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+\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. Some
+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.
+
+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-mscs}.
+
+
+\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}
+\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 \isa{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 \isa{let}, set theory does not use polymorphism. All terms in
+ZF have type~\tydx{i}, which is the type of individuals and has
+class~\cldx{term}. The type of first-order formulae, remember,
+is~\tydx{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 $\lnot(a\in b)$,
+which is equivalent to $a\notin b$. 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 \isa{Upair($A$,$B$)} denotes the set~$\{A,B\}$ and
+\isa{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(\isa{Upair}(A,B)) \\
+ \isa{cons}(a,B) & \equiv & \isa{Upair}(a,a) \un B
+\end{eqnarray*}
+The $\{a@1, \ldots\}$ notation abbreviates finite sets constructed in the
+obvious manner using~\isa{cons} and~$\emptyset$ (the empty set) \isasymin \begin{eqnarray*}
+ \{a,b,c\} & \equiv & \isa{cons}(a,\isa{cons}(b,\isa{cons}(c,\emptyset)))
+\end{eqnarray*}
+
+The constant \cdx{Pair} constructs ordered pairs, as in \isa{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{\isa{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}
+\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
+\[\begin{array}{rcl}
+ term & = & \hbox{expression of type~$i$} \\
+ & | & "let"~id~"="~term";"\dots";"~id~"="~term~"in"~term \\
+ & | & "if"~term~"then"~term~"else"~term \\
+ & | & "{\ttlbrace} " term\; ("," term)^* " {\ttrbrace}" \\
+ & | & "< " term\; ("," term)^* " >" \\
+ & | & "{\ttlbrace} " id ":" term " . " formula " {\ttrbrace}" \\
+ & | & "{\ttlbrace} " id " . " id ":" term ", " formula " {\ttrbrace}" \\
+ & | & "{\ttlbrace} " term " . " id ":" term " {\ttrbrace}" \\
+ & | & term " `` " term \\
+ & | & term " -`` " term \\
+ & | & term " ` " term \\
+ & | & term " * " term \\
+ & | & term " \isasyminter " term \\
+ & | & term " \isasymunion " term \\
+ & | & term " - " term \\
+ & | & term " -> " term \\
+ & | & "THE~~" id " . " formula\\
+ & | & "lam~~" id ":" term " . " term \\
+ & | & "INT~~" id ":" term " . " term \\
+ & | & "UN~~~" id ":" term " . " term \\
+ & | & "PROD~" id ":" term " . " term \\
+ & | & "SUM~~" id ":" term " . " term \\[2ex]
+ formula & = & \hbox{expression of type~$o$} \\
+ & | & term " : " term \\
+ & | & term " \ttilde: " term \\
+ & | & term " <= " term \\
+ & | & term " = " term \\
+ & | & term " \ttilde= " term \\
+ & | & "\ttilde\ " formula \\
+ & | & formula " \& " formula \\
+ & | & formula " | " formula \\
+ & | & formula " --> " formula \\
+ & | & formula " <-> " formula \\
+ & | & "ALL " id ":" term " . " formula \\
+ & | & "EX~~" id ":" term " . " formula \\
+ & | & "ALL~" id~id^* " . " formula \\
+ & | & "EX~~" id~id^* " . " formula \\
+ & | & "EX!~" id~id^* " . " formula
+ \end{array}
+\]
+\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 \isa{Collect($A$,$\lambda x. P[x]$)}, which consists of all $x\in A$ that
+satisfy~$P[x]$. Note that \isa{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
+\isa{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 \isa{map},
+since it applies a function to every element of a set. The syntax is
+\isa{\ttlbrace$b[x]$.\ $x$:$A$\ttrbrace}, which expands to
+\isa{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 \isa{UN $x$:$A$.\ $B[x]$} and \isa{INT $x$:$A$.\ $B[x]$}.
+Their meaning is expressed using \isa{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 \isa{Sigma($A$,$B$)} and \isa{Pi($A$,$B$)} we may
+write
+\isa{SUM $x$:$A$.\ $B[x]$} and \isa{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~\isa{op~*} and~\isa{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
+\isa{The($\lambda x. P[x]$)} or use the syntax \isa{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 \isa{Lambda($A$,$\lambda x. b[x]$)} or use the syntax \isa{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 \isa{Ball($A$,$P$)} and \isa{Bex($A$,$P$)} we may
+write
+\isa{ALL $x$:$A$.\ $P[x]$} and \isa{EX $x$:$A$.\ $P[x]$}.
+
+
+%%%% ZF.thy
+
+\begin{figure}
+\begin{alltt*}\isastyleminor
+\tdx{Let_def}: Let(s, f) == f(s)
+
+\tdx{Ball_def}: Ball(A,P) == {\isasymforall}x. x \isasymin A --> P(x)
+\tdx{Bex_def}: Bex(A,P) == {\isasymexists}x. x \isasymin A & P(x)
+
+\tdx{subset_def}: A \isasymsubseteq B == {\isasymforall}x \isasymin A. x \isasymin B
+\tdx{extension}: A = B <-> A \isasymsubseteq B & B \isasymsubseteq A
+
+\tdx{Union_iff}: A \isasymin Union(C) <-> ({\isasymexists}B \isasymin C. A \isasymin B)
+\tdx{Pow_iff}: A \isasymin Pow(B) <-> A \isasymsubseteq B
+\tdx{foundation}: A=0 | ({\isasymexists}x \isasymin A. {\isasymforall}y \isasymin x. y \isasymnotin A)
+
+\tdx{replacement}: ({\isasymforall}x \isasymin A. {\isasymforall}y z. P(x,y) & P(x,z) --> y=z) ==>
+ b \isasymin PrimReplace(A,P) <-> ({\isasymexists}x{\isasymin}A. P(x,b))
+\subcaption{The Zermelo-Fraenkel Axioms}
+
+\tdx{Replace_def}: Replace(A,P) ==
+ PrimReplace(A, \%x y. (\isasymexists!z. P(x,z)) & P(x,y))
+\tdx{RepFun_def}: RepFun(A,f) == {\ttlbrace}y . x \isasymin A, y=f(x)\ttrbrace
+\tdx{the_def}: The(P) == Union({\ttlbrace}y . x \isasymin {\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 \isasymin A, x=y & P(x){\ttrbrace}
+\tdx{Upair_def}: Upair(a,b) ==
+ {\ttlbrace}y. x\isasymin{}Pow(Pow(0)), x=0 & y=a | x=Pow(0) & y=b{\ttrbrace}
+\subcaption{Consequences of replacement}
+
+\tdx{Inter_def}: Inter(A) == {\ttlbrace}x \isasymin Union(A) . {\isasymforall}y \isasymin A. x \isasymin y{\ttrbrace}
+\tdx{Un_def}: A \isasymunion B == Union(Upair(A,B))
+\tdx{Int_def}: A \isasyminter B == Inter(Upair(A,B))
+\tdx{Diff_def}: A - B == {\ttlbrace}x \isasymin A . x \isasymnotin B{\ttrbrace}
+\subcaption{Union, intersection, difference}
+\end{alltt*}
+\caption{Rules and axioms of ZF} \label{zf-rules}
+\end{figure}
+
+
+\begin{figure}
+\begin{alltt*}\isastyleminor
+\tdx{cons_def}: cons(a,A) == Upair(a,a) \isasymunion A
+\tdx{succ_def}: succ(i) == cons(i,i)
+\tdx{infinity}: 0 \isasymin Inf & ({\isasymforall}y \isasymin Inf. succ(y) \isasymin 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. {\isasymexists}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) == {\isasymUnion}x \isasymin A. {\isasymUnion}y \isasymin B(x). {\ttlbrace}<x,y>{\ttrbrace}
+\subcaption{Ordered pairs and Cartesian products}
+
+\tdx{converse_def}: converse(r) == {\ttlbrace}z. w\isasymin{}r, {\isasymexists}x y. w=<x,y> & z=<y,x>{\ttrbrace}
+\tdx{domain_def}: domain(r) == {\ttlbrace}x. w \isasymin r, {\isasymexists}y. w=<x,y>{\ttrbrace}
+\tdx{range_def}: range(r) == domain(converse(r))
+\tdx{field_def}: field(r) == domain(r) \isasymunion range(r)
+\tdx{image_def}: r `` A == {\ttlbrace}y\isasymin{}range(r) . {\isasymexists}x \isasymin A. <x,y> \isasymin 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 \isasymin A{\ttrbrace}
+\tdx{apply_def}: f`a == THE y. <a,y> \isasymin f
+\tdx{Pi_def}: Pi(A,B) == {\ttlbrace}f\isasymin{}Pow(Sigma(A,B)). {\isasymforall}x\isasymin{}A. \isasymexists!y. <x,y>\isasymin{}f{\ttrbrace}
+\tdx{restrict_def}: restrict(f,A) == lam x \isasymin A. f`x
+\subcaption{Functions and general product}
+\end{alltt*}
+\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.
+
+The traditional replacement axiom asserts
+\[ y \in \isa{PrimReplace}(A,P) \bimp (\exists x\in A. P(x,y)) \]
+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 $y\in \isa{Replace}(A,P)$ if and only if there is some~$x$ such that
+$P(x,-)$ holds uniquely for~$y$. Because the equivalence is unconditional,
+\isa{Replace} is much easier to use than \isa{PrimReplace}; it defines the
+same set, if $P(x,y)$ is single-valued. The nice syntax for replacement
+expands to \isa{Replace}.
+
+Other consequences of replacement include replacement for
+meta-level functions
+(\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 \isa{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 $\isa{Pi}(A,B)$ 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.thy
+
+\begin{figure}
+\begin{alltt*}\isastyleminor
+\tdx{ballI}: [| !!x. x\isasymin{}A ==> P(x) |] ==> {\isasymforall}x\isasymin{}A. P(x)
+\tdx{bspec}: [| {\isasymforall}x\isasymin{}A. P(x); x\isasymin{}A |] ==> P(x)
+\tdx{ballE}: [| {\isasymforall}x\isasymin{}A. P(x); P(x) ==> Q; x \isasymnotin A ==> Q |] ==> Q
+
+\tdx{ball_cong}: [| A=A'; !!x. x\isasymin{}A' ==> P(x) <-> P'(x) |] ==>
+ ({\isasymforall}x\isasymin{}A. P(x)) <-> ({\isasymforall}x\isasymin{}A'. P'(x))
+
+\tdx{bexI}: [| P(x); x\isasymin{}A |] ==> {\isasymexists}x\isasymin{}A. P(x)
+\tdx{bexCI}: [| {\isasymforall}x\isasymin{}A. ~P(x) ==> P(a); a\isasymin{}A |] ==> {\isasymexists}x\isasymin{}A. P(x)
+\tdx{bexE}: [| {\isasymexists}x\isasymin{}A. P(x); !!x. [| x\isasymin{}A; P(x) |] ==> Q |] ==> Q
+
+\tdx{bex_cong}: [| A=A'; !!x. x\isasymin{}A' ==> P(x) <-> P'(x) |] ==>
+ ({\isasymexists}x\isasymin{}A. P(x)) <-> ({\isasymexists}x\isasymin{}A'. P'(x))
+\subcaption{Bounded quantifiers}
+
+\tdx{subsetI}: (!!x. x \isasymin A ==> x \isasymin B) ==> A \isasymsubseteq B
+\tdx{subsetD}: [| A \isasymsubseteq B; c \isasymin A |] ==> c \isasymin B
+\tdx{subsetCE}: [| A \isasymsubseteq B; c \isasymnotin A ==> P; c \isasymin B ==> P |] ==> P
+\tdx{subset_refl}: A \isasymsubseteq A
+\tdx{subset_trans}: [| A \isasymsubseteq B; B \isasymsubseteq C |] ==> A \isasymsubseteq C
+
+\tdx{equalityI}: [| A \isasymsubseteq B; B \isasymsubseteq A |] ==> A = B
+\tdx{equalityD1}: A = B ==> A \isasymsubseteq B
+\tdx{equalityD2}: A = B ==> B \isasymsubseteq A
+\tdx{equalityE}: [| A = B; [| A \isasymsubseteq B; B \isasymsubseteq A |] ==> P |] ==> P
+\subcaption{Subsets and extensionality}
+
+\tdx{emptyE}: a \isasymin 0 ==> P
+\tdx{empty_subsetI}: 0 \isasymsubseteq A
+\tdx{equals0I}: [| !!y. y \isasymin A ==> False |] ==> A=0
+\tdx{equals0D}: [| A=0; a \isasymin A |] ==> P
+
+\tdx{PowI}: A \isasymsubseteq B ==> A \isasymin Pow(B)
+\tdx{PowD}: A \isasymin Pow(B) ==> A \isasymsubseteq B
+\subcaption{The empty set; power sets}
+\end{alltt*}
+\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 \isa{PrimReplace} would be. The principle of
+separation is proved explicitly, although most proofs should use the
+natural deduction rules for \isa{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.
+
+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{alltt*}\isastyleminor
+\tdx{ReplaceI}: [| x\isasymin{}A; P(x,b); !!y. P(x,y) ==> y=b |] ==>
+ b\isasymin{}{\ttlbrace}y. x\isasymin{}A, P(x,y){\ttrbrace}
+
+\tdx{ReplaceE}: [| b\isasymin{}{\ttlbrace}y. x\isasymin{}A, P(x,y){\ttrbrace};
+ !!x. [| x\isasymin{}A; P(x,b); {\isasymforall}y. P(x,y)-->y=b |] ==> R
+ |] ==> R
+
+\tdx{RepFunI}: [| a\isasymin{}A |] ==> f(a)\isasymin{}{\ttlbrace}f(x). x\isasymin{}A{\ttrbrace}
+\tdx{RepFunE}: [| b\isasymin{}{\ttlbrace}f(x). x\isasymin{}A{\ttrbrace};
+ !!x.[| x\isasymin{}A; b=f(x) |] ==> P |] ==> P
+
+\tdx{separation}: a\isasymin{}{\ttlbrace}x\isasymin{}A. P(x){\ttrbrace} <-> a\isasymin{}A & P(a)
+\tdx{CollectI}: [| a\isasymin{}A; P(a) |] ==> a\isasymin{}{\ttlbrace}x\isasymin{}A. P(x){\ttrbrace}
+\tdx{CollectE}: [| a\isasymin{}{\ttlbrace}x\isasymin{}A. P(x){\ttrbrace}; [| a\isasymin{}A; P(a) |] ==> R |] ==> R
+\tdx{CollectD1}: a\isasymin{}{\ttlbrace}x\isasymin{}A. P(x){\ttrbrace} ==> a\isasymin{}A
+\tdx{CollectD2}: a\isasymin{}{\ttlbrace}x\isasymin{}A. P(x){\ttrbrace} ==> P(a)
+\end{alltt*}
+\caption{Replacement and separation} \label{zf-lemmas2}
+\end{figure}
+
+
+\begin{figure}
+\begin{alltt*}\isastyleminor
+\tdx{UnionI}: [| B\isasymin{}C; A\isasymin{}B |] ==> A\isasymin{}Union(C)
+\tdx{UnionE}: [| A\isasymin{}Union(C); !!B.[| A\isasymin{}B; B\isasymin{}C |] ==> R |] ==> R
+
+\tdx{InterI}: [| !!x. x\isasymin{}C ==> A\isasymin{}x; c\isasymin{}C |] ==> A\isasymin{}Inter(C)
+\tdx{InterD}: [| A\isasymin{}Inter(C); B\isasymin{}C |] ==> A\isasymin{}B
+\tdx{InterE}: [| A\isasymin{}Inter(C); A\isasymin{}B ==> R; B \isasymnotin C ==> R |] ==> R
+
+\tdx{UN_I}: [| a\isasymin{}A; b\isasymin{}B(a) |] ==> b\isasymin{}({\isasymUnion}x\isasymin{}A. B(x))
+\tdx{UN_E}: [| b\isasymin{}({\isasymUnion}x\isasymin{}A. B(x)); !!x.[| x\isasymin{}A; b\isasymin{}B(x) |] ==> R
+ |] ==> R
+
+\tdx{INT_I}: [| !!x. x\isasymin{}A ==> b\isasymin{}B(x); a\isasymin{}A |] ==> b\isasymin{}({\isasymInter}x\isasymin{}A. B(x))
+\tdx{INT_E}: [| b\isasymin{}({\isasymInter}x\isasymin{}A. B(x)); a\isasymin{}A |] ==> b\isasymin{}B(a)
+\end{alltt*}
+\caption{General union and intersection} \label{zf-lemmas3}
+\end{figure}
+
+
+%%% upair.thy
+
+\begin{figure}
+\begin{alltt*}\isastyleminor
+\tdx{pairing}: a\isasymin{}Upair(b,c) <-> (a=b | a=c)
+\tdx{UpairI1}: a\isasymin{}Upair(a,b)
+\tdx{UpairI2}: b\isasymin{}Upair(a,b)
+\tdx{UpairE}: [| a\isasymin{}Upair(b,c); a=b ==> P; a=c ==> P |] ==> P
+\end{alltt*}
+\caption{Unordered pairs} \label{zf-upair1}
+\end{figure}
+
+
+\begin{figure}
+\begin{alltt*}\isastyleminor
+\tdx{UnI1}: c\isasymin{}A ==> c\isasymin{}A \isasymunion B
+\tdx{UnI2}: c\isasymin{}B ==> c\isasymin{}A \isasymunion B
+\tdx{UnCI}: (c \isasymnotin B ==> c\isasymin{}A) ==> c\isasymin{}A \isasymunion B
+\tdx{UnE}: [| c\isasymin{}A \isasymunion B; c\isasymin{}A ==> P; c\isasymin{}B ==> P |] ==> P
+
+\tdx{IntI}: [| c\isasymin{}A; c\isasymin{}B |] ==> c\isasymin{}A \isasyminter B
+\tdx{IntD1}: c\isasymin{}A \isasyminter B ==> c\isasymin{}A
+\tdx{IntD2}: c\isasymin{}A \isasyminter B ==> c\isasymin{}B
+\tdx{IntE}: [| c\isasymin{}A \isasyminter B; [| c\isasymin{}A; c\isasymin{}B |] ==> P |] ==> P
+
+\tdx{DiffI}: [| c\isasymin{}A; c \isasymnotin B |] ==> c\isasymin{}A - B
+\tdx{DiffD1}: c\isasymin{}A - B ==> c\isasymin{}A
+\tdx{DiffD2}: c\isasymin{}A - B ==> c \isasymnotin B
+\tdx{DiffE}: [| c\isasymin{}A - B; [| c\isasymin{}A; c \isasymnotin B |] ==> P |] ==> P
+\end{alltt*}
+\caption{Union, intersection, difference} \label{zf-Un}
+\end{figure}
+
+
+\begin{figure}
+\begin{alltt*}\isastyleminor
+\tdx{consI1}: a\isasymin{}cons(a,B)
+\tdx{consI2}: a\isasymin{}B ==> a\isasymin{}cons(b,B)
+\tdx{consCI}: (a \isasymnotin B ==> a=b) ==> a\isasymin{}cons(b,B)
+\tdx{consE}: [| a\isasymin{}cons(b,A); a=b ==> P; a\isasymin{}A ==> P |] ==> P
+
+\tdx{singletonI}: a\isasymin{}{\ttlbrace}a{\ttrbrace}
+\tdx{singletonE}: [| a\isasymin{}{\ttlbrace}b{\ttrbrace}; a=b ==> P |] ==> P
+\end{alltt*}
+\caption{Finite and singleton sets} \label{zf-upair2}
+\end{figure}
+
+
+\begin{figure}
+\begin{alltt*}\isastyleminor
+\tdx{succI1}: i\isasymin{}succ(i)
+\tdx{succI2}: i\isasymin{}j ==> i\isasymin{}succ(j)
+\tdx{succCI}: (i \isasymnotin j ==> i=j) ==> i\isasymin{}succ(j)
+\tdx{succE}: [| i\isasymin{}succ(j); i=j ==> P; i\isasymin{}j ==> P |] ==> P
+\tdx{succ_neq_0}: [| succ(n)=0 |] ==> P
+\tdx{succ_inject}: succ(m) = succ(n) ==> m=n
+\end{alltt*}
+\caption{The successor function} \label{zf-succ}
+\end{figure}
+
+
+\begin{figure}
+\begin{alltt*}\isastyleminor
+\tdx{the_equality}: [| P(a); !!x. P(x) ==> x=a |] ==> (THE x. P(x))=a
+\tdx{theI}: \isasymexists! 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\isasymin{}b; b\isasymin{}a |] ==> P
+\tdx{mem_irrefl}: a\isasymin{}a ==> P
+\end{alltt*}
+\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 \isa{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~\isa{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~\isa{cons}. The proof that
+\isa{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.
+
+
+%%% subset.thy?
+
+\begin{figure}
+\begin{alltt*}\isastyleminor
+\tdx{Union_upper}: B\isasymin{}A ==> B \isasymsubseteq Union(A)
+\tdx{Union_least}: [| !!x. x\isasymin{}A ==> x \isasymsubseteq C |] ==> Union(A) \isasymsubseteq C
+
+\tdx{Inter_lower}: B\isasymin{}A ==> Inter(A) \isasymsubseteq B
+\tdx{Inter_greatest}: [| a\isasymin{}A; !!x. x\isasymin{}A ==> C \isasymsubseteq x |] ==> C\isasymsubseteq{}Inter(A)
+
+\tdx{Un_upper1}: A \isasymsubseteq A \isasymunion B
+\tdx{Un_upper2}: B \isasymsubseteq A \isasymunion B
+\tdx{Un_least}: [| A \isasymsubseteq C; B \isasymsubseteq C |] ==> A \isasymunion B \isasymsubseteq C
+
+\tdx{Int_lower1}: A \isasyminter B \isasymsubseteq A
+\tdx{Int_lower2}: A \isasyminter B \isasymsubseteq B
+\tdx{Int_greatest}: [| C \isasymsubseteq A; C \isasymsubseteq B |] ==> C \isasymsubseteq A \isasyminter B
+
+\tdx{Diff_subset}: A-B \isasymsubseteq A
+\tdx{Diff_contains}: [| C \isasymsubseteq A; C \isasyminter B = 0 |] ==> C \isasymsubseteq A-B
+
+\tdx{Collect_subset}: Collect(A,P) \isasymsubseteq A
+\end{alltt*}
+\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.
+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
+${\isa{Pow}(A)\cap \isa{Pow}(B)}= \isa{Pow}(A\cap B)$.
+
+%%% pair.thy
+
+\begin{figure}
+\begin{alltt*}\isastyleminor
+\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\isasymin{}A; b\isasymin{}B(a) |] ==> <a,b>\isasymin{}Sigma(A,B)
+
+\tdx{SigmaE}: [| c\isasymin{}Sigma(A,B);
+ !!x y.[| x\isasymin{}A; y\isasymin{}B(x); c=<x,y> |] ==> P |] ==> P
+
+\tdx{SigmaE2}: [| <a,b>\isasymin{}Sigma(A,B);
+ [| a\isasymin{}A; b\isasymin{}B(a) |] ==> P |] ==> P
+\end{alltt*}
+\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 --- in particular, 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 $\pair{a,b}\in \isa{Sigma}(A,B)$ 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 \isa{split(\%$x$ $y$.\ $t$)}
+\end{center}
+Nested patterns are translated recursively:
+{\tt\%<$x$,$y$,$z$>. $t$} $\leadsto$ {\tt\%<$x$,<$y$,$z$>>. $t$} $\leadsto$
+\isa{split(\%$x$.\%<$y$,$z$>. $t$)} $\leadsto$ \isa{split(\%$x$. split(\%$y$
+ $z$.\ $t$))}. The reverse translation is performed upon printing.
+\begin{warn}
+ The translation between patterns and \isa{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 \isa{(\%<x,y>.<y,x>)<a,b>} requires the theorem \isa{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:] \isa{let {\it pattern} = $t$ in $u$}
+\item[Choice:] \isa{THE~{\it pattern}~.~$P$}
+\item[Set operations:] \isa{\isasymUnion~{\it pattern}:$A$.~$B$}
+\item[Comprehension:] \isa{{\ttlbrace}~{\it pattern}:$A$~.~$P$~{\ttrbrace}}
+\end{description}
+
+
+%%% domrange.thy?
+
+\begin{figure}
+\begin{alltt*}\isastyleminor
+\tdx{domainI}: <a,b>\isasymin{}r ==> a\isasymin{}domain(r)
+\tdx{domainE}: [| a\isasymin{}domain(r); !!y. <a,y>\isasymin{}r ==> P |] ==> P
+\tdx{domain_subset}: domain(Sigma(A,B)) \isasymsubseteq A
+
+\tdx{rangeI}: <a,b>\isasymin{}r ==> b\isasymin{}range(r)
+\tdx{rangeE}: [| b\isasymin{}range(r); !!x. <x,b>\isasymin{}r ==> P |] ==> P
+\tdx{range_subset}: range(A*B) \isasymsubseteq B
+
+\tdx{fieldI1}: <a,b>\isasymin{}r ==> a\isasymin{}field(r)
+\tdx{fieldI2}: <a,b>\isasymin{}r ==> b\isasymin{}field(r)
+\tdx{fieldCI}: (<c,a> \isasymnotin r ==> <a,b>\isasymin{}r) ==> a\isasymin{}field(r)
+
+\tdx{fieldE}: [| a\isasymin{}field(r);
+ !!x. <a,x>\isasymin{}r ==> P;
+ !!x. <x,a>\isasymin{}r ==> P
+ |] ==> P
+
+\tdx{field_subset}: field(A*A) \isasymsubseteq A
+\end{alltt*}
+\caption{Domain, range and field of a relation} \label{zf-domrange}
+\end{figure}
+
+\begin{figure}
+\begin{alltt*}\isastyleminor
+\tdx{imageI}: [| <a,b>\isasymin{}r; a\isasymin{}A |] ==> b\isasymin{}r``A
+\tdx{imageE}: [| b\isasymin{}r``A; !!x.[| <x,b>\isasymin{}r; x\isasymin{}A |] ==> P |] ==> P
+
+\tdx{vimageI}: [| <a,b>\isasymin{}r; b\isasymin{}B |] ==> a\isasymin{}r-``B
+\tdx{vimageE}: [| a\isasymin{}r-``B; !!x.[| <a,x>\isasymin{}r; x\isasymin{}B |] ==> P |] ==> P
+\end{alltt*}
+\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.
+
+
+%%% func.thy
+
+\begin{figure}
+\begin{alltt*}\isastyleminor
+\tdx{fun_is_rel}: f\isasymin{}Pi(A,B) ==> f \isasymsubseteq Sigma(A,B)
+
+\tdx{apply_equality}: [| <a,b>\isasymin{}f; f\isasymin{}Pi(A,B) |] ==> f`a = b
+\tdx{apply_equality2}: [| <a,b>\isasymin{}f; <a,c>\isasymin{}f; f\isasymin{}Pi(A,B) |] ==> b=c
+
+\tdx{apply_type}: [| f\isasymin{}Pi(A,B); a\isasymin{}A |] ==> f`a\isasymin{}B(a)
+\tdx{apply_Pair}: [| f\isasymin{}Pi(A,B); a\isasymin{}A |] ==> <a,f`a>\isasymin{}f
+\tdx{apply_iff}: f\isasymin{}Pi(A,B) ==> <a,b>\isasymin{}f <-> a\isasymin{}A & f`a = b
+
+\tdx{fun_extension}: [| f\isasymin{}Pi(A,B); g\isasymin{}Pi(A,D);
+ !!x. x\isasymin{}A ==> f`x = g`x |] ==> f=g
+
+\tdx{domain_type}: [| <a,b>\isasymin{}f; f\isasymin{}Pi(A,B) |] ==> a\isasymin{}A
+\tdx{range_type}: [| <a,b>\isasymin{}f; f\isasymin{}Pi(A,B) |] ==> b\isasymin{}B(a)
+
+\tdx{Pi_type}: [| f\isasymin{}A->C; !!x. x\isasymin{}A ==> f`x\isasymin{}B(x) |] ==> f\isasymin{}Pi(A,B)
+\tdx{domain_of_fun}: f\isasymin{}Pi(A,B) ==> domain(f)=A
+\tdx{range_of_fun}: f\isasymin{}Pi(A,B) ==> f\isasymin{}A->range(f)
+
+\tdx{restrict}: a\isasymin{}A ==> restrict(f,A) ` a = f`a
+\tdx{restrict_type}: [| !!x. x\isasymin{}A ==> f`x\isasymin{}B(x) |] ==>
+ restrict(f,A)\isasymin{}Pi(A,B)
+\end{alltt*}
+\caption{Functions} \label{zf-func1}
+\end{figure}
+
+
+\begin{figure}
+\begin{alltt*}\isastyleminor
+\tdx{lamI}: a\isasymin{}A ==> <a,b(a)>\isasymin{}(lam x\isasymin{}A. b(x))
+\tdx{lamE}: [| p\isasymin{}(lam x\isasymin{}A. b(x)); !!x.[| x\isasymin{}A; p=<x,b(x)> |] ==> P
+ |] ==> P
+
+\tdx{lam_type}: [| !!x. x\isasymin{}A ==> b(x)\isasymin{}B(x) |] ==> (lam x\isasymin{}A. b(x))\isasymin{}Pi(A,B)
+
+\tdx{beta}: a\isasymin{}A ==> (lam x\isasymin{}A. b(x)) ` a = b(a)
+\tdx{eta}: f\isasymin{}Pi(A,B) ==> (lam x\isasymin{}A. f`x) = f
+\end{alltt*}
+\caption{$\lambda$-abstraction} \label{zf-lam}
+\end{figure}
+
+
+\begin{figure}
+\begin{alltt*}\isastyleminor
+\tdx{fun_empty}: 0\isasymin{}0->0
+\tdx{fun_single}: {\ttlbrace}<a,b>{\ttrbrace}\isasymin{}{\ttlbrace}a{\ttrbrace} -> {\ttlbrace}b{\ttrbrace}
+
+\tdx{fun_disjoint_Un}: [| f\isasymin{}A->B; g\isasymin{}C->D; A \isasyminter C = 0 |] ==>
+ (f \isasymunion g)\isasymin{}(A \isasymunion C) -> (B \isasymunion D)
+
+\tdx{fun_disjoint_apply1}: [| a\isasymin{}A; f\isasymin{}A->B; g\isasymin{}C->D; A\isasyminter{}C = 0 |] ==>
+ (f \isasymunion g)`a = f`a
+
+\tdx{fun_disjoint_apply2}: [| c\isasymin{}C; f\isasymin{}A->B; g\isasymin{}C->D; A\isasyminter{}C = 0 |] ==>
+ (f \isasymunion g)`c = g`c
+\end{alltt*}
+\caption{Constructing functions from smaller sets} \label{zf-func2}
+\end{figure}
+
+
+\subsection{Functions}
+Functions, represented by graphs, are notoriously difficult to reason
+about. The ZF theory provides many derived 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, $C=\isa{range}(f)$.
+
+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{alltt*}\isastyleminor
+\tdx{Int_absorb}: A \isasyminter A = A
+\tdx{Int_commute}: A \isasyminter B = B \isasyminter A
+\tdx{Int_assoc}: (A \isasyminter B) \isasyminter C = A \isasyminter (B \isasyminter C)
+\tdx{Int_Un_distrib}: (A \isasymunion B) \isasyminter C = (A \isasyminter C) \isasymunion (B \isasyminter C)
+
+\tdx{Un_absorb}: A \isasymunion A = A
+\tdx{Un_commute}: A \isasymunion B = B \isasymunion A
+\tdx{Un_assoc}: (A \isasymunion B) \isasymunion C = A \isasymunion (B \isasymunion C)
+\tdx{Un_Int_distrib}: (A \isasyminter B) \isasymunion C = (A \isasymunion C) \isasyminter (B \isasymunion C)
+
+\tdx{Diff_cancel}: A-A = 0
+\tdx{Diff_disjoint}: A \isasyminter (B-A) = 0
+\tdx{Diff_partition}: A \isasymsubseteq B ==> A \isasymunion (B-A) = B
+\tdx{double_complement}: [| A \isasymsubseteq B; B \isasymsubseteq C |] ==> (B - (C-A)) = A
+\tdx{Diff_Un}: A - (B \isasymunion C) = (A-B) \isasyminter (A-C)
+\tdx{Diff_Int}: A - (B \isasyminter C) = (A-B) \isasymunion (A-C)
+
+\tdx{Union_Un_distrib}: Union(A \isasymunion B) = Union(A) \isasymunion Union(B)
+\tdx{Inter_Un_distrib}: [| a \isasymin A; b \isasymin B |] ==>
+ Inter(A \isasymunion B) = Inter(A) \isasyminter Inter(B)
+
+\tdx{Int_Union_RepFun}: A \isasyminter Union(B) = ({\isasymUnion}C \isasymin B. A \isasyminter C)
+
+\tdx{Un_Inter_RepFun}: b \isasymin B ==>
+ A \isasymunion Inter(B) = ({\isasymInter}C \isasymin B. A \isasymunion C)
+
+\tdx{SUM_Un_distrib1}: (SUM x \isasymin A \isasymunion B. C(x)) =
+ (SUM x \isasymin A. C(x)) \isasymunion (SUM x \isasymin B. C(x))
+
+\tdx{SUM_Un_distrib2}: (SUM x \isasymin C. A(x) \isasymunion B(x)) =
+ (SUM x \isasymin C. A(x)) \isasymunion (SUM x \isasymin C. B(x))
+
+\tdx{SUM_Int_distrib1}: (SUM x \isasymin A \isasyminter B. C(x)) =
+ (SUM x \isasymin A. C(x)) \isasyminter (SUM x \isasymin B. C(x))
+
+\tdx{SUM_Int_distrib2}: (SUM x \isasymin C. A(x) \isasyminter B(x)) =
+ (SUM x \isasymin C. A(x)) \isasyminter (SUM x \isasymin C. B(x))
+\end{alltt*}
+\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 \isa{bool} \\
+% \cdx{not} & $i\To i$ & & negation for \isa{bool} \\
+% \sdx{and} & $[i,i]\To i$ & Left 70 & conjunction for \isa{bool} \\
+% \sdx{or} & $[i,i]\To i$ & Left 65 & disjunction for \isa{bool} \\
+% \sdx{xor} & $[i,i]\To i$ & Left 65 & exclusive-or for \isa{bool}
+%\end{constants}
+%
+\begin{alltt*}\isastyleminor
+\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 \isasymin bool
+\tdx{bool_0I}: 0 \isasymin bool
+\tdx{boolE}: [| c \isasymin 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{alltt*}
+\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 proofs.
+
+Figure~\ref{zf-equalities} presents commutative, associative, distributive,
+and idempotency laws of union and intersection, along with other equations.
+
+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
+\isa{bool}-valued functions, for example. The constant~\isa{1} is
+translated to \isa{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{alltt*}\isastyleminor
+\tdx{sum_def}: A+B == {\ttlbrace}0{\ttrbrace}*A \isasymunion {\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{InlI}: a \isasymin A ==> Inl(a) \isasymin A+B
+\tdx{InrI}: b \isasymin B ==> Inr(b) \isasymin 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{sum_iff}: u \isasymin A+B <-> ({\isasymexists}x\isasymin{}A. u=Inl(x)) | ({\isasymexists}y\isasymin{}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{alltt*}
+\caption{Disjoint unions} \label{zf-sum}
+\end{figure}
+
+
+\subsection{Disjoint unions}
+
+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{alltt*}\isastyleminor
+\tdx{QPair_def}: <a;b> == a+b
+\tdx{qsplit_def}: qsplit(c,p) == THE y. {\isasymexists}a b. p=<a;b> & y=c(a,b)
+\tdx{qfsplit_def}: qfsplit(R,z) == {\isasymexists}x y. z=<x;y> & R(x,y)
+\tdx{qconverse_def}: qconverse(r) == {\ttlbrace}z. w \isasymin r, {\isasymexists}x y. w=<x;y> & z=<y;x>{\ttrbrace}
+\tdx{QSigma_def}: QSigma(A,B) == {\isasymUnion}x \isasymin A. {\isasymUnion}y \isasymin B(x). {\ttlbrace}<x;y>{\ttrbrace}
+
+\tdx{qsum_def}: A <+> B == ({\ttlbrace}0{\ttrbrace} <*> A) \isasymunion ({\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{alltt*}
+\caption{Non-standard pairs, products and sums} \label{zf-qpair}
+\end{figure}
+
+
+\subsection{Non-standard ordered pairs}
+
+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-mscs}.
+
+\begin{figure}
+\begin{alltt*}\isastyleminor
+\tdx{bnd_mono_def}: bnd_mono(D,h) ==
+ h(D)\isasymsubseteq{}D & ({\isasymforall}W X. W\isasymsubseteq{}X --> X\isasymsubseteq{}D --> h(W)\isasymsubseteq{}h(X))
+
+\tdx{lfp_def}: lfp(D,h) == Inter({\ttlbrace}X \isasymin Pow(D). h(X) \isasymsubseteq X{\ttrbrace})
+\tdx{gfp_def}: gfp(D,h) == Union({\ttlbrace}X \isasymin Pow(D). X \isasymsubseteq h(X){\ttrbrace})
+
+
+\tdx{lfp_lowerbound}: [| h(A) \isasymsubseteq A; A \isasymsubseteq D |] ==> lfp(D,h) \isasymsubseteq A
+
+\tdx{lfp_subset}: lfp(D,h) \isasymsubseteq D
+
+\tdx{lfp_greatest}: [| bnd_mono(D,h);
+ !!X. [| h(X) \isasymsubseteq X; X \isasymsubseteq D |] ==> A \isasymsubseteq X
+ |] ==> A \isasymsubseteq lfp(D,h)
+
+\tdx{lfp_Tarski}: bnd_mono(D,h) ==> lfp(D,h) = h(lfp(D,h))
+
+\tdx{induct}: [| a \isasymin lfp(D,h); bnd_mono(D,h);
+ !!x. x \isasymin h(Collect(lfp(D,h),P)) ==> P(x)
+ |] ==> P(a)
+
+\tdx{lfp_mono}: [| bnd_mono(D,h); bnd_mono(E,i);
+ !!X. X \isasymsubseteq D ==> h(X) \isasymsubseteq i(X)
+ |] ==> lfp(D,h) \isasymsubseteq lfp(E,i)
+
+\tdx{gfp_upperbound}: [| A \isasymsubseteq h(A); A \isasymsubseteq D |] ==> A \isasymsubseteq gfp(D,h)
+
+\tdx{gfp_subset}: gfp(D,h) \isasymsubseteq D
+
+\tdx{gfp_least}: [| bnd_mono(D,h);
+ !!X. [| X \isasymsubseteq h(X); X \isasymsubseteq D |] ==> X \isasymsubseteq A
+ |] ==> gfp(D,h) \isasymsubseteq A
+
+\tdx{gfp_Tarski}: bnd_mono(D,h) ==> gfp(D,h) = h(gfp(D,h))
+
+\tdx{coinduct}: [| bnd_mono(D,h); a \isasymin X; X \isasymsubseteq h(X \isasymunion gfp(D,h)); X \isasymsubseteq D
+ |] ==> a \isasymin gfp(D,h)
+
+\tdx{gfp_mono}: [| bnd_mono(D,h); D \isasymsubseteq E;
+ !!X. X \isasymsubseteq D ==> h(X) \isasymsubseteq i(X)
+ |] ==> gfp(D,h) \isasymsubseteq gfp(E,i)
+\end{alltt*}
+\caption{Least and greatest fixedpoints} \label{zf-fixedpt}
+\end{figure}
+
+
+\subsection{Least and greatest fixedpoints}
+
+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.
+
+
+\subsection{Finite sets and lists}
+
+Theory \texttt{Finite} (Figure~\ref{zf-fin}) defines the finite set operator;
+$\isa{Fin}(A)$ 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{alltt*}\isastyleminor
+\tdx{Fin.emptyI} 0 \isasymin Fin(A)
+\tdx{Fin.consI} [| a \isasymin A; b \isasymin Fin(A) |] ==> cons(a,b) \isasymin Fin(A)
+
+\tdx{Fin_induct}
+ [| b \isasymin Fin(A);
+ P(0);
+ !!x y. [| x\isasymin{}A; y\isasymin{}Fin(A); x\isasymnotin{}y; P(y) |] ==> P(cons(x,y))
+ |] ==> P(b)
+
+\tdx{Fin_mono}: A \isasymsubseteq B ==> Fin(A) \isasymsubseteq Fin(B)
+\tdx{Fin_UnI}: [| b \isasymin Fin(A); c \isasymin Fin(A) |] ==> b \isasymunion c \isasymin Fin(A)
+\tdx{Fin_UnionI}: C \isasymin Fin(Fin(A)) ==> Union(C) \isasymin Fin(A)
+\tdx{Fin_subset}: [| c \isasymsubseteq b; b \isasymin Fin(A) |] ==> c \isasymin Fin(A)
+\end{alltt*}
+\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{alltt*}\isastyleminor
+\tdx{NilI}: Nil \isasymin list(A)
+\tdx{ConsI}: [| a \isasymin A; l \isasymin list(A) |] ==> Cons(a,l) \isasymin list(A)
+
+\tdx{List.induct}
+ [| l \isasymin list(A);
+ P(Nil);
+ !!x y. [| x \isasymin A; y \isasymin 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 \isasymnoteq Cons(a,l)
+
+\tdx{list_mono}: A \isasymsubseteq B ==> list(A) \isasymsubseteq list(B)
+
+\tdx{map_ident}: l\isasymin{}list(A) ==> map(\%u. u, l) = l
+\tdx{map_compose}: l\isasymin{}list(A) ==> map(h, map(j,l)) = map(\%u. h(j(u)), l)
+\tdx{map_app_distrib}: xs\isasymin{}list(A) ==> map(h, xs@ys) = map(h,xs)@map(h,ys)
+\tdx{map_type}
+ [| l\isasymin{}list(A); !!x. x\isasymin{}A ==> h(x)\isasymin{}B |] ==> map(h,l)\isasymin{}list(B)
+\tdx{map_flat}
+ ls: list(list(A)) ==> map(h, flat(ls)) = flat(map(map(h),ls))
+\end{alltt*}
+\caption{Lists} \label{zf-list}
+\end{figure}
+
+
+Figure~\ref{zf-list} presents the set of lists over~$A$, $\isa{list}(A)$. The
+definition employs Isabelle's datatype package, which defines the introduction
+and induction rules automatically, as well as the constructors, case operator
+(\isa{list\_case}) and recursion operator. The theory then defines the usual
+list functions by primitive recursion. See theory \texttt{List}.
+
+
+\subsection{Miscellaneous}
+
+\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{alltt*}\isastyleminor
+\tdx{comp_def}: r O s == {\ttlbrace}xz \isasymin domain(s)*range(r) .
+ {\isasymexists}x y z. xz=<x,z> & <x,y> \isasymin s & <y,z> \isasymin r{\ttrbrace}
+\tdx{id_def}: id(A) == (lam x \isasymin A. x)
+\tdx{inj_def}: inj(A,B) == {\ttlbrace} f\isasymin{}A->B. {\isasymforall}w\isasymin{}A. {\isasymforall}x\isasymin{}A. f`w=f`x --> w=x {\ttrbrace}
+\tdx{surj_def}: surj(A,B) == {\ttlbrace} f\isasymin{}A->B . {\isasymforall}y\isasymin{}B. {\isasymexists}x\isasymin{}A. f`x=y {\ttrbrace}
+\tdx{bij_def}: bij(A,B) == inj(A,B) \isasyminter surj(A,B)
+
+
+\tdx{left_inverse}: [| f\isasymin{}inj(A,B); a\isasymin{}A |] ==> converse(f)`(f`a) = a
+\tdx{right_inverse}: [| f\isasymin{}inj(A,B); b\isasymin{}range(f) |] ==>
+ f`(converse(f)`b) = b
+
+\tdx{inj_converse_inj}: f\isasymin{}inj(A,B) ==> converse(f) \isasymin inj(range(f),A)
+\tdx{bij_converse_bij}: f\isasymin{}bij(A,B) ==> converse(f) \isasymin bij(B,A)
+
+\tdx{comp_type}: [| s \isasymsubseteq A*B; r \isasymsubseteq B*C |] ==> (r O s) \isasymsubseteq A*C
+\tdx{comp_assoc}: (r O s) O t = r O (s O t)
+
+\tdx{left_comp_id}: r \isasymsubseteq A*B ==> id(B) O r = r
+\tdx{right_comp_id}: r \isasymsubseteq A*B ==> r O id(A) = r
+
+\tdx{comp_func}: [| g\isasymin{}A->B; f\isasymin{}B->C |] ==> (f O g) \isasymin A->C
+\tdx{comp_func_apply}: [| g\isasymin{}A->B; f\isasymin{}B->C; a\isasymin{}A |] ==> (f O g)`a = f`(g`a)
+
+\tdx{comp_inj}: [| g\isasymin{}inj(A,B); f\isasymin{}inj(B,C) |] ==> (f O g)\isasymin{}inj(A,C)
+\tdx{comp_surj}: [| g\isasymin{}surj(A,B); f\isasymin{}surj(B,C) |] ==> (f O g)\isasymin{}surj(A,C)
+\tdx{comp_bij}: [| g\isasymin{}bij(A,B); f\isasymin{}bij(B,C) |] ==> (f O g)\isasymin{}bij(A,C)
+
+\tdx{left_comp_inverse}: f\isasymin{}inj(A,B) ==> converse(f) O f = id(A)
+\tdx{right_comp_inverse}: f\isasymin{}surj(A,B) ==> f O converse(f) = id(B)
+
+\tdx{bij_disjoint_Un}:
+ [| f\isasymin{}bij(A,B); g\isasymin{}bij(C,D); A \isasyminter C = 0; B \isasyminter D = 0 |] ==>
+ (f \isasymunion g)\isasymin{}bij(A \isasymunion C, B \isasymunion D)
+
+\tdx{restrict_bij}: [| f\isasymin{}inj(A,B); C\isasymsubseteq{}A |] ==> restrict(f,C)\isasymin{}bij(C, f``C)
+\end{alltt*}
+\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.
+
+Theory \thydx{Univ} defines a `universe' $\isa{univ}(A)$, which is used by
+the datatype package. This set contains $A$ and the
+natural numbers. Vitally, it is closed under finite products:
+$\isa{univ}(A)\times\isa{univ}(A)\subseteq\isa{univ}(A)$. 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' $\isa{quniv}(A)$, which is used by
+the datatype package to construct codatatypes such as streams. It is
+analogous to $\isa{univ}(A)$ (and is defined in terms of it) but is closed
+under the non-standard product and sum.
+
+
+\section{Automatic Tools}
+
+ZF provides the simplifier and the classical reasoner. Moreover it supplies a
+specialized tool to infer `types' of terms.
+
+\subsection{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 $a\in A-B$.
+
+The default simpset used by \isa{simp} contains congruence rules for all of ZF's
+binding operators. It contains all the conversion rules, such as
+\isa{fst} and
+\isa{snd}, as well as the rewrites shown in Fig.\ts\ref{zf-simpdata}.
+
+Classical reasoner methods such as \isa{blast} and \isa{auto} refer to
+a rich collection of built-in axioms for all the set-theoretic
+primitives.
+
+
+\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 \lnot (a\in B)\\
+ \pair{a,b}\in \isa{Sigma}(A,B)
+ & \bimp & a\in A \conj b\in B(a)\\
+ a \in \isa{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}
+
+
+\subsection{Type-Checking Tactics}
+\index{type-checking tactics}
+
+Isabelle/ZF provides simple tactics to help automate those proofs that are
+essentially type-checking. Such proofs are built by applying rules such as
+these:
+\begin{ttbox}\isastyleminor
+[| ?P ==> ?a \isasymin ?A; ~?P ==> ?b \isasymin ?A |]
+==> (if ?P then ?a else ?b) \isasymin ?A
+
+[| ?m \isasymin nat; ?n \isasymin nat |] ==> ?m #+ ?n \isasymin nat
+
+?a \isasymin ?A ==> Inl(?a) \isasymin ?A + ?B
+\end{ttbox}
+In typical applications, the goal has the form $t\in\Var{A}$: in other words,
+we have a specific term~$t$ and need to infer its `type' by instantiating the
+set variable~$\Var{A}$. Neither the simplifier nor the classical reasoner
+does this job well. The if-then-else rule, and many similar ones, can make
+the classical reasoner loop. The simplifier refuses (on principle) to
+instantiate variables during rewriting, so goals such as \isa{i\#+j \isasymin \ ?A}
+are left unsolved.
+
+The simplifier calls the type-checker to solve rewritten subgoals: this stage
+can indeed instantiate variables. If you have defined new constants and
+proved type-checking rules for them, then declare the rules using
+the attribute \isa{TC} and the rest should be automatic. In
+particular, the simplifier will use type-checking to help satisfy
+conditional rewrite rules. Call the method \ttindex{typecheck} to
+break down all subgoals using type-checking rules. You can add new
+type-checking rules temporarily like this:
+\begin{isabelle}
+\isacommand{apply}\ (typecheck add:\ inj_is_fun)
+\end{isabelle}
+
+
+%Though the easiest way to invoke the type-checker is via the simplifier,
+%specialized applications may require more detailed knowledge of
+%the type-checking primitives. They are modelled on the simplifier's:
+%\begin{ttdescription}
+%\item[\ttindexbold{tcset}] is the type of tcsets: sets of type-checking rules.
+%
+%\item[\ttindexbold{addTCs}] is an infix operator to add type-checking rules to
+% a tcset.
+%
+%\item[\ttindexbold{delTCs}] is an infix operator to remove type-checking rules
+% from a tcset.
+%
+%\item[\ttindexbold{typecheck_tac}] is a tactic for attempting to prove all
+% subgoals using the rules given in its argument, a tcset.
+%\end{ttdescription}
+%
+%Tcsets, like simpsets, are associated with theories and are merged when
+%theories are merged. There are further primitives that use the default tcset.
+%\begin{ttdescription}
+%\item[\ttindexbold{tcset}] is a function to return the default tcset; use the
+% expression \isa{tcset()}.
+%
+%\item[\ttindexbold{AddTCs}] adds type-checking rules to the default tcset.
+%
+%\item[\ttindexbold{DelTCs}] removes type-checking rules from the default
+% tcset.
+%
+%\item[\ttindexbold{Typecheck_tac}] calls \isa{typecheck_tac} using the
+% default tcset.
+%\end{ttdescription}
+%
+%To supply some type-checking rules temporarily, using \isa{Addrules} and
+%later \isa{Delrules} is the simplest way. There is also a high-tech
+%approach. Call the simplifier with a new solver expressed using
+%\ttindexbold{type_solver_tac} and your temporary type-checking rules.
+%\begin{ttbox}\isastyleminor
+%by (asm_simp_tac
+% (simpset() setSolver type_solver_tac (tcset() addTCs prems)) 2);
+%\end{ttbox}
+
+
+\section{Natural number and integer arithmetic}
+
+\index{arithmetic|(}
+
+\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{alltt*}\isastyleminor
+\tdx{nat_def}: nat == lfp(lam r \isasymin Pow(Inf). {\ttlbrace}0{\ttrbrace} \isasymunion {\ttlbrace}succ(x). x \isasymin r{\ttrbrace}
+
+\tdx{nat_case_def}: nat_case(a,b,k) ==
+ THE y. k=0 & y=a | ({\isasymexists}x. k=succ(x) & y=b(x))
+
+\tdx{nat_0I}: 0 \isasymin nat
+\tdx{nat_succI}: n \isasymin nat ==> succ(n) \isasymin nat
+
+\tdx{nat_induct}:
+ [| n \isasymin nat; P(0); !!x. [| x \isasymin 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_natify}: 0 #+ n = natify(n)
+\tdx{add_succ}: succ(m) #+ n = succ(m #+ n)
+
+\tdx{mult_type}: m #* n \isasymin nat
+\tdx{mult_0}: 0 #* n = 0
+\tdx{mult_succ}: succ(m) #* n = n #+ (m #* n)
+\tdx{mult_commute}: m #* n = n #* m
+\tdx{add_mult_dist}: (m #+ n) #* k = (m #* k) #+ (n #* k)
+\tdx{mult_assoc}: (m #* n) #* k = m #* (n #* k)
+\tdx{mod_div_equality}: m \isasymin nat ==> (m div n)#*n #+ m mod n = m
+\end{alltt*}
+\caption{The natural numbers} \label{zf-nat}
+\end{figure}
+
+\index{natural numbers}
+
+Theory \thydx{Nat} defines the natural numbers and mathematical
+induction, along with a case analysis operator. The set of natural
+numbers, here called \isa{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 $a \bmod b +
+(a/b)\times b = a$.
+
+To minimize the need for tedious proofs of $t\in\isa{nat}$, the arithmetic
+operators coerce their arguments to be natural numbers. The function
+\cdx{natify} is defined such that $\isa{natify}(n) = n$ if $n$ is a natural
+number, $\isa{natify}(\isa{succ}(x)) =
+\isa{succ}(\isa{natify}(x))$ for all $x$, and finally
+$\isa{natify}(x)=0$ in all other cases. The benefit is that the addition,
+subtraction, multiplication, division and remainder operators always return
+natural numbers, regardless of their arguments. Algebraic laws (commutative,
+associative, distributive) are unconditional. Occurrences of \isa{natify}
+as operands of those operators are simplified away. Any remaining occurrences
+can either be tolerated or else eliminated by proving that the argument is a
+natural number.
+
+The simplifier automatically cancels common terms on the opposite sides of
+subtraction and of relations ($=$, $<$ and $\le$). Here is an example:
+\begin{isabelle}
+ 1. i \#+ j \#+ k \#- j < k \#+ l\isanewline
+\isacommand{apply}\ simp\isanewline
+ 1. natify(i) < natify(l)
+\end{isabelle}
+Given the assumptions \isa{i \isasymin nat} and \isa{l \isasymin nat}, both occurrences of
+\cdx{natify} would be simplified away.
+
+
+\begin{figure}\small
+\index{$*@{\tt\$*} symbol}
+\index{$+@{\tt\$+} symbol}
+\index{$-@{\tt\$-} symbol}
+\begin{constants}
+ \it symbol & \it meta-type & \it priority & \it description \\
+ \cdx{int} & $i$ & & set of integers \\
+ \tt \$* & $[i,i]\To i$ & Left 70 & multiplication \\
+ \tt \$+ & $[i,i]\To i$ & Left 65 & addition\\
+ \tt \$- & $[i,i]\To i$ & Left 65 & subtraction\\
+ \tt \$< & $[i,i]\To o$ & Left 50 & $<$ on integers\\
+ \tt \$<= & $[i,i]\To o$ & Left 50 & $\le$ on integers
+\end{constants}
+
+\begin{alltt*}\isastyleminor
+\tdx{zadd_0_intify}: 0 $+ n = intify(n)
+
+\tdx{zmult_type}: m $* n \isasymin int
+\tdx{zmult_0}: 0 $* n = 0
+\tdx{zmult_commute}: m $* n = n $* m
+\tdx{zadd_zmult_dist}: (m $+ n) $* k = (m $* k) $+ (n $* k)
+\tdx{zmult_assoc}: (m $* n) $* k = m $* (n $* k)
+\end{alltt*}
+\caption{The integers} \label{zf-int}
+\end{figure}
+
+
+\index{integers}
+
+Theory \thydx{Int} defines the integers, as equivalence classes of natural
+numbers. Figure~\ref{zf-int} presents a tidy collection of laws. In
+fact, a large library of facts is proved, including monotonicity laws for
+addition and multiplication, covering both positive and negative operands.
+
+As with the natural numbers, the need for typing proofs is minimized. All the
+operators defined in Fig.\ts\ref{zf-int} coerce their operands to integers by
+applying the function \cdx{intify}. This function is the identity on integers
+and maps other operands to zero.
+
+Decimal notation is provided for the integers. Numbers, written as
+\isa{\#$nnn$} or \isa{\#-$nnn$}, are represented internally in
+two's-complement binary. Expressions involving addition, subtraction and
+multiplication of numeral constants are evaluated (with acceptable efficiency)
+by simplification. The simplifier also collects similar terms, multiplying
+them by a numerical coefficient. It also cancels occurrences of the same
+terms on the other side of the relational operators. Example:
+\begin{isabelle}
+ 1. y \$+ z \$+ \#-3 \$* x \$+ y \$<= x \$* \#2 \$+
+z\isanewline
+\isacommand{apply}\ simp\isanewline
+ 1. \#2 \$* y \$<= \#5 \$* x
+\end{isabelle}
+For more information on the integers, please see the theories on directory
+\texttt{ZF/Integ}.
+
+\index{arithmetic|)}
+
+
+\section{Datatype definitions}
+\label{sec:ZF:datatype}
+\index{*datatype|(}
+
+The \ttindex{datatype} definition package of ZF constructs inductive datatypes
+similar to \ML's. 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 \isa{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 \isa{list}, which is built-in, and is
+defined by
+\begin{alltt*}\isastyleminor
+consts list :: "i=>i"
+datatype "list(A)" = Nil | Cons ("a \isasymin A", "l \isasymin list(A)")
+\end{alltt*}
+Note that the datatype operator must be declared as a constant first.
+However, the package declares the constructors. Here, \isa{Nil} gets type
+$i$ and \isa{Cons} gets type $[i,i]\To i$.
+
+Trees and forests can be modelled by the mutually recursive datatype
+definition
+\begin{alltt*}\isastyleminor
+consts
+ tree :: "i=>i"
+ forest :: "i=>i"
+ tree_forest :: "i=>i"
+datatype "tree(A)" = Tcons ("a{\isasymin}A", "f{\isasymin}forest(A)")
+and "forest(A)" = Fnil | Fcons ("t{\isasymin}tree(A)", "f{\isasymin}forest(A)")
+\end{alltt*}
+Here $\isa{tree}(A)$ is the set of trees over $A$, $\isa{forest}(A)$ is
+the set of forests over $A$, and $\isa{tree_forest}(A)$ is the union of
+the previous two sets. All three operators must be declared first.
+
+The datatype \isa{term}, which is defined by
+\begin{alltt*}\isastyleminor
+consts term :: "i=>i"
+datatype "term(A)" = Apply ("a \isasymin A", "l \isasymin list(term(A))")
+ monos list_mono
+ type_elims list_univ [THEN subsetD, elim_format]
+\end{alltt*}
+is an example of nested recursion. (The theorem \isa{list_mono} is proved
+in theory \isa{List}, and the \isa{term} example is developed in
+theory
+\thydx{Induct/Term}.)
+
+\subsubsection{Freeness of the constructors}
+
+Constructors satisfy {\em freeness} properties. Constructions are distinct,
+for example $\isa{Nil}\not=\isa{Cons}(a,l)$, and they are injective, for
+example $\isa{Cons}(a,l)=\isa{Cons}(a',l') \bimp a=a' \conj l=l'$.
+Because the number of freeness is quadratic in the number of constructors, the
+datatype package does not prove them. Instead, it ensures that simplification
+will prove them dynamically: when the simplifier encounters a formula
+asserting the equality of two datatype constructors, it performs freeness
+reasoning.
+
+Freeness reasoning can also be done using the classical reasoner, but it is
+more complicated. You have to add some safe elimination rules rules to the
+claset. For the \isa{list} datatype, they are called
+\isa{list.free_elims}. Occasionally this exposes the underlying
+representation of some constructor, which can be rectified using the command
+\isa{unfold list.con_defs [symmetric]}.
+
+
+\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
+\isa{list.induct} in Fig.\ts\ref{zf-list}. For mutually recursive
+datatypes, the induction rule is supplied in two forms. Consider datatype
+\isa{TF}. The rule \isa{tree_forest.induct} performs induction over a
+single predicate~\isa{P}, which is presumed to be defined for both trees
+and forests:
+\begin{alltt*}\isastyleminor
+[| x \isasymin tree_forest(A);
+ !!a f. [| a \isasymin A; f \isasymin forest(A); P(f) |] ==> P(Tcons(a, f));
+ P(Fnil);
+ !!f t. [| t \isasymin tree(A); P(t); f \isasymin forest(A); P(f) |]
+ ==> P(Fcons(t, f))
+|] ==> P(x)
+\end{alltt*}
+The rule \isa{tree_forest.mutual_induct} performs induction over two
+distinct predicates, \isa{P_tree} and \isa{P_forest}.
+\begin{alltt*}\isastyleminor
+[| !!a f.
+ [| a{\isasymin}A; f{\isasymin}forest(A); P_forest(f) |] ==> P_tree(Tcons(a,f));
+ P_forest(Fnil);
+ !!f t. [| t{\isasymin}tree(A); P_tree(t); f{\isasymin}forest(A); P_forest(f) |]
+ ==> P_forest(Fcons(t, f))
+|] ==> ({\isasymforall}za. za \isasymin tree(A) --> P_tree(za)) &
+ ({\isasymforall}za. za \isasymin forest(A) --> P_forest(za))
+\end{alltt*}
+
+For datatypes with nested recursion, such as the \isa{term} example from
+above, things are a bit more complicated. The rule \isa{term.induct}
+refers to the monotonic operator, \isa{list}:
+\begin{alltt*}\isastyleminor
+[| x \isasymin term(A);
+ !!a l. [| a\isasymin{}A; l\isasymin{}list(Collect(term(A), P)) |] ==> P(Apply(a,l))
+|] ==> P(x)
+\end{alltt*}
+The theory \isa{Induct/Term.thy} derives two higher-level induction rules,
+one of which is particularly useful for proving equations:
+\begin{alltt*}\isastyleminor
+[| t \isasymin term(A);
+ !!x zs. [| x \isasymin A; zs \isasymin list(term(A)); map(f, zs) = map(g, zs) |]
+ ==> f(Apply(x, zs)) = g(Apply(x, zs))
+|] ==> f(t) = g(t)
+\end{alltt*}
+How this can be generalized to other nested datatypes is a matter for future
+research.
+
+
+\subsubsection{The \isa{case} operator}
+
+The package defines an operator for performing case analysis over the
+datatype. For \isa{list}, it is called \isa{list_case} and satisfies
+the equations
+\begin{ttbox}\isastyleminor
+list_case(f_Nil, f_Cons, []) = f_Nil
+list_case(f_Nil, f_Cons, Cons(a, l)) = f_Cons(a, l)
+\end{ttbox}
+Here \isa{f_Nil} is the value to return if the argument is \isa{Nil} and
+\isa{f_Cons} is a function that computes the value to return if the
+argument has the form $\isa{Cons}(a,l)$. The function can be expressed as
+an abstraction, over patterns if desired (\S\ref{sec:pairs}).
+
+For mutually recursive datatypes, there is a single \isa{case} operator.
+In the tree/forest example, the constant \isa{tree_forest_case} handles all
+of the constructors of the two datatypes.
+
+
+\subsection{Defining datatypes}
+
+The theory syntax for datatype definitions is shown in the
+Isabelle/Isar reference manual. 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.
+
+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.
+
+\subsubsection{Specialized methods for datatypes}
+
+Induction and case-analysis can be invoked using these special-purpose
+methods:
+\begin{ttdescription}
+\item[\methdx{induct_tac} $x$] applies structural
+ induction on variable $x$ to subgoal~1, provided the type of $x$ is a
+ datatype. The induction variable should not occur among other assumptions
+ of the subgoal.
+\end{ttdescription}
+%
+% we also have the ind_cases method, but what does it do?
+In some situations, induction is overkill and a case distinction over all
+constructors of the datatype suffices.
+\begin{ttdescription}
+\item[\methdx{case_tac} $x$]
+ performs a 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 \isa{x \isasymin \ list(A)}. The tactics
+also work for the natural numbers (\isa{nat}) and disjoint sums, although
+these sets were not defined using the datatype package. (Disjoint sums are
+not recursive, so only \isa{case_tac} is available.)
+
+Structured Isar methods are also available. Below, $t$
+stands for the name of the datatype.
+\begin{ttdescription}
+\item[\methdx{induct} \isa{set:}\ $t$] is the Isar induction tactic.
+\item[\methdx{cases} \isa{set:}\ $t$] is the Isar case-analysis tactic.
+\end{ttdescription}
+
+
+\subsubsection{The theorems proved by a datatype declaration}
+
+Here are some more details for the technically minded. Processing the
+datatype declaration of a set~$t$ produces a name space~$t$ containing
+the following theorems:
+\begin{ttbox}\isastyleminor
+intros \textrm{the introduction rules}
+cases \textrm{the case analysis rule}
+induct \textrm{the standard induction rule}
+mutual_induct \textrm{the mutual induction rule, if needed}
+case_eqns \textrm{equations for the case operator}
+recursor_eqns \textrm{equations for the recursor}
+simps \textrm{the union of} case_eqns \textrm{and} recursor_eqns
+con_defs \textrm{definitions of the case operator and constructors}
+free_iffs \textrm{logical equivalences for proving freeness}
+free_elims \textrm{elimination rules for proving freeness}
+defs \textrm{datatype definition(s)}
+\end{ttbox}
+Furthermore there is the theorem $C$ for every constructor~$C$; for
+example, the \isa{list} datatype's introduction rules are bound to the
+identifiers \isa{Nil} and \isa{Cons}.
+
+For a codatatype, the component \isa{coinduct} is the coinduction rule,
+replacing the \isa{induct} component.
+
+See the theories \isa{Induct/Ntree} and \isa{Induct/Brouwer} for examples of
+infinitely branching datatypes. See theory \isa{Induct/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 $\isa{bt}(A)$ of binary trees over~$A$. The theory
+must contain these lines:
+\begin{alltt*}\isastyleminor
+consts bt :: "i=>i"
+datatype "bt(A)" = Lf | Br ("a\isasymin{}A", "t1\isasymin{}bt(A)", "t2\isasymin{}bt(A)")
+\end{alltt*}
+After loading the theory, we can prove some theorem.
+We begin by declaring the constructor's typechecking rules
+as simplification rules:
+\begin{isabelle}
+\isacommand{declare}\ bt.intros\ [simp]%
+\end{isabelle}
+
+Our first example is the theorem that no tree equals its
+left branch. To make the inductive hypothesis strong enough,
+the proof requires a quantified induction formula, but
+the \isa{rule\_format} attribute will remove the quantifiers
+before the theorem is stored.
+\begin{isabelle}
+\isacommand{lemma}\ Br\_neq\_left\ [rule\_format]:\ "l\isasymin bt(A)\ ==>\ \isasymforall x\ r.\ Br(x,l,r)\isasymnoteq{}l"\isanewline
+\ 1.\ l\ \isasymin \ bt(A)\ \isasymLongrightarrow \ \isasymforall x\ r.\ Br(x,\ l,\ r)\ \isasymnoteq \ l%
+\end{isabelle}
+This can be proved by the structural induction tactic:
+\begin{isabelle}
+\ \ \isacommand{apply}\ (induct\_tac\ l)\isanewline
+\ 1.\ \isasymforall x\ r.\ Br(x,\ Lf,\ r)\ \isasymnoteq \ Lf\isanewline
+\ 2.\ \isasymAnd a\ t1\ t2.\isanewline
+\isaindent{\ 2.\ \ \ \ }\isasymlbrakk a\ \isasymin \ A;\ t1\ \isasymin \ bt(A);\ \isasymforall x\ r.\ Br(x,\ t1,\ r)\ \isasymnoteq \ t1;\ t2\ \isasymin \ bt(A);\isanewline
+\isaindent{\ 2.\ \ \ \ \ \ \ }\isasymforall x\ r.\ Br(x,\ t2,\ r)\ \isasymnoteq \ t2\isasymrbrakk \isanewline
+\isaindent{\ 2.\ \ \ \ }\isasymLongrightarrow \ \isasymforall x\ r.\ Br(x,\ Br(a,\ t1,\ t2),\ r)\ \isasymnoteq \ Br(a,\ t1,\ t2)
+\end{isabelle}
+Both subgoals are proved using \isa{auto}, which performs the necessary
+freeness reasoning.
+\begin{isabelle}
+\ \ \isacommand{apply}\ auto\isanewline
+No\ subgoals!\isanewline
+\isacommand{done}
+\end{isabelle}
+
+An alternative proof uses Isar's fancy \isa{induct} method, which
+automatically quantifies over all free variables:
+
+\begin{isabelle}
+\isacommand{lemma}\ Br\_neq\_left':\ "l\ \isasymin \ bt(A)\ ==>\ (!!x\ r.\ Br(x,\ l,\ r)\ \isasymnoteq \ l)"\isanewline
+\ \ \isacommand{apply}\ (induct\ set:\ bt)\isanewline
+\ 1.\ \isasymAnd x\ r.\ Br(x,\ Lf,\ r)\ \isasymnoteq \ Lf\isanewline
+\ 2.\ \isasymAnd a\ t1\ t2\ x\ r.\isanewline
+\isaindent{\ 2.\ \ \ \ }\isasymlbrakk a\ \isasymin \ A;\ t1\ \isasymin \ bt(A);\ \isasymAnd x\ r.\ Br(x,\ t1,\ r)\ \isasymnoteq \ t1;\ t2\ \isasymin \ bt(A);\isanewline
+\isaindent{\ 2.\ \ \ \ \ \ \ }\isasymAnd x\ r.\ Br(x,\ t2,\ r)\ \isasymnoteq \ t2\isasymrbrakk \isanewline
+\isaindent{\ 2.\ \ \ \ }\isasymLongrightarrow \ Br(x,\ Br(a,\ t1,\ t2),\ r)\ \isasymnoteq \ Br(a,\ t1,\ t2)
+\end{isabelle}
+Compare the form of the induction hypotheses with the corresponding ones in
+the previous proof. As before, to conclude requires only \isa{auto}.
+
+When there are only a few constructors, we might prefer to prove the freenness
+theorems for each constructor. This is simple:
+\begin{isabelle}
+\isacommand{lemma}\ Br\_iff:\ "Br(a,l,r)\ =\ Br(a',l',r')\ <->\ a=a'\ \&\ l=l'\ \&\ r=r'"\isanewline
+\ \ \isacommand{by}\ (blast\ elim!:\ bt.free\_elims)
+\end{isabelle}
+Here we see a demonstration of freeness reasoning using
+\isa{bt.free\_elims}, but simpler still is just to apply \isa{auto}.
+
+An \ttindex{inductive\_cases} declaration generates instances of the
+case analysis rule that have been simplified using freeness
+reasoning.
+\begin{isabelle}
+\isacommand{inductive\_cases}\ Br\_in\_bt:\ "Br(a,\ l,\ r)\ \isasymin \ bt(A)"
+\end{isabelle}
+The theorem just created is
+\begin{isabelle}
+\isasymlbrakk Br(a,\ l,\ r)\ \isasymin \ bt(A);\ \isasymlbrakk a\ \isasymin \ A;\ l\ \isasymin \ bt(A);\ r\ \isasymin \ bt(A)\isasymrbrakk \ \isasymLongrightarrow \ Q\isasymrbrakk \ \isasymLongrightarrow \ Q.
+\end{isabelle}
+It is an elimination rule that from $\isa{Br}(a,l,r)\in\isa{bt}(A)$
+lets us infer $a\in A$, $l\in\isa{bt}(A)$ and
+$r\in\isa{bt}(A)$.
+
+
+\subsubsection{Mixfix syntax in datatypes}
+
+Mixfix syntax is sometimes convenient. The theory \isa{Induct/PropLog} makes a
+deep embedding of propositional logic:
+\begin{alltt*}\isastyleminor
+consts prop :: i
+datatype "prop" = Fls
+ | Var ("n \isasymin nat") ("#_" [100] 100)
+ | "=>" ("p \isasymin prop", "q \isasymin prop") (infixr 90)
+\end{alltt*}
+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{alltt*}\isastyleminor
+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{alltt*}
+The datatype package scales well. Even though all properties are proved
+rather than assumed, full processing of this definition takes around two seconds
+(on a 1.8GHz machine). The constructors have a balanced representation,
+related to binary notation, so freeness properties can be proved fast.
+\begin{isabelle}
+\isacommand{lemma}\ "C00 \isasymnoteq\ C01"\isanewline
+\ \ \isacommand{by}\ simp
+\end{isabelle}
+You need not derive such inequalities explicitly. The simplifier will
+dispose of them automatically.
+
+\index{*datatype|)}
+
+
+\subsection{Recursive function definitions}\label{sec:ZF:recursive}
+\index{recursive functions|see{recursion}}
+\index{*primrec|(}
+\index{recursion!primitive|(}
+
+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 axioms. Here is a dangerous way of defining a
+recursive function over binary trees:
+\begin{isabelle}
+\isacommand{consts}\ \ n\_nodes\ ::\ "i\ =>\ i"\isanewline
+\isacommand{axioms}\isanewline
+\ \ n\_nodes\_Lf:\ "n\_nodes(Lf)\ =\ 0"\isanewline
+\ \ n\_nodes\_Br:\ "n\_nodes(Br(a,l,r))\ =\ succ(n\_nodes(l)\ \#+\ n\_nodes(r))"
+\end{isabelle}
+Asserting axioms brings the danger of accidentally introducing
+contradictions. It should be avoided whenever possible.
+
+The \ttindex{primrec} declaration is a safe means of defining primitive
+recursive functions on datatypes:
+\begin{isabelle}
+\isacommand{consts}\ \ n\_nodes\ ::\ "i\ =>\ i"\isanewline
+\isacommand{primrec}\isanewline
+\ \ "n\_nodes(Lf)\ =\ 0"\isanewline
+\ \ "n\_nodes(Br(a,\ l,\ r))\ =\ succ(n\_nodes(l)\ \#+\ n\_nodes(r))"
+\end{isabelle}
+Isabelle will now derive the two equations from a low-level definition
+based upon well-founded recursion. If they do not define a legitimate
+recursion, then Isabelle will reject the declaration.
+
+
+\subsubsection{Syntax of recursive definitions}
+
+The general form of a primitive recursive definition is
+\begin{ttbox}\isastyleminor
+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
+\isa{rules} section of a theory, will be visible in proof scripts.
+
+The reduction rules become part of the default simpset, which
+leads to short proof scripts:
+\begin{isabelle}
+\isacommand{lemma}\ n\_nodes\_type\ [simp]:\ "t\ \isasymin \ bt(A)\ ==>\ n\_nodes(t)\ \isasymin \ nat"\isanewline
+\ \ \isacommand{by}\ (induct\_tac\ t,\ auto)
+\end{isabelle}
+
+You can even use the \isa{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. For example, let us
+define the tail-recursive version of \isa{n\_nodes}, using an
+accumulating argument for the counter. The second argument, $k$, varies in
+recursive calls.
+\begin{isabelle}
+\isacommand{consts}\ \ n\_nodes\_aux\ ::\ "i\ =>\ i"\isanewline
+\isacommand{primrec}\isanewline
+\ \ "n\_nodes\_aux(Lf)\ =\ (\isasymlambda k\ \isasymin \ nat.\ k)"\isanewline
+\ \ "n\_nodes\_aux(Br(a,l,r))\ =\ \isanewline
+\ \ \ \ \ \ (\isasymlambda k\ \isasymin \ nat.\ n\_nodes\_aux(r)\ `\ \ (n\_nodes\_aux(l)\ `\ succ(k)))"
+\end{isabelle}
+Now \isa{n\_nodes\_aux(t)\ `\ k} is our function in two arguments. We
+can prove a theorem relating it to \isa{n\_nodes}. Note the quantification
+over \isa{k\ \isasymin \ nat}:
+\begin{isabelle}
+\isacommand{lemma}\ n\_nodes\_aux\_eq\ [rule\_format]:\isanewline
+\ \ \ \ \ "t\ \isasymin \ bt(A)\ ==>\ \isasymforall k\ \isasymin \ nat.\ n\_nodes\_aux(t)`k\ =\ n\_nodes(t)\ \#+\ k"\isanewline
+\ \ \isacommand{by}\ (induct\_tac\ t,\ simp\_all)
+\end{isabelle}
+
+Now, we can use \isa{n\_nodes\_aux} to define a tail-recursive version
+of \isa{n\_nodes}:
+\begin{isabelle}
+\isacommand{constdefs}\isanewline
+\ \ n\_nodes\_tail\ ::\ "i\ =>\ i"\isanewline
+\ \ \ "n\_nodes\_tail(t)\ ==\ n\_nodes\_aux(t)\ `\ 0"
+\end{isabelle}
+It is easy to
+prove that \isa{n\_nodes\_tail} is equivalent to \isa{n\_nodes}:
+\begin{isabelle}
+\isacommand{lemma}\ "t\ \isasymin \ bt(A)\ ==>\ n\_nodes\_tail(t)\ =\ n\_nodes(t)"\isanewline
+\ \isacommand{by}\ (simp\ add:\ n\_nodes\_tail\_def\ n\_nodes\_aux\_eq)
+\end{isabelle}
+
+
+
+
+\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. It behaves identially to the analogous
+inductive definition except that instead of an induction rule there is
+a coinduction rule. Its treatment of coinduction 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}\isastyleminor
+inductive
+ domains {\it domain declarations}
+ intros {\it introduction rules}
+ monos {\it monotonicity theorems}
+ con_defs {\it constructor definitions}
+ type_intros {\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
+\isa{co\-inductive}.
+
+The \isa{monos}, \isa{con\_defs}, \isa{type\_intros} and \isa{type\_elims}
+sections are optional. If present, each is specified as a list of
+theorems, which may contain Isar attributes as usual.
+
+\begin{description}
+\item[\it domain declarations] are items of the form
+ {\it string\/}~\isa{\isasymsubseteq }~{\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 \isa{Induct/Primrec}.
+
+\item[\it type\_intros] 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 \isa{type\_intros} 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 \isa{mutual\_induct}.
+\end{itemize}
+
+
+\subsection{Example of an inductive definition}
+
+Below, we shall see how Isabelle/ZF defines the finite powerset
+operator. The first step is to declare the constant~\isa{Fin}. Then we
+must declare it inductively, with two introduction rules:
+\begin{isabelle}
+\isacommand{consts}\ \ Fin\ ::\ "i=>i"\isanewline
+\isacommand{inductive}\isanewline
+\ \ \isakeyword{domains}\ \ \ "Fin(A)"\ \isasymsubseteq\ "Pow(A)"\isanewline
+\ \ \isakeyword{intros}\isanewline
+\ \ \ \ emptyI:\ \ "0\ \isasymin\ Fin(A)"\isanewline
+\ \ \ \ consI:\ \ \ "[|\ a\ \isasymin\ A;\ \ b\ \isasymin\ Fin(A)\ |]\ ==>\ cons(a,b)\ \isasymin\ Fin(A)"\isanewline
+\ \ \isakeyword{type\_intros}\ \ empty\_subsetI\ cons\_subsetI\ PowI\isanewline
+\ \ \isakeyword{type\_elims}\ \ \ PowD\ [THEN\ revcut\_rl]\end{isabelle}
+The resulting theory contains a name space, called~\isa{Fin}.
+The \isa{Fin}$~A$ introduction rules can be referred to collectively as
+\isa{Fin.intros}, and also individually as \isa{Fin.emptyI} and
+\isa{Fin.consI}. The induction rule is \isa{Fin.induct}.
+
+The chief problem with making (co)inductive definitions involves type-checking
+the rules. Sometimes, additional theorems need to be supplied under
+\isa{type_intros} or \isa{type_elims}. If the package fails when trying
+to prove your introduction rules, then set the flag \ttindexbold{trace_induct}
+to \isa{true} and try again. (See the manual \emph{A Fixedpoint Approach
+ \ldots} for more discussion of type-checking.)
+
+In the example above, $\isa{Pow}(A)$ is given as the domain of
+$\isa{Fin}(A)$, 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
+\isa{type_intros} and \isa{type_elims} are omitted from the inductive
+definition above:
+\begin{alltt*}\isastyleminor
+Inductive definition Finite.Fin
+Fin(A) ==
+lfp(Pow(A),
+ \%X. {z\isasymin{}Pow(A) . z = 0 | ({\isasymexists}a b. z = cons(a,b) & a\isasymin{}A & b\isasymin{}X)})
+ Proving monotonicity...
+\ttbreak
+ Proving the introduction rules...
+The type-checking subgoal:
+0 \isasymin Fin(A)
+ 1. 0 \isasymin Pow(A)
+\ttbreak
+The subgoal after monos, type_elims:
+0 \isasymin Fin(A)
+ 1. 0 \isasymin Pow(A)
+*** prove_goal: tactic failed
+\end{alltt*}
+We see the need to supply theorems to let the package prove
+$\emptyset\in\isa{Pow}(A)$. Restoring the \isa{type_intros} but not the
+\isa{type_elims}, we again get an error message:
+\begin{alltt*}\isastyleminor
+The type-checking subgoal:
+0 \isasymin Fin(A)
+ 1. 0 \isasymin Pow(A)
+\ttbreak
+The subgoal after monos, type_elims:
+0 \isasymin Fin(A)
+ 1. 0 \isasymin Pow(A)
+\ttbreak
+The type-checking subgoal:
+cons(a, b) \isasymin Fin(A)
+ 1. [| a \isasymin A; b \isasymin Fin(A) |] ==> cons(a, b) \isasymin Pow(A)
+\ttbreak
+The subgoal after monos, type_elims:
+cons(a, b) \isasymin Fin(A)
+ 1. [| a \isasymin A; b \isasymin Pow(A) |] ==> cons(a, b) \isasymin Pow(A)
+*** prove_goal: tactic failed
+\end{alltt*}
+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 \isa{type_intros}. The solution actually used is
+to supply, under \isa{type_elims}, a rule that changes
+$b\in\isa{Pow}(A)$ to $b\subseteq A$; together with \isa{cons_subsetI}
+and \isa{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~\isa{Pow} in the introduction rule and the corresponding mention of the
+rule \isa{Pow\_mono} in the \isa{monos} list. If the desired rule has a
+universally quantified premise, usually the effect can be obtained using
+\isa{Pow}.
+\begin{isabelle}
+\isacommand{consts}\ \ acc\ ::\ "i\ =>\ i"\isanewline
+\isacommand{inductive}\isanewline
+\ \ \isakeyword{domains}\ "acc(r)"\ \isasymsubseteq \ "field(r)"\isanewline
+\ \ \isakeyword{intros}\isanewline
+\ \ \ \ vimage:\ \ "[|\ r-``\isacharbraceleft a\isacharbraceright\ \isasymin\ Pow(acc(r));\ a\ \isasymin \ field(r)\ |]
+\isanewline
+\ \ \ \ \ \ \ \ \ \ \ \ \ \ ==>\ a\ \isasymin \ acc(r)"\isanewline
+\ \ \isakeyword{monos}\ \ Pow\_mono
+\end{isabelle}
+
+Finally, here are some coinductive definitions. We begin by defining
+lazy (potentially infinite) lists as a codatatype:
+\begin{isabelle}
+\isacommand{consts}\ \ llist\ \ ::\ "i=>i"\isanewline
+\isacommand{codatatype}\isanewline
+\ \ "llist(A)"\ =\ LNil\ |\ LCons\ ("a\ \isasymin \ A",\ "l\ \isasymin \ llist(A)")\isanewline
+\end{isabelle}
+
+The notion of equality on such lists is modelled as a bisimulation:
+\begin{isabelle}
+\isacommand{consts}\ \ lleq\ ::\ "i=>i"\isanewline
+\isacommand{coinductive}\isanewline
+\ \ \isakeyword{domains}\ "lleq(A)"\ <=\ "llist(A)\ *\ llist(A)"\isanewline
+\ \ \isakeyword{intros}\isanewline
+\ \ \ \ LNil:\ \ "<LNil,\ LNil>\ \isasymin \ lleq(A)"\isanewline
+\ \ \ \ LCons:\ "[|\ a\ \isasymin \ A;\ <l,l'>\ \isasymin \ lleq(A)\ |]\ \isanewline
+\ \ \ \ \ \ \ \ \ \ \ \ ==>\ <LCons(a,l),\ LCons(a,l')>\ \isasymin \ lleq(A)"\isanewline
+\ \ \isakeyword{type\_intros}\ \ llist.intros
+\end{isabelle}
+This use of \isa{type_intros} is typical: the relation concerns the
+codatatype \isa{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 \isa{ZF/Induct}. The directory
+\isa{Coind} and the theory \isa{ZF/Induct/LList} contain coinductive
+definitions. Larger examples may be found on other subdirectories of
+\isa{ZF}, such as \isa{IMP}, and \isa{Resid}.
+
+
+\subsection{Theorems generated}
+
+Each (co)inductive set defined in a theory file generates a name space
+containing the following elements:
+\begin{ttbox}\isastyleminor
+intros \textrm{the introduction rules}
+elim \textrm{the elimination (case analysis) rule}
+induct \textrm{the standard induction rule}
+mutual_induct \textrm{the mutual induction rule, if needed}
+defs \textrm{definitions of inductive sets}
+bnd_mono \textrm{monotonicity property}
+dom_subset \textrm{inclusion in `bounding set'}
+\end{ttbox}
+Furthermore, each introduction rule is available under its declared
+name. For a codatatype, the component \isa{coinduct} is the coinduction rule,
+replacing the \isa{induct} component.
+
+Recall that the \ttindex{inductive\_cases} declaration generates
+simplified instances of the case analysis rule. It is as useful for
+inductive definitions as it is for datatypes. There are many examples
+in the theory
+\isa{Induct/Comb}, which is discussed at length
+elsewhere~\cite{paulson-generic}. The theory first defines the
+datatype
+\isa{comb} of combinators:
+\begin{alltt*}\isastyleminor
+consts comb :: i
+datatype "comb" = K
+ | S
+ | "#" ("p \isasymin comb", "q \isasymin comb") (infixl 90)
+\end{alltt*}
+The theory goes on to define contraction and parallel contraction
+inductively. Then the theory \isa{Induct/Comb.thy} defines special
+cases of contraction, such as this one:
+\begin{isabelle}
+\isacommand{inductive\_cases}\ K\_contractE [elim!]:\ "K -1-> r"
+\end{isabelle}
+The theorem just created is \isa{K -1-> r \ \isasymLongrightarrow \ Q},
+which expresses that the combinator \isa{K} cannot reduce to
+anything. (From the assumption \isa{K-1->r}, we can conclude any desired
+formula \isa{Q}\@.) Similar elimination rules for \isa{S} and application are also
+generated. The attribute \isa{elim!}\ shown above supplies the generated
+theorem to the classical reasoner. This mode of working allows
+effective reasoniung about operational semantics.
+
+\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 \isa{Trancl} defines the transitive closure of a relation
+ (as a least fixedpoint).
+
+ \item Theory \isa{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 \isa{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 \isa{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 \isa{Rel} defines the basic properties of relations, such as
+ reflexivity, symmetry and transitivity.
+
+\item Theory \isa{EquivClass} develops a theory of equivalence
+ classes, not using the Axiom of Choice.
+
+\item Theory \isa{Order} defines partial orderings, total orderings and
+ wellorderings.
+
+\item Theory \isa{OrderArith} defines orderings on sum and product sets.
+ These can be used to define ordinal arithmetic and have applications to
+ cardinal arithmetic.
+
+\item Theory \isa{OrderType} defines order types. Every wellordering is
+ equivalent to a unique ordinal, which is its order type.
+
+\item Theory \isa{Cardinal} defines equipollence and cardinal numbers.
+
+\item Theory \isa{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 \isa{AC} asserts the Axiom of Choice and proves some simple
+ equivalent forms.
+
+\item Theory \isa{Zorn} proves Hausdorff's Maximal Principle, Zorn's Lemma
+ and the Wellordering Theorem, following Abrial and
+ Laffitte~\cite{abrial93}.
+
+\item Theory \isa{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 \isa{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 \isa{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 \isa{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-fixedpt-milner}.
+\begin{itemize}
+\item File \isa{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 \isa{Ramsey} proves the finite exponent 2 version of
+ Ramsey's Theorem, following Basin and Kaufmann's
+ presentation~\cite{basin91}.
+
+\item Theory \isa{Integ} develops a theory of the integers as
+ equivalence classes of pairs of natural numbers.
+
+\item Theory \isa{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 \isa{Primes} defines the Greatest Common Divisor of two
+ natural numbers and and the ``divides'' relation.
+
+\item Theory \isa{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 0.3 seconds.
+
+\item Theory \isa{BT} defines the recursive data structure $\isa{bt}(A)$, labelled binary trees.
+
+\item Theory \isa{Term} defines a recursive data structure for terms
+ and term lists. These are simply finite branching trees.
+
+\item Theory \isa{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 \isa{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 \isa{ListN} inductively defines the lists of $n$
+ elements~\cite{paulin-tlca}.
+
+\item Theory \isa{Acc} inductively defines the accessible part of a
+ relation~\cite{paulin-tlca}.
+
+\item Theory \isa{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 \isa{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 ${\isa{Pow}(A)\cap \isa{Pow}(B)}= \isa{Pow}(A\cap B)$. 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 \isa{ZF/mono.ML}:
+\begin{isabelle}
+\tdx{Pow_mono}: A \isasymsubseteq B ==> Pow(A) \isasymsubseteq Pow(B)
+\end{isabelle}
+We enter the goal and make the first step, which breaks the equation into
+two inclusions by extensionality:\index{*equalityI theorem}
+\begin{isabelle}
+\isacommand{lemma}\ "Pow(A\ Int\ B)\ =\ Pow(A)\ Int\ Pow(B)"\isanewline
+\ 1.\ Pow(A\ \isasyminter \ B)\ =\ Pow(A)\ \isasyminter \ Pow(B)\isanewline
+\isacommand{apply}\ (rule\ equalityI)\isanewline
+\ 1.\ Pow(A\ \isasyminter \ B)\ \isasymsubseteq \ Pow(A)\ \isasyminter \ Pow(B)\isanewline
+\ 2.\ Pow(A)\ \isasyminter \ Pow(B)\ \isasymsubseteq \ Pow(A\ \isasyminter \ B)
+\end{isabelle}
+Both inclusions could be tackled straightforwardly using \isa{subsetI}.
+A shorter proof results from noting that intersection forms the greatest
+lower bound:\index{*Int_greatest theorem}
+\begin{isabelle}
+\isacommand{apply}\ (rule\ Int\_greatest)\isanewline
+\ 1.\ Pow(A\ \isasyminter \ B)\ \isasymsubseteq \ Pow(A)\isanewline
+\ 2.\ Pow(A\ \isasyminter \ B)\ \isasymsubseteq \ Pow(B)\isanewline
+\ 3.\ Pow(A)\ \isasyminter \ Pow(B)\ \isasymsubseteq \ Pow(A\ \isasyminter \ B)
+\end{isabelle}
+Subgoal~1 follows by applying the monotonicity of \isa{Pow} to $A\int
+B\subseteq A$; subgoal~2 follows similarly:
+\index{*Int_lower1 theorem}\index{*Int_lower2 theorem}
+\begin{isabelle}
+\isacommand{apply}\ (rule\ Int\_lower1\ [THEN\ Pow\_mono])\isanewline
+\ 1.\ Pow(A\ \isasyminter \ B)\ \isasymsubseteq \ Pow(B)\isanewline
+\ 2.\ Pow(A)\ \isasyminter \ Pow(B)\ \isasymsubseteq \ Pow(A\ \isasyminter \ B)
+\isanewline
+\isacommand{apply}\ (rule\ Int\_lower2\ [THEN\ Pow\_mono])\isanewline
+\ 1.\ Pow(A)\ \isasyminter \ Pow(B)\ \isasymsubseteq \ Pow(A\ \isasyminter \ B)
+\end{isabelle}
+We are left with the opposite inclusion, which we tackle in the
+straightforward way:\index{*subsetI theorem}
+\begin{isabelle}
+\isacommand{apply}\ (rule\ subsetI)\isanewline
+\ 1.\ \isasymAnd x.\ x\ \isasymin \ Pow(A)\ \isasyminter \ Pow(B)\ \isasymLongrightarrow \ x\ \isasymin \ Pow(A\ \isasyminter \ B)
+\end{isabelle}
+The subgoal is to show $x\in \isa{Pow}(A\cap B)$ assuming $x\in\isa{Pow}(A)\cap \isa{Pow}(B)$; eliminating this assumption produces two
+subgoals. The rule \tdx{IntE} treats the intersection like a conjunction
+instead of unfolding its definition.
+\begin{isabelle}
+\isacommand{apply}\ (erule\ IntE)\isanewline
+\ 1.\ \isasymAnd x.\ \isasymlbrakk x\ \isasymin \ Pow(A);\ x\ \isasymin \ Pow(B)\isasymrbrakk \ \isasymLongrightarrow \ x\ \isasymin \ Pow(A\ \isasyminter \ B)
+\end{isabelle}
+The next step replaces the \isa{Pow} by the subset
+relation~($\subseteq$).\index{*PowI theorem}
+\begin{isabelle}
+\isacommand{apply}\ (rule\ PowI)\isanewline
+\ 1.\ \isasymAnd x.\ \isasymlbrakk x\ \isasymin \ Pow(A);\ x\ \isasymin \ Pow(B)\isasymrbrakk \ \isasymLongrightarrow \ x\ \isasymsubseteq \ A\ \isasyminter \ B%
+\end{isabelle}
+We perform the same replacement in the assumptions. This is a good
+demonstration of the tactic \ttindex{drule}:\index{*PowD theorem}
+\begin{isabelle}
+\isacommand{apply}\ (drule\ PowD)+\isanewline
+\ 1.\ \isasymAnd x.\ \isasymlbrakk x\ \isasymsubseteq \ A;\ x\ \isasymsubseteq \ B\isasymrbrakk \ \isasymLongrightarrow \ x\ \isasymsubseteq \ A\ \isasyminter \ B%
+\end{isabelle}
+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{isabelle}
+\isacommand{apply}\ (rule\ Int\_greatest)\isanewline
+\ 1.\ \isasymAnd x.\ \isasymlbrakk x\ \isasymsubseteq \ A;\ x\ \isasymsubseteq \ B\isasymrbrakk \ \isasymLongrightarrow \ x\ \isasymsubseteq \ A\isanewline
+\ 2.\ \isasymAnd x.\ \isasymlbrakk x\ \isasymsubseteq \ A;\ x\ \isasymsubseteq \ B\isasymrbrakk \ \isasymLongrightarrow \ x\ \isasymsubseteq \ B%
+\end{isabelle}
+To conclude the proof, we clear up the trivial subgoals:
+\begin{isabelle}
+\isacommand{apply}\ (assumption+)\isanewline
+\isacommand{done}%
+\end{isabelle}
+
+We could have performed this proof instantly by calling
+\ttindex{blast}:
+\begin{isabelle}
+\isacommand{lemma}\ "Pow(A\ Int\ B)\ =\ Pow(A)\ Int\ Pow(B)"\isanewline
+\isacommand{by}
+\end{isabelle}
+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{isabelle}
+\isacommand{lemma}\ "C\isasymsubseteq D\ ==>\ Union(C)\
+\isasymsubseteq \ Union(D)"\isanewline
+\isacommand{apply}\ (rule\ subsetI)\isanewline
+\ 1.\ \isasymAnd x.\ \isasymlbrakk C\ \isasymsubseteq \ D;\ x\ \isasymin \ \isasymUnion C\isasymrbrakk \ \isasymLongrightarrow \ x\ \isasymin \ \isasymUnion D%
+\end{isabelle}
+Big union is like an existential quantifier --- the occurrence in the
+assumptions must be eliminated early, since it creates parameters.
+\index{*UnionE theorem}
+\begin{isabelle}
+\isacommand{apply}\ (erule\ UnionE)\isanewline
+\ 1.\ \isasymAnd x\ B.\ \isasymlbrakk C\ \isasymsubseteq \ D;\ x\ \isasymin \ B;\ B\ \isasymin \ C\isasymrbrakk \ \isasymLongrightarrow \ x\ \isasymin \ \isasymUnion D%
+\end{isabelle}
+Now we may apply \tdx{UnionI}, which creates an unknown involving the
+parameters. To show \isa{x\ \isasymin \ \isasymUnion D} it suffices to show that~\isa{x} belongs
+to some element, say~\isa{?B2(x,B)}, of~\isa{D}\@.
+\begin{isabelle}
+\isacommand{apply}\ (rule\ UnionI)\isanewline
+\ 1.\ \isasymAnd x\ B.\ \isasymlbrakk C\ \isasymsubseteq \ D;\ x\ \isasymin \ B;\ B\ \isasymin \ C\isasymrbrakk \ \isasymLongrightarrow \ ?B2(x,\ B)\ \isasymin \ D\isanewline
+\ 2.\ \isasymAnd x\ B.\ \isasymlbrakk C\ \isasymsubseteq \ D;\ x\ \isasymin \ B;\ B\ \isasymin \ C\isasymrbrakk \ \isasymLongrightarrow \ x\ \isasymin \ ?B2(x,\ B)
+\end{isabelle}
+Combining the rule \tdx{subsetD} with the assumption \isa{C\ \isasymsubseteq \ D} yields
+$\Var{a}\in C \Imp \Var{a}\in D$, which reduces subgoal~1. Note that
+\isa{erule} removes the subset assumption.
+\begin{isabelle}
+\isacommand{apply}\ (erule\ subsetD)\isanewline
+\ 1.\ \isasymAnd x\ B.\ \isasymlbrakk x\ \isasymin \ B;\ B\ \isasymin \ C\isasymrbrakk \ \isasymLongrightarrow \ ?B2(x,\ B)\ \isasymin \ C\isanewline
+\ 2.\ \isasymAnd x\ B.\ \isasymlbrakk C\ \isasymsubseteq \ D;\ x\ \isasymin \ B;\ B\ \isasymin \ C\isasymrbrakk \ \isasymLongrightarrow \ x\ \isasymin \ ?B2(x,\ B)
+\end{isabelle}
+The rest is routine. Observe how the first call to \isa{assumption}
+instantiates \isa{?B2(x,B)} to~\isa{B}\@.
+\begin{isabelle}
+\isacommand{apply}\ assumption\ \isanewline
+\ 1.\ \isasymAnd x\ B.\ \isasymlbrakk C\ \isasymsubseteq \ D;\ x\ \isasymin \ B;\ B\ \isasymin \ C\isasymrbrakk \ \isasymLongrightarrow \ x\ \isasymin \ B%
+\isanewline
+\isacommand{apply}\ assumption\ \isanewline
+No\ subgoals!\isanewline
+\isacommand{done}%
+\end{isabelle}
+Again, \isa{blast} can prove this theorem in one step.
+
+The theory \isa{ZF/equalities.thy} has many similar proofs. Reasoning about
+general intersection can be difficult because of its anomalous behaviour on
+the empty set. However, \isa{blast} copes well with these. Here is
+a typical example, borrowed from Devlin~\cite[page 12]{devlin79}:
+\[ a\in C \,\Imp\, \inter@{x\in C} \Bigl(A(x) \int B(x)\Bigr) =
+ \Bigl(\inter@{x\in C} A(x)\Bigr) \int
+ \Bigl(\inter@{x\in C} B(x)\Bigr) \]
+
+\section{Low-level reasoning about functions}
+The derived rules \isa{lamI}, \isa{lamE}, \isa{lam_type}, \isa{beta}
+and \isa{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{isabelle}
+\isacommand{lemma}\ "[|\ a\ \isasymin \ A;\ \ f\ \isasymin \ A->B;\ \ g\ \isasymin \ C->D;\ \ A\ \isasyminter \ C\ =\ 0\ |]
+\isanewline
+\ \ \ \ \ \ \ \ ==>\ (f\ \isasymunion \ g)`a\ =\ f`a"
+\end{isabelle}
+Using \tdx{apply_equality}, we reduce the equality to reasoning about
+ordered pairs. The second subgoal is to verify that \isa{f\ \isasymunion \ g} is a function, since
+\isa{Pi(?A,?B)} denotes a dependent function space.
+\begin{isabelle}
+\isacommand{apply}\ (rule\ apply\_equality)\isanewline
+\ 1.\ \isasymlbrakk a\ \isasymin \ A;\ f\ \isasymin \ A\ \isasymrightarrow \ B;\ g\ \isasymin \ C\ \isasymrightarrow \ D;\ A\ \isasyminter \ C\ =\ 0\isasymrbrakk \isanewline
+\isaindent{\ 1.\ }\isasymLongrightarrow \ \isasymlangle a,\ f\ `\ a\isasymrangle \ \isasymin \ f\ \isasymunion \ g\isanewline
+\ 2.\ \isasymlbrakk a\ \isasymin \ A;\ f\ \isasymin \ A\ \isasymrightarrow \ B;\ g\ \isasymin \ C\ \isasymrightarrow \ D;\ A\ \isasyminter \ C\ =\ 0\isasymrbrakk \isanewline
+\isaindent{\ 2.\ }\isasymLongrightarrow \ f\ \isasymunion \ g\ \isasymin \ Pi(?A,\ ?B)
+\end{isabelle}
+We must show that the pair belongs to~$f$ or~$g$; by~\tdx{UnI1} we
+choose~$f$:
+\begin{isabelle}
+\isacommand{apply}\ (rule\ UnI1)\isanewline
+\ 1.\ \isasymlbrakk a\ \isasymin \ A;\ f\ \isasymin \ A\ \isasymrightarrow \ B;\ g\ \isasymin \ C\ \isasymrightarrow \ D;\ A\ \isasyminter \ C\ =\ 0\isasymrbrakk \ \isasymLongrightarrow \ \isasymlangle a,\ f\ `\ a\isasymrangle \ \isasymin \ f\isanewline
+\ 2.\ \isasymlbrakk a\ \isasymin \ A;\ f\ \isasymin \ A\ \isasymrightarrow \ B;\ g\ \isasymin \ C\ \isasymrightarrow \ D;\ A\ \isasyminter \ C\ =\ 0\isasymrbrakk \isanewline
+\isaindent{\ 2.\ }\isasymLongrightarrow \ f\ \isasymunion \ g\ \isasymin \ Pi(?A,\ ?B)
+\end{isabelle}
+To show $\pair{a,f`a}\in f$ we use \tdx{apply_Pair}, which is
+essentially the converse of \tdx{apply_equality}:
+\begin{isabelle}
+\isacommand{apply}\ (rule\ apply\_Pair)\isanewline
+\ 1.\ \isasymlbrakk a\ \isasymin \ A;\ f\ \isasymin \ A\ \isasymrightarrow \ B;\ g\ \isasymin \ C\ \isasymrightarrow \ D;\ A\ \isasyminter \ C\ =\ 0\isasymrbrakk \ \isasymLongrightarrow \ f\ \isasymin \ Pi(?A2,?B2)\isanewline
+\ 2.\ \isasymlbrakk a\ \isasymin \ A;\ f\ \isasymin \ A\ \isasymrightarrow \ B;\ g\ \isasymin \ C\ \isasymrightarrow \ D;\ A\ \isasyminter \ C\ =\ 0\isasymrbrakk \ \isasymLongrightarrow \ a\ \isasymin \ ?A2\isanewline
+\ 3.\ \isasymlbrakk a\ \isasymin \ A;\ f\ \isasymin \ A\ \isasymrightarrow \ B;\ g\ \isasymin \ C\ \isasymrightarrow \ D;\ A\ \isasyminter \ C\ =\ 0\isasymrbrakk \isanewline
+\isaindent{\ 3.\ }\isasymLongrightarrow \ f\ \isasymunion \ g\ \isasymin \ Pi(?A,\ ?B)
+\end{isabelle}
+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} gets instantiated to~\isa{A}.
+\begin{isabelle}
+\isacommand{apply}\ assumption\ \isanewline
+\ 1.\ \isasymlbrakk a\ \isasymin \ A;\ f\ \isasymin \ A\ \isasymrightarrow \ B;\ g\ \isasymin \ C\ \isasymrightarrow \ D;\ A\ \isasyminter \ C\ =\ 0\isasymrbrakk \ \isasymLongrightarrow \ a\ \isasymin \ A\isanewline
+\ 2.\ \isasymlbrakk a\ \isasymin \ A;\ f\ \isasymin \ A\ \isasymrightarrow \ B;\ g\ \isasymin \ C\ \isasymrightarrow \ D;\ A\ \isasyminter \ C\ =\ 0\isasymrbrakk \isanewline
+\isaindent{\ 2.\ }\isasymLongrightarrow \ f\ \isasymunion \ g\ \isasymin \ Pi(?A,\ ?B)
+\isanewline
+\isacommand{apply}\ assumption\ \isanewline
+\ 1.\ \isasymlbrakk a\ \isasymin \ A;\ f\ \isasymin \ A\ \isasymrightarrow \ B;\ g\ \isasymin \ C\ \isasymrightarrow \ D;\ A\ \isasyminter \ C\ =\ 0\isasymrbrakk \isanewline
+\isaindent{\ 1.\ }\isasymLongrightarrow \ f\ \isasymunion \ g\ \isasymin \ Pi(?A,\ ?B)
+\end{isabelle}
+To construct functions of the form $f\un g$, we apply
+\tdx{fun_disjoint_Un}:
+\begin{isabelle}
+\isacommand{apply}\ (rule\ fun\_disjoint\_Un)\isanewline
+\ 1.\ \isasymlbrakk a\ \isasymin \ A;\ f\ \isasymin \ A\ \isasymrightarrow \ B;\ g\ \isasymin \ C\ \isasymrightarrow \ D;\ A\ \isasyminter \ C\ =\ 0\isasymrbrakk \ \isasymLongrightarrow \ f\ \isasymin \ ?A3\ \isasymrightarrow \ ?B3\isanewline
+\ 2.\ \isasymlbrakk a\ \isasymin \ A;\ f\ \isasymin \ A\ \isasymrightarrow \ B;\ g\ \isasymin \ C\ \isasymrightarrow \ D;\ A\ \isasyminter \ C\ =\ 0\isasymrbrakk \ \isasymLongrightarrow \ g\ \isasymin \ ?C3\ \isasymrightarrow \ ?D3\isanewline
+\ 3.\ \isasymlbrakk a\ \isasymin \ A;\ f\ \isasymin \ A\ \isasymrightarrow \ B;\ g\ \isasymin \ C\ \isasymrightarrow \ D;\ A\ \isasyminter \ C\ =\ 0\isasymrbrakk \ \isasymLongrightarrow \ ?A3\ \isasyminter \ ?C3\ =\ 0
+\end{isabelle}
+The remaining subgoals are instances of the assumptions. Again, observe how
+unknowns become instantiated:
+\begin{isabelle}
+\isacommand{apply}\ assumption\ \isanewline
+\ 1.\ \isasymlbrakk a\ \isasymin \ A;\ f\ \isasymin \ A\ \isasymrightarrow \ B;\ g\ \isasymin \ C\ \isasymrightarrow \ D;\ A\ \isasyminter \ C\ =\ 0\isasymrbrakk \ \isasymLongrightarrow \ g\ \isasymin \ ?C3\ \isasymrightarrow \ ?D3\isanewline
+\ 2.\ \isasymlbrakk a\ \isasymin \ A;\ f\ \isasymin \ A\ \isasymrightarrow \ B;\ g\ \isasymin \ C\ \isasymrightarrow \ D;\ A\ \isasyminter \ C\ =\ 0\isasymrbrakk \ \isasymLongrightarrow \ A\ \isasyminter \ ?C3\ =\ 0
+\isanewline
+\isacommand{apply}\ assumption\ \isanewline
+\ 1.\ \isasymlbrakk a\ \isasymin \ A;\ f\ \isasymin \ A\ \isasymrightarrow \ B;\ g\ \isasymin \ C\ \isasymrightarrow \ D;\ A\ \isasyminter \ C\ =\ 0\isasymrbrakk \ \isasymLongrightarrow \ A\ \isasyminter \ C\ =\ 0
+\isanewline
+\isacommand{apply}\ assumption\ \isanewline
+No\ subgoals!\isanewline
+\isacommand{done}
+\end{isabelle}
+See the theories \isa{ZF/func.thy} and \isa{ZF/WF.thy} for more
+examples of reasoning about functions.
+
+\index{set theory|)}