doc-src/IsarRef/Thy/HOL_Specific.thy
author krauss
Tue Oct 26 12:19:01 2010 +0200 (2010-10-26)
changeset 40170 751121d5ca35
parent 39608 76bc7e4999f8
child 40171 1fa547166a1d
permissions -rw-r--r--
remove outdated "(otherwise)" syntax from manual
     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 
   552 subsection {* Old-style recursive function definitions (TFL) *}
   553 
   554 text {*
   555   The old TFL commands @{command (HOL) "recdef"} and @{command (HOL)
   556   "recdef_tc"} for defining recursive are mostly obsolete; @{command
   557   (HOL) "function"} or @{command (HOL) "fun"} should be used instead.
   558 
   559   \begin{matharray}{rcl}
   560     @{command_def (HOL) "recdef"} & : & @{text "theory \<rightarrow> theory)"} \\
   561     @{command_def (HOL) "recdef_tc"}@{text "\<^sup>*"} & : & @{text "theory \<rightarrow> proof(prove)"} \\
   562   \end{matharray}
   563 
   564   \begin{rail}
   565     'recdef' ('(' 'permissive' ')')? \\ name term (prop +) hints?
   566     ;
   567     recdeftc thmdecl? tc
   568     ;
   569     hints: '(' 'hints' ( recdefmod * ) ')'
   570     ;
   571     recdefmod: (('recdef\_simp' | 'recdef\_cong' | 'recdef\_wf') (() | 'add' | 'del') ':' thmrefs) | clasimpmod
   572     ;
   573     tc: nameref ('(' nat ')')?
   574     ;
   575   \end{rail}
   576 
   577   \begin{description}
   578   
   579   \item @{command (HOL) "recdef"} defines general well-founded
   580   recursive functions (using the TFL package), see also
   581   \cite{isabelle-HOL}.  The ``@{text "(permissive)"}'' option tells
   582   TFL to recover from failed proof attempts, returning unfinished
   583   results.  The @{text recdef_simp}, @{text recdef_cong}, and @{text
   584   recdef_wf} hints refer to auxiliary rules to be used in the internal
   585   automated proof process of TFL.  Additional @{syntax clasimpmod}
   586   declarations (cf.\ \secref{sec:clasimp}) may be given to tune the
   587   context of the Simplifier (cf.\ \secref{sec:simplifier}) and
   588   Classical reasoner (cf.\ \secref{sec:classical}).
   589   
   590   \item @{command (HOL) "recdef_tc"}~@{text "c (i)"} recommences the
   591   proof for leftover termination condition number @{text i} (default
   592   1) as generated by a @{command (HOL) "recdef"} definition of
   593   constant @{text c}.
   594   
   595   Note that in most cases, @{command (HOL) "recdef"} is able to finish
   596   its internal proofs without manual intervention.
   597 
   598   \end{description}
   599 
   600   \medskip Hints for @{command (HOL) "recdef"} may be also declared
   601   globally, using the following attributes.
   602 
   603   \begin{matharray}{rcl}
   604     @{attribute_def (HOL) recdef_simp} & : & @{text attribute} \\
   605     @{attribute_def (HOL) recdef_cong} & : & @{text attribute} \\
   606     @{attribute_def (HOL) recdef_wf} & : & @{text attribute} \\
   607   \end{matharray}
   608 
   609   \begin{rail}
   610     ('recdef\_simp' | 'recdef\_cong' | 'recdef\_wf') (() | 'add' | 'del')
   611     ;
   612   \end{rail}
   613 *}
   614 
   615 
   616 section {* Inductive and coinductive definitions \label{sec:hol-inductive} *}
   617 
   618 text {*
   619   An \textbf{inductive definition} specifies the least predicate (or
   620   set) @{text R} closed under given rules: applying a rule to elements
   621   of @{text R} yields a result within @{text R}.  For example, a
   622   structural operational semantics is an inductive definition of an
   623   evaluation relation.
   624 
   625   Dually, a \textbf{coinductive definition} specifies the greatest
   626   predicate~/ set @{text R} that is consistent with given rules: every
   627   element of @{text R} can be seen as arising by applying a rule to
   628   elements of @{text R}.  An important example is using bisimulation
   629   relations to formalise equivalence of processes and infinite data
   630   structures.
   631 
   632   \medskip The HOL package is related to the ZF one, which is
   633   described in a separate paper,\footnote{It appeared in CADE
   634   \cite{paulson-CADE}; a longer version is distributed with Isabelle.}
   635   which you should refer to in case of difficulties.  The package is
   636   simpler than that of ZF thanks to implicit type-checking in HOL.
   637   The types of the (co)inductive predicates (or sets) determine the
   638   domain of the fixedpoint definition, and the package does not have
   639   to use inference rules for type-checking.
   640 
   641   \begin{matharray}{rcl}
   642     @{command_def (HOL) "inductive"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
   643     @{command_def (HOL) "inductive_set"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
   644     @{command_def (HOL) "coinductive"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
   645     @{command_def (HOL) "coinductive_set"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
   646     @{attribute_def (HOL) mono} & : & @{text attribute} \\
   647   \end{matharray}
   648 
   649   \begin{rail}
   650     ('inductive' | 'inductive\_set' | 'coinductive' | 'coinductive\_set') target? fixes ('for' fixes)? \\
   651     ('where' clauses)? ('monos' thmrefs)?
   652     ;
   653     clauses: (thmdecl? prop + '|')
   654     ;
   655     'mono' (() | 'add' | 'del')
   656     ;
   657   \end{rail}
   658 
   659   \begin{description}
   660 
   661   \item @{command (HOL) "inductive"} and @{command (HOL)
   662   "coinductive"} define (co)inductive predicates from the
   663   introduction rules given in the @{keyword "where"} part.  The
   664   optional @{keyword "for"} part contains a list of parameters of the
   665   (co)inductive predicates that remain fixed throughout the
   666   definition.  The optional @{keyword "monos"} section contains
   667   \emph{monotonicity theorems}, which are required for each operator
   668   applied to a recursive set in the introduction rules.  There
   669   \emph{must} be a theorem of the form @{text "A \<le> B \<Longrightarrow> M A \<le> M B"},
   670   for each premise @{text "M R\<^sub>i t"} in an introduction rule!
   671 
   672   \item @{command (HOL) "inductive_set"} and @{command (HOL)
   673   "coinductive_set"} are wrappers for to the previous commands,
   674   allowing the definition of (co)inductive sets.
   675 
   676   \item @{attribute (HOL) mono} declares monotonicity rules.  These
   677   rule are involved in the automated monotonicity proof of @{command
   678   (HOL) "inductive"}.
   679 
   680   \end{description}
   681 *}
   682 
   683 
   684 subsection {* Derived rules *}
   685 
   686 text {*
   687   Each (co)inductive definition @{text R} adds definitions to the
   688   theory and also proves some theorems:
   689 
   690   \begin{description}
   691 
   692   \item @{text R.intros} is the list of introduction rules as proven
   693   theorems, for the recursive predicates (or sets).  The rules are
   694   also available individually, using the names given them in the
   695   theory file;
   696 
   697   \item @{text R.cases} is the case analysis (or elimination) rule;
   698 
   699   \item @{text R.induct} or @{text R.coinduct} is the (co)induction
   700   rule.
   701 
   702   \end{description}
   703 
   704   When several predicates @{text "R\<^sub>1, \<dots>, R\<^sub>n"} are
   705   defined simultaneously, the list of introduction rules is called
   706   @{text "R\<^sub>1_\<dots>_R\<^sub>n.intros"}, the case analysis rules are
   707   called @{text "R\<^sub>1.cases, \<dots>, R\<^sub>n.cases"}, and the list
   708   of mutual induction rules is called @{text
   709   "R\<^sub>1_\<dots>_R\<^sub>n.inducts"}.
   710 *}
   711 
   712 
   713 subsection {* Monotonicity theorems *}
   714 
   715 text {*
   716   Each theory contains a default set of theorems that are used in
   717   monotonicity proofs.  New rules can be added to this set via the
   718   @{attribute (HOL) mono} attribute.  The HOL theory @{text Inductive}
   719   shows how this is done.  In general, the following monotonicity
   720   theorems may be added:
   721 
   722   \begin{itemize}
   723 
   724   \item Theorems of the form @{text "A \<le> B \<Longrightarrow> M A \<le> M B"}, for proving
   725   monotonicity of inductive definitions whose introduction rules have
   726   premises involving terms such as @{text "M R\<^sub>i t"}.
   727 
   728   \item Monotonicity theorems for logical operators, which are of the
   729   general form @{text "(\<dots> \<longrightarrow> \<dots>) \<Longrightarrow> \<dots> (\<dots> \<longrightarrow> \<dots>) \<Longrightarrow> \<dots> \<longrightarrow> \<dots>"}.  For example, in
   730   the case of the operator @{text "\<or>"}, the corresponding theorem is
   731   \[
   732   \infer{@{text "P\<^sub>1 \<or> P\<^sub>2 \<longrightarrow> Q\<^sub>1 \<or> Q\<^sub>2"}}{@{text "P\<^sub>1 \<longrightarrow> Q\<^sub>1"} & @{text "P\<^sub>2 \<longrightarrow> Q\<^sub>2"}}
   733   \]
   734 
   735   \item De Morgan style equations for reasoning about the ``polarity''
   736   of expressions, e.g.
   737   \[
   738   @{prop "\<not> \<not> P \<longleftrightarrow> P"} \qquad\qquad
   739   @{prop "\<not> (P \<and> Q) \<longleftrightarrow> \<not> P \<or> \<not> Q"}
   740   \]
   741 
   742   \item Equations for reducing complex operators to more primitive
   743   ones whose monotonicity can easily be proved, e.g.
   744   \[
   745   @{prop "(P \<longrightarrow> Q) \<longleftrightarrow> \<not> P \<or> Q"} \qquad\qquad
   746   @{prop "Ball A P \<equiv> \<forall>x. x \<in> A \<longrightarrow> P x"}
   747   \]
   748 
   749   \end{itemize}
   750 
   751   %FIXME: Example of an inductive definition
   752 *}
   753 
   754 
   755 section {* Arithmetic proof support *}
   756 
   757 text {*
   758   \begin{matharray}{rcl}
   759     @{method_def (HOL) arith} & : & @{text method} \\
   760     @{attribute_def (HOL) arith} & : & @{text attribute} \\
   761     @{attribute_def (HOL) arith_split} & : & @{text attribute} \\
   762   \end{matharray}
   763 
   764   The @{method (HOL) arith} method decides linear arithmetic problems
   765   (on types @{text nat}, @{text int}, @{text real}).  Any current
   766   facts are inserted into the goal before running the procedure.
   767 
   768   The @{attribute (HOL) arith} attribute declares facts that are
   769   always supplied to the arithmetic provers implicitly.
   770 
   771   The @{attribute (HOL) arith_split} attribute declares case split
   772   rules to be expanded before @{method (HOL) arith} is invoked.
   773 
   774   Note that a simpler (but faster) arithmetic prover is
   775   already invoked by the Simplifier.
   776 *}
   777 
   778 
   779 section {* Intuitionistic proof search *}
   780 
   781 text {*
   782   \begin{matharray}{rcl}
   783     @{method_def (HOL) iprover} & : & @{text method} \\
   784   \end{matharray}
   785 
   786   \begin{rail}
   787     'iprover' ( rulemod * )
   788     ;
   789   \end{rail}
   790 
   791   The @{method (HOL) iprover} method performs intuitionistic proof
   792   search, depending on specifically declared rules from the context,
   793   or given as explicit arguments.  Chained facts are inserted into the
   794   goal before commencing proof search.
   795 
   796   Rules need to be classified as @{attribute (Pure) intro},
   797   @{attribute (Pure) elim}, or @{attribute (Pure) dest}; here the
   798   ``@{text "!"}'' indicator refers to ``safe'' rules, which may be
   799   applied aggressively (without considering back-tracking later).
   800   Rules declared with ``@{text "?"}'' are ignored in proof search (the
   801   single-step @{method rule} method still observes these).  An
   802   explicit weight annotation may be given as well; otherwise the
   803   number of rule premises will be taken into account here.
   804 *}
   805 
   806 
   807 section {* Coherent Logic *}
   808 
   809 text {*
   810   \begin{matharray}{rcl}
   811     @{method_def (HOL) "coherent"} & : & @{text method} \\
   812   \end{matharray}
   813 
   814   \begin{rail}
   815     'coherent' thmrefs?
   816     ;
   817   \end{rail}
   818 
   819   The @{method (HOL) coherent} method solves problems of
   820   \emph{Coherent Logic} \cite{Bezem-Coquand:2005}, which covers
   821   applications in confluence theory, lattice theory and projective
   822   geometry.  See @{"file" "~~/src/HOL/ex/Coherent.thy"} for some
   823   examples.
   824 *}
   825 
   826 
   827 section {* Checking and refuting propositions *}
   828 
   829 text {*
   830   Identifying incorrect propositions usually involves evaluation of
   831   particular assignments and systematic counter example search.  This
   832   is supported by the following commands.
   833 
   834   \begin{matharray}{rcl}
   835     @{command_def (HOL) "value"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
   836     @{command_def (HOL) "quickcheck"}@{text "\<^sup>*"} & : & @{text "proof \<rightarrow>"} \\
   837     @{command_def (HOL) "quickcheck_params"} & : & @{text "theory \<rightarrow> theory"}
   838   \end{matharray}
   839 
   840   \begin{rail}
   841     'value' ( ( '[' name ']' ) ? ) modes? term
   842     ;
   843 
   844     'quickcheck' ( ( '[' args ']' ) ? ) nat?
   845     ;
   846 
   847     'quickcheck_params' ( ( '[' args ']' ) ? )
   848     ;
   849 
   850     modes: '(' (name + ) ')'
   851     ;
   852 
   853     args: ( name '=' value + ',' )
   854     ;
   855   \end{rail}
   856 
   857   \begin{description}
   858 
   859   \item @{command (HOL) "value"}~@{text t} evaluates and prints a
   860     term; optionally @{text modes} can be specified, which are
   861     appended to the current print mode (see also \cite{isabelle-ref}).
   862     Internally, the evaluation is performed by registered evaluators,
   863     which are invoked sequentially until a result is returned.
   864     Alternatively a specific evaluator can be selected using square
   865     brackets; typical evaluators use the current set of code equations
   866     to normalize and include @{text simp} for fully symbolic evaluation
   867     using the simplifier, @{text nbe} for \emph{normalization by evaluation}
   868     and \emph{code} for code generation in SML.
   869 
   870   \item @{command (HOL) "quickcheck"} tests the current goal for
   871     counter examples using a series of arbitrary assignments for its
   872     free variables; by default the first subgoal is tested, an other
   873     can be selected explicitly using an optional goal index.
   874     A number of configuration options are supported for
   875     @{command (HOL) "quickcheck"}, notably:
   876 
   877     \begin{description}
   878 
   879       \item[size] specifies the maximum size of the search space for
   880         assignment values.
   881 
   882       \item[iterations] sets how many sets of assignments are
   883         generated for each particular size.
   884 
   885       \item[no\_assms] specifies whether assumptions in
   886         structured proofs should be ignored.
   887 
   888     \end{description}
   889 
   890     These option can be given within square brackets.
   891 
   892   \item @{command (HOL) "quickcheck_params"} changes quickcheck
   893     configuration options persitently.
   894 
   895   \end{description}
   896 *}
   897 
   898 
   899 section {* Unstructured case analysis and induction \label{sec:hol-induct-tac} *}
   900 
   901 text {*
   902   The following tools of Isabelle/HOL support cases analysis and
   903   induction in unstructured tactic scripts; see also
   904   \secref{sec:cases-induct} for proper Isar versions of similar ideas.
   905 
   906   \begin{matharray}{rcl}
   907     @{method_def (HOL) case_tac}@{text "\<^sup>*"} & : & @{text method} \\
   908     @{method_def (HOL) induct_tac}@{text "\<^sup>*"} & : & @{text method} \\
   909     @{method_def (HOL) ind_cases}@{text "\<^sup>*"} & : & @{text method} \\
   910     @{command_def (HOL) "inductive_cases"}@{text "\<^sup>*"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
   911   \end{matharray}
   912 
   913   \begin{rail}
   914     'case\_tac' goalspec? term rule?
   915     ;
   916     'induct\_tac' goalspec? (insts * 'and') rule?
   917     ;
   918     'ind\_cases' (prop +) ('for' (name +)) ?
   919     ;
   920     'inductive\_cases' (thmdecl? (prop +) + 'and')
   921     ;
   922 
   923     rule: ('rule' ':' thmref)
   924     ;
   925   \end{rail}
   926 
   927   \begin{description}
   928 
   929   \item @{method (HOL) case_tac} and @{method (HOL) induct_tac} admit
   930   to reason about inductive types.  Rules are selected according to
   931   the declarations by the @{attribute cases} and @{attribute induct}
   932   attributes, cf.\ \secref{sec:cases-induct}.  The @{command (HOL)
   933   datatype} package already takes care of this.
   934 
   935   These unstructured tactics feature both goal addressing and dynamic
   936   instantiation.  Note that named rule cases are \emph{not} provided
   937   as would be by the proper @{method cases} and @{method induct} proof
   938   methods (see \secref{sec:cases-induct}).  Unlike the @{method
   939   induct} method, @{method induct_tac} does not handle structured rule
   940   statements, only the compact object-logic conclusion of the subgoal
   941   being addressed.
   942   
   943   \item @{method (HOL) ind_cases} and @{command (HOL)
   944   "inductive_cases"} provide an interface to the internal @{ML_text
   945   mk_cases} operation.  Rules are simplified in an unrestricted
   946   forward manner.
   947 
   948   While @{method (HOL) ind_cases} is a proof method to apply the
   949   result immediately as elimination rules, @{command (HOL)
   950   "inductive_cases"} provides case split theorems at the theory level
   951   for later use.  The @{keyword "for"} argument of the @{method (HOL)
   952   ind_cases} method allows to specify a list of variables that should
   953   be generalized before applying the resulting rule.
   954 
   955   \end{description}
   956 *}
   957 
   958 
   959 section {* Executable code *}
   960 
   961 text {*
   962   Isabelle/Pure provides two generic frameworks to support code
   963   generation from executable specifications.  Isabelle/HOL
   964   instantiates these mechanisms in a way that is amenable to end-user
   965   applications.
   966 
   967   \medskip One framework generates code from functional programs
   968   (including overloading using type classes) to SML \cite{SML}, OCaml
   969   \cite{OCaml}, Haskell \cite{haskell-revised-report} and Scala
   970   \cite{scala-overview-tech-report}.
   971   Conceptually, code generation is split up in three steps:
   972   \emph{selection} of code theorems, \emph{translation} into an
   973   abstract executable view and \emph{serialization} to a specific
   974   \emph{target language}.  Inductive specifications can be executed
   975   using the predicate compiler which operates within HOL.
   976   See \cite{isabelle-codegen} for an introduction.
   977 
   978   \begin{matharray}{rcl}
   979     @{command_def (HOL) "export_code"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
   980     @{attribute_def (HOL) code} & : & @{text attribute} \\
   981     @{command_def (HOL) "code_abort"} & : & @{text "theory \<rightarrow> theory"} \\
   982     @{command_def (HOL) "code_datatype"} & : & @{text "theory \<rightarrow> theory"} \\
   983     @{command_def (HOL) "print_codesetup"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
   984     @{attribute_def (HOL) code_inline} & : & @{text attribute} \\
   985     @{attribute_def (HOL) code_post} & : & @{text attribute} \\
   986     @{command_def (HOL) "print_codeproc"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
   987     @{command_def (HOL) "code_thms"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
   988     @{command_def (HOL) "code_deps"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
   989     @{command_def (HOL) "code_const"} & : & @{text "theory \<rightarrow> theory"} \\
   990     @{command_def (HOL) "code_type"} & : & @{text "theory \<rightarrow> theory"} \\
   991     @{command_def (HOL) "code_class"} & : & @{text "theory \<rightarrow> theory"} \\
   992     @{command_def (HOL) "code_instance"} & : & @{text "theory \<rightarrow> theory"} \\
   993     @{command_def (HOL) "code_reserved"} & : & @{text "theory \<rightarrow> theory"} \\
   994     @{command_def (HOL) "code_monad"} & : & @{text "theory \<rightarrow> theory"} \\
   995     @{command_def (HOL) "code_include"} & : & @{text "theory \<rightarrow> theory"} \\
   996     @{command_def (HOL) "code_modulename"} & : & @{text "theory \<rightarrow> theory"} \\
   997     @{command_def (HOL) "code_reflect"} & : & @{text "theory \<rightarrow> theory"}
   998   \end{matharray}
   999 
  1000   \begin{rail}
  1001      'export\_code' ( constexpr + ) \\
  1002        ( ( 'in' target ( 'module\_name' string ) ? \\
  1003         ( 'file' ( string | '-' ) ) ? ( '(' args ')' ) ?) + ) ?
  1004     ;
  1005 
  1006     const: term
  1007     ;
  1008 
  1009     constexpr: ( const | 'name.*' | '*' )
  1010     ;
  1011 
  1012     typeconstructor: nameref
  1013     ;
  1014 
  1015     class: nameref
  1016     ;
  1017 
  1018     target: 'SML' | 'OCaml' | 'Haskell' | 'Scala'
  1019     ;
  1020 
  1021     'code' ( 'del' | 'abstype' | 'abstract' ) ?
  1022     ;
  1023 
  1024     'code\_abort' ( const + )
  1025     ;
  1026 
  1027     'code\_datatype' ( const + )
  1028     ;
  1029 
  1030     'code_inline' ( 'del' ) ?
  1031     ;
  1032 
  1033     'code_post' ( 'del' ) ?
  1034     ;
  1035 
  1036     'code\_thms' ( constexpr + ) ?
  1037     ;
  1038 
  1039     'code\_deps' ( constexpr + ) ?
  1040     ;
  1041 
  1042     'code\_const' (const + 'and') \\
  1043       ( ( '(' target ( syntax ? + 'and' ) ')' ) + )
  1044     ;
  1045 
  1046     'code\_type' (typeconstructor + 'and') \\
  1047       ( ( '(' target ( syntax ? + 'and' ) ')' ) + )
  1048     ;
  1049 
  1050     'code\_class' (class + 'and') \\
  1051       ( ( '(' target \\ ( string ? + 'and' ) ')' ) + )
  1052     ;
  1053 
  1054     'code\_instance' (( typeconstructor '::' class ) + 'and') \\
  1055       ( ( '(' target ( '-' ? + 'and' ) ')' ) + )
  1056     ;
  1057 
  1058     'code\_reserved' target ( string + )
  1059     ;
  1060 
  1061     'code\_monad' const const target
  1062     ;
  1063 
  1064     'code\_include' target ( string ( string | '-') )
  1065     ;
  1066 
  1067     'code\_modulename' target ( ( string string ) + )
  1068     ;
  1069 
  1070     'code\_reflect' string ( 'datatypes' ( string '=' ( string + '|' ) + 'and' ) ) ? \\
  1071       ( 'functions' ( string + ) ) ? ( 'file' string ) ?
  1072     ;
  1073 
  1074     syntax: string | ( 'infix' | 'infixl' | 'infixr' ) nat string
  1075     ;
  1076 
  1077   \end{rail}
  1078 
  1079   \begin{description}
  1080 
  1081   \item @{command (HOL) "export_code"} generates code for a given list
  1082   of constants in the specified target language(s).  If no
  1083   serialization instruction is given, only abstract code is generated
  1084   internally.
  1085 
  1086   Constants may be specified by giving them literally, referring to
  1087   all executable contants within a certain theory by giving @{text
  1088   "name.*"}, or referring to \emph{all} executable constants currently
  1089   available by giving @{text "*"}.
  1090 
  1091   By default, for each involved theory one corresponding name space
  1092   module is generated.  Alternativly, a module name may be specified
  1093   after the @{keyword "module_name"} keyword; then \emph{all} code is
  1094   placed in this module.
  1095 
  1096   For \emph{SML}, \emph{OCaml} and \emph{Scala} the file specification
  1097   refers to a single file; for \emph{Haskell}, it refers to a whole
  1098   directory, where code is generated in multiple files reflecting the
  1099   module hierarchy.  Omitting the file specification denotes standard
  1100   output.
  1101 
  1102   Serializers take an optional list of arguments in parentheses.  For
  1103   \emph{SML} and \emph{OCaml}, ``@{text no_signatures}`` omits
  1104   explicit module signatures.
  1105   
  1106   For \emph{Haskell} a module name prefix may be given using the
  1107   ``@{text "root:"}'' argument; ``@{text string_classes}'' adds a
  1108   ``@{verbatim "deriving (Read, Show)"}'' clause to each appropriate
  1109   datatype declaration.
  1110 
  1111   \item @{attribute (HOL) code} explicitly selects (or with option
  1112   ``@{text "del"}'' deselects) a code equation for code generation.
  1113   Usually packages introducing code equations provide a reasonable
  1114   default setup for selection.  Variants @{text "code abstype"} and
  1115   @{text "code abstract"} declare abstract datatype certificates or
  1116   code equations on abstract datatype representations respectively.
  1117 
  1118   \item @{command (HOL) "code_abort"} declares constants which are not
  1119   required to have a definition by means of code equations; if needed
  1120   these are implemented by program abort instead.
  1121 
  1122   \item @{command (HOL) "code_datatype"} specifies a constructor set
  1123   for a logical type.
  1124 
  1125   \item @{command (HOL) "print_codesetup"} gives an overview on
  1126   selected code equations and code generator datatypes.
  1127 
  1128   \item @{attribute (HOL) code_inline} declares (or with option
  1129   ``@{text "del"}'' removes) inlining theorems which are applied as
  1130   rewrite rules to any code equation during preprocessing.
  1131 
  1132   \item @{attribute (HOL) code_post} declares (or with option ``@{text
  1133   "del"}'' removes) theorems which are applied as rewrite rules to any
  1134   result of an evaluation.
  1135 
  1136   \item @{command (HOL) "print_codeproc"} prints the setup of the code
  1137   generator preprocessor.
  1138 
  1139   \item @{command (HOL) "code_thms"} prints a list of theorems
  1140   representing the corresponding program containing all given
  1141   constants after preprocessing.
  1142 
  1143   \item @{command (HOL) "code_deps"} visualizes dependencies of
  1144   theorems representing the corresponding program containing all given
  1145   constants after preprocessing.
  1146 
  1147   \item @{command (HOL) "code_const"} associates a list of constants
  1148   with target-specific serializations; omitting a serialization
  1149   deletes an existing serialization.
  1150 
  1151   \item @{command (HOL) "code_type"} associates a list of type
  1152   constructors with target-specific serializations; omitting a
  1153   serialization deletes an existing serialization.
  1154 
  1155   \item @{command (HOL) "code_class"} associates a list of classes
  1156   with target-specific class names; omitting a serialization deletes
  1157   an existing serialization.  This applies only to \emph{Haskell}.
  1158 
  1159   \item @{command (HOL) "code_instance"} declares a list of type
  1160   constructor / class instance relations as ``already present'' for a
  1161   given target.  Omitting a ``@{text "-"}'' deletes an existing
  1162   ``already present'' declaration.  This applies only to
  1163   \emph{Haskell}.
  1164 
  1165   \item @{command (HOL) "code_reserved"} declares a list of names as
  1166   reserved for a given target, preventing it to be shadowed by any
  1167   generated code.
  1168 
  1169   \item @{command (HOL) "code_monad"} provides an auxiliary mechanism
  1170   to generate monadic code for Haskell.
  1171 
  1172   \item @{command (HOL) "code_include"} adds arbitrary named content
  1173   (``include'') to generated code.  A ``@{text "-"}'' as last argument
  1174   will remove an already added ``include''.
  1175 
  1176   \item @{command (HOL) "code_modulename"} declares aliasings from one
  1177   module name onto another.
  1178 
  1179   \item @{command (HOL) "code_reflect"} without a ``@{text "file"}''
  1180   argument compiles code into the system runtime environment and
  1181   modifies the code generator setup that future invocations of system
  1182   runtime code generation referring to one of the ``@{text
  1183   "datatypes"}'' or ``@{text "functions"}'' entities use these precompiled
  1184   entities.  With a ``@{text "file"}'' argument, the corresponding code
  1185   is generated into that specified file without modifying the code
  1186   generator setup.
  1187 
  1188   \end{description}
  1189 
  1190   The other framework generates code from both functional and
  1191   relational programs to SML.  See \cite{isabelle-HOL} for further
  1192   information (this actually covers the new-style theory format as
  1193   well).
  1194 
  1195   \begin{matharray}{rcl}
  1196     @{command_def (HOL) "code_module"} & : & @{text "theory \<rightarrow> theory"} \\
  1197     @{command_def (HOL) "code_library"} & : & @{text "theory \<rightarrow> theory"} \\
  1198     @{command_def (HOL) "consts_code"} & : & @{text "theory \<rightarrow> theory"} \\
  1199     @{command_def (HOL) "types_code"} & : & @{text "theory \<rightarrow> theory"} \\  
  1200     @{attribute_def (HOL) code} & : & @{text attribute} \\
  1201   \end{matharray}
  1202 
  1203   \begin{rail}
  1204   ( 'code\_module' | 'code\_library' ) modespec ? name ? \\
  1205     ( 'file' name ) ? ( 'imports' ( name + ) ) ? \\
  1206     'contains' ( ( name '=' term ) + | term + )
  1207   ;
  1208 
  1209   modespec: '(' ( name * ) ')'
  1210   ;
  1211 
  1212   'consts\_code' (codespec +)
  1213   ;
  1214 
  1215   codespec: const template attachment ?
  1216   ;
  1217 
  1218   'types\_code' (tycodespec +)
  1219   ;
  1220 
  1221   tycodespec: name template attachment ?
  1222   ;
  1223 
  1224   const: term
  1225   ;
  1226 
  1227   template: '(' string ')'
  1228   ;
  1229 
  1230   attachment: 'attach' modespec ? verblbrace text verbrbrace
  1231   ;
  1232 
  1233   'code' (name)?
  1234   ;
  1235   \end{rail}
  1236 
  1237 *}
  1238 
  1239 
  1240 section {* Definition by specification \label{sec:hol-specification} *}
  1241 
  1242 text {*
  1243   \begin{matharray}{rcl}
  1244     @{command_def (HOL) "specification"} & : & @{text "theory \<rightarrow> proof(prove)"} \\
  1245     @{command_def (HOL) "ax_specification"} & : & @{text "theory \<rightarrow> proof(prove)"} \\
  1246   \end{matharray}
  1247 
  1248   \begin{rail}
  1249   ('specification' | 'ax\_specification') '(' (decl +) ')' \\ (thmdecl? prop +)
  1250   ;
  1251   decl: ((name ':')? term '(' 'overloaded' ')'?)
  1252   \end{rail}
  1253 
  1254   \begin{description}
  1255 
  1256   \item @{command (HOL) "specification"}~@{text "decls \<phi>"} sets up a
  1257   goal stating the existence of terms with the properties specified to
  1258   hold for the constants given in @{text decls}.  After finishing the
  1259   proof, the theory will be augmented with definitions for the given
  1260   constants, as well as with theorems stating the properties for these
  1261   constants.
  1262 
  1263   \item @{command (HOL) "ax_specification"}~@{text "decls \<phi>"} sets up
  1264   a goal stating the existence of terms with the properties specified
  1265   to hold for the constants given in @{text decls}.  After finishing
  1266   the proof, the theory will be augmented with axioms expressing the
  1267   properties given in the first place.
  1268 
  1269   \item @{text decl} declares a constant to be defined by the
  1270   specification given.  The definition for the constant @{text c} is
  1271   bound to the name @{text c_def} unless a theorem name is given in
  1272   the declaration.  Overloaded constants should be declared as such.
  1273 
  1274   \end{description}
  1275 
  1276   Whether to use @{command (HOL) "specification"} or @{command (HOL)
  1277   "ax_specification"} is to some extent a matter of style.  @{command
  1278   (HOL) "specification"} introduces no new axioms, and so by
  1279   construction cannot introduce inconsistencies, whereas @{command
  1280   (HOL) "ax_specification"} does introduce axioms, but only after the
  1281   user has explicitly proven it to be safe.  A practical issue must be
  1282   considered, though: After introducing two constants with the same
  1283   properties using @{command (HOL) "specification"}, one can prove
  1284   that the two constants are, in fact, equal.  If this might be a
  1285   problem, one should use @{command (HOL) "ax_specification"}.
  1286 *}
  1287 
  1288 end