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-
-\chapter{Object-logic specific elements}\label{ch:logics}
-
-\section{HOL}
-
-\subsection{Primitive types}\label{sec:hol-typedef}
-
-\indexisarcmdof{HOL}{typedecl}\indexisarcmdof{HOL}{typedef}
-\begin{matharray}{rcl}
-  \isarcmd{typedecl} & : & \isartrans{theory}{theory} \\
-  \isarcmd{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 [$\isarkeyword{typedecl}~(\vec\alpha)t$] is similar to the original
-  $\isarkeyword{typedecl}$ of Isabelle/Pure (see \S\ref{sec:types-pure}), but
-  also declares type arity $t :: (type, \dots, type) type$, making $t$ an
-  actual HOL type constructor.
-  
-\item [$\isarkeyword{typedef}~(\vec\alpha)t = A$] sets up a goal stating
-  non-emptiness of the set $A$.  After finishing the proof, the theory will be
-  augmented by a Gordon/HOL-style type definition, which establishes a
-  bijection between the representing set $A$ and the new type $t$.
-  
-  Technically, $\isarkeyword{typedef}$ defines both a type $t$ 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 $Rep_t$, its inverse
-  $Abs_t$ (this may be changed via an explicit $\isarkeyword{morphisms}$
-  declaration).
-  
-  Theorems $Rep_t$, $Rep_t_inverse$, and $Abs_t_inverse$ provide the most
-  basic characterization as a corresponding injection/surjection pair (in both
-  directions).  Rules $Rep_t_inject$ and $Abs_t_inject$ provide a slightly
-  more convenient view on the injectivity part, suitable for automated proof
-  tools (e.g.\ in $simp$ or $iff$ declarations).  Rules
-  $Rep_t_cases/Rep_t_induct$, and $Abs_t_cases/Abs_t_induct$ provide
-  alternative views on surjectivity; these are already declared as set or type
-  rules for the generic $cases$ and $induct$ methods.
-  
-  An alternative name may be specified in parentheses; the default is to use
-  $t$ as indicated before.  The $open$ 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 $\isarkeyword{record}$ (see
-\S\ref{sec:hol-record}) or $\isarkeyword{datatype}$ (see
-\S\ref{sec:hol-datatype}).
-
-
-\subsection{Adhoc tuples}
-
-\indexisarattof{HOL}{split-format}
-\begin{matharray}{rcl}
-  split_format^* & : & \isaratt \\
-\end{matharray}
-
-\railalias{splitformat}{split\_format}
-\railterm{splitformat}
-
-\begin{rail}
-  splitformat (((name *) + 'and') | ('(' 'complete' ')'))
-  ;
-\end{rail}
-
-\begin{descr}
-  
-\item [$split_format~\vec p@1 \dots \vec p@n$] puts expressions of low-level
-  tuple types into canonical form as specified by the arguments given; $\vec
-  p@i$ refers to occurrences in premise $i$ of the rule.  The ``$(complete)$''
-  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}
-
-
-\subsection{Records}\label{sec:hol-record}
-
-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.
-
-
-\subsubsection{Basic concepts}
-
-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 & $\record{x = a\fs y = b}$ & $\record{x \ty A\fs y \ty B}$ \\
-  schematic & $\record{x = a\fs y = b\fs \more = m}$ &
-    $\record{x \ty A\fs y \ty B\fs \more \ty M}$ \\
-\end{tabular}
-\end{center}
-
-\noindent The ASCII representation of $\record{x = a}$ is \texttt{(| x = a |)}.
-
-A fixed record $\record{x = a\fs y = b}$ has field $x$ of value $a$ and field
-$y$ of value $b$.  The corresponding type is $\record{x \ty A\fs y \ty B}$,
-assuming that $a \ty A$ and $b \ty B$.
-
-A record scheme like $\record{x = a\fs y = b\fs \more = m}$ contains fields
-$x$ and $y$ as before, but also possibly further fields as indicated by the
-``$\more$'' notation (which is actually part of the syntax).  The improper
-field ``$\more$'' 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 $\record{x = a\fs y = b\fs z = c\fs \more =
-  m'}$, where $m'$ refers to a different more part.  Fixed records are special
-instances of record schemes, where ``$\more$'' is properly terminated by the
-$() :: unit$ element.  Actually, $\record{x = a\fs y = b}$ is just an
-abbreviation for $\record{x = a\fs y = b\fs \more = ()}$.
-
-\medskip
-
-Two key observations make extensible records in a simply typed language like
-HOL feasible:
-\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.
-
-
-\subsubsection{Record specifications}
-
-\indexisarcmdof{HOL}{record}
-\begin{matharray}{rcl}
-  \isarcmd{record} & : & \isartrans{theory}{theory} \\
-\end{matharray}
-
-\begin{rail}
-  'record' typespec '=' (type '+')? (constdecl +)
-  ;
-\end{rail}
-
-\begin{descr}
-\item [$\isarkeyword{record}~(\vec\alpha)t = \tau + \vec c :: \vec\sigma$]
-  defines extensible record type $(\vec\alpha)t$, derived from the optional
-  parent record $\tau$ by adding new field components $\vec c :: \vec\sigma$.
-
-  The type variables of $\tau$ and $\vec\sigma$ need to be covered by the
-  (distinct) parameters $\vec\alpha$.  Type constructor $t$ has to be new,
-  while $\tau$ needs to specify an instance of an existing record type.  At
-  least one new field $\vec c$ 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 $\tau$ is optional; if omitted $t$ 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, $(\vec\alpha) \, t$ is made a type abbreviation for the
-  fixed record type $\record{\vec c \ty \vec\sigma}$, likewise is
-  $(\vec\alpha, \zeta) \, t_scheme$ made an abbreviation for $\record{\vec c
-    \ty \vec\sigma\fs \more \ty \zeta}$.
-
-\end{descr}
-
-\subsubsection{Record operations}
-
-Any record definition of the form presented above produces certain standard
-operations.  Selectors and updates are provided for any field, including the
-improper one ``$more$''.  There are also cumulative record constructor
-functions.  To simplify the presentation below, we assume for now that
-$(\vec\alpha) \, t$ is a root record with fields $\vec c \ty \vec\sigma$.
-
-\medskip \textbf{Selectors} and \textbf{updates} are available for any field
-(including ``$more$''):
-\begin{matharray}{lll}
-  c@i & \ty & \record{\vec c \ty \vec \sigma, \more \ty \zeta} \To \sigma@i \\
-  c@i_update & \ty & \sigma@i \To \record{\vec c \ty \vec\sigma, \more \ty \zeta} \To
-    \record{\vec c \ty \vec\sigma, \more \ty \zeta}
-\end{matharray}
-
-There is special syntax for application of updates: $r \, \record{x \asn a}$
-abbreviates term $x_update \, a \, r$.  Further notation for repeated updates
-is also available: $r \, \record{x \asn a} \, \record{y \asn b} \, \record{z
-  \asn c}$ may be written $r \, \record{x \asn a\fs y \asn b\fs z \asn c}$.
-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 $\record{x \asn a\fs y \asn
-  b\fs z \asn c}$, 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}
-  t{\dtt}make & \ty & \vec\sigma \To \record{\vec c \ty \vec \sigma} \\
-\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 $(\vec\alpha) \, t$ has ancestor fields $\vec
-b \ty \vec\rho$, the above record operations will get the following types:
-\begin{matharray}{lll}
-  c@i & \ty & \record{\vec b \ty \vec\rho, \vec c \ty \vec\sigma, \more \ty
-    \zeta} \To \sigma@i \\
-  c@i_update & \ty & \sigma@i \To
-    \record{\vec b \ty \vec\rho, \vec c \ty \vec\sigma, \more \ty \zeta} \To
-    \record{\vec b \ty \vec\rho, \vec c \ty \vec\sigma, \more \ty \zeta} \\
-  t{\dtt}make & \ty & \vec\rho \To \vec\sigma \To
-    \record{\vec b \ty \vec\rho, \vec c \ty \vec \sigma} \\
-\end{matharray}
-\noindent
-
-\medskip Some further operations address the extension aspect of a derived
-record scheme specifically: $fields$ produces a record fragment consisting of
-exactly the new fields introduced here (the result may serve as a more part
-elsewhere); $extend$ takes a fixed record and adds a given more part;
-$truncate$ restricts a record scheme to a fixed record.
-
-\begin{matharray}{lll}
-  t{\dtt}fields & \ty & \vec\sigma \To \record{\vec c \ty \vec \sigma} \\
-  t{\dtt}extend & \ty & \record{\vec d \ty \vec \rho, \vec c \ty \vec\sigma} \To
-    \zeta \To \record{\vec d \ty \vec \rho, \vec c \ty \vec\sigma, \more \ty \zeta} \\
-  t{\dtt}truncate & \ty & \record{\vec d \ty \vec \rho, \vec c \ty \vec\sigma, \more \ty \zeta} \To
-    \record{\vec d \ty \vec \rho, \vec c \ty \vec\sigma} \\
-\end{matharray}
-
-\noindent Note that $t{\dtt}make$ and $t{\dtt}fields$ actually coincide for root records.
-
-
-\subsubsection{Derived rules and proof tools}
-
-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 $t$ 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 $simp$ method
-  without further arguments.  These rules are available as $t{\dtt}simps$,
-  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 $((x, y) = (x', y')) \equiv x=x'
-  \conj y=y'$ are declared to the Simplifier and Classical Reasoner as $iff$
-  rules.  These rules are available as $t{\dtt}iffs$.
-
-\item The introduction rule for record equality analogous to $x~r = x~r' \Imp
-  y~r = y~r' \Imp \dots \Imp r = r'$ is declared to the Simplifier, and as the
-  basic rule context as ``$intro?$''.  The rule is called $t{\dtt}equality$.
-
-\item Representations of arbitrary record expressions as canonical constructor
-  terms are provided both in $cases$ and $induct$ format (cf.\ the generic
-  proof methods of the same name, \S\ref{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 ``$(cases~r)$'' to a certain proof problem.
-
-\item The derived record operations $t{\dtt}make$, $t{\dtt}fields$,
-  $t{\dtt}extend$, $t{\dtt}truncate$ are \emph{not} treated automatically, but
-  usually need to be expanded by hand, using the collective fact
-  $t{\dtt}defs$.
-
-\end{enumerate}
-
-
-\subsection{Datatypes}\label{sec:hol-datatype}
-
-\indexisarcmdof{HOL}{datatype}\indexisarcmdof{HOL}{rep-datatype}
-\begin{matharray}{rcl}
-  \isarcmd{datatype} & : & \isartrans{theory}{theory} \\
-  \isarcmd{rep_datatype} & : & \isartrans{theory}{theory} \\
-\end{matharray}
-
-\railalias{repdatatype}{rep\_datatype}
-\railterm{repdatatype}
-
-\begin{rail}
-  'datatype' (dtspec + 'and')
-  ;
-  repdatatype (name *) dtrules
-  ;
-
-  dtspec: parname? typespec infix? '=' (cons + '|')
-  ;
-  cons: name (type *) mixfix?
-  ;
-  dtrules: 'distinct' thmrefs 'inject' thmrefs 'induction' thmrefs
-\end{rail}
-
-\begin{descr}
-\item [$\isarkeyword{datatype}$] defines inductive datatypes in HOL.
-\item [$\isarkeyword{rep_datatype}$] 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 \S\ref{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
-\texttt{case_tac} and \texttt{induct_tac} available, see
-\S\ref{sec:hol-induct-tac}; these admit to refer directly to the internal
-structure of subgoals (including internally bound parameters).
-
-
-\subsection{Recursive functions}\label{sec:recursion}
-
-\indexisarcmdof{HOL}{primrec}\indexisarcmdof{HOL}{fun}\indexisarcmdof{HOL}{function}\indexisarcmdof{HOL}{termination}
-
-\begin{matharray}{rcl}
-  \isarcmd{primrec} & : & \isarkeep{local{\dsh}theory} \\
-  \isarcmd{fun} & : & \isarkeep{local{\dsh}theory} \\
-  \isarcmd{function} & : & \isartrans{local{\dsh}theory}{proof(prove)} \\
-  \isarcmd{termination} & : & \isartrans{local{\dsh}theory}{proof(prove)} \\
-\end{matharray}
-
-\railalias{funopts}{function\_opts}
-
-\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 [$\isarkeyword{primrec}$] defines primitive recursive functions over
-  datatypes, see also \cite{isabelle-HOL}.
-
-\item [$\isarkeyword{function}$] 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 $f.psimps$) and induction rule (named
-  $f.pinduct$) are guarded by a generated domain predicate $f_dom$. 
-  The $\isarkeyword{termination}$ command can then be used to establish
-  that the function is total.
-
-\item [$\isarkeyword{fun}$] is a shorthand notation for
-  $\isarkeyword{function}~(\textit{sequential})$, followed by automated
-  proof attemts regarding pattern matching and termination. For
-  details, see \cite{isabelle-function}.
-
-\item [$\isarkeyword{termination}$~f] commences a termination proof
-  for the previously defined function $f$. If no name is given, it
-  refers to the most recent function definition. After the proof is
-  closed, the recursive equations and the induction principle is established.
-\end{descr}
-
-Recursive definitions introduced by both the $\isarkeyword{primrec}$
-and the $\isarkeyword{function}$ command accommodate reasoning by
-induction (cf.\ \S\ref{sec:cases-induct}): rule $c\mathord{.}induct$
-(where $c$ 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 $\isarkeyword{primrec}$ are
-that of the datatypes involved, while those of
-$\isarkeyword{function}$ are numbered (starting from $1$).
-
-The equations provided by these packages may be referred later as theorem list
-$f{\dtt}simps$, where $f$ is the (collective) name of the functions defined.
-Individual equations may be named explicitly as well.
-
-The $\isarkeyword{function}$ command accepts the following options:
-
-\begin{descr}
-\item [\emph{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 [\emph{in name}] gives the target for the definition.
-
-\item [\emph{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 [\emph{tailrec}] generates the unconstrained recursive equations
-  even without a termination proof, provided that the function is
-  tail-recursive. This currently only works 
-
-\item [\emph{default d}] allows to specify a default value for a
-  (partial) function, which will ensure that $f(x)=d(x)$ whenever $x
-  \notin \textit{f\_dom}$. This feature is experimental.
-\end{descr}
-
-\subsubsection{Proof methods related to recursive definitions}
-
-\indexisarmethof{HOL}{pat-completeness}
-\indexisarmethof{HOL}{relation}
-\indexisarmethof{HOL}{lexicographic-order}
-
-\begin{matharray}{rcl}
-  pat\_completeness & : & \isarmeth \\
-  relation & : & \isarmeth \\
-  lexicographic\_order & : & \isarmeth \\
-\end{matharray}
-
-\begin{rail}
-  'pat\_completeness'
-  ;
-  'relation' term
-  ;
-  'lexicographic\_order' clasimpmod
-\end{rail}
-
-\begin{descr}
-\item [\emph{pat\_completeness}] Specialized method to solve goals
-  regarding the completeness of pattern matching, as required by the
-  $\isarkeyword{function}$ package (cf.~\cite{isabelle-function}).
-
-\item [\emph{relation R}] Introduces a termination proof using the
-  relation $R$. The resulting proof state will contain goals
-  expressing that $R$ is wellfounded, and that the arguments
-  of recursive calls decrease with respect to $R$. Usually, this
-  method is used as the initial proof step of manual termination
-  proofs.
-
-\item [\emph{lexicographic\_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 \emph{auto} method, which it uses
-  internally to prove local descents. Hence, modifiers like
-  \emph{simp}, \emph{intro} etc.\ can be used to add ``hints'' for the
-  automated proofs. In case of failure, extensive information is
-  printed, which can help to analyse the failure (cf.~\cite{isabelle-function}).
-\end{descr}
-
-\subsubsection{Legacy recursion package}
-\indexisarcmdof{HOL}{recdef}\indexisarcmdof{HOL}{recdef-tc}
-
-The use of the legacy $\isarkeyword{recdef}$ command is now deprecated
-in favour of $\isarkeyword{function}$ and $\isarkeyword{fun}$.
-
-\begin{matharray}{rcl}
-  \isarcmd{recdef} & : & \isartrans{theory}{theory} \\
-  \isarcmd{recdef_tc}^* & : & \isartrans{theory}{proof(prove)} \\
-\end{matharray}
-
-\railalias{recdefsimp}{recdef\_simp}
-\railterm{recdefsimp}
-
-\railalias{recdefcong}{recdef\_cong}
-\railterm{recdefcong}
-
-\railalias{recdefwf}{recdef\_wf}
-\railterm{recdefwf}
-
-\railalias{recdeftc}{recdef\_tc}
-\railterm{recdeftc}
-
-\begin{rail}
-  'recdef' ('(' 'permissive' ')')? \\ name term (prop +) hints?
-  ;
-  recdeftc thmdecl? tc
-  ;
-  hints: '(' 'hints' (recdefmod *) ')'
-  ;
-  recdefmod: ((recdefsimp | recdefcong | recdefwf) (() | 'add' | 'del') ':' thmrefs) | clasimpmod
-  ;
-  tc: nameref ('(' nat ')')?
-  ;
-\end{rail}
-
-\begin{descr}
-  
-\item [$\isarkeyword{recdef}$] defines general well-founded recursive
-  functions (using the TFL package), see also \cite{isabelle-HOL}.  The
-  ``$(permissive)$'' option tells TFL to recover from failed proof attempts,
-  returning unfinished results.  The $recdef_simp$, $recdef_cong$, and
-  $recdef_wf$ hints refer to auxiliary rules to be used in the internal
-  automated proof process of TFL.  Additional $clasimpmod$ declarations (cf.\ 
-  \S\ref{sec:clasimp}) may be given to tune the context of the Simplifier
-  (cf.\ \S\ref{sec:simplifier}) and Classical reasoner (cf.\ 
-  \S\ref{sec:classical}).
-  
-\item [$\isarkeyword{recdef_tc}~c~(i)$] recommences the proof for leftover
-  termination condition number $i$ (default $1$) as generated by a
-  $\isarkeyword{recdef}$ definition of constant $c$.
-  
-  Note that in most cases, $\isarkeyword{recdef}$ is able to finish its
-  internal proofs without manual intervention.
-
-\end{descr}
-
-\medskip Hints for $\isarkeyword{recdef}$ may be also declared globally, using
-the following attributes.
-
-\indexisarattof{HOL}{recdef-simp}\indexisarattof{HOL}{recdef-cong}\indexisarattof{HOL}{recdef-wf}
-\begin{matharray}{rcl}
-  recdef_simp & : & \isaratt \\
-  recdef_cong & : & \isaratt \\
-  recdef_wf & : & \isaratt \\
-\end{matharray}
-
-\railalias{recdefsimp}{recdef\_simp}
-\railterm{recdefsimp}
-
-\railalias{recdefcong}{recdef\_cong}
-\railterm{recdefcong}
-
-\railalias{recdefwf}{recdef\_wf}
-\railterm{recdefwf}
-
-\begin{rail}
-  (recdefsimp | recdefcong | recdefwf) (() | 'add' | 'del')
-  ;
-\end{rail}
-
-\subsection{Definition by specification}\label{sec:hol-specification}
-
-\indexisarcmdof{HOL}{specification}
-\begin{matharray}{rcl}
-  \isarcmd{specification} & : & \isartrans{theory}{proof(prove)} \\
-  \isarcmd{ax_specification} & : & \isartrans{theory}{proof(prove)} \\
-\end{matharray}
-
-\begin{rail}
-('specification' | 'ax\_specification') '(' (decl +) ')' \\ (thmdecl? prop +)
-;
-decl: ((name ':')? term '(' 'overloaded' ')'?)
-\end{rail}
-
-\begin{descr}
-\item [$\isarkeyword{specification}~decls~\phi$] sets up a goal stating
-  the existence of terms with the properties specified to hold for the
-  constants given in $\mathit{decls}$.  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 [$\isarkeyword{ax_specification}~decls~\phi$] sets up a goal stating
-  the existence of terms with the properties specified to hold for the
-  constants given in $\mathit{decls}$.  After finishing the proof, the
-  theory will be augmented with axioms expressing the properties given
-  in the first place.
-\item[$decl$] declares a constant to be defined by the specification
-  given.  The definition for the constant $c$ is bound to the name
-  $c$\_def unless a theorem name is given in the declaration.
-  Overloaded constants should be declared as such.
-\end{descr}
-
-Whether to use $\isarkeyword{specification}$ or $\isarkeyword{ax_specification}$
-is to some extent a matter of style.  $\isarkeyword{specification}$ introduces no new axioms,
-and so by construction cannot introduce inconsistencies, whereas $\isarkeyword{ax_specification}$
-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 $\isarkeyword{specification}$, one can prove
-that the two constants are, in fact, equal.  If this might be a problem,
-one should use $\isarkeyword{ax_specification}$.
-
-\subsection{Inductive and coinductive definitions}\label{sec:hol-inductive}
-
-An {\bf inductive definition} specifies the least predicate (or set) $R$ closed under given
-rules.  (Applying a rule to elements of~$R$ yields a result within~$R$.)  For
-example, a structural operational semantics is an inductive definition of an
-evaluation relation.  Dually, a {\bf coinductive definition} specifies the
-greatest predicate (or set) $R$ consistent with given rules.  (Every element of~$R$ can be
-seen as arising by applying a rule to elements of~$R$.)  An important example
-is using bisimulation relations to formalise equivalence of processes and
-infinite data structures.
-
-This package is related to the ZF one, 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 ZF's thanks to HOL's extra-logical automatic type-checking.  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.
-
-\indexisarcmdof{HOL}{inductive}\indexisarcmdof{HOL}{inductive-set}\indexisarcmdof{HOL}{coinductive}\indexisarcmdof{HOL}{coinductive-set}\indexisarattof{HOL}{mono}
-\begin{matharray}{rcl}
-  \isarcmd{inductive} & : & \isarkeep{local{\dsh}theory} \\
-  \isarcmd{inductive_set} & : & \isarkeep{local{\dsh}theory} \\
-  \isarcmd{coinductive} & : & \isarkeep{local{\dsh}theory} \\
-  \isarcmd{coinductive_set} & : & \isarkeep{local{\dsh}theory} \\
-  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 [$\isarkeyword{inductive}$ and $\isarkeyword{coinductive}$] define
-  (co)inductive predicates from the introduction rules given in the \texttt{where} section.
-  The optional \texttt{for} section contains a list of parameters of the (co)inductive
-  predicates that remain fixed throughout the definition.
-  The optional \texttt{monos} section contains \textit{monotonicity theorems},
-  which are required for each operator applied to a recursive set in the introduction rules.
-  There {\bf must} be a theorem of the form $A \leq B \Imp M~A \leq M~B$, for each
-  premise $M~R@i~t$ in an introduction rule!
-\item [$\isarkeyword{inductive_set}$ and $\isarkeyword{coinductive_set}$] are wrappers
-  for to the previous commands, allowing the definition of (co)inductive sets.
-\item [$mono$] declares monotonicity rules.  These rule are involved in the
-  automated monotonicity proof of $\isarkeyword{inductive}$.
-\end{descr}
-
-\subsubsection{Derived rules}
-
-Each (co)inductive definition $R$ adds definitions to the theory and also
-proves some theorems:
-\begin{description}
-\item[$R{\dtt}intros$] is the list of introduction rules, now proved as theorems, for
-the recursive predicates (or sets).  The rules are also available individually,
-using the names given them in the theory file.
-\item[$R{\dtt}cases$] is the case analysis (or elimination) rule.
-\item[$R{\dtt}(co)induct$] is the (co)induction rule.
-\end{description}
-When several predicates $R@1$, $\ldots$, $R@n$ are defined simultaneously,
-the list of introduction rules is called $R@1_\ldots_R@n{\dtt}intros$, the
-case analysis rules are called $R@1{\dtt}cases$, $\ldots$, $R@n{\dtt}cases$, and
-the list of mutual induction rules is called $R@1_\ldots_R@n{\dtt}inducts$.
-
-\subsubsection{Monotonicity theorems}
-
-Each theory contains a default set of theorems that are used in monotonicity
-proofs. New rules can be added to this set via the $mono$ attribute.
-Theory \texttt{Inductive} shows how this is done. In general, the following
-monotonicity theorems may be added:
-\begin{itemize}
-\item Theorems of the form $A \leq B \Imp M~A \leq M~B$, for proving
-  monotonicity of inductive definitions whose introduction rules have premises
-  involving terms such as $M~R@i~t$.
-\item Monotonicity theorems for logical operators, which are of the general form
-  $\List{\cdots \to \cdots;~\ldots;~\cdots \to \cdots} \Imp
-    \cdots \to \cdots$.
-  For example, in the case of the operator $\lor$, the corresponding theorem is
-  \[
-  \infer{P@1 \lor P@2 \to Q@1 \lor Q@2}
-    {P@1 \to Q@1 & P@2 \to Q@2}
-  \]
-\item De Morgan style equations for reasoning about the ``polarity'' of expressions, e.g.
-  \[
-  (\lnot \lnot P) ~=~ P \qquad\qquad
-  (\lnot (P \land Q)) ~=~ (\lnot P \lor \lnot Q)
-  \]
-\item Equations for reducing complex operators to more primitive ones whose
-  monotonicity can easily be proved, e.g.
-  \[
-  (P \to Q) ~=~ (\lnot P \lor Q) \qquad\qquad
-  \mathtt{Ball}~A~P ~\equiv~ \forall x.~x \in A \to P~x
-  \]
-\end{itemize}
-
-%FIXME: Example of an inductive definition
-
-
-\subsection{Arithmetic proof support}
-
-\indexisarmethof{HOL}{arith}\indexisarattof{HOL}{arith-split}
-\begin{matharray}{rcl}
-  arith & : & \isarmeth \\
-  arith_split & : & \isaratt \\
-\end{matharray}
-
-\begin{rail}
-  'arith' '!'?
-  ;
-\end{rail}
-
-The $arith$ method decides linear arithmetic problems (on types $nat$, $int$,
-$real$).  Any current facts are inserted into the goal before running the
-procedure.  The ``!''~argument causes the full context of assumptions to be
-included.  The $arith_split$ 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.
-
-
-\subsection{Cases and induction: emulating tactic scripts}\label{sec:hol-induct-tac}
-
-The following important tactical tools of Isabelle/HOL have been ported to
-Isar.  These should be never used in proper proof texts!
-
-\indexisarmethof{HOL}{case-tac}\indexisarmethof{HOL}{induct-tac}
-\indexisarmethof{HOL}{ind-cases}\indexisarcmdof{HOL}{inductive-cases}
-\begin{matharray}{rcl}
-  case_tac^* & : & \isarmeth \\
-  induct_tac^* & : & \isarmeth \\
-  ind_cases^* & : & \isarmeth \\
-  \isarcmd{inductive_cases} & : & \isartrans{theory}{theory} \\
-\end{matharray}
-
-\railalias{casetac}{case\_tac}
-\railterm{casetac}
-
-\railalias{inducttac}{induct\_tac}
-\railterm{inducttac}
-
-\railalias{indcases}{ind\_cases}
-\railterm{indcases}
-
-\railalias{inductivecases}{inductive\_cases}
-\railterm{inductivecases}
-
-\begin{rail}
-  casetac goalspec? term rule?
-  ;
-  inducttac goalspec? (insts * 'and') rule?
-  ;
-  indcases (prop +) ('for' (name +)) ?
-  ;
-  inductivecases (thmdecl? (prop +) + 'and')
-  ;
-
-  rule: ('rule' ':' thmref)
-  ;
-\end{rail}
-
-\begin{descr}
-\item [$case_tac$ and $induct_tac$] admit to reason about inductive datatypes
-  only (unless an alternative rule is given explicitly).  Furthermore,
-  $case_tac$ does a classical case split on booleans; $induct_tac$ 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 $induct$ and $cases$ proof
-  methods (see \S\ref{sec:cases-induct}).
-  
-\item [$ind_cases$ and $\isarkeyword{inductive_cases}$] provide an interface
-  to the internal \texttt{mk_cases} operation.  Rules are simplified in an
-  unrestricted forward manner.
-
-  While $ind_cases$ is a proof method to apply the result immediately as
-  elimination rules, $\isarkeyword{inductive_cases}$ provides case split
-  theorems at the theory level for later use.
-  The \texttt{for} option of the $ind_cases$ method allows to specify a list
-  of variables that should be generalized before applying the resulting rule.
-\end{descr}
-
-
-\subsection{Executable code}
-
-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).
-
-\indexisarcmd{value}\indexisarcmd{code-module}\indexisarcmd{code-library}
-\indexisarcmd{consts-code}\indexisarcmd{types-code}
-\indexisaratt{code}
-
-\begin{matharray}{rcl}
-  \isarcmd{value}^* & : & \isarkeep{theory~|~proof} \\
-  \isarcmd{code_module} & : & \isartrans{theory}{theory} \\
-  \isarcmd{code_library} & : & \isartrans{theory}{theory} \\
-  \isarcmd{consts_code} & : & \isartrans{theory}{theory} \\
-  \isarcmd{types_code} & : & \isartrans{theory}{theory} \\  
-  code & : & \isaratt \\
-\end{matharray}
-
-\railalias{verblbrace}{\texttt{\ttlbrace*}}
-\railalias{verbrbrace}{\texttt{*\ttrbrace}}
-\railterm{verblbrace}
-\railterm{verbrbrace}
-
-\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 [$\isarkeyword{value}~t$] reads, evaluates and prints a term
-  using the code generator.
-\end{descr}
-
-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.
-
-\indexisarcmd{export-code}
-\indexisarcmd{code-thms}
-\indexisarcmd{code-deps}
-\indexisarcmd{code-datatype}
-\indexisarcmd{code-const}
-\indexisarcmd{code-type}
-\indexisarcmd{code-class}
-\indexisarcmd{code-instance}
-\indexisarcmd{code-monad}
-\indexisarcmd{code-reserved}
-\indexisarcmd{code-include}
-\indexisarcmd{code-modulename}
-\indexisarcmd{code-exception}
-\indexisarcmd{print-codesetup}
-\indexisaratt{code func}
-\indexisaratt{code inline}
-
-\begin{matharray}{rcl}
-  \isarcmd{export_code}^* & : & \isarkeep{theory~|~proof} \\
-  \isarcmd{code_thms}^* & : & \isarkeep{theory~|~proof} \\
-  \isarcmd{code_deps}^* & : & \isarkeep{theory~|~proof} \\
-  \isarcmd{code_datatype} & : & \isartrans{theory}{theory} \\
-  \isarcmd{code_const} & : & \isartrans{theory}{theory} \\
-  \isarcmd{code_type} & : & \isartrans{theory}{theory} \\
-  \isarcmd{code_class} & : & \isartrans{theory}{theory} \\
-  \isarcmd{code_instance} & : & \isartrans{theory}{theory} \\
-  \isarcmd{code_monad} & : & \isartrans{theory}{theory} \\
-  \isarcmd{code_reserved} & : & \isartrans{theory}{theory} \\
-  \isarcmd{code_include} & : & \isartrans{theory}{theory} \\
-  \isarcmd{code_modulename} & : & \isartrans{theory}{theory} \\
-  \isarcmd{code_exception} & : & \isartrans{theory}{theory} \\
-  \isarcmd{print_codesetup}^* & : & \isarkeep{theory~|~proof} \\
-  code\ func & : & \isaratt \\
-  code\ inline & : & \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;
-
-'print\_codesetup';
-
-'code\ func' ( 'del' ) ?;
-
-'code\ inline' ( 'del' ) ?;
-\end{rail}
-
-\begin{descr}
-
-\item [$\isarcmd{export_code}$] is the canonical interface for generating and
-  serializing code: for a given list of constants, code is generated for the specified
-  target language(s).  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 exeuctable contants within a certain theory named ``name''
-  by giving (``name.*''), or referring to \emph{all} executable
-  constants currently available (``*'').
-
-  By default, for each involved theory one corresponding name space module
-  is generated.  Alternativly, a module name may be specified after the
-  (``module_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 ``-'' 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 ``root:''
-  argument;  ``string\_classes'' adds a ``deriving (Read, Show)'' clause
-  to each appropriate datatype declaration.
-
-\item [$\isarcmd{code_thms}$] 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 [$\isarcmd{code_deps}$] 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 [$\isarcmd{code_datatype}$] specifies a constructor set for a logical type.
-
-\item [$\isarcmd{code_const}$] associates a list of constants
-  with target-specific serializations; omitting a serialization
-  deletes an existing serialization.
-
-\item [$\isarcmd{code_type}$] associates a list of type constructors
-  with target-specific serializations; omitting a serialization
-  deletes an existing serialization.
-
-\item [$\isarcmd{code_class}$] 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.  Applies only to \emph{Haskell}.
-
-\item [$\isarcmd{code_instance}$] declares a list of type constructor / class
-  instance relations as ``already present'' for a given target.
-  Omitting a ``-'' deletes an existing ``already present'' declaration.
-  Applies only to \emph{Haskell}.
-
-\item [$\isarcmd{code_monad}$] provides an auxiliary mechanism
-  to generate monadic code.
-
-\item [$\isarcmd{code_reserved}$] declares a list of names
-  as reserved for a given target, preventing it to be shadowed
-  by any generated code.
-
-\item [$\isarcmd{code_include}$] adds arbitrary named content (''include``)
-  to generated code. A as last argument ``-'' will remove an already added ''include``.
-
-\item [$\isarcmd{code_modulename}$] declares aliasings from one module name
-  onto another.
-
-\item [$\isarcmd{code_exception}$] declares constants which are not required
-  to have a definition by a defining equations; these are mapped on exceptions
-  instead.
-
-\item [$code\ func$] selects (or with option ''del``, deselects) explicitly
-  a defining equation for code generation.  Usually packages introducing
-  defining equations provide a resonable default setup for selection.
-
-\item [$code\ inline$] declares (or with option ''del``, removes)
-  inlining theorems which are applied as rewrite rules to any defining equation
-  during preprocessing.
-
-\item [$\isarcmd{print_codesetup}$] gives an overview on selected
-  defining equations, code generator datatypes and preprocessor setup.
-
-\end{descr}
-
-
-%%% Local Variables:
-%%% mode: latex
-%%% TeX-master: "isar-ref"
-%%% End:




