%% $Id$

 \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'\^etre\/}.  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 ($\neg$) \\

   \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 \\

   \tt\at & \cdx{Eps}  & $(\alpha\To bool)\To\alpha$ & 

         Hilbert description ($\varepsilon$) \\

   {\tt!~} or \sdx{ALL}  & \cdx{All}  & $(\alpha\To bool)\To bool$ & 

         universal quantifier ($\forall$) \\

   {\tt?~} or \sdx{EX}   & \cdx{Ex}   & $(\alpha\To bool)\To bool$ & 

         existential quantifier ($\exists$) \\

   {\tt?!} or \texttt{EX!}  & \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~$term$} \\

          & | & "\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 \\

          & | & "!~~~" 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}

 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 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.\ $\le$).

 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 $i \le j \Imp i \le j+k$ 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

 $(i::nat) \le j \Imp i \le j+k$, 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 \at $x$.\ $P[x]$}.

 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~$P\,x\,y$.

 \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 \texttt{false}, then~\texttt{ALL} and~\texttt{EX} are displayed.

 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.\ $\le$) $x$ 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

   $\le$ 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{case_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 $\neg P$, respectively.

 \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 \\

   \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} \\

          & | & "{\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 \\

          & | & "!~" id ":" term " . " formula 

          & | & "ALL " id ":" term " . " formula \\

          & | & "?~" id ":" term " . " formula 

          & | & "EX~~" 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

 $\neg(a\in b)$.  

 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 !~$x$:$A$.\ $P[x]$} 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 : {\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 (\neg \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{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.

 \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. 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{split_sum_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}, the details of which can be found in theory \texttt{NatDef}.

 %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{exhaust\_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[\S7]{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 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}) is made

   part of the standard 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: $t\dtt updates$.  Following a new trend in

 Isabelle system architecture, these names are \emph{not} bound at the {\ML}

 level, though.

 \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}.

 \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 $\tau = \sigma \rightarrow \tau'$, 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 $1 \le j \le n$ 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 \wedge \dots \wedge 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 \wedge 1 \leq i'' \leq n \wedge

        \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 \wedge 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 exhaustion 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[\ttindexbold{mutual_induct_tac}

   {\tt["}$x@1${\tt",}$\ldots${\tt,"}$x@n${\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{exhaust_tac} {\tt"}$u${\tt"} $i$]

  performs an exhaustive 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.  Exhaustion 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}

 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{exhaust_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 (mutual_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 (mutual_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.