%% $Id$
\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 higher-order logic.  Experience with the {\sc hol} system
has demonstrated that higher-order logic is useful for hardware verification;
beyond this, it is widely applicable in many areas of mathematics.  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.

Previous releases of Isabelle included a different 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.

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.  There is no
`apply' operator: function applications are written as simply~$f(a)$ rather
than $f{\tt`}a$.

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 {\tt show_types} set to {\tt 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{center}
\begin{tabular}{rrr} 
  \it name      &\it meta-type  & \it description \\ 
  \cdx{Trueprop}& $bool\To prop$                & coercion to $prop$\\
  \cdx{not}     & $bool\To bool$                & negation ($\neg$) \\
  \cdx{True}    & $bool$                        & tautology ($\top$) \\
  \cdx{False}   & $bool$                        & absurdity ($\bot$) \\
  \cdx{if}      & $[bool,\alpha,\alpha]\To\alpha::term$ & conditional \\
  \cdx{Inv}     & $(\alpha\To\beta)\To(\beta\To\alpha)$ & function inversion\\
  \cdx{Let}     & $[\alpha,\alpha\To\beta]\To\beta$ & let binder
\end{tabular}
\end{center}
\subcaption{Constants}

\begin{center}
\index{"@@{\tt\at} symbol}
\index{*"! symbol}\index{*"? symbol}
\index{*"?"! symbol}\index{*"E"X"! symbol}
\begin{tabular}{llrrr} 
  \it symbol &\it name     &\it meta-type & \it description \\
  \tt\at & \cdx{Eps}  & $(\alpha\To bool)\To\alpha::term$ & 
        Hilbert description ($\epsilon$) \\
  {\tt!~} or \sdx{ALL}  & \cdx{All}  & $(\alpha::term\To bool)\To bool$ & 
        universal quantifier ($\forall$) \\
  {\tt?~} or \sdx{EX}   & \cdx{Ex}   & $(\alpha::term\To bool)\To bool$ & 
        existential quantifier ($\exists$) \\
  {\tt?!} or {\tt EX!}  & \cdx{Ex1}  & $(\alpha::term\To bool)\To bool$ & 
        unique existence ($\exists!$)
\end{tabular}
\end{center}
\subcaption{Binders} 

\begin{center}
\index{*"= symbol}
\index{&@{\tt\&} symbol}
\index{*"| symbol}
\index{*"-"-"> symbol}
\begin{tabular}{rrrr} 
  \it symbol    & \it meta-type & \it priority & \it description \\ 
  \sdx{o}       & $[\beta\To\gamma,\alpha\To\beta]\To (\alpha\To\gamma)$ & 
        Right 50 & composition ($\circ$) \\
  \tt =         & $[\alpha::term,\alpha]\To bool$ & 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{tabular}
\end{center}
\subcaption{Infixes}
\caption{Syntax of {\tt HOL}} \label{hol-constants}
\end{figure}


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


\section{Syntax}
The type class of higher-order terms is called~\cldx{term}.  Type variables
range over this class by default.  The equality symbol and quantifiers are
polymorphic over class {\tt term}.

Class \cldx{ord} consists of all ordered types; the relations $<$ and
$\leq$ are polymorphic over this class, as are the functions
\cdx{mono}, \cdx{min} and \cdx{max}.  Three other
type classes --- \cldx{plus}, \cldx{minus} and \cldx{times} --- permit
overloading of the operators {\tt+}, {\tt-} and {\tt*}.  In particular,
{\tt-} is overloaded for set difference and subtraction.
\index{*"+ symbol}
\index{*"- symbol}
\index{*"* symbol}

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
$\neg(a=b)$.

\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,
  $\neg\neg P=P$ abbreviates $\neg\neg (P=P)$ and not $(\neg\neg P)=P$.
  When using $=$ to mean logical equivalence, enclose both operands in
  parentheses.
\end{warn}

\subsection{Types}\label{HOL-types}
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,term)term}.  Thus, $\sigma\To\tau$
belongs to class~{\tt term} if $\sigma$ and~$\tau$ do, allowing quantification
over functions.

Types in HOL must be non-empty; otherwise the quantifier rules would be
unsound.  I have commented on this elsewhere~\cite[\S7]{paulson-COLOG}.

\index{type definitions}
Gordon's {\sc hol} system supports {\bf type definitions}.  A type is
defined by exhibiting an existing type~$\sigma$, a predicate~$P::\sigma\To
bool$, and a theorem of the form $\exists x::\sigma.P(x)$.  Thus~$P$
specifies a non-empty subset of~$\sigma$, and the new type denotes this
subset.  New function constants are generated to establish an isomorphism
between the new type and the subset.  If type~$\sigma$ 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.  Melham~\cite{melham89} discusses type definitions at length, with
examples. 

Isabelle does not support type definitions at present.  Instead, they are
mimicked by explicit definitions of isomorphism functions.  The definitions
should be supported by theorems of the form $\exists x::\sigma.P(x)$, but
Isabelle cannot enforce this.


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

Existential quantification is defined by
\[ \exists x.P(x) \;\equiv\; P(\epsilon x.P(x)). \]
The unique existence quantifier, $\exists!x.P[x]$, 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~$P(x,y)$.

\index{*"! symbol}\index{*"? symbol}\index{HOL system@{\sc hol} system}
Quantifiers have two notations.  As in Gordon's {\sc hol} system, HOL
uses~{\tt!}\ and~{\tt?}\ to stand for $\forall$ and $\exists$.  The
existential quantifier must be followed by a space; thus {\tt?x} is an
unknown, while \verb'? x.f(x)=y' is a quantification.  Isabelle's usual
notation for quantifiers, \sdx{ALL} and \sdx{EX}, is also available.  Both
notations are accepted for input.  The {\ML} reference
\ttindexbold{HOL_quantifiers} governs the output notation.  If set to {\tt
  true}, then~{\tt!}\ and~{\tt?}\ are displayed; this is the default.  If set
to {\tt false}, then~{\tt ALL} and~{\tt EX} are displayed.

All these binders have priority 10. 


\subsection{The \sdx{let} and \sdx{case} constructions}
Local abbreviations can be introduced by a {\tt 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 {\tt case} constructs.  Therefore {\tt
  case} and \sdx{of} are reserved words.  However, so far this is mere
syntax and has no logical meaning.  By declaring translations, you can
cause instances of the {\tt case} construct to denote applications of
particular case operators.  The patterns supplied for $c@1$,~\ldots,~$c@n$
distinguish among the different case operators.  For an example, see the
case construct for lists on page~\pageref{hol-list} below.


\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 {\tt HOL} rules} \label{hol-rules}
\end{figure}


\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{Inv_def}    Inv   == (\%(f::'a=>'b) y. @x. f(x)=y)
\tdx{o_def}      op o  == (\%(f::'b=>'c) g (x::'a). f(g(x)))
\tdx{if_def}     if    == (\%P x y.@z::'a.(P=True --> z=x) & (P=False --> z=y))
\tdx{Let_def}    Let(s,f) == f(s)
\end{ttbox}
\caption{The {\tt HOL} definitions} \label{hol-defs}
\end{figure}


\section{Rules of inference}
Figure~\ref{hol-rules} shows the 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
  $\epsilon$-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 $\epsilon$-operator already makes the logic classical, as
    shown by Diaconescu; see Paulson~\cite{paulson-COLOG} for details.}
\end{ttdescription}

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.

Some of the rules mention type variables; for
example, {\tt refl} mentions the type variable~{\tt'a}.  This allows you to
instantiate type variables explicitly by calling {\tt res_inst_tac}.  By
default, explicit type variables have class \cldx{term}.

Include type constraints whenever you state a polymorphic goal.  Type
inference may otherwise make the goal more polymorphic than you intended,
with confusing results.

\begin{warn}
  If resolution fails for no obvious reason, try setting
  \ttindex{show_types} to {\tt true}, causing Isabelle to display types of
  terms.  Possibly set \ttindex{show_sorts} to {\tt true} as well, causing
  Isabelle to display 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 {\tt resolve_tac},
  possibly instantiating type variables.  Setting
  \ttindex{Unify.trace_types} to {\tt true} causes Isabelle to report
  omitted search paths during unification.\index{tracing!of unification}
\end{warn}


\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::'a)
\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)
\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::'a. P(x)) ==> !x. P(x)
\tdx{spec}      !x::'a.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::'a.P(x)
\tdx{exE}       [| ? x::'a.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,x,y) = x
\tdx{if_False}        if(False,x,y) = y
\tdx{if_P}            P ==> if(P,x,y) = x
\tdx{if_not_P}        ~ P ==> if(P,x,y) = y
\tdx{expand_if}       P(if(Q,x,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.


\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 \\
  \cdx{mono}    & $(\alpha\,set\To\beta\,set)\To bool$
        & monotonicity \\
  \cdx{inj}~~\cdx{surj}& $(\alpha\To\beta )\To bool$
        & injective/surjective \\
  \cdx{inj_onto}        & $[\alpha\To\beta ,\alpha\,set]\To bool$
        & injective over subset
\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 ($\inter$) \\
  \sdx{Un}      & $[\alpha\,set,\alpha\,set]\To\alpha\,set$
        & Left 65 & union ($\union$) \\
  \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 {\tt 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\\
  \{$a@1$, $\ldots$\}  &  insert($a@1$, $\ldots$\{\}) & \rm finite set \\
  \{$x$.$P[x]$\}        &  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 \\
  \tt ! $x$:$A$.$P[x]$ or \sdx{ALL} $x$:$A$.$P[x]$ & 
        Ball($A$,$\lambda x.P[x]$) & 
        \rm bounded $\forall$ \\
  \sdx{?} $x$:$A$.$P[x]$ or \sdx{EX}{\tt\ } $x$:$A$.$P[x]$ & 
        Bex($A$,$\lambda x.P[x]$) & \rm bounded $\exists$
\end{tabular}
\end{center}
\subcaption{Translations}

\dquotes
\[\begin{array}{rclcl}
    term & = & \hbox{other terms\ldots} \\
         & | & "\{\}" \\
         & | & "\{ " term\; ("," term)^* " \}" \\
         & | & "\{ " id " . " formula " \}" \\
         & | & term " `` " term \\
         & | & term " Int " term \\
         & | & term " Un " term \\
         & | & "INT~~"  id ":" term " . " term \\
         & | & "UN~~~"  id ":" term " . " term \\
         & | & "INT~~"  id~id^* " . " term \\
         & | & "UN~~~"  id~id^* " . " term \\[2ex]
 formula & = & \hbox{other formulae\ldots} \\
         & | & term " : " term \\
         & | & term " \ttilde: " term \\
         & | & term " <= " term \\
         & | & "!~" id ":" term " . " formula 
         & | & "ALL " id ":" term " . " formula \\
         & | & "?~" id ":" term " . " formula 
         & | & "EX~~" id ":" term " . " formula
  \end{array}
\]
\subcaption{Full Grammar}
\caption{Syntax of the theory {\tt 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.  Since Isabelle
does not support type definitions (as mentioned in \S\ref{HOL-types}), the
isomorphisms between the two types are declared explicitly.  Here they are
natural: {\tt 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 ($A\union B$
and $A\inter B$), the subset and membership relations, and the image
operator~{\tt``}\@.  Note that $a$\verb|~:|$b$ is translated to
$\neg(a\in b)$.  

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

The set \hbox{\tt\{$x$.$P[x]$\}} 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 {\tt Ball($A$,$P$)} and {\tt Bex($A$,$P$)} we may
write\index{*"! symbol}\index{*"? symbol}
\index{*ALL symbol}\index{*EX symbol} 
%
\hbox{\tt !~$x$:$A$.$P[x]$} and \hbox{\tt ?~$x$:$A$.$P[x]$}.  Isabelle's
usual quantifier symbols, \sdx{ALL} and \sdx{EX}, are also accepted
for input.  As with the primitive quantifiers, the {\ML} reference
\ttindex{HOL_quantifiers} specifies which notation to use for output.

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 : \{x.P(x)\}) = P(a)
\tdx{Collect_mem_eq}    \{x.x:A\} = A

\tdx{empty_def}         \{\}          == \{x.False\}
\tdx{insert_def}        insert(a,B) == \{x.x=a\} 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      == \{x.x:A | x:B\}
\tdx{Int_def}           A Int B     == \{x.x:A & x:B\}
\tdx{set_diff_def}      A - B       == \{x.x:A & x~:B\}
\tdx{Compl_def}         Compl(A)    == \{x. ~ x:A\}
\tdx{INTER_def}         INTER(A,B)  == \{y. ! x:A. y: B(x)\}
\tdx{UNION_def}         UNION(A,B)  == \{y. ? x:A. y: B(x)\}
\tdx{INTER1_def}        INTER1(B)   == INTER(\{x.True\}, B)
\tdx{UNION1_def}        UNION1(B)   == UNION(\{x.True\}, B)
\tdx{Inter_def}         Inter(S)    == (INT x:S. x)
\tdx{Union_def}         Union(S)    == (UN  x:S. x)
\tdx{Pow_def}           Pow(A)      == \{B. B <= A\}
\tdx{image_def}         f``A        == \{y. ? x:A. y=f(x)\}
\tdx{range_def}         range(f)    == \{y. ? x. y=f(x)\}
\tdx{mono_def}          mono(f)     == !A B. A <= B --> f(A) <= f(B)
\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_onto_def}      inj_onto(f,A) == !x:A. !y:A. f(x)=f(y) --> x=y
\end{ttbox}
\caption{Rules of the theory {\tt Set}} \label{hol-set-rules}
\end{figure}


\begin{figure} \underscoreon
\begin{ttbox}
\tdx{CollectI}        [| P(a) |] ==> a : \{x.P(x)\}
\tdx{CollectD}        [| a : \{x.P(x)\} |] ==> P(a)
\tdx{CollectE}        [| a : \{x.P(x)\};  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 : \{\} ==> 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
\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 {\tt 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 properties of functions: image~({\tt``}) and
{\tt range}, and predicates concerning monotonicity, injectiveness and
surjectiveness.  

The predicate \cdx{inj_onto} is used for simulating type definitions.
The statement ${\tt inj_onto}(f,A)$ 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{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

\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_ontoI}  (!!x y. [| f(x)=f(y); x:A; y:A |] ==> x=y) ==> inj_onto(f,A)
\tdx{inj_ontoD}  [| inj_onto(f,A);  f(x)=f(y);  x:A;  y:A |] ==> x=y

\tdx{inj_onto_inverseI}
    (!!x. x:A ==> g(f(x)) = x) ==> inj_onto(f,A)
\tdx{inj_onto_contraD}
    [| inj_onto(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) = \{x.False\}
\tdx{Compl_partition}   A Un  Compl(A) = \{x.True\}
\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 {\tt HOL/Set.ML} for proofs pertaining
to set theory.

Figure~\ref{hol-fun} presents derived inference rules involving functions.
They also include rules for \cdx{Inv}, which is defined in theory~{\tt
  HOL}; note that ${\tt Inv}(f)$ applies the Axiom of Choice to yield an
inverse of~$f$.  They also include natural deduction rules for the image
and range operators, and for the predicates {\tt inj} and {\tt inj_onto}.
Reasoning about function composition (the operator~\sdx{o}) and the
predicate~\cdx{surj} is done simply by expanding the definitions.  See
the file {\tt HOL/fun.ML} for a complete listing of the derived rules.

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 {\tt HOL/subset.ML}.

Figure~\ref{hol-equalities} presents many common set equalities.  They
include commutative, associative and distributive laws involving unions,
intersections and complements.  The proofs are mostly trivial, using the
classical reasoner; see file {\tt HOL/equalities.ML}.


\begin{figure}
\begin{constants}
  \it symbol    & \it meta-type &           & \it description \\ 
  \cdx{Pair}    & $[\alpha,\beta]\To \alpha\times\beta$
        & & ordered pairs $\langle a,b\rangle$ \\
  \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). \{<x,y>\}


\tdx{Pair_inject}  [| <a, b> = <a',b'>;  [| a=a';  b=b' |] ==> R |] ==> R
\tdx{fst_conv}     fst(<a,b>) = a
\tdx{snd_conv}     snd(<a,b>) = b
\tdx{split}        split(c, <a,b>) = c(a,b)

\tdx{surjective_pairing}  p = <fst(p),snd(p)>

\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} 


\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{sum_case_def}   sum_case == (\%f g p. @z. (!x. p=Inl(x) --> z=f(x)) &
                                        (!y. p=Inr(y) --> z=g(y)))

\tdx{Inl_not_Inr}    ~ Inl(a)=Inr(b)

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

\tdx{sumE}           [| !!x::'a. P(Inl(x));  !!y::'b. 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::'a. f(Inl(x)), \%y::'b. f(Inr(y)), s) = f(s)
\end{ttbox}
\caption{Type $\alpha+\beta$}\label{hol-sum}
\end{figure}


\section{Generic packages and classical reasoning}
HOL instantiates most of Isabelle's generic packages; see {\tt HOL/ROOT.ML}
for details.
\begin{itemize}
\item Because it includes a general substitution rule, HOL instantiates the
  tactic {\tt hyp_subst_tac}, which substitutes for an equality throughout a
  subgoal and its hypotheses.
\item 
It instantiates the simplifier, defining~\ttindexbold{HOL_ss} as the
simplification set for higher-order logic.  Equality~($=$), which also
expresses logical equivalence, may be used for rewriting.  See the file
{\tt HOL/simpdata.ML} for a complete listing of the simplification
rules. 
\item 
It instantiates the classical reasoner, as described below. 
\end{itemize}
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 set up as the structure {\tt Classical}.  This
structure is open, so {\ML} identifiers such as {\tt step_tac}, {\tt
  fast_tac}, {\tt best_tac}, etc., refer to it.  HOL defines the following
classical rule sets:
\begin{ttbox} 
prop_cs    : claset
HOL_cs     : claset
set_cs     : claset
\end{ttbox}
\begin{ttdescription}
\item[\ttindexbold{prop_cs}] contains the propositional rules, namely
those for~$\top$, $\bot$, $\conj$, $\disj$, $\neg$, $\imp$ and~$\bimp$,
along with the rule~{\tt refl}.

\item[\ttindexbold{HOL_cs}] extends {\tt prop_cs} with the safe rules
  {\tt allI} and~{\tt exE} and the unsafe rules {\tt allE}
  and~{\tt exI}, as well as rules for unique existence.  Search using
  this classical set is incomplete: quantified formulae are used at most
  once.

\item[\ttindexbold{set_cs}] extends {\tt HOL_cs} with rules for the bounded
  quantifiers, subsets, comprehensions, unions and intersections,
  complements, finite sets, images and ranges.
\end{ttdescription}
\noindent
See \iflabelundefined{chap:classical}{the {\em Reference Manual\/}}%
        {Chap.\ts\ref{chap:classical}} 
for more discussion of classical proof methods.


\section{Types}
The basic higher-order logic is augmented with a tremendous amount of
material, including support for recursive function and type definitions.  A
detailed discussion appears elsewhere~\cite{paulson-coind}.  The simpler
definitions are the same as those used the {\sc hol} system, but my
treatment of recursive types differs from Melham's~\cite{melham89}.  The
present section describes product, sum, natural number and list types.

\subsection{Product and sum types}\index{*"* type}\index{*"+ type}
Theory \thydx{Prod} defines the product type $\alpha\times\beta$, with
the ordered pair syntax {\tt<$a$,$b$>}.  Theory \thydx{Sum} defines the
sum type $\alpha+\beta$.  These use fairly standard constructions; see
Figs.\ts\ref{hol-prod} and~\ref{hol-sum}.  Because Isabelle does not
support abstract type definitions, the isomorphisms between these types and
their representations are made explicitly.

Most of the definitions are suppressed, but observe that the projections
and conditionals are defined as descriptions.  Their properties are easily
proved using \tdx{select_equality}.  

\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\\
  \cdx{pred_nat} & $(nat\times nat) set$ & & predecessor relation\\
  \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_case_def}  nat_case == (\%a f n. @z. (n=0 --> z=a) & 
                                       (!x. n=Suc(x) --> z=f(x)))
\tdx{pred_nat_def}  pred_nat == \{p. ? n. p = <n, Suc(n)>\} 
\tdx{less_def}      m<n      == <m,n>:pred_nat^+
\tdx{nat_rec_def}   nat_rec(n,c,d) == 
               wfrec(pred_nat, n, nat_case(\%g.c, \%m g. d(m,g(m))))

\tdx{add_def}   m+n     == nat_rec(m, n, \%u v.Suc(v))
\tdx{diff_def}  m-n     == nat_rec(n, m, \%u v. nat_rec(v, 0, \%x y.x))
\tdx{mult_def}  m*n     == nat_rec(m, 0, \%u v. n + v)
\tdx{mod_def}   m mod n == wfrec(trancl(pred_nat), m, \%j f. if(j<n,j,f(j-n)))
\tdx{quo_def}   m div n == wfrec(trancl(pred_nat), 
                        m, \%j f. if(j<n,0,Suc(f(j-n))))
\subcaption{Definitions}
\end{ttbox}
\caption{Defining {\tt nat}, the type of natural numbers} \label{hol-nat1}
\end{figure}


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

\tdx{pred_natI}      <n, Suc(n)> : pred_nat
\tdx{pred_natE}
    [| p : pred_nat;  !!x n. [| p = <n, Suc(n)> |] ==> R |] ==> R

\tdx{nat_case_0}     nat_case(a, f, 0) = a
\tdx{nat_case_Suc}   nat_case(a, f, Suc(k)) = f(k)

\tdx{wf_pred_nat}    wf(pred_nat)
\tdx{nat_rec_0}      nat_rec(0,c,h) = c
\tdx{nat_rec_Suc}    nat_rec(Suc(n), c, h) = h(n, nat_rec(n,c,h))
\subcaption{Case analysis and primitive recursion}

\tdx{less_trans}     [| i<j;  j<k |] ==> i<k
\tdx{lessI}          n < Suc(n)
\tdx{zero_less_Suc}  0 < Suc(n)

\tdx{less_not_sym}   n<m --> ~ m<n 
\tdx{less_not_refl}  ~ n<n
\tdx{not_less0}      ~ n<0

\tdx{Suc_less_eq}    (Suc(m) < Suc(n)) = (m<n)
\tdx{less_induct}    [| !!n. [| ! m. m<n --> P(m) |] ==> P(n) |]  ==>  P(n)

\tdx{less_linear}    m<n | m=n | n<m
\subcaption{The less-than relation}
\end{ttbox}
\caption{Derived rules for {\tt nat}} \label{hol-nat2}
\end{figure}


\subsection{The type of natural numbers, {\tt nat}}
The theory \thydx{Nat} defines the natural numbers in a roundabout but
traditional way.  The axiom of infinity postulates an 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.  As
usual, the isomorphisms between~\tydx{nat} and its representation are made
explicitly.

The definition makes use of a least fixed point operator \cdx{lfp},
defined using the Knaster-Tarski theorem.  This is used to define the
operator \cdx{trancl}, for taking the transitive closure of a relation.
Primitive recursion makes use of \cdx{wfrec}, an operator for recursion
along arbitrary well-founded relations.  The corresponding theories are
called {\tt Lfp}, {\tt Trancl} and {\tt WF}\@.  Elsewhere I have described
similar constructions in the context of set theory~\cite{paulson-set-II}.

Type~\tydx{nat} is postulated to belong to class~\cldx{ord}, which overloads
$<$ and $\leq$ on the natural numbers.  As of this writing, Isabelle provides
no means of verifying that such overloading is sensible; there is no means of
specifying the operators' properties and verifying that instances of the
operators satisfy those properties.  To be safe, the HOL theory includes no
polymorphic axioms asserting general properties of $<$ and~$\leq$.

Theory \thydx{Arith} develops arithmetic on the natural numbers.  It
defines addition, multiplication, subtraction, division, and remainder.
Many of their properties are proved: commutative, associative and
distributive laws, identity and cancellation laws, etc.  The most
interesting result is perhaps the theorem $a \bmod b + (a/b)\times b = a$.
Division and remainder are defined by repeated subtraction, which requires
well-founded rather than primitive recursion.  See Figs.\ts\ref{hol-nat1}
and~\ref{hol-nat2}.

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~$<$.

The tactic {\tt\ttindex{nat_ind_tac} "$n$" $i$} performs induction over the
variable~$n$ in subgoal~$i$.

\begin{figure}
\index{#@{\tt\#} symbol}
\index{"@@{\tt\at} symbol}
\begin{constants}
  \it symbol & \it meta-type & \it priority & \it description \\
  \cdx{Nil}     & $\alpha list$ & & 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{ttl}     & $\alpha list \To \alpha list$ & & total tail \\
  \tt\at  & $[\alpha list,\alpha list]\To \alpha list$ & Left 65 & append \\
  \sdx{mem}  & $[\alpha,\alpha list]\To bool$    &  Left 55   & membership\\
  \cdx{map}     & $(\alpha\To\beta) \To (\alpha list \To \beta list)$
        & & mapping functional\\
  \cdx{filter}  & $(\alpha \To bool) \To (\alpha list \To \alpha list)$
        & & filter functional\\
  \cdx{list_all}& $(\alpha \To bool) \To (\alpha list \To bool)$
        & & forall functional\\
  \cdx{list_rec}        & $[\alpha list, \beta, [\alpha ,\alpha list,
\beta]\To\beta] \To \beta$
        & & list recursor
\end{constants}
\subcaption{Constants and infixes}

\begin{center} \tt\frenchspacing
\begin{tabular}{rrr} 
  \it external        & \it internal  & \it description \\{}
  \sdx{[]}            & Nil           & \rm empty list \\{}
  [$x@1$, $\dots$, $x@n$]  &  $x@1$ \# $\cdots$ \# $x@n$ \# [] &
        \rm finite list \\{}
  [$x$:$l$. $P$]  & filter($\lambda x{.}P$, $l$) & 
        \rm list comprehension
\end{tabular}
\end{center}
\subcaption{Translations}

\begin{ttbox}
\tdx{list_induct}    [| P([]);  !!x xs. [| P(xs) |] ==> P(x#xs)) |]  ==> P(l)

\tdx{Cons_not_Nil}   (x # xs) ~= []
\tdx{Cons_Cons_eq}   ((x # xs) = (y # ys)) = (x=y & xs=ys)
\subcaption{Induction and freeness}
\end{ttbox}
\caption{The theory \thydx{List}} \label{hol-list}
\end{figure}

\begin{figure}
\begin{ttbox}\makeatother
\tdx{list_rec_Nil}    list_rec([],c,h) = c  
\tdx{list_rec_Cons}   list_rec(a#l, c, h) = h(a, l, list_rec(l,c,h))

\tdx{list_case_Nil}   list_case(c, h, []) = c 
\tdx{list_case_Cons}  list_case(c, h, x#xs) = h(x, xs)

\tdx{map_Nil}         map(f,[]) = []
\tdx{map_Cons}        map(f, x \# xs) = f(x) \# map(f,xs)

\tdx{null_Nil}        null([]) = True
\tdx{null_Cons}       null(x#xs) = False

\tdx{hd_Cons}         hd(x#xs) = x
\tdx{tl_Cons}         tl(x#xs) = xs

\tdx{ttl_Nil}         ttl([]) = []
\tdx{ttl_Cons}        ttl(x#xs) = xs

\tdx{append_Nil}      [] @ ys = ys
\tdx{append_Cons}     (x#xs) \at ys = x # xs \at ys

\tdx{mem_Nil}         x mem [] = False
\tdx{mem_Cons}        x mem (y#ys) = if(y=x, True, x mem ys)

\tdx{filter_Nil}      filter(P, []) = []
\tdx{filter_Cons}     filter(P,x#xs) = if(P(x), x#filter(P,xs), filter(P,xs))

\tdx{list_all_Nil}    list_all(P,[]) = True
\tdx{list_all_Cons}   list_all(P, x#xs) = (P(x) & list_all(P, xs))
\end{ttbox}
\caption{Rewrite rules for lists} \label{hol-list-simps}
\end{figure}


\subsection{The type constructor for lists, {\tt list}}
\index{*list type}

HOL's definition of lists is an example of an experimental method for handling
recursive data types.  Figure~\ref{hol-list} presents the theory \thydx{List}:
the basic list operations with their types and properties.

The \sdx{case} construct is defined by the following translation:
{\dquotes
\begin{eqnarray*}
  \begin{array}{r@{\;}l@{}l}
  "case " e " of" & "[]"    & " => " a\\
              "|" & x"\#"xs & " => " b
  \end{array} 
  & \equiv &
  "list_case"(a, \lambda x\;xs.b, e)
\end{eqnarray*}}%
The theory includes \cdx{list_rec}, a primitive recursion operator
for lists.  It is derived from well-founded recursion, a general principle
that can express arbitrary total recursive functions.

The simpset \ttindex{list_ss} contains, along with additional useful lemmas,
the basic rewrite rules that appear in Fig.\ts\ref{hol-list-simps}.

The tactic {\tt\ttindex{list_ind_tac} "$xs$" $i$} performs induction over the
variable~$xs$ in subgoal~$i$.


\section{Datatype declarations}
\index{*datatype|(}

\underscoreon

It is often necessary to extend a theory with \ML-like datatypes.  This
extension consists of the new type, declarations of its constructors and
rules that describe the new type. The theory definition section {\tt
  datatype} represents a compact way of doing this.


\subsection{Foundations}

A datatype declaration has the following general structure:
\[ \mbox{\tt datatype}~ (\alpha_1,\dots,\alpha_n)t ~=~
      C_1(\tau_{11},\dots,\tau_{1k_1}) ~\mid~ \dots ~\mid~
      C_m(\tau_{m1},\dots,\tau_{mk_m}) 
\]
where $\alpha_i$ are type variables, $C_i$ are distinct constructor names and
$\tau_{ij}$ are one of the following:
\begin{itemize}
\item type variables $\alpha_1,\dots,\alpha_n$,
\item types $(\beta_1,\dots,\beta_l)s$ where $s$ is a previously declared
  type or type synonym and $\{\beta_1,\dots,\beta_l\} \subseteq
  \{\alpha_1,\dots,\alpha_n\}$,
\item the newly defined type $(\alpha_1,\dots,\alpha_n)t$ \footnote{This
    makes it a recursive type. To ensure that the new type is not empty at
    least one constructor must consist of only non-recursive type
    components.}
\end{itemize}
If you would like one of the $\tau_{ij}$ to be a complex type expression
$\tau$ you need to declare a new type synonym $syn = \tau$ first and use
$syn$ in place of $\tau$. Of course this does not work if $\tau$ mentions the
recursive type itself, thus ruling out problematic cases like \[ \mbox{\tt
  datatype}~ t ~=~ C(t \To t) \] together with unproblematic ones like \[
\mbox{\tt datatype}~ t ~=~ C(t~list). \]

The constructors are automatically defined as functions of their respective
type:
\[ C_j : [\tau_{j1},\dots,\tau_{jk_j}] \To (\alpha_1,\dots,\alpha_n)t \]
These functions have certain {\em freeness} properties:
\begin{description}
\item[\tt distinct] They are distinct:
\[ C_i(x_1,\dots,x_{k_i}) \neq C_j(y_1,\dots,y_{k_j}) \qquad
   \mbox{for all}~ i \neq j.
\]
\item[\tt inject] They are injective:
\[ (C_j(x_1,\dots,x_{k_j}) = C_j(y_1,\dots,y_{k_j})) =
   (x_1 = y_1 \land \dots \land x_{k_j} = y_{k_j})
\]
\end{description}
Because the number of inequalities is quadratic in the number of
constructors, a different method is used if their number exceeds
a certain value, currently 4. In that case every constructor is mapped to a
natural number
\[
\begin{array}{lcl}
\mbox{\it t\_ord}(C_1(x_1,\dots,x_{k_1})) & = & 0 \\
& \vdots & \\
\mbox{\it t\_ord}(C_m(x_1,\dots,x_{k_m})) & = & m-1
\end{array}
\]
and distinctness of constructors is expressed by:
\[
\mbox{\it t\_ord}(x) \neq \mbox{\it t\_ord}(y) \Imp x \neq y.
\]
In addition a structural induction axiom {\tt induct} is provided: 
\[
\infer{P(x)}
{\begin{array}{lcl}
\Forall x_1\dots x_{k_1}.
  \List{P(x_{r_{11}}); \dots; P(x_{r_{1l_1}})} &
  \Imp  & P(C_1(x_1,\dots,x_{k_1})) \\
 & \vdots & \\
\Forall x_1\dots x_{k_m}.
  \List{P(x_{r_{m1}}); \dots; P(x_{r_{ml_m}})} &
  \Imp & P(C_m(x_1,\dots,x_{k_m}))
\end{array}}
\]
where $\{r_{j1},\dots,r_{jl_j}\} = \{i \in \{1,\dots k_j\} ~\mid~ \tau_{ji}
= (\alpha_1,\dots,\alpha_n)t \}$, i.e.\ the property $P$ can be assumed for
all arguments of the recursive type.

The type also comes with an \ML-like \sdx{case}-construct:
\[
\begin{array}{rrcl}
\mbox{\tt case}~e~\mbox{\tt of} & C_1(x_{11},\dots,x_{1k_1}) & \To & e_1 \\
                           \vdots \\
                           \mid & C_m(x_{m1},\dots,x_{mk_m}) & \To & e_m
\end{array}
\]
In contrast to \ML, {\em all} constructors must be present, their order is
fixed, and nested patterns are not supported.


\subsection{Defining datatypes}

A datatype is defined in a theory definition file using the keyword {\tt
  datatype}. The definition following {\tt datatype} must conform to the
syntax of {\em typedecl} specified in Fig.~\ref{datatype-grammar} and must
obey the rules in the previous section. As a result the theory is extended
with the new type, the constructors, and the theorems listed in the previous
section.

\begin{figure}
\begin{rail}
typedecl : typevarlist id '=' (cons + '|')
         ;
cons     : (id | string) ( () | '(' (typ + ',') ')' ) ( () | mixfix )
         ;
typ      : typevarlist id
           | tid
         ;
typevarlist : () | tid | '(' (tid + ',') ')'
         ;
\end{rail}
\caption{Syntax of datatype declarations}
\label{datatype-grammar}
\end{figure}

Reading the theory file produces a structure which, in addition to the usual
components, contains a structure named $t$ for each datatype $t$ defined in
the file.\footnote{Otherwise multiple datatypes in the same theory file would
  lead to name clashes.} Each structure $t$ contains the following elements:
\begin{ttbox}
val distinct : thm list
val inject : thm list
val induct : thm
val cases : thm list
val simps : thm list
val induct_tac : string -> int -> tactic
\end{ttbox}
{\tt distinct}, {\tt inject} and {\tt induct} contain the theorems described
above. For convenience {\tt distinct} contains inequalities in both
directions.
\begin{warn}
  If there are five or more constructors, the {\em t\_ord} scheme is used for
  {\tt distinct}.  In this case the theory {\tt Arith} must be contained
  in the current theory, if necessary by including it explicitly.
\end{warn}
The reduction rules of the {\tt case}-construct are in {\tt cases}.  All
theorems from {\tt distinct}, {\tt inject} and {\tt cases} are combined in
{\tt simps} for use with the simplifier. The tactic {\verb$induct_tac$~{\em
    var i}\/} applies structural induction over variable {\em var} to
subgoal {\em i}.


\subsection{Examples}

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

We want to define the type $\alpha~list$.\footnote{Of course there is a list
  type in HOL already. This is only an example.} To do this we have to build
a new theory that contains the type definition. We start from {\tt HOL}.
\begin{ttbox}
MyList = HOL +
  datatype 'a list = Nil | Cons ('a, 'a list)
end
\end{ttbox}
After loading the theory (\verb$use_thy "MyList"$), we can prove
$Cons(x,xs)\neq xs$.  First we build a suitable simpset for the simplifier:
\begin{ttbox}
val mylist_ss = HOL_ss addsimps MyList.list.simps;
goal MyList.thy "!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 (MyList.list.induct_tac "xs" 1);
{\out Level 1}
{\out ! x. Cons(x, xs) ~= xs}
{\out  1. ! x. Cons(x, Nil) ~= Nil}
{\out  2. !!a list.}
{\out        ! x. Cons(x, list) ~= list ==>}
{\out        ! x. Cons(x, Cons(a, list)) ~= Cons(a, list)}
\end{ttbox}
The first subgoal can be proved with the simplifier and the distinctness
axioms which are part of \verb$mylist_ss$.
\begin{ttbox}
by (simp_tac mylist_ss 1);
{\out Level 2}
{\out ! x. Cons(x, xs) ~= xs}
{\out  1. !!a list.}
{\out        ! x. Cons(x, list) ~= list ==>}
{\out        ! x. Cons(x, Cons(a, list)) ~= Cons(a, list)}
\end{ttbox}
Using the freeness axioms we can quickly prove the remaining goal.
\begin{ttbox}
by (asm_simp_tac mylist_ss 1);
{\out Level 3}
{\out ! x. Cons(x, xs) ~= xs}
{\out No subgoals!}
\end{ttbox}
Because both subgoals were proved by almost the same tactic we could have
done that in one step using
\begin{ttbox}
by (ALLGOALS (asm_simp_tac mylist_ss));
\end{ttbox}


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

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


\subsubsection{A datatype for weekdays}

This example shows a datatype that consists of more than four constructors:
\begin{ttbox}
Days = Arith +
  datatype days = Mo | Tu | We | Th | Fr | Sa | So
end
\end{ttbox}
Because there are more than four constructors, the theory must be based on
{\tt Arith}. Inequality is defined via a function \verb|days_ord|. Although
the expression \verb|Mo ~= Tu| is not directly contained in {\tt distinct},
it can be proved by the simplifier if \verb$arith_ss$ is used:
\begin{ttbox}
val days_ss = arith_ss addsimps Days.days.simps;

goal Days.thy "Mo ~= Tu";
by (simp_tac days_ss 1);
\end{ttbox}
Note that usually it is not necessary to derive these inequalities explicitly
because the simplifier will dispose of them automatically.

\subsection{Primitive recursive functions}
\index{primitive recursion|(}
\index{*primrec|(}

Datatypes come with a uniform way of defining functions, {\bf primitive
  recursion}. Although it is possible to define primitive recursive functions
by asserting their reduction rules as new axioms, e.g.\
\begin{ttbox}
Append = MyList +
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}
this carries with it the danger of accidentally asserting an inconsistency,
as in \verb$app([],ys) = us$. Therefore primitive recursive functions on
datatypes can be defined with a special syntax:
\begin{ttbox}
Append = MyList +
consts app :: "['a list,'a list] => 'a list"
primrec app MyList.list
   app_Nil   "app([],ys) = ys"
   app_Cons  "app(x#xs, ys) = x#app(xs,ys)"
end
\end{ttbox}
The system will now check that the two rules \verb$app_Nil$ and
\verb$app_Cons$ do indeed form a primitive recursive definition, thus
ensuring that consistency is maintained. For example
\begin{ttbox}
primrec app MyList.list
    app_Nil   "app([],ys) = us"
\end{ttbox}
is rejected:
\begin{ttbox}
Extra variables on rhs
\end{ttbox}
\bigskip

The general form of a primitive recursive definition is
\begin{ttbox}
primrec {\it function} {\it type}
    {\it reduction rules}
\end{ttbox}
where
\begin{itemize}
\item {\it function} is the name of the function, either as an {\it id} or a
  {\it string}. The function must already have been declared.
\item {\it type} is the name of the datatype, either as an {\it id} or in the
  long form {\it Thy.t}, where {\it Thy} is the name of the parent theory the
  datatype was declared in, and $t$ the name of the datatype. The long form
  is required if the {\tt datatype} and the {\tt primrec} sections are in
  different theories.
\item {\it reduction rules} specify one or more named equations of the form
  {\it id\/}~{\it string}, where the identifier gives the name of the rule in
  the result structure, and {\it string} is a reduction rule 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 exactly one reduction
  rule for each constructor.
\end{itemize}
A theory file may contain any number of {\tt primrec} sections which may be
intermixed with other declarations.

For the consistency-sensitive user it may be reassuring to know that {\tt
  primrec} does not assert the reduction rules as new axioms but derives them
as theorems from an explicit definition of the recursive function in terms of
a recursion operator on the datatype.

The primitive recursive function can also use infix or mixfix syntax:
\begin{ttbox}
Append = MyList +
consts "@"  :: "['a list,'a list] => 'a list"  (infixr 60)
primrec "op @" MyList.list
   app_Nil   "[] @ ys = ys"
   app_Cons  "(x#xs) @ ys = x#(xs @ ys)"
end
\end{ttbox}

The reduction rules become part of the ML structure \verb$Append$ and can
be used to prove theorems about the function:
\begin{ttbox}
val append_ss = HOL_ss addsimps [Append.app_Nil,Append.app_Cons];

goal Append.thy "(xs @ ys) @ zs = xs @ (ys @ zs)";
by (MyList.list.induct_tac "xs" 1);
by (ALLGOALS(asm_simp_tac append_ss));
\end{ttbox}

%Note that underdefined primitive recursive functions are allowed:
%\begin{ttbox}
%Tl = MyList +
%consts tl  :: "'a list => 'a list"
%primrec tl MyList.list
%   tl_Cons "tl(x#xs) = xs"
%end
%\end{ttbox}
%Nevertheless {\tt tl} is total, although we do not know what the result of
%\verb$tl([])$ is.

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


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

An {\bf inductive definition} specifies the least set closed under given
rules.  For example, a structural operational semantics is an inductive
definition of an evaluation relation.  Dually, a {\bf coinductive
  definition} specifies the greatest set consistent with given rules.  An
important example is using bisimulation relations to formalize 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 derived from the ZF one, described in a
separate paper,\footnote{It appeared in CADE~\cite{paulson-CADE} and 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
automatic type-checking.  The type of the (co)inductive determines the
domain of the fixedpoint definition, and the package does not 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 thy] is the new theory containing the recursive sets.

\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 elim] is the elimination rule.

\item[\tt mk\_cases] is a function to create simplified instances of {\tt
elim}, using freeness reasoning on some underlying datatype.
\end{description}

For an inductive definition, the result structure contains two induction rules,
{\tt induct} and \verb|mutual_induct|.  For a coinductive definition, it
contains the rule \verb|coinduct|.

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

\begin{figure}
\begin{ttbox}
sig
val thy          : theory
val defs         : thm list
val mono         : thm
val unfold       : thm
val intrs        : thm list
val elim         : thm
val mk_cases     : thm list -> string -> thm
{\it(Inductive definitions only)} 
val induct       : thm
val mutual_induct: thm
{\it(Coinductive definitions only)}
val coinduct    : thm
end
\end{ttbox}
\hrule
\caption{The 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
{\tt coinductive}.  

The {\tt monos} and {\tt con\_defs} sections are optional.  If present,
each is specified as a string, which must be a valid ML expression of type
{\tt thm list}.  It is simply inserted into the {\tt .thy.ML} file; if it
is ill-formed, it will trigger ML error messages.  You can then inspect the
file on your directory.

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

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

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

The package has a few notable restrictions:
\begin{itemize}
\item The theory must separately declare the recursive sets as
  constants.

\item The names of the recursive sets must be identifiers, not infix
operators.  

\item Side-conditions must not be conjunctions.  However, an introduction rule
may contain any number of side-conditions.

\item Side-conditions of the form $x=t$, where the variable~$x$ does not
  occur in~$t$, will be substituted through the rule \verb|mutual_induct|.
\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~{\tt Fin}.  Then we declare it
inductively, with two introduction rules:
\begin{ttbox}
consts Fin :: "'a set => 'a set set"
inductive "Fin(A)"
  intrs
    emptyI  "{} : Fin(A)"
    insertI "[| a: A;  b: Fin(A) |] ==> insert(a,b) : Fin(A)"
\end{ttbox}
The resulting theory structure contains a substructure, called~{\tt Fin}.
It contains the {\tt Fin}$(A)$ introduction rules as the list {\tt Fin.intrs},
and also individually as {\tt Fin.emptyI} and {\tt Fin.consI}.  The induction
rule is {\tt Fin.induct}.

For another example, here is a theory file defining the accessible part of a
relation.  The main thing to note is the use of~{\tt Pow} in the sole
introduction rule, and the corresponding mention of the rule
\verb|Pow_mono| in the {\tt monos} list.  The paper discusses a ZF version
of this example in more detail.
\begin{ttbox}
Acc = WF + 
consts pred :: "['b, ('a * 'b)set] => 'a set"   (*Set of predecessors*)
       acc  :: "('a * 'a)set => 'a set"         (*Accessible part*)
defs   pred_def  "pred(x,r) == {y. <y,x>:r}"
inductive "acc(r)"
  intrs
     pred "pred(a,r): Pow(acc(r)) ==> a: acc(r)"
  monos   "[Pow_mono]"
end
\end{ttbox}
The HOL distribution contains many other inductive definitions, such as the
theory {\tt HOL/ex/PropLog.thy} and the directory {\tt HOL/IMP}.  The
theory {\tt HOL/ex/LList.thy} contains coinductive definitions.

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


\section{The examples directories}
Directory {\tt HOL/Subst} contains Martin Coen's mechanisation 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}. 

Directory {\tt HOL/IMP} contains a mechanised version of a semantic
equivalence proof taken from Winskel~\cite{winskel93}.  It formalises the
denotational and operational semantics of a simple while-language, then
proves the two equivalent.  It contains several datatype and inductive
definitions, and demonstrates their use.

Directory {\tt HOL/ex} contains other examples and experimental proofs in HOL.
Here is an overview of the more interesting files.
\begin{itemize}
\item File {\tt 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 {\tt 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 {\tt mesontest.ML} contains test data for the {\sc meson} proof
  procedure.  These are mostly taken from Pelletier \cite{pelletier86}.

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

\item Theories {\tt InSort} and {\tt Qsort} prove correctness properties of
  insertion sort and quick sort.

\item The definition of lazy lists demonstrates methods for handling
  infinite data structures and coinduction in higher-order
  logic~\cite{paulson-coind}.  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.  Corecursion cannot easily define operations such
  as filter, which can compute indefinitely before yielding the next
  element (if any!) of the lazy list.  A coinduction principle is defined
  for proving equations on lazy lists.
  
\item Theory {\tt 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 {\tt Term} develops an experimental recursive type definition;
  the recursion goes through the type constructor~\tydx{list}.

\item Theory {\tt Simult} constructs mutually recursive sets of trees and
  forests, including induction and recursion rules.

\item Theory {\tt 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:
\[  \forall f::[\alpha,\alpha]\To bool. \exists S::\alpha\To bool.
    \forall x::\alpha. f(x) \not= S 
\] 
%
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}.  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 Set.thy "~ ?S : range(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
$\Var{S}\in {\tt range}(f)$ 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 ~ \{x. ?P7(x)\} : range(f)}
{\out  1. !!x. [| ?c3(x) : f(x); ?P7(?c3(x)) |] ==> False}
{\out  2. !!x. [| ~ ?c3(x) : \{x. ?P7(x)\}; ~ ?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 ~ \{x. ~ x : f(x)\} : range(f)}
{\out  1. !!x. [| ~ x : \{x. ~ x : f(x)\}; ~ 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 ~ \{x. ~ x : f(x)\} : range(f)}
{\out  1. !!x. [| ~ x : f(x); ~ False |] ==> ~ x : f(x)}
\ttbreak
by (assume_tac 1);
{\out Level 7}
{\out ~ \{x. ~ x : f(x)\} : range(f)}
{\out No subgoals!}
\end{ttbox}
How much creativity is required?  As it happens, Isabelle can prove this
theorem automatically.  The classical set \ttindex{set_cs} 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 (set_cs addSEs [equalityCE]) 1);
{\out Level 1}
{\out ~ \{x. ~ x : f(x)\} : range(f)}
{\out No subgoals!}
\end{ttbox}

\index{higher-order logic|)}
