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+++ b/doc-src/Logics/CHOL.tex Tue May 09 10:43:19 1995 +0200
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+%% $Id$
+\chapter{Higher-Order Logic}
+\index{higher-order logic|(}
+\index{CHOL system@{\sc chol} system}
+
+The theory~\thydx{CHOL} implements higher-order logic with curried
+function application. It is based on Gordon's~{\sc hol}
+system~\cite{mgordon-hol}, which itself is based on Church's original
+paper~\cite{church40}. Andrews's book~\cite{andrews86} is a full
+description of higher-order logic. Experience with the {\sc hol}
+system has demonstrated that higher-order logic is useful for hardware
+verification; beyond this, it is widely applicable in many areas of
+mathematics. It is weaker than {\ZF} set theory but for most
+applications this does not matter. If you prefer {\ML} to Lisp, you
+will probably prefer \CHOL\ to~{\ZF}.
+
+\CHOL\ is a modified version of Isabelle's \HOL\ and uses curried function
+application. Therefore the expression $f(a,b)$ (which in \HOL\ means
+``f applied to the two arguments $a$ and $b$'') means ``f applied to
+the pair $(a,b)$'' in \CHOL. N.B. that ordered pairs in \HOL\ are written as
+$<a,b>$ while in \CHOL\ the syntax $(a,b)$ is used. Previous
+releases of Isabelle also included a different version of~\HOL, with
+explicit type inference rules~\cite{paulson-COLOG}. This version no
+longer exists, but \thydx{ZF} supports a similar style of reasoning.
+
+\CHOL\ has a distinct feel, compared with {\ZF} and {\CTT}. It
+identifies object-level types with meta-level types, taking advantage of
+Isabelle's built-in type checker. It identifies object-level functions
+with meta-level functions, so it uses Isabelle's operations for abstraction
+and application. There is no `apply' operator: function applications are
+written as simply~$f~a$ rather than $f{\tt`}a$.
+
+These identifications allow Isabelle to support \CHOL\ particularly nicely,
+but they also mean that \CHOL\ requires more sophistication from the user
+--- in particular, an understanding of Isabelle's type system. Beginners
+should work with {\tt show_types} set to {\tt true}. Gain experience by
+working in first-order logic before attempting to use higher-order logic.
+This chapter assumes familiarity with~{\FOL{}}.
+
+
+\begin{figure}
+\begin{center}
+\begin{tabular}{rrr}
+ \it name &\it meta-type & \it description \\
+ \cdx{Trueprop}& $bool\To prop$ & coercion to $prop$\\
+ \cdx{not} & $bool\To bool$ & negation ($\neg$) \\
+ \cdx{True} & $bool$ & tautology ($\top$) \\
+ \cdx{False} & $bool$ & absurdity ($\bot$) \\
+ \cdx{If} & $[bool,\alpha,\alpha]\To\alpha::term$ & conditional \\
+ \cdx{Inv} & $(\alpha\To\beta)\To(\beta\To\alpha)$ & function inversion\\
+ \cdx{Let} & $[\alpha,\alpha\To\beta]\To\beta$ & let binder
+\end{tabular}
+\end{center}
+\subcaption{Constants}
+
+\begin{center}
+\index{"@@{\tt\at} symbol}
+\index{*"! symbol}\index{*"? symbol}
+\index{*"?"! symbol}\index{*"E"X"! symbol}
+\begin{tabular}{llrrr}
+ \it symbol &\it name &\it meta-type & \it description \\
+ \tt\at & \cdx{Eps} & $(\alpha\To bool)\To\alpha::term$ &
+ Hilbert description ($\epsilon$) \\
+ {\tt!~} or \sdx{ALL} & \cdx{All} & $(\alpha::term\To bool)\To bool$ &
+ universal quantifier ($\forall$) \\
+ {\tt?~} or \sdx{EX} & \cdx{Ex} & $(\alpha::term\To bool)\To bool$ &
+ existential quantifier ($\exists$) \\
+ {\tt?!} or {\tt EX!} & \cdx{Ex1} & $(\alpha::term\To bool)\To bool$ &
+ unique existence ($\exists!$)
+\end{tabular}
+\end{center}
+\subcaption{Binders}
+
+\begin{center}
+\index{*"= symbol}
+\index{&@{\tt\&} symbol}
+\index{*"| symbol}
+\index{*"-"-"> symbol}
+\begin{tabular}{rrrr}
+ \it symbol & \it meta-type & \it priority & \it description \\
+ \sdx{o} & $[\beta\To\gamma,\alpha\To\beta]\To (\alpha\To\gamma)$ &
+ Right 50 & composition ($\circ$) \\
+ \tt = & $[\alpha::term,\alpha]\To bool$ & Left 50 & equality ($=$) \\
+ \tt < & $[\alpha::ord,\alpha]\To bool$ & Left 50 & less than ($<$) \\
+ \tt <= & $[\alpha::ord,\alpha]\To bool$ & Left 50 &
+ less than or equals ($\leq$)\\
+ \tt \& & $[bool,bool]\To bool$ & Right 35 & conjunction ($\conj$) \\
+ \tt | & $[bool,bool]\To bool$ & Right 30 & disjunction ($\disj$) \\
+ \tt --> & $[bool,bool]\To bool$ & Right 25 & implication ($\imp$)
+\end{tabular}
+\end{center}
+\subcaption{Infixes}
+\caption{Syntax of {\tt HOL}} \label{chol-constants}
+\end{figure}
+
+
+\begin{figure}
+\index{*let symbol}
+\index{*in symbol}
+\dquotes
+\[\begin{array}{rclcl}
+ term & = & \hbox{expression of class~$term$} \\
+ & | & "\at~" id~id^* " . " formula \\
+ & | &
+ \multicolumn{3}{l}{"let"~id~"="~term";"\dots";"~id~"="~term~"in"~term} \\
+ & | &
+ \multicolumn{3}{l}{"if"~formula~"then"~term~"else"~term} \\[2ex]
+ formula & = & \hbox{expression of type~$bool$} \\
+ & | & term " = " term \\
+ & | & term " \ttilde= " term \\
+ & | & term " < " term \\
+ & | & term " <= " term \\
+ & | & "\ttilde\ " formula \\
+ & | & formula " \& " formula \\
+ & | & formula " | " formula \\
+ & | & formula " --> " formula \\
+ & | & "!~~~" id~id^* " . " formula
+ & | & "ALL~" id~id^* " . " formula \\
+ & | & "?~~~" id~id^* " . " formula
+ & | & "EX~~" id~id^* " . " formula \\
+ & | & "?!~~" id~id^* " . " formula
+ & | & "EX!~" id~id^* " . " formula
+ \end{array}
+\]
+\caption{Full grammar for \CHOL} \label{chol-grammar}
+\end{figure}
+
+
+\section{Syntax}
+The type class of higher-order terms is called~\cldx{term}. Type variables
+range over this class by default. The equality symbol and quantifiers are
+polymorphic over class {\tt term}.
+
+Class \cldx{ord} consists of all ordered types; the relations $<$ and
+$\leq$ are polymorphic over this class, as are the functions
+\cdx{mono}, \cdx{min} and \cdx{max}. Three other
+type classes --- \cldx{plus}, \cldx{minus} and \cldx{times} --- permit
+overloading of the operators {\tt+}, {\tt-} and {\tt*}. In particular,
+{\tt-} is overloaded for set difference and subtraction.
+\index{*"+ symbol}
+\index{*"- symbol}
+\index{*"* symbol}
+
+Figure~\ref{chol-constants} lists the constants (including infixes and
+binders), while Fig.\ts\ref{chol-grammar} presents the grammar of
+higher-order logic. Note that $a$\verb|~=|$b$ is translated to
+$\neg(a=b)$.
+
+\begin{warn}
+ \CHOL\ has no if-and-only-if connective; logical equivalence is expressed
+ using equality. But equality has a high priority, as befitting a
+ relation, while if-and-only-if typically has the lowest priority. Thus,
+ $\neg\neg P=P$ abbreviates $\neg\neg (P=P)$ and not $(\neg\neg P)=P$.
+ When using $=$ to mean logical equivalence, enclose both operands in
+ parentheses.
+\end{warn}
+
+\subsection{Types}\label{CHOL-types}
+The type of formulae, \tydx{bool}, belongs to class \cldx{term}; thus,
+formulae are terms. The built-in type~\tydx{fun}, which constructs function
+types, is overloaded with arity {\tt(term,term)term}. Thus, $\sigma\To\tau$
+belongs to class~{\tt term} if $\sigma$ and~$\tau$ do, allowing quantification
+over functions.
+
+Types in \CHOL\ must be non-empty; otherwise the quantifier rules would be
+unsound. I have commented on this elsewhere~\cite[\S7]{paulson-COLOG}.
+
+\index{type definitions}
+Gordon's {\sc hol} system supports {\bf type definitions}. A type is
+defined by exhibiting an existing type~$\sigma$, a predicate~$P::\sigma\To
+bool$, and a theorem of the form $\exists x::\sigma.P~x$. Thus~$P$
+specifies a non-empty subset of~$\sigma$, and the new type denotes this
+subset. New function constants are generated to establish an isomorphism
+between the new type and the subset. If type~$\sigma$ involves type
+variables $\alpha@1$, \ldots, $\alpha@n$, then the type definition creates
+a type constructor $(\alpha@1,\ldots,\alpha@n)ty$ rather than a particular
+type. Melham~\cite{melham89} discusses type definitions at length, with
+examples.
+
+Isabelle does not support type definitions at present. Instead, they are
+mimicked by explicit definitions of isomorphism functions. The definitions
+should be supported by theorems of the form $\exists x::\sigma.P~x$, but
+Isabelle cannot enforce this.
+
+
+\subsection{Binders}
+Hilbert's {\bf description} operator~$\epsilon x.P[x]$ stands for some~$a$
+satisfying~$P[a]$, if such exists. Since all terms in \CHOL\ denote
+something, a description is always meaningful, but we do not know its value
+unless $P[x]$ defines it uniquely. We may write descriptions as
+\cdx{Eps}($P$) or use the syntax
+\hbox{\tt \at $x$.$P[x]$}.
+
+Existential quantification is defined by
+\[ \exists x.P~x \;\equiv\; P(\epsilon x.P~x). \]
+The unique existence quantifier, $\exists!x.P[x]$, is defined in terms
+of~$\exists$ and~$\forall$. An Isabelle binder, it admits nested
+quantifications. For instance, $\exists!x y.P~x~y$ abbreviates
+$\exists!x. \exists!y.P~x~y$; note that this does not mean that there
+exists a unique pair $(x,y)$ satisfying~$P~x~y$.
+
+\index{*"! symbol}\index{*"? symbol}\index{CHOL system@{\sc hol} system}
+Quantifiers have two notations. As in Gordon's {\sc hol} system, \CHOL\
+uses~{\tt!}\ and~{\tt?}\ to stand for $\forall$ and $\exists$. The
+existential quantifier must be followed by a space; thus {\tt?x} is an
+unknown, while \verb'? x.f x=y' is a quantification. Isabelle's usual
+notation for quantifiers, \sdx{ALL} and \sdx{EX}, is also
+available. Both notations are accepted for input. The {\ML} reference
+\ttindexbold{HOL_quantifiers} governs the output notation. If set to {\tt
+true}, then~{\tt!}\ and~{\tt?}\ are displayed; this is the default. If set
+to {\tt false}, then~{\tt ALL} and~{\tt EX} are displayed.
+
+All these binders have priority 10.
+
+
+\subsection{The \sdx{let} and \sdx{case} constructions}
+Local abbreviations can be introduced by a {\tt let} construct whose
+syntax appears in Fig.\ts\ref{chol-grammar}. Internally it is translated into
+the constant~\cdx{Let}. It can be expanded by rewriting with its
+definition, \tdx{Let_def}.
+
+\CHOL\ also defines the basic syntax
+\[\dquotes"case"~e~"of"~c@1~"=>"~e@1~"|" \dots "|"~c@n~"=>"~e@n\]
+as a uniform means of expressing {\tt case} constructs. Therefore {\tt
+ case} and \sdx{of} are reserved words. However, so far this is mere
+syntax and has no logical meaning. By declaring translations, you can
+cause instances of the {\tt case} construct to denote applications of
+particular case operators. The patterns supplied for $c@1$,~\ldots,~$c@n$
+distinguish among the different case operators. For an example, see the
+case construct for lists on page~\pageref{chol-list} below.
+
+\begin{figure}
+\begin{ttbox}\makeatother
+\tdx{refl} t = (t::'a)
+\tdx{subst} [| s=t; P s |] ==> P(t::'a)
+\tdx{ext} (!!x::'a. (f x::'b) = g x) ==> (\%x.f x) = (\%x.g x)
+\tdx{impI} (P ==> Q) ==> P-->Q
+\tdx{mp} [| P-->Q; P |] ==> Q
+\tdx{iff} (P-->Q) --> (Q-->P) --> (P=Q)
+\tdx{selectI} P(x::'a) ==> P(@x.P x)
+\tdx{True_or_False} (P=True) | (P=False)
+\end{ttbox}
+\caption{The {\tt CHOL} rules} \label{chol-rules}
+\end{figure}
+
+
+\begin{figure}\hfuzz=4pt%suppress "Overfull \hbox" message
+\begin{ttbox}\makeatother
+\tdx{True_def} True == ((\%x::bool.x)=(\%x.x))
+\tdx{All_def} All == (\%P. P = (\%x.True))
+\tdx{Ex_def} Ex == (\%P. P(@x.P x))
+\tdx{False_def} False == (!P.P)
+\tdx{not_def} not == (\%P. P-->False)
+\tdx{and_def} op & == (\%P Q. !R. (P-->Q-->R) --> R)
+\tdx{or_def} op | == (\%P Q. !R. (P-->R) --> (Q-->R) --> R)
+\tdx{Ex1_def} Ex1 == (\%P. ? x. P x & (! y. P y --> y=x))
+
+\tdx{Inv_def} Inv == (\%(f::'a=>'b) y. @x. f x=y)
+\tdx{o_def} op o == (\%(f::'b=>'c) g (x::'a). f(g x))
+\tdx{if_def} If P x y == (\%P x y.@z::'a.(P=True --> z=x) & (P=False --> z=y))
+\tdx{Let_def} Let s f == f s
+\end{ttbox}
+\caption{The {\tt CHOL} definitions} \label{chol-defs}
+\end{figure}
+
+
+\section{Rules of inference}
+Figure~\ref{chol-rules} shows the inference rules of~\CHOL{}, with
+their~{\ML} names. Some of the rules deserve additional comments:
+\begin{ttdescription}
+\item[\tdx{ext}] expresses extensionality of functions.
+\item[\tdx{iff}] asserts that logically equivalent formulae are
+ equal.
+\item[\tdx{selectI}] gives the defining property of the Hilbert
+ $\epsilon$-operator. It is a form of the Axiom of Choice. The derived rule
+ \tdx{select_equality} (see below) is often easier to use.
+\item[\tdx{True_or_False}] makes the logic classical.\footnote{In
+ fact, the $\epsilon$-operator already makes the logic classical, as
+ shown by Diaconescu; see Paulson~\cite{paulson-COLOG} for details.}
+\end{ttdescription}
+
+\CHOL{} follows standard practice in higher-order logic: only a few
+connectives are taken as primitive, with the remainder defined obscurely
+(Fig.\ts\ref{chol-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~\CHOL{}, like all other Isabelle theories, uses
+meta-equality~({\tt==}) for definitions.
+
+Some of the rules mention type variables; for example, {\tt refl}
+mentions the type variable~{\tt'a}. This allows you to instantiate
+type variables explicitly by calling {\tt res_inst_tac}. By default,
+explicit type variables have class \cldx{term}.
+
+Include type constraints whenever you state a polymorphic goal. Type
+inference may otherwise make the goal more polymorphic than you intended,
+with confusing results.
+
+\begin{warn}
+ If resolution fails for no obvious reason, try setting
+ \ttindex{show_types} to {\tt true}, causing Isabelle to display types of
+ terms. Possibly set \ttindex{show_sorts} to {\tt true} as well, causing
+ Isabelle to display sorts.
+
+ \index{unification!incompleteness of}
+ Where function types are involved, Isabelle's unification code does not
+ guarantee to find instantiations for type variables automatically. Be
+ prepared to use \ttindex{res_inst_tac} instead of {\tt resolve_tac},
+ possibly instantiating type variables. Setting
+ \ttindex{Unify.trace_types} to {\tt true} causes Isabelle to report
+ omitted search paths during unification.\index{tracing!of unification}
+\end{warn}
+
+
+\begin{figure}
+\begin{ttbox}
+\tdx{sym} s=t ==> t=s
+\tdx{trans} [| r=s; s=t |] ==> r=t
+\tdx{ssubst} [| t=s; P s |] ==> P(t::'a)
+\tdx{box_equals} [| a=b; a=c; b=d |] ==> c=d
+\tdx{arg_cong} x=y ==> f x=f y
+\tdx{fun_cong} f=g ==> f x=g x
+\subcaption{Equality}
+
+\tdx{TrueI} True
+\tdx{FalseE} False ==> P
+
+\tdx{conjI} [| P; Q |] ==> P&Q
+\tdx{conjunct1} [| P&Q |] ==> P
+\tdx{conjunct2} [| P&Q |] ==> Q
+\tdx{conjE} [| P&Q; [| P; Q |] ==> R |] ==> R
+
+\tdx{disjI1} P ==> P|Q
+\tdx{disjI2} Q ==> P|Q
+\tdx{disjE} [| P | Q; P ==> R; Q ==> R |] ==> R
+
+\tdx{notI} (P ==> False) ==> ~ P
+\tdx{notE} [| ~ P; P |] ==> R
+\tdx{impE} [| P-->Q; P; Q ==> R |] ==> R
+\subcaption{Propositional logic}
+
+\tdx{iffI} [| P ==> Q; Q ==> P |] ==> P=Q
+\tdx{iffD1} [| P=Q; P |] ==> Q
+\tdx{iffD2} [| P=Q; Q |] ==> P
+\tdx{iffE} [| P=Q; [| P --> Q; Q --> P |] ==> R |] ==> R
+
+\tdx{eqTrueI} P ==> P=True
+\tdx{eqTrueE} P=True ==> P
+\subcaption{Logical equivalence}
+
+\end{ttbox}
+\caption{Derived rules for \CHOL} \label{chol-lemmas1}
+\end{figure}
+
+
+\begin{figure}
+\begin{ttbox}\makeatother
+\tdx{allI} (!!x::'a. P x) ==> !x. P x
+\tdx{spec} !x::'a.P x ==> P x
+\tdx{allE} [| !x.P x; P x ==> R |] ==> R
+\tdx{all_dupE} [| !x.P x; [| P x; !x.P x |] ==> R |] ==> R
+
+\tdx{exI} P x ==> ? x::'a.P x
+\tdx{exE} [| ? x::'a.P x; !!x. P x ==> Q |] ==> Q
+
+\tdx{ex1I} [| P a; !!x. P x ==> x=a |] ==> ?! x. P x
+\tdx{ex1E} [| ?! x.P x; !!x. [| P x; ! y. P y --> y=x |] ==> R
+ |] ==> R
+
+\tdx{select_equality} [| P a; !!x. P x ==> x=a |] ==> (@x.P x) = a
+\subcaption{Quantifiers and descriptions}
+
+\tdx{ccontr} (~P ==> False) ==> P
+\tdx{classical} (~P ==> P) ==> P
+\tdx{excluded_middle} ~P | P
+
+\tdx{disjCI} (~Q ==> P) ==> P|Q
+\tdx{exCI} (! x. ~ P x ==> P a) ==> ? x.P x
+\tdx{impCE} [| P-->Q; ~ P ==> R; Q ==> R |] ==> R
+\tdx{iffCE} [| P=Q; [| P;Q |] ==> R; [| ~P; ~Q |] ==> R |] ==> R
+\tdx{notnotD} ~~P ==> P
+\tdx{swap} ~P ==> (~Q ==> P) ==> Q
+\subcaption{Classical logic}
+
+\tdx{if_True} if True 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{expand_if} P(if Q then x else y) = ((Q --> P x) & (~Q --> P y))
+\subcaption{Conditionals}
+\end{ttbox}
+\caption{More derived rules} \label{chol-lemmas2}
+\end{figure}
+
+
+Some derived rules are shown in Figures~\ref{chol-lemmas1}
+and~\ref{chol-lemmas2}, with their {\ML} names. These include natural rules
+for the logical connectives, as well as sequent-style elimination rules for
+conjunctions, implications, and universal quantifiers.
+
+Note the equality rules: \tdx{ssubst} performs substitution in
+backward proofs, while \tdx{box_equals} supports reasoning by
+simplifying both sides of an equation.
+
+
+\begin{figure}
+\begin{center}
+\begin{tabular}{rrr}
+ \it name &\it meta-type & \it description \\
+\index{{}@\verb'{}' symbol}
+ \verb|{}| & $\alpha\,set$ & the empty set \\
+ \cdx{insert} & $[\alpha,\alpha\,set]\To \alpha\,set$
+ & insertion of element \\
+ \cdx{Collect} & $(\alpha\To bool)\To\alpha\,set$
+ & comprehension \\
+ \cdx{Compl} & $(\alpha\,set)\To\alpha\,set$
+ & complement \\
+ \cdx{INTER} & $[\alpha\,set,\alpha\To\beta\,set]\To\beta\,set$
+ & intersection over a set\\
+ \cdx{UNION} & $[\alpha\,set,\alpha\To\beta\,set]\To\beta\,set$
+ & union over a set\\
+ \cdx{Inter} & $(\alpha\,set)set\To\alpha\,set$
+ &set of sets intersection \\
+ \cdx{Union} & $(\alpha\,set)set\To\alpha\,set$
+ &set of sets union \\
+ \cdx{Pow} & $\alpha\,set \To (\alpha\,set)set$
+ & powerset \\[1ex]
+ \cdx{range} & $(\alpha\To\beta )\To\beta\,set$
+ & range of a function \\[1ex]
+ \cdx{Ball}~~\cdx{Bex} & $[\alpha\,set,\alpha\To bool]\To bool$
+ & bounded quantifiers \\
+ \cdx{mono} & $(\alpha\,set\To\beta\,set)\To bool$
+ & monotonicity \\
+ \cdx{inj}~~\cdx{surj}& $(\alpha\To\beta )\To bool$
+ & injective/surjective \\
+ \cdx{inj_onto} & $[\alpha\To\beta ,\alpha\,set]\To bool$
+ & injective over subset
+\end{tabular}
+\end{center}
+\subcaption{Constants}
+
+\begin{center}
+\begin{tabular}{llrrr}
+ \it symbol &\it name &\it meta-type & \it priority & \it description \\
+ \sdx{INT} & \cdx{INTER1} & $(\alpha\To\beta\,set)\To\beta\,set$ & 10 &
+ intersection over a type\\
+ \sdx{UN} & \cdx{UNION1} & $(\alpha\To\beta\,set)\To\beta\,set$ & 10 &
+ union over a type
+\end{tabular}
+\end{center}
+\subcaption{Binders}
+
+\begin{center}
+\index{*"`"` symbol}
+\index{*": symbol}
+\index{*"<"= symbol}
+\begin{tabular}{rrrr}
+ \it symbol & \it meta-type & \it priority & \it description \\
+ \tt `` & $[\alpha\To\beta ,\alpha\,set]\To (\beta\,set)$
+ & Left 90 & image \\
+ \sdx{Int} & $[\alpha\,set,\alpha\,set]\To\alpha\,set$
+ & Left 70 & intersection ($\inter$) \\
+ \sdx{Un} & $[\alpha\,set,\alpha\,set]\To\alpha\,set$
+ & Left 65 & union ($\union$) \\
+ \tt: & $[\alpha ,\alpha\,set]\To bool$
+ & Left 50 & membership ($\in$) \\
+ \tt <= & $[\alpha\,set,\alpha\,set]\To bool$
+ & Left 50 & subset ($\subseteq$)
+\end{tabular}
+\end{center}
+\subcaption{Infixes}
+\caption{Syntax of the theory {\tt Set}} \label{chol-set-syntax}
+\end{figure}
+
+
+\begin{figure}
+\begin{center} \tt\frenchspacing
+\index{*"! symbol}
+\begin{tabular}{rrr}
+ \it external & \it internal & \it description \\
+ $a$ \ttilde: $b$ & \ttilde($a$ : $b$) & \rm non-membership\\
+ \{$a@1$, $\ldots$\} & insert $a@1$ $\ldots$ \{\} & \rm finite set \\
+ \{$x$.$P[x]$\} & Collect($\lambda x.P[x]$) &
+ \rm comprehension \\
+ \sdx{INT} $x$:$A$.$B[x]$ & INTER $A$ $\lambda x.B[x]$ &
+ \rm intersection \\
+ \sdx{UN}{\tt\ } $x$:$A$.$B[x]$ & UNION $A$ $\lambda x.B[x]$ &
+ \rm union \\
+ \tt ! $x$:$A$.$P[x]$ or \sdx{ALL} $x$:$A$.$P[x]$ &
+ Ball $A$ $\lambda x.P[x]$ &
+ \rm bounded $\forall$ \\
+ \sdx{?} $x$:$A$.$P[x]$ or \sdx{EX}{\tt\ } $x$:$A$.$P[x]$ &
+ Bex $A$ $\lambda x.P[x]$ & \rm bounded $\exists$
+\end{tabular}
+\end{center}
+\subcaption{Translations}
+
+\dquotes
+\[\begin{array}{rclcl}
+ term & = & \hbox{other terms\ldots} \\
+ & | & "\{\}" \\
+ & | & "\{ " term\; ("," term)^* " \}" \\
+ & | & "\{ " id " . " formula " \}" \\
+ & | & term " `` " term \\
+ & | & term " Int " term \\
+ & | & term " Un " term \\
+ & | & "INT~~" id ":" term " . " term \\
+ & | & "UN~~~" id ":" term " . " term \\
+ & | & "INT~~" id~id^* " . " term \\
+ & | & "UN~~~" id~id^* " . " term \\[2ex]
+ formula & = & \hbox{other formulae\ldots} \\
+ & | & term " : " term \\
+ & | & term " \ttilde: " term \\
+ & | & term " <= " term \\
+ & | & "!~" id ":" term " . " formula
+ & | & "ALL " id ":" term " . " formula \\
+ & | & "?~" id ":" term " . " formula
+ & | & "EX~~" id ":" term " . " formula
+ \end{array}
+\]
+\subcaption{Full Grammar}
+\caption{Syntax of the theory {\tt Set} (continued)} \label{chol-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 \CHOL\ as they
+do in~{\ZF}. In \CHOL\ 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}
+\CHOL's set theory is called \thydx{Set}. The type $\alpha\,set$ is
+essentially the same as $\alpha\To bool$. The new type is defined for
+clarity and to avoid complications involving function types in unification.
+Since Isabelle does not support type definitions (as mentioned in
+\S\ref{CHOL-types}), the isomorphisms between the two types are declared
+explicitly. Here they are natural: {\tt Collect} maps $\alpha\To bool$ to
+$\alpha\,set$, while \hbox{\tt op :} maps in the other direction (ignoring
+argument order).
+
+Figure~\ref{chol-set-syntax} lists the constants, infixes, and syntax
+translations. Figure~\ref{chol-set-syntax2} presents the grammar of the new
+constructs. Infix operators include union and intersection ($A\union B$
+and $A\inter B$), the subset and membership relations, and the image
+operator~{\tt``}\@. Note that $a$\verb|~:|$b$ is translated to
+$\neg(a\in b)$.
+
+The {\tt\{\ldots\}} notation abbreviates finite sets constructed in the
+obvious manner using~{\tt insert} and~$\{\}$:
+\begin{eqnarray*}
+ \{a@1, \ldots, a@n\} & \equiv &
+ {\tt insert}~a@1~({\tt insert}\ldots({\tt insert}~a@n~\{\})\ldots)
+\end{eqnarray*}
+
+The set \hbox{\tt\{$x$.$P[x]$\}} consists of all $x$ (of suitable type)
+that satisfy~$P[x]$, where $P[x]$ is a formula that may contain free
+occurrences of~$x$. This syntax expands to \cdx{Collect}$(\lambda
+x.P[x])$. It defines sets by absolute comprehension, which is impossible
+in~{\ZF}; the type of~$x$ implicitly restricts the comprehension.
+
+The set theory defines two {\bf bounded quantifiers}:
+\begin{eqnarray*}
+ \forall x\in A.P[x] &\hbox{abbreviates}& \forall x. x\in A\imp P[x] \\
+ \exists x\in A.P[x] &\hbox{abbreviates}& \exists x. x\in A\conj P[x]
+\end{eqnarray*}
+The constants~\cdx{Ball} and~\cdx{Bex} are defined
+accordingly. Instead of {\tt Ball $A$ $P$} and {\tt Bex $A$ $P$} we may
+write\index{*"! symbol}\index{*"? symbol}
+\index{*ALL symbol}\index{*EX symbol}
+%
+\hbox{\tt !~$x$:$A$.$P[x]$} and \hbox{\tt ?~$x$:$A$.$P[x]$}. Isabelle's
+usual quantifier symbols, \sdx{ALL} and \sdx{EX}, are also accepted
+for input. As with the primitive quantifiers, the {\ML} reference
+\ttindex{HOL_quantifiers} specifies which notation to use for output.
+
+Unions and intersections over sets, namely $\bigcup@{x\in A}B[x]$ and
+$\bigcap@{x\in A}B[x]$, are written
+\sdx{UN}~\hbox{\tt$x$:$A$.$B[x]$} and
+\sdx{INT}~\hbox{\tt$x$:$A$.$B[x]$}.
+
+Unions and intersections over types, namely $\bigcup@x B[x]$ and $\bigcap@x
+B[x]$, are written \sdx{UN}~\hbox{\tt$x$.$B[x]$} and
+\sdx{INT}~\hbox{\tt$x$.$B[x]$}. They are equivalent to the previous
+union and intersection operators when $A$ is the universal set.
+
+The operators $\bigcup A$ and $\bigcap A$ act upon sets of sets. They are
+not binders, but are equal to $\bigcup@{x\in A}x$ and $\bigcap@{x\in A}x$,
+respectively.
+
+
+\begin{figure} \underscoreon
+\begin{ttbox}
+\tdx{mem_Collect_eq} (a : \{x.P x\}) = P a
+\tdx{Collect_mem_eq} \{x.x:A\} = A
+
+\tdx{empty_def} \{\} == \{x.False\}
+\tdx{insert_def} insert a B == \{x.x=a\} Un B
+\tdx{Ball_def} Ball A P == ! x. x:A --> P x
+\tdx{Bex_def} Bex A P == ? x. x:A & P x
+\tdx{subset_def} A <= B == ! x:A. x:B
+\tdx{Un_def} A Un B == \{x.x:A | x:B\}
+\tdx{Int_def} A Int B == \{x.x:A & x:B\}
+\tdx{set_diff_def} A - B == \{x.x:A & x~:B\}
+\tdx{Compl_def} Compl A == \{x. ~ x:A\}
+\tdx{INTER_def} INTER A B == \{y. ! x:A. y: B x\}
+\tdx{UNION_def} UNION A B == \{y. ? x:A. y: B x\}
+\tdx{INTER1_def} INTER1 B == INTER \{x.True\} B
+\tdx{UNION1_def} UNION1 B == UNION \{x.True\} B
+\tdx{Inter_def} Inter S == (INT x:S. x)
+\tdx{Union_def} Union S == (UN x:S. x)
+\tdx{Pow_def} Pow A == \{B. B <= A\}
+\tdx{image_def} f``A == \{y. ? x:A. y=f x\}
+\tdx{range_def} range f == \{y. ? x. y=f x\}
+\tdx{mono_def} mono f == !A B. A <= B --> f A <= f B
+\tdx{inj_def} inj f == ! x y. f x=f y --> x=y
+\tdx{surj_def} surj f == ! y. ? x. y=f x
+\tdx{inj_onto_def} inj_onto f A == !x:A. !y:A. f x=f y --> x=y
+\end{ttbox}
+\caption{Rules of the theory {\tt Set}} \label{chol-set-rules}
+\end{figure}
+
+
+\begin{figure} \underscoreon
+\begin{ttbox}
+\tdx{CollectI} [| P a |] ==> a : \{x.P x\}
+\tdx{CollectD} [| a : \{x.P x\} |] ==> P a
+\tdx{CollectE} [| a : \{x.P x\}; P a ==> W |] ==> W
+
+\tdx{ballI} [| !!x. x:A ==> P x |] ==> ! x:A. P x
+\tdx{bspec} [| ! x:A. P x; x:A |] ==> P x
+\tdx{ballE} [| ! x:A. P x; P x ==> Q; ~ x:A ==> Q |] ==> Q
+
+\tdx{bexI} [| P x; x:A |] ==> ? x:A. P x
+\tdx{bexCI} [| ! x:A. ~ P x ==> P a; a:A |] ==> ? x:A.P x
+\tdx{bexE} [| ? x:A. P x; !!x. [| x:A; P x |] ==> Q |] ==> Q
+\subcaption{Comprehension and Bounded quantifiers}
+
+\tdx{subsetI} (!!x.x:A ==> x:B) ==> A <= B
+\tdx{subsetD} [| A <= B; c:A |] ==> c:B
+\tdx{subsetCE} [| A <= B; ~ (c:A) ==> P; c:B ==> P |] ==> P
+
+\tdx{subset_refl} A <= A
+\tdx{subset_trans} [| A<=B; B<=C |] ==> A<=C
+
+\tdx{equalityI} [| A <= B; B <= A |] ==> A = B
+\tdx{equalityD1} A = B ==> A<=B
+\tdx{equalityD2} A = B ==> B<=A
+\tdx{equalityE} [| A = B; [| A<=B; B<=A |] ==> P |] ==> P
+
+\tdx{equalityCE} [| A = B; [| c:A; c:B |] ==> P;
+ [| ~ c:A; ~ c:B |] ==> P
+ |] ==> P
+\subcaption{The subset and equality relations}
+\end{ttbox}
+\caption{Derived rules for set theory} \label{chol-set1}
+\end{figure}
+
+
+\begin{figure} \underscoreon
+\begin{ttbox}
+\tdx{emptyE} a : \{\} ==> P
+
+\tdx{insertI1} a : insert a B
+\tdx{insertI2} a : B ==> a : insert b B
+\tdx{insertE} [| a : insert b A; a=b ==> P; a:A ==> P |] ==> P
+
+\tdx{ComplI} [| c:A ==> False |] ==> c : Compl A
+\tdx{ComplD} [| c : Compl A |] ==> ~ c:A
+
+\tdx{UnI1} c:A ==> c : A Un B
+\tdx{UnI2} c:B ==> c : A Un B
+\tdx{UnCI} (~c:B ==> c:A) ==> c : A Un B
+\tdx{UnE} [| c : A Un B; c:A ==> P; c:B ==> P |] ==> P
+
+\tdx{IntI} [| c:A; c:B |] ==> c : A Int B
+\tdx{IntD1} c : A Int B ==> c:A
+\tdx{IntD2} c : A Int B ==> c:B
+\tdx{IntE} [| c : A Int B; [| c:A; c:B |] ==> P |] ==> P
+
+\tdx{UN_I} [| a:A; b: B a |] ==> b: (UN x:A. B x)
+\tdx{UN_E} [| b: (UN x:A. B x); !!x.[| x:A; b:B x |] ==> R |] ==> R
+
+\tdx{INT_I} (!!x. x:A ==> b: B x) ==> b : (INT x:A. B x)
+\tdx{INT_D} [| b: (INT x:A. B x); a:A |] ==> b: B a
+\tdx{INT_E} [| b: (INT x:A. B x); b: B a ==> R; ~ a:A ==> R |] ==> R
+
+\tdx{UnionI} [| X:C; A:X |] ==> A : Union C
+\tdx{UnionE} [| A : Union C; !!X.[| A:X; X:C |] ==> R |] ==> R
+
+\tdx{InterI} [| !!X. X:C ==> A:X |] ==> A : Inter C
+\tdx{InterD} [| A : Inter C; X:C |] ==> A:X
+\tdx{InterE} [| A : Inter C; A:X ==> R; ~ X:C ==> R |] ==> R
+
+\tdx{PowI} A<=B ==> A: Pow B
+\tdx{PowD} A: Pow B ==> A<=B
+\end{ttbox}
+\caption{Further derived rules for set theory} \label{chol-set2}
+\end{figure}
+
+
+\subsection{Axioms and rules of set theory}
+Figure~\ref{chol-set-rules} presents the rules of theory \thydx{Set}. The
+axioms \tdx{mem_Collect_eq} and \tdx{Collect_mem_eq} assert
+that the functions {\tt Collect} and \hbox{\tt op :} are isomorphisms. Of
+course, \hbox{\tt op :} also serves as the membership relation.
+
+All the other axioms are definitions. They include the empty set, bounded
+quantifiers, unions, intersections, complements and the subset relation.
+They also include straightforward properties of functions: image~({\tt``}) and
+{\tt range}, and predicates concerning monotonicity, injectiveness and
+surjectiveness.
+
+The predicate \cdx{inj_onto} is used for simulating type definitions.
+The statement ${\tt inj_onto}~f~A$ asserts that $f$ is injective on the
+set~$A$, which specifies a subset of its domain type. In a type
+definition, $f$ is the abstraction function and $A$ is the set of valid
+representations; we should not expect $f$ to be injective outside of~$A$.
+
+\begin{figure} \underscoreon
+\begin{ttbox}
+\tdx{Inv_f_f} inj f ==> Inv f (f x) = x
+\tdx{f_Inv_f} y : range f ==> f(Inv f y) = y
+
+%\tdx{Inv_injective}
+% [| Inv f x=Inv f y; x: range f; y: range f |] ==> x=y
+%
+\tdx{imageI} [| x:A |] ==> f x : f``A
+\tdx{imageE} [| b : f``A; !!x.[| b=f x; x:A |] ==> P |] ==> P
+
+\tdx{rangeI} f x : range f
+\tdx{rangeE} [| b : range f; !!x.[| b=f x |] ==> P |] ==> P
+
+\tdx{monoI} [| !!A B. A <= B ==> f A <= f B |] ==> mono f
+\tdx{monoD} [| mono f; A <= B |] ==> f A <= f B
+
+\tdx{injI} [| !! x y. f x = f y ==> x=y |] ==> inj f
+\tdx{inj_inverseI} (!!x. g(f x) = x) ==> inj f
+\tdx{injD} [| inj f; f x = f y |] ==> x=y
+
+\tdx{inj_ontoI} (!!x y. [| f x=f y; x:A; y:A |] ==> x=y) ==> inj_onto f A
+\tdx{inj_ontoD} [| inj_onto f A; f x=f y; x:A; y:A |] ==> x=y
+
+\tdx{inj_onto_inverseI}
+ (!!x. x:A ==> g(f x) = x) ==> inj_onto f A
+\tdx{inj_onto_contraD}
+ [| inj_onto f A; x~=y; x:A; y:A |] ==> ~ f x=f y
+\end{ttbox}
+\caption{Derived rules involving functions} \label{chol-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{chol-subset}
+\end{figure}
+
+
+\begin{figure} \underscoreon \hfuzz=4pt%suppress "Overfull \hbox" message
+\begin{ttbox}
+\tdx{Int_absorb} A Int A = A
+\tdx{Int_commute} A Int B = B Int A
+\tdx{Int_assoc} (A Int B) Int C = A Int (B Int C)
+\tdx{Int_Un_distrib} (A Un B) Int C = (A Int C) Un (B Int C)
+
+\tdx{Un_absorb} A Un A = A
+\tdx{Un_commute} A Un B = B Un A
+\tdx{Un_assoc} (A Un B) Un C = A Un (B Un C)
+\tdx{Un_Int_distrib} (A Int B) Un C = (A Un C) Int (B Un C)
+
+\tdx{Compl_disjoint} A Int (Compl A) = \{x.False\}
+\tdx{Compl_partition} A Un (Compl A) = \{x.True\}
+\tdx{double_complement} Compl(Compl A) = A
+\tdx{Compl_Un} Compl(A Un B) = (Compl A) Int (Compl B)
+\tdx{Compl_Int} Compl(A Int B) = (Compl A) Un (Compl B)
+
+\tdx{Union_Un_distrib} Union(A Un B) = (Union A) Un (Union B)
+\tdx{Int_Union} A Int (Union B) = (UN C:B. A Int C)
+\tdx{Un_Union_image} (UN x:C.(A x) Un (B x)) = Union(A``C) Un Union(B``C)
+
+\tdx{Inter_Un_distrib} Inter(A Un B) = (Inter A) Int (Inter B)
+\tdx{Un_Inter} A Un (Inter B) = (INT C:B. A Un C)
+\tdx{Int_Inter_image} (INT x:C.(A x) Int (B x)) = Inter(A``C) Int Inter(B``C)
+\end{ttbox}
+\caption{Set equalities} \label{chol-equalities}
+\end{figure}
+
+
+Figures~\ref{chol-set1} and~\ref{chol-set2} present derived rules. Most are
+obvious and resemble rules of Isabelle's {\ZF} set theory. Certain rules,
+such as \tdx{subsetCE}, \tdx{bexCI} and \tdx{UnCI},
+are designed for classical reasoning; the rules \tdx{subsetD},
+\tdx{bexI}, \tdx{Un1} and~\tdx{Un2} are not
+strictly necessary but yield more natural proofs. Similarly,
+\tdx{equalityCE} supports classical reasoning about extensionality,
+after the fashion of \tdx{iffCE}. See the file {\tt CHOL/Set.ML} for
+proofs pertaining to set theory.
+
+Figure~\ref{chol-fun} presents derived inference rules involving functions.
+They also include rules for \cdx{Inv}, which is defined in theory~{\tt
+ CHOL}; note that ${\tt Inv}~f$ applies the Axiom of Choice to yield an
+inverse of~$f$. They also include natural deduction rules for the image
+and range operators, and for the predicates {\tt inj} and {\tt inj_onto}.
+Reasoning about function composition (the operator~\sdx{o}) and the
+predicate~\cdx{surj} is done simply by expanding the definitions. See
+the file {\tt CHOL/fun.ML} for a complete listing of the derived rules.
+
+Figure~\ref{chol-subset} presents lattice properties of the subset relation.
+Unions form least upper bounds; non-empty intersections form greatest lower
+bounds. Reasoning directly about subsets often yields clearer proofs than
+reasoning about the membership relation. See the file {\tt CHOL/subset.ML}.
+
+Figure~\ref{chol-equalities} presents many common set equalities. They
+include commutative, associative and distributive laws involving unions,
+intersections and complements. The proofs are mostly trivial, using the
+classical reasoner; see file {\tt CHOL/equalities.ML}.
+
+
+\begin{figure}
+\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. \{(x,y)\}
+
+
+\tdx{Pair_inject} [| (a, b) = (a',b'); [| a=a'; b=b' |] ==> R |] ==> R
+\tdx{fst_conv} fst (a,b) = a
+\tdx{snd_conv} snd (a,b) = b
+\tdx{split} split c (a,b) = c a b
+
+\tdx{surjective_pairing} p = (fst p,snd p)
+
+\tdx{SigmaI} [| a:A; b:B a |] ==> (a,b) : Sigma A B
+
+\tdx{SigmaE} [| c: Sigma A B;
+ !!x y.[| x:A; y:B x; c=(x,y) |] ==> P |] ==> P
+\end{ttbox}
+\caption{Type $\alpha\times\beta$}\label{chol-prod}
+\end{figure}
+
+
+\begin{figure}
+\begin{constants}
+ \it symbol & \it meta-type & & \it description \\
+ \cdx{Inl} & $\alpha \To \alpha+\beta$ & & first injection\\
+ \cdx{Inr} & $\beta \To \alpha+\beta$ & & second injection\\
+ \cdx{sum_case} & $[\alpha\To\gamma, \beta\To\gamma, \alpha+\beta] \To\gamma$
+ & & conditional
+\end{constants}
+\begin{ttbox}\makeatletter
+\tdx{sum_case_def} sum_case == (\%f g p. @z. (!x. p=Inl x --> z=f x) &
+ (!y. p=Inr y --> z=g y))
+
+\tdx{Inl_not_Inr} ~ Inl a=Inr b
+
+\tdx{inj_Inl} inj Inl
+\tdx{inj_Inr} inj Inr
+
+\tdx{sumE} [| !!x::'a. P(Inl x); !!y::'b. P(Inr y) |] ==> P s
+
+\tdx{sum_case_Inl} sum_case f g (Inl x) = f x
+\tdx{sum_case_Inr} sum_case f g (Inr x) = g x
+
+\tdx{surjective_sum} sum_case (\%x::'a. f(Inl x)) (\%y::'b. f(Inr y)) s = f s
+\end{ttbox}
+\caption{Type $\alpha+\beta$}\label{chol-sum}
+\end{figure}
+
+
+\section{Generic packages and classical reasoning}
+\CHOL\ instantiates most of Isabelle's generic packages;
+see {\tt CHOL/ROOT.ML} for details.
+\begin{itemize}
+\item
+Because it includes a general substitution rule, \CHOL\ instantiates the
+tactic {\tt hyp_subst_tac}, which substitutes for an equality
+throughout a subgoal and its hypotheses.
+\item
+It instantiates the simplifier, defining~\ttindexbold{HOL_ss} as the
+simplification set for higher-order logic. Equality~($=$), which also
+expresses logical equivalence, may be used for rewriting. See the file
+{\tt CHOL/simpdata.ML} for a complete listing of the simplification
+rules.
+\item
+It instantiates the classical reasoner, as described below.
+\end{itemize}
+\CHOL\ 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{chol-lemmas2} above.
+
+The classical reasoner is set up as the structure
+{\tt Classical}. This structure is open, so {\ML} identifiers such
+as {\tt step_tac}, {\tt fast_tac}, {\tt best_tac}, etc., refer to it.
+\HOL\ defines the following classical rule sets:
+\begin{ttbox}
+prop_cs : claset
+HOL_cs : claset
+set_cs : claset
+\end{ttbox}
+\begin{ttdescription}
+\item[\ttindexbold{prop_cs}] contains the propositional rules, namely
+those for~$\top$, $\bot$, $\conj$, $\disj$, $\neg$, $\imp$ and~$\bimp$,
+along with the rule~{\tt refl}.
+
+\item[\ttindexbold{HOL_cs}] extends {\tt prop_cs} with the safe rules
+ {\tt allI} and~{\tt exE} and the unsafe rules {\tt allE}
+ and~{\tt exI}, as well as rules for unique existence. Search using
+ this classical set is incomplete: quantified formulae are used at most
+ once.
+
+\item[\ttindexbold{set_cs}] extends {\tt HOL_cs} with rules for the bounded
+ quantifiers, subsets, comprehensions, unions and intersections,
+ complements, finite sets, images and ranges.
+\end{ttdescription}
+\noindent
+See \iflabelundefined{chap:classical}{the {\em Reference Manual\/}}%
+ {Chap.\ts\ref{chap:classical}}
+for more discussion of classical proof methods.
+
+
+\section{Types}
+The basic higher-order logic is augmented with a tremendous amount of
+material, including support for recursive function and type definitions. A
+detailed discussion appears elsewhere~\cite{paulson-coind}. The simpler
+definitions are the same as those used by the {\sc hol} system, but my
+treatment of recursive types differs from Melham's~\cite{melham89}. The
+present section describes product, sum, natural number and list types.
+
+\subsection{Product and sum types}\index{*"* type}\index{*"+ type}
+Theory \thydx{Prod} defines the product type $\alpha\times\beta$, with
+the ordered pair syntax {\tt($a$,$b$)}. Theory \thydx{Sum} defines the
+sum type $\alpha+\beta$. These use fairly standard constructions; see
+Figs.\ts\ref{chol-prod} and~\ref{chol-sum}. Because Isabelle does not
+support abstract type definitions, the isomorphisms between these types and
+their representations are made explicitly.
+
+Most of the definitions are suppressed, but observe that the projections
+and conditionals are defined as descriptions. Their properties are easily
+proved using \tdx{select_equality}.
+
+\begin{figure}
+\index{*"< symbol}
+\index{*"* symbol}
+\index{*div symbol}
+\index{*mod symbol}
+\index{*"+ symbol}
+\index{*"- symbol}
+\begin{constants}
+ \it symbol & \it meta-type & \it priority & \it description \\
+ \cdx{0} & $nat$ & & zero \\
+ \cdx{Suc} & $nat \To nat$ & & successor function\\
+ \cdx{nat_case} & $[\alpha, nat\To\alpha, nat] \To\alpha$
+ & & conditional\\
+ \cdx{nat_rec} & $[nat, \alpha, [nat, \alpha]\To\alpha] \To \alpha$
+ & & primitive recursor\\
+ \cdx{pred_nat} & $(nat\times nat) set$ & & predecessor relation\\
+ \tt * & $[nat,nat]\To nat$ & Left 70 & multiplication \\
+ \tt div & $[nat,nat]\To nat$ & Left 70 & division\\
+ \tt mod & $[nat,nat]\To nat$ & Left 70 & modulus\\
+ \tt + & $[nat,nat]\To nat$ & Left 65 & addition\\
+ \tt - & $[nat,nat]\To nat$ & Left 65 & subtraction
+\end{constants}
+\subcaption{Constants and infixes}
+
+\begin{ttbox}\makeatother
+\tdx{nat_case_def} nat_case == (\%a f n. @z. (n=0 --> z=a) &
+ (!x. n=Suc x --> z=f x))
+\tdx{pred_nat_def} pred_nat == \{p. ? n. p = (n, Suc n)\}
+\tdx{less_def} m<n == (m,n):pred_nat^+
+\tdx{nat_rec_def} nat_rec n c d ==
+ wfrec pred_nat n (nat_case (\%g.c) (\%m g. d m (g m)))
+
+\tdx{add_def} m+n == nat_rec m n (\%u v. Suc v)
+\tdx{diff_def} m-n == nat_rec n m (\%u v. nat_rec v 0 (\%x y.x))
+\tdx{mult_def} m*n == nat_rec m 0 (\%u v. n + v)
+\tdx{mod_def} m mod n == wfrec (trancl pred_nat)
+ m (\%j f. if j<n then j else f j-n))
+\tdx{quo_def} m div n == wfrec (trancl pred_nat),
+ m (\%j f. if j<n then 0 else Suc(f j-n))
+\subcaption{Definitions}
+\end{ttbox}
+\caption{Defining {\tt nat}, the type of natural numbers} \label{chol-nat1}
+\end{figure}
+
+
+\begin{figure} \underscoreon
+\begin{ttbox}
+\tdx{nat_induct} [| P 0; !!k. [| P k |] ==> P(Suc k) |] ==> P n
+
+\tdx{Suc_not_Zero} Suc m ~= 0
+\tdx{inj_Suc} inj Suc
+\tdx{n_not_Suc_n} n~=Suc n
+\subcaption{Basic properties}
+
+\tdx{pred_natI} (n, Suc n) : pred_nat
+\tdx{pred_natE}
+ [| p : pred_nat; !!x n. [| p = (n, Suc n) |] ==> R |] ==> R
+
+\tdx{nat_case_0} nat_case a f 0 = a
+\tdx{nat_case_Suc} nat_case a f (Suc k) = f k
+
+\tdx{wf_pred_nat} wf pred_nat
+\tdx{nat_rec_0} nat_rec 0 c h = c
+\tdx{nat_rec_Suc} nat_rec (Suc n) c h = h n (nat_rec n c h)
+\subcaption{Case analysis and primitive recursion}
+
+\tdx{less_trans} [| i<j; j<k |] ==> i<k
+\tdx{lessI} n < Suc n
+\tdx{zero_less_Suc} 0 < Suc n
+
+\tdx{less_not_sym} n<m --> ~ m<n
+\tdx{less_not_refl} ~ n<n
+\tdx{not_less0} ~ n<0
+
+\tdx{Suc_less_eq} (Suc m < Suc n) = (m<n)
+\tdx{less_induct} [| !!n. [| ! m. m<n --> P m |] ==> P n |] ==> P n
+
+\tdx{less_linear} m<n | m=n | n<m
+\subcaption{The less-than relation}
+\end{ttbox}
+\caption{Derived rules for {\tt nat}} \label{chol-nat2}
+\end{figure}
+
+
+\subsection{The type of natural numbers, {\tt nat}}
+The theory \thydx{Nat} defines the natural numbers in a roundabout but
+traditional way. The axiom of infinity postulates an type~\tydx{ind} of
+individuals, which is non-empty and closed under an injective operation.
+The natural numbers are inductively generated by choosing an arbitrary
+individual for~0 and using the injective operation to take successors. As
+usual, the isomorphisms between~\tydx{nat} and its representation are made
+explicitly.
+
+The definition makes use of a least fixed point operator \cdx{lfp},
+defined using the Knaster-Tarski theorem. This is used to define the
+operator \cdx{trancl}, for taking the transitive closure of a relation.
+Primitive recursion makes use of \cdx{wfrec}, an operator for recursion
+along arbitrary well-founded relations. The corresponding theories are
+called {\tt Lfp}, {\tt Trancl} and {\tt WF}\@. Elsewhere I have described
+similar constructions in the context of set theory~\cite{paulson-set-II}.
+
+Type~\tydx{nat} is postulated to belong to class~\cldx{ord}, which
+overloads $<$ and $\leq$ on the natural numbers. As of this writing,
+Isabelle provides no means of verifying that such overloading is sensible;
+there is no means of specifying the operators' properties and verifying
+that instances of the operators satisfy those properties. To be safe, the
+\CHOL\ theory includes no polymorphic axioms asserting general properties of
+$<$ and~$\leq$.
+
+Theory \thydx{Arith} develops arithmetic on the natural numbers. It
+defines addition, multiplication, subtraction, division, and remainder.
+Many of their properties are proved: commutative, associative and
+distributive laws, identity and cancellation laws, etc. The most
+interesting result is perhaps the theorem $a \bmod b + (a/b)\times b = a$.
+Division and remainder are defined by repeated subtraction, which requires
+well-founded rather than primitive recursion. See Figs.\ts\ref{chol-nat1}
+and~\ref{chol-nat2}.
+
+The predecessor relation, \cdx{pred_nat}, is shown to be well-founded.
+Recursion along this relation resembles primitive recursion, but is
+stronger because we are in higher-order logic; using primitive recursion to
+define a higher-order function, we can easily Ackermann's function, which
+is not primitive recursive \cite[page~104]{thompson91}.
+The transitive closure of \cdx{pred_nat} is~$<$. Many functions on the
+natural numbers are most easily expressed using recursion along~$<$.
+
+The tactic {\tt\ttindex{nat_ind_tac} "$n$" $i$} performs induction over the
+variable~$n$ in subgoal~$i$.
+
+\begin{figure}
+\index{#@{\tt\#} symbol}
+\index{"@@{\tt\at} symbol}
+\begin{constants}
+ \it symbol & \it meta-type & \it priority & \it description \\
+ \cdx{Nil} & $\alpha list$ & & empty list\\
+ \tt \# & $[\alpha,\alpha list]\To \alpha list$ & Right 65 &
+ list constructor \\
+ \cdx{null} & $\alpha list \To bool$ & & emptiness test\\
+ \cdx{hd} & $\alpha list \To \alpha$ & & head \\
+ \cdx{tl} & $\alpha list \To \alpha list$ & & tail \\
+ \cdx{ttl} & $\alpha list \To \alpha list$ & & total tail \\
+ \tt\at & $[\alpha list,\alpha list]\To \alpha list$ & Left 65 & append \\
+ \sdx{mem} & $[\alpha,\alpha list]\To bool$ & Left 55 & membership\\
+ \cdx{map} & $(\alpha\To\beta) \To (\alpha list \To \beta list)$
+ & & mapping functional\\
+ \cdx{filter} & $(\alpha \To bool) \To (\alpha list \To \alpha list)$
+ & & filter functional\\
+ \cdx{list_all}& $(\alpha \To bool) \To (\alpha list \To bool)$
+ & & forall functional\\
+ \cdx{list_rec} & $[\alpha list, \beta, [\alpha ,\alpha list,
+\beta]\To\beta] \To \beta$
+ & & list recursor
+\end{constants}
+\subcaption{Constants and infixes}
+
+\begin{center} \tt\frenchspacing
+\begin{tabular}{rrr}
+ \it external & \it internal & \it description \\{}
+ \sdx{[]} & Nil & \rm empty list \\{}
+ [$x@1$, $\dots$, $x@n$] & $x@1$ \# $\cdots$ \# $x@n$ \# [] &
+ \rm finite list \\{}
+ [$x$:$l$. $P$] & filter ($\lambda x{.}P$) $l$ &
+ \rm list comprehension
+\end{tabular}
+\end{center}
+\subcaption{Translations}
+
+\begin{ttbox}
+\tdx{list_induct} [| P []; !!x xs. [| P xs |] ==> P x#xs) |] ==> P l
+
+\tdx{Cons_not_Nil} (x # xs) ~= []
+\tdx{Cons_Cons_eq} ((x # xs) = (y # ys)) = (x=y & xs=ys)
+\subcaption{Induction and freeness}
+\end{ttbox}
+\caption{The theory \thydx{List}} \label{chol-list}
+\end{figure}
+
+\begin{figure}
+\begin{ttbox}\makeatother
+\tdx{list_rec_Nil} list_rec [] c h = c
+\tdx{list_rec_Cons} list_rec a#l c h = h a l (list_rec l c h)
+
+\tdx{list_case_Nil} list_case c h [] = c
+\tdx{list_case_Cons} list_case c h x#xs = h x xs
+
+\tdx{map_Nil} map f [] = []
+\tdx{map_Cons} map f x \# xs = f x \# map f xs
+
+\tdx{null_Nil} null [] = True
+\tdx{null_Cons} null x#xs = False
+
+\tdx{hd_Cons} hd x#xs = x
+\tdx{tl_Cons} tl x#xs = xs
+
+\tdx{ttl_Nil} ttl [] = []
+\tdx{ttl_Cons} ttl x#xs = xs
+
+\tdx{append_Nil} [] @ ys = ys
+\tdx{append_Cons} (x#xs) \at ys = x # xs \at ys
+
+\tdx{mem_Nil} x mem [] = False
+\tdx{mem_Cons} x mem (y#ys) = if y=x then True else x mem ys
+
+\tdx{filter_Nil} filter P [] = []
+\tdx{filter_Cons} filter P x#xs = if P x then x#filter P xs else filter P xs
+
+\tdx{list_all_Nil} list_all P [] = True
+\tdx{list_all_Cons} list_all P x#xs = (P x & list_all P xs)
+\end{ttbox}
+\caption{Rewrite rules for lists} \label{chol-list-simps}
+\end{figure}
+
+
+\subsection{The type constructor for lists, {\tt list}}
+\index{*list type}
+
+\CHOL's definition of lists is an example of an experimental method for
+handling recursive data types. Figure~\ref{chol-list} presents the theory
+\thydx{List}: the basic list operations with their types and properties.
+
+The \sdx{case} construct is defined by the following translation:
+{\dquotes
+\begin{eqnarray*}
+ \begin{array}{r@{\;}l@{}l}
+ "case " e " of" & "[]" & " => " a\\
+ "|" & x"\#"xs & " => " b
+ \end{array}
+ & \equiv &
+ "list_case"~a~(\lambda x\;xs.b)~e
+\end{eqnarray*}}%
+The theory includes \cdx{list_rec}, a primitive recursion operator
+for lists. It is derived from well-founded recursion, a general principle
+that can express arbitrary total recursive functions.
+
+The simpset \ttindex{list_ss} contains, along with additional useful lemmas,
+the basic rewrite rules that appear in Fig.\ts\ref{chol-list-simps}.
+
+The tactic {\tt\ttindex{list_ind_tac} "$xs$" $i$} performs induction over the
+variable~$xs$ in subgoal~$i$.
+
+
+\section{Datatype declarations}
+\index{*datatype|(}
+
+\underscoreon
+
+It is often necessary to extend a theory with \ML-like datatypes. This
+extension consists of the new type, declarations of its constructors and
+rules that describe the new type. The theory definition section {\tt
+ datatype} represents a compact way of doing this.
+
+
+\subsection{Foundations}
+
+A datatype declaration has the following general structure:
+\[ \mbox{\tt datatype}~ (\alpha_1,\dots,\alpha_n)t ~=~
+ C_1~\tau_{11}~\dots~\tau_{1k_1} ~\mid~ \dots ~\mid~
+ C_m~\tau_{m1}~\dots~\tau_{mk_m}
+\]
+where $\alpha_i$ are type variables, $C_i$ are distinct constructor names and
+$\tau_{ij}$ are one of the following:
+\begin{itemize}
+\item type variables $\alpha_1,\dots,\alpha_n$,
+\item types $(\beta_1,\dots,\beta_l)s$ where $s$ is a previously declared
+ type or type synonym and $\{\beta_1,\dots,\beta_l\} \subseteq
+ \{\alpha_1,\dots,\alpha_n\}$,
+\item the newly defined type $(\alpha_1,\dots,\alpha_n)t$ \footnote{This
+ makes it a recursive type. To ensure that the new type is not empty at
+ least one constructor must consist of only non-recursive type
+ components.}
+\end{itemize}
+If you would like one of the $\tau_{ij}$ to be a complex type expression
+$\tau$ you need to declare a new type synonym $syn = \tau$ first and use
+$syn$ in place of $\tau$. Of course this does not work if $\tau$ mentions the
+recursive type itself, thus ruling out problematic cases like \[ \mbox{\tt
+ datatype}~ t ~=~ C(t \To t) \] together with unproblematic ones like \[
+\mbox{\tt datatype}~ t ~=~ C(t~list). \]
+
+The constructors are automatically defined as functions of their respective
+type:
+\[ C_j : [\tau_{j1},\dots,\tau_{jk_j}] \To (\alpha_1,\dots,\alpha_n)t \]
+These functions have certain {\em freeness} properties:
+\begin{description}
+\item[\tt distinct] They are distinct:
+\[ C_i~x_1~\dots~x_{k_i} \neq C_j~y_1~\dots~y_{k_j} \qquad
+ \mbox{for all}~ i \neq j.
+\]
+\item[\tt inject] They are injective:
+\[ (C_j~x_1~\dots~x_{k_j} = C_j~y_1~\dots~y_{k_j}) =
+ (x_1 = y_1 \land \dots \land x_{k_j} = y_{k_j})
+\]
+\end{description}
+Because the number of inequalities is quadratic in the number of
+constructors, a different method is used if their number exceeds
+a certain value, currently 4. In that case every constructor is mapped to a
+natural number
+\[
+\begin{array}{lcl}
+\mbox{\it t\_ord}(C_1~x_1~\dots~x_{k_1}) & = & 0 \\
+& \vdots & \\
+\mbox{\it t\_ord}(C_m x_1~\dots~x_{k_m}) & = & m-1
+\end{array}
+\]
+and distinctness of constructors is expressed by:
+\[
+\mbox{\it t\_ord}~x \neq \mbox{\it t\_ord}~y \Imp x \neq y.
+\]
+In addition a structural induction axiom {\tt induct} is provided:
+\[
+\infer{P x}
+{\begin{array}{lcl}
+\Forall x_1\dots x_{k_1}.
+ \List{P~x_{r_{11}}; \dots; P~x_{r_{1l_1}}} &
+ \Imp & P(C_1~x_1~\dots~x_{k_1}) \\
+ & \vdots & \\
+\Forall x_1\dots x_{k_m}.
+ \List{P~x_{r_{m1}}; \dots; P~x_{r_{ml_m}}} &
+ \Imp & P(C_m~x_1~\dots~x_{k_m})
+\end{array}}
+\]
+where $\{r_{j1},\dots,r_{jl_j}\} = \{i \in \{1,\dots k_j\} ~\mid~ \tau_{ji}
+= (\alpha_1,\dots,\alpha_n)t \}$, i.e.\ the property $P$ can be assumed for
+all arguments of the recursive type.
+
+The type also comes with an \ML-like \sdx{case}-construct:
+\[
+\begin{array}{rrcl}
+\mbox{\tt case}~e~\mbox{\tt of} & C_1~x_{11}~\dots~x_{1k_1} & \To & e_1 \\
+ \vdots \\
+ \mid & C_m~x_{m1}~\dots~x_{mk_m} & \To & e_m
+\end{array}
+\]
+In contrast to \ML, {\em all} constructors must be present, their order is
+fixed, and nested patterns are not supported.
+
+
+\subsection{Defining datatypes}
+
+A datatype is defined in a theory definition file using the keyword {\tt
+ datatype}. The definition following {\tt datatype} must conform to the
+syntax of {\em typedecl} specified in Fig.~\ref{datatype-grammar} and must
+obey the rules in the previous section. As a result the theory is extended
+with the new type, the constructors, and the theorems listed in the previous
+section.
+
+\begin{figure}
+\begin{rail}
+typedecl : typevarlist id '=' (cons + '|')
+ ;
+cons : (id | string) ( () | '(' (typ + ',') ')' ) ( () | mixfix )
+ ;
+typ : typevarlist id
+ | tid
+ ;
+typevarlist : () | tid | '(' (tid + ',') ')'
+ ;
+\end{rail}
+\caption{Syntax of datatype declarations}
+\label{datatype-grammar}
+\end{figure}
+
+Reading the theory file produces a structure which, in addition to the usual
+components, contains a structure named $t$ for each datatype $t$ defined in
+the file.\footnote{Otherwise multiple datatypes in the same theory file would
+ lead to name clashes.} Each structure $t$ contains the following elements:
+\begin{ttbox}
+val distinct : thm list
+val inject : thm list
+val induct : thm
+val cases : thm list
+val simps : thm list
+val induct_tac : string -> int -> tactic
+\end{ttbox}
+{\tt distinct}, {\tt inject} and {\tt induct} contain the theorems described
+above. For convenience {\tt distinct} contains inequalities in both
+directions.
+\begin{warn}
+ If there are five or more constructors, the {\em t\_ord} scheme is used for
+ {\tt distinct}. In this case the theory {\tt Arith} must be contained
+ in the current theory, if necessary by including it explicitly.
+\end{warn}
+The reduction rules of the {\tt case}-construct are in {\tt cases}. All
+theorems from {\tt distinct}, {\tt inject} and {\tt cases} are combined in
+{\tt simps} for use with the simplifier. The tactic {\verb$induct_tac$~{\em
+ var i}\/} applies structural induction over variable {\em var} to
+subgoal {\em i}.
+
+
+\subsection{Examples}
+
+\subsubsection{The datatype $\alpha~list$}
+
+We want to define the type $\alpha~list$.\footnote{Of course there is a list
+ type in CHOL already. This is only an example.} To do this we have to build
+a new theory that contains the type definition. We start from {\tt CHOL}.
+\begin{ttbox}
+MyList = CHOL +
+ datatype 'a list = Nil | Cons 'a ('a list)
+end
+\end{ttbox}
+After loading the theory (\verb$use_thy "MyList"$), we can prove
+$Cons~x~xs\neq xs$. First we build a suitable simpset for the simplifier:
+\begin{ttbox}
+val mylist_ss = HOL_ss addsimps MyList.list.simps;
+goal MyList.thy "!x. Cons x xs ~= xs";
+{\out Level 0}
+{\out ! x. Cons x xs ~= xs}
+{\out 1. ! x. Cons x xs ~= xs}
+\end{ttbox}
+This can be proved by the structural induction tactic:
+\begin{ttbox}
+by (MyList.list.induct_tac "xs" 1);
+{\out Level 1}
+{\out ! x. Cons x xs ~= xs}
+{\out 1. ! x. Cons x Nil ~= Nil}
+{\out 2. !!a list.}
+{\out ! x. Cons x list ~= list ==>}
+{\out ! x. Cons x (Cons a list) ~= Cons a list}
+\end{ttbox}
+The first subgoal can be proved with the simplifier and the distinctness
+axioms which are part of \verb$mylist_ss$.
+\begin{ttbox}
+by (simp_tac mylist_ss 1);
+{\out Level 2}
+{\out ! x. Cons x xs ~= xs}
+{\out 1. !!a list.}
+{\out ! x. Cons x list ~= list ==>}
+{\out ! x. Cons x (Cons a list) ~= Cons a list}
+\end{ttbox}
+Using the freeness axioms we can quickly prove the remaining goal.
+\begin{ttbox}
+by (asm_simp_tac mylist_ss 1);
+{\out Level 3}
+{\out ! x. Cons x xs ~= xs}
+{\out No subgoals!}
+\end{ttbox}
+Because both subgoals were proved by almost the same tactic we could have
+done that in one step using
+\begin{ttbox}
+by (ALLGOALS (asm_simp_tac mylist_ss));
+\end{ttbox}
+
+
+\subsubsection{The datatype $\alpha~list$ with mixfix syntax}
+
+In this example we define the type $\alpha~list$ again but this time we want
+to write {\tt []} instead of {\tt Nil} and we want to use the infix operator
+\verb|#| instead of {\tt Cons}. To do this we simply add mixfix annotations
+after the constructor declarations as follows:
+\begin{ttbox}
+MyList = CHOL +
+ datatype 'a list = "[]" ("[]")
+ | "#" 'a ('a list) (infixr 70)
+end
+\end{ttbox}
+Now the theorem in the previous example can be written \verb|x#xs ~= xs|. The
+proof is the same.
+
+
+\subsubsection{A datatype for weekdays}
+
+This example shows a datatype that consists of more than four constructors:
+\begin{ttbox}
+Days = Arith +
+ datatype days = Mo | Tu | We | Th | Fr | Sa | So
+end
+\end{ttbox}
+Because there are more than four constructors, the theory must be based on
+{\tt Arith}. Inequality is defined via a function \verb|days_ord|. Although
+the expression \verb|Mo ~= Tu| is not directly contained in {\tt distinct},
+it can be proved by the simplifier if \verb$arith_ss$ is used:
+\begin{ttbox}
+val days_ss = arith_ss addsimps Days.days.simps;
+
+goal Days.thy "Mo ~= Tu";
+by (simp_tac days_ss 1);
+\end{ttbox}
+Note that usually it is not necessary to derive these inequalities explicitly
+because the simplifier will dispose of them automatically.
+
+\subsection{Primitive recursive functions}
+\index{primitive recursion|(}
+\index{*primrec|(}
+
+Datatypes come with a uniform way of defining functions, {\bf primitive
+ recursion}. Although it is possible to define primitive recursive functions
+by asserting their reduction rules as new axioms, e.g.\
+\begin{ttbox}
+Append = MyList +
+consts app :: "['a list,'a list] => 'a list"
+rules
+ app_Nil "app [] ys = ys"
+ app_Cons "app x#xs ys = x#app xs ys"
+end
+\end{ttbox}
+this carries with it the danger of accidentally asserting an inconsistency,
+as in \verb$app [] ys = us$. Therefore primitive recursive functions on
+datatypes can be defined with a special syntax:
+\begin{ttbox}
+Append = MyList +
+consts app :: "'['a list,'a list] => 'a list"
+primrec app MyList.list
+ app_Nil "app [] ys = ys"
+ app_Cons "app x#xs ys = x#app xs ys"
+end
+\end{ttbox}
+The system will now check that the two rules \verb$app_Nil$ and
+\verb$app_Cons$ do indeed form a primitive recursive definition, thus
+ensuring that consistency is maintained. For example
+\begin{ttbox}
+primrec app MyList.list
+ app_Nil "app [] ys = us"
+\end{ttbox}
+is rejected:
+\begin{ttbox}
+Extra variables on rhs
+\end{ttbox}
+\bigskip
+
+The general form of a primitive recursive definition is
+\begin{ttbox}
+primrec {\it function} {\it type}
+ {\it reduction rules}
+\end{ttbox}
+where
+\begin{itemize}
+\item {\it function} is the name of the function, either as an {\it id} or a
+ {\it string}. The function must already have been declared.
+\item {\it type} is the name of the datatype, either as an {\it id} or in the
+ long form {\it Thy.t}, where {\it Thy} is the name of the parent theory the
+ datatype was declared in, and $t$ the name of the datatype. The long form
+ is required if the {\tt datatype} and the {\tt primrec} sections are in
+ different theories.
+\item {\it reduction rules} specify one or more named equations of the form
+ {\it id\/}~{\it string}, where the identifier gives the name of the rule in
+ the result structure, and {\it string} is a reduction rule of the form \[
+ f~x_1~\dots~x_m~(C~y_1~\dots~y_k)~z_1~\dots~z_n = r \] such that $C$ is a
+ constructor of the datatype, $r$ contains only the free variables on the
+ left-hand side, and all recursive calls in $r$ are of the form
+ $f~\dots~y_i~\dots$ for some $i$. There must be exactly one reduction
+ rule for each constructor.
+\end{itemize}
+A theory file may contain any number of {\tt primrec} sections which may be
+intermixed with other declarations.
+
+For the consistency-sensitive user it may be reassuring to know that {\tt
+ primrec} does not assert the reduction rules as new axioms but derives them
+as theorems from an explicit definition of the recursive function in terms of
+a recursion operator on the datatype.
+
+The primitive recursive function can also use infix or mixfix syntax:
+\begin{ttbox}
+Append = MyList +
+consts "@" :: "['a list,'a list] => 'a list" (infixr 60)
+primrec "op @" MyList.list
+ app_Nil "[] @ ys = ys"
+ app_Cons "(x#xs) @ ys = x#(xs @ ys)"
+end
+\end{ttbox}
+
+The reduction rules become part of the ML structure \verb$Append$ and can
+be used to prove theorems about the function:
+\begin{ttbox}
+val append_ss = HOL_ss addsimps [Append.app_Nil,Append.app_Cons];
+
+goal Append.thy "(xs @ ys) @ zs = xs @ (ys @ zs)";
+by (MyList.list.induct_tac "xs" 1);
+by (ALLGOALS(asm_simp_tac append_ss));
+\end{ttbox}
+
+%Note that underdefined primitive recursive functions are allowed:
+%\begin{ttbox}
+%Tl = MyList +
+%consts tl :: "'a list => 'a list"
+%primrec tl MyList.list
+% tl_Cons "tl(x#xs) = xs"
+%end
+%\end{ttbox}
+%Nevertheless {\tt tl} is total, although we do not know what the result of
+%\verb$tl([])$ is.
+
+\index{primitive recursion|)}
+\index{*primrec|)}
+\index{*datatype|)}
+
+
+\section{Inductive and coinductive definitions}
+\index{*inductive|(}
+\index{*coinductive|(}
+
+An {\bf inductive definition} specifies the least set closed under given
+rules. For example, a structural operational semantics is an inductive
+definition of an evaluation relation. Dually, a {\bf coinductive
+ definition} specifies the greatest set closed under given rules. An
+important example is using bisimulation relations to formalize equivalence
+of processes and infinite data structures.
+
+A theory file may contain any number of inductive and coinductive
+definitions. They may be intermixed with other declarations; in
+particular, the (co)inductive sets {\bf must} be declared separately as
+constants, and may have mixfix syntax or be subject to syntax translations.
+
+Each (co)inductive definition adds definitions to the theory and also
+proves some theorems. Each definition creates an ML structure, which is a
+substructure of the main theory structure.
+
+This package is derived from the ZF one, described in a
+separate paper,\footnote{It appeared in CADE~\cite{paulson-CADE} and a
+ longer version is distributed with Isabelle.} which you should refer to
+in case of difficulties. The package is simpler than ZF's, thanks to CHOL's
+automatic type-checking. The type of the (co)inductive determines the
+domain of the fixedpoint definition, and the package does not use inference
+rules for type-checking.
+
+
+\subsection{The result structure}
+Many of the result structure's components have been discussed in the paper;
+others are self-explanatory.
+\begin{description}
+\item[\tt thy] is the new theory containing the recursive sets.
+
+\item[\tt defs] is the list of definitions of the recursive sets.
+
+\item[\tt mono] is a monotonicity theorem for the fixedpoint operator.
+
+\item[\tt unfold] is a fixedpoint equation for the recursive set (the union of
+the recursive sets, in the case of mutual recursion).
+
+\item[\tt intrs] is the list of introduction rules, now proved as theorems, for
+the recursive sets. The rules are also available individually, using the
+names given them in the theory file.
+
+\item[\tt elim] is the elimination rule.
+
+\item[\tt mk\_cases] is a function to create simplified instances of {\tt
+elim}, using freeness reasoning on some underlying datatype.
+\end{description}
+
+For an inductive definition, the result structure contains two induction rules,
+{\tt induct} and \verb|mutual_induct|. For a coinductive definition, it
+contains the rule \verb|coinduct|.
+
+Figure~\ref{def-result-fig} summarizes the two result signatures,
+specifying the types of all these components.
+
+\begin{figure}
+\begin{ttbox}
+sig
+val thy : theory
+val defs : thm list
+val mono : thm
+val unfold : thm
+val intrs : thm list
+val elim : thm
+val mk_cases : thm list -> string -> thm
+{\it(Inductive definitions only)}
+val induct : thm
+val mutual_induct: thm
+{\it(Coinductive definitions only)}
+val coinduct : thm
+end
+\end{ttbox}
+\hrule
+\caption{The result of a (co)inductive definition} \label{def-result-fig}
+\end{figure}
+
+\subsection{The syntax of a (co)inductive definition}
+An inductive definition has the form
+\begin{ttbox}
+inductive {\it inductive sets}
+ intrs {\it introduction rules}
+ monos {\it monotonicity theorems}
+ con_defs {\it constructor definitions}
+\end{ttbox}
+A coinductive definition is identical, except that it starts with the keyword
+{\tt coinductive}.
+
+The {\tt monos} and {\tt con\_defs} sections are optional. If present,
+each is specified as a string, which must be a valid ML expression of type
+{\tt thm list}. It is simply inserted into the {\tt .thy.ML} file; if it
+is ill-formed, it will trigger ML error messages. You can then inspect the
+file on your directory.
+
+\begin{itemize}
+\item The {\it inductive sets} are specified by one or more strings.
+
+\item The {\it introduction rules} specify one or more introduction rules in
+ the form {\it ident\/}~{\it string}, where the identifier gives the name of
+ the rule in the result structure.
+
+\item The {\it monotonicity theorems} are required for each operator
+ applied to a recursive set in the introduction rules. There {\bf must}
+ be a theorem of the form $A\subseteq B\Imp M(A)\subseteq M(B)$, for each
+ premise $t\in M(R_i)$ in an introduction rule!
+
+\item The {\it constructor definitions} contain definitions of constants
+ appearing in the introduction rules. In most cases it can be omitted.
+\end{itemize}
+
+The package has a few notable restrictions:
+\begin{itemize}
+\item The theory must separately declare the recursive sets as
+ constants.
+
+\item The names of the recursive sets must be identifiers, not infix
+operators.
+
+\item Side-conditions must not be conjunctions. However, an introduction rule
+may contain any number of side-conditions.
+
+\item Side-conditions of the form $x=t$, where the variable~$x$ does not
+ occur in~$t$, will be substituted through the rule \verb|mutual_induct|.
+\end{itemize}
+
+
+\subsection{Example of an inductive definition}
+Two declarations, included in a theory file, define the finite powerset
+operator. First we declare the constant~{\tt Fin}. Then we declare it
+inductively, with two introduction rules:
+\begin{ttbox}
+consts Fin :: "'a set => 'a set set"
+inductive "Fin A"
+ intrs
+ emptyI "{} : Fin A"
+ insertI "[| a: A; b: Fin A |] ==> insert a b : Fin A"
+\end{ttbox}
+The resulting theory structure contains a substructure, called~{\tt Fin}.
+It contains the {\tt Fin}$~A$ introduction rules as the list {\tt Fin.intrs},
+and also individually as {\tt Fin.emptyI} and {\tt Fin.consI}. The induction
+rule is {\tt Fin.induct}.
+
+For another example, here is a theory file defining the accessible part of a
+relation. The main thing to note is the use of~{\tt Pow} in the sole
+introduction rule, and the corresponding mention of the rule
+\verb|Pow_mono| in the {\tt monos} list. The paper discusses a ZF version
+of this example in more detail.
+\begin{ttbox}
+Acc = WF +
+consts pred :: "['b, ('a * 'b)set] => 'a set" (*Set of predecessors*)
+ acc :: "('a * 'a)set => 'a set" (*Accessible part*)
+defs pred_def "pred x r == {y. (y,x):r}"
+inductive "acc r"
+ intrs
+ pred "pred a r: Pow(acc r) ==> a: acc r"
+ monos "[Pow_mono]"
+end
+\end{ttbox}
+The CHOL distribution contains many other inductive definitions, such as the
+theory {\tt CHOL/ex/PropLog.thy} and the directory {\tt CHOL/IMP}. The
+theory {\tt CHOL/ex/LList.thy} contains coinductive definitions.
+
+\index{*coinductive|)} \index{*inductive|)} \underscoreoff
+
+
+\section{The examples directories}
+Directory {\tt CHOL/Subst} contains Martin Coen's mechanisation of a theory of
+substitutions and unifiers. It is based on Paulson's previous
+mechanisation in {\LCF}~\cite{paulson85} of Manna and Waldinger's
+theory~\cite{mw81}.
+
+Directory {\tt CHOL/IMP} contains a mechanised version of a semantic
+equivalence proof taken from Winskel~\cite{winskel93}. It formalises the
+denotational and operational semantics of a simple while-language, then
+proves the two equivalent. It contains several datatype and inductive
+definitions, and demonstrates their use.
+
+Directory {\tt CHOL/ex} contains other examples and experimental proofs in
+{\CHOL}. Here is an overview of the more interesting files.
+\begin{itemize}
+\item File {\tt cla.ML} demonstrates the classical reasoner on over sixty
+ predicate calculus theorems, ranging from simple tautologies to
+ moderately difficult problems involving equality and quantifiers.
+
+\item File {\tt meson.ML} contains an experimental implementation of the {\sc
+ meson} proof procedure, inspired by Plaisted~\cite{plaisted90}. It is
+ much more powerful than Isabelle's classical reasoner. But it is less
+ useful in practice because it works only for pure logic; it does not
+ accept derived rules for the set theory primitives, for example.
+
+\item File {\tt mesontest.ML} contains test data for the {\sc meson} proof
+ procedure. These are mostly taken from Pelletier \cite{pelletier86}.
+
+\item File {\tt set.ML} proves Cantor's Theorem, which is presented in
+ \S\ref{sec:chol-cantor} below, and the Schr\"oder-Bernstein Theorem.
+
+\item Theories {\tt InSort} and {\tt Qsort} prove correctness properties of
+ insertion sort and quick sort.
+
+\item The definition of lazy lists demonstrates methods for handling
+ infinite data structures and coinduction in higher-order
+ logic~\cite{paulson-coind}. Theory \thydx{LList} defines an operator for
+ corecursion on lazy lists, which is used to define a few simple functions
+ such as map and append. Corecursion cannot easily define operations such
+ as filter, which can compute indefinitely before yielding the next
+ element (if any!) of the lazy list. A coinduction principle is defined
+ for proving equations on lazy lists.
+
+\item Theory {\tt PropLog} proves the soundness and completeness of
+ classical propositional logic, given a truth table semantics. The only
+ connective is $\imp$. A Hilbert-style axiom system is specified, and its
+ set of theorems defined inductively. A similar proof in \ZF{} is
+ described elsewhere~\cite{paulson-set-II}.
+
+\item Theory {\tt Term} develops an experimental recursive type definition;
+ the recursion goes through the type constructor~\tydx{list}.
+
+\item Theory {\tt Simult} constructs mutually recursive sets of trees and
+ forests, including induction and recursion rules.
+
+\item Theory {\tt MT} contains Jacob Frost's formalization~\cite{frost93} of
+ Milner and Tofte's coinduction example~\cite{milner-coind}. This
+ substantial proof concerns the soundness of a type system for a simple
+ functional language. The semantics of recursion is given by a cyclic
+ environment, which makes a coinductive argument appropriate.
+\end{itemize}
+
+
+\goodbreak
+\section{Example: Cantor's Theorem}\label{sec:chol-cantor}
+Cantor's Theorem states that every set has more subsets than it has
+elements. It has become a favourite example in higher-order logic since
+it is so easily expressed:
+\[ \forall f::[\alpha,\alpha]\To bool. \exists S::\alpha\To bool.
+ \forall x::\alpha. f~x \not= S
+\]
+%
+Viewing types as sets, $\alpha\To bool$ represents the powerset
+of~$\alpha$. This version states that for every function from $\alpha$ to
+its powerset, some subset is outside its range.
+
+The Isabelle proof uses \CHOL's set theory, with the type $\alpha\,set$ and
+the operator \cdx{range}. The set~$S$ is given as an unknown instead of a
+quantified variable so that we may inspect the subset found by the proof.
+\begin{ttbox}
+goal Set.thy "~ ?S : range(f :: 'a=>'a set)";
+{\out Level 0}
+{\out ~ ?S : range f}
+{\out 1. ~ ?S : range f}
+\end{ttbox}
+The first two steps are routine. The rule \tdx{rangeE} replaces
+$\Var{S}\in {\tt range} f$ by $\Var{S}=f~x$ for some~$x$.
+\begin{ttbox}
+by (resolve_tac [notI] 1);
+{\out Level 1}
+{\out ~ ?S : range f}
+{\out 1. ?S : range f ==> False}
+\ttbreak
+by (eresolve_tac [rangeE] 1);
+{\out Level 2}
+{\out ~ ?S : range f}
+{\out 1. !!x. ?S = f x ==> False}
+\end{ttbox}
+Next, we apply \tdx{equalityCE}, reasoning that since $\Var{S}=f~x$,
+we have $\Var{c}\in \Var{S}$ if and only if $\Var{c}\in f~x$ for
+any~$\Var{c}$.
+\begin{ttbox}
+by (eresolve_tac [equalityCE] 1);
+{\out Level 3}
+{\out ~ ?S : range f}
+{\out 1. !!x. [| ?c3 x : ?S; ?c3 x : f x |] ==> False}
+{\out 2. !!x. [| ~ ?c3 x : ?S; ~ ?c3 x : f x |] ==> False}
+\end{ttbox}
+Now we use a bit of creativity. Suppose that~$\Var{S}$ has the form of a
+comprehension. Then $\Var{c}\in\{x.\Var{P}~x\}$ implies
+$\Var{P}~\Var{c}$. Destruct-resolution using \tdx{CollectD}
+instantiates~$\Var{S}$ and creates the new assumption.
+\begin{ttbox}
+by (dresolve_tac [CollectD] 1);
+{\out Level 4}
+{\out ~ \{x. ?P7 x\} : range f}
+{\out 1. !!x. [| ?c3 x : f x; ?P7(?c3 x) |] ==> False}
+{\out 2. !!x. [| ~ ?c3 x : \{x. ?P7 x\}; ~ ?c3 x : f x |] ==> False}
+\end{ttbox}
+Forcing a contradiction between the two assumptions of subgoal~1 completes
+the instantiation of~$S$. It is now the set $\{x. x\not\in f~x\}$, which
+is the standard diagonal construction.
+\begin{ttbox}
+by (contr_tac 1);
+{\out Level 5}
+{\out ~ \{x. ~ x : f x\} : range f}
+{\out 1. !!x. [| ~ x : \{x. ~ x : f x\}; ~ x : f x |] ==> False}
+\end{ttbox}
+The rest should be easy. To apply \tdx{CollectI} to the negated
+assumption, we employ \ttindex{swap_res_tac}:
+\begin{ttbox}
+by (swap_res_tac [CollectI] 1);
+{\out Level 6}
+{\out ~ \{x. ~ x : f x\} : range f}
+{\out 1. !!x. [| ~ x : f x; ~ False |] ==> ~ x : f x}
+\ttbreak
+by (assume_tac 1);
+{\out Level 7}
+{\out ~ \{x. ~ x : f x\} : range f}
+{\out No subgoals!}
+\end{ttbox}
+How much creativity is required? As it happens, Isabelle can prove this
+theorem automatically. The classical set \ttindex{set_cs} contains rules
+for most of the constructs of \CHOL's set theory. We must augment it with
+\tdx{equalityCE} to break up set equalities, and then apply best-first
+search. Depth-first search would diverge, but best-first search
+successfully navigates through the large search space.
+\index{search!best-first}
+\begin{ttbox}
+choplev 0;
+{\out Level 0}
+{\out ~ ?S : range f}
+{\out 1. ~ ?S : range f}
+\ttbreak
+by (best_tac (set_cs addSEs [equalityCE]) 1);
+{\out Level 1}
+{\out ~ \{x. ~ x : f x\} : range f}
+{\out No subgoals!}
+\end{ttbox}
+
+\index{higher-order logic|)}
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/Logics/HOL.tex Tue May 09 10:43:19 1995 +0200
@@ -0,0 +1,1898 @@
+%% $Id$
+\chapter{Higher-Order Logic}
+\index{higher-order logic|(}
+\index{CHOL system@{\sc chol} system}
+
+The theory~\thydx{CHOL} implements higher-order logic with curried
+function application. It is based on Gordon's~{\sc hol}
+system~\cite{mgordon-hol}, which itself is based on Church's original
+paper~\cite{church40}. Andrews's book~\cite{andrews86} is a full
+description of higher-order logic. Experience with the {\sc hol}
+system has demonstrated that higher-order logic is useful for hardware
+verification; beyond this, it is widely applicable in many areas of
+mathematics. It is weaker than {\ZF} set theory but for most
+applications this does not matter. If you prefer {\ML} to Lisp, you
+will probably prefer \CHOL\ to~{\ZF}.
+
+\CHOL\ is a modified version of Isabelle's \HOL\ and uses curried function
+application. Therefore the expression $f(a,b)$ (which in \HOL\ means
+``f applied to the two arguments $a$ and $b$'') means ``f applied to
+the pair $(a,b)$'' in \CHOL. N.B. that ordered pairs in \HOL\ are written as
+$<a,b>$ while in \CHOL\ the syntax $(a,b)$ is used. Previous
+releases of Isabelle also included a different version of~\HOL, with
+explicit type inference rules~\cite{paulson-COLOG}. This version no
+longer exists, but \thydx{ZF} supports a similar style of reasoning.
+
+\CHOL\ has a distinct feel, compared with {\ZF} and {\CTT}. It
+identifies object-level types with meta-level types, taking advantage of
+Isabelle's built-in type checker. It identifies object-level functions
+with meta-level functions, so it uses Isabelle's operations for abstraction
+and application. There is no `apply' operator: function applications are
+written as simply~$f~a$ rather than $f{\tt`}a$.
+
+These identifications allow Isabelle to support \CHOL\ particularly nicely,
+but they also mean that \CHOL\ requires more sophistication from the user
+--- in particular, an understanding of Isabelle's type system. Beginners
+should work with {\tt show_types} set to {\tt true}. Gain experience by
+working in first-order logic before attempting to use higher-order logic.
+This chapter assumes familiarity with~{\FOL{}}.
+
+
+\begin{figure}
+\begin{center}
+\begin{tabular}{rrr}
+ \it name &\it meta-type & \it description \\
+ \cdx{Trueprop}& $bool\To prop$ & coercion to $prop$\\
+ \cdx{not} & $bool\To bool$ & negation ($\neg$) \\
+ \cdx{True} & $bool$ & tautology ($\top$) \\
+ \cdx{False} & $bool$ & absurdity ($\bot$) \\
+ \cdx{If} & $[bool,\alpha,\alpha]\To\alpha::term$ & conditional \\
+ \cdx{Inv} & $(\alpha\To\beta)\To(\beta\To\alpha)$ & function inversion\\
+ \cdx{Let} & $[\alpha,\alpha\To\beta]\To\beta$ & let binder
+\end{tabular}
+\end{center}
+\subcaption{Constants}
+
+\begin{center}
+\index{"@@{\tt\at} symbol}
+\index{*"! symbol}\index{*"? symbol}
+\index{*"?"! symbol}\index{*"E"X"! symbol}
+\begin{tabular}{llrrr}
+ \it symbol &\it name &\it meta-type & \it description \\
+ \tt\at & \cdx{Eps} & $(\alpha\To bool)\To\alpha::term$ &
+ Hilbert description ($\epsilon$) \\
+ {\tt!~} or \sdx{ALL} & \cdx{All} & $(\alpha::term\To bool)\To bool$ &
+ universal quantifier ($\forall$) \\
+ {\tt?~} or \sdx{EX} & \cdx{Ex} & $(\alpha::term\To bool)\To bool$ &
+ existential quantifier ($\exists$) \\
+ {\tt?!} or {\tt EX!} & \cdx{Ex1} & $(\alpha::term\To bool)\To bool$ &
+ unique existence ($\exists!$)
+\end{tabular}
+\end{center}
+\subcaption{Binders}
+
+\begin{center}
+\index{*"= symbol}
+\index{&@{\tt\&} symbol}
+\index{*"| symbol}
+\index{*"-"-"> symbol}
+\begin{tabular}{rrrr}
+ \it symbol & \it meta-type & \it priority & \it description \\
+ \sdx{o} & $[\beta\To\gamma,\alpha\To\beta]\To (\alpha\To\gamma)$ &
+ Right 50 & composition ($\circ$) \\
+ \tt = & $[\alpha::term,\alpha]\To bool$ & Left 50 & equality ($=$) \\
+ \tt < & $[\alpha::ord,\alpha]\To bool$ & Left 50 & less than ($<$) \\
+ \tt <= & $[\alpha::ord,\alpha]\To bool$ & Left 50 &
+ less than or equals ($\leq$)\\
+ \tt \& & $[bool,bool]\To bool$ & Right 35 & conjunction ($\conj$) \\
+ \tt | & $[bool,bool]\To bool$ & Right 30 & disjunction ($\disj$) \\
+ \tt --> & $[bool,bool]\To bool$ & Right 25 & implication ($\imp$)
+\end{tabular}
+\end{center}
+\subcaption{Infixes}
+\caption{Syntax of {\tt HOL}} \label{chol-constants}
+\end{figure}
+
+
+\begin{figure}
+\index{*let symbol}
+\index{*in symbol}
+\dquotes
+\[\begin{array}{rclcl}
+ term & = & \hbox{expression of class~$term$} \\
+ & | & "\at~" id~id^* " . " formula \\
+ & | &
+ \multicolumn{3}{l}{"let"~id~"="~term";"\dots";"~id~"="~term~"in"~term} \\
+ & | &
+ \multicolumn{3}{l}{"if"~formula~"then"~term~"else"~term} \\[2ex]
+ formula & = & \hbox{expression of type~$bool$} \\
+ & | & term " = " term \\
+ & | & term " \ttilde= " term \\
+ & | & term " < " term \\
+ & | & term " <= " term \\
+ & | & "\ttilde\ " formula \\
+ & | & formula " \& " formula \\
+ & | & formula " | " formula \\
+ & | & formula " --> " formula \\
+ & | & "!~~~" id~id^* " . " formula
+ & | & "ALL~" id~id^* " . " formula \\
+ & | & "?~~~" id~id^* " . " formula
+ & | & "EX~~" id~id^* " . " formula \\
+ & | & "?!~~" id~id^* " . " formula
+ & | & "EX!~" id~id^* " . " formula
+ \end{array}
+\]
+\caption{Full grammar for \CHOL} \label{chol-grammar}
+\end{figure}
+
+
+\section{Syntax}
+The type class of higher-order terms is called~\cldx{term}. Type variables
+range over this class by default. The equality symbol and quantifiers are
+polymorphic over class {\tt term}.
+
+Class \cldx{ord} consists of all ordered types; the relations $<$ and
+$\leq$ are polymorphic over this class, as are the functions
+\cdx{mono}, \cdx{min} and \cdx{max}. Three other
+type classes --- \cldx{plus}, \cldx{minus} and \cldx{times} --- permit
+overloading of the operators {\tt+}, {\tt-} and {\tt*}. In particular,
+{\tt-} is overloaded for set difference and subtraction.
+\index{*"+ symbol}
+\index{*"- symbol}
+\index{*"* symbol}
+
+Figure~\ref{chol-constants} lists the constants (including infixes and
+binders), while Fig.\ts\ref{chol-grammar} presents the grammar of
+higher-order logic. Note that $a$\verb|~=|$b$ is translated to
+$\neg(a=b)$.
+
+\begin{warn}
+ \CHOL\ has no if-and-only-if connective; logical equivalence is expressed
+ using equality. But equality has a high priority, as befitting a
+ relation, while if-and-only-if typically has the lowest priority. Thus,
+ $\neg\neg P=P$ abbreviates $\neg\neg (P=P)$ and not $(\neg\neg P)=P$.
+ When using $=$ to mean logical equivalence, enclose both operands in
+ parentheses.
+\end{warn}
+
+\subsection{Types}\label{CHOL-types}
+The type of formulae, \tydx{bool}, belongs to class \cldx{term}; thus,
+formulae are terms. The built-in type~\tydx{fun}, which constructs function
+types, is overloaded with arity {\tt(term,term)term}. Thus, $\sigma\To\tau$
+belongs to class~{\tt term} if $\sigma$ and~$\tau$ do, allowing quantification
+over functions.
+
+Types in \CHOL\ must be non-empty; otherwise the quantifier rules would be
+unsound. I have commented on this elsewhere~\cite[\S7]{paulson-COLOG}.
+
+\index{type definitions}
+Gordon's {\sc hol} system supports {\bf type definitions}. A type is
+defined by exhibiting an existing type~$\sigma$, a predicate~$P::\sigma\To
+bool$, and a theorem of the form $\exists x::\sigma.P~x$. Thus~$P$
+specifies a non-empty subset of~$\sigma$, and the new type denotes this
+subset. New function constants are generated to establish an isomorphism
+between the new type and the subset. If type~$\sigma$ involves type
+variables $\alpha@1$, \ldots, $\alpha@n$, then the type definition creates
+a type constructor $(\alpha@1,\ldots,\alpha@n)ty$ rather than a particular
+type. Melham~\cite{melham89} discusses type definitions at length, with
+examples.
+
+Isabelle does not support type definitions at present. Instead, they are
+mimicked by explicit definitions of isomorphism functions. The definitions
+should be supported by theorems of the form $\exists x::\sigma.P~x$, but
+Isabelle cannot enforce this.
+
+
+\subsection{Binders}
+Hilbert's {\bf description} operator~$\epsilon x.P[x]$ stands for some~$a$
+satisfying~$P[a]$, if such exists. Since all terms in \CHOL\ denote
+something, a description is always meaningful, but we do not know its value
+unless $P[x]$ defines it uniquely. We may write descriptions as
+\cdx{Eps}($P$) or use the syntax
+\hbox{\tt \at $x$.$P[x]$}.
+
+Existential quantification is defined by
+\[ \exists x.P~x \;\equiv\; P(\epsilon x.P~x). \]
+The unique existence quantifier, $\exists!x.P[x]$, is defined in terms
+of~$\exists$ and~$\forall$. An Isabelle binder, it admits nested
+quantifications. For instance, $\exists!x y.P~x~y$ abbreviates
+$\exists!x. \exists!y.P~x~y$; note that this does not mean that there
+exists a unique pair $(x,y)$ satisfying~$P~x~y$.
+
+\index{*"! symbol}\index{*"? symbol}\index{CHOL system@{\sc hol} system}
+Quantifiers have two notations. As in Gordon's {\sc hol} system, \CHOL\
+uses~{\tt!}\ and~{\tt?}\ to stand for $\forall$ and $\exists$. The
+existential quantifier must be followed by a space; thus {\tt?x} is an
+unknown, while \verb'? x.f x=y' is a quantification. Isabelle's usual
+notation for quantifiers, \sdx{ALL} and \sdx{EX}, is also
+available. Both notations are accepted for input. The {\ML} reference
+\ttindexbold{HOL_quantifiers} governs the output notation. If set to {\tt
+true}, then~{\tt!}\ and~{\tt?}\ are displayed; this is the default. If set
+to {\tt false}, then~{\tt ALL} and~{\tt EX} are displayed.
+
+All these binders have priority 10.
+
+
+\subsection{The \sdx{let} and \sdx{case} constructions}
+Local abbreviations can be introduced by a {\tt let} construct whose
+syntax appears in Fig.\ts\ref{chol-grammar}. Internally it is translated into
+the constant~\cdx{Let}. It can be expanded by rewriting with its
+definition, \tdx{Let_def}.
+
+\CHOL\ also defines the basic syntax
+\[\dquotes"case"~e~"of"~c@1~"=>"~e@1~"|" \dots "|"~c@n~"=>"~e@n\]
+as a uniform means of expressing {\tt case} constructs. Therefore {\tt
+ case} and \sdx{of} are reserved words. However, so far this is mere
+syntax and has no logical meaning. By declaring translations, you can
+cause instances of the {\tt case} construct to denote applications of
+particular case operators. The patterns supplied for $c@1$,~\ldots,~$c@n$
+distinguish among the different case operators. For an example, see the
+case construct for lists on page~\pageref{chol-list} below.
+
+\begin{figure}
+\begin{ttbox}\makeatother
+\tdx{refl} t = (t::'a)
+\tdx{subst} [| s=t; P s |] ==> P(t::'a)
+\tdx{ext} (!!x::'a. (f x::'b) = g x) ==> (\%x.f x) = (\%x.g x)
+\tdx{impI} (P ==> Q) ==> P-->Q
+\tdx{mp} [| P-->Q; P |] ==> Q
+\tdx{iff} (P-->Q) --> (Q-->P) --> (P=Q)
+\tdx{selectI} P(x::'a) ==> P(@x.P x)
+\tdx{True_or_False} (P=True) | (P=False)
+\end{ttbox}
+\caption{The {\tt CHOL} rules} \label{chol-rules}
+\end{figure}
+
+
+\begin{figure}\hfuzz=4pt%suppress "Overfull \hbox" message
+\begin{ttbox}\makeatother
+\tdx{True_def} True == ((\%x::bool.x)=(\%x.x))
+\tdx{All_def} All == (\%P. P = (\%x.True))
+\tdx{Ex_def} Ex == (\%P. P(@x.P x))
+\tdx{False_def} False == (!P.P)
+\tdx{not_def} not == (\%P. P-->False)
+\tdx{and_def} op & == (\%P Q. !R. (P-->Q-->R) --> R)
+\tdx{or_def} op | == (\%P Q. !R. (P-->R) --> (Q-->R) --> R)
+\tdx{Ex1_def} Ex1 == (\%P. ? x. P x & (! y. P y --> y=x))
+
+\tdx{Inv_def} Inv == (\%(f::'a=>'b) y. @x. f x=y)
+\tdx{o_def} op o == (\%(f::'b=>'c) g (x::'a). f(g x))
+\tdx{if_def} If P x y == (\%P x y.@z::'a.(P=True --> z=x) & (P=False --> z=y))
+\tdx{Let_def} Let s f == f s
+\end{ttbox}
+\caption{The {\tt CHOL} definitions} \label{chol-defs}
+\end{figure}
+
+
+\section{Rules of inference}
+Figure~\ref{chol-rules} shows the inference rules of~\CHOL{}, with
+their~{\ML} names. Some of the rules deserve additional comments:
+\begin{ttdescription}
+\item[\tdx{ext}] expresses extensionality of functions.
+\item[\tdx{iff}] asserts that logically equivalent formulae are
+ equal.
+\item[\tdx{selectI}] gives the defining property of the Hilbert
+ $\epsilon$-operator. It is a form of the Axiom of Choice. The derived rule
+ \tdx{select_equality} (see below) is often easier to use.
+\item[\tdx{True_or_False}] makes the logic classical.\footnote{In
+ fact, the $\epsilon$-operator already makes the logic classical, as
+ shown by Diaconescu; see Paulson~\cite{paulson-COLOG} for details.}
+\end{ttdescription}
+
+\CHOL{} follows standard practice in higher-order logic: only a few
+connectives are taken as primitive, with the remainder defined obscurely
+(Fig.\ts\ref{chol-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~\CHOL{}, like all other Isabelle theories, uses
+meta-equality~({\tt==}) for definitions.
+
+Some of the rules mention type variables; for example, {\tt refl}
+mentions the type variable~{\tt'a}. This allows you to instantiate
+type variables explicitly by calling {\tt res_inst_tac}. By default,
+explicit type variables have class \cldx{term}.
+
+Include type constraints whenever you state a polymorphic goal. Type
+inference may otherwise make the goal more polymorphic than you intended,
+with confusing results.
+
+\begin{warn}
+ If resolution fails for no obvious reason, try setting
+ \ttindex{show_types} to {\tt true}, causing Isabelle to display types of
+ terms. Possibly set \ttindex{show_sorts} to {\tt true} as well, causing
+ Isabelle to display sorts.
+
+ \index{unification!incompleteness of}
+ Where function types are involved, Isabelle's unification code does not
+ guarantee to find instantiations for type variables automatically. Be
+ prepared to use \ttindex{res_inst_tac} instead of {\tt resolve_tac},
+ possibly instantiating type variables. Setting
+ \ttindex{Unify.trace_types} to {\tt true} causes Isabelle to report
+ omitted search paths during unification.\index{tracing!of unification}
+\end{warn}
+
+
+\begin{figure}
+\begin{ttbox}
+\tdx{sym} s=t ==> t=s
+\tdx{trans} [| r=s; s=t |] ==> r=t
+\tdx{ssubst} [| t=s; P s |] ==> P(t::'a)
+\tdx{box_equals} [| a=b; a=c; b=d |] ==> c=d
+\tdx{arg_cong} x=y ==> f x=f y
+\tdx{fun_cong} f=g ==> f x=g x
+\subcaption{Equality}
+
+\tdx{TrueI} True
+\tdx{FalseE} False ==> P
+
+\tdx{conjI} [| P; Q |] ==> P&Q
+\tdx{conjunct1} [| P&Q |] ==> P
+\tdx{conjunct2} [| P&Q |] ==> Q
+\tdx{conjE} [| P&Q; [| P; Q |] ==> R |] ==> R
+
+\tdx{disjI1} P ==> P|Q
+\tdx{disjI2} Q ==> P|Q
+\tdx{disjE} [| P | Q; P ==> R; Q ==> R |] ==> R
+
+\tdx{notI} (P ==> False) ==> ~ P
+\tdx{notE} [| ~ P; P |] ==> R
+\tdx{impE} [| P-->Q; P; Q ==> R |] ==> R
+\subcaption{Propositional logic}
+
+\tdx{iffI} [| P ==> Q; Q ==> P |] ==> P=Q
+\tdx{iffD1} [| P=Q; P |] ==> Q
+\tdx{iffD2} [| P=Q; Q |] ==> P
+\tdx{iffE} [| P=Q; [| P --> Q; Q --> P |] ==> R |] ==> R
+
+\tdx{eqTrueI} P ==> P=True
+\tdx{eqTrueE} P=True ==> P
+\subcaption{Logical equivalence}
+
+\end{ttbox}
+\caption{Derived rules for \CHOL} \label{chol-lemmas1}
+\end{figure}
+
+
+\begin{figure}
+\begin{ttbox}\makeatother
+\tdx{allI} (!!x::'a. P x) ==> !x. P x
+\tdx{spec} !x::'a.P x ==> P x
+\tdx{allE} [| !x.P x; P x ==> R |] ==> R
+\tdx{all_dupE} [| !x.P x; [| P x; !x.P x |] ==> R |] ==> R
+
+\tdx{exI} P x ==> ? x::'a.P x
+\tdx{exE} [| ? x::'a.P x; !!x. P x ==> Q |] ==> Q
+
+\tdx{ex1I} [| P a; !!x. P x ==> x=a |] ==> ?! x. P x
+\tdx{ex1E} [| ?! x.P x; !!x. [| P x; ! y. P y --> y=x |] ==> R
+ |] ==> R
+
+\tdx{select_equality} [| P a; !!x. P x ==> x=a |] ==> (@x.P x) = a
+\subcaption{Quantifiers and descriptions}
+
+\tdx{ccontr} (~P ==> False) ==> P
+\tdx{classical} (~P ==> P) ==> P
+\tdx{excluded_middle} ~P | P
+
+\tdx{disjCI} (~Q ==> P) ==> P|Q
+\tdx{exCI} (! x. ~ P x ==> P a) ==> ? x.P x
+\tdx{impCE} [| P-->Q; ~ P ==> R; Q ==> R |] ==> R
+\tdx{iffCE} [| P=Q; [| P;Q |] ==> R; [| ~P; ~Q |] ==> R |] ==> R
+\tdx{notnotD} ~~P ==> P
+\tdx{swap} ~P ==> (~Q ==> P) ==> Q
+\subcaption{Classical logic}
+
+\tdx{if_True} if True 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{expand_if} P(if Q then x else y) = ((Q --> P x) & (~Q --> P y))
+\subcaption{Conditionals}
+\end{ttbox}
+\caption{More derived rules} \label{chol-lemmas2}
+\end{figure}
+
+
+Some derived rules are shown in Figures~\ref{chol-lemmas1}
+and~\ref{chol-lemmas2}, with their {\ML} names. These include natural rules
+for the logical connectives, as well as sequent-style elimination rules for
+conjunctions, implications, and universal quantifiers.
+
+Note the equality rules: \tdx{ssubst} performs substitution in
+backward proofs, while \tdx{box_equals} supports reasoning by
+simplifying both sides of an equation.
+
+
+\begin{figure}
+\begin{center}
+\begin{tabular}{rrr}
+ \it name &\it meta-type & \it description \\
+\index{{}@\verb'{}' symbol}
+ \verb|{}| & $\alpha\,set$ & the empty set \\
+ \cdx{insert} & $[\alpha,\alpha\,set]\To \alpha\,set$
+ & insertion of element \\
+ \cdx{Collect} & $(\alpha\To bool)\To\alpha\,set$
+ & comprehension \\
+ \cdx{Compl} & $(\alpha\,set)\To\alpha\,set$
+ & complement \\
+ \cdx{INTER} & $[\alpha\,set,\alpha\To\beta\,set]\To\beta\,set$
+ & intersection over a set\\
+ \cdx{UNION} & $[\alpha\,set,\alpha\To\beta\,set]\To\beta\,set$
+ & union over a set\\
+ \cdx{Inter} & $(\alpha\,set)set\To\alpha\,set$
+ &set of sets intersection \\
+ \cdx{Union} & $(\alpha\,set)set\To\alpha\,set$
+ &set of sets union \\
+ \cdx{Pow} & $\alpha\,set \To (\alpha\,set)set$
+ & powerset \\[1ex]
+ \cdx{range} & $(\alpha\To\beta )\To\beta\,set$
+ & range of a function \\[1ex]
+ \cdx{Ball}~~\cdx{Bex} & $[\alpha\,set,\alpha\To bool]\To bool$
+ & bounded quantifiers \\
+ \cdx{mono} & $(\alpha\,set\To\beta\,set)\To bool$
+ & monotonicity \\
+ \cdx{inj}~~\cdx{surj}& $(\alpha\To\beta )\To bool$
+ & injective/surjective \\
+ \cdx{inj_onto} & $[\alpha\To\beta ,\alpha\,set]\To bool$
+ & injective over subset
+\end{tabular}
+\end{center}
+\subcaption{Constants}
+
+\begin{center}
+\begin{tabular}{llrrr}
+ \it symbol &\it name &\it meta-type & \it priority & \it description \\
+ \sdx{INT} & \cdx{INTER1} & $(\alpha\To\beta\,set)\To\beta\,set$ & 10 &
+ intersection over a type\\
+ \sdx{UN} & \cdx{UNION1} & $(\alpha\To\beta\,set)\To\beta\,set$ & 10 &
+ union over a type
+\end{tabular}
+\end{center}
+\subcaption{Binders}
+
+\begin{center}
+\index{*"`"` symbol}
+\index{*": symbol}
+\index{*"<"= symbol}
+\begin{tabular}{rrrr}
+ \it symbol & \it meta-type & \it priority & \it description \\
+ \tt `` & $[\alpha\To\beta ,\alpha\,set]\To (\beta\,set)$
+ & Left 90 & image \\
+ \sdx{Int} & $[\alpha\,set,\alpha\,set]\To\alpha\,set$
+ & Left 70 & intersection ($\inter$) \\
+ \sdx{Un} & $[\alpha\,set,\alpha\,set]\To\alpha\,set$
+ & Left 65 & union ($\union$) \\
+ \tt: & $[\alpha ,\alpha\,set]\To bool$
+ & Left 50 & membership ($\in$) \\
+ \tt <= & $[\alpha\,set,\alpha\,set]\To bool$
+ & Left 50 & subset ($\subseteq$)
+\end{tabular}
+\end{center}
+\subcaption{Infixes}
+\caption{Syntax of the theory {\tt Set}} \label{chol-set-syntax}
+\end{figure}
+
+
+\begin{figure}
+\begin{center} \tt\frenchspacing
+\index{*"! symbol}
+\begin{tabular}{rrr}
+ \it external & \it internal & \it description \\
+ $a$ \ttilde: $b$ & \ttilde($a$ : $b$) & \rm non-membership\\
+ \{$a@1$, $\ldots$\} & insert $a@1$ $\ldots$ \{\} & \rm finite set \\
+ \{$x$.$P[x]$\} & Collect($\lambda x.P[x]$) &
+ \rm comprehension \\
+ \sdx{INT} $x$:$A$.$B[x]$ & INTER $A$ $\lambda x.B[x]$ &
+ \rm intersection \\
+ \sdx{UN}{\tt\ } $x$:$A$.$B[x]$ & UNION $A$ $\lambda x.B[x]$ &
+ \rm union \\
+ \tt ! $x$:$A$.$P[x]$ or \sdx{ALL} $x$:$A$.$P[x]$ &
+ Ball $A$ $\lambda x.P[x]$ &
+ \rm bounded $\forall$ \\
+ \sdx{?} $x$:$A$.$P[x]$ or \sdx{EX}{\tt\ } $x$:$A$.$P[x]$ &
+ Bex $A$ $\lambda x.P[x]$ & \rm bounded $\exists$
+\end{tabular}
+\end{center}
+\subcaption{Translations}
+
+\dquotes
+\[\begin{array}{rclcl}
+ term & = & \hbox{other terms\ldots} \\
+ & | & "\{\}" \\
+ & | & "\{ " term\; ("," term)^* " \}" \\
+ & | & "\{ " id " . " formula " \}" \\
+ & | & term " `` " term \\
+ & | & term " Int " term \\
+ & | & term " Un " term \\
+ & | & "INT~~" id ":" term " . " term \\
+ & | & "UN~~~" id ":" term " . " term \\
+ & | & "INT~~" id~id^* " . " term \\
+ & | & "UN~~~" id~id^* " . " term \\[2ex]
+ formula & = & \hbox{other formulae\ldots} \\
+ & | & term " : " term \\
+ & | & term " \ttilde: " term \\
+ & | & term " <= " term \\
+ & | & "!~" id ":" term " . " formula
+ & | & "ALL " id ":" term " . " formula \\
+ & | & "?~" id ":" term " . " formula
+ & | & "EX~~" id ":" term " . " formula
+ \end{array}
+\]
+\subcaption{Full Grammar}
+\caption{Syntax of the theory {\tt Set} (continued)} \label{chol-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 \CHOL\ as they
+do in~{\ZF}. In \CHOL\ 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}
+\CHOL's set theory is called \thydx{Set}. The type $\alpha\,set$ is
+essentially the same as $\alpha\To bool$. The new type is defined for
+clarity and to avoid complications involving function types in unification.
+Since Isabelle does not support type definitions (as mentioned in
+\S\ref{CHOL-types}), the isomorphisms between the two types are declared
+explicitly. Here they are natural: {\tt Collect} maps $\alpha\To bool$ to
+$\alpha\,set$, while \hbox{\tt op :} maps in the other direction (ignoring
+argument order).
+
+Figure~\ref{chol-set-syntax} lists the constants, infixes, and syntax
+translations. Figure~\ref{chol-set-syntax2} presents the grammar of the new
+constructs. Infix operators include union and intersection ($A\union B$
+and $A\inter B$), the subset and membership relations, and the image
+operator~{\tt``}\@. Note that $a$\verb|~:|$b$ is translated to
+$\neg(a\in b)$.
+
+The {\tt\{\ldots\}} notation abbreviates finite sets constructed in the
+obvious manner using~{\tt insert} and~$\{\}$:
+\begin{eqnarray*}
+ \{a@1, \ldots, a@n\} & \equiv &
+ {\tt insert}~a@1~({\tt insert}\ldots({\tt insert}~a@n~\{\})\ldots)
+\end{eqnarray*}
+
+The set \hbox{\tt\{$x$.$P[x]$\}} consists of all $x$ (of suitable type)
+that satisfy~$P[x]$, where $P[x]$ is a formula that may contain free
+occurrences of~$x$. This syntax expands to \cdx{Collect}$(\lambda
+x.P[x])$. It defines sets by absolute comprehension, which is impossible
+in~{\ZF}; the type of~$x$ implicitly restricts the comprehension.
+
+The set theory defines two {\bf bounded quantifiers}:
+\begin{eqnarray*}
+ \forall x\in A.P[x] &\hbox{abbreviates}& \forall x. x\in A\imp P[x] \\
+ \exists x\in A.P[x] &\hbox{abbreviates}& \exists x. x\in A\conj P[x]
+\end{eqnarray*}
+The constants~\cdx{Ball} and~\cdx{Bex} are defined
+accordingly. Instead of {\tt Ball $A$ $P$} and {\tt Bex $A$ $P$} we may
+write\index{*"! symbol}\index{*"? symbol}
+\index{*ALL symbol}\index{*EX symbol}
+%
+\hbox{\tt !~$x$:$A$.$P[x]$} and \hbox{\tt ?~$x$:$A$.$P[x]$}. Isabelle's
+usual quantifier symbols, \sdx{ALL} and \sdx{EX}, are also accepted
+for input. As with the primitive quantifiers, the {\ML} reference
+\ttindex{HOL_quantifiers} specifies which notation to use for output.
+
+Unions and intersections over sets, namely $\bigcup@{x\in A}B[x]$ and
+$\bigcap@{x\in A}B[x]$, are written
+\sdx{UN}~\hbox{\tt$x$:$A$.$B[x]$} and
+\sdx{INT}~\hbox{\tt$x$:$A$.$B[x]$}.
+
+Unions and intersections over types, namely $\bigcup@x B[x]$ and $\bigcap@x
+B[x]$, are written \sdx{UN}~\hbox{\tt$x$.$B[x]$} and
+\sdx{INT}~\hbox{\tt$x$.$B[x]$}. They are equivalent to the previous
+union and intersection operators when $A$ is the universal set.
+
+The operators $\bigcup A$ and $\bigcap A$ act upon sets of sets. They are
+not binders, but are equal to $\bigcup@{x\in A}x$ and $\bigcap@{x\in A}x$,
+respectively.
+
+
+\begin{figure} \underscoreon
+\begin{ttbox}
+\tdx{mem_Collect_eq} (a : \{x.P x\}) = P a
+\tdx{Collect_mem_eq} \{x.x:A\} = A
+
+\tdx{empty_def} \{\} == \{x.False\}
+\tdx{insert_def} insert a B == \{x.x=a\} Un B
+\tdx{Ball_def} Ball A P == ! x. x:A --> P x
+\tdx{Bex_def} Bex A P == ? x. x:A & P x
+\tdx{subset_def} A <= B == ! x:A. x:B
+\tdx{Un_def} A Un B == \{x.x:A | x:B\}
+\tdx{Int_def} A Int B == \{x.x:A & x:B\}
+\tdx{set_diff_def} A - B == \{x.x:A & x~:B\}
+\tdx{Compl_def} Compl A == \{x. ~ x:A\}
+\tdx{INTER_def} INTER A B == \{y. ! x:A. y: B x\}
+\tdx{UNION_def} UNION A B == \{y. ? x:A. y: B x\}
+\tdx{INTER1_def} INTER1 B == INTER \{x.True\} B
+\tdx{UNION1_def} UNION1 B == UNION \{x.True\} B
+\tdx{Inter_def} Inter S == (INT x:S. x)
+\tdx{Union_def} Union S == (UN x:S. x)
+\tdx{Pow_def} Pow A == \{B. B <= A\}
+\tdx{image_def} f``A == \{y. ? x:A. y=f x\}
+\tdx{range_def} range f == \{y. ? x. y=f x\}
+\tdx{mono_def} mono f == !A B. A <= B --> f A <= f B
+\tdx{inj_def} inj f == ! x y. f x=f y --> x=y
+\tdx{surj_def} surj f == ! y. ? x. y=f x
+\tdx{inj_onto_def} inj_onto f A == !x:A. !y:A. f x=f y --> x=y
+\end{ttbox}
+\caption{Rules of the theory {\tt Set}} \label{chol-set-rules}
+\end{figure}
+
+
+\begin{figure} \underscoreon
+\begin{ttbox}
+\tdx{CollectI} [| P a |] ==> a : \{x.P x\}
+\tdx{CollectD} [| a : \{x.P x\} |] ==> P a
+\tdx{CollectE} [| a : \{x.P x\}; P a ==> W |] ==> W
+
+\tdx{ballI} [| !!x. x:A ==> P x |] ==> ! x:A. P x
+\tdx{bspec} [| ! x:A. P x; x:A |] ==> P x
+\tdx{ballE} [| ! x:A. P x; P x ==> Q; ~ x:A ==> Q |] ==> Q
+
+\tdx{bexI} [| P x; x:A |] ==> ? x:A. P x
+\tdx{bexCI} [| ! x:A. ~ P x ==> P a; a:A |] ==> ? x:A.P x
+\tdx{bexE} [| ? x:A. P x; !!x. [| x:A; P x |] ==> Q |] ==> Q
+\subcaption{Comprehension and Bounded quantifiers}
+
+\tdx{subsetI} (!!x.x:A ==> x:B) ==> A <= B
+\tdx{subsetD} [| A <= B; c:A |] ==> c:B
+\tdx{subsetCE} [| A <= B; ~ (c:A) ==> P; c:B ==> P |] ==> P
+
+\tdx{subset_refl} A <= A
+\tdx{subset_trans} [| A<=B; B<=C |] ==> A<=C
+
+\tdx{equalityI} [| A <= B; B <= A |] ==> A = B
+\tdx{equalityD1} A = B ==> A<=B
+\tdx{equalityD2} A = B ==> B<=A
+\tdx{equalityE} [| A = B; [| A<=B; B<=A |] ==> P |] ==> P
+
+\tdx{equalityCE} [| A = B; [| c:A; c:B |] ==> P;
+ [| ~ c:A; ~ c:B |] ==> P
+ |] ==> P
+\subcaption{The subset and equality relations}
+\end{ttbox}
+\caption{Derived rules for set theory} \label{chol-set1}
+\end{figure}
+
+
+\begin{figure} \underscoreon
+\begin{ttbox}
+\tdx{emptyE} a : \{\} ==> P
+
+\tdx{insertI1} a : insert a B
+\tdx{insertI2} a : B ==> a : insert b B
+\tdx{insertE} [| a : insert b A; a=b ==> P; a:A ==> P |] ==> P
+
+\tdx{ComplI} [| c:A ==> False |] ==> c : Compl A
+\tdx{ComplD} [| c : Compl A |] ==> ~ c:A
+
+\tdx{UnI1} c:A ==> c : A Un B
+\tdx{UnI2} c:B ==> c : A Un B
+\tdx{UnCI} (~c:B ==> c:A) ==> c : A Un B
+\tdx{UnE} [| c : A Un B; c:A ==> P; c:B ==> P |] ==> P
+
+\tdx{IntI} [| c:A; c:B |] ==> c : A Int B
+\tdx{IntD1} c : A Int B ==> c:A
+\tdx{IntD2} c : A Int B ==> c:B
+\tdx{IntE} [| c : A Int B; [| c:A; c:B |] ==> P |] ==> P
+
+\tdx{UN_I} [| a:A; b: B a |] ==> b: (UN x:A. B x)
+\tdx{UN_E} [| b: (UN x:A. B x); !!x.[| x:A; b:B x |] ==> R |] ==> R
+
+\tdx{INT_I} (!!x. x:A ==> b: B x) ==> b : (INT x:A. B x)
+\tdx{INT_D} [| b: (INT x:A. B x); a:A |] ==> b: B a
+\tdx{INT_E} [| b: (INT x:A. B x); b: B a ==> R; ~ a:A ==> R |] ==> R
+
+\tdx{UnionI} [| X:C; A:X |] ==> A : Union C
+\tdx{UnionE} [| A : Union C; !!X.[| A:X; X:C |] ==> R |] ==> R
+
+\tdx{InterI} [| !!X. X:C ==> A:X |] ==> A : Inter C
+\tdx{InterD} [| A : Inter C; X:C |] ==> A:X
+\tdx{InterE} [| A : Inter C; A:X ==> R; ~ X:C ==> R |] ==> R
+
+\tdx{PowI} A<=B ==> A: Pow B
+\tdx{PowD} A: Pow B ==> A<=B
+\end{ttbox}
+\caption{Further derived rules for set theory} \label{chol-set2}
+\end{figure}
+
+
+\subsection{Axioms and rules of set theory}
+Figure~\ref{chol-set-rules} presents the rules of theory \thydx{Set}. The
+axioms \tdx{mem_Collect_eq} and \tdx{Collect_mem_eq} assert
+that the functions {\tt Collect} and \hbox{\tt op :} are isomorphisms. Of
+course, \hbox{\tt op :} also serves as the membership relation.
+
+All the other axioms are definitions. They include the empty set, bounded
+quantifiers, unions, intersections, complements and the subset relation.
+They also include straightforward properties of functions: image~({\tt``}) and
+{\tt range}, and predicates concerning monotonicity, injectiveness and
+surjectiveness.
+
+The predicate \cdx{inj_onto} is used for simulating type definitions.
+The statement ${\tt inj_onto}~f~A$ asserts that $f$ is injective on the
+set~$A$, which specifies a subset of its domain type. In a type
+definition, $f$ is the abstraction function and $A$ is the set of valid
+representations; we should not expect $f$ to be injective outside of~$A$.
+
+\begin{figure} \underscoreon
+\begin{ttbox}
+\tdx{Inv_f_f} inj f ==> Inv f (f x) = x
+\tdx{f_Inv_f} y : range f ==> f(Inv f y) = y
+
+%\tdx{Inv_injective}
+% [| Inv f x=Inv f y; x: range f; y: range f |] ==> x=y
+%
+\tdx{imageI} [| x:A |] ==> f x : f``A
+\tdx{imageE} [| b : f``A; !!x.[| b=f x; x:A |] ==> P |] ==> P
+
+\tdx{rangeI} f x : range f
+\tdx{rangeE} [| b : range f; !!x.[| b=f x |] ==> P |] ==> P
+
+\tdx{monoI} [| !!A B. A <= B ==> f A <= f B |] ==> mono f
+\tdx{monoD} [| mono f; A <= B |] ==> f A <= f B
+
+\tdx{injI} [| !! x y. f x = f y ==> x=y |] ==> inj f
+\tdx{inj_inverseI} (!!x. g(f x) = x) ==> inj f
+\tdx{injD} [| inj f; f x = f y |] ==> x=y
+
+\tdx{inj_ontoI} (!!x y. [| f x=f y; x:A; y:A |] ==> x=y) ==> inj_onto f A
+\tdx{inj_ontoD} [| inj_onto f A; f x=f y; x:A; y:A |] ==> x=y
+
+\tdx{inj_onto_inverseI}
+ (!!x. x:A ==> g(f x) = x) ==> inj_onto f A
+\tdx{inj_onto_contraD}
+ [| inj_onto f A; x~=y; x:A; y:A |] ==> ~ f x=f y
+\end{ttbox}
+\caption{Derived rules involving functions} \label{chol-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{chol-subset}
+\end{figure}
+
+
+\begin{figure} \underscoreon \hfuzz=4pt%suppress "Overfull \hbox" message
+\begin{ttbox}
+\tdx{Int_absorb} A Int A = A
+\tdx{Int_commute} A Int B = B Int A
+\tdx{Int_assoc} (A Int B) Int C = A Int (B Int C)
+\tdx{Int_Un_distrib} (A Un B) Int C = (A Int C) Un (B Int C)
+
+\tdx{Un_absorb} A Un A = A
+\tdx{Un_commute} A Un B = B Un A
+\tdx{Un_assoc} (A Un B) Un C = A Un (B Un C)
+\tdx{Un_Int_distrib} (A Int B) Un C = (A Un C) Int (B Un C)
+
+\tdx{Compl_disjoint} A Int (Compl A) = \{x.False\}
+\tdx{Compl_partition} A Un (Compl A) = \{x.True\}
+\tdx{double_complement} Compl(Compl A) = A
+\tdx{Compl_Un} Compl(A Un B) = (Compl A) Int (Compl B)
+\tdx{Compl_Int} Compl(A Int B) = (Compl A) Un (Compl B)
+
+\tdx{Union_Un_distrib} Union(A Un B) = (Union A) Un (Union B)
+\tdx{Int_Union} A Int (Union B) = (UN C:B. A Int C)
+\tdx{Un_Union_image} (UN x:C.(A x) Un (B x)) = Union(A``C) Un Union(B``C)
+
+\tdx{Inter_Un_distrib} Inter(A Un B) = (Inter A) Int (Inter B)
+\tdx{Un_Inter} A Un (Inter B) = (INT C:B. A Un C)
+\tdx{Int_Inter_image} (INT x:C.(A x) Int (B x)) = Inter(A``C) Int Inter(B``C)
+\end{ttbox}
+\caption{Set equalities} \label{chol-equalities}
+\end{figure}
+
+
+Figures~\ref{chol-set1} and~\ref{chol-set2} present derived rules. Most are
+obvious and resemble rules of Isabelle's {\ZF} set theory. Certain rules,
+such as \tdx{subsetCE}, \tdx{bexCI} and \tdx{UnCI},
+are designed for classical reasoning; the rules \tdx{subsetD},
+\tdx{bexI}, \tdx{Un1} and~\tdx{Un2} are not
+strictly necessary but yield more natural proofs. Similarly,
+\tdx{equalityCE} supports classical reasoning about extensionality,
+after the fashion of \tdx{iffCE}. See the file {\tt CHOL/Set.ML} for
+proofs pertaining to set theory.
+
+Figure~\ref{chol-fun} presents derived inference rules involving functions.
+They also include rules for \cdx{Inv}, which is defined in theory~{\tt
+ CHOL}; note that ${\tt Inv}~f$ applies the Axiom of Choice to yield an
+inverse of~$f$. They also include natural deduction rules for the image
+and range operators, and for the predicates {\tt inj} and {\tt inj_onto}.
+Reasoning about function composition (the operator~\sdx{o}) and the
+predicate~\cdx{surj} is done simply by expanding the definitions. See
+the file {\tt CHOL/fun.ML} for a complete listing of the derived rules.
+
+Figure~\ref{chol-subset} presents lattice properties of the subset relation.
+Unions form least upper bounds; non-empty intersections form greatest lower
+bounds. Reasoning directly about subsets often yields clearer proofs than
+reasoning about the membership relation. See the file {\tt CHOL/subset.ML}.
+
+Figure~\ref{chol-equalities} presents many common set equalities. They
+include commutative, associative and distributive laws involving unions,
+intersections and complements. The proofs are mostly trivial, using the
+classical reasoner; see file {\tt CHOL/equalities.ML}.
+
+
+\begin{figure}
+\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. \{(x,y)\}
+
+
+\tdx{Pair_inject} [| (a, b) = (a',b'); [| a=a'; b=b' |] ==> R |] ==> R
+\tdx{fst_conv} fst (a,b) = a
+\tdx{snd_conv} snd (a,b) = b
+\tdx{split} split c (a,b) = c a b
+
+\tdx{surjective_pairing} p = (fst p,snd p)
+
+\tdx{SigmaI} [| a:A; b:B a |] ==> (a,b) : Sigma A B
+
+\tdx{SigmaE} [| c: Sigma A B;
+ !!x y.[| x:A; y:B x; c=(x,y) |] ==> P |] ==> P
+\end{ttbox}
+\caption{Type $\alpha\times\beta$}\label{chol-prod}
+\end{figure}
+
+
+\begin{figure}
+\begin{constants}
+ \it symbol & \it meta-type & & \it description \\
+ \cdx{Inl} & $\alpha \To \alpha+\beta$ & & first injection\\
+ \cdx{Inr} & $\beta \To \alpha+\beta$ & & second injection\\
+ \cdx{sum_case} & $[\alpha\To\gamma, \beta\To\gamma, \alpha+\beta] \To\gamma$
+ & & conditional
+\end{constants}
+\begin{ttbox}\makeatletter
+\tdx{sum_case_def} sum_case == (\%f g p. @z. (!x. p=Inl x --> z=f x) &
+ (!y. p=Inr y --> z=g y))
+
+\tdx{Inl_not_Inr} ~ Inl a=Inr b
+
+\tdx{inj_Inl} inj Inl
+\tdx{inj_Inr} inj Inr
+
+\tdx{sumE} [| !!x::'a. P(Inl x); !!y::'b. P(Inr y) |] ==> P s
+
+\tdx{sum_case_Inl} sum_case f g (Inl x) = f x
+\tdx{sum_case_Inr} sum_case f g (Inr x) = g x
+
+\tdx{surjective_sum} sum_case (\%x::'a. f(Inl x)) (\%y::'b. f(Inr y)) s = f s
+\end{ttbox}
+\caption{Type $\alpha+\beta$}\label{chol-sum}
+\end{figure}
+
+
+\section{Generic packages and classical reasoning}
+\CHOL\ instantiates most of Isabelle's generic packages;
+see {\tt CHOL/ROOT.ML} for details.
+\begin{itemize}
+\item
+Because it includes a general substitution rule, \CHOL\ instantiates the
+tactic {\tt hyp_subst_tac}, which substitutes for an equality
+throughout a subgoal and its hypotheses.
+\item
+It instantiates the simplifier, defining~\ttindexbold{HOL_ss} as the
+simplification set for higher-order logic. Equality~($=$), which also
+expresses logical equivalence, may be used for rewriting. See the file
+{\tt CHOL/simpdata.ML} for a complete listing of the simplification
+rules.
+\item
+It instantiates the classical reasoner, as described below.
+\end{itemize}
+\CHOL\ 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{chol-lemmas2} above.
+
+The classical reasoner is set up as the structure
+{\tt Classical}. This structure is open, so {\ML} identifiers such
+as {\tt step_tac}, {\tt fast_tac}, {\tt best_tac}, etc., refer to it.
+\HOL\ defines the following classical rule sets:
+\begin{ttbox}
+prop_cs : claset
+HOL_cs : claset
+set_cs : claset
+\end{ttbox}
+\begin{ttdescription}
+\item[\ttindexbold{prop_cs}] contains the propositional rules, namely
+those for~$\top$, $\bot$, $\conj$, $\disj$, $\neg$, $\imp$ and~$\bimp$,
+along with the rule~{\tt refl}.
+
+\item[\ttindexbold{HOL_cs}] extends {\tt prop_cs} with the safe rules
+ {\tt allI} and~{\tt exE} and the unsafe rules {\tt allE}
+ and~{\tt exI}, as well as rules for unique existence. Search using
+ this classical set is incomplete: quantified formulae are used at most
+ once.
+
+\item[\ttindexbold{set_cs}] extends {\tt HOL_cs} with rules for the bounded
+ quantifiers, subsets, comprehensions, unions and intersections,
+ complements, finite sets, images and ranges.
+\end{ttdescription}
+\noindent
+See \iflabelundefined{chap:classical}{the {\em Reference Manual\/}}%
+ {Chap.\ts\ref{chap:classical}}
+for more discussion of classical proof methods.
+
+
+\section{Types}
+The basic higher-order logic is augmented with a tremendous amount of
+material, including support for recursive function and type definitions. A
+detailed discussion appears elsewhere~\cite{paulson-coind}. The simpler
+definitions are the same as those used by the {\sc hol} system, but my
+treatment of recursive types differs from Melham's~\cite{melham89}. The
+present section describes product, sum, natural number and list types.
+
+\subsection{Product and sum types}\index{*"* type}\index{*"+ type}
+Theory \thydx{Prod} defines the product type $\alpha\times\beta$, with
+the ordered pair syntax {\tt($a$,$b$)}. Theory \thydx{Sum} defines the
+sum type $\alpha+\beta$. These use fairly standard constructions; see
+Figs.\ts\ref{chol-prod} and~\ref{chol-sum}. Because Isabelle does not
+support abstract type definitions, the isomorphisms between these types and
+their representations are made explicitly.
+
+Most of the definitions are suppressed, but observe that the projections
+and conditionals are defined as descriptions. Their properties are easily
+proved using \tdx{select_equality}.
+
+\begin{figure}
+\index{*"< symbol}
+\index{*"* symbol}
+\index{*div symbol}
+\index{*mod symbol}
+\index{*"+ symbol}
+\index{*"- symbol}
+\begin{constants}
+ \it symbol & \it meta-type & \it priority & \it description \\
+ \cdx{0} & $nat$ & & zero \\
+ \cdx{Suc} & $nat \To nat$ & & successor function\\
+ \cdx{nat_case} & $[\alpha, nat\To\alpha, nat] \To\alpha$
+ & & conditional\\
+ \cdx{nat_rec} & $[nat, \alpha, [nat, \alpha]\To\alpha] \To \alpha$
+ & & primitive recursor\\
+ \cdx{pred_nat} & $(nat\times nat) set$ & & predecessor relation\\
+ \tt * & $[nat,nat]\To nat$ & Left 70 & multiplication \\
+ \tt div & $[nat,nat]\To nat$ & Left 70 & division\\
+ \tt mod & $[nat,nat]\To nat$ & Left 70 & modulus\\
+ \tt + & $[nat,nat]\To nat$ & Left 65 & addition\\
+ \tt - & $[nat,nat]\To nat$ & Left 65 & subtraction
+\end{constants}
+\subcaption{Constants and infixes}
+
+\begin{ttbox}\makeatother
+\tdx{nat_case_def} nat_case == (\%a f n. @z. (n=0 --> z=a) &
+ (!x. n=Suc x --> z=f x))
+\tdx{pred_nat_def} pred_nat == \{p. ? n. p = (n, Suc n)\}
+\tdx{less_def} m<n == (m,n):pred_nat^+
+\tdx{nat_rec_def} nat_rec n c d ==
+ wfrec pred_nat n (nat_case (\%g.c) (\%m g. d m (g m)))
+
+\tdx{add_def} m+n == nat_rec m n (\%u v. Suc v)
+\tdx{diff_def} m-n == nat_rec n m (\%u v. nat_rec v 0 (\%x y.x))
+\tdx{mult_def} m*n == nat_rec m 0 (\%u v. n + v)
+\tdx{mod_def} m mod n == wfrec (trancl pred_nat)
+ m (\%j f. if j<n then j else f j-n))
+\tdx{quo_def} m div n == wfrec (trancl pred_nat),
+ m (\%j f. if j<n then 0 else Suc(f j-n))
+\subcaption{Definitions}
+\end{ttbox}
+\caption{Defining {\tt nat}, the type of natural numbers} \label{chol-nat1}
+\end{figure}
+
+
+\begin{figure} \underscoreon
+\begin{ttbox}
+\tdx{nat_induct} [| P 0; !!k. [| P k |] ==> P(Suc k) |] ==> P n
+
+\tdx{Suc_not_Zero} Suc m ~= 0
+\tdx{inj_Suc} inj Suc
+\tdx{n_not_Suc_n} n~=Suc n
+\subcaption{Basic properties}
+
+\tdx{pred_natI} (n, Suc n) : pred_nat
+\tdx{pred_natE}
+ [| p : pred_nat; !!x n. [| p = (n, Suc n) |] ==> R |] ==> R
+
+\tdx{nat_case_0} nat_case a f 0 = a
+\tdx{nat_case_Suc} nat_case a f (Suc k) = f k
+
+\tdx{wf_pred_nat} wf pred_nat
+\tdx{nat_rec_0} nat_rec 0 c h = c
+\tdx{nat_rec_Suc} nat_rec (Suc n) c h = h n (nat_rec n c h)
+\subcaption{Case analysis and primitive recursion}
+
+\tdx{less_trans} [| i<j; j<k |] ==> i<k
+\tdx{lessI} n < Suc n
+\tdx{zero_less_Suc} 0 < Suc n
+
+\tdx{less_not_sym} n<m --> ~ m<n
+\tdx{less_not_refl} ~ n<n
+\tdx{not_less0} ~ n<0
+
+\tdx{Suc_less_eq} (Suc m < Suc n) = (m<n)
+\tdx{less_induct} [| !!n. [| ! m. m<n --> P m |] ==> P n |] ==> P n
+
+\tdx{less_linear} m<n | m=n | n<m
+\subcaption{The less-than relation}
+\end{ttbox}
+\caption{Derived rules for {\tt nat}} \label{chol-nat2}
+\end{figure}
+
+
+\subsection{The type of natural numbers, {\tt nat}}
+The theory \thydx{Nat} defines the natural numbers in a roundabout but
+traditional way. The axiom of infinity postulates an type~\tydx{ind} of
+individuals, which is non-empty and closed under an injective operation.
+The natural numbers are inductively generated by choosing an arbitrary
+individual for~0 and using the injective operation to take successors. As
+usual, the isomorphisms between~\tydx{nat} and its representation are made
+explicitly.
+
+The definition makes use of a least fixed point operator \cdx{lfp},
+defined using the Knaster-Tarski theorem. This is used to define the
+operator \cdx{trancl}, for taking the transitive closure of a relation.
+Primitive recursion makes use of \cdx{wfrec}, an operator for recursion
+along arbitrary well-founded relations. The corresponding theories are
+called {\tt Lfp}, {\tt Trancl} and {\tt WF}\@. Elsewhere I have described
+similar constructions in the context of set theory~\cite{paulson-set-II}.
+
+Type~\tydx{nat} is postulated to belong to class~\cldx{ord}, which
+overloads $<$ and $\leq$ on the natural numbers. As of this writing,
+Isabelle provides no means of verifying that such overloading is sensible;
+there is no means of specifying the operators' properties and verifying
+that instances of the operators satisfy those properties. To be safe, the
+\CHOL\ theory includes no polymorphic axioms asserting general properties of
+$<$ and~$\leq$.
+
+Theory \thydx{Arith} develops arithmetic on the natural numbers. It
+defines addition, multiplication, subtraction, division, and remainder.
+Many of their properties are proved: commutative, associative and
+distributive laws, identity and cancellation laws, etc. The most
+interesting result is perhaps the theorem $a \bmod b + (a/b)\times b = a$.
+Division and remainder are defined by repeated subtraction, which requires
+well-founded rather than primitive recursion. See Figs.\ts\ref{chol-nat1}
+and~\ref{chol-nat2}.
+
+The predecessor relation, \cdx{pred_nat}, is shown to be well-founded.
+Recursion along this relation resembles primitive recursion, but is
+stronger because we are in higher-order logic; using primitive recursion to
+define a higher-order function, we can easily Ackermann's function, which
+is not primitive recursive \cite[page~104]{thompson91}.
+The transitive closure of \cdx{pred_nat} is~$<$. Many functions on the
+natural numbers are most easily expressed using recursion along~$<$.
+
+The tactic {\tt\ttindex{nat_ind_tac} "$n$" $i$} performs induction over the
+variable~$n$ in subgoal~$i$.
+
+\begin{figure}
+\index{#@{\tt\#} symbol}
+\index{"@@{\tt\at} symbol}
+\begin{constants}
+ \it symbol & \it meta-type & \it priority & \it description \\
+ \cdx{Nil} & $\alpha list$ & & empty list\\
+ \tt \# & $[\alpha,\alpha list]\To \alpha list$ & Right 65 &
+ list constructor \\
+ \cdx{null} & $\alpha list \To bool$ & & emptiness test\\
+ \cdx{hd} & $\alpha list \To \alpha$ & & head \\
+ \cdx{tl} & $\alpha list \To \alpha list$ & & tail \\
+ \cdx{ttl} & $\alpha list \To \alpha list$ & & total tail \\
+ \tt\at & $[\alpha list,\alpha list]\To \alpha list$ & Left 65 & append \\
+ \sdx{mem} & $[\alpha,\alpha list]\To bool$ & Left 55 & membership\\
+ \cdx{map} & $(\alpha\To\beta) \To (\alpha list \To \beta list)$
+ & & mapping functional\\
+ \cdx{filter} & $(\alpha \To bool) \To (\alpha list \To \alpha list)$
+ & & filter functional\\
+ \cdx{list_all}& $(\alpha \To bool) \To (\alpha list \To bool)$
+ & & forall functional\\
+ \cdx{list_rec} & $[\alpha list, \beta, [\alpha ,\alpha list,
+\beta]\To\beta] \To \beta$
+ & & list recursor
+\end{constants}
+\subcaption{Constants and infixes}
+
+\begin{center} \tt\frenchspacing
+\begin{tabular}{rrr}
+ \it external & \it internal & \it description \\{}
+ \sdx{[]} & Nil & \rm empty list \\{}
+ [$x@1$, $\dots$, $x@n$] & $x@1$ \# $\cdots$ \# $x@n$ \# [] &
+ \rm finite list \\{}
+ [$x$:$l$. $P$] & filter ($\lambda x{.}P$) $l$ &
+ \rm list comprehension
+\end{tabular}
+\end{center}
+\subcaption{Translations}
+
+\begin{ttbox}
+\tdx{list_induct} [| P []; !!x xs. [| P xs |] ==> P x#xs) |] ==> P l
+
+\tdx{Cons_not_Nil} (x # xs) ~= []
+\tdx{Cons_Cons_eq} ((x # xs) = (y # ys)) = (x=y & xs=ys)
+\subcaption{Induction and freeness}
+\end{ttbox}
+\caption{The theory \thydx{List}} \label{chol-list}
+\end{figure}
+
+\begin{figure}
+\begin{ttbox}\makeatother
+\tdx{list_rec_Nil} list_rec [] c h = c
+\tdx{list_rec_Cons} list_rec a#l c h = h a l (list_rec l c h)
+
+\tdx{list_case_Nil} list_case c h [] = c
+\tdx{list_case_Cons} list_case c h x#xs = h x xs
+
+\tdx{map_Nil} map f [] = []
+\tdx{map_Cons} map f x \# xs = f x \# map f xs
+
+\tdx{null_Nil} null [] = True
+\tdx{null_Cons} null x#xs = False
+
+\tdx{hd_Cons} hd x#xs = x
+\tdx{tl_Cons} tl x#xs = xs
+
+\tdx{ttl_Nil} ttl [] = []
+\tdx{ttl_Cons} ttl x#xs = xs
+
+\tdx{append_Nil} [] @ ys = ys
+\tdx{append_Cons} (x#xs) \at ys = x # xs \at ys
+
+\tdx{mem_Nil} x mem [] = False
+\tdx{mem_Cons} x mem (y#ys) = if y=x then True else x mem ys
+
+\tdx{filter_Nil} filter P [] = []
+\tdx{filter_Cons} filter P x#xs = if P x then x#filter P xs else filter P xs
+
+\tdx{list_all_Nil} list_all P [] = True
+\tdx{list_all_Cons} list_all P x#xs = (P x & list_all P xs)
+\end{ttbox}
+\caption{Rewrite rules for lists} \label{chol-list-simps}
+\end{figure}
+
+
+\subsection{The type constructor for lists, {\tt list}}
+\index{*list type}
+
+\CHOL's definition of lists is an example of an experimental method for
+handling recursive data types. Figure~\ref{chol-list} presents the theory
+\thydx{List}: the basic list operations with their types and properties.
+
+The \sdx{case} construct is defined by the following translation:
+{\dquotes
+\begin{eqnarray*}
+ \begin{array}{r@{\;}l@{}l}
+ "case " e " of" & "[]" & " => " a\\
+ "|" & x"\#"xs & " => " b
+ \end{array}
+ & \equiv &
+ "list_case"~a~(\lambda x\;xs.b)~e
+\end{eqnarray*}}%
+The theory includes \cdx{list_rec}, a primitive recursion operator
+for lists. It is derived from well-founded recursion, a general principle
+that can express arbitrary total recursive functions.
+
+The simpset \ttindex{list_ss} contains, along with additional useful lemmas,
+the basic rewrite rules that appear in Fig.\ts\ref{chol-list-simps}.
+
+The tactic {\tt\ttindex{list_ind_tac} "$xs$" $i$} performs induction over the
+variable~$xs$ in subgoal~$i$.
+
+
+\section{Datatype declarations}
+\index{*datatype|(}
+
+\underscoreon
+
+It is often necessary to extend a theory with \ML-like datatypes. This
+extension consists of the new type, declarations of its constructors and
+rules that describe the new type. The theory definition section {\tt
+ datatype} represents a compact way of doing this.
+
+
+\subsection{Foundations}
+
+A datatype declaration has the following general structure:
+\[ \mbox{\tt datatype}~ (\alpha_1,\dots,\alpha_n)t ~=~
+ C_1~\tau_{11}~\dots~\tau_{1k_1} ~\mid~ \dots ~\mid~
+ C_m~\tau_{m1}~\dots~\tau_{mk_m}
+\]
+where $\alpha_i$ are type variables, $C_i$ are distinct constructor names and
+$\tau_{ij}$ are one of the following:
+\begin{itemize}
+\item type variables $\alpha_1,\dots,\alpha_n$,
+\item types $(\beta_1,\dots,\beta_l)s$ where $s$ is a previously declared
+ type or type synonym and $\{\beta_1,\dots,\beta_l\} \subseteq
+ \{\alpha_1,\dots,\alpha_n\}$,
+\item the newly defined type $(\alpha_1,\dots,\alpha_n)t$ \footnote{This
+ makes it a recursive type. To ensure that the new type is not empty at
+ least one constructor must consist of only non-recursive type
+ components.}
+\end{itemize}
+If you would like one of the $\tau_{ij}$ to be a complex type expression
+$\tau$ you need to declare a new type synonym $syn = \tau$ first and use
+$syn$ in place of $\tau$. Of course this does not work if $\tau$ mentions the
+recursive type itself, thus ruling out problematic cases like \[ \mbox{\tt
+ datatype}~ t ~=~ C(t \To t) \] together with unproblematic ones like \[
+\mbox{\tt datatype}~ t ~=~ C(t~list). \]
+
+The constructors are automatically defined as functions of their respective
+type:
+\[ C_j : [\tau_{j1},\dots,\tau_{jk_j}] \To (\alpha_1,\dots,\alpha_n)t \]
+These functions have certain {\em freeness} properties:
+\begin{description}
+\item[\tt distinct] They are distinct:
+\[ C_i~x_1~\dots~x_{k_i} \neq C_j~y_1~\dots~y_{k_j} \qquad
+ \mbox{for all}~ i \neq j.
+\]
+\item[\tt inject] They are injective:
+\[ (C_j~x_1~\dots~x_{k_j} = C_j~y_1~\dots~y_{k_j}) =
+ (x_1 = y_1 \land \dots \land x_{k_j} = y_{k_j})
+\]
+\end{description}
+Because the number of inequalities is quadratic in the number of
+constructors, a different method is used if their number exceeds
+a certain value, currently 4. In that case every constructor is mapped to a
+natural number
+\[
+\begin{array}{lcl}
+\mbox{\it t\_ord}(C_1~x_1~\dots~x_{k_1}) & = & 0 \\
+& \vdots & \\
+\mbox{\it t\_ord}(C_m x_1~\dots~x_{k_m}) & = & m-1
+\end{array}
+\]
+and distinctness of constructors is expressed by:
+\[
+\mbox{\it t\_ord}~x \neq \mbox{\it t\_ord}~y \Imp x \neq y.
+\]
+In addition a structural induction axiom {\tt induct} is provided:
+\[
+\infer{P x}
+{\begin{array}{lcl}
+\Forall x_1\dots x_{k_1}.
+ \List{P~x_{r_{11}}; \dots; P~x_{r_{1l_1}}} &
+ \Imp & P(C_1~x_1~\dots~x_{k_1}) \\
+ & \vdots & \\
+\Forall x_1\dots x_{k_m}.
+ \List{P~x_{r_{m1}}; \dots; P~x_{r_{ml_m}}} &
+ \Imp & P(C_m~x_1~\dots~x_{k_m})
+\end{array}}
+\]
+where $\{r_{j1},\dots,r_{jl_j}\} = \{i \in \{1,\dots k_j\} ~\mid~ \tau_{ji}
+= (\alpha_1,\dots,\alpha_n)t \}$, i.e.\ the property $P$ can be assumed for
+all arguments of the recursive type.
+
+The type also comes with an \ML-like \sdx{case}-construct:
+\[
+\begin{array}{rrcl}
+\mbox{\tt case}~e~\mbox{\tt of} & C_1~x_{11}~\dots~x_{1k_1} & \To & e_1 \\
+ \vdots \\
+ \mid & C_m~x_{m1}~\dots~x_{mk_m} & \To & e_m
+\end{array}
+\]
+In contrast to \ML, {\em all} constructors must be present, their order is
+fixed, and nested patterns are not supported.
+
+
+\subsection{Defining datatypes}
+
+A datatype is defined in a theory definition file using the keyword {\tt
+ datatype}. The definition following {\tt datatype} must conform to the
+syntax of {\em typedecl} specified in Fig.~\ref{datatype-grammar} and must
+obey the rules in the previous section. As a result the theory is extended
+with the new type, the constructors, and the theorems listed in the previous
+section.
+
+\begin{figure}
+\begin{rail}
+typedecl : typevarlist id '=' (cons + '|')
+ ;
+cons : (id | string) ( () | '(' (typ + ',') ')' ) ( () | mixfix )
+ ;
+typ : typevarlist id
+ | tid
+ ;
+typevarlist : () | tid | '(' (tid + ',') ')'
+ ;
+\end{rail}
+\caption{Syntax of datatype declarations}
+\label{datatype-grammar}
+\end{figure}
+
+Reading the theory file produces a structure which, in addition to the usual
+components, contains a structure named $t$ for each datatype $t$ defined in
+the file.\footnote{Otherwise multiple datatypes in the same theory file would
+ lead to name clashes.} Each structure $t$ contains the following elements:
+\begin{ttbox}
+val distinct : thm list
+val inject : thm list
+val induct : thm
+val cases : thm list
+val simps : thm list
+val induct_tac : string -> int -> tactic
+\end{ttbox}
+{\tt distinct}, {\tt inject} and {\tt induct} contain the theorems described
+above. For convenience {\tt distinct} contains inequalities in both
+directions.
+\begin{warn}
+ If there are five or more constructors, the {\em t\_ord} scheme is used for
+ {\tt distinct}. In this case the theory {\tt Arith} must be contained
+ in the current theory, if necessary by including it explicitly.
+\end{warn}
+The reduction rules of the {\tt case}-construct are in {\tt cases}. All
+theorems from {\tt distinct}, {\tt inject} and {\tt cases} are combined in
+{\tt simps} for use with the simplifier. The tactic {\verb$induct_tac$~{\em
+ var i}\/} applies structural induction over variable {\em var} to
+subgoal {\em i}.
+
+
+\subsection{Examples}
+
+\subsubsection{The datatype $\alpha~list$}
+
+We want to define the type $\alpha~list$.\footnote{Of course there is a list
+ type in CHOL already. This is only an example.} To do this we have to build
+a new theory that contains the type definition. We start from {\tt CHOL}.
+\begin{ttbox}
+MyList = CHOL +
+ datatype 'a list = Nil | Cons 'a ('a list)
+end
+\end{ttbox}
+After loading the theory (\verb$use_thy "MyList"$), we can prove
+$Cons~x~xs\neq xs$. First we build a suitable simpset for the simplifier:
+\begin{ttbox}
+val mylist_ss = HOL_ss addsimps MyList.list.simps;
+goal MyList.thy "!x. Cons x xs ~= xs";
+{\out Level 0}
+{\out ! x. Cons x xs ~= xs}
+{\out 1. ! x. Cons x xs ~= xs}
+\end{ttbox}
+This can be proved by the structural induction tactic:
+\begin{ttbox}
+by (MyList.list.induct_tac "xs" 1);
+{\out Level 1}
+{\out ! x. Cons x xs ~= xs}
+{\out 1. ! x. Cons x Nil ~= Nil}
+{\out 2. !!a list.}
+{\out ! x. Cons x list ~= list ==>}
+{\out ! x. Cons x (Cons a list) ~= Cons a list}
+\end{ttbox}
+The first subgoal can be proved with the simplifier and the distinctness
+axioms which are part of \verb$mylist_ss$.
+\begin{ttbox}
+by (simp_tac mylist_ss 1);
+{\out Level 2}
+{\out ! x. Cons x xs ~= xs}
+{\out 1. !!a list.}
+{\out ! x. Cons x list ~= list ==>}
+{\out ! x. Cons x (Cons a list) ~= Cons a list}
+\end{ttbox}
+Using the freeness axioms we can quickly prove the remaining goal.
+\begin{ttbox}
+by (asm_simp_tac mylist_ss 1);
+{\out Level 3}
+{\out ! x. Cons x xs ~= xs}
+{\out No subgoals!}
+\end{ttbox}
+Because both subgoals were proved by almost the same tactic we could have
+done that in one step using
+\begin{ttbox}
+by (ALLGOALS (asm_simp_tac mylist_ss));
+\end{ttbox}
+
+
+\subsubsection{The datatype $\alpha~list$ with mixfix syntax}
+
+In this example we define the type $\alpha~list$ again but this time we want
+to write {\tt []} instead of {\tt Nil} and we want to use the infix operator
+\verb|#| instead of {\tt Cons}. To do this we simply add mixfix annotations
+after the constructor declarations as follows:
+\begin{ttbox}
+MyList = CHOL +
+ datatype 'a list = "[]" ("[]")
+ | "#" 'a ('a list) (infixr 70)
+end
+\end{ttbox}
+Now the theorem in the previous example can be written \verb|x#xs ~= xs|. The
+proof is the same.
+
+
+\subsubsection{A datatype for weekdays}
+
+This example shows a datatype that consists of more than four constructors:
+\begin{ttbox}
+Days = Arith +
+ datatype days = Mo | Tu | We | Th | Fr | Sa | So
+end
+\end{ttbox}
+Because there are more than four constructors, the theory must be based on
+{\tt Arith}. Inequality is defined via a function \verb|days_ord|. Although
+the expression \verb|Mo ~= Tu| is not directly contained in {\tt distinct},
+it can be proved by the simplifier if \verb$arith_ss$ is used:
+\begin{ttbox}
+val days_ss = arith_ss addsimps Days.days.simps;
+
+goal Days.thy "Mo ~= Tu";
+by (simp_tac days_ss 1);
+\end{ttbox}
+Note that usually it is not necessary to derive these inequalities explicitly
+because the simplifier will dispose of them automatically.
+
+\subsection{Primitive recursive functions}
+\index{primitive recursion|(}
+\index{*primrec|(}
+
+Datatypes come with a uniform way of defining functions, {\bf primitive
+ recursion}. Although it is possible to define primitive recursive functions
+by asserting their reduction rules as new axioms, e.g.\
+\begin{ttbox}
+Append = MyList +
+consts app :: "['a list,'a list] => 'a list"
+rules
+ app_Nil "app [] ys = ys"
+ app_Cons "app x#xs ys = x#app xs ys"
+end
+\end{ttbox}
+this carries with it the danger of accidentally asserting an inconsistency,
+as in \verb$app [] ys = us$. Therefore primitive recursive functions on
+datatypes can be defined with a special syntax:
+\begin{ttbox}
+Append = MyList +
+consts app :: "'['a list,'a list] => 'a list"
+primrec app MyList.list
+ app_Nil "app [] ys = ys"
+ app_Cons "app x#xs ys = x#app xs ys"
+end
+\end{ttbox}
+The system will now check that the two rules \verb$app_Nil$ and
+\verb$app_Cons$ do indeed form a primitive recursive definition, thus
+ensuring that consistency is maintained. For example
+\begin{ttbox}
+primrec app MyList.list
+ app_Nil "app [] ys = us"
+\end{ttbox}
+is rejected:
+\begin{ttbox}
+Extra variables on rhs
+\end{ttbox}
+\bigskip
+
+The general form of a primitive recursive definition is
+\begin{ttbox}
+primrec {\it function} {\it type}
+ {\it reduction rules}
+\end{ttbox}
+where
+\begin{itemize}
+\item {\it function} is the name of the function, either as an {\it id} or a
+ {\it string}. The function must already have been declared.
+\item {\it type} is the name of the datatype, either as an {\it id} or in the
+ long form {\it Thy.t}, where {\it Thy} is the name of the parent theory the
+ datatype was declared in, and $t$ the name of the datatype. The long form
+ is required if the {\tt datatype} and the {\tt primrec} sections are in
+ different theories.
+\item {\it reduction rules} specify one or more named equations of the form
+ {\it id\/}~{\it string}, where the identifier gives the name of the rule in
+ the result structure, and {\it string} is a reduction rule of the form \[
+ f~x_1~\dots~x_m~(C~y_1~\dots~y_k)~z_1~\dots~z_n = r \] such that $C$ is a
+ constructor of the datatype, $r$ contains only the free variables on the
+ left-hand side, and all recursive calls in $r$ are of the form
+ $f~\dots~y_i~\dots$ for some $i$. There must be exactly one reduction
+ rule for each constructor.
+\end{itemize}
+A theory file may contain any number of {\tt primrec} sections which may be
+intermixed with other declarations.
+
+For the consistency-sensitive user it may be reassuring to know that {\tt
+ primrec} does not assert the reduction rules as new axioms but derives them
+as theorems from an explicit definition of the recursive function in terms of
+a recursion operator on the datatype.
+
+The primitive recursive function can also use infix or mixfix syntax:
+\begin{ttbox}
+Append = MyList +
+consts "@" :: "['a list,'a list] => 'a list" (infixr 60)
+primrec "op @" MyList.list
+ app_Nil "[] @ ys = ys"
+ app_Cons "(x#xs) @ ys = x#(xs @ ys)"
+end
+\end{ttbox}
+
+The reduction rules become part of the ML structure \verb$Append$ and can
+be used to prove theorems about the function:
+\begin{ttbox}
+val append_ss = HOL_ss addsimps [Append.app_Nil,Append.app_Cons];
+
+goal Append.thy "(xs @ ys) @ zs = xs @ (ys @ zs)";
+by (MyList.list.induct_tac "xs" 1);
+by (ALLGOALS(asm_simp_tac append_ss));
+\end{ttbox}
+
+%Note that underdefined primitive recursive functions are allowed:
+%\begin{ttbox}
+%Tl = MyList +
+%consts tl :: "'a list => 'a list"
+%primrec tl MyList.list
+% tl_Cons "tl(x#xs) = xs"
+%end
+%\end{ttbox}
+%Nevertheless {\tt tl} is total, although we do not know what the result of
+%\verb$tl([])$ is.
+
+\index{primitive recursion|)}
+\index{*primrec|)}
+\index{*datatype|)}
+
+
+\section{Inductive and coinductive definitions}
+\index{*inductive|(}
+\index{*coinductive|(}
+
+An {\bf inductive definition} specifies the least set closed under given
+rules. For example, a structural operational semantics is an inductive
+definition of an evaluation relation. Dually, a {\bf coinductive
+ definition} specifies the greatest set closed under given rules. An
+important example is using bisimulation relations to formalize equivalence
+of processes and infinite data structures.
+
+A theory file may contain any number of inductive and coinductive
+definitions. They may be intermixed with other declarations; in
+particular, the (co)inductive sets {\bf must} be declared separately as
+constants, and may have mixfix syntax or be subject to syntax translations.
+
+Each (co)inductive definition adds definitions to the theory and also
+proves some theorems. Each definition creates an ML structure, which is a
+substructure of the main theory structure.
+
+This package is derived from the ZF one, described in a
+separate paper,\footnote{It appeared in CADE~\cite{paulson-CADE} and a
+ longer version is distributed with Isabelle.} which you should refer to
+in case of difficulties. The package is simpler than ZF's, thanks to CHOL's
+automatic type-checking. The type of the (co)inductive determines the
+domain of the fixedpoint definition, and the package does not use inference
+rules for type-checking.
+
+
+\subsection{The result structure}
+Many of the result structure's components have been discussed in the paper;
+others are self-explanatory.
+\begin{description}
+\item[\tt thy] is the new theory containing the recursive sets.
+
+\item[\tt defs] is the list of definitions of the recursive sets.
+
+\item[\tt mono] is a monotonicity theorem for the fixedpoint operator.
+
+\item[\tt unfold] is a fixedpoint equation for the recursive set (the union of
+the recursive sets, in the case of mutual recursion).
+
+\item[\tt intrs] is the list of introduction rules, now proved as theorems, for
+the recursive sets. The rules are also available individually, using the
+names given them in the theory file.
+
+\item[\tt elim] is the elimination rule.
+
+\item[\tt mk\_cases] is a function to create simplified instances of {\tt
+elim}, using freeness reasoning on some underlying datatype.
+\end{description}
+
+For an inductive definition, the result structure contains two induction rules,
+{\tt induct} and \verb|mutual_induct|. For a coinductive definition, it
+contains the rule \verb|coinduct|.
+
+Figure~\ref{def-result-fig} summarizes the two result signatures,
+specifying the types of all these components.
+
+\begin{figure}
+\begin{ttbox}
+sig
+val thy : theory
+val defs : thm list
+val mono : thm
+val unfold : thm
+val intrs : thm list
+val elim : thm
+val mk_cases : thm list -> string -> thm
+{\it(Inductive definitions only)}
+val induct : thm
+val mutual_induct: thm
+{\it(Coinductive definitions only)}
+val coinduct : thm
+end
+\end{ttbox}
+\hrule
+\caption{The result of a (co)inductive definition} \label{def-result-fig}
+\end{figure}
+
+\subsection{The syntax of a (co)inductive definition}
+An inductive definition has the form
+\begin{ttbox}
+inductive {\it inductive sets}
+ intrs {\it introduction rules}
+ monos {\it monotonicity theorems}
+ con_defs {\it constructor definitions}
+\end{ttbox}
+A coinductive definition is identical, except that it starts with the keyword
+{\tt coinductive}.
+
+The {\tt monos} and {\tt con\_defs} sections are optional. If present,
+each is specified as a string, which must be a valid ML expression of type
+{\tt thm list}. It is simply inserted into the {\tt .thy.ML} file; if it
+is ill-formed, it will trigger ML error messages. You can then inspect the
+file on your directory.
+
+\begin{itemize}
+\item The {\it inductive sets} are specified by one or more strings.
+
+\item The {\it introduction rules} specify one or more introduction rules in
+ the form {\it ident\/}~{\it string}, where the identifier gives the name of
+ the rule in the result structure.
+
+\item The {\it monotonicity theorems} are required for each operator
+ applied to a recursive set in the introduction rules. There {\bf must}
+ be a theorem of the form $A\subseteq B\Imp M(A)\subseteq M(B)$, for each
+ premise $t\in M(R_i)$ in an introduction rule!
+
+\item The {\it constructor definitions} contain definitions of constants
+ appearing in the introduction rules. In most cases it can be omitted.
+\end{itemize}
+
+The package has a few notable restrictions:
+\begin{itemize}
+\item The theory must separately declare the recursive sets as
+ constants.
+
+\item The names of the recursive sets must be identifiers, not infix
+operators.
+
+\item Side-conditions must not be conjunctions. However, an introduction rule
+may contain any number of side-conditions.
+
+\item Side-conditions of the form $x=t$, where the variable~$x$ does not
+ occur in~$t$, will be substituted through the rule \verb|mutual_induct|.
+\end{itemize}
+
+
+\subsection{Example of an inductive definition}
+Two declarations, included in a theory file, define the finite powerset
+operator. First we declare the constant~{\tt Fin}. Then we declare it
+inductively, with two introduction rules:
+\begin{ttbox}
+consts Fin :: "'a set => 'a set set"
+inductive "Fin A"
+ intrs
+ emptyI "{} : Fin A"
+ insertI "[| a: A; b: Fin A |] ==> insert a b : Fin A"
+\end{ttbox}
+The resulting theory structure contains a substructure, called~{\tt Fin}.
+It contains the {\tt Fin}$~A$ introduction rules as the list {\tt Fin.intrs},
+and also individually as {\tt Fin.emptyI} and {\tt Fin.consI}. The induction
+rule is {\tt Fin.induct}.
+
+For another example, here is a theory file defining the accessible part of a
+relation. The main thing to note is the use of~{\tt Pow} in the sole
+introduction rule, and the corresponding mention of the rule
+\verb|Pow_mono| in the {\tt monos} list. The paper discusses a ZF version
+of this example in more detail.
+\begin{ttbox}
+Acc = WF +
+consts pred :: "['b, ('a * 'b)set] => 'a set" (*Set of predecessors*)
+ acc :: "('a * 'a)set => 'a set" (*Accessible part*)
+defs pred_def "pred x r == {y. (y,x):r}"
+inductive "acc r"
+ intrs
+ pred "pred a r: Pow(acc r) ==> a: acc r"
+ monos "[Pow_mono]"
+end
+\end{ttbox}
+The CHOL distribution contains many other inductive definitions, such as the
+theory {\tt CHOL/ex/PropLog.thy} and the directory {\tt CHOL/IMP}. The
+theory {\tt CHOL/ex/LList.thy} contains coinductive definitions.
+
+\index{*coinductive|)} \index{*inductive|)} \underscoreoff
+
+
+\section{The examples directories}
+Directory {\tt CHOL/Subst} contains Martin Coen's mechanisation of a theory of
+substitutions and unifiers. It is based on Paulson's previous
+mechanisation in {\LCF}~\cite{paulson85} of Manna and Waldinger's
+theory~\cite{mw81}.
+
+Directory {\tt CHOL/IMP} contains a mechanised version of a semantic
+equivalence proof taken from Winskel~\cite{winskel93}. It formalises the
+denotational and operational semantics of a simple while-language, then
+proves the two equivalent. It contains several datatype and inductive
+definitions, and demonstrates their use.
+
+Directory {\tt CHOL/ex} contains other examples and experimental proofs in
+{\CHOL}. Here is an overview of the more interesting files.
+\begin{itemize}
+\item File {\tt cla.ML} demonstrates the classical reasoner on over sixty
+ predicate calculus theorems, ranging from simple tautologies to
+ moderately difficult problems involving equality and quantifiers.
+
+\item File {\tt meson.ML} contains an experimental implementation of the {\sc
+ meson} proof procedure, inspired by Plaisted~\cite{plaisted90}. It is
+ much more powerful than Isabelle's classical reasoner. But it is less
+ useful in practice because it works only for pure logic; it does not
+ accept derived rules for the set theory primitives, for example.
+
+\item File {\tt mesontest.ML} contains test data for the {\sc meson} proof
+ procedure. These are mostly taken from Pelletier \cite{pelletier86}.
+
+\item File {\tt set.ML} proves Cantor's Theorem, which is presented in
+ \S\ref{sec:chol-cantor} below, and the Schr\"oder-Bernstein Theorem.
+
+\item Theories {\tt InSort} and {\tt Qsort} prove correctness properties of
+ insertion sort and quick sort.
+
+\item The definition of lazy lists demonstrates methods for handling
+ infinite data structures and coinduction in higher-order
+ logic~\cite{paulson-coind}. Theory \thydx{LList} defines an operator for
+ corecursion on lazy lists, which is used to define a few simple functions
+ such as map and append. Corecursion cannot easily define operations such
+ as filter, which can compute indefinitely before yielding the next
+ element (if any!) of the lazy list. A coinduction principle is defined
+ for proving equations on lazy lists.
+
+\item Theory {\tt PropLog} proves the soundness and completeness of
+ classical propositional logic, given a truth table semantics. The only
+ connective is $\imp$. A Hilbert-style axiom system is specified, and its
+ set of theorems defined inductively. A similar proof in \ZF{} is
+ described elsewhere~\cite{paulson-set-II}.
+
+\item Theory {\tt Term} develops an experimental recursive type definition;
+ the recursion goes through the type constructor~\tydx{list}.
+
+\item Theory {\tt Simult} constructs mutually recursive sets of trees and
+ forests, including induction and recursion rules.
+
+\item Theory {\tt MT} contains Jacob Frost's formalization~\cite{frost93} of
+ Milner and Tofte's coinduction example~\cite{milner-coind}. This
+ substantial proof concerns the soundness of a type system for a simple
+ functional language. The semantics of recursion is given by a cyclic
+ environment, which makes a coinductive argument appropriate.
+\end{itemize}
+
+
+\goodbreak
+\section{Example: Cantor's Theorem}\label{sec:chol-cantor}
+Cantor's Theorem states that every set has more subsets than it has
+elements. It has become a favourite example in higher-order logic since
+it is so easily expressed:
+\[ \forall f::[\alpha,\alpha]\To bool. \exists S::\alpha\To bool.
+ \forall x::\alpha. f~x \not= S
+\]
+%
+Viewing types as sets, $\alpha\To bool$ represents the powerset
+of~$\alpha$. This version states that for every function from $\alpha$ to
+its powerset, some subset is outside its range.
+
+The Isabelle proof uses \CHOL's set theory, with the type $\alpha\,set$ and
+the operator \cdx{range}. The set~$S$ is given as an unknown instead of a
+quantified variable so that we may inspect the subset found by the proof.
+\begin{ttbox}
+goal Set.thy "~ ?S : range(f :: 'a=>'a set)";
+{\out Level 0}
+{\out ~ ?S : range f}
+{\out 1. ~ ?S : range f}
+\end{ttbox}
+The first two steps are routine. The rule \tdx{rangeE} replaces
+$\Var{S}\in {\tt range} f$ by $\Var{S}=f~x$ for some~$x$.
+\begin{ttbox}
+by (resolve_tac [notI] 1);
+{\out Level 1}
+{\out ~ ?S : range f}
+{\out 1. ?S : range f ==> False}
+\ttbreak
+by (eresolve_tac [rangeE] 1);
+{\out Level 2}
+{\out ~ ?S : range f}
+{\out 1. !!x. ?S = f x ==> False}
+\end{ttbox}
+Next, we apply \tdx{equalityCE}, reasoning that since $\Var{S}=f~x$,
+we have $\Var{c}\in \Var{S}$ if and only if $\Var{c}\in f~x$ for
+any~$\Var{c}$.
+\begin{ttbox}
+by (eresolve_tac [equalityCE] 1);
+{\out Level 3}
+{\out ~ ?S : range f}
+{\out 1. !!x. [| ?c3 x : ?S; ?c3 x : f x |] ==> False}
+{\out 2. !!x. [| ~ ?c3 x : ?S; ~ ?c3 x : f x |] ==> False}
+\end{ttbox}
+Now we use a bit of creativity. Suppose that~$\Var{S}$ has the form of a
+comprehension. Then $\Var{c}\in\{x.\Var{P}~x\}$ implies
+$\Var{P}~\Var{c}$. Destruct-resolution using \tdx{CollectD}
+instantiates~$\Var{S}$ and creates the new assumption.
+\begin{ttbox}
+by (dresolve_tac [CollectD] 1);
+{\out Level 4}
+{\out ~ \{x. ?P7 x\} : range f}
+{\out 1. !!x. [| ?c3 x : f x; ?P7(?c3 x) |] ==> False}
+{\out 2. !!x. [| ~ ?c3 x : \{x. ?P7 x\}; ~ ?c3 x : f x |] ==> False}
+\end{ttbox}
+Forcing a contradiction between the two assumptions of subgoal~1 completes
+the instantiation of~$S$. It is now the set $\{x. x\not\in f~x\}$, which
+is the standard diagonal construction.
+\begin{ttbox}
+by (contr_tac 1);
+{\out Level 5}
+{\out ~ \{x. ~ x : f x\} : range f}
+{\out 1. !!x. [| ~ x : \{x. ~ x : f x\}; ~ x : f x |] ==> False}
+\end{ttbox}
+The rest should be easy. To apply \tdx{CollectI} to the negated
+assumption, we employ \ttindex{swap_res_tac}:
+\begin{ttbox}
+by (swap_res_tac [CollectI] 1);
+{\out Level 6}
+{\out ~ \{x. ~ x : f x\} : range f}
+{\out 1. !!x. [| ~ x : f x; ~ False |] ==> ~ x : f x}
+\ttbreak
+by (assume_tac 1);
+{\out Level 7}
+{\out ~ \{x. ~ x : f x\} : range f}
+{\out No subgoals!}
+\end{ttbox}
+How much creativity is required? As it happens, Isabelle can prove this
+theorem automatically. The classical set \ttindex{set_cs} contains rules
+for most of the constructs of \CHOL's set theory. We must augment it with
+\tdx{equalityCE} to break up set equalities, and then apply best-first
+search. Depth-first search would diverge, but best-first search
+successfully navigates through the large search space.
+\index{search!best-first}
+\begin{ttbox}
+choplev 0;
+{\out Level 0}
+{\out ~ ?S : range f}
+{\out 1. ~ ?S : range f}
+\ttbreak
+by (best_tac (set_cs addSEs [equalityCE]) 1);
+{\out Level 1}
+{\out ~ \{x. ~ x : f x\} : range f}
+{\out No subgoals!}
+\end{ttbox}
+
+\index{higher-order logic|)}
--- a/doc-src/Logics/logics.tex Tue May 09 10:42:23 1995 +0200
+++ b/doc-src/Logics/logics.tex Tue May 09 10:43:19 1995 +0200
@@ -47,7 +47,7 @@
\include{intro}
\include{FOL}
\include{ZF}
-\include{HOL}
+\include{CHOL}
\include{LK}
%%\include{Modal}
\include{CTT}