xref: /seL4-l4v-10.1.1/isabelle/src/Doc/Logics_ZF/document/ZF.tex

\chapter{Zermelo-Fraenkel Set Theory}
\index{set theory|(}

The theory~\thydx{ZF} implements Zermelo-Fraenkel set
theory~\cite{halmos60,suppes72} as an extension of~\texttt{FOL}, classical
first-order logic.  The theory includes a collection of derived natural
deduction rules, for use with Isabelle's classical reasoner.  Some
of it is based on the work of No\"el~\cite{noel}.

A tremendous amount of set theory has been formally developed, including the
basic properties of relations, functions, ordinals and cardinals.  Significant
results have been proved, such as the Schr\"oder-Bernstein Theorem, the
Wellordering Theorem and a version of Ramsey's Theorem.  \texttt{ZF} provides
both the integers and the natural numbers.  General methods have been
developed for solving recursion equations over monotonic functors; these have
been applied to yield constructions of lists, trees, infinite lists, etc.

\texttt{ZF} has a flexible package for handling inductive definitions,
such as inference systems, and datatype definitions, such as lists and
trees.  Moreover it handles coinductive definitions, such as
bisimulation relations, and codatatype definitions, such as streams.  It
provides a streamlined syntax for defining primitive recursive functions over
datatypes. 

Published articles~\cite{paulson-set-I,paulson-set-II} describe \texttt{ZF}
less formally than this chapter.  Isabelle employs a novel treatment of
non-well-founded data structures within the standard {\sc zf} axioms including
the Axiom of Foundation~\cite{paulson-mscs}.


\section{Which version of axiomatic set theory?}
The two main axiom systems for set theory are Bernays-G\"odel~({\sc bg})
and Zermelo-Fraenkel~({\sc zf}).  Resolution theorem provers can use {\sc
  bg} because it is finite~\cite{boyer86,quaife92}.  {\sc zf} does not
have a finite axiom system because of its Axiom Scheme of Replacement.
This makes it awkward to use with many theorem provers, since instances
of the axiom scheme have to be invoked explicitly.  Since Isabelle has no
difficulty with axiom schemes, we may adopt either axiom system.

These two theories differ in their treatment of {\bf classes}, which are
collections that are `too big' to be sets.  The class of all sets,~$V$,
cannot be a set without admitting Russell's Paradox.  In {\sc bg}, both
classes and sets are individuals; $x\in V$ expresses that $x$ is a set.  In
{\sc zf}, all variables denote sets; classes are identified with unary
predicates.  The two systems define essentially the same sets and classes,
with similar properties.  In particular, a class cannot belong to another
class (let alone a set).

Modern set theorists tend to prefer {\sc zf} because they are mainly concerned
with sets, rather than classes.  {\sc bg} requires tiresome proofs that various
collections are sets; for instance, showing $x\in\{x\}$ requires showing that
$x$ is a set.


\begin{figure} \small
\begin{center}
\begin{tabular}{rrr} 
  \it name      &\it meta-type  & \it description \\ 
  \cdx{Let}     & $[\alpha,\alpha\To\beta]\To\beta$ & let binder\\
  \cdx{0}       & $i$           & empty set\\
  \cdx{cons}    & $[i,i]\To i$  & finite set constructor\\
  \cdx{Upair}   & $[i,i]\To i$  & unordered pairing\\
  \cdx{Pair}    & $[i,i]\To i$  & ordered pairing\\
  \cdx{Inf}     & $i$   & infinite set\\
  \cdx{Pow}     & $i\To i$      & powerset\\
  \cdx{Union} \cdx{Inter} & $i\To i$    & set union/intersection \\
  \cdx{split}   & $[[i,i]\To i, i] \To i$ & generalized projection\\
  \cdx{fst} \cdx{snd}   & $i\To i$      & projections\\
  \cdx{converse}& $i\To i$      & converse of a relation\\
  \cdx{succ}    & $i\To i$      & successor\\
  \cdx{Collect} & $[i,i\To o]\To i$     & separation\\
  \cdx{Replace} & $[i, [i,i]\To o] \To i$       & replacement\\
  \cdx{PrimReplace} & $[i, [i,i]\To o] \To i$   & primitive replacement\\
  \cdx{RepFun}  & $[i, i\To i] \To i$   & functional replacement\\
  \cdx{Pi} \cdx{Sigma}  & $[i,i\To i]\To i$     & general product/sum\\
  \cdx{domain}  & $i\To i$      & domain of a relation\\
  \cdx{range}   & $i\To i$      & range of a relation\\
  \cdx{field}   & $i\To i$      & field of a relation\\
  \cdx{Lambda}  & $[i, i\To i]\To i$    & $\lambda$-abstraction\\
  \cdx{restrict}& $[i, i] \To i$        & restriction of a function\\
  \cdx{The}     & $[i\To o]\To i$       & definite description\\
  \cdx{if}      & $[o,i,i]\To i$        & conditional\\
  \cdx{Ball} \cdx{Bex}  & $[i, i\To o]\To o$    & bounded quantifiers
\end{tabular}
\end{center}
\subcaption{Constants}

\begin{center}
\index{*"`"` symbol}
\index{*"-"`"` symbol}
\index{*"` symbol}\index{function applications}
\index{*"- symbol}
\index{*": symbol}
\index{*"<"= symbol}
\begin{tabular}{rrrr} 
  \it symbol  & \it meta-type & \it priority & \it description \\ 
  \tt ``        & $[i,i]\To i$  &  Left 90      & image \\
  \tt -``       & $[i,i]\To i$  &  Left 90      & inverse image \\
  \tt `         & $[i,i]\To i$  &  Left 90      & application \\
  \sdx{Int}     & $[i,i]\To i$  &  Left 70      & intersection ($\int$) \\
  \sdx{Un}      & $[i,i]\To i$  &  Left 65      & union ($\un$) \\
  \tt -         & $[i,i]\To i$  &  Left 65      & set difference ($-$) \\[1ex]
  \tt:          & $[i,i]\To o$  &  Left 50      & membership ($\in$) \\
  \tt <=        & $[i,i]\To o$  &  Left 50      & subset ($\subseteq$) 
\end{tabular}
\end{center}
\subcaption{Infixes}
\caption{Constants of ZF} \label{zf-constants}
\end{figure} 


\section{The syntax of set theory}
The language of set theory, as studied by logicians, has no constants.  The
traditional axioms merely assert the existence of empty sets, unions,
powersets, etc.; this would be intolerable for practical reasoning.  The
Isabelle theory declares constants for primitive sets.  It also extends
\texttt{FOL} with additional syntax for finite sets, ordered pairs,
comprehension, general union/intersection, general sums/products, and
bounded quantifiers.  In most other respects, Isabelle implements precisely
Zermelo-Fraenkel set theory.

Figure~\ref{zf-constants} lists the constants and infixes of~ZF, while
Figure~\ref{zf-trans} presents the syntax translations.  Finally,
Figure~\ref{zf-syntax} presents the full grammar for set theory, including the
constructs of FOL.

Local abbreviations can be introduced by a \isa{let} construct whose
syntax appears in Fig.\ts\ref{zf-syntax}.  Internally it is translated into
the constant~\cdx{Let}.  It can be expanded by rewriting with its
definition, \tdx{Let_def}.

Apart from \isa{let}, set theory does not use polymorphism.  All terms in
ZF have type~\tydx{i}, which is the type of individuals and has
class~\cldx{term}.  The type of first-order formulae, remember, 
is~\tydx{o}.

Infix operators include binary union and intersection ($A\un B$ and
$A\int B$), set difference ($A-B$), and the subset and membership
relations.  Note that $a$\verb|~:|$b$ is translated to $\lnot(a\in b)$,
which is equivalent to  $a\notin b$.  The
union and intersection operators ($\bigcup A$ and $\bigcap A$) form the
union or intersection of a set of sets; $\bigcup A$ means the same as
$\bigcup@{x\in A}x$.  Of these operators, only $\bigcup A$ is primitive.

The constant \cdx{Upair} constructs unordered pairs; thus \isa{Upair($A$,$B$)} denotes the set~$\{A,B\}$ and
\isa{Upair($A$,$A$)} denotes the singleton~$\{A\}$.  General union is
used to define binary union.  The Isabelle version goes on to define
the constant
\cdx{cons}:
\begin{eqnarray*}
   A\cup B              & \equiv &       \bigcup(\isa{Upair}(A,B)) \\
   \isa{cons}(a,B)      & \equiv &        \isa{Upair}(a,a) \un B
\end{eqnarray*}
The $\{a@1, \ldots\}$ notation abbreviates finite sets constructed in the
obvious manner using~\isa{cons} and~$\emptyset$ (the empty set) \isasymin \begin{eqnarray*}
 \{a,b,c\} & \equiv & \isa{cons}(a,\isa{cons}(b,\isa{cons}(c,\emptyset)))
\end{eqnarray*}

The constant \cdx{Pair} constructs ordered pairs, as in \isa{Pair($a$,$b$)}.  Ordered pairs may also be written within angle brackets,
as {\tt<$a$,$b$>}.  The $n$-tuple {\tt<$a@1$,\ldots,$a@{n-1}$,$a@n$>}
abbreviates the nest of pairs\par\nobreak
\centerline{\isa{Pair($a@1$,\ldots,Pair($a@{n-1}$,$a@n$)\ldots).}}

In ZF, a function is a set of pairs.  A ZF function~$f$ is simply an
individual as far as Isabelle is concerned: its Isabelle type is~$i$, not say
$i\To i$.  The infix operator~{\tt`} denotes the application of a function set
to its argument; we must write~$f{\tt`}x$, not~$f(x)$.  The syntax for image
is~$f{\tt``}A$ and that for inverse image is~$f{\tt-``}A$.


\begin{figure} 
\index{lambda abs@$\lambda$-abstractions}
\index{*"-"> symbol}
\index{*"* symbol}
\begin{center} \footnotesize\tt\frenchspacing
\begin{tabular}{rrr} 
  \it external          & \it internal  & \it description \\ 
  $a$ \ttilde: $b$      & \ttilde($a$ : $b$)    & \rm negated membership\\
  \ttlbrace$a@1$, $\ldots$, $a@n$\ttrbrace  &  cons($a@1$,$\ldots$,cons($a@n$,0)) &
        \rm finite set \\
  <$a@1$, $\ldots$, $a@{n-1}$, $a@n$> & 
        Pair($a@1$,\ldots,Pair($a@{n-1}$,$a@n$)\ldots) &
        \rm ordered $n$-tuple \\
  \ttlbrace$x$:$A . P[x]$\ttrbrace    &  Collect($A$,$\lambda x. P[x]$) &
        \rm separation \\
  \ttlbrace$y . x$:$A$, $Q[x,y]$\ttrbrace  &  Replace($A$,$\lambda x\,y. Q[x,y]$) &
        \rm replacement \\
  \ttlbrace$b[x] . x$:$A$\ttrbrace  &  RepFun($A$,$\lambda x. b[x]$) &
        \rm functional replacement \\
  \sdx{INT} $x$:$A . B[x]$      & Inter(\ttlbrace$B[x] . x$:$A$\ttrbrace) &
        \rm general intersection \\
  \sdx{UN}  $x$:$A . B[x]$      & Union(\ttlbrace$B[x] . x$:$A$\ttrbrace) &
        \rm general union \\
  \sdx{PROD} $x$:$A . B[x]$     & Pi($A$,$\lambda x. B[x]$) & 
        \rm general product \\
  \sdx{SUM}  $x$:$A . B[x]$     & Sigma($A$,$\lambda x. B[x]$) & 
        \rm general sum \\
  $A$ -> $B$            & Pi($A$,$\lambda x. B$) & 
        \rm function space \\
  $A$ * $B$             & Sigma($A$,$\lambda x. B$) & 
        \rm binary product \\
  \sdx{THE}  $x . P[x]$ & The($\lambda x. P[x]$) & 
        \rm definite description \\
  \sdx{lam}  $x$:$A . b[x]$     & Lambda($A$,$\lambda x. b[x]$) & 
        \rm $\lambda$-abstraction\\[1ex]
  \sdx{ALL} $x$:$A . P[x]$      & Ball($A$,$\lambda x. P[x]$) & 
        \rm bounded $\forall$ \\
  \sdx{EX}  $x$:$A . P[x]$      & Bex($A$,$\lambda x. P[x]$) & 
        \rm bounded $\exists$
\end{tabular}
\end{center}
\caption{Translations for ZF} \label{zf-trans}
\end{figure} 


\begin{figure} 
\index{*let symbol}
\index{*in symbol}
\dquotes
\[\begin{array}{rcl}
    term & = & \hbox{expression of type~$i$} \\
         & | & "let"~id~"="~term";"\dots";"~id~"="~term~"in"~term \\
         & | & "if"~term~"then"~term~"else"~term \\
         & | & "{\ttlbrace} " term\; ("," term)^* " {\ttrbrace}" \\
         & | & "< "  term\; ("," term)^* " >"  \\
         & | & "{\ttlbrace} " id ":" term " . " formula " {\ttrbrace}" \\
         & | & "{\ttlbrace} " id " . " id ":" term ", " formula " {\ttrbrace}" \\
         & | & "{\ttlbrace} " term " . " id ":" term " {\ttrbrace}" \\
         & | & term " `` " term \\
         & | & term " -`` " term \\
         & | & term " ` " term \\
         & | & term " * " term \\
         & | & term " \isasyminter " term \\
         & | & term " \isasymunion " term \\
         & | & term " - " term \\
         & | & term " -> " term \\
         & | & "THE~~"  id  " . " formula\\
         & | & "lam~~"  id ":" term " . " term \\
         & | & "INT~~"  id ":" term " . " term \\
         & | & "UN~~~"  id ":" term " . " term \\
         & | & "PROD~"  id ":" term " . " term \\
         & | & "SUM~~"  id ":" term " . " term \\[2ex]
 formula & = & \hbox{expression of type~$o$} \\
         & | & term " : " term \\
         & | & term " \ttilde: " term \\
         & | & term " <= " term \\
         & | & term " = " term \\
         & | & term " \ttilde= " term \\
         & | & "\ttilde\ " formula \\
         & | & formula " \& " formula \\
         & | & formula " | " formula \\
         & | & formula " --> " formula \\
         & | & formula " <-> " formula \\
         & | & "ALL " id ":" term " . " formula \\
         & | & "EX~~" id ":" term " . " formula \\
         & | & "ALL~" id~id^* " . " formula \\
         & | & "EX~~" id~id^* " . " formula \\
         & | & "EX!~" id~id^* " . " formula
  \end{array}
\]
\caption{Full grammar for ZF} \label{zf-syntax}
\end{figure} 


\section{Binding operators}
The constant \cdx{Collect} constructs sets by the principle of {\bf
  separation}.  The syntax for separation is
\hbox{\tt\ttlbrace$x$:$A$.\ $P[x]$\ttrbrace}, where $P[x]$ is a formula
that may contain free occurrences of~$x$.  It abbreviates the set \isa{Collect($A$,$\lambda x. P[x]$)}, which consists of all $x\in A$ that
satisfy~$P[x]$.  Note that \isa{Collect} is an unfortunate choice of
name: some set theories adopt a set-formation principle, related to
replacement, called collection.

The constant \cdx{Replace} constructs sets by the principle of {\bf
  replacement}.  The syntax
\hbox{\tt\ttlbrace$y$.\ $x$:$A$,$Q[x,y]$\ttrbrace} denotes the set 
\isa{Replace($A$,$\lambda x\,y. Q[x,y]$)}, which consists of all~$y$ such
that there exists $x\in A$ satisfying~$Q[x,y]$.  The Replacement Axiom
has the condition that $Q$ must be single-valued over~$A$: for
all~$x\in A$ there exists at most one $y$ satisfying~$Q[x,y]$.  A
single-valued binary predicate is also called a {\bf class function}.

The constant \cdx{RepFun} expresses a special case of replacement,
where $Q[x,y]$ has the form $y=b[x]$.  Such a $Q$ is trivially
single-valued, since it is just the graph of the meta-level
function~$\lambda x. b[x]$.  The resulting set consists of all $b[x]$
for~$x\in A$.  This is analogous to the \ML{} functional \isa{map},
since it applies a function to every element of a set.  The syntax is
\isa{\ttlbrace$b[x]$.\ $x$:$A$\ttrbrace}, which expands to 
\isa{RepFun($A$,$\lambda x. b[x]$)}.

\index{*INT symbol}\index{*UN symbol} 
General unions and intersections of indexed
families of sets, namely $\bigcup@{x\in A}B[x]$ and $\bigcap@{x\in A}B[x]$,
are written \isa{UN $x$:$A$.\ $B[x]$} and \isa{INT $x$:$A$.\ $B[x]$}.
Their meaning is expressed using \isa{RepFun} as
\[
\bigcup(\{B[x]. x\in A\}) \qquad\hbox{and}\qquad 
\bigcap(\{B[x]. x\in A\}). 
\]
General sums $\sum@{x\in A}B[x]$ and products $\prod@{x\in A}B[x]$ can be
constructed in set theory, where $B[x]$ is a family of sets over~$A$.  They
have as special cases $A\times B$ and $A\to B$, where $B$ is simply a set.
This is similar to the situation in Constructive Type Theory (set theory
has `dependent sets') and calls for similar syntactic conventions.  The
constants~\cdx{Sigma} and~\cdx{Pi} construct general sums and
products.  Instead of \isa{Sigma($A$,$B$)} and \isa{Pi($A$,$B$)} we may
write 
\isa{SUM $x$:$A$.\ $B[x]$} and \isa{PROD $x$:$A$.\ $B[x]$}.  
\index{*SUM symbol}\index{*PROD symbol}%
The special cases as \hbox{\tt$A$*$B$} and \hbox{\tt$A$->$B$} abbreviate
general sums and products over a constant family.\footnote{Unlike normal
infix operators, {\tt*} and {\tt->} merely define abbreviations; there are
no constants~\isa{op~*} and~\isa{op~->}.} Isabelle accepts these
abbreviations in parsing and uses them whenever possible for printing.

\index{*THE symbol} As mentioned above, whenever the axioms assert the
existence and uniqueness of a set, Isabelle's set theory declares a constant
for that set.  These constants can express the {\bf definite description}
operator~$\iota x. P[x]$, which stands for the unique~$a$ satisfying~$P[a]$,
if such exists.  Since all terms in ZF denote something, a description is
always meaningful, but we do not know its value unless $P[x]$ defines it
uniquely.  Using the constant~\cdx{The}, we may write descriptions as 
\isa{The($\lambda x. P[x]$)} or use the syntax \isa{THE $x$.\ $P[x]$}.

\index{*lam symbol}
Function sets may be written in $\lambda$-notation; $\lambda x\in A. b[x]$
stands for the set of all pairs $\pair{x,b[x]}$ for $x\in A$.  In order for
this to be a set, the function's domain~$A$ must be given.  Using the
constant~\cdx{Lambda}, we may express function sets as \isa{Lambda($A$,$\lambda x. b[x]$)} or use the syntax \isa{lam $x$:$A$.\ $b[x]$}.

Isabelle's set theory defines two {\bf bounded quantifiers}:
\begin{eqnarray*}
   \forall x\in A. P[x] &\hbox{abbreviates}& \forall x. x\in A\imp P[x] \\
   \exists x\in A. P[x] &\hbox{abbreviates}& \exists x. x\in A\conj P[x]
\end{eqnarray*}
The constants~\cdx{Ball} and~\cdx{Bex} are defined
accordingly.  Instead of \isa{Ball($A$,$P$)} and \isa{Bex($A$,$P$)} we may
write
\isa{ALL $x$:$A$.\ $P[x]$} and \isa{EX $x$:$A$.\ $P[x]$}.


%%%% ZF.thy

\begin{figure}
\begin{alltt*}\isastyleminor
\tdx{Let_def}:           Let(s, f) == f(s)

\tdx{Ball_def}:          Ball(A,P) == {\isasymforall}x. x \isasymin A --> P(x)
\tdx{Bex_def}:           Bex(A,P)  == {\isasymexists}x. x \isasymin A & P(x)

\tdx{subset_def}:        A \isasymsubseteq B  == {\isasymforall}x \isasymin A. x \isasymin B
\tdx{extension}:         A = B  <->  A \isasymsubseteq B & B \isasymsubseteq A

\tdx{Union_iff}:         A \isasymin Union(C) <-> ({\isasymexists}B \isasymin C. A \isasymin B)
\tdx{Pow_iff}:           A \isasymin Pow(B) <-> A \isasymsubseteq B
\tdx{foundation}:        A=0 | ({\isasymexists}x \isasymin A. {\isasymforall}y \isasymin x. y \isasymnotin A)

\tdx{replacement}:       ({\isasymforall}x \isasymin A. {\isasymforall}y z. P(x,y) & P(x,z) --> y=z) ==>
                   b \isasymin PrimReplace(A,P) <-> ({\isasymexists}x{\isasymin}A. P(x,b))
\subcaption{The Zermelo-Fraenkel Axioms}

\tdx{Replace_def}: Replace(A,P) == 
                   PrimReplace(A, \%x y. (\isasymexists!z. P(x,z)) & P(x,y))
\tdx{RepFun_def}:  RepFun(A,f)  == {\ttlbrace}y . x \isasymin A, y=f(x)\ttrbrace
\tdx{the_def}:     The(P)       == Union({\ttlbrace}y . x \isasymin {\ttlbrace}0{\ttrbrace}, P(y){\ttrbrace})
\tdx{if_def}:      if(P,a,b)    == THE z. P & z=a | ~P & z=b
\tdx{Collect_def}: Collect(A,P) == {\ttlbrace}y . x \isasymin A, x=y & P(x){\ttrbrace}
\tdx{Upair_def}:   Upair(a,b)   == 
               {\ttlbrace}y. x\isasymin{}Pow(Pow(0)), x=0 & y=a | x=Pow(0) & y=b{\ttrbrace}
\subcaption{Consequences of replacement}

\tdx{Inter_def}:   Inter(A) == {\ttlbrace}x \isasymin Union(A) . {\isasymforall}y \isasymin A. x \isasymin y{\ttrbrace}
\tdx{Un_def}:      A \isasymunion B  == Union(Upair(A,B))
\tdx{Int_def}:     A \isasyminter B  == Inter(Upair(A,B))
\tdx{Diff_def}:    A - B    == {\ttlbrace}x \isasymin A . x \isasymnotin B{\ttrbrace}
\subcaption{Union, intersection, difference}
\end{alltt*}
\caption{Rules and axioms of ZF} \label{zf-rules}
\end{figure}


\begin{figure}
\begin{alltt*}\isastyleminor
\tdx{cons_def}:    cons(a,A) == Upair(a,a) \isasymunion A
\tdx{succ_def}:    succ(i) == cons(i,i)
\tdx{infinity}:    0 \isasymin Inf & ({\isasymforall}y \isasymin Inf. succ(y) \isasymin Inf)
\subcaption{Finite and infinite sets}

\tdx{Pair_def}:      <a,b>      == {\ttlbrace}{\ttlbrace}a,a{\ttrbrace}, {\ttlbrace}a,b{\ttrbrace}{\ttrbrace}
\tdx{split_def}:     split(c,p) == THE y. {\isasymexists}a b. p=<a,b> & y=c(a,b)
\tdx{fst_def}:       fst(A)     == split(\%x y. x, p)
\tdx{snd_def}:       snd(A)     == split(\%x y. y, p)
\tdx{Sigma_def}:     Sigma(A,B) == {\isasymUnion}x \isasymin A. {\isasymUnion}y \isasymin B(x). {\ttlbrace}<x,y>{\ttrbrace}
\subcaption{Ordered pairs and Cartesian products}

\tdx{converse_def}: converse(r) == {\ttlbrace}z. w\isasymin{}r, {\isasymexists}x y. w=<x,y> & z=<y,x>{\ttrbrace}
\tdx{domain_def}:   domain(r)   == {\ttlbrace}x. w \isasymin r, {\isasymexists}y. w=<x,y>{\ttrbrace}
\tdx{range_def}:    range(r)    == domain(converse(r))
\tdx{field_def}:    field(r)    == domain(r) \isasymunion range(r)
\tdx{image_def}:    r `` A      == {\ttlbrace}y\isasymin{}range(r) . {\isasymexists}x \isasymin A. <x,y> \isasymin r{\ttrbrace}
\tdx{vimage_def}:   r -`` A     == converse(r)``A
\subcaption{Operations on relations}

\tdx{lam_def}:   Lambda(A,b) == {\ttlbrace}<x,b(x)> . x \isasymin A{\ttrbrace}
\tdx{apply_def}: f`a         == THE y. <a,y> \isasymin f
\tdx{Pi_def}: Pi(A,B) == {\ttlbrace}f\isasymin{}Pow(Sigma(A,B)). {\isasymforall}x\isasymin{}A. \isasymexists!y. <x,y>\isasymin{}f{\ttrbrace}
\tdx{restrict_def}:  restrict(f,A) == lam x \isasymin A. f`x
\subcaption{Functions and general product}
\end{alltt*}
\caption{Further definitions of ZF} \label{zf-defs}
\end{figure}



\section{The Zermelo-Fraenkel axioms}
The axioms appear in Fig.\ts \ref{zf-rules}.  They resemble those
presented by Suppes~\cite{suppes72}.  Most of the theory consists of
definitions.  In particular, bounded quantifiers and the subset relation
appear in other axioms.  Object-level quantifiers and implications have
been replaced by meta-level ones wherever possible, to simplify use of the
axioms.  

The traditional replacement axiom asserts
\[ y \in \isa{PrimReplace}(A,P) \bimp (\exists x\in A. P(x,y)) \]
subject to the condition that $P(x,y)$ is single-valued for all~$x\in A$.
The Isabelle theory defines \cdx{Replace} to apply
\cdx{PrimReplace} to the single-valued part of~$P$, namely
\[ (\exists!z. P(x,z)) \conj P(x,y). \]
Thus $y\in \isa{Replace}(A,P)$ if and only if there is some~$x$ such that
$P(x,-)$ holds uniquely for~$y$.  Because the equivalence is unconditional,
\isa{Replace} is much easier to use than \isa{PrimReplace}; it defines the
same set, if $P(x,y)$ is single-valued.  The nice syntax for replacement
expands to \isa{Replace}.

Other consequences of replacement include replacement for 
meta-level functions
(\cdx{RepFun}) and definite descriptions (\cdx{The}).
Axioms for separation (\cdx{Collect}) and unordered pairs
(\cdx{Upair}) are traditionally assumed, but they actually follow
from replacement~\cite[pages 237--8]{suppes72}.

The definitions of general intersection, etc., are straightforward.  Note
the definition of \isa{cons}, which underlies the finite set notation.
The axiom of infinity gives us a set that contains~0 and is closed under
successor (\cdx{succ}).  Although this set is not uniquely defined,
the theory names it (\cdx{Inf}) in order to simplify the
construction of the natural numbers.
                                             
Further definitions appear in Fig.\ts\ref{zf-defs}.  Ordered pairs are
defined in the standard way, $\pair{a,b}\equiv\{\{a\},\{a,b\}\}$.  Recall
that \cdx{Sigma}$(A,B)$ generalizes the Cartesian product of two
sets.  It is defined to be the union of all singleton sets
$\{\pair{x,y}\}$, for $x\in A$ and $y\in B(x)$.  This is a typical usage of
general union.

The projections \cdx{fst} and~\cdx{snd} are defined in terms of the
generalized projection \cdx{split}.  The latter has been borrowed from
Martin-L\"of's Type Theory, and is often easier to use than \cdx{fst}
and~\cdx{snd}.

Operations on relations include converse, domain, range, and image.  The
set $\isa{Pi}(A,B)$ generalizes the space of functions between two sets.
Note the simple definitions of $\lambda$-abstraction (using
\cdx{RepFun}) and application (using a definite description).  The
function \cdx{restrict}$(f,A)$ has the same values as~$f$, but only
over the domain~$A$.


%%%% zf.thy

\begin{figure}
\begin{alltt*}\isastyleminor
\tdx{ballI}:     [| !!x. x\isasymin{}A ==> P(x) |] ==> {\isasymforall}x\isasymin{}A. P(x)
\tdx{bspec}:     [| {\isasymforall}x\isasymin{}A. P(x);  x\isasymin{}A |] ==> P(x)
\tdx{ballE}:     [| {\isasymforall}x\isasymin{}A. P(x);  P(x) ==> Q;  x \isasymnotin A ==> Q |] ==> Q

\tdx{ball_cong}:  [| A=A';  !!x. x\isasymin{}A' ==> P(x) <-> P'(x) |] ==> 
             ({\isasymforall}x\isasymin{}A. P(x)) <-> ({\isasymforall}x\isasymin{}A'. P'(x))

\tdx{bexI}:      [| P(x);  x\isasymin{}A |] ==> {\isasymexists}x\isasymin{}A. P(x)
\tdx{bexCI}:     [| {\isasymforall}x\isasymin{}A. ~P(x) ==> P(a);  a\isasymin{}A |] ==> {\isasymexists}x\isasymin{}A. P(x)
\tdx{bexE}:      [| {\isasymexists}x\isasymin{}A. P(x);  !!x. [| x\isasymin{}A; P(x) |] ==> Q |] ==> Q

\tdx{bex_cong}:  [| A=A';  !!x. x\isasymin{}A' ==> P(x) <-> P'(x) |] ==> 
             ({\isasymexists}x\isasymin{}A. P(x)) <-> ({\isasymexists}x\isasymin{}A'. P'(x))
\subcaption{Bounded quantifiers}

\tdx{subsetI}:     (!!x. x \isasymin A ==> x \isasymin B) ==> A \isasymsubseteq B
\tdx{subsetD}:     [| A \isasymsubseteq B;  c \isasymin A |] ==> c \isasymin B
\tdx{subsetCE}:    [| A \isasymsubseteq B;  c \isasymnotin A ==> P;  c \isasymin B ==> P |] ==> P
\tdx{subset_refl}:  A \isasymsubseteq A
\tdx{subset_trans}: [| A \isasymsubseteq B;  B \isasymsubseteq C |] ==> A \isasymsubseteq C

\tdx{equalityI}:   [| A \isasymsubseteq B;  B \isasymsubseteq A |] ==> A = B
\tdx{equalityD1}:  A = B ==> A \isasymsubseteq B
\tdx{equalityD2}:  A = B ==> B \isasymsubseteq A
\tdx{equalityE}:   [| A = B;  [| A \isasymsubseteq B; B \isasymsubseteq A |] ==> P |]  ==>  P
\subcaption{Subsets and extensionality}

\tdx{emptyE}:        a \isasymin 0 ==> P
\tdx{empty_subsetI}:  0 \isasymsubseteq A
\tdx{equals0I}:      [| !!y. y \isasymin A ==> False |] ==> A=0
\tdx{equals0D}:      [| A=0;  a \isasymin A |] ==> P

\tdx{PowI}:          A \isasymsubseteq B ==> A \isasymin Pow(B)
\tdx{PowD}:          A \isasymin Pow(B)  ==>  A \isasymsubseteq B
\subcaption{The empty set; power sets}
\end{alltt*}
\caption{Basic derived rules for ZF} \label{zf-lemmas1}
\end{figure}


\section{From basic lemmas to function spaces}
Faced with so many definitions, it is essential to prove lemmas.  Even
trivial theorems like $A \int B = B \int A$ would be difficult to
prove from the definitions alone.  Isabelle's set theory derives many
rules using a natural deduction style.  Ideally, a natural deduction
rule should introduce or eliminate just one operator, but this is not
always practical.  For most operators, we may forget its definition
and use its derived rules instead.

\subsection{Fundamental lemmas}
Figure~\ref{zf-lemmas1} presents the derived rules for the most basic
operators.  The rules for the bounded quantifiers resemble those for the
ordinary quantifiers, but note that \tdx{ballE} uses a negated assumption
in the style of Isabelle's classical reasoner.  The \rmindex{congruence
  rules} \tdx{ball_cong} and \tdx{bex_cong} are required by Isabelle's
simplifier, but have few other uses.  Congruence rules must be specially
derived for all binding operators, and henceforth will not be shown.

Figure~\ref{zf-lemmas1} also shows rules for the subset and equality
relations (proof by extensionality), and rules about the empty set and the
power set operator.

Figure~\ref{zf-lemmas2} presents rules for replacement and separation.
The rules for \cdx{Replace} and \cdx{RepFun} are much simpler than
comparable rules for \isa{PrimReplace} would be.  The principle of
separation is proved explicitly, although most proofs should use the
natural deduction rules for \isa{Collect}.  The elimination rule
\tdx{CollectE} is equivalent to the two destruction rules
\tdx{CollectD1} and \tdx{CollectD2}, but each rule is suited to
particular circumstances.  Although too many rules can be confusing, there
is no reason to aim for a minimal set of rules.  

Figure~\ref{zf-lemmas3} presents rules for general union and intersection.
The empty intersection should be undefined.  We cannot have
$\bigcap(\emptyset)=V$ because $V$, the universal class, is not a set.  All
expressions denote something in ZF set theory; the definition of
intersection implies $\bigcap(\emptyset)=\emptyset$, but this value is
arbitrary.  The rule \tdx{InterI} must have a premise to exclude
the empty intersection.  Some of the laws governing intersections require
similar premises.


%the [p] gives better page breaking for the book
\begin{figure}[p]
\begin{alltt*}\isastyleminor
\tdx{ReplaceI}:   [| x\isasymin{}A;  P(x,b);  !!y. P(x,y) ==> y=b |] ==> 
            b\isasymin{}{\ttlbrace}y. x\isasymin{}A, P(x,y){\ttrbrace}

\tdx{ReplaceE}:   [| b\isasymin{}{\ttlbrace}y. x\isasymin{}A, P(x,y){\ttrbrace};  
               !!x. [| x\isasymin{}A; P(x,b); {\isasymforall}y. P(x,y)-->y=b |] ==> R 
            |] ==> R

\tdx{RepFunI}:    [| a\isasymin{}A |] ==> f(a)\isasymin{}{\ttlbrace}f(x). x\isasymin{}A{\ttrbrace}
\tdx{RepFunE}:    [| b\isasymin{}{\ttlbrace}f(x). x\isasymin{}A{\ttrbrace};  
                !!x.[| x\isasymin{}A;  b=f(x) |] ==> P |] ==> P

\tdx{separation}:  a\isasymin{}{\ttlbrace}x\isasymin{}A. P(x){\ttrbrace} <-> a\isasymin{}A & P(a)
\tdx{CollectI}:    [| a\isasymin{}A;  P(a) |] ==> a\isasymin{}{\ttlbrace}x\isasymin{}A. P(x){\ttrbrace}
\tdx{CollectE}:    [| a\isasymin{}{\ttlbrace}x\isasymin{}A. P(x){\ttrbrace};  [| a\isasymin{}A; P(a) |] ==> R |] ==> R
\tdx{CollectD1}:   a\isasymin{}{\ttlbrace}x\isasymin{}A. P(x){\ttrbrace} ==> a\isasymin{}A
\tdx{CollectD2}:   a\isasymin{}{\ttlbrace}x\isasymin{}A. P(x){\ttrbrace} ==> P(a)
\end{alltt*}
\caption{Replacement and separation} \label{zf-lemmas2}
\end{figure}


\begin{figure}
\begin{alltt*}\isastyleminor
\tdx{UnionI}: [| B\isasymin{}C;  A\isasymin{}B |] ==> A\isasymin{}Union(C)
\tdx{UnionE}: [| A\isasymin{}Union(C);  !!B.[| A\isasymin{}B;  B\isasymin{}C |] ==> R |] ==> R

\tdx{InterI}: [| !!x. x\isasymin{}C ==> A\isasymin{}x;  c\isasymin{}C |] ==> A\isasymin{}Inter(C)
\tdx{InterD}: [| A\isasymin{}Inter(C);  B\isasymin{}C |] ==> A\isasymin{}B
\tdx{InterE}: [| A\isasymin{}Inter(C);  A\isasymin{}B ==> R;  B \isasymnotin C ==> R |] ==> R

\tdx{UN_I}:   [| a\isasymin{}A;  b\isasymin{}B(a) |] ==> b\isasymin{}({\isasymUnion}x\isasymin{}A. B(x))
\tdx{UN_E}:   [| b\isasymin{}({\isasymUnion}x\isasymin{}A. B(x));  !!x.[| x\isasymin{}A;  b\isasymin{}B(x) |] ==> R 
           |] ==> R

\tdx{INT_I}:  [| !!x. x\isasymin{}A ==> b\isasymin{}B(x);  a\isasymin{}A |] ==> b\isasymin{}({\isasymInter}x\isasymin{}A. B(x))
\tdx{INT_E}:  [| b\isasymin{}({\isasymInter}x\isasymin{}A. B(x));  a\isasymin{}A |] ==> b\isasymin{}B(a)
\end{alltt*}
\caption{General union and intersection} \label{zf-lemmas3}
\end{figure}


%%% upair.thy

\begin{figure}
\begin{alltt*}\isastyleminor
\tdx{pairing}:   a\isasymin{}Upair(b,c) <-> (a=b | a=c)
\tdx{UpairI1}:   a\isasymin{}Upair(a,b)
\tdx{UpairI2}:   b\isasymin{}Upair(a,b)
\tdx{UpairE}:    [| a\isasymin{}Upair(b,c);  a=b ==> P;  a=c ==> P |] ==> P
\end{alltt*}
\caption{Unordered pairs} \label{zf-upair1}
\end{figure}


\begin{figure}
\begin{alltt*}\isastyleminor
\tdx{UnI1}:      c\isasymin{}A ==> c\isasymin{}A \isasymunion B
\tdx{UnI2}:      c\isasymin{}B ==> c\isasymin{}A \isasymunion B
\tdx{UnCI}:      (c \isasymnotin B ==> c\isasymin{}A) ==> c\isasymin{}A \isasymunion B
\tdx{UnE}:       [| c\isasymin{}A \isasymunion B;  c\isasymin{}A ==> P;  c\isasymin{}B ==> P |] ==> P

\tdx{IntI}:      [| c\isasymin{}A;  c\isasymin{}B |] ==> c\isasymin{}A \isasyminter B
\tdx{IntD1}:     c\isasymin{}A \isasyminter B ==> c\isasymin{}A
\tdx{IntD2}:     c\isasymin{}A \isasyminter B ==> c\isasymin{}B
\tdx{IntE}:      [| c\isasymin{}A \isasyminter B;  [| c\isasymin{}A; c\isasymin{}B |] ==> P |] ==> P

\tdx{DiffI}:     [| c\isasymin{}A;  c \isasymnotin B |] ==> c\isasymin{}A - B
\tdx{DiffD1}:    c\isasymin{}A - B ==> c\isasymin{}A
\tdx{DiffD2}:    c\isasymin{}A - B ==> c  \isasymnotin  B
\tdx{DiffE}:     [| c\isasymin{}A - B;  [| c\isasymin{}A; c \isasymnotin B |] ==> P |] ==> P
\end{alltt*}
\caption{Union, intersection, difference} \label{zf-Un}
\end{figure}


\begin{figure}
\begin{alltt*}\isastyleminor
\tdx{consI1}:    a\isasymin{}cons(a,B)
\tdx{consI2}:    a\isasymin{}B ==> a\isasymin{}cons(b,B)
\tdx{consCI}:    (a \isasymnotin B ==> a=b) ==> a\isasymin{}cons(b,B)
\tdx{consE}:     [| a\isasymin{}cons(b,A);  a=b ==> P;  a\isasymin{}A ==> P |] ==> P

\tdx{singletonI}:  a\isasymin{}{\ttlbrace}a{\ttrbrace}
\tdx{singletonE}:  [| a\isasymin{}{\ttlbrace}b{\ttrbrace}; a=b ==> P |] ==> P
\end{alltt*}
\caption{Finite and singleton sets} \label{zf-upair2}
\end{figure}


\begin{figure}
\begin{alltt*}\isastyleminor
\tdx{succI1}:    i\isasymin{}succ(i)
\tdx{succI2}:    i\isasymin{}j ==> i\isasymin{}succ(j)
\tdx{succCI}:    (i \isasymnotin j ==> i=j) ==> i\isasymin{}succ(j)
\tdx{succE}:     [| i\isasymin{}succ(j);  i=j ==> P;  i\isasymin{}j ==> P |] ==> P
\tdx{succ_neq_0}:  [| succ(n)=0 |] ==> P
\tdx{succ_inject}: succ(m) = succ(n) ==> m=n
\end{alltt*}
\caption{The successor function} \label{zf-succ}
\end{figure}


\begin{figure}
\begin{alltt*}\isastyleminor
\tdx{the_equality}: [| P(a); !!x. P(x) ==> x=a |] ==> (THE x. P(x))=a
\tdx{theI}:         \isasymexists! x. P(x) ==> P(THE x. P(x))

\tdx{if_P}:          P ==> (if P then a else b) = a
\tdx{if_not_P}:     ~P ==> (if P then a else b) = b

\tdx{mem_asym}:     [| a\isasymin{}b;  b\isasymin{}a |] ==> P
\tdx{mem_irrefl}:   a\isasymin{}a ==> P
\end{alltt*}
\caption{Descriptions; non-circularity} \label{zf-the}
\end{figure}


\subsection{Unordered pairs and finite sets}
Figure~\ref{zf-upair1} presents the principle of unordered pairing, along
with its derived rules.  Binary union and intersection are defined in terms
of ordered pairs (Fig.\ts\ref{zf-Un}).  Set difference is also included.  The
rule \tdx{UnCI} is useful for classical reasoning about unions,
like \isa{disjCI}\@; it supersedes \tdx{UnI1} and
\tdx{UnI2}, but these rules are often easier to work with.  For
intersection and difference we have both elimination and destruction rules.
Again, there is no reason to provide a minimal rule set.

Figure~\ref{zf-upair2} is concerned with finite sets: it presents rules
for~\isa{cons}, the finite set constructor, and rules for singleton
sets.  Figure~\ref{zf-succ} presents derived rules for the successor
function, which is defined in terms of~\isa{cons}.  The proof that 
\isa{succ} is injective appears to require the Axiom of Foundation.

Definite descriptions (\sdx{THE}) are defined in terms of the singleton
set~$\{0\}$, but their derived rules fortunately hide this
(Fig.\ts\ref{zf-the}).  The rule~\tdx{theI} is difficult to apply
because of the two occurrences of~$\Var{P}$.  However,
\tdx{the_equality} does not have this problem and the files contain
many examples of its use.

Finally, the impossibility of having both $a\in b$ and $b\in a$
(\tdx{mem_asym}) is proved by applying the Axiom of Foundation to
the set $\{a,b\}$.  The impossibility of $a\in a$ is a trivial consequence.


%%% subset.thy?

\begin{figure}
\begin{alltt*}\isastyleminor
\tdx{Union_upper}:    B\isasymin{}A ==> B \isasymsubseteq Union(A)
\tdx{Union_least}:    [| !!x. x\isasymin{}A ==> x \isasymsubseteq C |] ==> Union(A) \isasymsubseteq C

\tdx{Inter_lower}:    B\isasymin{}A ==> Inter(A) \isasymsubseteq B
\tdx{Inter_greatest}: [| a\isasymin{}A; !!x. x\isasymin{}A ==> C \isasymsubseteq x |] ==> C\isasymsubseteq{}Inter(A)

\tdx{Un_upper1}:      A \isasymsubseteq A \isasymunion B
\tdx{Un_upper2}:      B \isasymsubseteq A \isasymunion B
\tdx{Un_least}:       [| A \isasymsubseteq C;  B \isasymsubseteq C |] ==> A \isasymunion B \isasymsubseteq C

\tdx{Int_lower1}:     A \isasyminter B \isasymsubseteq A
\tdx{Int_lower2}:     A \isasyminter B \isasymsubseteq B
\tdx{Int_greatest}:   [| C \isasymsubseteq A;  C \isasymsubseteq B |] ==> C \isasymsubseteq A \isasyminter B

\tdx{Diff_subset}:    A-B \isasymsubseteq A
\tdx{Diff_contains}:  [| C \isasymsubseteq A;  C \isasyminter B = 0 |] ==> C \isasymsubseteq A-B

\tdx{Collect_subset}: Collect(A,P) \isasymsubseteq A
\end{alltt*}
\caption{Subset and lattice properties} \label{zf-subset}
\end{figure}


\subsection{Subset and lattice properties}
The subset relation is a complete lattice.  Unions form least upper bounds;
non-empty intersections form greatest lower bounds.  Figure~\ref{zf-subset}
shows the corresponding rules.  A few other laws involving subsets are
included. 
Reasoning directly about subsets often yields clearer proofs than
reasoning about the membership relation.  Section~\ref{sec:ZF-pow-example}
below presents an example of this, proving the equation 
${\isa{Pow}(A)\cap \isa{Pow}(B)}= \isa{Pow}(A\cap B)$.

%%% pair.thy

\begin{figure}
\begin{alltt*}\isastyleminor
\tdx{Pair_inject1}: <a,b> = <c,d> ==> a=c
\tdx{Pair_inject2}: <a,b> = <c,d> ==> b=d
\tdx{Pair_inject}:  [| <a,b> = <c,d>;  [| a=c; b=d |] ==> P |] ==> P
\tdx{Pair_neq_0}:   <a,b>=0 ==> P

\tdx{fst_conv}:     fst(<a,b>) = a
\tdx{snd_conv}:     snd(<a,b>) = b
\tdx{split}:        split(\%x y. c(x,y), <a,b>) = c(a,b)

\tdx{SigmaI}:     [| a\isasymin{}A;  b\isasymin{}B(a) |] ==> <a,b>\isasymin{}Sigma(A,B)

\tdx{SigmaE}:     [| c\isasymin{}Sigma(A,B);  
                !!x y.[| x\isasymin{}A; y\isasymin{}B(x); c=<x,y> |] ==> P |] ==> P

\tdx{SigmaE2}:    [| <a,b>\isasymin{}Sigma(A,B);    
                [| a\isasymin{}A;  b\isasymin{}B(a) |] ==> P   |] ==> P
\end{alltt*}
\caption{Ordered pairs; projections; general sums} \label{zf-pair}
\end{figure}


\subsection{Ordered pairs} \label{sec:pairs}

Figure~\ref{zf-pair} presents the rules governing ordered pairs,
projections and general sums --- in particular, that
$\{\{a\},\{a,b\}\}$ functions as an ordered pair.  This property is
expressed as two destruction rules,
\tdx{Pair_inject1} and \tdx{Pair_inject2}, and equivalently
as the elimination rule \tdx{Pair_inject}.

The rule \tdx{Pair_neq_0} asserts $\pair{a,b}\neq\emptyset$.  This
is a property of $\{\{a\},\{a,b\}\}$, and need not hold for other 
encodings of ordered pairs.  The non-standard ordered pairs mentioned below
satisfy $\pair{\emptyset;\emptyset}=\emptyset$.

The natural deduction rules \tdx{SigmaI} and \tdx{SigmaE}
assert that \cdx{Sigma}$(A,B)$ consists of all pairs of the form
$\pair{x,y}$, for $x\in A$ and $y\in B(x)$.  The rule \tdx{SigmaE2}
merely states that $\pair{a,b}\in \isa{Sigma}(A,B)$ implies $a\in A$ and
$b\in B(a)$.

In addition, it is possible to use tuples as patterns in abstractions:
\begin{center}
{\tt\%<$x$,$y$>. $t$} \quad stands for\quad \isa{split(\%$x$ $y$.\ $t$)}
\end{center}
Nested patterns are translated recursively:
{\tt\%<$x$,$y$,$z$>. $t$} $\leadsto$ {\tt\%<$x$,<$y$,$z$>>. $t$} $\leadsto$
\isa{split(\%$x$.\%<$y$,$z$>. $t$)} $\leadsto$ \isa{split(\%$x$. split(\%$y$
  $z$.\ $t$))}.  The reverse translation is performed upon printing.
\begin{warn}
  The translation between patterns and \isa{split} is performed automatically
  by the parser and printer.  Thus the internal and external form of a term
  may differ, which affects proofs.  For example the term \isa{(\%<x,y>.<y,x>)<a,b>} requires the theorem \isa{split} to rewrite to
  {\tt<b,a>}.
\end{warn}
In addition to explicit $\lambda$-abstractions, patterns can be used in any
variable binding construct which is internally described by a
$\lambda$-abstraction.  Here are some important examples:
\begin{description}
\item[Let:] \isa{let {\it pattern} = $t$ in $u$}
\item[Choice:] \isa{THE~{\it pattern}~.~$P$}
\item[Set operations:] \isa{\isasymUnion~{\it pattern}:$A$.~$B$}
\item[Comprehension:] \isa{{\ttlbrace}~{\it pattern}:$A$~.~$P$~{\ttrbrace}}
\end{description}


%%% domrange.thy?

\begin{figure}
\begin{alltt*}\isastyleminor
\tdx{domainI}:     <a,b>\isasymin{}r ==> a\isasymin{}domain(r)
\tdx{domainE}:     [| a\isasymin{}domain(r); !!y. <a,y>\isasymin{}r ==> P |] ==> P
\tdx{domain_subset}: domain(Sigma(A,B)) \isasymsubseteq A

\tdx{rangeI}:      <a,b>\isasymin{}r ==> b\isasymin{}range(r)
\tdx{rangeE}:      [| b\isasymin{}range(r); !!x. <x,b>\isasymin{}r ==> P |] ==> P
\tdx{range_subset}: range(A*B) \isasymsubseteq B

\tdx{fieldI1}:     <a,b>\isasymin{}r ==> a\isasymin{}field(r)
\tdx{fieldI2}:     <a,b>\isasymin{}r ==> b\isasymin{}field(r)
\tdx{fieldCI}:     (<c,a> \isasymnotin r ==> <a,b>\isasymin{}r) ==> a\isasymin{}field(r)

\tdx{fieldE}:      [| a\isasymin{}field(r); 
                !!x. <a,x>\isasymin{}r ==> P; 
                !!x. <x,a>\isasymin{}r ==> P      
             |] ==> P

\tdx{field_subset}:  field(A*A) \isasymsubseteq A
\end{alltt*}
\caption{Domain, range and field of a relation} \label{zf-domrange}
\end{figure}

\begin{figure}
\begin{alltt*}\isastyleminor
\tdx{imageI}:      [| <a,b>\isasymin{}r; a\isasymin{}A |] ==> b\isasymin{}r``A
\tdx{imageE}:      [| b\isasymin{}r``A; !!x.[| <x,b>\isasymin{}r; x\isasymin{}A |] ==> P |] ==> P

\tdx{vimageI}:     [| <a,b>\isasymin{}r; b\isasymin{}B |] ==> a\isasymin{}r-``B
\tdx{vimageE}:     [| a\isasymin{}r-``B; !!x.[| <a,x>\isasymin{}r;  x\isasymin{}B |] ==> P |] ==> P
\end{alltt*}
\caption{Image and inverse image} \label{zf-domrange2}
\end{figure}


\subsection{Relations}
Figure~\ref{zf-domrange} presents rules involving relations, which are sets
of ordered pairs.  The converse of a relation~$r$ is the set of all pairs
$\pair{y,x}$ such that $\pair{x,y}\in r$; if $r$ is a function, then
{\cdx{converse}$(r)$} is its inverse.  The rules for the domain
operation, namely \tdx{domainI} and~\tdx{domainE}, assert that
\cdx{domain}$(r)$ consists of all~$x$ such that $r$ contains
some pair of the form~$\pair{x,y}$.  The range operation is similar, and
the field of a relation is merely the union of its domain and range.  

Figure~\ref{zf-domrange2} presents rules for images and inverse images.
Note that these operations are generalisations of range and domain,
respectively. 


%%% func.thy

\begin{figure}
\begin{alltt*}\isastyleminor
\tdx{fun_is_rel}:     f\isasymin{}Pi(A,B) ==> f \isasymsubseteq Sigma(A,B)

\tdx{apply_equality}:  [| <a,b>\isasymin{}f; f\isasymin{}Pi(A,B) |] ==> f`a = b
\tdx{apply_equality2}: [| <a,b>\isasymin{}f; <a,c>\isasymin{}f; f\isasymin{}Pi(A,B) |] ==> b=c

\tdx{apply_type}:     [| f\isasymin{}Pi(A,B); a\isasymin{}A |] ==> f`a\isasymin{}B(a)
\tdx{apply_Pair}:     [| f\isasymin{}Pi(A,B); a\isasymin{}A |] ==> <a,f`a>\isasymin{}f
\tdx{apply_iff}:      f\isasymin{}Pi(A,B) ==> <a,b>\isasymin{}f <-> a\isasymin{}A & f`a = b

\tdx{fun_extension}:  [| f\isasymin{}Pi(A,B); g\isasymin{}Pi(A,D);
                   !!x. x\isasymin{}A ==> f`x = g`x     |] ==> f=g

\tdx{domain_type}:    [| <a,b>\isasymin{}f; f\isasymin{}Pi(A,B) |] ==> a\isasymin{}A
\tdx{range_type}:     [| <a,b>\isasymin{}f; f\isasymin{}Pi(A,B) |] ==> b\isasymin{}B(a)

\tdx{Pi_type}:        [| f\isasymin{}A->C; !!x. x\isasymin{}A ==> f`x\isasymin{}B(x) |] ==> f\isasymin{}Pi(A,B)
\tdx{domain_of_fun}:  f\isasymin{}Pi(A,B) ==> domain(f)=A
\tdx{range_of_fun}:   f\isasymin{}Pi(A,B) ==> f\isasymin{}A->range(f)

\tdx{restrict}:       a\isasymin{}A ==> restrict(f,A) ` a = f`a
\tdx{restrict_type}:  [| !!x. x\isasymin{}A ==> f`x\isasymin{}B(x) |] ==> 
                restrict(f,A)\isasymin{}Pi(A,B)
\end{alltt*}
\caption{Functions} \label{zf-func1}
\end{figure}


\begin{figure}
\begin{alltt*}\isastyleminor
\tdx{lamI}:     a\isasymin{}A ==> <a,b(a)>\isasymin{}(lam x\isasymin{}A. b(x))
\tdx{lamE}:     [| p\isasymin{}(lam x\isasymin{}A. b(x)); !!x.[| x\isasymin{}A; p=<x,b(x)> |] ==> P 
          |] ==>  P

\tdx{lam_type}: [| !!x. x\isasymin{}A ==> b(x)\isasymin{}B(x) |] ==> (lam x\isasymin{}A. b(x))\isasymin{}Pi(A,B)

\tdx{beta}:     a\isasymin{}A ==> (lam x\isasymin{}A. b(x)) ` a = b(a)
\tdx{eta}:      f\isasymin{}Pi(A,B) ==> (lam x\isasymin{}A. f`x) = f
\end{alltt*}
\caption{$\lambda$-abstraction} \label{zf-lam}
\end{figure}


\begin{figure}
\begin{alltt*}\isastyleminor
\tdx{fun_empty}:           0\isasymin{}0->0
\tdx{fun_single}:          {\ttlbrace}<a,b>{\ttrbrace}\isasymin{}{\ttlbrace}a{\ttrbrace} -> {\ttlbrace}b{\ttrbrace}

\tdx{fun_disjoint_Un}:     [| f\isasymin{}A->B; g\isasymin{}C->D; A \isasyminter C = 0  |] ==>  
                     (f \isasymunion g)\isasymin{}(A \isasymunion C) -> (B \isasymunion D)

\tdx{fun_disjoint_apply1}: [| a\isasymin{}A; f\isasymin{}A->B; g\isasymin{}C->D;  A\isasyminter{}C = 0 |] ==>  
                     (f \isasymunion g)`a = f`a

\tdx{fun_disjoint_apply2}: [| c\isasymin{}C; f\isasymin{}A->B; g\isasymin{}C->D;  A\isasyminter{}C = 0 |] ==>  
                     (f \isasymunion g)`c = g`c
\end{alltt*}
\caption{Constructing functions from smaller sets} \label{zf-func2}
\end{figure}


\subsection{Functions}
Functions, represented by graphs, are notoriously difficult to reason
about.  The ZF theory provides many derived rules, which overlap more
than they ought.  This section presents the more important rules.

Figure~\ref{zf-func1} presents the basic properties of \cdx{Pi}$(A,B)$,
the generalized function space.  For example, if $f$ is a function and
$\pair{a,b}\in f$, then $f`a=b$ (\tdx{apply_equality}).  Two functions
are equal provided they have equal domains and deliver equals results
(\tdx{fun_extension}).

By \tdx{Pi_type}, a function typing of the form $f\in A\to C$ can be
refined to the dependent typing $f\in\prod@{x\in A}B(x)$, given a suitable
family of sets $\{B(x)\}@{x\in A}$.  Conversely, by \tdx{range_of_fun},
any dependent typing can be flattened to yield a function type of the form
$A\to C$; here, $C=\isa{range}(f)$.

Among the laws for $\lambda$-abstraction, \tdx{lamI} and \tdx{lamE}
describe the graph of the generated function, while \tdx{beta} and
\tdx{eta} are the standard conversions.  We essentially have a
dependently-typed $\lambda$-calculus (Fig.\ts\ref{zf-lam}).

Figure~\ref{zf-func2} presents some rules that can be used to construct
functions explicitly.  We start with functions consisting of at most one
pair, and may form the union of two functions provided their domains are
disjoint.  


\begin{figure}
\begin{alltt*}\isastyleminor
\tdx{Int_absorb}:        A \isasyminter A = A
\tdx{Int_commute}:       A \isasyminter B = B \isasyminter A
\tdx{Int_assoc}:         (A \isasyminter B) \isasyminter C  =  A \isasyminter (B \isasyminter C)
\tdx{Int_Un_distrib}:    (A \isasymunion B) \isasyminter C  =  (A \isasyminter C) \isasymunion (B \isasyminter C)

\tdx{Un_absorb}:         A \isasymunion A = A
\tdx{Un_commute}:        A \isasymunion B = B \isasymunion A
\tdx{Un_assoc}:          (A \isasymunion B) \isasymunion C  =  A \isasymunion (B \isasymunion C)
\tdx{Un_Int_distrib}:    (A \isasyminter B) \isasymunion C  =  (A \isasymunion C) \isasyminter (B \isasymunion C)

\tdx{Diff_cancel}:       A-A = 0
\tdx{Diff_disjoint}:     A \isasyminter (B-A) = 0
\tdx{Diff_partition}:    A \isasymsubseteq B ==> A \isasymunion (B-A) = B
\tdx{double_complement}: [| A \isasymsubseteq B; B \isasymsubseteq C |] ==> (B - (C-A)) = A
\tdx{Diff_Un}:           A - (B \isasymunion C) = (A-B) \isasyminter (A-C)
\tdx{Diff_Int}:          A - (B \isasyminter C) = (A-B) \isasymunion (A-C)

\tdx{Union_Un_distrib}:  Union(A \isasymunion B) = Union(A) \isasymunion Union(B)
\tdx{Inter_Un_distrib}:  [| a \isasymin A;  b \isasymin B |] ==> 
                   Inter(A \isasymunion B) = Inter(A) \isasyminter Inter(B)

\tdx{Int_Union_RepFun}:  A \isasyminter Union(B) = ({\isasymUnion}C \isasymin B. A \isasyminter C)

\tdx{Un_Inter_RepFun}:   b \isasymin B ==> 
                   A \isasymunion Inter(B) = ({\isasymInter}C \isasymin B. A \isasymunion C)

\tdx{SUM_Un_distrib1}:   (SUM x \isasymin A \isasymunion B. C(x)) = 
                   (SUM x \isasymin A. C(x)) \isasymunion (SUM x \isasymin B. C(x))

\tdx{SUM_Un_distrib2}:   (SUM x \isasymin C. A(x) \isasymunion B(x)) =
                   (SUM x \isasymin C. A(x)) \isasymunion (SUM x \isasymin C. B(x))

\tdx{SUM_Int_distrib1}:  (SUM x \isasymin A \isasyminter B. C(x)) =
                   (SUM x \isasymin A. C(x)) \isasyminter (SUM x \isasymin B. C(x))

\tdx{SUM_Int_distrib2}:  (SUM x \isasymin C. A(x) \isasyminter B(x)) =
                   (SUM x \isasymin C. A(x)) \isasyminter (SUM x \isasymin C. B(x))
\end{alltt*}
\caption{Equalities} \label{zf-equalities}
\end{figure}


\begin{figure}
%\begin{constants} 
%  \cdx{1}       & $i$           &       & $\{\emptyset\}$       \\
%  \cdx{bool}    & $i$           &       & the set $\{\emptyset,1\}$     \\
%  \cdx{cond}   & $[i,i,i]\To i$ &       & conditional for \isa{bool}    \\
%  \cdx{not}    & $i\To i$       &       & negation for \isa{bool}       \\
%  \sdx{and}    & $[i,i]\To i$   & Left 70 & conjunction for \isa{bool}  \\
%  \sdx{or}     & $[i,i]\To i$   & Left 65 & disjunction for \isa{bool}  \\
%  \sdx{xor}    & $[i,i]\To i$   & Left 65 & exclusive-or for \isa{bool}
%\end{constants}
%
\begin{alltt*}\isastyleminor
\tdx{bool_def}:      bool == {\ttlbrace}0,1{\ttrbrace}
\tdx{cond_def}:      cond(b,c,d) == if b=1 then c else d
\tdx{not_def}:       not(b)  == cond(b,0,1)
\tdx{and_def}:       a and b == cond(a,b,0)
\tdx{or_def}:        a or b  == cond(a,1,b)
\tdx{xor_def}:       a xor b == cond(a,not(b),b)

\tdx{bool_1I}:       1 \isasymin bool
\tdx{bool_0I}:       0 \isasymin bool
\tdx{boolE}:         [| c \isasymin bool;  c=1 ==> P;  c=0 ==> P |] ==> P
\tdx{cond_1}:        cond(1,c,d) = c
\tdx{cond_0}:        cond(0,c,d) = d
\end{alltt*}
\caption{The booleans} \label{zf-bool}
\end{figure}


\section{Further developments}
The next group of developments is complex and extensive, and only
highlights can be covered here.  It involves many theories and proofs. 

Figure~\ref{zf-equalities} presents commutative, associative, distributive,
and idempotency laws of union and intersection, along with other equations.

Theory \thydx{Bool} defines $\{0,1\}$ as a set of booleans, with the usual
operators including a conditional (Fig.\ts\ref{zf-bool}).  Although ZF is a
first-order theory, you can obtain the effect of higher-order logic using
\isa{bool}-valued functions, for example.  The constant~\isa{1} is
translated to \isa{succ(0)}.

\begin{figure}
\index{*"+ symbol}
\begin{constants}
  \it symbol    & \it meta-type & \it priority & \it description \\ 
  \tt +         & $[i,i]\To i$  &  Right 65     & disjoint union operator\\
  \cdx{Inl}~~\cdx{Inr}  & $i\To i$      &       & injections\\
  \cdx{case}    & $[i\To i,i\To i, i]\To i$ &   & conditional for $A+B$
\end{constants}
\begin{alltt*}\isastyleminor
\tdx{sum_def}:   A+B == {\ttlbrace}0{\ttrbrace}*A \isasymunion {\ttlbrace}1{\ttrbrace}*B
\tdx{Inl_def}:   Inl(a) == <0,a>
\tdx{Inr_def}:   Inr(b) == <1,b>
\tdx{case_def}:  case(c,d,u) == split(\%y z. cond(y, d(z), c(z)), u)

\tdx{InlI}:      a \isasymin A ==> Inl(a) \isasymin A+B
\tdx{InrI}:      b \isasymin B ==> Inr(b) \isasymin A+B

\tdx{Inl_inject}:  Inl(a)=Inl(b) ==> a=b
\tdx{Inr_inject}:  Inr(a)=Inr(b) ==> a=b
\tdx{Inl_neq_Inr}: Inl(a)=Inr(b) ==> P

\tdx{sum_iff}:  u \isasymin A+B <-> ({\isasymexists}x\isasymin{}A. u=Inl(x)) | ({\isasymexists}y\isasymin{}B. u=Inr(y))

\tdx{case_Inl}:  case(c,d,Inl(a)) = c(a)
\tdx{case_Inr}:  case(c,d,Inr(b)) = d(b)
\end{alltt*}
\caption{Disjoint unions} \label{zf-sum}
\end{figure}


\subsection{Disjoint unions}

Theory \thydx{Sum} defines the disjoint union of two sets, with
injections and a case analysis operator (Fig.\ts\ref{zf-sum}).  Disjoint
unions play a role in datatype definitions, particularly when there is
mutual recursion~\cite{paulson-set-II}.

\begin{figure}
\begin{alltt*}\isastyleminor
\tdx{QPair_def}:      <a;b> == a+b
\tdx{qsplit_def}:     qsplit(c,p)  == THE y. {\isasymexists}a b. p=<a;b> & y=c(a,b)
\tdx{qfsplit_def}:    qfsplit(R,z) == {\isasymexists}x y. z=<x;y> & R(x,y)
\tdx{qconverse_def}:  qconverse(r) == {\ttlbrace}z. w \isasymin r, {\isasymexists}x y. w=<x;y> & z=<y;x>{\ttrbrace}
\tdx{QSigma_def}:     QSigma(A,B)  == {\isasymUnion}x \isasymin A. {\isasymUnion}y \isasymin B(x). {\ttlbrace}<x;y>{\ttrbrace}

\tdx{qsum_def}:       A <+> B      == ({\ttlbrace}0{\ttrbrace} <*> A) \isasymunion ({\ttlbrace}1{\ttrbrace} <*> B)
\tdx{QInl_def}:       QInl(a)      == <0;a>
\tdx{QInr_def}:       QInr(b)      == <1;b>
\tdx{qcase_def}:      qcase(c,d)   == qsplit(\%y z. cond(y, d(z), c(z)))
\end{alltt*}
\caption{Non-standard pairs, products and sums} \label{zf-qpair}
\end{figure}


\subsection{Non-standard ordered pairs}

Theory \thydx{QPair} defines a notion of ordered pair that admits
non-well-founded tupling (Fig.\ts\ref{zf-qpair}).  Such pairs are written
{\tt<$a$;$b$>}.  It also defines the eliminator \cdx{qsplit}, the
converse operator \cdx{qconverse}, and the summation operator
\cdx{QSigma}.  These are completely analogous to the corresponding
versions for standard ordered pairs.  The theory goes on to define a
non-standard notion of disjoint sum using non-standard pairs.  All of these
concepts satisfy the same properties as their standard counterparts; in
addition, {\tt<$a$;$b$>} is continuous.  The theory supports coinductive
definitions, for example of infinite lists~\cite{paulson-mscs}.

\begin{figure}
\begin{alltt*}\isastyleminor
\tdx{bnd_mono_def}:  bnd_mono(D,h) == 
               h(D)\isasymsubseteq{}D & ({\isasymforall}W X. W\isasymsubseteq{}X --> X\isasymsubseteq{}D --> h(W)\isasymsubseteq{}h(X))

\tdx{lfp_def}:       lfp(D,h) == Inter({\ttlbrace}X \isasymin Pow(D). h(X) \isasymsubseteq X{\ttrbrace})
\tdx{gfp_def}:       gfp(D,h) == Union({\ttlbrace}X \isasymin Pow(D). X \isasymsubseteq h(X){\ttrbrace})


\tdx{lfp_lowerbound}: [| h(A) \isasymsubseteq A;  A \isasymsubseteq D |] ==> lfp(D,h) \isasymsubseteq A

\tdx{lfp_subset}:    lfp(D,h) \isasymsubseteq D

\tdx{lfp_greatest}:  [| bnd_mono(D,h);  
                  !!X. [| h(X) \isasymsubseteq X;  X \isasymsubseteq D |] ==> A \isasymsubseteq X 
               |] ==> A \isasymsubseteq lfp(D,h)

\tdx{lfp_Tarski}:    bnd_mono(D,h) ==> lfp(D,h) = h(lfp(D,h))

\tdx{induct}:        [| a \isasymin lfp(D,h);  bnd_mono(D,h);
                  !!x. x \isasymin h(Collect(lfp(D,h),P)) ==> P(x)
               |] ==> P(a)

\tdx{lfp_mono}:      [| bnd_mono(D,h);  bnd_mono(E,i);
                  !!X. X \isasymsubseteq D ==> h(X) \isasymsubseteq i(X)  
               |] ==> lfp(D,h) \isasymsubseteq lfp(E,i)

\tdx{gfp_upperbound}: [| A \isasymsubseteq h(A);  A \isasymsubseteq D |] ==> A \isasymsubseteq gfp(D,h)

\tdx{gfp_subset}:    gfp(D,h) \isasymsubseteq D

\tdx{gfp_least}:     [| bnd_mono(D,h);  
                  !!X. [| X \isasymsubseteq h(X);  X \isasymsubseteq D |] ==> X \isasymsubseteq A
               |] ==> gfp(D,h) \isasymsubseteq A

\tdx{gfp_Tarski}:    bnd_mono(D,h) ==> gfp(D,h) = h(gfp(D,h))

\tdx{coinduct}:      [| bnd_mono(D,h); a \isasymin X; X \isasymsubseteq h(X \isasymunion gfp(D,h)); X \isasymsubseteq D 
               |] ==> a \isasymin gfp(D,h)

\tdx{gfp_mono}:      [| bnd_mono(D,h);  D \isasymsubseteq E;
                  !!X. X \isasymsubseteq D ==> h(X) \isasymsubseteq i(X)  
               |] ==> gfp(D,h) \isasymsubseteq gfp(E,i)
\end{alltt*}
\caption{Least and greatest fixedpoints} \label{zf-fixedpt}
\end{figure}


\subsection{Least and greatest fixedpoints}

The Knaster-Tarski Theorem states that every monotone function over a
complete lattice has a fixedpoint.  Theory \thydx{Fixedpt} proves the
Theorem only for a particular lattice, namely the lattice of subsets of a
set (Fig.\ts\ref{zf-fixedpt}).  The theory defines least and greatest
fixedpoint operators with corresponding induction and coinduction rules.
These are essential to many definitions that follow, including the natural
numbers and the transitive closure operator.  The (co)inductive definition
package also uses the fixedpoint operators~\cite{paulson-CADE}.  See
Davey and Priestley~\cite{davey-priestley} for more on the Knaster-Tarski
Theorem and my paper~\cite{paulson-set-II} for discussion of the Isabelle
proofs.

Monotonicity properties are proved for most of the set-forming operations:
union, intersection, Cartesian product, image, domain, range, etc.  These
are useful for applying the Knaster-Tarski Fixedpoint Theorem.  The proofs
themselves are trivial applications of Isabelle's classical reasoner. 


\subsection{Finite sets and lists}

Theory \texttt{Finite} (Figure~\ref{zf-fin}) defines the finite set operator;
$\isa{Fin}(A)$ is the set of all finite sets over~$A$.  The theory employs
Isabelle's inductive definition package, which proves various rules
automatically.  The induction rule shown is stronger than the one proved by
the package.  The theory also defines the set of all finite functions
between two given sets.

\begin{figure}
\begin{alltt*}\isastyleminor
\tdx{Fin.emptyI}      0 \isasymin Fin(A)
\tdx{Fin.consI}       [| a \isasymin A;  b \isasymin Fin(A) |] ==> cons(a,b) \isasymin Fin(A)

\tdx{Fin_induct}
    [| b \isasymin Fin(A);
       P(0);
       !!x y. [| x\isasymin{}A; y\isasymin{}Fin(A); x\isasymnotin{}y; P(y) |] ==> P(cons(x,y))
    |] ==> P(b)

\tdx{Fin_mono}:       A \isasymsubseteq B ==> Fin(A) \isasymsubseteq Fin(B)
\tdx{Fin_UnI}:        [| b \isasymin Fin(A);  c \isasymin Fin(A) |] ==> b \isasymunion c \isasymin Fin(A)
\tdx{Fin_UnionI}:     C \isasymin Fin(Fin(A)) ==> Union(C) \isasymin Fin(A)
\tdx{Fin_subset}:     [| c \isasymsubseteq b;  b \isasymin Fin(A) |] ==> c \isasymin Fin(A)
\end{alltt*}
\caption{The finite set operator} \label{zf-fin}
\end{figure}

\begin{figure}
\begin{constants}
  \it symbol  & \it meta-type & \it priority & \it description \\ 
  \cdx{list}    & $i\To i$      && lists over some set\\
  \cdx{list_case} & $[i, [i,i]\To i, i] \To i$  && conditional for $list(A)$ \\
  \cdx{map}     & $[i\To i, i] \To i$   &       & mapping functional\\
  \cdx{length}  & $i\To i$              &       & length of a list\\
  \cdx{rev}     & $i\To i$              &       & reverse of a list\\
  \tt \at       & $[i,i]\To i$  &  Right 60     & append for lists\\
  \cdx{flat}    & $i\To i$   &                  & append of list of lists
\end{constants}

\underscoreon %%because @ is used here
\begin{alltt*}\isastyleminor
\tdx{NilI}:       Nil \isasymin list(A)
\tdx{ConsI}:      [| a \isasymin A;  l \isasymin list(A) |] ==> Cons(a,l) \isasymin list(A)

\tdx{List.induct}
    [| l \isasymin list(A);
       P(Nil);
       !!x y. [| x \isasymin A;  y \isasymin list(A);  P(y) |] ==> P(Cons(x,y))
    |] ==> P(l)

\tdx{Cons_iff}:       Cons(a,l)=Cons(a',l') <-> a=a' & l=l'
\tdx{Nil_Cons_iff}:    Nil \isasymnoteq Cons(a,l)

\tdx{list_mono}:      A \isasymsubseteq B ==> list(A) \isasymsubseteq list(B)

\tdx{map_ident}:      l\isasymin{}list(A) ==> map(\%u. u, l) = l
\tdx{map_compose}:    l\isasymin{}list(A) ==> map(h, map(j,l)) = map(\%u. h(j(u)), l)
\tdx{map_app_distrib}: xs\isasymin{}list(A) ==> map(h, xs@ys) = map(h,xs)@map(h,ys)
\tdx{map_type}
    [| l\isasymin{}list(A); !!x. x\isasymin{}A ==> h(x)\isasymin{}B |] ==> map(h,l)\isasymin{}list(B)
\tdx{map_flat}
    ls: list(list(A)) ==> map(h, flat(ls)) = flat(map(map(h),ls))
\end{alltt*}
\caption{Lists} \label{zf-list}
\end{figure}


Figure~\ref{zf-list} presents the set of lists over~$A$, $\isa{list}(A)$.  The
definition employs Isabelle's datatype package, which defines the introduction
and induction rules automatically, as well as the constructors, case operator
(\isa{list\_case}) and recursion operator.  The theory then defines the usual
list functions by primitive recursion.  See theory \texttt{List}.


\subsection{Miscellaneous}

\begin{figure}
\begin{constants} 
  \it symbol  & \it meta-type & \it priority & \it description \\ 
  \sdx{O}       & $[i,i]\To i$  &  Right 60     & composition ($\circ$) \\
  \cdx{id}      & $i\To i$      &       & identity function \\
  \cdx{inj}     & $[i,i]\To i$  &       & injective function space\\
  \cdx{surj}    & $[i,i]\To i$  &       & surjective function space\\
  \cdx{bij}     & $[i,i]\To i$  &       & bijective function space
\end{constants}

\begin{alltt*}\isastyleminor
\tdx{comp_def}: r O s     == {\ttlbrace}xz \isasymin domain(s)*range(r) . 
                        {\isasymexists}x y z. xz=<x,z> & <x,y> \isasymin s & <y,z> \isasymin r{\ttrbrace}
\tdx{id_def}:   id(A)     == (lam x \isasymin A. x)
\tdx{inj_def}:  inj(A,B)  == {\ttlbrace} f\isasymin{}A->B. {\isasymforall}w\isasymin{}A. {\isasymforall}x\isasymin{}A. f`w=f`x --> w=x {\ttrbrace}
\tdx{surj_def}: surj(A,B) == {\ttlbrace} f\isasymin{}A->B . {\isasymforall}y\isasymin{}B. {\isasymexists}x\isasymin{}A. f`x=y {\ttrbrace}
\tdx{bij_def}:  bij(A,B)  == inj(A,B) \isasyminter surj(A,B)


\tdx{left_inverse}:    [| f\isasymin{}inj(A,B);  a\isasymin{}A |] ==> converse(f)`(f`a) = a
\tdx{right_inverse}:   [| f\isasymin{}inj(A,B);  b\isasymin{}range(f) |] ==> 
                 f`(converse(f)`b) = b

\tdx{inj_converse_inj}: f\isasymin{}inj(A,B) ==> converse(f) \isasymin inj(range(f),A)
\tdx{bij_converse_bij}: f\isasymin{}bij(A,B) ==> converse(f) \isasymin bij(B,A)

\tdx{comp_type}:     [| s \isasymsubseteq A*B;  r \isasymsubseteq B*C |] ==> (r O s) \isasymsubseteq A*C
\tdx{comp_assoc}:    (r O s) O t = r O (s O t)

\tdx{left_comp_id}:  r \isasymsubseteq A*B ==> id(B) O r = r
\tdx{right_comp_id}: r \isasymsubseteq A*B ==> r O id(A) = r

\tdx{comp_func}:     [| g\isasymin{}A->B; f\isasymin{}B->C |] ==> (f O g) \isasymin A->C
\tdx{comp_func_apply}: [| g\isasymin{}A->B; f\isasymin{}B->C; a\isasymin{}A |] ==> (f O g)`a = f`(g`a)

\tdx{comp_inj}:      [| g\isasymin{}inj(A,B);  f\isasymin{}inj(B,C)  |] ==> (f O g)\isasymin{}inj(A,C)
\tdx{comp_surj}:     [| g\isasymin{}surj(A,B); f\isasymin{}surj(B,C) |] ==> (f O g)\isasymin{}surj(A,C)
\tdx{comp_bij}:      [| g\isasymin{}bij(A,B); f\isasymin{}bij(B,C) |] ==> (f O g)\isasymin{}bij(A,C)

\tdx{left_comp_inverse}:    f\isasymin{}inj(A,B) ==> converse(f) O f = id(A)
\tdx{right_comp_inverse}:   f\isasymin{}surj(A,B) ==> f O converse(f) = id(B)

\tdx{bij_disjoint_Un}:  
    [| f\isasymin{}bij(A,B);  g\isasymin{}bij(C,D);  A \isasyminter C = 0;  B \isasyminter D = 0 |] ==> 
    (f \isasymunion g)\isasymin{}bij(A \isasymunion C, B \isasymunion D)

\tdx{restrict_bij}: [| f\isasymin{}inj(A,B); C\isasymsubseteq{}A |] ==> restrict(f,C)\isasymin{}bij(C, f``C)
\end{alltt*}
\caption{Permutations} \label{zf-perm}
\end{figure}

The theory \thydx{Perm} is concerned with permutations (bijections) and
related concepts.  These include composition of relations, the identity
relation, and three specialized function spaces: injective, surjective and
bijective.  Figure~\ref{zf-perm} displays many of their properties that
have been proved.  These results are fundamental to a treatment of
equipollence and cardinality.

Theory \thydx{Univ} defines a `universe' $\isa{univ}(A)$, which is used by
the datatype package.  This set contains $A$ and the
natural numbers.  Vitally, it is closed under finite products: 
$\isa{univ}(A)\times\isa{univ}(A)\subseteq\isa{univ}(A)$.  This theory also
defines the cumulative hierarchy of axiomatic set theory, which
traditionally is written $V@\alpha$ for an ordinal~$\alpha$.  The
`universe' is a simple generalization of~$V@\omega$.

Theory \thydx{QUniv} defines a `universe' $\isa{quniv}(A)$, which is used by
the datatype package to construct codatatypes such as streams.  It is
analogous to $\isa{univ}(A)$ (and is defined in terms of it) but is closed
under the non-standard product and sum.


\section{Automatic Tools}

ZF provides the simplifier and the classical reasoner.  Moreover it supplies a
specialized tool to infer `types' of terms.

\subsection{Simplification and Classical Reasoning}

ZF inherits simplification from FOL but adopts it for set theory.  The
extraction of rewrite rules takes the ZF primitives into account.  It can
strip bounded universal quantifiers from a formula; for example, ${\forall
  x\in A. f(x)=g(x)}$ yields the conditional rewrite rule $x\in A \Imp
f(x)=g(x)$.  Given $a\in\{x\in A. P(x)\}$ it extracts rewrite rules from $a\in
A$ and~$P(a)$.  It can also break down $a\in A\int B$ and $a\in A-B$.

The default simpset used by \isa{simp} contains congruence rules for all of ZF's
binding operators.  It contains all the conversion rules, such as
\isa{fst} and
\isa{snd}, as well as the rewrites shown in Fig.\ts\ref{zf-simpdata}.

Classical reasoner methods such as \isa{blast} and \isa{auto} refer to
a rich collection of built-in axioms for all the set-theoretic
primitives.


\begin{figure}
\begin{eqnarray*}
  a\in \emptyset        & \bimp &  \bot\\
  a \in A \un B      & \bimp &  a\in A \disj a\in B\\
  a \in A \int B      & \bimp &  a\in A \conj a\in B\\
  a \in A-B             & \bimp &  a\in A \conj \lnot (a\in B)\\
  \pair{a,b}\in \isa{Sigma}(A,B)
                        & \bimp &  a\in A \conj b\in B(a)\\
  a \in \isa{Collect}(A,P)      & \bimp &  a\in A \conj P(a)\\
  (\forall x \in \emptyset. P(x)) & \bimp &  \top\\
  (\forall x \in A. \top)       & \bimp &  \top
\end{eqnarray*}
\caption{Some rewrite rules for set theory} \label{zf-simpdata}
\end{figure}


\subsection{Type-Checking Tactics}
\index{type-checking tactics}

Isabelle/ZF provides simple tactics to help automate those proofs that are
essentially type-checking.  Such proofs are built by applying rules such as
these:
\begin{ttbox}\isastyleminor
[| ?P ==> ?a \isasymin ?A; ~?P ==> ?b \isasymin ?A |] 
==> (if ?P then ?a else ?b) \isasymin ?A

[| ?m \isasymin nat; ?n \isasymin nat |] ==> ?m #+ ?n \isasymin nat

?a \isasymin ?A ==> Inl(?a) \isasymin ?A + ?B  
\end{ttbox}
In typical applications, the goal has the form $t\in\Var{A}$: in other words,
we have a specific term~$t$ and need to infer its `type' by instantiating the
set variable~$\Var{A}$.  Neither the simplifier nor the classical reasoner
does this job well.  The if-then-else rule, and many similar ones, can make
the classical reasoner loop.  The simplifier refuses (on principle) to
instantiate variables during rewriting, so goals such as \isa{i\#+j \isasymin \ ?A}
are left unsolved.

The simplifier calls the type-checker to solve rewritten subgoals: this stage
can indeed instantiate variables.  If you have defined new constants and
proved type-checking rules for them, then declare the rules using
the attribute \isa{TC} and the rest should be automatic.  In
particular, the simplifier will use type-checking to help satisfy
conditional rewrite rules. Call the method \ttindex{typecheck} to
break down all subgoals using type-checking rules. You can add new
type-checking rules temporarily like this:
\begin{isabelle}
\isacommand{apply}\ (typecheck add:\ inj_is_fun)
\end{isabelle}


%Though the easiest way to invoke the type-checker is via the simplifier,
%specialized applications may require more detailed knowledge of
%the type-checking primitives.  They are modelled on the simplifier's:
%\begin{ttdescription}
%\item[\ttindexbold{tcset}] is the type of tcsets: sets of type-checking rules.
%
%\item[\ttindexbold{addTCs}] is an infix operator to add type-checking rules to
%  a tcset.
%  
%\item[\ttindexbold{delTCs}] is an infix operator to remove type-checking rules
%  from a tcset.
%
%\item[\ttindexbold{typecheck_tac}] is a tactic for attempting to prove all
%  subgoals using the rules given in its argument, a tcset.
%\end{ttdescription}
%
%Tcsets, like simpsets, are associated with theories and are merged when
%theories are merged.  There are further primitives that use the default tcset.
%\begin{ttdescription}
%\item[\ttindexbold{tcset}] is a function to return the default tcset; use the
%  expression \isa{tcset()}.
%
%\item[\ttindexbold{AddTCs}] adds type-checking rules to the default tcset.
%  
%\item[\ttindexbold{DelTCs}] removes type-checking rules from the default
%  tcset.
%
%\item[\ttindexbold{Typecheck_tac}] calls \isa{typecheck_tac} using the
%  default tcset.
%\end{ttdescription}
%
%To supply some type-checking rules temporarily, using \isa{Addrules} and
%later \isa{Delrules} is the simplest way.  There is also a high-tech
%approach.  Call the simplifier with a new solver expressed using
%\ttindexbold{type_solver_tac} and your temporary type-checking rules.
%\begin{ttbox}\isastyleminor
%by (asm_simp_tac 
%     (simpset() setSolver type_solver_tac (tcset() addTCs prems)) 2);
%\end{ttbox}


\section{Natural number and integer arithmetic}

\index{arithmetic|(}

\begin{figure}\small
\index{#*@{\tt\#*} symbol}
\index{*div symbol}
\index{*mod symbol}
\index{#+@{\tt\#+} symbol}
\index{#-@{\tt\#-} symbol}
\begin{constants}
  \it symbol  & \it meta-type & \it priority & \it description \\ 
  \cdx{nat}     & $i$                   &       & set of natural numbers \\
  \cdx{nat_case}& $[i,i\To i,i]\To i$     &     & conditional for $nat$\\
  \tt \#*       & $[i,i]\To i$  &  Left 70      & multiplication \\
  \tt div       & $[i,i]\To i$  &  Left 70      & division\\
  \tt mod       & $[i,i]\To i$  &  Left 70      & modulus\\
  \tt \#+       & $[i,i]\To i$  &  Left 65      & addition\\
  \tt \#-       & $[i,i]\To i$  &  Left 65      & subtraction
\end{constants}

\begin{alltt*}\isastyleminor
\tdx{nat_def}: nat == lfp(lam r \isasymin Pow(Inf). {\ttlbrace}0{\ttrbrace} \isasymunion {\ttlbrace}succ(x). x \isasymin r{\ttrbrace}

\tdx{nat_case_def}:  nat_case(a,b,k) == 
              THE y. k=0 & y=a | ({\isasymexists}x. k=succ(x) & y=b(x))

\tdx{nat_0I}:           0 \isasymin nat
\tdx{nat_succI}:        n \isasymin nat ==> succ(n) \isasymin nat

\tdx{nat_induct}:        
    [| n \isasymin nat;  P(0);  !!x. [| x \isasymin nat;  P(x) |] ==> P(succ(x)) 
    |] ==> P(n)

\tdx{nat_case_0}:       nat_case(a,b,0) = a
\tdx{nat_case_succ}:    nat_case(a,b,succ(m)) = b(m)

\tdx{add_0_natify}:     0 #+ n = natify(n)
\tdx{add_succ}:         succ(m) #+ n = succ(m #+ n)

\tdx{mult_type}:        m #* n \isasymin nat
\tdx{mult_0}:           0 #* n = 0
\tdx{mult_succ}:        succ(m) #* n = n #+ (m #* n)
\tdx{mult_commute}:     m #* n = n #* m
\tdx{add_mult_dist}:    (m #+ n) #* k = (m #* k) #+ (n #* k)
\tdx{mult_assoc}:       (m #* n) #* k = m #* (n #* k)
\tdx{mod_div_equality}: m \isasymin nat ==> (m div n)#*n #+ m mod n = m
\end{alltt*}
\caption{The natural numbers} \label{zf-nat}
\end{figure}

\index{natural numbers}

Theory \thydx{Nat} defines the natural numbers and mathematical
induction, along with a case analysis operator.  The set of natural
numbers, here called \isa{nat}, is known in set theory as the ordinal~$\omega$.

Theory \thydx{Arith} develops arithmetic on the natural numbers
(Fig.\ts\ref{zf-nat}).  Addition, multiplication and subtraction are defined
by primitive recursion.  Division and remainder are defined by repeated
subtraction, which requires well-founded recursion; the termination argument
relies on the divisor's being non-zero.  Many properties are proved:
commutative, associative and distributive laws, identity and cancellation
laws, etc.  The most interesting result is perhaps the theorem $a \bmod b +
(a/b)\times b = a$.

To minimize the need for tedious proofs of $t\in\isa{nat}$, the arithmetic
operators coerce their arguments to be natural numbers.  The function
\cdx{natify} is defined such that $\isa{natify}(n) = n$ if $n$ is a natural
number, $\isa{natify}(\isa{succ}(x)) =
\isa{succ}(\isa{natify}(x))$ for all $x$, and finally
$\isa{natify}(x)=0$ in all other cases.  The benefit is that the addition,
subtraction, multiplication, division and remainder operators always return
natural numbers, regardless of their arguments.  Algebraic laws (commutative,
associative, distributive) are unconditional.  Occurrences of \isa{natify}
as operands of those operators are simplified away.  Any remaining occurrences
can either be tolerated or else eliminated by proving that the argument is a
natural number.

The simplifier automatically cancels common terms on the opposite sides of
subtraction and of relations ($=$, $<$ and $\le$).  Here is an example:
\begin{isabelle}
 1. i \#+ j \#+ k \#- j < k \#+ l\isanewline
\isacommand{apply}\ simp\isanewline
 1. natify(i) < natify(l)
\end{isabelle}
Given the assumptions \isa{i \isasymin nat} and \isa{l \isasymin nat}, both occurrences of
\cdx{natify} would be simplified away.


\begin{figure}\small
\index{$*@{\tt\$*} symbol}
\index{$+@{\tt\$+} symbol}
\index{$-@{\tt\$-} symbol}
\begin{constants}
  \it symbol  & \it meta-type & \it priority & \it description \\ 
  \cdx{int}     & $i$                   &       & set of integers \\
  \tt \$*       & $[i,i]\To i$  &  Left 70      & multiplication \\
  \tt \$+       & $[i,i]\To i$  &  Left 65      & addition\\
  \tt \$-       & $[i,i]\To i$  &  Left 65      & subtraction\\
  \tt \$<       & $[i,i]\To o$  &  Left 50      & $<$ on integers\\
  \tt \$<=      & $[i,i]\To o$  &  Left 50      & $\le$ on integers
\end{constants}

\begin{alltt*}\isastyleminor
\tdx{zadd_0_intify}:    0 $+ n = intify(n)

\tdx{zmult_type}:       m $* n \isasymin int
\tdx{zmult_0}:          0 $* n = 0
\tdx{zmult_commute}:    m $* n = n $* m
\tdx{zadd_zmult_dist}:   (m $+ n) $* k = (m $* k) $+ (n $* k)
\tdx{zmult_assoc}:      (m $* n) $* k = m $* (n $* k)
\end{alltt*}
\caption{The integers} \label{zf-int}
\end{figure}


\index{integers}

Theory \thydx{Int} defines the integers, as equivalence classes of natural
numbers.   Figure~\ref{zf-int} presents a tidy collection of laws.  In
fact, a large library of facts is proved, including monotonicity laws for
addition and multiplication, covering both positive and negative operands.  

As with the natural numbers, the need for typing proofs is minimized.  All the
operators defined in Fig.\ts\ref{zf-int} coerce their operands to integers by
applying the function \cdx{intify}.  This function is the identity on integers
and maps other operands to zero.

Decimal notation is provided for the integers.  Numbers, written as
\isa{\#$nnn$} or \isa{\#-$nnn$}, are represented internally in
two's-complement binary.  Expressions involving addition, subtraction and
multiplication of numeral constants are evaluated (with acceptable efficiency)
by simplification.  The simplifier also collects similar terms, multiplying
them by a numerical coefficient.  It also cancels occurrences of the same
terms on the other side of the relational operators.  Example:
\begin{isabelle}
 1. y \$+ z \$+ \#-3 \$* x \$+ y \$<=  x \$* \#2 \$+
z\isanewline
\isacommand{apply}\ simp\isanewline
 1. \#2 \$* y \$<= \#5 \$* x
\end{isabelle}
For more information on the integers, please see the theories on directory
\texttt{ZF/Integ}. 

\index{arithmetic|)}


\section{Datatype definitions}
\label{sec:ZF:datatype}
\index{*datatype|(}

The \ttindex{datatype} definition package of ZF constructs inductive datatypes
similar to \ML's.  It can also construct coinductive datatypes
(codatatypes), which are non-well-founded structures such as streams.  It
defines the set using a fixed-point construction and proves induction rules,
as well as theorems for recursion and case combinators.  It supplies
mechanisms for reasoning about freeness.  The datatype package can handle both
mutual and indirect recursion.


\subsection{Basics}
\label{subsec:datatype:basics}

A \isa{datatype} definition has the following form:
\[
\begin{array}{llcl}
\mathtt{datatype} & t@1(A@1,\ldots,A@h) & = &
  constructor^1@1 ~\mid~ \ldots ~\mid~ constructor^1@{k@1} \\
 & & \vdots \\
\mathtt{and} & t@n(A@1,\ldots,A@h) & = &
  constructor^n@1~ ~\mid~ \ldots ~\mid~ constructor^n@{k@n}
\end{array}
\]
Here $t@1$, \ldots,~$t@n$ are identifiers and $A@1$, \ldots,~$A@h$ are
variables: the datatype's parameters.  Each constructor specification has the
form \dquotesoff
\[ C \hbox{\tt~( } \hbox{\tt"} x@1 \hbox{\tt:} T@1 \hbox{\tt"},\;
                   \ldots,\;
                   \hbox{\tt"} x@m \hbox{\tt:} T@m \hbox{\tt"}
     \hbox{\tt~)}
\]
Here $C$ is the constructor name, and variables $x@1$, \ldots,~$x@m$ are the
constructor arguments, belonging to the sets $T@1$, \ldots, $T@m$,
respectively.  Typically each $T@j$ is either a constant set, a datatype
parameter (one of $A@1$, \ldots, $A@h$) or a recursive occurrence of one of
the datatypes, say $t@i(A@1,\ldots,A@h)$.  More complex possibilities exist,
but they are much harder to realize.  Often, additional information must be
supplied in the form of theorems.

A datatype can occur recursively as the argument of some function~$F$.  This
is called a {\em nested} (or \emph{indirect}) occurrence.  It is only allowed
if the datatype package is given a theorem asserting that $F$ is monotonic.
If the datatype has indirect occurrences, then Isabelle/ZF does not support
recursive function definitions.

A simple example of a datatype is \isa{list}, which is built-in, and is
defined by
\begin{alltt*}\isastyleminor
consts     list :: "i=>i"
datatype  "list(A)" = Nil | Cons ("a \isasymin A", "l \isasymin list(A)")
\end{alltt*}
Note that the datatype operator must be declared as a constant first.
However, the package declares the constructors.  Here, \isa{Nil} gets type
$i$ and \isa{Cons} gets type $[i,i]\To i$.

Trees and forests can be modelled by the mutually recursive datatype
definition
\begin{alltt*}\isastyleminor
consts   
  tree :: "i=>i"
  forest :: "i=>i"
  tree_forest :: "i=>i"
datatype  "tree(A)"   = Tcons ("a{\isasymin}A",  "f{\isasymin}forest(A)")
and "forest(A)" = Fnil | Fcons ("t{\isasymin}tree(A)",  "f{\isasymin}forest(A)")
\end{alltt*}
Here $\isa{tree}(A)$ is the set of trees over $A$, $\isa{forest}(A)$ is
the set of forests over $A$, and  $\isa{tree_forest}(A)$ is the union of
the previous two sets.  All three operators must be declared first.

The datatype \isa{term}, which is defined by
\begin{alltt*}\isastyleminor
consts     term :: "i=>i"
datatype  "term(A)" = Apply ("a \isasymin A", "l \isasymin list(term(A))")
  monos list_mono
  type_elims list_univ [THEN subsetD, elim_format]
\end{alltt*}
is an example of nested recursion.  (The theorem \isa{list_mono} is proved
in theory \isa{List}, and the \isa{term} example is developed in
theory
\thydx{Induct/Term}.)

\subsubsection{Freeness of the constructors}

Constructors satisfy {\em freeness} properties.  Constructions are distinct,
for example $\isa{Nil}\not=\isa{Cons}(a,l)$, and they are injective, for
example $\isa{Cons}(a,l)=\isa{Cons}(a',l') \bimp a=a' \conj l=l'$.
Because the number of freeness is quadratic in the number of constructors, the
datatype package does not prove them.  Instead, it ensures that simplification
will prove them dynamically: when the simplifier encounters a formula
asserting the equality of two datatype constructors, it performs freeness
reasoning.  

Freeness reasoning can also be done using the classical reasoner, but it is
more complicated.  You have to add some safe elimination rules rules to the
claset.  For the \isa{list} datatype, they are called
\isa{list.free_elims}.  Occasionally this exposes the underlying
representation of some constructor, which can be rectified using the command
\isa{unfold list.con_defs [symmetric]}.


\subsubsection{Structural induction}

The datatype package also provides structural induction rules.  For datatypes
without mutual or nested recursion, the rule has the form exemplified by
\isa{list.induct} in Fig.\ts\ref{zf-list}.  For mutually recursive
datatypes, the induction rule is supplied in two forms.  Consider datatype
\isa{TF}.  The rule \isa{tree_forest.induct} performs induction over a
single predicate~\isa{P}, which is presumed to be defined for both trees
and forests:
\begin{alltt*}\isastyleminor
[| x \isasymin tree_forest(A);
   !!a f. [| a \isasymin A; f \isasymin forest(A); P(f) |] ==> P(Tcons(a, f)); 
   P(Fnil);
   !!f t. [| t \isasymin tree(A); P(t); f \isasymin forest(A); P(f) |]
          ==> P(Fcons(t, f)) 
|] ==> P(x)
\end{alltt*}
The rule \isa{tree_forest.mutual_induct} performs induction over two
distinct predicates, \isa{P_tree} and \isa{P_forest}.
\begin{alltt*}\isastyleminor
[| !!a f.
      [| a{\isasymin}A; f{\isasymin}forest(A); P_forest(f) |] ==> P_tree(Tcons(a,f));
   P_forest(Fnil);
   !!f t. [| t{\isasymin}tree(A); P_tree(t); f{\isasymin}forest(A); P_forest(f) |]
          ==> P_forest(Fcons(t, f)) 
|] ==> ({\isasymforall}za. za \isasymin tree(A) --> P_tree(za)) &
    ({\isasymforall}za. za \isasymin forest(A) --> P_forest(za))
\end{alltt*}

For datatypes with nested recursion, such as the \isa{term} example from
above, things are a bit more complicated.  The rule \isa{term.induct}
refers to the monotonic operator, \isa{list}:
\begin{alltt*}\isastyleminor
[| x \isasymin term(A);
   !!a l. [| a\isasymin{}A; l\isasymin{}list(Collect(term(A), P)) |] ==> P(Apply(a,l)) 
|] ==> P(x)
\end{alltt*}
The theory \isa{Induct/Term.thy} derives two higher-level induction rules,
one of which is particularly useful for proving equations:
\begin{alltt*}\isastyleminor
[| t \isasymin term(A);
   !!x zs. [| x \isasymin A; zs \isasymin list(term(A)); map(f, zs) = map(g, zs) |]
           ==> f(Apply(x, zs)) = g(Apply(x, zs)) 
|] ==> f(t) = g(t)  
\end{alltt*}
How this can be generalized to other nested datatypes is a matter for future
research.


\subsubsection{The \isa{case} operator}

The package defines an operator for performing case analysis over the
datatype.  For \isa{list}, it is called \isa{list_case} and satisfies
the equations
\begin{ttbox}\isastyleminor
list_case(f_Nil, f_Cons, []) = f_Nil
list_case(f_Nil, f_Cons, Cons(a, l)) = f_Cons(a, l)
\end{ttbox}
Here \isa{f_Nil} is the value to return if the argument is \isa{Nil} and
\isa{f_Cons} is a function that computes the value to return if the
argument has the form $\isa{Cons}(a,l)$.  The function can be expressed as
an abstraction, over patterns if desired (\S\ref{sec:pairs}).

For mutually recursive datatypes, there is a single \isa{case} operator.
In the tree/forest example, the constant \isa{tree_forest_case} handles all
of the constructors of the two datatypes.


\subsection{Defining datatypes}

The theory syntax for datatype definitions is shown in the
Isabelle/Isar reference manual.  In order to be well-formed, a
datatype definition has to obey the rules stated in the previous
section.  As a result the theory is extended with the new types, the
constructors, and the theorems listed in the previous section.

Codatatypes are declared like datatypes and are identical to them in every
respect except that they have a coinduction rule instead of an induction rule.
Note that while an induction rule has the effect of limiting the values
contained in the set, a coinduction rule gives a way of constructing new
values of the set.

Most of the theorems about datatypes become part of the default simpset.  You
never need to see them again because the simplifier applies them
automatically.  

\subsubsection{Specialized methods for datatypes}

Induction and case-analysis can be invoked using these special-purpose
methods:
\begin{ttdescription}
\item[\methdx{induct_tac} $x$] applies structural
  induction on variable $x$ to subgoal~1, provided the type of $x$ is a
  datatype.  The induction variable should not occur among other assumptions
  of the subgoal.
\end{ttdescription}
% 
% we also have the ind_cases method, but what does it do?
In some situations, induction is overkill and a case distinction over all
constructors of the datatype suffices.
\begin{ttdescription}
\item[\methdx{case_tac} $x$]
 performs a case analysis for the variable~$x$.
\end{ttdescription}

Both tactics can only be applied to a variable, whose typing must be given in
some assumption, for example the assumption \isa{x \isasymin \ list(A)}.  The tactics
also work for the natural numbers (\isa{nat}) and disjoint sums, although
these sets were not defined using the datatype package.  (Disjoint sums are
not recursive, so only \isa{case_tac} is available.)

Structured Isar methods are also available. Below, $t$ 
stands for the name of the datatype.
\begin{ttdescription}
\item[\methdx{induct} \isa{set:}\ $t$] is the Isar induction tactic.
\item[\methdx{cases} \isa{set:}\ $t$] is the Isar case-analysis tactic.
\end{ttdescription}


\subsubsection{The theorems proved by a datatype declaration}

Here are some more details for the technically minded.  Processing the
datatype declaration of a set~$t$ produces a name space~$t$ containing
the following theorems:
\begin{ttbox}\isastyleminor
intros          \textrm{the introduction rules}
cases           \textrm{the case analysis rule}
induct          \textrm{the standard induction rule}
mutual_induct   \textrm{the mutual induction rule, if needed}
case_eqns       \textrm{equations for the case operator}
recursor_eqns   \textrm{equations for the recursor}
simps           \textrm{the union of} case_eqns \textrm{and} recursor_eqns
con_defs        \textrm{definitions of the case operator and constructors}
free_iffs       \textrm{logical equivalences for proving freeness}
free_elims      \textrm{elimination rules for proving freeness}
defs            \textrm{datatype definition(s)}
\end{ttbox}
Furthermore there is the theorem $C$ for every constructor~$C$; for
example, the \isa{list} datatype's introduction rules are bound to the
identifiers \isa{Nil} and \isa{Cons}.

For a codatatype, the component \isa{coinduct} is the coinduction rule,
replacing the \isa{induct} component.

See the theories \isa{Induct/Ntree} and \isa{Induct/Brouwer} for examples of
infinitely branching datatypes.  See theory \isa{Induct/LList} for an example
of a codatatype.  Some of these theories illustrate the use of additional,
undocumented features of the datatype package.  Datatype definitions are
reduced to inductive definitions, and the advanced features should be
understood in that light.


\subsection{Examples}

\subsubsection{The datatype of binary trees}

Let us define the set $\isa{bt}(A)$ of binary trees over~$A$.  The theory
must contain these lines:
\begin{alltt*}\isastyleminor
consts   bt :: "i=>i"
datatype "bt(A)" = Lf | Br ("a\isasymin{}A", "t1\isasymin{}bt(A)", "t2\isasymin{}bt(A)")
\end{alltt*}
After loading the theory, we can prove some theorem.  
We begin by declaring the constructor's typechecking rules
as simplification rules:
\begin{isabelle}
\isacommand{declare}\ bt.intros\ [simp]%
\end{isabelle}

Our first example is the theorem that no tree equals its
left branch.  To make the inductive hypothesis strong enough, 
the proof requires a quantified induction formula, but 
the \isa{rule\_format} attribute will remove the quantifiers 
before the theorem is stored.
\begin{isabelle}
\isacommand{lemma}\ Br\_neq\_left\ [rule\_format]:\ "l\isasymin bt(A)\ ==>\ \isasymforall x\ r.\ Br(x,l,r)\isasymnoteq{}l"\isanewline
\ 1.\ l\ \isasymin \ bt(A)\ \isasymLongrightarrow \ \isasymforall x\ r.\ Br(x,\ l,\ r)\ \isasymnoteq \ l%
\end{isabelle}
This can be proved by the structural induction tactic:
\begin{isabelle}
\ \ \isacommand{apply}\ (induct\_tac\ l)\isanewline
\ 1.\ \isasymforall x\ r.\ Br(x,\ Lf,\ r)\ \isasymnoteq \ Lf\isanewline
\ 2.\ \isasymAnd a\ t1\ t2.\isanewline
\isaindent{\ 2.\ \ \ \ }\isasymlbrakk a\ \isasymin \ A;\ t1\ \isasymin \ bt(A);\ \isasymforall x\ r.\ Br(x,\ t1,\ r)\ \isasymnoteq \ t1;\ t2\ \isasymin \ bt(A);\isanewline
\isaindent{\ 2.\ \ \ \ \ \ \ }\isasymforall x\ r.\ Br(x,\ t2,\ r)\ \isasymnoteq \ t2\isasymrbrakk \isanewline
\isaindent{\ 2.\ \ \ \ }\isasymLongrightarrow \ \isasymforall x\ r.\ Br(x,\ Br(a,\ t1,\ t2),\ r)\ \isasymnoteq \ Br(a,\ t1,\ t2)
\end{isabelle}
Both subgoals are proved using \isa{auto}, which performs the necessary
freeness reasoning. 
\begin{isabelle}
\ \ \isacommand{apply}\ auto\isanewline
No\ subgoals!\isanewline
\isacommand{done}
\end{isabelle}

An alternative proof uses Isar's fancy \isa{induct} method, which 
automatically quantifies over all free variables:

\begin{isabelle}
\isacommand{lemma}\ Br\_neq\_left':\ "l\ \isasymin \ bt(A)\ ==>\ (!!x\ r.\ Br(x,\ l,\ r)\ \isasymnoteq \ l)"\isanewline
\ \ \isacommand{apply}\ (induct\ set:\ bt)\isanewline
\ 1.\ \isasymAnd x\ r.\ Br(x,\ Lf,\ r)\ \isasymnoteq \ Lf\isanewline
\ 2.\ \isasymAnd a\ t1\ t2\ x\ r.\isanewline
\isaindent{\ 2.\ \ \ \ }\isasymlbrakk a\ \isasymin \ A;\ t1\ \isasymin \ bt(A);\ \isasymAnd x\ r.\ Br(x,\ t1,\ r)\ \isasymnoteq \ t1;\ t2\ \isasymin \ bt(A);\isanewline
\isaindent{\ 2.\ \ \ \ \ \ \ }\isasymAnd x\ r.\ Br(x,\ t2,\ r)\ \isasymnoteq \ t2\isasymrbrakk \isanewline
\isaindent{\ 2.\ \ \ \ }\isasymLongrightarrow \ Br(x,\ Br(a,\ t1,\ t2),\ r)\ \isasymnoteq \ Br(a,\ t1,\ t2)
\end{isabelle}
Compare the form of the induction hypotheses with the corresponding ones in
the previous proof. As before, to conclude requires only \isa{auto}.

When there are only a few constructors, we might prefer to prove the freenness
theorems for each constructor.  This is simple:
\begin{isabelle}
\isacommand{lemma}\ Br\_iff:\ "Br(a,l,r)\ =\ Br(a',l',r')\ <->\ a=a'\ \&\ l=l'\ \&\ r=r'"\isanewline
\ \ \isacommand{by}\ (blast\ elim!:\ bt.free\_elims)
\end{isabelle}
Here we see a demonstration of freeness reasoning using
\isa{bt.free\_elims}, but simpler still is just to apply \isa{auto}.

An \ttindex{inductive\_cases} declaration generates instances of the
case analysis rule that have been simplified  using freeness
reasoning. 
\begin{isabelle}
\isacommand{inductive\_cases}\ Br\_in\_bt:\ "Br(a,\ l,\ r)\ \isasymin \ bt(A)"
\end{isabelle}
The theorem just created is 
\begin{isabelle}
\isasymlbrakk Br(a,\ l,\ r)\ \isasymin \ bt(A);\ \isasymlbrakk a\ \isasymin \ A;\ l\ \isasymin \ bt(A);\ r\ \isasymin \ bt(A)\isasymrbrakk \ \isasymLongrightarrow \ Q\isasymrbrakk \ \isasymLongrightarrow \ Q.
\end{isabelle}
It is an elimination rule that from $\isa{Br}(a,l,r)\in\isa{bt}(A)$
lets us infer $a\in A$, $l\in\isa{bt}(A)$ and
$r\in\isa{bt}(A)$.


\subsubsection{Mixfix syntax in datatypes}

Mixfix syntax is sometimes convenient.  The theory \isa{Induct/PropLog} makes a
deep embedding of propositional logic:
\begin{alltt*}\isastyleminor
consts     prop :: i
datatype  "prop" = Fls
                 | Var ("n \isasymin nat")                ("#_" [100] 100)
                 | "=>" ("p \isasymin prop", "q \isasymin prop")   (infixr 90)
\end{alltt*}
The second constructor has a special $\#n$ syntax, while the third constructor
is an infixed arrow.


\subsubsection{A giant enumeration type}

This example shows a datatype that consists of 60 constructors:
\begin{alltt*}\isastyleminor
consts  enum :: i
datatype
  "enum" = C00 | C01 | C02 | C03 | C04 | C05 | C06 | C07 | C08 | C09
         | C10 | C11 | C12 | C13 | C14 | C15 | C16 | C17 | C18 | C19
         | C20 | C21 | C22 | C23 | C24 | C25 | C26 | C27 | C28 | C29
         | C30 | C31 | C32 | C33 | C34 | C35 | C36 | C37 | C38 | C39
         | C40 | C41 | C42 | C43 | C44 | C45 | C46 | C47 | C48 | C49
         | C50 | C51 | C52 | C53 | C54 | C55 | C56 | C57 | C58 | C59
end
\end{alltt*}
The datatype package scales well.  Even though all properties are proved
rather than assumed, full processing of this definition takes around two seconds
(on a 1.8GHz machine).  The constructors have a balanced representation,
related to binary notation, so freeness properties can be proved fast.
\begin{isabelle}
\isacommand{lemma}\ "C00 \isasymnoteq\ C01"\isanewline
\ \ \isacommand{by}\ simp
\end{isabelle}
You need not derive such inequalities explicitly.  The simplifier will
dispose of them automatically.

\index{*datatype|)}


\subsection{Recursive function definitions}\label{sec:ZF:recursive}
\index{recursive functions|see{recursion}}
\index{*primrec|(}
\index{recursion!primitive|(}

Datatypes come with a uniform way of defining functions, {\bf primitive
  recursion}.  Such definitions rely on the recursion operator defined by the
datatype package.  Isabelle proves the desired recursion equations as
theorems.

In principle, one could introduce primitive recursive functions by asserting
their reduction rules as axioms.  Here is a dangerous way of defining a
recursive function over binary trees:
\begin{isabelle}
\isacommand{consts}\ \ n\_nodes\ ::\ "i\ =>\ i"\isanewline
\isacommand{axioms}\isanewline
\ \ n\_nodes\_Lf:\ "n\_nodes(Lf)\ =\ 0"\isanewline
\ \ n\_nodes\_Br:\ "n\_nodes(Br(a,l,r))\ =\ succ(n\_nodes(l)\ \#+\ n\_nodes(r))"
\end{isabelle}
Asserting axioms brings the danger of accidentally introducing
contradictions.  It should be avoided whenever possible.

The \ttindex{primrec} declaration is a safe means of defining primitive
recursive functions on datatypes:
\begin{isabelle}
\isacommand{consts}\ \ n\_nodes\ ::\ "i\ =>\ i"\isanewline
\isacommand{primrec}\isanewline
\ \ "n\_nodes(Lf)\ =\ 0"\isanewline
\ \ "n\_nodes(Br(a,\ l,\ r))\ =\ succ(n\_nodes(l)\ \#+\ n\_nodes(r))"
\end{isabelle}
Isabelle will now derive the two equations from a low-level definition  
based upon well-founded recursion.  If they do not define a legitimate
recursion, then Isabelle will reject the declaration.


\subsubsection{Syntax of recursive definitions}

The general form of a primitive recursive definition is
\begin{ttbox}\isastyleminor
primrec
    {\it reduction rules}
\end{ttbox}
where \textit{reduction rules} specify one or more equations of the form
\[ f \, x@1 \, \dots \, x@m \, (C \, y@1 \, \dots \, y@k) \, z@1 \,
\dots \, z@n = r \] such that $C$ is a constructor of the datatype, $r$
contains only the free variables on the left-hand side, and all recursive
calls in $r$ are of the form $f \, \dots \, y@i \, \dots$ for some $i$.  
There must be at most one reduction rule for each constructor.  The order is
immaterial.  For missing constructors, the function is defined to return zero.

All reduction rules are added to the default simpset.
If you would like to refer to some rule by name, then you must prefix
the rule with an identifier.  These identifiers, like those in the
\isa{rules} section of a theory, will be visible in proof scripts.

The reduction rules become part of the default simpset, which
leads to short proof scripts:
\begin{isabelle}
\isacommand{lemma}\ n\_nodes\_type\ [simp]:\ "t\ \isasymin \ bt(A)\ ==>\ n\_nodes(t)\ \isasymin \ nat"\isanewline
\ \ \isacommand{by}\ (induct\_tac\ t,\ auto)
\end{isabelle}

You can even use the \isa{primrec} form with non-recursive datatypes and
with codatatypes.  Recursion is not allowed, but it provides a convenient
syntax for defining functions by cases.


\subsubsection{Example: varying arguments}

All arguments, other than the recursive one, must be the same in each equation
and in each recursive call.  To get around this restriction, use explict
$\lambda$-abstraction and function application.  For example, let us
define the tail-recursive version of \isa{n\_nodes}, using an 
accumulating argument for the counter.  The second argument, $k$, varies in
recursive calls.
\begin{isabelle}
\isacommand{consts}\ \ n\_nodes\_aux\ ::\ "i\ =>\ i"\isanewline
\isacommand{primrec}\isanewline
\ \ "n\_nodes\_aux(Lf)\ =\ (\isasymlambda k\ \isasymin \ nat.\ k)"\isanewline
\ \ "n\_nodes\_aux(Br(a,l,r))\ =\ \isanewline
\ \ \ \ \ \ (\isasymlambda k\ \isasymin \ nat.\ n\_nodes\_aux(r)\ `\ \ (n\_nodes\_aux(l)\ `\ succ(k)))"
\end{isabelle}
Now \isa{n\_nodes\_aux(t)\ `\ k} is our function in two arguments. We
can prove a theorem relating it to \isa{n\_nodes}. Note the quantification
over \isa{k\ \isasymin \ nat}:
\begin{isabelle}
\isacommand{lemma}\ n\_nodes\_aux\_eq\ [rule\_format]:\isanewline
\ \ \ \ \ "t\ \isasymin \ bt(A)\ ==>\ \isasymforall k\ \isasymin \ nat.\ n\_nodes\_aux(t)`k\ =\ n\_nodes(t)\ \#+\ k"\isanewline
\ \ \isacommand{by}\ (induct\_tac\ t,\ simp\_all)
\end{isabelle}

Now, we can use \isa{n\_nodes\_aux} to define a tail-recursive version
of \isa{n\_nodes}:
\begin{isabelle}
\isacommand{constdefs}\isanewline
\ \ n\_nodes\_tail\ ::\ "i\ =>\ i"\isanewline
\ \ \ "n\_nodes\_tail(t)\ ==\ n\_nodes\_aux(t)\ `\ 0"
\end{isabelle}
It is easy to
prove that \isa{n\_nodes\_tail} is equivalent to \isa{n\_nodes}:
\begin{isabelle}
\isacommand{lemma}\ "t\ \isasymin \ bt(A)\ ==>\ n\_nodes\_tail(t)\ =\ n\_nodes(t)"\isanewline
\ \isacommand{by}\ (simp\ add:\ n\_nodes\_tail\_def\ n\_nodes\_aux\_eq)
\end{isabelle}




\index{recursion!primitive|)}
\index{*primrec|)}


\section{Inductive and coinductive definitions}
\index{*inductive|(}
\index{*coinductive|(}

An {\bf inductive definition} specifies the least set~$R$ closed under given
rules.  (Applying a rule to elements of~$R$ yields a result within~$R$.)  For
example, a structural operational semantics is an inductive definition of an
evaluation relation.  Dually, a {\bf coinductive definition} specifies the
greatest set~$R$ consistent with given rules.  (Every element of~$R$ can be
seen as arising by applying a rule to elements of~$R$.)  An important example
is using bisimulation relations to formalise equivalence of processes and
infinite data structures.

A theory file may contain any number of inductive and coinductive
definitions.  They may be intermixed with other declarations; in
particular, the (co)inductive sets {\bf must} be declared separately as
constants, and may have mixfix syntax or be subject to syntax translations.

Each (co)inductive definition adds definitions to the theory and also
proves some theorems.  It behaves identially to the analogous
inductive definition except that instead of an induction rule there is
a coinduction rule.  Its treatment of coinduction is described in
detail in a separate paper,%
\footnote{It appeared in CADE~\cite{paulson-CADE}; a longer version is
  distributed with Isabelle as \emph{A Fixedpoint Approach to 
 (Co)Inductive and (Co)Datatype Definitions}.}  %
which you might refer to for background information.


\subsection{The syntax of a (co)inductive definition}
An inductive definition has the form
\begin{ttbox}\isastyleminor
inductive
  domains     {\it domain declarations}
  intros      {\it introduction rules}
  monos       {\it monotonicity theorems}
  con_defs    {\it constructor definitions}
  type_intros {\it introduction rules for type-checking}
  type_elims  {\it elimination rules for type-checking}
\end{ttbox}
A coinductive definition is identical, but starts with the keyword
\isa{co\-inductive}.  

The \isa{monos}, \isa{con\_defs}, \isa{type\_intros} and \isa{type\_elims}
sections are optional.  If present, each is specified as a list of
theorems, which may contain Isar attributes as usual.

\begin{description}
\item[\it domain declarations] are items of the form
  {\it string\/}~\isa{\isasymsubseteq }~{\it string}, associating each recursive set with
  its domain.  (The domain is some existing set that is large enough to
  hold the new set being defined.)

\item[\it introduction rules] specify one or more introduction rules in
  the form {\it ident\/}~{\it string}, where the identifier gives the name of
  the rule in the result structure.

\item[\it monotonicity theorems] are required for each operator applied to
  a recursive set in the introduction rules.  There \textbf{must} be a theorem
  of the form $A\subseteq B\Imp M(A)\subseteq M(B)$, for each premise $t\in M(R_i)$
  in an introduction rule!

\item[\it constructor definitions] contain definitions of constants
  appearing in the introduction rules.  The (co)datatype package supplies
  the constructors' definitions here.  Most (co)inductive definitions omit
  this section; one exception is the primitive recursive functions example;
  see theory \isa{Induct/Primrec}.
  
\item[\it type\_intros] consists of introduction rules for type-checking the
  definition: for demonstrating that the new set is included in its domain.
  (The proof uses depth-first search.)

\item[\it type\_elims] consists of elimination rules for type-checking the
  definition.  They are presumed to be safe and are applied as often as
  possible prior to the \isa{type\_intros} search.
\end{description}

The package has a few restrictions:
\begin{itemize}
\item The theory must separately declare the recursive sets as
  constants.

\item The names of the recursive sets must be identifiers, not infix
operators.  

\item Side-conditions must not be conjunctions.  However, an introduction rule
may contain any number of side-conditions.

\item Side-conditions of the form $x=t$, where the variable~$x$ does not
  occur in~$t$, will be substituted through the rule \isa{mutual\_induct}.
\end{itemize}


\subsection{Example of an inductive definition}

Below, we shall see how Isabelle/ZF defines the finite powerset
operator.  The first step is to declare the constant~\isa{Fin}.  Then we
must declare it inductively, with two introduction rules:
\begin{isabelle}
\isacommand{consts}\ \ Fin\ ::\ "i=>i"\isanewline
\isacommand{inductive}\isanewline
\ \ \isakeyword{domains}\ \ \ "Fin(A)"\ \isasymsubseteq\ "Pow(A)"\isanewline
\ \ \isakeyword{intros}\isanewline
\ \ \ \ emptyI:\ \ "0\ \isasymin\ Fin(A)"\isanewline
\ \ \ \ consI:\ \ \ "[|\ a\ \isasymin\ A;\ \ b\ \isasymin\ Fin(A)\ |]\ ==>\ cons(a,b)\ \isasymin\ Fin(A)"\isanewline
\ \ \isakeyword{type\_intros}\ \ empty\_subsetI\ cons\_subsetI\ PowI\isanewline
\ \ \isakeyword{type\_elims}\ \ \ PowD\ [THEN\ revcut\_rl]\end{isabelle}
The resulting theory contains a name space, called~\isa{Fin}.
The \isa{Fin}$~A$ introduction rules can be referred to collectively as
\isa{Fin.intros}, and also individually as \isa{Fin.emptyI} and
\isa{Fin.consI}.  The induction rule is \isa{Fin.induct}.

The chief problem with making (co)inductive definitions involves type-checking
the rules.  Sometimes, additional theorems need to be supplied under
\isa{type_intros} or \isa{type_elims}.  If the package fails when trying
to prove your introduction rules, then set the flag \ttindexbold{trace_induct}
to \isa{true} and try again.  (See the manual \emph{A Fixedpoint Approach
  \ldots} for more discussion of type-checking.)

In the example above, $\isa{Pow}(A)$ is given as the domain of
$\isa{Fin}(A)$, for obviously every finite subset of~$A$ is a subset
of~$A$.  However, the inductive definition package can only prove that given a
few hints.
Here is the output that results (with the flag set) when the
\isa{type_intros} and \isa{type_elims} are omitted from the inductive
definition above:
\begin{alltt*}\isastyleminor
Inductive definition Finite.Fin
Fin(A) ==
lfp(Pow(A),
    \%X. {z\isasymin{}Pow(A) . z = 0 | ({\isasymexists}a b. z = cons(a,b) & a\isasymin{}A & b\isasymin{}X)})
  Proving monotonicity...
\ttbreak
  Proving the introduction rules...
The type-checking subgoal:
0 \isasymin Fin(A)
 1. 0 \isasymin Pow(A)
\ttbreak
The subgoal after monos, type_elims:
0 \isasymin Fin(A)
 1. 0 \isasymin Pow(A)
*** prove_goal: tactic failed
\end{alltt*}
We see the need to supply theorems to let the package prove
$\emptyset\in\isa{Pow}(A)$.  Restoring the \isa{type_intros} but not the
\isa{type_elims}, we again get an error message:
\begin{alltt*}\isastyleminor
The type-checking subgoal:
0 \isasymin Fin(A)
 1. 0 \isasymin Pow(A)
\ttbreak
The subgoal after monos, type_elims:
0 \isasymin Fin(A)
 1. 0 \isasymin Pow(A)
\ttbreak
The type-checking subgoal:
cons(a, b) \isasymin Fin(A)
 1. [| a \isasymin A; b \isasymin Fin(A) |] ==> cons(a, b) \isasymin Pow(A)
\ttbreak
The subgoal after monos, type_elims:
cons(a, b) \isasymin Fin(A)
 1. [| a \isasymin A; b \isasymin Pow(A) |] ==> cons(a, b) \isasymin Pow(A)
*** prove_goal: tactic failed
\end{alltt*}
The first rule has been type-checked, but the second one has failed.  The
simplest solution to such problems is to prove the failed subgoal separately
and to supply it under \isa{type_intros}.  The solution actually used is
to supply, under \isa{type_elims}, a rule that changes
$b\in\isa{Pow}(A)$ to $b\subseteq A$; together with \isa{cons_subsetI}
and \isa{PowI}, it is enough to complete the type-checking.



\subsection{Further examples}

An inductive definition may involve arbitrary monotonic operators.  Here is a
standard example: the accessible part of a relation.  Note the use
of~\isa{Pow} in the introduction rule and the corresponding mention of the
rule \isa{Pow\_mono} in the \isa{monos} list.  If the desired rule has a
universally quantified premise, usually the effect can be obtained using
\isa{Pow}.
\begin{isabelle}
\isacommand{consts}\ \ acc\ ::\ "i\ =>\ i"\isanewline
\isacommand{inductive}\isanewline
\ \ \isakeyword{domains}\ "acc(r)"\ \isasymsubseteq \ "field(r)"\isanewline
\ \ \isakeyword{intros}\isanewline
\ \ \ \ vimage:\ \ "[|\ r-``\isacharbraceleft a\isacharbraceright\ \isasymin\ Pow(acc(r));\ a\ \isasymin \ field(r)\ |]
\isanewline
\ \ \ \ \ \ \ \ \ \ \ \ \ \ ==>\ a\ \isasymin \ acc(r)"\isanewline
\ \ \isakeyword{monos}\ \ Pow\_mono
\end{isabelle}

Finally, here are some coinductive definitions.  We begin by defining
lazy (potentially infinite) lists as a codatatype:
\begin{isabelle}
\isacommand{consts}\ \ llist\ \ ::\ "i=>i"\isanewline
\isacommand{codatatype}\isanewline
\ \ "llist(A)"\ =\ LNil\ |\ LCons\ ("a\ \isasymin \ A",\ "l\ \isasymin \ llist(A)")\isanewline
\end{isabelle}

The notion of equality on such lists is modelled as a bisimulation:
\begin{isabelle}
\isacommand{consts}\ \ lleq\ ::\ "i=>i"\isanewline
\isacommand{coinductive}\isanewline
\ \ \isakeyword{domains}\ "lleq(A)"\ <=\ "llist(A)\ *\ llist(A)"\isanewline
\ \ \isakeyword{intros}\isanewline
\ \ \ \ LNil:\ \ "<LNil,\ LNil>\ \isasymin \ lleq(A)"\isanewline
\ \ \ \ LCons:\ "[|\ a\ \isasymin \ A;\ <l,l'>\ \isasymin \ lleq(A)\ |]\ \isanewline
\ \ \ \ \ \ \ \ \ \ \ \ ==>\ <LCons(a,l),\ LCons(a,l')>\ \isasymin \ lleq(A)"\isanewline
\ \ \isakeyword{type\_intros}\ \ llist.intros
\end{isabelle}
This use of \isa{type_intros} is typical: the relation concerns the
codatatype \isa{llist}, so naturally the introduction rules for that
codatatype will be required for type-checking the rules.

The Isabelle distribution contains many other inductive definitions.  Simple
examples are collected on subdirectory \isa{ZF/Induct}.  The directory
\isa{Coind} and the theory \isa{ZF/Induct/LList} contain coinductive
definitions.  Larger examples may be found on other subdirectories of
\isa{ZF}, such as \isa{IMP}, and \isa{Resid}.


\subsection{Theorems generated}

Each (co)inductive set defined in a theory file generates a name space
containing the following elements:
\begin{ttbox}\isastyleminor
intros        \textrm{the introduction rules}
elim          \textrm{the elimination (case analysis) rule}
induct        \textrm{the standard induction rule}
mutual_induct \textrm{the mutual induction rule, if needed}
defs          \textrm{definitions of inductive sets}
bnd_mono      \textrm{monotonicity property}
dom_subset    \textrm{inclusion in `bounding set'}
\end{ttbox}
Furthermore, each introduction rule is available under its declared
name. For a codatatype, the component \isa{coinduct} is the coinduction rule,
replacing the \isa{induct} component.

Recall that the \ttindex{inductive\_cases} declaration generates
simplified instances of the case analysis rule.  It is as useful for
inductive definitions as it is for datatypes.  There are many examples
in the theory
\isa{Induct/Comb}, which is discussed at length
elsewhere~\cite{paulson-generic}.  The theory first defines the
datatype
\isa{comb} of combinators:
\begin{alltt*}\isastyleminor
consts comb :: i
datatype  "comb" = K
                 | S
                 | "#" ("p \isasymin comb", "q \isasymin comb")   (infixl 90)
\end{alltt*}
The theory goes on to define contraction and parallel contraction
inductively.  Then the theory \isa{Induct/Comb.thy} defines special
cases of contraction, such as this one:
\begin{isabelle}
\isacommand{inductive\_cases}\ K\_contractE [elim!]:\ "K -1-> r"
\end{isabelle}
The theorem just created is \isa{K -1-> r \ \isasymLongrightarrow \ Q},
which expresses that the combinator \isa{K} cannot reduce to
anything.  (From the assumption \isa{K-1->r}, we can conclude any desired
formula \isa{Q}\@.)  Similar elimination rules for \isa{S} and application are also
generated. The attribute \isa{elim!}\ shown above supplies the generated
theorem to the classical reasoner.  This mode of working allows
effective reasoniung about operational semantics.

\index{*coinductive|)} \index{*inductive|)}



\section{The outer reaches of set theory}

The constructions of the natural numbers and lists use a suite of
operators for handling recursive function definitions.  I have described
the developments in detail elsewhere~\cite{paulson-set-II}.  Here is a brief
summary:
\begin{itemize}
  \item Theory \isa{Trancl} defines the transitive closure of a relation
    (as a least fixedpoint).

  \item Theory \isa{WF} proves the well-founded recursion theorem, using an
    elegant approach of Tobias Nipkow.  This theorem permits general
    recursive definitions within set theory.

  \item Theory \isa{Ord} defines the notions of transitive set and ordinal
    number.  It derives transfinite induction.  A key definition is {\bf
      less than}: $i<j$ if and only if $i$ and $j$ are both ordinals and
    $i\in j$.  As a special case, it includes less than on the natural
    numbers.
    
  \item Theory \isa{Epsilon} derives $\varepsilon$-induction and
    $\varepsilon$-recursion, which are generalisations of transfinite
    induction and recursion.  It also defines \cdx{rank}$(x)$, which is the
    least ordinal $\alpha$ such that $x$ is constructed at stage $\alpha$ of
    the cumulative hierarchy (thus $x\in V@{\alpha+1}$).
\end{itemize}

Other important theories lead to a theory of cardinal numbers.  They have
not yet been written up anywhere.  Here is a summary:
\begin{itemize}
\item Theory \isa{Rel} defines the basic properties of relations, such as
  reflexivity, symmetry and transitivity.

\item Theory \isa{EquivClass} develops a theory of equivalence
  classes, not using the Axiom of Choice.

\item Theory \isa{Order} defines partial orderings, total orderings and
  wellorderings.

\item Theory \isa{OrderArith} defines orderings on sum and product sets.
  These can be used to define ordinal arithmetic and have applications to
  cardinal arithmetic.

\item Theory \isa{OrderType} defines order types.  Every wellordering is
  equivalent to a unique ordinal, which is its order type.

\item Theory \isa{Cardinal} defines equipollence and cardinal numbers.
 
\item Theory \isa{CardinalArith} defines cardinal addition and
  multiplication, and proves their elementary laws.  It proves that there
  is no greatest cardinal.  It also proves a deep result, namely
  $\kappa\otimes\kappa=\kappa$ for every infinite cardinal~$\kappa$; see
  Kunen~\cite[page 29]{kunen80}.  None of these results assume the Axiom of
  Choice, which complicates their proofs considerably.  
\end{itemize}

The following developments involve the Axiom of Choice (AC):
\begin{itemize}
\item Theory \isa{AC} asserts the Axiom of Choice and proves some simple
  equivalent forms.

\item Theory \isa{Zorn} proves Hausdorff's Maximal Principle, Zorn's Lemma
  and the Wellordering Theorem, following Abrial and
  Laffitte~\cite{abrial93}.

\item Theory \isa{Cardinal\_AC} uses AC to prove simplified theorems about
  the cardinals.  It also proves a theorem needed to justify
  infinitely branching datatype declarations: if $\kappa$ is an infinite
  cardinal and $|X(\alpha)| \le \kappa$ for all $\alpha<\kappa$ then
  $|\union\sb{\alpha<\kappa} X(\alpha)| \le \kappa$.

\item Theory \isa{InfDatatype} proves theorems to justify infinitely
  branching datatypes.  Arbitrary index sets are allowed, provided their
  cardinalities have an upper bound.  The theory also justifies some
  unusual cases of finite branching, involving the finite powerset operator
  and the finite function space operator.
\end{itemize}



\section{The examples directories}
Directory \isa{HOL/IMP} contains a mechanised version of a semantic
equivalence proof taken from Winskel~\cite{winskel93}.  It formalises the
denotational and operational semantics of a simple while-language, then
proves the two equivalent.  It contains several datatype and inductive
definitions, and demonstrates their use.

The directory \isa{ZF/ex} contains further developments in ZF set theory.
Here is an overview; see the files themselves for more details.  I describe
much of this material in other
publications~\cite{paulson-set-I,paulson-set-II,paulson-fixedpt-milner}.
\begin{itemize}
\item File \isa{misc.ML} contains miscellaneous examples such as
  Cantor's Theorem, the Schr\"oder-Bernstein Theorem and the `Composition
  of homomorphisms' challenge~\cite{boyer86}.

\item Theory \isa{Ramsey} proves the finite exponent 2 version of
  Ramsey's Theorem, following Basin and Kaufmann's
  presentation~\cite{basin91}.

\item Theory \isa{Integ} develops a theory of the integers as
  equivalence classes of pairs of natural numbers.

\item Theory \isa{Primrec} develops some computation theory.  It
  inductively defines the set of primitive recursive functions and presents a
  proof that Ackermann's function is not primitive recursive.

\item Theory \isa{Primes} defines the Greatest Common Divisor of two
  natural numbers and and the ``divides'' relation.

\item Theory \isa{Bin} defines a datatype for two's complement binary
  integers, then proves rewrite rules to perform binary arithmetic.  For
  instance, $1359\times {-}2468 = {-}3354012$ takes 0.3 seconds.

\item Theory \isa{BT} defines the recursive data structure $\isa{bt}(A)$, labelled binary trees.

\item Theory \isa{Term} defines a recursive data structure for terms
  and term lists.  These are simply finite branching trees.

\item Theory \isa{TF} defines primitives for solving mutually
  recursive equations over sets.  It constructs sets of trees and forests
  as an example, including induction and recursion rules that handle the
  mutual recursion.

\item Theory \isa{Prop} proves soundness and completeness of
  propositional logic~\cite{paulson-set-II}.  This illustrates datatype
  definitions, inductive definitions, structural induction and rule
  induction.

\item Theory \isa{ListN} inductively defines the lists of $n$
  elements~\cite{paulin-tlca}.

\item Theory \isa{Acc} inductively defines the accessible part of a
  relation~\cite{paulin-tlca}.

\item Theory \isa{Comb} defines the datatype of combinators and
  inductively defines contraction and parallel contraction.  It goes on to
  prove the Church-Rosser Theorem.  This case study follows Camilleri and
  Melham~\cite{camilleri92}.

\item Theory \isa{LList} defines lazy lists and a coinduction
  principle for proving equations between them.
\end{itemize}


\section{A proof about powersets}\label{sec:ZF-pow-example}
To demonstrate high-level reasoning about subsets, let us prove the
equation ${\isa{Pow}(A)\cap \isa{Pow}(B)}= \isa{Pow}(A\cap B)$.  Compared
with first-order logic, set theory involves a maze of rules, and theorems
have many different proofs.  Attempting other proofs of the theorem might
be instructive.  This proof exploits the lattice properties of
intersection.  It also uses the monotonicity of the powerset operation,
from \isa{ZF/mono.ML}:
\begin{isabelle}
\tdx{Pow_mono}:     A \isasymsubseteq B ==> Pow(A) \isasymsubseteq Pow(B)
\end{isabelle}
We enter the goal and make the first step, which breaks the equation into
two inclusions by extensionality:\index{*equalityI theorem}
\begin{isabelle}
\isacommand{lemma}\ "Pow(A\ Int\ B)\ =\ Pow(A)\ Int\ Pow(B)"\isanewline
\ 1.\ Pow(A\ \isasyminter \ B)\ =\ Pow(A)\ \isasyminter \ Pow(B)\isanewline
\isacommand{apply}\ (rule\ equalityI)\isanewline
\ 1.\ Pow(A\ \isasyminter \ B)\ \isasymsubseteq \ Pow(A)\ \isasyminter \ Pow(B)\isanewline
\ 2.\ Pow(A)\ \isasyminter \ Pow(B)\ \isasymsubseteq \ Pow(A\ \isasyminter \ B)
\end{isabelle}
Both inclusions could be tackled straightforwardly using \isa{subsetI}.
A shorter proof results from noting that intersection forms the greatest
lower bound:\index{*Int_greatest theorem}
\begin{isabelle}
\isacommand{apply}\ (rule\ Int\_greatest)\isanewline
\ 1.\ Pow(A\ \isasyminter \ B)\ \isasymsubseteq \ Pow(A)\isanewline
\ 2.\ Pow(A\ \isasyminter \ B)\ \isasymsubseteq \ Pow(B)\isanewline
\ 3.\ Pow(A)\ \isasyminter \ Pow(B)\ \isasymsubseteq \ Pow(A\ \isasyminter \ B)
\end{isabelle}
Subgoal~1 follows by applying the monotonicity of \isa{Pow} to $A\int
B\subseteq A$; subgoal~2 follows similarly:
\index{*Int_lower1 theorem}\index{*Int_lower2 theorem}
\begin{isabelle}
\isacommand{apply}\ (rule\ Int\_lower1\ [THEN\ Pow\_mono])\isanewline
\ 1.\ Pow(A\ \isasyminter \ B)\ \isasymsubseteq \ Pow(B)\isanewline
\ 2.\ Pow(A)\ \isasyminter \ Pow(B)\ \isasymsubseteq \ Pow(A\ \isasyminter \ B)
\isanewline
\isacommand{apply}\ (rule\ Int\_lower2\ [THEN\ Pow\_mono])\isanewline
\ 1.\ Pow(A)\ \isasyminter \ Pow(B)\ \isasymsubseteq \ Pow(A\ \isasyminter \ B)
\end{isabelle}
We are left with the opposite inclusion, which we tackle in the
straightforward way:\index{*subsetI theorem}
\begin{isabelle}
\isacommand{apply}\ (rule\ subsetI)\isanewline
\ 1.\ \isasymAnd x.\ x\ \isasymin \ Pow(A)\ \isasyminter \ Pow(B)\ \isasymLongrightarrow \ x\ \isasymin \ Pow(A\ \isasyminter \ B)
\end{isabelle}
The subgoal is to show $x\in \isa{Pow}(A\cap B)$ assuming $x\in\isa{Pow}(A)\cap \isa{Pow}(B)$; eliminating this assumption produces two
subgoals.  The rule \tdx{IntE} treats the intersection like a conjunction
instead of unfolding its definition.
\begin{isabelle}
\isacommand{apply}\ (erule\ IntE)\isanewline
\ 1.\ \isasymAnd x.\ \isasymlbrakk x\ \isasymin \ Pow(A);\ x\ \isasymin \ Pow(B)\isasymrbrakk \ \isasymLongrightarrow \ x\ \isasymin \ Pow(A\ \isasyminter \ B)
\end{isabelle}
The next step replaces the \isa{Pow} by the subset
relation~($\subseteq$).\index{*PowI theorem}
\begin{isabelle}
\isacommand{apply}\ (rule\ PowI)\isanewline
\ 1.\ \isasymAnd x.\ \isasymlbrakk x\ \isasymin \ Pow(A);\ x\ \isasymin \ Pow(B)\isasymrbrakk \ \isasymLongrightarrow \ x\ \isasymsubseteq \ A\ \isasyminter \ B%
\end{isabelle}
We perform the same replacement in the assumptions.  This is a good
demonstration of the tactic \ttindex{drule}:\index{*PowD theorem}
\begin{isabelle}
\isacommand{apply}\ (drule\ PowD)+\isanewline
\ 1.\ \isasymAnd x.\ \isasymlbrakk x\ \isasymsubseteq \ A;\ x\ \isasymsubseteq \ B\isasymrbrakk \ \isasymLongrightarrow \ x\ \isasymsubseteq \ A\ \isasyminter \ B%
\end{isabelle}
The assumptions are that $x$ is a lower bound of both $A$ and~$B$, but
$A\int B$ is the greatest lower bound:\index{*Int_greatest theorem}
\begin{isabelle}
\isacommand{apply}\ (rule\ Int\_greatest)\isanewline
\ 1.\ \isasymAnd x.\ \isasymlbrakk x\ \isasymsubseteq \ A;\ x\ \isasymsubseteq \ B\isasymrbrakk \ \isasymLongrightarrow \ x\ \isasymsubseteq \ A\isanewline
\ 2.\ \isasymAnd x.\ \isasymlbrakk x\ \isasymsubseteq \ A;\ x\ \isasymsubseteq \ B\isasymrbrakk \ \isasymLongrightarrow \ x\ \isasymsubseteq \ B%
\end{isabelle}
To conclude the proof, we clear up the trivial subgoals:
\begin{isabelle}
\isacommand{apply}\ (assumption+)\isanewline
\isacommand{done}%
\end{isabelle}

We could have performed this proof instantly by calling
\ttindex{blast}:
\begin{isabelle}
\isacommand{lemma}\ "Pow(A\ Int\ B)\ =\ Pow(A)\ Int\ Pow(B)"\isanewline
\isacommand{by}
\end{isabelle}
Past researchers regarded this as a difficult proof, as indeed it is if all
the symbols are replaced by their definitions.
\goodbreak

\section{Monotonicity of the union operator}
For another example, we prove that general union is monotonic:
${C\subseteq D}$ implies $\bigcup(C)\subseteq \bigcup(D)$.  To begin, we
tackle the inclusion using \tdx{subsetI}:
\begin{isabelle}
\isacommand{lemma}\ "C\isasymsubseteq D\ ==>\ Union(C)\
\isasymsubseteq \ Union(D)"\isanewline
\isacommand{apply}\ (rule\ subsetI)\isanewline
\ 1.\ \isasymAnd x.\ \isasymlbrakk C\ \isasymsubseteq \ D;\ x\ \isasymin \ \isasymUnion C\isasymrbrakk \ \isasymLongrightarrow \ x\ \isasymin \ \isasymUnion D%
\end{isabelle}
Big union is like an existential quantifier --- the occurrence in the
assumptions must be eliminated early, since it creates parameters.
\index{*UnionE theorem}
\begin{isabelle}
\isacommand{apply}\ (erule\ UnionE)\isanewline
\ 1.\ \isasymAnd x\ B.\ \isasymlbrakk C\ \isasymsubseteq \ D;\ x\ \isasymin \ B;\ B\ \isasymin \ C\isasymrbrakk \ \isasymLongrightarrow \ x\ \isasymin \ \isasymUnion D%
\end{isabelle}
Now we may apply \tdx{UnionI}, which creates an unknown involving the
parameters.  To show \isa{x\ \isasymin \ \isasymUnion D} it suffices to show that~\isa{x} belongs
to some element, say~\isa{?B2(x,B)}, of~\isa{D}\@.
\begin{isabelle}
\isacommand{apply}\ (rule\ UnionI)\isanewline
\ 1.\ \isasymAnd x\ B.\ \isasymlbrakk C\ \isasymsubseteq \ D;\ x\ \isasymin \ B;\ B\ \isasymin \ C\isasymrbrakk \ \isasymLongrightarrow \ ?B2(x,\ B)\ \isasymin \ D\isanewline
\ 2.\ \isasymAnd x\ B.\ \isasymlbrakk C\ \isasymsubseteq \ D;\ x\ \isasymin \ B;\ B\ \isasymin \ C\isasymrbrakk \ \isasymLongrightarrow \ x\ \isasymin \ ?B2(x,\ B)
\end{isabelle}
Combining the rule \tdx{subsetD} with the assumption \isa{C\ \isasymsubseteq \ D} yields 
$\Var{a}\in C \Imp \Var{a}\in D$, which reduces subgoal~1.  Note that
\isa{erule} removes the subset assumption.
\begin{isabelle}
\isacommand{apply}\ (erule\ subsetD)\isanewline
\ 1.\ \isasymAnd x\ B.\ \isasymlbrakk x\ \isasymin \ B;\ B\ \isasymin \ C\isasymrbrakk \ \isasymLongrightarrow \ ?B2(x,\ B)\ \isasymin \ C\isanewline
\ 2.\ \isasymAnd x\ B.\ \isasymlbrakk C\ \isasymsubseteq \ D;\ x\ \isasymin \ B;\ B\ \isasymin \ C\isasymrbrakk \ \isasymLongrightarrow \ x\ \isasymin \ ?B2(x,\ B)
\end{isabelle}
The rest is routine.  Observe how the first call to \isa{assumption}
instantiates \isa{?B2(x,B)} to~\isa{B}\@.
\begin{isabelle}
\isacommand{apply}\ assumption\ \isanewline
\ 1.\ \isasymAnd x\ B.\ \isasymlbrakk C\ \isasymsubseteq \ D;\ x\ \isasymin \ B;\ B\ \isasymin \ C\isasymrbrakk \ \isasymLongrightarrow \ x\ \isasymin \ B%
\isanewline
\isacommand{apply}\ assumption\ \isanewline
No\ subgoals!\isanewline
\isacommand{done}%
\end{isabelle}
Again, \isa{blast} can prove this theorem in one step.

The theory \isa{ZF/equalities.thy} has many similar proofs.  Reasoning about
general intersection can be difficult because of its anomalous behaviour on
the empty set.  However, \isa{blast} copes well with these.  Here is
a typical example, borrowed from Devlin~\cite[page 12]{devlin79}:
\[ a\in C \,\Imp\, \inter@{x\in C} \Bigl(A(x) \int B(x)\Bigr) =        
       \Bigl(\inter@{x\in C} A(x)\Bigr)  \int  
       \Bigl(\inter@{x\in C} B(x)\Bigr)  \]

\section{Low-level reasoning about functions}
The derived rules \isa{lamI}, \isa{lamE}, \isa{lam_type}, \isa{beta}
and \isa{eta} support reasoning about functions in a
$\lambda$-calculus style.  This is generally easier than regarding
functions as sets of ordered pairs.  But sometimes we must look at the
underlying representation, as in the following proof
of~\tdx{fun_disjoint_apply1}.  This states that if $f$ and~$g$ are
functions with disjoint domains~$A$ and~$C$, and if $a\in A$, then
$(f\un g)`a = f`a$:
\begin{isabelle}
\isacommand{lemma}\ "[|\ a\ \isasymin \ A;\ \ f\ \isasymin \ A->B;\ \ g\ \isasymin \ C->D;\ \ A\ \isasyminter \ C\ =\ 0\ |]
\isanewline
\ \ \ \ \ \ \ \ ==>\ (f\ \isasymunion \ g)`a\ =\ f`a"
\end{isabelle}
Using \tdx{apply_equality}, we reduce the equality to reasoning about
ordered pairs.  The second subgoal is to verify that \isa{f\ \isasymunion \ g} is a function, since
\isa{Pi(?A,?B)} denotes a dependent function space.
\begin{isabelle}
\isacommand{apply}\ (rule\ apply\_equality)\isanewline
\ 1.\ \isasymlbrakk a\ \isasymin \ A;\ f\ \isasymin \ A\ \isasymrightarrow \ B;\ g\ \isasymin \ C\ \isasymrightarrow \ D;\ A\ \isasyminter \ C\ =\ 0\isasymrbrakk \isanewline
\isaindent{\ 1.\ }\isasymLongrightarrow \ \isasymlangle a,\ f\ `\ a\isasymrangle \ \isasymin \ f\ \isasymunion \ g\isanewline
\ 2.\ \isasymlbrakk a\ \isasymin \ A;\ f\ \isasymin \ A\ \isasymrightarrow \ B;\ g\ \isasymin \ C\ \isasymrightarrow \ D;\ A\ \isasyminter \ C\ =\ 0\isasymrbrakk \isanewline
\isaindent{\ 2.\ }\isasymLongrightarrow \ f\ \isasymunion \ g\ \isasymin \ Pi(?A,\ ?B)
\end{isabelle}
We must show that the pair belongs to~$f$ or~$g$; by~\tdx{UnI1} we
choose~$f$:
\begin{isabelle}
\isacommand{apply}\ (rule\ UnI1)\isanewline
\ 1.\ \isasymlbrakk a\ \isasymin \ A;\ f\ \isasymin \ A\ \isasymrightarrow \ B;\ g\ \isasymin \ C\ \isasymrightarrow \ D;\ A\ \isasyminter \ C\ =\ 0\isasymrbrakk \ \isasymLongrightarrow \ \isasymlangle a,\ f\ `\ a\isasymrangle \ \isasymin \ f\isanewline
\ 2.\ \isasymlbrakk a\ \isasymin \ A;\ f\ \isasymin \ A\ \isasymrightarrow \ B;\ g\ \isasymin \ C\ \isasymrightarrow \ D;\ A\ \isasyminter \ C\ =\ 0\isasymrbrakk \isanewline
\isaindent{\ 2.\ }\isasymLongrightarrow \ f\ \isasymunion \ g\ \isasymin \ Pi(?A,\ ?B)
\end{isabelle}
To show $\pair{a,f`a}\in f$ we use \tdx{apply_Pair}, which is
essentially the converse of \tdx{apply_equality}:
\begin{isabelle}
\isacommand{apply}\ (rule\ apply\_Pair)\isanewline
\ 1.\ \isasymlbrakk a\ \isasymin \ A;\ f\ \isasymin \ A\ \isasymrightarrow \ B;\ g\ \isasymin \ C\ \isasymrightarrow \ D;\ A\ \isasyminter \ C\ =\ 0\isasymrbrakk \ \isasymLongrightarrow \ f\ \isasymin \ Pi(?A2,?B2)\isanewline
\ 2.\ \isasymlbrakk a\ \isasymin \ A;\ f\ \isasymin \ A\ \isasymrightarrow \ B;\ g\ \isasymin \ C\ \isasymrightarrow \ D;\ A\ \isasyminter \ C\ =\ 0\isasymrbrakk \ \isasymLongrightarrow \ a\ \isasymin \ ?A2\isanewline
\ 3.\ \isasymlbrakk a\ \isasymin \ A;\ f\ \isasymin \ A\ \isasymrightarrow \ B;\ g\ \isasymin \ C\ \isasymrightarrow \ D;\ A\ \isasyminter \ C\ =\ 0\isasymrbrakk \isanewline
\isaindent{\ 3.\ }\isasymLongrightarrow \ f\ \isasymunion \ g\ \isasymin \ Pi(?A,\ ?B)
\end{isabelle}
Using the assumptions $f\in A\to B$ and $a\in A$, we solve the two subgoals
from \tdx{apply_Pair}.  Recall that a $\Pi$-set is merely a generalized
function space, and observe that~{\tt?A2} gets instantiated to~\isa{A}.
\begin{isabelle}
\isacommand{apply}\ assumption\ \isanewline
\ 1.\ \isasymlbrakk a\ \isasymin \ A;\ f\ \isasymin \ A\ \isasymrightarrow \ B;\ g\ \isasymin \ C\ \isasymrightarrow \ D;\ A\ \isasyminter \ C\ =\ 0\isasymrbrakk \ \isasymLongrightarrow \ a\ \isasymin \ A\isanewline
\ 2.\ \isasymlbrakk a\ \isasymin \ A;\ f\ \isasymin \ A\ \isasymrightarrow \ B;\ g\ \isasymin \ C\ \isasymrightarrow \ D;\ A\ \isasyminter \ C\ =\ 0\isasymrbrakk \isanewline
\isaindent{\ 2.\ }\isasymLongrightarrow \ f\ \isasymunion \ g\ \isasymin \ Pi(?A,\ ?B)
\isanewline
\isacommand{apply}\ assumption\ \isanewline
\ 1.\ \isasymlbrakk a\ \isasymin \ A;\ f\ \isasymin \ A\ \isasymrightarrow \ B;\ g\ \isasymin \ C\ \isasymrightarrow \ D;\ A\ \isasyminter \ C\ =\ 0\isasymrbrakk \isanewline
\isaindent{\ 1.\ }\isasymLongrightarrow \ f\ \isasymunion \ g\ \isasymin \ Pi(?A,\ ?B)
\end{isabelle}
To construct functions of the form $f\un g$, we apply
\tdx{fun_disjoint_Un}:
\begin{isabelle}
\isacommand{apply}\ (rule\ fun\_disjoint\_Un)\isanewline
\ 1.\ \isasymlbrakk a\ \isasymin \ A;\ f\ \isasymin \ A\ \isasymrightarrow \ B;\ g\ \isasymin \ C\ \isasymrightarrow \ D;\ A\ \isasyminter \ C\ =\ 0\isasymrbrakk \ \isasymLongrightarrow \ f\ \isasymin \ ?A3\ \isasymrightarrow \ ?B3\isanewline
\ 2.\ \isasymlbrakk a\ \isasymin \ A;\ f\ \isasymin \ A\ \isasymrightarrow \ B;\ g\ \isasymin \ C\ \isasymrightarrow \ D;\ A\ \isasyminter \ C\ =\ 0\isasymrbrakk \ \isasymLongrightarrow \ g\ \isasymin \ ?C3\ \isasymrightarrow \ ?D3\isanewline
\ 3.\ \isasymlbrakk a\ \isasymin \ A;\ f\ \isasymin \ A\ \isasymrightarrow \ B;\ g\ \isasymin \ C\ \isasymrightarrow \ D;\ A\ \isasyminter \ C\ =\ 0\isasymrbrakk \ \isasymLongrightarrow \ ?A3\ \isasyminter \ ?C3\ =\ 0
\end{isabelle}
The remaining subgoals are instances of the assumptions.  Again, observe how
unknowns become instantiated:
\begin{isabelle}
\isacommand{apply}\ assumption\ \isanewline
\ 1.\ \isasymlbrakk a\ \isasymin \ A;\ f\ \isasymin \ A\ \isasymrightarrow \ B;\ g\ \isasymin \ C\ \isasymrightarrow \ D;\ A\ \isasyminter \ C\ =\ 0\isasymrbrakk \ \isasymLongrightarrow \ g\ \isasymin \ ?C3\ \isasymrightarrow \ ?D3\isanewline
\ 2.\ \isasymlbrakk a\ \isasymin \ A;\ f\ \isasymin \ A\ \isasymrightarrow \ B;\ g\ \isasymin \ C\ \isasymrightarrow \ D;\ A\ \isasyminter \ C\ =\ 0\isasymrbrakk \ \isasymLongrightarrow \ A\ \isasyminter \ ?C3\ =\ 0
\isanewline
\isacommand{apply}\ assumption\ \isanewline
\ 1.\ \isasymlbrakk a\ \isasymin \ A;\ f\ \isasymin \ A\ \isasymrightarrow \ B;\ g\ \isasymin \ C\ \isasymrightarrow \ D;\ A\ \isasyminter \ C\ =\ 0\isasymrbrakk \ \isasymLongrightarrow \ A\ \isasyminter \ C\ =\ 0
\isanewline
\isacommand{apply}\ assumption\ \isanewline
No\ subgoals!\isanewline
\isacommand{done}
\end{isabelle}
See the theories \isa{ZF/func.thy} and \isa{ZF/WF.thy} for more
examples of reasoning about functions.

\index{set theory|)}