doc-src/IsarRef/Thy/document/HOL_Specific.tex
author wenzelm
Mon Jun 02 23:12:23 2008 +0200 (2008-06-02)
changeset 27047 2dcdea037385
parent 27042 8fcf19f2168b
child 27103 d8549f4d900b
permissions -rw-r--r--
updated generated file;
     1 %
     2 \begin{isabellebody}%
     3 \def\isabellecontext{HOL{\isacharunderscore}Specific}%
     4 %
     5 \isadelimtheory
     6 \isanewline
     7 \isanewline
     8 %
     9 \endisadelimtheory
    10 %
    11 \isatagtheory
    12 \isacommand{theory}\isamarkupfalse%
    13 \ HOL{\isacharunderscore}Specific\isanewline
    14 \isakeyword{imports}\ Main\isanewline
    15 \isakeyword{begin}%
    16 \endisatagtheory
    17 {\isafoldtheory}%
    18 %
    19 \isadelimtheory
    20 %
    21 \endisadelimtheory
    22 %
    23 \isamarkupchapter{Isabelle/HOL \label{ch:hol}%
    24 }
    25 \isamarkuptrue%
    26 %
    27 \isamarkupsection{Primitive types \label{sec:hol-typedef}%
    28 }
    29 \isamarkuptrue%
    30 %
    31 \begin{isamarkuptext}%
    32 \begin{matharray}{rcl}
    33     \indexdef{HOL}{command}{typedecl}\hypertarget{command.HOL.typedecl}{\hyperlink{command.HOL.typedecl}{\mbox{\isa{\isacommand{typedecl}}}}} & : & \isartrans{theory}{theory} \\
    34     \indexdef{HOL}{command}{typedef}\hypertarget{command.HOL.typedef}{\hyperlink{command.HOL.typedef}{\mbox{\isa{\isacommand{typedef}}}}} & : & \isartrans{theory}{proof(prove)} \\
    35   \end{matharray}
    36 
    37   \begin{rail}
    38     'typedecl' typespec infix?
    39     ;
    40     'typedef' altname? abstype '=' repset
    41     ;
    42 
    43     altname: '(' (name | 'open' | 'open' name) ')'
    44     ;
    45     abstype: typespec infix?
    46     ;
    47     repset: term ('morphisms' name name)?
    48     ;
    49   \end{rail}
    50 
    51   \begin{descr}
    52   
    53   \item [\hyperlink{command.HOL.typedecl}{\mbox{\isa{\isacommand{typedecl}}}}~\isa{{\isachardoublequote}{\isacharparenleft}{\isasymalpha}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymalpha}\isactrlsub n{\isacharparenright}\ t{\isachardoublequote}}] is similar to the original \hyperlink{command.typedecl}{\mbox{\isa{\isacommand{typedecl}}}} of
    54   Isabelle/Pure (see \secref{sec:types-pure}), but also declares type
    55   arity \isa{{\isachardoublequote}t\ {\isacharcolon}{\isacharcolon}\ {\isacharparenleft}type{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ type{\isacharparenright}\ type{\isachardoublequote}}, making \isa{t} an
    56   actual HOL type constructor.   %FIXME check, update
    57   
    58   \item [\hyperlink{command.HOL.typedef}{\mbox{\isa{\isacommand{typedef}}}}~\isa{{\isachardoublequote}{\isacharparenleft}{\isasymalpha}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymalpha}\isactrlsub n{\isacharparenright}\ t\ {\isacharequal}\ A{\isachardoublequote}}] sets up a goal stating non-emptiness of the set \isa{A}.
    59   After finishing the proof, the theory will be augmented by a
    60   Gordon/HOL-style type definition, which establishes a bijection
    61   between the representing set \isa{A} and the new type \isa{t}.
    62   
    63   Technically, \hyperlink{command.HOL.typedef}{\mbox{\isa{\isacommand{typedef}}}} defines both a type \isa{t} and a set (term constant) of the same name (an alternative base
    64   name may be given in parentheses).  The injection from type to set
    65   is called \isa{Rep{\isacharunderscore}t}, its inverse \isa{Abs{\isacharunderscore}t} (this may be
    66   changed via an explicit \hyperlink{keyword.HOL.morphisms}{\mbox{\isa{\isakeyword{morphisms}}}} declaration).
    67   
    68   Theorems \isa{Rep{\isacharunderscore}t}, \isa{Rep{\isacharunderscore}t{\isacharunderscore}inverse}, and \isa{Abs{\isacharunderscore}t{\isacharunderscore}inverse} provide the most basic characterization as a
    69   corresponding injection/surjection pair (in both directions).  Rules
    70   \isa{Rep{\isacharunderscore}t{\isacharunderscore}inject} and \isa{Abs{\isacharunderscore}t{\isacharunderscore}inject} provide a slightly
    71   more convenient view on the injectivity part, suitable for automated
    72   proof tools (e.g.\ in \hyperlink{attribute.simp}{\mbox{\isa{simp}}} or \hyperlink{attribute.iff}{\mbox{\isa{iff}}}
    73   declarations).  Rules \isa{Rep{\isacharunderscore}t{\isacharunderscore}cases}/\isa{Rep{\isacharunderscore}t{\isacharunderscore}induct}, and
    74   \isa{Abs{\isacharunderscore}t{\isacharunderscore}cases}/\isa{Abs{\isacharunderscore}t{\isacharunderscore}induct} provide alternative views
    75   on surjectivity; these are already declared as set or type rules for
    76   the generic \hyperlink{method.cases}{\mbox{\isa{cases}}} and \hyperlink{method.induct}{\mbox{\isa{induct}}} methods.
    77   
    78   An alternative name may be specified in parentheses; the default is
    79   to use \isa{t} as indicated before.  The ``\isa{{\isachardoublequote}{\isacharparenleft}open{\isacharparenright}{\isachardoublequote}}''
    80   declaration suppresses a separate constant definition for the
    81   representing set.
    82 
    83   \end{descr}
    84 
    85   Note that raw type declarations are rarely used in practice; the
    86   main application is with experimental (or even axiomatic!) theory
    87   fragments.  Instead of primitive HOL type definitions, user-level
    88   theories usually refer to higher-level packages such as \hyperlink{command.HOL.record}{\mbox{\isa{\isacommand{record}}}} (see \secref{sec:hol-record}) or \hyperlink{command.HOL.datatype}{\mbox{\isa{\isacommand{datatype}}}} (see \secref{sec:hol-datatype}).%
    89 \end{isamarkuptext}%
    90 \isamarkuptrue%
    91 %
    92 \isamarkupsection{Adhoc tuples%
    93 }
    94 \isamarkuptrue%
    95 %
    96 \begin{isamarkuptext}%
    97 \begin{matharray}{rcl}
    98     \hyperlink{attribute.HOL.split-format}{\mbox{\isa{split{\isacharunderscore}format}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isaratt \\
    99   \end{matharray}
   100 
   101   \begin{rail}
   102     'split\_format' (((name *) + 'and') | ('(' 'complete' ')'))
   103     ;
   104   \end{rail}
   105 
   106   \begin{descr}
   107   
   108   \item [\hyperlink{attribute.HOL.split-format}{\mbox{\isa{split{\isacharunderscore}format}}}~\isa{{\isachardoublequote}p\isactrlsub {\isadigit{1}}\ {\isasymdots}\ p\isactrlsub m\ {\isasymAND}\ {\isasymdots}\ {\isasymAND}\ q\isactrlsub {\isadigit{1}}\ {\isasymdots}\ q\isactrlsub n{\isachardoublequote}}] puts expressions of
   109   low-level tuple types into canonical form as specified by the
   110   arguments given; the \isa{i}-th collection of arguments refers to
   111   occurrences in premise \isa{i} of the rule.  The ``\isa{{\isachardoublequote}{\isacharparenleft}complete{\isacharparenright}{\isachardoublequote}}'' option causes \emph{all} arguments in function
   112   applications to be represented canonically according to their tuple
   113   type structure.
   114 
   115   Note that these operations tend to invent funny names for new local
   116   parameters to be introduced.
   117 
   118   \end{descr}%
   119 \end{isamarkuptext}%
   120 \isamarkuptrue%
   121 %
   122 \isamarkupsection{Records \label{sec:hol-record}%
   123 }
   124 \isamarkuptrue%
   125 %
   126 \begin{isamarkuptext}%
   127 In principle, records merely generalize the concept of tuples, where
   128   components may be addressed by labels instead of just position.  The
   129   logical infrastructure of records in Isabelle/HOL is slightly more
   130   advanced, though, supporting truly extensible record schemes.  This
   131   admits operations that are polymorphic with respect to record
   132   extension, yielding ``object-oriented'' effects like (single)
   133   inheritance.  See also \cite{NaraschewskiW-TPHOLs98} for more
   134   details on object-oriented verification and record subtyping in HOL.%
   135 \end{isamarkuptext}%
   136 \isamarkuptrue%
   137 %
   138 \isamarkupsubsection{Basic concepts%
   139 }
   140 \isamarkuptrue%
   141 %
   142 \begin{isamarkuptext}%
   143 Isabelle/HOL supports both \emph{fixed} and \emph{schematic} records
   144   at the level of terms and types.  The notation is as follows:
   145 
   146   \begin{center}
   147   \begin{tabular}{l|l|l}
   148     & record terms & record types \\ \hline
   149     fixed & \isa{{\isachardoublequote}{\isasymlparr}x\ {\isacharequal}\ a{\isacharcomma}\ y\ {\isacharequal}\ b{\isasymrparr}{\isachardoublequote}} & \isa{{\isachardoublequote}{\isasymlparr}x\ {\isacharcolon}{\isacharcolon}\ A{\isacharcomma}\ y\ {\isacharcolon}{\isacharcolon}\ B{\isasymrparr}{\isachardoublequote}} \\
   150     schematic & \isa{{\isachardoublequote}{\isasymlparr}x\ {\isacharequal}\ a{\isacharcomma}\ y\ {\isacharequal}\ b{\isacharcomma}\ {\isasymdots}\ {\isacharequal}\ m{\isasymrparr}{\isachardoublequote}} &
   151       \isa{{\isachardoublequote}{\isasymlparr}x\ {\isacharcolon}{\isacharcolon}\ A{\isacharcomma}\ y\ {\isacharcolon}{\isacharcolon}\ B{\isacharcomma}\ {\isasymdots}\ {\isacharcolon}{\isacharcolon}\ M{\isasymrparr}{\isachardoublequote}} \\
   152   \end{tabular}
   153   \end{center}
   154 
   155   \noindent The ASCII representation of \isa{{\isachardoublequote}{\isasymlparr}x\ {\isacharequal}\ a{\isasymrparr}{\isachardoublequote}} is \isa{{\isachardoublequote}{\isacharparenleft}{\isacharbar}\ x\ {\isacharequal}\ a\ {\isacharbar}{\isacharparenright}{\isachardoublequote}}.
   156 
   157   A fixed record \isa{{\isachardoublequote}{\isasymlparr}x\ {\isacharequal}\ a{\isacharcomma}\ y\ {\isacharequal}\ b{\isasymrparr}{\isachardoublequote}} has field \isa{x} of value
   158   \isa{a} and field \isa{y} of value \isa{b}.  The corresponding
   159   type is \isa{{\isachardoublequote}{\isasymlparr}x\ {\isacharcolon}{\isacharcolon}\ A{\isacharcomma}\ y\ {\isacharcolon}{\isacharcolon}\ B{\isasymrparr}{\isachardoublequote}}, assuming that \isa{{\isachardoublequote}a\ {\isacharcolon}{\isacharcolon}\ A{\isachardoublequote}}
   160   and \isa{{\isachardoublequote}b\ {\isacharcolon}{\isacharcolon}\ B{\isachardoublequote}}.
   161 
   162   A record scheme like \isa{{\isachardoublequote}{\isasymlparr}x\ {\isacharequal}\ a{\isacharcomma}\ y\ {\isacharequal}\ b{\isacharcomma}\ {\isasymdots}\ {\isacharequal}\ m{\isasymrparr}{\isachardoublequote}} contains fields
   163   \isa{x} and \isa{y} as before, but also possibly further fields
   164   as indicated by the ``\isa{{\isachardoublequote}{\isasymdots}{\isachardoublequote}}'' notation (which is actually part
   165   of the syntax).  The improper field ``\isa{{\isachardoublequote}{\isasymdots}{\isachardoublequote}}'' of a record
   166   scheme is called the \emph{more part}.  Logically it is just a free
   167   variable, which is occasionally referred to as ``row variable'' in
   168   the literature.  The more part of a record scheme may be
   169   instantiated by zero or more further components.  For example, the
   170   previous scheme may get instantiated to \isa{{\isachardoublequote}{\isasymlparr}x\ {\isacharequal}\ a{\isacharcomma}\ y\ {\isacharequal}\ b{\isacharcomma}\ z\ {\isacharequal}\ c{\isacharcomma}\ {\isasymdots}\ {\isacharequal}\ m{\isacharprime}{\isasymrparr}{\isachardoublequote}}, where \isa{m{\isacharprime}} refers to a different more part.
   171   Fixed records are special instances of record schemes, where
   172   ``\isa{{\isachardoublequote}{\isasymdots}{\isachardoublequote}}'' is properly terminated by the \isa{{\isachardoublequote}{\isacharparenleft}{\isacharparenright}\ {\isacharcolon}{\isacharcolon}\ unit{\isachardoublequote}}
   173   element.  In fact, \isa{{\isachardoublequote}{\isasymlparr}x\ {\isacharequal}\ a{\isacharcomma}\ y\ {\isacharequal}\ b{\isasymrparr}{\isachardoublequote}} is just an abbreviation
   174   for \isa{{\isachardoublequote}{\isasymlparr}x\ {\isacharequal}\ a{\isacharcomma}\ y\ {\isacharequal}\ b{\isacharcomma}\ {\isasymdots}\ {\isacharequal}\ {\isacharparenleft}{\isacharparenright}{\isasymrparr}{\isachardoublequote}}.
   175   
   176   \medskip Two key observations make extensible records in a simply
   177   typed language like HOL work out:
   178 
   179   \begin{enumerate}
   180 
   181   \item the more part is internalized, as a free term or type
   182   variable,
   183 
   184   \item field names are externalized, they cannot be accessed within
   185   the logic as first-class values.
   186 
   187   \end{enumerate}
   188 
   189   \medskip In Isabelle/HOL record types have to be defined explicitly,
   190   fixing their field names and types, and their (optional) parent
   191   record.  Afterwards, records may be formed using above syntax, while
   192   obeying the canonical order of fields as given by their declaration.
   193   The record package provides several standard operations like
   194   selectors and updates.  The common setup for various generic proof
   195   tools enable succinct reasoning patterns.  See also the Isabelle/HOL
   196   tutorial \cite{isabelle-hol-book} for further instructions on using
   197   records in practice.%
   198 \end{isamarkuptext}%
   199 \isamarkuptrue%
   200 %
   201 \isamarkupsubsection{Record specifications%
   202 }
   203 \isamarkuptrue%
   204 %
   205 \begin{isamarkuptext}%
   206 \begin{matharray}{rcl}
   207     \indexdef{HOL}{command}{record}\hypertarget{command.HOL.record}{\hyperlink{command.HOL.record}{\mbox{\isa{\isacommand{record}}}}} & : & \isartrans{theory}{theory} \\
   208   \end{matharray}
   209 
   210   \begin{rail}
   211     'record' typespec '=' (type '+')? (constdecl +)
   212     ;
   213   \end{rail}
   214 
   215   \begin{descr}
   216 
   217   \item [\hyperlink{command.HOL.record}{\mbox{\isa{\isacommand{record}}}}~\isa{{\isachardoublequote}{\isacharparenleft}{\isasymalpha}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymalpha}\isactrlsub m{\isacharparenright}\ t\ {\isacharequal}\ {\isasymtau}\ {\isacharplus}\ c\isactrlsub {\isadigit{1}}\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}\isactrlsub {\isadigit{1}}\ {\isasymdots}\ c\isactrlsub n\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}\isactrlsub n{\isachardoublequote}}] defines
   218   extensible record type \isa{{\isachardoublequote}{\isacharparenleft}{\isasymalpha}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymalpha}\isactrlsub m{\isacharparenright}\ t{\isachardoublequote}},
   219   derived from the optional parent record \isa{{\isachardoublequote}{\isasymtau}{\isachardoublequote}} by adding new
   220   field components \isa{{\isachardoublequote}c\isactrlsub i\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}\isactrlsub i{\isachardoublequote}} etc.
   221 
   222   The type variables of \isa{{\isachardoublequote}{\isasymtau}{\isachardoublequote}} and \isa{{\isachardoublequote}{\isasymsigma}\isactrlsub i{\isachardoublequote}} need to be
   223   covered by the (distinct) parameters \isa{{\isachardoublequote}{\isasymalpha}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymalpha}\isactrlsub m{\isachardoublequote}}.  Type constructor \isa{t} has to be new, while \isa{{\isasymtau}} needs to specify an instance of an existing record type.  At
   224   least one new field \isa{{\isachardoublequote}c\isactrlsub i{\isachardoublequote}} has to be specified.
   225   Basically, field names need to belong to a unique record.  This is
   226   not a real restriction in practice, since fields are qualified by
   227   the record name internally.
   228 
   229   The parent record specification \isa{{\isasymtau}} is optional; if omitted
   230   \isa{t} becomes a root record.  The hierarchy of all records
   231   declared within a theory context forms a forest structure, i.e.\ a
   232   set of trees starting with a root record each.  There is no way to
   233   merge multiple parent records!
   234 
   235   For convenience, \isa{{\isachardoublequote}{\isacharparenleft}{\isasymalpha}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymalpha}\isactrlsub m{\isacharparenright}\ t{\isachardoublequote}} is made a
   236   type abbreviation for the fixed record type \isa{{\isachardoublequote}{\isasymlparr}c\isactrlsub {\isadigit{1}}\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ c\isactrlsub n\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}\isactrlsub n{\isasymrparr}{\isachardoublequote}}, likewise is \isa{{\isachardoublequote}{\isacharparenleft}{\isasymalpha}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymalpha}\isactrlsub m{\isacharcomma}\ {\isasymzeta}{\isacharparenright}\ t{\isacharunderscore}scheme{\isachardoublequote}} made an abbreviation for
   237   \isa{{\isachardoublequote}{\isasymlparr}c\isactrlsub {\isadigit{1}}\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ c\isactrlsub n\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}\isactrlsub n{\isacharcomma}\ {\isasymdots}\ {\isacharcolon}{\isacharcolon}\ {\isasymzeta}{\isasymrparr}{\isachardoublequote}}.
   238 
   239   \end{descr}%
   240 \end{isamarkuptext}%
   241 \isamarkuptrue%
   242 %
   243 \isamarkupsubsection{Record operations%
   244 }
   245 \isamarkuptrue%
   246 %
   247 \begin{isamarkuptext}%
   248 Any record definition of the form presented above produces certain
   249   standard operations.  Selectors and updates are provided for any
   250   field, including the improper one ``\isa{more}''.  There are also
   251   cumulative record constructor functions.  To simplify the
   252   presentation below, we assume for now that \isa{{\isachardoublequote}{\isacharparenleft}{\isasymalpha}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymalpha}\isactrlsub m{\isacharparenright}\ t{\isachardoublequote}} is a root record with fields \isa{{\isachardoublequote}c\isactrlsub {\isadigit{1}}\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ c\isactrlsub n\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}\isactrlsub n{\isachardoublequote}}.
   253 
   254   \medskip \textbf{Selectors} and \textbf{updates} are available for
   255   any field (including ``\isa{more}''):
   256 
   257   \begin{matharray}{lll}
   258     \isa{{\isachardoublequote}c\isactrlsub i{\isachardoublequote}} & \isa{{\isachardoublequote}{\isacharcolon}{\isacharcolon}{\isachardoublequote}} & \isa{{\isachardoublequote}{\isasymlparr}\isactrlvec c\ {\isacharcolon}{\isacharcolon}\ \isactrlvec {\isasymsigma}{\isacharcomma}\ {\isasymdots}\ {\isacharcolon}{\isacharcolon}\ {\isasymzeta}{\isasymrparr}\ {\isasymRightarrow}\ {\isasymsigma}\isactrlsub i{\isachardoublequote}} \\
   259     \isa{{\isachardoublequote}c\isactrlsub i{\isacharunderscore}update{\isachardoublequote}} & \isa{{\isachardoublequote}{\isacharcolon}{\isacharcolon}{\isachardoublequote}} & \isa{{\isachardoublequote}{\isasymsigma}\isactrlsub i\ {\isasymRightarrow}\ {\isasymlparr}\isactrlvec c\ {\isacharcolon}{\isacharcolon}\ \isactrlvec {\isasymsigma}{\isacharcomma}\ {\isasymdots}\ {\isacharcolon}{\isacharcolon}\ {\isasymzeta}{\isasymrparr}\ {\isasymRightarrow}\ {\isasymlparr}\isactrlvec c\ {\isacharcolon}{\isacharcolon}\ \isactrlvec {\isasymsigma}{\isacharcomma}\ {\isasymdots}\ {\isacharcolon}{\isacharcolon}\ {\isasymzeta}{\isasymrparr}{\isachardoublequote}} \\
   260   \end{matharray}
   261 
   262   There is special syntax for application of updates: \isa{{\isachardoublequote}r{\isasymlparr}x\ {\isacharcolon}{\isacharequal}\ a{\isasymrparr}{\isachardoublequote}} abbreviates term \isa{{\isachardoublequote}x{\isacharunderscore}update\ a\ r{\isachardoublequote}}.  Further notation for
   263   repeated updates is also available: \isa{{\isachardoublequote}r{\isasymlparr}x\ {\isacharcolon}{\isacharequal}\ a{\isasymrparr}{\isasymlparr}y\ {\isacharcolon}{\isacharequal}\ b{\isasymrparr}{\isasymlparr}z\ {\isacharcolon}{\isacharequal}\ c{\isasymrparr}{\isachardoublequote}} may be written \isa{{\isachardoublequote}r{\isasymlparr}x\ {\isacharcolon}{\isacharequal}\ a{\isacharcomma}\ y\ {\isacharcolon}{\isacharequal}\ b{\isacharcomma}\ z\ {\isacharcolon}{\isacharequal}\ c{\isasymrparr}{\isachardoublequote}}.  Note that
   264   because of postfix notation the order of fields shown here is
   265   reverse than in the actual term.  Since repeated updates are just
   266   function applications, fields may be freely permuted in \isa{{\isachardoublequote}{\isasymlparr}x\ {\isacharcolon}{\isacharequal}\ a{\isacharcomma}\ y\ {\isacharcolon}{\isacharequal}\ b{\isacharcomma}\ z\ {\isacharcolon}{\isacharequal}\ c{\isasymrparr}{\isachardoublequote}}, as far as logical equality is concerned.
   267   Thus commutativity of independent updates can be proven within the
   268   logic for any two fields, but not as a general theorem.
   269 
   270   \medskip The \textbf{make} operation provides a cumulative record
   271   constructor function:
   272 
   273   \begin{matharray}{lll}
   274     \isa{{\isachardoublequote}t{\isachardot}make{\isachardoublequote}} & \isa{{\isachardoublequote}{\isacharcolon}{\isacharcolon}{\isachardoublequote}} & \isa{{\isachardoublequote}{\isasymsigma}\isactrlsub {\isadigit{1}}\ {\isasymRightarrow}\ {\isasymdots}\ {\isasymsigma}\isactrlsub n\ {\isasymRightarrow}\ {\isasymlparr}\isactrlvec c\ {\isacharcolon}{\isacharcolon}\ \isactrlvec {\isasymsigma}{\isasymrparr}{\isachardoublequote}} \\
   275   \end{matharray}
   276 
   277   \medskip We now reconsider the case of non-root records, which are
   278   derived of some parent.  In general, the latter may depend on
   279   another parent as well, resulting in a list of \emph{ancestor
   280   records}.  Appending the lists of fields of all ancestors results in
   281   a certain field prefix.  The record package automatically takes care
   282   of this by lifting operations over this context of ancestor fields.
   283   Assuming that \isa{{\isachardoublequote}{\isacharparenleft}{\isasymalpha}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymalpha}\isactrlsub m{\isacharparenright}\ t{\isachardoublequote}} has ancestor
   284   fields \isa{{\isachardoublequote}b\isactrlsub {\isadigit{1}}\ {\isacharcolon}{\isacharcolon}\ {\isasymrho}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ b\isactrlsub k\ {\isacharcolon}{\isacharcolon}\ {\isasymrho}\isactrlsub k{\isachardoublequote}},
   285   the above record operations will get the following types:
   286 
   287   \medskip
   288   \begin{tabular}{lll}
   289     \isa{{\isachardoublequote}c\isactrlsub i{\isachardoublequote}} & \isa{{\isachardoublequote}{\isacharcolon}{\isacharcolon}{\isachardoublequote}} & \isa{{\isachardoublequote}{\isasymlparr}\isactrlvec b\ {\isacharcolon}{\isacharcolon}\ \isactrlvec {\isasymrho}{\isacharcomma}\ \isactrlvec c\ {\isacharcolon}{\isacharcolon}\ \isactrlvec {\isasymsigma}{\isacharcomma}\ {\isasymdots}\ {\isacharcolon}{\isacharcolon}\ {\isasymzeta}{\isasymrparr}\ {\isasymRightarrow}\ {\isasymsigma}\isactrlsub i{\isachardoublequote}} \\
   290     \isa{{\isachardoublequote}c\isactrlsub i{\isacharunderscore}update{\isachardoublequote}} & \isa{{\isachardoublequote}{\isacharcolon}{\isacharcolon}{\isachardoublequote}} & \isa{{\isachardoublequote}{\isasymsigma}\isactrlsub i\ {\isasymRightarrow}\ {\isasymlparr}\isactrlvec b\ {\isacharcolon}{\isacharcolon}\ \isactrlvec {\isasymrho}{\isacharcomma}\ \isactrlvec c\ {\isacharcolon}{\isacharcolon}\ \isactrlvec {\isasymsigma}{\isacharcomma}\ {\isasymdots}\ {\isacharcolon}{\isacharcolon}\ {\isasymzeta}{\isasymrparr}\ {\isasymRightarrow}\ {\isasymlparr}\isactrlvec b\ {\isacharcolon}{\isacharcolon}\ \isactrlvec {\isasymrho}{\isacharcomma}\ \isactrlvec c\ {\isacharcolon}{\isacharcolon}\ \isactrlvec {\isasymsigma}{\isacharcomma}\ {\isasymdots}\ {\isacharcolon}{\isacharcolon}\ {\isasymzeta}{\isasymrparr}{\isachardoublequote}} \\
   291     \isa{{\isachardoublequote}t{\isachardot}make{\isachardoublequote}} & \isa{{\isachardoublequote}{\isacharcolon}{\isacharcolon}{\isachardoublequote}} & \isa{{\isachardoublequote}{\isasymrho}\isactrlsub {\isadigit{1}}\ {\isasymRightarrow}\ {\isasymdots}\ {\isasymrho}\isactrlsub k\ {\isasymRightarrow}\ {\isasymsigma}\isactrlsub {\isadigit{1}}\ {\isasymRightarrow}\ {\isasymdots}\ {\isasymsigma}\isactrlsub n\ {\isasymRightarrow}\ {\isasymlparr}\isactrlvec b\ {\isacharcolon}{\isacharcolon}\ \isactrlvec {\isasymrho}{\isacharcomma}\ \isactrlvec c\ {\isacharcolon}{\isacharcolon}\ \isactrlvec {\isasymsigma}{\isasymrparr}{\isachardoublequote}} \\
   292   \end{tabular}
   293   \medskip
   294 
   295   \noindent Some further operations address the extension aspect of a
   296   derived record scheme specifically: \isa{{\isachardoublequote}t{\isachardot}fields{\isachardoublequote}} produces a
   297   record fragment consisting of exactly the new fields introduced here
   298   (the result may serve as a more part elsewhere); \isa{{\isachardoublequote}t{\isachardot}extend{\isachardoublequote}}
   299   takes a fixed record and adds a given more part; \isa{{\isachardoublequote}t{\isachardot}truncate{\isachardoublequote}} restricts a record scheme to a fixed record.
   300 
   301   \medskip
   302   \begin{tabular}{lll}
   303     \isa{{\isachardoublequote}t{\isachardot}fields{\isachardoublequote}} & \isa{{\isachardoublequote}{\isacharcolon}{\isacharcolon}{\isachardoublequote}} & \isa{{\isachardoublequote}{\isasymsigma}\isactrlsub {\isadigit{1}}\ {\isasymRightarrow}\ {\isasymdots}\ {\isasymsigma}\isactrlsub n\ {\isasymRightarrow}\ {\isasymlparr}\isactrlvec c\ {\isacharcolon}{\isacharcolon}\ \isactrlvec {\isasymsigma}{\isasymrparr}{\isachardoublequote}} \\
   304     \isa{{\isachardoublequote}t{\isachardot}extend{\isachardoublequote}} & \isa{{\isachardoublequote}{\isacharcolon}{\isacharcolon}{\isachardoublequote}} & \isa{{\isachardoublequote}{\isasymlparr}\isactrlvec b\ {\isacharcolon}{\isacharcolon}\ \isactrlvec {\isasymrho}{\isacharcomma}\ \isactrlvec c\ {\isacharcolon}{\isacharcolon}\ \isactrlvec {\isasymsigma}{\isasymrparr}\ {\isasymRightarrow}\ {\isasymzeta}\ {\isasymRightarrow}\ {\isasymlparr}\isactrlvec b\ {\isacharcolon}{\isacharcolon}\ \isactrlvec {\isasymrho}{\isacharcomma}\ \isactrlvec c\ {\isacharcolon}{\isacharcolon}\ \isactrlvec {\isasymsigma}{\isacharcomma}\ {\isasymdots}\ {\isacharcolon}{\isacharcolon}\ {\isasymzeta}{\isasymrparr}{\isachardoublequote}} \\
   305     \isa{{\isachardoublequote}t{\isachardot}truncate{\isachardoublequote}} & \isa{{\isachardoublequote}{\isacharcolon}{\isacharcolon}{\isachardoublequote}} & \isa{{\isachardoublequote}{\isasymlparr}\isactrlvec b\ {\isacharcolon}{\isacharcolon}\ \isactrlvec {\isasymrho}{\isacharcomma}\ \isactrlvec c\ {\isacharcolon}{\isacharcolon}\ \isactrlvec {\isasymsigma}{\isacharcomma}\ {\isasymdots}\ {\isacharcolon}{\isacharcolon}\ {\isasymzeta}{\isasymrparr}\ {\isasymRightarrow}\ {\isasymlparr}\isactrlvec b\ {\isacharcolon}{\isacharcolon}\ \isactrlvec {\isasymrho}{\isacharcomma}\ \isactrlvec c\ {\isacharcolon}{\isacharcolon}\ \isactrlvec {\isasymsigma}{\isasymrparr}{\isachardoublequote}} \\
   306   \end{tabular}
   307   \medskip
   308 
   309   \noindent Note that \isa{{\isachardoublequote}t{\isachardot}make{\isachardoublequote}} and \isa{{\isachardoublequote}t{\isachardot}fields{\isachardoublequote}} coincide
   310   for root records.%
   311 \end{isamarkuptext}%
   312 \isamarkuptrue%
   313 %
   314 \isamarkupsubsection{Derived rules and proof tools%
   315 }
   316 \isamarkuptrue%
   317 %
   318 \begin{isamarkuptext}%
   319 The record package proves several results internally, declaring
   320   these facts to appropriate proof tools.  This enables users to
   321   reason about record structures quite conveniently.  Assume that
   322   \isa{t} is a record type as specified above.
   323 
   324   \begin{enumerate}
   325   
   326   \item Standard conversions for selectors or updates applied to
   327   record constructor terms are made part of the default Simplifier
   328   context; thus proofs by reduction of basic operations merely require
   329   the \hyperlink{method.simp}{\mbox{\isa{simp}}} method without further arguments.  These rules
   330   are available as \isa{{\isachardoublequote}t{\isachardot}simps{\isachardoublequote}}, too.
   331   
   332   \item Selectors applied to updated records are automatically reduced
   333   by an internal simplification procedure, which is also part of the
   334   standard Simplifier setup.
   335 
   336   \item Inject equations of a form analogous to \isa{{\isachardoublequote}{\isacharparenleft}x{\isacharcomma}\ y{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}x{\isacharprime}{\isacharcomma}\ y{\isacharprime}{\isacharparenright}\ {\isasymequiv}\ x\ {\isacharequal}\ x{\isacharprime}\ {\isasymand}\ y\ {\isacharequal}\ y{\isacharprime}{\isachardoublequote}} are declared to the Simplifier and Classical
   337   Reasoner as \hyperlink{attribute.iff}{\mbox{\isa{iff}}} rules.  These rules are available as
   338   \isa{{\isachardoublequote}t{\isachardot}iffs{\isachardoublequote}}.
   339 
   340   \item The introduction rule for record equality analogous to \isa{{\isachardoublequote}x\ r\ {\isacharequal}\ x\ r{\isacharprime}\ {\isasymLongrightarrow}\ y\ r\ {\isacharequal}\ y\ r{\isacharprime}\ {\isasymdots}\ {\isasymLongrightarrow}\ r\ {\isacharequal}\ r{\isacharprime}{\isachardoublequote}} is declared to the Simplifier,
   341   and as the basic rule context as ``\hyperlink{attribute.intro}{\mbox{\isa{intro}}}\isa{{\isachardoublequote}{\isacharquery}{\isachardoublequote}}''.
   342   The rule is called \isa{{\isachardoublequote}t{\isachardot}equality{\isachardoublequote}}.
   343 
   344   \item Representations of arbitrary record expressions as canonical
   345   constructor terms are provided both in \hyperlink{method.cases}{\mbox{\isa{cases}}} and \hyperlink{method.induct}{\mbox{\isa{induct}}} format (cf.\ the generic proof methods of the same name,
   346   \secref{sec:cases-induct}).  Several variations are available, for
   347   fixed records, record schemes, more parts etc.
   348   
   349   The generic proof methods are sufficiently smart to pick the most
   350   sensible rule according to the type of the indicated record
   351   expression: users just need to apply something like ``\isa{{\isachardoublequote}{\isacharparenleft}cases\ r{\isacharparenright}{\isachardoublequote}}'' to a certain proof problem.
   352 
   353   \item The derived record operations \isa{{\isachardoublequote}t{\isachardot}make{\isachardoublequote}}, \isa{{\isachardoublequote}t{\isachardot}fields{\isachardoublequote}}, \isa{{\isachardoublequote}t{\isachardot}extend{\isachardoublequote}}, \isa{{\isachardoublequote}t{\isachardot}truncate{\isachardoublequote}} are \emph{not}
   354   treated automatically, but usually need to be expanded by hand,
   355   using the collective fact \isa{{\isachardoublequote}t{\isachardot}defs{\isachardoublequote}}.
   356 
   357   \end{enumerate}%
   358 \end{isamarkuptext}%
   359 \isamarkuptrue%
   360 %
   361 \isamarkupsection{Datatypes \label{sec:hol-datatype}%
   362 }
   363 \isamarkuptrue%
   364 %
   365 \begin{isamarkuptext}%
   366 \begin{matharray}{rcl}
   367     \indexdef{HOL}{command}{datatype}\hypertarget{command.HOL.datatype}{\hyperlink{command.HOL.datatype}{\mbox{\isa{\isacommand{datatype}}}}} & : & \isartrans{theory}{theory} \\
   368     \indexdef{HOL}{command}{rep\_datatype}\hypertarget{command.HOL.rep-datatype}{\hyperlink{command.HOL.rep-datatype}{\mbox{\isa{\isacommand{rep{\isacharunderscore}datatype}}}}} & : & \isartrans{theory}{theory} \\
   369   \end{matharray}
   370 
   371   \begin{rail}
   372     'datatype' (dtspec + 'and')
   373     ;
   374     'rep\_datatype' (name *) dtrules
   375     ;
   376 
   377     dtspec: parname? typespec infix? '=' (cons + '|')
   378     ;
   379     cons: name (type *) mixfix?
   380     ;
   381     dtrules: 'distinct' thmrefs 'inject' thmrefs 'induction' thmrefs
   382   \end{rail}
   383 
   384   \begin{descr}
   385 
   386   \item [\hyperlink{command.HOL.datatype}{\mbox{\isa{\isacommand{datatype}}}}] defines inductive datatypes in
   387   HOL.
   388 
   389   \item [\hyperlink{command.HOL.rep-datatype}{\mbox{\isa{\isacommand{rep{\isacharunderscore}datatype}}}}] represents existing types as
   390   inductive ones, generating the standard infrastructure of derived
   391   concepts (primitive recursion etc.).
   392 
   393   \end{descr}
   394 
   395   The induction and exhaustion theorems generated provide case names
   396   according to the constructors involved, while parameters are named
   397   after the types (see also \secref{sec:cases-induct}).
   398 
   399   See \cite{isabelle-HOL} for more details on datatypes, but beware of
   400   the old-style theory syntax being used there!  Apart from proper
   401   proof methods for case-analysis and induction, there are also
   402   emulations of ML tactics \hyperlink{method.HOL.case-tac}{\mbox{\isa{case{\isacharunderscore}tac}}} and \hyperlink{method.HOL.induct-tac}{\mbox{\isa{induct{\isacharunderscore}tac}}} available, see \secref{sec:hol-induct-tac}; these admit
   403   to refer directly to the internal structure of subgoals (including
   404   internally bound parameters).%
   405 \end{isamarkuptext}%
   406 \isamarkuptrue%
   407 %
   408 \isamarkupsection{Recursive functions \label{sec:recursion}%
   409 }
   410 \isamarkuptrue%
   411 %
   412 \begin{isamarkuptext}%
   413 \begin{matharray}{rcl}
   414     \indexdef{HOL}{command}{primrec}\hypertarget{command.HOL.primrec}{\hyperlink{command.HOL.primrec}{\mbox{\isa{\isacommand{primrec}}}}} & : & \isarkeep{local{\dsh}theory} \\
   415     \indexdef{HOL}{command}{fun}\hypertarget{command.HOL.fun}{\hyperlink{command.HOL.fun}{\mbox{\isa{\isacommand{fun}}}}} & : & \isarkeep{local{\dsh}theory} \\
   416     \indexdef{HOL}{command}{function}\hypertarget{command.HOL.function}{\hyperlink{command.HOL.function}{\mbox{\isa{\isacommand{function}}}}} & : & \isartrans{local{\dsh}theory}{proof(prove)} \\
   417     \indexdef{HOL}{command}{termination}\hypertarget{command.HOL.termination}{\hyperlink{command.HOL.termination}{\mbox{\isa{\isacommand{termination}}}}} & : & \isartrans{local{\dsh}theory}{proof(prove)} \\
   418   \end{matharray}
   419 
   420   \begin{rail}
   421     'primrec' target? fixes 'where' equations
   422     ;
   423     equations: (thmdecl? prop + '|')
   424     ;
   425     ('fun' | 'function') target? functionopts? fixes 'where' clauses
   426     ;
   427     clauses: (thmdecl? prop ('(' 'otherwise' ')')? + '|')
   428     ;
   429     functionopts: '(' (('sequential' | 'domintros' | 'tailrec' | 'default' term) + ',') ')'
   430     ;
   431     'termination' ( term )?
   432   \end{rail}
   433 
   434   \begin{descr}
   435 
   436   \item [\hyperlink{command.HOL.primrec}{\mbox{\isa{\isacommand{primrec}}}}] defines primitive recursive
   437   functions over datatypes, see also \cite{isabelle-HOL}.
   438 
   439   \item [\hyperlink{command.HOL.function}{\mbox{\isa{\isacommand{function}}}}] defines functions by general
   440   wellfounded recursion. A detailed description with examples can be
   441   found in \cite{isabelle-function}. The function is specified by a
   442   set of (possibly conditional) recursive equations with arbitrary
   443   pattern matching. The command generates proof obligations for the
   444   completeness and the compatibility of patterns.
   445 
   446   The defined function is considered partial, and the resulting
   447   simplification rules (named \isa{{\isachardoublequote}f{\isachardot}psimps{\isachardoublequote}}) and induction rule
   448   (named \isa{{\isachardoublequote}f{\isachardot}pinduct{\isachardoublequote}}) are guarded by a generated domain
   449   predicate \isa{{\isachardoublequote}f{\isacharunderscore}dom{\isachardoublequote}}. The \hyperlink{command.HOL.termination}{\mbox{\isa{\isacommand{termination}}}}
   450   command can then be used to establish that the function is total.
   451 
   452   \item [\hyperlink{command.HOL.fun}{\mbox{\isa{\isacommand{fun}}}}] is a shorthand notation for
   453   ``\hyperlink{command.HOL.function}{\mbox{\isa{\isacommand{function}}}}~\isa{{\isachardoublequote}{\isacharparenleft}sequential{\isacharparenright}{\isachardoublequote}}, followed by
   454   automated proof attempts regarding pattern matching and termination.
   455   See \cite{isabelle-function} for further details.
   456 
   457   \item [\hyperlink{command.HOL.termination}{\mbox{\isa{\isacommand{termination}}}}~\isa{f}] commences a
   458   termination proof for the previously defined function \isa{f}.  If
   459   this is omitted, the command refers to the most recent function
   460   definition.  After the proof is closed, the recursive equations and
   461   the induction principle is established.
   462 
   463   \end{descr}
   464 
   465   %FIXME check
   466 
   467   Recursive definitions introduced by both the \hyperlink{command.HOL.primrec}{\mbox{\isa{\isacommand{primrec}}}} and the \hyperlink{command.HOL.function}{\mbox{\isa{\isacommand{function}}}} command accommodate
   468   reasoning by induction (cf.\ \secref{sec:cases-induct}): rule \isa{{\isachardoublequote}c{\isachardot}induct{\isachardoublequote}} (where \isa{c} is the name of the function definition)
   469   refers to a specific induction rule, with parameters named according
   470   to the user-specified equations.  Case names of \hyperlink{command.HOL.primrec}{\mbox{\isa{\isacommand{primrec}}}} are that of the datatypes involved, while those of
   471   \hyperlink{command.HOL.function}{\mbox{\isa{\isacommand{function}}}} are numbered (starting from 1).
   472 
   473   The equations provided by these packages may be referred later as
   474   theorem list \isa{{\isachardoublequote}f{\isachardot}simps{\isachardoublequote}}, where \isa{f} is the (collective)
   475   name of the functions defined.  Individual equations may be named
   476   explicitly as well.
   477 
   478   The \hyperlink{command.HOL.function}{\mbox{\isa{\isacommand{function}}}} command accepts the following
   479   options.
   480 
   481   \begin{descr}
   482 
   483   \item [\isa{sequential}] enables a preprocessor which
   484   disambiguates overlapping patterns by making them mutually disjoint.
   485   Earlier equations take precedence over later ones.  This allows to
   486   give the specification in a format very similar to functional
   487   programming.  Note that the resulting simplification and induction
   488   rules correspond to the transformed specification, not the one given
   489   originally. This usually means that each equation given by the user
   490   may result in several theroems.  Also note that this automatic
   491   transformation only works for ML-style datatype patterns.
   492 
   493   \item [\isa{domintros}] enables the automated generation of
   494   introduction rules for the domain predicate. While mostly not
   495   needed, they can be helpful in some proofs about partial functions.
   496 
   497   \item [\isa{tailrec}] generates the unconstrained recursive
   498   equations even without a termination proof, provided that the
   499   function is tail-recursive. This currently only works
   500 
   501   \item [\isa{{\isachardoublequote}default\ d{\isachardoublequote}}] allows to specify a default value for a
   502   (partial) function, which will ensure that \isa{{\isachardoublequote}f\ x\ {\isacharequal}\ d\ x{\isachardoublequote}}
   503   whenever \isa{{\isachardoublequote}x\ {\isasymnotin}\ f{\isacharunderscore}dom{\isachardoublequote}}.
   504 
   505   \end{descr}%
   506 \end{isamarkuptext}%
   507 \isamarkuptrue%
   508 %
   509 \isamarkupsubsection{Proof methods related to recursive definitions%
   510 }
   511 \isamarkuptrue%
   512 %
   513 \begin{isamarkuptext}%
   514 \begin{matharray}{rcl}
   515     \indexdef{HOL}{method}{pat\_completeness}\hypertarget{method.HOL.pat-completeness}{\hyperlink{method.HOL.pat-completeness}{\mbox{\isa{pat{\isacharunderscore}completeness}}}} & : & \isarmeth \\
   516     \indexdef{HOL}{method}{relation}\hypertarget{method.HOL.relation}{\hyperlink{method.HOL.relation}{\mbox{\isa{relation}}}} & : & \isarmeth \\
   517     \indexdef{HOL}{method}{lexicographic\_order}\hypertarget{method.HOL.lexicographic-order}{\hyperlink{method.HOL.lexicographic-order}{\mbox{\isa{lexicographic{\isacharunderscore}order}}}} & : & \isarmeth \\
   518   \end{matharray}
   519 
   520   \begin{rail}
   521     'relation' term
   522     ;
   523     'lexicographic\_order' (clasimpmod *)
   524     ;
   525   \end{rail}
   526 
   527   \begin{descr}
   528 
   529   \item [\hyperlink{method.HOL.pat-completeness}{\mbox{\isa{pat{\isacharunderscore}completeness}}}] is a specialized method to
   530   solve goals regarding the completeness of pattern matching, as
   531   required by the \hyperlink{command.HOL.function}{\mbox{\isa{\isacommand{function}}}} package (cf.\
   532   \cite{isabelle-function}).
   533 
   534   \item [\hyperlink{method.HOL.relation}{\mbox{\isa{relation}}}~\isa{R}] introduces a termination
   535   proof using the relation \isa{R}.  The resulting proof state will
   536   contain goals expressing that \isa{R} is wellfounded, and that the
   537   arguments of recursive calls decrease with respect to \isa{R}.
   538   Usually, this method is used as the initial proof step of manual
   539   termination proofs.
   540 
   541   \item [\hyperlink{method.HOL.lexicographic-order}{\mbox{\isa{lexicographic{\isacharunderscore}order}}}] attempts a fully
   542   automated termination proof by searching for a lexicographic
   543   combination of size measures on the arguments of the function. The
   544   method accepts the same arguments as the \hyperlink{method.auto}{\mbox{\isa{auto}}} method,
   545   which it uses internally to prove local descents.  The same context
   546   modifiers as for \hyperlink{method.auto}{\mbox{\isa{auto}}} are accepted, see
   547   \secref{sec:clasimp}.
   548 
   549   In case of failure, extensive information is printed, which can help
   550   to analyse the situation (cf.\ \cite{isabelle-function}).
   551 
   552   \end{descr}%
   553 \end{isamarkuptext}%
   554 \isamarkuptrue%
   555 %
   556 \isamarkupsubsection{Old-style recursive function definitions (TFL)%
   557 }
   558 \isamarkuptrue%
   559 %
   560 \begin{isamarkuptext}%
   561 The old TFL commands \hyperlink{command.HOL.recdef}{\mbox{\isa{\isacommand{recdef}}}} and \hyperlink{command.HOL.recdef-tc}{\mbox{\isa{\isacommand{recdef{\isacharunderscore}tc}}}} for defining recursive are mostly obsolete; \hyperlink{command.HOL.function}{\mbox{\isa{\isacommand{function}}}} or \hyperlink{command.HOL.fun}{\mbox{\isa{\isacommand{fun}}}} should be used instead.
   562 
   563   \begin{matharray}{rcl}
   564     \indexdef{HOL}{command}{recdef}\hypertarget{command.HOL.recdef}{\hyperlink{command.HOL.recdef}{\mbox{\isa{\isacommand{recdef}}}}} & : & \isartrans{theory}{theory} \\
   565     \indexdef{HOL}{command}{recdef\_tc}\hypertarget{command.HOL.recdef-tc}{\hyperlink{command.HOL.recdef-tc}{\mbox{\isa{\isacommand{recdef{\isacharunderscore}tc}}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isartrans{theory}{proof(prove)} \\
   566   \end{matharray}
   567 
   568   \begin{rail}
   569     'recdef' ('(' 'permissive' ')')? \\ name term (prop +) hints?
   570     ;
   571     recdeftc thmdecl? tc
   572     ;
   573     hints: '(' 'hints' (recdefmod *) ')'
   574     ;
   575     recdefmod: (('recdef\_simp' | 'recdef\_cong' | 'recdef\_wf') (() | 'add' | 'del') ':' thmrefs) | clasimpmod
   576     ;
   577     tc: nameref ('(' nat ')')?
   578     ;
   579   \end{rail}
   580 
   581   \begin{descr}
   582   
   583   \item [\hyperlink{command.HOL.recdef}{\mbox{\isa{\isacommand{recdef}}}}] defines general well-founded
   584   recursive functions (using the TFL package), see also
   585   \cite{isabelle-HOL}.  The ``\isa{{\isachardoublequote}{\isacharparenleft}permissive{\isacharparenright}{\isachardoublequote}}'' option tells
   586   TFL to recover from failed proof attempts, returning unfinished
   587   results.  The \isa{recdef{\isacharunderscore}simp}, \isa{recdef{\isacharunderscore}cong}, and \isa{recdef{\isacharunderscore}wf} hints refer to auxiliary rules to be used in the internal
   588   automated proof process of TFL.  Additional \hyperlink{syntax.clasimpmod}{\mbox{\isa{clasimpmod}}}
   589   declarations (cf.\ \secref{sec:clasimp}) may be given to tune the
   590   context of the Simplifier (cf.\ \secref{sec:simplifier}) and
   591   Classical reasoner (cf.\ \secref{sec:classical}).
   592   
   593   \item [\hyperlink{command.HOL.recdef-tc}{\mbox{\isa{\isacommand{recdef{\isacharunderscore}tc}}}}~\isa{{\isachardoublequote}c\ {\isacharparenleft}i{\isacharparenright}{\isachardoublequote}}] recommences the
   594   proof for leftover termination condition number \isa{i} (default
   595   1) as generated by a \hyperlink{command.HOL.recdef}{\mbox{\isa{\isacommand{recdef}}}} definition of
   596   constant \isa{c}.
   597   
   598   Note that in most cases, \hyperlink{command.HOL.recdef}{\mbox{\isa{\isacommand{recdef}}}} is able to finish
   599   its internal proofs without manual intervention.
   600 
   601   \end{descr}
   602 
   603   \medskip Hints for \hyperlink{command.HOL.recdef}{\mbox{\isa{\isacommand{recdef}}}} may be also declared
   604   globally, using the following attributes.
   605 
   606   \begin{matharray}{rcl}
   607     \indexdef{HOL}{attribute}{recdef\_simp}\hypertarget{attribute.HOL.recdef-simp}{\hyperlink{attribute.HOL.recdef-simp}{\mbox{\isa{recdef{\isacharunderscore}simp}}}} & : & \isaratt \\
   608     \indexdef{HOL}{attribute}{recdef\_cong}\hypertarget{attribute.HOL.recdef-cong}{\hyperlink{attribute.HOL.recdef-cong}{\mbox{\isa{recdef{\isacharunderscore}cong}}}} & : & \isaratt \\
   609     \indexdef{HOL}{attribute}{recdef\_wf}\hypertarget{attribute.HOL.recdef-wf}{\hyperlink{attribute.HOL.recdef-wf}{\mbox{\isa{recdef{\isacharunderscore}wf}}}} & : & \isaratt \\
   610   \end{matharray}
   611 
   612   \begin{rail}
   613     ('recdef\_simp' | 'recdef\_cong' | 'recdef\_wf') (() | 'add' | 'del')
   614     ;
   615   \end{rail}%
   616 \end{isamarkuptext}%
   617 \isamarkuptrue%
   618 %
   619 \isamarkupsection{Inductive and coinductive definitions \label{sec:hol-inductive}%
   620 }
   621 \isamarkuptrue%
   622 %
   623 \begin{isamarkuptext}%
   624 An \textbf{inductive definition} specifies the least predicate (or
   625   set) \isa{R} closed under given rules: applying a rule to elements
   626   of \isa{R} yields a result within \isa{R}.  For example, a
   627   structural operational semantics is an inductive definition of an
   628   evaluation relation.
   629 
   630   Dually, a \textbf{coinductive definition} specifies the greatest
   631   predicate~/ set \isa{R} that is consistent with given rules: every
   632   element of \isa{R} can be seen as arising by applying a rule to
   633   elements of \isa{R}.  An important example is using bisimulation
   634   relations to formalise equivalence of processes and infinite data
   635   structures.
   636 
   637   \medskip The HOL package is related to the ZF one, which is
   638   described in a separate paper,\footnote{It appeared in CADE
   639   \cite{paulson-CADE}; a longer version is distributed with Isabelle.}
   640   which you should refer to in case of difficulties.  The package is
   641   simpler than that of ZF thanks to implicit type-checking in HOL.
   642   The types of the (co)inductive predicates (or sets) determine the
   643   domain of the fixedpoint definition, and the package does not have
   644   to use inference rules for type-checking.
   645 
   646   \begin{matharray}{rcl}
   647     \indexdef{HOL}{command}{inductive}\hypertarget{command.HOL.inductive}{\hyperlink{command.HOL.inductive}{\mbox{\isa{\isacommand{inductive}}}}} & : & \isarkeep{local{\dsh}theory} \\
   648     \indexdef{HOL}{command}{inductive\_set}\hypertarget{command.HOL.inductive-set}{\hyperlink{command.HOL.inductive-set}{\mbox{\isa{\isacommand{inductive{\isacharunderscore}set}}}}} & : & \isarkeep{local{\dsh}theory} \\
   649     \indexdef{HOL}{command}{coinductive}\hypertarget{command.HOL.coinductive}{\hyperlink{command.HOL.coinductive}{\mbox{\isa{\isacommand{coinductive}}}}} & : & \isarkeep{local{\dsh}theory} \\
   650     \indexdef{HOL}{command}{coinductive\_set}\hypertarget{command.HOL.coinductive-set}{\hyperlink{command.HOL.coinductive-set}{\mbox{\isa{\isacommand{coinductive{\isacharunderscore}set}}}}} & : & \isarkeep{local{\dsh}theory} \\
   651     \indexdef{HOL}{attribute}{mono}\hypertarget{attribute.HOL.mono}{\hyperlink{attribute.HOL.mono}{\mbox{\isa{mono}}}} & : & \isaratt \\
   652   \end{matharray}
   653 
   654   \begin{rail}
   655     ('inductive' | 'inductive\_set' | 'coinductive' | 'coinductive\_set') target? fixes ('for' fixes)? \\
   656     ('where' clauses)? ('monos' thmrefs)?
   657     ;
   658     clauses: (thmdecl? prop + '|')
   659     ;
   660     'mono' (() | 'add' | 'del')
   661     ;
   662   \end{rail}
   663 
   664   \begin{descr}
   665 
   666   \item [\hyperlink{command.HOL.inductive}{\mbox{\isa{\isacommand{inductive}}}} and \hyperlink{command.HOL.coinductive}{\mbox{\isa{\isacommand{coinductive}}}}] define (co)inductive predicates from the
   667   introduction rules given in the \hyperlink{keyword.where}{\mbox{\isa{\isakeyword{where}}}} part.  The
   668   optional \hyperlink{keyword.for}{\mbox{\isa{\isakeyword{for}}}} part contains a list of parameters of the
   669   (co)inductive predicates that remain fixed throughout the
   670   definition.  The optional \hyperlink{keyword.monos}{\mbox{\isa{\isakeyword{monos}}}} section contains
   671   \emph{monotonicity theorems}, which are required for each operator
   672   applied to a recursive set in the introduction rules.  There
   673   \emph{must} be a theorem of the form \isa{{\isachardoublequote}A\ {\isasymle}\ B\ {\isasymLongrightarrow}\ M\ A\ {\isasymle}\ M\ B{\isachardoublequote}},
   674   for each premise \isa{{\isachardoublequote}M\ R\isactrlsub i\ t{\isachardoublequote}} in an introduction rule!
   675 
   676   \item [\hyperlink{command.HOL.inductive-set}{\mbox{\isa{\isacommand{inductive{\isacharunderscore}set}}}} and \hyperlink{command.HOL.coinductive-set}{\mbox{\isa{\isacommand{coinductive{\isacharunderscore}set}}}}] are wrappers for to the previous commands,
   677   allowing the definition of (co)inductive sets.
   678 
   679   \item [\hyperlink{attribute.HOL.mono}{\mbox{\isa{mono}}}] declares monotonicity rules.  These
   680   rule are involved in the automated monotonicity proof of \hyperlink{command.HOL.inductive}{\mbox{\isa{\isacommand{inductive}}}}.
   681 
   682   \end{descr}%
   683 \end{isamarkuptext}%
   684 \isamarkuptrue%
   685 %
   686 \isamarkupsubsection{Derived rules%
   687 }
   688 \isamarkuptrue%
   689 %
   690 \begin{isamarkuptext}%
   691 Each (co)inductive definition \isa{R} adds definitions to the
   692   theory and also proves some theorems:
   693 
   694   \begin{description}
   695 
   696   \item [\isa{R{\isachardot}intros}] is the list of introduction rules as proven
   697   theorems, for the recursive predicates (or sets).  The rules are
   698   also available individually, using the names given them in the
   699   theory file;
   700 
   701   \item [\isa{R{\isachardot}cases}] is the case analysis (or elimination) rule;
   702 
   703   \item [\isa{R{\isachardot}induct} or \isa{R{\isachardot}coinduct}] is the (co)induction
   704   rule.
   705 
   706   \end{description}
   707 
   708   When several predicates \isa{{\isachardoublequote}R\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ R\isactrlsub n{\isachardoublequote}} are
   709   defined simultaneously, the list of introduction rules is called
   710   \isa{{\isachardoublequote}R\isactrlsub {\isadigit{1}}{\isacharunderscore}{\isasymdots}{\isacharunderscore}R\isactrlsub n{\isachardot}intros{\isachardoublequote}}, the case analysis rules are
   711   called \isa{{\isachardoublequote}R\isactrlsub {\isadigit{1}}{\isachardot}cases{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ R\isactrlsub n{\isachardot}cases{\isachardoublequote}}, and the list
   712   of mutual induction rules is called \isa{{\isachardoublequote}R\isactrlsub {\isadigit{1}}{\isacharunderscore}{\isasymdots}{\isacharunderscore}R\isactrlsub n{\isachardot}inducts{\isachardoublequote}}.%
   713 \end{isamarkuptext}%
   714 \isamarkuptrue%
   715 %
   716 \isamarkupsubsection{Monotonicity theorems%
   717 }
   718 \isamarkuptrue%
   719 %
   720 \begin{isamarkuptext}%
   721 Each theory contains a default set of theorems that are used in
   722   monotonicity proofs.  New rules can be added to this set via the
   723   \hyperlink{attribute.HOL.mono}{\mbox{\isa{mono}}} attribute.  The HOL theory \isa{Inductive}
   724   shows how this is done.  In general, the following monotonicity
   725   theorems may be added:
   726 
   727   \begin{itemize}
   728 
   729   \item Theorems of the form \isa{{\isachardoublequote}A\ {\isasymle}\ B\ {\isasymLongrightarrow}\ M\ A\ {\isasymle}\ M\ B{\isachardoublequote}}, for proving
   730   monotonicity of inductive definitions whose introduction rules have
   731   premises involving terms such as \isa{{\isachardoublequote}M\ R\isactrlsub i\ t{\isachardoublequote}}.
   732 
   733   \item Monotonicity theorems for logical operators, which are of the
   734   general form \isa{{\isachardoublequote}{\isacharparenleft}{\isasymdots}\ {\isasymlongrightarrow}\ {\isasymdots}{\isacharparenright}\ {\isasymLongrightarrow}\ {\isasymdots}\ {\isacharparenleft}{\isasymdots}\ {\isasymlongrightarrow}\ {\isasymdots}{\isacharparenright}\ {\isasymLongrightarrow}\ {\isasymdots}\ {\isasymlongrightarrow}\ {\isasymdots}{\isachardoublequote}}.  For example, in
   735   the case of the operator \isa{{\isachardoublequote}{\isasymor}{\isachardoublequote}}, the corresponding theorem is
   736   \[
   737   \infer{\isa{{\isachardoublequote}P\isactrlsub {\isadigit{1}}\ {\isasymor}\ P\isactrlsub {\isadigit{2}}\ {\isasymlongrightarrow}\ Q\isactrlsub {\isadigit{1}}\ {\isasymor}\ Q\isactrlsub {\isadigit{2}}{\isachardoublequote}}}{\isa{{\isachardoublequote}P\isactrlsub {\isadigit{1}}\ {\isasymlongrightarrow}\ Q\isactrlsub {\isadigit{1}}{\isachardoublequote}} & \isa{{\isachardoublequote}P\isactrlsub {\isadigit{2}}\ {\isasymlongrightarrow}\ Q\isactrlsub {\isadigit{2}}{\isachardoublequote}}}
   738   \]
   739 
   740   \item De Morgan style equations for reasoning about the ``polarity''
   741   of expressions, e.g.
   742   \[
   743   \isa{{\isachardoublequote}{\isasymnot}\ {\isasymnot}\ P\ {\isasymlongleftrightarrow}\ P{\isachardoublequote}} \qquad\qquad
   744   \isa{{\isachardoublequote}{\isasymnot}\ {\isacharparenleft}P\ {\isasymand}\ Q{\isacharparenright}\ {\isasymlongleftrightarrow}\ {\isasymnot}\ P\ {\isasymor}\ {\isasymnot}\ Q{\isachardoublequote}}
   745   \]
   746 
   747   \item Equations for reducing complex operators to more primitive
   748   ones whose monotonicity can easily be proved, e.g.
   749   \[
   750   \isa{{\isachardoublequote}{\isacharparenleft}P\ {\isasymlongrightarrow}\ Q{\isacharparenright}\ {\isasymlongleftrightarrow}\ {\isasymnot}\ P\ {\isasymor}\ Q{\isachardoublequote}} \qquad\qquad
   751   \isa{{\isachardoublequote}Ball\ A\ P\ {\isasymequiv}\ {\isasymforall}x{\isachardot}\ x\ {\isasymin}\ A\ {\isasymlongrightarrow}\ P\ x{\isachardoublequote}}
   752   \]
   753 
   754   \end{itemize}
   755 
   756   %FIXME: Example of an inductive definition%
   757 \end{isamarkuptext}%
   758 \isamarkuptrue%
   759 %
   760 \isamarkupsection{Arithmetic proof support%
   761 }
   762 \isamarkuptrue%
   763 %
   764 \begin{isamarkuptext}%
   765 \begin{matharray}{rcl}
   766     \indexdef{HOL}{method}{arith}\hypertarget{method.HOL.arith}{\hyperlink{method.HOL.arith}{\mbox{\isa{arith}}}} & : & \isarmeth \\
   767     \indexdef{HOL}{attribute}{arith\_split}\hypertarget{attribute.HOL.arith-split}{\hyperlink{attribute.HOL.arith-split}{\mbox{\isa{arith{\isacharunderscore}split}}}} & : & \isaratt \\
   768   \end{matharray}
   769 
   770   The \hyperlink{method.HOL.arith}{\mbox{\isa{arith}}} method decides linear arithmetic problems
   771   (on types \isa{nat}, \isa{int}, \isa{real}).  Any current
   772   facts are inserted into the goal before running the procedure.
   773 
   774   The \hyperlink{attribute.HOL.arith-split}{\mbox{\isa{arith{\isacharunderscore}split}}} attribute declares case split
   775   rules to be expanded before the arithmetic procedure is invoked.
   776 
   777   Note that a simpler (but faster) version of arithmetic reasoning is
   778   already performed by the Simplifier.%
   779 \end{isamarkuptext}%
   780 \isamarkuptrue%
   781 %
   782 \isamarkupsection{Cases and induction: emulating tactic scripts \label{sec:hol-induct-tac}%
   783 }
   784 \isamarkuptrue%
   785 %
   786 \begin{isamarkuptext}%
   787 The following important tactical tools of Isabelle/HOL have been
   788   ported to Isar.  These should be never used in proper proof texts!
   789 
   790   \begin{matharray}{rcl}
   791     \indexdef{HOL}{method}{case\_tac}\hypertarget{method.HOL.case-tac}{\hyperlink{method.HOL.case-tac}{\mbox{\isa{case{\isacharunderscore}tac}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isarmeth \\
   792     \indexdef{HOL}{method}{induct\_tac}\hypertarget{method.HOL.induct-tac}{\hyperlink{method.HOL.induct-tac}{\mbox{\isa{induct{\isacharunderscore}tac}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isarmeth \\
   793     \indexdef{HOL}{method}{ind\_cases}\hypertarget{method.HOL.ind-cases}{\hyperlink{method.HOL.ind-cases}{\mbox{\isa{ind{\isacharunderscore}cases}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isarmeth \\
   794     \indexdef{HOL}{command}{inductive\_cases}\hypertarget{command.HOL.inductive-cases}{\hyperlink{command.HOL.inductive-cases}{\mbox{\isa{\isacommand{inductive{\isacharunderscore}cases}}}}} & : & \isartrans{theory}{theory} \\
   795   \end{matharray}
   796 
   797   \begin{rail}
   798     'case\_tac' goalspec? term rule?
   799     ;
   800     'induct\_tac' goalspec? (insts * 'and') rule?
   801     ;
   802     'ind\_cases' (prop +) ('for' (name +)) ?
   803     ;
   804     'inductive\_cases' (thmdecl? (prop +) + 'and')
   805     ;
   806 
   807     rule: ('rule' ':' thmref)
   808     ;
   809   \end{rail}
   810 
   811   \begin{descr}
   812 
   813   \item [\hyperlink{method.HOL.case-tac}{\mbox{\isa{case{\isacharunderscore}tac}}} and \hyperlink{method.HOL.induct-tac}{\mbox{\isa{induct{\isacharunderscore}tac}}}]
   814   admit to reason about inductive datatypes only (unless an
   815   alternative rule is given explicitly).  Furthermore, \hyperlink{method.HOL.case-tac}{\mbox{\isa{case{\isacharunderscore}tac}}} does a classical case split on booleans; \hyperlink{method.HOL.induct-tac}{\mbox{\isa{induct{\isacharunderscore}tac}}} allows only variables to be given as instantiation.
   816   These tactic emulations feature both goal addressing and dynamic
   817   instantiation.  Note that named rule cases are \emph{not} provided
   818   as would be by the proper \hyperlink{method.induct}{\mbox{\isa{induct}}} and \hyperlink{method.cases}{\mbox{\isa{cases}}} proof
   819   methods (see \secref{sec:cases-induct}).
   820   
   821   \item [\hyperlink{method.HOL.ind-cases}{\mbox{\isa{ind{\isacharunderscore}cases}}} and \hyperlink{command.HOL.inductive-cases}{\mbox{\isa{\isacommand{inductive{\isacharunderscore}cases}}}}] provide an interface to the internal \verb|mk_cases| operation.  Rules are simplified in an unrestricted
   822   forward manner.
   823 
   824   While \hyperlink{method.HOL.ind-cases}{\mbox{\isa{ind{\isacharunderscore}cases}}} is a proof method to apply the
   825   result immediately as elimination rules, \hyperlink{command.HOL.inductive-cases}{\mbox{\isa{\isacommand{inductive{\isacharunderscore}cases}}}} provides case split theorems at the theory level
   826   for later use.  The \hyperlink{keyword.for}{\mbox{\isa{\isakeyword{for}}}} argument of the \hyperlink{method.HOL.ind-cases}{\mbox{\isa{ind{\isacharunderscore}cases}}} method allows to specify a list of variables that should
   827   be generalized before applying the resulting rule.
   828 
   829   \end{descr}%
   830 \end{isamarkuptext}%
   831 \isamarkuptrue%
   832 %
   833 \isamarkupsection{Executable code%
   834 }
   835 \isamarkuptrue%
   836 %
   837 \begin{isamarkuptext}%
   838 Isabelle/Pure provides two generic frameworks to support code
   839   generation from executable specifications.  Isabelle/HOL
   840   instantiates these mechanisms in a way that is amenable to end-user
   841   applications.
   842 
   843   One framework generates code from both functional and relational
   844   programs to SML.  See \cite{isabelle-HOL} for further information
   845   (this actually covers the new-style theory format as well).
   846 
   847   \begin{matharray}{rcl}
   848     \indexdef{HOL}{command}{value}\hypertarget{command.HOL.value}{\hyperlink{command.HOL.value}{\mbox{\isa{\isacommand{value}}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isarkeep{theory~|~proof} \\
   849     \indexdef{HOL}{command}{code\_module}\hypertarget{command.HOL.code-module}{\hyperlink{command.HOL.code-module}{\mbox{\isa{\isacommand{code{\isacharunderscore}module}}}}} & : & \isartrans{theory}{theory} \\
   850     \indexdef{HOL}{command}{code\_library}\hypertarget{command.HOL.code-library}{\hyperlink{command.HOL.code-library}{\mbox{\isa{\isacommand{code{\isacharunderscore}library}}}}} & : & \isartrans{theory}{theory} \\
   851     \indexdef{HOL}{command}{consts\_code}\hypertarget{command.HOL.consts-code}{\hyperlink{command.HOL.consts-code}{\mbox{\isa{\isacommand{consts{\isacharunderscore}code}}}}} & : & \isartrans{theory}{theory} \\
   852     \indexdef{HOL}{command}{types\_code}\hypertarget{command.HOL.types-code}{\hyperlink{command.HOL.types-code}{\mbox{\isa{\isacommand{types{\isacharunderscore}code}}}}} & : & \isartrans{theory}{theory} \\  
   853     \indexdef{HOL}{attribute}{code}\hypertarget{attribute.HOL.code}{\hyperlink{attribute.HOL.code}{\mbox{\isa{code}}}} & : & \isaratt \\
   854   \end{matharray}
   855 
   856   \begin{rail}
   857   'value' term
   858   ;
   859 
   860   ( 'code\_module' | 'code\_library' ) modespec ? name ? \\
   861     ( 'file' name ) ? ( 'imports' ( name + ) ) ? \\
   862     'contains' ( ( name '=' term ) + | term + )
   863   ;
   864 
   865   modespec: '(' ( name * ) ')'
   866   ;
   867 
   868   'consts\_code' (codespec +)
   869   ;
   870 
   871   codespec: const template attachment ?
   872   ;
   873 
   874   'types\_code' (tycodespec +)
   875   ;
   876 
   877   tycodespec: name template attachment ?
   878   ;
   879 
   880   const: term
   881   ;
   882 
   883   template: '(' string ')'
   884   ;
   885 
   886   attachment: 'attach' modespec ? verblbrace text verbrbrace
   887   ;
   888 
   889   'code' (name)?
   890   ;
   891   \end{rail}
   892 
   893   \begin{descr}
   894 
   895   \item [\hyperlink{command.HOL.value}{\mbox{\isa{\isacommand{value}}}}~\isa{t}] evaluates and prints a
   896   term using the code generator.
   897 
   898   \end{descr}
   899 
   900   \medskip The other framework generates code from functional programs
   901   (including overloading using type classes) to SML \cite{SML}, OCaml
   902   \cite{OCaml} and Haskell \cite{haskell-revised-report}.
   903   Conceptually, code generation is split up in three steps:
   904   \emph{selection} of code theorems, \emph{translation} into an
   905   abstract executable view and \emph{serialization} to a specific
   906   \emph{target language}.  See \cite{isabelle-codegen} for an
   907   introduction on how to use it.
   908 
   909   \begin{matharray}{rcl}
   910     \indexdef{HOL}{command}{export\_code}\hypertarget{command.HOL.export-code}{\hyperlink{command.HOL.export-code}{\mbox{\isa{\isacommand{export{\isacharunderscore}code}}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isarkeep{theory~|~proof} \\
   911     \indexdef{HOL}{command}{code\_thms}\hypertarget{command.HOL.code-thms}{\hyperlink{command.HOL.code-thms}{\mbox{\isa{\isacommand{code{\isacharunderscore}thms}}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isarkeep{theory~|~proof} \\
   912     \indexdef{HOL}{command}{code\_deps}\hypertarget{command.HOL.code-deps}{\hyperlink{command.HOL.code-deps}{\mbox{\isa{\isacommand{code{\isacharunderscore}deps}}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isarkeep{theory~|~proof} \\
   913     \indexdef{HOL}{command}{code\_datatype}\hypertarget{command.HOL.code-datatype}{\hyperlink{command.HOL.code-datatype}{\mbox{\isa{\isacommand{code{\isacharunderscore}datatype}}}}} & : & \isartrans{theory}{theory} \\
   914     \indexdef{HOL}{command}{code\_const}\hypertarget{command.HOL.code-const}{\hyperlink{command.HOL.code-const}{\mbox{\isa{\isacommand{code{\isacharunderscore}const}}}}} & : & \isartrans{theory}{theory} \\
   915     \indexdef{HOL}{command}{code\_type}\hypertarget{command.HOL.code-type}{\hyperlink{command.HOL.code-type}{\mbox{\isa{\isacommand{code{\isacharunderscore}type}}}}} & : & \isartrans{theory}{theory} \\
   916     \indexdef{HOL}{command}{code\_class}\hypertarget{command.HOL.code-class}{\hyperlink{command.HOL.code-class}{\mbox{\isa{\isacommand{code{\isacharunderscore}class}}}}} & : & \isartrans{theory}{theory} \\
   917     \indexdef{HOL}{command}{code\_instance}\hypertarget{command.HOL.code-instance}{\hyperlink{command.HOL.code-instance}{\mbox{\isa{\isacommand{code{\isacharunderscore}instance}}}}} & : & \isartrans{theory}{theory} \\
   918     \indexdef{HOL}{command}{code\_monad}\hypertarget{command.HOL.code-monad}{\hyperlink{command.HOL.code-monad}{\mbox{\isa{\isacommand{code{\isacharunderscore}monad}}}}} & : & \isartrans{theory}{theory} \\
   919     \indexdef{HOL}{command}{code\_reserved}\hypertarget{command.HOL.code-reserved}{\hyperlink{command.HOL.code-reserved}{\mbox{\isa{\isacommand{code{\isacharunderscore}reserved}}}}} & : & \isartrans{theory}{theory} \\
   920     \indexdef{HOL}{command}{code\_include}\hypertarget{command.HOL.code-include}{\hyperlink{command.HOL.code-include}{\mbox{\isa{\isacommand{code{\isacharunderscore}include}}}}} & : & \isartrans{theory}{theory} \\
   921     \indexdef{HOL}{command}{code\_modulename}\hypertarget{command.HOL.code-modulename}{\hyperlink{command.HOL.code-modulename}{\mbox{\isa{\isacommand{code{\isacharunderscore}modulename}}}}} & : & \isartrans{theory}{theory} \\
   922     \indexdef{HOL}{command}{code\_exception}\hypertarget{command.HOL.code-exception}{\hyperlink{command.HOL.code-exception}{\mbox{\isa{\isacommand{code{\isacharunderscore}exception}}}}} & : & \isartrans{theory}{theory} \\
   923     \indexdef{HOL}{command}{print\_codesetup}\hypertarget{command.HOL.print-codesetup}{\hyperlink{command.HOL.print-codesetup}{\mbox{\isa{\isacommand{print{\isacharunderscore}codesetup}}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isarkeep{theory~|~proof} \\
   924     \indexdef{HOL}{attribute}{code}\hypertarget{attribute.HOL.code}{\hyperlink{attribute.HOL.code}{\mbox{\isa{code}}}} & : & \isaratt \\
   925   \end{matharray}
   926 
   927   \begin{rail}
   928     'export\_code' ( constexpr + ) ? \\
   929       ( ( 'in' target ( 'module\_name' string ) ? \\
   930         ( 'file' ( string | '-' ) ) ? ( '(' args ')' ) ?) + ) ?
   931     ;
   932 
   933     'code\_thms' ( constexpr + ) ?
   934     ;
   935 
   936     'code\_deps' ( constexpr + ) ?
   937     ;
   938 
   939     const: term
   940     ;
   941 
   942     constexpr: ( const | 'name.*' | '*' )
   943     ;
   944 
   945     typeconstructor: nameref
   946     ;
   947 
   948     class: nameref
   949     ;
   950 
   951     target: 'OCaml' | 'SML' | 'Haskell'
   952     ;
   953 
   954     'code\_datatype' const +
   955     ;
   956 
   957     'code\_const' (const + 'and') \\
   958       ( ( '(' target ( syntax ? + 'and' ) ')' ) + )
   959     ;
   960 
   961     'code\_type' (typeconstructor + 'and') \\
   962       ( ( '(' target ( syntax ? + 'and' ) ')' ) + )
   963     ;
   964 
   965     'code\_class' (class + 'and') \\
   966       ( ( '(' target \\
   967         ( ( string ('where' \\
   968           ( const ( '==' | equiv ) string ) + ) ? ) ? + 'and' ) ')' ) + )
   969     ;
   970 
   971     'code\_instance' (( typeconstructor '::' class ) + 'and') \\
   972       ( ( '(' target ( '-' ? + 'and' ) ')' ) + )
   973     ;
   974 
   975     'code\_monad' const const target
   976     ;
   977 
   978     'code\_reserved' target ( string + )
   979     ;
   980 
   981     'code\_include' target ( string ( string | '-') )
   982     ;
   983 
   984     'code\_modulename' target ( ( string string ) + )
   985     ;
   986 
   987     'code\_exception' ( const + )
   988     ;
   989 
   990     syntax: string | ( 'infix' | 'infixl' | 'infixr' ) nat string
   991     ;
   992 
   993     'code' ('func' | 'inline') ( 'del' )?
   994     ;
   995   \end{rail}
   996 
   997   \begin{descr}
   998 
   999   \item [\hyperlink{command.HOL.export-code}{\mbox{\isa{\isacommand{export{\isacharunderscore}code}}}}] is the canonical interface
  1000   for generating and serializing code: for a given list of constants,
  1001   code is generated for the specified target languages.  Abstract code
  1002   is cached incrementally.  If no constant is given, the currently
  1003   cached code is serialized.  If no serialization instruction is
  1004   given, only abstract code is cached.
  1005 
  1006   Constants may be specified by giving them literally, referring to
  1007   all executable contants within a certain theory by giving \isa{{\isachardoublequote}name{\isachardot}{\isacharasterisk}{\isachardoublequote}}, or referring to \emph{all} executable constants currently
  1008   available by giving \isa{{\isachardoublequote}{\isacharasterisk}{\isachardoublequote}}.
  1009 
  1010   By default, for each involved theory one corresponding name space
  1011   module is generated.  Alternativly, a module name may be specified
  1012   after the \hyperlink{keyword.module-name}{\mbox{\isa{\isakeyword{module{\isacharunderscore}name}}}} keyword; then \emph{all} code is
  1013   placed in this module.
  1014 
  1015   For \emph{SML} and \emph{OCaml}, the file specification refers to a
  1016   single file; for \emph{Haskell}, it refers to a whole directory,
  1017   where code is generated in multiple files reflecting the module
  1018   hierarchy.  The file specification ``\isa{{\isachardoublequote}{\isacharminus}{\isachardoublequote}}'' denotes standard
  1019   output.  For \emph{SML}, omitting the file specification compiles
  1020   code internally in the context of the current ML session.
  1021 
  1022   Serializers take an optional list of arguments in parentheses.  For
  1023   \emph{Haskell} a module name prefix may be given using the ``\isa{{\isachardoublequote}root{\isacharcolon}{\isachardoublequote}}'' argument; ``\isa{string{\isacharunderscore}classes}'' adds a ``\verb|deriving (Read, Show)|'' clause to each appropriate datatype
  1024   declaration.
  1025 
  1026   \item [\hyperlink{command.HOL.code-thms}{\mbox{\isa{\isacommand{code{\isacharunderscore}thms}}}}] prints a list of theorems
  1027   representing the corresponding program containing all given
  1028   constants; if no constants are given, the currently cached code
  1029   theorems are printed.
  1030 
  1031   \item [\hyperlink{command.HOL.code-deps}{\mbox{\isa{\isacommand{code{\isacharunderscore}deps}}}}] visualizes dependencies of
  1032   theorems representing the corresponding program containing all given
  1033   constants; if no constants are given, the currently cached code
  1034   theorems are visualized.
  1035 
  1036   \item [\hyperlink{command.HOL.code-datatype}{\mbox{\isa{\isacommand{code{\isacharunderscore}datatype}}}}] specifies a constructor set
  1037   for a logical type.
  1038 
  1039   \item [\hyperlink{command.HOL.code-const}{\mbox{\isa{\isacommand{code{\isacharunderscore}const}}}}] associates a list of constants
  1040   with target-specific serializations; omitting a serialization
  1041   deletes an existing serialization.
  1042 
  1043   \item [\hyperlink{command.HOL.code-type}{\mbox{\isa{\isacommand{code{\isacharunderscore}type}}}}] associates a list of type
  1044   constructors with target-specific serializations; omitting a
  1045   serialization deletes an existing serialization.
  1046 
  1047   \item [\hyperlink{command.HOL.code-class}{\mbox{\isa{\isacommand{code{\isacharunderscore}class}}}}] associates a list of classes
  1048   with target-specific class names; in addition, constants associated
  1049   with this class may be given target-specific names used for instance
  1050   declarations; omitting a serialization deletes an existing
  1051   serialization.  This applies only to \emph{Haskell}.
  1052 
  1053   \item [\hyperlink{command.HOL.code-instance}{\mbox{\isa{\isacommand{code{\isacharunderscore}instance}}}}] declares a list of type
  1054   constructor / class instance relations as ``already present'' for a
  1055   given target.  Omitting a ``\isa{{\isachardoublequote}{\isacharminus}{\isachardoublequote}}'' deletes an existing
  1056   ``already present'' declaration.  This applies only to
  1057   \emph{Haskell}.
  1058 
  1059   \item [\hyperlink{command.HOL.code-monad}{\mbox{\isa{\isacommand{code{\isacharunderscore}monad}}}}] provides an auxiliary
  1060   mechanism to generate monadic code.
  1061 
  1062   \item [\hyperlink{command.HOL.code-reserved}{\mbox{\isa{\isacommand{code{\isacharunderscore}reserved}}}}] declares a list of names as
  1063   reserved for a given target, preventing it to be shadowed by any
  1064   generated code.
  1065 
  1066   \item [\hyperlink{command.HOL.code-include}{\mbox{\isa{\isacommand{code{\isacharunderscore}include}}}}] adds arbitrary named content
  1067   (``include'') to generated code.  A as last argument ``\isa{{\isachardoublequote}{\isacharminus}{\isachardoublequote}}''
  1068   will remove an already added ``include''.
  1069 
  1070   \item [\hyperlink{command.HOL.code-modulename}{\mbox{\isa{\isacommand{code{\isacharunderscore}modulename}}}}] declares aliasings from
  1071   one module name onto another.
  1072 
  1073   \item [\hyperlink{command.HOL.code-exception}{\mbox{\isa{\isacommand{code{\isacharunderscore}exception}}}}] declares constants which
  1074   are not required to have a definition by a defining equations; these
  1075   are mapped on exceptions instead.
  1076 
  1077   \item [\hyperlink{attribute.HOL.code}{\mbox{\isa{code}}}~\isa{func}] explicitly selects (or
  1078   with option ``\isa{{\isachardoublequote}del{\isacharcolon}{\isachardoublequote}}'' deselects) a defining equation for
  1079   code generation.  Usually packages introducing defining equations
  1080   provide a resonable default setup for selection.
  1081 
  1082   \item [\hyperlink{attribute.HOL.code}{\mbox{\isa{code}}}\isa{inline}] declares (or with
  1083   option ``\isa{{\isachardoublequote}del{\isacharcolon}{\isachardoublequote}}'' removes) inlining theorems which are
  1084   applied as rewrite rules to any defining equation during
  1085   preprocessing.
  1086 
  1087   \item [\hyperlink{command.HOL.print-codesetup}{\mbox{\isa{\isacommand{print{\isacharunderscore}codesetup}}}}] gives an overview on
  1088   selected defining equations, code generator datatypes and
  1089   preprocessor setup.
  1090 
  1091   \end{descr}%
  1092 \end{isamarkuptext}%
  1093 \isamarkuptrue%
  1094 %
  1095 \isamarkupsection{Definition by specification \label{sec:hol-specification}%
  1096 }
  1097 \isamarkuptrue%
  1098 %
  1099 \begin{isamarkuptext}%
  1100 \begin{matharray}{rcl}
  1101     \indexdef{HOL}{command}{specification}\hypertarget{command.HOL.specification}{\hyperlink{command.HOL.specification}{\mbox{\isa{\isacommand{specification}}}}} & : & \isartrans{theory}{proof(prove)} \\
  1102     \indexdef{HOL}{command}{ax\_specification}\hypertarget{command.HOL.ax-specification}{\hyperlink{command.HOL.ax-specification}{\mbox{\isa{\isacommand{ax{\isacharunderscore}specification}}}}} & : & \isartrans{theory}{proof(prove)} \\
  1103   \end{matharray}
  1104 
  1105   \begin{rail}
  1106   ('specification' | 'ax\_specification') '(' (decl +) ')' \\ (thmdecl? prop +)
  1107   ;
  1108   decl: ((name ':')? term '(' 'overloaded' ')'?)
  1109   \end{rail}
  1110 
  1111   \begin{descr}
  1112 
  1113   \item [\hyperlink{command.HOL.specification}{\mbox{\isa{\isacommand{specification}}}}~\isa{{\isachardoublequote}decls\ {\isasymphi}{\isachardoublequote}}] sets up a
  1114   goal stating the existence of terms with the properties specified to
  1115   hold for the constants given in \isa{decls}.  After finishing the
  1116   proof, the theory will be augmented with definitions for the given
  1117   constants, as well as with theorems stating the properties for these
  1118   constants.
  1119 
  1120   \item [\hyperlink{command.HOL.ax-specification}{\mbox{\isa{\isacommand{ax{\isacharunderscore}specification}}}}~\isa{{\isachardoublequote}decls\ {\isasymphi}{\isachardoublequote}}] sets
  1121   up a goal stating the existence of terms with the properties
  1122   specified to hold for the constants given in \isa{decls}.  After
  1123   finishing the proof, the theory will be augmented with axioms
  1124   expressing the properties given in the first place.
  1125 
  1126   \item [\isa{decl}] declares a constant to be defined by the
  1127   specification given.  The definition for the constant \isa{c} is
  1128   bound to the name \isa{c{\isacharunderscore}def} unless a theorem name is given in
  1129   the declaration.  Overloaded constants should be declared as such.
  1130 
  1131   \end{descr}
  1132 
  1133   Whether to use \hyperlink{command.HOL.specification}{\mbox{\isa{\isacommand{specification}}}} or \hyperlink{command.HOL.ax-specification}{\mbox{\isa{\isacommand{ax{\isacharunderscore}specification}}}} is to some extent a matter of style.  \hyperlink{command.HOL.specification}{\mbox{\isa{\isacommand{specification}}}} introduces no new axioms, and so by
  1134   construction cannot introduce inconsistencies, whereas \hyperlink{command.HOL.ax-specification}{\mbox{\isa{\isacommand{ax{\isacharunderscore}specification}}}} does introduce axioms, but only after the
  1135   user has explicitly proven it to be safe.  A practical issue must be
  1136   considered, though: After introducing two constants with the same
  1137   properties using \hyperlink{command.HOL.specification}{\mbox{\isa{\isacommand{specification}}}}, one can prove
  1138   that the two constants are, in fact, equal.  If this might be a
  1139   problem, one should use \hyperlink{command.HOL.ax-specification}{\mbox{\isa{\isacommand{ax{\isacharunderscore}specification}}}}.%
  1140 \end{isamarkuptext}%
  1141 \isamarkuptrue%
  1142 %
  1143 \isadelimtheory
  1144 %
  1145 \endisadelimtheory
  1146 %
  1147 \isatagtheory
  1148 \isacommand{end}\isamarkupfalse%
  1149 %
  1150 \endisatagtheory
  1151 {\isafoldtheory}%
  1152 %
  1153 \isadelimtheory
  1154 %
  1155 \endisadelimtheory
  1156 \isanewline
  1157 \end{isabellebody}%
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