|
\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
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