doc-src/IsarRef/Thy/document/HOL_Specific.tex
author wenzelm
Thu May 08 23:07:15 2008 +0200 (2008-05-08)
changeset 26861 e6fe036ec21d
parent 26854 9b4aec46ad78
child 26895 d066f9db833b
permissions -rw-r--r--
updated generated file;
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%
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\begin{isabellebody}%
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\def\isabellecontext{HOL{\isacharunderscore}Specific}%
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%
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\isadelimtheory
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\isanewline
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\isanewline
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%
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\endisadelimtheory
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%
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\isatagtheory
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\isacommand{theory}\isamarkupfalse%
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\ HOL{\isacharunderscore}Specific\isanewline
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\isakeyword{imports}\ Main\isanewline
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\isakeyword{begin}%
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\endisatagtheory
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{\isafoldtheory}%
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%
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\isadelimtheory
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%
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\endisadelimtheory
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%
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\isamarkupchapter{Isabelle/HOL \label{ch:hol}%
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}
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\isamarkuptrue%
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%
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\isamarkupsection{Primitive types \label{sec:hol-typedef}%
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}
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\isamarkuptrue%
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%
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\begin{isamarkuptext}%
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\begin{matharray}{rcl}
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    \indexdef{HOL}{command}{typedecl}\mbox{\isa{\isacommand{typedecl}}} & : & \isartrans{theory}{theory} \\
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    \indexdef{HOL}{command}{typedef}\mbox{\isa{\isacommand{typedef}}} & : & \isartrans{theory}{proof(prove)} \\
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  \end{matharray}
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  \begin{rail}
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    'typedecl' typespec infix?
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    ;
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    'typedef' altname? abstype '=' repset
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    ;
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    altname: '(' (name | 'open' | 'open' name) ')'
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    ;
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    abstype: typespec infix?
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    ;
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    repset: term ('morphisms' name name)?
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    ;
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  \end{rail}
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  \begin{descr}
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  \item [\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 \mbox{\isa{\isacommand{typedecl}}} of
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  Isabelle/Pure (see \secref{sec:types-pure}), but also declares type
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  arity \isa{{\isachardoublequote}t\ {\isacharcolon}{\isacharcolon}\ {\isacharparenleft}type{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ type{\isacharparenright}\ type{\isachardoublequote}}, making \isa{t} an
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  actual HOL type constructor.   %FIXME check, update
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  \item [\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}.
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  After finishing the proof, the theory will be augmented by a
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  Gordon/HOL-style type definition, which establishes a bijection
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  between the representing set \isa{A} and the new type \isa{t}.
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  Technically, \mbox{\isa{\isacommand{typedef}}} defines both a type \isa{t} and a set (term constant) of the same name (an alternative base
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  name may be given in parentheses).  The injection from type to set
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  is called \isa{Rep{\isacharunderscore}t}, its inverse \isa{Abs{\isacharunderscore}t} (this may be
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  changed via an explicit \mbox{\isa{\isakeyword{morphisms}}} declaration).
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  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
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  corresponding injection/surjection pair (in both directions).  Rules
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  \isa{Rep{\isacharunderscore}t{\isacharunderscore}inject} and \isa{Abs{\isacharunderscore}t{\isacharunderscore}inject} provide a slightly
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  more convenient view on the injectivity part, suitable for automated
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  proof tools (e.g.\ in \mbox{\isa{simp}} or \mbox{\isa{iff}} declarations).
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  Rules \isa{Rep{\isacharunderscore}t{\isacharunderscore}cases}/\isa{Rep{\isacharunderscore}t{\isacharunderscore}induct}, and \isa{Abs{\isacharunderscore}t{\isacharunderscore}cases}/\isa{Abs{\isacharunderscore}t{\isacharunderscore}induct} provide alternative views on
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  surjectivity; these are already declared as set or type rules for
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  the generic \mbox{\isa{cases}} and \mbox{\isa{induct}} methods.
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  An alternative name may be specified in parentheses; the default is
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  to use \isa{t} as indicated before.  The ``\isa{{\isachardoublequote}{\isacharparenleft}open{\isacharparenright}{\isachardoublequote}}''
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  declaration suppresses a separate constant definition for the
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  representing set.
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  \end{descr}
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  Note that raw type declarations are rarely used in practice; the
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  main application is with experimental (or even axiomatic!) theory
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  fragments.  Instead of primitive HOL type definitions, user-level
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  theories usually refer to higher-level packages such as \mbox{\isa{\isacommand{record}}} (see \secref{sec:hol-record}) or \mbox{\isa{\isacommand{datatype}}} (see \secref{sec:hol-datatype}).%
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\end{isamarkuptext}%
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\isamarkuptrue%
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%
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\isamarkupsection{Adhoc tuples%
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}
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\isamarkuptrue%
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%
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\begin{isamarkuptext}%
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\begin{matharray}{rcl}
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    \mbox{\isa{split{\isacharunderscore}format}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isaratt \\
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  \end{matharray}
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  \begin{rail}
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    'split\_format' (((name *) + 'and') | ('(' 'complete' ')'))
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    ;
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  \end{rail}
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  \begin{descr}
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  \item [\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
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  low-level tuple types into canonical form as specified by the
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  arguments given; the \isa{i}-th collection of arguments refers to
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  occurrences in premise \isa{i} of the rule.  The ``\isa{{\isachardoublequote}{\isacharparenleft}complete{\isacharparenright}{\isachardoublequote}}'' option causes \emph{all} arguments in function
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  applications to be represented canonically according to their tuple
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  type structure.
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  Note that these operations tend to invent funny names for new local
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  parameters to be introduced.
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  \end{descr}%
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\end{isamarkuptext}%
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\isamarkuptrue%
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%
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\isamarkupsection{Records \label{sec:hol-record}%
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}
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\isamarkuptrue%
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%
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\begin{isamarkuptext}%
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In principle, records merely generalize the concept of tuples, where
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  components may be addressed by labels instead of just position.  The
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  logical infrastructure of records in Isabelle/HOL is slightly more
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  advanced, though, supporting truly extensible record schemes.  This
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  admits operations that are polymorphic with respect to record
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  extension, yielding ``object-oriented'' effects like (single)
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  inheritance.  See also \cite{NaraschewskiW-TPHOLs98} for more
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  details on object-oriented verification and record subtyping in HOL.%
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\end{isamarkuptext}%
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\isamarkuptrue%
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%
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\isamarkupsubsection{Basic concepts%
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}
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\isamarkuptrue%
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%
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\begin{isamarkuptext}%
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Isabelle/HOL supports both \emph{fixed} and \emph{schematic} records
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  at the level of terms and types.  The notation is as follows:
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  \begin{center}
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  \begin{tabular}{l|l|l}
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    & record terms & record types \\ \hline
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    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}} \\
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    schematic & \isa{{\isachardoublequote}{\isasymlparr}x\ {\isacharequal}\ a{\isacharcomma}\ y\ {\isacharequal}\ b{\isacharcomma}\ {\isasymdots}\ {\isacharequal}\ m{\isasymrparr}{\isachardoublequote}} &
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      \isa{{\isachardoublequote}{\isasymlparr}x\ {\isacharcolon}{\isacharcolon}\ A{\isacharcomma}\ y\ {\isacharcolon}{\isacharcolon}\ B{\isacharcomma}\ {\isasymdots}\ {\isacharcolon}{\isacharcolon}\ M{\isasymrparr}{\isachardoublequote}} \\
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  \end{tabular}
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  \end{center}
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  \noindent The ASCII representation of \isa{{\isachardoublequote}{\isasymlparr}x\ {\isacharequal}\ a{\isasymrparr}{\isachardoublequote}} is \isa{{\isachardoublequote}{\isacharparenleft}{\isacharbar}\ x\ {\isacharequal}\ a\ {\isacharbar}{\isacharparenright}{\isachardoublequote}}.
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  A fixed record \isa{{\isachardoublequote}{\isasymlparr}x\ {\isacharequal}\ a{\isacharcomma}\ y\ {\isacharequal}\ b{\isasymrparr}{\isachardoublequote}} has field \isa{x} of value
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  \isa{a} and field \isa{y} of value \isa{b}.  The corresponding
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  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}}
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  and \isa{{\isachardoublequote}b\ {\isacharcolon}{\isacharcolon}\ B{\isachardoublequote}}.
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  A record scheme like \isa{{\isachardoublequote}{\isasymlparr}x\ {\isacharequal}\ a{\isacharcomma}\ y\ {\isacharequal}\ b{\isacharcomma}\ {\isasymdots}\ {\isacharequal}\ m{\isasymrparr}{\isachardoublequote}} contains fields
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  \isa{x} and \isa{y} as before, but also possibly further fields
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  as indicated by the ``\isa{{\isachardoublequote}{\isasymdots}{\isachardoublequote}}'' notation (which is actually part
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  of the syntax).  The improper field ``\isa{{\isachardoublequote}{\isasymdots}{\isachardoublequote}}'' of a record
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  scheme is called the \emph{more part}.  Logically it is just a free
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  variable, which is occasionally referred to as ``row variable'' in
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  the literature.  The more part of a record scheme may be
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  instantiated by zero or more further components.  For example, the
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  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.
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  Fixed records are special instances of record schemes, where
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  ``\isa{{\isachardoublequote}{\isasymdots}{\isachardoublequote}}'' is properly terminated by the \isa{{\isachardoublequote}{\isacharparenleft}{\isacharparenright}\ {\isacharcolon}{\isacharcolon}\ unit{\isachardoublequote}}
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  element.  In fact, \isa{{\isachardoublequote}{\isasymlparr}x\ {\isacharequal}\ a{\isacharcomma}\ y\ {\isacharequal}\ b{\isasymrparr}{\isachardoublequote}} is just an abbreviation
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  for \isa{{\isachardoublequote}{\isasymlparr}x\ {\isacharequal}\ a{\isacharcomma}\ y\ {\isacharequal}\ b{\isacharcomma}\ {\isasymdots}\ {\isacharequal}\ {\isacharparenleft}{\isacharparenright}{\isasymrparr}{\isachardoublequote}}.
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  \medskip Two key observations make extensible records in a simply
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  typed language like HOL work out:
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  \begin{enumerate}
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  \item the more part is internalized, as a free term or type
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  variable,
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  \item field names are externalized, they cannot be accessed within
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  the logic as first-class values.
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  \end{enumerate}
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  \medskip In Isabelle/HOL record types have to be defined explicitly,
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  fixing their field names and types, and their (optional) parent
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  record.  Afterwards, records may be formed using above syntax, while
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  obeying the canonical order of fields as given by their declaration.
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  The record package provides several standard operations like
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  selectors and updates.  The common setup for various generic proof
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  tools enable succinct reasoning patterns.  See also the Isabelle/HOL
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  tutorial \cite{isabelle-hol-book} for further instructions on using
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  records in practice.%
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\end{isamarkuptext}%
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\isamarkuptrue%
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%
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\isamarkupsubsection{Record specifications%
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}
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\isamarkuptrue%
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%
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\begin{isamarkuptext}%
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\begin{matharray}{rcl}
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    \indexdef{HOL}{command}{record}\mbox{\isa{\isacommand{record}}} & : & \isartrans{theory}{theory} \\
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  \end{matharray}
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  \begin{rail}
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    'record' typespec '=' (type '+')? (constdecl +)
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    ;
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  \end{rail}
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  \begin{descr}
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  \item [\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
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  extensible record type \isa{{\isachardoublequote}{\isacharparenleft}{\isasymalpha}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymalpha}\isactrlsub m{\isacharparenright}\ t{\isachardoublequote}},
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  derived from the optional parent record \isa{{\isachardoublequote}{\isasymtau}{\isachardoublequote}} by adding new
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  field components \isa{{\isachardoublequote}c\isactrlsub i\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}\isactrlsub i{\isachardoublequote}} etc.
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  The type variables of \isa{{\isachardoublequote}{\isasymtau}{\isachardoublequote}} and \isa{{\isachardoublequote}{\isasymsigma}\isactrlsub i{\isachardoublequote}} need to be
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  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
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  least one new field \isa{{\isachardoublequote}c\isactrlsub i{\isachardoublequote}} has to be specified.
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  Basically, field names need to belong to a unique record.  This is
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  not a real restriction in practice, since fields are qualified by
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  the record name internally.
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  The parent record specification \isa{{\isasymtau}} is optional; if omitted
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  \isa{t} becomes a root record.  The hierarchy of all records
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  declared within a theory context forms a forest structure, i.e.\ a
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  set of trees starting with a root record each.  There is no way to
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  merge multiple parent records!
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  For convenience, \isa{{\isachardoublequote}{\isacharparenleft}{\isasymalpha}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymalpha}\isactrlsub m{\isacharparenright}\ t{\isachardoublequote}} is made a
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  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
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  \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}}.
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  \end{descr}%
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\end{isamarkuptext}%
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\isamarkuptrue%
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%
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\isamarkupsubsection{Record operations%
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}
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\isamarkuptrue%
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%
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\begin{isamarkuptext}%
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Any record definition of the form presented above produces certain
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  standard operations.  Selectors and updates are provided for any
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  field, including the improper one ``\isa{more}''.  There are also
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  cumulative record constructor functions.  To simplify the
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  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}}.
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  \medskip \textbf{Selectors} and \textbf{updates} are available for
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  any field (including ``\isa{more}''):
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  \begin{matharray}{lll}
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    \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}} \\
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    \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}} \\
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  \end{matharray}
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  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
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  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
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  because of postfix notation the order of fields shown here is
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  reverse than in the actual term.  Since repeated updates are just
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  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.
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  Thus commutativity of independent updates can be proven within the
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  logic for any two fields, but not as a general theorem.
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  \medskip The \textbf{make} operation provides a cumulative record
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  constructor function:
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  \begin{matharray}{lll}
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    \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}} \\
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  \end{matharray}
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  \medskip We now reconsider the case of non-root records, which are
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  derived of some parent.  In general, the latter may depend on
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  another parent as well, resulting in a list of \emph{ancestor
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  records}.  Appending the lists of fields of all ancestors results in
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  a certain field prefix.  The record package automatically takes care
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  of this by lifting operations over this context of ancestor fields.
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  Assuming that \isa{{\isachardoublequote}{\isacharparenleft}{\isasymalpha}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymalpha}\isactrlsub m{\isacharparenright}\ t{\isachardoublequote}} has ancestor
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  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}},
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  the above record operations will get the following types:
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  \medskip
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  \begin{tabular}{lll}
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    \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}} \\
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    \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}} \\
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    \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}} \\
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  \end{tabular}
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  \medskip
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   293
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  \noindent Some further operations address the extension aspect of a
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  derived record scheme specifically: \isa{{\isachardoublequote}t{\isachardot}fields{\isachardoublequote}} produces a
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  record fragment consisting of exactly the new fields introduced here
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  (the result may serve as a more part elsewhere); \isa{{\isachardoublequote}t{\isachardot}extend{\isachardoublequote}}
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  takes a fixed record and adds a given more part; \isa{{\isachardoublequote}t{\isachardot}truncate{\isachardoublequote}} restricts a record scheme to a fixed record.
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  \medskip
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  \begin{tabular}{lll}
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    \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}} \\
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    \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}} \\
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    \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}} \\
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  \end{tabular}
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  \medskip
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   307
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  \noindent Note that \isa{{\isachardoublequote}t{\isachardot}make{\isachardoublequote}} and \isa{{\isachardoublequote}t{\isachardot}fields{\isachardoublequote}} coincide
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  for root records.%
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\end{isamarkuptext}%
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\isamarkuptrue%
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%
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\isamarkupsubsection{Derived rules and proof tools%
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   314
}
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\isamarkuptrue%
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%
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\begin{isamarkuptext}%
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The record package proves several results internally, declaring
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  these facts to appropriate proof tools.  This enables users to
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  reason about record structures quite conveniently.  Assume that
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  \isa{t} is a record type as specified above.
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  \begin{enumerate}
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  \item Standard conversions for selectors or updates applied to
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  record constructor terms are made part of the default Simplifier
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  context; thus proofs by reduction of basic operations merely require
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  the \mbox{\isa{simp}} method without further arguments.  These rules
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  are available as \isa{{\isachardoublequote}t{\isachardot}simps{\isachardoublequote}}, too.
wenzelm@26849
   330
  
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  \item Selectors applied to updated records are automatically reduced
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  by an internal simplification procedure, which is also part of the
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  standard Simplifier setup.
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  \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
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  Reasoner as \mbox{\isa{iff}} rules.  These rules are available as
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  \isa{{\isachardoublequote}t{\isachardot}iffs{\isachardoublequote}}.
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  \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,
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  and as the basic rule context as ``\mbox{\isa{intro}}\isa{{\isachardoublequote}{\isacharquery}{\isachardoublequote}}''.
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  The rule is called \isa{{\isachardoublequote}t{\isachardot}equality{\isachardoublequote}}.
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   342
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  \item Representations of arbitrary record expressions as canonical
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  constructor terms are provided both in \mbox{\isa{cases}} and \mbox{\isa{induct}} format (cf.\ the generic proof methods of the same name,
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  \secref{sec:cases-induct}).  Several variations are available, for
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  fixed records, record schemes, more parts etc.
wenzelm@26849
   347
  
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  The generic proof methods are sufficiently smart to pick the most
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  sensible rule according to the type of the indicated record
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  expression: users just need to apply something like ``\isa{{\isachardoublequote}{\isacharparenleft}cases\ r{\isacharparenright}{\isachardoublequote}}'' to a certain proof problem.
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  \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}
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  treated automatically, but usually need to be expanded by hand,
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  using the collective fact \isa{{\isachardoublequote}t{\isachardot}defs{\isachardoublequote}}.
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   355
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   356
  \end{enumerate}%
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\end{isamarkuptext}%
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\isamarkuptrue%
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%
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   360
\isamarkupsection{Datatypes \label{sec:hol-datatype}%
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}
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   362
\isamarkuptrue%
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%
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\begin{isamarkuptext}%
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\begin{matharray}{rcl}
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    \indexdef{HOL}{command}{datatype}\mbox{\isa{\isacommand{datatype}}} & : & \isartrans{theory}{theory} \\
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    \indexdef{HOL}{command}{rep\_datatype}\mbox{\isa{\isacommand{rep{\isacharunderscore}datatype}}} & : & \isartrans{theory}{theory} \\
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  \end{matharray}
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  \begin{rail}
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    'datatype' (dtspec + 'and')
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    ;
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    'rep\_datatype' (name *) dtrules
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    ;
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    dtspec: parname? typespec infix? '=' (cons + '|')
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    ;
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    cons: name (type *) mixfix?
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   379
    ;
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    dtrules: 'distinct' thmrefs 'inject' thmrefs 'induction' thmrefs
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   381
  \end{rail}
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  \begin{descr}
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   384
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  \item [\mbox{\isa{\isacommand{datatype}}}] defines inductive datatypes in
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  HOL.
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   387
wenzelm@26849
   388
  \item [\mbox{\isa{\isacommand{rep{\isacharunderscore}datatype}}}] represents existing types as
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   389
  inductive ones, generating the standard infrastructure of derived
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  concepts (primitive recursion etc.).
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   391
wenzelm@26849
   392
  \end{descr}
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   393
wenzelm@26849
   394
  The induction and exhaustion theorems generated provide case names
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   395
  according to the constructors involved, while parameters are named
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  after the types (see also \secref{sec:cases-induct}).
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   397
wenzelm@26849
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  See \cite{isabelle-HOL} for more details on datatypes, but beware of
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  the old-style theory syntax being used there!  Apart from proper
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  proof methods for case-analysis and induction, there are also
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  emulations of ML tactics \mbox{\isa{case{\isacharunderscore}tac}} and \mbox{\isa{induct{\isacharunderscore}tac}} available, see \secref{sec:hol-induct-tac}; these admit
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  to refer directly to the internal structure of subgoals (including
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   403
  internally bound parameters).%
wenzelm@26849
   404
\end{isamarkuptext}%
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   405
\isamarkuptrue%
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   406
%
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   407
\isamarkupsection{Recursive functions \label{sec:recursion}%
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   408
}
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   409
\isamarkuptrue%
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   410
%
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   411
\begin{isamarkuptext}%
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   412
\begin{matharray}{rcl}
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   413
    \indexdef{HOL}{command}{primrec}\mbox{\isa{\isacommand{primrec}}} & : & \isarkeep{local{\dsh}theory} \\
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   414
    \indexdef{HOL}{command}{fun}\mbox{\isa{\isacommand{fun}}} & : & \isarkeep{local{\dsh}theory} \\
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   415
    \indexdef{HOL}{command}{function}\mbox{\isa{\isacommand{function}}} & : & \isartrans{local{\dsh}theory}{proof(prove)} \\
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    \indexdef{HOL}{command}{termination}\mbox{\isa{\isacommand{termination}}} & : & \isartrans{local{\dsh}theory}{proof(prove)} \\
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   417
  \end{matharray}
wenzelm@26849
   418
wenzelm@26849
   419
  \railalias{funopts}{function\_opts}  %FIXME ??
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   420
wenzelm@26849
   421
  \begin{rail}
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   422
    'primrec' target? fixes 'where' equations
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   423
    ;
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   424
    equations: (thmdecl? prop + '|')
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   425
    ;
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   426
    ('fun' | 'function') (funopts)? fixes 'where' clauses
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   427
    ;
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   428
    clauses: (thmdecl? prop ('(' 'otherwise' ')')? + '|')
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   429
    ;
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   430
    funopts: '(' (('sequential' | 'in' name | 'domintros' | 'tailrec' |
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   431
    'default' term) + ',') ')'
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   432
    ;
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    'termination' ( term )?
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   434
  \end{rail}
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   435
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   436
  \begin{descr}
wenzelm@26849
   437
wenzelm@26849
   438
  \item [\mbox{\isa{\isacommand{primrec}}}] defines primitive recursive
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   439
  functions over datatypes, see also \cite{isabelle-HOL}.
wenzelm@26849
   440
wenzelm@26849
   441
  \item [\mbox{\isa{\isacommand{function}}}] defines functions by general
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   442
  wellfounded recursion. A detailed description with examples can be
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  found in \cite{isabelle-function}. The function is specified by a
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  set of (possibly conditional) recursive equations with arbitrary
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  pattern matching. The command generates proof obligations for the
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  completeness and the compatibility of patterns.
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   447
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  The defined function is considered partial, and the resulting
wenzelm@26849
   449
  simplification rules (named \isa{{\isachardoublequote}f{\isachardot}psimps{\isachardoublequote}}) and induction rule
wenzelm@26849
   450
  (named \isa{{\isachardoublequote}f{\isachardot}pinduct{\isachardoublequote}}) are guarded by a generated domain
wenzelm@26849
   451
  predicate \isa{{\isachardoublequote}f{\isacharunderscore}dom{\isachardoublequote}}. The \mbox{\isa{\isacommand{termination}}}
wenzelm@26849
   452
  command can then be used to establish that the function is total.
wenzelm@26849
   453
wenzelm@26849
   454
  \item [\mbox{\isa{\isacommand{fun}}}] is a shorthand notation for
wenzelm@26849
   455
  ``\mbox{\isa{\isacommand{function}}}~\isa{{\isachardoublequote}{\isacharparenleft}sequential{\isacharparenright}{\isachardoublequote}}, followed by
wenzelm@26849
   456
  automated proof attempts regarding pattern matching and termination.
wenzelm@26849
   457
  See \cite{isabelle-function} for further details.
wenzelm@26849
   458
wenzelm@26849
   459
  \item [\mbox{\isa{\isacommand{termination}}}~\isa{f}] commences a
wenzelm@26849
   460
  termination proof for the previously defined function \isa{f}.  If
wenzelm@26849
   461
  this is omitted, the command refers to the most recent function
wenzelm@26849
   462
  definition.  After the proof is closed, the recursive equations and
wenzelm@26849
   463
  the induction principle is established.
wenzelm@26849
   464
wenzelm@26849
   465
  \end{descr}
wenzelm@26849
   466
wenzelm@26849
   467
  %FIXME check
wenzelm@26849
   468
wenzelm@26849
   469
  Recursive definitions introduced by both the \mbox{\isa{\isacommand{primrec}}} and the \mbox{\isa{\isacommand{function}}} command accommodate
wenzelm@26849
   470
  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)
wenzelm@26849
   471
  refers to a specific induction rule, with parameters named according
wenzelm@26849
   472
  to the user-specified equations.  Case names of \mbox{\isa{\isacommand{primrec}}} are that of the datatypes involved, while those of
wenzelm@26849
   473
  \mbox{\isa{\isacommand{function}}} are numbered (starting from 1).
wenzelm@26849
   474
wenzelm@26849
   475
  The equations provided by these packages may be referred later as
wenzelm@26849
   476
  theorem list \isa{{\isachardoublequote}f{\isachardot}simps{\isachardoublequote}}, where \isa{f} is the (collective)
wenzelm@26849
   477
  name of the functions defined.  Individual equations may be named
wenzelm@26849
   478
  explicitly as well.
wenzelm@26849
   479
wenzelm@26849
   480
  The \mbox{\isa{\isacommand{function}}} command accepts the following
wenzelm@26849
   481
  options.
wenzelm@26849
   482
wenzelm@26849
   483
  \begin{descr}
wenzelm@26849
   484
wenzelm@26849
   485
  \item [\isa{sequential}] enables a preprocessor which
wenzelm@26849
   486
  disambiguates overlapping patterns by making them mutually disjoint.
wenzelm@26849
   487
  Earlier equations take precedence over later ones.  This allows to
wenzelm@26849
   488
  give the specification in a format very similar to functional
wenzelm@26849
   489
  programming.  Note that the resulting simplification and induction
wenzelm@26849
   490
  rules correspond to the transformed specification, not the one given
wenzelm@26849
   491
  originally. This usually means that each equation given by the user
wenzelm@26849
   492
  may result in several theroems.  Also note that this automatic
wenzelm@26849
   493
  transformation only works for ML-style datatype patterns.
wenzelm@26849
   494
wenzelm@26849
   495
  \item [\isa{{\isachardoublequote}{\isasymIN}\ name{\isachardoublequote}}] gives the target for the definition.
wenzelm@26849
   496
  %FIXME ?!?
wenzelm@26849
   497
wenzelm@26849
   498
  \item [\isa{domintros}] enables the automated generation of
wenzelm@26849
   499
  introduction rules for the domain predicate. While mostly not
wenzelm@26849
   500
  needed, they can be helpful in some proofs about partial functions.
wenzelm@26849
   501
wenzelm@26849
   502
  \item [\isa{tailrec}] generates the unconstrained recursive
wenzelm@26849
   503
  equations even without a termination proof, provided that the
wenzelm@26849
   504
  function is tail-recursive. This currently only works
wenzelm@26849
   505
wenzelm@26849
   506
  \item [\isa{{\isachardoublequote}default\ d{\isachardoublequote}}] allows to specify a default value for a
wenzelm@26849
   507
  (partial) function, which will ensure that \isa{{\isachardoublequote}f\ x\ {\isacharequal}\ d\ x{\isachardoublequote}}
wenzelm@26849
   508
  whenever \isa{{\isachardoublequote}x\ {\isasymnotin}\ f{\isacharunderscore}dom{\isachardoublequote}}.
wenzelm@26849
   509
wenzelm@26849
   510
  \end{descr}%
wenzelm@26849
   511
\end{isamarkuptext}%
wenzelm@26849
   512
\isamarkuptrue%
wenzelm@26849
   513
%
wenzelm@26849
   514
\isamarkupsubsection{Proof methods related to recursive definitions%
wenzelm@26849
   515
}
wenzelm@26849
   516
\isamarkuptrue%
wenzelm@26849
   517
%
wenzelm@26849
   518
\begin{isamarkuptext}%
wenzelm@26849
   519
\begin{matharray}{rcl}
wenzelm@26854
   520
    \indexdef{HOL}{method}{pat\_completeness}\mbox{\isa{pat{\isacharunderscore}completeness}} & : & \isarmeth \\
wenzelm@26849
   521
    \indexdef{HOL}{method}{relation}\mbox{\isa{relation}} & : & \isarmeth \\
wenzelm@26854
   522
    \indexdef{HOL}{method}{lexicographic\_order}\mbox{\isa{lexicographic{\isacharunderscore}order}} & : & \isarmeth \\
wenzelm@26849
   523
  \end{matharray}
wenzelm@26849
   524
wenzelm@26849
   525
  \begin{rail}
wenzelm@26849
   526
    'relation' term
wenzelm@26849
   527
    ;
wenzelm@26849
   528
    'lexicographic\_order' (clasimpmod *)
wenzelm@26849
   529
    ;
wenzelm@26849
   530
  \end{rail}
wenzelm@26849
   531
wenzelm@26849
   532
  \begin{descr}
wenzelm@26849
   533
wenzelm@26849
   534
  \item [\mbox{\isa{pat{\isacharunderscore}completeness}}] is a specialized method to
wenzelm@26849
   535
  solve goals regarding the completeness of pattern matching, as
wenzelm@26849
   536
  required by the \mbox{\isa{\isacommand{function}}} package (cf.\
wenzelm@26849
   537
  \cite{isabelle-function}).
wenzelm@26849
   538
wenzelm@26849
   539
  \item [\mbox{\isa{relation}}~\isa{R}] introduces a termination
wenzelm@26849
   540
  proof using the relation \isa{R}.  The resulting proof state will
wenzelm@26849
   541
  contain goals expressing that \isa{R} is wellfounded, and that the
wenzelm@26849
   542
  arguments of recursive calls decrease with respect to \isa{R}.
wenzelm@26849
   543
  Usually, this method is used as the initial proof step of manual
wenzelm@26849
   544
  termination proofs.
wenzelm@26849
   545
wenzelm@26849
   546
  \item [\mbox{\isa{lexicographic{\isacharunderscore}order}}] attempts a fully
wenzelm@26849
   547
  automated termination proof by searching for a lexicographic
wenzelm@26849
   548
  combination of size measures on the arguments of the function. The
wenzelm@26849
   549
  method accepts the same arguments as the \mbox{\isa{auto}} method,
wenzelm@26849
   550
  which it uses internally to prove local descents.  The same context
wenzelm@26849
   551
  modifiers as for \mbox{\isa{auto}} are accepted, see
wenzelm@26849
   552
  \secref{sec:clasimp}.
wenzelm@26849
   553
wenzelm@26849
   554
  In case of failure, extensive information is printed, which can help
wenzelm@26849
   555
  to analyse the situation (cf.\ \cite{isabelle-function}).
wenzelm@26849
   556
wenzelm@26849
   557
  \end{descr}%
wenzelm@26849
   558
\end{isamarkuptext}%
wenzelm@26849
   559
\isamarkuptrue%
wenzelm@26849
   560
%
wenzelm@26849
   561
\isamarkupsubsection{Old-style recursive function definitions (TFL)%
wenzelm@26849
   562
}
wenzelm@26849
   563
\isamarkuptrue%
wenzelm@26849
   564
%
wenzelm@26849
   565
\begin{isamarkuptext}%
wenzelm@26849
   566
The old TFL commands \mbox{\isa{\isacommand{recdef}}} and \mbox{\isa{\isacommand{recdef{\isacharunderscore}tc}}} for defining recursive are mostly obsolete; \mbox{\isa{\isacommand{function}}} or \mbox{\isa{\isacommand{fun}}} should be used instead.
wenzelm@26849
   567
wenzelm@26849
   568
  \begin{matharray}{rcl}
wenzelm@26849
   569
    \indexdef{HOL}{command}{recdef}\mbox{\isa{\isacommand{recdef}}} & : & \isartrans{theory}{theory} \\
wenzelm@26854
   570
    \indexdef{HOL}{command}{recdef\_tc}\mbox{\isa{\isacommand{recdef{\isacharunderscore}tc}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isartrans{theory}{proof(prove)} \\
wenzelm@26849
   571
  \end{matharray}
wenzelm@26849
   572
wenzelm@26849
   573
  \begin{rail}
wenzelm@26849
   574
    'recdef' ('(' 'permissive' ')')? \\ name term (prop +) hints?
wenzelm@26849
   575
    ;
wenzelm@26849
   576
    recdeftc thmdecl? tc
wenzelm@26849
   577
    ;
wenzelm@26849
   578
    hints: '(' 'hints' (recdefmod *) ')'
wenzelm@26849
   579
    ;
wenzelm@26849
   580
    recdefmod: (('recdef\_simp' | 'recdef\_cong' | 'recdef\_wf') (() | 'add' | 'del') ':' thmrefs) | clasimpmod
wenzelm@26849
   581
    ;
wenzelm@26849
   582
    tc: nameref ('(' nat ')')?
wenzelm@26849
   583
    ;
wenzelm@26849
   584
  \end{rail}
wenzelm@26849
   585
wenzelm@26849
   586
  \begin{descr}
wenzelm@26849
   587
  
wenzelm@26849
   588
  \item [\mbox{\isa{\isacommand{recdef}}}] defines general well-founded
wenzelm@26849
   589
  recursive functions (using the TFL package), see also
wenzelm@26849
   590
  \cite{isabelle-HOL}.  The ``\isa{{\isachardoublequote}{\isacharparenleft}permissive{\isacharparenright}{\isachardoublequote}}'' option tells
wenzelm@26849
   591
  TFL to recover from failed proof attempts, returning unfinished
wenzelm@26849
   592
  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
wenzelm@26849
   593
  automated proof process of TFL.  Additional \mbox{\isa{clasimpmod}}
wenzelm@26849
   594
  declarations (cf.\ \secref{sec:clasimp}) may be given to tune the
wenzelm@26849
   595
  context of the Simplifier (cf.\ \secref{sec:simplifier}) and
wenzelm@26849
   596
  Classical reasoner (cf.\ \secref{sec:classical}).
wenzelm@26849
   597
  
wenzelm@26849
   598
  \item [\mbox{\isa{\isacommand{recdef{\isacharunderscore}tc}}}~\isa{{\isachardoublequote}c\ {\isacharparenleft}i{\isacharparenright}{\isachardoublequote}}] recommences the
wenzelm@26849
   599
  proof for leftover termination condition number \isa{i} (default
wenzelm@26849
   600
  1) as generated by a \mbox{\isa{\isacommand{recdef}}} definition of
wenzelm@26849
   601
  constant \isa{c}.
wenzelm@26849
   602
  
wenzelm@26849
   603
  Note that in most cases, \mbox{\isa{\isacommand{recdef}}} is able to finish
wenzelm@26849
   604
  its internal proofs without manual intervention.
wenzelm@26849
   605
wenzelm@26849
   606
  \end{descr}
wenzelm@26849
   607
wenzelm@26849
   608
  \medskip Hints for \mbox{\isa{\isacommand{recdef}}} may be also declared
wenzelm@26849
   609
  globally, using the following attributes.
wenzelm@26849
   610
wenzelm@26849
   611
  \begin{matharray}{rcl}
wenzelm@26854
   612
    \indexdef{HOL}{attribute}{recdef\_simp}\mbox{\isa{recdef{\isacharunderscore}simp}} & : & \isaratt \\
wenzelm@26854
   613
    \indexdef{HOL}{attribute}{recdef\_cong}\mbox{\isa{recdef{\isacharunderscore}cong}} & : & \isaratt \\
wenzelm@26854
   614
    \indexdef{HOL}{attribute}{recdef\_wf}\mbox{\isa{recdef{\isacharunderscore}wf}} & : & \isaratt \\
wenzelm@26849
   615
  \end{matharray}
wenzelm@26849
   616
wenzelm@26849
   617
  \begin{rail}
wenzelm@26849
   618
    ('recdef\_simp' | 'recdef\_cong' | 'recdef\_wf') (() | 'add' | 'del')
wenzelm@26849
   619
    ;
wenzelm@26849
   620
  \end{rail}%
wenzelm@26849
   621
\end{isamarkuptext}%
wenzelm@26849
   622
\isamarkuptrue%
wenzelm@26849
   623
%
wenzelm@26849
   624
\isamarkupsection{Definition by specification \label{sec:hol-specification}%
wenzelm@26849
   625
}
wenzelm@26849
   626
\isamarkuptrue%
wenzelm@26849
   627
%
wenzelm@26849
   628
\begin{isamarkuptext}%
wenzelm@26849
   629
\begin{matharray}{rcl}
wenzelm@26849
   630
    \indexdef{HOL}{command}{specification}\mbox{\isa{\isacommand{specification}}} & : & \isartrans{theory}{proof(prove)} \\
wenzelm@26854
   631
    \indexdef{HOL}{command}{ax\_specification}\mbox{\isa{\isacommand{ax{\isacharunderscore}specification}}} & : & \isartrans{theory}{proof(prove)} \\
wenzelm@26849
   632
  \end{matharray}
wenzelm@26849
   633
wenzelm@26849
   634
  \begin{rail}
wenzelm@26849
   635
  ('specification' | 'ax\_specification') '(' (decl +) ')' \\ (thmdecl? prop +)
wenzelm@26849
   636
  ;
wenzelm@26849
   637
  decl: ((name ':')? term '(' 'overloaded' ')'?)
wenzelm@26849
   638
  \end{rail}
wenzelm@26849
   639
wenzelm@26849
   640
  \begin{descr}
wenzelm@26849
   641
wenzelm@26849
   642
  \item [\mbox{\isa{\isacommand{specification}}}~\isa{{\isachardoublequote}decls\ {\isasymphi}{\isachardoublequote}}] sets up a
wenzelm@26849
   643
  goal stating the existence of terms with the properties specified to
wenzelm@26849
   644
  hold for the constants given in \isa{decls}.  After finishing the
wenzelm@26849
   645
  proof, the theory will be augmented with definitions for the given
wenzelm@26849
   646
  constants, as well as with theorems stating the properties for these
wenzelm@26849
   647
  constants.
wenzelm@26849
   648
wenzelm@26849
   649
  \item [\mbox{\isa{\isacommand{ax{\isacharunderscore}specification}}}~\isa{{\isachardoublequote}decls\ {\isasymphi}{\isachardoublequote}}] sets
wenzelm@26849
   650
  up a goal stating the existence of terms with the properties
wenzelm@26849
   651
  specified to hold for the constants given in \isa{decls}.  After
wenzelm@26849
   652
  finishing the proof, the theory will be augmented with axioms
wenzelm@26849
   653
  expressing the properties given in the first place.
wenzelm@26849
   654
wenzelm@26849
   655
  \item [\isa{decl}] declares a constant to be defined by the
wenzelm@26849
   656
  specification given.  The definition for the constant \isa{c} is
wenzelm@26849
   657
  bound to the name \isa{c{\isacharunderscore}def} unless a theorem name is given in
wenzelm@26849
   658
  the declaration.  Overloaded constants should be declared as such.
wenzelm@26849
   659
wenzelm@26849
   660
  \end{descr}
wenzelm@26849
   661
wenzelm@26849
   662
  Whether to use \mbox{\isa{\isacommand{specification}}} or \mbox{\isa{\isacommand{ax{\isacharunderscore}specification}}} is to some extent a matter of style.  \mbox{\isa{\isacommand{specification}}} introduces no new axioms, and so by
wenzelm@26849
   663
  construction cannot introduce inconsistencies, whereas \mbox{\isa{\isacommand{ax{\isacharunderscore}specification}}} does introduce axioms, but only after the
wenzelm@26849
   664
  user has explicitly proven it to be safe.  A practical issue must be
wenzelm@26849
   665
  considered, though: After introducing two constants with the same
wenzelm@26849
   666
  properties using \mbox{\isa{\isacommand{specification}}}, one can prove
wenzelm@26849
   667
  that the two constants are, in fact, equal.  If this might be a
wenzelm@26849
   668
  problem, one should use \mbox{\isa{\isacommand{ax{\isacharunderscore}specification}}}.%
wenzelm@26849
   669
\end{isamarkuptext}%
wenzelm@26849
   670
\isamarkuptrue%
wenzelm@26849
   671
%
wenzelm@26849
   672
\isamarkupsection{Inductive and coinductive definitions \label{sec:hol-inductive}%
wenzelm@26849
   673
}
wenzelm@26849
   674
\isamarkuptrue%
wenzelm@26849
   675
%
wenzelm@26849
   676
\begin{isamarkuptext}%
wenzelm@26849
   677
An \textbf{inductive definition} specifies the least predicate (or
wenzelm@26849
   678
  set) \isa{R} closed under given rules: applying a rule to elements
wenzelm@26849
   679
  of \isa{R} yields a result within \isa{R}.  For example, a
wenzelm@26849
   680
  structural operational semantics is an inductive definition of an
wenzelm@26849
   681
  evaluation relation.
wenzelm@26849
   682
wenzelm@26849
   683
  Dually, a \textbf{coinductive definition} specifies the greatest
wenzelm@26849
   684
  predicate~/ set \isa{R} that is consistent with given rules: every
wenzelm@26849
   685
  element of \isa{R} can be seen as arising by applying a rule to
wenzelm@26849
   686
  elements of \isa{R}.  An important example is using bisimulation
wenzelm@26849
   687
  relations to formalise equivalence of processes and infinite data
wenzelm@26849
   688
  structures.
wenzelm@26849
   689
wenzelm@26849
   690
  \medskip The HOL package is related to the ZF one, which is
wenzelm@26849
   691
  described in a separate paper,\footnote{It appeared in CADE
wenzelm@26849
   692
  \cite{paulson-CADE}; a longer version is distributed with Isabelle.}
wenzelm@26849
   693
  which you should refer to in case of difficulties.  The package is
wenzelm@26849
   694
  simpler than that of ZF thanks to implicit type-checking in HOL.
wenzelm@26849
   695
  The types of the (co)inductive predicates (or sets) determine the
wenzelm@26849
   696
  domain of the fixedpoint definition, and the package does not have
wenzelm@26849
   697
  to use inference rules for type-checking.
wenzelm@26849
   698
wenzelm@26849
   699
  \begin{matharray}{rcl}
wenzelm@26849
   700
    \indexdef{HOL}{command}{inductive}\mbox{\isa{\isacommand{inductive}}} & : & \isarkeep{local{\dsh}theory} \\
wenzelm@26854
   701
    \indexdef{HOL}{command}{inductive\_set}\mbox{\isa{\isacommand{inductive{\isacharunderscore}set}}} & : & \isarkeep{local{\dsh}theory} \\
wenzelm@26849
   702
    \indexdef{HOL}{command}{coinductive}\mbox{\isa{\isacommand{coinductive}}} & : & \isarkeep{local{\dsh}theory} \\
wenzelm@26854
   703
    \indexdef{HOL}{command}{coinductive\_set}\mbox{\isa{\isacommand{coinductive{\isacharunderscore}set}}} & : & \isarkeep{local{\dsh}theory} \\
wenzelm@26849
   704
    \indexdef{HOL}{attribute}{mono}\mbox{\isa{mono}} & : & \isaratt \\
wenzelm@26849
   705
  \end{matharray}
wenzelm@26849
   706
wenzelm@26849
   707
  \begin{rail}
wenzelm@26849
   708
    ('inductive' | 'inductive\_set' | 'coinductive' | 'coinductive\_set') target? fixes ('for' fixes)? \\
wenzelm@26849
   709
    ('where' clauses)? ('monos' thmrefs)?
wenzelm@26849
   710
    ;
wenzelm@26849
   711
    clauses: (thmdecl? prop + '|')
wenzelm@26849
   712
    ;
wenzelm@26849
   713
    'mono' (() | 'add' | 'del')
wenzelm@26849
   714
    ;
wenzelm@26849
   715
  \end{rail}
wenzelm@26849
   716
wenzelm@26849
   717
  \begin{descr}
wenzelm@26849
   718
wenzelm@26849
   719
  \item [\mbox{\isa{\isacommand{inductive}}} and \mbox{\isa{\isacommand{coinductive}}}] define (co)inductive predicates from the
wenzelm@26849
   720
  introduction rules given in the \mbox{\isa{\isakeyword{where}}} part.  The
wenzelm@26849
   721
  optional \mbox{\isa{\isakeyword{for}}} part contains a list of parameters of the
wenzelm@26849
   722
  (co)inductive predicates that remain fixed throughout the
wenzelm@26849
   723
  definition.  The optional \mbox{\isa{\isakeyword{monos}}} section contains
wenzelm@26849
   724
  \emph{monotonicity theorems}, which are required for each operator
wenzelm@26849
   725
  applied to a recursive set in the introduction rules.  There
wenzelm@26849
   726
  \emph{must} be a theorem of the form \isa{{\isachardoublequote}A\ {\isasymle}\ B\ {\isasymLongrightarrow}\ M\ A\ {\isasymle}\ M\ B{\isachardoublequote}},
wenzelm@26849
   727
  for each premise \isa{{\isachardoublequote}M\ R\isactrlsub i\ t{\isachardoublequote}} in an introduction rule!
wenzelm@26849
   728
wenzelm@26849
   729
  \item [\mbox{\isa{\isacommand{inductive{\isacharunderscore}set}}} and \mbox{\isa{\isacommand{coinductive{\isacharunderscore}set}}}] are wrappers for to the previous commands,
wenzelm@26849
   730
  allowing the definition of (co)inductive sets.
wenzelm@26849
   731
wenzelm@26849
   732
  \item [\mbox{\isa{mono}}] declares monotonicity rules.  These
wenzelm@26849
   733
  rule are involved in the automated monotonicity proof of \mbox{\isa{\isacommand{inductive}}}.
wenzelm@26849
   734
wenzelm@26849
   735
  \end{descr}%
wenzelm@26849
   736
\end{isamarkuptext}%
wenzelm@26849
   737
\isamarkuptrue%
wenzelm@26849
   738
%
wenzelm@26849
   739
\isamarkupsubsection{Derived rules%
wenzelm@26849
   740
}
wenzelm@26849
   741
\isamarkuptrue%
wenzelm@26849
   742
%
wenzelm@26849
   743
\begin{isamarkuptext}%
wenzelm@26849
   744
Each (co)inductive definition \isa{R} adds definitions to the
wenzelm@26849
   745
  theory and also proves some theorems:
wenzelm@26849
   746
wenzelm@26849
   747
  \begin{description}
wenzelm@26849
   748
wenzelm@26849
   749
  \item [\isa{R{\isachardot}intros}] is the list of introduction rules as proven
wenzelm@26849
   750
  theorems, for the recursive predicates (or sets).  The rules are
wenzelm@26849
   751
  also available individually, using the names given them in the
wenzelm@26849
   752
  theory file;
wenzelm@26849
   753
wenzelm@26849
   754
  \item [\isa{R{\isachardot}cases}] is the case analysis (or elimination) rule;
wenzelm@26849
   755
wenzelm@26849
   756
  \item [\isa{R{\isachardot}induct} or \isa{R{\isachardot}coinduct}] is the (co)induction
wenzelm@26849
   757
  rule.
wenzelm@26849
   758
wenzelm@26849
   759
  \end{description}
wenzelm@26849
   760
wenzelm@26849
   761
  When several predicates \isa{{\isachardoublequote}R\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ R\isactrlsub n{\isachardoublequote}} are
wenzelm@26849
   762
  defined simultaneously, the list of introduction rules is called
wenzelm@26849
   763
  \isa{{\isachardoublequote}R\isactrlsub {\isadigit{1}}{\isacharunderscore}{\isasymdots}{\isacharunderscore}R\isactrlsub n{\isachardot}intros{\isachardoublequote}}, the case analysis rules are
wenzelm@26849
   764
  called \isa{{\isachardoublequote}R\isactrlsub {\isadigit{1}}{\isachardot}cases{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ R\isactrlsub n{\isachardot}cases{\isachardoublequote}}, and the list
wenzelm@26849
   765
  of mutual induction rules is called \isa{{\isachardoublequote}R\isactrlsub {\isadigit{1}}{\isacharunderscore}{\isasymdots}{\isacharunderscore}R\isactrlsub n{\isachardot}inducts{\isachardoublequote}}.%
wenzelm@26849
   766
\end{isamarkuptext}%
wenzelm@26849
   767
\isamarkuptrue%
wenzelm@26849
   768
%
wenzelm@26849
   769
\isamarkupsubsection{Monotonicity theorems%
wenzelm@26849
   770
}
wenzelm@26849
   771
\isamarkuptrue%
wenzelm@26849
   772
%
wenzelm@26849
   773
\begin{isamarkuptext}%
wenzelm@26849
   774
Each theory contains a default set of theorems that are used in
wenzelm@26849
   775
  monotonicity proofs.  New rules can be added to this set via the
wenzelm@26849
   776
  \mbox{\isa{mono}} attribute.  The HOL theory \isa{Inductive}
wenzelm@26849
   777
  shows how this is done.  In general, the following monotonicity
wenzelm@26849
   778
  theorems may be added:
wenzelm@26849
   779
wenzelm@26849
   780
  \begin{itemize}
wenzelm@26849
   781
wenzelm@26849
   782
  \item Theorems of the form \isa{{\isachardoublequote}A\ {\isasymle}\ B\ {\isasymLongrightarrow}\ M\ A\ {\isasymle}\ M\ B{\isachardoublequote}}, for proving
wenzelm@26849
   783
  monotonicity of inductive definitions whose introduction rules have
wenzelm@26849
   784
  premises involving terms such as \isa{{\isachardoublequote}M\ R\isactrlsub i\ t{\isachardoublequote}}.
wenzelm@26849
   785
wenzelm@26849
   786
  \item Monotonicity theorems for logical operators, which are of the
wenzelm@26849
   787
  general form \isa{{\isachardoublequote}{\isacharparenleft}{\isasymdots}\ {\isasymlongrightarrow}\ {\isasymdots}{\isacharparenright}\ {\isasymLongrightarrow}\ {\isasymdots}\ {\isacharparenleft}{\isasymdots}\ {\isasymlongrightarrow}\ {\isasymdots}{\isacharparenright}\ {\isasymLongrightarrow}\ {\isasymdots}\ {\isasymlongrightarrow}\ {\isasymdots}{\isachardoublequote}}.  For example, in
wenzelm@26849
   788
  the case of the operator \isa{{\isachardoublequote}{\isasymor}{\isachardoublequote}}, the corresponding theorem is
wenzelm@26849
   789
  \[
wenzelm@26849
   790
  \infer{\isa{{\isachardoublequote}P\isactrlsub {\isadigit{1}}\ {\isasymor}\ P\isactrlsub {\isadigit{2}}\ {\isasymlongrightarrow}\ Q\isactrlsub {\isadigit{1}}\ {\isasymor}\ Q\isactrlsub {\isadigit{2}}{\isachardoublequote}}}{\isa{{\isachardoublequote}P\isactrlsub {\isadigit{1}}\ {\isasymlongrightarrow}\ Q\isactrlsub {\isadigit{1}}{\isachardoublequote}} & \isa{{\isachardoublequote}P\isactrlsub {\isadigit{2}}\ {\isasymlongrightarrow}\ Q\isactrlsub {\isadigit{2}}{\isachardoublequote}}}
wenzelm@26849
   791
  \]
wenzelm@26849
   792
wenzelm@26849
   793
  \item De Morgan style equations for reasoning about the ``polarity''
wenzelm@26849
   794
  of expressions, e.g.
wenzelm@26849
   795
  \[
wenzelm@26849
   796
  \isa{{\isachardoublequote}{\isasymnot}\ {\isasymnot}\ P\ {\isasymlongleftrightarrow}\ P{\isachardoublequote}} \qquad\qquad
wenzelm@26849
   797
  \isa{{\isachardoublequote}{\isasymnot}\ {\isacharparenleft}P\ {\isasymand}\ Q{\isacharparenright}\ {\isasymlongleftrightarrow}\ {\isasymnot}\ P\ {\isasymor}\ {\isasymnot}\ Q{\isachardoublequote}}
wenzelm@26849
   798
  \]
wenzelm@26849
   799
wenzelm@26849
   800
  \item Equations for reducing complex operators to more primitive
wenzelm@26849
   801
  ones whose monotonicity can easily be proved, e.g.
wenzelm@26849
   802
  \[
wenzelm@26849
   803
  \isa{{\isachardoublequote}{\isacharparenleft}P\ {\isasymlongrightarrow}\ Q{\isacharparenright}\ {\isasymlongleftrightarrow}\ {\isasymnot}\ P\ {\isasymor}\ Q{\isachardoublequote}} \qquad\qquad
wenzelm@26849
   804
  \isa{{\isachardoublequote}Ball\ A\ P\ {\isasymequiv}\ {\isasymforall}x{\isachardot}\ x\ {\isasymin}\ A\ {\isasymlongrightarrow}\ P\ x{\isachardoublequote}}
wenzelm@26849
   805
  \]
wenzelm@26849
   806
wenzelm@26849
   807
  \end{itemize}
wenzelm@26849
   808
wenzelm@26849
   809
  %FIXME: Example of an inductive definition%
wenzelm@26849
   810
\end{isamarkuptext}%
wenzelm@26849
   811
\isamarkuptrue%
wenzelm@26849
   812
%
wenzelm@26849
   813
\isamarkupsection{Arithmetic proof support%
wenzelm@26849
   814
}
wenzelm@26849
   815
\isamarkuptrue%
wenzelm@26849
   816
%
wenzelm@26849
   817
\begin{isamarkuptext}%
wenzelm@26849
   818
\begin{matharray}{rcl}
wenzelm@26849
   819
    \indexdef{HOL}{method}{arith}\mbox{\isa{arith}} & : & \isarmeth \\
wenzelm@26854
   820
    \indexdef{HOL}{method}{arith\_split}\mbox{\isa{arith{\isacharunderscore}split}} & : & \isaratt \\
wenzelm@26849
   821
  \end{matharray}
wenzelm@26849
   822
wenzelm@26849
   823
  The \mbox{\isa{arith}} method decides linear arithmetic problems
wenzelm@26849
   824
  (on types \isa{nat}, \isa{int}, \isa{real}).  Any current
wenzelm@26849
   825
  facts are inserted into the goal before running the procedure.
wenzelm@26849
   826
wenzelm@26849
   827
  The \mbox{\isa{arith{\isacharunderscore}split}} attribute declares case split rules
wenzelm@26849
   828
  to be expanded before the arithmetic procedure is invoked.
wenzelm@26849
   829
wenzelm@26849
   830
  Note that a simpler (but faster) version of arithmetic reasoning is
wenzelm@26849
   831
  already performed by the Simplifier.%
wenzelm@26849
   832
\end{isamarkuptext}%
wenzelm@26849
   833
\isamarkuptrue%
wenzelm@26849
   834
%
wenzelm@26849
   835
\isamarkupsection{Cases and induction: emulating tactic scripts \label{sec:hol-induct-tac}%
wenzelm@26849
   836
}
wenzelm@26849
   837
\isamarkuptrue%
wenzelm@26849
   838
%
wenzelm@26849
   839
\begin{isamarkuptext}%
wenzelm@26849
   840
The following important tactical tools of Isabelle/HOL have been
wenzelm@26849
   841
  ported to Isar.  These should be never used in proper proof texts!
wenzelm@26849
   842
wenzelm@26849
   843
  \begin{matharray}{rcl}
wenzelm@26854
   844
    \indexdef{HOL}{method}{case\_tac}\mbox{\isa{case{\isacharunderscore}tac}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isarmeth \\
wenzelm@26854
   845
    \indexdef{HOL}{method}{induct\_tac}\mbox{\isa{induct{\isacharunderscore}tac}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isarmeth \\
wenzelm@26854
   846
    \indexdef{HOL}{method}{ind\_cases}\mbox{\isa{ind{\isacharunderscore}cases}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isarmeth \\
wenzelm@26854
   847
    \indexdef{HOL}{command}{inductive\_cases}\mbox{\isa{\isacommand{inductive{\isacharunderscore}cases}}} & : & \isartrans{theory}{theory} \\
wenzelm@26849
   848
  \end{matharray}
wenzelm@26849
   849
wenzelm@26849
   850
  \begin{rail}
wenzelm@26849
   851
    'case\_tac' goalspec? term rule?
wenzelm@26849
   852
    ;
wenzelm@26849
   853
    'induct\_tac' goalspec? (insts * 'and') rule?
wenzelm@26849
   854
    ;
wenzelm@26849
   855
    'ind\_cases' (prop +) ('for' (name +)) ?
wenzelm@26849
   856
    ;
wenzelm@26849
   857
    'inductive\_cases' (thmdecl? (prop +) + 'and')
wenzelm@26849
   858
    ;
wenzelm@26849
   859
wenzelm@26849
   860
    rule: ('rule' ':' thmref)
wenzelm@26849
   861
    ;
wenzelm@26849
   862
  \end{rail}
wenzelm@26849
   863
wenzelm@26849
   864
  \begin{descr}
wenzelm@26849
   865
wenzelm@26849
   866
  \item [\mbox{\isa{case{\isacharunderscore}tac}} and \mbox{\isa{induct{\isacharunderscore}tac}}]
wenzelm@26849
   867
  admit to reason about inductive datatypes only (unless an
wenzelm@26849
   868
  alternative rule is given explicitly).  Furthermore, \mbox{\isa{case{\isacharunderscore}tac}} does a classical case split on booleans; \mbox{\isa{induct{\isacharunderscore}tac}} allows only variables to be given as instantiation.
wenzelm@26849
   869
  These tactic emulations feature both goal addressing and dynamic
wenzelm@26849
   870
  instantiation.  Note that named rule cases are \emph{not} provided
wenzelm@26849
   871
  as would be by the proper \mbox{\isa{induct}} and \mbox{\isa{cases}} proof
wenzelm@26849
   872
  methods (see \secref{sec:cases-induct}).
wenzelm@26849
   873
  
wenzelm@26861
   874
  \item [\mbox{\isa{ind{\isacharunderscore}cases}} and \mbox{\isa{\isacommand{inductive{\isacharunderscore}cases}}}] provide an interface to the internal \verb|mk_cases| operation.  Rules are simplified in an unrestricted
wenzelm@26861
   875
  forward manner.
wenzelm@26849
   876
wenzelm@26849
   877
  While \mbox{\isa{ind{\isacharunderscore}cases}} is a proof method to apply the
wenzelm@26849
   878
  result immediately as elimination rules, \mbox{\isa{\isacommand{inductive{\isacharunderscore}cases}}} provides case split theorems at the theory level
wenzelm@26849
   879
  for later use.  The \mbox{\isa{\isakeyword{for}}} argument of the \mbox{\isa{ind{\isacharunderscore}cases}} method allows to specify a list of variables that should
wenzelm@26849
   880
  be generalized before applying the resulting rule.
wenzelm@26849
   881
wenzelm@26849
   882
  \end{descr}%
wenzelm@26849
   883
\end{isamarkuptext}%
wenzelm@26849
   884
\isamarkuptrue%
wenzelm@26849
   885
%
wenzelm@26849
   886
\isamarkupsection{Executable code%
wenzelm@26849
   887
}
wenzelm@26849
   888
\isamarkuptrue%
wenzelm@26849
   889
%
wenzelm@26849
   890
\begin{isamarkuptext}%
wenzelm@26849
   891
Isabelle/Pure provides two generic frameworks to support code
wenzelm@26849
   892
  generation from executable specifications.  Isabelle/HOL
wenzelm@26849
   893
  instantiates these mechanisms in a way that is amenable to end-user
wenzelm@26849
   894
  applications.
wenzelm@26849
   895
wenzelm@26849
   896
  One framework generates code from both functional and relational
wenzelm@26849
   897
  programs to SML.  See \cite{isabelle-HOL} for further information
wenzelm@26849
   898
  (this actually covers the new-style theory format as well).
wenzelm@26849
   899
wenzelm@26849
   900
  \begin{matharray}{rcl}
wenzelm@26849
   901
    \indexdef{HOL}{command}{value}\mbox{\isa{\isacommand{value}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isarkeep{theory~|~proof} \\
wenzelm@26854
   902
    \indexdef{HOL}{command}{code\_module}\mbox{\isa{\isacommand{code{\isacharunderscore}module}}} & : & \isartrans{theory}{theory} \\
wenzelm@26854
   903
    \indexdef{HOL}{command}{code\_library}\mbox{\isa{\isacommand{code{\isacharunderscore}library}}} & : & \isartrans{theory}{theory} \\
wenzelm@26854
   904
    \indexdef{HOL}{command}{consts\_code}\mbox{\isa{\isacommand{consts{\isacharunderscore}code}}} & : & \isartrans{theory}{theory} \\
wenzelm@26854
   905
    \indexdef{HOL}{command}{types\_code}\mbox{\isa{\isacommand{types{\isacharunderscore}code}}} & : & \isartrans{theory}{theory} \\  
wenzelm@26849
   906
    \indexdef{HOL}{attribute}{code}\mbox{\isa{code}} & : & \isaratt \\
wenzelm@26849
   907
  \end{matharray}
wenzelm@26849
   908
wenzelm@26849
   909
  \begin{rail}
wenzelm@26849
   910
  'value' term
wenzelm@26849
   911
  ;
wenzelm@26849
   912
wenzelm@26849
   913
  ( 'code\_module' | 'code\_library' ) modespec ? name ? \\
wenzelm@26849
   914
    ( 'file' name ) ? ( 'imports' ( name + ) ) ? \\
wenzelm@26849
   915
    'contains' ( ( name '=' term ) + | term + )
wenzelm@26849
   916
  ;
wenzelm@26849
   917
wenzelm@26849
   918
  modespec: '(' ( name * ) ')'
wenzelm@26849
   919
  ;
wenzelm@26849
   920
wenzelm@26849
   921
  'consts\_code' (codespec +)
wenzelm@26849
   922
  ;
wenzelm@26849
   923
wenzelm@26849
   924
  codespec: const template attachment ?
wenzelm@26849
   925
  ;
wenzelm@26849
   926
wenzelm@26849
   927
  'types\_code' (tycodespec +)
wenzelm@26849
   928
  ;
wenzelm@26849
   929
wenzelm@26849
   930
  tycodespec: name template attachment ?
wenzelm@26849
   931
  ;
wenzelm@26849
   932
wenzelm@26849
   933
  const: term
wenzelm@26849
   934
  ;
wenzelm@26849
   935
wenzelm@26849
   936
  template: '(' string ')'
wenzelm@26849
   937
  ;
wenzelm@26849
   938
wenzelm@26849
   939
  attachment: 'attach' modespec ? verblbrace text verbrbrace
wenzelm@26849
   940
  ;
wenzelm@26849
   941
wenzelm@26849
   942
  'code' (name)?
wenzelm@26849
   943
  ;
wenzelm@26849
   944
  \end{rail}
wenzelm@26849
   945
wenzelm@26849
   946
  \begin{descr}
wenzelm@26849
   947
wenzelm@26849
   948
  \item [\mbox{\isa{\isacommand{value}}}~\isa{t}] evaluates and prints a
wenzelm@26849
   949
  term using the code generator.
wenzelm@26849
   950
wenzelm@26849
   951
  \end{descr}
wenzelm@26849
   952
wenzelm@26849
   953
  \medskip The other framework generates code from functional programs
wenzelm@26849
   954
  (including overloading using type classes) to SML \cite{SML}, OCaml
wenzelm@26849
   955
  \cite{OCaml} and Haskell \cite{haskell-revised-report}.
wenzelm@26849
   956
  Conceptually, code generation is split up in three steps:
wenzelm@26849
   957
  \emph{selection} of code theorems, \emph{translation} into an
wenzelm@26849
   958
  abstract executable view and \emph{serialization} to a specific
wenzelm@26849
   959
  \emph{target language}.  See \cite{isabelle-codegen} for an
wenzelm@26849
   960
  introduction on how to use it.
wenzelm@26849
   961
wenzelm@26849
   962
  \begin{matharray}{rcl}
wenzelm@26854
   963
    \indexdef{HOL}{command}{export\_code}\mbox{\isa{\isacommand{export{\isacharunderscore}code}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isarkeep{theory~|~proof} \\
wenzelm@26854
   964
    \indexdef{HOL}{command}{code\_thms}\mbox{\isa{\isacommand{code{\isacharunderscore}thms}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isarkeep{theory~|~proof} \\
wenzelm@26854
   965
    \indexdef{HOL}{command}{code\_deps}\mbox{\isa{\isacommand{code{\isacharunderscore}deps}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isarkeep{theory~|~proof} \\
wenzelm@26854
   966
    \indexdef{HOL}{command}{code\_datatype}\mbox{\isa{\isacommand{code{\isacharunderscore}datatype}}} & : & \isartrans{theory}{theory} \\
wenzelm@26854
   967
    \indexdef{HOL}{command}{code\_const}\mbox{\isa{\isacommand{code{\isacharunderscore}const}}} & : & \isartrans{theory}{theory} \\
wenzelm@26854
   968
    \indexdef{HOL}{command}{code\_type}\mbox{\isa{\isacommand{code{\isacharunderscore}type}}} & : & \isartrans{theory}{theory} \\
wenzelm@26854
   969
    \indexdef{HOL}{command}{code\_class}\mbox{\isa{\isacommand{code{\isacharunderscore}class}}} & : & \isartrans{theory}{theory} \\
wenzelm@26854
   970
    \indexdef{HOL}{command}{code\_instance}\mbox{\isa{\isacommand{code{\isacharunderscore}instance}}} & : & \isartrans{theory}{theory} \\
wenzelm@26854
   971
    \indexdef{HOL}{command}{code\_monad}\mbox{\isa{\isacommand{code{\isacharunderscore}monad}}} & : & \isartrans{theory}{theory} \\
wenzelm@26854
   972
    \indexdef{HOL}{command}{code\_reserved}\mbox{\isa{\isacommand{code{\isacharunderscore}reserved}}} & : & \isartrans{theory}{theory} \\
wenzelm@26854
   973
    \indexdef{HOL}{command}{code\_include}\mbox{\isa{\isacommand{code{\isacharunderscore}include}}} & : & \isartrans{theory}{theory} \\
wenzelm@26854
   974
    \indexdef{HOL}{command}{code\_modulename}\mbox{\isa{\isacommand{code{\isacharunderscore}modulename}}} & : & \isartrans{theory}{theory} \\
wenzelm@26854
   975
    \indexdef{HOL}{command}{code\_exception}\mbox{\isa{\isacommand{code{\isacharunderscore}exception}}} & : & \isartrans{theory}{theory} \\
wenzelm@26854
   976
    \indexdef{HOL}{command}{print\_codesetup}\mbox{\isa{\isacommand{print{\isacharunderscore}codesetup}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isarkeep{theory~|~proof} \\
wenzelm@26849
   977
    \indexdef{HOL}{attribute}{code}\mbox{\isa{code}} & : & \isaratt \\
wenzelm@26849
   978
  \end{matharray}
wenzelm@26849
   979
wenzelm@26849
   980
  \begin{rail}
wenzelm@26849
   981
    'export\_code' ( constexpr + ) ? \\
wenzelm@26849
   982
      ( ( 'in' target ( 'module\_name' string ) ? \\
wenzelm@26849
   983
        ( 'file' ( string | '-' ) ) ? ( '(' args ')' ) ?) + ) ?
wenzelm@26849
   984
    ;
wenzelm@26849
   985
wenzelm@26849
   986
    'code\_thms' ( constexpr + ) ?
wenzelm@26849
   987
    ;
wenzelm@26849
   988
wenzelm@26849
   989
    'code\_deps' ( constexpr + ) ?
wenzelm@26849
   990
    ;
wenzelm@26849
   991
wenzelm@26849
   992
    const: term
wenzelm@26849
   993
    ;
wenzelm@26849
   994
wenzelm@26849
   995
    constexpr: ( const | 'name.*' | '*' )
wenzelm@26849
   996
    ;
wenzelm@26849
   997
wenzelm@26849
   998
    typeconstructor: nameref
wenzelm@26849
   999
    ;
wenzelm@26849
  1000
wenzelm@26849
  1001
    class: nameref
wenzelm@26849
  1002
    ;
wenzelm@26849
  1003
wenzelm@26849
  1004
    target: 'OCaml' | 'SML' | 'Haskell'
wenzelm@26849
  1005
    ;
wenzelm@26849
  1006
wenzelm@26849
  1007
    'code\_datatype' const +
wenzelm@26849
  1008
    ;
wenzelm@26849
  1009
wenzelm@26849
  1010
    'code\_const' (const + 'and') \\
wenzelm@26849
  1011
      ( ( '(' target ( syntax ? + 'and' ) ')' ) + )
wenzelm@26849
  1012
    ;
wenzelm@26849
  1013
wenzelm@26849
  1014
    'code\_type' (typeconstructor + 'and') \\
wenzelm@26849
  1015
      ( ( '(' target ( syntax ? + 'and' ) ')' ) + )
wenzelm@26849
  1016
    ;
wenzelm@26849
  1017
wenzelm@26849
  1018
    'code\_class' (class + 'and') \\
wenzelm@26849
  1019
      ( ( '(' target \\
wenzelm@26849
  1020
        ( ( string ('where' \\
wenzelm@26849
  1021
          ( const ( '==' | equiv ) string ) + ) ? ) ? + 'and' ) ')' ) + )
wenzelm@26849
  1022
    ;
wenzelm@26849
  1023
wenzelm@26849
  1024
    'code\_instance' (( typeconstructor '::' class ) + 'and') \\
wenzelm@26849
  1025
      ( ( '(' target ( '-' ? + 'and' ) ')' ) + )
wenzelm@26849
  1026
    ;
wenzelm@26849
  1027
wenzelm@26849
  1028
    'code\_monad' const const target
wenzelm@26849
  1029
    ;
wenzelm@26849
  1030
wenzelm@26849
  1031
    'code\_reserved' target ( string + )
wenzelm@26849
  1032
    ;
wenzelm@26849
  1033
wenzelm@26849
  1034
    'code\_include' target ( string ( string | '-') )
wenzelm@26849
  1035
    ;
wenzelm@26849
  1036
wenzelm@26849
  1037
    'code\_modulename' target ( ( string string ) + )
wenzelm@26849
  1038
    ;
wenzelm@26849
  1039
wenzelm@26849
  1040
    'code\_exception' ( const + )
wenzelm@26849
  1041
    ;
wenzelm@26849
  1042
wenzelm@26849
  1043
    syntax: string | ( 'infix' | 'infixl' | 'infixr' ) nat string
wenzelm@26849
  1044
    ;
wenzelm@26849
  1045
wenzelm@26849
  1046
    'code' ('func' | 'inline') ( 'del' )?
wenzelm@26849
  1047
    ;
wenzelm@26849
  1048
  \end{rail}
wenzelm@26849
  1049
wenzelm@26849
  1050
  \begin{descr}
wenzelm@26849
  1051
wenzelm@26849
  1052
  \item [\mbox{\isa{\isacommand{export{\isacharunderscore}code}}}] is the canonical interface
wenzelm@26849
  1053
  for generating and serializing code: for a given list of constants,
wenzelm@26849
  1054
  code is generated for the specified target languages.  Abstract code
wenzelm@26849
  1055
  is cached incrementally.  If no constant is given, the currently
wenzelm@26849
  1056
  cached code is serialized.  If no serialization instruction is
wenzelm@26849
  1057
  given, only abstract code is cached.
wenzelm@26849
  1058
wenzelm@26849
  1059
  Constants may be specified by giving them literally, referring to
wenzelm@26849
  1060
  all executable contants within a certain theory by giving \isa{{\isachardoublequote}name{\isachardot}{\isacharasterisk}{\isachardoublequote}}, or referring to \emph{all} executable constants currently
wenzelm@26849
  1061
  available by giving \isa{{\isachardoublequote}{\isacharasterisk}{\isachardoublequote}}.
wenzelm@26849
  1062
wenzelm@26849
  1063
  By default, for each involved theory one corresponding name space
wenzelm@26849
  1064
  module is generated.  Alternativly, a module name may be specified
wenzelm@26849
  1065
  after the \mbox{\isa{\isakeyword{module{\isacharunderscore}name}}} keyword; then \emph{all} code is
wenzelm@26849
  1066
  placed in this module.
wenzelm@26849
  1067
wenzelm@26849
  1068
  For \emph{SML} and \emph{OCaml}, the file specification refers to a
wenzelm@26849
  1069
  single file; for \emph{Haskell}, it refers to a whole directory,
wenzelm@26849
  1070
  where code is generated in multiple files reflecting the module
wenzelm@26849
  1071
  hierarchy.  The file specification ``\isa{{\isachardoublequote}{\isacharminus}{\isachardoublequote}}'' denotes standard
wenzelm@26849
  1072
  output.  For \emph{SML}, omitting the file specification compiles
wenzelm@26849
  1073
  code internally in the context of the current ML session.
wenzelm@26849
  1074
wenzelm@26849
  1075
  Serializers take an optional list of arguments in parentheses.  For
wenzelm@26849
  1076
  \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
wenzelm@26849
  1077
  declaration.
wenzelm@26849
  1078
wenzelm@26849
  1079
  \item [\mbox{\isa{\isacommand{code{\isacharunderscore}thms}}}] prints a list of theorems
wenzelm@26849
  1080
  representing the corresponding program containing all given
wenzelm@26849
  1081
  constants; if no constants are given, the currently cached code
wenzelm@26849
  1082
  theorems are printed.
wenzelm@26849
  1083
wenzelm@26849
  1084
  \item [\mbox{\isa{\isacommand{code{\isacharunderscore}deps}}}] visualizes dependencies of
wenzelm@26849
  1085
  theorems representing the corresponding program containing all given
wenzelm@26849
  1086
  constants; if no constants are given, the currently cached code
wenzelm@26849
  1087
  theorems are visualized.
wenzelm@26849
  1088
wenzelm@26849
  1089
  \item [\mbox{\isa{\isacommand{code{\isacharunderscore}datatype}}}] specifies a constructor set
wenzelm@26849
  1090
  for a logical type.
wenzelm@26849
  1091
wenzelm@26849
  1092
  \item [\mbox{\isa{\isacommand{code{\isacharunderscore}const}}}] associates a list of constants
wenzelm@26849
  1093
  with target-specific serializations; omitting a serialization
wenzelm@26849
  1094
  deletes an existing serialization.
wenzelm@26849
  1095
wenzelm@26849
  1096
  \item [\mbox{\isa{\isacommand{code{\isacharunderscore}type}}}] associates a list of type
wenzelm@26849
  1097
  constructors with target-specific serializations; omitting a
wenzelm@26849
  1098
  serialization deletes an existing serialization.
wenzelm@26849
  1099
wenzelm@26849
  1100
  \item [\mbox{\isa{\isacommand{code{\isacharunderscore}class}}}] associates a list of classes
wenzelm@26849
  1101
  with target-specific class names; in addition, constants associated
wenzelm@26849
  1102
  with this class may be given target-specific names used for instance
wenzelm@26849
  1103
  declarations; omitting a serialization deletes an existing
wenzelm@26849
  1104
  serialization.  This applies only to \emph{Haskell}.
wenzelm@26849
  1105
wenzelm@26849
  1106
  \item [\mbox{\isa{\isacommand{code{\isacharunderscore}instance}}}] declares a list of type
wenzelm@26849
  1107
  constructor / class instance relations as ``already present'' for a
wenzelm@26849
  1108
  given target.  Omitting a ``\isa{{\isachardoublequote}{\isacharminus}{\isachardoublequote}}'' deletes an existing
wenzelm@26849
  1109
  ``already present'' declaration.  This applies only to
wenzelm@26849
  1110
  \emph{Haskell}.
wenzelm@26849
  1111
wenzelm@26849
  1112
  \item [\mbox{\isa{\isacommand{code{\isacharunderscore}monad}}}] provides an auxiliary
wenzelm@26849
  1113
  mechanism to generate monadic code.
wenzelm@26849
  1114
wenzelm@26849
  1115
  \item [\mbox{\isa{\isacommand{code{\isacharunderscore}reserved}}}] declares a list of names as
wenzelm@26849
  1116
  reserved for a given target, preventing it to be shadowed by any
wenzelm@26849
  1117
  generated code.
wenzelm@26849
  1118
wenzelm@26849
  1119
  \item [\mbox{\isa{\isacommand{code{\isacharunderscore}include}}}] adds arbitrary named content
wenzelm@26849
  1120
  (``include'') to generated code.  A as last argument ``\isa{{\isachardoublequote}{\isacharminus}{\isachardoublequote}}''
wenzelm@26849
  1121
  will remove an already added ``include''.
wenzelm@26849
  1122
wenzelm@26849
  1123
  \item [\mbox{\isa{\isacommand{code{\isacharunderscore}modulename}}}] declares aliasings from
wenzelm@26849
  1124
  one module name onto another.
wenzelm@26849
  1125
wenzelm@26849
  1126
  \item [\mbox{\isa{\isacommand{code{\isacharunderscore}exception}}}] declares constants which
wenzelm@26849
  1127
  are not required to have a definition by a defining equations; these
wenzelm@26849
  1128
  are mapped on exceptions instead.
wenzelm@26849
  1129
wenzelm@26849
  1130
  \item [\mbox{\isa{code}}~\isa{func}] explicitly selects (or
wenzelm@26849
  1131
  with option ``\isa{{\isachardoublequote}del{\isacharcolon}{\isachardoublequote}}'' deselects) a defining equation for
wenzelm@26849
  1132
  code generation.  Usually packages introducing defining equations
wenzelm@26849
  1133
  provide a resonable default setup for selection.
wenzelm@26849
  1134
wenzelm@26849
  1135
  \item [\mbox{\isa{code}}\isa{inline}] declares (or with
wenzelm@26849
  1136
  option ``\isa{{\isachardoublequote}del{\isacharcolon}{\isachardoublequote}}'' removes) inlining theorems which are
wenzelm@26849
  1137
  applied as rewrite rules to any defining equation during
wenzelm@26849
  1138
  preprocessing.
wenzelm@26849
  1139
wenzelm@26849
  1140
  \item [\mbox{\isa{\isacommand{print{\isacharunderscore}codesetup}}}] gives an overview on
wenzelm@26849
  1141
  selected defining equations, code generator datatypes and
wenzelm@26849
  1142
  preprocessor setup.
wenzelm@26849
  1143
wenzelm@26849
  1144
  \end{descr}%
wenzelm@26849
  1145
\end{isamarkuptext}%
wenzelm@26849
  1146
\isamarkuptrue%
wenzelm@26849
  1147
%
wenzelm@26849
  1148
\isadelimtheory
wenzelm@26849
  1149
%
wenzelm@26849
  1150
\endisadelimtheory
wenzelm@26849
  1151
%
wenzelm@26849
  1152
\isatagtheory
wenzelm@26840
  1153
\isacommand{end}\isamarkupfalse%
wenzelm@26840
  1154
%
wenzelm@26840
  1155
\endisatagtheory
wenzelm@26840
  1156
{\isafoldtheory}%
wenzelm@26840
  1157
%
wenzelm@26840
  1158
\isadelimtheory
wenzelm@26840
  1159
%
wenzelm@26840
  1160
\endisadelimtheory
wenzelm@26849
  1161
\isanewline
wenzelm@26849
  1162
\isanewline
wenzelm@26840
  1163
\end{isabellebody}%
wenzelm@26840
  1164
%%% Local Variables:
wenzelm@26840
  1165
%%% mode: latex
wenzelm@26840
  1166
%%% TeX-master: "root"
wenzelm@26840
  1167
%%% End: