%% $Id$

 \chapter{Zermelo-Fraenkel Set Theory}

 \index{set theory|(}

 The theory~\thydx{ZF} implements Zermelo-Fraenkel set

 theory~\cite{halmos60,suppes72} as an extension of~\texttt{FOL}, classical

 first-order logic.  The theory includes a collection of derived natural

 deduction rules, for use with Isabelle's classical reasoner.  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