wenzelm@26840: %
wenzelm@26840: \begin{isabellebody}%
wenzelm@26840: \def\isabellecontext{HOL{\isacharunderscore}Specific}%
wenzelm@26840: %
wenzelm@26840: \isadelimtheory
wenzelm@26840: \isanewline
wenzelm@26840: \isanewline
wenzelm@26840: %
wenzelm@26840: \endisadelimtheory
wenzelm@26840: %
wenzelm@26840: \isatagtheory
wenzelm@26840: \isacommand{theory}\isamarkupfalse%
wenzelm@26840: \ HOL{\isacharunderscore}Specific\isanewline
wenzelm@26849: \isakeyword{imports}\ Main\isanewline
wenzelm@26849: \isakeyword{begin}%
wenzelm@26849: \endisatagtheory
wenzelm@26849: {\isafoldtheory}%
wenzelm@26849: %
wenzelm@26849: \isadelimtheory
wenzelm@26849: %
wenzelm@26849: \endisadelimtheory
wenzelm@26849: %
wenzelm@26852: \isamarkupchapter{Isabelle/HOL \label{ch:hol}%
wenzelm@26849: }
wenzelm@26849: \isamarkuptrue%
wenzelm@26849: %
wenzelm@26849: \isamarkupsection{Primitive types \label{sec:hol-typedef}%
wenzelm@26849: }
wenzelm@26849: \isamarkuptrue%
wenzelm@26849: %
wenzelm@26849: \begin{isamarkuptext}%
wenzelm@26849: \begin{matharray}{rcl}
wenzelm@26902:     \indexdef{HOL}{command}{typedecl}\hypertarget{command.HOL.typedecl}{\hyperlink{command.HOL.typedecl}{\mbox{\isa{\isacommand{typedecl}}}}} & : & \isartrans{theory}{theory} \\
wenzelm@26902:     \indexdef{HOL}{command}{typedef}\hypertarget{command.HOL.typedef}{\hyperlink{command.HOL.typedef}{\mbox{\isa{\isacommand{typedef}}}}} & : & \isartrans{theory}{proof(prove)} \\
wenzelm@26849:   \end{matharray}
wenzelm@26849: 
wenzelm@26849:   \begin{rail}
wenzelm@26849:     'typedecl' typespec infix?
wenzelm@26849:     ;
wenzelm@26849:     'typedef' altname? abstype '=' repset
wenzelm@26849:     ;
wenzelm@26849: 
wenzelm@26849:     altname: '(' (name | 'open' | 'open' name) ')'
wenzelm@26849:     ;
wenzelm@26849:     abstype: typespec infix?
wenzelm@26849:     ;
wenzelm@26849:     repset: term ('morphisms' name name)?
wenzelm@26849:     ;
wenzelm@26849:   \end{rail}
wenzelm@26849: 
wenzelm@26849:   \begin{descr}
wenzelm@26849:   
wenzelm@26902:   \item [\hyperlink{command.HOL.typedecl}{\mbox{\isa{\isacommand{typedecl}}}}~\isa{{\isachardoublequote}{\isacharparenleft}{\isasymalpha}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymalpha}\isactrlsub n{\isacharparenright}\ t{\isachardoublequote}}] is similar to the original \hyperlink{command.typedecl}{\mbox{\isa{\isacommand{typedecl}}}} of
wenzelm@26849:   Isabelle/Pure (see \secref{sec:types-pure}), but also declares type
wenzelm@26849:   arity \isa{{\isachardoublequote}t\ {\isacharcolon}{\isacharcolon}\ {\isacharparenleft}type{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ type{\isacharparenright}\ type{\isachardoublequote}}, making \isa{t} an
wenzelm@26849:   actual HOL type constructor.   %FIXME check, update
wenzelm@26849:   
wenzelm@26902:   \item [\hyperlink{command.HOL.typedef}{\mbox{\isa{\isacommand{typedef}}}}~\isa{{\isachardoublequote}{\isacharparenleft}{\isasymalpha}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymalpha}\isactrlsub n{\isacharparenright}\ t\ {\isacharequal}\ A{\isachardoublequote}}] sets up a goal stating non-emptiness of the set \isa{A}.
wenzelm@26849:   After finishing the proof, the theory will be augmented by a
wenzelm@26849:   Gordon/HOL-style type definition, which establishes a bijection
wenzelm@26849:   between the representing set \isa{A} and the new type \isa{t}.
wenzelm@26849:   
wenzelm@26902:   Technically, \hyperlink{command.HOL.typedef}{\mbox{\isa{\isacommand{typedef}}}} defines both a type \isa{t} and a set (term constant) of the same name (an alternative base
wenzelm@26849:   name may be given in parentheses).  The injection from type to set
wenzelm@26849:   is called \isa{Rep{\isacharunderscore}t}, its inverse \isa{Abs{\isacharunderscore}t} (this may be
wenzelm@26902:   changed via an explicit \hyperlink{keyword.HOL.morphisms}{\mbox{\isa{\isakeyword{morphisms}}}} declaration).
wenzelm@26849:   
wenzelm@26849:   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
wenzelm@26849:   corresponding injection/surjection pair (in both directions).  Rules
wenzelm@26849:   \isa{Rep{\isacharunderscore}t{\isacharunderscore}inject} and \isa{Abs{\isacharunderscore}t{\isacharunderscore}inject} provide a slightly
wenzelm@26849:   more convenient view on the injectivity part, suitable for automated
wenzelm@26902:   proof tools (e.g.\ in \hyperlink{attribute.simp}{\mbox{\isa{simp}}} or \hyperlink{attribute.iff}{\mbox{\isa{iff}}}
wenzelm@26895:   declarations).  Rules \isa{Rep{\isacharunderscore}t{\isacharunderscore}cases}/\isa{Rep{\isacharunderscore}t{\isacharunderscore}induct}, and
wenzelm@26895:   \isa{Abs{\isacharunderscore}t{\isacharunderscore}cases}/\isa{Abs{\isacharunderscore}t{\isacharunderscore}induct} provide alternative views
wenzelm@26895:   on surjectivity; these are already declared as set or type rules for
wenzelm@26902:   the generic \hyperlink{method.cases}{\mbox{\isa{cases}}} and \hyperlink{method.induct}{\mbox{\isa{induct}}} methods.
wenzelm@26849:   
wenzelm@26849:   An alternative name may be specified in parentheses; the default is
wenzelm@26849:   to use \isa{t} as indicated before.  The ``\isa{{\isachardoublequote}{\isacharparenleft}open{\isacharparenright}{\isachardoublequote}}''
wenzelm@26849:   declaration suppresses a separate constant definition for the
wenzelm@26849:   representing set.
wenzelm@26849: 
wenzelm@26849:   \end{descr}
wenzelm@26849: 
wenzelm@26849:   Note that raw type declarations are rarely used in practice; the
wenzelm@26849:   main application is with experimental (or even axiomatic!) theory
wenzelm@26849:   fragments.  Instead of primitive HOL type definitions, user-level
wenzelm@26902:   theories usually refer to higher-level packages such as \hyperlink{command.HOL.record}{\mbox{\isa{\isacommand{record}}}} (see \secref{sec:hol-record}) or \hyperlink{command.HOL.datatype}{\mbox{\isa{\isacommand{datatype}}}} (see \secref{sec:hol-datatype}).%
wenzelm@26849: \end{isamarkuptext}%
wenzelm@26849: \isamarkuptrue%
wenzelm@26849: %
wenzelm@26849: \isamarkupsection{Adhoc tuples%
wenzelm@26849: }
wenzelm@26849: \isamarkuptrue%
wenzelm@26849: %
wenzelm@26849: \begin{isamarkuptext}%
wenzelm@26849: \begin{matharray}{rcl}
wenzelm@26907:     \hyperlink{attribute.HOL.split-format}{\mbox{\isa{split{\isacharunderscore}format}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isaratt \\
wenzelm@26849:   \end{matharray}
wenzelm@26849: 
wenzelm@26849:   \begin{rail}
wenzelm@26849:     'split\_format' (((name *) + 'and') | ('(' 'complete' ')'))
wenzelm@26849:     ;
wenzelm@26849:   \end{rail}
wenzelm@26849: 
wenzelm@26849:   \begin{descr}
wenzelm@26849:   
wenzelm@26907:   \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
wenzelm@26849:   low-level tuple types into canonical form as specified by the
wenzelm@26849:   arguments given; the \isa{i}-th collection of arguments refers to
wenzelm@26849:   occurrences in premise \isa{i} of the rule.  The ``\isa{{\isachardoublequote}{\isacharparenleft}complete{\isacharparenright}{\isachardoublequote}}'' option causes \emph{all} arguments in function
wenzelm@26849:   applications to be represented canonically according to their tuple
wenzelm@26849:   type structure.
wenzelm@26849: 
wenzelm@26849:   Note that these operations tend to invent funny names for new local
wenzelm@26849:   parameters to be introduced.
wenzelm@26849: 
wenzelm@26849:   \end{descr}%
wenzelm@26849: \end{isamarkuptext}%
wenzelm@26849: \isamarkuptrue%
wenzelm@26849: %
wenzelm@26849: \isamarkupsection{Records \label{sec:hol-record}%
wenzelm@26849: }
wenzelm@26849: \isamarkuptrue%
wenzelm@26849: %
wenzelm@26849: \begin{isamarkuptext}%
wenzelm@26849: In principle, records merely generalize the concept of tuples, where
wenzelm@26849:   components may be addressed by labels instead of just position.  The
wenzelm@26849:   logical infrastructure of records in Isabelle/HOL is slightly more
wenzelm@26849:   advanced, though, supporting truly extensible record schemes.  This
wenzelm@26849:   admits operations that are polymorphic with respect to record
wenzelm@26849:   extension, yielding ``object-oriented'' effects like (single)
wenzelm@26849:   inheritance.  See also \cite{NaraschewskiW-TPHOLs98} for more
wenzelm@26849:   details on object-oriented verification and record subtyping in HOL.%
wenzelm@26849: \end{isamarkuptext}%
wenzelm@26849: \isamarkuptrue%
wenzelm@26849: %
wenzelm@26849: \isamarkupsubsection{Basic concepts%
wenzelm@26849: }
wenzelm@26849: \isamarkuptrue%
wenzelm@26849: %
wenzelm@26849: \begin{isamarkuptext}%
wenzelm@26849: Isabelle/HOL supports both \emph{fixed} and \emph{schematic} records
wenzelm@26849:   at the level of terms and types.  The notation is as follows:
wenzelm@26849: 
wenzelm@26849:   \begin{center}
wenzelm@26849:   \begin{tabular}{l|l|l}
wenzelm@26849:     & record terms & record types \\ \hline
wenzelm@26849:     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}} \\
wenzelm@26849:     schematic & \isa{{\isachardoublequote}{\isasymlparr}x\ {\isacharequal}\ a{\isacharcomma}\ y\ {\isacharequal}\ b{\isacharcomma}\ {\isasymdots}\ {\isacharequal}\ m{\isasymrparr}{\isachardoublequote}} &
wenzelm@26849:       \isa{{\isachardoublequote}{\isasymlparr}x\ {\isacharcolon}{\isacharcolon}\ A{\isacharcomma}\ y\ {\isacharcolon}{\isacharcolon}\ B{\isacharcomma}\ {\isasymdots}\ {\isacharcolon}{\isacharcolon}\ M{\isasymrparr}{\isachardoublequote}} \\
wenzelm@26849:   \end{tabular}
wenzelm@26849:   \end{center}
wenzelm@26849: 
wenzelm@26849:   \noindent The ASCII representation of \isa{{\isachardoublequote}{\isasymlparr}x\ {\isacharequal}\ a{\isasymrparr}{\isachardoublequote}} is \isa{{\isachardoublequote}{\isacharparenleft}{\isacharbar}\ x\ {\isacharequal}\ a\ {\isacharbar}{\isacharparenright}{\isachardoublequote}}.
wenzelm@26849: 
wenzelm@26849:   A fixed record \isa{{\isachardoublequote}{\isasymlparr}x\ {\isacharequal}\ a{\isacharcomma}\ y\ {\isacharequal}\ b{\isasymrparr}{\isachardoublequote}} has field \isa{x} of value
wenzelm@26849:   \isa{a} and field \isa{y} of value \isa{b}.  The corresponding
wenzelm@26849:   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}}
wenzelm@26849:   and \isa{{\isachardoublequote}b\ {\isacharcolon}{\isacharcolon}\ B{\isachardoublequote}}.
wenzelm@26849: 
wenzelm@26849:   A record scheme like \isa{{\isachardoublequote}{\isasymlparr}x\ {\isacharequal}\ a{\isacharcomma}\ y\ {\isacharequal}\ b{\isacharcomma}\ {\isasymdots}\ {\isacharequal}\ m{\isasymrparr}{\isachardoublequote}} contains fields
wenzelm@26849:   \isa{x} and \isa{y} as before, but also possibly further fields
wenzelm@26849:   as indicated by the ``\isa{{\isachardoublequote}{\isasymdots}{\isachardoublequote}}'' notation (which is actually part
wenzelm@26849:   of the syntax).  The improper field ``\isa{{\isachardoublequote}{\isasymdots}{\isachardoublequote}}'' of a record
wenzelm@26849:   scheme is called the \emph{more part}.  Logically it is just a free
wenzelm@26849:   variable, which is occasionally referred to as ``row variable'' in
wenzelm@26849:   the literature.  The more part of a record scheme may be
wenzelm@26849:   instantiated by zero or more further components.  For example, the
wenzelm@26852:   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.
wenzelm@26849:   Fixed records are special instances of record schemes, where
wenzelm@26849:   ``\isa{{\isachardoublequote}{\isasymdots}{\isachardoublequote}}'' is properly terminated by the \isa{{\isachardoublequote}{\isacharparenleft}{\isacharparenright}\ {\isacharcolon}{\isacharcolon}\ unit{\isachardoublequote}}
wenzelm@26849:   element.  In fact, \isa{{\isachardoublequote}{\isasymlparr}x\ {\isacharequal}\ a{\isacharcomma}\ y\ {\isacharequal}\ b{\isasymrparr}{\isachardoublequote}} is just an abbreviation
wenzelm@26849:   for \isa{{\isachardoublequote}{\isasymlparr}x\ {\isacharequal}\ a{\isacharcomma}\ y\ {\isacharequal}\ b{\isacharcomma}\ {\isasymdots}\ {\isacharequal}\ {\isacharparenleft}{\isacharparenright}{\isasymrparr}{\isachardoublequote}}.
wenzelm@26849:   
wenzelm@26849:   \medskip Two key observations make extensible records in a simply
wenzelm@26849:   typed language like HOL work out:
wenzelm@26849: 
wenzelm@26849:   \begin{enumerate}
wenzelm@26849: 
wenzelm@26849:   \item the more part is internalized, as a free term or type
wenzelm@26849:   variable,
wenzelm@26849: 
wenzelm@26852:   \item field names are externalized, they cannot be accessed within
wenzelm@26852:   the logic as first-class values.
wenzelm@26849: 
wenzelm@26849:   \end{enumerate}
wenzelm@26849: 
wenzelm@26849:   \medskip In Isabelle/HOL record types have to be defined explicitly,
wenzelm@26849:   fixing their field names and types, and their (optional) parent
wenzelm@26849:   record.  Afterwards, records may be formed using above syntax, while
wenzelm@26849:   obeying the canonical order of fields as given by their declaration.
wenzelm@26849:   The record package provides several standard operations like
wenzelm@26849:   selectors and updates.  The common setup for various generic proof
wenzelm@26849:   tools enable succinct reasoning patterns.  See also the Isabelle/HOL
wenzelm@26849:   tutorial \cite{isabelle-hol-book} for further instructions on using
wenzelm@26849:   records in practice.%
wenzelm@26849: \end{isamarkuptext}%
wenzelm@26849: \isamarkuptrue%
wenzelm@26849: %
wenzelm@26849: \isamarkupsubsection{Record specifications%
wenzelm@26849: }
wenzelm@26849: \isamarkuptrue%
wenzelm@26849: %
wenzelm@26849: \begin{isamarkuptext}%
wenzelm@26849: \begin{matharray}{rcl}
wenzelm@26902:     \indexdef{HOL}{command}{record}\hypertarget{command.HOL.record}{\hyperlink{command.HOL.record}{\mbox{\isa{\isacommand{record}}}}} & : & \isartrans{theory}{theory} \\
wenzelm@26849:   \end{matharray}
wenzelm@26849: 
wenzelm@26849:   \begin{rail}
wenzelm@26849:     'record' typespec '=' (type '+')? (constdecl +)
wenzelm@26849:     ;
wenzelm@26849:   \end{rail}
wenzelm@26849: 
wenzelm@26849:   \begin{descr}
wenzelm@26849: 
wenzelm@26902:   \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
wenzelm@26849:   extensible record type \isa{{\isachardoublequote}{\isacharparenleft}{\isasymalpha}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymalpha}\isactrlsub m{\isacharparenright}\ t{\isachardoublequote}},
wenzelm@26849:   derived from the optional parent record \isa{{\isachardoublequote}{\isasymtau}{\isachardoublequote}} by adding new
wenzelm@26849:   field components \isa{{\isachardoublequote}c\isactrlsub i\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}\isactrlsub i{\isachardoublequote}} etc.
wenzelm@26849: 
wenzelm@26849:   The type variables of \isa{{\isachardoublequote}{\isasymtau}{\isachardoublequote}} and \isa{{\isachardoublequote}{\isasymsigma}\isactrlsub i{\isachardoublequote}} need to be
wenzelm@26849:   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
wenzelm@26849:   least one new field \isa{{\isachardoublequote}c\isactrlsub i{\isachardoublequote}} has to be specified.
wenzelm@26849:   Basically, field names need to belong to a unique record.  This is
wenzelm@26849:   not a real restriction in practice, since fields are qualified by
wenzelm@26849:   the record name internally.
wenzelm@26849: 
wenzelm@26849:   The parent record specification \isa{{\isasymtau}} is optional; if omitted
wenzelm@26849:   \isa{t} becomes a root record.  The hierarchy of all records
wenzelm@26849:   declared within a theory context forms a forest structure, i.e.\ a
wenzelm@26849:   set of trees starting with a root record each.  There is no way to
wenzelm@26849:   merge multiple parent records!
wenzelm@26849: 
wenzelm@26849:   For convenience, \isa{{\isachardoublequote}{\isacharparenleft}{\isasymalpha}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymalpha}\isactrlsub m{\isacharparenright}\ t{\isachardoublequote}} is made a
wenzelm@26849:   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
wenzelm@26849:   \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}}.
wenzelm@26849: 
wenzelm@26849:   \end{descr}%
wenzelm@26849: \end{isamarkuptext}%
wenzelm@26849: \isamarkuptrue%
wenzelm@26849: %
wenzelm@26849: \isamarkupsubsection{Record operations%
wenzelm@26849: }
wenzelm@26849: \isamarkuptrue%
wenzelm@26849: %
wenzelm@26849: \begin{isamarkuptext}%
wenzelm@26849: Any record definition of the form presented above produces certain
wenzelm@26849:   standard operations.  Selectors and updates are provided for any
wenzelm@26849:   field, including the improper one ``\isa{more}''.  There are also
wenzelm@26849:   cumulative record constructor functions.  To simplify the
wenzelm@26849:   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}}.
wenzelm@26849: 
wenzelm@26849:   \medskip \textbf{Selectors} and \textbf{updates} are available for
wenzelm@26849:   any field (including ``\isa{more}''):
wenzelm@26849: 
wenzelm@26849:   \begin{matharray}{lll}
wenzelm@26852:     \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}} \\
wenzelm@26852:     \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}} \\
wenzelm@26849:   \end{matharray}
wenzelm@26849: 
wenzelm@26849:   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
wenzelm@26849:   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
wenzelm@26849:   because of postfix notation the order of fields shown here is
wenzelm@26849:   reverse than in the actual term.  Since repeated updates are just
wenzelm@26849:   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.
wenzelm@26849:   Thus commutativity of independent updates can be proven within the
wenzelm@26849:   logic for any two fields, but not as a general theorem.
wenzelm@26849: 
wenzelm@26849:   \medskip The \textbf{make} operation provides a cumulative record
wenzelm@26849:   constructor function:
wenzelm@26849: 
wenzelm@26849:   \begin{matharray}{lll}
wenzelm@26852:     \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}} \\
wenzelm@26849:   \end{matharray}
wenzelm@26849: 
wenzelm@26849:   \medskip We now reconsider the case of non-root records, which are
wenzelm@26849:   derived of some parent.  In general, the latter may depend on
wenzelm@26849:   another parent as well, resulting in a list of \emph{ancestor
wenzelm@26849:   records}.  Appending the lists of fields of all ancestors results in
wenzelm@26849:   a certain field prefix.  The record package automatically takes care
wenzelm@26849:   of this by lifting operations over this context of ancestor fields.
wenzelm@26849:   Assuming that \isa{{\isachardoublequote}{\isacharparenleft}{\isasymalpha}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymalpha}\isactrlsub m{\isacharparenright}\ t{\isachardoublequote}} has ancestor
wenzelm@26849:   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}},
wenzelm@26849:   the above record operations will get the following types:
wenzelm@26849: 
wenzelm@26852:   \medskip
wenzelm@26852:   \begin{tabular}{lll}
wenzelm@26852:     \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}} \\
wenzelm@26852:     \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}} \\
wenzelm@26852:     \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}} \\
wenzelm@26852:   \end{tabular}
wenzelm@26852:   \medskip
wenzelm@26849: 
wenzelm@26852:   \noindent Some further operations address the extension aspect of a
wenzelm@26849:   derived record scheme specifically: \isa{{\isachardoublequote}t{\isachardot}fields{\isachardoublequote}} produces a
wenzelm@26849:   record fragment consisting of exactly the new fields introduced here
wenzelm@26849:   (the result may serve as a more part elsewhere); \isa{{\isachardoublequote}t{\isachardot}extend{\isachardoublequote}}
wenzelm@26849:   takes a fixed record and adds a given more part; \isa{{\isachardoublequote}t{\isachardot}truncate{\isachardoublequote}} restricts a record scheme to a fixed record.
wenzelm@26849: 
wenzelm@26852:   \medskip
wenzelm@26852:   \begin{tabular}{lll}
wenzelm@26852:     \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}} \\
wenzelm@26852:     \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}} \\
wenzelm@26852:     \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}} \\
wenzelm@26852:   \end{tabular}
wenzelm@26852:   \medskip
wenzelm@26849: 
wenzelm@26849:   \noindent Note that \isa{{\isachardoublequote}t{\isachardot}make{\isachardoublequote}} and \isa{{\isachardoublequote}t{\isachardot}fields{\isachardoublequote}} coincide
wenzelm@26849:   for root records.%
wenzelm@26849: \end{isamarkuptext}%
wenzelm@26849: \isamarkuptrue%
wenzelm@26849: %
wenzelm@26849: \isamarkupsubsection{Derived rules and proof tools%
wenzelm@26849: }
wenzelm@26849: \isamarkuptrue%
wenzelm@26849: %
wenzelm@26849: \begin{isamarkuptext}%
wenzelm@26849: The record package proves several results internally, declaring
wenzelm@26849:   these facts to appropriate proof tools.  This enables users to
wenzelm@26849:   reason about record structures quite conveniently.  Assume that
wenzelm@26849:   \isa{t} is a record type as specified above.
wenzelm@26849: 
wenzelm@26849:   \begin{enumerate}
wenzelm@26849:   
wenzelm@26849:   \item Standard conversions for selectors or updates applied to
wenzelm@26849:   record constructor terms are made part of the default Simplifier
wenzelm@26849:   context; thus proofs by reduction of basic operations merely require
wenzelm@26902:   the \hyperlink{method.simp}{\mbox{\isa{simp}}} method without further arguments.  These rules
wenzelm@26849:   are available as \isa{{\isachardoublequote}t{\isachardot}simps{\isachardoublequote}}, too.
wenzelm@26849:   
wenzelm@26849:   \item Selectors applied to updated records are automatically reduced
wenzelm@26849:   by an internal simplification procedure, which is also part of the
wenzelm@26849:   standard Simplifier setup.
wenzelm@26849: 
wenzelm@26849:   \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
wenzelm@26902:   Reasoner as \hyperlink{attribute.iff}{\mbox{\isa{iff}}} rules.  These rules are available as
wenzelm@26849:   \isa{{\isachardoublequote}t{\isachardot}iffs{\isachardoublequote}}.
wenzelm@26849: 
wenzelm@26849:   \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,
wenzelm@26902:   and as the basic rule context as ``\hyperlink{attribute.intro}{\mbox{\isa{intro}}}\isa{{\isachardoublequote}{\isacharquery}{\isachardoublequote}}''.
wenzelm@26849:   The rule is called \isa{{\isachardoublequote}t{\isachardot}equality{\isachardoublequote}}.
wenzelm@26849: 
wenzelm@26849:   \item Representations of arbitrary record expressions as canonical
wenzelm@26902:   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,
wenzelm@26849:   \secref{sec:cases-induct}).  Several variations are available, for
wenzelm@26849:   fixed records, record schemes, more parts etc.
wenzelm@26849:   
wenzelm@26849:   The generic proof methods are sufficiently smart to pick the most
wenzelm@26849:   sensible rule according to the type of the indicated record
wenzelm@26849:   expression: users just need to apply something like ``\isa{{\isachardoublequote}{\isacharparenleft}cases\ r{\isacharparenright}{\isachardoublequote}}'' to a certain proof problem.
wenzelm@26849: 
wenzelm@26849:   \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}
wenzelm@26849:   treated automatically, but usually need to be expanded by hand,
wenzelm@26849:   using the collective fact \isa{{\isachardoublequote}t{\isachardot}defs{\isachardoublequote}}.
wenzelm@26849: 
wenzelm@26849:   \end{enumerate}%
wenzelm@26849: \end{isamarkuptext}%
wenzelm@26849: \isamarkuptrue%
wenzelm@26849: %
wenzelm@26849: \isamarkupsection{Datatypes \label{sec:hol-datatype}%
wenzelm@26849: }
wenzelm@26849: \isamarkuptrue%
wenzelm@26849: %
wenzelm@26849: \begin{isamarkuptext}%
wenzelm@26849: \begin{matharray}{rcl}
wenzelm@26902:     \indexdef{HOL}{command}{datatype}\hypertarget{command.HOL.datatype}{\hyperlink{command.HOL.datatype}{\mbox{\isa{\isacommand{datatype}}}}} & : & \isartrans{theory}{theory} \\
haftmann@27452:   \indexdef{HOL}{command}{rep\_datatype}\hypertarget{command.HOL.rep-datatype}{\hyperlink{command.HOL.rep-datatype}{\mbox{\isa{\isacommand{rep{\isacharunderscore}datatype}}}}} & : & \isartrans{theory}{proof} \\
wenzelm@26849:   \end{matharray}
wenzelm@26849: 
wenzelm@26849:   \begin{rail}
wenzelm@26849:     'datatype' (dtspec + 'and')
wenzelm@26849:     ;
haftmann@27452:     'rep\_datatype' ('(' (name +) ')')? (term +)
wenzelm@26849:     ;
wenzelm@26849: 
wenzelm@26849:     dtspec: parname? typespec infix? '=' (cons + '|')
wenzelm@26849:     ;
wenzelm@26849:     cons: name (type *) mixfix?
wenzelm@26849:   \end{rail}
wenzelm@26849: 
wenzelm@26849:   \begin{descr}
wenzelm@26849: 
wenzelm@26902:   \item [\hyperlink{command.HOL.datatype}{\mbox{\isa{\isacommand{datatype}}}}] defines inductive datatypes in
wenzelm@26849:   HOL.
wenzelm@26849: 
wenzelm@26907:   \item [\hyperlink{command.HOL.rep-datatype}{\mbox{\isa{\isacommand{rep{\isacharunderscore}datatype}}}}] represents existing types as
wenzelm@26849:   inductive ones, generating the standard infrastructure of derived
wenzelm@26849:   concepts (primitive recursion etc.).
wenzelm@26849: 
wenzelm@26849:   \end{descr}
wenzelm@26849: 
wenzelm@26849:   The induction and exhaustion theorems generated provide case names
wenzelm@26849:   according to the constructors involved, while parameters are named
wenzelm@26849:   after the types (see also \secref{sec:cases-induct}).
wenzelm@26849: 
wenzelm@26849:   See \cite{isabelle-HOL} for more details on datatypes, but beware of
wenzelm@26849:   the old-style theory syntax being used there!  Apart from proper
wenzelm@26849:   proof methods for case-analysis and induction, there are also
wenzelm@26907:   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
wenzelm@26849:   to refer directly to the internal structure of subgoals (including
wenzelm@26849:   internally bound parameters).%
wenzelm@26849: \end{isamarkuptext}%
wenzelm@26849: \isamarkuptrue%
wenzelm@26849: %
wenzelm@26849: \isamarkupsection{Recursive functions \label{sec:recursion}%
wenzelm@26849: }
wenzelm@26849: \isamarkuptrue%
wenzelm@26849: %
wenzelm@26849: \begin{isamarkuptext}%
wenzelm@26849: \begin{matharray}{rcl}
wenzelm@26902:     \indexdef{HOL}{command}{primrec}\hypertarget{command.HOL.primrec}{\hyperlink{command.HOL.primrec}{\mbox{\isa{\isacommand{primrec}}}}} & : & \isarkeep{local{\dsh}theory} \\
wenzelm@26902:     \indexdef{HOL}{command}{fun}\hypertarget{command.HOL.fun}{\hyperlink{command.HOL.fun}{\mbox{\isa{\isacommand{fun}}}}} & : & \isarkeep{local{\dsh}theory} \\
wenzelm@26902:     \indexdef{HOL}{command}{function}\hypertarget{command.HOL.function}{\hyperlink{command.HOL.function}{\mbox{\isa{\isacommand{function}}}}} & : & \isartrans{local{\dsh}theory}{proof(prove)} \\
wenzelm@26902:     \indexdef{HOL}{command}{termination}\hypertarget{command.HOL.termination}{\hyperlink{command.HOL.termination}{\mbox{\isa{\isacommand{termination}}}}} & : & \isartrans{local{\dsh}theory}{proof(prove)} \\
wenzelm@26849:   \end{matharray}
wenzelm@26849: 
wenzelm@26849:   \begin{rail}
wenzelm@26849:     'primrec' target? fixes 'where' equations
wenzelm@26849:     ;
wenzelm@26849:     equations: (thmdecl? prop + '|')
wenzelm@26849:     ;
wenzelm@26987:     ('fun' | 'function') target? functionopts? fixes 'where' clauses
wenzelm@26849:     ;
wenzelm@26849:     clauses: (thmdecl? prop ('(' 'otherwise' ')')? + '|')
wenzelm@26849:     ;
wenzelm@26987:     functionopts: '(' (('sequential' | 'domintros' | 'tailrec' | 'default' term) + ',') ')'
wenzelm@26849:     ;
wenzelm@26849:     'termination' ( term )?
wenzelm@26849:   \end{rail}
wenzelm@26849: 
wenzelm@26849:   \begin{descr}
wenzelm@26849: 
wenzelm@26902:   \item [\hyperlink{command.HOL.primrec}{\mbox{\isa{\isacommand{primrec}}}}] defines primitive recursive
wenzelm@26849:   functions over datatypes, see also \cite{isabelle-HOL}.
wenzelm@26849: 
wenzelm@26902:   \item [\hyperlink{command.HOL.function}{\mbox{\isa{\isacommand{function}}}}] defines functions by general
wenzelm@26849:   wellfounded recursion. A detailed description with examples can be
wenzelm@26849:   found in \cite{isabelle-function}. The function is specified by a
wenzelm@26849:   set of (possibly conditional) recursive equations with arbitrary
wenzelm@26849:   pattern matching. The command generates proof obligations for the
wenzelm@26849:   completeness and the compatibility of patterns.
wenzelm@26849: 
wenzelm@26849:   The defined function is considered partial, and the resulting
wenzelm@26849:   simplification rules (named \isa{{\isachardoublequote}f{\isachardot}psimps{\isachardoublequote}}) and induction rule
wenzelm@26849:   (named \isa{{\isachardoublequote}f{\isachardot}pinduct{\isachardoublequote}}) are guarded by a generated domain
wenzelm@26902:   predicate \isa{{\isachardoublequote}f{\isacharunderscore}dom{\isachardoublequote}}. The \hyperlink{command.HOL.termination}{\mbox{\isa{\isacommand{termination}}}}
wenzelm@26849:   command can then be used to establish that the function is total.
wenzelm@26849: 
wenzelm@26902:   \item [\hyperlink{command.HOL.fun}{\mbox{\isa{\isacommand{fun}}}}] is a shorthand notation for
wenzelm@26902:   ``\hyperlink{command.HOL.function}{\mbox{\isa{\isacommand{function}}}}~\isa{{\isachardoublequote}{\isacharparenleft}sequential{\isacharparenright}{\isachardoublequote}}, followed by
wenzelm@26849:   automated proof attempts regarding pattern matching and termination.
wenzelm@26849:   See \cite{isabelle-function} for further details.
wenzelm@26849: 
wenzelm@26902:   \item [\hyperlink{command.HOL.termination}{\mbox{\isa{\isacommand{termination}}}}~\isa{f}] commences a
wenzelm@26849:   termination proof for the previously defined function \isa{f}.  If
wenzelm@26849:   this is omitted, the command refers to the most recent function
wenzelm@26849:   definition.  After the proof is closed, the recursive equations and
wenzelm@26849:   the induction principle is established.
wenzelm@26849: 
wenzelm@26849:   \end{descr}
wenzelm@26849: 
wenzelm@26849:   %FIXME check
wenzelm@26849: 
haftmann@27452:   Recursive definitions introduced by the \hyperlink{command.HOL.function}{\mbox{\isa{\isacommand{function}}}}
haftmann@27452:   command accommodate
wenzelm@26849:   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:   refers to a specific induction rule, with parameters named according
haftmann@27452:   to the user-specified equations.
haftmann@27452:   For the \hyperlink{command.HOL.primrec}{\mbox{\isa{\isacommand{primrec}}}} the induction principle coincides
haftmann@27452:   with structural recursion on the datatype the recursion is carried
haftmann@27452:   out.
haftmann@27452:   Case names of \hyperlink{command.HOL.primrec}{\mbox{\isa{\isacommand{primrec}}}} are that of the datatypes involved, while those of
wenzelm@26902:   \hyperlink{command.HOL.function}{\mbox{\isa{\isacommand{function}}}} are numbered (starting from 1).
wenzelm@26849: 
wenzelm@26849:   The equations provided by these packages may be referred later as
wenzelm@26849:   theorem list \isa{{\isachardoublequote}f{\isachardot}simps{\isachardoublequote}}, where \isa{f} is the (collective)
wenzelm@26849:   name of the functions defined.  Individual equations may be named
wenzelm@26849:   explicitly as well.
wenzelm@26849: 
wenzelm@26902:   The \hyperlink{command.HOL.function}{\mbox{\isa{\isacommand{function}}}} command accepts the following
wenzelm@26849:   options.
wenzelm@26849: 
wenzelm@26849:   \begin{descr}
wenzelm@26849: 
wenzelm@26849:   \item [\isa{sequential}] enables a preprocessor which
wenzelm@26849:   disambiguates overlapping patterns by making them mutually disjoint.
wenzelm@26849:   Earlier equations take precedence over later ones.  This allows to
wenzelm@26849:   give the specification in a format very similar to functional
wenzelm@26849:   programming.  Note that the resulting simplification and induction
wenzelm@26849:   rules correspond to the transformed specification, not the one given
wenzelm@26849:   originally. This usually means that each equation given by the user
wenzelm@26849:   may result in several theroems.  Also note that this automatic
wenzelm@26849:   transformation only works for ML-style datatype patterns.
wenzelm@26849: 
wenzelm@26849:   \item [\isa{domintros}] enables the automated generation of
wenzelm@26849:   introduction rules for the domain predicate. While mostly not
wenzelm@26849:   needed, they can be helpful in some proofs about partial functions.
wenzelm@26849: 
wenzelm@26849:   \item [\isa{tailrec}] generates the unconstrained recursive
wenzelm@26849:   equations even without a termination proof, provided that the
wenzelm@26849:   function is tail-recursive. This currently only works
wenzelm@26849: 
wenzelm@26849:   \item [\isa{{\isachardoublequote}default\ d{\isachardoublequote}}] allows to specify a default value for a
wenzelm@26849:   (partial) function, which will ensure that \isa{{\isachardoublequote}f\ x\ {\isacharequal}\ d\ x{\isachardoublequote}}
wenzelm@26849:   whenever \isa{{\isachardoublequote}x\ {\isasymnotin}\ f{\isacharunderscore}dom{\isachardoublequote}}.
wenzelm@26849: 
wenzelm@26849:   \end{descr}%
wenzelm@26849: \end{isamarkuptext}%
wenzelm@26849: \isamarkuptrue%
wenzelm@26849: %
wenzelm@26849: \isamarkupsubsection{Proof methods related to recursive definitions%
wenzelm@26849: }
wenzelm@26849: \isamarkuptrue%
wenzelm@26849: %
wenzelm@26849: \begin{isamarkuptext}%
wenzelm@26849: \begin{matharray}{rcl}
wenzelm@26907:     \indexdef{HOL}{method}{pat\_completeness}\hypertarget{method.HOL.pat-completeness}{\hyperlink{method.HOL.pat-completeness}{\mbox{\isa{pat{\isacharunderscore}completeness}}}} & : & \isarmeth \\
wenzelm@26902:     \indexdef{HOL}{method}{relation}\hypertarget{method.HOL.relation}{\hyperlink{method.HOL.relation}{\mbox{\isa{relation}}}} & : & \isarmeth \\
wenzelm@26907:     \indexdef{HOL}{method}{lexicographic\_order}\hypertarget{method.HOL.lexicographic-order}{\hyperlink{method.HOL.lexicographic-order}{\mbox{\isa{lexicographic{\isacharunderscore}order}}}} & : & \isarmeth \\
wenzelm@26849:   \end{matharray}
wenzelm@26849: 
wenzelm@26849:   \begin{rail}
wenzelm@26849:     'relation' term
wenzelm@26849:     ;
wenzelm@26849:     'lexicographic\_order' (clasimpmod *)
wenzelm@26849:     ;
wenzelm@26849:   \end{rail}
wenzelm@26849: 
wenzelm@26849:   \begin{descr}
wenzelm@26849: 
wenzelm@26907:   \item [\hyperlink{method.HOL.pat-completeness}{\mbox{\isa{pat{\isacharunderscore}completeness}}}] is a specialized method to
wenzelm@26849:   solve goals regarding the completeness of pattern matching, as
wenzelm@26902:   required by the \hyperlink{command.HOL.function}{\mbox{\isa{\isacommand{function}}}} package (cf.\
wenzelm@26849:   \cite{isabelle-function}).
wenzelm@26849: 
wenzelm@26902:   \item [\hyperlink{method.HOL.relation}{\mbox{\isa{relation}}}~\isa{R}] introduces a termination
wenzelm@26849:   proof using the relation \isa{R}.  The resulting proof state will
wenzelm@26849:   contain goals expressing that \isa{R} is wellfounded, and that the
wenzelm@26849:   arguments of recursive calls decrease with respect to \isa{R}.
wenzelm@26849:   Usually, this method is used as the initial proof step of manual
wenzelm@26849:   termination proofs.
wenzelm@26849: 
wenzelm@26907:   \item [\hyperlink{method.HOL.lexicographic-order}{\mbox{\isa{lexicographic{\isacharunderscore}order}}}] attempts a fully
wenzelm@26849:   automated termination proof by searching for a lexicographic
wenzelm@26849:   combination of size measures on the arguments of the function. The
wenzelm@26902:   method accepts the same arguments as the \hyperlink{method.auto}{\mbox{\isa{auto}}} method,
wenzelm@26849:   which it uses internally to prove local descents.  The same context
wenzelm@26902:   modifiers as for \hyperlink{method.auto}{\mbox{\isa{auto}}} are accepted, see
wenzelm@26849:   \secref{sec:clasimp}.
wenzelm@26849: 
wenzelm@26849:   In case of failure, extensive information is printed, which can help
wenzelm@26849:   to analyse the situation (cf.\ \cite{isabelle-function}).
wenzelm@26849: 
wenzelm@26849:   \end{descr}%
wenzelm@26849: \end{isamarkuptext}%
wenzelm@26849: \isamarkuptrue%
wenzelm@26849: %
wenzelm@26849: \isamarkupsubsection{Old-style recursive function definitions (TFL)%
wenzelm@26849: }
wenzelm@26849: \isamarkuptrue%
wenzelm@26849: %
wenzelm@26849: \begin{isamarkuptext}%
wenzelm@26907: 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: 
wenzelm@26849:   \begin{matharray}{rcl}
wenzelm@26902:     \indexdef{HOL}{command}{recdef}\hypertarget{command.HOL.recdef}{\hyperlink{command.HOL.recdef}{\mbox{\isa{\isacommand{recdef}}}}} & : & \isartrans{theory}{theory} \\
wenzelm@26907:     \indexdef{HOL}{command}{recdef\_tc}\hypertarget{command.HOL.recdef-tc}{\hyperlink{command.HOL.recdef-tc}{\mbox{\isa{\isacommand{recdef{\isacharunderscore}tc}}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isartrans{theory}{proof(prove)} \\
wenzelm@26849:   \end{matharray}
wenzelm@26849: 
wenzelm@26849:   \begin{rail}
wenzelm@26849:     'recdef' ('(' 'permissive' ')')? \\ name term (prop +) hints?
wenzelm@26849:     ;
wenzelm@26849:     recdeftc thmdecl? tc
wenzelm@26849:     ;
wenzelm@26849:     hints: '(' 'hints' (recdefmod *) ')'
wenzelm@26849:     ;
wenzelm@26849:     recdefmod: (('recdef\_simp' | 'recdef\_cong' | 'recdef\_wf') (() | 'add' | 'del') ':' thmrefs) | clasimpmod
wenzelm@26849:     ;
wenzelm@26849:     tc: nameref ('(' nat ')')?
wenzelm@26849:     ;
wenzelm@26849:   \end{rail}
wenzelm@26849: 
wenzelm@26849:   \begin{descr}
wenzelm@26849:   
wenzelm@26902:   \item [\hyperlink{command.HOL.recdef}{\mbox{\isa{\isacommand{recdef}}}}] defines general well-founded
wenzelm@26849:   recursive functions (using the TFL package), see also
wenzelm@26849:   \cite{isabelle-HOL}.  The ``\isa{{\isachardoublequote}{\isacharparenleft}permissive{\isacharparenright}{\isachardoublequote}}'' option tells
wenzelm@26849:   TFL to recover from failed proof attempts, returning unfinished
wenzelm@26849:   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:   automated proof process of TFL.  Additional \hyperlink{syntax.clasimpmod}{\mbox{\isa{clasimpmod}}}
wenzelm@26849:   declarations (cf.\ \secref{sec:clasimp}) may be given to tune the
wenzelm@26849:   context of the Simplifier (cf.\ \secref{sec:simplifier}) and
wenzelm@26849:   Classical reasoner (cf.\ \secref{sec:classical}).
wenzelm@26849:   
wenzelm@26907:   \item [\hyperlink{command.HOL.recdef-tc}{\mbox{\isa{\isacommand{recdef{\isacharunderscore}tc}}}}~\isa{{\isachardoublequote}c\ {\isacharparenleft}i{\isacharparenright}{\isachardoublequote}}] recommences the
wenzelm@26849:   proof for leftover termination condition number \isa{i} (default
wenzelm@26902:   1) as generated by a \hyperlink{command.HOL.recdef}{\mbox{\isa{\isacommand{recdef}}}} definition of
wenzelm@26849:   constant \isa{c}.
wenzelm@26849:   
wenzelm@26902:   Note that in most cases, \hyperlink{command.HOL.recdef}{\mbox{\isa{\isacommand{recdef}}}} is able to finish
wenzelm@26849:   its internal proofs without manual intervention.
wenzelm@26849: 
wenzelm@26849:   \end{descr}
wenzelm@26849: 
wenzelm@26902:   \medskip Hints for \hyperlink{command.HOL.recdef}{\mbox{\isa{\isacommand{recdef}}}} may be also declared
wenzelm@26849:   globally, using the following attributes.
wenzelm@26849: 
wenzelm@26849:   \begin{matharray}{rcl}
wenzelm@26907:     \indexdef{HOL}{attribute}{recdef\_simp}\hypertarget{attribute.HOL.recdef-simp}{\hyperlink{attribute.HOL.recdef-simp}{\mbox{\isa{recdef{\isacharunderscore}simp}}}} & : & \isaratt \\
wenzelm@26907:     \indexdef{HOL}{attribute}{recdef\_cong}\hypertarget{attribute.HOL.recdef-cong}{\hyperlink{attribute.HOL.recdef-cong}{\mbox{\isa{recdef{\isacharunderscore}cong}}}} & : & \isaratt \\
wenzelm@26907:     \indexdef{HOL}{attribute}{recdef\_wf}\hypertarget{attribute.HOL.recdef-wf}{\hyperlink{attribute.HOL.recdef-wf}{\mbox{\isa{recdef{\isacharunderscore}wf}}}} & : & \isaratt \\
wenzelm@26849:   \end{matharray}
wenzelm@26849: 
wenzelm@26849:   \begin{rail}
wenzelm@26849:     ('recdef\_simp' | 'recdef\_cong' | 'recdef\_wf') (() | 'add' | 'del')
wenzelm@26849:     ;
wenzelm@26849:   \end{rail}%
wenzelm@26849: \end{isamarkuptext}%
wenzelm@26849: \isamarkuptrue%
wenzelm@26849: %
wenzelm@26849: \isamarkupsection{Inductive and coinductive definitions \label{sec:hol-inductive}%
wenzelm@26849: }
wenzelm@26849: \isamarkuptrue%
wenzelm@26849: %
wenzelm@26849: \begin{isamarkuptext}%
wenzelm@26849: An \textbf{inductive definition} specifies the least predicate (or
wenzelm@26849:   set) \isa{R} closed under given rules: applying a rule to elements
wenzelm@26849:   of \isa{R} yields a result within \isa{R}.  For example, a
wenzelm@26849:   structural operational semantics is an inductive definition of an
wenzelm@26849:   evaluation relation.
wenzelm@26849: 
wenzelm@26849:   Dually, a \textbf{coinductive definition} specifies the greatest
wenzelm@26849:   predicate~/ set \isa{R} that is consistent with given rules: every
wenzelm@26849:   element of \isa{R} can be seen as arising by applying a rule to
wenzelm@26849:   elements of \isa{R}.  An important example is using bisimulation
wenzelm@26849:   relations to formalise equivalence of processes and infinite data
wenzelm@26849:   structures.
wenzelm@26849: 
wenzelm@26849:   \medskip The HOL package is related to the ZF one, which is
wenzelm@26849:   described in a separate paper,\footnote{It appeared in CADE
wenzelm@26849:   \cite{paulson-CADE}; a longer version is distributed with Isabelle.}
wenzelm@26849:   which you should refer to in case of difficulties.  The package is
wenzelm@26849:   simpler than that of ZF thanks to implicit type-checking in HOL.
wenzelm@26849:   The types of the (co)inductive predicates (or sets) determine the
wenzelm@26849:   domain of the fixedpoint definition, and the package does not have
wenzelm@26849:   to use inference rules for type-checking.
wenzelm@26849: 
wenzelm@26849:   \begin{matharray}{rcl}
wenzelm@26902:     \indexdef{HOL}{command}{inductive}\hypertarget{command.HOL.inductive}{\hyperlink{command.HOL.inductive}{\mbox{\isa{\isacommand{inductive}}}}} & : & \isarkeep{local{\dsh}theory} \\
wenzelm@26907:     \indexdef{HOL}{command}{inductive\_set}\hypertarget{command.HOL.inductive-set}{\hyperlink{command.HOL.inductive-set}{\mbox{\isa{\isacommand{inductive{\isacharunderscore}set}}}}} & : & \isarkeep{local{\dsh}theory} \\
wenzelm@26902:     \indexdef{HOL}{command}{coinductive}\hypertarget{command.HOL.coinductive}{\hyperlink{command.HOL.coinductive}{\mbox{\isa{\isacommand{coinductive}}}}} & : & \isarkeep{local{\dsh}theory} \\
wenzelm@26907:     \indexdef{HOL}{command}{coinductive\_set}\hypertarget{command.HOL.coinductive-set}{\hyperlink{command.HOL.coinductive-set}{\mbox{\isa{\isacommand{coinductive{\isacharunderscore}set}}}}} & : & \isarkeep{local{\dsh}theory} \\
wenzelm@26902:     \indexdef{HOL}{attribute}{mono}\hypertarget{attribute.HOL.mono}{\hyperlink{attribute.HOL.mono}{\mbox{\isa{mono}}}} & : & \isaratt \\
wenzelm@26849:   \end{matharray}
wenzelm@26849: 
wenzelm@26849:   \begin{rail}
wenzelm@26849:     ('inductive' | 'inductive\_set' | 'coinductive' | 'coinductive\_set') target? fixes ('for' fixes)? \\
wenzelm@26849:     ('where' clauses)? ('monos' thmrefs)?
wenzelm@26849:     ;
wenzelm@26849:     clauses: (thmdecl? prop + '|')
wenzelm@26849:     ;
wenzelm@26849:     'mono' (() | 'add' | 'del')
wenzelm@26849:     ;
wenzelm@26849:   \end{rail}
wenzelm@26849: 
wenzelm@26849:   \begin{descr}
wenzelm@26849: 
wenzelm@26902:   \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:   introduction rules given in the \hyperlink{keyword.where}{\mbox{\isa{\isakeyword{where}}}} part.  The
wenzelm@26902:   optional \hyperlink{keyword.for}{\mbox{\isa{\isakeyword{for}}}} part contains a list of parameters of the
wenzelm@26849:   (co)inductive predicates that remain fixed throughout the
wenzelm@26902:   definition.  The optional \hyperlink{keyword.monos}{\mbox{\isa{\isakeyword{monos}}}} section contains
wenzelm@26849:   \emph{monotonicity theorems}, which are required for each operator
wenzelm@26849:   applied to a recursive set in the introduction rules.  There
wenzelm@26849:   \emph{must} be a theorem of the form \isa{{\isachardoublequote}A\ {\isasymle}\ B\ {\isasymLongrightarrow}\ M\ A\ {\isasymle}\ M\ B{\isachardoublequote}},
wenzelm@26849:   for each premise \isa{{\isachardoublequote}M\ R\isactrlsub i\ t{\isachardoublequote}} in an introduction rule!
wenzelm@26849: 
wenzelm@26907:   \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:   allowing the definition of (co)inductive sets.
wenzelm@26849: 
wenzelm@26902:   \item [\hyperlink{attribute.HOL.mono}{\mbox{\isa{mono}}}] declares monotonicity rules.  These
wenzelm@26902:   rule are involved in the automated monotonicity proof of \hyperlink{command.HOL.inductive}{\mbox{\isa{\isacommand{inductive}}}}.
wenzelm@26849: 
wenzelm@26849:   \end{descr}%
wenzelm@26849: \end{isamarkuptext}%
wenzelm@26849: \isamarkuptrue%
wenzelm@26849: %
wenzelm@26849: \isamarkupsubsection{Derived rules%
wenzelm@26849: }
wenzelm@26849: \isamarkuptrue%
wenzelm@26849: %
wenzelm@26849: \begin{isamarkuptext}%
wenzelm@26849: Each (co)inductive definition \isa{R} adds definitions to the
wenzelm@26849:   theory and also proves some theorems:
wenzelm@26849: 
wenzelm@26849:   \begin{description}
wenzelm@26849: 
wenzelm@26849:   \item [\isa{R{\isachardot}intros}] is the list of introduction rules as proven
wenzelm@26849:   theorems, for the recursive predicates (or sets).  The rules are
wenzelm@26849:   also available individually, using the names given them in the
wenzelm@26849:   theory file;
wenzelm@26849: 
wenzelm@26849:   \item [\isa{R{\isachardot}cases}] is the case analysis (or elimination) rule;
wenzelm@26849: 
wenzelm@26849:   \item [\isa{R{\isachardot}induct} or \isa{R{\isachardot}coinduct}] is the (co)induction
wenzelm@26849:   rule.
wenzelm@26849: 
wenzelm@26849:   \end{description}
wenzelm@26849: 
wenzelm@26849:   When several predicates \isa{{\isachardoublequote}R\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ R\isactrlsub n{\isachardoublequote}} are
wenzelm@26849:   defined simultaneously, the list of introduction rules is called
wenzelm@26849:   \isa{{\isachardoublequote}R\isactrlsub {\isadigit{1}}{\isacharunderscore}{\isasymdots}{\isacharunderscore}R\isactrlsub n{\isachardot}intros{\isachardoublequote}}, the case analysis rules are
wenzelm@26849:   called \isa{{\isachardoublequote}R\isactrlsub {\isadigit{1}}{\isachardot}cases{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ R\isactrlsub n{\isachardot}cases{\isachardoublequote}}, and the list
wenzelm@26849:   of mutual induction rules is called \isa{{\isachardoublequote}R\isactrlsub {\isadigit{1}}{\isacharunderscore}{\isasymdots}{\isacharunderscore}R\isactrlsub n{\isachardot}inducts{\isachardoublequote}}.%
wenzelm@26849: \end{isamarkuptext}%
wenzelm@26849: \isamarkuptrue%
wenzelm@26849: %
wenzelm@26849: \isamarkupsubsection{Monotonicity theorems%
wenzelm@26849: }
wenzelm@26849: \isamarkuptrue%
wenzelm@26849: %
wenzelm@26849: \begin{isamarkuptext}%
wenzelm@26849: Each theory contains a default set of theorems that are used in
wenzelm@26849:   monotonicity proofs.  New rules can be added to this set via the
wenzelm@26902:   \hyperlink{attribute.HOL.mono}{\mbox{\isa{mono}}} attribute.  The HOL theory \isa{Inductive}
wenzelm@26849:   shows how this is done.  In general, the following monotonicity
wenzelm@26849:   theorems may be added:
wenzelm@26849: 
wenzelm@26849:   \begin{itemize}
wenzelm@26849: 
wenzelm@26849:   \item Theorems of the form \isa{{\isachardoublequote}A\ {\isasymle}\ B\ {\isasymLongrightarrow}\ M\ A\ {\isasymle}\ M\ B{\isachardoublequote}}, for proving
wenzelm@26849:   monotonicity of inductive definitions whose introduction rules have
wenzelm@26849:   premises involving terms such as \isa{{\isachardoublequote}M\ R\isactrlsub i\ t{\isachardoublequote}}.
wenzelm@26849: 
wenzelm@26849:   \item Monotonicity theorems for logical operators, which are of the
wenzelm@26849:   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:   the case of the operator \isa{{\isachardoublequote}{\isasymor}{\isachardoublequote}}, the corresponding theorem is
wenzelm@26849:   \[
wenzelm@26849:   \infer{\isa{{\isachardoublequote}P\isactrlsub {\isadigit{1}}\ {\isasymor}\ P\isactrlsub {\isadigit{2}}\ {\isasymlongrightarrow}\ Q\isactrlsub {\isadigit{1}}\ {\isasymor}\ Q\isactrlsub {\isadigit{2}}{\isachardoublequote}}}{\isa{{\isachardoublequote}P\isactrlsub {\isadigit{1}}\ {\isasymlongrightarrow}\ Q\isactrlsub {\isadigit{1}}{\isachardoublequote}} & \isa{{\isachardoublequote}P\isactrlsub {\isadigit{2}}\ {\isasymlongrightarrow}\ Q\isactrlsub {\isadigit{2}}{\isachardoublequote}}}
wenzelm@26849:   \]
wenzelm@26849: 
wenzelm@26849:   \item De Morgan style equations for reasoning about the ``polarity''
wenzelm@26849:   of expressions, e.g.
wenzelm@26849:   \[
wenzelm@26849:   \isa{{\isachardoublequote}{\isasymnot}\ {\isasymnot}\ P\ {\isasymlongleftrightarrow}\ P{\isachardoublequote}} \qquad\qquad
wenzelm@26849:   \isa{{\isachardoublequote}{\isasymnot}\ {\isacharparenleft}P\ {\isasymand}\ Q{\isacharparenright}\ {\isasymlongleftrightarrow}\ {\isasymnot}\ P\ {\isasymor}\ {\isasymnot}\ Q{\isachardoublequote}}
wenzelm@26849:   \]
wenzelm@26849: 
wenzelm@26849:   \item Equations for reducing complex operators to more primitive
wenzelm@26849:   ones whose monotonicity can easily be proved, e.g.
wenzelm@26849:   \[
wenzelm@26849:   \isa{{\isachardoublequote}{\isacharparenleft}P\ {\isasymlongrightarrow}\ Q{\isacharparenright}\ {\isasymlongleftrightarrow}\ {\isasymnot}\ P\ {\isasymor}\ Q{\isachardoublequote}} \qquad\qquad
wenzelm@26849:   \isa{{\isachardoublequote}Ball\ A\ P\ {\isasymequiv}\ {\isasymforall}x{\isachardot}\ x\ {\isasymin}\ A\ {\isasymlongrightarrow}\ P\ x{\isachardoublequote}}
wenzelm@26849:   \]
wenzelm@26849: 
wenzelm@26849:   \end{itemize}
wenzelm@26849: 
wenzelm@26849:   %FIXME: Example of an inductive definition%
wenzelm@26849: \end{isamarkuptext}%
wenzelm@26849: \isamarkuptrue%
wenzelm@26849: %
wenzelm@26849: \isamarkupsection{Arithmetic proof support%
wenzelm@26849: }
wenzelm@26849: \isamarkuptrue%
wenzelm@26849: %
wenzelm@26849: \begin{isamarkuptext}%
wenzelm@26849: \begin{matharray}{rcl}
wenzelm@26902:     \indexdef{HOL}{method}{arith}\hypertarget{method.HOL.arith}{\hyperlink{method.HOL.arith}{\mbox{\isa{arith}}}} & : & \isarmeth \\
wenzelm@26907:     \indexdef{HOL}{attribute}{arith\_split}\hypertarget{attribute.HOL.arith-split}{\hyperlink{attribute.HOL.arith-split}{\mbox{\isa{arith{\isacharunderscore}split}}}} & : & \isaratt \\
wenzelm@26849:   \end{matharray}
wenzelm@26849: 
wenzelm@26902:   The \hyperlink{method.HOL.arith}{\mbox{\isa{arith}}} method decides linear arithmetic problems
wenzelm@26849:   (on types \isa{nat}, \isa{int}, \isa{real}).  Any current
wenzelm@26849:   facts are inserted into the goal before running the procedure.
wenzelm@26849: 
wenzelm@26907:   The \hyperlink{attribute.HOL.arith-split}{\mbox{\isa{arith{\isacharunderscore}split}}} attribute declares case split
wenzelm@26895:   rules to be expanded before the arithmetic procedure is invoked.
wenzelm@26849: 
wenzelm@26849:   Note that a simpler (but faster) version of arithmetic reasoning is
wenzelm@26849:   already performed by the Simplifier.%
wenzelm@26849: \end{isamarkuptext}%
wenzelm@26849: \isamarkuptrue%
wenzelm@26849: %
wenzelm@28603: \isamarkupsection{Invoking automated reasoning tools -- The Sledgehammer%
wenzelm@28603: }
wenzelm@28603: \isamarkuptrue%
wenzelm@28603: %
wenzelm@28603: \begin{isamarkuptext}%
wenzelm@28603: Isabelle/HOL includes a generic \emph{ATP manager} that allows
wenzelm@28603:   external automated reasoning tools to crunch a pending goal.
wenzelm@28603:   Supported provers include E\footnote{\url{http://www.eprover.org}},
wenzelm@28603:   SPASS\footnote{\url{http://www.spass-prover.org/}}, and Vampire.
wenzelm@28603:   There is also a wrapper to invoke provers remotely via the
wenzelm@28603:   SystemOnTPTP\footnote{\url{http://www.cs.miami.edu/~tptp/cgi-bin/SystemOnTPTP}}
wenzelm@28603:   web service.
wenzelm@28603: 
wenzelm@28603:   The problem passed to external provers consists of the goal together
wenzelm@28603:   with a smart selection of lemmas from the current theory context.
wenzelm@28603:   The result of a successful proof search is some source text that
wenzelm@28603:   usually reconstructs the proof within Isabelle, without requiring
wenzelm@28603:   external provers again.  The Metis
wenzelm@28603:   prover\footnote{\url{http://www.gilith.com/software/metis/}} that is
wenzelm@28603:   integrated into Isabelle/HOL is being used here.
wenzelm@28603: 
wenzelm@28603:   In this mode of operation, heavy means of automated reasoning are
wenzelm@28603:   used as a strong relevance filter, while the main proof checking
wenzelm@28603:   works via explicit inferences going through the Isabelle kernel.
wenzelm@28603:   Moreover, rechecking Isabelle proof texts with already specified
wenzelm@28603:   auxiliary facts is much faster than performing fully automated
wenzelm@28603:   search over and over again.
wenzelm@28603: 
wenzelm@28603:   \begin{matharray}{rcl}
wenzelm@28603:     \indexdef{HOL}{command}{sledgehammer}\hypertarget{command.HOL.sledgehammer}{\hyperlink{command.HOL.sledgehammer}{\mbox{\isa{\isacommand{sledgehammer}}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isarkeep{proof} \\
wenzelm@28603:     \indexdef{HOL}{command}{print\_atps}\hypertarget{command.HOL.print-atps}{\hyperlink{command.HOL.print-atps}{\mbox{\isa{\isacommand{print{\isacharunderscore}atps}}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isarkeep{theory~|~proof} \\
wenzelm@28603:     \indexdef{HOL}{command}{atp\_info}\hypertarget{command.HOL.atp-info}{\hyperlink{command.HOL.atp-info}{\mbox{\isa{\isacommand{atp{\isacharunderscore}info}}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isarkeep{\cdot} \\
wenzelm@28603:     \indexdef{HOL}{command}{atp\_kill}\hypertarget{command.HOL.atp-kill}{\hyperlink{command.HOL.atp-kill}{\mbox{\isa{\isacommand{atp{\isacharunderscore}kill}}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isarkeep{\cdot} \\
wenzelm@28603:     \indexdef{HOL}{method}{metis}\hypertarget{method.HOL.metis}{\hyperlink{method.HOL.metis}{\mbox{\isa{metis}}}} & : & \isarmeth \\
wenzelm@28603:   \end{matharray}
wenzelm@28603: 
wenzelm@28603:   \begin{rail}
wenzelm@28603:   'sledgehammer' (nameref *)
wenzelm@28603:   ;
wenzelm@28603: 
wenzelm@28603:   'metis' thmrefs
wenzelm@28603:   ;
wenzelm@28603:   \end{rail}
wenzelm@28603: 
wenzelm@28603:   \begin{descr}
wenzelm@28603: 
wenzelm@28603:   \item [\hyperlink{command.HOL.sledgehammer}{\mbox{\isa{\isacommand{sledgehammer}}}}~\isa{{\isachardoublequote}prover\isactrlsub {\isadigit{1}}\ {\isasymdots}\ prover\isactrlsub n{\isachardoublequote}}] invokes the specified automated theorem provers on
wenzelm@28603:   the first subgoal.  Provers are run in parallel, the first
wenzelm@28603:   successful result is displayed, and the other attempts are
wenzelm@28603:   terminated.
wenzelm@28603: 
wenzelm@28603:   Provers are defined in the theory context, see also \hyperlink{command.HOL.print-atps}{\mbox{\isa{\isacommand{print{\isacharunderscore}atps}}}}.  If no provers are given as arguments to \hyperlink{command.HOL.sledgehammer}{\mbox{\isa{\isacommand{sledgehammer}}}}, the system refers to the default defined as
wenzelm@28603:   ``ATP provers'' preference by the user interface.
wenzelm@28603: 
wenzelm@28603:   There are additional preferences for timeout (default: 60 seconds),
wenzelm@28603:   and the maximum number of independent prover processes (default: 5);
wenzelm@28603:   excessive provers are automatically terminated.
wenzelm@28603: 
wenzelm@28603:   \item [\hyperlink{command.HOL.print-atps}{\mbox{\isa{\isacommand{print{\isacharunderscore}atps}}}}] prints the list of automated
wenzelm@28603:   theorem provers available to the \hyperlink{command.HOL.sledgehammer}{\mbox{\isa{\isacommand{sledgehammer}}}}
wenzelm@28603:   command.
wenzelm@28603: 
wenzelm@28603:   \item [\hyperlink{command.HOL.atp-info}{\mbox{\isa{\isacommand{atp{\isacharunderscore}info}}}}] prints information about presently
wenzelm@28603:   running provers, including elapsed runtime, and the remaining time
wenzelm@28603:   until timeout.
wenzelm@28603: 
wenzelm@28603:   \item [\hyperlink{command.HOL.atp-kill}{\mbox{\isa{\isacommand{atp{\isacharunderscore}kill}}}}] terminates all presently running
wenzelm@28603:   provers.
wenzelm@28603: 
wenzelm@28603:   \item [\hyperlink{method.HOL.metis}{\mbox{\isa{metis}}}~\isa{{\isachardoublequote}facts{\isachardoublequote}}] invokes the Metis
wenzelm@28603:   prover with the given facts.  Metis is an automated proof tool of
wenzelm@28603:   medium strength, but is fully integrated into Isabelle/HOL, with
wenzelm@28603:   explicit inferences going through the kernel.  Thus its results are
wenzelm@28603:   guaranteed to be ``correct by construction''.
wenzelm@28603: 
wenzelm@28603:   Note that all facts used with Metis need to be specified as explicit
wenzelm@28603:   arguments.  There are no rule declarations as for other Isabelle
wenzelm@28603:   provers, like \hyperlink{method.blast}{\mbox{\isa{blast}}} or \hyperlink{method.fast}{\mbox{\isa{fast}}}.
wenzelm@28603: 
wenzelm@28603:   \end{descr}%
wenzelm@28603: \end{isamarkuptext}%
wenzelm@28603: \isamarkuptrue%
wenzelm@28603: %
wenzelm@27124: \isamarkupsection{Unstructured cases analysis and induction \label{sec:hol-induct-tac}%
wenzelm@26849: }
wenzelm@26849: \isamarkuptrue%
wenzelm@26849: %
wenzelm@26849: \begin{isamarkuptext}%
wenzelm@27124: The following tools of Isabelle/HOL support cases analysis and
wenzelm@27124:   induction in unstructured tactic scripts; see also
wenzelm@27124:   \secref{sec:cases-induct} for proper Isar versions of similar ideas.
wenzelm@26849: 
wenzelm@26849:   \begin{matharray}{rcl}
wenzelm@26907:     \indexdef{HOL}{method}{case\_tac}\hypertarget{method.HOL.case-tac}{\hyperlink{method.HOL.case-tac}{\mbox{\isa{case{\isacharunderscore}tac}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isarmeth \\
wenzelm@26907:     \indexdef{HOL}{method}{induct\_tac}\hypertarget{method.HOL.induct-tac}{\hyperlink{method.HOL.induct-tac}{\mbox{\isa{induct{\isacharunderscore}tac}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isarmeth \\
wenzelm@26907:     \indexdef{HOL}{method}{ind\_cases}\hypertarget{method.HOL.ind-cases}{\hyperlink{method.HOL.ind-cases}{\mbox{\isa{ind{\isacharunderscore}cases}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isarmeth \\
wenzelm@27124:     \indexdef{HOL}{command}{inductive\_cases}\hypertarget{command.HOL.inductive-cases}{\hyperlink{command.HOL.inductive-cases}{\mbox{\isa{\isacommand{inductive{\isacharunderscore}cases}}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isartrans{theory}{theory} \\
wenzelm@26849:   \end{matharray}
wenzelm@26849: 
wenzelm@26849:   \begin{rail}
wenzelm@26849:     'case\_tac' goalspec? term rule?
wenzelm@26849:     ;
wenzelm@26849:     'induct\_tac' goalspec? (insts * 'and') rule?
wenzelm@26849:     ;
wenzelm@26849:     'ind\_cases' (prop +) ('for' (name +)) ?
wenzelm@26849:     ;
wenzelm@26849:     'inductive\_cases' (thmdecl? (prop +) + 'and')
wenzelm@26849:     ;
wenzelm@26849: 
wenzelm@26849:     rule: ('rule' ':' thmref)
wenzelm@26849:     ;
wenzelm@26849:   \end{rail}
wenzelm@26849: 
wenzelm@26849:   \begin{descr}
wenzelm@26849: 
wenzelm@26907:   \item [\hyperlink{method.HOL.case-tac}{\mbox{\isa{case{\isacharunderscore}tac}}} and \hyperlink{method.HOL.induct-tac}{\mbox{\isa{induct{\isacharunderscore}tac}}}]
wenzelm@27124:   admit to reason about inductive types.  Rules are selected according
wenzelm@27124:   to the declarations by the \hyperlink{attribute.cases}{\mbox{\isa{cases}}} and \hyperlink{attribute.induct}{\mbox{\isa{induct}}} attributes, cf.\ \secref{sec:cases-induct}.  The \hyperlink{command.HOL.datatype}{\mbox{\isa{\isacommand{datatype}}}} package already takes care of this.
wenzelm@27124: 
wenzelm@27124:   These unstructured tactics feature both goal addressing and dynamic
wenzelm@26849:   instantiation.  Note that named rule cases are \emph{not} provided
wenzelm@27124:   as would be by the proper \hyperlink{method.cases}{\mbox{\isa{cases}}} and \hyperlink{method.induct}{\mbox{\isa{induct}}} proof
wenzelm@27124:   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:   statements, only the compact object-logic conclusion of the subgoal
wenzelm@27124:   being addressed.
wenzelm@26849:   
wenzelm@26907:   \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:   forward manner.
wenzelm@26849: 
wenzelm@26907:   While \hyperlink{method.HOL.ind-cases}{\mbox{\isa{ind{\isacharunderscore}cases}}} is a proof method to apply the
wenzelm@26907:   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:   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:   be generalized before applying the resulting rule.
wenzelm@26849: 
wenzelm@26849:   \end{descr}%
wenzelm@26849: \end{isamarkuptext}%
wenzelm@26849: \isamarkuptrue%
wenzelm@26849: %
wenzelm@26849: \isamarkupsection{Executable code%
wenzelm@26849: }
wenzelm@26849: \isamarkuptrue%
wenzelm@26849: %
wenzelm@26849: \begin{isamarkuptext}%
wenzelm@26849: Isabelle/Pure provides two generic frameworks to support code
wenzelm@26849:   generation from executable specifications.  Isabelle/HOL
wenzelm@26849:   instantiates these mechanisms in a way that is amenable to end-user
wenzelm@26849:   applications.
wenzelm@26849: 
wenzelm@26849:   One framework generates code from both functional and relational
wenzelm@26849:   programs to SML.  See \cite{isabelle-HOL} for further information
wenzelm@26849:   (this actually covers the new-style theory format as well).
wenzelm@26849: 
wenzelm@26849:   \begin{matharray}{rcl}
wenzelm@26902:     \indexdef{HOL}{command}{value}\hypertarget{command.HOL.value}{\hyperlink{command.HOL.value}{\mbox{\isa{\isacommand{value}}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isarkeep{theory~|~proof} \\
wenzelm@26907:     \indexdef{HOL}{command}{code\_module}\hypertarget{command.HOL.code-module}{\hyperlink{command.HOL.code-module}{\mbox{\isa{\isacommand{code{\isacharunderscore}module}}}}} & : & \isartrans{theory}{theory} \\
wenzelm@26907:     \indexdef{HOL}{command}{code\_library}\hypertarget{command.HOL.code-library}{\hyperlink{command.HOL.code-library}{\mbox{\isa{\isacommand{code{\isacharunderscore}library}}}}} & : & \isartrans{theory}{theory} \\
wenzelm@26907:     \indexdef{HOL}{command}{consts\_code}\hypertarget{command.HOL.consts-code}{\hyperlink{command.HOL.consts-code}{\mbox{\isa{\isacommand{consts{\isacharunderscore}code}}}}} & : & \isartrans{theory}{theory} \\
wenzelm@26907:     \indexdef{HOL}{command}{types\_code}\hypertarget{command.HOL.types-code}{\hyperlink{command.HOL.types-code}{\mbox{\isa{\isacommand{types{\isacharunderscore}code}}}}} & : & \isartrans{theory}{theory} \\  
wenzelm@26902:     \indexdef{HOL}{attribute}{code}\hypertarget{attribute.HOL.code}{\hyperlink{attribute.HOL.code}{\mbox{\isa{code}}}} & : & \isaratt \\
wenzelm@26849:   \end{matharray}
wenzelm@26849: 
wenzelm@26849:   \begin{rail}
wenzelm@26849:   'value' term
wenzelm@26849:   ;
wenzelm@26849: 
wenzelm@26849:   ( 'code\_module' | 'code\_library' ) modespec ? name ? \\
wenzelm@26849:     ( 'file' name ) ? ( 'imports' ( name + ) ) ? \\
wenzelm@26849:     'contains' ( ( name '=' term ) + | term + )
wenzelm@26849:   ;
wenzelm@26849: 
wenzelm@26849:   modespec: '(' ( name * ) ')'
wenzelm@26849:   ;
wenzelm@26849: 
wenzelm@26849:   'consts\_code' (codespec +)
wenzelm@26849:   ;
wenzelm@26849: 
wenzelm@26849:   codespec: const template attachment ?
wenzelm@26849:   ;
wenzelm@26849: 
wenzelm@26849:   'types\_code' (tycodespec +)
wenzelm@26849:   ;
wenzelm@26849: 
wenzelm@26849:   tycodespec: name template attachment ?
wenzelm@26849:   ;
wenzelm@26849: 
wenzelm@26849:   const: term
wenzelm@26849:   ;
wenzelm@26849: 
wenzelm@26849:   template: '(' string ')'
wenzelm@26849:   ;
wenzelm@26849: 
wenzelm@26849:   attachment: 'attach' modespec ? verblbrace text verbrbrace
wenzelm@26849:   ;
wenzelm@26849: 
wenzelm@26849:   'code' (name)?
wenzelm@26849:   ;
wenzelm@26849:   \end{rail}
wenzelm@26849: 
wenzelm@26849:   \begin{descr}
wenzelm@26849: 
wenzelm@26902:   \item [\hyperlink{command.HOL.value}{\mbox{\isa{\isacommand{value}}}}~\isa{t}] evaluates and prints a
wenzelm@26849:   term using the code generator.
wenzelm@26849: 
wenzelm@26849:   \end{descr}
wenzelm@26849: 
wenzelm@26849:   \medskip The other framework generates code from functional programs
wenzelm@26849:   (including overloading using type classes) to SML \cite{SML}, OCaml
wenzelm@26849:   \cite{OCaml} and Haskell \cite{haskell-revised-report}.
wenzelm@26849:   Conceptually, code generation is split up in three steps:
wenzelm@26849:   \emph{selection} of code theorems, \emph{translation} into an
wenzelm@26849:   abstract executable view and \emph{serialization} to a specific
wenzelm@26849:   \emph{target language}.  See \cite{isabelle-codegen} for an
wenzelm@26849:   introduction on how to use it.
wenzelm@26849: 
wenzelm@26849:   \begin{matharray}{rcl}
wenzelm@26907:     \indexdef{HOL}{command}{export\_code}\hypertarget{command.HOL.export-code}{\hyperlink{command.HOL.export-code}{\mbox{\isa{\isacommand{export{\isacharunderscore}code}}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isarkeep{theory~|~proof} \\
wenzelm@26907:     \indexdef{HOL}{command}{code\_thms}\hypertarget{command.HOL.code-thms}{\hyperlink{command.HOL.code-thms}{\mbox{\isa{\isacommand{code{\isacharunderscore}thms}}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isarkeep{theory~|~proof} \\
wenzelm@26907:     \indexdef{HOL}{command}{code\_deps}\hypertarget{command.HOL.code-deps}{\hyperlink{command.HOL.code-deps}{\mbox{\isa{\isacommand{code{\isacharunderscore}deps}}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isarkeep{theory~|~proof} \\
wenzelm@26907:     \indexdef{HOL}{command}{code\_datatype}\hypertarget{command.HOL.code-datatype}{\hyperlink{command.HOL.code-datatype}{\mbox{\isa{\isacommand{code{\isacharunderscore}datatype}}}}} & : & \isartrans{theory}{theory} \\
wenzelm@26907:     \indexdef{HOL}{command}{code\_const}\hypertarget{command.HOL.code-const}{\hyperlink{command.HOL.code-const}{\mbox{\isa{\isacommand{code{\isacharunderscore}const}}}}} & : & \isartrans{theory}{theory} \\
wenzelm@26907:     \indexdef{HOL}{command}{code\_type}\hypertarget{command.HOL.code-type}{\hyperlink{command.HOL.code-type}{\mbox{\isa{\isacommand{code{\isacharunderscore}type}}}}} & : & \isartrans{theory}{theory} \\
wenzelm@26907:     \indexdef{HOL}{command}{code\_class}\hypertarget{command.HOL.code-class}{\hyperlink{command.HOL.code-class}{\mbox{\isa{\isacommand{code{\isacharunderscore}class}}}}} & : & \isartrans{theory}{theory} \\
wenzelm@26907:     \indexdef{HOL}{command}{code\_instance}\hypertarget{command.HOL.code-instance}{\hyperlink{command.HOL.code-instance}{\mbox{\isa{\isacommand{code{\isacharunderscore}instance}}}}} & : & \isartrans{theory}{theory} \\
wenzelm@26907:     \indexdef{HOL}{command}{code\_monad}\hypertarget{command.HOL.code-monad}{\hyperlink{command.HOL.code-monad}{\mbox{\isa{\isacommand{code{\isacharunderscore}monad}}}}} & : & \isartrans{theory}{theory} \\
wenzelm@26907:     \indexdef{HOL}{command}{code\_reserved}\hypertarget{command.HOL.code-reserved}{\hyperlink{command.HOL.code-reserved}{\mbox{\isa{\isacommand{code{\isacharunderscore}reserved}}}}} & : & \isartrans{theory}{theory} \\
wenzelm@26907:     \indexdef{HOL}{command}{code\_include}\hypertarget{command.HOL.code-include}{\hyperlink{command.HOL.code-include}{\mbox{\isa{\isacommand{code{\isacharunderscore}include}}}}} & : & \isartrans{theory}{theory} \\
wenzelm@26907:     \indexdef{HOL}{command}{code\_modulename}\hypertarget{command.HOL.code-modulename}{\hyperlink{command.HOL.code-modulename}{\mbox{\isa{\isacommand{code{\isacharunderscore}modulename}}}}} & : & \isartrans{theory}{theory} \\
haftmann@27103:     \indexdef{HOL}{command}{code\_abort}\hypertarget{command.HOL.code-abort}{\hyperlink{command.HOL.code-abort}{\mbox{\isa{\isacommand{code{\isacharunderscore}abort}}}}} & : & \isartrans{theory}{theory} \\
wenzelm@26907:     \indexdef{HOL}{command}{print\_codesetup}\hypertarget{command.HOL.print-codesetup}{\hyperlink{command.HOL.print-codesetup}{\mbox{\isa{\isacommand{print{\isacharunderscore}codesetup}}}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isarkeep{theory~|~proof} \\
wenzelm@26902:     \indexdef{HOL}{attribute}{code}\hypertarget{attribute.HOL.code}{\hyperlink{attribute.HOL.code}{\mbox{\isa{code}}}} & : & \isaratt \\
wenzelm@26849:   \end{matharray}
wenzelm@26849: 
wenzelm@26849:   \begin{rail}
wenzelm@26849:     'export\_code' ( constexpr + ) ? \\
wenzelm@26849:       ( ( 'in' target ( 'module\_name' string ) ? \\
wenzelm@26849:         ( 'file' ( string | '-' ) ) ? ( '(' args ')' ) ?) + ) ?
wenzelm@26849:     ;
wenzelm@26849: 
wenzelm@26849:     'code\_thms' ( constexpr + ) ?
wenzelm@26849:     ;
wenzelm@26849: 
wenzelm@26849:     'code\_deps' ( constexpr + ) ?
wenzelm@26849:     ;
wenzelm@26849: 
wenzelm@26849:     const: term
wenzelm@26849:     ;
wenzelm@26849: 
wenzelm@26849:     constexpr: ( const | 'name.*' | '*' )
wenzelm@26849:     ;
wenzelm@26849: 
wenzelm@26849:     typeconstructor: nameref
wenzelm@26849:     ;
wenzelm@26849: 
wenzelm@26849:     class: nameref
wenzelm@26849:     ;
wenzelm@26849: 
wenzelm@26849:     target: 'OCaml' | 'SML' | 'Haskell'
wenzelm@26849:     ;
wenzelm@26849: 
wenzelm@26849:     'code\_datatype' const +
wenzelm@26849:     ;
wenzelm@26849: 
wenzelm@26849:     'code\_const' (const + 'and') \\
wenzelm@26849:       ( ( '(' target ( syntax ? + 'and' ) ')' ) + )
wenzelm@26849:     ;
wenzelm@26849: 
wenzelm@26849:     'code\_type' (typeconstructor + 'and') \\
wenzelm@26849:       ( ( '(' target ( syntax ? + 'and' ) ')' ) + )
wenzelm@26849:     ;
wenzelm@26849: 
wenzelm@26849:     'code\_class' (class + 'and') \\
wenzelm@26849:       ( ( '(' target \\
wenzelm@26849:         ( ( string ('where' \\
wenzelm@26849:           ( const ( '==' | equiv ) string ) + ) ? ) ? + 'and' ) ')' ) + )
wenzelm@26849:     ;
wenzelm@26849: 
wenzelm@26849:     'code\_instance' (( typeconstructor '::' class ) + 'and') \\
wenzelm@26849:       ( ( '(' target ( '-' ? + 'and' ) ')' ) + )
wenzelm@26849:     ;
wenzelm@26849: 
wenzelm@26849:     'code\_monad' const const target
wenzelm@26849:     ;
wenzelm@26849: 
wenzelm@26849:     'code\_reserved' target ( string + )
wenzelm@26849:     ;
wenzelm@26849: 
wenzelm@26849:     'code\_include' target ( string ( string | '-') )
wenzelm@26849:     ;
wenzelm@26849: 
wenzelm@26849:     'code\_modulename' target ( ( string string ) + )
wenzelm@26849:     ;
wenzelm@26849: 
haftmann@27452:     'code\_abort' ( const + )
wenzelm@26849:     ;
wenzelm@26849: 
wenzelm@26849:     syntax: string | ( 'infix' | 'infixl' | 'infixr' ) nat string
wenzelm@26849:     ;
wenzelm@26849: 
haftmann@28562:     'code' ( 'inline' ) ? ( 'del' ) ?
wenzelm@26849:     ;
wenzelm@26849:   \end{rail}
wenzelm@26849: 
wenzelm@26849:   \begin{descr}
wenzelm@26849: 
wenzelm@26907:   \item [\hyperlink{command.HOL.export-code}{\mbox{\isa{\isacommand{export{\isacharunderscore}code}}}}] is the canonical interface
wenzelm@26849:   for generating and serializing code: for a given list of constants,
wenzelm@26849:   code is generated for the specified target languages.  Abstract code
wenzelm@26849:   is cached incrementally.  If no constant is given, the currently
wenzelm@26849:   cached code is serialized.  If no serialization instruction is
wenzelm@26849:   given, only abstract code is cached.
wenzelm@26849: 
wenzelm@26849:   Constants may be specified by giving them literally, referring to
wenzelm@26849:   all executable contants within a certain theory by giving \isa{{\isachardoublequote}name{\isachardot}{\isacharasterisk}{\isachardoublequote}}, or referring to \emph{all} executable constants currently
wenzelm@26849:   available by giving \isa{{\isachardoublequote}{\isacharasterisk}{\isachardoublequote}}.
wenzelm@26849: 
wenzelm@26849:   By default, for each involved theory one corresponding name space
wenzelm@26849:   module is generated.  Alternativly, a module name may be specified
wenzelm@26907:   after the \hyperlink{keyword.module-name}{\mbox{\isa{\isakeyword{module{\isacharunderscore}name}}}} keyword; then \emph{all} code is
wenzelm@26849:   placed in this module.
wenzelm@26849: 
wenzelm@26849:   For \emph{SML} and \emph{OCaml}, the file specification refers to a
wenzelm@26849:   single file; for \emph{Haskell}, it refers to a whole directory,
wenzelm@26849:   where code is generated in multiple files reflecting the module
wenzelm@26849:   hierarchy.  The file specification ``\isa{{\isachardoublequote}{\isacharminus}{\isachardoublequote}}'' denotes standard
wenzelm@26849:   output.  For \emph{SML}, omitting the file specification compiles
wenzelm@26849:   code internally in the context of the current ML session.
wenzelm@26849: 
wenzelm@26849:   Serializers take an optional list of arguments in parentheses.  For
wenzelm@26849:   \emph{Haskell} a module name prefix may be given using the ``\isa{{\isachardoublequote}root{\isacharcolon}{\isachardoublequote}}'' argument; ``\isa{string{\isacharunderscore}classes}'' adds a ``\verb|deriving (Read, Show)|'' clause to each appropriate datatype
wenzelm@26849:   declaration.
wenzelm@26849: 
wenzelm@26907:   \item [\hyperlink{command.HOL.code-thms}{\mbox{\isa{\isacommand{code{\isacharunderscore}thms}}}}] prints a list of theorems
wenzelm@26849:   representing the corresponding program containing all given
wenzelm@26849:   constants; if no constants are given, the currently cached code
wenzelm@26849:   theorems are printed.
wenzelm@26849: 
wenzelm@26907:   \item [\hyperlink{command.HOL.code-deps}{\mbox{\isa{\isacommand{code{\isacharunderscore}deps}}}}] visualizes dependencies of
wenzelm@26849:   theorems representing the corresponding program containing all given
wenzelm@26849:   constants; if no constants are given, the currently cached code
wenzelm@26849:   theorems are visualized.
wenzelm@26849: 
wenzelm@26907:   \item [\hyperlink{command.HOL.code-datatype}{\mbox{\isa{\isacommand{code{\isacharunderscore}datatype}}}}] specifies a constructor set
wenzelm@26849:   for a logical type.
wenzelm@26849: 
wenzelm@26907:   \item [\hyperlink{command.HOL.code-const}{\mbox{\isa{\isacommand{code{\isacharunderscore}const}}}}] associates a list of constants
wenzelm@26849:   with target-specific serializations; omitting a serialization
wenzelm@26849:   deletes an existing serialization.
wenzelm@26849: 
wenzelm@26907:   \item [\hyperlink{command.HOL.code-type}{\mbox{\isa{\isacommand{code{\isacharunderscore}type}}}}] associates a list of type
wenzelm@26849:   constructors with target-specific serializations; omitting a
wenzelm@26849:   serialization deletes an existing serialization.
wenzelm@26849: 
wenzelm@26907:   \item [\hyperlink{command.HOL.code-class}{\mbox{\isa{\isacommand{code{\isacharunderscore}class}}}}] associates a list of classes
wenzelm@26849:   with target-specific class names; in addition, constants associated
wenzelm@26849:   with this class may be given target-specific names used for instance
wenzelm@26849:   declarations; omitting a serialization deletes an existing
wenzelm@26849:   serialization.  This applies only to \emph{Haskell}.
wenzelm@26849: 
wenzelm@26907:   \item [\hyperlink{command.HOL.code-instance}{\mbox{\isa{\isacommand{code{\isacharunderscore}instance}}}}] declares a list of type
wenzelm@26849:   constructor / class instance relations as ``already present'' for a
wenzelm@26849:   given target.  Omitting a ``\isa{{\isachardoublequote}{\isacharminus}{\isachardoublequote}}'' deletes an existing
wenzelm@26849:   ``already present'' declaration.  This applies only to
wenzelm@26849:   \emph{Haskell}.
wenzelm@26849: 
wenzelm@26907:   \item [\hyperlink{command.HOL.code-monad}{\mbox{\isa{\isacommand{code{\isacharunderscore}monad}}}}] provides an auxiliary
wenzelm@27834:   mechanism to generate monadic code for Haskell.
wenzelm@26849: 
wenzelm@26907:   \item [\hyperlink{command.HOL.code-reserved}{\mbox{\isa{\isacommand{code{\isacharunderscore}reserved}}}}] declares a list of names as
wenzelm@26849:   reserved for a given target, preventing it to be shadowed by any
wenzelm@26849:   generated code.
wenzelm@26849: 
wenzelm@26907:   \item [\hyperlink{command.HOL.code-include}{\mbox{\isa{\isacommand{code{\isacharunderscore}include}}}}] adds arbitrary named content
wenzelm@27834:   (``include'') to generated code.  A ``\isa{{\isachardoublequote}{\isacharminus}{\isachardoublequote}}'' as last argument
wenzelm@26849:   will remove an already added ``include''.
wenzelm@26849: 
wenzelm@26907:   \item [\hyperlink{command.HOL.code-modulename}{\mbox{\isa{\isacommand{code{\isacharunderscore}modulename}}}}] declares aliasings from
wenzelm@26849:   one module name onto another.
wenzelm@26849: 
haftmann@27103:   \item [\hyperlink{command.HOL.code-abort}{\mbox{\isa{\isacommand{code{\isacharunderscore}abort}}}}] declares constants which
haftmann@27452:   are not required to have a definition by means of defining equations;
haftmann@27103:   if needed these are implemented by program abort instead.
wenzelm@26849: 
haftmann@28562:   \item [\hyperlink{attribute.HOL.code}{\mbox{\isa{code}}}] explicitly selects (or
haftmann@28562:   with option ``\isa{{\isachardoublequote}del{\isachardoublequote}}'' deselects) a defining equation for
wenzelm@26849:   code generation.  Usually packages introducing defining equations
haftmann@27452:   provide a reasonable default setup for selection.
wenzelm@26849: 
wenzelm@26902:   \item [\hyperlink{attribute.HOL.code}{\mbox{\isa{code}}}\isa{inline}] declares (or with
haftmann@28562:   option ``\isa{{\isachardoublequote}del{\isachardoublequote}}'' removes) inlining theorems which are
wenzelm@26849:   applied as rewrite rules to any defining equation during
wenzelm@26849:   preprocessing.
wenzelm@26849: 
wenzelm@26907:   \item [\hyperlink{command.HOL.print-codesetup}{\mbox{\isa{\isacommand{print{\isacharunderscore}codesetup}}}}] gives an overview on
wenzelm@26849:   selected defining equations, code generator datatypes and
wenzelm@26849:   preprocessor setup.
wenzelm@26849: 
wenzelm@26849:   \end{descr}%
wenzelm@26849: \end{isamarkuptext}%
wenzelm@26849: \isamarkuptrue%
wenzelm@26849: %
wenzelm@27047: \isamarkupsection{Definition by specification \label{sec:hol-specification}%
wenzelm@27047: }
wenzelm@27047: \isamarkuptrue%
wenzelm@27047: %
wenzelm@27047: \begin{isamarkuptext}%
wenzelm@27047: \begin{matharray}{rcl}
wenzelm@27047:     \indexdef{HOL}{command}{specification}\hypertarget{command.HOL.specification}{\hyperlink{command.HOL.specification}{\mbox{\isa{\isacommand{specification}}}}} & : & \isartrans{theory}{proof(prove)} \\
wenzelm@27047:     \indexdef{HOL}{command}{ax\_specification}\hypertarget{command.HOL.ax-specification}{\hyperlink{command.HOL.ax-specification}{\mbox{\isa{\isacommand{ax{\isacharunderscore}specification}}}}} & : & \isartrans{theory}{proof(prove)} \\
wenzelm@27047:   \end{matharray}
wenzelm@27047: 
wenzelm@27047:   \begin{rail}
wenzelm@27047:   ('specification' | 'ax\_specification') '(' (decl +) ')' \\ (thmdecl? prop +)
wenzelm@27047:   ;
wenzelm@27047:   decl: ((name ':')? term '(' 'overloaded' ')'?)
wenzelm@27047:   \end{rail}
wenzelm@27047: 
wenzelm@27047:   \begin{descr}
wenzelm@27047: 
wenzelm@27047:   \item [\hyperlink{command.HOL.specification}{\mbox{\isa{\isacommand{specification}}}}~\isa{{\isachardoublequote}decls\ {\isasymphi}{\isachardoublequote}}] sets up a
wenzelm@27047:   goal stating the existence of terms with the properties specified to
wenzelm@27047:   hold for the constants given in \isa{decls}.  After finishing the
wenzelm@27047:   proof, the theory will be augmented with definitions for the given
wenzelm@27047:   constants, as well as with theorems stating the properties for these
wenzelm@27047:   constants.
wenzelm@27047: 
wenzelm@27047:   \item [\hyperlink{command.HOL.ax-specification}{\mbox{\isa{\isacommand{ax{\isacharunderscore}specification}}}}~\isa{{\isachardoublequote}decls\ {\isasymphi}{\isachardoublequote}}] sets
wenzelm@27047:   up a goal stating the existence of terms with the properties
wenzelm@27047:   specified to hold for the constants given in \isa{decls}.  After
wenzelm@27047:   finishing the proof, the theory will be augmented with axioms
wenzelm@27047:   expressing the properties given in the first place.
wenzelm@27047: 
wenzelm@27047:   \item [\isa{decl}] declares a constant to be defined by the
wenzelm@27047:   specification given.  The definition for the constant \isa{c} is
wenzelm@27047:   bound to the name \isa{c{\isacharunderscore}def} unless a theorem name is given in
wenzelm@27047:   the declaration.  Overloaded constants should be declared as such.
wenzelm@27047: 
wenzelm@27047:   \end{descr}
wenzelm@27047: 
wenzelm@27047:   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:   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:   user has explicitly proven it to be safe.  A practical issue must be
wenzelm@27047:   considered, though: After introducing two constants with the same
wenzelm@27047:   properties using \hyperlink{command.HOL.specification}{\mbox{\isa{\isacommand{specification}}}}, one can prove
wenzelm@27047:   that the two constants are, in fact, equal.  If this might be a
wenzelm@27047:   problem, one should use \hyperlink{command.HOL.ax-specification}{\mbox{\isa{\isacommand{ax{\isacharunderscore}specification}}}}.%
wenzelm@27047: \end{isamarkuptext}%
wenzelm@27047: \isamarkuptrue%
wenzelm@27047: %
wenzelm@26849: \isadelimtheory
wenzelm@26849: %
wenzelm@26849: \endisadelimtheory
wenzelm@26849: %
wenzelm@26849: \isatagtheory
wenzelm@26840: \isacommand{end}\isamarkupfalse%
wenzelm@26840: %
wenzelm@26840: \endisatagtheory
wenzelm@26840: {\isafoldtheory}%
wenzelm@26840: %
wenzelm@26840: \isadelimtheory
wenzelm@26840: %
wenzelm@26840: \endisadelimtheory
wenzelm@26849: \isanewline
wenzelm@26840: \end{isabellebody}%
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