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+%% $Id$
+\chapter{Zermelo-Fraenkel Set Theory}
+\index{set theory|(}
+
+The theory~\thydx{ZF} implements Zermelo-Fraenkel set
+theory~\cite{halmos60,suppes72} as an extension of~\texttt{FOL}, classical
+first-order logic.  The theory includes a collection of derived natural
+deduction rules, for use with Isabelle's classical reasoner.  Much
+of it is based on the work of No\"el~\cite{noel}.
+
+A tremendous amount of set theory has been formally developed, including the
+basic properties of relations, functions, ordinals and cardinals.  Significant
+results have been proved, such as the Schr\"oder-Bernstein Theorem, the
+Wellordering Theorem and a version of Ramsey's Theorem.  \texttt{ZF} provides
+both the integers and the natural numbers.  General methods have been
+developed for solving recursion equations over monotonic functors; these have
+been applied to yield constructions of lists, trees, infinite lists, etc.
+
+\texttt{ZF} has a flexible package for handling inductive definitions,
+such as inference systems, and datatype definitions, such as lists and
+trees.  Moreover it handles coinductive definitions, such as
+bisimulation relations, and codatatype definitions, such as streams.  It
+provides a streamlined syntax for defining primitive recursive functions over
+datatypes. 
+
+Because {\ZF} is an extension of {\FOL}, it provides the same
+packages, namely \texttt{hyp_subst_tac}, the simplifier, and the
+classical reasoner.  The default simpset and claset are usually
+satisfactory.
+
+Published articles~\cite{paulson-set-I,paulson-set-II} describe \texttt{ZF}
+less formally than this chapter.  Isabelle employs a novel treatment of
+non-well-founded data structures within the standard {\sc zf} axioms including
+the Axiom of Foundation~\cite{paulson-final}.
+
+
+\section{Which version of axiomatic set theory?}
+The two main axiom systems for set theory are Bernays-G\"odel~({\sc bg})
+and Zermelo-Fraenkel~({\sc zf}).  Resolution theorem provers can use {\sc
+  bg} because it is finite~\cite{boyer86,quaife92}.  {\sc zf} does not
+have a finite axiom system because of its Axiom Scheme of Replacement.
+This makes it awkward to use with many theorem provers, since instances
+of the axiom scheme have to be invoked explicitly.  Since Isabelle has no
+difficulty with axiom schemes, we may adopt either axiom system.
+
+These two theories differ in their treatment of {\bf classes}, which are
+collections that are `too big' to be sets.  The class of all sets,~$V$,
+cannot be a set without admitting Russell's Paradox.  In {\sc bg}, both
+classes and sets are individuals; $x\in V$ expresses that $x$ is a set.  In
+{\sc zf}, all variables denote sets; classes are identified with unary
+predicates.  The two systems define essentially the same sets and classes,
+with similar properties.  In particular, a class cannot belong to another
+class (let alone a set).
+
+Modern set theorists tend to prefer {\sc zf} because they are mainly concerned
+with sets, rather than classes.  {\sc bg} requires tiresome proofs that various
+collections are sets; for instance, showing $x\in\{x\}$ requires showing that
+$x$ is a set.
+
+
+\begin{figure} \small
+\begin{center}
+\begin{tabular}{rrr} 
+  \it name      &\it meta-type  & \it description \\ 
+  \cdx{Let}     & $[\alpha,\alpha\To\beta]\To\beta$ & let binder\\
+  \cdx{0}       & $i$           & empty set\\
+  \cdx{cons}    & $[i,i]\To i$  & finite set constructor\\
+  \cdx{Upair}   & $[i,i]\To i$  & unordered pairing\\
+  \cdx{Pair}    & $[i,i]\To i$  & ordered pairing\\
+  \cdx{Inf}     & $i$   & infinite set\\
+  \cdx{Pow}     & $i\To i$      & powerset\\
+  \cdx{Union} \cdx{Inter} & $i\To i$    & set union/intersection \\
+  \cdx{split}   & $[[i,i]\To i, i] \To i$ & generalized projection\\
+  \cdx{fst} \cdx{snd}   & $i\To i$      & projections\\
+  \cdx{converse}& $i\To i$      & converse of a relation\\
+  \cdx{succ}    & $i\To i$      & successor\\
+  \cdx{Collect} & $[i,i\To o]\To i$     & separation\\
+  \cdx{Replace} & $[i, [i,i]\To o] \To i$       & replacement\\
+  \cdx{PrimReplace} & $[i, [i,i]\To o] \To i$   & primitive replacement\\
+  \cdx{RepFun}  & $[i, i\To i] \To i$   & functional replacement\\
+  \cdx{Pi} \cdx{Sigma}  & $[i,i\To i]\To i$     & general product/sum\\
+  \cdx{domain}  & $i\To i$      & domain of a relation\\
+  \cdx{range}   & $i\To i$      & range of a relation\\
+  \cdx{field}   & $i\To i$      & field of a relation\\
+  \cdx{Lambda}  & $[i, i\To i]\To i$    & $\lambda$-abstraction\\
+  \cdx{restrict}& $[i, i] \To i$        & restriction of a function\\
+  \cdx{The}     & $[i\To o]\To i$       & definite description\\
+  \cdx{if}      & $[o,i,i]\To i$        & conditional\\
+  \cdx{Ball} \cdx{Bex}  & $[i, i\To o]\To o$    & bounded quantifiers
+\end{tabular}
+\end{center}
+\subcaption{Constants}
+
+\begin{center}
+\index{*"`"` symbol}
+\index{*"-"`"` symbol}
+\index{*"` symbol}\index{function applications!in \ZF}
+\index{*"- symbol}
+\index{*": symbol}
+\index{*"<"= symbol}
+\begin{tabular}{rrrr} 
+  \it symbol  & \it meta-type & \it priority & \it description \\ 
+  \tt ``        & $[i,i]\To i$  &  Left 90      & image \\
+  \tt -``       & $[i,i]\To i$  &  Left 90      & inverse image \\
+  \tt `         & $[i,i]\To i$  &  Left 90      & application \\
+  \sdx{Int}     & $[i,i]\To i$  &  Left 70      & intersection ($\int$) \\
+  \sdx{Un}      & $[i,i]\To i$  &  Left 65      & union ($\un$) \\
+  \tt -         & $[i,i]\To i$  &  Left 65      & set difference ($-$) \\[1ex]
+  \tt:          & $[i,i]\To o$  &  Left 50      & membership ($\in$) \\
+  \tt <=        & $[i,i]\To o$  &  Left 50      & subset ($\subseteq$) 
+\end{tabular}
+\end{center}
+\subcaption{Infixes}
+\caption{Constants of {\ZF}} \label{zf-constants}
+\end{figure} 
+
+
+\section{The syntax of set theory}
+The language of set theory, as studied by logicians, has no constants.  The
+traditional axioms merely assert the existence of empty sets, unions,
+powersets, etc.; this would be intolerable for practical reasoning.  The
+Isabelle theory declares constants for primitive sets.  It also extends
+\texttt{FOL} with additional syntax for finite sets, ordered pairs,
+comprehension, general union/intersection, general sums/products, and
+bounded quantifiers.  In most other respects, Isabelle implements precisely
+Zermelo-Fraenkel set theory.
+
+Figure~\ref{zf-constants} lists the constants and infixes of~\ZF, while
+Figure~\ref{zf-trans} presents the syntax translations.  Finally,
+Figure~\ref{zf-syntax} presents the full grammar for set theory, including
+the constructs of \FOL.
+
+Local abbreviations can be introduced by a \texttt{let} construct whose
+syntax appears in Fig.\ts\ref{zf-syntax}.  Internally it is translated into
+the constant~\cdx{Let}.  It can be expanded by rewriting with its
+definition, \tdx{Let_def}.
+
+Apart from \texttt{let}, set theory does not use polymorphism.  All terms in
+{\ZF} have type~\tydx{i}, which is the type of individuals and has class~{\tt
+  term}.  The type of first-order formulae, remember, is~\textit{o}.
+
+Infix operators include binary union and intersection ($A\un B$ and
+$A\int B$), set difference ($A-B$), and the subset and membership
+relations.  Note that $a$\verb|~:|$b$ is translated to $\neg(a\in b)$.  The
+union and intersection operators ($\bigcup A$ and $\bigcap A$) form the
+union or intersection of a set of sets; $\bigcup A$ means the same as
+$\bigcup@{x\in A}x$.  Of these operators, only $\bigcup A$ is primitive.
+
+The constant \cdx{Upair} constructs unordered pairs; thus {\tt
+  Upair($A$,$B$)} denotes the set~$\{A,B\}$ and \texttt{Upair($A$,$A$)}
+denotes the singleton~$\{A\}$.  General union is used to define binary
+union.  The Isabelle version goes on to define the constant
+\cdx{cons}:
+\begin{eqnarray*}
+   A\cup B              & \equiv &       \bigcup(\texttt{Upair}(A,B)) \\
+   \texttt{cons}(a,B)      & \equiv &        \texttt{Upair}(a,a) \un B
+\end{eqnarray*}
+The $\{a@1, \ldots\}$ notation abbreviates finite sets constructed in the
+obvious manner using~\texttt{cons} and~$\emptyset$ (the empty set):
+\begin{eqnarray*}
+ \{a,b,c\} & \equiv & \texttt{cons}(a,\texttt{cons}(b,\texttt{cons}(c,\emptyset)))
+\end{eqnarray*}
+
+The constant \cdx{Pair} constructs ordered pairs, as in {\tt
+Pair($a$,$b$)}.  Ordered pairs may also be written within angle brackets,
+as {\tt<$a$,$b$>}.  The $n$-tuple {\tt<$a@1$,\ldots,$a@{n-1}$,$a@n$>}
+abbreviates the nest of pairs\par\nobreak
+\centerline{\texttt{Pair($a@1$,\ldots,Pair($a@{n-1}$,$a@n$)\ldots).}}
+
+In {\ZF}, a function is a set of pairs.  A {\ZF} function~$f$ is simply an
+individual as far as Isabelle is concerned: its Isabelle type is~$i$, not
+say $i\To i$.  The infix operator~{\tt`} denotes the application of a
+function set to its argument; we must write~$f{\tt`}x$, not~$f(x)$.  The
+syntax for image is~$f{\tt``}A$ and that for inverse image is~$f{\tt-``}A$.
+
+
+\begin{figure} 
+\index{lambda abs@$\lambda$-abstractions!in \ZF}
+\index{*"-"> symbol}
+\index{*"* symbol}
+\begin{center} \footnotesize\tt\frenchspacing
+\begin{tabular}{rrr} 
+  \it external          & \it internal  & \it description \\ 
+  $a$ \ttilde: $b$      & \ttilde($a$ : $b$)    & \rm negated membership\\
+  \ttlbrace$a@1$, $\ldots$, $a@n$\ttrbrace  &  cons($a@1$,$\ldots$,cons($a@n$,0)) &
+        \rm finite set \\
+  <$a@1$, $\ldots$, $a@{n-1}$, $a@n$> & 
+        Pair($a@1$,\ldots,Pair($a@{n-1}$,$a@n$)\ldots) &
+        \rm ordered $n$-tuple \\
+  \ttlbrace$x$:$A . P[x]$\ttrbrace    &  Collect($A$,$\lambda x. P[x]$) &
+        \rm separation \\
+  \ttlbrace$y . x$:$A$, $Q[x,y]$\ttrbrace  &  Replace($A$,$\lambda x\,y. Q[x,y]$) &
+        \rm replacement \\
+  \ttlbrace$b[x] . x$:$A$\ttrbrace  &  RepFun($A$,$\lambda x. b[x]$) &
+        \rm functional replacement \\
+  \sdx{INT} $x$:$A . B[x]$      & Inter(\ttlbrace$B[x] . x$:$A$\ttrbrace) &
+        \rm general intersection \\
+  \sdx{UN}  $x$:$A . B[x]$      & Union(\ttlbrace$B[x] . x$:$A$\ttrbrace) &
+        \rm general union \\
+  \sdx{PROD} $x$:$A . B[x]$     & Pi($A$,$\lambda x. B[x]$) & 
+        \rm general product \\
+  \sdx{SUM}  $x$:$A . B[x]$     & Sigma($A$,$\lambda x. B[x]$) & 
+        \rm general sum \\
+  $A$ -> $B$            & Pi($A$,$\lambda x. B$) & 
+        \rm function space \\
+  $A$ * $B$             & Sigma($A$,$\lambda x. B$) & 
+        \rm binary product \\
+  \sdx{THE}  $x . P[x]$ & The($\lambda x. P[x]$) & 
+        \rm definite description \\
+  \sdx{lam}  $x$:$A . b[x]$     & Lambda($A$,$\lambda x. b[x]$) & 
+        \rm $\lambda$-abstraction\\[1ex]
+  \sdx{ALL} $x$:$A . P[x]$      & Ball($A$,$\lambda x. P[x]$) & 
+        \rm bounded $\forall$ \\
+  \sdx{EX}  $x$:$A . P[x]$      & Bex($A$,$\lambda x. P[x]$) & 
+        \rm bounded $\exists$
+\end{tabular}
+\end{center}
+\caption{Translations for {\ZF}} \label{zf-trans}
+\end{figure} 
+
+
+\begin{figure} 
+\index{*let symbol}
+\index{*in symbol}
+\dquotes
+\[\begin{array}{rcl}
+    term & = & \hbox{expression of type~$i$} \\
+         & | & "let"~id~"="~term";"\dots";"~id~"="~term~"in"~term \\
+         & | & "if"~term~"then"~term~"else"~term \\
+         & | & "{\ttlbrace} " term\; ("," term)^* " {\ttrbrace}" \\
+         & | & "< "  term\; ("," term)^* " >"  \\
+         & | & "{\ttlbrace} " id ":" term " . " formula " {\ttrbrace}" \\
+         & | & "{\ttlbrace} " id " . " id ":" term ", " formula " {\ttrbrace}" \\
+         & | & "{\ttlbrace} " term " . " id ":" term " {\ttrbrace}" \\
+         & | & term " `` " term \\
+         & | & term " -`` " term \\
+         & | & term " ` " term \\
+         & | & term " * " term \\
+         & | & term " Int " term \\
+         & | & term " Un " term \\
+         & | & term " - " term \\
+         & | & term " -> " term \\
+         & | & "THE~~"  id  " . " formula\\
+         & | & "lam~~"  id ":" term " . " term \\
+         & | & "INT~~"  id ":" term " . " term \\
+         & | & "UN~~~"  id ":" term " . " term \\
+         & | & "PROD~"  id ":" term " . " term \\
+         & | & "SUM~~"  id ":" term " . " term \\[2ex]
+ formula & = & \hbox{expression of type~$o$} \\
+         & | & term " : " term \\
+         & | & term " \ttilde: " term \\
+         & | & term " <= " term \\
+         & | & term " = " term \\
+         & | & term " \ttilde= " term \\
+         & | & "\ttilde\ " formula \\
+         & | & formula " \& " formula \\
+         & | & formula " | " formula \\
+         & | & formula " --> " formula \\
+         & | & formula " <-> " formula \\
+         & | & "ALL " id ":" term " . " formula \\
+         & | & "EX~~" id ":" term " . " formula \\
+         & | & "ALL~" id~id^* " . " formula \\
+         & | & "EX~~" id~id^* " . " formula \\
+         & | & "EX!~" id~id^* " . " formula
+  \end{array}
+\]
+\caption{Full grammar for {\ZF}} \label{zf-syntax}
+\end{figure} 
+
+
+\section{Binding operators}
+The constant \cdx{Collect} constructs sets by the principle of {\bf
+  separation}.  The syntax for separation is
+\hbox{\tt\ttlbrace$x$:$A$.\ $P[x]$\ttrbrace}, where $P[x]$ is a formula
+that may contain free occurrences of~$x$.  It abbreviates the set {\tt
+  Collect($A$,$\lambda x. P[x]$)}, which consists of all $x\in A$ that
+satisfy~$P[x]$.  Note that \texttt{Collect} is an unfortunate choice of
+name: some set theories adopt a set-formation principle, related to
+replacement, called collection.
+
+The constant \cdx{Replace} constructs sets by the principle of {\bf
+  replacement}.  The syntax
+\hbox{\tt\ttlbrace$y$.\ $x$:$A$,$Q[x,y]$\ttrbrace} denotes the set {\tt
+  Replace($A$,$\lambda x\,y. Q[x,y]$)}, which consists of all~$y$ such
+that there exists $x\in A$ satisfying~$Q[x,y]$.  The Replacement Axiom
+has the condition that $Q$ must be single-valued over~$A$: for
+all~$x\in A$ there exists at most one $y$ satisfying~$Q[x,y]$.  A
+single-valued binary predicate is also called a {\bf class function}.
+
+The constant \cdx{RepFun} expresses a special case of replacement,
+where $Q[x,y]$ has the form $y=b[x]$.  Such a $Q$ is trivially
+single-valued, since it is just the graph of the meta-level
+function~$\lambda x. b[x]$.  The resulting set consists of all $b[x]$
+for~$x\in A$.  This is analogous to the \ML{} functional \texttt{map},
+since it applies a function to every element of a set.  The syntax is
+\hbox{\tt\ttlbrace$b[x]$.\ $x$:$A$\ttrbrace}, which expands to {\tt
+  RepFun($A$,$\lambda x. b[x]$)}.
+
+\index{*INT symbol}\index{*UN symbol} 
+General unions and intersections of indexed
+families of sets, namely $\bigcup@{x\in A}B[x]$ and $\bigcap@{x\in A}B[x]$,
+are written \hbox{\tt UN $x$:$A$.\ $B[x]$} and \hbox{\tt INT $x$:$A$.\ $B[x]$}.
+Their meaning is expressed using \texttt{RepFun} as
+\[
+\bigcup(\{B[x]. x\in A\}) \qquad\hbox{and}\qquad 
+\bigcap(\{B[x]. x\in A\}). 
+\]
+General sums $\sum@{x\in A}B[x]$ and products $\prod@{x\in A}B[x]$ can be
+constructed in set theory, where $B[x]$ is a family of sets over~$A$.  They
+have as special cases $A\times B$ and $A\to B$, where $B$ is simply a set.
+This is similar to the situation in Constructive Type Theory (set theory
+has `dependent sets') and calls for similar syntactic conventions.  The
+constants~\cdx{Sigma} and~\cdx{Pi} construct general sums and
+products.  Instead of \texttt{Sigma($A$,$B$)} and \texttt{Pi($A$,$B$)} we may
+write 
+\hbox{\tt SUM $x$:$A$.\ $B[x]$} and \hbox{\tt PROD $x$:$A$.\ $B[x]$}.  
+\index{*SUM symbol}\index{*PROD symbol}%
+The special cases as \hbox{\tt$A$*$B$} and \hbox{\tt$A$->$B$} abbreviate
+general sums and products over a constant family.\footnote{Unlike normal
+infix operators, {\tt*} and {\tt->} merely define abbreviations; there are
+no constants~\texttt{op~*} and~\hbox{\tt op~->}.} Isabelle accepts these
+abbreviations in parsing and uses them whenever possible for printing.
+
+\index{*THE symbol} 
+As mentioned above, whenever the axioms assert the existence and uniqueness
+of a set, Isabelle's set theory declares a constant for that set.  These
+constants can express the {\bf definite description} operator~$\iota
+x. P[x]$, which stands for the unique~$a$ satisfying~$P[a]$, if such exists.
+Since all terms in {\ZF} denote something, a description is always
+meaningful, but we do not know its value unless $P[x]$ defines it uniquely.
+Using the constant~\cdx{The}, we may write descriptions as {\tt
+  The($\lambda x. P[x]$)} or use the syntax \hbox{\tt THE $x$.\ $P[x]$}.
+
+\index{*lam symbol}
+Function sets may be written in $\lambda$-notation; $\lambda x\in A. b[x]$
+stands for the set of all pairs $\pair{x,b[x]}$ for $x\in A$.  In order for
+this to be a set, the function's domain~$A$ must be given.  Using the
+constant~\cdx{Lambda}, we may express function sets as {\tt
+Lambda($A$,$\lambda x. b[x]$)} or use the syntax \hbox{\tt lam $x$:$A$.\ $b[x]$}.
+
+Isabelle's set theory defines two {\bf bounded quantifiers}:
+\begin{eqnarray*}
+   \forall x\in A. P[x] &\hbox{abbreviates}& \forall x. x\in A\imp P[x] \\
+   \exists x\in A. P[x] &\hbox{abbreviates}& \exists x. x\in A\conj P[x]
+\end{eqnarray*}
+The constants~\cdx{Ball} and~\cdx{Bex} are defined
+accordingly.  Instead of \texttt{Ball($A$,$P$)} and \texttt{Bex($A$,$P$)} we may
+write
+\hbox{\tt ALL $x$:$A$.\ $P[x]$} and \hbox{\tt EX $x$:$A$.\ $P[x]$}.
+
+
+%%%% ZF.thy
+
+\begin{figure}
+\begin{ttbox}
+\tdx{Let_def}            Let(s, f) == f(s)
+
+\tdx{Ball_def}           Ball(A,P) == ALL x. x:A --> P(x)
+\tdx{Bex_def}            Bex(A,P)  == EX x. x:A & P(x)
+
+\tdx{subset_def}         A <= B  == ALL x:A. x:B
+\tdx{extension}          A = B  <->  A <= B & B <= A
+
+\tdx{Union_iff}          A : Union(C) <-> (EX B:C. A:B)
+\tdx{Pow_iff}            A : Pow(B) <-> A <= B
+\tdx{foundation}         A=0 | (EX x:A. ALL y:x. ~ y:A)
+
+\tdx{replacement}        (ALL x:A. ALL y z. P(x,y) & P(x,z) --> y=z) ==>
+                   b : PrimReplace(A,P) <-> (EX x:A. P(x,b))
+\subcaption{The Zermelo-Fraenkel Axioms}
+
+\tdx{Replace_def}  Replace(A,P) == 
+                   PrimReplace(A, \%x y. (EX!z. P(x,z)) & P(x,y))
+\tdx{RepFun_def}   RepFun(A,f)  == {\ttlbrace}y . x:A, y=f(x)\ttrbrace
+\tdx{the_def}      The(P)       == Union({\ttlbrace}y . x:{\ttlbrace}0{\ttrbrace}, P(y){\ttrbrace})
+\tdx{if_def}       if(P,a,b)    == THE z. P & z=a | ~P & z=b
+\tdx{Collect_def}  Collect(A,P) == {\ttlbrace}y . x:A, x=y & P(x){\ttrbrace}
+\tdx{Upair_def}    Upair(a,b)   == 
+                 {\ttlbrace}y. x:Pow(Pow(0)), (x=0 & y=a) | (x=Pow(0) & y=b){\ttrbrace}
+\subcaption{Consequences of replacement}
+
+\tdx{Inter_def}    Inter(A) == {\ttlbrace}x:Union(A) . ALL y:A. x:y{\ttrbrace}
+\tdx{Un_def}       A Un  B  == Union(Upair(A,B))
+\tdx{Int_def}      A Int B  == Inter(Upair(A,B))
+\tdx{Diff_def}     A - B    == {\ttlbrace}x:A . x~:B{\ttrbrace}
+\subcaption{Union, intersection, difference}
+\end{ttbox}
+\caption{Rules and axioms of {\ZF}} \label{zf-rules}
+\end{figure}
+
+
+\begin{figure}
+\begin{ttbox}
+\tdx{cons_def}     cons(a,A) == Upair(a,a) Un A
+\tdx{succ_def}     succ(i) == cons(i,i)
+\tdx{infinity}     0:Inf & (ALL y:Inf. succ(y): Inf)
+\subcaption{Finite and infinite sets}
+
+\tdx{Pair_def}       <a,b>      == {\ttlbrace}{\ttlbrace}a,a{\ttrbrace}, {\ttlbrace}a,b{\ttrbrace}{\ttrbrace}
+\tdx{split_def}      split(c,p) == THE y. EX a b. p=<a,b> & y=c(a,b)
+\tdx{fst_def}        fst(A)     == split(\%x y. x, p)
+\tdx{snd_def}        snd(A)     == split(\%x y. y, p)
+\tdx{Sigma_def}      Sigma(A,B) == UN x:A. UN y:B(x). {\ttlbrace}<x,y>{\ttrbrace}
+\subcaption{Ordered pairs and Cartesian products}
+
+\tdx{converse_def}   converse(r) == {\ttlbrace}z. w:r, EX x y. w=<x,y> & z=<y,x>{\ttrbrace}
+\tdx{domain_def}     domain(r)   == {\ttlbrace}x. w:r, EX y. w=<x,y>{\ttrbrace}
+\tdx{range_def}      range(r)    == domain(converse(r))
+\tdx{field_def}      field(r)    == domain(r) Un range(r)
+\tdx{image_def}      r `` A      == {\ttlbrace}y : range(r) . EX x:A. <x,y> : r{\ttrbrace}
+\tdx{vimage_def}     r -`` A     == converse(r)``A
+\subcaption{Operations on relations}
+
+\tdx{lam_def}    Lambda(A,b) == {\ttlbrace}<x,b(x)> . x:A{\ttrbrace}
+\tdx{apply_def}  f`a         == THE y. <a,y> : f
+\tdx{Pi_def}     Pi(A,B) == {\ttlbrace}f: Pow(Sigma(A,B)). ALL x:A. EX! y. <x,y>: f{\ttrbrace}
+\tdx{restrict_def}   restrict(f,A) == lam x:A. f`x
+\subcaption{Functions and general product}
+\end{ttbox}
+\caption{Further definitions of {\ZF}} \label{zf-defs}
+\end{figure}
+
+
+
+\section{The Zermelo-Fraenkel axioms}
+The axioms appear in Fig.\ts \ref{zf-rules}.  They resemble those
+presented by Suppes~\cite{suppes72}.  Most of the theory consists of
+definitions.  In particular, bounded quantifiers and the subset relation
+appear in other axioms.  Object-level quantifiers and implications have
+been replaced by meta-level ones wherever possible, to simplify use of the
+axioms.  See the file \texttt{ZF/ZF.thy} for details.
+
+The traditional replacement axiom asserts
+\[ y \in \texttt{PrimReplace}(A,P) \bimp (\exists x\in A. P(x,y)) \]
+subject to the condition that $P(x,y)$ is single-valued for all~$x\in A$.
+The Isabelle theory defines \cdx{Replace} to apply
+\cdx{PrimReplace} to the single-valued part of~$P$, namely
+\[ (\exists!z. P(x,z)) \conj P(x,y). \]
+Thus $y\in \texttt{Replace}(A,P)$ if and only if there is some~$x$ such that
+$P(x,-)$ holds uniquely for~$y$.  Because the equivalence is unconditional,
+\texttt{Replace} is much easier to use than \texttt{PrimReplace}; it defines the
+same set, if $P(x,y)$ is single-valued.  The nice syntax for replacement
+expands to \texttt{Replace}.
+
+Other consequences of replacement include functional replacement
+(\cdx{RepFun}) and definite descriptions (\cdx{The}).
+Axioms for separation (\cdx{Collect}) and unordered pairs
+(\cdx{Upair}) are traditionally assumed, but they actually follow
+from replacement~\cite[pages 237--8]{suppes72}.
+
+The definitions of general intersection, etc., are straightforward.  Note
+the definition of \texttt{cons}, which underlies the finite set notation.
+The axiom of infinity gives us a set that contains~0 and is closed under
+successor (\cdx{succ}).  Although this set is not uniquely defined,
+the theory names it (\cdx{Inf}) in order to simplify the
+construction of the natural numbers.
+                                             
+Further definitions appear in Fig.\ts\ref{zf-defs}.  Ordered pairs are
+defined in the standard way, $\pair{a,b}\equiv\{\{a\},\{a,b\}\}$.  Recall
+that \cdx{Sigma}$(A,B)$ generalizes the Cartesian product of two
+sets.  It is defined to be the union of all singleton sets
+$\{\pair{x,y}\}$, for $x\in A$ and $y\in B(x)$.  This is a typical usage of
+general union.
+
+The projections \cdx{fst} and~\cdx{snd} are defined in terms of the
+generalized projection \cdx{split}.  The latter has been borrowed from
+Martin-L\"of's Type Theory, and is often easier to use than \cdx{fst}
+and~\cdx{snd}.
+
+Operations on relations include converse, domain, range, and image.  The
+set ${\tt Pi}(A,B)$ generalizes the space of functions between two sets.
+Note the simple definitions of $\lambda$-abstraction (using
+\cdx{RepFun}) and application (using a definite description).  The
+function \cdx{restrict}$(f,A)$ has the same values as~$f$, but only
+over the domain~$A$.
+
+
+%%%% zf.ML
+
+\begin{figure}
+\begin{ttbox}
+\tdx{ballI}       [| !!x. x:A ==> P(x) |] ==> ALL x:A. P(x)
+\tdx{bspec}       [| ALL x:A. P(x);  x: A |] ==> P(x)
+\tdx{ballE}       [| ALL x:A. P(x);  P(x) ==> Q;  ~ x:A ==> Q |] ==> Q
+
+\tdx{ball_cong}   [| A=A';  !!x. x:A' ==> P(x) <-> P'(x) |] ==> 
+            (ALL x:A. P(x)) <-> (ALL x:A'. P'(x))
+
+\tdx{bexI}        [| P(x);  x: A |] ==> EX x:A. P(x)
+\tdx{bexCI}       [| ALL x:A. ~P(x) ==> P(a);  a: A |] ==> EX x:A. P(x)
+\tdx{bexE}        [| EX x:A. P(x);  !!x. [| x:A; P(x) |] ==> Q |] ==> Q
+
+\tdx{bex_cong}    [| A=A';  !!x. x:A' ==> P(x) <-> P'(x) |] ==> 
+            (EX x:A. P(x)) <-> (EX x:A'. P'(x))
+\subcaption{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
+\subcaption{Subsets and extensionality}
+
+\tdx{emptyE}          a:0 ==> P
+\tdx{empty_subsetI}   0 <= A
+\tdx{equals0I}        [| !!y. y:A ==> False |] ==> A=0
+\tdx{equals0D}        [| A=0;  a:A |] ==> P
+
+\tdx{PowI}            A <= B ==> A : Pow(B)
+\tdx{PowD}            A : Pow(B)  ==>  A<=B
+\subcaption{The empty set; power sets}
+\end{ttbox}
+\caption{Basic derived rules for {\ZF}} \label{zf-lemmas1}
+\end{figure}
+
+
+\section{From basic lemmas to function spaces}
+Faced with so many definitions, it is essential to prove lemmas.  Even
+trivial theorems like $A \int B = B \int A$ would be difficult to
+prove from the definitions alone.  Isabelle's set theory derives many
+rules using a natural deduction style.  Ideally, a natural deduction
+rule should introduce or eliminate just one operator, but this is not
+always practical.  For most operators, we may forget its definition
+and use its derived rules instead.
+
+\subsection{Fundamental lemmas}
+Figure~\ref{zf-lemmas1} presents the derived rules for the most basic
+operators.  The rules for the bounded quantifiers resemble those for the
+ordinary quantifiers, but note that \tdx{ballE} uses a negated assumption
+in the style of Isabelle's classical reasoner.  The \rmindex{congruence
+  rules} \tdx{ball_cong} and \tdx{bex_cong} are required by Isabelle's
+simplifier, but have few other uses.  Congruence rules must be specially
+derived for all binding operators, and henceforth will not be shown.
+
+Figure~\ref{zf-lemmas1} also shows rules for the subset and equality
+relations (proof by extensionality), and rules about the empty set and the
+power set operator.
+
+Figure~\ref{zf-lemmas2} presents rules for replacement and separation.
+The rules for \cdx{Replace} and \cdx{RepFun} are much simpler than
+comparable rules for \texttt{PrimReplace} would be.  The principle of
+separation is proved explicitly, although most proofs should use the
+natural deduction rules for \texttt{Collect}.  The elimination rule
+\tdx{CollectE} is equivalent to the two destruction rules
+\tdx{CollectD1} and \tdx{CollectD2}, but each rule is suited to
+particular circumstances.  Although too many rules can be confusing, there
+is no reason to aim for a minimal set of rules.  See the file
+\texttt{ZF/ZF.ML} for a complete listing.
+
+Figure~\ref{zf-lemmas3} presents rules for general union and intersection.
+The empty intersection should be undefined.  We cannot have
+$\bigcap(\emptyset)=V$ because $V$, the universal class, is not a set.  All
+expressions denote something in {\ZF} set theory; the definition of
+intersection implies $\bigcap(\emptyset)=\emptyset$, but this value is
+arbitrary.  The rule \tdx{InterI} must have a premise to exclude
+the empty intersection.  Some of the laws governing intersections require
+similar premises.
+
+
+%the [p] gives better page breaking for the book
+\begin{figure}[p]
+\begin{ttbox}
+\tdx{ReplaceI}      [| x: A;  P(x,b);  !!y. P(x,y) ==> y=b |] ==> 
+              b : {\ttlbrace}y. x:A, P(x,y){\ttrbrace}
+
+\tdx{ReplaceE}      [| b : {\ttlbrace}y. x:A, P(x,y){\ttrbrace};  
+                 !!x. [| x: A;  P(x,b);  ALL y. P(x,y)-->y=b |] ==> R 
+              |] ==> R
+
+\tdx{RepFunI}       [| a : A |] ==> f(a) : {\ttlbrace}f(x). x:A{\ttrbrace}
+\tdx{RepFunE}       [| b : {\ttlbrace}f(x). x:A{\ttrbrace};  
+                 !!x.[| x:A;  b=f(x) |] ==> P |] ==> P
+
+\tdx{separation}     a : {\ttlbrace}x:A. P(x){\ttrbrace} <-> a:A & P(a)
+\tdx{CollectI}       [| a:A;  P(a) |] ==> a : {\ttlbrace}x:A. P(x){\ttrbrace}
+\tdx{CollectE}       [| a : {\ttlbrace}x:A. P(x){\ttrbrace};  [| a:A; P(a) |] ==> R |] ==> R
+\tdx{CollectD1}      a : {\ttlbrace}x:A. P(x){\ttrbrace} ==> a:A
+\tdx{CollectD2}      a : {\ttlbrace}x:A. P(x){\ttrbrace} ==> P(a)
+\end{ttbox}
+\caption{Replacement and separation} \label{zf-lemmas2}
+\end{figure}
+
+
+\begin{figure}
+\begin{ttbox}
+\tdx{UnionI}    [| B: C;  A: B |] ==> A: Union(C)
+\tdx{UnionE}    [| A : Union(C);  !!B.[| A: B;  B: C |] ==> R |] ==> R
+
+\tdx{InterI}    [| !!x. x: C ==> A: x;  c:C |] ==> A : Inter(C)
+\tdx{InterD}    [| A : Inter(C);  B : C |] ==> A : B
+\tdx{InterE}    [| A : Inter(C);  A:B ==> R;  ~ B:C ==> R |] ==> R
+
+\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);  a: A |] ==> b: (INT x:A. B(x))
+\tdx{INT_E}     [| b : (INT x:A. B(x));  a: A |] ==> b : B(a)
+\end{ttbox}
+\caption{General union and intersection} \label{zf-lemmas3}
+\end{figure}
+
+
+%%% upair.ML
+
+\begin{figure}
+\begin{ttbox}
+\tdx{pairing}      a:Upair(b,c) <-> (a=b | a=c)
+\tdx{UpairI1}      a : Upair(a,b)
+\tdx{UpairI2}      b : Upair(a,b)
+\tdx{UpairE}       [| a : Upair(b,c);  a = b ==> P;  a = c ==> P |] ==> P
+\end{ttbox}
+\caption{Unordered pairs} \label{zf-upair1}
+\end{figure}
+
+
+\begin{figure}
+\begin{ttbox}
+\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{DiffI}        [| c : A;  ~ c : B |] ==> c : A - B
+\tdx{DiffD1}       c : A - B ==> c : A
+\tdx{DiffD2}       c : A - B ==> c ~: B
+\tdx{DiffE}        [| c : A - B;  [| c:A; ~ c:B |] ==> P |] ==> P
+\end{ttbox}
+\caption{Union, intersection, difference} \label{zf-Un}
+\end{figure}
+
+
+\begin{figure}
+\begin{ttbox}
+\tdx{consI1}       a : cons(a,B)
+\tdx{consI2}       a : B ==> a : cons(b,B)
+\tdx{consCI}       (~ a:B ==> a=b) ==> a: cons(b,B)
+\tdx{consE}        [| a : cons(b,A);  a=b ==> P;  a:A ==> P |] ==> P
+
+\tdx{singletonI}   a : {\ttlbrace}a{\ttrbrace}
+\tdx{singletonE}   [| a : {\ttlbrace}b{\ttrbrace}; a=b ==> P |] ==> P
+\end{ttbox}
+\caption{Finite and singleton sets} \label{zf-upair2}
+\end{figure}
+
+
+\begin{figure}
+\begin{ttbox}
+\tdx{succI1}       i : succ(i)
+\tdx{succI2}       i : j ==> i : succ(j)
+\tdx{succCI}       (~ i:j ==> i=j) ==> i: succ(j)
+\tdx{succE}        [| i : succ(j);  i=j ==> P;  i:j ==> P |] ==> P
+\tdx{succ_neq_0}   [| succ(n)=0 |] ==> P
+\tdx{succ_inject}  succ(m) = succ(n) ==> m=n
+\end{ttbox}
+\caption{The successor function} \label{zf-succ}
+\end{figure}
+
+
+\begin{figure}
+\begin{ttbox}
+\tdx{the_equality}     [| P(a);  !!x. P(x) ==> x=a |] ==> (THE x. P(x)) = a
+\tdx{theI}             EX! x. P(x) ==> P(THE x. P(x))
+
+\tdx{if_P}              P ==> (if P then a else b) = a
+\tdx{if_not_P}         ~P ==> (if P then a else b) = b
+
+\tdx{mem_asym}         [| a:b;  b:a |] ==> P
+\tdx{mem_irrefl}       a:a ==> P
+\end{ttbox}
+\caption{Descriptions; non-circularity} \label{zf-the}
+\end{figure}
+
+
+\subsection{Unordered pairs and finite sets}
+Figure~\ref{zf-upair1} presents the principle of unordered pairing, along
+with its derived rules.  Binary union and intersection are defined in terms
+of ordered pairs (Fig.\ts\ref{zf-Un}).  Set difference is also included.  The
+rule \tdx{UnCI} is useful for classical reasoning about unions,
+like \texttt{disjCI}\@; it supersedes \tdx{UnI1} and
+\tdx{UnI2}, but these rules are often easier to work with.  For
+intersection and difference we have both elimination and destruction rules.
+Again, there is no reason to provide a minimal rule set.
+
+Figure~\ref{zf-upair2} is concerned with finite sets: it presents rules
+for~\texttt{cons}, the finite set constructor, and rules for singleton
+sets.  Figure~\ref{zf-succ} presents derived rules for the successor
+function, which is defined in terms of~\texttt{cons}.  The proof that {\tt
+  succ} is injective appears to require the Axiom of Foundation.
+
+Definite descriptions (\sdx{THE}) are defined in terms of the singleton
+set~$\{0\}$, but their derived rules fortunately hide this
+(Fig.\ts\ref{zf-the}).  The rule~\tdx{theI} is difficult to apply
+because of the two occurrences of~$\Var{P}$.  However,
+\tdx{the_equality} does not have this problem and the files contain
+many examples of its use.
+
+Finally, the impossibility of having both $a\in b$ and $b\in a$
+(\tdx{mem_asym}) is proved by applying the Axiom of Foundation to
+the set $\{a,b\}$.  The impossibility of $a\in a$ is a trivial consequence.
+
+See the file \texttt{ZF/upair.ML} for full proofs of the rules discussed in
+this section.
+
+
+%%% subset.ML
+
+\begin{figure}
+\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}    [| a:A;  !!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
+
+\tdx{Diff_subset}       A-B <= A
+\tdx{Diff_contains}     [| C<=A;  C Int B = 0 |] ==> C <= A-B
+
+\tdx{Collect_subset}    Collect(A,P) <= A
+\end{ttbox}
+\caption{Subset and lattice properties} \label{zf-subset}
+\end{figure}
+
+
+\subsection{Subset and lattice properties}
+The subset relation is a complete lattice.  Unions form least upper bounds;
+non-empty intersections form greatest lower bounds.  Figure~\ref{zf-subset}
+shows the corresponding rules.  A few other laws involving subsets are
+included.  Proofs are in the file \texttt{ZF/subset.ML}.
+
+Reasoning directly about subsets often yields clearer proofs than
+reasoning about the membership relation.  Section~\ref{sec:ZF-pow-example}
+below presents an example of this, proving the equation ${{\tt Pow}(A)\cap
+  {\tt Pow}(B)}= {\tt Pow}(A\cap B)$.
+
+%%% pair.ML
+
+\begin{figure}
+\begin{ttbox}
+\tdx{Pair_inject1}    <a,b> = <c,d> ==> a=c
+\tdx{Pair_inject2}    <a,b> = <c,d> ==> b=d
+\tdx{Pair_inject}     [| <a,b> = <c,d>;  [| a=c; b=d |] ==> P |] ==> P
+\tdx{Pair_neq_0}      <a,b>=0 ==> P
+
+\tdx{fst_conv}        fst(<a,b>) = a
+\tdx{snd_conv}        snd(<a,b>) = b
+\tdx{split}           split(\%x y. c(x,y), <a,b>) = c(a,b)
+
+\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
+
+\tdx{SigmaE2}         [| <a,b> : Sigma(A,B);    
+                   [| a:A;  b:B(a) |] ==> P   |] ==> P
+\end{ttbox}
+\caption{Ordered pairs; projections; general sums} \label{zf-pair}
+\end{figure}
+
+
+\subsection{Ordered pairs} \label{sec:pairs}
+
+Figure~\ref{zf-pair} presents the rules governing ordered pairs,
+projections and general sums.  File \texttt{ZF/pair.ML} contains the
+full (and tedious) proof that $\{\{a\},\{a,b\}\}$ functions as an ordered
+pair.  This property is expressed as two destruction rules,
+\tdx{Pair_inject1} and \tdx{Pair_inject2}, and equivalently
+as the elimination rule \tdx{Pair_inject}.
+
+The rule \tdx{Pair_neq_0} asserts $\pair{a,b}\neq\emptyset$.  This
+is a property of $\{\{a\},\{a,b\}\}$, and need not hold for other 
+encodings of ordered pairs.  The non-standard ordered pairs mentioned below
+satisfy $\pair{\emptyset;\emptyset}=\emptyset$.
+
+The natural deduction rules \tdx{SigmaI} and \tdx{SigmaE}
+assert that \cdx{Sigma}$(A,B)$ consists of all pairs of the form
+$\pair{x,y}$, for $x\in A$ and $y\in B(x)$.  The rule \tdx{SigmaE2}
+merely states that $\pair{a,b}\in \texttt{Sigma}(A,B)$ implies $a\in A$ and
+$b\in B(a)$.
+
+In addition, it is possible to use tuples as patterns in abstractions:
+\begin{center}
+{\tt\%<$x$,$y$>. $t$} \quad stands for\quad \texttt{split(\%$x$ $y$.\ $t$)}
+\end{center}
+Nested patterns are translated recursively:
+{\tt\%<$x$,$y$,$z$>. $t$} $\leadsto$ {\tt\%<$x$,<$y$,$z$>>. $t$} $\leadsto$
+\texttt{split(\%$x$.\%<$y$,$z$>. $t$)} $\leadsto$ \texttt{split(\%$x$. split(\%$y$
+  $z$.\ $t$))}.  The reverse translation is performed upon printing.
+\begin{warn}
+  The translation between patterns and \texttt{split} is performed automatically
+  by the parser and printer.  Thus the internal and external form of a term
+  may differ, which affects proofs.  For example the term {\tt
+    (\%<x,y>.<y,x>)<a,b>} requires the theorem \texttt{split} to rewrite to
+  {\tt<b,a>}.
+\end{warn}
+In addition to explicit $\lambda$-abstractions, patterns can be used in any
+variable binding construct which is internally described by a
+$\lambda$-abstraction.  Here are some important examples:
+\begin{description}
+\item[Let:] \texttt{let {\it pattern} = $t$ in $u$}
+\item[Choice:] \texttt{THE~{\it pattern}~.~$P$}
+\item[Set operations:] \texttt{UN~{\it pattern}:$A$.~$B$}
+\item[Comprehension:] \texttt{{\ttlbrace}~{\it pattern}:$A$~.~$P$~{\ttrbrace}}
+\end{description}
+
+
+%%% domrange.ML
+
+\begin{figure}
+\begin{ttbox}
+\tdx{domainI}        <a,b>: r ==> a : domain(r)
+\tdx{domainE}        [| a : domain(r);  !!y. <a,y>: r ==> P |] ==> P
+\tdx{domain_subset}  domain(Sigma(A,B)) <= A
+
+\tdx{rangeI}         <a,b>: r ==> b : range(r)
+\tdx{rangeE}         [| b : range(r);  !!x. <x,b>: r ==> P |] ==> P
+\tdx{range_subset}   range(A*B) <= B
+
+\tdx{fieldI1}        <a,b>: r ==> a : field(r)
+\tdx{fieldI2}        <a,b>: r ==> b : field(r)
+\tdx{fieldCI}        (~ <c,a>:r ==> <a,b>: r) ==> a : field(r)
+
+\tdx{fieldE}         [| a : field(r);  
+                  !!x. <a,x>: r ==> P;  
+                  !!x. <x,a>: r ==> P      
+               |] ==> P
+
+\tdx{field_subset}   field(A*A) <= A
+\end{ttbox}
+\caption{Domain, range and field of a relation} \label{zf-domrange}
+\end{figure}
+
+\begin{figure}
+\begin{ttbox}
+\tdx{imageI}         [| <a,b>: r;  a:A |] ==> b : r``A
+\tdx{imageE}         [| b: r``A;  !!x.[| <x,b>: r;  x:A |] ==> P |] ==> P
+
+\tdx{vimageI}        [| <a,b>: r;  b:B |] ==> a : r-``B
+\tdx{vimageE}        [| a: r-``B;  !!x.[| <a,x>: r;  x:B |] ==> P |] ==> P
+\end{ttbox}
+\caption{Image and inverse image} \label{zf-domrange2}
+\end{figure}
+
+
+\subsection{Relations}
+Figure~\ref{zf-domrange} presents rules involving relations, which are sets
+of ordered pairs.  The converse of a relation~$r$ is the set of all pairs
+$\pair{y,x}$ such that $\pair{x,y}\in r$; if $r$ is a function, then
+{\cdx{converse}$(r)$} is its inverse.  The rules for the domain
+operation, namely \tdx{domainI} and~\tdx{domainE}, assert that
+\cdx{domain}$(r)$ consists of all~$x$ such that $r$ contains
+some pair of the form~$\pair{x,y}$.  The range operation is similar, and
+the field of a relation is merely the union of its domain and range.  
+
+Figure~\ref{zf-domrange2} presents rules for images and inverse images.
+Note that these operations are generalisations of range and domain,
+respectively.  See the file \texttt{ZF/domrange.ML} for derivations of the
+rules.
+
+
+%%% func.ML
+
+\begin{figure}
+\begin{ttbox}
+\tdx{fun_is_rel}      f: Pi(A,B) ==> f <= Sigma(A,B)
+
+\tdx{apply_equality}  [| <a,b>: f;  f: Pi(A,B) |] ==> f`a = b
+\tdx{apply_equality2} [| <a,b>: f;  <a,c>: f;  f: Pi(A,B) |] ==> b=c
+
+\tdx{apply_type}      [| f: Pi(A,B);  a:A |] ==> f`a : B(a)
+\tdx{apply_Pair}      [| f: Pi(A,B);  a:A |] ==> <a,f`a>: f
+\tdx{apply_iff}       f: Pi(A,B) ==> <a,b>: f <-> a:A & f`a = b
+
+\tdx{fun_extension}   [| f : Pi(A,B);  g: Pi(A,D);
+                   !!x. x:A ==> f`x = g`x     |] ==> f=g
+
+\tdx{domain_type}     [| <a,b> : f;  f: Pi(A,B) |] ==> a : A
+\tdx{range_type}      [| <a,b> : f;  f: Pi(A,B) |] ==> b : B(a)
+
+\tdx{Pi_type}         [| f: A->C;  !!x. x:A ==> f`x: B(x) |] ==> f: Pi(A,B)
+\tdx{domain_of_fun}   f: Pi(A,B) ==> domain(f)=A
+\tdx{range_of_fun}    f: Pi(A,B) ==> f: A->range(f)
+
+\tdx{restrict}        a : A ==> restrict(f,A) ` a = f`a
+\tdx{restrict_type}   [| !!x. x:A ==> f`x: B(x) |] ==> 
+                restrict(f,A) : Pi(A,B)
+\end{ttbox}
+\caption{Functions} \label{zf-func1}
+\end{figure}
+
+
+\begin{figure}
+\begin{ttbox}
+\tdx{lamI}         a:A ==> <a,b(a)> : (lam x:A. b(x))
+\tdx{lamE}         [| p: (lam x:A. b(x));  !!x.[| x:A; p=<x,b(x)> |] ==> P 
+             |] ==>  P
+
+\tdx{lam_type}     [| !!x. x:A ==> b(x): B(x) |] ==> (lam x:A. b(x)) : Pi(A,B)
+
+\tdx{beta}         a : A ==> (lam x:A. b(x)) ` a = b(a)
+\tdx{eta}          f : Pi(A,B) ==> (lam x:A. f`x) = f
+\end{ttbox}
+\caption{$\lambda$-abstraction} \label{zf-lam}
+\end{figure}
+
+
+\begin{figure}
+\begin{ttbox}
+\tdx{fun_empty}            0: 0->0
+\tdx{fun_single}           {\ttlbrace}<a,b>{\ttrbrace} : {\ttlbrace}a{\ttrbrace} -> {\ttlbrace}b{\ttrbrace}
+
+\tdx{fun_disjoint_Un}      [| f: A->B;  g: C->D;  A Int C = 0  |] ==>  
+                     (f Un g) : (A Un C) -> (B Un D)
+
+\tdx{fun_disjoint_apply1}  [| a:A;  f: A->B;  g: C->D;  A Int C = 0 |] ==>  
+                     (f Un g)`a = f`a
+
+\tdx{fun_disjoint_apply2}  [| c:C;  f: A->B;  g: C->D;  A Int C = 0 |] ==>  
+                     (f Un g)`c = g`c
+\end{ttbox}
+\caption{Constructing functions from smaller sets} \label{zf-func2}
+\end{figure}
+
+
+\subsection{Functions}
+Functions, represented by graphs, are notoriously difficult to reason
+about.  The file \texttt{ZF/func.ML} derives many rules, which overlap more
+than they ought.  This section presents the more important rules.
+
+Figure~\ref{zf-func1} presents the basic properties of \cdx{Pi}$(A,B)$,
+the generalized function space.  For example, if $f$ is a function and
+$\pair{a,b}\in f$, then $f`a=b$ (\tdx{apply_equality}).  Two functions
+are equal provided they have equal domains and deliver equals results
+(\tdx{fun_extension}).
+
+By \tdx{Pi_type}, a function typing of the form $f\in A\to C$ can be
+refined to the dependent typing $f\in\prod@{x\in A}B(x)$, given a suitable
+family of sets $\{B(x)\}@{x\in A}$.  Conversely, by \tdx{range_of_fun},
+any dependent typing can be flattened to yield a function type of the form
+$A\to C$; here, $C={\tt range}(f)$.
+
+Among the laws for $\lambda$-abstraction, \tdx{lamI} and \tdx{lamE}
+describe the graph of the generated function, while \tdx{beta} and
+\tdx{eta} are the standard conversions.  We essentially have a
+dependently-typed $\lambda$-calculus (Fig.\ts\ref{zf-lam}).
+
+Figure~\ref{zf-func2} presents some rules that can be used to construct
+functions explicitly.  We start with functions consisting of at most one
+pair, and may form the union of two functions provided their domains are
+disjoint.  
+
+
+\begin{figure}
+\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{Diff_cancel}        A-A = 0
+\tdx{Diff_disjoint}      A Int (B-A) = 0
+\tdx{Diff_partition}     A<=B ==> A Un (B-A) = B
+\tdx{double_complement}  [| A<=B; B<= C |] ==> (B - (C-A)) = A
+\tdx{Diff_Un}            A - (B Un C) = (A-B) Int (A-C)
+\tdx{Diff_Int}           A - (B Int C) = (A-B) Un (A-C)
+
+\tdx{Union_Un_distrib}   Union(A Un B) = Union(A) Un Union(B)
+\tdx{Inter_Un_distrib}   [| a:A;  b:B |] ==> 
+                   Inter(A Un B) = Inter(A) Int Inter(B)
+
+\tdx{Int_Union_RepFun}   A Int Union(B) = (UN C:B. A Int C)
+
+\tdx{Un_Inter_RepFun}    b:B ==> 
+                   A Un Inter(B) = (INT C:B. A Un C)
+
+\tdx{SUM_Un_distrib1}    (SUM x:A Un B. C(x)) = 
+                   (SUM x:A. C(x)) Un (SUM x:B. C(x))
+
+\tdx{SUM_Un_distrib2}    (SUM x:C. A(x) Un B(x)) =
+                   (SUM x:C. A(x))  Un  (SUM x:C. B(x))
+
+\tdx{SUM_Int_distrib1}   (SUM x:A Int B. C(x)) =
+                   (SUM x:A. C(x)) Int (SUM x:B. C(x))
+
+\tdx{SUM_Int_distrib2}   (SUM x:C. A(x) Int B(x)) =
+                   (SUM x:C. A(x)) Int (SUM x:C. B(x))
+\end{ttbox}
+\caption{Equalities} \label{zf-equalities}
+\end{figure}
+
+
+\begin{figure}
+%\begin{constants} 
+%  \cdx{1}       & $i$           &       & $\{\emptyset\}$       \\
+%  \cdx{bool}    & $i$           &       & the set $\{\emptyset,1\}$     \\
+%  \cdx{cond}   & $[i,i,i]\To i$ &       & conditional for \texttt{bool}    \\
+%  \cdx{not}    & $i\To i$       &       & negation for \texttt{bool}       \\
+%  \sdx{and}    & $[i,i]\To i$   & Left 70 & conjunction for \texttt{bool}  \\
+%  \sdx{or}     & $[i,i]\To i$   & Left 65 & disjunction for \texttt{bool}  \\
+%  \sdx{xor}    & $[i,i]\To i$   & Left 65 & exclusive-or for \texttt{bool}
+%\end{constants}
+%
+\begin{ttbox}
+\tdx{bool_def}       bool == {\ttlbrace}0,1{\ttrbrace}
+\tdx{cond_def}       cond(b,c,d) == if b=1 then c else d
+\tdx{not_def}        not(b)  == cond(b,0,1)
+\tdx{and_def}        a and b == cond(a,b,0)
+\tdx{or_def}         a or b  == cond(a,1,b)
+\tdx{xor_def}        a xor b == cond(a,not(b),b)
+
+\tdx{bool_1I}        1 : bool
+\tdx{bool_0I}        0 : bool
+\tdx{boolE}          [| c: bool;  c=1 ==> P;  c=0 ==> P |] ==> P
+\tdx{cond_1}         cond(1,c,d) = c
+\tdx{cond_0}         cond(0,c,d) = d
+\end{ttbox}
+\caption{The booleans} \label{zf-bool}
+\end{figure}
+
+
+\section{Further developments}
+The next group of developments is complex and extensive, and only
+highlights can be covered here.  It involves many theories and ML files of
+proofs. 
+
+Figure~\ref{zf-equalities} presents commutative, associative, distributive,
+and idempotency laws of union and intersection, along with other equations.
+See file \texttt{ZF/equalities.ML}.
+
+Theory \thydx{Bool} defines $\{0,1\}$ as a set of booleans, with the usual
+operators including a conditional (Fig.\ts\ref{zf-bool}).  Although {\ZF} is a
+first-order theory, you can obtain the effect of higher-order logic using
+\texttt{bool}-valued functions, for example.  The constant~\texttt{1} is
+translated to \texttt{succ(0)}.
+
+\begin{figure}
+\index{*"+ symbol}
+\begin{constants}
+  \it symbol    & \it meta-type & \it priority & \it description \\ 
+  \tt +         & $[i,i]\To i$  &  Right 65     & disjoint union operator\\
+  \cdx{Inl}~~\cdx{Inr}  & $i\To i$      &       & injections\\
+  \cdx{case}    & $[i\To i,i\To i, i]\To i$ &   & conditional for $A+B$
+\end{constants}
+\begin{ttbox}
+\tdx{sum_def}        A+B == {\ttlbrace}0{\ttrbrace}*A Un {\ttlbrace}1{\ttrbrace}*B
+\tdx{Inl_def}        Inl(a) == <0,a>
+\tdx{Inr_def}        Inr(b) == <1,b>
+\tdx{case_def}       case(c,d,u) == split(\%y z. cond(y, d(z), c(z)), u)
+
+\tdx{sum_InlI}       a : A ==> Inl(a) : A+B
+\tdx{sum_InrI}       b : B ==> Inr(b) : A+B
+
+\tdx{Inl_inject}     Inl(a)=Inl(b) ==> a=b
+\tdx{Inr_inject}     Inr(a)=Inr(b) ==> a=b
+\tdx{Inl_neq_Inr}    Inl(a)=Inr(b) ==> P
+
+\tdx{sumE2}   u: A+B ==> (EX x. x:A & u=Inl(x)) | (EX y. y:B & u=Inr(y))
+
+\tdx{case_Inl}       case(c,d,Inl(a)) = c(a)
+\tdx{case_Inr}       case(c,d,Inr(b)) = d(b)
+\end{ttbox}
+\caption{Disjoint unions} \label{zf-sum}
+\end{figure}
+
+
+Theory \thydx{Sum} defines the disjoint union of two sets, with
+injections and a case analysis operator (Fig.\ts\ref{zf-sum}).  Disjoint
+unions play a role in datatype definitions, particularly when there is
+mutual recursion~\cite{paulson-set-II}.
+
+\begin{figure}
+\begin{ttbox}
+\tdx{QPair_def}       <a;b> == a+b
+\tdx{qsplit_def}      qsplit(c,p)  == THE y. EX a b. p=<a;b> & y=c(a,b)
+\tdx{qfsplit_def}     qfsplit(R,z) == EX x y. z=<x;y> & R(x,y)
+\tdx{qconverse_def}   qconverse(r) == {\ttlbrace}z. w:r, EX x y. w=<x;y> & z=<y;x>{\ttrbrace}
+\tdx{QSigma_def}      QSigma(A,B)  == UN x:A. UN y:B(x). {\ttlbrace}<x;y>{\ttrbrace}
+
+\tdx{qsum_def}        A <+> B      == ({\ttlbrace}0{\ttrbrace} <*> A) Un ({\ttlbrace}1{\ttrbrace} <*> B)
+\tdx{QInl_def}        QInl(a)      == <0;a>
+\tdx{QInr_def}        QInr(b)      == <1;b>
+\tdx{qcase_def}       qcase(c,d)   == qsplit(\%y z. cond(y, d(z), c(z)))
+\end{ttbox}
+\caption{Non-standard pairs, products and sums} \label{zf-qpair}
+\end{figure}
+
+Theory \thydx{QPair} defines a notion of ordered pair that admits
+non-well-founded tupling (Fig.\ts\ref{zf-qpair}).  Such pairs are written
+{\tt<$a$;$b$>}.  It also defines the eliminator \cdx{qsplit}, the
+converse operator \cdx{qconverse}, and the summation operator
+\cdx{QSigma}.  These are completely analogous to the corresponding
+versions for standard ordered pairs.  The theory goes on to define a
+non-standard notion of disjoint sum using non-standard pairs.  All of these
+concepts satisfy the same properties as their standard counterparts; in
+addition, {\tt<$a$;$b$>} is continuous.  The theory supports coinductive
+definitions, for example of infinite lists~\cite{paulson-final}.
+
+\begin{figure}
+\begin{ttbox}
+\tdx{bnd_mono_def}   bnd_mono(D,h) == 
+                 h(D)<=D & (ALL W X. W<=X --> X<=D --> h(W) <= h(X))
+
+\tdx{lfp_def}        lfp(D,h) == Inter({\ttlbrace}X: Pow(D). h(X) <= X{\ttrbrace})
+\tdx{gfp_def}        gfp(D,h) == Union({\ttlbrace}X: Pow(D). X <= h(X){\ttrbrace})
+
+
+\tdx{lfp_lowerbound} [| h(A) <= A;  A<=D |] ==> lfp(D,h) <= A
+
+\tdx{lfp_subset}     lfp(D,h) <= D
+
+\tdx{lfp_greatest}   [| bnd_mono(D,h);  
+                  !!X. [| h(X) <= X;  X<=D |] ==> A<=X 
+               |] ==> A <= lfp(D,h)
+
+\tdx{lfp_Tarski}     bnd_mono(D,h) ==> lfp(D,h) = h(lfp(D,h))
+
+\tdx{induct}         [| a : lfp(D,h);  bnd_mono(D,h);
+                  !!x. x : h(Collect(lfp(D,h),P)) ==> P(x)
+               |] ==> P(a)
+
+\tdx{lfp_mono}       [| bnd_mono(D,h);  bnd_mono(E,i);
+                  !!X. X<=D ==> h(X) <= i(X)  
+               |] ==> lfp(D,h) <= lfp(E,i)
+
+\tdx{gfp_upperbound} [| A <= h(A);  A<=D |] ==> A <= gfp(D,h)
+
+\tdx{gfp_subset}     gfp(D,h) <= D
+
+\tdx{gfp_least}      [| bnd_mono(D,h);  
+                  !!X. [| X <= h(X);  X<=D |] ==> X<=A
+               |] ==> gfp(D,h) <= A
+
+\tdx{gfp_Tarski}     bnd_mono(D,h) ==> gfp(D,h) = h(gfp(D,h))
+
+\tdx{coinduct}       [| bnd_mono(D,h); a: X; X <= h(X Un gfp(D,h)); X <= D 
+               |] ==> a : gfp(D,h)
+
+\tdx{gfp_mono}       [| bnd_mono(D,h);  D <= E;
+                  !!X. X<=D ==> h(X) <= i(X)  
+               |] ==> gfp(D,h) <= gfp(E,i)
+\end{ttbox}
+\caption{Least and greatest fixedpoints} \label{zf-fixedpt}
+\end{figure}
+
+The Knaster-Tarski Theorem states that every monotone function over a
+complete lattice has a fixedpoint.  Theory \thydx{Fixedpt} proves the
+Theorem only for a particular lattice, namely the lattice of subsets of a
+set (Fig.\ts\ref{zf-fixedpt}).  The theory defines least and greatest
+fixedpoint operators with corresponding induction and coinduction rules.
+These are essential to many definitions that follow, including the natural
+numbers and the transitive closure operator.  The (co)inductive definition
+package also uses the fixedpoint operators~\cite{paulson-CADE}.  See
+Davey and Priestley~\cite{davey&priestley} for more on the Knaster-Tarski
+Theorem and my paper~\cite{paulson-set-II} for discussion of the Isabelle
+proofs.
+
+Monotonicity properties are proved for most of the set-forming operations:
+union, intersection, Cartesian product, image, domain, range, etc.  These
+are useful for applying the Knaster-Tarski Fixedpoint Theorem.  The proofs
+themselves are trivial applications of Isabelle's classical reasoner.  See
+file \texttt{ZF/mono.ML}.
+
+
+\begin{figure}
+\begin{constants} 
+  \it symbol  & \it meta-type & \it priority & \it description \\ 
+  \sdx{O}       & $[i,i]\To i$  &  Right 60     & composition ($\circ$) \\
+  \cdx{id}      & $i\To i$      &       & identity function \\
+  \cdx{inj}     & $[i,i]\To i$  &       & injective function space\\
+  \cdx{surj}    & $[i,i]\To i$  &       & surjective function space\\
+  \cdx{bij}     & $[i,i]\To i$  &       & bijective function space
+\end{constants}
+
+\begin{ttbox}
+\tdx{comp_def}  r O s     == {\ttlbrace}xz : domain(s)*range(r) . 
+                        EX x y z. xz=<x,z> & <x,y>:s & <y,z>:r{\ttrbrace}
+\tdx{id_def}    id(A)     == (lam x:A. x)
+\tdx{inj_def}   inj(A,B)  == {\ttlbrace} f: A->B. ALL w:A. ALL x:A. f`w=f`x --> w=x {\ttrbrace}
+\tdx{surj_def}  surj(A,B) == {\ttlbrace} f: A->B . ALL y:B. EX x:A. f`x=y {\ttrbrace}
+\tdx{bij_def}   bij(A,B)  == inj(A,B) Int surj(A,B)
+
+
+\tdx{left_inverse}     [| f: inj(A,B);  a: A |] ==> converse(f)`(f`a) = a
+\tdx{right_inverse}    [| f: inj(A,B);  b: range(f) |] ==> 
+                 f`(converse(f)`b) = b
+
+\tdx{inj_converse_inj} f: inj(A,B) ==> converse(f): inj(range(f), A)
+\tdx{bij_converse_bij} f: bij(A,B) ==> converse(f): bij(B,A)
+
+\tdx{comp_type}        [| s<=A*B;  r<=B*C |] ==> (r O s) <= A*C
+\tdx{comp_assoc}       (r O s) O t = r O (s O t)
+
+\tdx{left_comp_id}     r<=A*B ==> id(B) O r = r
+\tdx{right_comp_id}    r<=A*B ==> r O id(A) = r
+
+\tdx{comp_func}        [| g:A->B; f:B->C |] ==> (f O g):A->C
+\tdx{comp_func_apply}  [| g:A->B; f:B->C; a:A |] ==> (f O g)`a = f`(g`a)
+
+\tdx{comp_inj}         [| g:inj(A,B);  f:inj(B,C)  |] ==> (f O g):inj(A,C)
+\tdx{comp_surj}        [| g:surj(A,B); f:surj(B,C) |] ==> (f O g):surj(A,C)
+\tdx{comp_bij}         [| g:bij(A,B); f:bij(B,C) |] ==> (f O g):bij(A,C)
+
+\tdx{left_comp_inverse}     f: inj(A,B) ==> converse(f) O f = id(A)
+\tdx{right_comp_inverse}    f: surj(A,B) ==> f O converse(f) = id(B)
+
+\tdx{bij_disjoint_Un}   
+    [| f: bij(A,B);  g: bij(C,D);  A Int C = 0;  B Int D = 0 |] ==> 
+    (f Un g) : bij(A Un C, B Un D)
+
+\tdx{restrict_bij}  [| f:inj(A,B);  C<=A |] ==> restrict(f,C): bij(C, f``C)
+\end{ttbox}
+\caption{Permutations} \label{zf-perm}
+\end{figure}
+
+The theory \thydx{Perm} is concerned with permutations (bijections) and
+related concepts.  These include composition of relations, the identity
+relation, and three specialized function spaces: injective, surjective and
+bijective.  Figure~\ref{zf-perm} displays many of their properties that
+have been proved.  These results are fundamental to a treatment of
+equipollence and cardinality.
+
+\begin{figure}\small
+\index{#*@{\tt\#*} symbol}
+\index{*div symbol}
+\index{*mod symbol}
+\index{#+@{\tt\#+} symbol}
+\index{#-@{\tt\#-} symbol}
+\begin{constants}
+  \it symbol  & \it meta-type & \it priority & \it description \\ 
+  \cdx{nat}     & $i$                   &       & set of natural numbers \\
+  \cdx{nat_case}& $[i,i\To i,i]\To i$     &     & conditional for $nat$\\
+  \tt \#*       & $[i,i]\To i$  &  Left 70      & multiplication \\
+  \tt div       & $[i,i]\To i$  &  Left 70      & division\\
+  \tt mod       & $[i,i]\To i$  &  Left 70      & modulus\\
+  \tt \#+       & $[i,i]\To i$  &  Left 65      & addition\\
+  \tt \#-       & $[i,i]\To i$  &  Left 65      & subtraction
+\end{constants}
+
+\begin{ttbox}
+\tdx{nat_def}  nat == lfp(lam r: Pow(Inf). {\ttlbrace}0{\ttrbrace} Un {\ttlbrace}succ(x). x:r{\ttrbrace}
+
+\tdx{mod_def}  m mod n == transrec(m, \%j f. if j:n then j else f`(j#-n))
+\tdx{div_def}  m div n == transrec(m, \%j f. if j:n then 0 else succ(f`(j#-n)))
+
+\tdx{nat_case_def}  nat_case(a,b,k) == 
+              THE y. k=0 & y=a | (EX x. k=succ(x) & y=b(x))
+
+\tdx{nat_0I}        0 : nat
+\tdx{nat_succI}     n : nat ==> succ(n) : nat
+
+\tdx{nat_induct}        
+    [| n: nat;  P(0);  !!x. [| x: nat;  P(x) |] ==> P(succ(x)) 
+    |] ==> P(n)
+
+\tdx{nat_case_0}    nat_case(a,b,0) = a
+\tdx{nat_case_succ} nat_case(a,b,succ(m)) = b(m)
+
+\tdx{add_0}        0 #+ n = n
+\tdx{add_succ}     succ(m) #+ n = succ(m #+ n)
+
+\tdx{mult_type}     [| m:nat;  n:nat |] ==> m #* n : nat
+\tdx{mult_0}        0 #* n = 0
+\tdx{mult_succ}     succ(m) #* n = n #+ (m #* n)
+\tdx{mult_commute}  [| m:nat; n:nat |] ==> m #* n = n #* m
+\tdx{add_mult_dist} [| m:nat; k:nat |] ==> (m #+ n) #* k = (m #* k){\thinspace}#+{\thinspace}(n #* k)
+\tdx{mult_assoc}
+    [| m:nat;  n:nat;  k:nat |] ==> (m #* n) #* k = m #* (n #* k)
+\tdx{mod_quo_equality}
+    [| 0:n;  m:nat;  n:nat |] ==> (m div n)#*n #+ m mod n = m
+\end{ttbox}
+\caption{The natural numbers} \label{zf-nat}
+\end{figure}
+
+Theory \thydx{Nat} defines the natural numbers and mathematical
+induction, along with a case analysis operator.  The set of natural
+numbers, here called \texttt{nat}, is known in set theory as the ordinal~$\omega$.
+
+Theory \thydx{Arith} develops arithmetic on the natural numbers
+(Fig.\ts\ref{zf-nat}).  Addition, multiplication and subtraction are defined
+by primitive recursion.  Division and remainder are defined by repeated
+subtraction, which requires well-founded recursion; the termination argument
+relies on the divisor's being non-zero.  Many properties are proved:
+commutative, associative and distributive laws, identity and cancellation
+laws, etc.  The most interesting result is perhaps the theorem $a \bmod b +
+(a/b)\times b = a$.
+
+Theory \thydx{Univ} defines a `universe' $\texttt{univ}(A)$, which is used by
+the datatype package.  This set contains $A$ and the
+natural numbers.  Vitally, it is closed under finite products: ${\tt
+  univ}(A)\times{\tt univ}(A)\subseteq{\tt univ}(A)$.  This theory also
+defines the cumulative hierarchy of axiomatic set theory, which
+traditionally is written $V@\alpha$ for an ordinal~$\alpha$.  The
+`universe' is a simple generalization of~$V@\omega$.
+
+Theory \thydx{QUniv} defines a `universe' ${\tt quniv}(A)$, which is used by
+the datatype package to construct codatatypes such as streams.  It is
+analogous to ${\tt univ}(A)$ (and is defined in terms of it) but is closed
+under the non-standard product and sum.
+
+Theory \texttt{Finite} (Figure~\ref{zf-fin}) defines the finite set operator;
+${\tt Fin}(A)$ is the set of all finite sets over~$A$.  The theory employs
+Isabelle's inductive definition package, which proves various rules
+automatically.  The induction rule shown is stronger than the one proved by
+the package.  The theory also defines the set of all finite functions
+between two given sets.
+
+\begin{figure}
+\begin{ttbox}
+\tdx{Fin.emptyI}      0 : Fin(A)
+\tdx{Fin.consI}       [| a: A;  b: Fin(A) |] ==> cons(a,b) : Fin(A)
+
+\tdx{Fin_induct}
+    [| b: Fin(A);
+       P(0);
+       !!x y. [| x: A;  y: Fin(A);  x~:y;  P(y) |] ==> P(cons(x,y))
+    |] ==> P(b)
+
+\tdx{Fin_mono}        A<=B ==> Fin(A) <= Fin(B)
+\tdx{Fin_UnI}         [| b: Fin(A);  c: Fin(A) |] ==> b Un c : Fin(A)
+\tdx{Fin_UnionI}      C : Fin(Fin(A)) ==> Union(C) : Fin(A)
+\tdx{Fin_subset}      [| c<=b;  b: Fin(A) |] ==> c: Fin(A)
+\end{ttbox}
+\caption{The finite set operator} \label{zf-fin}
+\end{figure}
+
+\begin{figure}
+\begin{constants}
+  \it symbol  & \it meta-type & \it priority & \it description \\ 
+  \cdx{list}    & $i\To i$      && lists over some set\\
+  \cdx{list_case} & $[i, [i,i]\To i, i] \To i$  && conditional for $list(A)$ \\
+  \cdx{map}     & $[i\To i, i] \To i$   &       & mapping functional\\
+  \cdx{length}  & $i\To i$              &       & length of a list\\
+  \cdx{rev}     & $i\To i$              &       & reverse of a list\\
+  \tt \at       & $[i,i]\To i$  &  Right 60     & append for lists\\
+  \cdx{flat}    & $i\To i$   &                  & append of list of lists
+\end{constants}
+
+\underscoreon %%because @ is used here
+\begin{ttbox}
+\tdx{NilI}            Nil : list(A)
+\tdx{ConsI}           [| a: A;  l: list(A) |] ==> Cons(a,l) : list(A)
+
+\tdx{List.induct}
+    [| l: list(A);
+       P(Nil);
+       !!x y. [| x: A;  y: list(A);  P(y) |] ==> P(Cons(x,y))
+    |] ==> P(l)
+
+\tdx{Cons_iff}        Cons(a,l)=Cons(a',l') <-> a=a' & l=l'
+\tdx{Nil_Cons_iff}    ~ Nil=Cons(a,l)
+
+\tdx{list_mono}       A<=B ==> list(A) <= list(B)
+
+\tdx{map_ident}       l: list(A) ==> map(\%u. u, l) = l
+\tdx{map_compose}     l: list(A) ==> map(h, map(j,l)) = map(\%u. h(j(u)), l)
+\tdx{map_app_distrib} xs: list(A) ==> map(h, xs@ys) = map(h,xs) @ map(h,ys)
+\tdx{map_type}
+    [| l: list(A);  !!x. x: A ==> h(x): B |] ==> map(h,l) : list(B)
+\tdx{map_flat}
+    ls: list(list(A)) ==> map(h, flat(ls)) = flat(map(map(h),ls))
+\end{ttbox}
+\caption{Lists} \label{zf-list}
+\end{figure}
+
+
+Figure~\ref{zf-list} presents the set of lists over~$A$, ${\tt list}(A)$.  The
+definition employs Isabelle's datatype package, which defines the introduction
+and induction rules automatically, as well as the constructors, case operator
+(\verb|list_case|) and recursion operator.  The theory then defines the usual
+list functions by primitive recursion.  See theory \texttt{List}.
+
+
+\section{Simplification and classical reasoning}
+
+{\ZF} inherits simplification from {\FOL} but adopts it for set theory.  The
+extraction of rewrite rules takes the {\ZF} primitives into account.  It can
+strip bounded universal quantifiers from a formula; for example, ${\forall
+  x\in A. f(x)=g(x)}$ yields the conditional rewrite rule $x\in A \Imp
+f(x)=g(x)$.  Given $a\in\{x\in A. P(x)\}$ it extracts rewrite rules from $a\in
+A$ and~$P(a)$.  It can also break down $a\in A\int B$ and $a\in A-B$.
+
+Simplification tactics tactics such as \texttt{Asm_simp_tac} and
+\texttt{Full_simp_tac} use the default simpset (\texttt{simpset()}), which
+works for most purposes.  A small simplification set for set theory is
+called~\ttindexbold{ZF_ss}, and you can even use \ttindex{FOL_ss} as a minimal
+starting point.  \texttt{ZF_ss} contains congruence rules for all the binding
+operators of {\ZF}\@.  It contains all the conversion rules, such as
+\texttt{fst} and \texttt{snd}, as well as the rewrites shown in
+Fig.\ts\ref{zf-simpdata}.  See the file \texttt{ZF/simpdata.ML} for a fuller
+list.
+
+As for the classical reasoner, tactics such as \texttt{Blast_tac} and {\tt
+  Best_tac} refer to the default claset (\texttt{claset()}).  This works for
+most purposes.  Named clasets include \ttindexbold{ZF_cs} (basic set theory)
+and \ttindexbold{le_cs} (useful for reasoning about the relations $<$ and
+$\le$).  You can use \ttindex{FOL_cs} as a minimal basis for building your own
+clasets.  See \iflabelundefined{chap:classical}{the {\em Reference Manual\/}}%
+{Chap.\ts\ref{chap:classical}} for more discussion of classical proof methods.
+
+
+\begin{figure}
+\begin{eqnarray*}
+  a\in \emptyset        & \bimp &  \bot\\
+  a \in A \un B      & \bimp &  a\in A \disj a\in B\\
+  a \in A \int B      & \bimp &  a\in A \conj a\in B\\
+  a \in A-B             & \bimp &  a\in A \conj \neg (a\in B)\\
+  \pair{a,b}\in {\tt Sigma}(A,B)
+                        & \bimp &  a\in A \conj b\in B(a)\\
+  a \in {\tt Collect}(A,P)      & \bimp &  a\in A \conj P(a)\\
+  (\forall x \in \emptyset. P(x)) & \bimp &  \top\\
+  (\forall x \in A. \top)       & \bimp &  \top
+\end{eqnarray*}
+\caption{Some rewrite rules for set theory} \label{zf-simpdata}
+\end{figure}
+
+
+\section{Datatype definitions}
+\label{sec:ZF:datatype}
+\index{*datatype|(}
+
+The \ttindex{datatype} definition package of \ZF\ constructs inductive
+datatypes similar to those of \ML.  It can also construct coinductive
+datatypes (codatatypes), which are non-well-founded structures such as
+streams.  It defines the set using a fixed-point construction and proves
+induction rules, as well as theorems for recursion and case combinators.  It
+supplies mechanisms for reasoning about freeness.  The datatype package can
+handle both mutual and indirect recursion.
+
+
+\subsection{Basics}
+\label{subsec:datatype:basics}
+
+A \texttt{datatype} definition has the following form:
+\[
+\begin{array}{llcl}
+\mathtt{datatype} & t@1(A@1,\ldots,A@h) & = &
+  constructor^1@1 ~\mid~ \ldots ~\mid~ constructor^1@{k@1} \\
+ & & \vdots \\
+\mathtt{and} & t@n(A@1,\ldots,A@h) & = &
+  constructor^n@1~ ~\mid~ \ldots ~\mid~ constructor^n@{k@n}
+\end{array}
+\]
+Here $t@1$, \ldots,~$t@n$ are identifiers and $A@1$, \ldots,~$A@h$ are
+variables: the datatype's parameters.  Each constructor specification has the
+form \dquotesoff
+\[ C \hbox{\tt~( } \hbox{\tt"} x@1 \hbox{\tt:} T@1 \hbox{\tt"},\;
+                   \ldots,\;
+                   \hbox{\tt"} x@m \hbox{\tt:} T@m \hbox{\tt"}
+     \hbox{\tt~)}
+\]
+Here $C$ is the constructor name, and variables $x@1$, \ldots,~$x@m$ are the
+constructor arguments, belonging to the sets $T@1$, \ldots, $T@m$,
+respectively.  Typically each $T@j$ is either a constant set, a datatype
+parameter (one of $A@1$, \ldots, $A@h$) or a recursive occurrence of one of
+the datatypes, say $t@i(A@1,\ldots,A@h)$.  More complex possibilities exist,
+but they are much harder to realize.  Often, additional information must be
+supplied in the form of theorems.
+
+A datatype can occur recursively as the argument of some function~$F$.  This
+is called a {\em nested} (or \emph{indirect}) occurrence.  It is only allowed
+if the datatype package is given a theorem asserting that $F$ is monotonic.
+If the datatype has indirect occurrences, then Isabelle/ZF does not support
+recursive function definitions.
+
+A simple example of a datatype is \texttt{list}, which is built-in, and is
+defined by
+\begin{ttbox}
+consts     list :: i=>i
+datatype  "list(A)" = Nil | Cons ("a:A", "l: list(A)")
+\end{ttbox}
+Note that the datatype operator must be declared as a constant first.
+However, the package declares the constructors.  Here, \texttt{Nil} gets type
+$i$ and \texttt{Cons} gets type $[i,i]\To i$.
+
+Trees and forests can be modelled by the mutually recursive datatype
+definition
+\begin{ttbox}
+consts     tree, forest, tree_forest :: i=>i
+datatype  "tree(A)"   = Tcons ("a: A",  "f: forest(A)")
+and       "forest(A)" = Fnil  |  Fcons ("t: tree(A)",  "f: forest(A)")
+\end{ttbox}
+Here $\texttt{tree}(A)$ is the set of trees over $A$, $\texttt{forest}(A)$ is
+the set of forests over $A$, and  $\texttt{tree_forest}(A)$ is the union of
+the previous two sets.  All three operators must be declared first.
+
+The datatype \texttt{term}, which is defined by
+\begin{ttbox}
+consts     term :: i=>i
+datatype  "term(A)" = Apply ("a: A", "l: list(term(A))")
+  monos "[list_mono]"
+\end{ttbox}
+is an example of nested recursion.  (The theorem \texttt{list_mono} is proved
+in file \texttt{List.ML}, and the \texttt{term} example is devaloped in theory
+\thydx{ex/Term}.)
+
+\subsubsection{Freeness of the constructors}
+
+Constructors satisfy {\em freeness} properties.  Constructions are distinct,
+for example $\texttt{Nil}\not=\texttt{Cons}(a,l)$, and they are injective, for
+example $\texttt{Cons}(a,l)=\texttt{Cons}(a',l') \bimp a=a' \conj l=l'$.
+Because the number of freeness is quadratic in the number of constructors, the
+datatype package does not prove them, but instead provides several means of
+proving them dynamically.  For the \texttt{list} datatype, freeness reasoning
+can be done in two ways: by simplifying with the theorems
+\texttt{list.free_iffs} or by invoking the classical reasoner with
+\texttt{list.free_SEs} as safe elimination rules.  Occasionally this exposes
+the underlying representation of some constructor, which can be rectified
+using the command \hbox{\tt fold_tac list.con_defs}.
+
+\subsubsection{Structural induction}
+
+The datatype package also provides structural induction rules.  For datatypes
+without mutual or nested recursion, the rule has the form exemplified by
+\texttt{list.induct} in Fig.\ts\ref{zf-list}.  For mutually recursive
+datatypes, the induction rule is supplied in two forms.  Consider datatype
+\texttt{TF}.  The rule \texttt{tree_forest.induct} performs induction over a
+single predicate~\texttt{P}, which is presumed to be defined for both trees
+and forests:
+\begin{ttbox}
+[| x : tree_forest(A);
+   !!a f. [| a : A; f : forest(A); P(f) |] ==> P(Tcons(a, f)); P(Fnil);
+   !!f t. [| t : tree(A); P(t); f : forest(A); P(f) |]
+          ==> P(Fcons(t, f)) 
+|] ==> P(x)
+\end{ttbox}
+The rule \texttt{tree_forest.mutual_induct} performs induction over two
+distinct predicates, \texttt{P_tree} and \texttt{P_forest}.
+\begin{ttbox}
+[| !!a f.
+      [| a : A; f : forest(A); P_forest(f) |] ==> P_tree(Tcons(a, f));
+   P_forest(Fnil);
+   !!f t. [| t : tree(A); P_tree(t); f : forest(A); P_forest(f) |]
+          ==> P_forest(Fcons(t, f)) 
+|] ==> (ALL za. za : tree(A) --> P_tree(za)) &
+    (ALL za. za : forest(A) --> P_forest(za))
+\end{ttbox}
+
+For datatypes with nested recursion, such as the \texttt{term} example from
+above, things are a bit more complicated.  The rule \texttt{term.induct}
+refers to the monotonic operator, \texttt{list}:
+\begin{ttbox}
+[| x : term(A);
+   !!a l. [| a : A; l : list(Collect(term(A), P)) |] ==> P(Apply(a, l)) 
+|] ==> P(x)
+\end{ttbox}
+The file \texttt{ex/Term.ML} derives two higher-level induction rules, one of
+which is particularly useful for proving equations:
+\begin{ttbox}
+[| t : term(A);
+   !!x zs. [| x : A; zs : list(term(A)); map(f, zs) = map(g, zs) |]
+           ==> f(Apply(x, zs)) = g(Apply(x, zs)) 
+|] ==> f(t) = g(t)  
+\end{ttbox}
+How this can be generalized to other nested datatypes is a matter for future
+research.
+
+
+\subsubsection{The \texttt{case} operator}
+
+The package defines an operator for performing case analysis over the
+datatype.  For \texttt{list}, it is called \texttt{list_case} and satisfies
+the equations
+\begin{ttbox}
+list_case(f_Nil, f_Cons, []) = f_Nil
+list_case(f_Nil, f_Cons, Cons(a, l)) = f_Cons(a, l)
+\end{ttbox}
+Here \texttt{f_Nil} is the value to return if the argument is \texttt{Nil} and
+\texttt{f_Cons} is a function that computes the value to return if the
+argument has the form $\texttt{Cons}(a,l)$.  The function can be expressed as
+an abstraction, over patterns if desired (\S\ref{sec:pairs}).
+
+For mutually recursive datatypes, there is a single \texttt{case} operator.
+In the tree/forest example, the constant \texttt{tree_forest_case} handles all
+of the constructors of the two datatypes.
+
+
+
+
+\subsection{Defining datatypes}
+
+The theory syntax for datatype definitions is shown in
+Fig.~\ref{datatype-grammar}.  In order to be well-formed, a datatype
+definition has to obey the rules stated in the previous section.  As a result
+the theory is extended with the new types, the constructors, and the theorems
+listed in the previous section.  The quotation marks are necessary because
+they enclose general Isabelle formul\ae.
+
+\begin{figure}
+\begin{rail}
+datatype : ( 'datatype' | 'codatatype' ) datadecls;
+
+datadecls: ( '"' id arglist '"' '=' (constructor + '|') ) + 'and'
+         ;
+constructor : name ( () | consargs )  ( () | ( '(' mixfix ')' ) )
+         ;
+consargs : '(' ('"' var ':' term '"' + ',') ')'
+         ;
+\end{rail}
+\caption{Syntax of datatype declarations}
+\label{datatype-grammar}
+\end{figure}
+
+Codatatypes are declared like datatypes and are identical to them in every
+respect except that they have a coinduction rule instead of an induction rule.
+Note that while an induction rule has the effect of limiting the values
+contained in the set, a coinduction rule gives a way of constructing new
+values of the set.
+
+Most of the theorems about datatypes become part of the default simpset.  You
+never need to see them again because the simplifier applies them
+automatically.  Add freeness properties (\texttt{free_iffs}) to the simpset
+when you want them.  Induction or exhaustion are usually invoked by hand,
+usually via these special-purpose tactics:
+\begin{ttdescription}
+\item[\ttindexbold{induct_tac} {\tt"}$x${\tt"} $i$] applies structural
+  induction on variable $x$ to subgoal $i$, provided the type of $x$ is a
+  datatype.  The induction variable should not occur among other assumptions
+  of the subgoal.
+\end{ttdescription}
+In some cases, induction is overkill and a case distinction over all
+constructors of the datatype suffices.
+\begin{ttdescription}
+\item[\ttindexbold{exhaust_tac} {\tt"}$x${\tt"} $i$]
+ performs an exhaustive case analysis for the variable~$x$.
+\end{ttdescription}
+
+Both tactics can only be applied to a variable, whose typing must be given in
+some assumption, for example the assumption \texttt{x:\ list(A)}.  The tactics
+also work for the natural numbers (\texttt{nat}) and disjoint sums, although
+these sets were not defined using the datatype package.  (Disjoint sums are
+not recursive, so only \texttt{exhaust_tac} is available.)
+
+\bigskip
+Here are some more details for the technically minded.  Processing the
+theory file produces an \ML\ structure which, in addition to the usual
+components, contains a structure named $t$ for each datatype $t$ defined in
+the file.  Each structure $t$ contains the following elements:
+\begin{ttbox}
+val intrs         : thm list  \textrm{the introduction rules}
+val elim          : thm       \textrm{the elimination (case analysis) rule}
+val induct        : thm       \textrm{the standard induction rule}
+val mutual_induct : thm       \textrm{the mutual induction rule, or \texttt{True}}
+val case_eqns     : thm list  \textrm{equations for the case operator}
+val recursor_eqns : thm list  \textrm{equations for the recursor}
+val con_defs      : thm list  \textrm{definitions of the case operator and constructors}
+val free_iffs     : thm list  \textrm{logical equivalences for proving freeness}
+val free_SEs      : thm list  \textrm{elimination rules for proving freeness}
+val mk_free       : string -> thm  \textrm{A function for proving freeness theorems}
+val mk_cases      : thm list -> string -> thm  \textrm{case analysis, see below}
+val defs          : thm list  \textrm{definitions of operators}
+val bnd_mono      : thm list  \textrm{monotonicity property}
+val dom_subset    : thm list  \textrm{inclusion in `bounding set'}
+\end{ttbox}
+Furthermore there is the theorem $C$\texttt{_I} for every constructor~$C$; for
+example, the \texttt{list} datatype's introduction rules are bound to the
+identifiers \texttt{Nil_I} and \texttt{Cons_I}.
+
+For a codatatype, the component \texttt{coinduct} is the coinduction rule,
+replacing the \texttt{induct} component.
+
+See the theories \texttt{ex/Ntree} and \texttt{ex/Brouwer} for examples of
+infinitely branching datatypes.  See theory \texttt{ex/LList} for an example
+of a codatatype.  Some of these theories illustrate the use of additional,
+undocumented features of the datatype package.  Datatype definitions are
+reduced to inductive definitions, and the advanced features should be
+understood in that light.
+
+
+\subsection{Examples}
+
+\subsubsection{The datatype of binary trees}
+
+Let us define the set $\texttt{bt}(A)$ of binary trees over~$A$.  The theory
+must contain these lines:
+\begin{ttbox}
+consts   bt :: i=>i
+datatype "bt(A)"  =  Lf  |  Br ("a: A",  "t1: bt(A)",  "t2: bt(A)")
+\end{ttbox}
+After loading the theory, we can prove, for example, that no tree equals its
+left branch.  To ease the induction, we state the goal using quantifiers.
+\begin{ttbox}
+Goal "l : bt(A) ==> ALL x r. Br(x,l,r) ~= l";
+{\out Level 0}
+{\out l : bt(A) ==> ALL x r. Br(x, l, r) ~= l}
+{\out  1. l : bt(A) ==> ALL x r. Br(x, l, r) ~= l}
+\end{ttbox}
+This can be proved by the structural induction tactic:
+\begin{ttbox}
+by (induct_tac "l" 1);
+{\out Level 1}
+{\out l : bt(A) ==> ALL x r. Br(x, l, r) ~= l}
+{\out  1. ALL x r. Br(x, Lf, r) ~= Lf}
+{\out  2. !!a t1 t2.}
+{\out        [| a : A; t1 : bt(A); ALL x r. Br(x, t1, r) ~= t1; t2 : bt(A);}
+{\out           ALL x r. Br(x, t2, r) ~= t2 |]}
+{\out        ==> ALL x r. Br(x, Br(a, t1, t2), r) ~= Br(a, t1, t2)}
+\end{ttbox}
+Both subgoals are proved using the simplifier.  Tactic
+\texttt{asm_full_simp_tac} is used, rewriting the assumptions.
+This is because simplification using the freeness properties can unfold the
+definition of constructor~\texttt{Br}, so we arrange that all occurrences are
+unfolded. 
+\begin{ttbox}
+by (ALLGOALS (asm_full_simp_tac (simpset() addsimps bt.free_iffs)));
+{\out Level 2}
+{\out l : bt(A) ==> ALL x r. Br(x, l, r) ~= l}
+{\out No subgoals!}
+\end{ttbox}
+To remove the quantifiers from the induction formula, we save the theorem using
+\ttindex{qed_spec_mp}.
+\begin{ttbox}
+qed_spec_mp "Br_neq_left";
+{\out val Br_neq_left = "?l : bt(?A) ==> Br(?x, ?l, ?r) ~= ?l" : thm}
+\end{ttbox}
+
+When there are only a few constructors, we might prefer to prove the freenness
+theorems for each constructor.  This is trivial, using the function given us
+for that purpose:
+\begin{ttbox}
+val Br_iff = bt.mk_free "Br(a,l,r)=Br(a',l',r') <-> a=a' & l=l' & r=r'";
+{\out val Br_iff =}
+{\out   "Br(?a, ?l, ?r) = Br(?a', ?l', ?r') <->}
+{\out                     ?a = ?a' & ?l = ?l' & ?r = ?r'" : thm}
+\end{ttbox}
+
+The purpose of \ttindex{mk_cases} is to generate simplified instances of the
+elimination (case analysis) rule.  Its theorem list argument is a list of
+constructor definitions, which it uses for freeness reasoning.  For example,
+this instance of the elimination rule propagates type-checking information
+from the premise $\texttt{Br}(a,l,r)\in\texttt{bt}(A)$:
+\begin{ttbox}
+val BrE = bt.mk_cases bt.con_defs "Br(a,l,r) : bt(A)";
+{\out val BrE =}
+{\out   "[| Br(?a, ?l, ?r) : bt(?A);}
+{\out       [| ?a : ?A; ?l : bt(?A); ?r : bt(?A) |] ==> ?Q |] ==> ?Q" : thm}
+\end{ttbox}
+
+
+\subsubsection{Mixfix syntax in datatypes}
+
+Mixfix syntax is sometimes convenient.  The theory \texttt{ex/PropLog} makes a
+deep embedding of propositional logic:
+\begin{ttbox}
+consts     prop :: i
+datatype  "prop" = Fls
+                 | Var ("n: nat")                ("#_" [100] 100)
+                 | "=>" ("p: prop", "q: prop")   (infixr 90)
+\end{ttbox}
+The second constructor has a special $\#n$ syntax, while the third constructor
+is an infixed arrow.
+
+
+\subsubsection{A giant enumeration type}
+
+This example shows a datatype that consists of 60 constructors:
+\begin{ttbox}
+consts  enum :: i
+datatype
+  "enum" = C00 | C01 | C02 | C03 | C04 | C05 | C06 | C07 | C08 | C09
+         | C10 | C11 | C12 | C13 | C14 | C15 | C16 | C17 | C18 | C19
+         | C20 | C21 | C22 | C23 | C24 | C25 | C26 | C27 | C28 | C29
+         | C30 | C31 | C32 | C33 | C34 | C35 | C36 | C37 | C38 | C39
+         | C40 | C41 | C42 | C43 | C44 | C45 | C46 | C47 | C48 | C49
+         | C50 | C51 | C52 | C53 | C54 | C55 | C56 | C57 | C58 | C59
+end
+\end{ttbox}
+The datatype package scales well.  Even though all properties are proved
+rather than assumed, full processing of this definition takes under 15 seconds
+(on a 300 MHz Pentium).  The constructors have a balanced representation,
+essentially binary notation, so freeness properties can be proved fast.
+\begin{ttbox}
+Goal "C00 ~= C01";
+by (simp_tac (simpset() addsimps enum.free_iffs) 1);
+\end{ttbox}
+You need not derive such inequalities explicitly.  The simplifier will dispose
+of them automatically, given the theorem list \texttt{free_iffs}.
+
+\index{*datatype|)}
+
+
+\subsection{Recursive function definitions}\label{sec:ZF:recursive}
+\index{recursive functions|see{recursion}}
+\index{*primrec|(}
+
+Datatypes come with a uniform way of defining functions, {\bf primitive
+  recursion}.  Such definitions rely on the recursion operator defined by the
+datatype package.  Isabelle proves the desired recursion equations as
+theorems.
+
+In principle, one could introduce primitive recursive functions by asserting
+their reduction rules as new axioms.  Here is a dangerous way of defining the
+append function for lists:
+\begin{ttbox}\slshape
+consts  "\at" :: [i,i]=>i                        (infixr 60)
+rules 
+   app_Nil   "[] \at ys = ys"
+   app_Cons  "(Cons(a,l)) \at ys = Cons(a, l \at ys)"
+\end{ttbox}
+Asserting axioms brings the danger of accidentally asserting nonsense.  It
+should be avoided at all costs!
+
+The \ttindex{primrec} declaration is a safe means of defining primitive
+recursive functions on datatypes:
+\begin{ttbox}
+consts  "\at" :: [i,i]=>i                        (infixr 60)
+primrec 
+   "[] \at ys = ys"
+   "(Cons(a,l)) \at ys = Cons(a, l \at ys)"
+\end{ttbox}
+Isabelle will now check that the two rules do indeed form a primitive
+recursive definition.  For example, the declaration
+\begin{ttbox}
+primrec
+   "[] \at ys = us"
+\end{ttbox}
+is rejected with an error message ``\texttt{Extra variables on rhs}''.
+
+
+\subsubsection{Syntax of recursive definitions}
+
+The general form of a primitive recursive definition is
+\begin{ttbox}
+primrec
+    {\it reduction rules}
+\end{ttbox}
+where \textit{reduction rules} specify one or more equations of the form
+\[ f \, x@1 \, \dots \, x@m \, (C \, y@1 \, \dots \, y@k) \, z@1 \,
+\dots \, z@n = r \] such that $C$ is a constructor of the datatype, $r$
+contains only the free variables on the left-hand side, and all recursive
+calls in $r$ are of the form $f \, \dots \, y@i \, \dots$ for some $i$.  
+There must be at most one reduction rule for each constructor.  The order is
+immaterial.  For missing constructors, the function is defined to return zero.
+
+All reduction rules are added to the default simpset.
+If you would like to refer to some rule by name, then you must prefix
+the rule with an identifier.  These identifiers, like those in the
+\texttt{rules} section of a theory, will be visible at the \ML\ level.
+
+The reduction rules for {\tt\at} become part of the default simpset, which
+leads to short proof scripts:
+\begin{ttbox}\underscoreon
+Goal "xs: list(A) ==> (xs @ ys) @ zs = xs @ (ys @ zs)";
+by (induct\_tac "xs" 1);
+by (ALLGOALS Asm\_simp\_tac);
+\end{ttbox}
+
+You can even use the \texttt{primrec} form with non-recursive datatypes and
+with codatatypes.  Recursion is not allowed, but it provides a convenient
+syntax for defining functions by cases.
+
+
+\subsubsection{Example: varying arguments}
+
+All arguments, other than the recursive one, must be the same in each equation
+and in each recursive call.  To get around this restriction, use explict
+$\lambda$-abstraction and function application.  Here is an example, drawn
+from the theory \texttt{Resid/Substitution}.  The type of redexes is declared
+as follows:
+\begin{ttbox}
+consts  redexes :: i
+datatype
+  "redexes" = Var ("n: nat")            
+            | Fun ("t: redexes")
+            | App ("b:bool" ,"f:redexes" , "a:redexes")
+\end{ttbox}
+
+The function \texttt{lift} takes a second argument, $k$, which varies in
+recursive calls.
+\begin{ttbox}
+primrec
+  "lift(Var(i)) = (lam k:nat. if i<k then Var(i) else Var(succ(i)))"
+  "lift(Fun(t)) = (lam k:nat. Fun(lift(t) ` succ(k)))"
+  "lift(App(b,f,a)) = (lam k:nat. App(b, lift(f)`k, lift(a)`k))"
+\end{ttbox}
+Now \texttt{lift(r)`k} satisfies the required recursion equations.
+
+\index{recursion!primitive|)}
+\index{*primrec|)}
+
+
+\section{Inductive and coinductive definitions}
+\index{*inductive|(}
+\index{*coinductive|(}
+
+An {\bf inductive definition} specifies the least set~$R$ closed under given
+rules.  (Applying a rule to elements of~$R$ yields a result within~$R$.)  For
+example, a structural operational semantics is an inductive definition of an
+evaluation relation.  Dually, a {\bf coinductive definition} specifies the
+greatest set~$R$ consistent with given rules.  (Every element of~$R$ can be
+seen as arising by applying a rule to elements of~$R$.)  An important example
+is using bisimulation relations to formalise equivalence of processes and
+infinite data structures.
+
+A theory file may contain any number of inductive and coinductive
+definitions.  They may be intermixed with other declarations; in
+particular, the (co)inductive sets {\bf must} be declared separately as
+constants, and may have mixfix syntax or be subject to syntax translations.
+
+Each (co)inductive definition adds definitions to the theory and also
+proves some theorems.  Each definition creates an \ML\ structure, which is a
+substructure of the main theory structure.
+This package is described in detail in a separate paper,%
+\footnote{It appeared in CADE~\cite{paulson-CADE}; a longer version is
+  distributed with Isabelle as \emph{A Fixedpoint Approach to 
+ (Co)Inductive and (Co)Datatype Definitions}.}  %
+which you might refer to for background information.
+
+
+\subsection{The syntax of a (co)inductive definition}
+An inductive definition has the form
+\begin{ttbox}
+inductive
+  domains    {\it domain declarations}
+  intrs      {\it introduction rules}
+  monos      {\it monotonicity theorems}
+  con_defs   {\it constructor definitions}
+  type_intrs {\it introduction rules for type-checking}
+  type_elims {\it elimination rules for type-checking}
+\end{ttbox}
+A coinductive definition is identical, but starts with the keyword
+{\tt coinductive}.  
+
+The {\tt monos}, {\tt con\_defs}, {\tt type\_intrs} and {\tt type\_elims}
+sections are optional.  If present, each is specified either as a list of
+identifiers or as a string.  If the latter, then the string must be a valid
+\textsc{ml} expression of type {\tt thm list}.  The string is simply inserted
+into the {\tt _thy.ML} file; if it is ill-formed, it will trigger \textsc{ml}
+error messages.  You can then inspect the file on the temporary directory.
+
+\begin{description}
+\item[\it domain declarations] consist of one or more items of the form
+  {\it string\/}~{\tt <=}~{\it string}, associating each recursive set with
+  its domain.  (The domain is some existing set that is large enough to
+  hold the new set being defined.)
+
+\item[\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[\it monotonicity theorems] are required for each operator applied to
+  a recursive set in the introduction rules.  There \textbf{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[\it constructor definitions] contain definitions of constants
+  appearing in the introduction rules.  The (co)datatype package supplies
+  the constructors' definitions here.  Most (co)inductive definitions omit
+  this section; one exception is the primitive recursive functions example;
+  see theory \texttt{ex/Primrec}.
+  
+\item[\it type\_intrs] consists of introduction rules for type-checking the
+  definition: for demonstrating that the new set is included in its domain.
+  (The proof uses depth-first search.)
+
+\item[\it type\_elims] consists of elimination rules for type-checking the
+  definition.  They are presumed to be safe and are applied as often as
+  possible prior to the {\tt type\_intrs} search.
+\end{description}
+
+The package has a few 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~\texttt{Fin}.  Then we declare it
+inductively, with two introduction rules:
+\begin{ttbox}
+consts  Fin :: i=>i
+
+inductive
+  domains   "Fin(A)" <= "Pow(A)"
+  intrs
+    emptyI  "0 : Fin(A)"
+    consI   "[| a: A;  b: Fin(A) |] ==> cons(a,b) : Fin(A)"
+  type_intrs empty_subsetI, cons_subsetI, PowI
+  type_elims "[make_elim PowD]"
+\end{ttbox}
+The resulting theory structure contains a substructure, called~\texttt{Fin}.
+It contains the \texttt{Fin}$~A$ introduction rules as the list
+\texttt{Fin.intrs}, and also individually as \texttt{Fin.emptyI} and
+\texttt{Fin.consI}.  The induction rule is \texttt{Fin.induct}.
+
+The chief problem with making (co)inductive definitions involves type-checking
+the rules.  Sometimes, additional theorems need to be supplied under
+\texttt{type_intrs} or \texttt{type_elims}.  If the package fails when trying
+to prove your introduction rules, then set the flag \ttindexbold{trace_induct}
+to \texttt{true} and try again.  (See the manual \emph{A Fixedpoint Approach
+  \ldots} for more discussion of type-checking.)
+
+In the example above, $\texttt{Pow}(A)$ is given as the domain of
+$\texttt{Fin}(A)$, for obviously every finite subset of~$A$ is a subset
+of~$A$.  However, the inductive definition package can only prove that given a
+few hints.
+Here is the output that results (with the flag set) when the
+\texttt{type_intrs} and \texttt{type_elims} are omitted from the inductive
+definition above:
+\begin{ttbox}
+Inductive definition Finite.Fin
+Fin(A) ==
+lfp(Pow(A),
+    \%X. {z: Pow(A) . z = 0 | (EX a b. z = cons(a, b) & a : A & b : X)})
+  Proving monotonicity...
+\ttbreak
+  Proving the introduction rules...
+The typechecking subgoal:
+0 : Fin(A)
+ 1. 0 : Pow(A)
+\ttbreak
+The subgoal after monos, type_elims:
+0 : Fin(A)
+ 1. 0 : Pow(A)
+*** prove_goal: tactic failed
+\end{ttbox}
+We see the need to supply theorems to let the package prove
+$\emptyset\in\texttt{Pow}(A)$.  Restoring the \texttt{type_intrs} but not the
+\texttt{type_elims}, we again get an error message:
+\begin{ttbox}
+The typechecking subgoal:
+0 : Fin(A)
+ 1. 0 : Pow(A)
+\ttbreak
+The subgoal after monos, type_elims:
+0 : Fin(A)
+ 1. 0 : Pow(A)
+\ttbreak
+The typechecking subgoal:
+cons(a, b) : Fin(A)
+ 1. [| a : A; b : Fin(A) |] ==> cons(a, b) : Pow(A)
+\ttbreak
+The subgoal after monos, type_elims:
+cons(a, b) : Fin(A)
+ 1. [| a : A; b : Pow(A) |] ==> cons(a, b) : Pow(A)
+*** prove_goal: tactic failed
+\end{ttbox}
+The first rule has been type-checked, but the second one has failed.  The
+simplest solution to such problems is to prove the failed subgoal separately
+and to supply it under \texttt{type_intrs}.  The solution actually used is
+to supply, under \texttt{type_elims}, a rule that changes
+$b\in\texttt{Pow}(A)$ to $b\subseteq A$; together with \texttt{cons_subsetI}
+and \texttt{PowI}, it is enough to complete the type-checking.
+
+
+
+\subsection{Further examples}
+
+An inductive definition may involve arbitrary monotonic operators.  Here is a
+standard example: the accessible part of a relation.  Note the use
+of~\texttt{Pow} in the introduction rule and the corresponding mention of the
+rule \verb|Pow_mono| in the \texttt{monos} list.  If the desired rule has a
+universally quantified premise, usually the effect can be obtained using
+\texttt{Pow}.
+\begin{ttbox}
+consts  acc :: i=>i
+inductive
+  domains "acc(r)" <= "field(r)"
+  intrs
+    vimage  "[| r-``{a}: Pow(acc(r)); a: field(r) |] ==> a: acc(r)"
+  monos      Pow_mono
+\end{ttbox}
+
+Finally, here is a coinductive definition.  It captures (as a bisimulation)
+the notion of equality on lazy lists, which are first defined as a codatatype:
+\begin{ttbox}
+consts  llist :: i=>i
+codatatype  "llist(A)" = LNil | LCons ("a: A", "l: llist(A)")
+\ttbreak
+
+consts  lleq :: i=>i
+coinductive
+  domains "lleq(A)" <= "llist(A) * llist(A)"
+  intrs
+    LNil  "<LNil, LNil> : lleq(A)"
+    LCons "[| a:A; <l,l'>: lleq(A) |] 
+           ==> <LCons(a,l), LCons(a,l')>: lleq(A)"
+  type_intrs  "llist.intrs"
+\end{ttbox}
+This use of \texttt{type_intrs} is typical: the relation concerns the
+codatatype \texttt{llist}, so naturally the introduction rules for that
+codatatype will be required for type-checking the rules.
+
+The Isabelle distribution contains many other inductive definitions.  Simple
+examples are collected on subdirectory \texttt{ZF/ex}.  The directory
+\texttt{Coind} and the theory \texttt{ZF/ex/LList} contain coinductive
+definitions.  Larger examples may be found on other subdirectories of
+\texttt{ZF}, such as \texttt{IMP}, and \texttt{Resid}.
+
+
+\subsection{The result structure}
+
+Each (co)inductive set defined in a theory file generates an \ML\ substructure
+having the same name.  The the substructure contains the following elements:
+
+\begin{ttbox}
+val intrs         : thm list  \textrm{the introduction rules}
+val elim          : thm       \textrm{the elimination (case analysis) rule}
+val mk_cases      : thm list -> string -> thm  \textrm{case analysis, see below}
+val induct        : thm       \textrm{the standard induction rule}
+val mutual_induct : thm       \textrm{the mutual induction rule, or \texttt{True}}
+val defs          : thm list  \textrm{definitions of operators}
+val bnd_mono      : thm list  \textrm{monotonicity property}
+val dom_subset    : thm list  \textrm{inclusion in `bounding set'}
+\end{ttbox}
+Furthermore there is the theorem $C$\texttt{_I} for every constructor~$C$; for
+example, the \texttt{list} datatype's introduction rules are bound to the
+identifiers \texttt{Nil_I} and \texttt{Cons_I}.
+
+For a codatatype, the component \texttt{coinduct} is the coinduction rule,
+replacing the \texttt{induct} component.
+
+Recall that \ttindex{mk_cases} generates simplified instances of the
+elimination (case analysis) rule.  It is as useful for inductive definitions
+as it is for datatypes.  There are many examples in the theory
+\texttt{ex/Comb}, which is discussed at length
+elsewhere~\cite{paulson-generic}.  The theory first defines the datatype
+\texttt{comb} of combinators:
+\begin{ttbox}
+consts comb :: i
+datatype  "comb" = K
+                 | S
+                 | "#" ("p: comb", "q: comb")   (infixl 90)
+\end{ttbox}
+The theory goes on to define contraction and parallel contraction
+inductively.  Then the file \texttt{ex/Comb.ML} defines special cases of
+contraction using \texttt{mk_cases}:
+\begin{ttbox}
+val K_contractE = contract.mk_cases comb.con_defs "K -1-> r";
+{\out val K_contractE = "K -1-> ?r ==> ?Q" : thm}
+\end{ttbox}
+We can read this as saying that the combinator \texttt{K} cannot reduce to
+anything.  Similar elimination rules for \texttt{S} and application are also
+generated and are supplied to the classical reasoner.  Note that
+\texttt{comb.con_defs} is given to \texttt{mk_cases} to allow freeness
+reasoning on datatype \texttt{comb}.
+
+\index{*coinductive|)} \index{*inductive|)}
+
+
+
+
+\section{The outer reaches of set theory}
+
+The constructions of the natural numbers and lists use a suite of
+operators for handling recursive function definitions.  I have described
+the developments in detail elsewhere~\cite{paulson-set-II}.  Here is a brief
+summary:
+\begin{itemize}
+  \item Theory \texttt{Trancl} defines the transitive closure of a relation
+    (as a least fixedpoint).
+
+  \item Theory \texttt{WF} proves the Well-Founded Recursion Theorem, using an
+    elegant approach of Tobias Nipkow.  This theorem permits general
+    recursive definitions within set theory.
+
+  \item Theory \texttt{Ord} defines the notions of transitive set and ordinal
+    number.  It derives transfinite induction.  A key definition is {\bf
+      less than}: $i<j$ if and only if $i$ and $j$ are both ordinals and
+    $i\in j$.  As a special case, it includes less than on the natural
+    numbers.
+    
+  \item Theory \texttt{Epsilon} derives $\varepsilon$-induction and
+    $\varepsilon$-recursion, which are generalisations of transfinite
+    induction and recursion.  It also defines \cdx{rank}$(x)$, which
+    is the least ordinal $\alpha$ such that $x$ is constructed at
+    stage $\alpha$ of the cumulative hierarchy (thus $x\in
+    V@{\alpha+1}$).
+\end{itemize}
+
+Other important theories lead to a theory of cardinal numbers.  They have
+not yet been written up anywhere.  Here is a summary:
+\begin{itemize}
+\item Theory \texttt{Rel} defines the basic properties of relations, such as
+  (ir)reflexivity, (a)symmetry, and transitivity.
+
+\item Theory \texttt{EquivClass} develops a theory of equivalence
+  classes, not using the Axiom of Choice.
+
+\item Theory \texttt{Order} defines partial orderings, total orderings and
+  wellorderings.
+
+\item Theory \texttt{OrderArith} defines orderings on sum and product sets.
+  These can be used to define ordinal arithmetic and have applications to
+  cardinal arithmetic.
+
+\item Theory \texttt{OrderType} defines order types.  Every wellordering is
+  equivalent to a unique ordinal, which is its order type.
+
+\item Theory \texttt{Cardinal} defines equipollence and cardinal numbers.
+ 
+\item Theory \texttt{CardinalArith} defines cardinal addition and
+  multiplication, and proves their elementary laws.  It proves that there
+  is no greatest cardinal.  It also proves a deep result, namely
+  $\kappa\otimes\kappa=\kappa$ for every infinite cardinal~$\kappa$; see
+  Kunen~\cite[page 29]{kunen80}.  None of these results assume the Axiom of
+  Choice, which complicates their proofs considerably.  
+\end{itemize}
+
+The following developments involve the Axiom of Choice (AC):
+\begin{itemize}
+\item Theory \texttt{AC} asserts the Axiom of Choice and proves some simple
+  equivalent forms.
+
+\item Theory \texttt{Zorn} proves Hausdorff's Maximal Principle, Zorn's Lemma
+  and the Wellordering Theorem, following Abrial and
+  Laffitte~\cite{abrial93}.
+
+\item Theory \verb|Cardinal_AC| uses AC to prove simplified theorems about
+  the cardinals.  It also proves a theorem needed to justify
+  infinitely branching datatype declarations: if $\kappa$ is an infinite
+  cardinal and $|X(\alpha)| \le \kappa$ for all $\alpha<\kappa$ then
+  $|\union\sb{\alpha<\kappa} X(\alpha)| \le \kappa$.
+
+\item Theory \texttt{InfDatatype} proves theorems to justify infinitely
+  branching datatypes.  Arbitrary index sets are allowed, provided their
+  cardinalities have an upper bound.  The theory also justifies some
+  unusual cases of finite branching, involving the finite powerset operator
+  and the finite function space operator.
+\end{itemize}
+
+
+
+\section{The examples directories}
+Directory \texttt{HOL/IMP} contains a mechanised version of a semantic
+equivalence proof taken from Winskel~\cite{winskel93}.  It formalises the
+denotational and operational semantics of a simple while-language, then
+proves the two equivalent.  It contains several datatype and inductive
+definitions, and demonstrates their use.
+
+The directory \texttt{ZF/ex} contains further developments in {\ZF} set
+theory.  Here is an overview; see the files themselves for more details.  I
+describe much of this material in other
+publications~\cite{paulson-set-I,paulson-set-II,paulson-CADE}. 
+\begin{itemize}
+\item File \texttt{misc.ML} contains miscellaneous examples such as
+  Cantor's Theorem, the Schr\"oder-Bernstein Theorem and the `Composition
+  of homomorphisms' challenge~\cite{boyer86}.
+
+\item Theory \texttt{Ramsey} proves the finite exponent 2 version of
+  Ramsey's Theorem, following Basin and Kaufmann's
+  presentation~\cite{basin91}.
+
+\item Theory \texttt{Integ} develops a theory of the integers as
+  equivalence classes of pairs of natural numbers.
+
+\item Theory \texttt{Primrec} develops some computation theory.  It
+  inductively defines the set of primitive recursive functions and presents a
+  proof that Ackermann's function is not primitive recursive.
+
+\item Theory \texttt{Primes} defines the Greatest Common Divisor of two
+  natural numbers and and the ``divides'' relation.
+
+\item Theory \texttt{Bin} defines a datatype for two's complement binary
+  integers, then proves rewrite rules to perform binary arithmetic.  For
+  instance, $1359\times {-}2468 = {-}3354012$ takes under 14 seconds.
+
+\item Theory \texttt{BT} defines the recursive data structure ${\tt
+    bt}(A)$, labelled binary trees.
+
+\item Theory \texttt{Term} defines a recursive data structure for terms
+  and term lists.  These are simply finite branching trees.
+
+\item Theory \texttt{TF} defines primitives for solving mutually
+  recursive equations over sets.  It constructs sets of trees and forests
+  as an example, including induction and recursion rules that handle the
+  mutual recursion.
+
+\item Theory \texttt{Prop} proves soundness and completeness of
+  propositional logic~\cite{paulson-set-II}.  This illustrates datatype
+  definitions, inductive definitions, structural induction and rule
+  induction.
+
+\item Theory \texttt{ListN} inductively defines the lists of $n$
+  elements~\cite{paulin92}.
+
+\item Theory \texttt{Acc} inductively defines the accessible part of a
+  relation~\cite{paulin92}.
+
+\item Theory \texttt{Comb} defines the datatype of combinators and
+  inductively defines contraction and parallel contraction.  It goes on to
+  prove the Church-Rosser Theorem.  This case study follows Camilleri and
+  Melham~\cite{camilleri92}.
+
+\item Theory \texttt{LList} defines lazy lists and a coinduction
+  principle for proving equations between them.
+\end{itemize}
+
+
+\section{A proof about powersets}\label{sec:ZF-pow-example}
+To demonstrate high-level reasoning about subsets, let us prove the
+equation ${{\tt Pow}(A)\cap {\tt Pow}(B)}= {\tt Pow}(A\cap B)$.  Compared
+with first-order logic, set theory involves a maze of rules, and theorems
+have many different proofs.  Attempting other proofs of the theorem might
+be instructive.  This proof exploits the lattice properties of
+intersection.  It also uses the monotonicity of the powerset operation,
+from \texttt{ZF/mono.ML}:
+\begin{ttbox}
+\tdx{Pow_mono}      A<=B ==> Pow(A) <= Pow(B)
+\end{ttbox}
+We enter the goal and make the first step, which breaks the equation into
+two inclusions by extensionality:\index{*equalityI theorem}
+\begin{ttbox}
+Goal "Pow(A Int B) = Pow(A) Int Pow(B)";
+{\out Level 0}
+{\out Pow(A Int B) = Pow(A) Int Pow(B)}
+{\out  1. Pow(A Int B) = Pow(A) Int Pow(B)}
+\ttbreak
+by (resolve_tac [equalityI] 1);
+{\out Level 1}
+{\out Pow(A Int B) = Pow(A) Int Pow(B)}
+{\out  1. Pow(A Int B) <= Pow(A) Int Pow(B)}
+{\out  2. Pow(A) Int Pow(B) <= Pow(A Int B)}
+\end{ttbox}
+Both inclusions could be tackled straightforwardly using \texttt{subsetI}.
+A shorter proof results from noting that intersection forms the greatest
+lower bound:\index{*Int_greatest theorem}
+\begin{ttbox}
+by (resolve_tac [Int_greatest] 1);
+{\out Level 2}
+{\out Pow(A Int B) = Pow(A) Int Pow(B)}
+{\out  1. Pow(A Int B) <= Pow(A)}
+{\out  2. Pow(A Int B) <= Pow(B)}
+{\out  3. Pow(A) Int Pow(B) <= Pow(A Int B)}
+\end{ttbox}
+Subgoal~1 follows by applying the monotonicity of \texttt{Pow} to $A\int
+B\subseteq A$; subgoal~2 follows similarly:
+\index{*Int_lower1 theorem}\index{*Int_lower2 theorem}
+\begin{ttbox}
+by (resolve_tac [Int_lower1 RS Pow_mono] 1);
+{\out Level 3}
+{\out Pow(A Int B) = Pow(A) Int Pow(B)}
+{\out  1. Pow(A Int B) <= Pow(B)}
+{\out  2. Pow(A) Int Pow(B) <= Pow(A Int B)}
+\ttbreak
+by (resolve_tac [Int_lower2 RS Pow_mono] 1);
+{\out Level 4}
+{\out Pow(A Int B) = Pow(A) Int Pow(B)}
+{\out  1. Pow(A) Int Pow(B) <= Pow(A Int B)}
+\end{ttbox}
+We are left with the opposite inclusion, which we tackle in the
+straightforward way:\index{*subsetI theorem}
+\begin{ttbox}
+by (resolve_tac [subsetI] 1);
+{\out Level 5}
+{\out Pow(A Int B) = Pow(A) Int Pow(B)}
+{\out  1. !!x. x : Pow(A) Int Pow(B) ==> x : Pow(A Int B)}
+\end{ttbox}
+The subgoal is to show $x\in {\tt Pow}(A\cap B)$ assuming $x\in{\tt
+Pow}(A)\cap {\tt Pow}(B)$; eliminating this assumption produces two
+subgoals.  The rule \tdx{IntE} treats the intersection like a conjunction
+instead of unfolding its definition.
+\begin{ttbox}
+by (eresolve_tac [IntE] 1);
+{\out Level 6}
+{\out Pow(A Int B) = Pow(A) Int Pow(B)}
+{\out  1. !!x. [| x : Pow(A); x : Pow(B) |] ==> x : Pow(A Int B)}
+\end{ttbox}
+The next step replaces the \texttt{Pow} by the subset
+relation~($\subseteq$).\index{*PowI theorem}
+\begin{ttbox}
+by (resolve_tac [PowI] 1);
+{\out Level 7}
+{\out Pow(A Int B) = Pow(A) Int Pow(B)}
+{\out  1. !!x. [| x : Pow(A); x : Pow(B) |] ==> x <= A Int B}
+\end{ttbox}
+We perform the same replacement in the assumptions.  This is a good
+demonstration of the tactic \ttindex{dresolve_tac}:\index{*PowD theorem}
+\begin{ttbox}
+by (REPEAT (dresolve_tac [PowD] 1));
+{\out Level 8}
+{\out Pow(A Int B) = Pow(A) Int Pow(B)}
+{\out  1. !!x. [| x <= A; x <= B |] ==> x <= A Int B}
+\end{ttbox}
+The assumptions are that $x$ is a lower bound of both $A$ and~$B$, but
+$A\int B$ is the greatest lower bound:\index{*Int_greatest theorem}
+\begin{ttbox}
+by (resolve_tac [Int_greatest] 1);
+{\out Level 9}
+{\out Pow(A Int B) = Pow(A) Int Pow(B)}
+{\out  1. !!x. [| x <= A; x <= B |] ==> x <= A}
+{\out  2. !!x. [| x <= A; x <= B |] ==> x <= B}
+\end{ttbox}
+To conclude the proof, we clear up the trivial subgoals:
+\begin{ttbox}
+by (REPEAT (assume_tac 1));
+{\out Level 10}
+{\out Pow(A Int B) = Pow(A) Int Pow(B)}
+{\out No subgoals!}
+\end{ttbox}
+\medskip
+We could have performed this proof in one step by applying
+\ttindex{Blast_tac}.  Let us
+go back to the start:
+\begin{ttbox}
+choplev 0;
+{\out Level 0}
+{\out Pow(A Int B) = Pow(A) Int Pow(B)}
+{\out  1. Pow(A Int B) = Pow(A) Int Pow(B)}
+by (Blast_tac 1);
+{\out Depth = 0}
+{\out Depth = 1}
+{\out Depth = 2}
+{\out Depth = 3}
+{\out Level 1}
+{\out Pow(A Int B) = Pow(A) Int Pow(B)}
+{\out No subgoals!}
+\end{ttbox}
+Past researchers regarded this as a difficult proof, as indeed it is if all
+the symbols are replaced by their definitions.
+\goodbreak
+
+\section{Monotonicity of the union operator}
+For another example, we prove that general union is monotonic:
+${C\subseteq D}$ implies $\bigcup(C)\subseteq \bigcup(D)$.  To begin, we
+tackle the inclusion using \tdx{subsetI}:
+\begin{ttbox}
+Goal "C<=D ==> Union(C) <= Union(D)";
+{\out Level 0}
+{\out C <= D ==> Union(C) <= Union(D)}
+{\out  1. C <= D ==> Union(C) <= Union(D)}
+\ttbreak
+by (resolve_tac [subsetI] 1);
+{\out Level 1}
+{\out C <= D ==> Union(C) <= Union(D)}
+{\out  1. !!x. [| C <= D; x : Union(C) |] ==> x : Union(D)}
+\end{ttbox}
+Big union is like an existential quantifier --- the occurrence in the
+assumptions must be eliminated early, since it creates parameters.
+\index{*UnionE theorem}
+\begin{ttbox}
+by (eresolve_tac [UnionE] 1);
+{\out Level 2}
+{\out C <= D ==> Union(C) <= Union(D)}
+{\out  1. !!x B. [| C <= D; x : B; B : C |] ==> x : Union(D)}
+\end{ttbox}
+Now we may apply \tdx{UnionI}, which creates an unknown involving the
+parameters.  To show $x\in \bigcup(D)$ it suffices to show that $x$ belongs
+to some element, say~$\Var{B2}(x,B)$, of~$D$.
+\begin{ttbox}
+by (resolve_tac [UnionI] 1);
+{\out Level 3}
+{\out C <= D ==> Union(C) <= Union(D)}
+{\out  1. !!x B. [| C <= D; x : B; B : C |] ==> ?B2(x,B) : D}
+{\out  2. !!x B. [| C <= D; x : B; B : C |] ==> x : ?B2(x,B)}
+\end{ttbox}
+Combining \tdx{subsetD} with the assumption $C\subseteq D$ yields 
+$\Var{a}\in C \Imp \Var{a}\in D$, which reduces subgoal~1.  Note that
+\texttt{eresolve_tac} has removed that assumption.
+\begin{ttbox}
+by (eresolve_tac [subsetD] 1);
+{\out Level 4}
+{\out C <= D ==> Union(C) <= Union(D)}
+{\out  1. !!x B. [| x : B; B : C |] ==> ?B2(x,B) : C}
+{\out  2. !!x B. [| C <= D; x : B; B : C |] ==> x : ?B2(x,B)}
+\end{ttbox}
+The rest is routine.  Observe how~$\Var{B2}(x,B)$ is instantiated.
+\begin{ttbox}
+by (assume_tac 1);
+{\out Level 5}
+{\out C <= D ==> Union(C) <= Union(D)}
+{\out  1. !!x B. [| C <= D; x : B; B : C |] ==> x : B}
+by (assume_tac 1);
+{\out Level 6}
+{\out C <= D ==> Union(C) <= Union(D)}
+{\out No subgoals!}
+\end{ttbox}
+Again, \ttindex{Blast_tac} can prove the theorem in one step.
+\begin{ttbox}
+by (Blast_tac 1);
+{\out Depth = 0}
+{\out Depth = 1}
+{\out Depth = 2}
+{\out Level 1}
+{\out C <= D ==> Union(C) <= Union(D)}
+{\out No subgoals!}
+\end{ttbox}
+
+The file \texttt{ZF/equalities.ML} has many similar proofs.  Reasoning about
+general intersection can be difficult because of its anomalous behaviour on
+the empty set.  However, \ttindex{Blast_tac} copes well with these.  Here is
+a typical example, borrowed from Devlin~\cite[page 12]{devlin79}:
+\begin{ttbox}
+a:C ==> (INT x:C. A(x) Int B(x)) = (INT x:C. A(x)) Int (INT x:C. B(x))
+\end{ttbox}
+In traditional notation this is
+\[ a\in C \,\Imp\, \inter@{x\in C} \Bigl(A(x) \int B(x)\Bigr) =        
+       \Bigl(\inter@{x\in C} A(x)\Bigr)  \int  
+       \Bigl(\inter@{x\in C} B(x)\Bigr)  \]
+
+\section{Low-level reasoning about functions}
+The derived rules \texttt{lamI}, \texttt{lamE}, \texttt{lam_type}, \texttt{beta}
+and \texttt{eta} support reasoning about functions in a
+$\lambda$-calculus style.  This is generally easier than regarding
+functions as sets of ordered pairs.  But sometimes we must look at the
+underlying representation, as in the following proof
+of~\tdx{fun_disjoint_apply1}.  This states that if $f$ and~$g$ are
+functions with disjoint domains~$A$ and~$C$, and if $a\in A$, then
+$(f\un g)`a = f`a$:
+\begin{ttbox}
+Goal "[| a:A;  f: A->B;  g: C->D;  A Int C = 0 |] ==>  \ttback
+\ttback    (f Un g)`a = f`a";
+{\out Level 0}
+{\out [| a : A; f : A -> B; g : C -> D; A Int C = 0 |]}
+{\out ==> (f Un g) ` a = f ` a}
+{\out  1. [| a : A; f : A -> B; g : C -> D; A Int C = 0 |]}
+{\out     ==> (f Un g) ` a = f ` a}
+\end{ttbox}
+Using \tdx{apply_equality}, we reduce the equality to reasoning about
+ordered pairs.  The second subgoal is to verify that $f\un g$ is a function.
+To save space, the assumptions will be abbreviated below.
+\begin{ttbox}
+by (resolve_tac [apply_equality] 1);
+{\out Level 1}
+{\out [| \ldots |] ==> (f Un g) ` a = f ` a}
+{\out  1. [| \ldots |] ==> <a,f ` a> : f Un g}
+{\out  2. [| \ldots |] ==> f Un g : (PROD x:?A. ?B(x))}
+\end{ttbox}
+We must show that the pair belongs to~$f$ or~$g$; by~\tdx{UnI1} we
+choose~$f$:
+\begin{ttbox}
+by (resolve_tac [UnI1] 1);
+{\out Level 2}
+{\out [| \ldots |] ==> (f Un g) ` a = f ` a}
+{\out  1. [| \ldots |] ==> <a,f ` a> : f}
+{\out  2. [| \ldots |] ==> f Un g : (PROD x:?A. ?B(x))}
+\end{ttbox}
+To show $\pair{a,f`a}\in f$ we use \tdx{apply_Pair}, which is
+essentially the converse of \tdx{apply_equality}:
+\begin{ttbox}
+by (resolve_tac [apply_Pair] 1);
+{\out Level 3}
+{\out [| \ldots |] ==> (f Un g) ` a = f ` a}
+{\out  1. [| \ldots |] ==> f : (PROD x:?A2. ?B2(x))}
+{\out  2. [| \ldots |] ==> a : ?A2}
+{\out  3. [| \ldots |] ==> f Un g : (PROD x:?A. ?B(x))}
+\end{ttbox}
+Using the assumptions $f\in A\to B$ and $a\in A$, we solve the two subgoals
+from \tdx{apply_Pair}.  Recall that a $\Pi$-set is merely a generalized
+function space, and observe that~{\tt?A2} is instantiated to~\texttt{A}.
+\begin{ttbox}
+by (assume_tac 1);
+{\out Level 4}
+{\out [| \ldots |] ==> (f Un g) ` a = f ` a}
+{\out  1. [| \ldots |] ==> a : A}
+{\out  2. [| \ldots |] ==> f Un g : (PROD x:?A. ?B(x))}
+by (assume_tac 1);
+{\out Level 5}
+{\out [| \ldots |] ==> (f Un g) ` a = f ` a}
+{\out  1. [| \ldots |] ==> f Un g : (PROD x:?A. ?B(x))}
+\end{ttbox}
+To construct functions of the form $f\un g$, we apply
+\tdx{fun_disjoint_Un}:
+\begin{ttbox}
+by (resolve_tac [fun_disjoint_Un] 1);
+{\out Level 6}
+{\out [| \ldots |] ==> (f Un g) ` a = f ` a}
+{\out  1. [| \ldots |] ==> f : ?A3 -> ?B3}
+{\out  2. [| \ldots |] ==> g : ?C3 -> ?D3}
+{\out  3. [| \ldots |] ==> ?A3 Int ?C3 = 0}
+\end{ttbox}
+The remaining subgoals are instances of the assumptions.  Again, observe how
+unknowns are instantiated:
+\begin{ttbox}
+by (assume_tac 1);
+{\out Level 7}
+{\out [| \ldots |] ==> (f Un g) ` a = f ` a}
+{\out  1. [| \ldots |] ==> g : ?C3 -> ?D3}
+{\out  2. [| \ldots |] ==> A Int ?C3 = 0}
+by (assume_tac 1);
+{\out Level 8}
+{\out [| \ldots |] ==> (f Un g) ` a = f ` a}
+{\out  1. [| \ldots |] ==> A Int C = 0}
+by (assume_tac 1);
+{\out Level 9}
+{\out [| \ldots |] ==> (f Un g) ` a = f ` a}
+{\out No subgoals!}
+\end{ttbox}
+See the files \texttt{ZF/func.ML} and \texttt{ZF/WF.ML} for more
+examples of reasoning about functions.
+
+\index{set theory|)}