--- a/doc-src/IsarRef/Thy/document/HOL_Specific.tex Thu May 08 12:27:19 2008 +0200
+++ b/doc-src/IsarRef/Thy/document/HOL_Specific.tex Thu May 08 12:29:18 2008 +0200
@@ -11,18 +11,1153 @@
\isatagtheory
\isacommand{theory}\isamarkupfalse%
\ HOL{\isacharunderscore}Specific\isanewline
-\isakeyword{imports}\ HOL\isanewline
-\isakeyword{begin}\isanewline
-\isanewline
+\isakeyword{imports}\ Main\isanewline
+\isakeyword{begin}%
+\endisatagtheory
+{\isafoldtheory}%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+%
+\isamarkupchapter{HOL specific elements \label{ch:logics}%
+}
+\isamarkuptrue%
+%
+\isamarkupsection{Primitive types \label{sec:hol-typedef}%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+\begin{matharray}{rcl}
+ \indexdef{HOL}{command}{typedecl}\mbox{\isa{\isacommand{typedecl}}} & : & \isartrans{theory}{theory} \\
+ \indexdef{HOL}{command}{typedef}\mbox{\isa{\isacommand{typedef}}} & : & \isartrans{theory}{proof(prove)} \\
+ \end{matharray}
+
+ \begin{rail}
+ 'typedecl' typespec infix?
+ ;
+ 'typedef' altname? abstype '=' repset
+ ;
+
+ altname: '(' (name | 'open' | 'open' name) ')'
+ ;
+ abstype: typespec infix?
+ ;
+ repset: term ('morphisms' name name)?
+ ;
+ \end{rail}
+
+ \begin{descr}
+
+ \item [\mbox{\isa{\isacommand{typedecl}}}~\isa{{\isachardoublequote}{\isacharparenleft}{\isasymalpha}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymalpha}\isactrlsub n{\isacharparenright}\ t{\isachardoublequote}}] is similar to the original \mbox{\isa{\isacommand{typedecl}}} of
+ Isabelle/Pure (see \secref{sec:types-pure}), but also declares type
+ arity \isa{{\isachardoublequote}t\ {\isacharcolon}{\isacharcolon}\ {\isacharparenleft}type{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ type{\isacharparenright}\ type{\isachardoublequote}}, making \isa{t} an
+ actual HOL type constructor. %FIXME check, update
+
+ \item [\mbox{\isa{\isacommand{typedef}}}~\isa{{\isachardoublequote}{\isacharparenleft}{\isasymalpha}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymalpha}\isactrlsub n{\isacharparenright}\ t\ {\isacharequal}\ A{\isachardoublequote}}] sets up a goal stating non-emptiness of the set \isa{A}.
+ After finishing the proof, the theory will be augmented by a
+ Gordon/HOL-style type definition, which establishes a bijection
+ between the representing set \isa{A} and the new type \isa{t}.
+
+ Technically, \mbox{\isa{\isacommand{typedef}}} defines both a type \isa{t} and a set (term constant) of the same name (an alternative base
+ name may be given in parentheses). The injection from type to set
+ is called \isa{Rep{\isacharunderscore}t}, its inverse \isa{Abs{\isacharunderscore}t} (this may be
+ changed via an explicit \mbox{\isa{\isakeyword{morphisms}}} declaration).
+
+ 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
+ corresponding injection/surjection pair (in both directions). Rules
+ \isa{Rep{\isacharunderscore}t{\isacharunderscore}inject} and \isa{Abs{\isacharunderscore}t{\isacharunderscore}inject} provide a slightly
+ more convenient view on the injectivity part, suitable for automated
+ proof tools (e.g.\ in \mbox{\isa{simp}} or \mbox{\isa{iff}} declarations).
+ Rules \isa{Rep{\isacharunderscore}t{\isacharunderscore}cases}/\isa{Rep{\isacharunderscore}t{\isacharunderscore}induct}, and \isa{Abs{\isacharunderscore}t{\isacharunderscore}cases}/\isa{Abs{\isacharunderscore}t{\isacharunderscore}induct} provide alternative views on
+ surjectivity; these are already declared as set or type rules for
+ the generic \mbox{\isa{cases}} and \mbox{\isa{induct}} methods.
+
+ An alternative name may be specified in parentheses; the default is
+ to use \isa{t} as indicated before. The ``\isa{{\isachardoublequote}{\isacharparenleft}open{\isacharparenright}{\isachardoublequote}}''
+ declaration suppresses a separate constant definition for the
+ representing set.
+
+ \end{descr}
+
+ Note that raw type declarations are rarely used in practice; the
+ main application is with experimental (or even axiomatic!) theory
+ fragments. Instead of primitive HOL type definitions, user-level
+ theories usually refer to higher-level packages such as \mbox{\isa{\isacommand{record}}} (see \secref{sec:hol-record}) or \mbox{\isa{\isacommand{datatype}}} (see \secref{sec:hol-datatype}).%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsection{Adhoc tuples%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+\begin{matharray}{rcl}
+ \mbox{\isa{split{\isacharunderscore}format}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isaratt \\
+ \end{matharray}
+
+ \begin{rail}
+ 'split\_format' (((name *) + 'and') | ('(' 'complete' ')'))
+ ;
+ \end{rail}
+
+ \begin{descr}
+
+ \item [\mbox{\isa{split{\isacharunderscore}format}}~\isa{{\isachardoublequote}p\isactrlsub {\isadigit{1}}\ {\isasymdots}\ p\isactrlsub m\ {\isasymAND}\ {\isasymdots}\ {\isasymAND}\ q\isactrlsub {\isadigit{1}}\ {\isasymdots}\ q\isactrlsub n{\isachardoublequote}}] puts expressions of
+ low-level tuple types into canonical form as specified by the
+ arguments given; the \isa{i}-th collection of arguments refers to
+ occurrences in premise \isa{i} of the rule. The ``\isa{{\isachardoublequote}{\isacharparenleft}complete{\isacharparenright}{\isachardoublequote}}'' option causes \emph{all} arguments in function
+ applications to be represented canonically according to their tuple
+ type structure.
+
+ Note that these operations tend to invent funny names for new local
+ parameters to be introduced.
+
+ \end{descr}%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsection{Records \label{sec:hol-record}%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+In principle, records merely generalize the concept of tuples, where
+ components may be addressed by labels instead of just position. The
+ logical infrastructure of records in Isabelle/HOL is slightly more
+ advanced, though, supporting truly extensible record schemes. This
+ admits operations that are polymorphic with respect to record
+ extension, yielding ``object-oriented'' effects like (single)
+ inheritance. See also \cite{NaraschewskiW-TPHOLs98} for more
+ details on object-oriented verification and record subtyping in HOL.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsubsection{Basic concepts%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+Isabelle/HOL supports both \emph{fixed} and \emph{schematic} records
+ at the level of terms and types. The notation is as follows:
+
+ \begin{center}
+ \begin{tabular}{l|l|l}
+ & record terms & record types \\ \hline
+ 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}} \\
+ schematic & \isa{{\isachardoublequote}{\isasymlparr}x\ {\isacharequal}\ a{\isacharcomma}\ y\ {\isacharequal}\ b{\isacharcomma}\ {\isasymdots}\ {\isacharequal}\ m{\isasymrparr}{\isachardoublequote}} &
+ \isa{{\isachardoublequote}{\isasymlparr}x\ {\isacharcolon}{\isacharcolon}\ A{\isacharcomma}\ y\ {\isacharcolon}{\isacharcolon}\ B{\isacharcomma}\ {\isasymdots}\ {\isacharcolon}{\isacharcolon}\ M{\isasymrparr}{\isachardoublequote}} \\
+ \end{tabular}
+ \end{center}
+
+ \noindent The ASCII representation of \isa{{\isachardoublequote}{\isasymlparr}x\ {\isacharequal}\ a{\isasymrparr}{\isachardoublequote}} is \isa{{\isachardoublequote}{\isacharparenleft}{\isacharbar}\ x\ {\isacharequal}\ a\ {\isacharbar}{\isacharparenright}{\isachardoublequote}}.
+
+ A fixed record \isa{{\isachardoublequote}{\isasymlparr}x\ {\isacharequal}\ a{\isacharcomma}\ y\ {\isacharequal}\ b{\isasymrparr}{\isachardoublequote}} has field \isa{x} of value
+ \isa{a} and field \isa{y} of value \isa{b}. The corresponding
+ 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}}
+ and \isa{{\isachardoublequote}b\ {\isacharcolon}{\isacharcolon}\ B{\isachardoublequote}}.
+
+ A record scheme like \isa{{\isachardoublequote}{\isasymlparr}x\ {\isacharequal}\ a{\isacharcomma}\ y\ {\isacharequal}\ b{\isacharcomma}\ {\isasymdots}\ {\isacharequal}\ m{\isasymrparr}{\isachardoublequote}} contains fields
+ \isa{x} and \isa{y} as before, but also possibly further fields
+ as indicated by the ``\isa{{\isachardoublequote}{\isasymdots}{\isachardoublequote}}'' notation (which is actually part
+ of the syntax). The improper field ``\isa{{\isachardoublequote}{\isasymdots}{\isachardoublequote}}'' of a record
+ scheme is called the \emph{more part}. Logically it is just a free
+ variable, which is occasionally referred to as ``row variable'' in
+ the literature. The more part of a record scheme may be
+ instantiated by zero or more further components. For example, the
+ 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}{\isachardoublequote}}, where \isa{m{\isacharprime}} refers to a different more part.
+ Fixed records are special instances of record schemes, where
+ ``\isa{{\isachardoublequote}{\isasymdots}{\isachardoublequote}}'' is properly terminated by the \isa{{\isachardoublequote}{\isacharparenleft}{\isacharparenright}\ {\isacharcolon}{\isacharcolon}\ unit{\isachardoublequote}}
+ element. In fact, \isa{{\isachardoublequote}{\isasymlparr}x\ {\isacharequal}\ a{\isacharcomma}\ y\ {\isacharequal}\ b{\isasymrparr}{\isachardoublequote}} is just an abbreviation
+ for \isa{{\isachardoublequote}{\isasymlparr}x\ {\isacharequal}\ a{\isacharcomma}\ y\ {\isacharequal}\ b{\isacharcomma}\ {\isasymdots}\ {\isacharequal}\ {\isacharparenleft}{\isacharparenright}{\isasymrparr}{\isachardoublequote}}.
+
+ \medskip Two key observations make extensible records in a simply
+ typed language like HOL work out:
+
+ \begin{enumerate}
+
+ \item the more part is internalized, as a free term or type
+ variable,
+
+ \item field names are externalized, they cannot be accessed within the logic
+ as first-class values.
+
+ \end{enumerate}
+
+ \medskip In Isabelle/HOL record types have to be defined explicitly,
+ fixing their field names and types, and their (optional) parent
+ record. Afterwards, records may be formed using above syntax, while
+ obeying the canonical order of fields as given by their declaration.
+ The record package provides several standard operations like
+ selectors and updates. The common setup for various generic proof
+ tools enable succinct reasoning patterns. See also the Isabelle/HOL
+ tutorial \cite{isabelle-hol-book} for further instructions on using
+ records in practice.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsubsection{Record specifications%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+\begin{matharray}{rcl}
+ \indexdef{HOL}{command}{record}\mbox{\isa{\isacommand{record}}} & : & \isartrans{theory}{theory} \\
+ \end{matharray}
+
+ \begin{rail}
+ 'record' typespec '=' (type '+')? (constdecl +)
+ ;
+ \end{rail}
+
+ \begin{descr}
+
+ \item [\mbox{\isa{\isacommand{record}}}~\isa{{\isachardoublequote}{\isacharparenleft}{\isasymalpha}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymalpha}\isactrlsub m{\isacharparenright}\ t\ {\isacharequal}\ {\isasymtau}\ {\isacharplus}\ c\isactrlsub {\isadigit{1}}\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}\isactrlsub {\isadigit{1}}\ {\isasymdots}\ c\isactrlsub n\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}\isactrlsub n{\isachardoublequote}}] defines
+ extensible record type \isa{{\isachardoublequote}{\isacharparenleft}{\isasymalpha}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymalpha}\isactrlsub m{\isacharparenright}\ t{\isachardoublequote}},
+ derived from the optional parent record \isa{{\isachardoublequote}{\isasymtau}{\isachardoublequote}} by adding new
+ field components \isa{{\isachardoublequote}c\isactrlsub i\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}\isactrlsub i{\isachardoublequote}} etc.
+
+ The type variables of \isa{{\isachardoublequote}{\isasymtau}{\isachardoublequote}} and \isa{{\isachardoublequote}{\isasymsigma}\isactrlsub i{\isachardoublequote}} need to be
+ 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
+ least one new field \isa{{\isachardoublequote}c\isactrlsub i{\isachardoublequote}} has to be specified.
+ Basically, field names need to belong to a unique record. This is
+ not a real restriction in practice, since fields are qualified by
+ the record name internally.
+
+ The parent record specification \isa{{\isasymtau}} is optional; if omitted
+ \isa{t} becomes a root record. The hierarchy of all records
+ declared within a theory context forms a forest structure, i.e.\ a
+ set of trees starting with a root record each. There is no way to
+ merge multiple parent records!
+
+ For convenience, \isa{{\isachardoublequote}{\isacharparenleft}{\isasymalpha}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymalpha}\isactrlsub m{\isacharparenright}\ t{\isachardoublequote}} is made a
+ 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
+ \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}}.
+
+ \end{descr}%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsubsection{Record operations%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+Any record definition of the form presented above produces certain
+ standard operations. Selectors and updates are provided for any
+ field, including the improper one ``\isa{more}''. There are also
+ cumulative record constructor functions. To simplify the
+ 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}}.
+
+ \medskip \textbf{Selectors} and \textbf{updates} are available for
+ any field (including ``\isa{more}''):
+
+ \begin{matharray}{lll}
+ \isa{{\isachardoublequote}c\isactrlsub i{\isachardoublequote}} & \isa{{\isachardoublequote}{\isacharcolon}{\isacharcolon}{\isachardoublequote}} & \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}\ {\isasymRightarrow}\ {\isasymsigma}\isactrlsub i{\isachardoublequote}} \\
+ \isa{{\isachardoublequote}c\isactrlsub i{\isacharunderscore}update{\isachardoublequote}} & \isa{{\isachardoublequote}{\isacharcolon}{\isacharcolon}{\isachardoublequote}} & \isa{{\isachardoublequote}{\isasymsigma}\isactrlsub i\ {\isasymRightarrow}\ {\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}\ {\isasymRightarrow}\ {\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}} \\
+ \end{matharray}
+
+ 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
+ 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
+ because of postfix notation the order of fields shown here is
+ reverse than in the actual term. Since repeated updates are just
+ 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.
+ Thus commutativity of independent updates can be proven within the
+ logic for any two fields, but not as a general theorem.
+
+ \medskip The \textbf{make} operation provides a cumulative record
+ constructor function:
+
+ \begin{matharray}{lll}
+ \isa{{\isachardoublequote}t{\isachardot}make{\isachardoublequote}} & \isa{{\isachardoublequote}{\isacharcolon}{\isacharcolon}{\isachardoublequote}} & \isa{{\isachardoublequote}{\isasymsigma}\isactrlsub {\isadigit{1}}\ {\isasymRightarrow}\ {\isasymdots}\ {\isasymsigma}\isactrlsub n\ {\isasymRightarrow}\ {\isasymlparr}c\isactrlsub {\isadigit{1}}\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ c\isactrlsub n\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}\isactrlsub n{\isasymrparr}{\isachardoublequote}} \\
+ \end{matharray}
+
+ \medskip We now reconsider the case of non-root records, which are
+ derived of some parent. In general, the latter may depend on
+ another parent as well, resulting in a list of \emph{ancestor
+ records}. Appending the lists of fields of all ancestors results in
+ a certain field prefix. The record package automatically takes care
+ of this by lifting operations over this context of ancestor fields.
+ Assuming that \isa{{\isachardoublequote}{\isacharparenleft}{\isasymalpha}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ {\isasymalpha}\isactrlsub m{\isacharparenright}\ t{\isachardoublequote}} has ancestor
+ 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}},
+ the above record operations will get the following types:
+
+ \begin{matharray}{lll}
+ \isa{{\isachardoublequote}c\isactrlsub i{\isachardoublequote}} & \isa{{\isachardoublequote}{\isacharcolon}{\isacharcolon}{\isachardoublequote}} & \isa{{\isachardoublequote}{\isasymlparr}b\isactrlsub {\isadigit{1}}\ {\isacharcolon}{\isacharcolon}\ {\isasymrho}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ b\isactrlsub k\ {\isacharcolon}{\isacharcolon}\ {\isasymrho}\isactrlsub k{\isacharcomma}\ 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}\ {\isasymRightarrow}\ {\isasymsigma}\isactrlsub i{\isachardoublequote}} \\
+ \isa{{\isachardoublequote}c\isactrlsub i{\isacharunderscore}update{\isachardoublequote}} & \isa{{\isachardoublequote}{\isacharcolon}{\isacharcolon}{\isachardoublequote}} & \isa{{\isachardoublequote}{\isasymsigma}\isactrlsub i\ {\isasymRightarrow}\ {\isasymlparr}b\isactrlsub {\isadigit{1}}\ {\isacharcolon}{\isacharcolon}\ {\isasymrho}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ b\isactrlsub k\ {\isacharcolon}{\isacharcolon}\ {\isasymrho}\isactrlsub k{\isacharcomma}\ 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}\ {\isasymRightarrow}\ {\isasymlparr}b\isactrlsub {\isadigit{1}}\ {\isacharcolon}{\isacharcolon}\ {\isasymrho}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ b\isactrlsub k\ {\isacharcolon}{\isacharcolon}\ {\isasymrho}\isactrlsub k{\isacharcomma}\ 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}} \\
+ \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}b\isactrlsub {\isadigit{1}}\ {\isacharcolon}{\isacharcolon}\ {\isasymrho}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ b\isactrlsub k\ {\isacharcolon}{\isacharcolon}\ {\isasymrho}\isactrlsub k{\isacharcomma}\ c\isactrlsub {\isadigit{1}}\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ c\isactrlsub n\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}\isactrlsub n{\isachardoublequote}} \\
+ \end{matharray}
+ \noindent
+
+ \medskip Some further operations address the extension aspect of a
+ derived record scheme specifically: \isa{{\isachardoublequote}t{\isachardot}fields{\isachardoublequote}} produces a
+ record fragment consisting of exactly the new fields introduced here
+ (the result may serve as a more part elsewhere); \isa{{\isachardoublequote}t{\isachardot}extend{\isachardoublequote}}
+ takes a fixed record and adds a given more part; \isa{{\isachardoublequote}t{\isachardot}truncate{\isachardoublequote}} restricts a record scheme to a fixed record.
+
+ \begin{matharray}{lll}
+ \isa{{\isachardoublequote}t{\isachardot}fields{\isachardoublequote}} & \isa{{\isachardoublequote}{\isacharcolon}{\isacharcolon}{\isachardoublequote}} & \isa{{\isachardoublequote}{\isasymsigma}\isactrlsub {\isadigit{1}}\ {\isasymRightarrow}\ {\isasymdots}\ {\isasymsigma}\isactrlsub n\ {\isasymRightarrow}\ {\isasymlparr}c\isactrlsub {\isadigit{1}}\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ c\isactrlsub n\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}\isactrlsub n{\isachardoublequote}} \\
+ \isa{{\isachardoublequote}t{\isachardot}extend{\isachardoublequote}} & \isa{{\isachardoublequote}{\isacharcolon}{\isacharcolon}{\isachardoublequote}} & \isa{{\isachardoublequote}{\isasymlparr}b\isactrlsub {\isadigit{1}}\ {\isacharcolon}{\isacharcolon}\ {\isasymrho}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ b\isactrlsub k\ {\isacharcolon}{\isacharcolon}\ {\isasymrho}\isactrlsub k{\isacharcomma}\ c\isactrlsub {\isadigit{1}}\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ c\isactrlsub n\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}\isactrlsub n{\isasymrparr}\ {\isasymRightarrow}\ {\isasymzeta}\ {\isasymRightarrow}\ {\isasymlparr}b\isactrlsub {\isadigit{1}}\ {\isacharcolon}{\isacharcolon}\ {\isasymrho}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ b\isactrlsub k\ {\isacharcolon}{\isacharcolon}\ {\isasymrho}\isactrlsub k{\isacharcomma}\ 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}} \\
+ \isa{{\isachardoublequote}t{\isachardot}truncate{\isachardoublequote}} & \isa{{\isachardoublequote}{\isacharcolon}{\isacharcolon}{\isachardoublequote}} & \isa{{\isachardoublequote}{\isasymlparr}b\isactrlsub {\isadigit{1}}\ {\isacharcolon}{\isacharcolon}\ {\isasymrho}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ b\isactrlsub k\ {\isacharcolon}{\isacharcolon}\ {\isasymrho}\isactrlsub k{\isacharcomma}\ 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}\ {\isasymRightarrow}\ {\isasymlparr}b\isactrlsub {\isadigit{1}}\ {\isacharcolon}{\isacharcolon}\ {\isasymrho}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ b\isactrlsub k\ {\isacharcolon}{\isacharcolon}\ {\isasymrho}\isactrlsub k{\isacharcomma}\ c\isactrlsub {\isadigit{1}}\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ c\isactrlsub n\ {\isacharcolon}{\isacharcolon}\ {\isasymsigma}\isactrlsub n{\isasymrparr}{\isachardoublequote}} \\
+ \end{matharray}
+
+ \noindent Note that \isa{{\isachardoublequote}t{\isachardot}make{\isachardoublequote}} and \isa{{\isachardoublequote}t{\isachardot}fields{\isachardoublequote}} coincide
+ for root records.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsubsection{Derived rules and proof tools%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+The record package proves several results internally, declaring
+ these facts to appropriate proof tools. This enables users to
+ reason about record structures quite conveniently. Assume that
+ \isa{t} is a record type as specified above.
+
+ \begin{enumerate}
+
+ \item Standard conversions for selectors or updates applied to
+ record constructor terms are made part of the default Simplifier
+ context; thus proofs by reduction of basic operations merely require
+ the \mbox{\isa{simp}} method without further arguments. These rules
+ are available as \isa{{\isachardoublequote}t{\isachardot}simps{\isachardoublequote}}, too.
+
+ \item Selectors applied to updated records are automatically reduced
+ by an internal simplification procedure, which is also part of the
+ standard Simplifier setup.
+
+ \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
+ Reasoner as \mbox{\isa{iff}} rules. These rules are available as
+ \isa{{\isachardoublequote}t{\isachardot}iffs{\isachardoublequote}}.
+
+ \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,
+ and as the basic rule context as ``\mbox{\isa{intro}}\isa{{\isachardoublequote}{\isacharquery}{\isachardoublequote}}''.
+ The rule is called \isa{{\isachardoublequote}t{\isachardot}equality{\isachardoublequote}}.
+
+ \item Representations of arbitrary record expressions as canonical
+ constructor terms are provided both in \mbox{\isa{cases}} and \mbox{\isa{induct}} format (cf.\ the generic proof methods of the same name,
+ \secref{sec:cases-induct}). Several variations are available, for
+ fixed records, record schemes, more parts etc.
+
+ The generic proof methods are sufficiently smart to pick the most
+ sensible rule according to the type of the indicated record
+ expression: users just need to apply something like ``\isa{{\isachardoublequote}{\isacharparenleft}cases\ r{\isacharparenright}{\isachardoublequote}}'' to a certain proof problem.
+
+ \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}
+ treated automatically, but usually need to be expanded by hand,
+ using the collective fact \isa{{\isachardoublequote}t{\isachardot}defs{\isachardoublequote}}.
+
+ \end{enumerate}%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsection{Datatypes \label{sec:hol-datatype}%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+\begin{matharray}{rcl}
+ \indexdef{HOL}{command}{datatype}\mbox{\isa{\isacommand{datatype}}} & : & \isartrans{theory}{theory} \\
+ \indexdef{HOL}{command}{rep-datatype}\mbox{\isa{\isacommand{rep{\isacharunderscore}datatype}}} & : & \isartrans{theory}{theory} \\
+ \end{matharray}
+
+ \begin{rail}
+ 'datatype' (dtspec + 'and')
+ ;
+ 'rep\_datatype' (name *) dtrules
+ ;
+
+ dtspec: parname? typespec infix? '=' (cons + '|')
+ ;
+ cons: name (type *) mixfix?
+ ;
+ dtrules: 'distinct' thmrefs 'inject' thmrefs 'induction' thmrefs
+ \end{rail}
+
+ \begin{descr}
+
+ \item [\mbox{\isa{\isacommand{datatype}}}] defines inductive datatypes in
+ HOL.
+
+ \item [\mbox{\isa{\isacommand{rep{\isacharunderscore}datatype}}}] represents existing types as
+ inductive ones, generating the standard infrastructure of derived
+ concepts (primitive recursion etc.).
+
+ \end{descr}
+
+ The induction and exhaustion theorems generated provide case names
+ according to the constructors involved, while parameters are named
+ after the types (see also \secref{sec:cases-induct}).
+
+ See \cite{isabelle-HOL} for more details on datatypes, but beware of
+ the old-style theory syntax being used there! Apart from proper
+ proof methods for case-analysis and induction, there are also
+ emulations of ML tactics \mbox{\isa{case{\isacharunderscore}tac}} and \mbox{\isa{induct{\isacharunderscore}tac}} available, see \secref{sec:hol-induct-tac}; these admit
+ to refer directly to the internal structure of subgoals (including
+ internally bound parameters).%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsection{Recursive functions \label{sec:recursion}%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+\begin{matharray}{rcl}
+ \indexdef{HOL}{command}{primrec}\mbox{\isa{\isacommand{primrec}}} & : & \isarkeep{local{\dsh}theory} \\
+ \indexdef{HOL}{command}{fun}\mbox{\isa{\isacommand{fun}}} & : & \isarkeep{local{\dsh}theory} \\
+ \indexdef{HOL}{command}{function}\mbox{\isa{\isacommand{function}}} & : & \isartrans{local{\dsh}theory}{proof(prove)} \\
+ \indexdef{HOL}{command}{termination}\mbox{\isa{\isacommand{termination}}} & : & \isartrans{local{\dsh}theory}{proof(prove)} \\
+ \end{matharray}
+
+ \railalias{funopts}{function\_opts} %FIXME ??
+
+ \begin{rail}
+ 'primrec' target? fixes 'where' equations
+ ;
+ equations: (thmdecl? prop + '|')
+ ;
+ ('fun' | 'function') (funopts)? fixes 'where' clauses
+ ;
+ clauses: (thmdecl? prop ('(' 'otherwise' ')')? + '|')
+ ;
+ funopts: '(' (('sequential' | 'in' name | 'domintros' | 'tailrec' |
+ 'default' term) + ',') ')'
+ ;
+ 'termination' ( term )?
+ \end{rail}
+
+ \begin{descr}
+
+ \item [\mbox{\isa{\isacommand{primrec}}}] defines primitive recursive
+ functions over datatypes, see also \cite{isabelle-HOL}.
+
+ \item [\mbox{\isa{\isacommand{function}}}] defines functions by general
+ wellfounded recursion. A detailed description with examples can be
+ found in \cite{isabelle-function}. The function is specified by a
+ set of (possibly conditional) recursive equations with arbitrary
+ pattern matching. The command generates proof obligations for the
+ completeness and the compatibility of patterns.
+
+ The defined function is considered partial, and the resulting
+ simplification rules (named \isa{{\isachardoublequote}f{\isachardot}psimps{\isachardoublequote}}) and induction rule
+ (named \isa{{\isachardoublequote}f{\isachardot}pinduct{\isachardoublequote}}) are guarded by a generated domain
+ predicate \isa{{\isachardoublequote}f{\isacharunderscore}dom{\isachardoublequote}}. The \mbox{\isa{\isacommand{termination}}}
+ command can then be used to establish that the function is total.
+
+ \item [\mbox{\isa{\isacommand{fun}}}] is a shorthand notation for
+ ``\mbox{\isa{\isacommand{function}}}~\isa{{\isachardoublequote}{\isacharparenleft}sequential{\isacharparenright}{\isachardoublequote}}, followed by
+ automated proof attempts regarding pattern matching and termination.
+ See \cite{isabelle-function} for further details.
+
+ \item [\mbox{\isa{\isacommand{termination}}}~\isa{f}] commences a
+ termination proof for the previously defined function \isa{f}. If
+ this is omitted, the command refers to the most recent function
+ definition. After the proof is closed, the recursive equations and
+ the induction principle is established.
+
+ \end{descr}
+
+ %FIXME check
+
+ Recursive definitions introduced by both the \mbox{\isa{\isacommand{primrec}}} and the \mbox{\isa{\isacommand{function}}} command accommodate
+ 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)
+ refers to a specific induction rule, with parameters named according
+ to the user-specified equations. Case names of \mbox{\isa{\isacommand{primrec}}} are that of the datatypes involved, while those of
+ \mbox{\isa{\isacommand{function}}} are numbered (starting from 1).
+
+ The equations provided by these packages may be referred later as
+ theorem list \isa{{\isachardoublequote}f{\isachardot}simps{\isachardoublequote}}, where \isa{f} is the (collective)
+ name of the functions defined. Individual equations may be named
+ explicitly as well.
+
+ The \mbox{\isa{\isacommand{function}}} command accepts the following
+ options.
+
+ \begin{descr}
+
+ \item [\isa{sequential}] enables a preprocessor which
+ disambiguates overlapping patterns by making them mutually disjoint.
+ Earlier equations take precedence over later ones. This allows to
+ give the specification in a format very similar to functional
+ programming. Note that the resulting simplification and induction
+ rules correspond to the transformed specification, not the one given
+ originally. This usually means that each equation given by the user
+ may result in several theroems. Also note that this automatic
+ transformation only works for ML-style datatype patterns.
+
+ \item [\isa{{\isachardoublequote}{\isasymIN}\ name{\isachardoublequote}}] gives the target for the definition.
+ %FIXME ?!?
+
+ \item [\isa{domintros}] enables the automated generation of
+ introduction rules for the domain predicate. While mostly not
+ needed, they can be helpful in some proofs about partial functions.
+
+ \item [\isa{tailrec}] generates the unconstrained recursive
+ equations even without a termination proof, provided that the
+ function is tail-recursive. This currently only works
+
+ \item [\isa{{\isachardoublequote}default\ d{\isachardoublequote}}] allows to specify a default value for a
+ (partial) function, which will ensure that \isa{{\isachardoublequote}f\ x\ {\isacharequal}\ d\ x{\isachardoublequote}}
+ whenever \isa{{\isachardoublequote}x\ {\isasymnotin}\ f{\isacharunderscore}dom{\isachardoublequote}}.
+
+ \end{descr}%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsubsection{Proof methods related to recursive definitions%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+\begin{matharray}{rcl}
+ \indexdef{HOL}{method}{pat-completeness}\mbox{\isa{pat{\isacharunderscore}completeness}} & : & \isarmeth \\
+ \indexdef{HOL}{method}{relation}\mbox{\isa{relation}} & : & \isarmeth \\
+ \indexdef{HOL}{method}{lexicographic-order}\mbox{\isa{lexicographic{\isacharunderscore}order}} & : & \isarmeth \\
+ \end{matharray}
+
+ \begin{rail}
+ 'relation' term
+ ;
+ 'lexicographic\_order' (clasimpmod *)
+ ;
+ \end{rail}
+
+ \begin{descr}
+
+ \item [\mbox{\isa{pat{\isacharunderscore}completeness}}] is a specialized method to
+ solve goals regarding the completeness of pattern matching, as
+ required by the \mbox{\isa{\isacommand{function}}} package (cf.\
+ \cite{isabelle-function}).
+
+ \item [\mbox{\isa{relation}}~\isa{R}] introduces a termination
+ proof using the relation \isa{R}. The resulting proof state will
+ contain goals expressing that \isa{R} is wellfounded, and that the
+ arguments of recursive calls decrease with respect to \isa{R}.
+ Usually, this method is used as the initial proof step of manual
+ termination proofs.
+
+ \item [\mbox{\isa{lexicographic{\isacharunderscore}order}}] attempts a fully
+ automated termination proof by searching for a lexicographic
+ combination of size measures on the arguments of the function. The
+ method accepts the same arguments as the \mbox{\isa{auto}} method,
+ which it uses internally to prove local descents. The same context
+ modifiers as for \mbox{\isa{auto}} are accepted, see
+ \secref{sec:clasimp}.
+
+ In case of failure, extensive information is printed, which can help
+ to analyse the situation (cf.\ \cite{isabelle-function}).
+
+ \end{descr}%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsubsection{Old-style recursive function definitions (TFL)%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+The old TFL commands \mbox{\isa{\isacommand{recdef}}} and \mbox{\isa{\isacommand{recdef{\isacharunderscore}tc}}} for defining recursive are mostly obsolete; \mbox{\isa{\isacommand{function}}} or \mbox{\isa{\isacommand{fun}}} should be used instead.
+
+ \begin{matharray}{rcl}
+ \indexdef{HOL}{command}{recdef}\mbox{\isa{\isacommand{recdef}}} & : & \isartrans{theory}{theory} \\
+ \indexdef{HOL}{command}{recdef-tc}\mbox{\isa{\isacommand{recdef{\isacharunderscore}tc}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isartrans{theory}{proof(prove)} \\
+ \end{matharray}
+
+ \begin{rail}
+ 'recdef' ('(' 'permissive' ')')? \\ name term (prop +) hints?
+ ;
+ recdeftc thmdecl? tc
+ ;
+ hints: '(' 'hints' (recdefmod *) ')'
+ ;
+ recdefmod: (('recdef\_simp' | 'recdef\_cong' | 'recdef\_wf') (() | 'add' | 'del') ':' thmrefs) | clasimpmod
+ ;
+ tc: nameref ('(' nat ')')?
+ ;
+ \end{rail}
+
+ \begin{descr}
+
+ \item [\mbox{\isa{\isacommand{recdef}}}] defines general well-founded
+ recursive functions (using the TFL package), see also
+ \cite{isabelle-HOL}. The ``\isa{{\isachardoublequote}{\isacharparenleft}permissive{\isacharparenright}{\isachardoublequote}}'' option tells
+ TFL to recover from failed proof attempts, returning unfinished
+ 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
+ automated proof process of TFL. Additional \mbox{\isa{clasimpmod}}
+ declarations (cf.\ \secref{sec:clasimp}) may be given to tune the
+ context of the Simplifier (cf.\ \secref{sec:simplifier}) and
+ Classical reasoner (cf.\ \secref{sec:classical}).
+
+ \item [\mbox{\isa{\isacommand{recdef{\isacharunderscore}tc}}}~\isa{{\isachardoublequote}c\ {\isacharparenleft}i{\isacharparenright}{\isachardoublequote}}] recommences the
+ proof for leftover termination condition number \isa{i} (default
+ 1) as generated by a \mbox{\isa{\isacommand{recdef}}} definition of
+ constant \isa{c}.
+
+ Note that in most cases, \mbox{\isa{\isacommand{recdef}}} is able to finish
+ its internal proofs without manual intervention.
+
+ \end{descr}
+
+ \medskip Hints for \mbox{\isa{\isacommand{recdef}}} may be also declared
+ globally, using the following attributes.
+
+ \begin{matharray}{rcl}
+ \indexdef{HOL}{attribute}{recdef-simp}\mbox{\isa{recdef{\isacharunderscore}simp}} & : & \isaratt \\
+ \indexdef{HOL}{attribute}{recdef-cong}\mbox{\isa{recdef{\isacharunderscore}cong}} & : & \isaratt \\
+ \indexdef{HOL}{attribute}{recdef-wf}\mbox{\isa{recdef{\isacharunderscore}wf}} & : & \isaratt \\
+ \end{matharray}
+
+ \begin{rail}
+ ('recdef\_simp' | 'recdef\_cong' | 'recdef\_wf') (() | 'add' | 'del')
+ ;
+ \end{rail}%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsection{Definition by specification \label{sec:hol-specification}%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+\begin{matharray}{rcl}
+ \indexdef{HOL}{command}{specification}\mbox{\isa{\isacommand{specification}}} & : & \isartrans{theory}{proof(prove)} \\
+ \indexdef{HOL}{command}{ax-specification}\mbox{\isa{\isacommand{ax{\isacharunderscore}specification}}} & : & \isartrans{theory}{proof(prove)} \\
+ \end{matharray}
+
+ \begin{rail}
+ ('specification' | 'ax\_specification') '(' (decl +) ')' \\ (thmdecl? prop +)
+ ;
+ decl: ((name ':')? term '(' 'overloaded' ')'?)
+ \end{rail}
+
+ \begin{descr}
+
+ \item [\mbox{\isa{\isacommand{specification}}}~\isa{{\isachardoublequote}decls\ {\isasymphi}{\isachardoublequote}}] sets up a
+ goal stating the existence of terms with the properties specified to
+ hold for the constants given in \isa{decls}. After finishing the
+ proof, the theory will be augmented with definitions for the given
+ constants, as well as with theorems stating the properties for these
+ constants.
+
+ \item [\mbox{\isa{\isacommand{ax{\isacharunderscore}specification}}}~\isa{{\isachardoublequote}decls\ {\isasymphi}{\isachardoublequote}}] sets
+ up a goal stating the existence of terms with the properties
+ specified to hold for the constants given in \isa{decls}. After
+ finishing the proof, the theory will be augmented with axioms
+ expressing the properties given in the first place.
+
+ \item [\isa{decl}] declares a constant to be defined by the
+ specification given. The definition for the constant \isa{c} is
+ bound to the name \isa{c{\isacharunderscore}def} unless a theorem name is given in
+ the declaration. Overloaded constants should be declared as such.
+
+ \end{descr}
+
+ Whether to use \mbox{\isa{\isacommand{specification}}} or \mbox{\isa{\isacommand{ax{\isacharunderscore}specification}}} is to some extent a matter of style. \mbox{\isa{\isacommand{specification}}} introduces no new axioms, and so by
+ construction cannot introduce inconsistencies, whereas \mbox{\isa{\isacommand{ax{\isacharunderscore}specification}}} does introduce axioms, but only after the
+ user has explicitly proven it to be safe. A practical issue must be
+ considered, though: After introducing two constants with the same
+ properties using \mbox{\isa{\isacommand{specification}}}, one can prove
+ that the two constants are, in fact, equal. If this might be a
+ problem, one should use \mbox{\isa{\isacommand{ax{\isacharunderscore}specification}}}.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsection{Inductive and coinductive definitions \label{sec:hol-inductive}%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+An \textbf{inductive definition} specifies the least predicate (or
+ set) \isa{R} closed under given rules: applying a rule to elements
+ of \isa{R} yields a result within \isa{R}. For example, a
+ structural operational semantics is an inductive definition of an
+ evaluation relation.
+
+ Dually, a \textbf{coinductive definition} specifies the greatest
+ predicate~/ set \isa{R} that is consistent with given rules: every
+ element of \isa{R} can be seen as arising by applying a rule to
+ elements of \isa{R}. An important example is using bisimulation
+ relations to formalise equivalence of processes and infinite data
+ structures.
+
+ \medskip The HOL package is related to the ZF one, which is
+ described in a separate paper,\footnote{It appeared in CADE
+ \cite{paulson-CADE}; a longer version is distributed with Isabelle.}
+ which you should refer to in case of difficulties. The package is
+ simpler than that of ZF thanks to implicit type-checking in HOL.
+ The types of the (co)inductive predicates (or sets) determine the
+ domain of the fixedpoint definition, and the package does not have
+ to use inference rules for type-checking.
+
+ \begin{matharray}{rcl}
+ \indexdef{HOL}{command}{inductive}\mbox{\isa{\isacommand{inductive}}} & : & \isarkeep{local{\dsh}theory} \\
+ \indexdef{HOL}{command}{inductive-set}\mbox{\isa{\isacommand{inductive{\isacharunderscore}set}}} & : & \isarkeep{local{\dsh}theory} \\
+ \indexdef{HOL}{command}{coinductive}\mbox{\isa{\isacommand{coinductive}}} & : & \isarkeep{local{\dsh}theory} \\
+ \indexdef{HOL}{command}{coinductive-set}\mbox{\isa{\isacommand{coinductive{\isacharunderscore}set}}} & : & \isarkeep{local{\dsh}theory} \\
+ \indexdef{HOL}{attribute}{mono}\mbox{\isa{mono}} & : & \isaratt \\
+ \end{matharray}
+
+ \begin{rail}
+ ('inductive' | 'inductive\_set' | 'coinductive' | 'coinductive\_set') target? fixes ('for' fixes)? \\
+ ('where' clauses)? ('monos' thmrefs)?
+ ;
+ clauses: (thmdecl? prop + '|')
+ ;
+ 'mono' (() | 'add' | 'del')
+ ;
+ \end{rail}
+
+ \begin{descr}
+
+ \item [\mbox{\isa{\isacommand{inductive}}} and \mbox{\isa{\isacommand{coinductive}}}] define (co)inductive predicates from the
+ introduction rules given in the \mbox{\isa{\isakeyword{where}}} part. The
+ optional \mbox{\isa{\isakeyword{for}}} part contains a list of parameters of the
+ (co)inductive predicates that remain fixed throughout the
+ definition. The optional \mbox{\isa{\isakeyword{monos}}} section contains
+ \emph{monotonicity theorems}, which are required for each operator
+ applied to a recursive set in the introduction rules. There
+ \emph{must} be a theorem of the form \isa{{\isachardoublequote}A\ {\isasymle}\ B\ {\isasymLongrightarrow}\ M\ A\ {\isasymle}\ M\ B{\isachardoublequote}},
+ for each premise \isa{{\isachardoublequote}M\ R\isactrlsub i\ t{\isachardoublequote}} in an introduction rule!
+
+ \item [\mbox{\isa{\isacommand{inductive{\isacharunderscore}set}}} and \mbox{\isa{\isacommand{coinductive{\isacharunderscore}set}}}] are wrappers for to the previous commands,
+ allowing the definition of (co)inductive sets.
+
+ \item [\mbox{\isa{mono}}] declares monotonicity rules. These
+ rule are involved in the automated monotonicity proof of \mbox{\isa{\isacommand{inductive}}}.
+
+ \end{descr}%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsubsection{Derived rules%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+Each (co)inductive definition \isa{R} adds definitions to the
+ theory and also proves some theorems:
+
+ \begin{description}
+
+ \item [\isa{R{\isachardot}intros}] is the list of introduction rules as proven
+ theorems, for the recursive predicates (or sets). The rules are
+ also available individually, using the names given them in the
+ theory file;
+
+ \item [\isa{R{\isachardot}cases}] is the case analysis (or elimination) rule;
+
+ \item [\isa{R{\isachardot}induct} or \isa{R{\isachardot}coinduct}] is the (co)induction
+ rule.
+
+ \end{description}
+
+ When several predicates \isa{{\isachardoublequote}R\isactrlsub {\isadigit{1}}{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ R\isactrlsub n{\isachardoublequote}} are
+ defined simultaneously, the list of introduction rules is called
+ \isa{{\isachardoublequote}R\isactrlsub {\isadigit{1}}{\isacharunderscore}{\isasymdots}{\isacharunderscore}R\isactrlsub n{\isachardot}intros{\isachardoublequote}}, the case analysis rules are
+ called \isa{{\isachardoublequote}R\isactrlsub {\isadigit{1}}{\isachardot}cases{\isacharcomma}\ {\isasymdots}{\isacharcomma}\ R\isactrlsub n{\isachardot}cases{\isachardoublequote}}, and the list
+ of mutual induction rules is called \isa{{\isachardoublequote}R\isactrlsub {\isadigit{1}}{\isacharunderscore}{\isasymdots}{\isacharunderscore}R\isactrlsub n{\isachardot}inducts{\isachardoublequote}}.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsubsection{Monotonicity theorems%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+Each theory contains a default set of theorems that are used in
+ monotonicity proofs. New rules can be added to this set via the
+ \mbox{\isa{mono}} attribute. The HOL theory \isa{Inductive}
+ shows how this is done. In general, the following monotonicity
+ theorems may be added:
+
+ \begin{itemize}
+
+ \item Theorems of the form \isa{{\isachardoublequote}A\ {\isasymle}\ B\ {\isasymLongrightarrow}\ M\ A\ {\isasymle}\ M\ B{\isachardoublequote}}, for proving
+ monotonicity of inductive definitions whose introduction rules have
+ premises involving terms such as \isa{{\isachardoublequote}M\ R\isactrlsub i\ t{\isachardoublequote}}.
+
+ \item Monotonicity theorems for logical operators, which are of the
+ general form \isa{{\isachardoublequote}{\isacharparenleft}{\isasymdots}\ {\isasymlongrightarrow}\ {\isasymdots}{\isacharparenright}\ {\isasymLongrightarrow}\ {\isasymdots}\ {\isacharparenleft}{\isasymdots}\ {\isasymlongrightarrow}\ {\isasymdots}{\isacharparenright}\ {\isasymLongrightarrow}\ {\isasymdots}\ {\isasymlongrightarrow}\ {\isasymdots}{\isachardoublequote}}. For example, in
+ the case of the operator \isa{{\isachardoublequote}{\isasymor}{\isachardoublequote}}, the corresponding theorem is
+ \[
+ \infer{\isa{{\isachardoublequote}P\isactrlsub {\isadigit{1}}\ {\isasymor}\ P\isactrlsub {\isadigit{2}}\ {\isasymlongrightarrow}\ Q\isactrlsub {\isadigit{1}}\ {\isasymor}\ Q\isactrlsub {\isadigit{2}}{\isachardoublequote}}}{\isa{{\isachardoublequote}P\isactrlsub {\isadigit{1}}\ {\isasymlongrightarrow}\ Q\isactrlsub {\isadigit{1}}{\isachardoublequote}} & \isa{{\isachardoublequote}P\isactrlsub {\isadigit{2}}\ {\isasymlongrightarrow}\ Q\isactrlsub {\isadigit{2}}{\isachardoublequote}}}
+ \]
+
+ \item De Morgan style equations for reasoning about the ``polarity''
+ of expressions, e.g.
+ \[
+ \isa{{\isachardoublequote}{\isasymnot}\ {\isasymnot}\ P\ {\isasymlongleftrightarrow}\ P{\isachardoublequote}} \qquad\qquad
+ \isa{{\isachardoublequote}{\isasymnot}\ {\isacharparenleft}P\ {\isasymand}\ Q{\isacharparenright}\ {\isasymlongleftrightarrow}\ {\isasymnot}\ P\ {\isasymor}\ {\isasymnot}\ Q{\isachardoublequote}}
+ \]
+
+ \item Equations for reducing complex operators to more primitive
+ ones whose monotonicity can easily be proved, e.g.
+ \[
+ \isa{{\isachardoublequote}{\isacharparenleft}P\ {\isasymlongrightarrow}\ Q{\isacharparenright}\ {\isasymlongleftrightarrow}\ {\isasymnot}\ P\ {\isasymor}\ Q{\isachardoublequote}} \qquad\qquad
+ \isa{{\isachardoublequote}Ball\ A\ P\ {\isasymequiv}\ {\isasymforall}x{\isachardot}\ x\ {\isasymin}\ A\ {\isasymlongrightarrow}\ P\ x{\isachardoublequote}}
+ \]
+
+ \end{itemize}
+
+ %FIXME: Example of an inductive definition%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsection{Arithmetic proof support%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+\begin{matharray}{rcl}
+ \indexdef{HOL}{method}{arith}\mbox{\isa{arith}} & : & \isarmeth \\
+ \indexdef{HOL}{method}{arith-split}\mbox{\isa{arith{\isacharunderscore}split}} & : & \isaratt \\
+ \end{matharray}
+
+ The \mbox{\isa{arith}} method decides linear arithmetic problems
+ (on types \isa{nat}, \isa{int}, \isa{real}). Any current
+ facts are inserted into the goal before running the procedure.
+
+ The \mbox{\isa{arith{\isacharunderscore}split}} attribute declares case split rules
+ to be expanded before the arithmetic procedure is invoked.
+
+ Note that a simpler (but faster) version of arithmetic reasoning is
+ already performed by the Simplifier.%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsection{Cases and induction: emulating tactic scripts \label{sec:hol-induct-tac}%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+The following important tactical tools of Isabelle/HOL have been
+ ported to Isar. These should be never used in proper proof texts!
+
+ \begin{matharray}{rcl}
+ \indexdef{HOL}{method}{case-tac}\mbox{\isa{case{\isacharunderscore}tac}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isarmeth \\
+ \indexdef{HOL}{method}{induct-tac}\mbox{\isa{induct{\isacharunderscore}tac}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isarmeth \\
+ \indexdef{HOL}{method}{ind-cases}\mbox{\isa{ind{\isacharunderscore}cases}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isarmeth \\
+ \indexdef{HOL}{command}{inductive-cases}\mbox{\isa{\isacommand{inductive{\isacharunderscore}cases}}} & : & \isartrans{theory}{theory} \\
+ \end{matharray}
+
+ \begin{rail}
+ 'case\_tac' goalspec? term rule?
+ ;
+ 'induct\_tac' goalspec? (insts * 'and') rule?
+ ;
+ 'ind\_cases' (prop +) ('for' (name +)) ?
+ ;
+ 'inductive\_cases' (thmdecl? (prop +) + 'and')
+ ;
+
+ rule: ('rule' ':' thmref)
+ ;
+ \end{rail}
+
+ \begin{descr}
+
+ \item [\mbox{\isa{case{\isacharunderscore}tac}} and \mbox{\isa{induct{\isacharunderscore}tac}}]
+ admit to reason about inductive datatypes only (unless an
+ alternative rule is given explicitly). Furthermore, \mbox{\isa{case{\isacharunderscore}tac}} does a classical case split on booleans; \mbox{\isa{induct{\isacharunderscore}tac}} allows only variables to be given as instantiation.
+ These tactic emulations feature both goal addressing and dynamic
+ instantiation. Note that named rule cases are \emph{not} provided
+ as would be by the proper \mbox{\isa{induct}} and \mbox{\isa{cases}} proof
+ methods (see \secref{sec:cases-induct}).
+
+ \item [\mbox{\isa{ind{\isacharunderscore}cases}} and \mbox{\isa{\isacommand{inductive{\isacharunderscore}cases}}}] provide an interface to the internal
+ \texttt{mk_cases} operation. Rules are simplified in an
+ unrestricted forward manner.
+
+ While \mbox{\isa{ind{\isacharunderscore}cases}} is a proof method to apply the
+ result immediately as elimination rules, \mbox{\isa{\isacommand{inductive{\isacharunderscore}cases}}} provides case split theorems at the theory level
+ for later use. The \mbox{\isa{\isakeyword{for}}} argument of the \mbox{\isa{ind{\isacharunderscore}cases}} method allows to specify a list of variables that should
+ be generalized before applying the resulting rule.
+
+ \end{descr}%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isamarkupsection{Executable code%
+}
+\isamarkuptrue%
+%
+\begin{isamarkuptext}%
+Isabelle/Pure provides two generic frameworks to support code
+ generation from executable specifications. Isabelle/HOL
+ instantiates these mechanisms in a way that is amenable to end-user
+ applications.
+
+ One framework generates code from both functional and relational
+ programs to SML. See \cite{isabelle-HOL} for further information
+ (this actually covers the new-style theory format as well).
+
+ \begin{matharray}{rcl}
+ \indexdef{HOL}{command}{value}\mbox{\isa{\isacommand{value}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isarkeep{theory~|~proof} \\
+ \indexdef{HOL}{command}{code-module}\mbox{\isa{\isacommand{code{\isacharunderscore}module}}} & : & \isartrans{theory}{theory} \\
+ \indexdef{HOL}{command}{code-library}\mbox{\isa{\isacommand{code{\isacharunderscore}library}}} & : & \isartrans{theory}{theory} \\
+ \indexdef{HOL}{command}{consts-code}\mbox{\isa{\isacommand{consts{\isacharunderscore}code}}} & : & \isartrans{theory}{theory} \\
+ \indexdef{HOL}{command}{types-code}\mbox{\isa{\isacommand{types{\isacharunderscore}code}}} & : & \isartrans{theory}{theory} \\
+ \indexdef{HOL}{attribute}{code}\mbox{\isa{code}} & : & \isaratt \\
+ \end{matharray}
+
+ \begin{rail}
+ 'value' term
+ ;
+
+ ( 'code\_module' | 'code\_library' ) modespec ? name ? \\
+ ( 'file' name ) ? ( 'imports' ( name + ) ) ? \\
+ 'contains' ( ( name '=' term ) + | term + )
+ ;
+
+ modespec: '(' ( name * ) ')'
+ ;
+
+ 'consts\_code' (codespec +)
+ ;
+
+ codespec: const template attachment ?
+ ;
+
+ 'types\_code' (tycodespec +)
+ ;
+
+ tycodespec: name template attachment ?
+ ;
+
+ const: term
+ ;
+
+ template: '(' string ')'
+ ;
+
+ attachment: 'attach' modespec ? verblbrace text verbrbrace
+ ;
+
+ 'code' (name)?
+ ;
+ \end{rail}
+
+ \begin{descr}
+
+ \item [\mbox{\isa{\isacommand{value}}}~\isa{t}] evaluates and prints a
+ term using the code generator.
+
+ \end{descr}
+
+ \medskip The other framework generates code from functional programs
+ (including overloading using type classes) to SML \cite{SML}, OCaml
+ \cite{OCaml} and Haskell \cite{haskell-revised-report}.
+ Conceptually, code generation is split up in three steps:
+ \emph{selection} of code theorems, \emph{translation} into an
+ abstract executable view and \emph{serialization} to a specific
+ \emph{target language}. See \cite{isabelle-codegen} for an
+ introduction on how to use it.
+
+ \begin{matharray}{rcl}
+ \indexdef{HOL}{command}{export-code}\mbox{\isa{\isacommand{export{\isacharunderscore}code}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isarkeep{theory~|~proof} \\
+ \indexdef{HOL}{command}{code-thms}\mbox{\isa{\isacommand{code{\isacharunderscore}thms}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isarkeep{theory~|~proof} \\
+ \indexdef{HOL}{command}{code-deps}\mbox{\isa{\isacommand{code{\isacharunderscore}deps}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isarkeep{theory~|~proof} \\
+ \indexdef{HOL}{command}{code-datatype}\mbox{\isa{\isacommand{code{\isacharunderscore}datatype}}} & : & \isartrans{theory}{theory} \\
+ \indexdef{HOL}{command}{code-const}\mbox{\isa{\isacommand{code{\isacharunderscore}const}}} & : & \isartrans{theory}{theory} \\
+ \indexdef{HOL}{command}{code-type}\mbox{\isa{\isacommand{code{\isacharunderscore}type}}} & : & \isartrans{theory}{theory} \\
+ \indexdef{HOL}{command}{code-class}\mbox{\isa{\isacommand{code{\isacharunderscore}class}}} & : & \isartrans{theory}{theory} \\
+ \indexdef{HOL}{command}{code-instance}\mbox{\isa{\isacommand{code{\isacharunderscore}instance}}} & : & \isartrans{theory}{theory} \\
+ \indexdef{HOL}{command}{code-monad}\mbox{\isa{\isacommand{code{\isacharunderscore}monad}}} & : & \isartrans{theory}{theory} \\
+ \indexdef{HOL}{command}{code-reserved}\mbox{\isa{\isacommand{code{\isacharunderscore}reserved}}} & : & \isartrans{theory}{theory} \\
+ \indexdef{HOL}{command}{code-include}\mbox{\isa{\isacommand{code{\isacharunderscore}include}}} & : & \isartrans{theory}{theory} \\
+ \indexdef{HOL}{command}{code-modulename}\mbox{\isa{\isacommand{code{\isacharunderscore}modulename}}} & : & \isartrans{theory}{theory} \\
+ \indexdef{HOL}{command}{code-exception}\mbox{\isa{\isacommand{code{\isacharunderscore}exception}}} & : & \isartrans{theory}{theory} \\
+ \indexdef{HOL}{command}{print-codesetup}\mbox{\isa{\isacommand{print{\isacharunderscore}codesetup}}}\isa{{\isachardoublequote}\isactrlsup {\isacharasterisk}{\isachardoublequote}} & : & \isarkeep{theory~|~proof} \\
+ \indexdef{HOL}{attribute}{code}\mbox{\isa{code}} & : & \isaratt \\
+ \end{matharray}
+
+ \begin{rail}
+ 'export\_code' ( constexpr + ) ? \\
+ ( ( 'in' target ( 'module\_name' string ) ? \\
+ ( 'file' ( string | '-' ) ) ? ( '(' args ')' ) ?) + ) ?
+ ;
+
+ 'code\_thms' ( constexpr + ) ?
+ ;
+
+ 'code\_deps' ( constexpr + ) ?
+ ;
+
+ const: term
+ ;
+
+ constexpr: ( const | 'name.*' | '*' )
+ ;
+
+ typeconstructor: nameref
+ ;
+
+ class: nameref
+ ;
+
+ target: 'OCaml' | 'SML' | 'Haskell'
+ ;
+
+ 'code\_datatype' const +
+ ;
+
+ 'code\_const' (const + 'and') \\
+ ( ( '(' target ( syntax ? + 'and' ) ')' ) + )
+ ;
+
+ 'code\_type' (typeconstructor + 'and') \\
+ ( ( '(' target ( syntax ? + 'and' ) ')' ) + )
+ ;
+
+ 'code\_class' (class + 'and') \\
+ ( ( '(' target \\
+ ( ( string ('where' \\
+ ( const ( '==' | equiv ) string ) + ) ? ) ? + 'and' ) ')' ) + )
+ ;
+
+ 'code\_instance' (( typeconstructor '::' class ) + 'and') \\
+ ( ( '(' target ( '-' ? + 'and' ) ')' ) + )
+ ;
+
+ 'code\_monad' const const target
+ ;
+
+ 'code\_reserved' target ( string + )
+ ;
+
+ 'code\_include' target ( string ( string | '-') )
+ ;
+
+ 'code\_modulename' target ( ( string string ) + )
+ ;
+
+ 'code\_exception' ( const + )
+ ;
+
+ syntax: string | ( 'infix' | 'infixl' | 'infixr' ) nat string
+ ;
+
+ 'code' ('func' | 'inline') ( 'del' )?
+ ;
+ \end{rail}
+
+ \begin{descr}
+
+ \item [\mbox{\isa{\isacommand{export{\isacharunderscore}code}}}] is the canonical interface
+ for generating and serializing code: for a given list of constants,
+ code is generated for the specified target languages. Abstract code
+ is cached incrementally. If no constant is given, the currently
+ cached code is serialized. If no serialization instruction is
+ given, only abstract code is cached.
+
+ Constants may be specified by giving them literally, referring to
+ all executable contants within a certain theory by giving \isa{{\isachardoublequote}name{\isachardot}{\isacharasterisk}{\isachardoublequote}}, or referring to \emph{all} executable constants currently
+ available by giving \isa{{\isachardoublequote}{\isacharasterisk}{\isachardoublequote}}.
+
+ By default, for each involved theory one corresponding name space
+ module is generated. Alternativly, a module name may be specified
+ after the \mbox{\isa{\isakeyword{module{\isacharunderscore}name}}} keyword; then \emph{all} code is
+ placed in this module.
+
+ For \emph{SML} and \emph{OCaml}, the file specification refers to a
+ single file; for \emph{Haskell}, it refers to a whole directory,
+ where code is generated in multiple files reflecting the module
+ hierarchy. The file specification ``\isa{{\isachardoublequote}{\isacharminus}{\isachardoublequote}}'' denotes standard
+ output. For \emph{SML}, omitting the file specification compiles
+ code internally in the context of the current ML session.
+
+ Serializers take an optional list of arguments in parentheses. For
+ \emph{Haskell} a module name prefix may be given using the ``\isa{{\isachardoublequote}root{\isacharcolon}{\isachardoublequote}}'' argument; ``\isa{string{\isacharunderscore}classes}'' adds a ``\verb|deriving (Read, Show)|'' clause to each appropriate datatype
+ declaration.
+
+ \item [\mbox{\isa{\isacommand{code{\isacharunderscore}thms}}}] prints a list of theorems
+ representing the corresponding program containing all given
+ constants; if no constants are given, the currently cached code
+ theorems are printed.
+
+ \item [\mbox{\isa{\isacommand{code{\isacharunderscore}deps}}}] visualizes dependencies of
+ theorems representing the corresponding program containing all given
+ constants; if no constants are given, the currently cached code
+ theorems are visualized.
+
+ \item [\mbox{\isa{\isacommand{code{\isacharunderscore}datatype}}}] specifies a constructor set
+ for a logical type.
+
+ \item [\mbox{\isa{\isacommand{code{\isacharunderscore}const}}}] associates a list of constants
+ with target-specific serializations; omitting a serialization
+ deletes an existing serialization.
+
+ \item [\mbox{\isa{\isacommand{code{\isacharunderscore}type}}}] associates a list of type
+ constructors with target-specific serializations; omitting a
+ serialization deletes an existing serialization.
+
+ \item [\mbox{\isa{\isacommand{code{\isacharunderscore}class}}}] associates a list of classes
+ with target-specific class names; in addition, constants associated
+ with this class may be given target-specific names used for instance
+ declarations; omitting a serialization deletes an existing
+ serialization. This applies only to \emph{Haskell}.
+
+ \item [\mbox{\isa{\isacommand{code{\isacharunderscore}instance}}}] declares a list of type
+ constructor / class instance relations as ``already present'' for a
+ given target. Omitting a ``\isa{{\isachardoublequote}{\isacharminus}{\isachardoublequote}}'' deletes an existing
+ ``already present'' declaration. This applies only to
+ \emph{Haskell}.
+
+ \item [\mbox{\isa{\isacommand{code{\isacharunderscore}monad}}}] provides an auxiliary
+ mechanism to generate monadic code.
+
+ \item [\mbox{\isa{\isacommand{code{\isacharunderscore}reserved}}}] declares a list of names as
+ reserved for a given target, preventing it to be shadowed by any
+ generated code.
+
+ \item [\mbox{\isa{\isacommand{code{\isacharunderscore}include}}}] adds arbitrary named content
+ (``include'') to generated code. A as last argument ``\isa{{\isachardoublequote}{\isacharminus}{\isachardoublequote}}''
+ will remove an already added ``include''.
+
+ \item [\mbox{\isa{\isacommand{code{\isacharunderscore}modulename}}}] declares aliasings from
+ one module name onto another.
+
+ \item [\mbox{\isa{\isacommand{code{\isacharunderscore}exception}}}] declares constants which
+ are not required to have a definition by a defining equations; these
+ are mapped on exceptions instead.
+
+ \item [\mbox{\isa{code}}~\isa{func}] explicitly selects (or
+ with option ``\isa{{\isachardoublequote}del{\isacharcolon}{\isachardoublequote}}'' deselects) a defining equation for
+ code generation. Usually packages introducing defining equations
+ provide a resonable default setup for selection.
+
+ \item [\mbox{\isa{code}}\isa{inline}] declares (or with
+ option ``\isa{{\isachardoublequote}del{\isacharcolon}{\isachardoublequote}}'' removes) inlining theorems which are
+ applied as rewrite rules to any defining equation during
+ preprocessing.
+
+ \item [\mbox{\isa{\isacommand{print{\isacharunderscore}codesetup}}}] gives an overview on
+ selected defining equations, code generator datatypes and
+ preprocessor setup.
+
+ \end{descr}%
+\end{isamarkuptext}%
+\isamarkuptrue%
+%
+\isadelimtheory
+%
+\endisadelimtheory
+%
+\isatagtheory
\isacommand{end}\isamarkupfalse%
%
\endisatagtheory
{\isafoldtheory}%
%
\isadelimtheory
-\isanewline
%
\endisadelimtheory
+\isanewline
+\isanewline
\end{isabellebody}%
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