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+%% $Id$
+%%%See grant/bra/Lib/ZF.tex for lfp figure
+\chapter{Zermelo-Fraenkel set theory}
+The directory~\ttindexbold{ZF} implements Zermelo-Fraenkel set
+theory~\cite{halmos60,suppes72} as an extension of~\ttindex{FOL}, classical
+first-order logic. The theory includes a collection of derived natural
+deduction rules, for use with Isabelle's classical reasoning module. Much
+of it is based on the work of No\"el~\cite{noel}. The theory has the {\ML}
+identifier \ttindexbold{ZF.thy}. However, many further theories
+are defined, introducing the natural numbers, etc.
+
+A tremendous amount of set theory has been formally developed, including
+the basic properties of relations, functions and ordinals. Significant
+results have been proved, such as the Schr\"oder-Bernstein Theorem and the
+Recursion Theorem. General methods have been developed for solving
+recursion equations over monotonic functors; these have been applied to
+yield constructions of lists and trees. Thus, we may even regard set
+theory as a computational logic. It admits recursive definitions of
+functions and types. It has similarities with Martin-L\"of type theory,
+although of course it is classical.
+
+Because {\ZF} is an extension of {\FOL}, it provides the same packages,
+namely \ttindex{hyp_subst_tac}, the simplifier, and the classical reasoning
+module. The main simplification set is called \ttindexbold{ZF_ss}.
+Several classical rule sets are defined, including \ttindexbold{lemmas_cs},
+\ttindexbold{upair_cs} and~\ttindexbold{ZF_cs}. See the files on directory
+{\tt ZF} for details.
+
+
+\section{Which version of axiomatic set theory?}
+Resolution theorem provers can work in set theory, using the
+Bernays-G\"odel axiom system~(BG) because it is
+finite~\cite{boyer86,quaife92}. {\ZF} does not have a finite axiom system
+(because of its Axiom Scheme of Replacement) and is therefore unsuitable
+for classical resolution. 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 BG, both classes
+and sets are individuals; $x\in V$ expresses that $x$ is a set. In {\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 {\ZF} because they are mainly concerned
+with sets, rather than classes. BG requires tiresome proofs that various
+collections are sets; for instance, showing $x\in\{x\}$ requires showing that
+$x$ is a set. {\ZF} does not have this problem.
+
+
+\begin{figure}
+\begin{center}
+\begin{tabular}{rrr}
+ \it name &\it meta-type & \it description \\
+ \idx{0} & $i$ & empty set\\
+ \idx{cons} & $[i,i]\To i$ & finite set constructor\\
+ \idx{Upair} & $[i,i]\To i$ & unordered pairing\\
+ \idx{Pair} & $[i,i]\To i$ & ordered pairing\\
+ \idx{Inf} & $i$ & infinite set\\
+ \idx{Pow} & $i\To i$ & powerset\\
+ \idx{Union} \idx{Inter} & $i\To i$ & set union/intersection \\
+ \idx{split} & $[i, [i,i]\To i] \To i$ & generalized projection\\
+ \idx{fst} \idx{snd} & $i\To i$ & projections\\
+ \idx{converse}& $i\To i$ & converse of a relation\\
+ \idx{succ} & $i\To i$ & successor\\
+ \idx{Collect} & $[i,i\To o]\To i$ & separation\\
+ \idx{Replace} & $[i, [i,i]\To o] \To i$ & replacement\\
+ \idx{PrimReplace} & $[i, [i,i]\To o] \To i$ & primitive replacement\\
+ \idx{RepFun} & $[i, i\To i] \To i$ & functional replacement\\
+ \idx{Pi} \idx{Sigma} & $[i,i\To i]\To i$ & general product/sum\\
+ \idx{domain} & $i\To i$ & domain of a relation\\
+ \idx{range} & $i\To i$ & range of a relation\\
+ \idx{field} & $i\To i$ & field of a relation\\
+ \idx{Lambda} & $[i, i\To i]\To i$ & $\lambda$-abstraction\\
+ \idx{restrict}& $[i, i] \To i$ & restriction of a function\\
+ \idx{The} & $[i\To o]\To i$ & definite description\\
+ \idx{if} & $[o,i,i]\To i$ & conditional\\
+ \idx{Ball} \idx{Bex} & $[i, i\To o]\To o$ & bounded quantifiers
+\end{tabular}
+\end{center}
+\subcaption{Constants}
+
+\begin{center}
+\indexbold{*"`"`}
+\indexbold{*"-"`"`}
+\indexbold{*"`}
+\indexbold{*"-}
+\indexbold{*":}
+\indexbold{*"<"=}
+\begin{tabular}{rrrr}
+ \it symbol & \it meta-type & \it precedence & \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 \\
+ \idx{Int} & $[i,i]\To i$ & Left 70 & intersection ($\inter$) \\
+ \idx{Un} & $[i,i]\To i$ & Left 65 & union ($\union$) \\
+ \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
+{\tt 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.
+
+Set theory does not use polymorphism. All terms in {\ZF} have type~{\it
+i}, which is the type of individuals and lies in class {\it logic}.
+The type of first-order formulae,
+remember, is~{\it o}.
+
+Infix operators include union and intersection ($A\union B$ and $A\inter
+B$), and the subset and membership relations. Note that $a$\verb|~:|$b$ is
+translated to \verb|~(|$a$:$b$\verb|)|. 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 \ttindexbold{Upair} constructs unordered pairs; thus {\tt
+Upair($A$,$B$)} denotes the set~$\{A,B\}$ and {\tt Upair($A$,$A$)} denotes
+the singleton~$\{A\}$. As usual in {\ZF}, general union is used to define
+binary union. The Isabelle version goes on to define the constant
+\ttindexbold{cons}:
+\begin{eqnarray*}
+ A\cup B & \equiv & \bigcup({\tt Upair}(A,B)) \\
+ {\tt cons}(a,B) & \equiv & {\tt Upair}(a,a) \union B
+\end{eqnarray*}
+The {\tt\{\ldots\}} notation abbreviates finite sets constructed in the
+obvious manner using~{\tt cons} and~$\emptyset$ (the empty set):
+\begin{eqnarray*}
+ \{a,b,c\} & \equiv & {\tt cons}(a,{\tt cons}(b,{\tt cons}(c,\emptyset)))
+\end{eqnarray*}
+
+The constant \ttindexbold{Pair} constructs ordered pairs, as in {\tt
+Pair($a$,$b$)}. Ordered pairs may also be written within angle brackets,
+as {\tt<$a$,$b$>}.
+
+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}
+\indexbold{*"-">}
+\indexbold{*"*}
+\begin{center} \tt\frenchspacing
+\begin{tabular}{rrr}
+ \it external & \it internal & \it description \\
+ $a$ \ttilde: $b$ & \ttilde($a$ : $b$) & \rm negated membership\\
+ \{$a@1$, $\ldots$, $a@n$\} & cons($a@1$,$\cdots$,cons($a@n$,0)) &
+ \rm finite set \\
+ <$a$, $b$> & Pair($a$,$b$) & \rm ordered pair \\
+ <$a$, $b$, $c$> & <$a$, <$b$, $c$>> & \rm nested pairs (any depth) \\
+ \{$x$:$A . P[x]$\} & Collect($A$,$\lambda x.P[x]$) &
+ \rm separation \\
+ \{$y . x$:$A$, $Q[x,y]$\} & Replace($A$,$\lambda x\,y.Q[x,y]$) &
+ \rm replacement \\
+ \{$b[x] . x$:$A$\} & RepFun($A$,$\lambda x.b[x]$) &
+ \rm functional replacement \\
+ \idx{INT} $x$:$A . B[x]$ & Inter(\{$B[x] . x$:$A$\}) &
+ \rm general intersection \\
+ \idx{UN} $x$:$A . B[x]$ & Union(\{$B[x] . x$:$A$\}) &
+ \rm general union \\
+ \idx{PROD} $x$:$A . B[x]$ & Pi($A$,$\lambda x.B[x]$) &
+ \rm general product \\
+ \idx{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 \\
+ \idx{THE} $x . P[x]$ & The($\lambda x.P[x]$) &
+ \rm definite description \\
+ \idx{lam} $x$:$A . b[x]$ & Lambda($A$,$\lambda x.b[x]$) &
+ \rm $\lambda$-abstraction\\[1ex]
+ \idx{ALL} $x$:$A . P[x]$ & Ball($A$,$\lambda x.P[x]$) &
+ \rm bounded $\forall$ \\
+ \idx{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}
+\dquotes
+\[\begin{array}{rcl}
+ term & = & \hbox{expression of type~$i$} \\
+ & | & "\{ " term\; ("," term)^* " \}" \\
+ & | & "< " term ", " term " >" \\
+ & | & "\{ " id ":" term " . " formula " \}" \\
+ & | & "\{ " id " . " id ":" term "," formula " \}" \\
+ & | & "\{ " term " . " id ":" term " \}" \\
+ & | & term " `` " term \\
+ & | & term " -`` " term \\
+ & | & term " ` " term \\
+ & | & term " * " term \\
+ & | & term " Int " term \\
+ & | & term " Un " term \\
+ & | & term " - " term \\
+ & | & term " -> " term \\
+ & | & "THE~~" id " . " formula\\
+ & | & "lam~~" id ":" term " . " term \\
+ & | & "INT~~" id ":" term " . " term \\
+ & | & "UN~~~" id ":" term " . " term \\
+ & | & "PROD~" id ":" term " . " term \\
+ & | & "SUM~~" id ":" term " . " term \\[2ex]
+ formula & = & \hbox{expression of type~$o$} \\
+ & | & term " : " term \\
+ & | & term " <= " term \\
+ & | & term " = " term \\
+ & | & "\ttilde\ " formula \\
+ & | & formula " \& " formula \\
+ & | & formula " | " formula \\
+ & | & formula " --> " formula \\
+ & | & formula " <-> " formula \\
+ & | & "ALL " id ":" term " . " formula \\
+ & | & "EX~~" id ":" term " . " formula \\
+ & | & "ALL~" id~id^* " . " formula \\
+ & | & "EX~~" id~id^* " . " formula \\
+ & | & "EX!~" id~id^* " . " formula
+ \end{array}
+\]
+\caption{Full grammar for {\ZF}} \label{ZF-syntax}
+\end{figure}
+
+
+\section{Binding operators}
+The constant \ttindexbold{Collect} constructs sets by the principle of {\bf
+ separation}. The syntax for separation is \hbox{\tt\{$x$:$A$.$P[x]$\}},
+where $P[x]$ is a formula that may contain free occurrences of~$x$. It
+abbreviates the set {\tt Collect($A$,$\lambda x.P$[x])}, which consists of
+all $x\in A$ that satisfy~$P[x]$. Note that {\tt Collect} is an
+unfortunate choice of name: some set theories adopt a set-formation
+principle, related to replacement, called collection.
+
+The constant \ttindexbold{Replace} constructs sets by the principle of {\bf
+ replacement}. The syntax for replacement is
+\hbox{\tt\{$y$.$x$:$A$,$Q[x,y]$\}}. It denotes the set {\tt
+ Replace($A$,$\lambda x\,y.Q$[x,y])} consisting 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 \ttindexbold{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 syntax is \hbox{\tt\{$b[x]$.$x$:$A$\}},
+denoting set {\tt RepFun($A$,$\lambda x.b[x]$)} of all $b[x]$ for~$x\in A$.
+This is analogous to the \ML{} functional {\tt map}, since it applies a
+function to every element of a set.
+
+\indexbold{*INT}\indexbold{*UN}
+General unions and intersections of families, namely $\bigcup@{x\in A}B[x]$ and
+$\bigcap@{x\in A}B[x]$, are written \hbox{\tt UN $x$:$A$.$B[x]$} and
+\hbox{\tt INT $x$:$A$.$B[x]$}. Their meaning is expressed using {\tt
+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~\ttindexbold{Sigma} and~\ttindexbold{Pi} construct general sums and
+products. Instead of {\tt Sigma($A$,$B$)} and {\tt Pi($A$,$B$)} we may write
+\hbox{\tt SUM $x$:$A$.$B[x]$} and \hbox{\tt PROD $x$:$A$.$B[x]$}.
+\indexbold{*SUM}\indexbold{*PROD}%
+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~{\tt op~*} and~\hbox{\tt op~->}.} Isabelle accepts these
+abbreviations in parsing and uses them whenever possible for printing.
+
+\indexbold{*THE}
+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~\ttindexbold{The}, we may write descriptions as {\tt
+ The($\lambda x.P[x]$)} or use the syntax \hbox{\tt THE $x$.$P[x]$}.
+
+\indexbold{*lam}
+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~\ttindexbold{Lambda}, we may express function sets as {\tt
+Lambda($A$,$\lambda x.b[x]$)} or use the syntax \hbox{\tt lam $x$:$A$.$b[x]$}.
+
+Isabelle's set theory defines two {\bf bounded quantifiers}:
+\begin{eqnarray*}
+ \forall x\in A.P[x] &\hbox{which abbreviates}& \forall x. x\in A\imp P[x] \\
+ \exists x\in A.P[x] &\hbox{which abbreviates}& \exists x. x\in A\conj P[x]
+\end{eqnarray*}
+The constants~\ttindexbold{Ball} and~\ttindexbold{Bex} are defined
+accordingly. Instead of {\tt Ball($A$,$P$)} and {\tt Bex($A$,$P$)} we may
+write
+\hbox{\tt ALL $x$:$A$.$P[x]$} and \hbox{\tt EX $x$:$A$.$P[x]$}.
+
+
+%%%% zf.thy
+
+\begin{figure}
+\begin{ttbox}
+\idx{Ball_def} Ball(A,P) == ALL x. x:A --> P(x)
+\idx{Bex_def} Bex(A,P) == EX x. x:A & P(x)
+
+\idx{subset_def} A <= B == ALL x:A. x:B
+\idx{extension} A = B <-> A <= B & B <= A
+
+\idx{union_iff} A : Union(C) <-> (EX B:C. A:B)
+\idx{power_set} A : Pow(B) <-> A <= B
+\idx{foundation} A=0 | (EX x:A. ALL y:x. ~ y:A)
+
+\idx{replacement} (ALL x:A. ALL y z. P(x,y) & P(x,z) --> y=z) ==>
+ b : PrimReplace(A,P) <-> (EX x:A. P(x,b))
+\subcaption{The Zermelo-Fraenkel Axioms}
+
+\idx{Replace_def} Replace(A,P) ==
+ PrimReplace(A, %x y. (EX!z.P(x,z)) & P(x,y))
+\idx{RepFun_def} RepFun(A,f) == \{y . x:A, y=f(x)\}
+\idx{the_def} The(P) == Union(\{y . x:\{0\}, P(y)\})
+\idx{if_def} if(P,a,b) == THE z. P & z=a | ~P & z=b
+\idx{Collect_def} Collect(A,P) == \{y . x:A, x=y & P(x)\}
+\idx{Upair_def} Upair(a,b) ==
+ \{y. x:Pow(Pow(0)), (x=0 & y=a) | (x=Pow(0) & y=b)\}
+\subcaption{Consequences of replacement}
+
+\idx{Inter_def} Inter(A) == \{ x:Union(A) . ALL y:A. x:y\}
+\idx{Un_def} A Un B == Union(Upair(A,B))
+\idx{Int_def} A Int B == Inter(Upair(A,B))
+\idx{Diff_def} A - B == \{ x:A . ~(x:B) \}
+\subcaption{Union, intersection, difference}
+
+\idx{cons_def} cons(a,A) == Upair(a,a) Un A
+\idx{succ_def} succ(i) == cons(i,i)
+\idx{infinity} 0:Inf & (ALL y:Inf. succ(y): Inf)
+\subcaption{Finite and infinite sets}
+\end{ttbox}
+\caption{Rules and axioms of {\ZF}} \label{ZF-rules}
+\end{figure}
+
+
+\begin{figure}
+\begin{ttbox}
+\idx{Pair_def} == \{\{a,a\}, \{a,b\}\}
+\idx{split_def} split(p,c) == THE y. EX a b. p= & y=c(a,b)
+\idx{fst_def} fst(A) == split(p, %x y.x)
+\idx{snd_def} snd(A) == split(p, %x y.y)
+\idx{Sigma_def} Sigma(A,B) == UN x:A. UN y:B(x). \{\}
+\subcaption{Ordered pairs and Cartesian products}
+
+\idx{converse_def} converse(r) == \{z. w:r, EX x y. w= & z=\}
+\idx{domain_def} domain(r) == \{x. w:r, EX y. w=\}
+\idx{range_def} range(r) == domain(converse(r))
+\idx{field_def} field(r) == domain(r) Un range(r)
+\idx{image_def} r `` A == \{y : range(r) . EX x:A. : r\}
+\idx{vimage_def} r -`` A == converse(r)``A
+\subcaption{Operations on relations}
+
+\idx{lam_def} Lambda(A,b) == \{ . x:A\}
+\idx{apply_def} f`a == THE y. : f
+\idx{Pi_def} Pi(A,B) == \{f: Pow(Sigma(A,B)). ALL x:A. EX! y. : f\}
+\idx{restrict_def} restrict(f,A) == lam x:A.f`x
+\subcaption{Functions and general product}
+\end{ttbox}
+\caption{Further definitions of {\ZF}} \label{ZF-defs}
+\end{figure}
+
+
+%%%% zf.ML
+
+\begin{figure}
+\begin{ttbox}
+\idx{ballI} [| !!x. x:A ==> P(x) |] ==> ALL x:A. P(x)
+\idx{bspec} [| ALL x:A. P(x); x: A |] ==> P(x)
+\idx{ballE} [| ALL x:A. P(x); P(x) ==> Q; ~ x:A ==> Q |] ==> Q
+
+\idx{ball_cong} [| A=A'; !!x. x:A' ==> P(x) <-> P'(x) |] ==>
+ (ALL x:A. P(x)) <-> (ALL x:A'. P'(x))
+
+\idx{bexI} [| P(x); x: A |] ==> EX x:A. P(x)
+\idx{bexCI} [| ALL x:A. ~P(x) ==> P(a); a: A |] ==> EX x:A.P(x)
+\idx{bexE} [| EX x:A. P(x); !!x. [| x:A; P(x) |] ==> Q |] ==> Q
+
+\idx{bex_cong} [| A=A'; !!x. x:A' ==> P(x) <-> P'(x) |] ==>
+ (EX x:A. P(x)) <-> (EX x:A'. P'(x))
+\subcaption{Bounded quantifiers}
+
+\idx{subsetI} (!!x.x:A ==> x:B) ==> A <= B
+\idx{subsetD} [| A <= B; c:A |] ==> c:B
+\idx{subsetCE} [| A <= B; ~(c:A) ==> P; c:B ==> P |] ==> P
+\idx{subset_refl} A <= A
+\idx{subset_trans} [| A<=B; B<=C |] ==> A<=C
+
+\idx{equalityI} [| A <= B; B <= A |] ==> A = B
+\idx{equalityD1} A = B ==> A<=B
+\idx{equalityD2} A = B ==> B<=A
+\idx{equalityE} [| A = B; [| A<=B; B<=A |] ==> P |] ==> P
+\subcaption{Subsets and extensionality}
+
+\idx{emptyE} a:0 ==> P
+\idx{empty_subsetI} 0 <= A
+\idx{equals0I} [| !!y. y:A ==> False |] ==> A=0
+\idx{equals0D} [| A=0; a:A |] ==> P
+
+\idx{PowI} A <= B ==> A : Pow(B)
+\idx{PowD} A : Pow(B) ==> A<=B
+\subcaption{The empty set; power sets}
+\end{ttbox}
+\caption{Basic derived rules for {\ZF}} \label{ZF-lemmas1}
+\end{figure}
+
+
+
+\begin{figure}
+\begin{ttbox}
+\idx{ReplaceI} [| x: A; P(x,b); !!y. P(x,y) ==> y=b |] ==>
+ b : \{y. x:A, P(x,y)\}
+
+\idx{ReplaceE} [| b : \{y. x:A, P(x,y)\};
+ !!x. [| x: A; P(x,b); ALL y. P(x,y)-->y=b |] ==> R
+ |] ==> R
+
+\idx{RepFunI} [| a : A |] ==> f(a) : \{f(x). x:A\}
+\idx{RepFunE} [| b : \{f(x). x:A\};
+ !!x.[| x:A; b=f(x) |] ==> P |] ==> P
+
+\idx{separation} a : \{x:A. P(x)\} <-> a:A & P(a)
+\idx{CollectI} [| a:A; P(a) |] ==> a : \{x:A. P(x)\}
+\idx{CollectE} [| a : \{x:A. P(x)\}; [| a:A; P(a) |] ==> R |] ==> R
+\idx{CollectD1} a : \{x:A. P(x)\} ==> a:A
+\idx{CollectD2} a : \{x:A. P(x)\} ==> P(a)
+\end{ttbox}
+\caption{Replacement and separation} \label{ZF-lemmas2}
+\end{figure}
+
+
+\begin{figure}
+\begin{ttbox}
+\idx{UnionI} [| B: C; A: B |] ==> A: Union(C)
+\idx{UnionE} [| A : Union(C); !!B.[| A: B; B: C |] ==> R |] ==> R
+
+\idx{InterI} [| !!x. x: C ==> A: x; c:C |] ==> A : Inter(C)
+\idx{InterD} [| A : Inter(C); B : C |] ==> A : B
+\idx{InterE} [| A : Inter(C); A:B ==> R; ~ B:C ==> R |] ==> R
+
+\idx{UN_I} [| a: A; b: B(a) |] ==> b: (UN x:A. B(x))
+\idx{UN_E} [| b : (UN x:A. B(x)); !!x.[| x: A; b: B(x) |] ==> R
+ |] ==> R
+
+\idx{INT_I} [| !!x. x: A ==> b: B(x); a: A |] ==> b: (INT x:A. B(x))
+\idx{INT_E} [| b : (INT x:A. B(x)); a: A |] ==> b : B(a)
+\end{ttbox}
+\caption{General Union and Intersection} \label{ZF-lemmas3}
+\end{figure}
+
+
+\section{The Zermelo-Fraenkel axioms}
+The axioms appear in Figure~\ref{ZF-rules}. They resemble those
+presented by Suppes~\cite{suppes72}. Most of the theory consists of
+definitions. In particular, bounded quantifiers and the subset relation
+appear in other axioms. Object-level quantifiers and implications have
+been replaced by meta-level ones wherever possible, to simplify use of the
+axioms. See the file \ttindexbold{ZF/zf.thy} for details.
+
+The traditional replacement axiom asserts
+\[ y \in {\tt PrimReplace}(A,P) \bimp (\exists x\in A. P(x,y)) \]
+subject to the condition that $P(x,y)$ is single-valued for all~$x\in A$.
+The Isabelle theory defines \ttindex{Replace} to apply
+\ttindexbold{PrimReplace} to the single-valued part of~$P$, namely
+\[ (\exists!z.P(x,z)) \conj P(x,y). \]
+Thus $y\in {\tt Replace}(A,P)$ if and only if there is some~$x$ such that
+$P(x,-)$ holds uniquely for~$y$. Because the equivalence is unconditional,
+{\tt Replace} is much easier to use than {\tt PrimReplace}; it defines the
+same set, if $P(x,y)$ is single-valued. The nice syntax for replacement
+expands to {\tt Replace}.
+
+Other consequences of replacement include functional replacement
+(\ttindexbold{RepFun}) and definite descriptions (\ttindexbold{The}).
+Axioms for separation (\ttindexbold{Collect}) and unordered pairs
+(\ttindexbold{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 \ttindex{cons}, which underlies the finite set notation.
+The axiom of infinity gives us a set that contains~0 and is closed under
+successor (\ttindexbold{succ}). Although this set is not uniquely defined,
+the theory names it (\ttindexbold{Inf}) in order to simplify the
+construction of the natural numbers.
+
+Further definitions appear in Figure~\ref{ZF-defs}. Ordered pairs are
+defined in the standard way, $\pair{a,b}\equiv\{\{a\},\{a,b\}\}$. Recall
+that \ttindexbold{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 involve definite descriptions. The \ttindex{split}
+operation is like the similar operation in Martin-L\"of Type Theory, and is
+often easier to use than \ttindex{fst} and~\ttindex{snd}. It is defined
+using a description for convenience, but could equivalently be defined by
+\begin{ttbox}
+split(p,c) == c(fst(p),snd(p))
+\end{ttbox}
+Operations on relations include converse, domain, range, and image. The
+set ${\tt Pi}(A,B)$ generalizes the space of functions between two sets.
+Note the simple definitions of $\lambda$-abstraction (using
+\ttindex{RepFun}) and application (using a definite description). The
+function \ttindex{restrict}$(f,A)$ has the same values as~$f$, but only
+over the domain~$A$.
+
+No axiom of choice is provided. It is traditional to include this axiom
+only where it is needed --- mainly in the theory of cardinal numbers, which
+Isabelle does not formalize at present.
+
+
+\section{From basic lemmas to function spaces}
+Faced with so many definitions, it is essential to prove lemmas. Even
+trivial theorems like $A\inter B=B\inter A$ 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 \ttindex{BallE} uses a negated
+assumption in the style of Isabelle's classical module. The congruence rules
+\ttindex{ball_cong} and \ttindex{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 \ttindex{Replace} and \ttindex{RepFun} are much simpler than
+comparable rules for {\tt PrimReplace} would be. The principle of
+separation is proved explicitly, although most proofs should use the
+natural deduction rules for \ttindex{Collect}. The elimination rule
+\ttindex{CollectE} is equivalent to the two destruction rules
+\ttindex{CollectD1} and \ttindex{CollectD2}, but each rule is suited to
+particular circumstances. Although too many rules can be confusing, there
+is no reason to aim for a minimal set of rules. See the file
+\ttindexbold{ZF/zf.ML} for a complete listing.
+
+Figure~\ref{ZF-lemmas3} presents rules for general union and intersection.
+The empty intersection should be undefined. We cannot have
+$\bigcap(\emptyset)=V$ because $V$, the universal class, is not a set. All
+expressions denote something in {\ZF} set theory; the definition of
+intersection implies $\bigcap(\emptyset)=\emptyset$, but this value is
+arbitrary. The rule \ttindexbold{InterI} must have a premise to exclude
+the empty intersection. Some of the laws governing intersections require
+similar premises.
+
+
+%%% upair.ML
+
+\begin{figure}
+\begin{ttbox}
+\idx{pairing} a:Upair(b,c) <-> (a=b | a=c)
+\idx{UpairI1} a : Upair(a,b)
+\idx{UpairI2} b : Upair(a,b)
+\idx{UpairE} [| a : Upair(b,c); a = b ==> P; a = c ==> P |] ==> P
+\subcaption{Unordered pairs}
+
+\idx{UnI1} c : A ==> c : A Un B
+\idx{UnI2} c : B ==> c : A Un B
+\idx{UnCI} (~c : B ==> c : A) ==> c : A Un B
+\idx{UnE} [| c : A Un B; c:A ==> P; c:B ==> P |] ==> P
+
+\idx{IntI} [| c : A; c : B |] ==> c : A Int B
+\idx{IntD1} c : A Int B ==> c : A
+\idx{IntD2} c : A Int B ==> c : B
+\idx{IntE} [| c : A Int B; [| c:A; c:B |] ==> P |] ==> P
+
+\idx{DiffI} [| c : A; ~ c : B |] ==> c : A - B
+\idx{DiffD1} c : A - B ==> c : A
+\idx{DiffD2} [| c : A - B; c : B |] ==> P
+\idx{DiffE} [| c : A - B; [| c:A; ~ c:B |] ==> P |] ==> P
+\subcaption{Union, intersection, difference}
+\end{ttbox}
+\caption{Unordered pairs and their consequences} \label{ZF-upair1}
+\end{figure}
+
+
+\begin{figure}
+\begin{ttbox}
+\idx{consI1} a : cons(a,B)
+\idx{consI2} a : B ==> a : cons(b,B)
+\idx{consCI} (~ a:B ==> a=b) ==> a: cons(b,B)
+\idx{consE} [| a : cons(b,A); a=b ==> P; a:A ==> P |] ==> P
+
+\idx{singletonI} a : \{a\}
+\idx{singletonE} [| a : \{b\}; a=b ==> P |] ==> P
+\subcaption{Finite and singleton sets}
+
+\idx{succI1} i : succ(i)
+\idx{succI2} i : j ==> i : succ(j)
+\idx{succCI} (~ i:j ==> i=j) ==> i: succ(j)
+\idx{succE} [| i : succ(j); i=j ==> P; i:j ==> P |] ==> P
+\idx{succ_neq_0} [| succ(n)=0 |] ==> P
+\idx{succ_inject} succ(m) = succ(n) ==> m=n
+\subcaption{The successor function}
+
+\idx{the_equality} [| P(a); !!x. P(x) ==> x=a |] ==> (THE x. P(x)) = a
+\idx{theI} EX! x. P(x) ==> P(THE x. P(x))
+
+\idx{if_P} P ==> if(P,a,b) = a
+\idx{if_not_P} ~P ==> if(P,a,b) = b
+
+\idx{mem_anti_sym} [| a:b; b:a |] ==> P
+\idx{mem_anti_refl} a:a ==> P
+\subcaption{Descriptions; non-circularity}
+\end{ttbox}
+\caption{Finite sets and their consequences} \label{ZF-upair2}
+\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, and set difference is included for completeness. The
+rule \ttindexbold{UnCI} is useful for classical reasoning about unions,
+like {\tt disjCI}\@; it supersedes \ttindexbold{UnI1} and
+\ttindexbold{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~\ttindex{cons}, the finite set constructor, and rules for singleton
+sets. Because the successor function is defined in terms of~{\tt cons},
+its derived rules appear here.
+
+Definite descriptions (\ttindex{THE}) are defined in terms of the singleton
+set $\{0\}$, but their derived rules fortunately hide this. The
+rule~\ttindex{theI} can be difficult to apply, because $\Var{P}$ must be
+instantiated correctly. However, \ttindex{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 $b\in a$
+(\ttindex{mem_anti_sym}) is proved by applying the axiom of foundation to
+the set $\{a,b\}$. The impossibility of $a\in a$ is a trivial consequence.
+
+See the file \ttindexbold{ZF/upair.ML} for full details.
+
+
+%%% subset.ML
+
+\begin{figure}
+\begin{ttbox}
+\idx{Union_upper} B:A ==> B <= Union(A)
+\idx{Union_least} [| !!x. x:A ==> x<=C |] ==> Union(A) <= C
+
+\idx{Inter_lower} B:A ==> Inter(A) <= B
+\idx{Inter_greatest} [| a:A; !!x. x:A ==> C<=x |] ==> C <= Inter(A)
+
+\idx{Un_upper1} A <= A Un B
+\idx{Un_upper2} B <= A Un B
+\idx{Un_least} [| A<=C; B<=C |] ==> A Un B <= C
+
+\idx{Int_lower1} A Int B <= A
+\idx{Int_lower2} A Int B <= B
+\idx{Int_greatest} [| C<=A; C<=B |] ==> C <= A Int B
+
+\idx{Diff_subset} A-B <= A
+\idx{Diff_contains} [| C<=A; C Int B = 0 |] ==> C <= A-B
+
+\idx{Collect_subset} Collect(A,P) <= A
+\end{ttbox}
+\caption{Subset and lattice properties} \label{ZF-subset}
+\end{figure}
+
+
+\subsection{Subset and lattice properties}
+Figure~\ref{ZF-subset} shows that the subset relation is a complete
+lattice. Unions form least upper bounds; non-empty intersections form
+greatest lower bounds. A few other laws involving subsets are included.
+See the file \ttindexbold{ZF/subset.ML}.
+
+%%% pair.ML
+
+\begin{figure}
+\begin{ttbox}
+\idx{Pair_inject1} = ==> a=c
+\idx{Pair_inject2} = ==> b=d
+\idx{Pair_inject} [| = ; [| a=c; b=d |] ==> P |] ==> P
+\idx{Pair_neq_0} =0 ==> P
+
+\idx{fst} fst() = a
+\idx{snd} snd() = b
+\idx{split} split(, %x y.c(x,y)) = c(a,b)
+
+\idx{SigmaI} [| a:A; b:B(a) |] ==> : Sigma(A,B)
+
+\idx{SigmaE} [| c: Sigma(A,B);
+ !!x y.[| x:A; y:B(x); c= |] ==> P |] ==> P
+
+\idx{SigmaE2} [| : Sigma(A,B);
+ [| a:A; b:B(a) |] ==> P |] ==> P
+\end{ttbox}
+\caption{Ordered pairs; projections; general sums} \label{ZF-pair}
+\end{figure}
+
+
+\subsection{Ordered pairs}
+Figure~\ref{ZF-pair} presents the rules governing ordered pairs,
+projections and general sums. File \ttindexbold{ZF/pair.ML} contains the
+full (and tedious) proof that $\{\{a\},\{a,b\}\}$ functions as an ordered
+pair. This property is expressed as two destruction rules,
+\ttindexbold{Pair_inject1} and \ttindexbold{Pair_inject2}, and equivalently
+as the elimination rule \ttindexbold{Pair_inject}.
+
+Note the rule \ttindexbold{Pair_neq_0}, which asserts
+$\pair{a,b}\neq\emptyset$. This is no arbitrary property of
+$\{\{a\},\{a,b\}\}$, but one that we can reasonably expect to hold for any
+encoding of ordered pairs. It turns out to be useful for constructing
+Lisp-style S-expressions in set theory.
+
+The natural deduction rules \ttindexbold{SigmaI} and \ttindexbold{SigmaE}
+assert that \ttindex{Sigma}$(A,B)$ consists of all pairs of the form
+$\pair{x,y}$, for $x\in A$ and $y\in B(x)$. The rule \ttindexbold{SigmaE2}
+merely states that $\pair{a,b}\in {\tt Sigma}(A,B)$ implies $a\in A$ and
+$b\in B(a)$.
+
+
+%%% domrange.ML
+
+\begin{figure}
+\begin{ttbox}
+\idx{domainI} : r ==> a : domain(r)
+\idx{domainE} [| a : domain(r); !!y. : r ==> P |] ==> P
+\idx{domain_subset} domain(Sigma(A,B)) <= A
+
+\idx{rangeI} : r ==> b : range(r)
+\idx{rangeE} [| b : range(r); !!x. : r ==> P |] ==> P
+\idx{range_subset} range(A*B) <= B
+
+\idx{fieldI1} : r ==> a : field(r)
+\idx{fieldI2} : r ==> b : field(r)
+\idx{fieldCI} (~ :r ==> : r) ==> a : field(r)
+
+\idx{fieldE} [| a : field(r);
+ !!x. : r ==> P;
+ !!x. : r ==> P
+ |] ==> P
+
+\idx{field_subset} field(A*A) <= A
+\subcaption{Domain, range and field of a Relation}
+
+\idx{imageI} [| : r; a:A |] ==> b : r``A
+\idx{imageE} [| b: r``A; !!x.[| : r; x:A |] ==> P |] ==> P
+
+\idx{vimageI} [| : r; b:B |] ==> a : r-``B
+\idx{vimageE} [| a: r-``B; !!x.[| : r; x:B |] ==> P |] ==> P
+\subcaption{Image and inverse image}
+\end{ttbox}
+\caption{Relations} \label{ZF-domrange}
+\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
+{\ttindex{converse}$(r)$} is its inverse. The rules for the domain
+operation, \ttindex{domainI} and~\ttindex{domainE}, assert that
+\ttindex{domain}$(r)$ consists of every element~$a$ 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. Note
+that image and inverse image are generalizations of range and domain,
+respectively. See the file
+\ttindexbold{ZF/domrange.ML} for derivations of the rules.
+
+
+%%% func.ML
+
+\begin{figure}
+\begin{ttbox}
+\idx{fun_is_rel} f: Pi(A,B) ==> f <= Sigma(A,B)
+
+\idx{apply_equality} [| : f; f: Pi(A,B) |] ==> f`a = b
+\idx{apply_equality2} [| : f; : f; f: Pi(A,B) |] ==> b=c
+
+\idx{apply_type} [| f: Pi(A,B); a:A |] ==> f`a : B(a)
+\idx{apply_Pair} [| f: Pi(A,B); a:A |] ==> : f
+\idx{apply_iff} f: Pi(A,B) ==> : f <-> a:A & f`a = b
+
+\idx{fun_extension} [| f : Pi(A,B); g: Pi(A,D);
+ !!x. x:A ==> f`x = g`x |] ==> f=g
+
+\idx{domain_type} [| : f; f: Pi(A,B) |] ==> a : A
+\idx{range_type} [| : f; f: Pi(A,B) |] ==> b : B(a)
+
+\idx{Pi_type} [| f: A->C; !!x. x:A ==> f`x: B(x) |] ==> f: Pi(A,B)
+\idx{domain_of_fun} f: Pi(A,B) ==> domain(f)=A
+\idx{range_of_fun} f: Pi(A,B) ==> f: A->range(f)
+
+\idx{restrict} a : A ==> restrict(f,A) ` a = f`a
+\idx{restrict_type} [| !!x. x:A ==> f`x: B(x) |] ==>
+ restrict(f,A) : Pi(A,B)
+
+\idx{lamI} a:A ==> : (lam x:A. b(x))
+\idx{lamE} [| p: (lam x:A. b(x)); !!x.[| x:A; p= |] ==> P
+ |] ==> P
+
+\idx{lam_type} [| !!x. x:A ==> b(x): B(x) |] ==> (lam x:A.b(x)) : Pi(A,B)
+
+\idx{beta} a : A ==> (lam x:A.b(x)) ` a = b(a)
+\idx{eta} f : Pi(A,B) ==> (lam x:A. f`x) = f
+
+\idx{lam_theI} (!!x. x:A ==> EX! y. Q(x,y)) ==> EX h. ALL x:A. Q(x, h`x)
+\end{ttbox}
+\caption{Functions and $\lambda$-abstraction} \label{ZF-func1}
+\end{figure}
+
+
+\begin{figure}
+\begin{ttbox}
+\idx{fun_empty} 0: 0->0
+\idx{fun_single} \{\} : \{a\} -> \{b\}
+
+\idx{fun_disjoint_Un} [| f: A->B; g: C->D; A Int C = 0 |] ==>
+ (f Un g) : (A Un C) -> (B Un D)
+
+\idx{fun_disjoint_apply1} [| a:A; f: A->B; g: C->D; A Int C = 0 |] ==>
+ (f Un g)`a = f`a
+
+\idx{fun_disjoint_apply2} [| c:C; f: A->B; g: C->D; A Int C = 0 |] ==>
+ (f Un g)`c = g`c
+\end{ttbox}
+\caption{Constructing functions from smaller sets} \label{ZF-func2}
+\end{figure}
+
+
+\subsection{Functions}
+Functions, represented by graphs, are notoriously difficult to reason
+about. The file \ttindexbold{ZF/func.ML} derives many rules, which overlap
+more than they ought. One day these rules will be tidied up; this section
+presents only the more important ones.
+
+Figure~\ref{ZF-func1} presents the basic properties of \ttindex{Pi}$(A,B)$,
+the generalized function space. For example, if $f$ is a function and
+$\pair{a,b}\in f$, then $f`a=b$ (\ttindex{apply_equality}). Two functions
+are equal provided they have equal domains and deliver equals results
+(\ttindex{fun_extension}).
+
+By \ttindex{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 \ttindex{range_of_fun},
+any dependent typing can be flattened to yield a function type of the form
+$A\to C$; here, $C={\tt range}(f)$.
+
+Among the laws for $\lambda$-abstraction, \ttindex{lamI} and \ttindex{lamE}
+describe the graph of the generated function, while \ttindex{beta} and
+\ttindex{eta} are the standard conversions. We essentially have a
+dependently-typed $\lambda$-calculus.
+
+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{center}
+\begin{tabular}{rrr}
+ \it name &\it meta-type & \it description \\
+ \idx{id} & $i$ & identity function \\
+ \idx{inj} & $[i,i]\To i$ & injective function space\\
+ \idx{surj} & $[i,i]\To i$ & surjective function space\\
+ \idx{bij} & $[i,i]\To i$ & bijective function space
+ \\[1ex]
+ \idx{1} & $i$ & $\{\emptyset\}$ \\
+ \idx{bool} & $i$ & the set $\{\emptyset,1\}$ \\
+ \idx{cond} & $[i,i,i]\To i$ & conditional for {\tt bool}
+ \\[1ex]
+ \idx{Inl}~~\idx{Inr} & $i\To i$ & injections\\
+ \idx{case} & $[i,i\To i,i\To i]\To i$ & conditional for $+$
+ \\[1ex]
+ \idx{nat} & $i$ & set of natural numbers \\
+ \idx{nat_case}& $[i,i,i\To i]\To i$ & conditional for $nat$\\
+ \idx{rec} & $[i,i,[i,i]\To i]\To i$ & recursor for $nat$
+ \\[1ex]
+ \idx{list} & $i\To i$ & lists over some set\\
+ \idx{list_case} & $[i, i, [i,i]\To i] \To i$ & conditional for $list(A)$ \\
+ \idx{list_rec} & $[i, i, [i,i,i]\To i] \To i$ & recursor for $list(A)$ \\
+ \idx{map} & $[i\To i, i] \To i$ & mapping functional\\
+ \idx{length} & $i\To i$ & length of a list\\
+ \idx{rev} & $i\To i$ & reverse of a list\\
+ \idx{flat} & $i\To i$ & flatting a list of lists\\
+\end{tabular}
+\end{center}
+\subcaption{Constants}
+
+\begin{center}
+\indexbold{*"+}
+\index{#*@{\tt\#*}|bold}
+\index{*div|bold}
+\index{*mod|bold}
+\index{#+@{\tt\#+}|bold}
+\index{#-@{\tt\#-}|bold}
+\begin{tabular}{rrrr}
+ \idx{O} & $[i,i]\To i$ & Right 60 & composition ($\circ$) \\
+ \tt + & $[i,i]\To i$ & Right 65 & disjoint union \\
+ \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\\
+ \tt \@ & $[i,i]\To i$ & Right 60 & append for lists
+\end{tabular}
+\end{center}
+\subcaption{Infixes}
+\caption{Further constants for {\ZF}} \label{ZF-further-constants}
+\end{figure}
+
+
+\begin{figure}
+\begin{ttbox}
+\idx{Int_absorb} A Int A = A
+\idx{Int_commute} A Int B = B Int A
+\idx{Int_assoc} (A Int B) Int C = A Int (B Int C)
+\idx{Int_Un_distrib} (A Un B) Int C = (A Int C) Un (B Int C)
+
+\idx{Un_absorb} A Un A = A
+\idx{Un_commute} A Un B = B Un A
+\idx{Un_assoc} (A Un B) Un C = A Un (B Un C)
+\idx{Un_Int_distrib} (A Int B) Un C = (A Un C) Int (B Un C)
+
+\idx{Diff_cancel} A-A = 0
+\idx{Diff_disjoint} A Int (B-A) = 0
+\idx{Diff_partition} A<=B ==> A Un (B-A) = B
+\idx{double_complement} [| A<=B; B<= C |] ==> (B - (C-A)) = A
+\idx{Diff_Un} A - (B Un C) = (A-B) Int (A-C)
+\idx{Diff_Int} A - (B Int C) = (A-B) Un (A-C)
+
+\idx{Union_Un_distrib} Union(A Un B) = Union(A) Un Union(B)
+\idx{Inter_Un_distrib} [| a:A; b:B |] ==>
+ Inter(A Un B) = Inter(A) Int Inter(B)
+
+\idx{Int_Union_RepFun} A Int Union(B) = (UN C:B. A Int C)
+
+\idx{Un_Inter_RepFun} b:B ==>
+ A Un Inter(B) = (INT C:B. A Un C)
+
+\idx{SUM_Un_distrib1} (SUM x:A Un B. C(x)) =
+ (SUM x:A. C(x)) Un (SUM x:B. C(x))
+
+\idx{SUM_Un_distrib2} (SUM x:C. A(x) Un B(x)) =
+ (SUM x:C. A(x)) Un (SUM x:C. B(x))
+
+\idx{SUM_Int_distrib1} (SUM x:A Int B. C(x)) =
+ (SUM x:A. C(x)) Int (SUM x:B. C(x))
+
+\idx{SUM_Int_distrib2} (SUM x:C. A(x) Int B(x)) =
+ (SUM x:C. A(x)) Int (SUM x:C. B(x))
+\end{ttbox}
+\caption{Equalities} \label{zf-equalities}
+\end{figure}
+
+\begin{figure}
+\begin{ttbox}
+\idx{comp_def} r O s == \{xz : domain(s)*range(r) .
+ EX x y z. xz= & :s & :r\}
+\idx{id_def} id(A) == (lam x:A. x)
+\idx{inj_def} inj(A,B) == \{ f: A->B. ALL w:A. ALL x:A. f`w=f`x --> w=x\}
+\idx{surj_def} surj(A,B) == \{ f: A->B . ALL y:B. EX x:A. f`x=y\}
+\idx{bij_def} bij(A,B) == inj(A,B) Int surj(A,B)
+\subcaption{Definitions}
+
+\idx{left_inverse} [| f: inj(A,B); a: A |] ==> converse(f)`(f`a) = a
+\idx{right_inverse} [| f: inj(A,B); b: range(f) |] ==>
+ f`(converse(f)`b) = b
+
+\idx{inj_converse_inj} f: inj(A,B) ==> converse(f): inj(range(f), A)
+\idx{bij_converse_bij} f: bij(A,B) ==> converse(f): bij(B,A)
+
+\idx{comp_type} [| s<=A*B; r<=B*C |] ==> (r O s) <= A*C
+\idx{comp_assoc} (r O s) O t = r O (s O t)
+
+\idx{left_comp_id} r<=A*B ==> id(B) O r = r
+\idx{right_comp_id} r<=A*B ==> r O id(A) = r
+
+\idx{comp_func} [| g:A->B; f:B->C |] ==> (f O g):A->C
+\idx{comp_func_apply} [| g:A->B; f:B->C; a:A |] ==> (f O g)`a = f`(g`a)
+
+\idx{comp_inj} [| g:inj(A,B); f:inj(B,C) |] ==> (f O g):inj(A,C)
+\idx{comp_surj} [| g:surj(A,B); f:surj(B,C) |] ==> (f O g):surj(A,C)
+\idx{comp_bij} [| g:bij(A,B); f:bij(B,C) |] ==> (f O g):bij(A,C)
+
+\idx{left_comp_inverse} f: inj(A,B) ==> converse(f) O f = id(A)
+\idx{right_comp_inverse} f: surj(A,B) ==> f O converse(f) = id(B)
+
+\idx{bij_disjoint_Un}
+ [| f: bij(A,B); g: bij(C,D); A Int C = 0; B Int D = 0 |] ==>
+ (f Un g) : bij(A Un C, B Un D)
+
+\idx{restrict_bij} [| f:inj(A,B); C<=A |] ==> restrict(f,C): bij(C, f``C)
+\end{ttbox}
+\caption{Permutations} \label{zf-perm}
+\end{figure}
+
+\begin{figure}
+\begin{ttbox}
+\idx{one_def} 1 == succ(0)
+\idx{bool_def} bool == {0,1}
+\idx{cond_def} cond(b,c,d) == if(b=1,c,d)
+
+\idx{sum_def} A+B == \{0\}*A Un \{1\}*B
+\idx{Inl_def} Inl(a) == <0,a>
+\idx{Inr_def} Inr(b) == <1,b>
+\idx{case_def} case(u,c,d) == split(u, %y z. cond(y, d(z), c(z)))
+\subcaption{Definitions}
+
+\idx{bool_1I} 1 : bool
+\idx{bool_0I} 0 : bool
+
+\idx{boolE} [| c: bool; P(1); P(0) |] ==> P(c)
+\idx{cond_1} cond(1,c,d) = c
+\idx{cond_0} cond(0,c,d) = d
+
+\idx{sum_InlI} a : A ==> Inl(a) : A+B
+\idx{sum_InrI} b : B ==> Inr(b) : A+B
+
+\idx{Inl_inject} Inl(a)=Inl(b) ==> a=b
+\idx{Inr_inject} Inr(a)=Inr(b) ==> a=b
+\idx{Inl_neq_Inr} Inl(a)=Inr(b) ==> P
+
+\idx{sumE2} u: A+B ==> (EX x. x:A & u=Inl(x)) | (EX y. y:B & u=Inr(y))
+
+\idx{case_Inl} case(Inl(a),c,d) = c(a)
+\idx{case_Inr} case(Inr(b),c,d) = d(b)
+\end{ttbox}
+\caption{Booleans and disjoint unions} \label{zf-sum}
+\end{figure}
+
+\begin{figure}
+\begin{ttbox}
+\idx{nat_def} nat == lfp(lam r: Pow(Inf). \{0\} Un \{succ(x). x:r\}
+
+\idx{nat_case_def} nat_case(n,a,b) ==
+ THE y. n=0 & y=a | (EX x. n=succ(x) & y=b(x))
+
+\idx{rec_def} rec(k,a,b) ==
+ transrec(k, %n f. nat_case(n, a, %m. b(m, f`m)))
+
+\idx{add_def} m#+n == rec(m, n, %u v.succ(v))
+\idx{diff_def} m#-n == rec(n, m, %u v. rec(v, 0, %x y.x))
+\idx{mult_def} m#*n == rec(m, 0, %u v. n #+ v)
+\idx{mod_def} m mod n == transrec(m, %j f. if(j:n, j, f`(j#-n)))
+\idx{quo_def} m div n == transrec(m, %j f. if(j:n, 0, succ(f`(j#-n))))
+\subcaption{Definitions}
+
+\idx{nat_0I} 0 : nat
+\idx{nat_succI} n : nat ==> succ(n) : nat
+
+\idx{nat_induct}
+ [| n: nat; P(0); !!x. [| x: nat; P(x) |] ==> P(succ(x))
+ |] ==> P(n)
+
+\idx{nat_case_0} nat_case(0,a,b) = a
+\idx{nat_case_succ} nat_case(succ(m),a,b) = b(m)
+
+\idx{rec_0} rec(0,a,b) = a
+\idx{rec_succ} rec(succ(m),a,b) = b(m, rec(m,a,b))
+
+\idx{mult_type} [| m:nat; n:nat |] ==> m #* n : nat
+\idx{mult_0} 0 #* n = 0
+\idx{mult_succ} succ(m) #* n = n #+ (m #* n)
+\idx{mult_commute} [| m:nat; n:nat |] ==> m #* n = n #* m
+\idx{add_mult_dist}
+ [| m:nat; k:nat |] ==> (m #+ n) #* k = (m #* k) #+ (n #* k)
+\idx{mult_assoc}
+ [| m:nat; n:nat; k:nat |] ==> (m #* n) #* k = m #* (n #* k)
+
+\idx{mod_quo_equality}
+ [| 0:n; m:nat; n:nat |] ==> (m div n)#*n #+ m mod n = m
+\end{ttbox}
+\caption{The natural numbers} \label{zf-nat}
+\end{figure}
+
+\begin{figure}\underscoreon %%because @ is used here
+\begin{ttbox}
+\idx{list_def} list(A) == lfp(univ(A), %X. {0} Un A*X)
+
+\idx{list_case_def} list_case(l,c,h) ==
+ THE z. l=0 & z=c | (EX x y. l = & z=h(x,y))
+
+\idx{list_rec_def} list_rec(l,c,h) ==
+ Vrec(l, %l g.list_case(l, c, %x xs. h(x, xs, g`xs)))
+
+\idx{map_def} map(f,l) == list_rec(l, 0, %x xs r. )
+\idx{length_def} length(l) == list_rec(l, 0, %x xs r. succ(r))
+\idx{app_def} xs@ys == list_rec(xs, ys, %x xs r. )
+\idx{rev_def} rev(l) == list_rec(l, 0, %x xs r. r @ )
+\idx{flat_def} flat(ls) == list_rec(ls, 0, %l ls r. l @ r)
+\subcaption{Definitions}
+
+\idx{list_0I} 0 : list(A)
+\idx{list_PairI} [| a: A; l: list(A) |] ==> : list(A)
+
+\idx{list_induct}
+ [| l: list(A);
+ P(0);
+ !!x y. [| x: A; y: list(A); P(y) |] ==> P()
+ |] ==> P(l)
+
+\idx{list_case_0} list_case(0,c,h) = c
+\idx{list_case_Pair} list_case(, c, h) = h(a,l)
+
+\idx{list_rec_0} list_rec(0,c,h) = c
+\idx{list_rec_Pair} list_rec(, c, h) = h(a, l, list_rec(l,c,h))
+
+\idx{map_ident} l: list(A) ==> map(%u.u, l) = l
+\idx{map_compose} l: list(A) ==> map(h, map(j,l)) = map(%u.h(j(u)), l)
+\idx{map_app_distrib} xs: list(A) ==> map(h, xs@ys) = map(h,xs) @ map(h,ys)
+\idx{map_type}
+ [| l: list(A); !!x. x: A ==> h(x): B |] ==> map(h,l) : list(B)
+\idx{map_flat}
+ ls: list(list(A)) ==> map(h, flat(ls)) = flat(map(map(h),ls))
+\end{ttbox}
+\caption{Lists} \label{zf-list}
+\end{figure}
+
+\section{Further developments}
+The next group of developments is complex and extensive, and only
+highlights can be covered here. Figure~\ref{ZF-further-constants} lists
+some of the further constants and infixes that are declared in the various
+theory extensions.
+
+Figure~\ref{zf-sum} defines $\{0,1\}$ as a set of booleans, with a
+conditional operator. It also defines the disjoint union of two sets, with
+injections and a case analysis operator. See files
+\ttindexbold{ZF/bool.ML} and \ttindexbold{ZF/sum.ML}.
+
+Monotonicity properties of most of the set-forming operations are proved.
+These are useful for applying the Knaster-Tarski Fixedpoint Theorem.
+See file \ttindexbold{ZF/mono.ML}.
+
+Figure~\ref{zf-equalities} presents commutative, associative, distributive,
+and idempotency laws of union and intersection, along with other equations.
+See file \ttindexbold{ZF/equalities.ML}.
+
+Figure~\ref{zf-perm} defines composition and injective, surjective and
+bijective function spaces, with proofs of many of their properties.
+See file \ttindexbold{ZF/perm.ML}.
+
+Figure~\ref{zf-nat} presents the natural numbers, with induction and a
+primitive recursion operator. See file \ttindexbold{ZF/nat.ML}. File
+\ttindexbold{ZF/arith.ML} develops arithmetic on the natural numbers. It
+defines addition, multiplication, subtraction, division, and remainder,
+proving the theorem $a \bmod b + (a/b)\times b = a$. Division and
+remainder are defined by repeated subtraction, which requires well-founded
+rather than primitive recursion.
+
+Figure~\ref{zf-list} presents defines the set of lists over~$A$, ${\tt
+list}(A)$ as the least solution of the equation ${\tt list}(A)\equiv \{0\}
+\union (A\times{\tt list}(A))$. Structural induction and recursion are
+derived, with some of the usual list functions. See file
+\ttindexbold{ZF/list.ML}.
+
+The constructions of the natural numbers and lists make use of a suite of
+operators for handling recursive definitions. The developments are
+summarized below:
+\begin{description}
+\item[\ttindexbold{ZF/lfp.ML}]
+proves the Knaster-Tarski Fixedpoint Theorem in the lattice of subsets of a
+set. The file defines a least fixedpoint operator with corresponding
+induction rules. These are used repeatedly in the sequel to define sets
+inductively. As a simple application, the file contains a short proof of
+the Schr\"oder-Bernstein Theorem.
+
+\item[\ttindexbold{ZF/trancl.ML}]
+defines the transitive closure of a relation (as a least fixedpoint).
+
+\item[\ttindexbold{ZF/wf.ML}]
+proves the Well-Founded Recursion Theorem, using an elegant
+approach of Tobias Nipkow. This theorem permits general recursive
+definitions within set theory.
+
+\item[\ttindexbold{ZF/ordinal.ML}]
+defines the notions of transitive set and ordinal number. It derives
+transfinite induction.
+
+\item[\ttindexbold{ZF/epsilon.ML}]
+derives $\epsilon$-induction and $\epsilon$-recursion, which are
+generalizations of transfinite induction. It also defines
+\ttindexbold{rank}$(x)$, which is the least ordinal $\alpha$ such that $x$
+is constructed at stage $\alpha$ of the cumulative hierarchy (thus $x\in
+V@{\alpha+1}$).
+
+\item[\ttindexbold{ZF/univ.ML}]
+defines a ``universe'' ${\tt univ}(A)$, for constructing sets inductively.
+This set contains $A$ and the natural numbers. Vitally, it is
+closed under finite products:
+${\tt univ}(A)\times{\tt univ}(A)\subseteq{\tt univ}(A)$. This file also
+defines set theory's cumulative hierarchy, traditionally written $V@\alpha$
+where $\alpha$ is any ordinal.
+\end{description}
+
+
+\begin{figure}
+\begin{eqnarray*}
+ a\in a & \bimp & \bot\\
+ a\in \emptyset & \bimp & \bot\\
+ a \in A \union B & \bimp & a\in A \disj a\in B\\
+ a \in A \inter B & \bimp & a\in A \conj a\in B\\
+ a \in A-B & \bimp & a\in A \conj \neg (a\in B)\\
+ a \in {\tt cons}(b,B) & \bimp & a=b \disj a\in B\\
+ i \in {\tt succ}(j) & \bimp & i=j \disj i\in j\\
+ \pair{a,b}\in {\tt Sigma}(A,B)
+ & \bimp & a\in A \conj b\in B(a)\\
+ a \in {\tt Collect}(A,P) & \bimp & a\in A \conj P(a)\\
+ (\forall x \in A. \top) & \bimp & \top
+\end{eqnarray*}
+\caption{Rewrite rules for set theory} \label{ZF-simpdata}
+\end{figure}
+
+
+\section{Simplification rules}
+{\ZF} does not merely inherit simplification from \FOL, but instantiates
+the rewriting package new. File \ttindexbold{ZF/simpdata.ML} contains the
+details; here is a summary of the key differences:
+\begin{itemize}
+\item
+\ttindexbold{mk_rew_rules} is given as a function that can
+strip bounded universal quantifiers from a formula. For example, $\forall
+x\in A.f(x)=g(x)$ yields the conditional rewrite rule $x\in A \Imp
+f(x)=g(x)$.
+\item
+\ttindexbold{ZF_ss} contains congruence rules for all the operators of
+{\ZF}, including the binding operators. It contains all the conversion
+rules, such as \ttindex{fst} and \ttindex{snd}, as well as the
+rewrites shown in Figure~\ref{ZF-simpdata}.
+\item
+\ttindexbold{FOL_ss} is redeclared with the same {\FOL} rules as the
+previous version, so that it may still be used.
+\end{itemize}
+
+
+\section{The examples directory}
+This directory contains further developments in {\ZF} set theory. Here is
+an overview; see the files themselves for more details.
+\begin{description}
+\item[\ttindexbold{ZF/ex/misc.ML}]
+contains miscellaneous examples such as Cantor's Theorem and the
+``Composition of homomorphisms'' challenge.
+
+\item[\ttindexbold{ZF/ex/ramsey.ML}]
+proves the finite exponent 2 version of Ramsey's Theorem.
+
+\item[\ttindexbold{ZF/ex/bt.ML}]
+defines the recursive data structure ${\tt bt}(A)$, labelled binary trees.
+
+\item[\ttindexbold{ZF/ex/sexp.ML}]
+defines the set of Lisp $S$-expressions over~$A$, ${\tt sexp}(A)$. These
+are unlabelled binary trees whose leaves contain elements of $(A)$.
+
+\item[\ttindexbold{ZF/ex/term.ML}]
+defines a recursive data structure for terms and term lists.
+
+\item[\ttindexbold{ZF/ex/simult.ML}]
+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[\ttindexbold{ZF/ex/finite.ML}]
+inductively defines a finite powerset operator.
+
+\item[\ttindexbold{ZF/ex/prop-log.ML}]
+proves soundness and completeness of propositional logic. This illustrates
+the main forms of induction.
+\end{description}
+
+
+\section{A proof about powersets}
+To demonstrate high-level reasoning about subsets, let us prove the equation
+${Pow(A)\cap Pow(B)}= Pow(A\cap B)$. 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 {\tt ZF/mono.ML}:
+\begin{ttbox}
+\idx{Pow_mono} A<=B ==> Pow(A) <= Pow(B)
+\end{ttbox}
+We enter the goal and make the first step, which breaks the equation into
+two inclusions by extensionality:\index{equalityI}
+\begin{ttbox}
+goal ZF.thy "Pow(A Int B) = Pow(A) Int Pow(B)";
+{\out Level 0}
+{\out Pow(A Int B) = Pow(A) Int Pow(B)}
+{\out 1. Pow(A Int B) = Pow(A) Int Pow(B)}
+by (resolve_tac [equalityI] 1);
+{\out Level 1}
+{\out Pow(A Int B) = Pow(A) Int Pow(B)}
+{\out 1. Pow(A Int B) <= Pow(A) Int Pow(B)}
+{\out 2. Pow(A) Int Pow(B) <= Pow(A Int B)}
+\end{ttbox}
+Both inclusions could be tackled straightforwardly using {\tt subsetI}.
+A shorter proof results from noting that intersection forms the greatest
+lower bound:\index{*Int_greatest}
+\begin{ttbox}
+by (resolve_tac [Int_greatest] 1);
+{\out Level 2}
+{\out Pow(A Int B) = Pow(A) Int Pow(B)}
+{\out 1. Pow(A Int B) <= Pow(A)}
+{\out 2. Pow(A Int B) <= Pow(B)}
+{\out 3. Pow(A) Int Pow(B) <= Pow(A Int B)}
+\end{ttbox}
+Subgoal~1 follows by applying the monotonicity of {\tt Pow} to $A\inter
+B\subseteq A$; subgoal~2 follows similarly:
+\index{*Int_lower1}\index{*Int_lower2}
+\begin{ttbox}
+by (resolve_tac [Int_lower1 RS Pow_mono] 1);
+{\out Level 3}
+{\out Pow(A Int B) = Pow(A) Int Pow(B)}
+{\out 1. Pow(A Int B) <= Pow(B)}
+{\out 2. Pow(A) Int Pow(B) <= Pow(A Int B)}
+by (resolve_tac [Int_lower2 RS Pow_mono] 1);
+{\out Level 4}
+{\out Pow(A Int B) = Pow(A) Int Pow(B)}
+{\out 1. Pow(A) Int Pow(B) <= Pow(A Int B)}
+\end{ttbox}
+We are left with the opposite inclusion, which we tackle in the
+straightforward way:\index{*subsetI}
+\begin{ttbox}
+by (resolve_tac [subsetI] 1);
+{\out Level 5}
+{\out Pow(A Int B) = Pow(A) Int Pow(B)}
+{\out 1. !!x. x : Pow(A) Int Pow(B) ==> x : Pow(A Int B)}
+\end{ttbox}
+The subgoal is to show $x\in {\tt Pow}(A\cap B)$ assuming $x\in{\tt
+Pow}(A)\cap {\tt Pow}(B)$. Eliminating this assumption produces two
+subgoals, since intersection is like conjunction.\index{*IntE}
+\begin{ttbox}
+by (eresolve_tac [IntE] 1);
+{\out Level 6}
+{\out Pow(A Int B) = Pow(A) Int Pow(B)}
+{\out 1. !!x. [| x : Pow(A); x : Pow(B) |] ==> x : Pow(A Int B)}
+\end{ttbox}
+The next step replaces the {\tt Pow} by the subset
+relation~($\subseteq$).\index{*PowI}
+\begin{ttbox}
+by (resolve_tac [PowI] 1);
+{\out Level 7}
+{\out Pow(A Int B) = Pow(A) Int Pow(B)}
+{\out 1. !!x. [| x : Pow(A); x : Pow(B) |] ==> x <= A Int B}
+\end{ttbox}
+We perform the same replacement in the assumptions:\index{*PowD}
+\begin{ttbox}
+by (REPEAT (dresolve_tac [PowD] 1));
+{\out Level 8}
+{\out Pow(A Int B) = Pow(A) Int Pow(B)}
+{\out 1. !!x. [| x <= A; x <= B |] ==> x <= A Int B}
+\end{ttbox}
+Here, $x$ is a lower bound of $A$ and~$B$, but $A\inter B$ is the greatest
+lower bound:\index{*Int_greatest}
+\begin{ttbox}
+by (resolve_tac [Int_greatest] 1);
+{\out Level 9}
+{\out Pow(A Int B) = Pow(A) Int Pow(B)}
+{\out 1. !!x. [| x <= A; x <= B |] ==> x <= A}
+{\out 2. !!x. [| x <= A; x <= B |] ==> x <= B}
+by (REPEAT (assume_tac 1));
+{\out Level 10}
+{\out Pow(A Int B) = Pow(A) Int Pow(B)}
+{\out No subgoals!}
+\end{ttbox}
+We could have performed this proof in one step by applying
+\ttindex{fast_tac} with the classical rule set \ttindex{ZF_cs}. But we
+must add \ttindex{equalityI} as an introduction rule, since extensionality
+is not used by default:
+\begin{ttbox}
+choplev 0;
+{\out Level 0}
+{\out Pow(A Int B) = Pow(A) Int Pow(B)}
+{\out 1. Pow(A Int B) = Pow(A) Int Pow(B)}
+by (fast_tac (ZF_cs addIs [equalityI]) 1);
+{\out Level 1}
+{\out Pow(A Int B) = Pow(A) Int Pow(B)}
+{\out No subgoals!}
+\end{ttbox}
+
+
+\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 \ttindex{subsetI}:
+\begin{ttbox}
+val [prem] = goal ZF.thy "C<=D ==> Union(C) <= Union(D)";
+{\out Level 0}
+{\out Union(C) <= Union(D)}
+{\out 1. Union(C) <= Union(D)}
+by (resolve_tac [subsetI] 1);
+{\out Level 1}
+{\out Union(C) <= Union(D)}
+{\out 1. !!x. x : Union(C) ==> x : Union(D)}
+\end{ttbox}
+Big union is like an existential quantifier --- the occurrence in the
+assumptions must be eliminated early, since it creates parameters.
+\index{*UnionE}
+\begin{ttbox}
+by (eresolve_tac [UnionE] 1);
+{\out Level 2}
+{\out Union(C) <= Union(D)}
+{\out 1. !!x B. [| x : B; B : C |] ==> x : Union(D)}
+\end{ttbox}
+Now we may apply \ttindex{UnionI}, which creates an unknown involving the
+parameters. To show $x\in \bigcup(D)$ it suffices to show that $x$ belongs
+to some element, say~$\Var{B2}(x,B)$, of~$D$.
+\begin{ttbox}
+by (resolve_tac [UnionI] 1);
+{\out Level 3}
+{\out Union(C) <= Union(D)}
+{\out 1. !!x B. [| x : B; B : C |] ==> ?B2(x,B) : D}
+{\out 2. !!x B. [| x : B; B : C |] ==> x : ?B2(x,B)}
+\end{ttbox}
+Combining \ttindex{subsetD} with the premise $C\subseteq D$ yields
+$\Var{a}\in C \Imp \Var{a}\in D$, which reduces subgoal~1:
+\begin{ttbox}
+by (resolve_tac [prem RS subsetD] 1);
+{\out Level 4}
+{\out Union(C) <= Union(D)}
+{\out 1. !!x B. [| x : B; B : C |] ==> ?B2(x,B) : C}
+{\out 2. !!x B. [| x : B; B : C |] ==> x : ?B2(x,B)}
+\end{ttbox}
+The rest is routine. Note how~$\Var{B2}(x,B)$ is instantiated.
+\begin{ttbox}
+by (assume_tac 1);
+{\out Level 5}
+{\out Union(C) <= Union(D)}
+{\out 1. !!x B. [| x : B; B : C |] ==> x : B}
+by (assume_tac 1);
+{\out Level 6}
+{\out Union(C) <= Union(D)}
+{\out No subgoals!}
+\end{ttbox}
+Again, \ttindex{fast_tac} with \ttindex{ZF_cs} can do this proof in one
+step, provided we somehow supply it with~{\tt prem}. We can either add
+this premise to the assumptions using \ttindex{cut_facts_tac}, or add
+\hbox{\tt prem RS subsetD} to \ttindex{ZF_cs} as an introduction rule.
+
+The file \ttindex{ZF/equalities.ML} has many similar proofs.
+Reasoning about general intersection can be difficult because of its anomalous
+behavior on the empty set. However, \ttindex{fast_tac} copes well with
+these. Here is a typical example:
+\begin{ttbox}
+a:C ==> (INT x:C. A(x) Int B(x)) = (INT x:C.A(x)) Int (INT x:C.B(x))
+\end{ttbox}
+In traditional notation this is
+\[ a\in C \,\Imp\, \bigcap@{x\in C} \Bigl(A(x) \inter B(x)\Bigr) =
+ \Bigl(\bigcap@{x\in C} A(x)\Bigr) \inter
+ \Bigl(\bigcap@{x\in C} B(x)\Bigr) \]
+
+\section{Low-level reasoning about functions}
+The derived rules {\tt lamI}, {\tt lamE}, {\tt lam_type}, {\tt beta}
+and {\tt 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~\ttindex{fun_disjoint_apply1}. This states that if $f$ and~$g$ are
+functions with disjoint domains~$A$ and~$C$, and if $a\in A$, then
+$(f\union g)`a = f`a$:
+\begin{ttbox}
+val prems = goal ZF.thy
+ "[| a:A; f: A->B; g: C->D; A Int C = 0 |] ==> \ttback
+\ttback (f Un g)`a = f`a";
+{\out Level 0}
+{\out (f Un g) ` a = f ` a}
+{\out 1. (f Un g) ` a = f ` a}
+\end{ttbox}
+Using \ttindex{apply_equality}, we reduce the equality to reasoning about
+ordered pairs.
+\begin{ttbox}
+by (resolve_tac [apply_equality] 1);
+{\out Level 1}
+{\out (f Un g) ` a = f ` a}
+{\out 1. : f Un g}
+{\out 2. f Un g : (PROD x:?A. ?B(x))}
+\end{ttbox}
+We must show that the pair belongs to~$f$ or~$g$; by~\ttindex{UnI1} we
+choose~$f$:
+\begin{ttbox}
+by (resolve_tac [UnI1] 1);
+{\out Level 2}
+{\out (f Un g) ` a = f ` a}
+{\out 1. : f}
+{\out 2. f Un g : (PROD x:?A. ?B(x))}
+\end{ttbox}
+To show $\pair{a,f`a}\in f$ we use \ttindex{apply_Pair}, which is
+essentially the converse of \ttindex{apply_equality}:
+\begin{ttbox}
+by (resolve_tac [apply_Pair] 1);
+{\out Level 3}
+{\out (f Un g) ` a = f ` a}
+{\out 1. f : (PROD x:?A2. ?B2(x))}
+{\out 2. a : ?A2}
+{\out 3. f Un g : (PROD x:?A. ?B(x))}
+\end{ttbox}
+Using the premises $f\in A\to B$ and $a\in A$, we solve the two subgoals
+from \ttindex{apply_Pair}. Recall that a $\Pi$-set is merely a generalized
+function space, and observe that~{\tt?A2} is instantiated to~{\tt A}.
+\begin{ttbox}
+by (resolve_tac prems 1);
+{\out Level 4}
+{\out (f Un g) ` a = f ` a}
+{\out 1. a : A}
+{\out 2. f Un g : (PROD x:?A. ?B(x))}
+by (resolve_tac prems 1);
+{\out Level 5}
+{\out (f Un g) ` a = f ` a}
+{\out 1. f Un g : (PROD x:?A. ?B(x))}
+\end{ttbox}
+To construct functions of the form $f\union g$, we apply
+\ttindex{fun_disjoint_Un}:
+\begin{ttbox}
+by (resolve_tac [fun_disjoint_Un] 1);
+{\out Level 6}
+{\out (f Un g) ` a = f ` a}
+{\out 1. f : ?A3 -> ?B3}
+{\out 2. g : ?C3 -> ?D3}
+{\out 3. ?A3 Int ?C3 = 0}
+\end{ttbox}
+The remaining subgoals are instances of the premises. Again, observe how
+unknowns are instantiated:
+\begin{ttbox}
+by (resolve_tac prems 1);
+{\out Level 7}
+{\out (f Un g) ` a = f ` a}
+{\out 1. g : ?C3 -> ?D3}
+{\out 2. A Int ?C3 = 0}
+by (resolve_tac prems 1);
+{\out Level 8}
+{\out (f Un g) ` a = f ` a}
+{\out 1. A Int C = 0}
+by (resolve_tac prems 1);
+{\out Level 9}
+{\out (f Un g) ` a = f ` a}
+{\out No subgoals!}
+\end{ttbox}
+See the files \ttindex{ZF/func.ML} and \ttindex{ZF/wf.ML} for more
+examples of reasoning about functions.