doc-src/IsarRef/Thy/document/HOL_Specific.tex
author wenzelm
Fri Oct 29 11:49:56 2010 +0200 (2010-10-29)
changeset 40255 9ffbc25e1606
parent 40254 6d1ebaa7a4ba
child 40364 55a1693affb6
permissions -rw-r--r--
eliminated obsolete \_ escapes in rail environments;
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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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  \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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    ;
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    'rep_datatype' ('(' (name +) ')')? (term +)
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    ;
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    dtspec: parname? typespec mixfix? '=' (cons + '|')
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    ;
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    cons: name ( type * ) mixfix?
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  \end{rail}
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  \begin{description}
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  \item \hyperlink{command.HOL.datatype}{\mbox{\isa{\isacommand{datatype}}}} defines inductive datatypes in
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  HOL.
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  \item \hyperlink{command.HOL.rep-datatype}{\mbox{\isa{\isacommand{rep{\isacharunderscore}datatype}}}} represents existing types as
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  inductive ones, generating the standard infrastructure of derived
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  concepts (primitive recursion etc.).
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  \end{description}
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  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
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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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%
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\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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    \indexdef{HOL}{command}{fun}\hypertarget{command.HOL.fun}{\hyperlink{command.HOL.fun}{\mbox{\isa{\isacommand{fun}}}}} & : & \isa{{\isachardoublequote}local{\isacharunderscore}theory\ {\isasymrightarrow}\ local{\isacharunderscore}theory{\isachardoublequote}} \\
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    \indexdef{HOL}{command}{function}\hypertarget{command.HOL.function}{\hyperlink{command.HOL.function}{\mbox{\isa{\isacommand{function}}}}} & : & \isa{{\isachardoublequote}local{\isacharunderscore}theory\ {\isasymrightarrow}\ proof{\isacharparenleft}prove{\isacharparenright}{\isachardoublequote}} \\
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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}
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   410
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  \begin{rail}
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    'primrec' target? fixes 'where' equations
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    ;
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    ('fun' | 'function') target? functionopts? fixes \\ 'where' equations
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   415
    ;
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    equations: (thmdecl? prop + '|')
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    ;
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    functionopts: '(' (('sequential' | 'domintros' | 'tailrec' | 'default' term) + ',') ')'
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    ;
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    'termination' ( term )?
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  \end{rail}
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   422
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  \begin{description}
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   424
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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}.
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   427
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   428
  \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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  set of (possibly conditional) recursive equations with arbitrary
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  pattern matching. The command generates proof obligations for the
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  completeness and the compatibility of patterns.
wenzelm@26849
   434
wenzelm@26849
   435
  The defined function is considered partial, and the resulting
wenzelm@26849
   436
  simplification rules (named \isa{{\isachardoublequote}f{\isachardot}psimps{\isachardoublequote}}) and induction rule
wenzelm@26849
   437
  (named \isa{{\isachardoublequote}f{\isachardot}pinduct{\isachardoublequote}}) are guarded by a generated domain
wenzelm@26902
   438
  predicate \isa{{\isachardoublequote}f{\isacharunderscore}dom{\isachardoublequote}}. The \hyperlink{command.HOL.termination}{\mbox{\isa{\isacommand{termination}}}}
wenzelm@26849
   439
  command can then be used to establish that the function is total.
wenzelm@26849
   440
wenzelm@28788
   441
  \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
   442
  proof attempts regarding pattern matching and termination.  See
wenzelm@28788
   443
  \cite{isabelle-function} for further details.
wenzelm@26849
   444
wenzelm@28788
   445
  \item \hyperlink{command.HOL.termination}{\mbox{\isa{\isacommand{termination}}}}~\isa{f} commences a
wenzelm@26849
   446
  termination proof for the previously defined function \isa{f}.  If
wenzelm@26849
   447
  this is omitted, the command refers to the most recent function
wenzelm@26849
   448
  definition.  After the proof is closed, the recursive equations and
wenzelm@26849
   449
  the induction principle is established.
wenzelm@26849
   450
wenzelm@28788
   451
  \end{description}
wenzelm@26849
   452
haftmann@27452
   453
  Recursive definitions introduced by the \hyperlink{command.HOL.function}{\mbox{\isa{\isacommand{function}}}}
haftmann@27452
   454
  command accommodate
wenzelm@26849
   455
  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
   456
  refers to a specific induction rule, with parameters named according
krauss@33857
   457
  to the user-specified equations. Cases are numbered (starting from 1).
krauss@33857
   458
krauss@33857
   459
  For \hyperlink{command.HOL.primrec}{\mbox{\isa{\isacommand{primrec}}}}, the induction principle coincides
haftmann@27452
   460
  with structural recursion on the datatype the recursion is carried
haftmann@27452
   461
  out.
wenzelm@26849
   462
wenzelm@26849
   463
  The equations provided by these packages may be referred later as
wenzelm@26849
   464
  theorem list \isa{{\isachardoublequote}f{\isachardot}simps{\isachardoublequote}}, where \isa{f} is the (collective)
wenzelm@26849
   465
  name of the functions defined.  Individual equations may be named
wenzelm@26849
   466
  explicitly as well.
wenzelm@26849
   467
wenzelm@26902
   468
  The \hyperlink{command.HOL.function}{\mbox{\isa{\isacommand{function}}}} command accepts the following
wenzelm@26849
   469
  options.
wenzelm@26849
   470
wenzelm@28788
   471
  \begin{description}
wenzelm@26849
   472
wenzelm@28788
   473
  \item \isa{sequential} enables a preprocessor which disambiguates
wenzelm@28788
   474
  overlapping patterns by making them mutually disjoint.  Earlier
wenzelm@28788
   475
  equations take precedence over later ones.  This allows to give the
wenzelm@28788
   476
  specification in a format very similar to functional programming.
wenzelm@28788
   477
  Note that the resulting simplification and induction rules
wenzelm@28788
   478
  correspond to the transformed specification, not the one given
wenzelm@26849
   479
  originally. This usually means that each equation given by the user
hoelzl@36139
   480
  may result in several theorems.  Also note that this automatic
wenzelm@26849
   481
  transformation only works for ML-style datatype patterns.
wenzelm@26849
   482
wenzelm@28788
   483
  \item \isa{domintros} enables the automated generation of
wenzelm@26849
   484
  introduction rules for the domain predicate. While mostly not
wenzelm@26849
   485
  needed, they can be helpful in some proofs about partial functions.
wenzelm@26849
   486
wenzelm@28788
   487
  \item \isa{tailrec} generates the unconstrained recursive
wenzelm@26849
   488
  equations even without a termination proof, provided that the
wenzelm@26849
   489
  function is tail-recursive. This currently only works
wenzelm@26849
   490
wenzelm@28788
   491
  \item \isa{{\isachardoublequote}default\ d{\isachardoublequote}} allows to specify a default value for a
wenzelm@26849
   492
  (partial) function, which will ensure that \isa{{\isachardoublequote}f\ x\ {\isacharequal}\ d\ x{\isachardoublequote}}
wenzelm@26849
   493
  whenever \isa{{\isachardoublequote}x\ {\isasymnotin}\ f{\isacharunderscore}dom{\isachardoublequote}}.
wenzelm@26849
   494
wenzelm@28788
   495
  \end{description}%
wenzelm@26849
   496
\end{isamarkuptext}%
wenzelm@26849
   497
\isamarkuptrue%
wenzelm@26849
   498
%
wenzelm@26849
   499
\isamarkupsubsection{Proof methods related to recursive definitions%
wenzelm@26849
   500
}
wenzelm@26849
   501
\isamarkuptrue%
wenzelm@26849
   502
%
wenzelm@26849
   503
\begin{isamarkuptext}%
wenzelm@26849
   504
\begin{matharray}{rcl}
wenzelm@28788
   505
    \indexdef{HOL}{method}{pat\_completeness}\hypertarget{method.HOL.pat-completeness}{\hyperlink{method.HOL.pat-completeness}{\mbox{\isa{pat{\isacharunderscore}completeness}}}} & : & \isa{method} \\
wenzelm@28788
   506
    \indexdef{HOL}{method}{relation}\hypertarget{method.HOL.relation}{\hyperlink{method.HOL.relation}{\mbox{\isa{relation}}}} & : & \isa{method} \\
wenzelm@28788
   507
    \indexdef{HOL}{method}{lexicographic\_order}\hypertarget{method.HOL.lexicographic-order}{\hyperlink{method.HOL.lexicographic-order}{\mbox{\isa{lexicographic{\isacharunderscore}order}}}} & : & \isa{method} \\
krauss@33858
   508
    \indexdef{HOL}{method}{size\_change}\hypertarget{method.HOL.size-change}{\hyperlink{method.HOL.size-change}{\mbox{\isa{size{\isacharunderscore}change}}}} & : & \isa{method} \\
wenzelm@26849
   509
  \end{matharray}
wenzelm@26849
   510
wenzelm@26849
   511
  \begin{rail}
wenzelm@26849
   512
    'relation' term
wenzelm@26849
   513
    ;
wenzelm@40255
   514
    'lexicographic_order' ( clasimpmod * )
wenzelm@26849
   515
    ;
wenzelm@40255
   516
    'size_change' ( orders ( clasimpmod * ) )
krauss@33858
   517
    ;
krauss@33858
   518
    orders: ( 'max' | 'min' | 'ms' ) *
wenzelm@26849
   519
  \end{rail}
wenzelm@26849
   520
wenzelm@28788
   521
  \begin{description}
wenzelm@26849
   522
wenzelm@28788
   523
  \item \hyperlink{method.HOL.pat-completeness}{\mbox{\isa{pat{\isacharunderscore}completeness}}} is a specialized method to
wenzelm@26849
   524
  solve goals regarding the completeness of pattern matching, as
wenzelm@26902
   525
  required by the \hyperlink{command.HOL.function}{\mbox{\isa{\isacommand{function}}}} package (cf.\
wenzelm@26849
   526
  \cite{isabelle-function}).
wenzelm@26849
   527
wenzelm@28788
   528
  \item \hyperlink{method.HOL.relation}{\mbox{\isa{relation}}}~\isa{R} introduces a termination
wenzelm@26849
   529
  proof using the relation \isa{R}.  The resulting proof state will
wenzelm@26849
   530
  contain goals expressing that \isa{R} is wellfounded, and that the
wenzelm@26849
   531
  arguments of recursive calls decrease with respect to \isa{R}.
wenzelm@26849
   532
  Usually, this method is used as the initial proof step of manual
wenzelm@26849
   533
  termination proofs.
wenzelm@26849
   534
wenzelm@28788
   535
  \item \hyperlink{method.HOL.lexicographic-order}{\mbox{\isa{lexicographic{\isacharunderscore}order}}} attempts a fully
wenzelm@26849
   536
  automated termination proof by searching for a lexicographic
wenzelm@26849
   537
  combination of size measures on the arguments of the function. The
wenzelm@26902
   538
  method accepts the same arguments as the \hyperlink{method.auto}{\mbox{\isa{auto}}} method,
wenzelm@26849
   539
  which it uses internally to prove local descents.  The same context
wenzelm@26902
   540
  modifiers as for \hyperlink{method.auto}{\mbox{\isa{auto}}} are accepted, see
wenzelm@26849
   541
  \secref{sec:clasimp}.
wenzelm@26849
   542
wenzelm@26849
   543
  In case of failure, extensive information is printed, which can help
wenzelm@26849
   544
  to analyse the situation (cf.\ \cite{isabelle-function}).
wenzelm@26849
   545
krauss@33858
   546
  \item \hyperlink{method.HOL.size-change}{\mbox{\isa{size{\isacharunderscore}change}}} also works on termination goals,
krauss@33858
   547
  using a variation of the size-change principle, together with a
krauss@33858
   548
  graph decomposition technique (see \cite{krauss_phd} for details).
krauss@33858
   549
  Three kinds of orders are used internally: \isa{max}, \isa{min},
krauss@33858
   550
  and \isa{ms} (multiset), which is only available when the theory
krauss@33858
   551
  \isa{Multiset} is loaded. When no order kinds are given, they are
krauss@33858
   552
  tried in order. The search for a termination proof uses SAT solving
krauss@33858
   553
  internally.
krauss@33858
   554
krauss@33858
   555
 For local descent proofs, the same context modifiers as for \hyperlink{method.auto}{\mbox{\isa{auto}}} are accepted, see \secref{sec:clasimp}.
krauss@33858
   556
wenzelm@28788
   557
  \end{description}%
wenzelm@26849
   558
\end{isamarkuptext}%
wenzelm@26849
   559
\isamarkuptrue%
wenzelm@26849
   560
%
krauss@40171
   561
\isamarkupsubsection{Functions with explicit partiality%
krauss@40171
   562
}
krauss@40171
   563
\isamarkuptrue%
krauss@40171
   564
%
krauss@40171
   565
\begin{isamarkuptext}%
krauss@40171
   566
\begin{matharray}{rcl}
krauss@40171
   567
    \indexdef{HOL}{command}{partial\_function}\hypertarget{command.HOL.partial-function}{\hyperlink{command.HOL.partial-function}{\mbox{\isa{\isacommand{partial{\isacharunderscore}function}}}}} & : & \isa{{\isachardoublequote}local{\isacharunderscore}theory\ {\isasymrightarrow}\ local{\isacharunderscore}theory{\isachardoublequote}} \\
krauss@40171
   568
    \indexdef{HOL}{attribute}{partial\_function\_mono}\hypertarget{attribute.HOL.partial-function-mono}{\hyperlink{attribute.HOL.partial-function-mono}{\mbox{\isa{partial{\isacharunderscore}function{\isacharunderscore}mono}}}} & : & \isa{attribute} \\
krauss@40171
   569
  \end{matharray}
krauss@40171
   570
krauss@40171
   571
  \begin{rail}
krauss@40171
   572
    'partial_function' target? '(' mode ')' fixes \\ 'where' thmdecl? prop
krauss@40171
   573
  \end{rail}
krauss@40171
   574
krauss@40171
   575
  \begin{description}
krauss@40171
   576
krauss@40171
   577
  \item \hyperlink{command.HOL.partial-function}{\mbox{\isa{\isacommand{partial{\isacharunderscore}function}}}} defines recursive
krauss@40171
   578
  functions based on fixpoints in complete partial orders. No
krauss@40171
   579
  termination proof is required from the user or constructed
krauss@40171
   580
  internally. Instead, the possibility of non-termination is modelled
krauss@40171
   581
  explicitly in the result type, which contains an explicit bottom
krauss@40171
   582
  element.
krauss@40171
   583
krauss@40171
   584
  Pattern matching and mutual recursion are currently not supported.
krauss@40171
   585
  Thus, the specification consists of a single function described by a
krauss@40171
   586
  single recursive equation.
krauss@40171
   587
krauss@40171
   588
  There are no fixed syntactic restrictions on the body of the
krauss@40171
   589
  function, but the induced functional must be provably monotonic
krauss@40171
   590
  wrt.\ the underlying order.  The monotonicitity proof is performed
krauss@40171
   591
  internally, and the definition is rejected when it fails. The proof
krauss@40171
   592
  can be influenced by declaring hints using the
krauss@40171
   593
  \hyperlink{attribute.HOL.partial-function-mono}{\mbox{\isa{partial{\isacharunderscore}function{\isacharunderscore}mono}}} attribute.
krauss@40171
   594
krauss@40171
   595
  The mandatory \isa{mode} argument specifies the mode of operation
krauss@40171
   596
  of the command, which directly corresponds to a complete partial
krauss@40171
   597
  order on the result type. By default, the following modes are
krauss@40171
   598
  defined: 
krauss@40171
   599
krauss@40171
   600
  \begin{description}
krauss@40171
   601
  \item \isa{option} defines functions that map into the \isa{option} type. Here, the value \isa{None} is used to model a
krauss@40171
   602
  non-terminating computation. Monotonicity requires that if \isa{None} is returned by a recursive call, then the overall result
krauss@40171
   603
  must also be \isa{None}. This is best achieved through the use of
krauss@40171
   604
  the monadic operator \isa{{\isachardoublequote}Option{\isachardot}bind{\isachardoublequote}}.
krauss@40171
   605
  
krauss@40171
   606
  \item \isa{tailrec} defines functions with an arbitrary result
krauss@40171
   607
  type and uses the slightly degenerated partial order where \isa{{\isachardoublequote}undefined{\isachardoublequote}} is the bottom element.  Now, monotonicity requires that
krauss@40171
   608
  if \isa{undefined} is returned by a recursive call, then the
krauss@40171
   609
  overall result must also be \isa{undefined}. In practice, this is
krauss@40171
   610
  only satisfied when each recursive call is a tail call, whose result
krauss@40171
   611
  is directly returned. Thus, this mode of operation allows the
krauss@40171
   612
  definition of arbitrary tail-recursive functions.
krauss@40171
   613
  \end{description}
krauss@40171
   614
krauss@40171
   615
  Experienced users may define new modes by instantiating the locale
krauss@40171
   616
  \isa{{\isachardoublequote}partial{\isacharunderscore}function{\isacharunderscore}definitions{\isachardoublequote}} appropriately.
krauss@40171
   617
krauss@40171
   618
  \item \hyperlink{attribute.HOL.partial-function-mono}{\mbox{\isa{partial{\isacharunderscore}function{\isacharunderscore}mono}}} declares rules for
krauss@40171
   619
  use in the internal monononicity proofs of partial function
krauss@40171
   620
  definitions.
krauss@40171
   621
krauss@40171
   622
  \end{description}%
krauss@40171
   623
\end{isamarkuptext}%
krauss@40171
   624
\isamarkuptrue%
krauss@40171
   625
%
wenzelm@26849
   626
\isamarkupsubsection{Old-style recursive function definitions (TFL)%
wenzelm@26849
   627
}
wenzelm@26849
   628
\isamarkuptrue%
wenzelm@26849
   629
%
wenzelm@26849
   630
\begin{isamarkuptext}%
wenzelm@26907
   631
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
   632
wenzelm@26849
   633
  \begin{matharray}{rcl}
wenzelm@28788
   634
    \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
   635
    \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
   636
  \end{matharray}
wenzelm@26849
   637
wenzelm@26849
   638
  \begin{rail}
wenzelm@26849
   639
    'recdef' ('(' 'permissive' ')')? \\ name term (prop +) hints?
wenzelm@26849
   640
    ;
wenzelm@26849
   641
    recdeftc thmdecl? tc
wenzelm@26849
   642
    ;
haftmann@31913
   643
    hints: '(' 'hints' ( recdefmod * ) ')'
wenzelm@26849
   644
    ;
wenzelm@40255
   645
    recdefmod: (('recdef_simp' | 'recdef_cong' | 'recdef_wf') (() | 'add' | 'del') ':' thmrefs) | clasimpmod
wenzelm@26849
   646
    ;
wenzelm@26849
   647
    tc: nameref ('(' nat ')')?
wenzelm@26849
   648
    ;
wenzelm@26849
   649
  \end{rail}
wenzelm@26849
   650
wenzelm@28788
   651
  \begin{description}
wenzelm@26849
   652
  
wenzelm@28788
   653
  \item \hyperlink{command.HOL.recdef}{\mbox{\isa{\isacommand{recdef}}}} defines general well-founded
wenzelm@26849
   654
  recursive functions (using the TFL package), see also
wenzelm@26849
   655
  \cite{isabelle-HOL}.  The ``\isa{{\isachardoublequote}{\isacharparenleft}permissive{\isacharparenright}{\isachardoublequote}}'' option tells
wenzelm@26849
   656
  TFL to recover from failed proof attempts, returning unfinished
wenzelm@26849
   657
  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
   658
  automated proof process of TFL.  Additional \hyperlink{syntax.clasimpmod}{\mbox{\isa{clasimpmod}}}
wenzelm@26849
   659
  declarations (cf.\ \secref{sec:clasimp}) may be given to tune the
wenzelm@26849
   660
  context of the Simplifier (cf.\ \secref{sec:simplifier}) and
wenzelm@26849
   661
  Classical reasoner (cf.\ \secref{sec:classical}).
wenzelm@26849
   662
  
wenzelm@28788
   663
  \item \hyperlink{command.HOL.recdef-tc}{\mbox{\isa{\isacommand{recdef{\isacharunderscore}tc}}}}~\isa{{\isachardoublequote}c\ {\isacharparenleft}i{\isacharparenright}{\isachardoublequote}} recommences the
wenzelm@26849
   664
  proof for leftover termination condition number \isa{i} (default
wenzelm@26902
   665
  1) as generated by a \hyperlink{command.HOL.recdef}{\mbox{\isa{\isacommand{recdef}}}} definition of
wenzelm@26849
   666
  constant \isa{c}.
wenzelm@26849
   667
  
wenzelm@26902
   668
  Note that in most cases, \hyperlink{command.HOL.recdef}{\mbox{\isa{\isacommand{recdef}}}} is able to finish
wenzelm@26849
   669
  its internal proofs without manual intervention.
wenzelm@26849
   670
wenzelm@28788
   671
  \end{description}
wenzelm@26849
   672
wenzelm@26902
   673
  \medskip Hints for \hyperlink{command.HOL.recdef}{\mbox{\isa{\isacommand{recdef}}}} may be also declared
wenzelm@26849
   674
  globally, using the following attributes.
wenzelm@26849
   675
wenzelm@26849
   676
  \begin{matharray}{rcl}
wenzelm@28788
   677
    \indexdef{HOL}{attribute}{recdef\_simp}\hypertarget{attribute.HOL.recdef-simp}{\hyperlink{attribute.HOL.recdef-simp}{\mbox{\isa{recdef{\isacharunderscore}simp}}}} & : & \isa{attribute} \\
wenzelm@28788
   678
    \indexdef{HOL}{attribute}{recdef\_cong}\hypertarget{attribute.HOL.recdef-cong}{\hyperlink{attribute.HOL.recdef-cong}{\mbox{\isa{recdef{\isacharunderscore}cong}}}} & : & \isa{attribute} \\
wenzelm@28788
   679
    \indexdef{HOL}{attribute}{recdef\_wf}\hypertarget{attribute.HOL.recdef-wf}{\hyperlink{attribute.HOL.recdef-wf}{\mbox{\isa{recdef{\isacharunderscore}wf}}}} & : & \isa{attribute} \\
wenzelm@26849
   680
  \end{matharray}
wenzelm@26849
   681
wenzelm@26849
   682
  \begin{rail}
wenzelm@40255
   683
    ('recdef_simp' | 'recdef_cong' | 'recdef_wf') (() | 'add' | 'del')
wenzelm@26849
   684
    ;
wenzelm@26849
   685
  \end{rail}%
wenzelm@26849
   686
\end{isamarkuptext}%
wenzelm@26849
   687
\isamarkuptrue%
wenzelm@26849
   688
%
wenzelm@26849
   689
\isamarkupsection{Inductive and coinductive definitions \label{sec:hol-inductive}%
wenzelm@26849
   690
}
wenzelm@26849
   691
\isamarkuptrue%
wenzelm@26849
   692
%
wenzelm@26849
   693
\begin{isamarkuptext}%
wenzelm@26849
   694
An \textbf{inductive definition} specifies the least predicate (or
wenzelm@26849
   695
  set) \isa{R} closed under given rules: applying a rule to elements
wenzelm@26849
   696
  of \isa{R} yields a result within \isa{R}.  For example, a
wenzelm@26849
   697
  structural operational semantics is an inductive definition of an
wenzelm@26849
   698
  evaluation relation.
wenzelm@26849
   699
wenzelm@26849
   700
  Dually, a \textbf{coinductive definition} specifies the greatest
wenzelm@26849
   701
  predicate~/ set \isa{R} that is consistent with given rules: every
wenzelm@26849
   702
  element of \isa{R} can be seen as arising by applying a rule to
wenzelm@26849
   703
  elements of \isa{R}.  An important example is using bisimulation
wenzelm@26849
   704
  relations to formalise equivalence of processes and infinite data
wenzelm@26849
   705
  structures.
wenzelm@26849
   706
wenzelm@26849
   707
  \medskip The HOL package is related to the ZF one, which is
wenzelm@26849
   708
  described in a separate paper,\footnote{It appeared in CADE
wenzelm@26849
   709
  \cite{paulson-CADE}; a longer version is distributed with Isabelle.}
wenzelm@26849
   710
  which you should refer to in case of difficulties.  The package is
wenzelm@26849
   711
  simpler than that of ZF thanks to implicit type-checking in HOL.
wenzelm@26849
   712
  The types of the (co)inductive predicates (or sets) determine the
wenzelm@26849
   713
  domain of the fixedpoint definition, and the package does not have
wenzelm@26849
   714
  to use inference rules for type-checking.
wenzelm@26849
   715
wenzelm@26849
   716
  \begin{matharray}{rcl}
wenzelm@28788
   717
    \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
   718
    \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
   719
    \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
   720
    \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
   721
    \indexdef{HOL}{attribute}{mono}\hypertarget{attribute.HOL.mono}{\hyperlink{attribute.HOL.mono}{\mbox{\isa{mono}}}} & : & \isa{attribute} \\
wenzelm@26849
   722
  \end{matharray}
wenzelm@26849
   723
wenzelm@26849
   724
  \begin{rail}
wenzelm@40255
   725
    ('inductive' | 'inductive_set' | 'coinductive' | 'coinductive_set') target? fixes ('for' fixes)? \\
wenzelm@26849
   726
    ('where' clauses)? ('monos' thmrefs)?
wenzelm@26849
   727
    ;
wenzelm@26849
   728
    clauses: (thmdecl? prop + '|')
wenzelm@26849
   729
    ;
wenzelm@26849
   730
    'mono' (() | 'add' | 'del')
wenzelm@26849
   731
    ;
wenzelm@26849
   732
  \end{rail}
wenzelm@26849
   733
wenzelm@28788
   734
  \begin{description}
wenzelm@26849
   735
wenzelm@28788
   736
  \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
   737
  introduction rules given in the \hyperlink{keyword.where}{\mbox{\isa{\isakeyword{where}}}} part.  The
wenzelm@26902
   738
  optional \hyperlink{keyword.for}{\mbox{\isa{\isakeyword{for}}}} part contains a list of parameters of the
wenzelm@26849
   739
  (co)inductive predicates that remain fixed throughout the
wenzelm@26902
   740
  definition.  The optional \hyperlink{keyword.monos}{\mbox{\isa{\isakeyword{monos}}}} section contains
wenzelm@26849
   741
  \emph{monotonicity theorems}, which are required for each operator
wenzelm@26849
   742
  applied to a recursive set in the introduction rules.  There
wenzelm@26849
   743
  \emph{must} be a theorem of the form \isa{{\isachardoublequote}A\ {\isasymle}\ B\ {\isasymLongrightarrow}\ M\ A\ {\isasymle}\ M\ B{\isachardoublequote}},
wenzelm@26849
   744
  for each premise \isa{{\isachardoublequote}M\ R\isactrlsub i\ t{\isachardoublequote}} in an introduction rule!
wenzelm@26849
   745
wenzelm@28788
   746
  \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
   747
  allowing the definition of (co)inductive sets.
wenzelm@26849
   748
wenzelm@28788
   749
  \item \hyperlink{attribute.HOL.mono}{\mbox{\isa{mono}}} declares monotonicity rules.  These
wenzelm@26902
   750
  rule are involved in the automated monotonicity proof of \hyperlink{command.HOL.inductive}{\mbox{\isa{\isacommand{inductive}}}}.
wenzelm@26849
   751
wenzelm@28788
   752
  \end{description}%
wenzelm@26849
   753
\end{isamarkuptext}%
wenzelm@26849
   754
\isamarkuptrue%
wenzelm@26849
   755
%
wenzelm@26849
   756
\isamarkupsubsection{Derived rules%
wenzelm@26849
   757
}
wenzelm@26849
   758
\isamarkuptrue%
wenzelm@26849
   759
%
wenzelm@26849
   760
\begin{isamarkuptext}%
wenzelm@26849
   761
Each (co)inductive definition \isa{R} adds definitions to the
wenzelm@26849
   762
  theory and also proves some theorems:
wenzelm@26849
   763
wenzelm@26849
   764
  \begin{description}
wenzelm@26849
   765
wenzelm@28788
   766
  \item \isa{R{\isachardot}intros} is the list of introduction rules as proven
wenzelm@26849
   767
  theorems, for the recursive predicates (or sets).  The rules are
wenzelm@26849
   768
  also available individually, using the names given them in the
wenzelm@26849
   769
  theory file;
wenzelm@26849
   770
wenzelm@28788
   771
  \item \isa{R{\isachardot}cases} is the case analysis (or elimination) rule;
wenzelm@26849
   772
wenzelm@28788
   773
  \item \isa{R{\isachardot}induct} or \isa{R{\isachardot}coinduct} is the (co)induction
wenzelm@26849
   774
  rule.
wenzelm@26849
   775
wenzelm@26849
   776
  \end{description}
wenzelm@26849
   777
wenzelm@26849
   778
  When several predicates \isa{{\isachardoublequote}R\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ R\isactrlsub n{\isachardoublequote}} are
wenzelm@26849
   779
  defined simultaneously, the list of introduction rules is called
wenzelm@26849
   780
  \isa{{\isachardoublequote}R\isactrlsub {\isadigit{1}}{\isacharunderscore}{\isasymdots}{\isacharunderscore}R\isactrlsub n{\isachardot}intros{\isachardoublequote}}, the case analysis rules are
wenzelm@26849
   781
  called \isa{{\isachardoublequote}R\isactrlsub {\isadigit{1}}{\isachardot}cases{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ R\isactrlsub n{\isachardot}cases{\isachardoublequote}}, and the list
wenzelm@26849
   782
  of mutual induction rules is called \isa{{\isachardoublequote}R\isactrlsub {\isadigit{1}}{\isacharunderscore}{\isasymdots}{\isacharunderscore}R\isactrlsub n{\isachardot}inducts{\isachardoublequote}}.%
wenzelm@26849
   783
\end{isamarkuptext}%
wenzelm@26849
   784
\isamarkuptrue%
wenzelm@26849
   785
%
wenzelm@26849
   786
\isamarkupsubsection{Monotonicity theorems%
wenzelm@26849
   787
}
wenzelm@26849
   788
\isamarkuptrue%
wenzelm@26849
   789
%
wenzelm@26849
   790
\begin{isamarkuptext}%
wenzelm@26849
   791
Each theory contains a default set of theorems that are used in
wenzelm@26849
   792
  monotonicity proofs.  New rules can be added to this set via the
wenzelm@26902
   793
  \hyperlink{attribute.HOL.mono}{\mbox{\isa{mono}}} attribute.  The HOL theory \isa{Inductive}
wenzelm@26849
   794
  shows how this is done.  In general, the following monotonicity
wenzelm@26849
   795
  theorems may be added:
wenzelm@26849
   796
wenzelm@26849
   797
  \begin{itemize}
wenzelm@26849
   798
wenzelm@26849
   799
  \item Theorems of the form \isa{{\isachardoublequote}A\ {\isasymle}\ B\ {\isasymLongrightarrow}\ M\ A\ {\isasymle}\ M\ B{\isachardoublequote}}, for proving
wenzelm@26849
   800
  monotonicity of inductive definitions whose introduction rules have
wenzelm@26849
   801
  premises involving terms such as \isa{{\isachardoublequote}M\ R\isactrlsub i\ t{\isachardoublequote}}.
wenzelm@26849
   802
wenzelm@26849
   803
  \item Monotonicity theorems for logical operators, which are of the
wenzelm@26849
   804
  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
   805
  the case of the operator \isa{{\isachardoublequote}{\isasymor}{\isachardoublequote}}, the corresponding theorem is
wenzelm@26849
   806
  \[
wenzelm@26849
   807
  \infer{\isa{{\isachardoublequote}P\isactrlsub {\isadigit{1}}\ {\isasymor}\ P\isactrlsub {\isadigit{2}}\ {\isasymlongrightarrow}\ Q\isactrlsub {\isadigit{1}}\ {\isasymor}\ Q\isactrlsub {\isadigit{2}}{\isachardoublequote}}}{\isa{{\isachardoublequote}P\isactrlsub {\isadigit{1}}\ {\isasymlongrightarrow}\ Q\isactrlsub {\isadigit{1}}{\isachardoublequote}} & \isa{{\isachardoublequote}P\isactrlsub {\isadigit{2}}\ {\isasymlongrightarrow}\ Q\isactrlsub {\isadigit{2}}{\isachardoublequote}}}
wenzelm@26849
   808
  \]
wenzelm@26849
   809
wenzelm@26849
   810
  \item De Morgan style equations for reasoning about the ``polarity''
wenzelm@26849
   811
  of expressions, e.g.
wenzelm@26849
   812
  \[
wenzelm@26849
   813
  \isa{{\isachardoublequote}{\isasymnot}\ {\isasymnot}\ P\ {\isasymlongleftrightarrow}\ P{\isachardoublequote}} \qquad\qquad
wenzelm@26849
   814
  \isa{{\isachardoublequote}{\isasymnot}\ {\isacharparenleft}P\ {\isasymand}\ Q{\isacharparenright}\ {\isasymlongleftrightarrow}\ {\isasymnot}\ P\ {\isasymor}\ {\isasymnot}\ Q{\isachardoublequote}}
wenzelm@26849
   815
  \]
wenzelm@26849
   816
wenzelm@26849
   817
  \item Equations for reducing complex operators to more primitive
wenzelm@26849
   818
  ones whose monotonicity can easily be proved, e.g.
wenzelm@26849
   819
  \[
wenzelm@26849
   820
  \isa{{\isachardoublequote}{\isacharparenleft}P\ {\isasymlongrightarrow}\ Q{\isacharparenright}\ {\isasymlongleftrightarrow}\ {\isasymnot}\ P\ {\isasymor}\ Q{\isachardoublequote}} \qquad\qquad
wenzelm@26849
   821
  \isa{{\isachardoublequote}Ball\ A\ P\ {\isasymequiv}\ {\isasymforall}x{\isachardot}\ x\ {\isasymin}\ A\ {\isasymlongrightarrow}\ P\ x{\isachardoublequote}}
wenzelm@26849
   822
  \]
wenzelm@26849
   823
wenzelm@26849
   824
  \end{itemize}
wenzelm@26849
   825
wenzelm@26849
   826
  %FIXME: Example of an inductive definition%
wenzelm@26849
   827
\end{isamarkuptext}%
wenzelm@26849
   828
\isamarkuptrue%
wenzelm@26849
   829
%
wenzelm@26849
   830
\isamarkupsection{Arithmetic proof support%
wenzelm@26849
   831
}
wenzelm@26849
   832
\isamarkuptrue%
wenzelm@26849
   833
%
wenzelm@26849
   834
\begin{isamarkuptext}%
wenzelm@26849
   835
\begin{matharray}{rcl}
wenzelm@28788
   836
    \indexdef{HOL}{method}{arith}\hypertarget{method.HOL.arith}{\hyperlink{method.HOL.arith}{\mbox{\isa{arith}}}} & : & \isa{method} \\
nipkow@30863
   837
    \indexdef{HOL}{attribute}{arith}\hypertarget{attribute.HOL.arith}{\hyperlink{attribute.HOL.arith}{\mbox{\isa{arith}}}} & : & \isa{attribute} \\
wenzelm@28788
   838
    \indexdef{HOL}{attribute}{arith\_split}\hypertarget{attribute.HOL.arith-split}{\hyperlink{attribute.HOL.arith-split}{\mbox{\isa{arith{\isacharunderscore}split}}}} & : & \isa{attribute} \\
wenzelm@26849
   839
  \end{matharray}
wenzelm@26849
   840
wenzelm@26902
   841
  The \hyperlink{method.HOL.arith}{\mbox{\isa{arith}}} method decides linear arithmetic problems
wenzelm@26849
   842
  (on types \isa{nat}, \isa{int}, \isa{real}).  Any current
wenzelm@26849
   843
  facts are inserted into the goal before running the procedure.
wenzelm@26849
   844
nipkow@30863
   845
  The \hyperlink{attribute.HOL.arith}{\mbox{\isa{arith}}} attribute declares facts that are
nipkow@30863
   846
  always supplied to the arithmetic provers implicitly.
wenzelm@26849
   847
nipkow@30863
   848
  The \hyperlink{attribute.HOL.arith-split}{\mbox{\isa{arith{\isacharunderscore}split}}} attribute declares case split
wenzelm@30865
   849
  rules to be expanded before \hyperlink{method.HOL.arith}{\mbox{\isa{arith}}} is invoked.
nipkow@30863
   850
nipkow@30863
   851
  Note that a simpler (but faster) arithmetic prover is
nipkow@30863
   852
  already invoked by the Simplifier.%
wenzelm@26849
   853
\end{isamarkuptext}%
wenzelm@26849
   854
\isamarkuptrue%
wenzelm@26849
   855
%
wenzelm@30172
   856
\isamarkupsection{Intuitionistic proof search%
wenzelm@30172
   857
}
wenzelm@30172
   858
\isamarkuptrue%
wenzelm@30172
   859
%
wenzelm@30172
   860
\begin{isamarkuptext}%
wenzelm@30172
   861
\begin{matharray}{rcl}
wenzelm@30172
   862
    \indexdef{HOL}{method}{iprover}\hypertarget{method.HOL.iprover}{\hyperlink{method.HOL.iprover}{\mbox{\isa{iprover}}}} & : & \isa{method} \\
wenzelm@30172
   863
  \end{matharray}
wenzelm@30172
   864
wenzelm@30172
   865
  \begin{rail}
wenzelm@35613
   866
    'iprover' ( rulemod * )
wenzelm@30172
   867
    ;
wenzelm@30172
   868
  \end{rail}
wenzelm@30172
   869
wenzelm@30172
   870
  The \hyperlink{method.HOL.iprover}{\mbox{\isa{iprover}}} method performs intuitionistic proof
wenzelm@30172
   871
  search, depending on specifically declared rules from the context,
wenzelm@30172
   872
  or given as explicit arguments.  Chained facts are inserted into the
wenzelm@35613
   873
  goal before commencing proof search.
wenzelm@35613
   874
wenzelm@30172
   875
  Rules need to be classified as \hyperlink{attribute.Pure.intro}{\mbox{\isa{intro}}},
wenzelm@30172
   876
  \hyperlink{attribute.Pure.elim}{\mbox{\isa{elim}}}, or \hyperlink{attribute.Pure.dest}{\mbox{\isa{dest}}}; here the
wenzelm@30172
   877
  ``\isa{{\isachardoublequote}{\isacharbang}{\isachardoublequote}}'' indicator refers to ``safe'' rules, which may be
wenzelm@30172
   878
  applied aggressively (without considering back-tracking later).
wenzelm@30172
   879
  Rules declared with ``\isa{{\isachardoublequote}{\isacharquery}{\isachardoublequote}}'' are ignored in proof search (the
wenzelm@30172
   880
  single-step \hyperlink{method.rule}{\mbox{\isa{rule}}} method still observes these).  An
wenzelm@30172
   881
  explicit weight annotation may be given as well; otherwise the
wenzelm@30172
   882
  number of rule premises will be taken into account here.%
wenzelm@30172
   883
\end{isamarkuptext}%
wenzelm@30172
   884
\isamarkuptrue%
wenzelm@30172
   885
%
wenzelm@30172
   886
\isamarkupsection{Coherent Logic%
wenzelm@30172
   887
}
wenzelm@30172
   888
\isamarkuptrue%
wenzelm@30172
   889
%
wenzelm@30172
   890
\begin{isamarkuptext}%
wenzelm@30172
   891
\begin{matharray}{rcl}
wenzelm@30172
   892
    \indexdef{HOL}{method}{coherent}\hypertarget{method.HOL.coherent}{\hyperlink{method.HOL.coherent}{\mbox{\isa{coherent}}}} & : & \isa{method} \\
wenzelm@30172
   893
  \end{matharray}
wenzelm@30172
   894
wenzelm@30172
   895
  \begin{rail}
wenzelm@30172
   896
    'coherent' thmrefs?
wenzelm@30172
   897
    ;
wenzelm@30172
   898
  \end{rail}
wenzelm@30172
   899
wenzelm@30172
   900
  The \hyperlink{method.HOL.coherent}{\mbox{\isa{coherent}}} method solves problems of
wenzelm@30172
   901
  \emph{Coherent Logic} \cite{Bezem-Coquand:2005}, which covers
wenzelm@30172
   902
  applications in confluence theory, lattice theory and projective
wenzelm@30172
   903
  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
   904
  examples.%
wenzelm@30172
   905
\end{isamarkuptext}%
wenzelm@30172
   906
\isamarkuptrue%
wenzelm@30172
   907
%
haftmann@31913
   908
\isamarkupsection{Checking and refuting propositions%
haftmann@31913
   909
}
haftmann@31913
   910
\isamarkuptrue%
haftmann@31913
   911
%
haftmann@31913
   912
\begin{isamarkuptext}%
haftmann@31913
   913
Identifying incorrect propositions usually involves evaluation of
haftmann@31913
   914
  particular assignments and systematic counter example search.  This
haftmann@31913
   915
  is supported by the following commands.
haftmann@31913
   916
haftmann@31913
   917
  \begin{matharray}{rcl}
haftmann@31913
   918
    \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
   919
    \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
   920
    \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
   921
  \end{matharray}
haftmann@31913
   922
haftmann@31913
   923
  \begin{rail}
haftmann@31913
   924
    'value' ( ( '[' name ']' ) ? ) modes? term
haftmann@31913
   925
    ;
haftmann@31913
   926
haftmann@31913
   927
    'quickcheck' ( ( '[' args ']' ) ? ) nat?
haftmann@31913
   928
    ;
haftmann@31913
   929
haftmann@31913
   930
    'quickcheck_params' ( ( '[' args ']' ) ? )
haftmann@31913
   931
    ;
haftmann@31913
   932
haftmann@31913
   933
    modes: '(' (name + ) ')'
haftmann@31913
   934
    ;
haftmann@31913
   935
haftmann@31913
   936
    args: ( name '=' value + ',' )
haftmann@31913
   937
    ;
haftmann@31913
   938
  \end{rail}
haftmann@31913
   939
haftmann@31913
   940
  \begin{description}
haftmann@31913
   941
haftmann@31913
   942
  \item \hyperlink{command.HOL.value}{\mbox{\isa{\isacommand{value}}}}~\isa{t} evaluates and prints a
haftmann@31913
   943
    term; optionally \isa{modes} can be specified, which are
haftmann@31913
   944
    appended to the current print mode (see also \cite{isabelle-ref}).
haftmann@31913
   945
    Internally, the evaluation is performed by registered evaluators,
haftmann@31913
   946
    which are invoked sequentially until a result is returned.
haftmann@31913
   947
    Alternatively a specific evaluator can be selected using square
haftmann@37444
   948
    brackets; typical evaluators use the current set of code equations
haftmann@37444
   949
    to normalize and include \isa{simp} for fully symbolic evaluation
haftmann@37444
   950
    using the simplifier, \isa{nbe} for \emph{normalization by evaluation}
haftmann@37444
   951
    and \emph{code} for code generation in SML.
haftmann@31913
   952
haftmann@31913
   953
  \item \hyperlink{command.HOL.quickcheck}{\mbox{\isa{\isacommand{quickcheck}}}} tests the current goal for
haftmann@31913
   954
    counter examples using a series of arbitrary assignments for its
haftmann@31913
   955
    free variables; by default the first subgoal is tested, an other
haftmann@31913
   956
    can be selected explicitly using an optional goal index.
haftmann@31913
   957
    A number of configuration options are supported for
haftmann@31913
   958
    \hyperlink{command.HOL.quickcheck}{\mbox{\isa{\isacommand{quickcheck}}}}, notably:
haftmann@31913
   959
haftmann@31913
   960
    \begin{description}
haftmann@31913
   961
wenzelm@40254
   962
    \item[\isa{size}] specifies the maximum size of the search space
wenzelm@40254
   963
    for assignment values.
wenzelm@40254
   964
wenzelm@40254
   965
    \item[\isa{iterations}] sets how many sets of assignments are
wenzelm@40254
   966
    generated for each particular size.
wenzelm@40254
   967
wenzelm@40254
   968
    \item[\isa{no{\isacharunderscore}assms}] specifies whether assumptions in
wenzelm@40254
   969
    structured proofs should be ignored.
wenzelm@40254
   970
wenzelm@40254
   971
    \item[\isa{timeout}] sets the time limit in seconds.
haftmann@31913
   972
wenzelm@40254
   973
    \item[\isa{default{\isacharunderscore}type}] sets the type(s) generally used to
wenzelm@40254
   974
    instantiate type variables.
wenzelm@40254
   975
wenzelm@40254
   976
    \item[\isa{report}] if set quickcheck reports how many tests
wenzelm@40254
   977
    fulfilled the preconditions.
haftmann@31913
   978
wenzelm@40254
   979
    \item[\isa{quiet}] if not set quickcheck informs about the
wenzelm@40254
   980
    current size for assignment values.
wenzelm@40254
   981
wenzelm@40254
   982
    \item[\isa{expect}] can be used to check if the user's
wenzelm@40254
   983
    expectation was met (\isa{no{\isacharunderscore}expectation}, \isa{no{\isacharunderscore}counterexample}, or \isa{counterexample}).
wenzelm@35351
   984
haftmann@31913
   985
    \end{description}
haftmann@31913
   986
haftmann@31913
   987
    These option can be given within square brackets.
haftmann@31913
   988
haftmann@31913
   989
  \item \hyperlink{command.HOL.quickcheck-params}{\mbox{\isa{\isacommand{quickcheck{\isacharunderscore}params}}}} changes quickcheck
haftmann@31913
   990
    configuration options persitently.
haftmann@31913
   991
haftmann@31913
   992
  \end{description}%
haftmann@31913
   993
\end{isamarkuptext}%
haftmann@31913
   994
\isamarkuptrue%
haftmann@31913
   995
%
wenzelm@28788
   996
\isamarkupsection{Unstructured case analysis and induction \label{sec:hol-induct-tac}%
wenzelm@26849
   997
}
wenzelm@26849
   998
\isamarkuptrue%
wenzelm@26849
   999
%
wenzelm@26849
  1000
\begin{isamarkuptext}%
wenzelm@27124
  1001
The following tools of Isabelle/HOL support cases analysis and
wenzelm@27124
  1002
  induction in unstructured tactic scripts; see also
wenzelm@27124
  1003
  \secref{sec:cases-induct} for proper Isar versions of similar ideas.
wenzelm@26849
  1004
wenzelm@26849
  1005
  \begin{matharray}{rcl}
wenzelm@28788
  1006
    \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
  1007
    \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
  1008
    \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
  1009
    \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
  1010
  \end{matharray}
wenzelm@26849
  1011
wenzelm@26849
  1012
  \begin{rail}
wenzelm@40255
  1013
    'case_tac' goalspec? term rule?
wenzelm@26849
  1014
    ;
wenzelm@40255
  1015
    'induct_tac' goalspec? (insts * 'and') rule?
wenzelm@26849
  1016
    ;
wenzelm@40255
  1017
    'ind_cases' (prop +) ('for' (name +)) ?
wenzelm@26849
  1018
    ;
wenzelm@40255
  1019
    'inductive_cases' (thmdecl? (prop +) + 'and')
wenzelm@26849
  1020
    ;
wenzelm@26849
  1021
wenzelm@26849
  1022
    rule: ('rule' ':' thmref)
wenzelm@26849
  1023
    ;
wenzelm@26849
  1024
  \end{rail}
wenzelm@26849
  1025
wenzelm@28788
  1026
  \begin{description}
wenzelm@26849
  1027
wenzelm@28788
  1028
  \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
  1029
  to reason about inductive types.  Rules are selected according to
wenzelm@28788
  1030
  the declarations by the \hyperlink{attribute.cases}{\mbox{\isa{cases}}} and \hyperlink{attribute.induct}{\mbox{\isa{induct}}}
wenzelm@28788
  1031
  attributes, cf.\ \secref{sec:cases-induct}.  The \hyperlink{command.HOL.datatype}{\mbox{\isa{\isacommand{datatype}}}} package already takes care of this.
wenzelm@27124
  1032
wenzelm@27124
  1033
  These unstructured tactics feature both goal addressing and dynamic
wenzelm@26849
  1034
  instantiation.  Note that named rule cases are \emph{not} provided
wenzelm@27124
  1035
  as would be by the proper \hyperlink{method.cases}{\mbox{\isa{cases}}} and \hyperlink{method.induct}{\mbox{\isa{induct}}} proof
wenzelm@27124
  1036
  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
  1037
  statements, only the compact object-logic conclusion of the subgoal
wenzelm@27124
  1038
  being addressed.
wenzelm@26849
  1039
  
wenzelm@28788
  1040
  \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
  1041
  forward manner.
wenzelm@26849
  1042
wenzelm@26907
  1043
  While \hyperlink{method.HOL.ind-cases}{\mbox{\isa{ind{\isacharunderscore}cases}}} is a proof method to apply the
wenzelm@26907
  1044
  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
  1045
  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
  1046
  be generalized before applying the resulting rule.
wenzelm@26849
  1047
wenzelm@28788
  1048
  \end{description}%
wenzelm@26849
  1049
\end{isamarkuptext}%
wenzelm@26849
  1050
\isamarkuptrue%
wenzelm@26849
  1051
%
wenzelm@26849
  1052
\isamarkupsection{Executable code%
wenzelm@26849
  1053
}
wenzelm@26849
  1054
\isamarkuptrue%
wenzelm@26849
  1055
%
wenzelm@26849
  1056
\begin{isamarkuptext}%
wenzelm@26849
  1057
Isabelle/Pure provides two generic frameworks to support code
wenzelm@26849
  1058
  generation from executable specifications.  Isabelle/HOL
wenzelm@26849
  1059
  instantiates these mechanisms in a way that is amenable to end-user
wenzelm@26849
  1060
  applications.
wenzelm@26849
  1061
haftmann@37422
  1062
  \medskip One framework generates code from functional programs
haftmann@37422
  1063
  (including overloading using type classes) to SML \cite{SML}, OCaml
haftmann@38813
  1064
  \cite{OCaml}, Haskell \cite{haskell-revised-report} and Scala
haftmann@38813
  1065
  \cite{scala-overview-tech-report}.
haftmann@37422
  1066
  Conceptually, code generation is split up in three steps:
haftmann@37422
  1067
  \emph{selection} of code theorems, \emph{translation} into an
haftmann@37422
  1068
  abstract executable view and \emph{serialization} to a specific
haftmann@37422
  1069
  \emph{target language}.  Inductive specifications can be executed
haftmann@37422
  1070
  using the predicate compiler which operates within HOL.
haftmann@37422
  1071
  See \cite{isabelle-codegen} for an introduction.
haftmann@37422
  1072
haftmann@37422
  1073
  \begin{matharray}{rcl}
haftmann@37422
  1074
    \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}} \\
haftmann@37422
  1075
    \indexdef{HOL}{attribute}{code}\hypertarget{attribute.HOL.code}{\hyperlink{attribute.HOL.code}{\mbox{\isa{code}}}} & : & \isa{attribute} \\
haftmann@37422
  1076
    \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}} \\
haftmann@37422
  1077
    \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}} \\
haftmann@37422
  1078
    \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@37422
  1079
    \indexdef{HOL}{attribute}{code\_inline}\hypertarget{attribute.HOL.code-inline}{\hyperlink{attribute.HOL.code-inline}{\mbox{\isa{code{\isacharunderscore}inline}}}} & : & \isa{attribute} \\
haftmann@37422
  1080
    \indexdef{HOL}{attribute}{code\_post}\hypertarget{attribute.HOL.code-post}{\hyperlink{attribute.HOL.code-post}{\mbox{\isa{code{\isacharunderscore}post}}}} & : & \isa{attribute} \\
haftmann@37422
  1081
    \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}} \\
haftmann@37422
  1082
    \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}} \\
haftmann@37422
  1083
    \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}} \\
haftmann@37422
  1084
    \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}} \\
haftmann@37422
  1085
    \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}} \\
haftmann@37422
  1086
    \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}} \\
haftmann@37422
  1087
    \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}} \\
haftmann@37422
  1088
    \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}} \\
haftmann@37422
  1089
    \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}} \\
haftmann@37422
  1090
    \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}} \\
haftmann@37422
  1091
    \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}} \\
haftmann@39608
  1092
    \indexdef{HOL}{command}{code\_reflect}\hypertarget{command.HOL.code-reflect}{\hyperlink{command.HOL.code-reflect}{\mbox{\isa{\isacommand{code{\isacharunderscore}reflect}}}}} & : & \isa{{\isachardoublequote}theory\ {\isasymrightarrow}\ theory{\isachardoublequote}}
haftmann@37422
  1093
  \end{matharray}
haftmann@37422
  1094
haftmann@37422
  1095
  \begin{rail}
wenzelm@40255
  1096
     'export_code' ( constexpr + ) \\
wenzelm@40255
  1097
       ( ( 'in' target ( 'module_name' string ) ? \\
haftmann@37820
  1098
        ( 'file' ( string | '-' ) ) ? ( '(' args ')' ) ?) + ) ?
haftmann@37422
  1099
    ;
haftmann@37422
  1100
haftmann@37422
  1101
    const: term
haftmann@37422
  1102
    ;
haftmann@37422
  1103
haftmann@37422
  1104
    constexpr: ( const | 'name.*' | '*' )
haftmann@37422
  1105
    ;
haftmann@37422
  1106
haftmann@37422
  1107
    typeconstructor: nameref
haftmann@37422
  1108
    ;
haftmann@37422
  1109
haftmann@37422
  1110
    class: nameref
haftmann@37422
  1111
    ;
haftmann@37422
  1112
haftmann@38813
  1113
    target: 'SML' | 'OCaml' | 'Haskell' | 'Scala'
haftmann@37422
  1114
    ;
haftmann@37422
  1115
haftmann@38462
  1116
    'code' ( 'del' | 'abstype' | 'abstract' ) ?
haftmann@37422
  1117
    ;
haftmann@37422
  1118
wenzelm@40255
  1119
    'code_abort' ( const + )
haftmann@37422
  1120
    ;
haftmann@37422
  1121
wenzelm@40255
  1122
    'code_datatype' ( const + )
haftmann@37422
  1123
    ;
haftmann@37422
  1124
haftmann@37422
  1125
    'code_inline' ( 'del' ) ?
haftmann@37422
  1126
    ;
haftmann@37422
  1127
haftmann@37422
  1128
    'code_post' ( 'del' ) ?
haftmann@37422
  1129
    ;
haftmann@37422
  1130
wenzelm@40255
  1131
    'code_thms' ( constexpr + ) ?
haftmann@37422
  1132
    ;
haftmann@37422
  1133
wenzelm@40255
  1134
    'code_deps' ( constexpr + ) ?
haftmann@37422
  1135
    ;
haftmann@37422
  1136
wenzelm@40255
  1137
    'code_const' (const + 'and') \\
haftmann@37422
  1138
      ( ( '(' target ( syntax ? + 'and' ) ')' ) + )
haftmann@37422
  1139
    ;
haftmann@37422
  1140
wenzelm@40255
  1141
    'code_type' (typeconstructor + 'and') \\
haftmann@37422
  1142
      ( ( '(' target ( syntax ? + 'and' ) ')' ) + )
haftmann@37422
  1143
    ;
haftmann@37422
  1144
wenzelm@40255
  1145
    'code_class' (class + 'and') \\
haftmann@37422
  1146
      ( ( '(' target \\ ( string ? + 'and' ) ')' ) + )
haftmann@37422
  1147
    ;
haftmann@37422
  1148
wenzelm@40255
  1149
    'code_instance' (( typeconstructor '::' class ) + 'and') \\
haftmann@37422
  1150
      ( ( '(' target ( '-' ? + 'and' ) ')' ) + )
haftmann@37422
  1151
    ;
haftmann@37422
  1152
wenzelm@40255
  1153
    'code_reserved' target ( string + )
haftmann@37422
  1154
    ;
haftmann@37422
  1155
wenzelm@40255
  1156
    'code_monad' const const target
haftmann@37422
  1157
    ;
haftmann@37422
  1158
wenzelm@40255
  1159
    'code_include' target ( string ( string | '-') )
haftmann@37422
  1160
    ;
haftmann@37422
  1161
wenzelm@40255
  1162
    'code_modulename' target ( ( string string ) + )
haftmann@37422
  1163
    ;
haftmann@37422
  1164
wenzelm@40255
  1165
    'code_reflect' string ( 'datatypes' ( string '=' ( string + '|' ) + 'and' ) ) ? \\
haftmann@39608
  1166
      ( 'functions' ( string + ) ) ? ( 'file' string ) ?
haftmann@39608
  1167
    ;
haftmann@39608
  1168
haftmann@37422
  1169
    syntax: string | ( 'infix' | 'infixl' | 'infixr' ) nat string
haftmann@37422
  1170
    ;
haftmann@37422
  1171
haftmann@37422
  1172
  \end{rail}
haftmann@37422
  1173
haftmann@37422
  1174
  \begin{description}
haftmann@37422
  1175
haftmann@37422
  1176
  \item \hyperlink{command.HOL.export-code}{\mbox{\isa{\isacommand{export{\isacharunderscore}code}}}} generates code for a given list
haftmann@39608
  1177
  of constants in the specified target language(s).  If no
haftmann@39608
  1178
  serialization instruction is given, only abstract code is generated
haftmann@39608
  1179
  internally.
haftmann@37422
  1180
haftmann@37422
  1181
  Constants may be specified by giving them literally, referring to
haftmann@37422
  1182
  all executable contants within a certain theory by giving \isa{{\isachardoublequote}name{\isachardot}{\isacharasterisk}{\isachardoublequote}}, or referring to \emph{all} executable constants currently
haftmann@37422
  1183
  available by giving \isa{{\isachardoublequote}{\isacharasterisk}{\isachardoublequote}}.
haftmann@37422
  1184
haftmann@37422
  1185
  By default, for each involved theory one corresponding name space
haftmann@37422
  1186
  module is generated.  Alternativly, a module name may be specified
haftmann@37422
  1187
  after the \hyperlink{keyword.module-name}{\mbox{\isa{\isakeyword{module{\isacharunderscore}name}}}} keyword; then \emph{all} code is
haftmann@37422
  1188
  placed in this module.
haftmann@37422
  1189
haftmann@39608
  1190
  For \emph{SML}, \emph{OCaml} and \emph{Scala} the file specification
haftmann@39608
  1191
  refers to a single file; for \emph{Haskell}, it refers to a whole
haftmann@39608
  1192
  directory, where code is generated in multiple files reflecting the
haftmann@39608
  1193
  module hierarchy.  Omitting the file specification denotes standard
haftmann@37749
  1194
  output.
haftmann@37422
  1195
haftmann@37422
  1196
  Serializers take an optional list of arguments in parentheses.  For
haftmann@37422
  1197
  \emph{SML} and \emph{OCaml}, ``\isa{no{\isacharunderscore}signatures}`` omits
haftmann@37422
  1198
  explicit module signatures.
haftmann@37422
  1199
  
haftmann@39608
  1200
  For \emph{Haskell} a module name prefix may be given using the
haftmann@39608
  1201
  ``\isa{{\isachardoublequote}root{\isacharcolon}{\isachardoublequote}}'' argument; ``\isa{string{\isacharunderscore}classes}'' adds a
haftmann@39608
  1202
  ``\verb|deriving (Read, Show)|'' clause to each appropriate
haftmann@39608
  1203
  datatype declaration.
haftmann@37422
  1204
haftmann@37422
  1205
  \item \hyperlink{attribute.HOL.code}{\mbox{\isa{code}}} explicitly selects (or with option
haftmann@38462
  1206
  ``\isa{{\isachardoublequote}del{\isachardoublequote}}'' deselects) a code equation for code generation.
haftmann@38462
  1207
  Usually packages introducing code equations provide a reasonable
haftmann@38462
  1208
  default setup for selection.  Variants \isa{{\isachardoublequote}code\ abstype{\isachardoublequote}} and
haftmann@38462
  1209
  \isa{{\isachardoublequote}code\ abstract{\isachardoublequote}} declare abstract datatype certificates or
haftmann@38462
  1210
  code equations on abstract datatype representations respectively.
haftmann@37422
  1211
haftmann@37422
  1212
  \item \hyperlink{command.HOL.code-abort}{\mbox{\isa{\isacommand{code{\isacharunderscore}abort}}}} declares constants which are not
haftmann@39608
  1213
  required to have a definition by means of code equations; if needed
haftmann@39608
  1214
  these are implemented by program abort instead.
haftmann@37422
  1215
haftmann@37422
  1216
  \item \hyperlink{command.HOL.code-datatype}{\mbox{\isa{\isacommand{code{\isacharunderscore}datatype}}}} specifies a constructor set
haftmann@37422
  1217
  for a logical type.
haftmann@37422
  1218
haftmann@37422
  1219
  \item \hyperlink{command.HOL.print-codesetup}{\mbox{\isa{\isacommand{print{\isacharunderscore}codesetup}}}} gives an overview on
haftmann@37422
  1220
  selected code equations and code generator datatypes.
haftmann@37422
  1221
haftmann@39608
  1222
  \item \hyperlink{attribute.HOL.code-inline}{\mbox{\isa{code{\isacharunderscore}inline}}} declares (or with option
haftmann@39608
  1223
  ``\isa{{\isachardoublequote}del{\isachardoublequote}}'' removes) inlining theorems which are applied as
haftmann@39608
  1224
  rewrite rules to any code equation during preprocessing.
haftmann@37422
  1225
haftmann@39608
  1226
  \item \hyperlink{attribute.HOL.code-post}{\mbox{\isa{code{\isacharunderscore}post}}} declares (or with option ``\isa{{\isachardoublequote}del{\isachardoublequote}}'' removes) theorems which are applied as rewrite rules to any
haftmann@39608
  1227
  result of an evaluation.
haftmann@37422
  1228
haftmann@39608
  1229
  \item \hyperlink{command.HOL.print-codeproc}{\mbox{\isa{\isacommand{print{\isacharunderscore}codeproc}}}} prints the setup of the code
haftmann@39608
  1230
  generator preprocessor.
haftmann@37422
  1231
haftmann@37422
  1232
  \item \hyperlink{command.HOL.code-thms}{\mbox{\isa{\isacommand{code{\isacharunderscore}thms}}}} prints a list of theorems
haftmann@37422
  1233
  representing the corresponding program containing all given
haftmann@37422
  1234
  constants after preprocessing.
haftmann@37422
  1235
haftmann@37422
  1236
  \item \hyperlink{command.HOL.code-deps}{\mbox{\isa{\isacommand{code{\isacharunderscore}deps}}}} visualizes dependencies of
haftmann@37422
  1237
  theorems representing the corresponding program containing all given
haftmann@37422
  1238
  constants after preprocessing.
haftmann@37422
  1239
haftmann@37422
  1240
  \item \hyperlink{command.HOL.code-const}{\mbox{\isa{\isacommand{code{\isacharunderscore}const}}}} associates a list of constants
haftmann@37422
  1241
  with target-specific serializations; omitting a serialization
haftmann@37422
  1242
  deletes an existing serialization.
haftmann@37422
  1243
haftmann@37422
  1244
  \item \hyperlink{command.HOL.code-type}{\mbox{\isa{\isacommand{code{\isacharunderscore}type}}}} associates a list of type
haftmann@37422
  1245
  constructors with target-specific serializations; omitting a
haftmann@37422
  1246
  serialization deletes an existing serialization.
haftmann@37422
  1247
haftmann@37422
  1248
  \item \hyperlink{command.HOL.code-class}{\mbox{\isa{\isacommand{code{\isacharunderscore}class}}}} associates a list of classes
haftmann@37422
  1249
  with target-specific class names; omitting a serialization deletes
haftmann@37422
  1250
  an existing serialization.  This applies only to \emph{Haskell}.
haftmann@37422
  1251
haftmann@37422
  1252
  \item \hyperlink{command.HOL.code-instance}{\mbox{\isa{\isacommand{code{\isacharunderscore}instance}}}} declares a list of type
haftmann@37422
  1253
  constructor / class instance relations as ``already present'' for a
haftmann@37422
  1254
  given target.  Omitting a ``\isa{{\isachardoublequote}{\isacharminus}{\isachardoublequote}}'' deletes an existing
haftmann@37422
  1255
  ``already present'' declaration.  This applies only to
haftmann@37422
  1256
  \emph{Haskell}.
haftmann@37422
  1257
haftmann@37422
  1258
  \item \hyperlink{command.HOL.code-reserved}{\mbox{\isa{\isacommand{code{\isacharunderscore}reserved}}}} declares a list of names as
haftmann@37422
  1259
  reserved for a given target, preventing it to be shadowed by any
haftmann@37422
  1260
  generated code.
haftmann@37422
  1261
haftmann@37422
  1262
  \item \hyperlink{command.HOL.code-monad}{\mbox{\isa{\isacommand{code{\isacharunderscore}monad}}}} provides an auxiliary mechanism
haftmann@37422
  1263
  to generate monadic code for Haskell.
haftmann@37422
  1264
haftmann@37422
  1265
  \item \hyperlink{command.HOL.code-include}{\mbox{\isa{\isacommand{code{\isacharunderscore}include}}}} adds arbitrary named content
haftmann@37422
  1266
  (``include'') to generated code.  A ``\isa{{\isachardoublequote}{\isacharminus}{\isachardoublequote}}'' as last argument
haftmann@37422
  1267
  will remove an already added ``include''.
haftmann@37422
  1268
haftmann@37422
  1269
  \item \hyperlink{command.HOL.code-modulename}{\mbox{\isa{\isacommand{code{\isacharunderscore}modulename}}}} declares aliasings from one
haftmann@37422
  1270
  module name onto another.
haftmann@37422
  1271
haftmann@39608
  1272
  \item \hyperlink{command.HOL.code-reflect}{\mbox{\isa{\isacommand{code{\isacharunderscore}reflect}}}} without a ``\isa{{\isachardoublequote}file{\isachardoublequote}}''
haftmann@39608
  1273
  argument compiles code into the system runtime environment and
haftmann@39608
  1274
  modifies the code generator setup that future invocations of system
haftmann@39608
  1275
  runtime code generation referring to one of the ``\isa{{\isachardoublequote}datatypes{\isachardoublequote}}'' or ``\isa{{\isachardoublequote}functions{\isachardoublequote}}'' entities use these precompiled
haftmann@39608
  1276
  entities.  With a ``\isa{{\isachardoublequote}file{\isachardoublequote}}'' argument, the corresponding code
haftmann@39608
  1277
  is generated into that specified file without modifying the code
haftmann@39608
  1278
  generator setup.
haftmann@39608
  1279
haftmann@37422
  1280
  \end{description}
haftmann@37422
  1281
haftmann@39608
  1282
  The other framework generates code from both functional and
haftmann@39608
  1283
  relational programs to SML.  See \cite{isabelle-HOL} for further
haftmann@39608
  1284
  information (this actually covers the new-style theory format as
haftmann@39608
  1285
  well).
wenzelm@26849
  1286
wenzelm@26849
  1287
  \begin{matharray}{rcl}
wenzelm@28788
  1288
    \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
  1289
    \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
  1290
    \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
  1291
    \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
  1292
    \indexdef{HOL}{attribute}{code}\hypertarget{attribute.HOL.code}{\hyperlink{attribute.HOL.code}{\mbox{\isa{code}}}} & : & \isa{attribute} \\
wenzelm@26849
  1293
  \end{matharray}
wenzelm@26849
  1294
wenzelm@26849
  1295
  \begin{rail}
wenzelm@40255
  1296
  ( 'code_module' | 'code_library' ) modespec ? name ? \\
wenzelm@26849
  1297
    ( 'file' name ) ? ( 'imports' ( name + ) ) ? \\
wenzelm@26849
  1298
    'contains' ( ( name '=' term ) + | term + )
wenzelm@26849
  1299
  ;
wenzelm@26849
  1300
wenzelm@26849
  1301
  modespec: '(' ( name * ) ')'
wenzelm@26849
  1302
  ;
wenzelm@26849
  1303
wenzelm@40255
  1304
  'consts_code' (codespec +)
wenzelm@26849
  1305
  ;
wenzelm@26849
  1306
wenzelm@26849
  1307
  codespec: const template attachment ?
wenzelm@26849
  1308
  ;
wenzelm@26849
  1309
wenzelm@40255
  1310
  'types_code' (tycodespec +)
wenzelm@26849
  1311
  ;
wenzelm@26849
  1312
wenzelm@26849
  1313
  tycodespec: name template attachment ?
wenzelm@26849
  1314
  ;
wenzelm@26849
  1315
wenzelm@26849
  1316
  const: term
wenzelm@26849
  1317
  ;
wenzelm@26849
  1318
wenzelm@26849
  1319
  template: '(' string ')'
wenzelm@26849
  1320
  ;
wenzelm@26849
  1321
wenzelm@26849
  1322
  attachment: 'attach' modespec ? verblbrace text verbrbrace
wenzelm@26849
  1323
  ;
wenzelm@26849
  1324
wenzelm@26849
  1325
  'code' (name)?
wenzelm@26849
  1326
  ;
haftmann@37422
  1327
  \end{rail}%
wenzelm@26849
  1328
\end{isamarkuptext}%
wenzelm@26849
  1329
\isamarkuptrue%
wenzelm@26849
  1330
%
wenzelm@27047
  1331
\isamarkupsection{Definition by specification \label{sec:hol-specification}%
wenzelm@27047
  1332
}
wenzelm@27047
  1333
\isamarkuptrue%
wenzelm@27047
  1334
%
wenzelm@27047
  1335
\begin{isamarkuptext}%
wenzelm@27047
  1336
\begin{matharray}{rcl}
wenzelm@28788
  1337
    \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
  1338
    \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
  1339
  \end{matharray}
wenzelm@27047
  1340
wenzelm@27047
  1341
  \begin{rail}
wenzelm@40255
  1342
  ('specification' | 'ax_specification') '(' (decl +) ')' \\ (thmdecl? prop +)
wenzelm@27047
  1343
  ;
wenzelm@27047
  1344
  decl: ((name ':')? term '(' 'overloaded' ')'?)
wenzelm@27047
  1345
  \end{rail}
wenzelm@27047
  1346
wenzelm@28788
  1347
  \begin{description}
wenzelm@27047
  1348
wenzelm@28788
  1349
  \item \hyperlink{command.HOL.specification}{\mbox{\isa{\isacommand{specification}}}}~\isa{{\isachardoublequote}decls\ {\isasymphi}{\isachardoublequote}} sets up a
wenzelm@27047
  1350
  goal stating the existence of terms with the properties specified to
wenzelm@27047
  1351
  hold for the constants given in \isa{decls}.  After finishing the
wenzelm@27047
  1352
  proof, the theory will be augmented with definitions for the given
wenzelm@27047
  1353
  constants, as well as with theorems stating the properties for these
wenzelm@27047
  1354
  constants.
wenzelm@27047
  1355
wenzelm@28788
  1356
  \item \hyperlink{command.HOL.ax-specification}{\mbox{\isa{\isacommand{ax{\isacharunderscore}specification}}}}~\isa{{\isachardoublequote}decls\ {\isasymphi}{\isachardoublequote}} sets up
wenzelm@28788
  1357
  a goal stating the existence of terms with the properties specified
wenzelm@28788
  1358
  to hold for the constants given in \isa{decls}.  After finishing
wenzelm@28788
  1359
  the proof, the theory will be augmented with axioms expressing the
wenzelm@28788
  1360
  properties given in the first place.
wenzelm@27047
  1361
wenzelm@28788
  1362
  \item \isa{decl} declares a constant to be defined by the
wenzelm@27047
  1363
  specification given.  The definition for the constant \isa{c} is
wenzelm@27047
  1364
  bound to the name \isa{c{\isacharunderscore}def} unless a theorem name is given in
wenzelm@27047
  1365
  the declaration.  Overloaded constants should be declared as such.
wenzelm@27047
  1366
wenzelm@28788
  1367
  \end{description}
wenzelm@27047
  1368
wenzelm@27047
  1369
  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
  1370
  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
  1371
  user has explicitly proven it to be safe.  A practical issue must be
wenzelm@27047
  1372
  considered, though: After introducing two constants with the same
wenzelm@27047
  1373
  properties using \hyperlink{command.HOL.specification}{\mbox{\isa{\isacommand{specification}}}}, one can prove
wenzelm@27047
  1374
  that the two constants are, in fact, equal.  If this might be a
wenzelm@27047
  1375
  problem, one should use \hyperlink{command.HOL.ax-specification}{\mbox{\isa{\isacommand{ax{\isacharunderscore}specification}}}}.%
wenzelm@27047
  1376
\end{isamarkuptext}%
wenzelm@27047
  1377
\isamarkuptrue%
wenzelm@27047
  1378
%
wenzelm@26849
  1379
\isadelimtheory
wenzelm@26849
  1380
%
wenzelm@26849
  1381
\endisadelimtheory
wenzelm@26849
  1382
%
wenzelm@26849
  1383
\isatagtheory
wenzelm@26840
  1384
\isacommand{end}\isamarkupfalse%
wenzelm@26840
  1385
%
wenzelm@26840
  1386
\endisatagtheory
wenzelm@26840
  1387
{\isafoldtheory}%
wenzelm@26840
  1388
%
wenzelm@26840
  1389
\isadelimtheory
wenzelm@26840
  1390
%
wenzelm@26840
  1391
\endisadelimtheory
wenzelm@26849
  1392
\isanewline
wenzelm@26840
  1393
\end{isabellebody}%
wenzelm@26840
  1394
%%% Local Variables:
wenzelm@26840
  1395
%%% mode: latex
wenzelm@26840
  1396
%%% TeX-master: "root"
wenzelm@26840
  1397
%%% End: