\chapter{Isabelle/HOL Tools and Packages}\label{ch:hol-tools}
\section{Primitive types}
\indexisarcmd{typedecl}\indexisarcmd{typedef}
\begin{matharray}{rcl}
\isarcmd{typedecl} & : & \isartrans{theory}{theory} \\
\isarcmd{typedef} & : & \isartrans{theory}{proof(prove)} \\
\end{matharray}
\begin{rail}
'typedecl' typespec infix? comment?
;
'typedef' parname? typespec infix? \\ '=' term comment?
;
\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 $t :: (term, \dots, term) term$, 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. See \cite{isabelle-HOL}
for more information. Note that user-level theories usually do not directly
refer to the HOL $\isarkeyword{typedef}$ primitive, but use more advanced
packages such as $\isarkeyword{record}$ (see \S\ref{sec:record}) and
$\isarkeyword{datatype}$ (see \S\ref{sec:datatype}).
\end{descr}
\section{Records}\label{sec:record}
%FIXME record_split method
\indexisarcmd{record}
\begin{matharray}{rcl}
\isarcmd{record} & : & \isartrans{theory}{theory} \\
\end{matharray}
\begin{rail}
'record' typespec '=' (type '+')? (field +)
;
field: name '::' type comment?
;
\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 $\vec c :: \vec\sigma$.
See \cite{isabelle-HOL,NaraschewskiW-TPHOLs98} for more information only
simply-typed extensible records.
\end{descr}
\section{Datatypes}\label{sec:datatype}
\indexisarcmd{datatype}\indexisarcmd{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' (parname? typespec infix? \\ '=' (constructor + '|') + 'and')
;
repdatatype (name * ) \\ 'distinct' thmrefs 'inject' thmrefs 'induction' thmrefs
;
constructor: name (type * ) mixfix? comment?
;
\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}
See \cite{isabelle-HOL} for more details on datatypes. Note that the theory
syntax above has been slightly simplified over the old version, usually
requiring more quotes and less parentheses.
\section{Recursive functions}
\indexisarcmd{primrec}\indexisarcmd{recdef}
\begin{matharray}{rcl}
\isarcmd{primrec} & : & \isartrans{theory}{theory} \\
\isarcmd{recdef} & : & \isartrans{theory}{theory} \\
%FIXME
% \isarcmd{defer_recdef} & : & \isartrans{theory}{theory} \\
\end{matharray}
\begin{rail}
'primrec' parname? (thmdecl? prop comment? + )
;
'recdef' name term (term comment? +) \\ ('congs' thmrefs)? ('simpset' name)?
;
\end{rail}
\begin{descr}
\item [$\isarkeyword{primrec}$] defines primitive recursive functions over
datatypes.
\item [$\isarkeyword{recdef}$] defines general well-founded recursive
functions (using the TFL package).
\end{descr}
See \cite{isabelle-HOL} for more information on both mechanisms.
\section{(Co)Inductive sets}
\indexisarcmd{inductive}\indexisarcmd{coinductive}\indexisarcmd{inductive-cases}
\begin{matharray}{rcl}
\isarcmd{inductive} & : & \isartrans{theory}{theory} \\
\isarcmd{coinductive} & : & \isartrans{theory}{theory} \\
\isarcmd{inductive_cases} & : & \isartrans{theory}{theory} \\
\end{matharray}
\railalias{condefs}{con\_defs}
\railalias{indcases}{inductive\_cases}
\railterm{condefs,indcases}
\begin{rail}
('inductive' | 'coinductive') (term comment? +) \\
'intrs' attributes? (thmdecl? prop comment? +) \\
'monos' thmrefs comment? \\ condefs thmrefs comment?
;
indcases thmdef? nameref ':' \\ (prop +) comment?
;
\end{rail}
\begin{descr}
\item [$\isarkeyword{inductive}$ and $\isarkeyword{coinductive}$] define
(co)inductive sets from the given introduction rules.
\item [$\isarkeyword{inductive_cases}$] creates simplified instances of
elimination rules of (co)inductive sets.
\end{descr}
See \cite{isabelle-HOL} for more information. Note that
$\isarkeyword{inductive_cases}$ corresponds to the \texttt{mk_cases} ML
function.
\section{Proof by induction}
\indexisarmeth{induct}
\begin{matharray}{rcl}
induct & : & \isarmeth \\
\end{matharray}
The $induct$ method provides a uniform interface to induction over datatypes,
inductive sets, recursive functions etc. Basically, it is just an interface
to the $rule$ method applied to appropriate instances of the corresponding
induction rules.
\begin{rail}
'induct' (inst * 'and') kind?
;
inst: term term?
;
kind: ('type' | 'set' | 'function' | 'rule') ':' nameref
;
\end{rail}
\begin{descr}
\item [$induct~insts~kind$] abbreviates method $rule~R$, where $R$ is the
induction rule specified by $kind$ and instantiated by $insts$. The rule is
either that of some type, set, or recursive function (defined via TFL), or
given explicitly.
The instantiation basically consists of a list of $P$ $x$ (induction
predicate and variable) specifications, where $P$ is optional. If $kind$ is
omitted, the default is to pick a datatype induction rule according to the
type of some induction variable, which may not be omitted that case.
\end{descr}
\section{Arithmetic}
\indexisarmeth{arith}
\begin{matharray}{rcl}
arith & : & \isarmeth \\
\end{matharray}
The $arith$ method decides linear arithmetic problems (on types $nat$, $int$,
$real$). Note that a simpler (but faster) version of arithmetic reasoning is
already performed by the Simplifier.
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