doc-src/ZF/ZF.tex

(reverted to previous version)

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−%% $Id$
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−\chapter{Zermelo-Fraenkel Set Theory}
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−\index{set theory|(}
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−
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−The theory~\thydx{ZF} implements Zermelo-Fraenkel set
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−theory~\cite{halmos60,suppes72} as an extension of~\texttt{FOL}, classical
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−first-order logic.  The theory includes a collection of derived natural
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−deduction rules, for use with Isabelle's classical reasoner.  Some
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−of it is based on the work of No\"el~\cite{noel}.
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−
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−A tremendous amount of set theory has been formally developed, including the
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−basic properties of relations, functions, ordinals and cardinals.  Significant
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−results have been proved, such as the Schr\"oder-Bernstein Theorem, the
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−Wellordering Theorem and a version of Ramsey's Theorem.  \texttt{ZF} provides
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−both the integers and the natural numbers.  General methods have been
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−developed for solving recursion equations over monotonic functors; these have
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−been applied to yield constructions of lists, trees, infinite lists, etc.
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−
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−\texttt{ZF} has a flexible package for handling inductive definitions,
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−such as inference systems, and datatype definitions, such as lists and
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−trees.  Moreover it handles coinductive definitions, such as
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−bisimulation relations, and codatatype definitions, such as streams.  It
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−provides a streamlined syntax for defining primitive recursive functions over
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−datatypes. 
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−
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−Published articles~\cite{paulson-set-I,paulson-set-II} describe \texttt{ZF}
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−less formally than this chapter.  Isabelle employs a novel treatment of
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−non-well-founded data structures within the standard {\sc zf} axioms including
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−the Axiom of Foundation~\cite{paulson-mscs}.
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−
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−
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−\section{Which version of axiomatic set theory?}
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−The two main axiom systems for set theory are Bernays-G\"odel~({\sc bg})
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−and Zermelo-Fraenkel~({\sc zf}).  Resolution theorem provers can use {\sc
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−  bg} because it is finite~\cite{boyer86,quaife92}.  {\sc zf} does not
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−have a finite axiom system because of its Axiom Scheme of Replacement.
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−This makes it awkward to use with many theorem provers, since instances
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−of the axiom scheme have to be invoked explicitly.  Since Isabelle has no
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−difficulty with axiom schemes, we may adopt either axiom system.
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−
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−These two theories differ in their treatment of {\bf classes}, which are
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−collections that are `too big' to be sets.  The class of all sets,~$V$,
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−cannot be a set without admitting Russell's Paradox.  In {\sc bg}, both
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−classes and sets are individuals; $x\in V$ expresses that $x$ is a set.  In
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−{\sc zf}, all variables denote sets; classes are identified with unary
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−predicates.  The two systems define essentially the same sets and classes,
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−with similar properties.  In particular, a class cannot belong to another
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−class (let alone a set).
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−
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−Modern set theorists tend to prefer {\sc zf} because they are mainly concerned
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−with sets, rather than classes.  {\sc bg} requires tiresome proofs that various
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−collections are sets; for instance, showing $x\in\{x\}$ requires showing that
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−$x$ is a set.
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−
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−
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−\begin{figure} \small
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−\begin{center}
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−\begin{tabular}{rrr} 
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−  \it name      &\it meta-type  & \it description \\ 
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−  \cdx{Let}     & $[\alpha,\alpha\To\beta]\To\beta$ & let binder\\
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−  \cdx{0}       & $i$           & empty set\\
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−  \cdx{cons}    & $[i,i]\To i$  & finite set constructor\\
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−  \cdx{Upair}   & $[i,i]\To i$  & unordered pairing\\
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−  \cdx{Pair}    & $[i,i]\To i$  & ordered pairing\\
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−  \cdx{Inf}     & $i$   & infinite set\\
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−  \cdx{Pow}     & $i\To i$      & powerset\\
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−  \cdx{Union} \cdx{Inter} & $i\To i$    & set union/intersection \\
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−  \cdx{split}   & $[[i,i]\To i, i] \To i$ & generalized projection\\
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−  \cdx{fst} \cdx{snd}   & $i\To i$      & projections\\
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−  \cdx{converse}& $i\To i$      & converse of a relation\\
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−  \cdx{succ}    & $i\To i$      & successor\\
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−  \cdx{Collect} & $[i,i\To o]\To i$     & separation\\
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−  \cdx{Replace} & $[i, [i,i]\To o] \To i$       & replacement\\
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−  \cdx{PrimReplace} & $[i, [i,i]\To o] \To i$   & primitive replacement\\
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−  \cdx{RepFun}  & $[i, i\To i] \To i$   & functional replacement\\
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−  \cdx{Pi} \cdx{Sigma}  & $[i,i\To i]\To i$     & general product/sum\\
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−  \cdx{domain}  & $i\To i$      & domain of a relation\\
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−  \cdx{range}   & $i\To i$      & range of a relation\\
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−  \cdx{field}   & $i\To i$      & field of a relation\\
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−  \cdx{Lambda}  & $[i, i\To i]\To i$    & $\lambda$-abstraction\\
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−  \cdx{restrict}& $[i, i] \To i$        & restriction of a function\\
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−  \cdx{The}     & $[i\To o]\To i$       & definite description\\
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−  \cdx{if}      & $[o,i,i]\To i$        & conditional\\
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−  \cdx{Ball} \cdx{Bex}  & $[i, i\To o]\To o$    & bounded quantifiers
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−\end{tabular}
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−\end{center}
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−\subcaption{Constants}
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−
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−\begin{center}
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−\index{*"`"` symbol}
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−\index{*"-"`"` symbol}
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−\index{*"` symbol}\index{function applications}
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−\index{*"- symbol}
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−\index{*": symbol}
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−\index{*"<"= symbol}
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−\begin{tabular}{rrrr} 
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−  \it symbol  & \it meta-type & \it priority & \it description \\ 
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−  \tt ``        & $[i,i]\To i$  &  Left 90      & image \\
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−  \tt -``       & $[i,i]\To i$  &  Left 90      & inverse image \\
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−  \tt `         & $[i,i]\To i$  &  Left 90      & application \\
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−  \sdx{Int}     & $[i,i]\To i$  &  Left 70      & intersection ($\int$) \\
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−  \sdx{Un}      & $[i,i]\To i$  &  Left 65      & union ($\un$) \\
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−  \tt -         & $[i,i]\To i$  &  Left 65      & set difference ($-$) \\[1ex]
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−  \tt:          & $[i,i]\To o$  &  Left 50      & membership ($\in$) \\
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−  \tt <=        & $[i,i]\To o$  &  Left 50      & subset ($\subseteq$) 
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−\end{tabular}
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−\end{center}
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−\subcaption{Infixes}
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−\caption{Constants of ZF} \label{zf-constants}
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−\end{figure} 
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−
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−
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−\section{The syntax of set theory}
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−The language of set theory, as studied by logicians, has no constants.  The
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−traditional axioms merely assert the existence of empty sets, unions,
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−powersets, etc.; this would be intolerable for practical reasoning.  The
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−Isabelle theory declares constants for primitive sets.  It also extends
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−\texttt{FOL} with additional syntax for finite sets, ordered pairs,
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−comprehension, general union/intersection, general sums/products, and
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−bounded quantifiers.  In most other respects, Isabelle implements precisely
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−Zermelo-Fraenkel set theory.
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−
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−Figure~\ref{zf-constants} lists the constants and infixes of~ZF, while
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−Figure~\ref{zf-trans} presents the syntax translations.  Finally,
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−Figure~\ref{zf-syntax} presents the full grammar for set theory, including the
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−constructs of FOL.
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−
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−Local abbreviations can be introduced by a \isa{let} construct whose
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−syntax appears in Fig.\ts\ref{zf-syntax}.  Internally it is translated into
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−the constant~\cdx{Let}.  It can be expanded by rewriting with its
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−definition, \tdx{Let_def}.
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−
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−Apart from \isa{let}, set theory does not use polymorphism.  All terms in
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−ZF have type~\tydx{i}, which is the type of individuals and has
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−class~\cldx{term}.  The type of first-order formulae, remember, 
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−is~\tydx{o}.
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−
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−Infix operators include binary union and intersection ($A\un B$ and
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−$A\int B$), set difference ($A-B$), and the subset and membership
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−relations.  Note that $a$\verb|~:|$b$ is translated to $\lnot(a\in b)$,
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−which is equivalent to  $a\notin b$.  The
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−union and intersection operators ($\bigcup A$ and $\bigcap A$) form the
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−union or intersection of a set of sets; $\bigcup A$ means the same as
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−$\bigcup@{x\in A}x$.  Of these operators, only $\bigcup A$ is primitive.
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−
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−The constant \cdx{Upair} constructs unordered pairs; thus \isa{Upair($A$,$B$)} denotes the set~$\{A,B\}$ and
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−\isa{Upair($A$,$A$)} denotes the singleton~$\{A\}$.  General union is
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−used to define binary union.  The Isabelle version goes on to define
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−the constant
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−\cdx{cons}:
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−\begin{eqnarray*}
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−   A\cup B              & \equiv &       \bigcup(\isa{Upair}(A,B)) \\
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−   \isa{cons}(a,B)      & \equiv &        \isa{Upair}(a,a) \un B
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−\end{eqnarray*}
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−The $\{a@1, \ldots\}$ notation abbreviates finite sets constructed in the
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−obvious manner using~\isa{cons} and~$\emptyset$ (the empty set) \isasymin \begin{eqnarray*}
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− \{a,b,c\} & \equiv & \isa{cons}(a,\isa{cons}(b,\isa{cons}(c,\emptyset)))
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−\end{eqnarray*}
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−
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−The constant \cdx{Pair} constructs ordered pairs, as in \isa{Pair($a$,$b$)}.  Ordered pairs may also be written within angle brackets,
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−as {\tt<$a$,$b$>}.  The $n$-tuple {\tt<$a@1$,\ldots,$a@{n-1}$,$a@n$>}
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−abbreviates the nest of pairs\par\nobreak
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−\centerline{\isa{Pair($a@1$,\ldots,Pair($a@{n-1}$,$a@n$)\ldots).}}
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−
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−In ZF, a function is a set of pairs.  A ZF function~$f$ is simply an
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−individual as far as Isabelle is concerned: its Isabelle type is~$i$, not say
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−$i\To i$.  The infix operator~{\tt`} denotes the application of a function set
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−to its argument; we must write~$f{\tt`}x$, not~$f(x)$.  The syntax for image
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−is~$f{\tt``}A$ and that for inverse image is~$f{\tt-``}A$.
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−
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−
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−\begin{figure} 
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−\index{lambda abs@$\lambda$-abstractions}
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−\index{*"-"> symbol}
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−\index{*"* symbol}
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−\begin{center} \footnotesize\tt\frenchspacing
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−\begin{tabular}{rrr} 
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−  \it external          & \it internal  & \it description \\ 
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−  $a$ \ttilde: $b$      & \ttilde($a$ : $b$)    & \rm negated membership\\
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−  \ttlbrace$a@1$, $\ldots$, $a@n$\ttrbrace  &  cons($a@1$,$\ldots$,cons($a@n$,0)) &
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−        \rm finite set \\
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−  <$a@1$, $\ldots$, $a@{n-1}$, $a@n$> & 
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−        Pair($a@1$,\ldots,Pair($a@{n-1}$,$a@n$)\ldots) &
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−        \rm ordered $n$-tuple \\
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−  \ttlbrace$x$:$A . P[x]$\ttrbrace    &  Collect($A$,$\lambda x. P[x]$) &
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−        \rm separation \\
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−  \ttlbrace$y . x$:$A$, $Q[x,y]$\ttrbrace  &  Replace($A$,$\lambda x\,y. Q[x,y]$) &
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−        \rm replacement \\
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−  \ttlbrace$b[x] . x$:$A$\ttrbrace  &  RepFun($A$,$\lambda x. b[x]$) &
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−        \rm functional replacement \\
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−  \sdx{INT} $x$:$A . B[x]$      & Inter(\ttlbrace$B[x] . x$:$A$\ttrbrace) &
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−        \rm general intersection \\
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−  \sdx{UN}  $x$:$A . B[x]$      & Union(\ttlbrace$B[x] . x$:$A$\ttrbrace) &
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−        \rm general union \\
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−  \sdx{PROD} $x$:$A . B[x]$     & Pi($A$,$\lambda x. B[x]$) & 
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−        \rm general product \\
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−  \sdx{SUM}  $x$:$A . B[x]$     & Sigma($A$,$\lambda x. B[x]$) & 
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−        \rm general sum \\
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−  $A$ -> $B$            & Pi($A$,$\lambda x. B$) & 
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−        \rm function space \\
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−  $A$ * $B$             & Sigma($A$,$\lambda x. B$) & 
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−        \rm binary product \\
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−  \sdx{THE}  $x . P[x]$ & The($\lambda x. P[x]$) & 
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−        \rm definite description \\
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−  \sdx{lam}  $x$:$A . b[x]$     & Lambda($A$,$\lambda x. b[x]$) & 
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−        \rm $\lambda$-abstraction\\[1ex]
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−  \sdx{ALL} $x$:$A . P[x]$      & Ball($A$,$\lambda x. P[x]$) & 
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−        \rm bounded $\forall$ \\
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−  \sdx{EX}  $x$:$A . P[x]$      & Bex($A$,$\lambda x. P[x]$) & 
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−        \rm bounded $\exists$
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−\end{tabular}
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−\end{center}
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−\caption{Translations for ZF} \label{zf-trans}
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−\end{figure} 
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−
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−
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−\begin{figure} 
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−\index{*let symbol}
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−\index{*in symbol}
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−\dquotes
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−\[\begin{array}{rcl}
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−    term & = & \hbox{expression of type~$i$} \\
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−         & | & "let"~id~"="~term";"\dots";"~id~"="~term~"in"~term \\
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−         & | & "if"~term~"then"~term~"else"~term \\
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−         & | & "{\ttlbrace} " term\; ("," term)^* " {\ttrbrace}" \\
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−         & | & "< "  term\; ("," term)^* " >"  \\
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−         & | & "{\ttlbrace} " id ":" term " . " formula " {\ttrbrace}" \\
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−         & | & "{\ttlbrace} " id " . " id ":" term ", " formula " {\ttrbrace}" \\
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−         & | & "{\ttlbrace} " term " . " id ":" term " {\ttrbrace}" \\
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−         & | & term " `` " term \\
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−         & | & term " -`` " term \\
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−         & | & term " ` " term \\
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−         & | & term " * " term \\
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−         & | & term " \isasyminter " term \\
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−         & | & term " \isasymunion " term \\
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−         & | & term " - " term \\
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−         & | & term " -> " term \\
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−         & | & "THE~~"  id  " . " formula\\
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−         & | & "lam~~"  id ":" term " . " term \\
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−         & | & "INT~~"  id ":" term " . " term \\
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−         & | & "UN~~~"  id ":" term " . " term \\
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−         & | & "PROD~"  id ":" term " . " term \\
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−         & | & "SUM~~"  id ":" term " . " term \\[2ex]
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− formula & = & \hbox{expression of type~$o$} \\
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−         & | & term " : " term \\
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−         & | & term " \ttilde: " term \\
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−         & | & term " <= " term \\
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−         & | & term " = " term \\
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−         & | & term " \ttilde= " term \\
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−         & | & "\ttilde\ " formula \\
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−         & | & formula " \& " formula \\
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−         & | & formula " | " formula \\
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−         & | & formula " --> " formula \\
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−         & | & formula " <-> " formula \\
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−         & | & "ALL " id ":" term " . " formula \\
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−         & | & "EX~~" id ":" term " . " formula \\
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−         & | & "ALL~" id~id^* " . " formula \\
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−         & | & "EX~~" id~id^* " . " formula \\
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−         & | & "EX!~" id~id^* " . " formula
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−  \end{array}
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−\]
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−\caption{Full grammar for ZF} \label{zf-syntax}
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−\end{figure} 
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−
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−
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−\section{Binding operators}
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−The constant \cdx{Collect} constructs sets by the principle of {\bf
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−  separation}.  The syntax for separation is
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−\hbox{\tt\ttlbrace$x$:$A$.\ $P[x]$\ttrbrace}, where $P[x]$ is a formula
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−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
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−satisfy~$P[x]$.  Note that \isa{Collect} is an unfortunate choice of
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−name: some set theories adopt a set-formation principle, related to
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−replacement, called collection.
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−
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−The constant \cdx{Replace} constructs sets by the principle of {\bf
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−  replacement}.  The syntax
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−\hbox{\tt\ttlbrace$y$.\ $x$:$A$,$Q[x,y]$\ttrbrace} denotes the set 
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−\isa{Replace($A$,$\lambda x\,y. Q[x,y]$)}, which consists of all~$y$ such
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−that there exists $x\in A$ satisfying~$Q[x,y]$.  The Replacement Axiom
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−has the condition that $Q$ must be single-valued over~$A$: for
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−all~$x\in A$ there exists at most one $y$ satisfying~$Q[x,y]$.  A
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−single-valued binary predicate is also called a {\bf class function}.
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−
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−The constant \cdx{RepFun} expresses a special case of replacement,
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−where $Q[x,y]$ has the form $y=b[x]$.  Such a $Q$ is trivially
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−single-valued, since it is just the graph of the meta-level
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−function~$\lambda x. b[x]$.  The resulting set consists of all $b[x]$
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−for~$x\in A$.  This is analogous to the \ML{} functional \isa{map},
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−since it applies a function to every element of a set.  The syntax is
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−\isa{\ttlbrace$b[x]$.\ $x$:$A$\ttrbrace}, which expands to 
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−\isa{RepFun($A$,$\lambda x. b[x]$)}.
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−
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−\index{*INT symbol}\index{*UN symbol} 
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−General unions and intersections of indexed
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−families of sets, namely $\bigcup@{x\in A}B[x]$ and $\bigcap@{x\in A}B[x]$,
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−are written \isa{UN $x$:$A$.\ $B[x]$} and \isa{INT $x$:$A$.\ $B[x]$}.
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−Their meaning is expressed using \isa{RepFun} as
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−\[
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−\bigcup(\{B[x]. x\in A\}) \qquad\hbox{and}\qquad 
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−\bigcap(\{B[x]. x\in A\}). 
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−\]
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−General sums $\sum@{x\in A}B[x]$ and products $\prod@{x\in A}B[x]$ can be
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−constructed in set theory, where $B[x]$ is a family of sets over~$A$.  They
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−have as special cases $A\times B$ and $A\to B$, where $B$ is simply a set.
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−This is similar to the situation in Constructive Type Theory (set theory
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−has `dependent sets') and calls for similar syntactic conventions.  The
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−constants~\cdx{Sigma} and~\cdx{Pi} construct general sums and
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−products.  Instead of \isa{Sigma($A$,$B$)} and \isa{Pi($A$,$B$)} we may
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−write 
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−\isa{SUM $x$:$A$.\ $B[x]$} and \isa{PROD $x$:$A$.\ $B[x]$}.  
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−\index{*SUM symbol}\index{*PROD symbol}%
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−The special cases as \hbox{\tt$A$*$B$} and \hbox{\tt$A$->$B$} abbreviate
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−general sums and products over a constant family.\footnote{Unlike normal
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−infix operators, {\tt*} and {\tt->} merely define abbreviations; there are
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−no constants~\isa{op~*} and~\isa{op~->}.} Isabelle accepts these
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−abbreviations in parsing and uses them whenever possible for printing.
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−
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−\index{*THE symbol} As mentioned above, whenever the axioms assert the
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−existence and uniqueness of a set, Isabelle's set theory declares a constant
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−for that set.  These constants can express the {\bf definite description}
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−operator~$\iota x. P[x]$, which stands for the unique~$a$ satisfying~$P[a]$,
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−if such exists.  Since all terms in ZF denote something, a description is
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−always meaningful, but we do not know its value unless $P[x]$ defines it
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−uniquely.  Using the constant~\cdx{The}, we may write descriptions as 
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−\isa{The($\lambda x. P[x]$)} or use the syntax \isa{THE $x$.\ $P[x]$}.
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−
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−\index{*lam symbol}
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−Function sets may be written in $\lambda$-notation; $\lambda x\in A. b[x]$
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−stands for the set of all pairs $\pair{x,b[x]}$ for $x\in A$.  In order for
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−this to be a set, the function's domain~$A$ must be given.  Using the
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−constant~\cdx{Lambda}, we may express function sets as \isa{Lambda($A$,$\lambda x. b[x]$)} or use the syntax \isa{lam $x$:$A$.\ $b[x]$}.
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−
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−Isabelle's set theory defines two {\bf bounded quantifiers}:
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−\begin{eqnarray*}
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−   \forall x\in A. P[x] &\hbox{abbreviates}& \forall x. x\in A\imp P[x] \\
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−   \exists x\in A. P[x] &\hbox{abbreviates}& \exists x. x\in A\conj P[x]
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−\end{eqnarray*}
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−The constants~\cdx{Ball} and~\cdx{Bex} are defined
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−accordingly.  Instead of \isa{Ball($A$,$P$)} and \isa{Bex($A$,$P$)} we may
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−write
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−\isa{ALL $x$:$A$.\ $P[x]$} and \isa{EX $x$:$A$.\ $P[x]$}.
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−
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−
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−%%%% ZF.thy
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−
+
−\begin{figure}
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−\begin{alltt*}\isastyleminor
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−\tdx{Let_def}:           Let(s, f) == f(s)
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−
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−\tdx{Ball_def}:          Ball(A,P) == {\isasymforall}x. x \isasymin A --> P(x)
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−\tdx{Bex_def}:           Bex(A,P)  == {\isasymexists}x. x \isasymin A & P(x)
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−
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−\tdx{subset_def}:        A \isasymsubseteq B  == {\isasymforall}x \isasymin A. x \isasymin B
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−\tdx{extension}:         A = B  <->  A \isasymsubseteq B & B \isasymsubseteq A
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−
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−\tdx{Union_iff}:         A \isasymin Union(C) <-> ({\isasymexists}B \isasymin C. A \isasymin B)
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−\tdx{Pow_iff}:           A \isasymin Pow(B) <-> A \isasymsubseteq B
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−\tdx{foundation}:        A=0 | ({\isasymexists}x \isasymin A. {\isasymforall}y \isasymin x. y \isasymnotin A)
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−
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−\tdx{replacement}:       ({\isasymforall}x \isasymin A. {\isasymforall}y z. P(x,y) & P(x,z) --> y=z) ==>
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−                   b \isasymin PrimReplace(A,P) <-> ({\isasymexists}x{\isasymin}A. P(x,b))
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−\subcaption{The Zermelo-Fraenkel Axioms}
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−
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−\tdx{Replace_def}: Replace(A,P) == 
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−                   PrimReplace(A, \%x y. (\isasymexists!z. P(x,z)) & P(x,y))
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−\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
+
−\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
+
−Fig.~\ref{datatype-grammar}.  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.  The quotation marks are necessary because
+
−they enclose general Isabelle formul\ae.
+
−
+
−\begin{figure}
+
−\begin{rail}
+
−datatype : ( 'datatype' | 'codatatype' ) datadecls;
+
−
+
−datadecls: ( '"' id arglist '"' '=' (constructor + '|') ) + 'and'
+
−         ;
+
−constructor : name ( () | consargs )  ( () | ( '(' mixfix ')' ) )
+
−         ;
+
−consargs : '(' ('"' var ' : ' term '"' + ',') ')'
+
−         ;
+
−\end{rail}
+
−\caption{Syntax of datatype declarations}
+
−\label{datatype-grammar}
+
−\end{figure}
+
−
+
−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|)}