doc-src/IsarRef/Thy/document/HOL_Specific.tex
author wenzelm
Thu Apr 15 20:56:04 2010 +0200 (2010-04-15)
changeset 36158 d2ad76e374d3
parent 36139 0c2538afe8e8
child 36453 2f383885d8f8
permissions -rw-r--r--
HOL record: explicitly allow sort constraints;
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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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%
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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{Typedef axiomatization \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}{typedef}\hypertarget{command.HOL.typedef}{\hyperlink{command.HOL.typedef}{\mbox{\isa{\isacommand{typedef}}}}} & : & \isa{{\isachardoublequote}local{\isacharunderscore}theory\ {\isasymrightarrow}\ proof{\isacharparenleft}prove{\isacharparenright}{\isachardoublequote}} \\
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  \end{matharray}
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  \begin{rail}
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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: typespecsorts mixfix?
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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.typedef}{\mbox{\isa{\isacommand{typedef}}}}~\isa{{\isachardoublequote}{\isacharparenleft}{\isasymalpha}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymalpha}\isactrlsub n{\isacharparenright}\ t\ {\isacharequal}\ A{\isachardoublequote}}
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  axiomatizes a Gordon/HOL-style type definition in the background
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  theory of the current context, depending on a non-emptiness result
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  of the set \isa{A} (which needs to be proven interactively).
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  The raw type may not depend on parameters or assumptions of the
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  context --- this is logically impossible in Isabelle/HOL --- but the
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  non-emptiness property can be local, potentially resulting in
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  multiple interpretations in target contexts.  Thus the established
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  bijection between the representing set \isa{A} and the new type
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  \isa{t} may semantically depend on local assumptions.
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  By default, \hyperlink{command.HOL.typedef}{\mbox{\isa{\isacommand{typedef}}}} defines both a type \isa{t}
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  and a set (term constant) of the same name, unless an alternative
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  base name is given in parentheses, or the ``\isa{{\isachardoublequote}{\isacharparenleft}open{\isacharparenright}{\isachardoublequote}}''
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  declaration is used to suppress a separate constant definition
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  altogether.  The injection from type to set is called \isa{Rep{\isacharunderscore}t},
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  its inverse \isa{Abs{\isacharunderscore}t} --- this may be changed via an explicit
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  \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 for the set definition (and other derived
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  entities) may be specified in parentheses; the default is to use
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  \isa{t} as indicated before.
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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{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' typespecsorts '=' (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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   301
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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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  \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
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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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   369
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    dtspec: parname? typespec mixfix? '=' (cons + '|')
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    ;
haftmann@31913
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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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   384
  \end{description}
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   385
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  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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   389
wenzelm@26849
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  See \cite{isabelle-HOL} for more details on datatypes, but beware of
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  the old-style theory syntax being used there!  Apart from proper
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  proof methods for case-analysis and induction, there are also
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  emulations of ML tactics \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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\end{isamarkuptext}%
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   397
\isamarkuptrue%
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   398
%
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   399
\isamarkupsection{Recursive functions \label{sec:recursion}%
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}
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   401
\isamarkuptrue%
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   402
%
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   403
\begin{isamarkuptext}%
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   404
\begin{matharray}{rcl}
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   405
    \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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   406
    \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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   407
    \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}} \\
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    \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}} \\
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  \end{matharray}
wenzelm@26849
   410
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  \begin{rail}
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    'primrec' target? fixes 'where' equations
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    ;
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    equations: (thmdecl? prop + '|')
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    ;
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    ('fun' | 'function') target? functionopts? fixes 'where' clauses
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    ;
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    clauses: (thmdecl? prop ('(' 'otherwise' ')')? + '|')
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   419
    ;
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    functionopts: '(' (('sequential' | 'domintros' | 'tailrec' | 'default' term) + ',') ')'
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    ;
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    'termination' ( term )?
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   423
  \end{rail}
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  \begin{description}
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   426
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  \item \hyperlink{command.HOL.primrec}{\mbox{\isa{\isacommand{primrec}}}} defines primitive recursive
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  functions over datatypes, see also \cite{isabelle-HOL}.
wenzelm@26849
   429
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   430
  \item \hyperlink{command.HOL.function}{\mbox{\isa{\isacommand{function}}}} defines functions by general
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  wellfounded recursion. A detailed description with examples can be
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  found in \cite{isabelle-function}. The function is specified by a
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   433
  set of (possibly conditional) recursive equations with arbitrary
wenzelm@26849
   434
  pattern matching. The command generates proof obligations for the
wenzelm@26849
   435
  completeness and the compatibility of patterns.
wenzelm@26849
   436
wenzelm@26849
   437
  The defined function is considered partial, and the resulting
wenzelm@26849
   438
  simplification rules (named \isa{{\isachardoublequote}f{\isachardot}psimps{\isachardoublequote}}) and induction rule
wenzelm@26849
   439
  (named \isa{{\isachardoublequote}f{\isachardot}pinduct{\isachardoublequote}}) are guarded by a generated domain
wenzelm@26902
   440
  predicate \isa{{\isachardoublequote}f{\isacharunderscore}dom{\isachardoublequote}}. The \hyperlink{command.HOL.termination}{\mbox{\isa{\isacommand{termination}}}}
wenzelm@26849
   441
  command can then be used to establish that the function is total.
wenzelm@26849
   442
wenzelm@28788
   443
  \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
   444
  proof attempts regarding pattern matching and termination.  See
wenzelm@28788
   445
  \cite{isabelle-function} for further details.
wenzelm@26849
   446
wenzelm@28788
   447
  \item \hyperlink{command.HOL.termination}{\mbox{\isa{\isacommand{termination}}}}~\isa{f} commences a
wenzelm@26849
   448
  termination proof for the previously defined function \isa{f}.  If
wenzelm@26849
   449
  this is omitted, the command refers to the most recent function
wenzelm@26849
   450
  definition.  After the proof is closed, the recursive equations and
wenzelm@26849
   451
  the induction principle is established.
wenzelm@26849
   452
wenzelm@28788
   453
  \end{description}
wenzelm@26849
   454
haftmann@27452
   455
  Recursive definitions introduced by the \hyperlink{command.HOL.function}{\mbox{\isa{\isacommand{function}}}}
haftmann@27452
   456
  command accommodate
wenzelm@26849
   457
  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
   458
  refers to a specific induction rule, with parameters named according
krauss@33857
   459
  to the user-specified equations. Cases are numbered (starting from 1).
krauss@33857
   460
krauss@33857
   461
  For \hyperlink{command.HOL.primrec}{\mbox{\isa{\isacommand{primrec}}}}, the induction principle coincides
haftmann@27452
   462
  with structural recursion on the datatype the recursion is carried
haftmann@27452
   463
  out.
wenzelm@26849
   464
wenzelm@26849
   465
  The equations provided by these packages may be referred later as
wenzelm@26849
   466
  theorem list \isa{{\isachardoublequote}f{\isachardot}simps{\isachardoublequote}}, where \isa{f} is the (collective)
wenzelm@26849
   467
  name of the functions defined.  Individual equations may be named
wenzelm@26849
   468
  explicitly as well.
wenzelm@26849
   469
wenzelm@26902
   470
  The \hyperlink{command.HOL.function}{\mbox{\isa{\isacommand{function}}}} command accepts the following
wenzelm@26849
   471
  options.
wenzelm@26849
   472
wenzelm@28788
   473
  \begin{description}
wenzelm@26849
   474
wenzelm@28788
   475
  \item \isa{sequential} enables a preprocessor which disambiguates
wenzelm@28788
   476
  overlapping patterns by making them mutually disjoint.  Earlier
wenzelm@28788
   477
  equations take precedence over later ones.  This allows to give the
wenzelm@28788
   478
  specification in a format very similar to functional programming.
wenzelm@28788
   479
  Note that the resulting simplification and induction rules
wenzelm@28788
   480
  correspond to the transformed specification, not the one given
wenzelm@26849
   481
  originally. This usually means that each equation given by the user
hoelzl@36139
   482
  may result in several theorems.  Also note that this automatic
wenzelm@26849
   483
  transformation only works for ML-style datatype patterns.
wenzelm@26849
   484
wenzelm@28788
   485
  \item \isa{domintros} enables the automated generation of
wenzelm@26849
   486
  introduction rules for the domain predicate. While mostly not
wenzelm@26849
   487
  needed, they can be helpful in some proofs about partial functions.
wenzelm@26849
   488
wenzelm@28788
   489
  \item \isa{tailrec} generates the unconstrained recursive
wenzelm@26849
   490
  equations even without a termination proof, provided that the
wenzelm@26849
   491
  function is tail-recursive. This currently only works
wenzelm@26849
   492
wenzelm@28788
   493
  \item \isa{{\isachardoublequote}default\ d{\isachardoublequote}} allows to specify a default value for a
wenzelm@26849
   494
  (partial) function, which will ensure that \isa{{\isachardoublequote}f\ x\ {\isacharequal}\ d\ x{\isachardoublequote}}
wenzelm@26849
   495
  whenever \isa{{\isachardoublequote}x\ {\isasymnotin}\ f{\isacharunderscore}dom{\isachardoublequote}}.
wenzelm@26849
   496
wenzelm@28788
   497
  \end{description}%
wenzelm@26849
   498
\end{isamarkuptext}%
wenzelm@26849
   499
\isamarkuptrue%
wenzelm@26849
   500
%
wenzelm@26849
   501
\isamarkupsubsection{Proof methods related to recursive definitions%
wenzelm@26849
   502
}
wenzelm@26849
   503
\isamarkuptrue%
wenzelm@26849
   504
%
wenzelm@26849
   505
\begin{isamarkuptext}%
wenzelm@26849
   506
\begin{matharray}{rcl}
wenzelm@28788
   507
    \indexdef{HOL}{method}{pat\_completeness}\hypertarget{method.HOL.pat-completeness}{\hyperlink{method.HOL.pat-completeness}{\mbox{\isa{pat{\isacharunderscore}completeness}}}} & : & \isa{method} \\
wenzelm@28788
   508
    \indexdef{HOL}{method}{relation}\hypertarget{method.HOL.relation}{\hyperlink{method.HOL.relation}{\mbox{\isa{relation}}}} & : & \isa{method} \\
wenzelm@28788
   509
    \indexdef{HOL}{method}{lexicographic\_order}\hypertarget{method.HOL.lexicographic-order}{\hyperlink{method.HOL.lexicographic-order}{\mbox{\isa{lexicographic{\isacharunderscore}order}}}} & : & \isa{method} \\
krauss@33858
   510
    \indexdef{HOL}{method}{size\_change}\hypertarget{method.HOL.size-change}{\hyperlink{method.HOL.size-change}{\mbox{\isa{size{\isacharunderscore}change}}}} & : & \isa{method} \\
wenzelm@26849
   511
  \end{matharray}
wenzelm@26849
   512
wenzelm@26849
   513
  \begin{rail}
wenzelm@26849
   514
    'relation' term
wenzelm@26849
   515
    ;
haftmann@31913
   516
    'lexicographic\_order' ( clasimpmod * )
wenzelm@26849
   517
    ;
krauss@33858
   518
    'size\_change' ( orders ( clasimpmod * ) )
krauss@33858
   519
    ;
krauss@33858
   520
    orders: ( 'max' | 'min' | 'ms' ) *
wenzelm@26849
   521
  \end{rail}
wenzelm@26849
   522
wenzelm@28788
   523
  \begin{description}
wenzelm@26849
   524
wenzelm@28788
   525
  \item \hyperlink{method.HOL.pat-completeness}{\mbox{\isa{pat{\isacharunderscore}completeness}}} is a specialized method to
wenzelm@26849
   526
  solve goals regarding the completeness of pattern matching, as
wenzelm@26902
   527
  required by the \hyperlink{command.HOL.function}{\mbox{\isa{\isacommand{function}}}} package (cf.\
wenzelm@26849
   528
  \cite{isabelle-function}).
wenzelm@26849
   529
wenzelm@28788
   530
  \item \hyperlink{method.HOL.relation}{\mbox{\isa{relation}}}~\isa{R} introduces a termination
wenzelm@26849
   531
  proof using the relation \isa{R}.  The resulting proof state will
wenzelm@26849
   532
  contain goals expressing that \isa{R} is wellfounded, and that the
wenzelm@26849
   533
  arguments of recursive calls decrease with respect to \isa{R}.
wenzelm@26849
   534
  Usually, this method is used as the initial proof step of manual
wenzelm@26849
   535
  termination proofs.
wenzelm@26849
   536
wenzelm@28788
   537
  \item \hyperlink{method.HOL.lexicographic-order}{\mbox{\isa{lexicographic{\isacharunderscore}order}}} attempts a fully
wenzelm@26849
   538
  automated termination proof by searching for a lexicographic
wenzelm@26849
   539
  combination of size measures on the arguments of the function. The
wenzelm@26902
   540
  method accepts the same arguments as the \hyperlink{method.auto}{\mbox{\isa{auto}}} method,
wenzelm@26849
   541
  which it uses internally to prove local descents.  The same context
wenzelm@26902
   542
  modifiers as for \hyperlink{method.auto}{\mbox{\isa{auto}}} are accepted, see
wenzelm@26849
   543
  \secref{sec:clasimp}.
wenzelm@26849
   544
wenzelm@26849
   545
  In case of failure, extensive information is printed, which can help
wenzelm@26849
   546
  to analyse the situation (cf.\ \cite{isabelle-function}).
wenzelm@26849
   547
krauss@33858
   548
  \item \hyperlink{method.HOL.size-change}{\mbox{\isa{size{\isacharunderscore}change}}} also works on termination goals,
krauss@33858
   549
  using a variation of the size-change principle, together with a
krauss@33858
   550
  graph decomposition technique (see \cite{krauss_phd} for details).
krauss@33858
   551
  Three kinds of orders are used internally: \isa{max}, \isa{min},
krauss@33858
   552
  and \isa{ms} (multiset), which is only available when the theory
krauss@33858
   553
  \isa{Multiset} is loaded. When no order kinds are given, they are
krauss@33858
   554
  tried in order. The search for a termination proof uses SAT solving
krauss@33858
   555
  internally.
krauss@33858
   556
krauss@33858
   557
 For local descent proofs, the same context modifiers as for \hyperlink{method.auto}{\mbox{\isa{auto}}} are accepted, see \secref{sec:clasimp}.
krauss@33858
   558
wenzelm@28788
   559
  \end{description}%
wenzelm@26849
   560
\end{isamarkuptext}%
wenzelm@26849
   561
\isamarkuptrue%
wenzelm@26849
   562
%
wenzelm@26849
   563
\isamarkupsubsection{Old-style recursive function definitions (TFL)%
wenzelm@26849
   564
}
wenzelm@26849
   565
\isamarkuptrue%
wenzelm@26849
   566
%
wenzelm@26849
   567
\begin{isamarkuptext}%
wenzelm@26907
   568
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
   569
wenzelm@26849
   570
  \begin{matharray}{rcl}
wenzelm@28788
   571
    \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
   572
    \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
   573
  \end{matharray}
wenzelm@26849
   574
wenzelm@26849
   575
  \begin{rail}
wenzelm@26849
   576
    'recdef' ('(' 'permissive' ')')? \\ name term (prop +) hints?
wenzelm@26849
   577
    ;
wenzelm@26849
   578
    recdeftc thmdecl? tc
wenzelm@26849
   579
    ;
haftmann@31913
   580
    hints: '(' 'hints' ( recdefmod * ) ')'
wenzelm@26849
   581
    ;
wenzelm@26849
   582
    recdefmod: (('recdef\_simp' | 'recdef\_cong' | 'recdef\_wf') (() | 'add' | 'del') ':' thmrefs) | clasimpmod
wenzelm@26849
   583
    ;
wenzelm@26849
   584
    tc: nameref ('(' nat ')')?
wenzelm@26849
   585
    ;
wenzelm@26849
   586
  \end{rail}
wenzelm@26849
   587
wenzelm@28788
   588
  \begin{description}
wenzelm@26849
   589
  
wenzelm@28788
   590
  \item \hyperlink{command.HOL.recdef}{\mbox{\isa{\isacommand{recdef}}}} defines general well-founded
wenzelm@26849
   591
  recursive functions (using the TFL package), see also
wenzelm@26849
   592
  \cite{isabelle-HOL}.  The ``\isa{{\isachardoublequote}{\isacharparenleft}permissive{\isacharparenright}{\isachardoublequote}}'' option tells
wenzelm@26849
   593
  TFL to recover from failed proof attempts, returning unfinished
wenzelm@26849
   594
  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
   595
  automated proof process of TFL.  Additional \hyperlink{syntax.clasimpmod}{\mbox{\isa{clasimpmod}}}
wenzelm@26849
   596
  declarations (cf.\ \secref{sec:clasimp}) may be given to tune the
wenzelm@26849
   597
  context of the Simplifier (cf.\ \secref{sec:simplifier}) and
wenzelm@26849
   598
  Classical reasoner (cf.\ \secref{sec:classical}).
wenzelm@26849
   599
  
wenzelm@28788
   600
  \item \hyperlink{command.HOL.recdef-tc}{\mbox{\isa{\isacommand{recdef{\isacharunderscore}tc}}}}~\isa{{\isachardoublequote}c\ {\isacharparenleft}i{\isacharparenright}{\isachardoublequote}} recommences the
wenzelm@26849
   601
  proof for leftover termination condition number \isa{i} (default
wenzelm@26902
   602
  1) as generated by a \hyperlink{command.HOL.recdef}{\mbox{\isa{\isacommand{recdef}}}} definition of
wenzelm@26849
   603
  constant \isa{c}.
wenzelm@26849
   604
  
wenzelm@26902
   605
  Note that in most cases, \hyperlink{command.HOL.recdef}{\mbox{\isa{\isacommand{recdef}}}} is able to finish
wenzelm@26849
   606
  its internal proofs without manual intervention.
wenzelm@26849
   607
wenzelm@28788
   608
  \end{description}
wenzelm@26849
   609
wenzelm@26902
   610
  \medskip Hints for \hyperlink{command.HOL.recdef}{\mbox{\isa{\isacommand{recdef}}}} may be also declared
wenzelm@26849
   611
  globally, using the following attributes.
wenzelm@26849
   612
wenzelm@26849
   613
  \begin{matharray}{rcl}
wenzelm@28788
   614
    \indexdef{HOL}{attribute}{recdef\_simp}\hypertarget{attribute.HOL.recdef-simp}{\hyperlink{attribute.HOL.recdef-simp}{\mbox{\isa{recdef{\isacharunderscore}simp}}}} & : & \isa{attribute} \\
wenzelm@28788
   615
    \indexdef{HOL}{attribute}{recdef\_cong}\hypertarget{attribute.HOL.recdef-cong}{\hyperlink{attribute.HOL.recdef-cong}{\mbox{\isa{recdef{\isacharunderscore}cong}}}} & : & \isa{attribute} \\
wenzelm@28788
   616
    \indexdef{HOL}{attribute}{recdef\_wf}\hypertarget{attribute.HOL.recdef-wf}{\hyperlink{attribute.HOL.recdef-wf}{\mbox{\isa{recdef{\isacharunderscore}wf}}}} & : & \isa{attribute} \\
wenzelm@26849
   617
  \end{matharray}
wenzelm@26849
   618
wenzelm@26849
   619
  \begin{rail}
wenzelm@26849
   620
    ('recdef\_simp' | 'recdef\_cong' | 'recdef\_wf') (() | 'add' | 'del')
wenzelm@26849
   621
    ;
wenzelm@26849
   622
  \end{rail}%
wenzelm@26849
   623
\end{isamarkuptext}%
wenzelm@26849
   624
\isamarkuptrue%
wenzelm@26849
   625
%
wenzelm@26849
   626
\isamarkupsection{Inductive and coinductive definitions \label{sec:hol-inductive}%
wenzelm@26849
   627
}
wenzelm@26849
   628
\isamarkuptrue%
wenzelm@26849
   629
%
wenzelm@26849
   630
\begin{isamarkuptext}%
wenzelm@26849
   631
An \textbf{inductive definition} specifies the least predicate (or
wenzelm@26849
   632
  set) \isa{R} closed under given rules: applying a rule to elements
wenzelm@26849
   633
  of \isa{R} yields a result within \isa{R}.  For example, a
wenzelm@26849
   634
  structural operational semantics is an inductive definition of an
wenzelm@26849
   635
  evaluation relation.
wenzelm@26849
   636
wenzelm@26849
   637
  Dually, a \textbf{coinductive definition} specifies the greatest
wenzelm@26849
   638
  predicate~/ set \isa{R} that is consistent with given rules: every
wenzelm@26849
   639
  element of \isa{R} can be seen as arising by applying a rule to
wenzelm@26849
   640
  elements of \isa{R}.  An important example is using bisimulation
wenzelm@26849
   641
  relations to formalise equivalence of processes and infinite data
wenzelm@26849
   642
  structures.
wenzelm@26849
   643
wenzelm@26849
   644
  \medskip The HOL package is related to the ZF one, which is
wenzelm@26849
   645
  described in a separate paper,\footnote{It appeared in CADE
wenzelm@26849
   646
  \cite{paulson-CADE}; a longer version is distributed with Isabelle.}
wenzelm@26849
   647
  which you should refer to in case of difficulties.  The package is
wenzelm@26849
   648
  simpler than that of ZF thanks to implicit type-checking in HOL.
wenzelm@26849
   649
  The types of the (co)inductive predicates (or sets) determine the
wenzelm@26849
   650
  domain of the fixedpoint definition, and the package does not have
wenzelm@26849
   651
  to use inference rules for type-checking.
wenzelm@26849
   652
wenzelm@26849
   653
  \begin{matharray}{rcl}
wenzelm@28788
   654
    \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
   655
    \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
   656
    \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
   657
    \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
   658
    \indexdef{HOL}{attribute}{mono}\hypertarget{attribute.HOL.mono}{\hyperlink{attribute.HOL.mono}{\mbox{\isa{mono}}}} & : & \isa{attribute} \\
wenzelm@26849
   659
  \end{matharray}
wenzelm@26849
   660
wenzelm@26849
   661
  \begin{rail}
wenzelm@26849
   662
    ('inductive' | 'inductive\_set' | 'coinductive' | 'coinductive\_set') target? fixes ('for' fixes)? \\
wenzelm@26849
   663
    ('where' clauses)? ('monos' thmrefs)?
wenzelm@26849
   664
    ;
wenzelm@26849
   665
    clauses: (thmdecl? prop + '|')
wenzelm@26849
   666
    ;
wenzelm@26849
   667
    'mono' (() | 'add' | 'del')
wenzelm@26849
   668
    ;
wenzelm@26849
   669
  \end{rail}
wenzelm@26849
   670
wenzelm@28788
   671
  \begin{description}
wenzelm@26849
   672
wenzelm@28788
   673
  \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
   674
  introduction rules given in the \hyperlink{keyword.where}{\mbox{\isa{\isakeyword{where}}}} part.  The
wenzelm@26902
   675
  optional \hyperlink{keyword.for}{\mbox{\isa{\isakeyword{for}}}} part contains a list of parameters of the
wenzelm@26849
   676
  (co)inductive predicates that remain fixed throughout the
wenzelm@26902
   677
  definition.  The optional \hyperlink{keyword.monos}{\mbox{\isa{\isakeyword{monos}}}} section contains
wenzelm@26849
   678
  \emph{monotonicity theorems}, which are required for each operator
wenzelm@26849
   679
  applied to a recursive set in the introduction rules.  There
wenzelm@26849
   680
  \emph{must} be a theorem of the form \isa{{\isachardoublequote}A\ {\isasymle}\ B\ {\isasymLongrightarrow}\ M\ A\ {\isasymle}\ M\ B{\isachardoublequote}},
wenzelm@26849
   681
  for each premise \isa{{\isachardoublequote}M\ R\isactrlsub i\ t{\isachardoublequote}} in an introduction rule!
wenzelm@26849
   682
wenzelm@28788
   683
  \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
   684
  allowing the definition of (co)inductive sets.
wenzelm@26849
   685
wenzelm@28788
   686
  \item \hyperlink{attribute.HOL.mono}{\mbox{\isa{mono}}} declares monotonicity rules.  These
wenzelm@26902
   687
  rule are involved in the automated monotonicity proof of \hyperlink{command.HOL.inductive}{\mbox{\isa{\isacommand{inductive}}}}.
wenzelm@26849
   688
wenzelm@28788
   689
  \end{description}%
wenzelm@26849
   690
\end{isamarkuptext}%
wenzelm@26849
   691
\isamarkuptrue%
wenzelm@26849
   692
%
wenzelm@26849
   693
\isamarkupsubsection{Derived rules%
wenzelm@26849
   694
}
wenzelm@26849
   695
\isamarkuptrue%
wenzelm@26849
   696
%
wenzelm@26849
   697
\begin{isamarkuptext}%
wenzelm@26849
   698
Each (co)inductive definition \isa{R} adds definitions to the
wenzelm@26849
   699
  theory and also proves some theorems:
wenzelm@26849
   700
wenzelm@26849
   701
  \begin{description}
wenzelm@26849
   702
wenzelm@28788
   703
  \item \isa{R{\isachardot}intros} is the list of introduction rules as proven
wenzelm@26849
   704
  theorems, for the recursive predicates (or sets).  The rules are
wenzelm@26849
   705
  also available individually, using the names given them in the
wenzelm@26849
   706
  theory file;
wenzelm@26849
   707
wenzelm@28788
   708
  \item \isa{R{\isachardot}cases} is the case analysis (or elimination) rule;
wenzelm@26849
   709
wenzelm@28788
   710
  \item \isa{R{\isachardot}induct} or \isa{R{\isachardot}coinduct} is the (co)induction
wenzelm@26849
   711
  rule.
wenzelm@26849
   712
wenzelm@26849
   713
  \end{description}
wenzelm@26849
   714
wenzelm@26849
   715
  When several predicates \isa{{\isachardoublequote}R\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ R\isactrlsub n{\isachardoublequote}} are
wenzelm@26849
   716
  defined simultaneously, the list of introduction rules is called
wenzelm@26849
   717
  \isa{{\isachardoublequote}R\isactrlsub {\isadigit{1}}{\isacharunderscore}{\isasymdots}{\isacharunderscore}R\isactrlsub n{\isachardot}intros{\isachardoublequote}}, the case analysis rules are
wenzelm@26849
   718
  called \isa{{\isachardoublequote}R\isactrlsub {\isadigit{1}}{\isachardot}cases{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ R\isactrlsub n{\isachardot}cases{\isachardoublequote}}, and the list
wenzelm@26849
   719
  of mutual induction rules is called \isa{{\isachardoublequote}R\isactrlsub {\isadigit{1}}{\isacharunderscore}{\isasymdots}{\isacharunderscore}R\isactrlsub n{\isachardot}inducts{\isachardoublequote}}.%
wenzelm@26849
   720
\end{isamarkuptext}%
wenzelm@26849
   721
\isamarkuptrue%
wenzelm@26849
   722
%
wenzelm@26849
   723
\isamarkupsubsection{Monotonicity theorems%
wenzelm@26849
   724
}
wenzelm@26849
   725
\isamarkuptrue%
wenzelm@26849
   726
%
wenzelm@26849
   727
\begin{isamarkuptext}%
wenzelm@26849
   728
Each theory contains a default set of theorems that are used in
wenzelm@26849
   729
  monotonicity proofs.  New rules can be added to this set via the
wenzelm@26902
   730
  \hyperlink{attribute.HOL.mono}{\mbox{\isa{mono}}} attribute.  The HOL theory \isa{Inductive}
wenzelm@26849
   731
  shows how this is done.  In general, the following monotonicity
wenzelm@26849
   732
  theorems may be added:
wenzelm@26849
   733
wenzelm@26849
   734
  \begin{itemize}
wenzelm@26849
   735
wenzelm@26849
   736
  \item Theorems of the form \isa{{\isachardoublequote}A\ {\isasymle}\ B\ {\isasymLongrightarrow}\ M\ A\ {\isasymle}\ M\ B{\isachardoublequote}}, for proving
wenzelm@26849
   737
  monotonicity of inductive definitions whose introduction rules have
wenzelm@26849
   738
  premises involving terms such as \isa{{\isachardoublequote}M\ R\isactrlsub i\ t{\isachardoublequote}}.
wenzelm@26849
   739
wenzelm@26849
   740
  \item Monotonicity theorems for logical operators, which are of the
wenzelm@26849
   741
  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
   742
  the case of the operator \isa{{\isachardoublequote}{\isasymor}{\isachardoublequote}}, the corresponding theorem is
wenzelm@26849
   743
  \[
wenzelm@26849
   744
  \infer{\isa{{\isachardoublequote}P\isactrlsub {\isadigit{1}}\ {\isasymor}\ P\isactrlsub {\isadigit{2}}\ {\isasymlongrightarrow}\ Q\isactrlsub {\isadigit{1}}\ {\isasymor}\ Q\isactrlsub {\isadigit{2}}{\isachardoublequote}}}{\isa{{\isachardoublequote}P\isactrlsub {\isadigit{1}}\ {\isasymlongrightarrow}\ Q\isactrlsub {\isadigit{1}}{\isachardoublequote}} & \isa{{\isachardoublequote}P\isactrlsub {\isadigit{2}}\ {\isasymlongrightarrow}\ Q\isactrlsub {\isadigit{2}}{\isachardoublequote}}}
wenzelm@26849
   745
  \]
wenzelm@26849
   746
wenzelm@26849
   747
  \item De Morgan style equations for reasoning about the ``polarity''
wenzelm@26849
   748
  of expressions, e.g.
wenzelm@26849
   749
  \[
wenzelm@26849
   750
  \isa{{\isachardoublequote}{\isasymnot}\ {\isasymnot}\ P\ {\isasymlongleftrightarrow}\ P{\isachardoublequote}} \qquad\qquad
wenzelm@26849
   751
  \isa{{\isachardoublequote}{\isasymnot}\ {\isacharparenleft}P\ {\isasymand}\ Q{\isacharparenright}\ {\isasymlongleftrightarrow}\ {\isasymnot}\ P\ {\isasymor}\ {\isasymnot}\ Q{\isachardoublequote}}
wenzelm@26849
   752
  \]
wenzelm@26849
   753
wenzelm@26849
   754
  \item Equations for reducing complex operators to more primitive
wenzelm@26849
   755
  ones whose monotonicity can easily be proved, e.g.
wenzelm@26849
   756
  \[
wenzelm@26849
   757
  \isa{{\isachardoublequote}{\isacharparenleft}P\ {\isasymlongrightarrow}\ Q{\isacharparenright}\ {\isasymlongleftrightarrow}\ {\isasymnot}\ P\ {\isasymor}\ Q{\isachardoublequote}} \qquad\qquad
wenzelm@26849
   758
  \isa{{\isachardoublequote}Ball\ A\ P\ {\isasymequiv}\ {\isasymforall}x{\isachardot}\ x\ {\isasymin}\ A\ {\isasymlongrightarrow}\ P\ x{\isachardoublequote}}
wenzelm@26849
   759
  \]
wenzelm@26849
   760
wenzelm@26849
   761
  \end{itemize}
wenzelm@26849
   762
wenzelm@26849
   763
  %FIXME: Example of an inductive definition%
wenzelm@26849
   764
\end{isamarkuptext}%
wenzelm@26849
   765
\isamarkuptrue%
wenzelm@26849
   766
%
wenzelm@26849
   767
\isamarkupsection{Arithmetic proof support%
wenzelm@26849
   768
}
wenzelm@26849
   769
\isamarkuptrue%
wenzelm@26849
   770
%
wenzelm@26849
   771
\begin{isamarkuptext}%
wenzelm@26849
   772
\begin{matharray}{rcl}
wenzelm@28788
   773
    \indexdef{HOL}{method}{arith}\hypertarget{method.HOL.arith}{\hyperlink{method.HOL.arith}{\mbox{\isa{arith}}}} & : & \isa{method} \\
nipkow@30863
   774
    \indexdef{HOL}{attribute}{arith}\hypertarget{attribute.HOL.arith}{\hyperlink{attribute.HOL.arith}{\mbox{\isa{arith}}}} & : & \isa{attribute} \\
wenzelm@28788
   775
    \indexdef{HOL}{attribute}{arith\_split}\hypertarget{attribute.HOL.arith-split}{\hyperlink{attribute.HOL.arith-split}{\mbox{\isa{arith{\isacharunderscore}split}}}} & : & \isa{attribute} \\
wenzelm@26849
   776
  \end{matharray}
wenzelm@26849
   777
wenzelm@26902
   778
  The \hyperlink{method.HOL.arith}{\mbox{\isa{arith}}} method decides linear arithmetic problems
wenzelm@26849
   779
  (on types \isa{nat}, \isa{int}, \isa{real}).  Any current
wenzelm@26849
   780
  facts are inserted into the goal before running the procedure.
wenzelm@26849
   781
nipkow@30863
   782
  The \hyperlink{attribute.HOL.arith}{\mbox{\isa{arith}}} attribute declares facts that are
nipkow@30863
   783
  always supplied to the arithmetic provers implicitly.
wenzelm@26849
   784
nipkow@30863
   785
  The \hyperlink{attribute.HOL.arith-split}{\mbox{\isa{arith{\isacharunderscore}split}}} attribute declares case split
wenzelm@30865
   786
  rules to be expanded before \hyperlink{method.HOL.arith}{\mbox{\isa{arith}}} is invoked.
nipkow@30863
   787
nipkow@30863
   788
  Note that a simpler (but faster) arithmetic prover is
nipkow@30863
   789
  already invoked by the Simplifier.%
wenzelm@26849
   790
\end{isamarkuptext}%
wenzelm@26849
   791
\isamarkuptrue%
wenzelm@26849
   792
%
wenzelm@30172
   793
\isamarkupsection{Intuitionistic proof search%
wenzelm@30172
   794
}
wenzelm@30172
   795
\isamarkuptrue%
wenzelm@30172
   796
%
wenzelm@30172
   797
\begin{isamarkuptext}%
wenzelm@30172
   798
\begin{matharray}{rcl}
wenzelm@30172
   799
    \indexdef{HOL}{method}{iprover}\hypertarget{method.HOL.iprover}{\hyperlink{method.HOL.iprover}{\mbox{\isa{iprover}}}} & : & \isa{method} \\
wenzelm@30172
   800
  \end{matharray}
wenzelm@30172
   801
wenzelm@30172
   802
  \begin{rail}
wenzelm@35613
   803
    'iprover' ( rulemod * )
wenzelm@30172
   804
    ;
wenzelm@30172
   805
  \end{rail}
wenzelm@30172
   806
wenzelm@30172
   807
  The \hyperlink{method.HOL.iprover}{\mbox{\isa{iprover}}} method performs intuitionistic proof
wenzelm@30172
   808
  search, depending on specifically declared rules from the context,
wenzelm@30172
   809
  or given as explicit arguments.  Chained facts are inserted into the
wenzelm@35613
   810
  goal before commencing proof search.
wenzelm@35613
   811
wenzelm@30172
   812
  Rules need to be classified as \hyperlink{attribute.Pure.intro}{\mbox{\isa{intro}}},
wenzelm@30172
   813
  \hyperlink{attribute.Pure.elim}{\mbox{\isa{elim}}}, or \hyperlink{attribute.Pure.dest}{\mbox{\isa{dest}}}; here the
wenzelm@30172
   814
  ``\isa{{\isachardoublequote}{\isacharbang}{\isachardoublequote}}'' indicator refers to ``safe'' rules, which may be
wenzelm@30172
   815
  applied aggressively (without considering back-tracking later).
wenzelm@30172
   816
  Rules declared with ``\isa{{\isachardoublequote}{\isacharquery}{\isachardoublequote}}'' are ignored in proof search (the
wenzelm@30172
   817
  single-step \hyperlink{method.rule}{\mbox{\isa{rule}}} method still observes these).  An
wenzelm@30172
   818
  explicit weight annotation may be given as well; otherwise the
wenzelm@30172
   819
  number of rule premises will be taken into account here.%
wenzelm@30172
   820
\end{isamarkuptext}%
wenzelm@30172
   821
\isamarkuptrue%
wenzelm@30172
   822
%
wenzelm@30172
   823
\isamarkupsection{Coherent Logic%
wenzelm@30172
   824
}
wenzelm@30172
   825
\isamarkuptrue%
wenzelm@30172
   826
%
wenzelm@30172
   827
\begin{isamarkuptext}%
wenzelm@30172
   828
\begin{matharray}{rcl}
wenzelm@30172
   829
    \indexdef{HOL}{method}{coherent}\hypertarget{method.HOL.coherent}{\hyperlink{method.HOL.coherent}{\mbox{\isa{coherent}}}} & : & \isa{method} \\
wenzelm@30172
   830
  \end{matharray}
wenzelm@30172
   831
wenzelm@30172
   832
  \begin{rail}
wenzelm@30172
   833
    'coherent' thmrefs?
wenzelm@30172
   834
    ;
wenzelm@30172
   835
  \end{rail}
wenzelm@30172
   836
wenzelm@30172
   837
  The \hyperlink{method.HOL.coherent}{\mbox{\isa{coherent}}} method solves problems of
wenzelm@30172
   838
  \emph{Coherent Logic} \cite{Bezem-Coquand:2005}, which covers
wenzelm@30172
   839
  applications in confluence theory, lattice theory and projective
wenzelm@30172
   840
  geometry.  See \hyperlink{file.~~/src/HOL/ex/Coherent.thy}{\mbox{\isa{\isatt{{\isachartilde}{\isachartilde}{\isacharslash}src{\isacharslash}HOL{\isacharslash}ex{\isacharslash}Coherent{\isachardot}thy}}}} for some
wenzelm@30172
   841
  examples.%
wenzelm@30172
   842
\end{isamarkuptext}%
wenzelm@30172
   843
\isamarkuptrue%
wenzelm@30172
   844
%
haftmann@31913
   845
\isamarkupsection{Checking and refuting propositions%
haftmann@31913
   846
}
haftmann@31913
   847
\isamarkuptrue%
haftmann@31913
   848
%
haftmann@31913
   849
\begin{isamarkuptext}%
haftmann@31913
   850
Identifying incorrect propositions usually involves evaluation of
haftmann@31913
   851
  particular assignments and systematic counter example search.  This
haftmann@31913
   852
  is supported by the following commands.
haftmann@31913
   853
haftmann@31913
   854
  \begin{matharray}{rcl}
haftmann@31913
   855
    \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}} \\
haftmann@31913
   856
    \indexdef{HOL}{command}{quickcheck}\hypertarget{command.HOL.quickcheck}{\hyperlink{command.HOL.quickcheck}{\mbox{\isa{\isacommand{quickcheck}}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isa{{\isachardoublequote}proof\ {\isasymrightarrow}{\isachardoublequote}} \\
haftmann@31913
   857
    \indexdef{HOL}{command}{quickcheck\_params}\hypertarget{command.HOL.quickcheck-params}{\hyperlink{command.HOL.quickcheck-params}{\mbox{\isa{\isacommand{quickcheck{\isacharunderscore}params}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ theory{\isachardoublequote}}
haftmann@31913
   858
  \end{matharray}
haftmann@31913
   859
haftmann@31913
   860
  \begin{rail}
haftmann@31913
   861
    'value' ( ( '[' name ']' ) ? ) modes? term
haftmann@31913
   862
    ;
haftmann@31913
   863
haftmann@31913
   864
    'quickcheck' ( ( '[' args ']' ) ? ) nat?
haftmann@31913
   865
    ;
haftmann@31913
   866
haftmann@31913
   867
    'quickcheck_params' ( ( '[' args ']' ) ? )
haftmann@31913
   868
    ;
haftmann@31913
   869
haftmann@31913
   870
    modes: '(' (name + ) ')'
haftmann@31913
   871
    ;
haftmann@31913
   872
haftmann@31913
   873
    args: ( name '=' value + ',' )
haftmann@31913
   874
    ;
haftmann@31913
   875
  \end{rail}
haftmann@31913
   876
haftmann@31913
   877
  \begin{description}
haftmann@31913
   878
haftmann@31913
   879
  \item \hyperlink{command.HOL.value}{\mbox{\isa{\isacommand{value}}}}~\isa{t} evaluates and prints a
haftmann@31913
   880
    term; optionally \isa{modes} can be specified, which are
haftmann@31913
   881
    appended to the current print mode (see also \cite{isabelle-ref}).
haftmann@31913
   882
    Internally, the evaluation is performed by registered evaluators,
haftmann@31913
   883
    which are invoked sequentially until a result is returned.
haftmann@31913
   884
    Alternatively a specific evaluator can be selected using square
haftmann@31913
   885
    brackets; available evaluators include \isa{nbe} for
haftmann@31913
   886
    \emph{normalization by evaluation} and \emph{code} for code
haftmann@31913
   887
    generation in SML.
haftmann@31913
   888
haftmann@31913
   889
  \item \hyperlink{command.HOL.quickcheck}{\mbox{\isa{\isacommand{quickcheck}}}} tests the current goal for
haftmann@31913
   890
    counter examples using a series of arbitrary assignments for its
haftmann@31913
   891
    free variables; by default the first subgoal is tested, an other
haftmann@31913
   892
    can be selected explicitly using an optional goal index.
haftmann@31913
   893
    A number of configuration options are supported for
haftmann@31913
   894
    \hyperlink{command.HOL.quickcheck}{\mbox{\isa{\isacommand{quickcheck}}}}, notably:
haftmann@31913
   895
haftmann@31913
   896
    \begin{description}
haftmann@31913
   897
haftmann@31913
   898
      \item[size] specifies the maximum size of the search space for
haftmann@31913
   899
        assignment values.
haftmann@31913
   900
haftmann@31913
   901
      \item[iterations] sets how many sets of assignments are
haftmann@31913
   902
        generated for each particular size.
haftmann@31913
   903
wenzelm@35351
   904
      \item[no\_assms] specifies whether assumptions in
wenzelm@35351
   905
        structured proofs should be ignored.
wenzelm@35351
   906
haftmann@31913
   907
    \end{description}
haftmann@31913
   908
haftmann@31913
   909
    These option can be given within square brackets.
haftmann@31913
   910
haftmann@31913
   911
  \item \hyperlink{command.HOL.quickcheck-params}{\mbox{\isa{\isacommand{quickcheck{\isacharunderscore}params}}}} changes quickcheck
haftmann@31913
   912
    configuration options persitently.
haftmann@31913
   913
haftmann@31913
   914
  \end{description}%
haftmann@31913
   915
\end{isamarkuptext}%
haftmann@31913
   916
\isamarkuptrue%
haftmann@31913
   917
%
wenzelm@35744
   918
\isamarkupsection{Invoking automated reasoning tools --- The Sledgehammer%
wenzelm@28603
   919
}
wenzelm@28603
   920
\isamarkuptrue%
wenzelm@28603
   921
%
wenzelm@28603
   922
\begin{isamarkuptext}%
wenzelm@28603
   923
Isabelle/HOL includes a generic \emph{ATP manager} that allows
wenzelm@28603
   924
  external automated reasoning tools to crunch a pending goal.
wenzelm@28603
   925
  Supported provers include E\footnote{\url{http://www.eprover.org}},
wenzelm@28603
   926
  SPASS\footnote{\url{http://www.spass-prover.org/}}, and Vampire.
wenzelm@28603
   927
  There is also a wrapper to invoke provers remotely via the
wenzelm@28603
   928
  SystemOnTPTP\footnote{\url{http://www.cs.miami.edu/~tptp/cgi-bin/SystemOnTPTP}}
wenzelm@28603
   929
  web service.
wenzelm@28603
   930
wenzelm@28603
   931
  The problem passed to external provers consists of the goal together
wenzelm@28603
   932
  with a smart selection of lemmas from the current theory context.
wenzelm@28603
   933
  The result of a successful proof search is some source text that
wenzelm@28603
   934
  usually reconstructs the proof within Isabelle, without requiring
wenzelm@28603
   935
  external provers again.  The Metis
wenzelm@28603
   936
  prover\footnote{\url{http://www.gilith.com/software/metis/}} that is
wenzelm@28603
   937
  integrated into Isabelle/HOL is being used here.
wenzelm@28603
   938
wenzelm@28603
   939
  In this mode of operation, heavy means of automated reasoning are
wenzelm@28603
   940
  used as a strong relevance filter, while the main proof checking
wenzelm@28603
   941
  works via explicit inferences going through the Isabelle kernel.
wenzelm@28603
   942
  Moreover, rechecking Isabelle proof texts with already specified
wenzelm@28603
   943
  auxiliary facts is much faster than performing fully automated
wenzelm@28603
   944
  search over and over again.
wenzelm@28603
   945
wenzelm@28603
   946
  \begin{matharray}{rcl}
wenzelm@28788
   947
    \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
   948
    \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
   949
    \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
   950
    \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
   951
    \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
   952
    \indexdef{HOL}{method}{metis}\hypertarget{method.HOL.metis}{\hyperlink{method.HOL.metis}{\mbox{\isa{metis}}}} & : & \isa{method} \\
wenzelm@28603
   953
  \end{matharray}
wenzelm@28603
   954
wenzelm@28603
   955
  \begin{rail}
haftmann@31913
   956
  'sledgehammer' ( nameref * )
wenzelm@28603
   957
  ;
wenzelm@29113
   958
  'atp\_messages' ('(' nat ')')?
wenzelm@29115
   959
  ;
wenzelm@28603
   960
wenzelm@28603
   961
  'metis' thmrefs
wenzelm@28603
   962
  ;
wenzelm@28603
   963
  \end{rail}
wenzelm@28603
   964
wenzelm@28788
   965
  \begin{description}
wenzelm@28603
   966
wenzelm@28788
   967
  \item \hyperlink{command.HOL.sledgehammer}{\mbox{\isa{\isacommand{sledgehammer}}}}~\isa{{\isachardoublequote}prover\isactrlsub {\isadigit{1}}\ {\isasymdots}\ prover\isactrlsub n{\isachardoublequote}}
wenzelm@28788
   968
  invokes the specified automated theorem provers on the first
wenzelm@28788
   969
  subgoal.  Provers are run in parallel, the first successful result
wenzelm@28788
   970
  is displayed, and the other attempts are terminated.
wenzelm@28603
   971
wenzelm@28603
   972
  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
   973
  ``ATP provers'' preference by the user interface.
wenzelm@28603
   974
wenzelm@28603
   975
  There are additional preferences for timeout (default: 60 seconds),
wenzelm@28603
   976
  and the maximum number of independent prover processes (default: 5);
wenzelm@28603
   977
  excessive provers are automatically terminated.
wenzelm@28603
   978
wenzelm@28788
   979
  \item \hyperlink{command.HOL.print-atps}{\mbox{\isa{\isacommand{print{\isacharunderscore}atps}}}} prints the list of automated
wenzelm@28603
   980
  theorem provers available to the \hyperlink{command.HOL.sledgehammer}{\mbox{\isa{\isacommand{sledgehammer}}}}
wenzelm@28603
   981
  command.
wenzelm@28603
   982
wenzelm@28788
   983
  \item \hyperlink{command.HOL.atp-info}{\mbox{\isa{\isacommand{atp{\isacharunderscore}info}}}} prints information about presently
wenzelm@28603
   984
  running provers, including elapsed runtime, and the remaining time
wenzelm@28603
   985
  until timeout.
wenzelm@28603
   986
wenzelm@28788
   987
  \item \hyperlink{command.HOL.atp-kill}{\mbox{\isa{\isacommand{atp{\isacharunderscore}kill}}}} terminates all presently running
wenzelm@28603
   988
  provers.
wenzelm@28603
   989
wenzelm@29113
   990
  \item \hyperlink{command.HOL.atp-messages}{\mbox{\isa{\isacommand{atp{\isacharunderscore}messages}}}} displays recent messages issued
wenzelm@29113
   991
  by automated theorem provers.  This allows to examine results that
wenzelm@29113
   992
  might have got lost due to the asynchronous nature of default
wenzelm@29115
   993
  \hyperlink{command.HOL.sledgehammer}{\mbox{\isa{\isacommand{sledgehammer}}}} output.  An optional message limit may
wenzelm@29115
   994
  be specified (default 5).
wenzelm@29113
   995
wenzelm@28788
   996
  \item \hyperlink{method.HOL.metis}{\mbox{\isa{metis}}}~\isa{{\isachardoublequote}facts{\isachardoublequote}} invokes the Metis prover
wenzelm@28788
   997
  with the given facts.  Metis is an automated proof tool of medium
wenzelm@28788
   998
  strength, but is fully integrated into Isabelle/HOL, with explicit
wenzelm@28788
   999
  inferences going through the kernel.  Thus its results are
wenzelm@28603
  1000
  guaranteed to be ``correct by construction''.
wenzelm@28603
  1001
wenzelm@28603
  1002
  Note that all facts used with Metis need to be specified as explicit
wenzelm@28603
  1003
  arguments.  There are no rule declarations as for other Isabelle
wenzelm@28603
  1004
  provers, like \hyperlink{method.blast}{\mbox{\isa{blast}}} or \hyperlink{method.fast}{\mbox{\isa{fast}}}.
wenzelm@28603
  1005
wenzelm@28788
  1006
  \end{description}%
wenzelm@28603
  1007
\end{isamarkuptext}%
wenzelm@28603
  1008
\isamarkuptrue%
wenzelm@28603
  1009
%
wenzelm@28788
  1010
\isamarkupsection{Unstructured case analysis and induction \label{sec:hol-induct-tac}%
wenzelm@26849
  1011
}
wenzelm@26849
  1012
\isamarkuptrue%
wenzelm@26849
  1013
%
wenzelm@26849
  1014
\begin{isamarkuptext}%
wenzelm@27124
  1015
The following tools of Isabelle/HOL support cases analysis and
wenzelm@27124
  1016
  induction in unstructured tactic scripts; see also
wenzelm@27124
  1017
  \secref{sec:cases-induct} for proper Isar versions of similar ideas.
wenzelm@26849
  1018
wenzelm@26849
  1019
  \begin{matharray}{rcl}
wenzelm@28788
  1020
    \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
  1021
    \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
  1022
    \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
  1023
    \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
  1024
  \end{matharray}
wenzelm@26849
  1025
wenzelm@26849
  1026
  \begin{rail}
wenzelm@26849
  1027
    'case\_tac' goalspec? term rule?
wenzelm@26849
  1028
    ;
wenzelm@26849
  1029
    'induct\_tac' goalspec? (insts * 'and') rule?
wenzelm@26849
  1030
    ;
wenzelm@26849
  1031
    'ind\_cases' (prop +) ('for' (name +)) ?
wenzelm@26849
  1032
    ;
wenzelm@26849
  1033
    'inductive\_cases' (thmdecl? (prop +) + 'and')
wenzelm@26849
  1034
    ;
wenzelm@26849
  1035
wenzelm@26849
  1036
    rule: ('rule' ':' thmref)
wenzelm@26849
  1037
    ;
wenzelm@26849
  1038
  \end{rail}
wenzelm@26849
  1039
wenzelm@28788
  1040
  \begin{description}
wenzelm@26849
  1041
wenzelm@28788
  1042
  \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
  1043
  to reason about inductive types.  Rules are selected according to
wenzelm@28788
  1044
  the declarations by the \hyperlink{attribute.cases}{\mbox{\isa{cases}}} and \hyperlink{attribute.induct}{\mbox{\isa{induct}}}
wenzelm@28788
  1045
  attributes, cf.\ \secref{sec:cases-induct}.  The \hyperlink{command.HOL.datatype}{\mbox{\isa{\isacommand{datatype}}}} package already takes care of this.
wenzelm@27124
  1046
wenzelm@27124
  1047
  These unstructured tactics feature both goal addressing and dynamic
wenzelm@26849
  1048
  instantiation.  Note that named rule cases are \emph{not} provided
wenzelm@27124
  1049
  as would be by the proper \hyperlink{method.cases}{\mbox{\isa{cases}}} and \hyperlink{method.induct}{\mbox{\isa{induct}}} proof
wenzelm@27124
  1050
  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
  1051
  statements, only the compact object-logic conclusion of the subgoal
wenzelm@27124
  1052
  being addressed.
wenzelm@26849
  1053
  
wenzelm@28788
  1054
  \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
  1055
  forward manner.
wenzelm@26849
  1056
wenzelm@26907
  1057
  While \hyperlink{method.HOL.ind-cases}{\mbox{\isa{ind{\isacharunderscore}cases}}} is a proof method to apply the
wenzelm@26907
  1058
  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
  1059
  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
  1060
  be generalized before applying the resulting rule.
wenzelm@26849
  1061
wenzelm@28788
  1062
  \end{description}%
wenzelm@26849
  1063
\end{isamarkuptext}%
wenzelm@26849
  1064
\isamarkuptrue%
wenzelm@26849
  1065
%
wenzelm@26849
  1066
\isamarkupsection{Executable code%
wenzelm@26849
  1067
}
wenzelm@26849
  1068
\isamarkuptrue%
wenzelm@26849
  1069
%
wenzelm@26849
  1070
\begin{isamarkuptext}%
wenzelm@26849
  1071
Isabelle/Pure provides two generic frameworks to support code
wenzelm@26849
  1072
  generation from executable specifications.  Isabelle/HOL
wenzelm@26849
  1073
  instantiates these mechanisms in a way that is amenable to end-user
wenzelm@26849
  1074
  applications.
wenzelm@26849
  1075
wenzelm@26849
  1076
  One framework generates code from both functional and relational
wenzelm@26849
  1077
  programs to SML.  See \cite{isabelle-HOL} for further information
wenzelm@26849
  1078
  (this actually covers the new-style theory format as well).
wenzelm@26849
  1079
wenzelm@26849
  1080
  \begin{matharray}{rcl}
wenzelm@28788
  1081
    \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
  1082
    \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
  1083
    \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
  1084
    \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
  1085
    \indexdef{HOL}{attribute}{code}\hypertarget{attribute.HOL.code}{\hyperlink{attribute.HOL.code}{\mbox{\isa{code}}}} & : & \isa{attribute} \\
wenzelm@26849
  1086
  \end{matharray}
wenzelm@26849
  1087
wenzelm@26849
  1088
  \begin{rail}
wenzelm@26849
  1089
  ( 'code\_module' | 'code\_library' ) modespec ? name ? \\
wenzelm@26849
  1090
    ( 'file' name ) ? ( 'imports' ( name + ) ) ? \\
wenzelm@26849
  1091
    'contains' ( ( name '=' term ) + | term + )
wenzelm@26849
  1092
  ;
wenzelm@26849
  1093
wenzelm@26849
  1094
  modespec: '(' ( name * ) ')'
wenzelm@26849
  1095
  ;
wenzelm@26849
  1096
wenzelm@26849
  1097
  'consts\_code' (codespec +)
wenzelm@26849
  1098
  ;
wenzelm@26849
  1099
wenzelm@26849
  1100
  codespec: const template attachment ?
wenzelm@26849
  1101
  ;
wenzelm@26849
  1102
wenzelm@26849
  1103
  'types\_code' (tycodespec +)
wenzelm@26849
  1104
  ;
wenzelm@26849
  1105
wenzelm@26849
  1106
  tycodespec: name template attachment ?
wenzelm@26849
  1107
  ;
wenzelm@26849
  1108
wenzelm@26849
  1109
  const: term
wenzelm@26849
  1110
  ;
wenzelm@26849
  1111
wenzelm@26849
  1112
  template: '(' string ')'
wenzelm@26849
  1113
  ;
wenzelm@26849
  1114
wenzelm@26849
  1115
  attachment: 'attach' modespec ? verblbrace text verbrbrace
wenzelm@26849
  1116
  ;
wenzelm@26849
  1117
wenzelm@26849
  1118
  'code' (name)?
wenzelm@26849
  1119
  ;
wenzelm@26849
  1120
  \end{rail}
wenzelm@26849
  1121
wenzelm@26849
  1122
  \medskip The other framework generates code from functional programs
wenzelm@26849
  1123
  (including overloading using type classes) to SML \cite{SML}, OCaml
wenzelm@26849
  1124
  \cite{OCaml} and Haskell \cite{haskell-revised-report}.
wenzelm@26849
  1125
  Conceptually, code generation is split up in three steps:
wenzelm@26849
  1126
  \emph{selection} of code theorems, \emph{translation} into an
wenzelm@26849
  1127
  abstract executable view and \emph{serialization} to a specific
wenzelm@26849
  1128
  \emph{target language}.  See \cite{isabelle-codegen} for an
wenzelm@26849
  1129
  introduction on how to use it.
wenzelm@26849
  1130
wenzelm@26849
  1131
  \begin{matharray}{rcl}
wenzelm@28788
  1132
    \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
  1133
    \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
  1134
    \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
  1135
    \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
  1136
    \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
  1137
    \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
  1138
    \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
  1139
    \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
  1140
    \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
  1141
    \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
  1142
    \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
  1143
    \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
  1144
    \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
  1145
    \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}} \\
haftmann@31254
  1146
    \indexdef{HOL}{command}{print\_codeproc}\hypertarget{command.HOL.print-codeproc}{\hyperlink{command.HOL.print-codeproc}{\mbox{\isa{\isacommand{print{\isacharunderscore}codeproc}}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isa{{\isachardoublequote}context\ {\isasymrightarrow}{\isachardoublequote}} \\
wenzelm@28788
  1147
    \indexdef{HOL}{attribute}{code}\hypertarget{attribute.HOL.code}{\hyperlink{attribute.HOL.code}{\mbox{\isa{code}}}} & : & \isa{attribute} \\
wenzelm@26849
  1148
  \end{matharray}
wenzelm@26849
  1149
wenzelm@26849
  1150
  \begin{rail}
haftmann@34172
  1151
    'export\_code' ( constexpr + ) \\
wenzelm@26849
  1152
      ( ( 'in' target ( 'module\_name' string ) ? \\
wenzelm@26849
  1153
        ( 'file' ( string | '-' ) ) ? ( '(' args ')' ) ?) + ) ?
wenzelm@26849
  1154
    ;
wenzelm@26849
  1155
wenzelm@26849
  1156
    'code\_thms' ( constexpr + ) ?
wenzelm@26849
  1157
    ;
wenzelm@26849
  1158
wenzelm@26849
  1159
    'code\_deps' ( constexpr + ) ?
wenzelm@26849
  1160
    ;
wenzelm@26849
  1161
wenzelm@26849
  1162
    const: term
wenzelm@26849
  1163
    ;
wenzelm@26849
  1164
wenzelm@26849
  1165
    constexpr: ( const | 'name.*' | '*' )
wenzelm@26849
  1166
    ;
wenzelm@26849
  1167
wenzelm@26849
  1168
    typeconstructor: nameref
wenzelm@26849
  1169
    ;
wenzelm@26849
  1170
wenzelm@26849
  1171
    class: nameref
wenzelm@26849
  1172
    ;
wenzelm@26849
  1173
wenzelm@26849
  1174
    target: 'OCaml' | 'SML' | 'Haskell'
wenzelm@26849
  1175
    ;
wenzelm@26849
  1176
wenzelm@26849
  1177
    'code\_datatype' const +
wenzelm@26849
  1178
    ;
wenzelm@26849
  1179
wenzelm@26849
  1180
    'code\_const' (const + 'and') \\
wenzelm@26849
  1181
      ( ( '(' target ( syntax ? + 'and' ) ')' ) + )
wenzelm@26849
  1182
    ;
wenzelm@26849
  1183
wenzelm@26849
  1184
    'code\_type' (typeconstructor + 'and') \\
wenzelm@26849
  1185
      ( ( '(' target ( syntax ? + 'and' ) ')' ) + )
wenzelm@26849
  1186
    ;
wenzelm@26849
  1187
wenzelm@26849
  1188
    'code\_class' (class + 'and') \\
haftmann@28687
  1189
      ( ( '(' target \\ ( string ? + 'and' ) ')' ) + )
wenzelm@26849
  1190
    ;
wenzelm@26849
  1191
wenzelm@26849
  1192
    'code\_instance' (( typeconstructor '::' class ) + 'and') \\
wenzelm@26849
  1193
      ( ( '(' target ( '-' ? + 'and' ) ')' ) + )
wenzelm@26849
  1194
    ;
wenzelm@26849
  1195
wenzelm@26849
  1196
    'code\_monad' const const target
wenzelm@26849
  1197
    ;
wenzelm@26849
  1198
wenzelm@26849
  1199
    'code\_reserved' target ( string + )
wenzelm@26849
  1200
    ;
wenzelm@26849
  1201
wenzelm@26849
  1202
    'code\_include' target ( string ( string | '-') )
wenzelm@26849
  1203
    ;
wenzelm@26849
  1204
wenzelm@26849
  1205
    'code\_modulename' target ( ( string string ) + )
wenzelm@26849
  1206
    ;
wenzelm@26849
  1207
haftmann@27452
  1208
    'code\_abort' ( const + )
wenzelm@26849
  1209
    ;
wenzelm@26849
  1210
wenzelm@26849
  1211
    syntax: string | ( 'infix' | 'infixl' | 'infixr' ) nat string
wenzelm@26849
  1212
    ;
wenzelm@26849
  1213
haftmann@31998
  1214
    'code' ( 'del' ) ?
haftmann@31998
  1215
    ;
haftmann@31998
  1216
haftmann@31998
  1217
    'code_unfold' ( 'del' ) ?
haftmann@31998
  1218
    ;
haftmann@31998
  1219
haftmann@31998
  1220
    'code_post' ( 'del' ) ?
wenzelm@26849
  1221
    ;
wenzelm@26849
  1222
  \end{rail}
wenzelm@26849
  1223
wenzelm@28788
  1224
  \begin{description}
wenzelm@26849
  1225
wenzelm@28788
  1226
  \item \hyperlink{command.HOL.export-code}{\mbox{\isa{\isacommand{export{\isacharunderscore}code}}}} is the canonical interface for
wenzelm@28788
  1227
  generating and serializing code: for a given list of constants, code
haftmann@34172
  1228
  is generated for the specified target languages.  If no serialization
haftmann@34172
  1229
  instruction is given, only abstract code is generated internally.
wenzelm@26849
  1230
wenzelm@26849
  1231
  Constants may be specified by giving them literally, referring to
wenzelm@26849
  1232
  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
  1233
  available by giving \isa{{\isachardoublequote}{\isacharasterisk}{\isachardoublequote}}.
wenzelm@26849
  1234
wenzelm@26849
  1235
  By default, for each involved theory one corresponding name space
wenzelm@26849
  1236
  module is generated.  Alternativly, a module name may be specified
wenzelm@26907
  1237
  after the \hyperlink{keyword.module-name}{\mbox{\isa{\isakeyword{module{\isacharunderscore}name}}}} keyword; then \emph{all} code is
wenzelm@26849
  1238
  placed in this module.
wenzelm@26849
  1239
wenzelm@26849
  1240
  For \emph{SML} and \emph{OCaml}, the file specification refers to a
wenzelm@26849
  1241
  single file; for \emph{Haskell}, it refers to a whole directory,
wenzelm@26849
  1242
  where code is generated in multiple files reflecting the module
wenzelm@26849
  1243
  hierarchy.  The file specification ``\isa{{\isachardoublequote}{\isacharminus}{\isachardoublequote}}'' denotes standard
wenzelm@26849
  1244
  output.  For \emph{SML}, omitting the file specification compiles
wenzelm@26849
  1245
  code internally in the context of the current ML session.
wenzelm@26849
  1246
wenzelm@26849
  1247
  Serializers take an optional list of arguments in parentheses.  For
haftmann@34172
  1248
  \emph{SML} and \emph{OCaml}, ``\isa{no{\isacharunderscore}signatures}`` omits
haftmann@34172
  1249
  explicit module signatures.
haftmann@34172
  1250
  
haftmann@34172
  1251
  For \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
  1252
  declaration.
wenzelm@26849
  1253
wenzelm@28788
  1254
  \item \hyperlink{command.HOL.code-thms}{\mbox{\isa{\isacommand{code{\isacharunderscore}thms}}}} prints a list of theorems
wenzelm@26849
  1255
  representing the corresponding program containing all given
haftmann@34172
  1256
  constants.
wenzelm@26849
  1257
wenzelm@28788
  1258
  \item \hyperlink{command.HOL.code-deps}{\mbox{\isa{\isacommand{code{\isacharunderscore}deps}}}} visualizes dependencies of
wenzelm@26849
  1259
  theorems representing the corresponding program containing all given
haftmann@34172
  1260
  constants.
wenzelm@26849
  1261
wenzelm@28788
  1262
  \item \hyperlink{command.HOL.code-datatype}{\mbox{\isa{\isacommand{code{\isacharunderscore}datatype}}}} specifies a constructor set
wenzelm@26849
  1263
  for a logical type.
wenzelm@26849
  1264
wenzelm@28788
  1265
  \item \hyperlink{command.HOL.code-const}{\mbox{\isa{\isacommand{code{\isacharunderscore}const}}}} associates a list of constants
wenzelm@26849
  1266
  with target-specific serializations; omitting a serialization
wenzelm@26849
  1267
  deletes an existing serialization.
wenzelm@26849
  1268
wenzelm@28788
  1269
  \item \hyperlink{command.HOL.code-type}{\mbox{\isa{\isacommand{code{\isacharunderscore}type}}}} associates a list of type
wenzelm@26849
  1270
  constructors with target-specific serializations; omitting a
wenzelm@26849
  1271
  serialization deletes an existing serialization.
wenzelm@26849
  1272
wenzelm@28788
  1273
  \item \hyperlink{command.HOL.code-class}{\mbox{\isa{\isacommand{code{\isacharunderscore}class}}}} associates a list of classes
wenzelm@28788
  1274
  with target-specific class names; omitting a serialization deletes
wenzelm@28788
  1275
  an existing serialization.  This applies only to \emph{Haskell}.
wenzelm@26849
  1276
wenzelm@28788
  1277
  \item \hyperlink{command.HOL.code-instance}{\mbox{\isa{\isacommand{code{\isacharunderscore}instance}}}} declares a list of type
wenzelm@26849
  1278
  constructor / class instance relations as ``already present'' for a
wenzelm@26849
  1279
  given target.  Omitting a ``\isa{{\isachardoublequote}{\isacharminus}{\isachardoublequote}}'' deletes an existing
wenzelm@26849
  1280
  ``already present'' declaration.  This applies only to
wenzelm@26849
  1281
  \emph{Haskell}.
wenzelm@26849
  1282
wenzelm@28788
  1283
  \item \hyperlink{command.HOL.code-monad}{\mbox{\isa{\isacommand{code{\isacharunderscore}monad}}}} provides an auxiliary mechanism
wenzelm@28788
  1284
  to generate monadic code for Haskell.
wenzelm@26849
  1285
wenzelm@28788
  1286
  \item \hyperlink{command.HOL.code-reserved}{\mbox{\isa{\isacommand{code{\isacharunderscore}reserved}}}} declares a list of names as
wenzelm@26849
  1287
  reserved for a given target, preventing it to be shadowed by any
wenzelm@26849
  1288
  generated code.
wenzelm@26849
  1289
wenzelm@28788
  1290
  \item \hyperlink{command.HOL.code-include}{\mbox{\isa{\isacommand{code{\isacharunderscore}include}}}} adds arbitrary named content
wenzelm@27834
  1291
  (``include'') to generated code.  A ``\isa{{\isachardoublequote}{\isacharminus}{\isachardoublequote}}'' as last argument
wenzelm@26849
  1292
  will remove an already added ``include''.
wenzelm@26849
  1293
wenzelm@28788
  1294
  \item \hyperlink{command.HOL.code-modulename}{\mbox{\isa{\isacommand{code{\isacharunderscore}modulename}}}} declares aliasings from one
wenzelm@28788
  1295
  module name onto another.
wenzelm@26849
  1296
wenzelm@28788
  1297
  \item \hyperlink{command.HOL.code-abort}{\mbox{\isa{\isacommand{code{\isacharunderscore}abort}}}} declares constants which are not
haftmann@29560
  1298
  required to have a definition by means of code equations; if
wenzelm@28788
  1299
  needed these are implemented by program abort instead.
wenzelm@26849
  1300
wenzelm@28788
  1301
  \item \hyperlink{attribute.HOL.code}{\mbox{\isa{code}}} explicitly selects (or with option
haftmann@29560
  1302
  ``\isa{{\isachardoublequote}del{\isachardoublequote}}'' deselects) a code equation for code
haftmann@29560
  1303
  generation.  Usually packages introducing code equations provide
wenzelm@28788
  1304
  a reasonable default setup for selection.
wenzelm@26849
  1305
haftmann@31998
  1306
  \item \hyperlink{attribute.HOL.code-inline}{\mbox{\isa{code{\isacharunderscore}inline}}} declares (or with
haftmann@28562
  1307
  option ``\isa{{\isachardoublequote}del{\isachardoublequote}}'' removes) inlining theorems which are
haftmann@29560
  1308
  applied as rewrite rules to any code equation during
wenzelm@26849
  1309
  preprocessing.
wenzelm@26849
  1310
haftmann@31998
  1311
  \item \hyperlink{attribute.HOL.code-post}{\mbox{\isa{code{\isacharunderscore}post}}} declares (or with
haftmann@31998
  1312
  option ``\isa{{\isachardoublequote}del{\isachardoublequote}}'' removes) theorems which are
haftmann@31998
  1313
  applied as rewrite rules to any result of an evaluation.
haftmann@31998
  1314
wenzelm@28788
  1315
  \item \hyperlink{command.HOL.print-codesetup}{\mbox{\isa{\isacommand{print{\isacharunderscore}codesetup}}}} gives an overview on
haftmann@31254
  1316
  selected code equations and code generator datatypes.
haftmann@31254
  1317
haftmann@31254
  1318
  \item \hyperlink{command.HOL.print-codeproc}{\mbox{\isa{\isacommand{print{\isacharunderscore}codeproc}}}} prints the setup
haftmann@31254
  1319
  of the code generator preprocessor.
wenzelm@26849
  1320
wenzelm@28788
  1321
  \end{description}%
wenzelm@26849
  1322
\end{isamarkuptext}%
wenzelm@26849
  1323
\isamarkuptrue%
wenzelm@26849
  1324
%
wenzelm@27047
  1325
\isamarkupsection{Definition by specification \label{sec:hol-specification}%
wenzelm@27047
  1326
}
wenzelm@27047
  1327
\isamarkuptrue%
wenzelm@27047
  1328
%
wenzelm@27047
  1329
\begin{isamarkuptext}%
wenzelm@27047
  1330
\begin{matharray}{rcl}
wenzelm@28788
  1331
    \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
  1332
    \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
  1333
  \end{matharray}
wenzelm@27047
  1334
wenzelm@27047
  1335
  \begin{rail}
wenzelm@27047
  1336
  ('specification' | 'ax\_specification') '(' (decl +) ')' \\ (thmdecl? prop +)
wenzelm@27047
  1337
  ;
wenzelm@27047
  1338
  decl: ((name ':')? term '(' 'overloaded' ')'?)
wenzelm@27047
  1339
  \end{rail}
wenzelm@27047
  1340
wenzelm@28788
  1341
  \begin{description}
wenzelm@27047
  1342
wenzelm@28788
  1343
  \item \hyperlink{command.HOL.specification}{\mbox{\isa{\isacommand{specification}}}}~\isa{{\isachardoublequote}decls\ {\isasymphi}{\isachardoublequote}} sets up a
wenzelm@27047
  1344
  goal stating the existence of terms with the properties specified to
wenzelm@27047
  1345
  hold for the constants given in \isa{decls}.  After finishing the
wenzelm@27047
  1346
  proof, the theory will be augmented with definitions for the given
wenzelm@27047
  1347
  constants, as well as with theorems stating the properties for these
wenzelm@27047
  1348
  constants.
wenzelm@27047
  1349
wenzelm@28788
  1350
  \item \hyperlink{command.HOL.ax-specification}{\mbox{\isa{\isacommand{ax{\isacharunderscore}specification}}}}~\isa{{\isachardoublequote}decls\ {\isasymphi}{\isachardoublequote}} sets up
wenzelm@28788
  1351
  a goal stating the existence of terms with the properties specified
wenzelm@28788
  1352
  to hold for the constants given in \isa{decls}.  After finishing
wenzelm@28788
  1353
  the proof, the theory will be augmented with axioms expressing the
wenzelm@28788
  1354
  properties given in the first place.
wenzelm@27047
  1355
wenzelm@28788
  1356
  \item \isa{decl} declares a constant to be defined by the
wenzelm@27047
  1357
  specification given.  The definition for the constant \isa{c} is
wenzelm@27047
  1358
  bound to the name \isa{c{\isacharunderscore}def} unless a theorem name is given in
wenzelm@27047
  1359
  the declaration.  Overloaded constants should be declared as such.
wenzelm@27047
  1360
wenzelm@28788
  1361
  \end{description}
wenzelm@27047
  1362
wenzelm@27047
  1363
  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
  1364
  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
  1365
  user has explicitly proven it to be safe.  A practical issue must be
wenzelm@27047
  1366
  considered, though: After introducing two constants with the same
wenzelm@27047
  1367
  properties using \hyperlink{command.HOL.specification}{\mbox{\isa{\isacommand{specification}}}}, one can prove
wenzelm@27047
  1368
  that the two constants are, in fact, equal.  If this might be a
wenzelm@27047
  1369
  problem, one should use \hyperlink{command.HOL.ax-specification}{\mbox{\isa{\isacommand{ax{\isacharunderscore}specification}}}}.%
wenzelm@27047
  1370
\end{isamarkuptext}%
wenzelm@27047
  1371
\isamarkuptrue%
wenzelm@27047
  1372
%
wenzelm@26849
  1373
\isadelimtheory
wenzelm@26849
  1374
%
wenzelm@26849
  1375
\endisadelimtheory
wenzelm@26849
  1376
%
wenzelm@26849
  1377
\isatagtheory
wenzelm@26840
  1378
\isacommand{end}\isamarkupfalse%
wenzelm@26840
  1379
%
wenzelm@26840
  1380
\endisatagtheory
wenzelm@26840
  1381
{\isafoldtheory}%
wenzelm@26840
  1382
%
wenzelm@26840
  1383
\isadelimtheory
wenzelm@26840
  1384
%
wenzelm@26840
  1385
\endisadelimtheory
wenzelm@26849
  1386
\isanewline
wenzelm@26840
  1387
\end{isabellebody}%
wenzelm@26840
  1388
%%% Local Variables:
wenzelm@26840
  1389
%%% mode: latex
wenzelm@26840
  1390
%%% TeX-master: "root"
wenzelm@26840
  1391
%%% End: