%% $Id$
\chapter{Higher-Order Logic}
\index{higher-order logic|(}
\index{HOL system@{\sc hol} system}

The theory~\thydx{HOL} implements higher-order logic.  It is based on
Gordon's~{\sc hol} system~\cite{mgordon-hol}, which itself is based on
Church's original paper~\cite{church40}.  Andrews's
book~\cite{andrews86} is a full description of the original
Church-style higher-order logic.  Experience with the {\sc hol} system
has demonstrated that higher-order logic is widely applicable in many
areas of mathematics and computer science, not just hardware
verification, {\sc hol}'s original \textit{raison d'{\^e}tre\/}.  It is
weaker than {\ZF} set theory but for most applications this does not
matter.  If you prefer {\ML} to Lisp, you will probably prefer \HOL\ 
to~{\ZF}.

The syntax of \HOL\footnote{Earlier versions of Isabelle's \HOL\ used a
different syntax.  Ancient releases of Isabelle included still another version
of~\HOL, with explicit type inference rules~\cite{paulson-COLOG}.  This
version no longer exists, but \thydx{ZF} supports a similar style of
reasoning.} follows $\lambda$-calculus and functional programming.  Function
application is curried.  To apply the function~$f$ of type
$\tau@1\To\tau@2\To\tau@3$ to the arguments~$a$ and~$b$ in \HOL, you simply
write $f\,a\,b$.  There is no `apply' operator as in \thydx{ZF}.  Note that
$f(a,b)$ means ``$f$ applied to the pair $(a,b)$'' in \HOL.  We write ordered
pairs as $(a,b)$, not $\langle a,b\rangle$ as in {\ZF}.

\HOL\ has a distinct feel, compared with {\ZF} and {\CTT}.  It
identifies object-level types with meta-level types, taking advantage of
Isabelle's built-in type-checker.  It identifies object-level functions
with meta-level functions, so it uses Isabelle's operations for abstraction
and application.

These identifications allow Isabelle to support \HOL\ particularly
nicely, but they also mean that \HOL\ requires more sophistication
from the user --- in particular, an understanding of Isabelle's type
system.  Beginners should work with \texttt{show_types} (or even
\texttt{show_sorts}) set to \texttt{true}.
%  Gain experience by
%working in first-order logic before attempting to use higher-order logic.
%This chapter assumes familiarity with~{\FOL{}}.


\begin{figure}
\begin{constants}
  \it name      &\it meta-type  & \it description \\
  \cdx{Trueprop}& $bool\To prop$                & coercion to $prop$\\
  \cdx{Not}     & $bool\To bool$                & negation ($\lnot$) \\
  \cdx{True}    & $bool$                        & tautology ($\top$) \\
  \cdx{False}   & $bool$                        & absurdity ($\bot$) \\
  \cdx{If}      & $[bool,\alpha,\alpha]\To\alpha$ & conditional \\
  \cdx{Let}     & $[\alpha,\alpha\To\beta]\To\beta$ & let binder
\end{constants}
\subcaption{Constants}

\begin{constants}
\index{"@@{\tt\at} symbol}
\index{*"! symbol}\index{*"? symbol}
\index{*"?"! symbol}\index{*"E"X"! symbol}
  \it symbol &\it name     &\it meta-type & \it description \\
  \sdx{SOME} or \tt\at & \cdx{Eps}  & $(\alpha\To bool)\To\alpha$ & 
        Hilbert description ($\varepsilon$) \\
  \sdx{ALL} or {\tt!~} & \cdx{All}  & $(\alpha\To bool)\To bool$ & 
        universal quantifier ($\forall$) \\
  \sdx{EX} or {\tt?~}  & \cdx{Ex}   & $(\alpha\To bool)\To bool$ & 
        existential quantifier ($\exists$) \\
  \texttt{EX!} or {\tt?!} & \cdx{Ex1}  & $(\alpha\To bool)\To bool$ & 
        unique existence ($\exists!$)\\
  \texttt{LEAST}  & \cdx{Least}  & $(\alpha::ord \To bool)\To\alpha$ & 
        least element
\end{constants}
\subcaption{Binders} 

\begin{constants}
\index{*"= symbol}
\index{&@{\tt\&} symbol}
\index{*"| symbol}
\index{*"-"-"> symbol}
  \it symbol    & \it meta-type & \it priority & \it description \\ 
  \sdx{o}       & $[\beta\To\gamma,\alpha\To\beta]\To (\alpha\To\gamma)$ & 
        Left 55 & composition ($\circ$) \\
  \tt =         & $[\alpha,\alpha]\To bool$ & Left 50 & equality ($=$) \\
  \tt <         & $[\alpha::ord,\alpha]\To bool$ & Left 50 & less than ($<$) \\
  \tt <=        & $[\alpha::ord,\alpha]\To bool$ & Left 50 & 
                less than or equals ($\leq$)\\
  \tt \&        & $[bool,bool]\To bool$ & Right 35 & conjunction ($\conj$) \\
  \tt |         & $[bool,bool]\To bool$ & Right 30 & disjunction ($\disj$) \\
  \tt -->       & $[bool,bool]\To bool$ & Right 25 & implication ($\imp$)
\end{constants}
\subcaption{Infixes}
\caption{Syntax of \texttt{HOL}} \label{hol-constants}
\end{figure}


\begin{figure}
\index{*let symbol}
\index{*in symbol}
\dquotes
\[\begin{array}{rclcl}
    term & = & \hbox{expression of class~$term$} \\
         & | & "SOME~" id " . " formula
         & | & "\at~" id " . " formula \\
         & | & 
    \multicolumn{3}{l}{"let"~id~"="~term";"\dots";"~id~"="~term~"in"~term} \\
         & | & 
    \multicolumn{3}{l}{"if"~formula~"then"~term~"else"~term} \\
         & | & "LEAST"~ id " . " formula \\[2ex]
 formula & = & \hbox{expression of type~$bool$} \\
         & | & term " = " term \\
         & | & term " \ttilde= " term \\
         & | & term " < " term \\
         & | & term " <= " term \\
         & | & "\ttilde\ " formula \\
         & | & formula " \& " formula \\
         & | & formula " | " formula \\
         & | & formula " --> " formula \\
         & | & "ALL~" id~id^* " . " formula
         & | & "!~~~" id~id^* " . " formula \\
         & | & "EX~~" id~id^* " . " formula 
         & | & "?~~~" id~id^* " . " formula \\
         & | & "EX!~" id~id^* " . " formula
         & | & "?!~~" id~id^* " . " formula \\
  \end{array}
\]
\caption{Full grammar for \HOL} \label{hol-grammar}
\end{figure} 


\section{Syntax}

Figure~\ref{hol-constants} lists the constants (including infixes and
binders), while Fig.\ts\ref{hol-grammar} presents the grammar of
higher-order logic.  Note that $a$\verb|~=|$b$ is translated to
$\lnot(a=b)$.

\begin{warn}
  \HOL\ has no if-and-only-if connective; logical equivalence is expressed
  using equality.  But equality has a high priority, as befitting a
  relation, while if-and-only-if typically has the lowest priority.  Thus,
  $\lnot\lnot P=P$ abbreviates $\lnot\lnot (P=P)$ and not $(\lnot\lnot P)=P$.
  When using $=$ to mean logical equivalence, enclose both operands in
  parentheses.
\end{warn}

\subsection{Types and classes}
The universal type class of higher-order terms is called~\cldx{term}.
By default, explicit type variables have class \cldx{term}.  In
particular the equality symbol and quantifiers are polymorphic over
class \texttt{term}.

The type of formulae, \tydx{bool}, belongs to class \cldx{term}; thus,
formulae are terms.  The built-in type~\tydx{fun}, which constructs
function types, is overloaded with arity {\tt(term,\thinspace
  term)\thinspace term}.  Thus, $\sigma\To\tau$ belongs to class~{\tt
  term} if $\sigma$ and~$\tau$ do, allowing quantification over
functions.

\HOL\ offers various methods for introducing new types.
See~{\S}\ref{sec:HOL:Types} and~{\S}\ref{sec:HOL:datatype}.

Theory \thydx{Ord} defines the syntactic class \cldx{ord} of order
signatures; the relations $<$ and $\leq$ are polymorphic over this
class, as are the functions \cdx{mono}, \cdx{min} and \cdx{max}, and
the \cdx{LEAST} operator. \thydx{Ord} also defines a subclass
\cldx{order} of \cldx{ord} which axiomatizes partially ordered types
(w.r.t.\ $\leq$).

Three other syntactic type classes --- \cldx{plus}, \cldx{minus} and
\cldx{times} --- permit overloading of the operators {\tt+},\index{*"+
  symbol} {\tt-}\index{*"- symbol} and {\tt*}.\index{*"* symbol} In
particular, {\tt-} is instantiated for set difference and subtraction
on natural numbers.

If you state a goal containing overloaded functions, you may need to include
type constraints.  Type inference may otherwise make the goal more
polymorphic than you intended, with confusing results.  For example, the
variables $i$, $j$ and $k$ in the goal $i \leq j \Imp i \leq j+k$ have type
$\alpha::\{ord,plus\}$, although you may have expected them to have some
numeric type, e.g. $nat$.  Instead you should have stated the goal as
$(i::nat) \leq j \Imp i \leq j+k$, which causes all three variables to have
type $nat$.

\begin{warn}
  If resolution fails for no obvious reason, try setting
  \ttindex{show_types} to \texttt{true}, causing Isabelle to display
  types of terms.  Possibly set \ttindex{show_sorts} to \texttt{true} as
  well, causing Isabelle to display type classes and sorts.

  \index{unification!incompleteness of}
  Where function types are involved, Isabelle's unification code does not
  guarantee to find instantiations for type variables automatically.  Be
  prepared to use \ttindex{res_inst_tac} instead of \texttt{resolve_tac},
  possibly instantiating type variables.  Setting
  \ttindex{Unify.trace_types} to \texttt{true} causes Isabelle to report
  omitted search paths during unification.\index{tracing!of unification}
\end{warn}


\subsection{Binders}

Hilbert's {\bf description} operator~$\varepsilon x. P[x]$ stands for
some~$x$ satisfying~$P$, if such exists.  Since all terms in \HOL\ 
denote something, a description is always meaningful, but we do not
know its value unless $P$ defines it uniquely.  We may write
descriptions as \cdx{Eps}($\lambda x. P[x]$) or use the syntax
\hbox{\tt SOME~$x$.~$P[x]$}.

Existential quantification is defined by
\[ \exists x. P~x \;\equiv\; P(\varepsilon x. P~x). \]
The unique existence quantifier, $\exists!x. P$, is defined in terms
of~$\exists$ and~$\forall$.  An Isabelle binder, it admits nested
quantifications.  For instance, $\exists!x\,y. P\,x\,y$ abbreviates
$\exists!x. \exists!y. P\,x\,y$; note that this does not mean that there
exists a unique pair $(x,y)$ satisfying~$P\,x\,y$.

\medskip

\index{*"! symbol}\index{*"? symbol}\index{HOL system@{\sc hol} system} The
basic Isabelle/HOL binders have two notations.  Apart from the usual
\texttt{ALL} and \texttt{EX} for $\forall$ and $\exists$, Isabelle/HOL also
supports the original notation of Gordon's {\sc hol} system: \texttt{!}\ 
and~\texttt{?}.  In the latter case, the existential quantifier \emph{must} be
followed by a space; thus {\tt?x} is an unknown, while \verb'? x. f x=y' is a
quantification.  Both notations are accepted for input.  The print mode
``\ttindexbold{HOL}'' governs the output notation.  If enabled (e.g.\ by
passing option \texttt{-m HOL} to the \texttt{isabelle} executable),
then~{\tt!}\ and~{\tt?}\ are displayed.

\medskip

If $\tau$ is a type of class \cldx{ord}, $P$ a formula and $x$ a
variable of type $\tau$, then the term \cdx{LEAST}~$x. P[x]$ is defined
to be the least (w.r.t.\ $\leq$) $x$ such that $P~x$ holds (see
Fig.~\ref{hol-defs}).  The definition uses Hilbert's $\varepsilon$
choice operator, so \texttt{Least} is always meaningful, but may yield
nothing useful in case there is not a unique least element satisfying
$P$.\footnote{Class $ord$ does not require much of its instances, so
  $\leq$ need not be a well-ordering, not even an order at all!}

\medskip All these binders have priority 10.

\begin{warn}
The low priority of binders means that they need to be enclosed in
parenthesis when they occur in the context of other operations.  For example,
instead of $P \land \forall x. Q$ you need to write $P \land (\forall x. Q)$.
\end{warn}


\subsection{The let and case constructions}
Local abbreviations can be introduced by a \texttt{let} construct whose
syntax appears in Fig.\ts\ref{hol-grammar}.  Internally it is translated into
the constant~\cdx{Let}.  It can be expanded by rewriting with its
definition, \tdx{Let_def}.

\HOL\ also defines the basic syntax
\[\dquotes"case"~e~"of"~c@1~"=>"~e@1~"|" \dots "|"~c@n~"=>"~e@n\] 
as a uniform means of expressing \texttt{case} constructs.  Therefore \texttt{case}
and \sdx{of} are reserved words.  Initially, this is mere syntax and has no
logical meaning.  By declaring translations, you can cause instances of the
\texttt{case} construct to denote applications of particular case operators.
This is what happens automatically for each \texttt{datatype} definition
(see~{\S}\ref{sec:HOL:datatype}).

\begin{warn}
Both \texttt{if} and \texttt{case} constructs have as low a priority as
quantifiers, which requires additional enclosing parentheses in the context
of most other operations.  For example, instead of $f~x = {\tt if\dots
then\dots else}\dots$ you need to write $f~x = ({\tt if\dots then\dots
else\dots})$.
\end{warn}

\section{Rules of inference}

\begin{figure}
\begin{ttbox}\makeatother
\tdx{refl}           t = (t::'a)
\tdx{subst}          [| s = t; P s |] ==> P (t::'a)
\tdx{ext}            (!!x::'a. (f x :: 'b) = g x) ==> (\%x. f x) = (\%x. g x)
\tdx{impI}           (P ==> Q) ==> P-->Q
\tdx{mp}             [| P-->Q;  P |] ==> Q
\tdx{iff}            (P-->Q) --> (Q-->P) --> (P=Q)
\tdx{selectI}        P(x::'a) ==> P(@x. P x)
\tdx{True_or_False}  (P=True) | (P=False)
\end{ttbox}
\caption{The \texttt{HOL} rules} \label{hol-rules}
\end{figure}

Figure~\ref{hol-rules} shows the primitive inference rules of~\HOL{},
with their~{\ML} names.  Some of the rules deserve additional
comments:
\begin{ttdescription}
\item[\tdx{ext}] expresses extensionality of functions.
\item[\tdx{iff}] asserts that logically equivalent formulae are
  equal.
\item[\tdx{selectI}] gives the defining property of the Hilbert
  $\varepsilon$-operator.  It is a form of the Axiom of Choice.  The derived rule
  \tdx{select_equality} (see below) is often easier to use.
\item[\tdx{True_or_False}] makes the logic classical.\footnote{In
    fact, the $\varepsilon$-operator already makes the logic classical, as
    shown by Diaconescu; see Paulson~\cite{paulson-COLOG} for details.}
\end{ttdescription}


\begin{figure}\hfuzz=4pt%suppress "Overfull \hbox" message
\begin{ttbox}\makeatother
\tdx{True_def}   True     == ((\%x::bool. x)=(\%x. x))
\tdx{All_def}    All      == (\%P. P = (\%x. True))
\tdx{Ex_def}     Ex       == (\%P. P(@x. P x))
\tdx{False_def}  False    == (!P. P)
\tdx{not_def}    not      == (\%P. P-->False)
\tdx{and_def}    op &     == (\%P Q. !R. (P-->Q-->R) --> R)
\tdx{or_def}     op |     == (\%P Q. !R. (P-->R) --> (Q-->R) --> R)
\tdx{Ex1_def}    Ex1      == (\%P. ? x. P x & (! y. P y --> y=x))

\tdx{o_def}      op o     == (\%(f::'b=>'c) g x::'a. f(g x))
\tdx{if_def}     If P x y ==
              (\%P x y. @z::'a.(P=True --> z=x) & (P=False --> z=y))
\tdx{Let_def}    Let s f  == f s
\tdx{Least_def}  Least P  == @x. P(x) & (ALL y. P(y) --> x <= y)"
\end{ttbox}
\caption{The \texttt{HOL} definitions} \label{hol-defs}
\end{figure}


\HOL{} follows standard practice in higher-order logic: only a few
connectives are taken as primitive, with the remainder defined obscurely
(Fig.\ts\ref{hol-defs}).  Gordon's {\sc hol} system expresses the
corresponding definitions \cite[page~270]{mgordon-hol} using
object-equality~({\tt=}), which is possible because equality in
higher-order logic may equate formulae and even functions over formulae.
But theory~\HOL{}, like all other Isabelle theories, uses
meta-equality~({\tt==}) for definitions.
\begin{warn}
The definitions above should never be expanded and are shown for completeness
only.  Instead users should reason in terms of the derived rules shown below
or, better still, using high-level tactics
(see~{\S}\ref{sec:HOL:generic-packages}).
\end{warn}

Some of the rules mention type variables; for example, \texttt{refl}
mentions the type variable~{\tt'a}.  This allows you to instantiate
type variables explicitly by calling \texttt{res_inst_tac}.


\begin{figure}
\begin{ttbox}
\tdx{sym}         s=t ==> t=s
\tdx{trans}       [| r=s; s=t |] ==> r=t
\tdx{ssubst}      [| t=s; P s |] ==> P t
\tdx{box_equals}  [| a=b;  a=c;  b=d |] ==> c=d  
\tdx{arg_cong}    x = y ==> f x = f y
\tdx{fun_cong}    f = g ==> f x = g x
\tdx{cong}        [| f = g; x = y |] ==> f x = g y
\tdx{not_sym}     t ~= s ==> s ~= t
\subcaption{Equality}

\tdx{TrueI}       True 
\tdx{FalseE}      False ==> P

\tdx{conjI}       [| P; Q |] ==> P&Q
\tdx{conjunct1}   [| P&Q |] ==> P
\tdx{conjunct2}   [| P&Q |] ==> Q 
\tdx{conjE}       [| P&Q;  [| P; Q |] ==> R |] ==> R

\tdx{disjI1}      P ==> P|Q
\tdx{disjI2}      Q ==> P|Q
\tdx{disjE}       [| P | Q; P ==> R; Q ==> R |] ==> R

\tdx{notI}        (P ==> False) ==> ~ P
\tdx{notE}        [| ~ P;  P |] ==> R
\tdx{impE}        [| P-->Q;  P;  Q ==> R |] ==> R
\subcaption{Propositional logic}

\tdx{iffI}        [| P ==> Q;  Q ==> P |] ==> P=Q
\tdx{iffD1}       [| P=Q; P |] ==> Q
\tdx{iffD2}       [| P=Q; Q |] ==> P
\tdx{iffE}        [| P=Q; [| P --> Q; Q --> P |] ==> R |] ==> R
%
%\tdx{eqTrueI}     P ==> P=True 
%\tdx{eqTrueE}     P=True ==> P 
\subcaption{Logical equivalence}

\end{ttbox}
\caption{Derived rules for \HOL} \label{hol-lemmas1}
\end{figure}


\begin{figure}
\begin{ttbox}\makeatother
\tdx{allI}      (!!x. P x) ==> !x. P x
\tdx{spec}      !x. P x ==> P x
\tdx{allE}      [| !x. P x;  P x ==> R |] ==> R
\tdx{all_dupE}  [| !x. P x;  [| P x; !x. P x |] ==> R |] ==> R

\tdx{exI}       P x ==> ? x. P x
\tdx{exE}       [| ? x. P x; !!x. P x ==> Q |] ==> Q

\tdx{ex1I}      [| P a;  !!x. P x ==> x=a |] ==> ?! x. P x
\tdx{ex1E}      [| ?! x. P x;  !!x. [| P x;  ! y. P y --> y=x |] ==> R 
          |] ==> R

\tdx{select_equality} [| P a;  !!x. P x ==> x=a |] ==> (@x. P x) = a
\subcaption{Quantifiers and descriptions}

\tdx{ccontr}          (~P ==> False) ==> P
\tdx{classical}       (~P ==> P) ==> P
\tdx{excluded_middle} ~P | P

\tdx{disjCI}          (~Q ==> P) ==> P|Q
\tdx{exCI}            (! x. ~ P x ==> P a) ==> ? x. P x
\tdx{impCE}           [| P-->Q; ~ P ==> R; Q ==> R |] ==> R
\tdx{iffCE}           [| P=Q;  [| P;Q |] ==> R;  [| ~P; ~Q |] ==> R |] ==> R
\tdx{notnotD}         ~~P ==> P
\tdx{swap}            ~P ==> (~Q ==> P) ==> Q
\subcaption{Classical logic}

%\tdx{if_True}         (if True then x else y) = x
%\tdx{if_False}        (if False then x else y) = y
\tdx{if_P}            P ==> (if P then x else y) = x
\tdx{if_not_P}        ~ P ==> (if P then x else y) = y
\tdx{split_if}        P(if Q then x else y) = ((Q --> P x) & (~Q --> P y))
\subcaption{Conditionals}
\end{ttbox}
\caption{More derived rules} \label{hol-lemmas2}
\end{figure}

Some derived rules are shown in Figures~\ref{hol-lemmas1}
and~\ref{hol-lemmas2}, with their {\ML} names.  These include natural rules
for the logical connectives, as well as sequent-style elimination rules for
conjunctions, implications, and universal quantifiers.  

Note the equality rules: \tdx{ssubst} performs substitution in
backward proofs, while \tdx{box_equals} supports reasoning by
simplifying both sides of an equation.

The following simple tactics are occasionally useful:
\begin{ttdescription}
\item[\ttindexbold{strip_tac} $i$] applies \texttt{allI} and \texttt{impI}
  repeatedly to remove all outermost universal quantifiers and implications
  from subgoal $i$.
\item[\ttindexbold{cases_tac} {\tt"}$P${\tt"} $i$] performs case distinction
  on $P$ for subgoal $i$: the latter is replaced by two identical subgoals
  with the added assumptions $P$ and $\lnot P$, respectively.
\item[\ttindexbold{smp_tac} $j$ $i$] applies $j$ times \texttt{spec} and then
  \texttt{mp} in subgoal $i$, which is typically useful when forward-chaining 
  from an induction hypothesis. As a generalization of \texttt{mp_tac}, 
  if there are assumptions $\forall \vec{x}. P \vec{x} \imp Q \vec{x}$ and 
  $P \vec{a}$, ($\vec{x}$ being a vector of $j$ variables)
  then it replaces the universally quantified implication by $Q \vec{a}$. 
  It may instantiate unknowns. It fails if it can do nothing.
\end{ttdescription}


\begin{figure} 
\begin{center}
\begin{tabular}{rrr}
  \it name      &\it meta-type  & \it description \\ 
\index{{}@\verb'{}' symbol}
  \verb|{}|     & $\alpha\,set$         & the empty set \\
  \cdx{insert}  & $[\alpha,\alpha\,set]\To \alpha\,set$
        & insertion of element \\
  \cdx{Collect} & $(\alpha\To bool)\To\alpha\,set$
        & comprehension \\
  \cdx{Compl}   & $\alpha\,set\To\alpha\,set$
        & complement \\
  \cdx{INTER} & $[\alpha\,set,\alpha\To\beta\,set]\To\beta\,set$
        & intersection over a set\\
  \cdx{UNION} & $[\alpha\,set,\alpha\To\beta\,set]\To\beta\,set$
        & union over a set\\
  \cdx{Inter} & $(\alpha\,set)set\To\alpha\,set$
        &set of sets intersection \\
  \cdx{Union} & $(\alpha\,set)set\To\alpha\,set$
        &set of sets union \\
  \cdx{Pow}   & $\alpha\,set \To (\alpha\,set)set$
        & powerset \\[1ex]
  \cdx{range}   & $(\alpha\To\beta )\To\beta\,set$
        & range of a function \\[1ex]
  \cdx{Ball}~~\cdx{Bex} & $[\alpha\,set,\alpha\To bool]\To bool$
        & bounded quantifiers
\end{tabular}
\end{center}
\subcaption{Constants}

\begin{center}
\begin{tabular}{llrrr} 
  \it symbol &\it name     &\it meta-type & \it priority & \it description \\
  \sdx{INT}  & \cdx{INTER1}  & $(\alpha\To\beta\,set)\To\beta\,set$ & 10 & 
        intersection over a type\\
  \sdx{UN}  & \cdx{UNION1}  & $(\alpha\To\beta\,set)\To\beta\,set$ & 10 & 
        union over a type
\end{tabular}
\end{center}
\subcaption{Binders} 

\begin{center}
\index{*"`"` symbol}
\index{*": symbol}
\index{*"<"= symbol}
\begin{tabular}{rrrr} 
  \it symbol    & \it meta-type & \it priority & \it description \\ 
  \tt ``        & $[\alpha\To\beta ,\alpha\,set]\To  \beta\,set$
        & Left 90 & image \\
  \sdx{Int}     & $[\alpha\,set,\alpha\,set]\To\alpha\,set$
        & Left 70 & intersection ($\int$) \\
  \sdx{Un}      & $[\alpha\,set,\alpha\,set]\To\alpha\,set$
        & Left 65 & union ($\un$) \\
  \tt:          & $[\alpha ,\alpha\,set]\To bool$       
        & Left 50 & membership ($\in$) \\
  \tt <=        & $[\alpha\,set,\alpha\,set]\To bool$
        & Left 50 & subset ($\subseteq$) 
\end{tabular}
\end{center}
\subcaption{Infixes}
\caption{Syntax of the theory \texttt{Set}} \label{hol-set-syntax}
\end{figure} 


\begin{figure} 
\begin{center} \tt\frenchspacing
\index{*"! symbol}
\begin{tabular}{rrr} 
  \it external          & \it internal  & \it description \\ 
  $a$ \ttilde: $b$      & \ttilde($a$ : $b$)    & \rm non-membership\\
  {\ttlbrace}$a@1$, $\ldots${\ttrbrace}  &  insert $a@1$ $\ldots$ {\ttlbrace}{\ttrbrace} & \rm finite set \\
  {\ttlbrace}$x$. $P[x]${\ttrbrace}        &  Collect($\lambda x. P[x]$) &
        \rm comprehension \\
  \sdx{INT} $x$:$A$. $B[x]$      & INTER $A$ $\lambda x. B[x]$ &
        \rm intersection \\
  \sdx{UN}{\tt\ }  $x$:$A$. $B[x]$      & UNION $A$ $\lambda x. B[x]$ &
        \rm union \\
  \sdx{ALL} $x$:$A$. $P[x]$ or \sdx{!} $x$:$A$. $P[x]$ &
        Ball $A$ $\lambda x. P[x]$ & 
        \rm bounded $\forall$ \\
  \sdx{EX}{\tt\ } $x$:$A$. $P[x]$ or \sdx{?} $x$:$A$. $P[x]$ & 
        Bex $A$ $\lambda x. P[x]$ & \rm bounded $\exists$
\end{tabular}
\end{center}
\subcaption{Translations}

\dquotes
\[\begin{array}{rclcl}
    term & = & \hbox{other terms\ldots} \\
         & | & "{\ttlbrace}{\ttrbrace}" \\
         & | & "{\ttlbrace} " term\; ("," term)^* " {\ttrbrace}" \\
         & | & "{\ttlbrace} " id " . " formula " {\ttrbrace}" \\
         & | & term " `` " term \\
         & | & term " Int " term \\
         & | & term " Un " term \\
         & | & "INT~~"  id ":" term " . " term \\
         & | & "UN~~~"  id ":" term " . " term \\
         & | & "INT~~"  id~id^* " . " term \\
         & | & "UN~~~"  id~id^* " . " term \\[2ex]
 formula & = & \hbox{other formulae\ldots} \\
         & | & term " : " term \\
         & | & term " \ttilde: " term \\
         & | & term " <= " term \\
         & | & "ALL " id ":" term " . " formula
         & | & "!~" id ":" term " . " formula \\
         & | & "EX~~" id ":" term " . " formula
         & | & "?~" id ":" term " . " formula \\
  \end{array}
\]
\subcaption{Full Grammar}
\caption{Syntax of the theory \texttt{Set} (continued)} \label{hol-set-syntax2}
\end{figure} 


\section{A formulation of set theory}
Historically, higher-order logic gives a foundation for Russell and
Whitehead's theory of classes.  Let us use modern terminology and call them
{\bf sets}, but note that these sets are distinct from those of {\ZF} set
theory, and behave more like {\ZF} classes.
\begin{itemize}
\item
Sets are given by predicates over some type~$\sigma$.  Types serve to
define universes for sets, but type-checking is still significant.
\item
There is a universal set (for each type).  Thus, sets have complements, and
may be defined by absolute comprehension.
\item
Although sets may contain other sets as elements, the containing set must
have a more complex type.
\end{itemize}
Finite unions and intersections have the same behaviour in \HOL\ as they
do in~{\ZF}.  In \HOL\ the intersection of the empty set is well-defined,
denoting the universal set for the given type.

\subsection{Syntax of set theory}\index{*set type}
\HOL's set theory is called \thydx{Set}.  The type $\alpha\,set$ is
essentially the same as $\alpha\To bool$.  The new type is defined for
clarity and to avoid complications involving function types in unification.
The isomorphisms between the two types are declared explicitly.  They are
very natural: \texttt{Collect} maps $\alpha\To bool$ to $\alpha\,set$, while
\hbox{\tt op :} maps in the other direction (ignoring argument order).

Figure~\ref{hol-set-syntax} lists the constants, infixes, and syntax
translations.  Figure~\ref{hol-set-syntax2} presents the grammar of the new
constructs.  Infix operators include union and intersection ($A\un B$
and $A\int B$), the subset and membership relations, and the image
operator~{\tt``}\@.  Note that $a$\verb|~:|$b$ is translated to
$\lnot(a\in b)$.  

The $\{a@1,\ldots\}$ notation abbreviates finite sets constructed in
the obvious manner using~\texttt{insert} and~$\{\}$:
\begin{eqnarray*}
  \{a, b, c\} & \equiv &
  \texttt{insert} \, a \, ({\tt insert} \, b \, ({\tt insert} \, c \, \{\}))
\end{eqnarray*}

The set \hbox{\tt{\ttlbrace}$x$.\ $P[x]${\ttrbrace}} consists of all $x$ (of suitable type)
that satisfy~$P[x]$, where $P[x]$ is a formula that may contain free
occurrences of~$x$.  This syntax expands to \cdx{Collect}$(\lambda
x. P[x])$.  It defines sets by absolute comprehension, which is impossible
in~{\ZF}; the type of~$x$ implicitly restricts the comprehension.

The set theory defines two {\bf bounded quantifiers}:
\begin{eqnarray*}
   \forall x\in A. P[x] &\hbox{abbreviates}& \forall x. x\in A\imp P[x] \\
   \exists x\in A. P[x] &\hbox{abbreviates}& \exists x. x\in A\conj P[x]
\end{eqnarray*}
The constants~\cdx{Ball} and~\cdx{Bex} are defined
accordingly.  Instead of \texttt{Ball $A$ $P$} and \texttt{Bex $A$ $P$} we may
write\index{*"! symbol}\index{*"? symbol}
\index{*ALL symbol}\index{*EX symbol} 
%
\hbox{\tt ALL~$x$:$A$.\ $P[x]$} and \hbox{\tt EX~$x$:$A$.\ $P[x]$}.  The
original notation of Gordon's {\sc hol} system is supported as well: \sdx{!}\ 
and \sdx{?}.

Unions and intersections over sets, namely $\bigcup@{x\in A}B[x]$ and
$\bigcap@{x\in A}B[x]$, are written 
\sdx{UN}~\hbox{\tt$x$:$A$.\ $B[x]$} and
\sdx{INT}~\hbox{\tt$x$:$A$.\ $B[x]$}.  

Unions and intersections over types, namely $\bigcup@x B[x]$ and $\bigcap@x
B[x]$, are written \sdx{UN}~\hbox{\tt$x$.\ $B[x]$} and
\sdx{INT}~\hbox{\tt$x$.\ $B[x]$}.  They are equivalent to the previous
union and intersection operators when $A$ is the universal set.

The operators $\bigcup A$ and $\bigcap A$ act upon sets of sets.  They are
not binders, but are equal to $\bigcup@{x\in A}x$ and $\bigcap@{x\in A}x$,
respectively.



\begin{figure} \underscoreon
\begin{ttbox}
\tdx{mem_Collect_eq}    (a : {\ttlbrace}x. P x{\ttrbrace}) = P a
\tdx{Collect_mem_eq}    {\ttlbrace}x. x:A{\ttrbrace} = A

\tdx{empty_def}         {\ttlbrace}{\ttrbrace}          == {\ttlbrace}x. False{\ttrbrace}
\tdx{insert_def}        insert a B  == {\ttlbrace}x. x=a{\ttrbrace} Un B
\tdx{Ball_def}          Ball A P    == ! x. x:A --> P x
\tdx{Bex_def}           Bex A P     == ? x. x:A & P x
\tdx{subset_def}        A <= B      == ! x:A. x:B
\tdx{Un_def}            A Un B      == {\ttlbrace}x. x:A | x:B{\ttrbrace}
\tdx{Int_def}           A Int B     == {\ttlbrace}x. x:A & x:B{\ttrbrace}
\tdx{set_diff_def}      A - B       == {\ttlbrace}x. x:A & x~:B{\ttrbrace}
\tdx{Compl_def}         Compl A     == {\ttlbrace}x. ~ x:A{\ttrbrace}
\tdx{INTER_def}         INTER A B   == {\ttlbrace}y. ! x:A. y: B x{\ttrbrace}
\tdx{UNION_def}         UNION A B   == {\ttlbrace}y. ? x:A. y: B x{\ttrbrace}
\tdx{INTER1_def}        INTER1 B    == INTER {\ttlbrace}x. True{\ttrbrace} B 
\tdx{UNION1_def}        UNION1 B    == UNION {\ttlbrace}x. True{\ttrbrace} B 
\tdx{Inter_def}         Inter S     == (INT x:S. x)
\tdx{Union_def}         Union S     == (UN  x:S. x)
\tdx{Pow_def}           Pow A       == {\ttlbrace}B. B <= A{\ttrbrace}
\tdx{image_def}         f``A        == {\ttlbrace}y. ? x:A. y=f x{\ttrbrace}
\tdx{range_def}         range f     == {\ttlbrace}y. ? x. y=f x{\ttrbrace}
\end{ttbox}
\caption{Rules of the theory \texttt{Set}} \label{hol-set-rules}
\end{figure}


\begin{figure} \underscoreon
\begin{ttbox}
\tdx{CollectI}        [| P a |] ==> a : {\ttlbrace}x. P x{\ttrbrace}
\tdx{CollectD}        [| a : {\ttlbrace}x. P x{\ttrbrace} |] ==> P a
\tdx{CollectE}        [| a : {\ttlbrace}x. P x{\ttrbrace};  P a ==> W |] ==> W

\tdx{ballI}           [| !!x. x:A ==> P x |] ==> ! x:A. P x
\tdx{bspec}           [| ! x:A. P x;  x:A |] ==> P x
\tdx{ballE}           [| ! x:A. P x;  P x ==> Q;  ~ x:A ==> Q |] ==> Q

\tdx{bexI}            [| P x;  x:A |] ==> ? x:A. P x
\tdx{bexCI}           [| ! x:A. ~ P x ==> P a;  a:A |] ==> ? x:A. P x
\tdx{bexE}            [| ? x:A. P x;  !!x. [| x:A; P x |] ==> Q  |] ==> Q
\subcaption{Comprehension and Bounded quantifiers}

\tdx{subsetI}         (!!x. x:A ==> x:B) ==> A <= B
\tdx{subsetD}         [| A <= B;  c:A |] ==> c:B
\tdx{subsetCE}        [| A <= B;  ~ (c:A) ==> P;  c:B ==> P |] ==> P

\tdx{subset_refl}     A <= A
\tdx{subset_trans}    [| A<=B;  B<=C |] ==> A<=C

\tdx{equalityI}       [| A <= B;  B <= A |] ==> A = B
\tdx{equalityD1}      A = B ==> A<=B
\tdx{equalityD2}      A = B ==> B<=A
\tdx{equalityE}       [| A = B;  [| A<=B; B<=A |] ==> P |]  ==>  P

\tdx{equalityCE}      [| A = B;  [| c:A; c:B |] ==> P;  
                           [| ~ c:A; ~ c:B |] ==> P 
                |]  ==>  P
\subcaption{The subset and equality relations}
\end{ttbox}
\caption{Derived rules for set theory} \label{hol-set1}
\end{figure}


\begin{figure} \underscoreon
\begin{ttbox}
\tdx{emptyE}   a : {\ttlbrace}{\ttrbrace} ==> P

\tdx{insertI1} a : insert a B
\tdx{insertI2} a : B ==> a : insert b B
\tdx{insertE}  [| a : insert b A;  a=b ==> P;  a:A ==> P |] ==> P

\tdx{ComplI}   [| c:A ==> False |] ==> c : Compl A
\tdx{ComplD}   [| c : Compl A |] ==> ~ c:A

\tdx{UnI1}     c:A ==> c : A Un B
\tdx{UnI2}     c:B ==> c : A Un B
\tdx{UnCI}     (~c:B ==> c:A) ==> c : A Un B
\tdx{UnE}      [| c : A Un B;  c:A ==> P;  c:B ==> P |] ==> P

\tdx{IntI}     [| c:A;  c:B |] ==> c : A Int B
\tdx{IntD1}    c : A Int B ==> c:A
\tdx{IntD2}    c : A Int B ==> c:B
\tdx{IntE}     [| c : A Int B;  [| c:A; c:B |] ==> P |] ==> P

\tdx{UN_I}     [| a:A;  b: B a |] ==> b: (UN x:A. B x)
\tdx{UN_E}     [| b: (UN x:A. B x);  !!x.[| x:A;  b:B x |] ==> R |] ==> R

\tdx{INT_I}    (!!x. x:A ==> b: B x) ==> b : (INT x:A. B x)
\tdx{INT_D}    [| b: (INT x:A. B x);  a:A |] ==> b: B a
\tdx{INT_E}    [| b: (INT x:A. B x);  b: B a ==> R;  ~ a:A ==> R |] ==> R

\tdx{UnionI}   [| X:C;  A:X |] ==> A : Union C
\tdx{UnionE}   [| A : Union C;  !!X.[| A:X;  X:C |] ==> R |] ==> R

\tdx{InterI}   [| !!X. X:C ==> A:X |] ==> A : Inter C
\tdx{InterD}   [| A : Inter C;  X:C |] ==> A:X
\tdx{InterE}   [| A : Inter C;  A:X ==> R;  ~ X:C ==> R |] ==> R

\tdx{PowI}     A<=B ==> A: Pow B
\tdx{PowD}     A: Pow B ==> A<=B

\tdx{imageI}   [| x:A |] ==> f x : f``A
\tdx{imageE}   [| b : f``A;  !!x.[| b=f x;  x:A |] ==> P |] ==> P

\tdx{rangeI}   f x : range f
\tdx{rangeE}   [| b : range f;  !!x.[| b=f x |] ==> P |] ==> P
\end{ttbox}
\caption{Further derived rules for set theory} \label{hol-set2}
\end{figure}


\subsection{Axioms and rules of set theory}
Figure~\ref{hol-set-rules} presents the rules of theory \thydx{Set}.  The
axioms \tdx{mem_Collect_eq} and \tdx{Collect_mem_eq} assert
that the functions \texttt{Collect} and \hbox{\tt op :} are isomorphisms.  Of
course, \hbox{\tt op :} also serves as the membership relation.

All the other axioms are definitions.  They include the empty set, bounded
quantifiers, unions, intersections, complements and the subset relation.
They also include straightforward constructions on functions: image~({\tt``})
and \texttt{range}.

%The predicate \cdx{inj_on} is used for simulating type definitions.
%The statement ${\tt inj_on}~f~A$ asserts that $f$ is injective on the
%set~$A$, which specifies a subset of its domain type.  In a type
%definition, $f$ is the abstraction function and $A$ is the set of valid
%representations; we should not expect $f$ to be injective outside of~$A$.

%\begin{figure} \underscoreon
%\begin{ttbox}
%\tdx{Inv_f_f}    inj f ==> Inv f (f x) = x
%\tdx{f_Inv_f}    y : range f ==> f(Inv f y) = y
%
%\tdx{Inv_injective}
%    [| Inv f x=Inv f y; x: range f;  y: range f |] ==> x=y
%
%
%\tdx{monoI}      [| !!A B. A <= B ==> f A <= f B |] ==> mono f
%\tdx{monoD}      [| mono f;  A <= B |] ==> f A <= f B
%
%\tdx{injI}       [| !! x y. f x = f y ==> x=y |] ==> inj f
%\tdx{inj_inverseI}              (!!x. g(f x) = x) ==> inj f
%\tdx{injD}       [| inj f; f x = f y |] ==> x=y
%
%\tdx{inj_onI}  (!!x y. [| f x=f y; x:A; y:A |] ==> x=y) ==> inj_on f A
%\tdx{inj_onD}  [| inj_on f A;  f x=f y;  x:A;  y:A |] ==> x=y
%
%\tdx{inj_on_inverseI}
%    (!!x. x:A ==> g(f x) = x) ==> inj_on f A
%\tdx{inj_on_contraD}
%    [| inj_on f A;  x~=y;  x:A;  y:A |] ==> ~ f x=f y
%\end{ttbox}
%\caption{Derived rules involving functions} \label{hol-fun}
%\end{figure}


\begin{figure} \underscoreon
\begin{ttbox}
\tdx{Union_upper}     B:A ==> B <= Union A
\tdx{Union_least}     [| !!X. X:A ==> X<=C |] ==> Union A <= C

\tdx{Inter_lower}     B:A ==> Inter A <= B
\tdx{Inter_greatest}  [| !!X. X:A ==> C<=X |] ==> C <= Inter A

\tdx{Un_upper1}       A <= A Un B
\tdx{Un_upper2}       B <= A Un B
\tdx{Un_least}        [| A<=C;  B<=C |] ==> A Un B <= C

\tdx{Int_lower1}      A Int B <= A
\tdx{Int_lower2}      A Int B <= B
\tdx{Int_greatest}    [| C<=A;  C<=B |] ==> C <= A Int B
\end{ttbox}
\caption{Derived rules involving subsets} \label{hol-subset}
\end{figure}


\begin{figure} \underscoreon   \hfuzz=4pt%suppress "Overfull \hbox" message
\begin{ttbox}
\tdx{Int_absorb}        A Int A = A
\tdx{Int_commute}       A Int B = B Int A
\tdx{Int_assoc}         (A Int B) Int C  =  A Int (B Int C)
\tdx{Int_Un_distrib}    (A Un B)  Int C  =  (A Int C) Un (B Int C)

\tdx{Un_absorb}         A Un A = A
\tdx{Un_commute}        A Un B = B Un A
\tdx{Un_assoc}          (A Un B)  Un C  =  A Un (B Un C)
\tdx{Un_Int_distrib}    (A Int B) Un C  =  (A Un C) Int (B Un C)

\tdx{Compl_disjoint}    A Int (Compl A) = {\ttlbrace}x. False{\ttrbrace}
\tdx{Compl_partition}   A Un  (Compl A) = {\ttlbrace}x. True{\ttrbrace}
\tdx{double_complement} Compl(Compl A) = A
\tdx{Compl_Un}          Compl(A Un B)  = (Compl A) Int (Compl B)
\tdx{Compl_Int}         Compl(A Int B) = (Compl A) Un (Compl B)

\tdx{Union_Un_distrib}  Union(A Un B) = (Union A) Un (Union B)
\tdx{Int_Union}         A Int (Union B) = (UN C:B. A Int C)
\tdx{Un_Union_image}    (UN x:C.(A x) Un (B x)) = Union(A``C) Un Union(B``C)

\tdx{Inter_Un_distrib}  Inter(A Un B) = (Inter A) Int (Inter B)
\tdx{Un_Inter}          A Un (Inter B) = (INT C:B. A Un C)
\tdx{Int_Inter_image}   (INT x:C.(A x) Int (B x)) = Inter(A``C) Int Inter(B``C)
\end{ttbox}
\caption{Set equalities} \label{hol-equalities}
\end{figure}


Figures~\ref{hol-set1} and~\ref{hol-set2} present derived rules.  Most are
obvious and resemble rules of Isabelle's {\ZF} set theory.  Certain rules,
such as \tdx{subsetCE}, \tdx{bexCI} and \tdx{UnCI},
are designed for classical reasoning; the rules \tdx{subsetD},
\tdx{bexI}, \tdx{Un1} and~\tdx{Un2} are not
strictly necessary but yield more natural proofs.  Similarly,
\tdx{equalityCE} supports classical reasoning about extensionality,
after the fashion of \tdx{iffCE}.  See the file \texttt{HOL/Set.ML} for
proofs pertaining to set theory.

Figure~\ref{hol-subset} presents lattice properties of the subset relation.
Unions form least upper bounds; non-empty intersections form greatest lower
bounds.  Reasoning directly about subsets often yields clearer proofs than
reasoning about the membership relation.  See the file \texttt{HOL/subset.ML}.

Figure~\ref{hol-equalities} presents many common set equalities.  They
include commutative, associative and distributive laws involving unions,
intersections and complements.  For a complete listing see the file {\tt
HOL/equalities.ML}.

\begin{warn}
\texttt{Blast_tac} proves many set-theoretic theorems automatically.
Hence you seldom need to refer to the theorems above.
\end{warn}

\begin{figure}
\begin{center}
\begin{tabular}{rrr}
  \it name      &\it meta-type  & \it description \\ 
  \cdx{inj}~~\cdx{surj}& $(\alpha\To\beta )\To bool$
        & injective/surjective \\
  \cdx{inj_on}        & $[\alpha\To\beta ,\alpha\,set]\To bool$
        & injective over subset\\
  \cdx{inv} & $(\alpha\To\beta)\To(\beta\To\alpha)$ & inverse function
\end{tabular}
\end{center}

\underscoreon
\begin{ttbox}
\tdx{inj_def}         inj f      == ! x y. f x=f y --> x=y
\tdx{surj_def}        surj f     == ! y. ? x. y=f x
\tdx{inj_on_def}      inj_on f A == !x:A. !y:A. f x=f y --> x=y
\tdx{inv_def}         inv f      == (\%y. @x. f(x)=y)
\end{ttbox}
\caption{Theory \thydx{Fun}} \label{fig:HOL:Fun}
\end{figure}

\subsection{Properties of functions}\nopagebreak
Figure~\ref{fig:HOL:Fun} presents a theory of simple properties of functions.
Note that ${\tt inv}~f$ uses Hilbert's $\varepsilon$ to yield an inverse
of~$f$.  See the file \texttt{HOL/Fun.ML} for a complete listing of the derived
rules.  Reasoning about function composition (the operator~\sdx{o}) and the
predicate~\cdx{surj} is done simply by expanding the definitions.

There is also a large collection of monotonicity theorems for constructions
on sets in the file \texttt{HOL/mono.ML}.


\section{Generic packages}
\label{sec:HOL:generic-packages}

\HOL\ instantiates most of Isabelle's generic packages, making available the
simplifier and the classical reasoner.

\subsection{Simplification and substitution}

Simplification tactics tactics such as \texttt{Asm_simp_tac} and \texttt{Full_simp_tac} use the default simpset
(\texttt{simpset()}), which works for most purposes.  A quite minimal
simplification set for higher-order logic is~\ttindexbold{HOL_ss};
even more frugal is \ttindexbold{HOL_basic_ss}.  Equality~($=$), which
also expresses logical equivalence, may be used for rewriting.  See
the file \texttt{HOL/simpdata.ML} for a complete listing of the basic
simplification rules.

See \iflabelundefined{chap:classical}{the {\em Reference Manual\/}}%
{Chaps.\ts\ref{substitution} and~\ref{simp-chap}} for details of substitution
and simplification.

\begin{warn}\index{simplification!of conjunctions}%
  Reducing $a=b\conj P(a)$ to $a=b\conj P(b)$ is sometimes advantageous.  The
  left part of a conjunction helps in simplifying the right part.  This effect
  is not available by default: it can be slow.  It can be obtained by
  including \ttindex{conj_cong} in a simpset, \verb$addcongs [conj_cong]$.
\end{warn}

If the simplifier cannot use a certain rewrite rule --- either because
of nontermination or because its left-hand side is too flexible ---
then you might try \texttt{stac}:
\begin{ttdescription}
\item[\ttindexbold{stac} $thm$ $i,$] where $thm$ is of the form $lhs = rhs$,
  replaces in subgoal $i$ instances of $lhs$ by corresponding instances of
  $rhs$.  In case of multiple instances of $lhs$ in subgoal $i$, backtracking
  may be necessary to select the desired ones.

If $thm$ is a conditional equality, the instantiated condition becomes an
additional (first) subgoal.
\end{ttdescription}

 \HOL{} provides the tactic \ttindex{hyp_subst_tac}, which substitutes
  for an equality throughout a subgoal and its hypotheses.  This tactic uses
  \HOL's general substitution rule.

\subsubsection{Case splitting}
\label{subsec:HOL:case:splitting}

\HOL{} also provides convenient means for case splitting during
rewriting. Goals containing a subterm of the form \texttt{if}~$b$~{\tt
then\dots else\dots} often require a case distinction on $b$. This is
expressed by the theorem \tdx{split_if}:
$$
\Var{P}(\mbox{\tt if}~\Var{b}~{\tt then}~\Var{x}~\mbox{\tt else}~\Var{y})~=~
((\Var{b} \to \Var{P}(\Var{x})) \land (\lnot \Var{b} \to \Var{P}(\Var{y})))
\eqno{(*)}
$$
For example, a simple instance of $(*)$ is
\[
x \in (\mbox{\tt if}~x \in A~{\tt then}~A~\mbox{\tt else}~\{x\})~=~
((x \in A \to x \in A) \land (x \notin A \to x \in \{x\}))
\]
Because $(*)$ is too general as a rewrite rule for the simplifier (the
left-hand side is not a higher-order pattern in the sense of
\iflabelundefined{chap:simplification}{the {\em Reference Manual\/}}%
{Chap.\ts\ref{chap:simplification}}), there is a special infix function 
\ttindexbold{addsplits} of type \texttt{simpset * thm list -> simpset}
(analogous to \texttt{addsimps}) that adds rules such as $(*)$ to a
simpset, as in
\begin{ttbox}
by(simp_tac (simpset() addsplits [split_if]) 1);
\end{ttbox}
The effect is that after each round of simplification, one occurrence of
\texttt{if} is split acording to \texttt{split_if}, until all occurences of
\texttt{if} have been eliminated.

It turns out that using \texttt{split_if} is almost always the right thing to
do. Hence \texttt{split_if} is already included in the default simpset. If
you want to delete it from a simpset, use \ttindexbold{delsplits}, which is
the inverse of \texttt{addsplits}:
\begin{ttbox}
by(simp_tac (simpset() delsplits [split_if]) 1);
\end{ttbox}

In general, \texttt{addsplits} accepts rules of the form
\[
\Var{P}(c~\Var{x@1}~\dots~\Var{x@n})~=~ rhs
\]
where $c$ is a constant and $rhs$ is arbitrary. Note that $(*)$ is of the
right form because internally the left-hand side is
$\Var{P}(\mathtt{If}~\Var{b}~\Var{x}~~\Var{y})$. Important further examples
are splitting rules for \texttt{case} expressions (see~{\S}\ref{subsec:list}
and~{\S}\ref{subsec:datatype:basics}).

Analogous to \texttt{Addsimps} and \texttt{Delsimps}, there are also
imperative versions of \texttt{addsplits} and \texttt{delsplits}
\begin{ttbox}
\ttindexbold{Addsplits}: thm list -> unit
\ttindexbold{Delsplits}: thm list -> unit
\end{ttbox}
for adding splitting rules to, and deleting them from the current simpset.

\subsection{Classical reasoning}

\HOL\ derives classical introduction rules for $\disj$ and~$\exists$, as
well as classical elimination rules for~$\imp$ and~$\bimp$, and the swap
rule; recall Fig.\ts\ref{hol-lemmas2} above.

The classical reasoner is installed.  Tactics such as \texttt{Blast_tac} and
{\tt Best_tac} refer to the default claset (\texttt{claset()}), which works
for most purposes.  Named clasets include \ttindexbold{prop_cs}, which
includes the propositional rules, and \ttindexbold{HOL_cs}, which also
includes quantifier rules.  See the file \texttt{HOL/cladata.ML} for lists of
the classical rules,
and \iflabelundefined{chap:classical}{the {\em Reference Manual\/}}%
{Chap.\ts\ref{chap:classical}} for more discussion of classical proof methods.


\section{Calling the decision procedure SVC}\label{sec:HOL:SVC}

\index{SVC decision procedure|(}

The Stanford Validity Checker (SVC) is a tool that can check the validity of
certain types of formulae.  If it is installed on your machine, then
Isabelle/HOL can be configured to call it through the tactic
\ttindex{svc_tac}.  It is ideal for large tautologies and complex problems in
linear arithmetic.  Subexpressions that SVC cannot handle are automatically
replaced by variables, so you can call the tactic on any subgoal.  See the
file \texttt{HOL/ex/svc_test.ML} for examples.
\begin{ttbox} 
svc_tac   : int -> tactic
Svc.trace : bool ref      \hfill{\bf initially false}
\end{ttbox}

\begin{ttdescription}
\item[\ttindexbold{svc_tac} $i$] attempts to prove subgoal~$i$ by translating
  it into a formula recognized by~SVC\@.  If it succeeds then the subgoal is
  removed.  It fails if SVC is unable to prove the subgoal.  It crashes with
  an error message if SVC appears not to be installed.  Numeric variables may
  have types \texttt{nat}, \texttt{int} or \texttt{real}.

\item[\ttindexbold{Svc.trace}] is a flag that, if set, causes \texttt{svc_tac}
  to trace its operations: abstraction of the subgoal, translation to SVC
  syntax, SVC's response.
\end{ttdescription}

Here is an example, with tracing turned on:
\begin{ttbox}
set Svc.trace;
{\out val it : bool = true}
Goal "(#3::nat)*a <= #2 + #4*b + #6*c  & #11 <= #2*a + b + #2*c & \ttback
\ttback     a + #3*b <= #5 + #2*c  --> #2 + #3*b <= #2*a + #6*c";

by (svc_tac 1);
{\out Subgoal abstracted to}
{\out #3 * a <= #2 + #4 * b + #6 * c &}
{\out #11 <= #2 * a + b + #2 * c & a + #3 * b <= #5 + #2 * c -->}
{\out #2 + #3 * b <= #2 * a + #6 * c}
{\out Calling SVC:}
{\out (=> (<= 0  (F_c) )  (=> (<= 0  (F_b) )  (=> (<= 0  (F_a) )}
{\out   (=> (AND (<= {* 3  (F_a) }  {+ {+ 2  {* 4  (F_b) } }  }
{\out {* 6  (F_c) } } )  (AND (<= 11  {+ {+ {* 2  (F_a) }  (F_b) }}
{\out   {* 2  (F_c) } } )  (<= {+ (F_a)  {* 3  (F_b) } }  {+ 5  }
{\out {* 2  (F_c) } } ) ) )  (< {+ 2  {* 3  (F_b) } }  {+ 1  {+ }
{\out {* 2  (F_a) }  {* 6  (F_c) } } } ) ) ) ) ) }
{\out SVC Returns:}
{\out VALID}
{\out Level 1}
{\out #3 * a <= #2 + #4 * b + #6 * c &}
{\out #11 <= #2 * a + b + #2 * c & a + #3 * b <= #5 + #2 * c -->}
{\out #2 + #3 * b <= #2 * a + #6 * c}
{\out No subgoals!}
\end{ttbox}


\begin{warn}
Calling \ttindex{svc_tac} entails an above-average risk of
unsoundness.  Isabelle does not check SVC's result independently.  Moreover,
the tactic translates the submitted formula using code that lies outside
Isabelle's inference core.  Theorems that depend upon results proved using SVC
(and other oracles) are displayed with the annotation \texttt{[!]} attached.
You can also use \texttt{\#der (rep_thm $th$)} to examine the proof object of
theorem~$th$, as described in the \emph{Reference Manual}.  
\end{warn}

To start, first download SVC from the Internet at URL
\begin{ttbox}
http://agamemnon.stanford.edu/~levitt/vc/index.html
\end{ttbox}
and install it using the instructions supplied.  SVC requires two environment
variables:
\begin{ttdescription}
\item[\ttindexbold{SVC_HOME}] is an absolute pathname to the SVC
    distribution directory. 
    
  \item[\ttindexbold{SVC_MACHINE}] identifies the type of computer and
    operating system.  
\end{ttdescription}
You can set these environment variables either using the Unix shell or through
an Isabelle \texttt{settings} file.  Isabelle assumes SVC to be installed if
\texttt{SVC_HOME} is defined.

\paragraph*{Acknowledgement.}  This interface uses code supplied by S{\o}ren
Heilmann.


\index{SVC decision procedure|)}




\section{Types}\label{sec:HOL:Types}
This section describes \HOL's basic predefined types ($\alpha \times
\beta$, $\alpha + \beta$, $nat$ and $\alpha \; list$) and ways for
introducing new types in general.  The most important type
construction, the \texttt{datatype}, is treated separately in
{\S}\ref{sec:HOL:datatype}.


\subsection{Product and sum types}\index{*"* type}\index{*"+ type}
\label{subsec:prod-sum}

\begin{figure}[htbp]
\begin{constants}
  \it symbol    & \it meta-type &           & \it description \\ 
  \cdx{Pair}    & $[\alpha,\beta]\To \alpha\times\beta$
        & & ordered pairs $(a,b)$ \\
  \cdx{fst}     & $\alpha\times\beta \To \alpha$        & & first projection\\
  \cdx{snd}     & $\alpha\times\beta \To \beta$         & & second projection\\
  \cdx{split}   & $[[\alpha,\beta]\To\gamma, \alpha\times\beta] \To \gamma$ 
        & & generalized projection\\
  \cdx{Sigma}  & 
        $[\alpha\,set, \alpha\To\beta\,set]\To(\alpha\times\beta)set$ &
        & general sum of sets
\end{constants}
\begin{ttbox}\makeatletter
%\tdx{fst_def}      fst p     == @a. ? b. p = (a,b)
%\tdx{snd_def}      snd p     == @b. ? a. p = (a,b)
%\tdx{split_def}    split c p == c (fst p) (snd p)
\tdx{Sigma_def}    Sigma A B == UN x:A. UN y:B x. {\ttlbrace}(x,y){\ttrbrace}

\tdx{Pair_eq}      ((a,b) = (a',b')) = (a=a' & b=b')
\tdx{Pair_inject}  [| (a, b) = (a',b');  [| a=a';  b=b' |] ==> R |] ==> R
\tdx{PairE}        [| !!x y. p = (x,y) ==> Q |] ==> Q

\tdx{fst_conv}     fst (a,b) = a
\tdx{snd_conv}     snd (a,b) = b
\tdx{surjective_pairing}  p = (fst p,snd p)

\tdx{split}        split c (a,b) = c a b
\tdx{split_split}  R(split c p) = (! x y. p = (x,y) --> R(c x y))

\tdx{SigmaI}    [| a:A;  b:B a |] ==> (a,b) : Sigma A B
\tdx{SigmaE}    [| c:Sigma A B; !!x y.[| x:A; y:B x; c=(x,y) |] ==> P |] ==> P
\end{ttbox}
\caption{Type $\alpha\times\beta$}\label{hol-prod}
\end{figure} 

Theory \thydx{Prod} (Fig.\ts\ref{hol-prod}) defines the product type
$\alpha\times\beta$, with the ordered pair syntax $(a, b)$.  General
tuples are simulated by pairs nested to the right:
\begin{center}
\begin{tabular}{c|c}
external & internal \\
\hline
$\tau@1 \times \dots \times \tau@n$ & $\tau@1 \times (\dots (\tau@{n-1} \times \tau@n)\dots)$ \\
\hline
$(t@1,\dots,t@n)$ & $(t@1,(\dots,(t@{n-1},t@n)\dots)$ \\
\end{tabular}
\end{center}
In addition, it is possible to use tuples
as patterns in abstractions:
\begin{center}
{\tt\%($x$,$y$). $t$} \quad stands for\quad \texttt{split(\%$x$\thinspace$y$.\ $t$)} 
\end{center}
Nested patterns are also supported.  They are translated stepwise:
{\tt\%($x$,$y$,$z$). $t$} $\leadsto$ {\tt\%($x$,($y$,$z$)). $t$} $\leadsto$
{\tt split(\%$x$.\%($y$,$z$). $t$)} $\leadsto$ \texttt{split(\%$x$. split(\%$y$
  $z$.\ $t$))}.  The reverse translation is performed upon printing.
\begin{warn}
  The translation between patterns and \texttt{split} is performed automatically
  by the parser and printer.  Thus the internal and external form of a term
  may differ, which can affects proofs.  For example the term {\tt
  (\%(x,y).(y,x))(a,b)} requires the theorem \texttt{split} (which is in the
  default simpset) to rewrite to {\tt(b,a)}.
\end{warn}
In addition to explicit $\lambda$-abstractions, patterns can be used in any
variable binding construct which is internally described by a
$\lambda$-abstraction.  Some important examples are
\begin{description}
\item[Let:] \texttt{let {\it pattern} = $t$ in $u$}
\item[Quantifiers:] \texttt{!~{\it pattern}:$A$.~$P$}
\item[Choice:] {\underscoreon \tt @~{\it pattern}~.~$P$}
\item[Set operations:] \texttt{UN~{\it pattern}:$A$.~$B$}
\item[Sets:] \texttt{{\ttlbrace}~{\it pattern}~.~$P$~{\ttrbrace}}
\end{description}

There is a simple tactic which supports reasoning about patterns:
\begin{ttdescription}
\item[\ttindexbold{split_all_tac} $i$] replaces in subgoal $i$ all
  {\tt!!}-quantified variables of product type by individual variables for
  each component.  A simple example:
\begin{ttbox}
{\out 1. !!p. (\%(x,y,z). (x, y, z)) p = p}
by(split_all_tac 1);
{\out 1. !!x xa ya. (\%(x,y,z). (x, y, z)) (x, xa, ya) = (x, xa, ya)}
\end{ttbox}
\end{ttdescription}

Theory \texttt{Prod} also introduces the degenerate product type \texttt{unit}
which contains only a single element named {\tt()} with the property
\begin{ttbox}
\tdx{unit_eq}       u = ()
\end{ttbox}
\bigskip

Theory \thydx{Sum} (Fig.~\ref{hol-sum}) defines the sum type $\alpha+\beta$
which associates to the right and has a lower priority than $*$: $\tau@1 +
\tau@2 + \tau@3*\tau@4$ means $\tau@1 + (\tau@2 + (\tau@3*\tau@4))$.

The definition of products and sums in terms of existing types is not
shown.  The constructions are fairly standard and can be found in the
respective theory files. Although the sum and product types are
constructed manually for foundational reasons, they are represented as
actual datatypes later (see {\S}\ref{subsec:datatype:rep_datatype}).
Therefore, the theory \texttt{Datatype} should be used instead of
\texttt{Sum} or \texttt{Prod}.

\begin{figure}
\begin{constants}
  \it symbol    & \it meta-type &           & \it description \\ 
  \cdx{Inl}     & $\alpha \To \alpha+\beta$    & & first injection\\
  \cdx{Inr}     & $\beta \To \alpha+\beta$     & & second injection\\
  \cdx{sum_case} & $[\alpha\To\gamma, \beta\To\gamma, \alpha+\beta] \To\gamma$
        & & conditional
\end{constants}
\begin{ttbox}\makeatletter
\tdx{Inl_not_Inr}    Inl a ~= Inr b

\tdx{inj_Inl}        inj Inl
\tdx{inj_Inr}        inj Inr

\tdx{sumE}           [| !!x. P(Inl x);  !!y. P(Inr y) |] ==> P s

\tdx{sum_case_Inl}   sum_case f g (Inl x) = f x
\tdx{sum_case_Inr}   sum_case f g (Inr x) = g x

\tdx{surjective_sum} sum_case (\%x. f(Inl x)) (\%y. f(Inr y)) s = f s
\tdx{sum.split_case} R(sum_case f g s) = ((! x. s = Inl(x) --> R(f(x))) &
                                     (! y. s = Inr(y) --> R(g(y))))
\end{ttbox}
\caption{Type $\alpha+\beta$}\label{hol-sum}
\end{figure}

\begin{figure}
\index{*"< symbol}
\index{*"* symbol}
\index{*div symbol}
\index{*mod symbol}
\index{*"+ symbol}
\index{*"- symbol}
\begin{constants}
  \it symbol    & \it meta-type & \it priority & \it description \\ 
  \cdx{0}       & $nat$         & & zero \\
  \cdx{Suc}     & $nat \To nat$ & & successor function\\
% \cdx{nat_case} & $[\alpha, nat\To\alpha, nat] \To\alpha$ & & conditional\\
% \cdx{nat_rec} & $[nat, \alpha, [nat, \alpha]\To\alpha] \To \alpha$
%        & & primitive recursor\\
  \tt *         & $[nat,nat]\To nat$    &  Left 70      & multiplication \\
  \tt div       & $[nat,nat]\To nat$    &  Left 70      & division\\
  \tt mod       & $[nat,nat]\To nat$    &  Left 70      & modulus\\
  \tt +         & $[nat,nat]\To nat$    &  Left 65      & addition\\
  \tt -         & $[nat,nat]\To nat$    &  Left 65      & subtraction
\end{constants}
\subcaption{Constants and infixes}

\begin{ttbox}\makeatother
\tdx{nat_induct}     [| P 0; !!n. P n ==> P(Suc n) |]  ==> P n

\tdx{Suc_not_Zero}   Suc m ~= 0
\tdx{inj_Suc}        inj Suc
\tdx{n_not_Suc_n}    n~=Suc n
\subcaption{Basic properties}
\end{ttbox}
\caption{The type of natural numbers, \tydx{nat}} \label{hol-nat1}
\end{figure}


\begin{figure}
\begin{ttbox}\makeatother
              0+n           = n
              (Suc m)+n     = Suc(m+n)

              m-0           = m
              0-n           = n
              Suc(m)-Suc(n) = m-n

              0*n           = 0
              Suc(m)*n      = n + m*n

\tdx{mod_less}      m<n ==> m mod n = m
\tdx{mod_geq}       [| 0<n;  ~m<n |] ==> m mod n = (m-n) mod n

\tdx{div_less}      m<n ==> m div n = 0
\tdx{div_geq}       [| 0<n;  ~m<n |] ==> m div n = Suc((m-n) div n)
\end{ttbox}
\caption{Recursion equations for the arithmetic operators} \label{hol-nat2}
\end{figure}

\subsection{The type of natural numbers, \textit{nat}}
\index{nat@{\textit{nat}} type|(}

The theory \thydx{NatDef} defines the natural numbers in a roundabout but
traditional way.  The axiom of infinity postulates a type~\tydx{ind} of
individuals, which is non-empty and closed under an injective operation.  The
natural numbers are inductively generated by choosing an arbitrary individual
for~0 and using the injective operation to take successors.  This is a least
fixedpoint construction.  For details see the file \texttt{NatDef.thy}.

Type~\tydx{nat} is an instance of class~\cldx{ord}, which makes the
overloaded functions of this class (esp.\ \cdx{<} and \cdx{<=}, but also
\cdx{min}, \cdx{max} and \cdx{LEAST}) available on \tydx{nat}.  Theory
\thydx{Nat} builds on \texttt{NatDef} and shows that {\tt<=} is a partial order,
so \tydx{nat} is also an instance of class \cldx{order}.

Theory \thydx{Arith} develops arithmetic on the natural numbers.  It defines
addition, multiplication and subtraction.  Theory \thydx{Divides} defines
division, remainder and the ``divides'' relation.  The numerous theorems
proved include commutative, associative, distributive, identity and
cancellation laws.  See Figs.\ts\ref{hol-nat1} and~\ref{hol-nat2}.  The
recursion equations for the operators \texttt{+}, \texttt{-} and \texttt{*} on
\texttt{nat} are part of the default simpset.

Functions on \tydx{nat} can be defined by primitive or well-founded recursion;
see {\S}\ref{sec:HOL:recursive}.  A simple example is addition.
Here, \texttt{op +} is the name of the infix operator~\texttt{+}, following
the standard convention.
\begin{ttbox}
\sdx{primrec}
      "0 + n = n"
  "Suc m + n = Suc (m + n)"
\end{ttbox}
There is also a \sdx{case}-construct
of the form
\begin{ttbox}
case \(e\) of 0 => \(a\) | Suc \(m\) => \(b\)
\end{ttbox}
Note that Isabelle insists on precisely this format; you may not even change
the order of the two cases.
Both \texttt{primrec} and \texttt{case} are realized by a recursion operator
\cdx{nat_rec}, which is available because \textit{nat} is represented as
a datatype (see {\S}\ref{subsec:datatype:rep_datatype}).

%The predecessor relation, \cdx{pred_nat}, is shown to be well-founded.
%Recursion along this relation resembles primitive recursion, but is
%stronger because we are in higher-order logic; using primitive recursion to
%define a higher-order function, we can easily Ackermann's function, which
%is not primitive recursive \cite[page~104]{thompson91}.
%The transitive closure of \cdx{pred_nat} is~$<$.  Many functions on the
%natural numbers are most easily expressed using recursion along~$<$.

Tactic {\tt\ttindex{induct_tac} "$n$" $i$} performs induction on variable~$n$
in subgoal~$i$ using theorem \texttt{nat_induct}.  There is also the derived
theorem \tdx{less_induct}:
\begin{ttbox}
[| !!n. [| ! m. m<n --> P m |] ==> P n |]  ==>  P n
\end{ttbox}


Reasoning about arithmetic inequalities can be tedious.  Fortunately HOL
provides a decision procedure for quantifier-free linear arithmetic (i.e.\ 
only addition and subtraction). The simplifier invokes a weak version of this
decision procedure automatically. If this is not sufficent, you can invoke
the full procedure \ttindex{arith_tac} explicitly.  It copes with arbitrary
formulae involving {\tt=}, {\tt<}, {\tt<=}, {\tt+}, {\tt-}, {\tt Suc}, {\tt
  min}, {\tt max} and numerical constants; other subterms are treated as
atomic; subformulae not involving type $nat$ are ignored; quantified
subformulae are ignored unless they are positive universal or negative
existential. Note that the running time is exponential in the number of
occurrences of {\tt min}, {\tt max}, and {\tt-} because they require case
distinctions. Note also that \texttt{arith_tac} is not complete: if
divisibility plays a role, it may fail to prove a valid formula, for example
$m+m \neq n+n+1$. Fortunately such examples are rare in practice.

If \texttt{arith_tac} fails you, try to find relevant arithmetic results in
the library.  The theory \texttt{NatDef} contains theorems about {\tt<} and
{\tt<=}, the theory \texttt{Arith} contains theorems about \texttt{+},
\texttt{-} and \texttt{*}, and theory \texttt{Divides} contains theorems about
\texttt{div} and \texttt{mod}.  Use the \texttt{find}-functions to locate them
(see the {\em Reference Manual\/}).

\begin{figure}
\index{#@{\tt[]} symbol}
\index{#@{\tt\#} symbol}
\index{"@@{\tt\at} symbol}
\index{*"! symbol}
\begin{constants}
  \it symbol & \it meta-type & \it priority & \it description \\
  \tt[]    & $\alpha\,list$ & & empty list\\
  \tt \#   & $[\alpha,\alpha\,list]\To \alpha\,list$ & Right 65 & 
        list constructor \\
  \cdx{null}    & $\alpha\,list \To bool$ & & emptiness test\\
  \cdx{hd}      & $\alpha\,list \To \alpha$ & & head \\
  \cdx{tl}      & $\alpha\,list \To \alpha\,list$ & & tail \\
  \cdx{last}    & $\alpha\,list \To \alpha$ & & last element \\
  \cdx{butlast} & $\alpha\,list \To \alpha\,list$ & & drop last element \\
  \tt\at  & $[\alpha\,list,\alpha\,list]\To \alpha\,list$ & Left 65 & append \\
  \cdx{map}     & $(\alpha\To\beta) \To (\alpha\,list \To \beta\,list)$
        & & apply to all\\
  \cdx{filter}  & $(\alpha \To bool) \To (\alpha\,list \To \alpha\,list)$
        & & filter functional\\
  \cdx{set}& $\alpha\,list \To \alpha\,set$ & & elements\\
  \sdx{mem}  & $\alpha \To \alpha\,list \To bool$  &  Left 55   & membership\\
  \cdx{foldl}   & $(\beta\To\alpha\To\beta) \To \beta \To \alpha\,list \To \beta$ &
  & iteration \\
  \cdx{concat}   & $(\alpha\,list)list\To \alpha\,list$ & & concatenation \\
  \cdx{rev}     & $\alpha\,list \To \alpha\,list$ & & reverse \\
  \cdx{length}  & $\alpha\,list \To nat$ & & length \\
  \tt! & $\alpha\,list \To nat \To \alpha$ & Left 100 & indexing \\
  \cdx{take}, \cdx{drop} & $nat \To \alpha\,list \To \alpha\,list$ &&
    take or drop a prefix \\
  \cdx{takeWhile},\\
  \cdx{dropWhile} &
    $(\alpha \To bool) \To \alpha\,list \To \alpha\,list$ &&
    take or drop a prefix
\end{constants}
\subcaption{Constants and infixes}

\begin{center} \tt\frenchspacing
\begin{tabular}{rrr} 
  \it external        & \it internal  & \it description \\{}
  [$x@1$, $\dots$, $x@n$]  &  $x@1$ \# $\cdots$ \# $x@n$ \# [] &
        \rm finite list \\{}
  [$x$:$l$. $P$]  & filter ($\lambda x{.}P$) $l$ & 
        \rm list comprehension
\end{tabular}
\end{center}
\subcaption{Translations}
\caption{The theory \thydx{List}} \label{hol-list}
\end{figure}


\begin{figure}
\begin{ttbox}\makeatother
null [] = True
null (x#xs) = False

hd (x#xs) = x
tl (x#xs) = xs
tl [] = []

[] @ ys = ys
(x#xs) @ ys = x # xs @ ys

map f [] = []
map f (x#xs) = f x # map f xs

filter P [] = []
filter P (x#xs) = (if P x then x#filter P xs else filter P xs)

set [] = \ttlbrace\ttrbrace
set (x#xs) = insert x (set xs)

x mem [] = False
x mem (y#ys) = (if y=x then True else x mem ys)

foldl f a [] = a
foldl f a (x#xs) = foldl f (f a x) xs

concat([]) = []
concat(x#xs) = x @ concat(xs)

rev([]) = []
rev(x#xs) = rev(xs) @ [x]

length([]) = 0
length(x#xs) = Suc(length(xs))

xs!0 = hd xs
xs!(Suc n) = (tl xs)!n

take n [] = []
take n (x#xs) = (case n of 0 => [] | Suc(m) => x # take m xs)

drop n [] = []
drop n (x#xs) = (case n of 0 => x#xs | Suc(m) => drop m xs)

takeWhile P [] = []
takeWhile P (x#xs) = (if P x then x#takeWhile P xs else [])

dropWhile P [] = []
dropWhile P (x#xs) = (if P x then dropWhile P xs else xs)
\end{ttbox}
\caption{Recursions equations for list processing functions}
\label{fig:HOL:list-simps}
\end{figure}
\index{nat@{\textit{nat}} type|)}


\subsection{The type constructor for lists, \textit{list}}
\label{subsec:list}
\index{list@{\textit{list}} type|(}

Figure~\ref{hol-list} presents the theory \thydx{List}: the basic list
operations with their types and syntax.  Type $\alpha \; list$ is
defined as a \texttt{datatype} with the constructors {\tt[]} and {\tt\#}.
As a result the generic structural induction and case analysis tactics
\texttt{induct\_tac} and \texttt{cases\_tac} also become available for
lists.  A \sdx{case} construct of the form
\begin{center}\tt
case $e$ of [] => $a$  |  \(x\)\#\(xs\) => b
\end{center}
is defined by translation.  For details see~{\S}\ref{sec:HOL:datatype}. There
is also a case splitting rule \tdx{split_list_case}
\[
\begin{array}{l}
P(\mathtt{case}~e~\mathtt{of}~\texttt{[] =>}~a ~\texttt{|}~
               x\texttt{\#}xs~\texttt{=>}~f~x~xs) ~= \\
((e = \texttt{[]} \to P(a)) \land
 (\forall x~ xs. e = x\texttt{\#}xs \to P(f~x~xs)))
\end{array}
\]
which can be fed to \ttindex{addsplits} just like
\texttt{split_if} (see~{\S}\ref{subsec:HOL:case:splitting}).

\texttt{List} provides a basic library of list processing functions defined by
primitive recursion (see~{\S}\ref{sec:HOL:primrec}).  The recursion equations
are shown in Fig.\ts\ref{fig:HOL:list-simps}.

\index{list@{\textit{list}} type|)}


\subsection{Introducing new types} \label{sec:typedef}

The \HOL-methodology dictates that all extensions to a theory should
be \textbf{definitional}.  The type definition mechanism that
meets this criterion is \ttindex{typedef}.  Note that \emph{type synonyms},
which are inherited from {\Pure} and described elsewhere, are just
syntactic abbreviations that have no logical meaning.

\begin{warn}
  Types in \HOL\ must be non-empty; otherwise the quantifier rules would be
  unsound, because $\exists x. x=x$ is a theorem \cite[{\S}7]{paulson-COLOG}.
\end{warn}
A \bfindex{type definition} identifies the new type with a subset of
an existing type.  More precisely, the new type is defined by
exhibiting an existing type~$\tau$, a set~$A::\tau\,set$, and a
theorem of the form $x:A$.  Thus~$A$ is a non-empty subset of~$\tau$,
and the new type denotes this subset.  New functions are defined that
establish an isomorphism between the new type and the subset.  If
type~$\tau$ involves type variables $\alpha@1$, \ldots, $\alpha@n$,
then the type definition creates a type constructor
$(\alpha@1,\ldots,\alpha@n)ty$ rather than a particular type.

\begin{figure}[htbp]
\begin{rail}
typedef  : 'typedef' ( () | '(' name ')') type '=' set witness;

type    : typevarlist name ( () | '(' infix ')' );
set     : string;
witness : () | '(' id ')';
\end{rail}
\caption{Syntax of type definitions}
\label{fig:HOL:typedef}
\end{figure}

The syntax for type definitions is shown in Fig.~\ref{fig:HOL:typedef}.  For
the definition of `typevarlist' and `infix' see
\iflabelundefined{chap:classical}
{the appendix of the {\em Reference Manual\/}}%
{Appendix~\ref{app:TheorySyntax}}.  The remaining nonterminals have the
following meaning:
\begin{description}
\item[\it type:] the new type constructor $(\alpha@1,\dots,\alpha@n)ty$ with
  optional infix annotation.
\item[\it name:] an alphanumeric name $T$ for the type constructor
  $ty$, in case $ty$ is a symbolic name.  Defaults to $ty$.
\item[\it set:] the representing subset $A$.
\item[\it witness:] name of a theorem of the form $a:A$ proving
  non-emptiness.  It can be omitted in case Isabelle manages to prove
  non-emptiness automatically.
\end{description}
If all context conditions are met (no duplicate type variables in
`typevarlist', no extra type variables in `set', and no free term variables
in `set'), the following components are added to the theory:
\begin{itemize}
\item a type $ty :: (term,\dots,term)term$
\item constants
\begin{eqnarray*}
T &::& \tau\;set \\
Rep_T &::& (\alpha@1,\dots,\alpha@n)ty \To \tau \\
Abs_T &::& \tau \To (\alpha@1,\dots,\alpha@n)ty
\end{eqnarray*}
\item a definition and three axioms
\[
\begin{array}{ll}
T{\tt_def} & T \equiv A \\
{\tt Rep_}T & Rep_T\,x \in T \\
{\tt Rep_}T{\tt_inverse} & Abs_T\,(Rep_T\,x) = x \\
{\tt Abs_}T{\tt_inverse} & y \in T \Imp Rep_T\,(Abs_T\,y) = y
\end{array}
\]
stating that $(\alpha@1,\dots,\alpha@n)ty$ is isomorphic to $A$ by $Rep_T$
and its inverse $Abs_T$.
\end{itemize}
Below are two simple examples of \HOL\ type definitions.  Non-emptiness
is proved automatically here.
\begin{ttbox}
typedef unit = "{\ttlbrace}True{\ttrbrace}"

typedef (prod)
  ('a, 'b) "*"    (infixr 20)
      = "{\ttlbrace}f . EX (a::'a) (b::'b). f = (\%x y. x = a & y = b){\ttrbrace}"
\end{ttbox}

Type definitions permit the introduction of abstract data types in a safe
way, namely by providing models based on already existing types.  Given some
abstract axiomatic description $P$ of a type, this involves two steps:
\begin{enumerate}
\item Find an appropriate type $\tau$ and subset $A$ which has the desired
  properties $P$, and make a type definition based on this representation.
\item Prove that $P$ holds for $ty$ by lifting $P$ from the representation.
\end{enumerate}
You can now forget about the representation and work solely in terms of the
abstract properties $P$.

\begin{warn}
If you introduce a new type (constructor) $ty$ axiomatically, i.e.\ by
declaring the type and its operations and by stating the desired axioms, you
should make sure the type has a non-empty model.  You must also have a clause
\par
\begin{ttbox}
arities \(ty\) :: (term,\thinspace\(\dots\),{\thinspace}term){\thinspace}term
\end{ttbox}
in your theory file to tell Isabelle that $ty$ is in class \texttt{term}, the
class of all \HOL\ types.
\end{warn}


\section{Records}

At a first approximation, records are just a minor generalisation of tuples,
where components may be addressed by labels instead of just position (think of
{\ML}, for example).  The version of records offered by Isabelle/HOL is
slightly more advanced, though, supporting \emph{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.


\subsection{Basics}

Isabelle/HOL supports fixed and schematic records both at the level of terms
and types.  The concrete syntax 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 \textsc{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
\emph{row variable} in the literature.  The more part of a record scheme may
be instantiated by zero or more further components.  For example, above scheme
might 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

There are two key features that make extensible records in a simply typed
language like HOL feasible:
\begin{enumerate}
\item the more part is internalised, as a free term or type variable,
\item field names are externalised, 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 (see
{\S}\ref{sec:HOL:record-def}).  Afterwards, records may be formed using above
syntax, while obeying the canonical order of fields as given by their
declaration.  The record package also provides several operations like
selectors and updates (see {\S}\ref{sec:HOL:record-ops}), together with
characteristic properties (see {\S}\ref{sec:HOL:record-thms}).

There is an example theory demonstrating most basic aspects of extensible
records (see theory \texttt{HOL/ex/Points} in the Isabelle sources).


\subsection{Defining records}\label{sec:HOL:record-def}

The theory syntax for record type definitions is shown in
Fig.~\ref{fig:HOL:record}.  For the definition of `typevarlist' and `type' see
\iflabelundefined{chap:classical}
{the appendix of the {\em Reference Manual\/}}%
{Appendix~\ref{app:TheorySyntax}}.

\begin{figure}[htbp]
\begin{rail}
record  : 'record' typevarlist name '=' parent (field +);

parent  : ( () | type '+');
field   : name '::' type;
\end{rail}
\caption{Syntax of record type definitions}
\label{fig:HOL:record}
\end{figure}

A general \ttindex{record} specification is of the following form:
\[
\mathtt{record}~(\alpha@1, \dots, \alpha@n) \, t ~=~
  (\tau@1, \dots, \tau@m) \, s ~+~ c@1 :: \sigma@1 ~ \dots ~ c@l :: \sigma@l
\]
where $\vec\alpha@n$ are distinct type variables, and $\vec\tau@m$,
$\vec\sigma@l$ are types containing at most variables from $\vec\alpha@n$.
Type constructor $t$ has to be new, while $s$ has to specify an existing
record type.  Furthermore, the $\vec c@l$ have to be distinct field names.
There has to be at least one field.

In principle, field names may never be shared with other records.  This is no
actual restriction in practice, since $\vec c@l$ are internally declared
within a separate name space qualified by the name $t$ of the record.

\medskip

Above definition introduces a new record type $(\vec\alpha@n) \, t$ by
extending an existing one $(\vec\tau@m) \, s$ by new fields $\vec c@l \ty
\vec\sigma@l$.  The parent record specification is optional, by omitting it
$t$ becomes a \emph{root record}.  The hierarchy of all records declared
within a theory forms a forest structure, i.e.\ a set of trees, where any of
these is rooted by some root record.

For convenience, $(\vec\alpha@n) \, t$ is made a type abbreviation for the
fixed record type $\record{\vec c@l \ty \vec\sigma@l}$, and $(\vec\alpha@n,
\zeta) \, t_scheme$ is made an abbreviation for $\record{\vec c@l \ty
  \vec\sigma@l\fs \more \ty \zeta}$.

\medskip

The following simple example defines a root record type $point$ with fields $x
\ty nat$ and $y \ty nat$, and record type $cpoint$ by extending $point$ with
an additional $colour$ component.

\begin{ttbox}
  record point =
    x :: nat
    y :: nat

  record cpoint = point +
    colour :: string
\end{ttbox}


\subsection{Record operations}\label{sec:HOL:record-ops}

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 first assume that $(\vec\alpha@n) \, t$
is a root record with fields $\vec c@l \ty \vec\sigma@l$.

\medskip

\textbf{Selectors} and \textbf{updates} are available for any field (including
``$more$'') as follows:
\begin{matharray}{lll}
  c@i & \ty & \record{\vec c@l \ty \vec \sigma@l, \more \ty \zeta} \To \sigma@i \\
  c@i_update & \ty & \sigma@i \To \record{\vec c@l \ty \vec \sigma@l, \more \ty \zeta} \To
    \record{\vec c@l \ty \vec \sigma@l, \more \ty \zeta}
\end{matharray}

There is some special syntax for updates: $r \, \record{x \asn a}$ abbreviates
term $x_update \, a \, r$.  Repeated updates are also supported: $r \,
\record{x \asn a} \, \record{y \asn b} \, \record{z \asn c}$ may be written as
$r \, \record{x \asn a\fs y \asn b\fs z \asn c}$.  Note that because of
postfix notation the order of fields shown here is reverse than in the actual
term.  This might lead to confusion in conjunction with proof tools like
ordered rewriting.

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 the logic
is concerned.  Thus commutativity of updates can be proven within the logic
for any two fields, but not as a general theorem: fields are not first-class
values.

\medskip

\textbf{Make} operations provide cumulative record constructor functions:
\begin{matharray}{lll}
  make & \ty & \vec\sigma@l \To \record{\vec c@l \ty \vec \sigma@l} \\
  make_scheme & \ty & \vec\sigma@l \To
    \zeta \To \record{\vec c@l \ty \vec \sigma@l, \more \ty \zeta} \\
\end{matharray}
\noindent
These functions are curried.  The corresponding definitions in terms of actual
record terms are part of the standard simpset.  Thus $point\dtt make\,a\,b$
rewrites to $\record{x = a\fs y = b}$.

\medskip

Any of above selector, update and make operations are declared within a local
name space prefixed by the name $t$ of the record.  In case that different
records share base names of fields, one has to qualify names explicitly (e.g.\ 
$t\dtt c@i_update$).  This is recommended especially for operations like
$make$ or $update_more$ that always have the same base name.  Just use $t\dtt
make$ etc.\ to avoid confusion.

\bigskip

We reconsider the case of non-root records, which are derived of some parent
record.  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 $(\vec\alpha@n) \, t$ has ancestor fields
$\vec d@k \ty \vec\rho@k$, selectors will get the following types:
\begin{matharray}{lll}
  c@i & \ty & \record{\vec d@k \ty \vec\rho@k, \vec c@l \ty \vec \sigma@l, \more \ty \zeta}
    \To \sigma@i
\end{matharray}
\noindent
Update and make operations are analogous.


\subsection{Proof tools}\label{sec:HOL:record-thms}

The record package provides the following proof rules for any record type $t$.
\begin{enumerate}
  
\item Standard conversions (selectors or updates applied to record constructor
  terms, make function definitions) are part of the standard simpset (via
  \texttt{addsimps}).
  
\item Selectors applied to updated records are automatically reduced by
  simplification procedure \ttindex{record_simproc}, which is part of the
  default simpset.
  
\item Inject equations of a form analogous to $((x, y) = (x', y')) \equiv x=x'
  \conj y=y'$ are made part of the standard simpset and claset (via
  \texttt{addIffs}).
  
\item A tactic for record field splitting (\ttindex{record_split_tac}) may be
  made part of the claset (via \texttt{addSWrapper}).  This tactic is based on
  rules analogous to $(\All x PROP~P~x) \equiv (\All{a~b} PROP~P(a, b))$ for
  any field.
\end{enumerate}

The first two kinds of rules are stored within the theory as $t\dtt simps$ and
$t\dtt iffs$, respectively.  In some situations it might be appropriate to
expand the definitions of updates: $t\dtt update_defs$.  Note that these names
are \emph{not} bound at the {\ML} level.

\medskip

The example theory \texttt{HOL/ex/Points} demonstrates typical proofs
concerning records.  The basic idea is to make \ttindex{record_split_tac}
expand quantified record variables and then simplify by the conversion rules.
By using a combination of the simplifier and classical prover together with
the default simpset and claset, record problems should be solved with a single
stroke of \texttt{Auto_tac} or \texttt{Force_tac}.  Most of the time, plain
\texttt{Simp_tac} should be sufficient, though.


\section{Datatype definitions}
\label{sec:HOL:datatype}
\index{*datatype|(}

Inductive datatypes, similar to those of \ML, frequently appear in
applications of Isabelle/HOL.  In principle, such types could be defined by
hand via \texttt{typedef} (see {\S}\ref{sec:typedef}), but this would be far too
tedious.  The \ttindex{datatype} definition package of Isabelle/HOL (cf.\ 
\cite{Berghofer-Wenzel:1999:TPHOL}) automates such chores.  It generates an
appropriate \texttt{typedef} based on a least fixed-point construction, and
proves freeness theorems and induction rules, as well as theorems for
recursion and case combinators.  The user just has to give a simple
specification of new inductive types using a notation similar to {\ML} or
Haskell.

The current datatype package can handle both mutual and indirect recursion.
It also offers to represent existing types as datatypes giving the advantage
of a more uniform view on standard theories.


\subsection{Basics}
\label{subsec:datatype:basics}

A general \texttt{datatype} definition is of the following form:
\[
\begin{array}{llcl}
\mathtt{datatype} & (\alpha@1,\ldots,\alpha@h)t@1 & = &
  C^1@1~\tau^1@{1,1}~\ldots~\tau^1@{1,m^1@1} ~\mid~ \ldots ~\mid~
    C^1@{k@1}~\tau^1@{k@1,1}~\ldots~\tau^1@{k@1,m^1@{k@1}} \\
 & & \vdots \\
\mathtt{and} & (\alpha@1,\ldots,\alpha@h)t@n & = &
  C^n@1~\tau^n@{1,1}~\ldots~\tau^n@{1,m^n@1} ~\mid~ \ldots ~\mid~
    C^n@{k@n}~\tau^n@{k@n,1}~\ldots~\tau^n@{k@n,m^n@{k@n}}
\end{array}
\]
where $\alpha@i$ are type variables, $C^j@i$ are distinct constructor
names and $\tau^j@{i,i'}$ are {\em admissible} types containing at
most the type variables $\alpha@1, \ldots, \alpha@h$. A type $\tau$
occurring in a \texttt{datatype} definition is {\em admissible} iff
\begin{itemize}
\item $\tau$ is non-recursive, i.e.\ $\tau$ does not contain any of the
newly defined type constructors $t@1,\ldots,t@n$, or
\item $\tau = (\alpha@1,\ldots,\alpha@h)t@{j'}$ where $1 \leq j' \leq n$, or
\item $\tau = (\tau'@1,\ldots,\tau'@{h'})t'$, where $t'$ is
the type constructor of an already existing datatype and $\tau'@1,\ldots,\tau'@{h'}$
are admissible types.
\item $\tau = \sigma \to \tau'$, where $\tau'$ is an admissible
type and $\sigma$ is non-recursive (i.e. the occurrences of the newly defined
types are {\em strictly positive})
\end{itemize}
If some $(\alpha@1,\ldots,\alpha@h)t@{j'}$ occurs in a type $\tau^j@{i,i'}$
of the form
\[
(\ldots,\ldots ~ (\alpha@1,\ldots,\alpha@h)t@{j'} ~ \ldots,\ldots)t'
\]
this is called a {\em nested} (or \emph{indirect}) occurrence. A very simple
example of a datatype is the type \texttt{list}, which can be defined by
\begin{ttbox}
datatype 'a list = Nil
                 | Cons 'a ('a list)
\end{ttbox}
Arithmetic expressions \texttt{aexp} and boolean expressions \texttt{bexp} can be modelled
by the mutually recursive datatype definition
\begin{ttbox}
datatype 'a aexp = If_then_else ('a bexp) ('a aexp) ('a aexp)
                 | Sum ('a aexp) ('a aexp)
                 | Diff ('a aexp) ('a aexp)
                 | Var 'a
                 | Num nat
and      'a bexp = Less ('a aexp) ('a aexp)
                 | And ('a bexp) ('a bexp)
                 | Or ('a bexp) ('a bexp)
\end{ttbox}
The datatype \texttt{term}, which is defined by
\begin{ttbox}
datatype ('a, 'b) term = Var 'a
                       | App 'b ((('a, 'b) term) list)
\end{ttbox}
is an example for a datatype with nested recursion. Using nested recursion
involving function spaces, we may also define infinitely branching datatypes, e.g.
\begin{ttbox}
datatype 'a tree = Atom 'a | Branch "nat => 'a tree"
\end{ttbox}

\medskip

Types in HOL must be non-empty. Each of the new datatypes
$(\alpha@1,\ldots,\alpha@h)t@j$ with $1 \leq j \leq n$ is non-empty iff it has a
constructor $C^j@i$ with the following property: for all argument types
$\tau^j@{i,i'}$ of the form $(\alpha@1,\ldots,\alpha@h)t@{j'}$ the datatype
$(\alpha@1,\ldots,\alpha@h)t@{j'}$ is non-empty.

If there are no nested occurrences of the newly defined datatypes, obviously
at least one of the newly defined datatypes $(\alpha@1,\ldots,\alpha@h)t@j$
must have a constructor $C^j@i$ without recursive arguments, a \emph{base
  case}, to ensure that the new types are non-empty. If there are nested
occurrences, a datatype can even be non-empty without having a base case
itself. Since \texttt{list} is a non-empty datatype, \texttt{datatype t = C (t
  list)} is non-empty as well.


\subsubsection{Freeness of the constructors}

The datatype constructors are automatically defined as functions of their
respective type:
\[ C^j@i :: [\tau^j@{i,1},\dots,\tau^j@{i,m^j@i}] \To (\alpha@1,\dots,\alpha@h)t@j \]
These functions have certain {\em freeness} properties.  They construct
distinct values:
\[
C^j@i~x@1~\dots~x@{m^j@i} \neq C^j@{i'}~y@1~\dots~y@{m^j@{i'}} \qquad
\mbox{for all}~ i \neq i'.
\]
The constructor functions are injective:
\[
(C^j@i~x@1~\dots~x@{m^j@i} = C^j@i~y@1~\dots~y@{m^j@i}) =
(x@1 = y@1 \land \dots \land x@{m^j@i} = y@{m^j@i})
\]
Since the number of distinctness inequalities is quadratic in the number of
constructors, the datatype package avoids proving them separately if there are
too many constructors. Instead, specific inequalities are proved by a suitable
simplification procedure on demand.\footnote{This procedure, which is already part
of the default simpset, may be referred to by the ML identifier
\texttt{DatatypePackage.distinct_simproc}.}

\subsubsection{Structural induction}

The datatype package also provides structural induction rules.  For
datatypes without nested recursion, this is of the following form:
\[
\infer{P@1~x@1 \land \dots \land P@n~x@n}
  {\begin{array}{lcl}
     \Forall x@1 \dots x@{m^1@1}.
       \List{P@{s^1@{1,1}}~x@{r^1@{1,1}}; \dots;
         P@{s^1@{1,l^1@1}}~x@{r^1@{1,l^1@1}}} & \Imp &
           P@1~\left(C^1@1~x@1~\dots~x@{m^1@1}\right) \\
     & \vdots \\
     \Forall x@1 \dots x@{m^1@{k@1}}.
       \List{P@{s^1@{k@1,1}}~x@{r^1@{k@1,1}}; \dots;
         P@{s^1@{k@1,l^1@{k@1}}}~x@{r^1@{k@1,l^1@{k@1}}}} & \Imp &
           P@1~\left(C^1@{k@1}~x@1~\ldots~x@{m^1@{k@1}}\right) \\
     & \vdots \\
     \Forall x@1 \dots x@{m^n@1}.
       \List{P@{s^n@{1,1}}~x@{r^n@{1,1}}; \dots;
         P@{s^n@{1,l^n@1}}~x@{r^n@{1,l^n@1}}} & \Imp &
           P@n~\left(C^n@1~x@1~\ldots~x@{m^n@1}\right) \\
     & \vdots \\
     \Forall x@1 \dots x@{m^n@{k@n}}.
       \List{P@{s^n@{k@n,1}}~x@{r^n@{k@n,1}}; \dots
         P@{s^n@{k@n,l^n@{k@n}}}~x@{r^n@{k@n,l^n@{k@n}}}} & \Imp &
           P@n~\left(C^n@{k@n}~x@1~\ldots~x@{m^n@{k@n}}\right)
   \end{array}}
\]
where
\[
\begin{array}{rcl}
Rec^j@i & := &
   \left\{\left(r^j@{i,1},s^j@{i,1}\right),\ldots,
     \left(r^j@{i,l^j@i},s^j@{i,l^j@i}\right)\right\} = \\[2ex]
&& \left\{(i',i'')~\left|~
     1\leq i' \leq m^j@i \land 1 \leq i'' \leq n \land
       \tau^j@{i,i'} = (\alpha@1,\ldots,\alpha@h)t@{i''}\right.\right\}
\end{array}
\]
i.e.\ the properties $P@j$ can be assumed for all recursive arguments.

For datatypes with nested recursion, such as the \texttt{term} example from
above, things are a bit more complicated.  Conceptually, Isabelle/HOL unfolds
a definition like
\begin{ttbox}
datatype ('a, 'b) term = Var 'a
                       | App 'b ((('a, 'b) term) list)
\end{ttbox}
to an equivalent definition without nesting:
\begin{ttbox}
datatype ('a, 'b) term      = Var
                            | App 'b (('a, 'b) term_list)
and      ('a, 'b) term_list = Nil'
                            | Cons' (('a,'b) term) (('a,'b) term_list)
\end{ttbox}
Note however, that the type \texttt{('a,'b) term_list} and the constructors {\tt
  Nil'} and \texttt{Cons'} are not really introduced.  One can directly work with
the original (isomorphic) type \texttt{(('a, 'b) term) list} and its existing
constructors \texttt{Nil} and \texttt{Cons}. Thus, the structural induction rule for
\texttt{term} gets the form
\[
\infer{P@1~x@1 \land P@2~x@2}
  {\begin{array}{l}
     \Forall x.~P@1~(\mathtt{Var}~x) \\
     \Forall x@1~x@2.~P@2~x@2 \Imp P@1~(\mathtt{App}~x@1~x@2) \\
     P@2~\mathtt{Nil} \\
     \Forall x@1~x@2. \List{P@1~x@1; P@2~x@2} \Imp P@2~(\mathtt{Cons}~x@1~x@2)
   \end{array}}
\]
Note that there are two predicates $P@1$ and $P@2$, one for the type \texttt{('a,'b) term}
and one for the type \texttt{(('a, 'b) term) list}.

For a datatype with function types such as \texttt{'a tree}, the induction rule
is of the form
\[
\infer{P~t}
  {\Forall a.~P~(\mathtt{Atom}~a) &
   \Forall ts.~(\forall x.~P~(ts~x)) \Imp P~(\mathtt{Branch}~ts)}
\]

\medskip In principle, inductive types are already fully determined by
freeness and structural induction.  For convenience in applications,
the following derived constructions are automatically provided for any
datatype.

\subsubsection{The \sdx{case} construct}

The type comes with an \ML-like \texttt{case}-construct:
\[
\begin{array}{rrcl}
\mbox{\tt case}~e~\mbox{\tt of} & C^j@1~x@{1,1}~\dots~x@{1,m^j@1} & \To & e@1 \\
                           \vdots \\
                           \mid & C^j@{k@j}~x@{k@j,1}~\dots~x@{k@j,m^j@{k@j}} & \To & e@{k@j}
\end{array}
\]
where the $x@{i,j}$ are either identifiers or nested tuple patterns as in
{\S}\ref{subsec:prod-sum}.
\begin{warn}
  All constructors must be present, their order is fixed, and nested patterns
  are not supported (with the exception of tuples).  Violating this
  restriction results in strange error messages.
\end{warn}

To perform case distinction on a goal containing a \texttt{case}-construct,
the theorem $t@j.$\texttt{split} is provided:
\[
\begin{array}{@{}rcl@{}}
P(t@j_\mathtt{case}~f@1~\dots~f@{k@j}~e) &\!\!\!=&
\!\!\! ((\forall x@1 \dots x@{m^j@1}. e = C^j@1~x@1\dots x@{m^j@1} \to
                             P(f@1~x@1\dots x@{m^j@1})) \\
&&\!\!\! ~\land~ \dots ~\land \\
&&~\!\!\! (\forall x@1 \dots x@{m^j@{k@j}}. e = C^j@{k@j}~x@1\dots x@{m^j@{k@j}} \to
                             P(f@{k@j}~x@1\dots x@{m^j@{k@j}})))
\end{array}
\]
where $t@j$\texttt{_case} is the internal name of the \texttt{case}-construct.
This theorem can be added to a simpset via \ttindex{addsplits}
(see~{\S}\ref{subsec:HOL:case:splitting}).

\subsubsection{The function \cdx{size}}\label{sec:HOL:size}

Theory \texttt{Arith} declares a generic function \texttt{size} of type
$\alpha\To nat$.  Each datatype defines a particular instance of \texttt{size}
by overloading according to the following scheme:
%%% FIXME: This formula is too big and is completely unreadable
\[
size(C^j@i~x@1~\dots~x@{m^j@i}) = \!
\left\{
\begin{array}{ll}
0 & \!\mbox{if $Rec^j@i = \emptyset$} \\
1+\sum\limits@{h=1}^{l^j@i}size~x@{r^j@{i,h}} &
 \!\mbox{if $Rec^j@i = \left\{\left(r^j@{i,1},s^j@{i,1}\right),\ldots,
  \left(r^j@{i,l^j@i},s^j@{i,l^j@i}\right)\right\}$}
\end{array}
\right.
\]
where $Rec^j@i$ is defined above.  Viewing datatypes as generalised trees, the
size of a leaf is 0 and the size of a node is the sum of the sizes of its
subtrees ${}+1$.

\subsection{Defining datatypes}

The theory syntax for datatype definitions is shown in
Fig.~\ref{datatype-grammar}.  In order to be well-formed, a datatype
definition has to obey the rules stated in the previous section.  As a result
the theory is extended with the new types, the constructors, and the theorems
listed in the previous section.

\begin{figure}
\begin{rail}
datatype : 'datatype' typedecls;

typedecls: ( newtype '=' (cons + '|') ) + 'and'
         ;
newtype  : typevarlist id ( () | '(' infix ')' )
         ;
cons     : name (argtype *) ( () | ( '(' mixfix ')' ) )
         ;
argtype  : id | tid | ('(' typevarlist id ')')
         ;
\end{rail}
\caption{Syntax of datatype declarations}
\label{datatype-grammar}
\end{figure}

Most of the theorems about datatypes become part of the default simpset and
you never need to see them again because the simplifier applies them
automatically.  Only induction or case distinction are usually invoked by hand.
\begin{ttdescription}
\item[\ttindexbold{induct_tac} {\tt"}$x${\tt"} $i$]
 applies structural induction on variable $x$ to subgoal $i$, provided the
 type of $x$ is a datatype.
\item[\texttt{induct_tac}
  {\tt"}$x@1$ $\ldots$ $x@n${\tt"} $i$] applies simultaneous
  structural induction on the variables $x@1,\ldots,x@n$ to subgoal $i$.  This
  is the canonical way to prove properties of mutually recursive datatypes
  such as \texttt{aexp} and \texttt{bexp}, or datatypes with nested recursion such as
  \texttt{term}.
\end{ttdescription}
In some cases, induction is overkill and a case distinction over all
constructors of the datatype suffices.
\begin{ttdescription}
\item[\ttindexbold{cases_tac} {\tt"}$u${\tt"} $i$]
 performs a case analysis for the term $u$ whose type  must be a datatype.
 If the datatype has $k@j$ constructors  $C^j@1$, \dots $C^j@{k@j}$, subgoal
 $i$ is replaced by $k@j$ new subgoals which  contain the additional
 assumption $u = C^j@{i'}~x@1~\dots~x@{m^j@{i'}}$ for  $i'=1$, $\dots$,~$k@j$.
\end{ttdescription}

Note that induction is only allowed on free variables that should not occur
among the premises of the subgoal. Case distinction applies to arbitrary terms.

\bigskip


For the technically minded, we exhibit some more details.  Processing the
theory file produces an \ML\ structure which, in addition to the usual
components, contains a structure named $t$ for each datatype $t$ defined in
the file.  Each structure $t$ contains the following elements:
\begin{ttbox}
val distinct : thm list
val inject : thm list
val induct : thm
val exhaust : thm
val cases : thm list
val split : thm
val split_asm : thm
val recs : thm list
val size : thm list
val simps : thm list
\end{ttbox}
\texttt{distinct}, \texttt{inject}, \texttt{induct}, \texttt{size}
and \texttt{split} contain the theorems
described above.  For user convenience, \texttt{distinct} contains
inequalities in both directions.  The reduction rules of the {\tt
  case}-construct are in \texttt{cases}.  All theorems from {\tt
  distinct}, \texttt{inject} and \texttt{cases} are combined in \texttt{simps}.
In case of mutually recursive datatypes, \texttt{recs}, \texttt{size}, \texttt{induct}
and \texttt{simps} are contained in a separate structure named $t@1_\ldots_t@n$.


\subsection{Representing existing types as datatypes}\label{subsec:datatype:rep_datatype}

For foundational reasons, some basic types such as \texttt{nat}, \texttt{*}, {\tt
  +}, \texttt{bool} and \texttt{unit} are not defined in a \texttt{datatype} section,
but by more primitive means using \texttt{typedef}. To be able to use the
tactics \texttt{induct_tac} and \texttt{cases_tac} and to define functions by
primitive recursion on these types, such types may be represented as actual
datatypes.  This is done by specifying an induction rule, as well as theorems
stating the distinctness and injectivity of constructors in a {\tt
  rep_datatype} section.  For type \texttt{nat} this works as follows:
\begin{ttbox}
rep_datatype nat
  distinct Suc_not_Zero, Zero_not_Suc
  inject   Suc_Suc_eq
  induct   nat_induct
\end{ttbox}
The datatype package automatically derives additional theorems for recursion
and case combinators from these rules.  Any of the basic HOL types mentioned
above are represented as datatypes.  Try an induction on \texttt{bool}
today.


\subsection{Examples}

\subsubsection{The datatype $\alpha~mylist$}

We want to define a type $\alpha~mylist$. To do this we have to build a new
theory that contains the type definition.  We start from the theory
\texttt{Datatype} instead of \texttt{Main} in order to avoid clashes with the
\texttt{List} theory of Isabelle/HOL.
\begin{ttbox}
MyList = Datatype +
  datatype 'a mylist = Nil | Cons 'a ('a mylist)
end
\end{ttbox}
After loading the theory, we can prove $Cons~x~xs\neq xs$, for example.  To
ease the induction applied below, we state the goal with $x$ quantified at the
object-level.  This will be stripped later using \ttindex{qed_spec_mp}.
\begin{ttbox}
Goal "!x. Cons x xs ~= xs";
{\out Level 0}
{\out ! x. Cons x xs ~= xs}
{\out  1. ! x. Cons x xs ~= xs}
\end{ttbox}
This can be proved by the structural induction tactic:
\begin{ttbox}
by (induct_tac "xs" 1);
{\out Level 1}
{\out ! x. Cons x xs ~= xs}
{\out  1. ! x. Cons x Nil ~= Nil}
{\out  2. !!a mylist.}
{\out        ! x. Cons x mylist ~= mylist ==>}
{\out        ! x. Cons x (Cons a mylist) ~= Cons a mylist}
\end{ttbox}
The first subgoal can be proved using the simplifier.  Isabelle/HOL has
already added the freeness properties of lists to the default simplification
set.
\begin{ttbox}
by (Simp_tac 1);
{\out Level 2}
{\out ! x. Cons x xs ~= xs}
{\out  1. !!a mylist.}
{\out        ! x. Cons x mylist ~= mylist ==>}
{\out        ! x. Cons x (Cons a mylist) ~= Cons a mylist}
\end{ttbox}
Similarly, we prove the remaining goal.
\begin{ttbox}
by (Asm_simp_tac 1);
{\out Level 3}
{\out ! x. Cons x xs ~= xs}
{\out No subgoals!}
\ttbreak
qed_spec_mp "not_Cons_self";
{\out val not_Cons_self = "Cons x xs ~= xs" : thm}
\end{ttbox}
Because both subgoals could have been proved by \texttt{Asm_simp_tac}
we could have done that in one step:
\begin{ttbox}
by (ALLGOALS Asm_simp_tac);
\end{ttbox}


\subsubsection{The datatype $\alpha~mylist$ with mixfix syntax}

In this example we define the type $\alpha~mylist$ again but this time
we want to write \texttt{[]} for \texttt{Nil} and we want to use infix
notation \verb|#| for \texttt{Cons}.  To do this we simply add mixfix
annotations after the constructor declarations as follows:
\begin{ttbox}
MyList = Datatype +
  datatype 'a mylist =
    Nil ("[]")  |
    Cons 'a ('a mylist)  (infixr "#" 70)
end
\end{ttbox}
Now the theorem in the previous example can be written \verb|x#xs ~= xs|.


\subsubsection{A datatype for weekdays}

This example shows a datatype that consists of 7 constructors:
\begin{ttbox}
Days = Main +
  datatype days = Mon | Tue | Wed | Thu | Fri | Sat | Sun
end
\end{ttbox}
Because there are more than 6 constructors, inequality is expressed via a function
\verb|days_ord|.  The theorem \verb|Mon ~= Tue| is not directly
contained among the distinctness theorems, but the simplifier can
prove it thanks to rewrite rules inherited from theory \texttt{Arith}:
\begin{ttbox}
Goal "Mon ~= Tue";
by (Simp_tac 1);
\end{ttbox}
You need not derive such inequalities explicitly: the simplifier will dispose
of them automatically.
\index{*datatype|)}


\section{Recursive function definitions}\label{sec:HOL:recursive}
\index{recursive functions|see{recursion}}

Isabelle/HOL provides two main mechanisms of defining recursive functions.
\begin{enumerate}
\item \textbf{Primitive recursion} is available only for datatypes, and it is
  somewhat restrictive.  Recursive calls are only allowed on the argument's
  immediate constituents.  On the other hand, it is the form of recursion most
  often wanted, and it is easy to use.
  
\item \textbf{Well-founded recursion} requires that you supply a well-founded
  relation that governs the recursion.  Recursive calls are only allowed if
  they make the argument decrease under the relation.  Complicated recursion
  forms, such as nested recursion, can be dealt with.  Termination can even be
  proved at a later time, though having unsolved termination conditions around
  can make work difficult.%
  \footnote{This facility is based on Konrad Slind's TFL
    package~\cite{slind-tfl}.  Thanks are due to Konrad for implementing TFL
    and assisting with its installation.}
\end{enumerate}

Following good HOL tradition, these declarations do not assert arbitrary
axioms.  Instead, they define the function using a recursion operator.  Both
HOL and ZF derive the theory of well-founded recursion from first
principles~\cite{paulson-set-II}.  Primitive recursion over some datatype
relies on the recursion operator provided by the datatype package.  With
either form of function definition, Isabelle proves the desired recursion
equations as theorems.


\subsection{Primitive recursive functions}
\label{sec:HOL:primrec}
\index{recursion!primitive|(}
\index{*primrec|(}

Datatypes come with a uniform way of defining functions, {\bf primitive
  recursion}.  In principle, one could introduce primitive recursive functions
by asserting their reduction rules as new axioms, but this is not recommended:
\begin{ttbox}\slshape
Append = Main +
consts app :: ['a list, 'a list] => 'a list
rules 
   app_Nil   "app [] ys = ys"
   app_Cons  "app (x#xs) ys = x#app xs ys"
end
\end{ttbox}
Asserting axioms brings the danger of accidentally asserting nonsense, as
in \verb$app [] ys = us$.

The \ttindex{primrec} declaration is a safe means of defining primitive
recursive functions on datatypes:
\begin{ttbox}
Append = Main +
consts app :: ['a list, 'a list] => 'a list
primrec
   "app [] ys = ys"
   "app (x#xs) ys = x#app xs ys"
end
\end{ttbox}
Isabelle will now check that the two rules do indeed form a primitive
recursive definition.  For example
\begin{ttbox}
primrec
    "app [] ys = us"
\end{ttbox}
is rejected with an error message ``\texttt{Extra variables on rhs}''.

\bigskip

The general form of a primitive recursive definition is
\begin{ttbox}
primrec
    {\it reduction rules}
\end{ttbox}
where \textit{reduction rules} specify one or more equations of the form
\[ f \, x@1 \, \dots \, x@m \, (C \, y@1 \, \dots \, y@k) \, z@1 \,
\dots \, z@n = r \] such that $C$ is a constructor of the datatype, $r$
contains only the free variables on the left-hand side, and all recursive
calls in $r$ are of the form $f \, \dots \, y@i \, \dots$ for some $i$.  There
must be at most one reduction rule for each constructor.  The order is
immaterial.  For missing constructors, the function is defined to return a
default value.  

If you would like to refer to some rule by name, then you must prefix
the rule with an identifier.  These identifiers, like those in the
\texttt{rules} section of a theory, will be visible at the \ML\ level.

The primitive recursive function can have infix or mixfix syntax:
\begin{ttbox}\underscoreon
consts "@"  :: ['a list, 'a list] => 'a list  (infixr 60)
primrec
   "[] @ ys = ys"
   "(x#xs) @ ys = x#(xs @ ys)"
\end{ttbox}

The reduction rules become part of the default simpset, which
leads to short proof scripts:
\begin{ttbox}\underscoreon
Goal "(xs @ ys) @ zs = xs @ (ys @ zs)";
by (induct\_tac "xs" 1);
by (ALLGOALS Asm\_simp\_tac);
\end{ttbox}

\subsubsection{Example: Evaluation of expressions}
Using mutual primitive recursion, we can define evaluation functions \texttt{evala}
and \texttt{eval_bexp} for the datatypes of arithmetic and boolean expressions mentioned in
{\S}\ref{subsec:datatype:basics}:
\begin{ttbox}
consts
  evala :: "['a => nat, 'a aexp] => nat"
  evalb :: "['a => nat, 'a bexp] => bool"

primrec
  "evala env (If_then_else b a1 a2) =
     (if evalb env b then evala env a1 else evala env a2)"
  "evala env (Sum a1 a2) = evala env a1 + evala env a2"
  "evala env (Diff a1 a2) = evala env a1 - evala env a2"
  "evala env (Var v) = env v"
  "evala env (Num n) = n"

  "evalb env (Less a1 a2) = (evala env a1 < evala env a2)"
  "evalb env (And b1 b2) = (evalb env b1 & evalb env b2)"
  "evalb env (Or b1 b2) = (evalb env b1 & evalb env b2)"
\end{ttbox}
Since the value of an expression depends on the value of its variables,
the functions \texttt{evala} and \texttt{evalb} take an additional
parameter, an {\em environment} of type \texttt{'a => nat}, which maps
variables to their values.

Similarly, we may define substitution functions \texttt{substa}
and \texttt{substb} for expressions: The mapping \texttt{f} of type
\texttt{'a => 'a aexp} given as a parameter is lifted canonically
on the types \texttt{'a aexp} and \texttt{'a bexp}:
\begin{ttbox}
consts
  substa :: "['a => 'b aexp, 'a aexp] => 'b aexp"
  substb :: "['a => 'b aexp, 'a bexp] => 'b bexp"

primrec
  "substa f (If_then_else b a1 a2) =
     If_then_else (substb f b) (substa f a1) (substa f a2)"
  "substa f (Sum a1 a2) = Sum (substa f a1) (substa f a2)"
  "substa f (Diff a1 a2) = Diff (substa f a1) (substa f a2)"
  "substa f (Var v) = f v"
  "substa f (Num n) = Num n"

  "substb f (Less a1 a2) = Less (substa f a1) (substa f a2)"
  "substb f (And b1 b2) = And (substb f b1) (substb f b2)"
  "substb f (Or b1 b2) = Or (substb f b1) (substb f b2)"
\end{ttbox}
In textbooks about semantics one often finds {\em substitution theorems},
which express the relationship between substitution and evaluation. For
\texttt{'a aexp} and \texttt{'a bexp}, we can prove such a theorem by mutual
induction, followed by simplification:
\begin{ttbox}
Goal
  "evala env (substa (Var(v := a')) a) =
     evala (env(v := evala env a')) a &
   evalb env (substb (Var(v := a')) b) =
     evalb (env(v := evala env a')) b";
by (induct_tac "a b" 1);
by (ALLGOALS Asm_full_simp_tac);
\end{ttbox}

\subsubsection{Example: A substitution function for terms}
Functions on datatypes with nested recursion, such as the type
\texttt{term} mentioned in {\S}\ref{subsec:datatype:basics}, are
also defined by mutual primitive recursion. A substitution
function \texttt{subst_term} on type \texttt{term}, similar to the functions
\texttt{substa} and \texttt{substb} described above, can
be defined as follows:
\begin{ttbox}
consts
  subst_term :: "['a => ('a, 'b) term, ('a, 'b) term] => ('a, 'b) term"
  subst_term_list ::
    "['a => ('a, 'b) term, ('a, 'b) term list] => ('a, 'b) term list"

primrec
  "subst_term f (Var a) = f a"
  "subst_term f (App b ts) = App b (subst_term_list f ts)"

  "subst_term_list f [] = []"
  "subst_term_list f (t # ts) =
     subst_term f t # subst_term_list f ts"
\end{ttbox}
The recursion scheme follows the structure of the unfolded definition of type
\texttt{term} shown in {\S}\ref{subsec:datatype:basics}. To prove properties of
this substitution function, mutual induction is needed:
\begin{ttbox}
Goal
  "(subst_term ((subst_term f1) o f2) t) =
     (subst_term f1 (subst_term f2 t)) &
   (subst_term_list ((subst_term f1) o f2) ts) =
     (subst_term_list f1 (subst_term_list f2 ts))";
by (induct_tac "t ts" 1);
by (ALLGOALS Asm_full_simp_tac);
\end{ttbox}

\subsubsection{Example: A map function for infinitely branching trees}
Defining functions on infinitely branching datatypes by primitive
recursion is just as easy. For example, we can define a function
\texttt{map_tree} on \texttt{'a tree} as follows:
\begin{ttbox}
consts
  map_tree :: "('a => 'b) => 'a tree => 'b tree"

primrec
  "map_tree f (Atom a) = Atom (f a)"
  "map_tree f (Branch ts) = Branch (\%x. map_tree f (ts x))"
\end{ttbox}
Note that all occurrences of functions such as \texttt{ts} in the
\texttt{primrec} clauses must be applied to an argument. In particular,
\texttt{map_tree f o ts} is not allowed.

\index{recursion!primitive|)}
\index{*primrec|)}


\subsection{General recursive functions}
\label{sec:HOL:recdef}
\index{recursion!general|(}
\index{*recdef|(}

Using \texttt{recdef}, you can declare functions involving nested recursion
and pattern-matching.  Recursion need not involve datatypes and there are few
syntactic restrictions.  Termination is proved by showing that each recursive
call makes the argument smaller in a suitable sense, which you specify by
supplying a well-founded relation.

Here is a simple example, the Fibonacci function.  The first line declares
\texttt{fib} to be a constant.  The well-founded relation is simply~$<$ (on
the natural numbers).  Pattern-matching is used here: \texttt{1} is a
macro for \texttt{Suc~0}.
\begin{ttbox}
consts fib  :: "nat => nat"
recdef fib "less_than"
    "fib 0 = 0"
    "fib 1 = 1"
    "fib (Suc(Suc x)) = (fib x + fib (Suc x))"
\end{ttbox}

With \texttt{recdef}, function definitions may be incomplete, and patterns may
overlap, as in functional programming.  The \texttt{recdef} package
disambiguates overlapping patterns by taking the order of rules into account.
For missing patterns, the function is defined to return a default value.

%For example, here is a declaration of the list function \cdx{hd}:
%\begin{ttbox}
%consts hd :: 'a list => 'a
%recdef hd "\{\}"
%    "hd (x#l) = x"
%\end{ttbox}
%Because this function is not recursive, we may supply the empty well-founded
%relation, $\{\}$.

The well-founded relation defines a notion of ``smaller'' for the function's
argument type.  The relation $\prec$ is \textbf{well-founded} provided it
admits no infinitely decreasing chains
\[ \cdots\prec x@n\prec\cdots\prec x@1. \]
If the function's argument has type~$\tau$, then $\prec$ has to be a relation
over~$\tau$: it must have type $(\tau\times\tau)set$.

Proving well-foundedness can be tricky, so Isabelle/HOL provides a collection
of operators for building well-founded relations.  The package recognises
these operators and automatically proves that the constructed relation is
well-founded.  Here are those operators, in order of importance:
\begin{itemize}
\item \texttt{less_than} is ``less than'' on the natural numbers.
  (It has type $(nat\times nat)set$, while $<$ has type $[nat,nat]\To bool$.
  
\item $\mathop{\mathtt{measure}} f$, where $f$ has type $\tau\To nat$, is the
  relation~$\prec$ on type~$\tau$ such that $x\prec y$ iff $f(x)<f(y)$.
  Typically, $f$ takes the recursive function's arguments (as a tuple) and
  returns a result expressed in terms of the function \texttt{size}.  It is
  called a \textbf{measure function}.  Recall that \texttt{size} is overloaded
  and is defined on all datatypes (see {\S}\ref{sec:HOL:size}).
                                                    
\item $\mathop{\mathtt{inv_image}} f\;R$ is a generalisation of
  \texttt{measure}.  It specifies a relation such that $x\prec y$ iff $f(x)$
  is less than $f(y)$ according to~$R$, which must itself be a well-founded
  relation.

\item $R@1\texttt{**}R@2$ is the lexicographic product of two relations.  It
  is a relation on pairs and satisfies $(x@1,x@2)\prec(y@1,y@2)$ iff $x@1$
  is less than $y@1$ according to~$R@1$ or $x@1=y@1$ and $x@2$
  is less than $y@2$ according to~$R@2$.

\item \texttt{finite_psubset} is the proper subset relation on finite sets.
\end{itemize}

We can use \texttt{measure} to declare Euclid's algorithm for the greatest
common divisor.  The measure function, $\lambda(m,n). n$, specifies that the
recursion terminates because argument~$n$ decreases.
\begin{ttbox}
recdef gcd "measure ((\%(m,n). n) ::nat*nat=>nat)"
    "gcd (m, n) = (if n=0 then m else gcd(n, m mod n))"
\end{ttbox}

The general form of a well-founded recursive definition is
\begin{ttbox}
recdef {\it function} {\it rel}
    congs   {\it congruence rules}      {\bf(optional)}
    simpset {\it simplification set}      {\bf(optional)}
   {\it reduction rules}
\end{ttbox}
where
\begin{itemize}
\item \textit{function} is the name of the function, either as an \textit{id}
  or a \textit{string}.  
  
\item \textit{rel} is a {\HOL} expression for the well-founded termination
  relation.
  
\item \textit{congruence rules} are required only in highly exceptional
  circumstances.
  
\item The \textit{simplification set} is used to prove that the supplied
  relation is well-founded.  It is also used to prove the \textbf{termination
    conditions}: assertions that arguments of recursive calls decrease under
  \textit{rel}.  By default, simplification uses \texttt{simpset()}, which
  is sufficient to prove well-foundedness for the built-in relations listed
  above. 
  
\item \textit{reduction rules} specify one or more recursion equations.  Each
  left-hand side must have the form $f\,t$, where $f$ is the function and $t$
  is a tuple of distinct variables.  If more than one equation is present then
  $f$ is defined by pattern-matching on components of its argument whose type
  is a \texttt{datatype}.  

  Unlike with \texttt{primrec}, the reduction rules are not added to the
  default simpset, and individual rules may not be labelled with identifiers.
  However, the identifier $f$\texttt{.rules} is visible at the \ML\ level
  as a list of theorems.
\end{itemize}

With the definition of \texttt{gcd} shown above, Isabelle/HOL is unable to
prove one termination condition.  It remains as a precondition of the
recursion theorems.
\begin{ttbox}
gcd.rules;
{\out ["! m n. n ~= 0 --> m mod n < n}
{\out   ==> gcd (?m, ?n) = (if ?n = 0 then ?m else gcd (?n, ?m mod ?n))"] }
{\out : thm list}
\end{ttbox}
The theory \texttt{HOL/ex/Primes} illustrates how to prove termination
conditions afterwards.  The function \texttt{Tfl.tgoalw} is like the standard
function \texttt{goalw}, which sets up a goal to prove, but its argument
should be the identifier $f$\texttt{.rules} and its effect is to set up a
proof of the termination conditions:
\begin{ttbox}
Tfl.tgoalw thy [] gcd.rules;
{\out Level 0}
{\out ! m n. n ~= 0 --> m mod n < n}
{\out  1. ! m n. n ~= 0 --> m mod n < n}
\end{ttbox}
This subgoal has a one-step proof using \texttt{simp_tac}.  Once the theorem
is proved, it can be used to eliminate the termination conditions from
elements of \texttt{gcd.rules}.  Theory \texttt{HOL/Subst/Unify} is a much
more complicated example of this process, where the termination conditions can
only be proved by complicated reasoning involving the recursive function
itself.

Isabelle/HOL can prove the \texttt{gcd} function's termination condition
automatically if supplied with the right simpset.
\begin{ttbox}
recdef gcd "measure ((\%(m,n). n) ::nat*nat=>nat)"
  simpset "simpset() addsimps [mod_less_divisor, zero_less_eq]"
    "gcd (m, n) = (if n=0 then m else gcd(n, m mod n))"
\end{ttbox}

A \texttt{recdef} definition also returns an induction rule specialised for
the recursive function.  For the \texttt{gcd} function above, the induction
rule is
\begin{ttbox}
gcd.induct;
{\out "(!!m n. n ~= 0 --> ?P n (m mod n) ==> ?P m n) ==> ?P ?u ?v" : thm}
\end{ttbox}
This rule should be used to reason inductively about the \texttt{gcd}
function.  It usually makes the induction hypothesis available at all
recursive calls, leading to very direct proofs.  If any termination conditions
remain unproved, they will become additional premises of this rule.

\index{recursion!general|)}
\index{*recdef|)}


\section{Inductive and coinductive definitions}
\index{*inductive|(}
\index{*coinductive|(}

An {\bf inductive definition} specifies the least 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 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.

A theory file may contain any number of inductive and coinductive
definitions.  They may be intermixed with other declarations; in
particular, the (co)inductive sets {\bf must} be declared separately as
constants, and may have mixfix syntax or be subject to syntax translations.

Each (co)inductive definition adds definitions to the theory and also
proves some theorems.  Each definition creates an \ML\ structure, which is a
substructure of the main theory structure.

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 sets determine the domain of the fixedpoint definition,
and the package does not have to use inference rules for type-checking.


\subsection{The result structure}
Many of the result structure's components have been discussed in the paper;
others are self-explanatory.
\begin{description}
\item[\tt defs] is the list of definitions of the recursive sets.

\item[\tt mono] is a monotonicity theorem for the fixedpoint operator.

\item[\tt unfold] is a fixedpoint equation for the recursive set (the union of
the recursive sets, in the case of mutual recursion).

\item[\tt intrs] is the list of introduction rules, now proved as theorems, for
the recursive sets.  The rules are also available individually, using the
names given them in the theory file. 

\item[\tt elims] is the list of elimination rule.

\item[\tt elim] is the head of the list \texttt{elims}.
  
\item[\tt mk_cases] is a function to create simplified instances of {\tt
elim} using freeness reasoning on underlying datatypes.
\end{description}

For an inductive definition, the result structure contains the
rule \texttt{induct}.  For a
coinductive definition, it contains the rule \verb|coinduct|.

Figure~\ref{def-result-fig} summarises the two result signatures,
specifying the types of all these components.

\begin{figure}
\begin{ttbox}
sig
val defs         : thm list
val mono         : thm
val unfold       : thm
val intrs        : thm list
val elims        : thm list
val elim         : thm
val mk_cases     : string -> thm
{\it(Inductive definitions only)} 
val induct       : thm
{\it(coinductive definitions only)}
val coinduct     : thm
end
\end{ttbox}
\hrule
\caption{The {\ML} result of a (co)inductive definition} \label{def-result-fig}
\end{figure}

\subsection{The syntax of a (co)inductive definition}
An inductive definition has the form
\begin{ttbox}
inductive    {\it inductive sets}
  intrs      {\it introduction rules}
  monos      {\it monotonicity theorems}
  con_defs   {\it constructor definitions}
\end{ttbox}
A coinductive definition is identical, except that it starts with the keyword
\texttt{coinductive}.  

The \texttt{monos} and \texttt{con_defs} sections are optional.  If present,
each is specified by a list of identifiers.

\begin{itemize}
\item The \textit{inductive sets} are specified by one or more strings.

\item The \textit{introduction rules} specify one or more introduction rules in
  the form \textit{ident\/}~\textit{string}, where the identifier gives the name of
  the rule in the result structure.

\item The \textit{monotonicity theorems} are required for each operator
  applied to a recursive set in the introduction rules.  There {\bf must}
  be a theorem of the form $A\subseteq B\Imp M(A)\subseteq M(B)$, for each
  premise $t\in M(R@i)$ in an introduction rule!

\item The \textit{constructor definitions} contain definitions of constants
  appearing in the introduction rules.  In most cases it can be omitted.
\end{itemize}


\subsection{*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 \texttt{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 $A\subseteq B\Imp M(A)\subseteq M(B)$, for proving
  monotonicity of inductive definitions whose introduction rules have premises
  involving terms such as $t\in M(R@i)$.
\item Monotonicity theorems for logical operators, which are of the general form
  $\List{\cdots \rightarrow \cdots;~\ldots;~\cdots \rightarrow \cdots} \Imp
    \cdots \rightarrow \cdots$.
  For example, in the case of the operator $\vee$, the corresponding theorem is
  \[
  \infer{P@1 \vee P@2 \rightarrow Q@1 \vee Q@2}
    {P@1 \rightarrow Q@1 & P@2 \rightarrow Q@2}
  \]
\item De Morgan style equations for reasoning about the ``polarity'' of expressions, e.g.
  \[
  (\neg \neg P) ~=~ P \qquad\qquad
  (\neg (P \wedge Q)) ~=~ (\neg P \vee \neg Q)
  \]
\item Equations for reducing complex operators to more primitive ones whose
  monotonicity can easily be proved, e.g.
  \[
  (P \rightarrow Q) ~=~ (\neg P \vee Q) \qquad\qquad
  \mathtt{Ball}~A~P ~\equiv~ \forall x.~x \in A \rightarrow P~x
  \]
\end{itemize}

\subsection{Example of an inductive definition}
Two declarations, included in a theory file, define the finite powerset
operator.  First we declare the constant~\texttt{Fin}.  Then we declare it
inductively, with two introduction rules:
\begin{ttbox}
consts Fin :: 'a set => 'a set set
inductive "Fin A"
  intrs
    emptyI  "{\ttlbrace}{\ttrbrace} : Fin A"
    insertI "[| a: A;  b: Fin A |] ==> insert a b : Fin A"
\end{ttbox}
The resulting theory structure contains a substructure, called~\texttt{Fin}.
It contains the \texttt{Fin}$~A$ introduction rules as the list \texttt{Fin.intrs},
and also individually as \texttt{Fin.emptyI} and \texttt{Fin.consI}.  The induction
rule is \texttt{Fin.induct}.

For another example, here is a theory file defining the accessible
part of a relation.  The paper
\cite{paulson-CADE} discusses a \ZF\ version of this example in more
detail.
\begin{ttbox}
Acc = WF + Inductive +

consts acc :: "('a * 'a)set => 'a set"   (* accessible part *)

inductive "acc r"
  intrs
    accI "ALL y. (y, x) : r --> y : acc r ==> x : acc r"

end
\end{ttbox}
The Isabelle distribution contains many other inductive definitions.  Simple
examples are collected on subdirectory \texttt{HOL/Induct}.  The theory
\texttt{HOL/Induct/LList} contains coinductive definitions.  Larger examples
may be found on other subdirectories of \texttt{HOL}, such as \texttt{IMP},
\texttt{Lambda} and \texttt{Auth}.

\index{*coinductive|)} \index{*inductive|)}


\section{The examples directories}

Directory \texttt{HOL/Auth} contains theories for proving the correctness of
cryptographic protocols~\cite{paulson-jcs}.  The approach is based upon
operational semantics rather than the more usual belief logics.  On the same
directory are proofs for some standard examples, such as the Needham-Schroeder
public-key authentication protocol and the Otway-Rees
protocol.

Directory \texttt{HOL/IMP} contains a formalization of various denotational,
operational and axiomatic semantics of a simple while-language, the necessary
equivalence proofs, soundness and completeness of the Hoare rules with
respect to the denotational semantics, and soundness and completeness of a
verification condition generator.  Much of development is taken from
Winskel~\cite{winskel93}.  For details see~\cite{nipkow-IMP}.

Directory \texttt{HOL/Hoare} contains a user friendly surface syntax for Hoare
logic, including a tactic for generating verification-conditions.

Directory \texttt{HOL/MiniML} contains a formalization of the type system of
the core functional language Mini-ML and a correctness proof for its type
inference algorithm $\cal W$~\cite{milner78,nipkow-W}.

Directory \texttt{HOL/Lambda} contains a formalization of untyped
$\lambda$-calculus in de~Bruijn notation and Church-Rosser proofs for $\beta$
and $\eta$ reduction~\cite{Nipkow-CR}.

Directory \texttt{HOL/Subst} contains Martin Coen's mechanization of a theory of
substitutions and unifiers.  It is based on Paulson's previous
mechanisation in {\LCF}~\cite{paulson85} of Manna and Waldinger's
theory~\cite{mw81}.  It demonstrates a complicated use of \texttt{recdef},
with nested recursion.

Directory \texttt{HOL/Induct} presents simple examples of (co)inductive
definitions and datatypes.
\begin{itemize}
\item Theory \texttt{PropLog} proves the soundness and completeness of
  classical propositional logic, given a truth table semantics.  The only
  connective is $\imp$.  A Hilbert-style axiom system is specified, and its
  set of theorems defined inductively.  A similar proof in \ZF{} is
  described elsewhere~\cite{paulson-set-II}.

\item Theory \texttt{Term} defines the datatype \texttt{term}.

\item Theory \texttt{ABexp} defines arithmetic and boolean expressions
 as mutually recursive datatypes.

\item The definition of lazy lists demonstrates methods for handling
  infinite data structures and coinduction in higher-order
  logic~\cite{paulson-coind}.%
\footnote{To be precise, these lists are \emph{potentially infinite} rather
  than lazy.  Lazy implies a particular operational semantics.}
  Theory \thydx{LList} defines an operator for
  corecursion on lazy lists, which is used to define a few simple functions
  such as map and append.   A coinduction principle is defined
  for proving equations on lazy lists.
  
\item Theory \thydx{LFilter} defines the filter functional for lazy lists.
  This functional is notoriously difficult to define because finding the next
  element meeting the predicate requires possibly unlimited search.  It is not
  computable, but can be expressed using a combination of induction and
  corecursion.  

\item Theory \thydx{Exp} illustrates the use of iterated inductive definitions
  to express a programming language semantics that appears to require mutual
  induction.  Iterated induction allows greater modularity.
\end{itemize}

Directory \texttt{HOL/ex} contains other examples and experimental proofs in
{\HOL}.  
\begin{itemize}
\item Theory \texttt{Recdef} presents many examples of using \texttt{recdef}
  to define recursive functions.  Another example is \texttt{Fib}, which
  defines the Fibonacci function.

\item Theory \texttt{Primes} defines the Greatest Common Divisor of two
  natural numbers and proves a key lemma of the Fundamental Theorem of
  Arithmetic: if $p$ is prime and $p$ divides $m\times n$ then $p$ divides~$m$
  or $p$ divides~$n$.

\item Theory \texttt{Primrec} develops some computation theory.  It
  inductively defines the set of primitive recursive functions and presents a
  proof that Ackermann's function is not primitive recursive.

\item File \texttt{cla.ML} demonstrates the classical reasoner on over sixty
  predicate calculus theorems, ranging from simple tautologies to
  moderately difficult problems involving equality and quantifiers.

\item File \texttt{meson.ML} contains an experimental implementation of the {\sc
    meson} proof procedure, inspired by Plaisted~\cite{plaisted90}.  It is
  much more powerful than Isabelle's classical reasoner.  But it is less
  useful in practice because it works only for pure logic; it does not
  accept derived rules for the set theory primitives, for example.

\item File \texttt{mesontest.ML} contains test data for the {\sc meson} proof
  procedure.  These are mostly taken from Pelletier \cite{pelletier86}.

\item File \texttt{set.ML} proves Cantor's Theorem, which is presented in
  {\S}\ref{sec:hol-cantor} below, and the Schr{\"o}der-Bernstein Theorem.

\item Theory \texttt{MT} contains Jacob Frost's formalization~\cite{frost93} of
  Milner and Tofte's coinduction example~\cite{milner-coind}.  This
  substantial proof concerns the soundness of a type system for a simple
  functional language.  The semantics of recursion is given by a cyclic
  environment, which makes a coinductive argument appropriate.
\end{itemize}


\goodbreak
\section{Example: Cantor's Theorem}\label{sec:hol-cantor}
Cantor's Theorem states that every set has more subsets than it has
elements.  It has become a favourite example in higher-order logic since
it is so easily expressed:
\[  \forall f::\alpha \To \alpha \To bool. \exists S::\alpha\To bool.
    \forall x::\alpha. f~x \not= S 
\] 
%
Viewing types as sets, $\alpha\To bool$ represents the powerset
of~$\alpha$.  This version states that for every function from $\alpha$ to
its powerset, some subset is outside its range.  

The Isabelle proof uses \HOL's set theory, with the type $\alpha\,set$ and
the operator \cdx{range}.
\begin{ttbox}
context Set.thy;
\end{ttbox}
The set~$S$ is given as an unknown instead of a
quantified variable so that we may inspect the subset found by the proof.
\begin{ttbox}
Goal "?S ~: range\thinspace(f :: 'a=>'a set)";
{\out Level 0}
{\out ?S ~: range f}
{\out  1. ?S ~: range f}
\end{ttbox}
The first two steps are routine.  The rule \tdx{rangeE} replaces
$\Var{S}\in \texttt{range} \, f$ by $\Var{S}=f~x$ for some~$x$.
\begin{ttbox}
by (resolve_tac [notI] 1);
{\out Level 1}
{\out ?S ~: range f}
{\out  1. ?S : range f ==> False}
\ttbreak
by (eresolve_tac [rangeE] 1);
{\out Level 2}
{\out ?S ~: range f}
{\out  1. !!x. ?S = f x ==> False}
\end{ttbox}
Next, we apply \tdx{equalityCE}, reasoning that since $\Var{S}=f~x$,
we have $\Var{c}\in \Var{S}$ if and only if $\Var{c}\in f~x$ for
any~$\Var{c}$.
\begin{ttbox}
by (eresolve_tac [equalityCE] 1);
{\out Level 3}
{\out ?S ~: range f}
{\out  1. !!x. [| ?c3 x : ?S; ?c3 x : f x |] ==> False}
{\out  2. !!x. [| ?c3 x ~: ?S; ?c3 x ~: f x |] ==> False}
\end{ttbox}
Now we use a bit of creativity.  Suppose that~$\Var{S}$ has the form of a
comprehension.  Then $\Var{c}\in\{x.\Var{P}~x\}$ implies
$\Var{P}~\Var{c}$.   Destruct-resolution using \tdx{CollectD}
instantiates~$\Var{S}$ and creates the new assumption.
\begin{ttbox}
by (dresolve_tac [CollectD] 1);
{\out Level 4}
{\out {\ttlbrace}x. ?P7 x{\ttrbrace} ~: range f}
{\out  1. !!x. [| ?c3 x : f x; ?P7(?c3 x) |] ==> False}
{\out  2. !!x. [| ?c3 x ~: {\ttlbrace}x. ?P7 x{\ttrbrace}; ?c3 x ~: f x |] ==> False}
\end{ttbox}
Forcing a contradiction between the two assumptions of subgoal~1
completes the instantiation of~$S$.  It is now the set $\{x. x\not\in
f~x\}$, which is the standard diagonal construction.
\begin{ttbox}
by (contr_tac 1);
{\out Level 5}
{\out {\ttlbrace}x. x ~: f x{\ttrbrace} ~: range f}
{\out  1. !!x. [| x ~: {\ttlbrace}x. x ~: f x{\ttrbrace}; x ~: f x |] ==> False}
\end{ttbox}
The rest should be easy.  To apply \tdx{CollectI} to the negated
assumption, we employ \ttindex{swap_res_tac}:
\begin{ttbox}
by (swap_res_tac [CollectI] 1);
{\out Level 6}
{\out {\ttlbrace}x. x ~: f x{\ttrbrace} ~: range f}
{\out  1. !!x. [| x ~: f x; ~ False |] ==> x ~: f x}
\ttbreak
by (assume_tac 1);
{\out Level 7}
{\out {\ttlbrace}x. x ~: f x{\ttrbrace} ~: range f}
{\out No subgoals!}
\end{ttbox}
How much creativity is required?  As it happens, Isabelle can prove this
theorem automatically.  The default classical set \texttt{claset()} contains rules
for most of the constructs of \HOL's set theory.  We must augment it with
\tdx{equalityCE} to break up set equalities, and then apply best-first
search.  Depth-first search would diverge, but best-first search
successfully navigates through the large search space.
\index{search!best-first}
\begin{ttbox}
choplev 0;
{\out Level 0}
{\out ?S ~: range f}
{\out  1. ?S ~: range f}
\ttbreak
by (best_tac (claset() addSEs [equalityCE]) 1);
{\out Level 1}
{\out {\ttlbrace}x. x ~: f x{\ttrbrace} ~: range f}
{\out No subgoals!}
\end{ttbox}
If you run this example interactively, make sure your current theory contains
theory \texttt{Set}, for example by executing \ttindex{context}~{\tt Set.thy}.
Otherwise the default claset may not contain the rules for set theory.
\index{higher-order logic|)}

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