doc-src/IsarRef/Thy/document/HOL_Specific.tex
author wenzelm
Thu May 15 18:12:43 2008 +0200 (2008-05-15)
changeset 26907 75466ad27dd7
parent 26902 8db1e960d636
child 26987 978cefd606ad
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   \railalias{funopts}{function\_opts}  %FIXME ??
   421 
   422   \begin{rail}
   423     'primrec' target? fixes 'where' equations
   424     ;
   425     equations: (thmdecl? prop + '|')
   426     ;
   427     ('fun' | 'function') (funopts)? fixes 'where' clauses
   428     ;
   429     clauses: (thmdecl? prop ('(' 'otherwise' ')')? + '|')
   430     ;
   431     funopts: '(' (('sequential' | 'in' name | 'domintros' | 'tailrec' |
   432     'default' term) + ',') ')'
   433     ;
   434     'termination' ( term )?
   435   \end{rail}
   436 
   437   \begin{descr}
   438 
   439   \item [\hyperlink{command.HOL.primrec}{\mbox{\isa{\isacommand{primrec}}}}] defines primitive recursive
   440   functions over datatypes, see also \cite{isabelle-HOL}.
   441 
   442   \item [\hyperlink{command.HOL.function}{\mbox{\isa{\isacommand{function}}}}] defines functions by general
   443   wellfounded recursion. A detailed description with examples can be
   444   found in \cite{isabelle-function}. The function is specified by a
   445   set of (possibly conditional) recursive equations with arbitrary
   446   pattern matching. The command generates proof obligations for the
   447   completeness and the compatibility of patterns.
   448 
   449   The defined function is considered partial, and the resulting
   450   simplification rules (named \isa{{\isachardoublequote}f{\isachardot}psimps{\isachardoublequote}}) and induction rule
   451   (named \isa{{\isachardoublequote}f{\isachardot}pinduct{\isachardoublequote}}) are guarded by a generated domain
   452   predicate \isa{{\isachardoublequote}f{\isacharunderscore}dom{\isachardoublequote}}. The \hyperlink{command.HOL.termination}{\mbox{\isa{\isacommand{termination}}}}
   453   command can then be used to establish that the function is total.
   454 
   455   \item [\hyperlink{command.HOL.fun}{\mbox{\isa{\isacommand{fun}}}}] is a shorthand notation for
   456   ``\hyperlink{command.HOL.function}{\mbox{\isa{\isacommand{function}}}}~\isa{{\isachardoublequote}{\isacharparenleft}sequential{\isacharparenright}{\isachardoublequote}}, followed by
   457   automated proof attempts regarding pattern matching and termination.
   458   See \cite{isabelle-function} for further details.
   459 
   460   \item [\hyperlink{command.HOL.termination}{\mbox{\isa{\isacommand{termination}}}}~\isa{f}] commences a
   461   termination proof for the previously defined function \isa{f}.  If
   462   this is omitted, the command refers to the most recent function
   463   definition.  After the proof is closed, the recursive equations and
   464   the induction principle is established.
   465 
   466   \end{descr}
   467 
   468   %FIXME check
   469 
   470   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
   471   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)
   472   refers to a specific induction rule, with parameters named according
   473   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
   474   \hyperlink{command.HOL.function}{\mbox{\isa{\isacommand{function}}}} are numbered (starting from 1).
   475 
   476   The equations provided by these packages may be referred later as
   477   theorem list \isa{{\isachardoublequote}f{\isachardot}simps{\isachardoublequote}}, where \isa{f} is the (collective)
   478   name of the functions defined.  Individual equations may be named
   479   explicitly as well.
   480 
   481   The \hyperlink{command.HOL.function}{\mbox{\isa{\isacommand{function}}}} command accepts the following
   482   options.
   483 
   484   \begin{descr}
   485 
   486   \item [\isa{sequential}] enables a preprocessor which
   487   disambiguates overlapping patterns by making them mutually disjoint.
   488   Earlier equations take precedence over later ones.  This allows to
   489   give the specification in a format very similar to functional
   490   programming.  Note that the resulting simplification and induction
   491   rules correspond to the transformed specification, not the one given
   492   originally. This usually means that each equation given by the user
   493   may result in several theroems.  Also note that this automatic
   494   transformation only works for ML-style datatype patterns.
   495 
   496   \item [\isa{{\isachardoublequote}{\isasymIN}\ name{\isachardoublequote}}] gives the target for the definition.
   497   %FIXME ?!?
   498 
   499   \item [\isa{domintros}] enables the automated generation of
   500   introduction rules for the domain predicate. While mostly not
   501   needed, they can be helpful in some proofs about partial functions.
   502 
   503   \item [\isa{tailrec}] generates the unconstrained recursive
   504   equations even without a termination proof, provided that the
   505   function is tail-recursive. This currently only works
   506 
   507   \item [\isa{{\isachardoublequote}default\ d{\isachardoublequote}}] allows to specify a default value for a
   508   (partial) function, which will ensure that \isa{{\isachardoublequote}f\ x\ {\isacharequal}\ d\ x{\isachardoublequote}}
   509   whenever \isa{{\isachardoublequote}x\ {\isasymnotin}\ f{\isacharunderscore}dom{\isachardoublequote}}.
   510 
   511   \end{descr}%
   512 \end{isamarkuptext}%
   513 \isamarkuptrue%
   514 %
   515 \isamarkupsubsection{Proof methods related to recursive definitions%
   516 }
   517 \isamarkuptrue%
   518 %
   519 \begin{isamarkuptext}%
   520 \begin{matharray}{rcl}
   521     \indexdef{HOL}{method}{pat\_completeness}\hypertarget{method.HOL.pat-completeness}{\hyperlink{method.HOL.pat-completeness}{\mbox{\isa{pat{\isacharunderscore}completeness}}}} & : & \isarmeth \\
   522     \indexdef{HOL}{method}{relation}\hypertarget{method.HOL.relation}{\hyperlink{method.HOL.relation}{\mbox{\isa{relation}}}} & : & \isarmeth \\
   523     \indexdef{HOL}{method}{lexicographic\_order}\hypertarget{method.HOL.lexicographic-order}{\hyperlink{method.HOL.lexicographic-order}{\mbox{\isa{lexicographic{\isacharunderscore}order}}}} & : & \isarmeth \\
   524   \end{matharray}
   525 
   526   \begin{rail}
   527     'relation' term
   528     ;
   529     'lexicographic\_order' (clasimpmod *)
   530     ;
   531   \end{rail}
   532 
   533   \begin{descr}
   534 
   535   \item [\hyperlink{method.HOL.pat-completeness}{\mbox{\isa{pat{\isacharunderscore}completeness}}}] is a specialized method to
   536   solve goals regarding the completeness of pattern matching, as
   537   required by the \hyperlink{command.HOL.function}{\mbox{\isa{\isacommand{function}}}} package (cf.\
   538   \cite{isabelle-function}).
   539 
   540   \item [\hyperlink{method.HOL.relation}{\mbox{\isa{relation}}}~\isa{R}] introduces a termination
   541   proof using the relation \isa{R}.  The resulting proof state will
   542   contain goals expressing that \isa{R} is wellfounded, and that the
   543   arguments of recursive calls decrease with respect to \isa{R}.
   544   Usually, this method is used as the initial proof step of manual
   545   termination proofs.
   546 
   547   \item [\hyperlink{method.HOL.lexicographic-order}{\mbox{\isa{lexicographic{\isacharunderscore}order}}}] attempts a fully
   548   automated termination proof by searching for a lexicographic
   549   combination of size measures on the arguments of the function. The
   550   method accepts the same arguments as the \hyperlink{method.auto}{\mbox{\isa{auto}}} method,
   551   which it uses internally to prove local descents.  The same context
   552   modifiers as for \hyperlink{method.auto}{\mbox{\isa{auto}}} are accepted, see
   553   \secref{sec:clasimp}.
   554 
   555   In case of failure, extensive information is printed, which can help
   556   to analyse the situation (cf.\ \cite{isabelle-function}).
   557 
   558   \end{descr}%
   559 \end{isamarkuptext}%
   560 \isamarkuptrue%
   561 %
   562 \isamarkupsubsection{Old-style recursive function definitions (TFL)%
   563 }
   564 \isamarkuptrue%
   565 %
   566 \begin{isamarkuptext}%
   567 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.
   568 
   569   \begin{matharray}{rcl}
   570     \indexdef{HOL}{command}{recdef}\hypertarget{command.HOL.recdef}{\hyperlink{command.HOL.recdef}{\mbox{\isa{\isacommand{recdef}}}}} & : & \isartrans{theory}{theory} \\
   571     \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)} \\
   572   \end{matharray}
   573 
   574   \begin{rail}
   575     'recdef' ('(' 'permissive' ')')? \\ name term (prop +) hints?
   576     ;
   577     recdeftc thmdecl? tc
   578     ;
   579     hints: '(' 'hints' (recdefmod *) ')'
   580     ;
   581     recdefmod: (('recdef\_simp' | 'recdef\_cong' | 'recdef\_wf') (() | 'add' | 'del') ':' thmrefs) | clasimpmod
   582     ;
   583     tc: nameref ('(' nat ')')?
   584     ;
   585   \end{rail}
   586 
   587   \begin{descr}
   588   
   589   \item [\hyperlink{command.HOL.recdef}{\mbox{\isa{\isacommand{recdef}}}}] defines general well-founded
   590   recursive functions (using the TFL package), see also
   591   \cite{isabelle-HOL}.  The ``\isa{{\isachardoublequote}{\isacharparenleft}permissive{\isacharparenright}{\isachardoublequote}}'' option tells
   592   TFL to recover from failed proof attempts, returning unfinished
   593   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
   594   automated proof process of TFL.  Additional \hyperlink{syntax.clasimpmod}{\mbox{\isa{clasimpmod}}}
   595   declarations (cf.\ \secref{sec:clasimp}) may be given to tune the
   596   context of the Simplifier (cf.\ \secref{sec:simplifier}) and
   597   Classical reasoner (cf.\ \secref{sec:classical}).
   598   
   599   \item [\hyperlink{command.HOL.recdef-tc}{\mbox{\isa{\isacommand{recdef{\isacharunderscore}tc}}}}~\isa{{\isachardoublequote}c\ {\isacharparenleft}i{\isacharparenright}{\isachardoublequote}}] recommences the
   600   proof for leftover termination condition number \isa{i} (default
   601   1) as generated by a \hyperlink{command.HOL.recdef}{\mbox{\isa{\isacommand{recdef}}}} definition of
   602   constant \isa{c}.
   603   
   604   Note that in most cases, \hyperlink{command.HOL.recdef}{\mbox{\isa{\isacommand{recdef}}}} is able to finish
   605   its internal proofs without manual intervention.
   606 
   607   \end{descr}
   608 
   609   \medskip Hints for \hyperlink{command.HOL.recdef}{\mbox{\isa{\isacommand{recdef}}}} may be also declared
   610   globally, using the following attributes.
   611 
   612   \begin{matharray}{rcl}
   613     \indexdef{HOL}{attribute}{recdef\_simp}\hypertarget{attribute.HOL.recdef-simp}{\hyperlink{attribute.HOL.recdef-simp}{\mbox{\isa{recdef{\isacharunderscore}simp}}}} & : & \isaratt \\
   614     \indexdef{HOL}{attribute}{recdef\_cong}\hypertarget{attribute.HOL.recdef-cong}{\hyperlink{attribute.HOL.recdef-cong}{\mbox{\isa{recdef{\isacharunderscore}cong}}}} & : & \isaratt \\
   615     \indexdef{HOL}{attribute}{recdef\_wf}\hypertarget{attribute.HOL.recdef-wf}{\hyperlink{attribute.HOL.recdef-wf}{\mbox{\isa{recdef{\isacharunderscore}wf}}}} & : & \isaratt \\
   616   \end{matharray}
   617 
   618   \begin{rail}
   619     ('recdef\_simp' | 'recdef\_cong' | 'recdef\_wf') (() | 'add' | 'del')
   620     ;
   621   \end{rail}%
   622 \end{isamarkuptext}%
   623 \isamarkuptrue%
   624 %
   625 \isamarkupsection{Definition by specification \label{sec:hol-specification}%
   626 }
   627 \isamarkuptrue%
   628 %
   629 \begin{isamarkuptext}%
   630 \begin{matharray}{rcl}
   631     \indexdef{HOL}{command}{specification}\hypertarget{command.HOL.specification}{\hyperlink{command.HOL.specification}{\mbox{\isa{\isacommand{specification}}}}} & : & \isartrans{theory}{proof(prove)} \\
   632     \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)} \\
   633   \end{matharray}
   634 
   635   \begin{rail}
   636   ('specification' | 'ax\_specification') '(' (decl +) ')' \\ (thmdecl? prop +)
   637   ;
   638   decl: ((name ':')? term '(' 'overloaded' ')'?)
   639   \end{rail}
   640 
   641   \begin{descr}
   642 
   643   \item [\hyperlink{command.HOL.specification}{\mbox{\isa{\isacommand{specification}}}}~\isa{{\isachardoublequote}decls\ {\isasymphi}{\isachardoublequote}}] sets up a
   644   goal stating the existence of terms with the properties specified to
   645   hold for the constants given in \isa{decls}.  After finishing the
   646   proof, the theory will be augmented with definitions for the given
   647   constants, as well as with theorems stating the properties for these
   648   constants.
   649 
   650   \item [\hyperlink{command.HOL.ax-specification}{\mbox{\isa{\isacommand{ax{\isacharunderscore}specification}}}}~\isa{{\isachardoublequote}decls\ {\isasymphi}{\isachardoublequote}}] sets
   651   up a goal stating the existence of terms with the properties
   652   specified to hold for the constants given in \isa{decls}.  After
   653   finishing the proof, the theory will be augmented with axioms
   654   expressing the properties given in the first place.
   655 
   656   \item [\isa{decl}] declares a constant to be defined by the
   657   specification given.  The definition for the constant \isa{c} is
   658   bound to the name \isa{c{\isacharunderscore}def} unless a theorem name is given in
   659   the declaration.  Overloaded constants should be declared as such.
   660 
   661   \end{descr}
   662 
   663   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
   664   construction cannot introduce inconsistencies, whereas \hyperlink{command.HOL.ax-specification}{\mbox{\isa{\isacommand{ax{\isacharunderscore}specification}}}} does introduce axioms, but only after the
   665   user has explicitly proven it to be safe.  A practical issue must be
   666   considered, though: After introducing two constants with the same
   667   properties using \hyperlink{command.HOL.specification}{\mbox{\isa{\isacommand{specification}}}}, one can prove
   668   that the two constants are, in fact, equal.  If this might be a
   669   problem, one should use \hyperlink{command.HOL.ax-specification}{\mbox{\isa{\isacommand{ax{\isacharunderscore}specification}}}}.%
   670 \end{isamarkuptext}%
   671 \isamarkuptrue%
   672 %
   673 \isamarkupsection{Inductive and coinductive definitions \label{sec:hol-inductive}%
   674 }
   675 \isamarkuptrue%
   676 %
   677 \begin{isamarkuptext}%
   678 An \textbf{inductive definition} specifies the least predicate (or
   679   set) \isa{R} closed under given rules: applying a rule to elements
   680   of \isa{R} yields a result within \isa{R}.  For example, a
   681   structural operational semantics is an inductive definition of an
   682   evaluation relation.
   683 
   684   Dually, a \textbf{coinductive definition} specifies the greatest
   685   predicate~/ set \isa{R} that is consistent with given rules: every
   686   element of \isa{R} can be seen as arising by applying a rule to
   687   elements of \isa{R}.  An important example is using bisimulation
   688   relations to formalise equivalence of processes and infinite data
   689   structures.
   690 
   691   \medskip The HOL package is related to the ZF one, which is
   692   described in a separate paper,\footnote{It appeared in CADE
   693   \cite{paulson-CADE}; a longer version is distributed with Isabelle.}
   694   which you should refer to in case of difficulties.  The package is
   695   simpler than that of ZF thanks to implicit type-checking in HOL.
   696   The types of the (co)inductive predicates (or sets) determine the
   697   domain of the fixedpoint definition, and the package does not have
   698   to use inference rules for type-checking.
   699 
   700   \begin{matharray}{rcl}
   701     \indexdef{HOL}{command}{inductive}\hypertarget{command.HOL.inductive}{\hyperlink{command.HOL.inductive}{\mbox{\isa{\isacommand{inductive}}}}} & : & \isarkeep{local{\dsh}theory} \\
   702     \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} \\
   703     \indexdef{HOL}{command}{coinductive}\hypertarget{command.HOL.coinductive}{\hyperlink{command.HOL.coinductive}{\mbox{\isa{\isacommand{coinductive}}}}} & : & \isarkeep{local{\dsh}theory} \\
   704     \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} \\
   705     \indexdef{HOL}{attribute}{mono}\hypertarget{attribute.HOL.mono}{\hyperlink{attribute.HOL.mono}{\mbox{\isa{mono}}}} & : & \isaratt \\
   706   \end{matharray}
   707 
   708   \begin{rail}
   709     ('inductive' | 'inductive\_set' | 'coinductive' | 'coinductive\_set') target? fixes ('for' fixes)? \\
   710     ('where' clauses)? ('monos' thmrefs)?
   711     ;
   712     clauses: (thmdecl? prop + '|')
   713     ;
   714     'mono' (() | 'add' | 'del')
   715     ;
   716   \end{rail}
   717 
   718   \begin{descr}
   719 
   720   \item [\hyperlink{command.HOL.inductive}{\mbox{\isa{\isacommand{inductive}}}} and \hyperlink{command.HOL.coinductive}{\mbox{\isa{\isacommand{coinductive}}}}] define (co)inductive predicates from the
   721   introduction rules given in the \hyperlink{keyword.where}{\mbox{\isa{\isakeyword{where}}}} part.  The
   722   optional \hyperlink{keyword.for}{\mbox{\isa{\isakeyword{for}}}} part contains a list of parameters of the
   723   (co)inductive predicates that remain fixed throughout the
   724   definition.  The optional \hyperlink{keyword.monos}{\mbox{\isa{\isakeyword{monos}}}} section contains
   725   \emph{monotonicity theorems}, which are required for each operator
   726   applied to a recursive set in the introduction rules.  There
   727   \emph{must} be a theorem of the form \isa{{\isachardoublequote}A\ {\isasymle}\ B\ {\isasymLongrightarrow}\ M\ A\ {\isasymle}\ M\ B{\isachardoublequote}},
   728   for each premise \isa{{\isachardoublequote}M\ R\isactrlsub i\ t{\isachardoublequote}} in an introduction rule!
   729 
   730   \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,
   731   allowing the definition of (co)inductive sets.
   732 
   733   \item [\hyperlink{attribute.HOL.mono}{\mbox{\isa{mono}}}] declares monotonicity rules.  These
   734   rule are involved in the automated monotonicity proof of \hyperlink{command.HOL.inductive}{\mbox{\isa{\isacommand{inductive}}}}.
   735 
   736   \end{descr}%
   737 \end{isamarkuptext}%
   738 \isamarkuptrue%
   739 %
   740 \isamarkupsubsection{Derived rules%
   741 }
   742 \isamarkuptrue%
   743 %
   744 \begin{isamarkuptext}%
   745 Each (co)inductive definition \isa{R} adds definitions to the
   746   theory and also proves some theorems:
   747 
   748   \begin{description}
   749 
   750   \item [\isa{R{\isachardot}intros}] is the list of introduction rules as proven
   751   theorems, for the recursive predicates (or sets).  The rules are
   752   also available individually, using the names given them in the
   753   theory file;
   754 
   755   \item [\isa{R{\isachardot}cases}] is the case analysis (or elimination) rule;
   756 
   757   \item [\isa{R{\isachardot}induct} or \isa{R{\isachardot}coinduct}] is the (co)induction
   758   rule.
   759 
   760   \end{description}
   761 
   762   When several predicates \isa{{\isachardoublequote}R\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ R\isactrlsub n{\isachardoublequote}} are
   763   defined simultaneously, the list of introduction rules is called
   764   \isa{{\isachardoublequote}R\isactrlsub {\isadigit{1}}{\isacharunderscore}{\isasymdots}{\isacharunderscore}R\isactrlsub n{\isachardot}intros{\isachardoublequote}}, the case analysis rules are
   765   called \isa{{\isachardoublequote}R\isactrlsub {\isadigit{1}}{\isachardot}cases{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ R\isactrlsub n{\isachardot}cases{\isachardoublequote}}, and the list
   766   of mutual induction rules is called \isa{{\isachardoublequote}R\isactrlsub {\isadigit{1}}{\isacharunderscore}{\isasymdots}{\isacharunderscore}R\isactrlsub n{\isachardot}inducts{\isachardoublequote}}.%
   767 \end{isamarkuptext}%
   768 \isamarkuptrue%
   769 %
   770 \isamarkupsubsection{Monotonicity theorems%
   771 }
   772 \isamarkuptrue%
   773 %
   774 \begin{isamarkuptext}%
   775 Each theory contains a default set of theorems that are used in
   776   monotonicity proofs.  New rules can be added to this set via the
   777   \hyperlink{attribute.HOL.mono}{\mbox{\isa{mono}}} attribute.  The HOL theory \isa{Inductive}
   778   shows how this is done.  In general, the following monotonicity
   779   theorems may be added:
   780 
   781   \begin{itemize}
   782 
   783   \item Theorems of the form \isa{{\isachardoublequote}A\ {\isasymle}\ B\ {\isasymLongrightarrow}\ M\ A\ {\isasymle}\ M\ B{\isachardoublequote}}, for proving
   784   monotonicity of inductive definitions whose introduction rules have
   785   premises involving terms such as \isa{{\isachardoublequote}M\ R\isactrlsub i\ t{\isachardoublequote}}.
   786 
   787   \item Monotonicity theorems for logical operators, which are of the
   788   general form \isa{{\isachardoublequote}{\isacharparenleft}{\isasymdots}\ {\isasymlongrightarrow}\ {\isasymdots}{\isacharparenright}\ {\isasymLongrightarrow}\ {\isasymdots}\ {\isacharparenleft}{\isasymdots}\ {\isasymlongrightarrow}\ {\isasymdots}{\isacharparenright}\ {\isasymLongrightarrow}\ {\isasymdots}\ {\isasymlongrightarrow}\ {\isasymdots}{\isachardoublequote}}.  For example, in
   789   the case of the operator \isa{{\isachardoublequote}{\isasymor}{\isachardoublequote}}, the corresponding theorem is
   790   \[
   791   \infer{\isa{{\isachardoublequote}P\isactrlsub {\isadigit{1}}\ {\isasymor}\ P\isactrlsub {\isadigit{2}}\ {\isasymlongrightarrow}\ Q\isactrlsub {\isadigit{1}}\ {\isasymor}\ Q\isactrlsub {\isadigit{2}}{\isachardoublequote}}}{\isa{{\isachardoublequote}P\isactrlsub {\isadigit{1}}\ {\isasymlongrightarrow}\ Q\isactrlsub {\isadigit{1}}{\isachardoublequote}} & \isa{{\isachardoublequote}P\isactrlsub {\isadigit{2}}\ {\isasymlongrightarrow}\ Q\isactrlsub {\isadigit{2}}{\isachardoublequote}}}
   792   \]
   793 
   794   \item De Morgan style equations for reasoning about the ``polarity''
   795   of expressions, e.g.
   796   \[
   797   \isa{{\isachardoublequote}{\isasymnot}\ {\isasymnot}\ P\ {\isasymlongleftrightarrow}\ P{\isachardoublequote}} \qquad\qquad
   798   \isa{{\isachardoublequote}{\isasymnot}\ {\isacharparenleft}P\ {\isasymand}\ Q{\isacharparenright}\ {\isasymlongleftrightarrow}\ {\isasymnot}\ P\ {\isasymor}\ {\isasymnot}\ Q{\isachardoublequote}}
   799   \]
   800 
   801   \item Equations for reducing complex operators to more primitive
   802   ones whose monotonicity can easily be proved, e.g.
   803   \[
   804   \isa{{\isachardoublequote}{\isacharparenleft}P\ {\isasymlongrightarrow}\ Q{\isacharparenright}\ {\isasymlongleftrightarrow}\ {\isasymnot}\ P\ {\isasymor}\ Q{\isachardoublequote}} \qquad\qquad
   805   \isa{{\isachardoublequote}Ball\ A\ P\ {\isasymequiv}\ {\isasymforall}x{\isachardot}\ x\ {\isasymin}\ A\ {\isasymlongrightarrow}\ P\ x{\isachardoublequote}}
   806   \]
   807 
   808   \end{itemize}
   809 
   810   %FIXME: Example of an inductive definition%
   811 \end{isamarkuptext}%
   812 \isamarkuptrue%
   813 %
   814 \isamarkupsection{Arithmetic proof support%
   815 }
   816 \isamarkuptrue%
   817 %
   818 \begin{isamarkuptext}%
   819 \begin{matharray}{rcl}
   820     \indexdef{HOL}{method}{arith}\hypertarget{method.HOL.arith}{\hyperlink{method.HOL.arith}{\mbox{\isa{arith}}}} & : & \isarmeth \\
   821     \indexdef{HOL}{attribute}{arith\_split}\hypertarget{attribute.HOL.arith-split}{\hyperlink{attribute.HOL.arith-split}{\mbox{\isa{arith{\isacharunderscore}split}}}} & : & \isaratt \\
   822   \end{matharray}
   823 
   824   The \hyperlink{method.HOL.arith}{\mbox{\isa{arith}}} method decides linear arithmetic problems
   825   (on types \isa{nat}, \isa{int}, \isa{real}).  Any current
   826   facts are inserted into the goal before running the procedure.
   827 
   828   The \hyperlink{attribute.HOL.arith-split}{\mbox{\isa{arith{\isacharunderscore}split}}} attribute declares case split
   829   rules to be expanded before the arithmetic procedure is invoked.
   830 
   831   Note that a simpler (but faster) version of arithmetic reasoning is
   832   already performed by the Simplifier.%
   833 \end{isamarkuptext}%
   834 \isamarkuptrue%
   835 %
   836 \isamarkupsection{Cases and induction: emulating tactic scripts \label{sec:hol-induct-tac}%
   837 }
   838 \isamarkuptrue%
   839 %
   840 \begin{isamarkuptext}%
   841 The following important tactical tools of Isabelle/HOL have been
   842   ported to Isar.  These should be never used in proper proof texts!
   843 
   844   \begin{matharray}{rcl}
   845     \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 \\
   846     \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 \\
   847     \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 \\
   848     \indexdef{HOL}{command}{inductive\_cases}\hypertarget{command.HOL.inductive-cases}{\hyperlink{command.HOL.inductive-cases}{\mbox{\isa{\isacommand{inductive{\isacharunderscore}cases}}}}} & : & \isartrans{theory}{theory} \\
   849   \end{matharray}
   850 
   851   \begin{rail}
   852     'case\_tac' goalspec? term rule?
   853     ;
   854     'induct\_tac' goalspec? (insts * 'and') rule?
   855     ;
   856     'ind\_cases' (prop +) ('for' (name +)) ?
   857     ;
   858     'inductive\_cases' (thmdecl? (prop +) + 'and')
   859     ;
   860 
   861     rule: ('rule' ':' thmref)
   862     ;
   863   \end{rail}
   864 
   865   \begin{descr}
   866 
   867   \item [\hyperlink{method.HOL.case-tac}{\mbox{\isa{case{\isacharunderscore}tac}}} and \hyperlink{method.HOL.induct-tac}{\mbox{\isa{induct{\isacharunderscore}tac}}}]
   868   admit to reason about inductive datatypes only (unless an
   869   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.
   870   These tactic emulations feature both goal addressing and dynamic
   871   instantiation.  Note that named rule cases are \emph{not} provided
   872   as would be by the proper \hyperlink{method.induct}{\mbox{\isa{induct}}} and \hyperlink{method.cases}{\mbox{\isa{cases}}} proof
   873   methods (see \secref{sec:cases-induct}).
   874   
   875   \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
   876   forward manner.
   877 
   878   While \hyperlink{method.HOL.ind-cases}{\mbox{\isa{ind{\isacharunderscore}cases}}} is a proof method to apply the
   879   result immediately as elimination rules, \hyperlink{command.HOL.inductive-cases}{\mbox{\isa{\isacommand{inductive{\isacharunderscore}cases}}}} provides case split theorems at the theory level
   880   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
   881   be generalized before applying the resulting rule.
   882 
   883   \end{descr}%
   884 \end{isamarkuptext}%
   885 \isamarkuptrue%
   886 %
   887 \isamarkupsection{Executable code%
   888 }
   889 \isamarkuptrue%
   890 %
   891 \begin{isamarkuptext}%
   892 Isabelle/Pure provides two generic frameworks to support code
   893   generation from executable specifications.  Isabelle/HOL
   894   instantiates these mechanisms in a way that is amenable to end-user
   895   applications.
   896 
   897   One framework generates code from both functional and relational
   898   programs to SML.  See \cite{isabelle-HOL} for further information
   899   (this actually covers the new-style theory format as well).
   900 
   901   \begin{matharray}{rcl}
   902     \indexdef{HOL}{command}{value}\hypertarget{command.HOL.value}{\hyperlink{command.HOL.value}{\mbox{\isa{\isacommand{value}}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isarkeep{theory~|~proof} \\
   903     \indexdef{HOL}{command}{code\_module}\hypertarget{command.HOL.code-module}{\hyperlink{command.HOL.code-module}{\mbox{\isa{\isacommand{code{\isacharunderscore}module}}}}} & : & \isartrans{theory}{theory} \\
   904     \indexdef{HOL}{command}{code\_library}\hypertarget{command.HOL.code-library}{\hyperlink{command.HOL.code-library}{\mbox{\isa{\isacommand{code{\isacharunderscore}library}}}}} & : & \isartrans{theory}{theory} \\
   905     \indexdef{HOL}{command}{consts\_code}\hypertarget{command.HOL.consts-code}{\hyperlink{command.HOL.consts-code}{\mbox{\isa{\isacommand{consts{\isacharunderscore}code}}}}} & : & \isartrans{theory}{theory} \\
   906     \indexdef{HOL}{command}{types\_code}\hypertarget{command.HOL.types-code}{\hyperlink{command.HOL.types-code}{\mbox{\isa{\isacommand{types{\isacharunderscore}code}}}}} & : & \isartrans{theory}{theory} \\  
   907     \indexdef{HOL}{attribute}{code}\hypertarget{attribute.HOL.code}{\hyperlink{attribute.HOL.code}{\mbox{\isa{code}}}} & : & \isaratt \\
   908   \end{matharray}
   909 
   910   \begin{rail}
   911   'value' term
   912   ;
   913 
   914   ( 'code\_module' | 'code\_library' ) modespec ? name ? \\
   915     ( 'file' name ) ? ( 'imports' ( name + ) ) ? \\
   916     'contains' ( ( name '=' term ) + | term + )
   917   ;
   918 
   919   modespec: '(' ( name * ) ')'
   920   ;
   921 
   922   'consts\_code' (codespec +)
   923   ;
   924 
   925   codespec: const template attachment ?
   926   ;
   927 
   928   'types\_code' (tycodespec +)
   929   ;
   930 
   931   tycodespec: name template attachment ?
   932   ;
   933 
   934   const: term
   935   ;
   936 
   937   template: '(' string ')'
   938   ;
   939 
   940   attachment: 'attach' modespec ? verblbrace text verbrbrace
   941   ;
   942 
   943   'code' (name)?
   944   ;
   945   \end{rail}
   946 
   947   \begin{descr}
   948 
   949   \item [\hyperlink{command.HOL.value}{\mbox{\isa{\isacommand{value}}}}~\isa{t}] evaluates and prints a
   950   term using the code generator.
   951 
   952   \end{descr}
   953 
   954   \medskip The other framework generates code from functional programs
   955   (including overloading using type classes) to SML \cite{SML}, OCaml
   956   \cite{OCaml} and Haskell \cite{haskell-revised-report}.
   957   Conceptually, code generation is split up in three steps:
   958   \emph{selection} of code theorems, \emph{translation} into an
   959   abstract executable view and \emph{serialization} to a specific
   960   \emph{target language}.  See \cite{isabelle-codegen} for an
   961   introduction on how to use it.
   962 
   963   \begin{matharray}{rcl}
   964     \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} \\
   965     \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} \\
   966     \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} \\
   967     \indexdef{HOL}{command}{code\_datatype}\hypertarget{command.HOL.code-datatype}{\hyperlink{command.HOL.code-datatype}{\mbox{\isa{\isacommand{code{\isacharunderscore}datatype}}}}} & : & \isartrans{theory}{theory} \\
   968     \indexdef{HOL}{command}{code\_const}\hypertarget{command.HOL.code-const}{\hyperlink{command.HOL.code-const}{\mbox{\isa{\isacommand{code{\isacharunderscore}const}}}}} & : & \isartrans{theory}{theory} \\
   969     \indexdef{HOL}{command}{code\_type}\hypertarget{command.HOL.code-type}{\hyperlink{command.HOL.code-type}{\mbox{\isa{\isacommand{code{\isacharunderscore}type}}}}} & : & \isartrans{theory}{theory} \\
   970     \indexdef{HOL}{command}{code\_class}\hypertarget{command.HOL.code-class}{\hyperlink{command.HOL.code-class}{\mbox{\isa{\isacommand{code{\isacharunderscore}class}}}}} & : & \isartrans{theory}{theory} \\
   971     \indexdef{HOL}{command}{code\_instance}\hypertarget{command.HOL.code-instance}{\hyperlink{command.HOL.code-instance}{\mbox{\isa{\isacommand{code{\isacharunderscore}instance}}}}} & : & \isartrans{theory}{theory} \\
   972     \indexdef{HOL}{command}{code\_monad}\hypertarget{command.HOL.code-monad}{\hyperlink{command.HOL.code-monad}{\mbox{\isa{\isacommand{code{\isacharunderscore}monad}}}}} & : & \isartrans{theory}{theory} \\
   973     \indexdef{HOL}{command}{code\_reserved}\hypertarget{command.HOL.code-reserved}{\hyperlink{command.HOL.code-reserved}{\mbox{\isa{\isacommand{code{\isacharunderscore}reserved}}}}} & : & \isartrans{theory}{theory} \\
   974     \indexdef{HOL}{command}{code\_include}\hypertarget{command.HOL.code-include}{\hyperlink{command.HOL.code-include}{\mbox{\isa{\isacommand{code{\isacharunderscore}include}}}}} & : & \isartrans{theory}{theory} \\
   975     \indexdef{HOL}{command}{code\_modulename}\hypertarget{command.HOL.code-modulename}{\hyperlink{command.HOL.code-modulename}{\mbox{\isa{\isacommand{code{\isacharunderscore}modulename}}}}} & : & \isartrans{theory}{theory} \\
   976     \indexdef{HOL}{command}{code\_exception}\hypertarget{command.HOL.code-exception}{\hyperlink{command.HOL.code-exception}{\mbox{\isa{\isacommand{code{\isacharunderscore}exception}}}}} & : & \isartrans{theory}{theory} \\
   977     \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} \\
   978     \indexdef{HOL}{attribute}{code}\hypertarget{attribute.HOL.code}{\hyperlink{attribute.HOL.code}{\mbox{\isa{code}}}} & : & \isaratt \\
   979   \end{matharray}
   980 
   981   \begin{rail}
   982     'export\_code' ( constexpr + ) ? \\
   983       ( ( 'in' target ( 'module\_name' string ) ? \\
   984         ( 'file' ( string | '-' ) ) ? ( '(' args ')' ) ?) + ) ?
   985     ;
   986 
   987     'code\_thms' ( constexpr + ) ?
   988     ;
   989 
   990     'code\_deps' ( constexpr + ) ?
   991     ;
   992 
   993     const: term
   994     ;
   995 
   996     constexpr: ( const | 'name.*' | '*' )
   997     ;
   998 
   999     typeconstructor: nameref
  1000     ;
  1001 
  1002     class: nameref
  1003     ;
  1004 
  1005     target: 'OCaml' | 'SML' | 'Haskell'
  1006     ;
  1007 
  1008     'code\_datatype' const +
  1009     ;
  1010 
  1011     'code\_const' (const + 'and') \\
  1012       ( ( '(' target ( syntax ? + 'and' ) ')' ) + )
  1013     ;
  1014 
  1015     'code\_type' (typeconstructor + 'and') \\
  1016       ( ( '(' target ( syntax ? + 'and' ) ')' ) + )
  1017     ;
  1018 
  1019     'code\_class' (class + 'and') \\
  1020       ( ( '(' target \\
  1021         ( ( string ('where' \\
  1022           ( const ( '==' | equiv ) string ) + ) ? ) ? + 'and' ) ')' ) + )
  1023     ;
  1024 
  1025     'code\_instance' (( typeconstructor '::' class ) + 'and') \\
  1026       ( ( '(' target ( '-' ? + 'and' ) ')' ) + )
  1027     ;
  1028 
  1029     'code\_monad' const const target
  1030     ;
  1031 
  1032     'code\_reserved' target ( string + )
  1033     ;
  1034 
  1035     'code\_include' target ( string ( string | '-') )
  1036     ;
  1037 
  1038     'code\_modulename' target ( ( string string ) + )
  1039     ;
  1040 
  1041     'code\_exception' ( const + )
  1042     ;
  1043 
  1044     syntax: string | ( 'infix' | 'infixl' | 'infixr' ) nat string
  1045     ;
  1046 
  1047     'code' ('func' | 'inline') ( 'del' )?
  1048     ;
  1049   \end{rail}
  1050 
  1051   \begin{descr}
  1052 
  1053   \item [\hyperlink{command.HOL.export-code}{\mbox{\isa{\isacommand{export{\isacharunderscore}code}}}}] is the canonical interface
  1054   for generating and serializing code: for a given list of constants,
  1055   code is generated for the specified target languages.  Abstract code
  1056   is cached incrementally.  If no constant is given, the currently
  1057   cached code is serialized.  If no serialization instruction is
  1058   given, only abstract code is cached.
  1059 
  1060   Constants may be specified by giving them literally, referring to
  1061   all executable contants within a certain theory by giving \isa{{\isachardoublequote}name{\isachardot}{\isacharasterisk}{\isachardoublequote}}, or referring to \emph{all} executable constants currently
  1062   available by giving \isa{{\isachardoublequote}{\isacharasterisk}{\isachardoublequote}}.
  1063 
  1064   By default, for each involved theory one corresponding name space
  1065   module is generated.  Alternativly, a module name may be specified
  1066   after the \hyperlink{keyword.module-name}{\mbox{\isa{\isakeyword{module{\isacharunderscore}name}}}} keyword; then \emph{all} code is
  1067   placed in this module.
  1068 
  1069   For \emph{SML} and \emph{OCaml}, the file specification refers to a
  1070   single file; for \emph{Haskell}, it refers to a whole directory,
  1071   where code is generated in multiple files reflecting the module
  1072   hierarchy.  The file specification ``\isa{{\isachardoublequote}{\isacharminus}{\isachardoublequote}}'' denotes standard
  1073   output.  For \emph{SML}, omitting the file specification compiles
  1074   code internally in the context of the current ML session.
  1075 
  1076   Serializers take an optional list of arguments in parentheses.  For
  1077   \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
  1078   declaration.
  1079 
  1080   \item [\hyperlink{command.HOL.code-thms}{\mbox{\isa{\isacommand{code{\isacharunderscore}thms}}}}] prints a list of theorems
  1081   representing the corresponding program containing all given
  1082   constants; if no constants are given, the currently cached code
  1083   theorems are printed.
  1084 
  1085   \item [\hyperlink{command.HOL.code-deps}{\mbox{\isa{\isacommand{code{\isacharunderscore}deps}}}}] visualizes dependencies of
  1086   theorems representing the corresponding program containing all given
  1087   constants; if no constants are given, the currently cached code
  1088   theorems are visualized.
  1089 
  1090   \item [\hyperlink{command.HOL.code-datatype}{\mbox{\isa{\isacommand{code{\isacharunderscore}datatype}}}}] specifies a constructor set
  1091   for a logical type.
  1092 
  1093   \item [\hyperlink{command.HOL.code-const}{\mbox{\isa{\isacommand{code{\isacharunderscore}const}}}}] associates a list of constants
  1094   with target-specific serializations; omitting a serialization
  1095   deletes an existing serialization.
  1096 
  1097   \item [\hyperlink{command.HOL.code-type}{\mbox{\isa{\isacommand{code{\isacharunderscore}type}}}}] associates a list of type
  1098   constructors with target-specific serializations; omitting a
  1099   serialization deletes an existing serialization.
  1100 
  1101   \item [\hyperlink{command.HOL.code-class}{\mbox{\isa{\isacommand{code{\isacharunderscore}class}}}}] associates a list of classes
  1102   with target-specific class names; in addition, constants associated
  1103   with this class may be given target-specific names used for instance
  1104   declarations; omitting a serialization deletes an existing
  1105   serialization.  This applies only to \emph{Haskell}.
  1106 
  1107   \item [\hyperlink{command.HOL.code-instance}{\mbox{\isa{\isacommand{code{\isacharunderscore}instance}}}}] declares a list of type
  1108   constructor / class instance relations as ``already present'' for a
  1109   given target.  Omitting a ``\isa{{\isachardoublequote}{\isacharminus}{\isachardoublequote}}'' deletes an existing
  1110   ``already present'' declaration.  This applies only to
  1111   \emph{Haskell}.
  1112 
  1113   \item [\hyperlink{command.HOL.code-monad}{\mbox{\isa{\isacommand{code{\isacharunderscore}monad}}}}] provides an auxiliary
  1114   mechanism to generate monadic code.
  1115 
  1116   \item [\hyperlink{command.HOL.code-reserved}{\mbox{\isa{\isacommand{code{\isacharunderscore}reserved}}}}] declares a list of names as
  1117   reserved for a given target, preventing it to be shadowed by any
  1118   generated code.
  1119 
  1120   \item [\hyperlink{command.HOL.code-include}{\mbox{\isa{\isacommand{code{\isacharunderscore}include}}}}] adds arbitrary named content
  1121   (``include'') to generated code.  A as last argument ``\isa{{\isachardoublequote}{\isacharminus}{\isachardoublequote}}''
  1122   will remove an already added ``include''.
  1123 
  1124   \item [\hyperlink{command.HOL.code-modulename}{\mbox{\isa{\isacommand{code{\isacharunderscore}modulename}}}}] declares aliasings from
  1125   one module name onto another.
  1126 
  1127   \item [\hyperlink{command.HOL.code-exception}{\mbox{\isa{\isacommand{code{\isacharunderscore}exception}}}}] declares constants which
  1128   are not required to have a definition by a defining equations; these
  1129   are mapped on exceptions instead.
  1130 
  1131   \item [\hyperlink{attribute.HOL.code}{\mbox{\isa{code}}}~\isa{func}] explicitly selects (or
  1132   with option ``\isa{{\isachardoublequote}del{\isacharcolon}{\isachardoublequote}}'' deselects) a defining equation for
  1133   code generation.  Usually packages introducing defining equations
  1134   provide a resonable default setup for selection.
  1135 
  1136   \item [\hyperlink{attribute.HOL.code}{\mbox{\isa{code}}}\isa{inline}] declares (or with
  1137   option ``\isa{{\isachardoublequote}del{\isacharcolon}{\isachardoublequote}}'' removes) inlining theorems which are
  1138   applied as rewrite rules to any defining equation during
  1139   preprocessing.
  1140 
  1141   \item [\hyperlink{command.HOL.print-codesetup}{\mbox{\isa{\isacommand{print{\isacharunderscore}codesetup}}}}] gives an overview on
  1142   selected defining equations, code generator datatypes and
  1143   preprocessor setup.
  1144 
  1145   \end{descr}%
  1146 \end{isamarkuptext}%
  1147 \isamarkuptrue%
  1148 %
  1149 \isadelimtheory
  1150 %
  1151 \endisadelimtheory
  1152 %
  1153 \isatagtheory
  1154 \isacommand{end}\isamarkupfalse%
  1155 %
  1156 \endisatagtheory
  1157 {\isafoldtheory}%
  1158 %
  1159 \isadelimtheory
  1160 %
  1161 \endisadelimtheory
  1162 \isanewline
  1163 \isanewline
  1164 \end{isabellebody}%
  1165 %%% Local Variables:
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  1167 %%% TeX-master: "root"
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