doc-src/IsarRef/Thy/document/HOL_Specific.tex
author wenzelm
Mon Dec 15 21:55:21 2008 +0100 (2008-12-15)
changeset 29115 6fb7be34506e
parent 29113 fb31b7a6c858
child 29560 fa6c5d62adf5
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}\hypertarget{command.HOL.typedecl}{\hyperlink{command.HOL.typedecl}{\mbox{\isa{\isacommand{typedecl}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ theory{\isachardoublequote}} \\
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    \indexdef{HOL}{command}{typedef}\hypertarget{command.HOL.typedef}{\hyperlink{command.HOL.typedef}{\mbox{\isa{\isacommand{typedef}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ proof{\isacharparenleft}prove{\isacharparenright}{\isachardoublequote}} \\
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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{description}
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  \item \hyperlink{command.HOL.typedecl}{\mbox{\isa{\isacommand{typedecl}}}}~\isa{{\isachardoublequote}{\isacharparenleft}{\isasymalpha}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymalpha}\isactrlsub n{\isacharparenright}\ t{\isachardoublequote}} is similar
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  to the original \hyperlink{command.typedecl}{\mbox{\isa{\isacommand{typedecl}}}} of Isabelle/Pure (see
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  \secref{sec:types-pure}), but also declares type arity \isa{{\isachardoublequote}t\ {\isacharcolon}{\isacharcolon}\ {\isacharparenleft}type{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ type{\isacharparenright}\ type{\isachardoublequote}}, making \isa{t} an actual HOL type
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  constructor.  %FIXME check, update
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  \item \hyperlink{command.HOL.typedef}{\mbox{\isa{\isacommand{typedef}}}}~\isa{{\isachardoublequote}{\isacharparenleft}{\isasymalpha}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymalpha}\isactrlsub n{\isacharparenright}\ t\ {\isacharequal}\ A{\isachardoublequote}} sets up
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  a goal stating non-emptiness of the set \isa{A}.  After finishing
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  the proof, the theory will be augmented by a Gordon/HOL-style type
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  definition, which establishes a bijection between the representing
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  set \isa{A} and the new type \isa{t}.
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  Technically, \hyperlink{command.HOL.typedef}{\mbox{\isa{\isacommand{typedef}}}} defines both a type \isa{t} and a set (term constant) of the same name (an alternative base
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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 \hyperlink{keyword.HOL.morphisms}{\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 \hyperlink{attribute.simp}{\mbox{\isa{simp}}} or \hyperlink{attribute.iff}{\mbox{\isa{iff}}}
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  declarations).  Rules \isa{Rep{\isacharunderscore}t{\isacharunderscore}cases}/\isa{Rep{\isacharunderscore}t{\isacharunderscore}induct}, and
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  \isa{Abs{\isacharunderscore}t{\isacharunderscore}cases}/\isa{Abs{\isacharunderscore}t{\isacharunderscore}induct} provide alternative views
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  on surjectivity; these are already declared as set or type rules for
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  the generic \hyperlink{method.cases}{\mbox{\isa{cases}}} and \hyperlink{method.induct}{\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{description}
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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 \hyperlink{command.HOL.record}{\mbox{\isa{\isacommand{record}}}} (see \secref{sec:hol-record}) or \hyperlink{command.HOL.datatype}{\mbox{\isa{\isacommand{datatype}}}} (see \secref{sec:hol-datatype}).%
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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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    \hyperlink{attribute.HOL.split-format}{\mbox{\isa{split{\isacharunderscore}format}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isa{attribute} \\
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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{description}
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  \item \hyperlink{attribute.HOL.split-format}{\mbox{\isa{split{\isacharunderscore}format}}}~\isa{{\isachardoublequote}p\isactrlsub {\isadigit{1}}\ {\isasymdots}\ p\isactrlsub m\ {\isasymAND}\ {\isasymdots}\ {\isasymAND}\ q\isactrlsub {\isadigit{1}}\ {\isasymdots}\ q\isactrlsub n{\isachardoublequote}} puts expressions of low-level tuple types into
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  canonical form as specified by the arguments given; the \isa{i}-th
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  collection of arguments refers to occurrences in premise \isa{i}
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  of the rule.  The ``\isa{{\isachardoublequote}{\isacharparenleft}complete{\isacharparenright}{\isachardoublequote}}'' option causes \emph{all}
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  arguments in function applications to be represented canonically
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  according to their tuple 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{description}%
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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}\hypertarget{command.HOL.record}{\hyperlink{command.HOL.record}{\mbox{\isa{\isacommand{record}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ theory{\isachardoublequote}} \\
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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{description}
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  \item \hyperlink{command.HOL.record}{\mbox{\isa{\isacommand{record}}}}~\isa{{\isachardoublequote}{\isacharparenleft}{\isasymalpha}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymalpha}\isactrlsub m{\isacharparenright}\ t\ {\isacharequal}\ {\isasymtau}\ {\isacharplus}\ c\isactrlsub {\isadigit{1}}\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}\isactrlsub {\isadigit{1}}\ {\isasymdots}\ c\isactrlsub n\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}\isactrlsub n{\isachardoublequote}} defines 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{description}%
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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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  \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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  \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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}
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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 \hyperlink{method.simp}{\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
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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 \hyperlink{attribute.iff}{\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 ``\hyperlink{attribute.intro}{\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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   343
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  \item Representations of arbitrary record expressions as canonical
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  constructor terms are provided both in \hyperlink{method.cases}{\mbox{\isa{cases}}} and \hyperlink{method.induct}{\mbox{\isa{induct}}} format (cf.\ the generic proof methods of the same name,
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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
   348
  
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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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  \end{enumerate}%
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\end{isamarkuptext}%
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\isamarkuptrue%
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%
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\isamarkupsection{Datatypes \label{sec:hol-datatype}%
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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}{datatype}\hypertarget{command.HOL.datatype}{\hyperlink{command.HOL.datatype}{\mbox{\isa{\isacommand{datatype}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ theory{\isachardoublequote}} \\
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  \indexdef{HOL}{command}{rep\_datatype}\hypertarget{command.HOL.rep-datatype}{\hyperlink{command.HOL.rep-datatype}{\mbox{\isa{\isacommand{rep{\isacharunderscore}datatype}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ proof{\isacharparenleft}prove{\isacharparenright}{\isachardoublequote}} \\
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  \end{matharray}
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  \begin{rail}
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    'datatype' (dtspec + 'and')
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    ;
haftmann@27452
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    'rep\_datatype' ('(' (name +) ')')? (term +)
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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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  \end{rail}
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  \begin{description}
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  \item \hyperlink{command.HOL.datatype}{\mbox{\isa{\isacommand{datatype}}}} defines inductive datatypes in
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  HOL.
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  \item \hyperlink{command.HOL.rep-datatype}{\mbox{\isa{\isacommand{rep{\isacharunderscore}datatype}}}} represents existing types as
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  inductive ones, generating the standard infrastructure of derived
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  concepts (primitive recursion etc.).
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  \end{description}
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   393
  The induction and exhaustion theorems generated provide case names
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  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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   396
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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 \hyperlink{method.HOL.case-tac}{\mbox{\isa{case{\isacharunderscore}tac}}} and \hyperlink{method.HOL.induct-tac}{\mbox{\isa{induct{\isacharunderscore}tac}}} available, see \secref{sec:hol-induct-tac}; these admit
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  to refer directly to the internal structure of subgoals (including
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  internally bound parameters).%
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   403
\end{isamarkuptext}%
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\isamarkuptrue%
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%
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\isamarkupsection{Recursive functions \label{sec:recursion}%
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   407
}
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\isamarkuptrue%
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%
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   410
\begin{isamarkuptext}%
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\begin{matharray}{rcl}
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    \indexdef{HOL}{command}{primrec}\hypertarget{command.HOL.primrec}{\hyperlink{command.HOL.primrec}{\mbox{\isa{\isacommand{primrec}}}}} & : & \isa{{\isachardoublequote}local{\isacharunderscore}theory\ {\isasymrightarrow}\ local{\isacharunderscore}theory{\isachardoublequote}} \\
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    \indexdef{HOL}{command}{fun}\hypertarget{command.HOL.fun}{\hyperlink{command.HOL.fun}{\mbox{\isa{\isacommand{fun}}}}} & : & \isa{{\isachardoublequote}local{\isacharunderscore}theory\ {\isasymrightarrow}\ local{\isacharunderscore}theory{\isachardoublequote}} \\
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    \indexdef{HOL}{command}{function}\hypertarget{command.HOL.function}{\hyperlink{command.HOL.function}{\mbox{\isa{\isacommand{function}}}}} & : & \isa{{\isachardoublequote}local{\isacharunderscore}theory\ {\isasymrightarrow}\ proof{\isacharparenleft}prove{\isacharparenright}{\isachardoublequote}} \\
wenzelm@28788
   415
    \indexdef{HOL}{command}{termination}\hypertarget{command.HOL.termination}{\hyperlink{command.HOL.termination}{\mbox{\isa{\isacommand{termination}}}}} & : & \isa{{\isachardoublequote}local{\isacharunderscore}theory\ {\isasymrightarrow}\ proof{\isacharparenleft}prove{\isacharparenright}{\isachardoublequote}} \\
wenzelm@26849
   416
  \end{matharray}
wenzelm@26849
   417
wenzelm@26849
   418
  \begin{rail}
wenzelm@26849
   419
    'primrec' target? fixes 'where' equations
wenzelm@26849
   420
    ;
wenzelm@26849
   421
    equations: (thmdecl? prop + '|')
wenzelm@26849
   422
    ;
wenzelm@26987
   423
    ('fun' | 'function') target? functionopts? fixes 'where' clauses
wenzelm@26849
   424
    ;
wenzelm@26849
   425
    clauses: (thmdecl? prop ('(' 'otherwise' ')')? + '|')
wenzelm@26849
   426
    ;
wenzelm@26987
   427
    functionopts: '(' (('sequential' | 'domintros' | 'tailrec' | 'default' term) + ',') ')'
wenzelm@26849
   428
    ;
wenzelm@26849
   429
    'termination' ( term )?
wenzelm@26849
   430
  \end{rail}
wenzelm@26849
   431
wenzelm@28788
   432
  \begin{description}
wenzelm@26849
   433
wenzelm@28788
   434
  \item \hyperlink{command.HOL.primrec}{\mbox{\isa{\isacommand{primrec}}}} defines primitive recursive
wenzelm@26849
   435
  functions over datatypes, see also \cite{isabelle-HOL}.
wenzelm@26849
   436
wenzelm@28788
   437
  \item \hyperlink{command.HOL.function}{\mbox{\isa{\isacommand{function}}}} defines functions by general
wenzelm@26849
   438
  wellfounded recursion. A detailed description with examples can be
wenzelm@26849
   439
  found in \cite{isabelle-function}. The function is specified by a
wenzelm@26849
   440
  set of (possibly conditional) recursive equations with arbitrary
wenzelm@26849
   441
  pattern matching. The command generates proof obligations for the
wenzelm@26849
   442
  completeness and the compatibility of patterns.
wenzelm@26849
   443
wenzelm@26849
   444
  The defined function is considered partial, and the resulting
wenzelm@26849
   445
  simplification rules (named \isa{{\isachardoublequote}f{\isachardot}psimps{\isachardoublequote}}) and induction rule
wenzelm@26849
   446
  (named \isa{{\isachardoublequote}f{\isachardot}pinduct{\isachardoublequote}}) are guarded by a generated domain
wenzelm@26902
   447
  predicate \isa{{\isachardoublequote}f{\isacharunderscore}dom{\isachardoublequote}}. The \hyperlink{command.HOL.termination}{\mbox{\isa{\isacommand{termination}}}}
wenzelm@26849
   448
  command can then be used to establish that the function is total.
wenzelm@26849
   449
wenzelm@28788
   450
  \item \hyperlink{command.HOL.fun}{\mbox{\isa{\isacommand{fun}}}} is a shorthand notation for ``\hyperlink{command.HOL.function}{\mbox{\isa{\isacommand{function}}}}~\isa{{\isachardoublequote}{\isacharparenleft}sequential{\isacharparenright}{\isachardoublequote}}, followed by automated
wenzelm@28788
   451
  proof attempts regarding pattern matching and termination.  See
wenzelm@28788
   452
  \cite{isabelle-function} for further details.
wenzelm@26849
   453
wenzelm@28788
   454
  \item \hyperlink{command.HOL.termination}{\mbox{\isa{\isacommand{termination}}}}~\isa{f} commences a
wenzelm@26849
   455
  termination proof for the previously defined function \isa{f}.  If
wenzelm@26849
   456
  this is omitted, the command refers to the most recent function
wenzelm@26849
   457
  definition.  After the proof is closed, the recursive equations and
wenzelm@26849
   458
  the induction principle is established.
wenzelm@26849
   459
wenzelm@28788
   460
  \end{description}
wenzelm@26849
   461
wenzelm@26849
   462
  %FIXME check
wenzelm@26849
   463
haftmann@27452
   464
  Recursive definitions introduced by the \hyperlink{command.HOL.function}{\mbox{\isa{\isacommand{function}}}}
haftmann@27452
   465
  command accommodate
wenzelm@26849
   466
  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
   467
  refers to a specific induction rule, with parameters named according
haftmann@27452
   468
  to the user-specified equations.
haftmann@27452
   469
  For the \hyperlink{command.HOL.primrec}{\mbox{\isa{\isacommand{primrec}}}} the induction principle coincides
haftmann@27452
   470
  with structural recursion on the datatype the recursion is carried
haftmann@27452
   471
  out.
haftmann@27452
   472
  Case names of \hyperlink{command.HOL.primrec}{\mbox{\isa{\isacommand{primrec}}}} are that of the datatypes involved, while those of
wenzelm@26902
   473
  \hyperlink{command.HOL.function}{\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@26902
   480
  The \hyperlink{command.HOL.function}{\mbox{\isa{\isacommand{function}}}} command accepts the following
wenzelm@26849
   481
  options.
wenzelm@26849
   482
wenzelm@28788
   483
  \begin{description}
wenzelm@26849
   484
wenzelm@28788
   485
  \item \isa{sequential} enables a preprocessor which disambiguates
wenzelm@28788
   486
  overlapping patterns by making them mutually disjoint.  Earlier
wenzelm@28788
   487
  equations take precedence over later ones.  This allows to give the
wenzelm@28788
   488
  specification in a format very similar to functional programming.
wenzelm@28788
   489
  Note that the resulting simplification and induction rules
wenzelm@28788
   490
  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@28788
   495
  \item \isa{domintros} enables the automated generation of
wenzelm@26849
   496
  introduction rules for the domain predicate. While mostly not
wenzelm@26849
   497
  needed, they can be helpful in some proofs about partial functions.
wenzelm@26849
   498
wenzelm@28788
   499
  \item \isa{tailrec} generates the unconstrained recursive
wenzelm@26849
   500
  equations even without a termination proof, provided that the
wenzelm@26849
   501
  function is tail-recursive. This currently only works
wenzelm@26849
   502
wenzelm@28788
   503
  \item \isa{{\isachardoublequote}default\ d{\isachardoublequote}} allows to specify a default value for a
wenzelm@26849
   504
  (partial) function, which will ensure that \isa{{\isachardoublequote}f\ x\ {\isacharequal}\ d\ x{\isachardoublequote}}
wenzelm@26849
   505
  whenever \isa{{\isachardoublequote}x\ {\isasymnotin}\ f{\isacharunderscore}dom{\isachardoublequote}}.
wenzelm@26849
   506
wenzelm@28788
   507
  \end{description}%
wenzelm@26849
   508
\end{isamarkuptext}%
wenzelm@26849
   509
\isamarkuptrue%
wenzelm@26849
   510
%
wenzelm@26849
   511
\isamarkupsubsection{Proof methods related to recursive definitions%
wenzelm@26849
   512
}
wenzelm@26849
   513
\isamarkuptrue%
wenzelm@26849
   514
%
wenzelm@26849
   515
\begin{isamarkuptext}%
wenzelm@26849
   516
\begin{matharray}{rcl}
wenzelm@28788
   517
    \indexdef{HOL}{method}{pat\_completeness}\hypertarget{method.HOL.pat-completeness}{\hyperlink{method.HOL.pat-completeness}{\mbox{\isa{pat{\isacharunderscore}completeness}}}} & : & \isa{method} \\
wenzelm@28788
   518
    \indexdef{HOL}{method}{relation}\hypertarget{method.HOL.relation}{\hyperlink{method.HOL.relation}{\mbox{\isa{relation}}}} & : & \isa{method} \\
wenzelm@28788
   519
    \indexdef{HOL}{method}{lexicographic\_order}\hypertarget{method.HOL.lexicographic-order}{\hyperlink{method.HOL.lexicographic-order}{\mbox{\isa{lexicographic{\isacharunderscore}order}}}} & : & \isa{method} \\
wenzelm@26849
   520
  \end{matharray}
wenzelm@26849
   521
wenzelm@26849
   522
  \begin{rail}
wenzelm@26849
   523
    'relation' term
wenzelm@26849
   524
    ;
wenzelm@26849
   525
    'lexicographic\_order' (clasimpmod *)
wenzelm@26849
   526
    ;
wenzelm@26849
   527
  \end{rail}
wenzelm@26849
   528
wenzelm@28788
   529
  \begin{description}
wenzelm@26849
   530
wenzelm@28788
   531
  \item \hyperlink{method.HOL.pat-completeness}{\mbox{\isa{pat{\isacharunderscore}completeness}}} is a specialized method to
wenzelm@26849
   532
  solve goals regarding the completeness of pattern matching, as
wenzelm@26902
   533
  required by the \hyperlink{command.HOL.function}{\mbox{\isa{\isacommand{function}}}} package (cf.\
wenzelm@26849
   534
  \cite{isabelle-function}).
wenzelm@26849
   535
wenzelm@28788
   536
  \item \hyperlink{method.HOL.relation}{\mbox{\isa{relation}}}~\isa{R} introduces a termination
wenzelm@26849
   537
  proof using the relation \isa{R}.  The resulting proof state will
wenzelm@26849
   538
  contain goals expressing that \isa{R} is wellfounded, and that the
wenzelm@26849
   539
  arguments of recursive calls decrease with respect to \isa{R}.
wenzelm@26849
   540
  Usually, this method is used as the initial proof step of manual
wenzelm@26849
   541
  termination proofs.
wenzelm@26849
   542
wenzelm@28788
   543
  \item \hyperlink{method.HOL.lexicographic-order}{\mbox{\isa{lexicographic{\isacharunderscore}order}}} attempts a fully
wenzelm@26849
   544
  automated termination proof by searching for a lexicographic
wenzelm@26849
   545
  combination of size measures on the arguments of the function. The
wenzelm@26902
   546
  method accepts the same arguments as the \hyperlink{method.auto}{\mbox{\isa{auto}}} method,
wenzelm@26849
   547
  which it uses internally to prove local descents.  The same context
wenzelm@26902
   548
  modifiers as for \hyperlink{method.auto}{\mbox{\isa{auto}}} are accepted, see
wenzelm@26849
   549
  \secref{sec:clasimp}.
wenzelm@26849
   550
wenzelm@26849
   551
  In case of failure, extensive information is printed, which can help
wenzelm@26849
   552
  to analyse the situation (cf.\ \cite{isabelle-function}).
wenzelm@26849
   553
wenzelm@28788
   554
  \end{description}%
wenzelm@26849
   555
\end{isamarkuptext}%
wenzelm@26849
   556
\isamarkuptrue%
wenzelm@26849
   557
%
wenzelm@26849
   558
\isamarkupsubsection{Old-style recursive function definitions (TFL)%
wenzelm@26849
   559
}
wenzelm@26849
   560
\isamarkuptrue%
wenzelm@26849
   561
%
wenzelm@26849
   562
\begin{isamarkuptext}%
wenzelm@26907
   563
The old TFL commands \hyperlink{command.HOL.recdef}{\mbox{\isa{\isacommand{recdef}}}} and \hyperlink{command.HOL.recdef-tc}{\mbox{\isa{\isacommand{recdef{\isacharunderscore}tc}}}} for defining recursive are mostly obsolete; \hyperlink{command.HOL.function}{\mbox{\isa{\isacommand{function}}}} or \hyperlink{command.HOL.fun}{\mbox{\isa{\isacommand{fun}}}} should be used instead.
wenzelm@26849
   564
wenzelm@26849
   565
  \begin{matharray}{rcl}
wenzelm@28788
   566
    \indexdef{HOL}{command}{recdef}\hypertarget{command.HOL.recdef}{\hyperlink{command.HOL.recdef}{\mbox{\isa{\isacommand{recdef}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ theory{\isacharparenright}{\isachardoublequote}} \\
wenzelm@28788
   567
    \indexdef{HOL}{command}{recdef\_tc}\hypertarget{command.HOL.recdef-tc}{\hyperlink{command.HOL.recdef-tc}{\mbox{\isa{\isacommand{recdef{\isacharunderscore}tc}}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ proof{\isacharparenleft}prove{\isacharparenright}{\isachardoublequote}} \\
wenzelm@26849
   568
  \end{matharray}
wenzelm@26849
   569
wenzelm@26849
   570
  \begin{rail}
wenzelm@26849
   571
    'recdef' ('(' 'permissive' ')')? \\ name term (prop +) hints?
wenzelm@26849
   572
    ;
wenzelm@26849
   573
    recdeftc thmdecl? tc
wenzelm@26849
   574
    ;
wenzelm@26849
   575
    hints: '(' 'hints' (recdefmod *) ')'
wenzelm@26849
   576
    ;
wenzelm@26849
   577
    recdefmod: (('recdef\_simp' | 'recdef\_cong' | 'recdef\_wf') (() | 'add' | 'del') ':' thmrefs) | clasimpmod
wenzelm@26849
   578
    ;
wenzelm@26849
   579
    tc: nameref ('(' nat ')')?
wenzelm@26849
   580
    ;
wenzelm@26849
   581
  \end{rail}
wenzelm@26849
   582
wenzelm@28788
   583
  \begin{description}
wenzelm@26849
   584
  
wenzelm@28788
   585
  \item \hyperlink{command.HOL.recdef}{\mbox{\isa{\isacommand{recdef}}}} defines general well-founded
wenzelm@26849
   586
  recursive functions (using the TFL package), see also
wenzelm@26849
   587
  \cite{isabelle-HOL}.  The ``\isa{{\isachardoublequote}{\isacharparenleft}permissive{\isacharparenright}{\isachardoublequote}}'' option tells
wenzelm@26849
   588
  TFL to recover from failed proof attempts, returning unfinished
wenzelm@26849
   589
  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@26902
   590
  automated proof process of TFL.  Additional \hyperlink{syntax.clasimpmod}{\mbox{\isa{clasimpmod}}}
wenzelm@26849
   591
  declarations (cf.\ \secref{sec:clasimp}) may be given to tune the
wenzelm@26849
   592
  context of the Simplifier (cf.\ \secref{sec:simplifier}) and
wenzelm@26849
   593
  Classical reasoner (cf.\ \secref{sec:classical}).
wenzelm@26849
   594
  
wenzelm@28788
   595
  \item \hyperlink{command.HOL.recdef-tc}{\mbox{\isa{\isacommand{recdef{\isacharunderscore}tc}}}}~\isa{{\isachardoublequote}c\ {\isacharparenleft}i{\isacharparenright}{\isachardoublequote}} recommences the
wenzelm@26849
   596
  proof for leftover termination condition number \isa{i} (default
wenzelm@26902
   597
  1) as generated by a \hyperlink{command.HOL.recdef}{\mbox{\isa{\isacommand{recdef}}}} definition of
wenzelm@26849
   598
  constant \isa{c}.
wenzelm@26849
   599
  
wenzelm@26902
   600
  Note that in most cases, \hyperlink{command.HOL.recdef}{\mbox{\isa{\isacommand{recdef}}}} is able to finish
wenzelm@26849
   601
  its internal proofs without manual intervention.
wenzelm@26849
   602
wenzelm@28788
   603
  \end{description}
wenzelm@26849
   604
wenzelm@26902
   605
  \medskip Hints for \hyperlink{command.HOL.recdef}{\mbox{\isa{\isacommand{recdef}}}} may be also declared
wenzelm@26849
   606
  globally, using the following attributes.
wenzelm@26849
   607
wenzelm@26849
   608
  \begin{matharray}{rcl}
wenzelm@28788
   609
    \indexdef{HOL}{attribute}{recdef\_simp}\hypertarget{attribute.HOL.recdef-simp}{\hyperlink{attribute.HOL.recdef-simp}{\mbox{\isa{recdef{\isacharunderscore}simp}}}} & : & \isa{attribute} \\
wenzelm@28788
   610
    \indexdef{HOL}{attribute}{recdef\_cong}\hypertarget{attribute.HOL.recdef-cong}{\hyperlink{attribute.HOL.recdef-cong}{\mbox{\isa{recdef{\isacharunderscore}cong}}}} & : & \isa{attribute} \\
wenzelm@28788
   611
    \indexdef{HOL}{attribute}{recdef\_wf}\hypertarget{attribute.HOL.recdef-wf}{\hyperlink{attribute.HOL.recdef-wf}{\mbox{\isa{recdef{\isacharunderscore}wf}}}} & : & \isa{attribute} \\
wenzelm@26849
   612
  \end{matharray}
wenzelm@26849
   613
wenzelm@26849
   614
  \begin{rail}
wenzelm@26849
   615
    ('recdef\_simp' | 'recdef\_cong' | 'recdef\_wf') (() | 'add' | 'del')
wenzelm@26849
   616
    ;
wenzelm@26849
   617
  \end{rail}%
wenzelm@26849
   618
\end{isamarkuptext}%
wenzelm@26849
   619
\isamarkuptrue%
wenzelm@26849
   620
%
wenzelm@26849
   621
\isamarkupsection{Inductive and coinductive definitions \label{sec:hol-inductive}%
wenzelm@26849
   622
}
wenzelm@26849
   623
\isamarkuptrue%
wenzelm@26849
   624
%
wenzelm@26849
   625
\begin{isamarkuptext}%
wenzelm@26849
   626
An \textbf{inductive definition} specifies the least predicate (or
wenzelm@26849
   627
  set) \isa{R} closed under given rules: applying a rule to elements
wenzelm@26849
   628
  of \isa{R} yields a result within \isa{R}.  For example, a
wenzelm@26849
   629
  structural operational semantics is an inductive definition of an
wenzelm@26849
   630
  evaluation relation.
wenzelm@26849
   631
wenzelm@26849
   632
  Dually, a \textbf{coinductive definition} specifies the greatest
wenzelm@26849
   633
  predicate~/ set \isa{R} that is consistent with given rules: every
wenzelm@26849
   634
  element of \isa{R} can be seen as arising by applying a rule to
wenzelm@26849
   635
  elements of \isa{R}.  An important example is using bisimulation
wenzelm@26849
   636
  relations to formalise equivalence of processes and infinite data
wenzelm@26849
   637
  structures.
wenzelm@26849
   638
wenzelm@26849
   639
  \medskip The HOL package is related to the ZF one, which is
wenzelm@26849
   640
  described in a separate paper,\footnote{It appeared in CADE
wenzelm@26849
   641
  \cite{paulson-CADE}; a longer version is distributed with Isabelle.}
wenzelm@26849
   642
  which you should refer to in case of difficulties.  The package is
wenzelm@26849
   643
  simpler than that of ZF thanks to implicit type-checking in HOL.
wenzelm@26849
   644
  The types of the (co)inductive predicates (or sets) determine the
wenzelm@26849
   645
  domain of the fixedpoint definition, and the package does not have
wenzelm@26849
   646
  to use inference rules for type-checking.
wenzelm@26849
   647
wenzelm@26849
   648
  \begin{matharray}{rcl}
wenzelm@28788
   649
    \indexdef{HOL}{command}{inductive}\hypertarget{command.HOL.inductive}{\hyperlink{command.HOL.inductive}{\mbox{\isa{\isacommand{inductive}}}}} & : & \isa{{\isachardoublequote}local{\isacharunderscore}theory\ {\isasymrightarrow}\ local{\isacharunderscore}theory{\isachardoublequote}} \\
wenzelm@28788
   650
    \indexdef{HOL}{command}{inductive\_set}\hypertarget{command.HOL.inductive-set}{\hyperlink{command.HOL.inductive-set}{\mbox{\isa{\isacommand{inductive{\isacharunderscore}set}}}}} & : & \isa{{\isachardoublequote}local{\isacharunderscore}theory\ {\isasymrightarrow}\ local{\isacharunderscore}theory{\isachardoublequote}} \\
wenzelm@28788
   651
    \indexdef{HOL}{command}{coinductive}\hypertarget{command.HOL.coinductive}{\hyperlink{command.HOL.coinductive}{\mbox{\isa{\isacommand{coinductive}}}}} & : & \isa{{\isachardoublequote}local{\isacharunderscore}theory\ {\isasymrightarrow}\ local{\isacharunderscore}theory{\isachardoublequote}} \\
wenzelm@28788
   652
    \indexdef{HOL}{command}{coinductive\_set}\hypertarget{command.HOL.coinductive-set}{\hyperlink{command.HOL.coinductive-set}{\mbox{\isa{\isacommand{coinductive{\isacharunderscore}set}}}}} & : & \isa{{\isachardoublequote}local{\isacharunderscore}theory\ {\isasymrightarrow}\ local{\isacharunderscore}theory{\isachardoublequote}} \\
wenzelm@28788
   653
    \indexdef{HOL}{attribute}{mono}\hypertarget{attribute.HOL.mono}{\hyperlink{attribute.HOL.mono}{\mbox{\isa{mono}}}} & : & \isa{attribute} \\
wenzelm@26849
   654
  \end{matharray}
wenzelm@26849
   655
wenzelm@26849
   656
  \begin{rail}
wenzelm@26849
   657
    ('inductive' | 'inductive\_set' | 'coinductive' | 'coinductive\_set') target? fixes ('for' fixes)? \\
wenzelm@26849
   658
    ('where' clauses)? ('monos' thmrefs)?
wenzelm@26849
   659
    ;
wenzelm@26849
   660
    clauses: (thmdecl? prop + '|')
wenzelm@26849
   661
    ;
wenzelm@26849
   662
    'mono' (() | 'add' | 'del')
wenzelm@26849
   663
    ;
wenzelm@26849
   664
  \end{rail}
wenzelm@26849
   665
wenzelm@28788
   666
  \begin{description}
wenzelm@26849
   667
wenzelm@28788
   668
  \item \hyperlink{command.HOL.inductive}{\mbox{\isa{\isacommand{inductive}}}} and \hyperlink{command.HOL.coinductive}{\mbox{\isa{\isacommand{coinductive}}}} define (co)inductive predicates from the
wenzelm@26902
   669
  introduction rules given in the \hyperlink{keyword.where}{\mbox{\isa{\isakeyword{where}}}} part.  The
wenzelm@26902
   670
  optional \hyperlink{keyword.for}{\mbox{\isa{\isakeyword{for}}}} part contains a list of parameters of the
wenzelm@26849
   671
  (co)inductive predicates that remain fixed throughout the
wenzelm@26902
   672
  definition.  The optional \hyperlink{keyword.monos}{\mbox{\isa{\isakeyword{monos}}}} section contains
wenzelm@26849
   673
  \emph{monotonicity theorems}, which are required for each operator
wenzelm@26849
   674
  applied to a recursive set in the introduction rules.  There
wenzelm@26849
   675
  \emph{must} be a theorem of the form \isa{{\isachardoublequote}A\ {\isasymle}\ B\ {\isasymLongrightarrow}\ M\ A\ {\isasymle}\ M\ B{\isachardoublequote}},
wenzelm@26849
   676
  for each premise \isa{{\isachardoublequote}M\ R\isactrlsub i\ t{\isachardoublequote}} in an introduction rule!
wenzelm@26849
   677
wenzelm@28788
   678
  \item \hyperlink{command.HOL.inductive-set}{\mbox{\isa{\isacommand{inductive{\isacharunderscore}set}}}} and \hyperlink{command.HOL.coinductive-set}{\mbox{\isa{\isacommand{coinductive{\isacharunderscore}set}}}} are wrappers for to the previous commands,
wenzelm@26849
   679
  allowing the definition of (co)inductive sets.
wenzelm@26849
   680
wenzelm@28788
   681
  \item \hyperlink{attribute.HOL.mono}{\mbox{\isa{mono}}} declares monotonicity rules.  These
wenzelm@26902
   682
  rule are involved in the automated monotonicity proof of \hyperlink{command.HOL.inductive}{\mbox{\isa{\isacommand{inductive}}}}.
wenzelm@26849
   683
wenzelm@28788
   684
  \end{description}%
wenzelm@26849
   685
\end{isamarkuptext}%
wenzelm@26849
   686
\isamarkuptrue%
wenzelm@26849
   687
%
wenzelm@26849
   688
\isamarkupsubsection{Derived rules%
wenzelm@26849
   689
}
wenzelm@26849
   690
\isamarkuptrue%
wenzelm@26849
   691
%
wenzelm@26849
   692
\begin{isamarkuptext}%
wenzelm@26849
   693
Each (co)inductive definition \isa{R} adds definitions to the
wenzelm@26849
   694
  theory and also proves some theorems:
wenzelm@26849
   695
wenzelm@26849
   696
  \begin{description}
wenzelm@26849
   697
wenzelm@28788
   698
  \item \isa{R{\isachardot}intros} is the list of introduction rules as proven
wenzelm@26849
   699
  theorems, for the recursive predicates (or sets).  The rules are
wenzelm@26849
   700
  also available individually, using the names given them in the
wenzelm@26849
   701
  theory file;
wenzelm@26849
   702
wenzelm@28788
   703
  \item \isa{R{\isachardot}cases} is the case analysis (or elimination) rule;
wenzelm@26849
   704
wenzelm@28788
   705
  \item \isa{R{\isachardot}induct} or \isa{R{\isachardot}coinduct} is the (co)induction
wenzelm@26849
   706
  rule.
wenzelm@26849
   707
wenzelm@26849
   708
  \end{description}
wenzelm@26849
   709
wenzelm@26849
   710
  When several predicates \isa{{\isachardoublequote}R\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ R\isactrlsub n{\isachardoublequote}} are
wenzelm@26849
   711
  defined simultaneously, the list of introduction rules is called
wenzelm@26849
   712
  \isa{{\isachardoublequote}R\isactrlsub {\isadigit{1}}{\isacharunderscore}{\isasymdots}{\isacharunderscore}R\isactrlsub n{\isachardot}intros{\isachardoublequote}}, the case analysis rules are
wenzelm@26849
   713
  called \isa{{\isachardoublequote}R\isactrlsub {\isadigit{1}}{\isachardot}cases{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ R\isactrlsub n{\isachardot}cases{\isachardoublequote}}, and the list
wenzelm@26849
   714
  of mutual induction rules is called \isa{{\isachardoublequote}R\isactrlsub {\isadigit{1}}{\isacharunderscore}{\isasymdots}{\isacharunderscore}R\isactrlsub n{\isachardot}inducts{\isachardoublequote}}.%
wenzelm@26849
   715
\end{isamarkuptext}%
wenzelm@26849
   716
\isamarkuptrue%
wenzelm@26849
   717
%
wenzelm@26849
   718
\isamarkupsubsection{Monotonicity theorems%
wenzelm@26849
   719
}
wenzelm@26849
   720
\isamarkuptrue%
wenzelm@26849
   721
%
wenzelm@26849
   722
\begin{isamarkuptext}%
wenzelm@26849
   723
Each theory contains a default set of theorems that are used in
wenzelm@26849
   724
  monotonicity proofs.  New rules can be added to this set via the
wenzelm@26902
   725
  \hyperlink{attribute.HOL.mono}{\mbox{\isa{mono}}} attribute.  The HOL theory \isa{Inductive}
wenzelm@26849
   726
  shows how this is done.  In general, the following monotonicity
wenzelm@26849
   727
  theorems may be added:
wenzelm@26849
   728
wenzelm@26849
   729
  \begin{itemize}
wenzelm@26849
   730
wenzelm@26849
   731
  \item Theorems of the form \isa{{\isachardoublequote}A\ {\isasymle}\ B\ {\isasymLongrightarrow}\ M\ A\ {\isasymle}\ M\ B{\isachardoublequote}}, for proving
wenzelm@26849
   732
  monotonicity of inductive definitions whose introduction rules have
wenzelm@26849
   733
  premises involving terms such as \isa{{\isachardoublequote}M\ R\isactrlsub i\ t{\isachardoublequote}}.
wenzelm@26849
   734
wenzelm@26849
   735
  \item Monotonicity theorems for logical operators, which are of the
wenzelm@26849
   736
  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
   737
  the case of the operator \isa{{\isachardoublequote}{\isasymor}{\isachardoublequote}}, the corresponding theorem is
wenzelm@26849
   738
  \[
wenzelm@26849
   739
  \infer{\isa{{\isachardoublequote}P\isactrlsub {\isadigit{1}}\ {\isasymor}\ P\isactrlsub {\isadigit{2}}\ {\isasymlongrightarrow}\ Q\isactrlsub {\isadigit{1}}\ {\isasymor}\ Q\isactrlsub {\isadigit{2}}{\isachardoublequote}}}{\isa{{\isachardoublequote}P\isactrlsub {\isadigit{1}}\ {\isasymlongrightarrow}\ Q\isactrlsub {\isadigit{1}}{\isachardoublequote}} & \isa{{\isachardoublequote}P\isactrlsub {\isadigit{2}}\ {\isasymlongrightarrow}\ Q\isactrlsub {\isadigit{2}}{\isachardoublequote}}}
wenzelm@26849
   740
  \]
wenzelm@26849
   741
wenzelm@26849
   742
  \item De Morgan style equations for reasoning about the ``polarity''
wenzelm@26849
   743
  of expressions, e.g.
wenzelm@26849
   744
  \[
wenzelm@26849
   745
  \isa{{\isachardoublequote}{\isasymnot}\ {\isasymnot}\ P\ {\isasymlongleftrightarrow}\ P{\isachardoublequote}} \qquad\qquad
wenzelm@26849
   746
  \isa{{\isachardoublequote}{\isasymnot}\ {\isacharparenleft}P\ {\isasymand}\ Q{\isacharparenright}\ {\isasymlongleftrightarrow}\ {\isasymnot}\ P\ {\isasymor}\ {\isasymnot}\ Q{\isachardoublequote}}
wenzelm@26849
   747
  \]
wenzelm@26849
   748
wenzelm@26849
   749
  \item Equations for reducing complex operators to more primitive
wenzelm@26849
   750
  ones whose monotonicity can easily be proved, e.g.
wenzelm@26849
   751
  \[
wenzelm@26849
   752
  \isa{{\isachardoublequote}{\isacharparenleft}P\ {\isasymlongrightarrow}\ Q{\isacharparenright}\ {\isasymlongleftrightarrow}\ {\isasymnot}\ P\ {\isasymor}\ Q{\isachardoublequote}} \qquad\qquad
wenzelm@26849
   753
  \isa{{\isachardoublequote}Ball\ A\ P\ {\isasymequiv}\ {\isasymforall}x{\isachardot}\ x\ {\isasymin}\ A\ {\isasymlongrightarrow}\ P\ x{\isachardoublequote}}
wenzelm@26849
   754
  \]
wenzelm@26849
   755
wenzelm@26849
   756
  \end{itemize}
wenzelm@26849
   757
wenzelm@26849
   758
  %FIXME: Example of an inductive definition%
wenzelm@26849
   759
\end{isamarkuptext}%
wenzelm@26849
   760
\isamarkuptrue%
wenzelm@26849
   761
%
wenzelm@26849
   762
\isamarkupsection{Arithmetic proof support%
wenzelm@26849
   763
}
wenzelm@26849
   764
\isamarkuptrue%
wenzelm@26849
   765
%
wenzelm@26849
   766
\begin{isamarkuptext}%
wenzelm@26849
   767
\begin{matharray}{rcl}
wenzelm@28788
   768
    \indexdef{HOL}{method}{arith}\hypertarget{method.HOL.arith}{\hyperlink{method.HOL.arith}{\mbox{\isa{arith}}}} & : & \isa{method} \\
wenzelm@28788
   769
    \indexdef{HOL}{attribute}{arith\_split}\hypertarget{attribute.HOL.arith-split}{\hyperlink{attribute.HOL.arith-split}{\mbox{\isa{arith{\isacharunderscore}split}}}} & : & \isa{attribute} \\
wenzelm@26849
   770
  \end{matharray}
wenzelm@26849
   771
wenzelm@26902
   772
  The \hyperlink{method.HOL.arith}{\mbox{\isa{arith}}} method decides linear arithmetic problems
wenzelm@26849
   773
  (on types \isa{nat}, \isa{int}, \isa{real}).  Any current
wenzelm@26849
   774
  facts are inserted into the goal before running the procedure.
wenzelm@26849
   775
wenzelm@26907
   776
  The \hyperlink{attribute.HOL.arith-split}{\mbox{\isa{arith{\isacharunderscore}split}}} attribute declares case split
wenzelm@26895
   777
  rules to be expanded before the arithmetic procedure is invoked.
wenzelm@26849
   778
wenzelm@26849
   779
  Note that a simpler (but faster) version of arithmetic reasoning is
wenzelm@26849
   780
  already performed by the Simplifier.%
wenzelm@26849
   781
\end{isamarkuptext}%
wenzelm@26849
   782
\isamarkuptrue%
wenzelm@26849
   783
%
wenzelm@28603
   784
\isamarkupsection{Invoking automated reasoning tools -- The Sledgehammer%
wenzelm@28603
   785
}
wenzelm@28603
   786
\isamarkuptrue%
wenzelm@28603
   787
%
wenzelm@28603
   788
\begin{isamarkuptext}%
wenzelm@28603
   789
Isabelle/HOL includes a generic \emph{ATP manager} that allows
wenzelm@28603
   790
  external automated reasoning tools to crunch a pending goal.
wenzelm@28603
   791
  Supported provers include E\footnote{\url{http://www.eprover.org}},
wenzelm@28603
   792
  SPASS\footnote{\url{http://www.spass-prover.org/}}, and Vampire.
wenzelm@28603
   793
  There is also a wrapper to invoke provers remotely via the
wenzelm@28603
   794
  SystemOnTPTP\footnote{\url{http://www.cs.miami.edu/~tptp/cgi-bin/SystemOnTPTP}}
wenzelm@28603
   795
  web service.
wenzelm@28603
   796
wenzelm@28603
   797
  The problem passed to external provers consists of the goal together
wenzelm@28603
   798
  with a smart selection of lemmas from the current theory context.
wenzelm@28603
   799
  The result of a successful proof search is some source text that
wenzelm@28603
   800
  usually reconstructs the proof within Isabelle, without requiring
wenzelm@28603
   801
  external provers again.  The Metis
wenzelm@28603
   802
  prover\footnote{\url{http://www.gilith.com/software/metis/}} that is
wenzelm@28603
   803
  integrated into Isabelle/HOL is being used here.
wenzelm@28603
   804
wenzelm@28603
   805
  In this mode of operation, heavy means of automated reasoning are
wenzelm@28603
   806
  used as a strong relevance filter, while the main proof checking
wenzelm@28603
   807
  works via explicit inferences going through the Isabelle kernel.
wenzelm@28603
   808
  Moreover, rechecking Isabelle proof texts with already specified
wenzelm@28603
   809
  auxiliary facts is much faster than performing fully automated
wenzelm@28603
   810
  search over and over again.
wenzelm@28603
   811
wenzelm@28603
   812
  \begin{matharray}{rcl}
wenzelm@28788
   813
    \indexdef{HOL}{command}{sledgehammer}\hypertarget{command.HOL.sledgehammer}{\hyperlink{command.HOL.sledgehammer}{\mbox{\isa{\isacommand{sledgehammer}}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isa{{\isachardoublequote}proof\ {\isasymrightarrow}{\isachardoublequote}} \\
wenzelm@28788
   814
    \indexdef{HOL}{command}{print\_atps}\hypertarget{command.HOL.print-atps}{\hyperlink{command.HOL.print-atps}{\mbox{\isa{\isacommand{print{\isacharunderscore}atps}}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isa{{\isachardoublequote}context\ {\isasymrightarrow}{\isachardoublequote}} \\
wenzelm@28788
   815
    \indexdef{HOL}{command}{atp\_info}\hypertarget{command.HOL.atp-info}{\hyperlink{command.HOL.atp-info}{\mbox{\isa{\isacommand{atp{\isacharunderscore}info}}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isa{{\isachardoublequote}any\ {\isasymrightarrow}{\isachardoublequote}} \\
wenzelm@28788
   816
    \indexdef{HOL}{command}{atp\_kill}\hypertarget{command.HOL.atp-kill}{\hyperlink{command.HOL.atp-kill}{\mbox{\isa{\isacommand{atp{\isacharunderscore}kill}}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isa{{\isachardoublequote}any\ {\isasymrightarrow}{\isachardoublequote}} \\
wenzelm@29113
   817
    \indexdef{HOL}{command}{atp\_messages}\hypertarget{command.HOL.atp-messages}{\hyperlink{command.HOL.atp-messages}{\mbox{\isa{\isacommand{atp{\isacharunderscore}messages}}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isa{{\isachardoublequote}any\ {\isasymrightarrow}{\isachardoublequote}} \\
wenzelm@28788
   818
    \indexdef{HOL}{method}{metis}\hypertarget{method.HOL.metis}{\hyperlink{method.HOL.metis}{\mbox{\isa{metis}}}} & : & \isa{method} \\
wenzelm@28603
   819
  \end{matharray}
wenzelm@28603
   820
wenzelm@28603
   821
  \begin{rail}
wenzelm@28603
   822
  'sledgehammer' (nameref *)
wenzelm@28603
   823
  ;
wenzelm@29113
   824
  'atp\_messages' ('(' nat ')')?
wenzelm@29115
   825
  ;
wenzelm@28603
   826
wenzelm@28603
   827
  'metis' thmrefs
wenzelm@28603
   828
  ;
wenzelm@28603
   829
  \end{rail}
wenzelm@28603
   830
wenzelm@28788
   831
  \begin{description}
wenzelm@28603
   832
wenzelm@28788
   833
  \item \hyperlink{command.HOL.sledgehammer}{\mbox{\isa{\isacommand{sledgehammer}}}}~\isa{{\isachardoublequote}prover\isactrlsub {\isadigit{1}}\ {\isasymdots}\ prover\isactrlsub n{\isachardoublequote}}
wenzelm@28788
   834
  invokes the specified automated theorem provers on the first
wenzelm@28788
   835
  subgoal.  Provers are run in parallel, the first successful result
wenzelm@28788
   836
  is displayed, and the other attempts are terminated.
wenzelm@28603
   837
wenzelm@28603
   838
  Provers are defined in the theory context, see also \hyperlink{command.HOL.print-atps}{\mbox{\isa{\isacommand{print{\isacharunderscore}atps}}}}.  If no provers are given as arguments to \hyperlink{command.HOL.sledgehammer}{\mbox{\isa{\isacommand{sledgehammer}}}}, the system refers to the default defined as
wenzelm@28603
   839
  ``ATP provers'' preference by the user interface.
wenzelm@28603
   840
wenzelm@28603
   841
  There are additional preferences for timeout (default: 60 seconds),
wenzelm@28603
   842
  and the maximum number of independent prover processes (default: 5);
wenzelm@28603
   843
  excessive provers are automatically terminated.
wenzelm@28603
   844
wenzelm@28788
   845
  \item \hyperlink{command.HOL.print-atps}{\mbox{\isa{\isacommand{print{\isacharunderscore}atps}}}} prints the list of automated
wenzelm@28603
   846
  theorem provers available to the \hyperlink{command.HOL.sledgehammer}{\mbox{\isa{\isacommand{sledgehammer}}}}
wenzelm@28603
   847
  command.
wenzelm@28603
   848
wenzelm@28788
   849
  \item \hyperlink{command.HOL.atp-info}{\mbox{\isa{\isacommand{atp{\isacharunderscore}info}}}} prints information about presently
wenzelm@28603
   850
  running provers, including elapsed runtime, and the remaining time
wenzelm@28603
   851
  until timeout.
wenzelm@28603
   852
wenzelm@28788
   853
  \item \hyperlink{command.HOL.atp-kill}{\mbox{\isa{\isacommand{atp{\isacharunderscore}kill}}}} terminates all presently running
wenzelm@28603
   854
  provers.
wenzelm@28603
   855
wenzelm@29113
   856
  \item \hyperlink{command.HOL.atp-messages}{\mbox{\isa{\isacommand{atp{\isacharunderscore}messages}}}} displays recent messages issued
wenzelm@29113
   857
  by automated theorem provers.  This allows to examine results that
wenzelm@29113
   858
  might have got lost due to the asynchronous nature of default
wenzelm@29115
   859
  \hyperlink{command.HOL.sledgehammer}{\mbox{\isa{\isacommand{sledgehammer}}}} output.  An optional message limit may
wenzelm@29115
   860
  be specified (default 5).
wenzelm@29113
   861
wenzelm@28788
   862
  \item \hyperlink{method.HOL.metis}{\mbox{\isa{metis}}}~\isa{{\isachardoublequote}facts{\isachardoublequote}} invokes the Metis prover
wenzelm@28788
   863
  with the given facts.  Metis is an automated proof tool of medium
wenzelm@28788
   864
  strength, but is fully integrated into Isabelle/HOL, with explicit
wenzelm@28788
   865
  inferences going through the kernel.  Thus its results are
wenzelm@28603
   866
  guaranteed to be ``correct by construction''.
wenzelm@28603
   867
wenzelm@28603
   868
  Note that all facts used with Metis need to be specified as explicit
wenzelm@28603
   869
  arguments.  There are no rule declarations as for other Isabelle
wenzelm@28603
   870
  provers, like \hyperlink{method.blast}{\mbox{\isa{blast}}} or \hyperlink{method.fast}{\mbox{\isa{fast}}}.
wenzelm@28603
   871
wenzelm@28788
   872
  \end{description}%
wenzelm@28603
   873
\end{isamarkuptext}%
wenzelm@28603
   874
\isamarkuptrue%
wenzelm@28603
   875
%
wenzelm@28788
   876
\isamarkupsection{Unstructured case analysis and induction \label{sec:hol-induct-tac}%
wenzelm@26849
   877
}
wenzelm@26849
   878
\isamarkuptrue%
wenzelm@26849
   879
%
wenzelm@26849
   880
\begin{isamarkuptext}%
wenzelm@27124
   881
The following tools of Isabelle/HOL support cases analysis and
wenzelm@27124
   882
  induction in unstructured tactic scripts; see also
wenzelm@27124
   883
  \secref{sec:cases-induct} for proper Isar versions of similar ideas.
wenzelm@26849
   884
wenzelm@26849
   885
  \begin{matharray}{rcl}
wenzelm@28788
   886
    \indexdef{HOL}{method}{case\_tac}\hypertarget{method.HOL.case-tac}{\hyperlink{method.HOL.case-tac}{\mbox{\isa{case{\isacharunderscore}tac}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isa{method} \\
wenzelm@28788
   887
    \indexdef{HOL}{method}{induct\_tac}\hypertarget{method.HOL.induct-tac}{\hyperlink{method.HOL.induct-tac}{\mbox{\isa{induct{\isacharunderscore}tac}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isa{method} \\
wenzelm@28788
   888
    \indexdef{HOL}{method}{ind\_cases}\hypertarget{method.HOL.ind-cases}{\hyperlink{method.HOL.ind-cases}{\mbox{\isa{ind{\isacharunderscore}cases}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isa{method} \\
wenzelm@28788
   889
    \indexdef{HOL}{command}{inductive\_cases}\hypertarget{command.HOL.inductive-cases}{\hyperlink{command.HOL.inductive-cases}{\mbox{\isa{\isacommand{inductive{\isacharunderscore}cases}}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isa{{\isachardoublequote}local{\isacharunderscore}theory\ {\isasymrightarrow}\ local{\isacharunderscore}theory{\isachardoublequote}} \\
wenzelm@26849
   890
  \end{matharray}
wenzelm@26849
   891
wenzelm@26849
   892
  \begin{rail}
wenzelm@26849
   893
    'case\_tac' goalspec? term rule?
wenzelm@26849
   894
    ;
wenzelm@26849
   895
    'induct\_tac' goalspec? (insts * 'and') rule?
wenzelm@26849
   896
    ;
wenzelm@26849
   897
    'ind\_cases' (prop +) ('for' (name +)) ?
wenzelm@26849
   898
    ;
wenzelm@26849
   899
    'inductive\_cases' (thmdecl? (prop +) + 'and')
wenzelm@26849
   900
    ;
wenzelm@26849
   901
wenzelm@26849
   902
    rule: ('rule' ':' thmref)
wenzelm@26849
   903
    ;
wenzelm@26849
   904
  \end{rail}
wenzelm@26849
   905
wenzelm@28788
   906
  \begin{description}
wenzelm@26849
   907
wenzelm@28788
   908
  \item \hyperlink{method.HOL.case-tac}{\mbox{\isa{case{\isacharunderscore}tac}}} and \hyperlink{method.HOL.induct-tac}{\mbox{\isa{induct{\isacharunderscore}tac}}} admit
wenzelm@28788
   909
  to reason about inductive types.  Rules are selected according to
wenzelm@28788
   910
  the declarations by the \hyperlink{attribute.cases}{\mbox{\isa{cases}}} and \hyperlink{attribute.induct}{\mbox{\isa{induct}}}
wenzelm@28788
   911
  attributes, cf.\ \secref{sec:cases-induct}.  The \hyperlink{command.HOL.datatype}{\mbox{\isa{\isacommand{datatype}}}} package already takes care of this.
wenzelm@27124
   912
wenzelm@27124
   913
  These unstructured tactics feature both goal addressing and dynamic
wenzelm@26849
   914
  instantiation.  Note that named rule cases are \emph{not} provided
wenzelm@27124
   915
  as would be by the proper \hyperlink{method.cases}{\mbox{\isa{cases}}} and \hyperlink{method.induct}{\mbox{\isa{induct}}} proof
wenzelm@27124
   916
  methods (see \secref{sec:cases-induct}).  Unlike the \hyperlink{method.induct}{\mbox{\isa{induct}}} method, \hyperlink{method.induct-tac}{\mbox{\isa{induct{\isacharunderscore}tac}}} does not handle structured rule
wenzelm@27124
   917
  statements, only the compact object-logic conclusion of the subgoal
wenzelm@27124
   918
  being addressed.
wenzelm@26849
   919
  
wenzelm@28788
   920
  \item \hyperlink{method.HOL.ind-cases}{\mbox{\isa{ind{\isacharunderscore}cases}}} and \hyperlink{command.HOL.inductive-cases}{\mbox{\isa{\isacommand{inductive{\isacharunderscore}cases}}}} provide an interface to the internal \verb|mk_cases| operation.  Rules are simplified in an unrestricted
wenzelm@26861
   921
  forward manner.
wenzelm@26849
   922
wenzelm@26907
   923
  While \hyperlink{method.HOL.ind-cases}{\mbox{\isa{ind{\isacharunderscore}cases}}} is a proof method to apply the
wenzelm@26907
   924
  result immediately as elimination rules, \hyperlink{command.HOL.inductive-cases}{\mbox{\isa{\isacommand{inductive{\isacharunderscore}cases}}}} provides case split theorems at the theory level
wenzelm@26907
   925
  for later use.  The \hyperlink{keyword.for}{\mbox{\isa{\isakeyword{for}}}} argument of the \hyperlink{method.HOL.ind-cases}{\mbox{\isa{ind{\isacharunderscore}cases}}} method allows to specify a list of variables that should
wenzelm@26849
   926
  be generalized before applying the resulting rule.
wenzelm@26849
   927
wenzelm@28788
   928
  \end{description}%
wenzelm@26849
   929
\end{isamarkuptext}%
wenzelm@26849
   930
\isamarkuptrue%
wenzelm@26849
   931
%
wenzelm@26849
   932
\isamarkupsection{Executable code%
wenzelm@26849
   933
}
wenzelm@26849
   934
\isamarkuptrue%
wenzelm@26849
   935
%
wenzelm@26849
   936
\begin{isamarkuptext}%
wenzelm@26849
   937
Isabelle/Pure provides two generic frameworks to support code
wenzelm@26849
   938
  generation from executable specifications.  Isabelle/HOL
wenzelm@26849
   939
  instantiates these mechanisms in a way that is amenable to end-user
wenzelm@26849
   940
  applications.
wenzelm@26849
   941
wenzelm@26849
   942
  One framework generates code from both functional and relational
wenzelm@26849
   943
  programs to SML.  See \cite{isabelle-HOL} for further information
wenzelm@26849
   944
  (this actually covers the new-style theory format as well).
wenzelm@26849
   945
wenzelm@26849
   946
  \begin{matharray}{rcl}
wenzelm@28788
   947
    \indexdef{HOL}{command}{value}\hypertarget{command.HOL.value}{\hyperlink{command.HOL.value}{\mbox{\isa{\isacommand{value}}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isa{{\isachardoublequote}context\ {\isasymrightarrow}{\isachardoublequote}} \\
wenzelm@28788
   948
    \indexdef{HOL}{command}{code\_module}\hypertarget{command.HOL.code-module}{\hyperlink{command.HOL.code-module}{\mbox{\isa{\isacommand{code{\isacharunderscore}module}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ theory{\isachardoublequote}} \\
wenzelm@28788
   949
    \indexdef{HOL}{command}{code\_library}\hypertarget{command.HOL.code-library}{\hyperlink{command.HOL.code-library}{\mbox{\isa{\isacommand{code{\isacharunderscore}library}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ theory{\isachardoublequote}} \\
wenzelm@28788
   950
    \indexdef{HOL}{command}{consts\_code}\hypertarget{command.HOL.consts-code}{\hyperlink{command.HOL.consts-code}{\mbox{\isa{\isacommand{consts{\isacharunderscore}code}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ theory{\isachardoublequote}} \\
wenzelm@28788
   951
    \indexdef{HOL}{command}{types\_code}\hypertarget{command.HOL.types-code}{\hyperlink{command.HOL.types-code}{\mbox{\isa{\isacommand{types{\isacharunderscore}code}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ theory{\isachardoublequote}} \\  
wenzelm@28788
   952
    \indexdef{HOL}{attribute}{code}\hypertarget{attribute.HOL.code}{\hyperlink{attribute.HOL.code}{\mbox{\isa{code}}}} & : & \isa{attribute} \\
wenzelm@26849
   953
  \end{matharray}
wenzelm@26849
   954
wenzelm@26849
   955
  \begin{rail}
wenzelm@26849
   956
  'value' term
wenzelm@26849
   957
  ;
wenzelm@26849
   958
wenzelm@26849
   959
  ( 'code\_module' | 'code\_library' ) modespec ? name ? \\
wenzelm@26849
   960
    ( 'file' name ) ? ( 'imports' ( name + ) ) ? \\
wenzelm@26849
   961
    'contains' ( ( name '=' term ) + | term + )
wenzelm@26849
   962
  ;
wenzelm@26849
   963
wenzelm@26849
   964
  modespec: '(' ( name * ) ')'
wenzelm@26849
   965
  ;
wenzelm@26849
   966
wenzelm@26849
   967
  'consts\_code' (codespec +)
wenzelm@26849
   968
  ;
wenzelm@26849
   969
wenzelm@26849
   970
  codespec: const template attachment ?
wenzelm@26849
   971
  ;
wenzelm@26849
   972
wenzelm@26849
   973
  'types\_code' (tycodespec +)
wenzelm@26849
   974
  ;
wenzelm@26849
   975
wenzelm@26849
   976
  tycodespec: name template attachment ?
wenzelm@26849
   977
  ;
wenzelm@26849
   978
wenzelm@26849
   979
  const: term
wenzelm@26849
   980
  ;
wenzelm@26849
   981
wenzelm@26849
   982
  template: '(' string ')'
wenzelm@26849
   983
  ;
wenzelm@26849
   984
wenzelm@26849
   985
  attachment: 'attach' modespec ? verblbrace text verbrbrace
wenzelm@26849
   986
  ;
wenzelm@26849
   987
wenzelm@26849
   988
  'code' (name)?
wenzelm@26849
   989
  ;
wenzelm@26849
   990
  \end{rail}
wenzelm@26849
   991
wenzelm@28788
   992
  \begin{description}
wenzelm@26849
   993
wenzelm@28788
   994
  \item \hyperlink{command.HOL.value}{\mbox{\isa{\isacommand{value}}}}~\isa{t} evaluates and prints a term
wenzelm@28788
   995
  using the code generator.
wenzelm@26849
   996
wenzelm@28788
   997
  \end{description}
wenzelm@26849
   998
wenzelm@26849
   999
  \medskip The other framework generates code from functional programs
wenzelm@26849
  1000
  (including overloading using type classes) to SML \cite{SML}, OCaml
wenzelm@26849
  1001
  \cite{OCaml} and Haskell \cite{haskell-revised-report}.
wenzelm@26849
  1002
  Conceptually, code generation is split up in three steps:
wenzelm@26849
  1003
  \emph{selection} of code theorems, \emph{translation} into an
wenzelm@26849
  1004
  abstract executable view and \emph{serialization} to a specific
wenzelm@26849
  1005
  \emph{target language}.  See \cite{isabelle-codegen} for an
wenzelm@26849
  1006
  introduction on how to use it.
wenzelm@26849
  1007
wenzelm@26849
  1008
  \begin{matharray}{rcl}
wenzelm@28788
  1009
    \indexdef{HOL}{command}{export\_code}\hypertarget{command.HOL.export-code}{\hyperlink{command.HOL.export-code}{\mbox{\isa{\isacommand{export{\isacharunderscore}code}}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isa{{\isachardoublequote}context\ {\isasymrightarrow}{\isachardoublequote}} \\
wenzelm@28788
  1010
    \indexdef{HOL}{command}{code\_thms}\hypertarget{command.HOL.code-thms}{\hyperlink{command.HOL.code-thms}{\mbox{\isa{\isacommand{code{\isacharunderscore}thms}}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isa{{\isachardoublequote}context\ {\isasymrightarrow}{\isachardoublequote}} \\
wenzelm@28788
  1011
    \indexdef{HOL}{command}{code\_deps}\hypertarget{command.HOL.code-deps}{\hyperlink{command.HOL.code-deps}{\mbox{\isa{\isacommand{code{\isacharunderscore}deps}}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isa{{\isachardoublequote}context\ {\isasymrightarrow}{\isachardoublequote}} \\
wenzelm@28788
  1012
    \indexdef{HOL}{command}{code\_datatype}\hypertarget{command.HOL.code-datatype}{\hyperlink{command.HOL.code-datatype}{\mbox{\isa{\isacommand{code{\isacharunderscore}datatype}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ theory{\isachardoublequote}} \\
wenzelm@28788
  1013
    \indexdef{HOL}{command}{code\_const}\hypertarget{command.HOL.code-const}{\hyperlink{command.HOL.code-const}{\mbox{\isa{\isacommand{code{\isacharunderscore}const}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ theory{\isachardoublequote}} \\
wenzelm@28788
  1014
    \indexdef{HOL}{command}{code\_type}\hypertarget{command.HOL.code-type}{\hyperlink{command.HOL.code-type}{\mbox{\isa{\isacommand{code{\isacharunderscore}type}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ theory{\isachardoublequote}} \\
wenzelm@28788
  1015
    \indexdef{HOL}{command}{code\_class}\hypertarget{command.HOL.code-class}{\hyperlink{command.HOL.code-class}{\mbox{\isa{\isacommand{code{\isacharunderscore}class}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ theory{\isachardoublequote}} \\
wenzelm@28788
  1016
    \indexdef{HOL}{command}{code\_instance}\hypertarget{command.HOL.code-instance}{\hyperlink{command.HOL.code-instance}{\mbox{\isa{\isacommand{code{\isacharunderscore}instance}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ theory{\isachardoublequote}} \\
wenzelm@28788
  1017
    \indexdef{HOL}{command}{code\_monad}\hypertarget{command.HOL.code-monad}{\hyperlink{command.HOL.code-monad}{\mbox{\isa{\isacommand{code{\isacharunderscore}monad}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ theory{\isachardoublequote}} \\
wenzelm@28788
  1018
    \indexdef{HOL}{command}{code\_reserved}\hypertarget{command.HOL.code-reserved}{\hyperlink{command.HOL.code-reserved}{\mbox{\isa{\isacommand{code{\isacharunderscore}reserved}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ theory{\isachardoublequote}} \\
wenzelm@28788
  1019
    \indexdef{HOL}{command}{code\_include}\hypertarget{command.HOL.code-include}{\hyperlink{command.HOL.code-include}{\mbox{\isa{\isacommand{code{\isacharunderscore}include}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ theory{\isachardoublequote}} \\
wenzelm@28788
  1020
    \indexdef{HOL}{command}{code\_modulename}\hypertarget{command.HOL.code-modulename}{\hyperlink{command.HOL.code-modulename}{\mbox{\isa{\isacommand{code{\isacharunderscore}modulename}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ theory{\isachardoublequote}} \\
wenzelm@28788
  1021
    \indexdef{HOL}{command}{code\_abort}\hypertarget{command.HOL.code-abort}{\hyperlink{command.HOL.code-abort}{\mbox{\isa{\isacommand{code{\isacharunderscore}abort}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ theory{\isachardoublequote}} \\
wenzelm@28788
  1022
    \indexdef{HOL}{command}{print\_codesetup}\hypertarget{command.HOL.print-codesetup}{\hyperlink{command.HOL.print-codesetup}{\mbox{\isa{\isacommand{print{\isacharunderscore}codesetup}}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isa{{\isachardoublequote}context\ {\isasymrightarrow}{\isachardoublequote}} \\
wenzelm@28788
  1023
    \indexdef{HOL}{attribute}{code}\hypertarget{attribute.HOL.code}{\hyperlink{attribute.HOL.code}{\mbox{\isa{code}}}} & : & \isa{attribute} \\
wenzelm@26849
  1024
  \end{matharray}
wenzelm@26849
  1025
wenzelm@26849
  1026
  \begin{rail}
wenzelm@26849
  1027
    'export\_code' ( constexpr + ) ? \\
wenzelm@26849
  1028
      ( ( 'in' target ( 'module\_name' string ) ? \\
wenzelm@26849
  1029
        ( 'file' ( string | '-' ) ) ? ( '(' args ')' ) ?) + ) ?
wenzelm@26849
  1030
    ;
wenzelm@26849
  1031
wenzelm@26849
  1032
    'code\_thms' ( constexpr + ) ?
wenzelm@26849
  1033
    ;
wenzelm@26849
  1034
wenzelm@26849
  1035
    'code\_deps' ( constexpr + ) ?
wenzelm@26849
  1036
    ;
wenzelm@26849
  1037
wenzelm@26849
  1038
    const: term
wenzelm@26849
  1039
    ;
wenzelm@26849
  1040
wenzelm@26849
  1041
    constexpr: ( const | 'name.*' | '*' )
wenzelm@26849
  1042
    ;
wenzelm@26849
  1043
wenzelm@26849
  1044
    typeconstructor: nameref
wenzelm@26849
  1045
    ;
wenzelm@26849
  1046
wenzelm@26849
  1047
    class: nameref
wenzelm@26849
  1048
    ;
wenzelm@26849
  1049
wenzelm@26849
  1050
    target: 'OCaml' | 'SML' | 'Haskell'
wenzelm@26849
  1051
    ;
wenzelm@26849
  1052
wenzelm@26849
  1053
    'code\_datatype' const +
wenzelm@26849
  1054
    ;
wenzelm@26849
  1055
wenzelm@26849
  1056
    'code\_const' (const + 'and') \\
wenzelm@26849
  1057
      ( ( '(' target ( syntax ? + 'and' ) ')' ) + )
wenzelm@26849
  1058
    ;
wenzelm@26849
  1059
wenzelm@26849
  1060
    'code\_type' (typeconstructor + 'and') \\
wenzelm@26849
  1061
      ( ( '(' target ( syntax ? + 'and' ) ')' ) + )
wenzelm@26849
  1062
    ;
wenzelm@26849
  1063
wenzelm@26849
  1064
    'code\_class' (class + 'and') \\
haftmann@28687
  1065
      ( ( '(' target \\ ( string ? + 'and' ) ')' ) + )
wenzelm@26849
  1066
    ;
wenzelm@26849
  1067
wenzelm@26849
  1068
    'code\_instance' (( typeconstructor '::' class ) + 'and') \\
wenzelm@26849
  1069
      ( ( '(' target ( '-' ? + 'and' ) ')' ) + )
wenzelm@26849
  1070
    ;
wenzelm@26849
  1071
wenzelm@26849
  1072
    'code\_monad' const const target
wenzelm@26849
  1073
    ;
wenzelm@26849
  1074
wenzelm@26849
  1075
    'code\_reserved' target ( string + )
wenzelm@26849
  1076
    ;
wenzelm@26849
  1077
wenzelm@26849
  1078
    'code\_include' target ( string ( string | '-') )
wenzelm@26849
  1079
    ;
wenzelm@26849
  1080
wenzelm@26849
  1081
    'code\_modulename' target ( ( string string ) + )
wenzelm@26849
  1082
    ;
wenzelm@26849
  1083
haftmann@27452
  1084
    'code\_abort' ( const + )
wenzelm@26849
  1085
    ;
wenzelm@26849
  1086
wenzelm@26849
  1087
    syntax: string | ( 'infix' | 'infixl' | 'infixr' ) nat string
wenzelm@26849
  1088
    ;
wenzelm@26849
  1089
haftmann@28562
  1090
    'code' ( 'inline' ) ? ( 'del' ) ?
wenzelm@26849
  1091
    ;
wenzelm@26849
  1092
  \end{rail}
wenzelm@26849
  1093
wenzelm@28788
  1094
  \begin{description}
wenzelm@26849
  1095
wenzelm@28788
  1096
  \item \hyperlink{command.HOL.export-code}{\mbox{\isa{\isacommand{export{\isacharunderscore}code}}}} is the canonical interface for
wenzelm@28788
  1097
  generating and serializing code: for a given list of constants, code
wenzelm@28788
  1098
  is generated for the specified target languages.  Abstract code is
wenzelm@28788
  1099
  cached incrementally.  If no constant is given, the currently cached
wenzelm@28788
  1100
  code is serialized.  If no serialization instruction is given, only
wenzelm@28788
  1101
  abstract code is cached.
wenzelm@26849
  1102
wenzelm@26849
  1103
  Constants may be specified by giving them literally, referring to
wenzelm@26849
  1104
  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
  1105
  available by giving \isa{{\isachardoublequote}{\isacharasterisk}{\isachardoublequote}}.
wenzelm@26849
  1106
wenzelm@26849
  1107
  By default, for each involved theory one corresponding name space
wenzelm@26849
  1108
  module is generated.  Alternativly, a module name may be specified
wenzelm@26907
  1109
  after the \hyperlink{keyword.module-name}{\mbox{\isa{\isakeyword{module{\isacharunderscore}name}}}} keyword; then \emph{all} code is
wenzelm@26849
  1110
  placed in this module.
wenzelm@26849
  1111
wenzelm@26849
  1112
  For \emph{SML} and \emph{OCaml}, the file specification refers to a
wenzelm@26849
  1113
  single file; for \emph{Haskell}, it refers to a whole directory,
wenzelm@26849
  1114
  where code is generated in multiple files reflecting the module
wenzelm@26849
  1115
  hierarchy.  The file specification ``\isa{{\isachardoublequote}{\isacharminus}{\isachardoublequote}}'' denotes standard
wenzelm@26849
  1116
  output.  For \emph{SML}, omitting the file specification compiles
wenzelm@26849
  1117
  code internally in the context of the current ML session.
wenzelm@26849
  1118
wenzelm@26849
  1119
  Serializers take an optional list of arguments in parentheses.  For
wenzelm@26849
  1120
  \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
  1121
  declaration.
wenzelm@26849
  1122
wenzelm@28788
  1123
  \item \hyperlink{command.HOL.code-thms}{\mbox{\isa{\isacommand{code{\isacharunderscore}thms}}}} prints a list of theorems
wenzelm@26849
  1124
  representing the corresponding program containing all given
wenzelm@26849
  1125
  constants; if no constants are given, the currently cached code
wenzelm@26849
  1126
  theorems are printed.
wenzelm@26849
  1127
wenzelm@28788
  1128
  \item \hyperlink{command.HOL.code-deps}{\mbox{\isa{\isacommand{code{\isacharunderscore}deps}}}} visualizes dependencies of
wenzelm@26849
  1129
  theorems representing the corresponding program containing all given
wenzelm@26849
  1130
  constants; if no constants are given, the currently cached code
wenzelm@26849
  1131
  theorems are visualized.
wenzelm@26849
  1132
wenzelm@28788
  1133
  \item \hyperlink{command.HOL.code-datatype}{\mbox{\isa{\isacommand{code{\isacharunderscore}datatype}}}} specifies a constructor set
wenzelm@26849
  1134
  for a logical type.
wenzelm@26849
  1135
wenzelm@28788
  1136
  \item \hyperlink{command.HOL.code-const}{\mbox{\isa{\isacommand{code{\isacharunderscore}const}}}} associates a list of constants
wenzelm@26849
  1137
  with target-specific serializations; omitting a serialization
wenzelm@26849
  1138
  deletes an existing serialization.
wenzelm@26849
  1139
wenzelm@28788
  1140
  \item \hyperlink{command.HOL.code-type}{\mbox{\isa{\isacommand{code{\isacharunderscore}type}}}} associates a list of type
wenzelm@26849
  1141
  constructors with target-specific serializations; omitting a
wenzelm@26849
  1142
  serialization deletes an existing serialization.
wenzelm@26849
  1143
wenzelm@28788
  1144
  \item \hyperlink{command.HOL.code-class}{\mbox{\isa{\isacommand{code{\isacharunderscore}class}}}} associates a list of classes
wenzelm@28788
  1145
  with target-specific class names; omitting a serialization deletes
wenzelm@28788
  1146
  an existing serialization.  This applies only to \emph{Haskell}.
wenzelm@26849
  1147
wenzelm@28788
  1148
  \item \hyperlink{command.HOL.code-instance}{\mbox{\isa{\isacommand{code{\isacharunderscore}instance}}}} declares a list of type
wenzelm@26849
  1149
  constructor / class instance relations as ``already present'' for a
wenzelm@26849
  1150
  given target.  Omitting a ``\isa{{\isachardoublequote}{\isacharminus}{\isachardoublequote}}'' deletes an existing
wenzelm@26849
  1151
  ``already present'' declaration.  This applies only to
wenzelm@26849
  1152
  \emph{Haskell}.
wenzelm@26849
  1153
wenzelm@28788
  1154
  \item \hyperlink{command.HOL.code-monad}{\mbox{\isa{\isacommand{code{\isacharunderscore}monad}}}} provides an auxiliary mechanism
wenzelm@28788
  1155
  to generate monadic code for Haskell.
wenzelm@26849
  1156
wenzelm@28788
  1157
  \item \hyperlink{command.HOL.code-reserved}{\mbox{\isa{\isacommand{code{\isacharunderscore}reserved}}}} declares a list of names as
wenzelm@26849
  1158
  reserved for a given target, preventing it to be shadowed by any
wenzelm@26849
  1159
  generated code.
wenzelm@26849
  1160
wenzelm@28788
  1161
  \item \hyperlink{command.HOL.code-include}{\mbox{\isa{\isacommand{code{\isacharunderscore}include}}}} adds arbitrary named content
wenzelm@27834
  1162
  (``include'') to generated code.  A ``\isa{{\isachardoublequote}{\isacharminus}{\isachardoublequote}}'' as last argument
wenzelm@26849
  1163
  will remove an already added ``include''.
wenzelm@26849
  1164
wenzelm@28788
  1165
  \item \hyperlink{command.HOL.code-modulename}{\mbox{\isa{\isacommand{code{\isacharunderscore}modulename}}}} declares aliasings from one
wenzelm@28788
  1166
  module name onto another.
wenzelm@26849
  1167
wenzelm@28788
  1168
  \item \hyperlink{command.HOL.code-abort}{\mbox{\isa{\isacommand{code{\isacharunderscore}abort}}}} declares constants which are not
wenzelm@28788
  1169
  required to have a definition by means of defining equations; if
wenzelm@28788
  1170
  needed these are implemented by program abort instead.
wenzelm@26849
  1171
wenzelm@28788
  1172
  \item \hyperlink{attribute.HOL.code}{\mbox{\isa{code}}} explicitly selects (or with option
wenzelm@28788
  1173
  ``\isa{{\isachardoublequote}del{\isachardoublequote}}'' deselects) a defining equation for code
wenzelm@28788
  1174
  generation.  Usually packages introducing defining equations provide
wenzelm@28788
  1175
  a reasonable default setup for selection.
wenzelm@26849
  1176
wenzelm@28788
  1177
  \item \hyperlink{attribute.HOL.code}{\mbox{\isa{code}}}~\isa{inline} declares (or with
haftmann@28562
  1178
  option ``\isa{{\isachardoublequote}del{\isachardoublequote}}'' removes) inlining theorems which are
wenzelm@26849
  1179
  applied as rewrite rules to any defining equation during
wenzelm@26849
  1180
  preprocessing.
wenzelm@26849
  1181
wenzelm@28788
  1182
  \item \hyperlink{command.HOL.print-codesetup}{\mbox{\isa{\isacommand{print{\isacharunderscore}codesetup}}}} gives an overview on
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  1183
  selected defining equations, code generator datatypes and
wenzelm@26849
  1184
  preprocessor setup.
wenzelm@26849
  1185
wenzelm@28788
  1186
  \end{description}%
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  1187
\end{isamarkuptext}%
wenzelm@26849
  1188
\isamarkuptrue%
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  1189
%
wenzelm@27047
  1190
\isamarkupsection{Definition by specification \label{sec:hol-specification}%
wenzelm@27047
  1191
}
wenzelm@27047
  1192
\isamarkuptrue%
wenzelm@27047
  1193
%
wenzelm@27047
  1194
\begin{isamarkuptext}%
wenzelm@27047
  1195
\begin{matharray}{rcl}
wenzelm@28788
  1196
    \indexdef{HOL}{command}{specification}\hypertarget{command.HOL.specification}{\hyperlink{command.HOL.specification}{\mbox{\isa{\isacommand{specification}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ proof{\isacharparenleft}prove{\isacharparenright}{\isachardoublequote}} \\
wenzelm@28788
  1197
    \indexdef{HOL}{command}{ax\_specification}\hypertarget{command.HOL.ax-specification}{\hyperlink{command.HOL.ax-specification}{\mbox{\isa{\isacommand{ax{\isacharunderscore}specification}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ proof{\isacharparenleft}prove{\isacharparenright}{\isachardoublequote}} \\
wenzelm@27047
  1198
  \end{matharray}
wenzelm@27047
  1199
wenzelm@27047
  1200
  \begin{rail}
wenzelm@27047
  1201
  ('specification' | 'ax\_specification') '(' (decl +) ')' \\ (thmdecl? prop +)
wenzelm@27047
  1202
  ;
wenzelm@27047
  1203
  decl: ((name ':')? term '(' 'overloaded' ')'?)
wenzelm@27047
  1204
  \end{rail}
wenzelm@27047
  1205
wenzelm@28788
  1206
  \begin{description}
wenzelm@27047
  1207
wenzelm@28788
  1208
  \item \hyperlink{command.HOL.specification}{\mbox{\isa{\isacommand{specification}}}}~\isa{{\isachardoublequote}decls\ {\isasymphi}{\isachardoublequote}} sets up a
wenzelm@27047
  1209
  goal stating the existence of terms with the properties specified to
wenzelm@27047
  1210
  hold for the constants given in \isa{decls}.  After finishing the
wenzelm@27047
  1211
  proof, the theory will be augmented with definitions for the given
wenzelm@27047
  1212
  constants, as well as with theorems stating the properties for these
wenzelm@27047
  1213
  constants.
wenzelm@27047
  1214
wenzelm@28788
  1215
  \item \hyperlink{command.HOL.ax-specification}{\mbox{\isa{\isacommand{ax{\isacharunderscore}specification}}}}~\isa{{\isachardoublequote}decls\ {\isasymphi}{\isachardoublequote}} sets up
wenzelm@28788
  1216
  a goal stating the existence of terms with the properties specified
wenzelm@28788
  1217
  to hold for the constants given in \isa{decls}.  After finishing
wenzelm@28788
  1218
  the proof, the theory will be augmented with axioms expressing the
wenzelm@28788
  1219
  properties given in the first place.
wenzelm@27047
  1220
wenzelm@28788
  1221
  \item \isa{decl} declares a constant to be defined by the
wenzelm@27047
  1222
  specification given.  The definition for the constant \isa{c} is
wenzelm@27047
  1223
  bound to the name \isa{c{\isacharunderscore}def} unless a theorem name is given in
wenzelm@27047
  1224
  the declaration.  Overloaded constants should be declared as such.
wenzelm@27047
  1225
wenzelm@28788
  1226
  \end{description}
wenzelm@27047
  1227
wenzelm@27047
  1228
  Whether to use \hyperlink{command.HOL.specification}{\mbox{\isa{\isacommand{specification}}}} or \hyperlink{command.HOL.ax-specification}{\mbox{\isa{\isacommand{ax{\isacharunderscore}specification}}}} is to some extent a matter of style.  \hyperlink{command.HOL.specification}{\mbox{\isa{\isacommand{specification}}}} introduces no new axioms, and so by
wenzelm@27047
  1229
  construction cannot introduce inconsistencies, whereas \hyperlink{command.HOL.ax-specification}{\mbox{\isa{\isacommand{ax{\isacharunderscore}specification}}}} does introduce axioms, but only after the
wenzelm@27047
  1230
  user has explicitly proven it to be safe.  A practical issue must be
wenzelm@27047
  1231
  considered, though: After introducing two constants with the same
wenzelm@27047
  1232
  properties using \hyperlink{command.HOL.specification}{\mbox{\isa{\isacommand{specification}}}}, one can prove
wenzelm@27047
  1233
  that the two constants are, in fact, equal.  If this might be a
wenzelm@27047
  1234
  problem, one should use \hyperlink{command.HOL.ax-specification}{\mbox{\isa{\isacommand{ax{\isacharunderscore}specification}}}}.%
wenzelm@27047
  1235
\end{isamarkuptext}%
wenzelm@27047
  1236
\isamarkuptrue%
wenzelm@27047
  1237
%
wenzelm@26849
  1238
\isadelimtheory
wenzelm@26849
  1239
%
wenzelm@26849
  1240
\endisadelimtheory
wenzelm@26849
  1241
%
wenzelm@26849
  1242
\isatagtheory
wenzelm@26840
  1243
\isacommand{end}\isamarkupfalse%
wenzelm@26840
  1244
%
wenzelm@26840
  1245
\endisatagtheory
wenzelm@26840
  1246
{\isafoldtheory}%
wenzelm@26840
  1247
%
wenzelm@26840
  1248
\isadelimtheory
wenzelm@26840
  1249
%
wenzelm@26840
  1250
\endisadelimtheory
wenzelm@26849
  1251
\isanewline
wenzelm@26840
  1252
\end{isabellebody}%
wenzelm@26840
  1253
%%% Local Variables:
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  1254
%%% mode: latex
wenzelm@26840
  1255
%%% TeX-master: "root"
wenzelm@26840
  1256
%%% End: