} & : & @{text attribute} \\
  1060     @{command_def (HOL) "code_abort"} & : & @{text "theory \<rightarrow> theory"} \\
  1061     @{command_def (HOL) "code_datatype"} & : & @{text "theory \<rightarrow> theory"} \\
  1062     @{command_def (HOL) "print_codesetup"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
  1063     @{attribute_def (HOL) code_inline} & : & @{text attribute} \\
  1064     @{attribute_def (HOL) code_post} & : & @{text attribute} \\
  1065     @{command_def (HOL) "print_codeproc"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
  1066     @{command_def (HOL) "code_thms"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
  1067     @{command_def (HOL) "code_deps"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
  1068     @{command_def (HOL) "code_const"} & : & @{text "theory \<rightarrow> theory"} \\
  1069     @{command_def (HOL) "code_type"} & : & @{text "theory \<rightarrow> theory"} \\
  1070     @{command_def (HOL) "code_class"} & : & @{text "theory \<rightarrow> theory"} \\
  1071     @{command_def (HOL) "code_instance"} & : & @{text "theory \<rightarrow> theory"} \\
  1072     @{command_def (HOL) "code_reserved"} & : & @{text "theory \<rightarrow> theory"} \\
  1073     @{command_def (HOL) "code_monad"} & : & @{text "theory \<rightarrow> theory"} \\
  1074     @{command_def (HOL) "code_include"} & : & @{text "theory \<rightarrow> theory"} \\
  1075     @{command_def (HOL) "code_modulename"} & : & @{text "theory \<rightarrow> theory"} \\
  1076     @{command_def (HOL) "code_reflect"} & : & @{text "theory \<rightarrow> theory"}
  1077   \end{matharray}
  1078 
  1079   \begin{rail}
  1080      'export\_code' ( constexpr + ) \\
  1081        ( ( 'in' target ( 'module\_name' string ) ? \\
  1082         ( 'file' ( string | '-' ) ) ? ( '(' args ')' ) ?) + ) ?
  1083     ;
  1084 
  1085     const: term
  1086     ;
  1087 
  1088     constexpr: ( const | 'name.*' | '*' )
  1089     ;
  1090 
  1091     typeconstructor: nameref
  1092     ;
  1093 
  1094     class: nameref
  1095     ;
  1096 
  1097     target: 'SML' | 'OCaml' | 'Haskell' | 'Scala'
  1098     ;
  1099 
  1100     'code' ( 'del' | 'abstype' | 'abstract' ) ?
  1101     ;
  1102 
  1103     'code\_abort' ( const + )
  1104     ;
  1105 
  1106     'code\_datatype' ( const + )
  1107     ;
  1108 
  1109     'code_inline' ( 'del' ) ?
  1110     ;
  1111 
  1112     'code_post' ( 'del' ) ?
  1113     ;
  1114 
  1115     'code\_thms' ( constexpr + ) ?
  1116     ;
  1117 
  1118     'code\_deps' ( constexpr + ) ?
  1119     ;
  1120 
  1121     'code\_const' (const + 'and') \\
  1122       ( ( '(' target ( syntax ? + 'and' ) ')' ) + )
  1123     ;
  1124 
  1125     'code\_type' (typeconstructor + 'and') \\
  1126       ( ( '(' target ( syntax ? + 'and' ) ')' ) + )
  1127     ;
  1128 
  1129     'code\_class' (class + 'and') \\
  1130       ( ( '(' target \\ ( string ? + 'and' ) ')' ) + )
  1131     ;
  1132 
  1133     'code\_instance' (( typeconstructor '::' class ) + 'and') \\
  1134       ( ( '(' target ( '-' ? + 'and' ) ')' ) + )
  1135     ;
  1136 
  1137     'code\_reserved' target ( string + )
  1138     ;
  1139 
  1140     'code\_monad' const const target
  1141     ;
  1142 
  1143     'code\_include' target ( string ( string | '-') )
  1144     ;
  1145 
  1146     'code\_modulename' target ( ( string string ) + )
  1147     ;
  1148 
  1149     'code\_reflect' string ( 'datatypes' ( string '=' ( string + '|' ) + 'and' ) ) ? \\
  1150       ( 'functions' ( string + ) ) ? ( 'file' string ) ?
  1151     ;
  1152 
  1153     syntax: string | ( 'infix' | 'infixl' | 'infixr' ) nat string
  1154     ;
  1155 
  1156   \end{rail}
  1157 
  1158   \begin{description}
  1159 
  1160   \item @{command (HOL) "export_code"} generates code for a given list
  1161   of constants in the specified target language(s).  If no
  1162   serialization instruction is given, only abstract code is generated
  1163   internally.
  1164 
  1165   Constants may be specified by giving them literally, referring to
  1166   all executable contants within a certain theory by giving @{text
  1167   "name.*"}, or referring to \emph{all} executable constants currently
  1168   available by giving @{text "*"}.
  1169 
  1170   By default, for each involved theory one corresponding name space
  1171   module is generated.  Alternativly, a module name may be specified
  1172   after the @{keyword "module_name"} keyword; then \emph{all} code is
  1173   placed in this module.
  1174 
  1175   For \emph{SML}, \emph{OCaml} and \emph{Scala} the file specification
  1176   refers to a single file; for \emph{Haskell}, it refers to a whole
  1177   directory, where code is generated in multiple files reflecting the
  1178   module hierarchy.  Omitting the file specification denotes standard
  1179   output.
  1180 
  1181   Serializers take an optional list of arguments in parentheses.  For
  1182   \emph{SML} and \emph{OCaml}, ``@{text no_signatures}`` omits
  1183   explicit module signatures.
  1184   
  1185   For \emph{Haskell} a module name prefix may be given using the
  1186   ``@{text "root:"}'' argument; ``@{text string_classes}'' adds a
  1187   ``@{verbatim "deriving (Read, Show)"}'' clause to each appropriate
  1188   datatype declaration.
  1189 
  1190   \item @{attribute (HOL) code} explicitly selects (or with option
  1191   ``@{text "del"}'' deselects) a code equation for code generation.
  1192   Usually packages introducing code equations provide a reasonable
  1193   default setup for selection.  Variants @{text "code abstype"} and
  1194   @{text "code abstract"} declare abstract datatype certificates or
  1195   code equations on abstract datatype representations respectively.
  1196 
  1197   \item @{command (HOL) "code_abort"} declares constants which are not
  1198   required to have a definition by means of code equations; if needed
  1199   these are implemented by program abort instead.
  1200 
  1201   \item @{command (HOL) "code_datatype"} specifies a constructor set
  1202   for a logical type.
  1203 
  1204   \item @{command (HOL) "print_codesetup"} gives an overview on
  1205   selected code equations and code generator datatypes.
  1206 
  1207   \item @{attribute (HOL) code_inline} declares (or with option
  1208   ``@{text "del"}'' removes) inlining theorems which are applied as
  1209   rewrite rules to any code equation during preprocessing.
  1210 
  1211   \item @{attribute (HOL) code_post} declares (or with option ``@{text
  1212   "del"}'' removes) theorems which are applied as rewrite rules to any
  1213   result of an evaluation.
  1214 
  1215   \item @{command (HOL) "print_codeproc"} prints the setup of the code
  1216   generator preprocessor.
  1217 
  1218   \item @{command (HOL) "code_thms"} prints a list of theorems
  1219   representing the corresponding program containing all given
  1220   constants after preprocessing.
  1221 
  1222   \item @{command (HOL) "code_deps"} visualizes dependencies of
  1223   theorems representing the corresponding program containing all given
  1224   constants after preprocessing.
  1225 
  1226   \item @{command (HOL) "code_const"} associates a list of constants
  1227   with target-specific serializations; omitting a serialization
  1228   deletes an existing serialization.
  1229 
  1230   \item @{command (HOL) "code_type"} associates a list of type
  1231   constructors with target-specific serializations; omitting a
  1232   serialization deletes an existing serialization.
  1233 
  1234   \item @{command (HOL) "code_class"} associates a list of classes
  1235   with target-specific class names; omitting a serialization deletes
  1236   an existing serialization.  This applies only to \emph{Haskell}.
  1237 
  1238   \item @{command (HOL) "code_instance"} declares a list of type
  1239   constructor / class instance relations as ``already present'' for a
  1240   given target.  Omitting a ``@{text "-"}'' deletes an existing
  1241   ``already present'' declaration.  This applies only to
  1242   \emph{Haskell}.
  1243 
  1244   \item @{command (HOL) "code_reserved"} declares a list of names as
  1245   reserved for a given target, preventing it to be shadowed by any
  1246   generated code.
  1247 
  1248   \item @{command (HOL) "code_monad"} provides an auxiliary mechanism
  1249   to generate monadic code for Haskell.
  1250 
  1251   \item @{command (HOL) "code_include"} adds arbitrary named content
  1252   (``include'') to generated code.  A ``@{text "-"}'' as last argument
  1253   will remove an already added ``include''.
  1254 
  1255   \item @{command (HOL) "code_modulename"} declares aliasings from one
  1256   module name onto another.
  1257 
  1258   \item @{command (HOL) "code_reflect"} without a ``@{text "file"}''
  1259   argument compiles code into the system runtime environment and
  1260   modifies the code generator setup that future invocations of system
  1261   runtime code generation referring to one of the ``@{text
  1262   "datatypes"}'' or ``@{text "functions"}'' entities use these precompiled
  1263   entities.  With a ``@{text "file"}'' argument, the corresponding code
  1264   is generated into that specified file without modifying the code
  1265   generator setup.
  1266 
  1267   \end{description}
  1268 
  1269   The other framework generates code from both functional and
  1270   relational programs to SML.  See \cite{isabelle-HOL} for further
  1271   information (this actually covers the new-style theory format as
  1272   well).
  1273 
  1274   \begin{matharray}{rcl}
  1275     @{command_def (HOL) "code_module"} & : & @{text "theory \<rightarrow> theory"} \\
  1276     @{command_def (HOL) "code_library"} & : & @{text "theory \<rightarrow> theory"} \\
  1277     @{command_def (HOL) "consts_code"} & : & @{text "theory \<rightarrow> theory"} \\
  1278     @{command_def (HOL) "types_code"} & : & @{text "theory \<rightarrow> theory"} \\  
  1279     @{attribute_def (HOL) code} & : & @{text attribute} \\
  1280   \end{matharray}
  1281 
  1282   \begin{rail}
  1283   ( 'code\_module' | 'code\_library' ) modespec ? name ? \\
  1284     ( 'file' name ) ? ( 'imports' ( name + ) ) ? \\
  1285     'contains' ( ( name '=' term ) + | term + )
  1286   ;
  1287 
  1288   modespec: '(' ( name * ) ')'
  1289   ;
  1290 
  1291   'consts\_code' (codespec +)
  1292   ;
  1293 
  1294   codespec: const template attachment ?
  1295   ;
  1296 
  1297   'types\_code' (tycodespec +)
  1298   ;
  1299 
  1300   tycodespec: name template attachment ?
  1301   ;
  1302 
  1303   const: term
  1304   ;
  1305 
  1306   template: '(' string ')'
  1307   ;
  1308 
  1309   attachment: 'attach' modespec ? verblbrace text verbrbrace
  1310   ;
  1311 
  1312   'code' (name)?
  1313   ;
  1314   \end{rail}
  1315 
  1316 *}
  1317 
  1318 
  1319 section {* Definition by specification \label{sec:hol-specification} *}
  1320 
  1321 text {*
  1322   \begin{matharray}{rcl}
  1323     @{command_def (HOL) "specification"} & : & @{text "theory \<rightarrow> proof(prove)"} \\
  1324     @{command_def (HOL) "ax_specification"} & : & @{text "theory \<rightarrow> proof(prove)"} \\
  1325   \end{matharray}
  1326 
  1327   \begin{rail}
  1328   ('specification' | 'ax\_specification') '(' (decl +) ')' \\ (thmdecl? prop +)
  1329   ;
  1330   decl: ((name ':')? term '(' 'overloaded' ')'?)
  1331   \end{rail}
  1332 
  1333   \begin{description}
  1334 
  1335   \item @{command (HOL) "specification"}~@{text "decls \<phi>"} sets up a
  1336   goal stating the existence of terms with the properties specified to
  1337   hold for the constants given in @{text decls}.  After finishing the
  1338   proof, the theory will be augmented with definitions for the given
  1339   constants, as well as with theorems stating the properties for these
  1340   constants.
  1341 
  1342   \item @{command (HOL) "ax_specification"}~@{text "decls \<phi>"} sets up
  1343   a goal stating the existence of terms with the properties specified
  1344   to hold for the constants given in @{text decls}.  After finishing
  1345   the proof, the theory will be augmented with axioms expressing the
  1346   properties given in the first place.
  1347 
  1348   \item @{text decl} declares a constant to be defined by the
  1349   specification given.  The definition for the constant @{text c} is
  1350   bound to the name @{text c_def} unless a theorem name is given in
  1351   the declaration.  Overloaded constants should be declared as such.
  1352 
  1353   \end{description}
  1354 
  1355   Whether to use @{command (HOL) "specification"} or @{command (HOL)
  1356   "ax_specification"} is to some extent a matter of style.  @{command
  1357   (HOL) "specification"} introduces no new axioms, and so by
  1358   construction cannot introduce inconsistencies, whereas @{command
  1359   (HOL) "ax_specification"} does introduce axioms, but only after the
  1360   user has explicitly proven it to be safe.  A practical issue must be
  1361   considered, though: After introducing two constants with the same
  1362   properties using @{command (HOL) "specification"}, one can prove
  1363   that the two constants are, in fact, equal.  If this might be a
  1364   problem, one should use @{command (HOL) "ax_specification"}.
  1365 *}
  1366 
  1367 end