doc-src/IsarRef/Thy/document/HOL_Specific.tex
author haftmann
Fri Oct 24 17:48:40 2008 +0200 (2008-10-24)
changeset 28687 150a8a1eae1a
parent 28603 b40800eef8a7
child 28788 ff9d8a8932e4
permissions -rw-r--r--
simplified syntax for class parameters
     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}{proof} \\
   369   \end{matharray}
   370 
   371   \begin{rail}
   372     'datatype' (dtspec + 'and')
   373     ;
   374     'rep\_datatype' ('(' (name +) ')')? (term +)
   375     ;
   376 
   377     dtspec: parname? typespec infix? '=' (cons + '|')
   378     ;
   379     cons: name (type *) mixfix?
   380   \end{rail}
   381 
   382   \begin{descr}
   383 
   384   \item [\hyperlink{command.HOL.datatype}{\mbox{\isa{\isacommand{datatype}}}}] defines inductive datatypes in
   385   HOL.
   386 
   387   \item [\hyperlink{command.HOL.rep-datatype}{\mbox{\isa{\isacommand{rep{\isacharunderscore}datatype}}}}] represents existing types as
   388   inductive ones, generating the standard infrastructure of derived
   389   concepts (primitive recursion etc.).
   390 
   391   \end{descr}
   392 
   393   The induction and exhaustion theorems generated provide case names
   394   according to the constructors involved, while parameters are named
   395   after the types (see also \secref{sec:cases-induct}).
   396 
   397   See \cite{isabelle-HOL} for more details on datatypes, but beware of
   398   the old-style theory syntax being used there!  Apart from proper
   399   proof methods for case-analysis and induction, there are also
   400   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
   401   to refer directly to the internal structure of subgoals (including
   402   internally bound parameters).%
   403 \end{isamarkuptext}%
   404 \isamarkuptrue%
   405 %
   406 \isamarkupsection{Recursive functions \label{sec:recursion}%
   407 }
   408 \isamarkuptrue%
   409 %
   410 \begin{isamarkuptext}%
   411 \begin{matharray}{rcl}
   412     \indexdef{HOL}{command}{primrec}\hypertarget{command.HOL.primrec}{\hyperlink{command.HOL.primrec}{\mbox{\isa{\isacommand{primrec}}}}} & : & \isarkeep{local{\dsh}theory} \\
   413     \indexdef{HOL}{command}{fun}\hypertarget{command.HOL.fun}{\hyperlink{command.HOL.fun}{\mbox{\isa{\isacommand{fun}}}}} & : & \isarkeep{local{\dsh}theory} \\
   414     \indexdef{HOL}{command}{function}\hypertarget{command.HOL.function}{\hyperlink{command.HOL.function}{\mbox{\isa{\isacommand{function}}}}} & : & \isartrans{local{\dsh}theory}{proof(prove)} \\
   415     \indexdef{HOL}{command}{termination}\hypertarget{command.HOL.termination}{\hyperlink{command.HOL.termination}{\mbox{\isa{\isacommand{termination}}}}} & : & \isartrans{local{\dsh}theory}{proof(prove)} \\
   416   \end{matharray}
   417 
   418   \begin{rail}
   419     'primrec' target? fixes 'where' equations
   420     ;
   421     equations: (thmdecl? prop + '|')
   422     ;
   423     ('fun' | 'function') target? functionopts? fixes 'where' clauses
   424     ;
   425     clauses: (thmdecl? prop ('(' 'otherwise' ')')? + '|')
   426     ;
   427     functionopts: '(' (('sequential' | 'domintros' | 'tailrec' | 'default' term) + ',') ')'
   428     ;
   429     'termination' ( term )?
   430   \end{rail}
   431 
   432   \begin{descr}
   433 
   434   \item [\hyperlink{command.HOL.primrec}{\mbox{\isa{\isacommand{primrec}}}}] defines primitive recursive
   435   functions over datatypes, see also \cite{isabelle-HOL}.
   436 
   437   \item [\hyperlink{command.HOL.function}{\mbox{\isa{\isacommand{function}}}}] defines functions by general
   438   wellfounded recursion. A detailed description with examples can be
   439   found in \cite{isabelle-function}. The function is specified by a
   440   set of (possibly conditional) recursive equations with arbitrary
   441   pattern matching. The command generates proof obligations for the
   442   completeness and the compatibility of patterns.
   443 
   444   The defined function is considered partial, and the resulting
   445   simplification rules (named \isa{{\isachardoublequote}f{\isachardot}psimps{\isachardoublequote}}) and induction rule
   446   (named \isa{{\isachardoublequote}f{\isachardot}pinduct{\isachardoublequote}}) are guarded by a generated domain
   447   predicate \isa{{\isachardoublequote}f{\isacharunderscore}dom{\isachardoublequote}}. The \hyperlink{command.HOL.termination}{\mbox{\isa{\isacommand{termination}}}}
   448   command can then be used to establish that the function is total.
   449 
   450   \item [\hyperlink{command.HOL.fun}{\mbox{\isa{\isacommand{fun}}}}] is a shorthand notation for
   451   ``\hyperlink{command.HOL.function}{\mbox{\isa{\isacommand{function}}}}~\isa{{\isachardoublequote}{\isacharparenleft}sequential{\isacharparenright}{\isachardoublequote}}, followed by
   452   automated proof attempts regarding pattern matching and termination.
   453   See \cite{isabelle-function} for further details.
   454 
   455   \item [\hyperlink{command.HOL.termination}{\mbox{\isa{\isacommand{termination}}}}~\isa{f}] commences a
   456   termination proof for the previously defined function \isa{f}.  If
   457   this is omitted, the command refers to the most recent function
   458   definition.  After the proof is closed, the recursive equations and
   459   the induction principle is established.
   460 
   461   \end{descr}
   462 
   463   %FIXME check
   464 
   465   Recursive definitions introduced by the \hyperlink{command.HOL.function}{\mbox{\isa{\isacommand{function}}}}
   466   command accommodate
   467   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)
   468   refers to a specific induction rule, with parameters named according
   469   to the user-specified equations.
   470   For the \hyperlink{command.HOL.primrec}{\mbox{\isa{\isacommand{primrec}}}} the induction principle coincides
   471   with structural recursion on the datatype the recursion is carried
   472   out.
   473   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{domintros}] enables the automated generation of
   497   introduction rules for the domain predicate. While mostly not
   498   needed, they can be helpful in some proofs about partial functions.
   499 
   500   \item [\isa{tailrec}] generates the unconstrained recursive
   501   equations even without a termination proof, provided that the
   502   function is tail-recursive. This currently only works
   503 
   504   \item [\isa{{\isachardoublequote}default\ d{\isachardoublequote}}] allows to specify a default value for a
   505   (partial) function, which will ensure that \isa{{\isachardoublequote}f\ x\ {\isacharequal}\ d\ x{\isachardoublequote}}
   506   whenever \isa{{\isachardoublequote}x\ {\isasymnotin}\ f{\isacharunderscore}dom{\isachardoublequote}}.
   507 
   508   \end{descr}%
   509 \end{isamarkuptext}%
   510 \isamarkuptrue%
   511 %
   512 \isamarkupsubsection{Proof methods related to recursive definitions%
   513 }
   514 \isamarkuptrue%
   515 %
   516 \begin{isamarkuptext}%
   517 \begin{matharray}{rcl}
   518     \indexdef{HOL}{method}{pat\_completeness}\hypertarget{method.HOL.pat-completeness}{\hyperlink{method.HOL.pat-completeness}{\mbox{\isa{pat{\isacharunderscore}completeness}}}} & : & \isarmeth \\
   519     \indexdef{HOL}{method}{relation}\hypertarget{method.HOL.relation}{\hyperlink{method.HOL.relation}{\mbox{\isa{relation}}}} & : & \isarmeth \\
   520     \indexdef{HOL}{method}{lexicographic\_order}\hypertarget{method.HOL.lexicographic-order}{\hyperlink{method.HOL.lexicographic-order}{\mbox{\isa{lexicographic{\isacharunderscore}order}}}} & : & \isarmeth \\
   521   \end{matharray}
   522 
   523   \begin{rail}
   524     'relation' term
   525     ;
   526     'lexicographic\_order' (clasimpmod *)
   527     ;
   528   \end{rail}
   529 
   530   \begin{descr}
   531 
   532   \item [\hyperlink{method.HOL.pat-completeness}{\mbox{\isa{pat{\isacharunderscore}completeness}}}] is a specialized method to
   533   solve goals regarding the completeness of pattern matching, as
   534   required by the \hyperlink{command.HOL.function}{\mbox{\isa{\isacommand{function}}}} package (cf.\
   535   \cite{isabelle-function}).
   536 
   537   \item [\hyperlink{method.HOL.relation}{\mbox{\isa{relation}}}~\isa{R}] introduces a termination
   538   proof using the relation \isa{R}.  The resulting proof state will
   539   contain goals expressing that \isa{R} is wellfounded, and that the
   540   arguments of recursive calls decrease with respect to \isa{R}.
   541   Usually, this method is used as the initial proof step of manual
   542   termination proofs.
   543 
   544   \item [\hyperlink{method.HOL.lexicographic-order}{\mbox{\isa{lexicographic{\isacharunderscore}order}}}] attempts a fully
   545   automated termination proof by searching for a lexicographic
   546   combination of size measures on the arguments of the function. The
   547   method accepts the same arguments as the \hyperlink{method.auto}{\mbox{\isa{auto}}} method,
   548   which it uses internally to prove local descents.  The same context
   549   modifiers as for \hyperlink{method.auto}{\mbox{\isa{auto}}} are accepted, see
   550   \secref{sec:clasimp}.
   551 
   552   In case of failure, extensive information is printed, which can help
   553   to analyse the situation (cf.\ \cite{isabelle-function}).
   554 
   555   \end{descr}%
   556 \end{isamarkuptext}%
   557 \isamarkuptrue%
   558 %
   559 \isamarkupsubsection{Old-style recursive function definitions (TFL)%
   560 }
   561 \isamarkuptrue%
   562 %
   563 \begin{isamarkuptext}%
   564 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.
   565 
   566   \begin{matharray}{rcl}
   567     \indexdef{HOL}{command}{recdef}\hypertarget{command.HOL.recdef}{\hyperlink{command.HOL.recdef}{\mbox{\isa{\isacommand{recdef}}}}} & : & \isartrans{theory}{theory} \\
   568     \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)} \\
   569   \end{matharray}
   570 
   571   \begin{rail}
   572     'recdef' ('(' 'permissive' ')')? \\ name term (prop +) hints?
   573     ;
   574     recdeftc thmdecl? tc
   575     ;
   576     hints: '(' 'hints' (recdefmod *) ')'
   577     ;
   578     recdefmod: (('recdef\_simp' | 'recdef\_cong' | 'recdef\_wf') (() | 'add' | 'del') ':' thmrefs) | clasimpmod
   579     ;
   580     tc: nameref ('(' nat ')')?
   581     ;
   582   \end{rail}
   583 
   584   \begin{descr}
   585   
   586   \item [\hyperlink{command.HOL.recdef}{\mbox{\isa{\isacommand{recdef}}}}] defines general well-founded
   587   recursive functions (using the TFL package), see also
   588   \cite{isabelle-HOL}.  The ``\isa{{\isachardoublequote}{\isacharparenleft}permissive{\isacharparenright}{\isachardoublequote}}'' option tells
   589   TFL to recover from failed proof attempts, returning unfinished
   590   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
   591   automated proof process of TFL.  Additional \hyperlink{syntax.clasimpmod}{\mbox{\isa{clasimpmod}}}
   592   declarations (cf.\ \secref{sec:clasimp}) may be given to tune the
   593   context of the Simplifier (cf.\ \secref{sec:simplifier}) and
   594   Classical reasoner (cf.\ \secref{sec:classical}).
   595   
   596   \item [\hyperlink{command.HOL.recdef-tc}{\mbox{\isa{\isacommand{recdef{\isacharunderscore}tc}}}}~\isa{{\isachardoublequote}c\ {\isacharparenleft}i{\isacharparenright}{\isachardoublequote}}] recommences the
   597   proof for leftover termination condition number \isa{i} (default
   598   1) as generated by a \hyperlink{command.HOL.recdef}{\mbox{\isa{\isacommand{recdef}}}} definition of
   599   constant \isa{c}.
   600   
   601   Note that in most cases, \hyperlink{command.HOL.recdef}{\mbox{\isa{\isacommand{recdef}}}} is able to finish
   602   its internal proofs without manual intervention.
   603 
   604   \end{descr}
   605 
   606   \medskip Hints for \hyperlink{command.HOL.recdef}{\mbox{\isa{\isacommand{recdef}}}} may be also declared
   607   globally, using the following attributes.
   608 
   609   \begin{matharray}{rcl}
   610     \indexdef{HOL}{attribute}{recdef\_simp}\hypertarget{attribute.HOL.recdef-simp}{\hyperlink{attribute.HOL.recdef-simp}{\mbox{\isa{recdef{\isacharunderscore}simp}}}} & : & \isaratt \\
   611     \indexdef{HOL}{attribute}{recdef\_cong}\hypertarget{attribute.HOL.recdef-cong}{\hyperlink{attribute.HOL.recdef-cong}{\mbox{\isa{recdef{\isacharunderscore}cong}}}} & : & \isaratt \\
   612     \indexdef{HOL}{attribute}{recdef\_wf}\hypertarget{attribute.HOL.recdef-wf}{\hyperlink{attribute.HOL.recdef-wf}{\mbox{\isa{recdef{\isacharunderscore}wf}}}} & : & \isaratt \\
   613   \end{matharray}
   614 
   615   \begin{rail}
   616     ('recdef\_simp' | 'recdef\_cong' | 'recdef\_wf') (() | 'add' | 'del')
   617     ;
   618   \end{rail}%
   619 \end{isamarkuptext}%
   620 \isamarkuptrue%
   621 %
   622 \isamarkupsection{Inductive and coinductive definitions \label{sec:hol-inductive}%
   623 }
   624 \isamarkuptrue%
   625 %
   626 \begin{isamarkuptext}%
   627 An \textbf{inductive definition} specifies the least predicate (or
   628   set) \isa{R} closed under given rules: applying a rule to elements
   629   of \isa{R} yields a result within \isa{R}.  For example, a
   630   structural operational semantics is an inductive definition of an
   631   evaluation relation.
   632 
   633   Dually, a \textbf{coinductive definition} specifies the greatest
   634   predicate~/ set \isa{R} that is consistent with given rules: every
   635   element of \isa{R} can be seen as arising by applying a rule to
   636   elements of \isa{R}.  An important example is using bisimulation
   637   relations to formalise equivalence of processes and infinite data
   638   structures.
   639 
   640   \medskip The HOL package is related to the ZF one, which is
   641   described in a separate paper,\footnote{It appeared in CADE
   642   \cite{paulson-CADE}; a longer version is distributed with Isabelle.}
   643   which you should refer to in case of difficulties.  The package is
   644   simpler than that of ZF thanks to implicit type-checking in HOL.
   645   The types of the (co)inductive predicates (or sets) determine the
   646   domain of the fixedpoint definition, and the package does not have
   647   to use inference rules for type-checking.
   648 
   649   \begin{matharray}{rcl}
   650     \indexdef{HOL}{command}{inductive}\hypertarget{command.HOL.inductive}{\hyperlink{command.HOL.inductive}{\mbox{\isa{\isacommand{inductive}}}}} & : & \isarkeep{local{\dsh}theory} \\
   651     \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} \\
   652     \indexdef{HOL}{command}{coinductive}\hypertarget{command.HOL.coinductive}{\hyperlink{command.HOL.coinductive}{\mbox{\isa{\isacommand{coinductive}}}}} & : & \isarkeep{local{\dsh}theory} \\
   653     \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} \\
   654     \indexdef{HOL}{attribute}{mono}\hypertarget{attribute.HOL.mono}{\hyperlink{attribute.HOL.mono}{\mbox{\isa{mono}}}} & : & \isaratt \\
   655   \end{matharray}
   656 
   657   \begin{rail}
   658     ('inductive' | 'inductive\_set' | 'coinductive' | 'coinductive\_set') target? fixes ('for' fixes)? \\
   659     ('where' clauses)? ('monos' thmrefs)?
   660     ;
   661     clauses: (thmdecl? prop + '|')
   662     ;
   663     'mono' (() | 'add' | 'del')
   664     ;
   665   \end{rail}
   666 
   667   \begin{descr}
   668 
   669   \item [\hyperlink{command.HOL.inductive}{\mbox{\isa{\isacommand{inductive}}}} and \hyperlink{command.HOL.coinductive}{\mbox{\isa{\isacommand{coinductive}}}}] define (co)inductive predicates from the
   670   introduction rules given in the \hyperlink{keyword.where}{\mbox{\isa{\isakeyword{where}}}} part.  The
   671   optional \hyperlink{keyword.for}{\mbox{\isa{\isakeyword{for}}}} part contains a list of parameters of the
   672   (co)inductive predicates that remain fixed throughout the
   673   definition.  The optional \hyperlink{keyword.monos}{\mbox{\isa{\isakeyword{monos}}}} section contains
   674   \emph{monotonicity theorems}, which are required for each operator
   675   applied to a recursive set in the introduction rules.  There
   676   \emph{must} be a theorem of the form \isa{{\isachardoublequote}A\ {\isasymle}\ B\ {\isasymLongrightarrow}\ M\ A\ {\isasymle}\ M\ B{\isachardoublequote}},
   677   for each premise \isa{{\isachardoublequote}M\ R\isactrlsub i\ t{\isachardoublequote}} in an introduction rule!
   678 
   679   \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,
   680   allowing the definition of (co)inductive sets.
   681 
   682   \item [\hyperlink{attribute.HOL.mono}{\mbox{\isa{mono}}}] declares monotonicity rules.  These
   683   rule are involved in the automated monotonicity proof of \hyperlink{command.HOL.inductive}{\mbox{\isa{\isacommand{inductive}}}}.
   684 
   685   \end{descr}%
   686 \end{isamarkuptext}%
   687 \isamarkuptrue%
   688 %
   689 \isamarkupsubsection{Derived rules%
   690 }
   691 \isamarkuptrue%
   692 %
   693 \begin{isamarkuptext}%
   694 Each (co)inductive definition \isa{R} adds definitions to the
   695   theory and also proves some theorems:
   696 
   697   \begin{description}
   698 
   699   \item [\isa{R{\isachardot}intros}] is the list of introduction rules as proven
   700   theorems, for the recursive predicates (or sets).  The rules are
   701   also available individually, using the names given them in the
   702   theory file;
   703 
   704   \item [\isa{R{\isachardot}cases}] is the case analysis (or elimination) rule;
   705 
   706   \item [\isa{R{\isachardot}induct} or \isa{R{\isachardot}coinduct}] is the (co)induction
   707   rule.
   708 
   709   \end{description}
   710 
   711   When several predicates \isa{{\isachardoublequote}R\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ R\isactrlsub n{\isachardoublequote}} are
   712   defined simultaneously, the list of introduction rules is called
   713   \isa{{\isachardoublequote}R\isactrlsub {\isadigit{1}}{\isacharunderscore}{\isasymdots}{\isacharunderscore}R\isactrlsub n{\isachardot}intros{\isachardoublequote}}, the case analysis rules are
   714   called \isa{{\isachardoublequote}R\isactrlsub {\isadigit{1}}{\isachardot}cases{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ R\isactrlsub n{\isachardot}cases{\isachardoublequote}}, and the list
   715   of mutual induction rules is called \isa{{\isachardoublequote}R\isactrlsub {\isadigit{1}}{\isacharunderscore}{\isasymdots}{\isacharunderscore}R\isactrlsub n{\isachardot}inducts{\isachardoublequote}}.%
   716 \end{isamarkuptext}%
   717 \isamarkuptrue%
   718 %
   719 \isamarkupsubsection{Monotonicity theorems%
   720 }
   721 \isamarkuptrue%
   722 %
   723 \begin{isamarkuptext}%
   724 Each theory contains a default set of theorems that are used in
   725   monotonicity proofs.  New rules can be added to this set via the
   726   \hyperlink{attribute.HOL.mono}{\mbox{\isa{mono}}} attribute.  The HOL theory \isa{Inductive}
   727   shows how this is done.  In general, the following monotonicity
   728   theorems may be added:
   729 
   730   \begin{itemize}
   731 
   732   \item Theorems of the form \isa{{\isachardoublequote}A\ {\isasymle}\ B\ {\isasymLongrightarrow}\ M\ A\ {\isasymle}\ M\ B{\isachardoublequote}}, for proving
   733   monotonicity of inductive definitions whose introduction rules have
   734   premises involving terms such as \isa{{\isachardoublequote}M\ R\isactrlsub i\ t{\isachardoublequote}}.
   735 
   736   \item Monotonicity theorems for logical operators, which are of the
   737   general form \isa{{\isachardoublequote}{\isacharparenleft}{\isasymdots}\ {\isasymlongrightarrow}\ {\isasymdots}{\isacharparenright}\ {\isasymLongrightarrow}\ {\isasymdots}\ {\isacharparenleft}{\isasymdots}\ {\isasymlongrightarrow}\ {\isasymdots}{\isacharparenright}\ {\isasymLongrightarrow}\ {\isasymdots}\ {\isasymlongrightarrow}\ {\isasymdots}{\isachardoublequote}}.  For example, in
   738   the case of the operator \isa{{\isachardoublequote}{\isasymor}{\isachardoublequote}}, the corresponding theorem is
   739   \[
   740   \infer{\isa{{\isachardoublequote}P\isactrlsub {\isadigit{1}}\ {\isasymor}\ P\isactrlsub {\isadigit{2}}\ {\isasymlongrightarrow}\ Q\isactrlsub {\isadigit{1}}\ {\isasymor}\ Q\isactrlsub {\isadigit{2}}{\isachardoublequote}}}{\isa{{\isachardoublequote}P\isactrlsub {\isadigit{1}}\ {\isasymlongrightarrow}\ Q\isactrlsub {\isadigit{1}}{\isachardoublequote}} & \isa{{\isachardoublequote}P\isactrlsub {\isadigit{2}}\ {\isasymlongrightarrow}\ Q\isactrlsub {\isadigit{2}}{\isachardoublequote}}}
   741   \]
   742 
   743   \item De Morgan style equations for reasoning about the ``polarity''
   744   of expressions, e.g.
   745   \[
   746   \isa{{\isachardoublequote}{\isasymnot}\ {\isasymnot}\ P\ {\isasymlongleftrightarrow}\ P{\isachardoublequote}} \qquad\qquad
   747   \isa{{\isachardoublequote}{\isasymnot}\ {\isacharparenleft}P\ {\isasymand}\ Q{\isacharparenright}\ {\isasymlongleftrightarrow}\ {\isasymnot}\ P\ {\isasymor}\ {\isasymnot}\ Q{\isachardoublequote}}
   748   \]
   749 
   750   \item Equations for reducing complex operators to more primitive
   751   ones whose monotonicity can easily be proved, e.g.
   752   \[
   753   \isa{{\isachardoublequote}{\isacharparenleft}P\ {\isasymlongrightarrow}\ Q{\isacharparenright}\ {\isasymlongleftrightarrow}\ {\isasymnot}\ P\ {\isasymor}\ Q{\isachardoublequote}} \qquad\qquad
   754   \isa{{\isachardoublequote}Ball\ A\ P\ {\isasymequiv}\ {\isasymforall}x{\isachardot}\ x\ {\isasymin}\ A\ {\isasymlongrightarrow}\ P\ x{\isachardoublequote}}
   755   \]
   756 
   757   \end{itemize}
   758 
   759   %FIXME: Example of an inductive definition%
   760 \end{isamarkuptext}%
   761 \isamarkuptrue%
   762 %
   763 \isamarkupsection{Arithmetic proof support%
   764 }
   765 \isamarkuptrue%
   766 %
   767 \begin{isamarkuptext}%
   768 \begin{matharray}{rcl}
   769     \indexdef{HOL}{method}{arith}\hypertarget{method.HOL.arith}{\hyperlink{method.HOL.arith}{\mbox{\isa{arith}}}} & : & \isarmeth \\
   770     \indexdef{HOL}{attribute}{arith\_split}\hypertarget{attribute.HOL.arith-split}{\hyperlink{attribute.HOL.arith-split}{\mbox{\isa{arith{\isacharunderscore}split}}}} & : & \isaratt \\
   771   \end{matharray}
   772 
   773   The \hyperlink{method.HOL.arith}{\mbox{\isa{arith}}} method decides linear arithmetic problems
   774   (on types \isa{nat}, \isa{int}, \isa{real}).  Any current
   775   facts are inserted into the goal before running the procedure.
   776 
   777   The \hyperlink{attribute.HOL.arith-split}{\mbox{\isa{arith{\isacharunderscore}split}}} attribute declares case split
   778   rules to be expanded before the arithmetic procedure is invoked.
   779 
   780   Note that a simpler (but faster) version of arithmetic reasoning is
   781   already performed by the Simplifier.%
   782 \end{isamarkuptext}%
   783 \isamarkuptrue%
   784 %
   785 \isamarkupsection{Invoking automated reasoning tools -- The Sledgehammer%
   786 }
   787 \isamarkuptrue%
   788 %
   789 \begin{isamarkuptext}%
   790 Isabelle/HOL includes a generic \emph{ATP manager} that allows
   791   external automated reasoning tools to crunch a pending goal.
   792   Supported provers include E\footnote{\url{http://www.eprover.org}},
   793   SPASS\footnote{\url{http://www.spass-prover.org/}}, and Vampire.
   794   There is also a wrapper to invoke provers remotely via the
   795   SystemOnTPTP\footnote{\url{http://www.cs.miami.edu/~tptp/cgi-bin/SystemOnTPTP}}
   796   web service.
   797 
   798   The problem passed to external provers consists of the goal together
   799   with a smart selection of lemmas from the current theory context.
   800   The result of a successful proof search is some source text that
   801   usually reconstructs the proof within Isabelle, without requiring
   802   external provers again.  The Metis
   803   prover\footnote{\url{http://www.gilith.com/software/metis/}} that is
   804   integrated into Isabelle/HOL is being used here.
   805 
   806   In this mode of operation, heavy means of automated reasoning are
   807   used as a strong relevance filter, while the main proof checking
   808   works via explicit inferences going through the Isabelle kernel.
   809   Moreover, rechecking Isabelle proof texts with already specified
   810   auxiliary facts is much faster than performing fully automated
   811   search over and over again.
   812 
   813   \begin{matharray}{rcl}
   814     \indexdef{HOL}{command}{sledgehammer}\hypertarget{command.HOL.sledgehammer}{\hyperlink{command.HOL.sledgehammer}{\mbox{\isa{\isacommand{sledgehammer}}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isarkeep{proof} \\
   815     \indexdef{HOL}{command}{print\_atps}\hypertarget{command.HOL.print-atps}{\hyperlink{command.HOL.print-atps}{\mbox{\isa{\isacommand{print{\isacharunderscore}atps}}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isarkeep{theory~|~proof} \\
   816     \indexdef{HOL}{command}{atp\_info}\hypertarget{command.HOL.atp-info}{\hyperlink{command.HOL.atp-info}{\mbox{\isa{\isacommand{atp{\isacharunderscore}info}}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isarkeep{\cdot} \\
   817     \indexdef{HOL}{command}{atp\_kill}\hypertarget{command.HOL.atp-kill}{\hyperlink{command.HOL.atp-kill}{\mbox{\isa{\isacommand{atp{\isacharunderscore}kill}}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isarkeep{\cdot} \\
   818     \indexdef{HOL}{method}{metis}\hypertarget{method.HOL.metis}{\hyperlink{method.HOL.metis}{\mbox{\isa{metis}}}} & : & \isarmeth \\
   819   \end{matharray}
   820 
   821   \begin{rail}
   822   'sledgehammer' (nameref *)
   823   ;
   824 
   825   'metis' thmrefs
   826   ;
   827   \end{rail}
   828 
   829   \begin{descr}
   830 
   831   \item [\hyperlink{command.HOL.sledgehammer}{\mbox{\isa{\isacommand{sledgehammer}}}}~\isa{{\isachardoublequote}prover\isactrlsub {\isadigit{1}}\ {\isasymdots}\ prover\isactrlsub n{\isachardoublequote}}] invokes the specified automated theorem provers on
   832   the first subgoal.  Provers are run in parallel, the first
   833   successful result is displayed, and the other attempts are
   834   terminated.
   835 
   836   Provers are defined in the theory context, see also \hyperlink{command.HOL.print-atps}{\mbox{\isa{\isacommand{print{\isacharunderscore}atps}}}}.  If no provers are given as arguments to \hyperlink{command.HOL.sledgehammer}{\mbox{\isa{\isacommand{sledgehammer}}}}, the system refers to the default defined as
   837   ``ATP provers'' preference by the user interface.
   838 
   839   There are additional preferences for timeout (default: 60 seconds),
   840   and the maximum number of independent prover processes (default: 5);
   841   excessive provers are automatically terminated.
   842 
   843   \item [\hyperlink{command.HOL.print-atps}{\mbox{\isa{\isacommand{print{\isacharunderscore}atps}}}}] prints the list of automated
   844   theorem provers available to the \hyperlink{command.HOL.sledgehammer}{\mbox{\isa{\isacommand{sledgehammer}}}}
   845   command.
   846 
   847   \item [\hyperlink{command.HOL.atp-info}{\mbox{\isa{\isacommand{atp{\isacharunderscore}info}}}}] prints information about presently
   848   running provers, including elapsed runtime, and the remaining time
   849   until timeout.
   850 
   851   \item [\hyperlink{command.HOL.atp-kill}{\mbox{\isa{\isacommand{atp{\isacharunderscore}kill}}}}] terminates all presently running
   852   provers.
   853 
   854   \item [\hyperlink{method.HOL.metis}{\mbox{\isa{metis}}}~\isa{{\isachardoublequote}facts{\isachardoublequote}}] invokes the Metis
   855   prover with the given facts.  Metis is an automated proof tool of
   856   medium strength, but is fully integrated into Isabelle/HOL, with
   857   explicit inferences going through the kernel.  Thus its results are
   858   guaranteed to be ``correct by construction''.
   859 
   860   Note that all facts used with Metis need to be specified as explicit
   861   arguments.  There are no rule declarations as for other Isabelle
   862   provers, like \hyperlink{method.blast}{\mbox{\isa{blast}}} or \hyperlink{method.fast}{\mbox{\isa{fast}}}.
   863 
   864   \end{descr}%
   865 \end{isamarkuptext}%
   866 \isamarkuptrue%
   867 %
   868 \isamarkupsection{Unstructured cases analysis and induction \label{sec:hol-induct-tac}%
   869 }
   870 \isamarkuptrue%
   871 %
   872 \begin{isamarkuptext}%
   873 The following tools of Isabelle/HOL support cases analysis and
   874   induction in unstructured tactic scripts; see also
   875   \secref{sec:cases-induct} for proper Isar versions of similar ideas.
   876 
   877   \begin{matharray}{rcl}
   878     \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 \\
   879     \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 \\
   880     \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 \\
   881     \indexdef{HOL}{command}{inductive\_cases}\hypertarget{command.HOL.inductive-cases}{\hyperlink{command.HOL.inductive-cases}{\mbox{\isa{\isacommand{inductive{\isacharunderscore}cases}}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isartrans{theory}{theory} \\
   882   \end{matharray}
   883 
   884   \begin{rail}
   885     'case\_tac' goalspec? term rule?
   886     ;
   887     'induct\_tac' goalspec? (insts * 'and') rule?
   888     ;
   889     'ind\_cases' (prop +) ('for' (name +)) ?
   890     ;
   891     'inductive\_cases' (thmdecl? (prop +) + 'and')
   892     ;
   893 
   894     rule: ('rule' ':' thmref)
   895     ;
   896   \end{rail}
   897 
   898   \begin{descr}
   899 
   900   \item [\hyperlink{method.HOL.case-tac}{\mbox{\isa{case{\isacharunderscore}tac}}} and \hyperlink{method.HOL.induct-tac}{\mbox{\isa{induct{\isacharunderscore}tac}}}]
   901   admit to reason about inductive types.  Rules are selected according
   902   to the declarations by the \hyperlink{attribute.cases}{\mbox{\isa{cases}}} and \hyperlink{attribute.induct}{\mbox{\isa{induct}}} attributes, cf.\ \secref{sec:cases-induct}.  The \hyperlink{command.HOL.datatype}{\mbox{\isa{\isacommand{datatype}}}} package already takes care of this.
   903 
   904   These unstructured tactics feature both goal addressing and dynamic
   905   instantiation.  Note that named rule cases are \emph{not} provided
   906   as would be by the proper \hyperlink{method.cases}{\mbox{\isa{cases}}} and \hyperlink{method.induct}{\mbox{\isa{induct}}} proof
   907   methods (see \secref{sec:cases-induct}).  Unlike the \hyperlink{method.induct}{\mbox{\isa{induct}}} method, \hyperlink{method.induct-tac}{\mbox{\isa{induct{\isacharunderscore}tac}}} does not handle structured rule
   908   statements, only the compact object-logic conclusion of the subgoal
   909   being addressed.
   910   
   911   \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
   912   forward manner.
   913 
   914   While \hyperlink{method.HOL.ind-cases}{\mbox{\isa{ind{\isacharunderscore}cases}}} is a proof method to apply the
   915   result immediately as elimination rules, \hyperlink{command.HOL.inductive-cases}{\mbox{\isa{\isacommand{inductive{\isacharunderscore}cases}}}} provides case split theorems at the theory level
   916   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
   917   be generalized before applying the resulting rule.
   918 
   919   \end{descr}%
   920 \end{isamarkuptext}%
   921 \isamarkuptrue%
   922 %
   923 \isamarkupsection{Executable code%
   924 }
   925 \isamarkuptrue%
   926 %
   927 \begin{isamarkuptext}%
   928 Isabelle/Pure provides two generic frameworks to support code
   929   generation from executable specifications.  Isabelle/HOL
   930   instantiates these mechanisms in a way that is amenable to end-user
   931   applications.
   932 
   933   One framework generates code from both functional and relational
   934   programs to SML.  See \cite{isabelle-HOL} for further information
   935   (this actually covers the new-style theory format as well).
   936 
   937   \begin{matharray}{rcl}
   938     \indexdef{HOL}{command}{value}\hypertarget{command.HOL.value}{\hyperlink{command.HOL.value}{\mbox{\isa{\isacommand{value}}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isarkeep{theory~|~proof} \\
   939     \indexdef{HOL}{command}{code\_module}\hypertarget{command.HOL.code-module}{\hyperlink{command.HOL.code-module}{\mbox{\isa{\isacommand{code{\isacharunderscore}module}}}}} & : & \isartrans{theory}{theory} \\
   940     \indexdef{HOL}{command}{code\_library}\hypertarget{command.HOL.code-library}{\hyperlink{command.HOL.code-library}{\mbox{\isa{\isacommand{code{\isacharunderscore}library}}}}} & : & \isartrans{theory}{theory} \\
   941     \indexdef{HOL}{command}{consts\_code}\hypertarget{command.HOL.consts-code}{\hyperlink{command.HOL.consts-code}{\mbox{\isa{\isacommand{consts{\isacharunderscore}code}}}}} & : & \isartrans{theory}{theory} \\
   942     \indexdef{HOL}{command}{types\_code}\hypertarget{command.HOL.types-code}{\hyperlink{command.HOL.types-code}{\mbox{\isa{\isacommand{types{\isacharunderscore}code}}}}} & : & \isartrans{theory}{theory} \\  
   943     \indexdef{HOL}{attribute}{code}\hypertarget{attribute.HOL.code}{\hyperlink{attribute.HOL.code}{\mbox{\isa{code}}}} & : & \isaratt \\
   944   \end{matharray}
   945 
   946   \begin{rail}
   947   'value' term
   948   ;
   949 
   950   ( 'code\_module' | 'code\_library' ) modespec ? name ? \\
   951     ( 'file' name ) ? ( 'imports' ( name + ) ) ? \\
   952     'contains' ( ( name '=' term ) + | term + )
   953   ;
   954 
   955   modespec: '(' ( name * ) ')'
   956   ;
   957 
   958   'consts\_code' (codespec +)
   959   ;
   960 
   961   codespec: const template attachment ?
   962   ;
   963 
   964   'types\_code' (tycodespec +)
   965   ;
   966 
   967   tycodespec: name template attachment ?
   968   ;
   969 
   970   const: term
   971   ;
   972 
   973   template: '(' string ')'
   974   ;
   975 
   976   attachment: 'attach' modespec ? verblbrace text verbrbrace
   977   ;
   978 
   979   'code' (name)?
   980   ;
   981   \end{rail}
   982 
   983   \begin{descr}
   984 
   985   \item [\hyperlink{command.HOL.value}{\mbox{\isa{\isacommand{value}}}}~\isa{t}] evaluates and prints a
   986   term using the code generator.
   987 
   988   \end{descr}
   989 
   990   \medskip The other framework generates code from functional programs
   991   (including overloading using type classes) to SML \cite{SML}, OCaml
   992   \cite{OCaml} and Haskell \cite{haskell-revised-report}.
   993   Conceptually, code generation is split up in three steps:
   994   \emph{selection} of code theorems, \emph{translation} into an
   995   abstract executable view and \emph{serialization} to a specific
   996   \emph{target language}.  See \cite{isabelle-codegen} for an
   997   introduction on how to use it.
   998 
   999   \begin{matharray}{rcl}
  1000     \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} \\
  1001     \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} \\
  1002     \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} \\
  1003     \indexdef{HOL}{command}{code\_datatype}\hypertarget{command.HOL.code-datatype}{\hyperlink{command.HOL.code-datatype}{\mbox{\isa{\isacommand{code{\isacharunderscore}datatype}}}}} & : & \isartrans{theory}{theory} \\
  1004     \indexdef{HOL}{command}{code\_const}\hypertarget{command.HOL.code-const}{\hyperlink{command.HOL.code-const}{\mbox{\isa{\isacommand{code{\isacharunderscore}const}}}}} & : & \isartrans{theory}{theory} \\
  1005     \indexdef{HOL}{command}{code\_type}\hypertarget{command.HOL.code-type}{\hyperlink{command.HOL.code-type}{\mbox{\isa{\isacommand{code{\isacharunderscore}type}}}}} & : & \isartrans{theory}{theory} \\
  1006     \indexdef{HOL}{command}{code\_class}\hypertarget{command.HOL.code-class}{\hyperlink{command.HOL.code-class}{\mbox{\isa{\isacommand{code{\isacharunderscore}class}}}}} & : & \isartrans{theory}{theory} \\
  1007     \indexdef{HOL}{command}{code\_instance}\hypertarget{command.HOL.code-instance}{\hyperlink{command.HOL.code-instance}{\mbox{\isa{\isacommand{code{\isacharunderscore}instance}}}}} & : & \isartrans{theory}{theory} \\
  1008     \indexdef{HOL}{command}{code\_monad}\hypertarget{command.HOL.code-monad}{\hyperlink{command.HOL.code-monad}{\mbox{\isa{\isacommand{code{\isacharunderscore}monad}}}}} & : & \isartrans{theory}{theory} \\
  1009     \indexdef{HOL}{command}{code\_reserved}\hypertarget{command.HOL.code-reserved}{\hyperlink{command.HOL.code-reserved}{\mbox{\isa{\isacommand{code{\isacharunderscore}reserved}}}}} & : & \isartrans{theory}{theory} \\
  1010     \indexdef{HOL}{command}{code\_include}\hypertarget{command.HOL.code-include}{\hyperlink{command.HOL.code-include}{\mbox{\isa{\isacommand{code{\isacharunderscore}include}}}}} & : & \isartrans{theory}{theory} \\
  1011     \indexdef{HOL}{command}{code\_modulename}\hypertarget{command.HOL.code-modulename}{\hyperlink{command.HOL.code-modulename}{\mbox{\isa{\isacommand{code{\isacharunderscore}modulename}}}}} & : & \isartrans{theory}{theory} \\
  1012     \indexdef{HOL}{command}{code\_abort}\hypertarget{command.HOL.code-abort}{\hyperlink{command.HOL.code-abort}{\mbox{\isa{\isacommand{code{\isacharunderscore}abort}}}}} & : & \isartrans{theory}{theory} \\
  1013     \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} \\
  1014     \indexdef{HOL}{attribute}{code}\hypertarget{attribute.HOL.code}{\hyperlink{attribute.HOL.code}{\mbox{\isa{code}}}} & : & \isaratt \\
  1015   \end{matharray}
  1016 
  1017   \begin{rail}
  1018     'export\_code' ( constexpr + ) ? \\
  1019       ( ( 'in' target ( 'module\_name' string ) ? \\
  1020         ( 'file' ( string | '-' ) ) ? ( '(' args ')' ) ?) + ) ?
  1021     ;
  1022 
  1023     'code\_thms' ( constexpr + ) ?
  1024     ;
  1025 
  1026     'code\_deps' ( constexpr + ) ?
  1027     ;
  1028 
  1029     const: term
  1030     ;
  1031 
  1032     constexpr: ( const | 'name.*' | '*' )
  1033     ;
  1034 
  1035     typeconstructor: nameref
  1036     ;
  1037 
  1038     class: nameref
  1039     ;
  1040 
  1041     target: 'OCaml' | 'SML' | 'Haskell'
  1042     ;
  1043 
  1044     'code\_datatype' const +
  1045     ;
  1046 
  1047     'code\_const' (const + 'and') \\
  1048       ( ( '(' target ( syntax ? + 'and' ) ')' ) + )
  1049     ;
  1050 
  1051     'code\_type' (typeconstructor + 'and') \\
  1052       ( ( '(' target ( syntax ? + 'and' ) ')' ) + )
  1053     ;
  1054 
  1055     'code\_class' (class + 'and') \\
  1056       ( ( '(' target \\ ( string ? + 'and' ) ')' ) + )
  1057     ;
  1058 
  1059     'code\_instance' (( typeconstructor '::' class ) + 'and') \\
  1060       ( ( '(' target ( '-' ? + 'and' ) ')' ) + )
  1061     ;
  1062 
  1063     'code\_monad' const const target
  1064     ;
  1065 
  1066     'code\_reserved' target ( string + )
  1067     ;
  1068 
  1069     'code\_include' target ( string ( string | '-') )
  1070     ;
  1071 
  1072     'code\_modulename' target ( ( string string ) + )
  1073     ;
  1074 
  1075     'code\_abort' ( const + )
  1076     ;
  1077 
  1078     syntax: string | ( 'infix' | 'infixl' | 'infixr' ) nat string
  1079     ;
  1080 
  1081     'code' ( 'inline' ) ? ( 'del' ) ?
  1082     ;
  1083   \end{rail}
  1084 
  1085   \begin{descr}
  1086 
  1087   \item [\hyperlink{command.HOL.export-code}{\mbox{\isa{\isacommand{export{\isacharunderscore}code}}}}] is the canonical interface
  1088   for generating and serializing code: for a given list of constants,
  1089   code is generated for the specified target languages.  Abstract code
  1090   is cached incrementally.  If no constant is given, the currently
  1091   cached code is serialized.  If no serialization instruction is
  1092   given, only abstract code is cached.
  1093 
  1094   Constants may be specified by giving them literally, referring to
  1095   all executable contants within a certain theory by giving \isa{{\isachardoublequote}name{\isachardot}{\isacharasterisk}{\isachardoublequote}}, or referring to \emph{all} executable constants currently
  1096   available by giving \isa{{\isachardoublequote}{\isacharasterisk}{\isachardoublequote}}.
  1097 
  1098   By default, for each involved theory one corresponding name space
  1099   module is generated.  Alternativly, a module name may be specified
  1100   after the \hyperlink{keyword.module-name}{\mbox{\isa{\isakeyword{module{\isacharunderscore}name}}}} keyword; then \emph{all} code is
  1101   placed in this module.
  1102 
  1103   For \emph{SML} and \emph{OCaml}, the file specification refers to a
  1104   single file; for \emph{Haskell}, it refers to a whole directory,
  1105   where code is generated in multiple files reflecting the module
  1106   hierarchy.  The file specification ``\isa{{\isachardoublequote}{\isacharminus}{\isachardoublequote}}'' denotes standard
  1107   output.  For \emph{SML}, omitting the file specification compiles
  1108   code internally in the context of the current ML session.
  1109 
  1110   Serializers take an optional list of arguments in parentheses.  For
  1111   \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
  1112   declaration.
  1113 
  1114   \item [\hyperlink{command.HOL.code-thms}{\mbox{\isa{\isacommand{code{\isacharunderscore}thms}}}}] prints a list of theorems
  1115   representing the corresponding program containing all given
  1116   constants; if no constants are given, the currently cached code
  1117   theorems are printed.
  1118 
  1119   \item [\hyperlink{command.HOL.code-deps}{\mbox{\isa{\isacommand{code{\isacharunderscore}deps}}}}] visualizes dependencies of
  1120   theorems representing the corresponding program containing all given
  1121   constants; if no constants are given, the currently cached code
  1122   theorems are visualized.
  1123 
  1124   \item [\hyperlink{command.HOL.code-datatype}{\mbox{\isa{\isacommand{code{\isacharunderscore}datatype}}}}] specifies a constructor set
  1125   for a logical type.
  1126 
  1127   \item [\hyperlink{command.HOL.code-const}{\mbox{\isa{\isacommand{code{\isacharunderscore}const}}}}] associates a list of constants
  1128   with target-specific serializations; omitting a serialization
  1129   deletes an existing serialization.
  1130 
  1131   \item [\hyperlink{command.HOL.code-type}{\mbox{\isa{\isacommand{code{\isacharunderscore}type}}}}] associates a list of type
  1132   constructors with target-specific serializations; omitting a
  1133   serialization deletes an existing serialization.
  1134 
  1135   \item [\hyperlink{command.HOL.code-class}{\mbox{\isa{\isacommand{code{\isacharunderscore}class}}}}] associates a list of classes
  1136   with target-specific class names; omitting a
  1137   serialization deletes an existing serialization.
  1138   This applies only to \emph{Haskell}.
  1139 
  1140   \item [\hyperlink{command.HOL.code-instance}{\mbox{\isa{\isacommand{code{\isacharunderscore}instance}}}}] declares a list of type
  1141   constructor / class instance relations as ``already present'' for a
  1142   given target.  Omitting a ``\isa{{\isachardoublequote}{\isacharminus}{\isachardoublequote}}'' deletes an existing
  1143   ``already present'' declaration.  This applies only to
  1144   \emph{Haskell}.
  1145 
  1146   \item [\hyperlink{command.HOL.code-monad}{\mbox{\isa{\isacommand{code{\isacharunderscore}monad}}}}] provides an auxiliary
  1147   mechanism to generate monadic code for Haskell.
  1148 
  1149   \item [\hyperlink{command.HOL.code-reserved}{\mbox{\isa{\isacommand{code{\isacharunderscore}reserved}}}}] declares a list of names as
  1150   reserved for a given target, preventing it to be shadowed by any
  1151   generated code.
  1152 
  1153   \item [\hyperlink{command.HOL.code-include}{\mbox{\isa{\isacommand{code{\isacharunderscore}include}}}}] adds arbitrary named content
  1154   (``include'') to generated code.  A ``\isa{{\isachardoublequote}{\isacharminus}{\isachardoublequote}}'' as last argument
  1155   will remove an already added ``include''.
  1156 
  1157   \item [\hyperlink{command.HOL.code-modulename}{\mbox{\isa{\isacommand{code{\isacharunderscore}modulename}}}}] declares aliasings from
  1158   one module name onto another.
  1159 
  1160   \item [\hyperlink{command.HOL.code-abort}{\mbox{\isa{\isacommand{code{\isacharunderscore}abort}}}}] declares constants which
  1161   are not required to have a definition by means of defining equations;
  1162   if needed these are implemented by program abort instead.
  1163 
  1164   \item [\hyperlink{attribute.HOL.code}{\mbox{\isa{code}}}] explicitly selects (or
  1165   with option ``\isa{{\isachardoublequote}del{\isachardoublequote}}'' deselects) a defining equation for
  1166   code generation.  Usually packages introducing defining equations
  1167   provide a reasonable default setup for selection.
  1168 
  1169   \item [\hyperlink{attribute.HOL.code}{\mbox{\isa{code}}}\isa{inline}] declares (or with
  1170   option ``\isa{{\isachardoublequote}del{\isachardoublequote}}'' removes) inlining theorems which are
  1171   applied as rewrite rules to any defining equation during
  1172   preprocessing.
  1173 
  1174   \item [\hyperlink{command.HOL.print-codesetup}{\mbox{\isa{\isacommand{print{\isacharunderscore}codesetup}}}}] gives an overview on
  1175   selected defining equations, code generator datatypes and
  1176   preprocessor setup.
  1177 
  1178   \end{descr}%
  1179 \end{isamarkuptext}%
  1180 \isamarkuptrue%
  1181 %
  1182 \isamarkupsection{Definition by specification \label{sec:hol-specification}%
  1183 }
  1184 \isamarkuptrue%
  1185 %
  1186 \begin{isamarkuptext}%
  1187 \begin{matharray}{rcl}
  1188     \indexdef{HOL}{command}{specification}\hypertarget{command.HOL.specification}{\hyperlink{command.HOL.specification}{\mbox{\isa{\isacommand{specification}}}}} & : & \isartrans{theory}{proof(prove)} \\
  1189     \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)} \\
  1190   \end{matharray}
  1191 
  1192   \begin{rail}
  1193   ('specification' | 'ax\_specification') '(' (decl +) ')' \\ (thmdecl? prop +)
  1194   ;
  1195   decl: ((name ':')? term '(' 'overloaded' ')'?)
  1196   \end{rail}
  1197 
  1198   \begin{descr}
  1199 
  1200   \item [\hyperlink{command.HOL.specification}{\mbox{\isa{\isacommand{specification}}}}~\isa{{\isachardoublequote}decls\ {\isasymphi}{\isachardoublequote}}] sets up a
  1201   goal stating the existence of terms with the properties specified to
  1202   hold for the constants given in \isa{decls}.  After finishing the
  1203   proof, the theory will be augmented with definitions for the given
  1204   constants, as well as with theorems stating the properties for these
  1205   constants.
  1206 
  1207   \item [\hyperlink{command.HOL.ax-specification}{\mbox{\isa{\isacommand{ax{\isacharunderscore}specification}}}}~\isa{{\isachardoublequote}decls\ {\isasymphi}{\isachardoublequote}}] sets
  1208   up a goal stating the existence of terms with the properties
  1209   specified to hold for the constants given in \isa{decls}.  After
  1210   finishing the proof, the theory will be augmented with axioms
  1211   expressing the properties given in the first place.
  1212 
  1213   \item [\isa{decl}] declares a constant to be defined by the
  1214   specification given.  The definition for the constant \isa{c} is
  1215   bound to the name \isa{c{\isacharunderscore}def} unless a theorem name is given in
  1216   the declaration.  Overloaded constants should be declared as such.
  1217 
  1218   \end{descr}
  1219 
  1220   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
  1221   construction cannot introduce inconsistencies, whereas \hyperlink{command.HOL.ax-specification}{\mbox{\isa{\isacommand{ax{\isacharunderscore}specification}}}} does introduce axioms, but only after the
  1222   user has explicitly proven it to be safe.  A practical issue must be
  1223   considered, though: After introducing two constants with the same
  1224   properties using \hyperlink{command.HOL.specification}{\mbox{\isa{\isacommand{specification}}}}, one can prove
  1225   that the two constants are, in fact, equal.  If this might be a
  1226   problem, one should use \hyperlink{command.HOL.ax-specification}{\mbox{\isa{\isacommand{ax{\isacharunderscore}specification}}}}.%
  1227 \end{isamarkuptext}%
  1228 \isamarkuptrue%
  1229 %
  1230 \isadelimtheory
  1231 %
  1232 \endisadelimtheory
  1233 %
  1234 \isatagtheory
  1235 \isacommand{end}\isamarkupfalse%
  1236 %
  1237 \endisatagtheory
  1238 {\isafoldtheory}%
  1239 %
  1240 \isadelimtheory
  1241 %
  1242 \endisadelimtheory
  1243 \isanewline
  1244 \end{isabellebody}%
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