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doc-src/ZF/ZF.tex

author | paulson |

Tue, 19 Aug 2003 13:53:58 +0200 | |

changeset 14152 | 12f6f18e7afc |

parent 9836 | 56b632fd1dcd |

child 14154 | 3bc0128e2c74 |

permissions | -rw-r--r-- |

For the Isar version of the ZF logics manual

%% $Id$ \chapter{Zermelo-Fraenkel Set Theory} \index{set theory|(} The theory~\thydx{ZF} implements Zermelo-Fraenkel set theory~\cite{halmos60,suppes72} as an extension of~\texttt{FOL}, classical first-order logic. The theory includes a collection of derived natural deduction rules, for use with Isabelle's classical reasoner. Much of it is based on the work of No\"el~\cite{noel}. A tremendous amount of set theory has been formally developed, including the basic properties of relations, functions, ordinals and cardinals. Significant results have been proved, such as the Schr\"oder-Bernstein Theorem, the Wellordering Theorem and a version of Ramsey's Theorem. \texttt{ZF} provides both the integers and the natural numbers. General methods have been developed for solving recursion equations over monotonic functors; these have been applied to yield constructions of lists, trees, infinite lists, etc. \texttt{ZF} has a flexible package for handling inductive definitions, such as inference systems, and datatype definitions, such as lists and trees. Moreover it handles coinductive definitions, such as bisimulation relations, and codatatype definitions, such as streams. It provides a streamlined syntax for defining primitive recursive functions over datatypes. Because ZF is an extension of FOL, it provides the same packages, namely \texttt{hyp_subst_tac}, the simplifier, and the classical reasoner. The default simpset and claset are usually satisfactory. Published articles~\cite{paulson-set-I,paulson-set-II} describe \texttt{ZF} less formally than this chapter. Isabelle employs a novel treatment of non-well-founded data structures within the standard {\sc zf} axioms including the Axiom of Foundation~\cite{paulson-mscs}. \section{Which version of axiomatic set theory?} The two main axiom systems for set theory are Bernays-G\"odel~({\sc bg}) and Zermelo-Fraenkel~({\sc zf}). Resolution theorem provers can use {\sc bg} because it is finite~\cite{boyer86,quaife92}. {\sc zf} does not have a finite axiom system because of its Axiom Scheme of Replacement. This makes it awkward to use with many theorem provers, since instances of the axiom scheme have to be invoked explicitly. Since Isabelle has no difficulty with axiom schemes, we may adopt either axiom system. These two theories differ in their treatment of {\bf classes}, which are collections that are `too big' to be sets. The class of all sets,~$V$, cannot be a set without admitting Russell's Paradox. In {\sc bg}, both classes and sets are individuals; $x\in V$ expresses that $x$ is a set. In {\sc zf}, all variables denote sets; classes are identified with unary predicates. The two systems define essentially the same sets and classes, with similar properties. In particular, a class cannot belong to another class (let alone a set). Modern set theorists tend to prefer {\sc zf} because they are mainly concerned with sets, rather than classes. {\sc bg} requires tiresome proofs that various collections are sets; for instance, showing $x\in\{x\}$ requires showing that $x$ is a set. \begin{figure} \small \begin{center} \begin{tabular}{rrr} \it name &\it meta-type & \it description \\ \cdx{Let} & $[\alpha,\alpha\To\beta]\To\beta$ & let binder\\ \cdx{0} & $i$ & empty set\\ \cdx{cons} & $[i,i]\To i$ & finite set constructor\\ \cdx{Upair} & $[i,i]\To i$ & unordered pairing\\ \cdx{Pair} & $[i,i]\To i$ & ordered pairing\\ \cdx{Inf} & $i$ & infinite set\\ \cdx{Pow} & $i\To i$ & powerset\\ \cdx{Union} \cdx{Inter} & $i\To i$ & set union/intersection \\ \cdx{split} & $[[i,i]\To i, i] \To i$ & generalized projection\\ \cdx{fst} \cdx{snd} & $i\To i$ & projections\\ \cdx{converse}& $i\To i$ & converse of a relation\\ \cdx{succ} & $i\To i$ & successor\\ \cdx{Collect} & $[i,i\To o]\To i$ & separation\\ \cdx{Replace} & $[i, [i,i]\To o] \To i$ & replacement\\ \cdx{PrimReplace} & $[i, [i,i]\To o] \To i$ & primitive replacement\\ \cdx{RepFun} & $[i, i\To i] \To i$ & functional replacement\\ \cdx{Pi} \cdx{Sigma} & $[i,i\To i]\To i$ & general product/sum\\ \cdx{domain} & $i\To i$ & domain of a relation\\ \cdx{range} & $i\To i$ & range of a relation\\ \cdx{field} & $i\To i$ & field of a relation\\ \cdx{Lambda} & $[i, i\To i]\To i$ & $\lambda$-abstraction\\ \cdx{restrict}& $[i, i] \To i$ & restriction of a function\\ \cdx{The} & $[i\To o]\To i$ & definite description\\ \cdx{if} & $[o,i,i]\To i$ & conditional\\ \cdx{Ball} \cdx{Bex} & $[i, i\To o]\To o$ & bounded quantifiers \end{tabular} \end{center} \subcaption{Constants} \begin{center} \index{*"`"` symbol} \index{*"-"`"` symbol} \index{*"` symbol}\index{function applications} \index{*"- symbol} \index{*": symbol} \index{*"<"= symbol} \begin{tabular}{rrrr} \it symbol & \it meta-type & \it priority & \it description \\ \tt `` & $[i,i]\To i$ & Left 90 & image \\ \tt -`` & $[i,i]\To i$ & Left 90 & inverse image \\ \tt ` & $[i,i]\To i$ & Left 90 & application \\ \sdx{Int} & $[i,i]\To i$ & Left 70 & intersection ($\int$) \\ \sdx{Un} & $[i,i]\To i$ & Left 65 & union ($\un$) \\ \tt - & $[i,i]\To i$ & Left 65 & set difference ($-$) \\[1ex] \tt: & $[i,i]\To o$ & Left 50 & membership ($\in$) \\ \tt <= & $[i,i]\To o$ & Left 50 & subset ($\subseteq$) \end{tabular} \end{center} \subcaption{Infixes} \caption{Constants of ZF} \label{zf-constants} \end{figure} \section{The syntax of set theory} The language of set theory, as studied by logicians, has no constants. The traditional axioms merely assert the existence of empty sets, unions, powersets, etc.; this would be intolerable for practical reasoning. The Isabelle theory declares constants for primitive sets. It also extends \texttt{FOL} with additional syntax for finite sets, ordered pairs, comprehension, general union/intersection, general sums/products, and bounded quantifiers. In most other respects, Isabelle implements precisely Zermelo-Fraenkel set theory. Figure~\ref{zf-constants} lists the constants and infixes of~ZF, while Figure~\ref{zf-trans} presents the syntax translations. Finally, Figure~\ref{zf-syntax} presents the full grammar for set theory, including the constructs of FOL. Local abbreviations can be introduced by a \texttt{let} construct whose syntax appears in Fig.\ts\ref{zf-syntax}. Internally it is translated into the constant~\cdx{Let}. It can be expanded by rewriting with its definition, \tdx{Let_def}. Apart from \texttt{let}, set theory does not use polymorphism. All terms in ZF have type~\tydx{i}, which is the type of individuals and has class~{\tt term}. The type of first-order formulae, remember, is~\textit{o}. Infix operators include binary union and intersection ($A\un B$ and $A\int B$), set difference ($A-B$), and the subset and membership relations. Note that $a$\verb|~:|$b$ is translated to $\neg(a\in b)$. The union and intersection operators ($\bigcup A$ and $\bigcap A$) form the union or intersection of a set of sets; $\bigcup A$ means the same as $\bigcup@{x\in A}x$. Of these operators, only $\bigcup A$ is primitive. The constant \cdx{Upair} constructs unordered pairs; thus {\tt Upair($A$,$B$)} denotes the set~$\{A,B\}$ and \texttt{Upair($A$,$A$)} denotes the singleton~$\{A\}$. General union is used to define binary union. The Isabelle version goes on to define the constant \cdx{cons}: \begin{eqnarray*} A\cup B & \equiv & \bigcup(\texttt{Upair}(A,B)) \\ \texttt{cons}(a,B) & \equiv & \texttt{Upair}(a,a) \un B \end{eqnarray*} The $\{a@1, \ldots\}$ notation abbreviates finite sets constructed in the obvious manner using~\texttt{cons} and~$\emptyset$ (the empty set): \begin{eqnarray*} \{a,b,c\} & \equiv & \texttt{cons}(a,\texttt{cons}(b,\texttt{cons}(c,\emptyset))) \end{eqnarray*} The constant \cdx{Pair} constructs ordered pairs, as in {\tt Pair($a$,$b$)}. Ordered pairs may also be written within angle brackets, as {\tt<$a$,$b$>}. The $n$-tuple {\tt<$a@1$,\ldots,$a@{n-1}$,$a@n$>} abbreviates the nest of pairs\par\nobreak \centerline{\texttt{Pair($a@1$,\ldots,Pair($a@{n-1}$,$a@n$)\ldots).}} In ZF, a function is a set of pairs. A ZF function~$f$ is simply an individual as far as Isabelle is concerned: its Isabelle type is~$i$, not say $i\To i$. The infix operator~{\tt`} denotes the application of a function set to its argument; we must write~$f{\tt`}x$, not~$f(x)$. The syntax for image is~$f{\tt``}A$ and that for inverse image is~$f{\tt-``}A$. \begin{figure} \index{lambda abs@$\lambda$-abstractions} \index{*"-"> symbol} \index{*"* symbol} \begin{center} \footnotesize\tt\frenchspacing \begin{tabular}{rrr} \it external & \it internal & \it description \\ $a$ \ttilde: $b$ & \ttilde($a$ : $b$) & \rm negated membership\\ \ttlbrace$a@1$, $\ldots$, $a@n$\ttrbrace & cons($a@1$,$\ldots$,cons($a@n$,0)) & \rm finite set \\ <$a@1$, $\ldots$, $a@{n-1}$, $a@n$> & Pair($a@1$,\ldots,Pair($a@{n-1}$,$a@n$)\ldots) & \rm ordered $n$-tuple \\ \ttlbrace$x$:$A . P[x]$\ttrbrace & Collect($A$,$\lambda x. P[x]$) & \rm separation \\ \ttlbrace$y . x$:$A$, $Q[x,y]$\ttrbrace & Replace($A$,$\lambda x\,y. Q[x,y]$) & \rm replacement \\ \ttlbrace$b[x] . x$:$A$\ttrbrace & RepFun($A$,$\lambda x. b[x]$) & \rm functional replacement \\ \sdx{INT} $x$:$A . B[x]$ & Inter(\ttlbrace$B[x] . x$:$A$\ttrbrace) & \rm general intersection \\ \sdx{UN} $x$:$A . B[x]$ & Union(\ttlbrace$B[x] . x$:$A$\ttrbrace) & \rm general union \\ \sdx{PROD} $x$:$A . B[x]$ & Pi($A$,$\lambda x. B[x]$) & \rm general product \\ \sdx{SUM} $x$:$A . B[x]$ & Sigma($A$,$\lambda x. B[x]$) & \rm general sum \\ $A$ -> $B$ & Pi($A$,$\lambda x. B$) & \rm function space \\ $A$ * $B$ & Sigma($A$,$\lambda x. B$) & \rm binary product \\ \sdx{THE} $x . P[x]$ & The($\lambda x. P[x]$) & \rm definite description \\ \sdx{lam} $x$:$A . b[x]$ & Lambda($A$,$\lambda x. b[x]$) & \rm $\lambda$-abstraction\\[1ex] \sdx{ALL} $x$:$A . P[x]$ & Ball($A$,$\lambda x. P[x]$) & \rm bounded $\forall$ \\ \sdx{EX} $x$:$A . P[x]$ & Bex($A$,$\lambda x. P[x]$) & \rm bounded $\exists$ \end{tabular} \end{center} \caption{Translations for ZF} \label{zf-trans} \end{figure} \begin{figure} \index{*let symbol} \index{*in symbol} \dquotes \[\begin{array}{rcl} term & = & \hbox{expression of type~$i$} \\ & | & "let"~id~"="~term";"\dots";"~id~"="~term~"in"~term \\ & | & "if"~term~"then"~term~"else"~term \\ & | & "{\ttlbrace} " term\; ("," term)^* " {\ttrbrace}" \\ & | & "< " term\; ("," term)^* " >" \\ & | & "{\ttlbrace} " id ":" term " . " formula " {\ttrbrace}" \\ & | & "{\ttlbrace} " id " . " id ":" term ", " formula " {\ttrbrace}" \\ & | & "{\ttlbrace} " term " . " id ":" term " {\ttrbrace}" \\ & | & term " `` " term \\ & | & term " -`` " term \\ & | & term " ` " term \\ & | & term " * " term \\ & | & term " Int " term \\ & | & term " Un " term \\ & | & term " - " term \\ & | & term " -> " term \\ & | & "THE~~" id " . " formula\\ & | & "lam~~" id ":" term " . " term \\ & | & "INT~~" id ":" term " . " term \\ & | & "UN~~~" id ":" term " . " term \\ & | & "PROD~" id ":" term " . " term \\ & | & "SUM~~" id ":" term " . " term \\[2ex] formula & = & \hbox{expression of type~$o$} \\ & | & term " : " term \\ & | & term " \ttilde: " term \\ & | & term " <= " term \\ & | & term " = " term \\ & | & term " \ttilde= " term \\ & | & "\ttilde\ " formula \\ & | & formula " \& " formula \\ & | & formula " | " formula \\ & | & formula " --> " formula \\ & | & formula " <-> " formula \\ & | & "ALL " id ":" term " . " formula \\ & | & "EX~~" id ":" term " . " formula \\ & | & "ALL~" id~id^* " . " formula \\ & | & "EX~~" id~id^* " . " formula \\ & | & "EX!~" id~id^* " . " formula \end{array} \] \caption{Full grammar for ZF} \label{zf-syntax} \end{figure} \section{Binding operators} The constant \cdx{Collect} constructs sets by the principle of {\bf separation}. The syntax for separation is \hbox{\tt\ttlbrace$x$:$A$.\ $P[x]$\ttrbrace}, where $P[x]$ is a formula that may contain free occurrences of~$x$. It abbreviates the set {\tt Collect($A$,$\lambda x. P[x]$)}, which consists of all $x\in A$ that satisfy~$P[x]$. Note that \texttt{Collect} is an unfortunate choice of name: some set theories adopt a set-formation principle, related to replacement, called collection. The constant \cdx{Replace} constructs sets by the principle of {\bf replacement}. The syntax \hbox{\tt\ttlbrace$y$.\ $x$:$A$,$Q[x,y]$\ttrbrace} denotes the set {\tt Replace($A$,$\lambda x\,y. Q[x,y]$)}, which consists of all~$y$ such that there exists $x\in A$ satisfying~$Q[x,y]$. The Replacement Axiom has the condition that $Q$ must be single-valued over~$A$: for all~$x\in A$ there exists at most one $y$ satisfying~$Q[x,y]$. A single-valued binary predicate is also called a {\bf class function}. The constant \cdx{RepFun} expresses a special case of replacement, where $Q[x,y]$ has the form $y=b[x]$. Such a $Q$ is trivially single-valued, since it is just the graph of the meta-level function~$\lambda x. b[x]$. The resulting set consists of all $b[x]$ for~$x\in A$. This is analogous to the \ML{} functional \texttt{map}, since it applies a function to every element of a set. The syntax is \hbox{\tt\ttlbrace$b[x]$.\ $x$:$A$\ttrbrace}, which expands to {\tt RepFun($A$,$\lambda x. b[x]$)}. \index{*INT symbol}\index{*UN symbol} General unions and intersections of indexed families of sets, namely $\bigcup@{x\in A}B[x]$ and $\bigcap@{x\in A}B[x]$, are written \hbox{\tt UN $x$:$A$.\ $B[x]$} and \hbox{\tt INT $x$:$A$.\ $B[x]$}. Their meaning is expressed using \texttt{RepFun} as \[ \bigcup(\{B[x]. x\in A\}) \qquad\hbox{and}\qquad \bigcap(\{B[x]. x\in A\}). \] General sums $\sum@{x\in A}B[x]$ and products $\prod@{x\in A}B[x]$ can be constructed in set theory, where $B[x]$ is a family of sets over~$A$. They have as special cases $A\times B$ and $A\to B$, where $B$ is simply a set. This is similar to the situation in Constructive Type Theory (set theory has `dependent sets') and calls for similar syntactic conventions. The constants~\cdx{Sigma} and~\cdx{Pi} construct general sums and products. Instead of \texttt{Sigma($A$,$B$)} and \texttt{Pi($A$,$B$)} we may write \hbox{\tt SUM $x$:$A$.\ $B[x]$} and \hbox{\tt PROD $x$:$A$.\ $B[x]$}. \index{*SUM symbol}\index{*PROD symbol}% The special cases as \hbox{\tt$A$*$B$} and \hbox{\tt$A$->$B$} abbreviate general sums and products over a constant family.\footnote{Unlike normal infix operators, {\tt*} and {\tt->} merely define abbreviations; there are no constants~\texttt{op~*} and~\hbox{\tt op~->}.} Isabelle accepts these abbreviations in parsing and uses them whenever possible for printing. \index{*THE symbol} As mentioned above, whenever the axioms assert the existence and uniqueness of a set, Isabelle's set theory declares a constant for that set. These constants can express the {\bf definite description} operator~$\iota x. P[x]$, which stands for the unique~$a$ satisfying~$P[a]$, if such exists. Since all terms in ZF denote something, a description is always meaningful, but we do not know its value unless $P[x]$ defines it uniquely. Using the constant~\cdx{The}, we may write descriptions as {\tt The($\lambda x. P[x]$)} or use the syntax \hbox{\tt THE $x$.\ $P[x]$}. \index{*lam symbol} Function sets may be written in $\lambda$-notation; $\lambda x\in A. b[x]$ stands for the set of all pairs $\pair{x,b[x]}$ for $x\in A$. In order for this to be a set, the function's domain~$A$ must be given. Using the constant~\cdx{Lambda}, we may express function sets as {\tt Lambda($A$,$\lambda x. b[x]$)} or use the syntax \hbox{\tt lam $x$:$A$.\ $b[x]$}. Isabelle's set theory defines two {\bf bounded quantifiers}: \begin{eqnarray*} \forall x\in A. P[x] &\hbox{abbreviates}& \forall x. x\in A\imp P[x] \\ \exists x\in A. P[x] &\hbox{abbreviates}& \exists x. x\in A\conj P[x] \end{eqnarray*} The constants~\cdx{Ball} and~\cdx{Bex} are defined accordingly. Instead of \texttt{Ball($A$,$P$)} and \texttt{Bex($A$,$P$)} we may write \hbox{\tt ALL $x$:$A$.\ $P[x]$} and \hbox{\tt EX $x$:$A$.\ $P[x]$}. %%%% ZF.thy \begin{figure} \begin{ttbox} \tdx{Let_def} Let(s, f) == f(s) \tdx{Ball_def} Ball(A,P) == ALL x. x:A --> P(x) \tdx{Bex_def} Bex(A,P) == EX x. x:A & P(x) \tdx{subset_def} A <= B == ALL x:A. x:B \tdx{extension} A = B <-> A <= B & B <= A \tdx{Union_iff} A : Union(C) <-> (EX B:C. A:B) \tdx{Pow_iff} A : Pow(B) <-> A <= B \tdx{foundation} A=0 | (EX x:A. ALL y:x. ~ y:A) \tdx{replacement} (ALL x:A. ALL y z. P(x,y) & P(x,z) --> y=z) ==> b : PrimReplace(A,P) <-> (EX x:A. P(x,b)) \subcaption{The Zermelo-Fraenkel Axioms} \tdx{Replace_def} Replace(A,P) == PrimReplace(A, \%x y. (EX!z. P(x,z)) & P(x,y)) \tdx{RepFun_def} RepFun(A,f) == {\ttlbrace}y . x:A, y=f(x)\ttrbrace \tdx{the_def} The(P) == Union({\ttlbrace}y . x:{\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:A, x=y & P(x){\ttrbrace} \tdx{Upair_def} Upair(a,b) == {\ttlbrace}y. x:Pow(Pow(0)), (x=0 & y=a) | (x=Pow(0) & y=b){\ttrbrace} \subcaption{Consequences of replacement} \tdx{Inter_def} Inter(A) == {\ttlbrace}x:Union(A) . ALL y:A. x:y{\ttrbrace} \tdx{Un_def} A Un B == Union(Upair(A,B)) \tdx{Int_def} A Int B == Inter(Upair(A,B)) \tdx{Diff_def} A - B == {\ttlbrace}x:A . x~:B{\ttrbrace} \subcaption{Union, intersection, difference} \end{ttbox} \caption{Rules and axioms of ZF} \label{zf-rules} \end{figure} \begin{figure} \begin{ttbox} \tdx{cons_def} cons(a,A) == Upair(a,a) Un A \tdx{succ_def} succ(i) == cons(i,i) \tdx{infinity} 0:Inf & (ALL y:Inf. succ(y): 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. EX 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) == UN x:A. UN y:B(x). {\ttlbrace}<x,y>{\ttrbrace} \subcaption{Ordered pairs and Cartesian products} \tdx{converse_def} converse(r) == {\ttlbrace}z. w:r, EX x y. w=<x,y> & z=<y,x>{\ttrbrace} \tdx{domain_def} domain(r) == {\ttlbrace}x. w:r, EX y. w=<x,y>{\ttrbrace} \tdx{range_def} range(r) == domain(converse(r)) \tdx{field_def} field(r) == domain(r) Un range(r) \tdx{image_def} r `` A == {\ttlbrace}y : range(r) . EX x:A. <x,y> : 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:A{\ttrbrace} \tdx{apply_def} f`a == THE y. <a,y> : f \tdx{Pi_def} Pi(A,B) == {\ttlbrace}f: Pow(Sigma(A,B)). ALL x:A. EX! y. <x,y>: f{\ttrbrace} \tdx{restrict_def} restrict(f,A) == lam x:A. f`x \subcaption{Functions and general product} \end{ttbox} \caption{Further definitions of ZF} \label{zf-defs} \end{figure} \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. See the file \texttt{ZF/ZF.thy} for details. The traditional replacement axiom asserts \[ y \in \texttt{PrimReplace}(A,P) \bimp (\exists x\in A. P(x,y)) \] 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 $y\in \texttt{Replace}(A,P)$ if and only if there is some~$x$ such that $P(x,-)$ holds uniquely for~$y$. Because the equivalence is unconditional, \texttt{Replace} is much easier to use than \texttt{PrimReplace}; it defines the same set, if $P(x,y)$ is single-valued. The nice syntax for replacement expands to \texttt{Replace}. Other consequences of replacement include functional replacement (\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 \texttt{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 ${\tt Pi}(A,B)$ 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.ML \begin{figure} \begin{ttbox} \tdx{ballI} [| !!x. x:A ==> P(x) |] ==> ALL x:A. P(x) \tdx{bspec} [| ALL x:A. P(x); x: A |] ==> P(x) \tdx{ballE} [| ALL x:A. P(x); P(x) ==> Q; ~ x:A ==> Q |] ==> Q \tdx{ball_cong} [| A=A'; !!x. x:A' ==> P(x) <-> P'(x) |] ==> (ALL x:A. P(x)) <-> (ALL x:A'. P'(x)) \tdx{bexI} [| P(x); x: A |] ==> EX x:A. P(x) \tdx{bexCI} [| ALL x:A. ~P(x) ==> P(a); a: A |] ==> EX x:A. P(x) \tdx{bexE} [| EX x:A. P(x); !!x. [| x:A; P(x) |] ==> Q |] ==> Q \tdx{bex_cong} [| A=A'; !!x. x:A' ==> P(x) <-> P'(x) |] ==> (EX x:A. P(x)) <-> (EX x:A'. P'(x)) \subcaption{Bounded quantifiers} \tdx{subsetI} (!!x. x:A ==> x:B) ==> A <= B \tdx{subsetD} [| A <= B; c:A |] ==> c:B \tdx{subsetCE} [| A <= B; ~(c:A) ==> P; c:B ==> P |] ==> P \tdx{subset_refl} A <= A \tdx{subset_trans} [| A<=B; B<=C |] ==> A<=C \tdx{equalityI} [| A <= B; B <= A |] ==> A = B \tdx{equalityD1} A = B ==> A<=B \tdx{equalityD2} A = B ==> B<=A \tdx{equalityE} [| A = B; [| A<=B; B<=A |] ==> P |] ==> P \subcaption{Subsets and extensionality} \tdx{emptyE} a:0 ==> P \tdx{empty_subsetI} 0 <= A \tdx{equals0I} [| !!y. y:A ==> False |] ==> A=0 \tdx{equals0D} [| A=0; a:A |] ==> P \tdx{PowI} A <= B ==> A : Pow(B) \tdx{PowD} A : Pow(B) ==> A<=B \subcaption{The empty set; power sets} \end{ttbox} \caption{Basic derived rules for ZF} \label{zf-lemmas1} \end{figure} \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 \texttt{PrimReplace} would be. The principle of separation is proved explicitly, although most proofs should use the natural deduction rules for \texttt{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. See the file \texttt{ZF/ZF.ML} for a complete listing. Figure~\ref{zf-lemmas3} presents rules for general union and intersection. The empty intersection should be undefined. We cannot have $\bigcap(\emptyset)=V$ because $V$, the universal class, is not a set. All expressions denote something in ZF set theory; the definition of intersection implies $\bigcap(\emptyset)=\emptyset$, but this value is arbitrary. The rule \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{ttbox} \tdx{ReplaceI} [| x: A; P(x,b); !!y. P(x,y) ==> y=b |] ==> b : {\ttlbrace}y. x:A, P(x,y){\ttrbrace} \tdx{ReplaceE} [| b : {\ttlbrace}y. x:A, P(x,y){\ttrbrace}; !!x. [| x: A; P(x,b); ALL y. P(x,y)-->y=b |] ==> R |] ==> R \tdx{RepFunI} [| a : A |] ==> f(a) : {\ttlbrace}f(x). x:A{\ttrbrace} \tdx{RepFunE} [| b : {\ttlbrace}f(x). x:A{\ttrbrace}; !!x.[| x:A; b=f(x) |] ==> P |] ==> P \tdx{separation} a : {\ttlbrace}x:A. P(x){\ttrbrace} <-> a:A & P(a) \tdx{CollectI} [| a:A; P(a) |] ==> a : {\ttlbrace}x:A. P(x){\ttrbrace} \tdx{CollectE} [| a : {\ttlbrace}x:A. P(x){\ttrbrace}; [| a:A; P(a) |] ==> R |] ==> R \tdx{CollectD1} a : {\ttlbrace}x:A. P(x){\ttrbrace} ==> a:A \tdx{CollectD2} a : {\ttlbrace}x:A. P(x){\ttrbrace} ==> P(a) \end{ttbox} \caption{Replacement and separation} \label{zf-lemmas2} \end{figure} \begin{figure} \begin{ttbox} \tdx{UnionI} [| B: C; A: B |] ==> A: Union(C) \tdx{UnionE} [| A : Union(C); !!B.[| A: B; B: C |] ==> R |] ==> R \tdx{InterI} [| !!x. x: C ==> A: x; c:C |] ==> A : Inter(C) \tdx{InterD} [| A : Inter(C); B : C |] ==> A : B \tdx{InterE} [| A : Inter(C); A:B ==> R; ~ B:C ==> R |] ==> R \tdx{UN_I} [| a: A; b: B(a) |] ==> b: (UN x:A. B(x)) \tdx{UN_E} [| b : (UN x:A. B(x)); !!x.[| x: A; b: B(x) |] ==> R |] ==> R \tdx{INT_I} [| !!x. x: A ==> b: B(x); a: A |] ==> b: (INT x:A. B(x)) \tdx{INT_E} [| b : (INT x:A. B(x)); a: A |] ==> b : B(a) \end{ttbox} \caption{General union and intersection} \label{zf-lemmas3} \end{figure} %%% upair.ML \begin{figure} \begin{ttbox} \tdx{pairing} a:Upair(b,c) <-> (a=b | a=c) \tdx{UpairI1} a : Upair(a,b) \tdx{UpairI2} b : Upair(a,b) \tdx{UpairE} [| a : Upair(b,c); a = b ==> P; a = c ==> P |] ==> P \end{ttbox} \caption{Unordered pairs} \label{zf-upair1} \end{figure} \begin{figure} \begin{ttbox} \tdx{UnI1} c : A ==> c : A Un B \tdx{UnI2} c : B ==> c : A Un B \tdx{UnCI} (~c : B ==> c : A) ==> c : A Un B \tdx{UnE} [| c : A Un B; c:A ==> P; c:B ==> P |] ==> P \tdx{IntI} [| c : A; c : B |] ==> c : A Int B \tdx{IntD1} c : A Int B ==> c : A \tdx{IntD2} c : A Int B ==> c : B \tdx{IntE} [| c : A Int B; [| c:A; c:B |] ==> P |] ==> P \tdx{DiffI} [| c : A; ~ c : B |] ==> c : A - B \tdx{DiffD1} c : A - B ==> c : A \tdx{DiffD2} c : A - B ==> c ~: B \tdx{DiffE} [| c : A - B; [| c:A; ~ c:B |] ==> P |] ==> P \end{ttbox} \caption{Union, intersection, difference} \label{zf-Un} \end{figure} \begin{figure} \begin{ttbox} \tdx{consI1} a : cons(a,B) \tdx{consI2} a : B ==> a : cons(b,B) \tdx{consCI} (~ a:B ==> a=b) ==> a: cons(b,B) \tdx{consE} [| a : cons(b,A); a=b ==> P; a:A ==> P |] ==> P \tdx{singletonI} a : {\ttlbrace}a{\ttrbrace} \tdx{singletonE} [| a : {\ttlbrace}b{\ttrbrace}; a=b ==> P |] ==> P \end{ttbox} \caption{Finite and singleton sets} \label{zf-upair2} \end{figure} \begin{figure} \begin{ttbox} \tdx{succI1} i : succ(i) \tdx{succI2} i : j ==> i : succ(j) \tdx{succCI} (~ i:j ==> i=j) ==> i: succ(j) \tdx{succE} [| i : succ(j); i=j ==> P; i:j ==> P |] ==> P \tdx{succ_neq_0} [| succ(n)=0 |] ==> P \tdx{succ_inject} succ(m) = succ(n) ==> m=n \end{ttbox} \caption{The successor function} \label{zf-succ} \end{figure} \begin{figure} \begin{ttbox} \tdx{the_equality} [| P(a); !!x. P(x) ==> x=a |] ==> (THE x. P(x)) = a \tdx{theI} EX! 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:b; b:a |] ==> P \tdx{mem_irrefl} a:a ==> P \end{ttbox} \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 \texttt{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~\texttt{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~\texttt{cons}. The proof that {\tt 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. See the file \texttt{ZF/upair.ML} for full proofs of the rules discussed in this section. %%% subset.ML \begin{figure} \begin{ttbox} \tdx{Union_upper} B:A ==> B <= Union(A) \tdx{Union_least} [| !!x. x:A ==> x<=C |] ==> Union(A) <= C \tdx{Inter_lower} B:A ==> Inter(A) <= B \tdx{Inter_greatest} [| a:A; !!x. x:A ==> C<=x |] ==> C <= Inter(A) \tdx{Un_upper1} A <= A Un B \tdx{Un_upper2} B <= A Un B \tdx{Un_least} [| A<=C; B<=C |] ==> A Un B <= C \tdx{Int_lower1} A Int B <= A \tdx{Int_lower2} A Int B <= B \tdx{Int_greatest} [| C<=A; C<=B |] ==> C <= A Int B \tdx{Diff_subset} A-B <= A \tdx{Diff_contains} [| C<=A; C Int B = 0 |] ==> C <= A-B \tdx{Collect_subset} Collect(A,P) <= A \end{ttbox} \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. Proofs are in the file \texttt{ZF/subset.ML}. 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 ${{\tt Pow}(A)\cap {\tt Pow}(B)}= {\tt Pow}(A\cap B)$. %%% pair.ML \begin{figure} \begin{ttbox} \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:A; b:B(a) |] ==> <a,b> : Sigma(A,B) \tdx{SigmaE} [| c: Sigma(A,B); !!x y.[| x:A; y:B(x); c=<x,y> |] ==> P |] ==> P \tdx{SigmaE2} [| <a,b> : Sigma(A,B); [| a:A; b:B(a) |] ==> P |] ==> P \end{ttbox} \caption{Ordered pairs; projections; general sums} \label{zf-pair} \end{figure} \subsection{Ordered pairs} \label{sec:pairs} Figure~\ref{zf-pair} presents the rules governing ordered pairs, projections and general sums. File \texttt{ZF/pair.ML} contains the full (and tedious) proof that $\{\{a\},\{a,b\}\}$ functions as an ordered pair. This property is expressed as two destruction rules, \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 $\pair{a,b}\in \texttt{Sigma}(A,B)$ 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 \texttt{split(\%$x$ $y$.\ $t$)} \end{center} Nested patterns are translated recursively: {\tt\%<$x$,$y$,$z$>. $t$} $\leadsto$ {\tt\%<$x$,<$y$,$z$>>. $t$} $\leadsto$ \texttt{split(\%$x$.\%<$y$,$z$>. $t$)} $\leadsto$ \texttt{split(\%$x$. split(\%$y$ $z$.\ $t$))}. The reverse translation is performed upon printing. \begin{warn} The translation between patterns and \texttt{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 {\tt (\%<x,y>.<y,x>)<a,b>} requires the theorem \texttt{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:] \texttt{let {\it pattern} = $t$ in $u$} \item[Choice:] \texttt{THE~{\it pattern}~.~$P$} \item[Set operations:] \texttt{UN~{\it pattern}:$A$.~$B$} \item[Comprehension:] \texttt{{\ttlbrace}~{\it pattern}:$A$~.~$P$~{\ttrbrace}} \end{description} %%% domrange.ML \begin{figure} \begin{ttbox} \tdx{domainI} <a,b>: r ==> a : domain(r) \tdx{domainE} [| a : domain(r); !!y. <a,y>: r ==> P |] ==> P \tdx{domain_subset} domain(Sigma(A,B)) <= A \tdx{rangeI} <a,b>: r ==> b : range(r) \tdx{rangeE} [| b : range(r); !!x. <x,b>: r ==> P |] ==> P \tdx{range_subset} range(A*B) <= B \tdx{fieldI1} <a,b>: r ==> a : field(r) \tdx{fieldI2} <a,b>: r ==> b : field(r) \tdx{fieldCI} (~ <c,a>:r ==> <a,b>: r) ==> a : field(r) \tdx{fieldE} [| a : field(r); !!x. <a,x>: r ==> P; !!x. <x,a>: r ==> P |] ==> P \tdx{field_subset} field(A*A) <= A \end{ttbox} \caption{Domain, range and field of a relation} \label{zf-domrange} \end{figure} \begin{figure} \begin{ttbox} \tdx{imageI} [| <a,b>: r; a:A |] ==> b : r``A \tdx{imageE} [| b: r``A; !!x.[| <x,b>: r; x:A |] ==> P |] ==> P \tdx{vimageI} [| <a,b>: r; b:B |] ==> a : r-``B \tdx{vimageE} [| a: r-``B; !!x.[| <a,x>: r; x:B |] ==> P |] ==> P \end{ttbox} \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. See the file \texttt{ZF/domrange.ML} for derivations of the rules. %%% func.ML \begin{figure} \begin{ttbox} \tdx{fun_is_rel} f: Pi(A,B) ==> f <= Sigma(A,B) \tdx{apply_equality} [| <a,b>: f; f: Pi(A,B) |] ==> f`a = b \tdx{apply_equality2} [| <a,b>: f; <a,c>: f; f: Pi(A,B) |] ==> b=c \tdx{apply_type} [| f: Pi(A,B); a:A |] ==> f`a : B(a) \tdx{apply_Pair} [| f: Pi(A,B); a:A |] ==> <a,f`a>: f \tdx{apply_iff} f: Pi(A,B) ==> <a,b>: f <-> a:A & f`a = b \tdx{fun_extension} [| f : Pi(A,B); g: Pi(A,D); !!x. x:A ==> f`x = g`x |] ==> f=g \tdx{domain_type} [| <a,b> : f; f: Pi(A,B) |] ==> a : A \tdx{range_type} [| <a,b> : f; f: Pi(A,B) |] ==> b : B(a) \tdx{Pi_type} [| f: A->C; !!x. x:A ==> f`x: B(x) |] ==> f: Pi(A,B) \tdx{domain_of_fun} f: Pi(A,B) ==> domain(f)=A \tdx{range_of_fun} f: Pi(A,B) ==> f: A->range(f) \tdx{restrict} a : A ==> restrict(f,A) ` a = f`a \tdx{restrict_type} [| !!x. x:A ==> f`x: B(x) |] ==> restrict(f,A) : Pi(A,B) \end{ttbox} \caption{Functions} \label{zf-func1} \end{figure} \begin{figure} \begin{ttbox} \tdx{lamI} a:A ==> <a,b(a)> : (lam x:A. b(x)) \tdx{lamE} [| p: (lam x:A. b(x)); !!x.[| x:A; p=<x,b(x)> |] ==> P |] ==> P \tdx{lam_type} [| !!x. x:A ==> b(x): B(x) |] ==> (lam x:A. b(x)) : Pi(A,B) \tdx{beta} a : A ==> (lam x:A. b(x)) ` a = b(a) \tdx{eta} f : Pi(A,B) ==> (lam x:A. f`x) = f \end{ttbox} \caption{$\lambda$-abstraction} \label{zf-lam} \end{figure} \begin{figure} \begin{ttbox} \tdx{fun_empty} 0: 0->0 \tdx{fun_single} {\ttlbrace}<a,b>{\ttrbrace} : {\ttlbrace}a{\ttrbrace} -> {\ttlbrace}b{\ttrbrace} \tdx{fun_disjoint_Un} [| f: A->B; g: C->D; A Int C = 0 |] ==> (f Un g) : (A Un C) -> (B Un D) \tdx{fun_disjoint_apply1} [| a:A; f: A->B; g: C->D; A Int C = 0 |] ==> (f Un g)`a = f`a \tdx{fun_disjoint_apply2} [| c:C; f: A->B; g: C->D; A Int C = 0 |] ==> (f Un g)`c = g`c \end{ttbox} \caption{Constructing functions from smaller sets} \label{zf-func2} \end{figure} \subsection{Functions} Functions, represented by graphs, are notoriously difficult to reason about. The file \texttt{ZF/func.ML} derives many 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, $C={\tt range}(f)$. 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{ttbox} \tdx{Int_absorb} A Int A = A \tdx{Int_commute} A Int B = B Int A \tdx{Int_assoc} (A Int B) Int C = A Int (B Int C) \tdx{Int_Un_distrib} (A Un B) Int C = (A Int C) Un (B Int C) \tdx{Un_absorb} A Un A = A \tdx{Un_commute} A Un B = B Un A \tdx{Un_assoc} (A Un B) Un C = A Un (B Un C) \tdx{Un_Int_distrib} (A Int B) Un C = (A Un C) Int (B Un C) \tdx{Diff_cancel} A-A = 0 \tdx{Diff_disjoint} A Int (B-A) = 0 \tdx{Diff_partition} A<=B ==> A Un (B-A) = B \tdx{double_complement} [| A<=B; B<= C |] ==> (B - (C-A)) = A \tdx{Diff_Un} A - (B Un C) = (A-B) Int (A-C) \tdx{Diff_Int} A - (B Int C) = (A-B) Un (A-C) \tdx{Union_Un_distrib} Union(A Un B) = Union(A) Un Union(B) \tdx{Inter_Un_distrib} [| a:A; b:B |] ==> Inter(A Un B) = Inter(A) Int Inter(B) \tdx{Int_Union_RepFun} A Int Union(B) = (UN C:B. A Int C) \tdx{Un_Inter_RepFun} b:B ==> A Un Inter(B) = (INT C:B. A Un C) \tdx{SUM_Un_distrib1} (SUM x:A Un B. C(x)) = (SUM x:A. C(x)) Un (SUM x:B. C(x)) \tdx{SUM_Un_distrib2} (SUM x:C. A(x) Un B(x)) = (SUM x:C. A(x)) Un (SUM x:C. B(x)) \tdx{SUM_Int_distrib1} (SUM x:A Int B. C(x)) = (SUM x:A. C(x)) Int (SUM x:B. C(x)) \tdx{SUM_Int_distrib2} (SUM x:C. A(x) Int B(x)) = (SUM x:C. A(x)) Int (SUM x:C. B(x)) \end{ttbox} \caption{Equalities} \label{zf-equalities} \end{figure} \begin{figure} %\begin{constants} % \cdx{1} & $i$ & & $\{\emptyset\}$ \\ % \cdx{bool} & $i$ & & the set $\{\emptyset,1\}$ \\ % \cdx{cond} & $[i,i,i]\To i$ & & conditional for \texttt{bool} \\ % \cdx{not} & $i\To i$ & & negation for \texttt{bool} \\ % \sdx{and} & $[i,i]\To i$ & Left 70 & conjunction for \texttt{bool} \\ % \sdx{or} & $[i,i]\To i$ & Left 65 & disjunction for \texttt{bool} \\ % \sdx{xor} & $[i,i]\To i$ & Left 65 & exclusive-or for \texttt{bool} %\end{constants} % \begin{ttbox} \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 : bool \tdx{bool_0I} 0 : bool \tdx{boolE} [| c: 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{ttbox} \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 ML files of proofs. Figure~\ref{zf-equalities} presents commutative, associative, distributive, and idempotency laws of union and intersection, along with other equations. See file \texttt{ZF/equalities.ML}. 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 \texttt{bool}-valued functions, for example. The constant~\texttt{1} is translated to \texttt{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{ttbox} \tdx{sum_def} A+B == {\ttlbrace}0{\ttrbrace}*A Un {\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{sum_InlI} a : A ==> Inl(a) : A+B \tdx{sum_InrI} b : B ==> Inr(b) : 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{sumE2} u: A+B ==> (EX x. x:A & u=Inl(x)) | (EX y. y: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{ttbox} \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{ttbox} \tdx{QPair_def} <a;b> == a+b \tdx{qsplit_def} qsplit(c,p) == THE y. EX a b. p=<a;b> & y=c(a,b) \tdx{qfsplit_def} qfsplit(R,z) == EX x y. z=<x;y> & R(x,y) \tdx{qconverse_def} qconverse(r) == {\ttlbrace}z. w:r, EX x y. w=<x;y> & z=<y;x>{\ttrbrace} \tdx{QSigma_def} QSigma(A,B) == UN x:A. UN y:B(x). {\ttlbrace}<x;y>{\ttrbrace} \tdx{qsum_def} A <+> B == ({\ttlbrace}0{\ttrbrace} <*> A) Un ({\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{ttbox} \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{ttbox} \tdx{bnd_mono_def} bnd_mono(D,h) == h(D)<=D & (ALL W X. W<=X --> X<=D --> h(W) <= h(X)) \tdx{lfp_def} lfp(D,h) == Inter({\ttlbrace}X: Pow(D). h(X) <= X{\ttrbrace}) \tdx{gfp_def} gfp(D,h) == Union({\ttlbrace}X: Pow(D). X <= h(X){\ttrbrace}) \tdx{lfp_lowerbound} [| h(A) <= A; A<=D |] ==> lfp(D,h) <= A \tdx{lfp_subset} lfp(D,h) <= D \tdx{lfp_greatest} [| bnd_mono(D,h); !!X. [| h(X) <= X; X<=D |] ==> A<=X |] ==> A <= lfp(D,h) \tdx{lfp_Tarski} bnd_mono(D,h) ==> lfp(D,h) = h(lfp(D,h)) \tdx{induct} [| a : lfp(D,h); bnd_mono(D,h); !!x. x : h(Collect(lfp(D,h),P)) ==> P(x) |] ==> P(a) \tdx{lfp_mono} [| bnd_mono(D,h); bnd_mono(E,i); !!X. X<=D ==> h(X) <= i(X) |] ==> lfp(D,h) <= lfp(E,i) \tdx{gfp_upperbound} [| A <= h(A); A<=D |] ==> A <= gfp(D,h) \tdx{gfp_subset} gfp(D,h) <= D \tdx{gfp_least} [| bnd_mono(D,h); !!X. [| X <= h(X); X<=D |] ==> X<=A |] ==> gfp(D,h) <= A \tdx{gfp_Tarski} bnd_mono(D,h) ==> gfp(D,h) = h(gfp(D,h)) \tdx{coinduct} [| bnd_mono(D,h); a: X; X <= h(X Un gfp(D,h)); X <= D |] ==> a : gfp(D,h) \tdx{gfp_mono} [| bnd_mono(D,h); D <= E; !!X. X<=D ==> h(X) <= i(X) |] ==> gfp(D,h) <= gfp(E,i) \end{ttbox} \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. See file \texttt{ZF/mono.ML}. \subsection{Finite sets and lists} Theory \texttt{Finite} (Figure~\ref{zf-fin}) defines the finite set operator; ${\tt Fin}(A)$ 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{ttbox} \tdx{Fin.emptyI} 0 : Fin(A) \tdx{Fin.consI} [| a: A; b: Fin(A) |] ==> cons(a,b) : Fin(A) \tdx{Fin_induct} [| b: Fin(A); P(0); !!x y. [| x: A; y: Fin(A); x~:y; P(y) |] ==> P(cons(x,y)) |] ==> P(b) \tdx{Fin_mono} A<=B ==> Fin(A) <= Fin(B) \tdx{Fin_UnI} [| b: Fin(A); c: Fin(A) |] ==> b Un c : Fin(A) \tdx{Fin_UnionI} C : Fin(Fin(A)) ==> Union(C) : Fin(A) \tdx{Fin_subset} [| c<=b; b: Fin(A) |] ==> c: Fin(A) \end{ttbox} \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{ttbox} \tdx{NilI} Nil : list(A) \tdx{ConsI} [| a: A; l: list(A) |] ==> Cons(a,l) : list(A) \tdx{List.induct} [| l: list(A); P(Nil); !!x y. [| x: A; y: 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=Cons(a,l) \tdx{list_mono} A<=B ==> list(A) <= list(B) \tdx{map_ident} l: list(A) ==> map(\%u. u, l) = l \tdx{map_compose} l: list(A) ==> map(h, map(j,l)) = map(\%u. h(j(u)), l) \tdx{map_app_distrib} xs: list(A) ==> map(h, xs@ys) = map(h,xs) @ map(h,ys) \tdx{map_type} [| l: list(A); !!x. x: A ==> h(x): B |] ==> map(h,l) : list(B) \tdx{map_flat} ls: list(list(A)) ==> map(h, flat(ls)) = flat(map(map(h),ls)) \end{ttbox} \caption{Lists} \label{zf-list} \end{figure} Figure~\ref{zf-list} presents the set of lists over~$A$, ${\tt list}(A)$. The definition employs Isabelle's datatype package, which defines the introduction and induction rules automatically, as well as the constructors, case operator (\verb|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{ttbox} \tdx{comp_def} r O s == {\ttlbrace}xz : domain(s)*range(r) . EX x y z. xz=<x,z> & <x,y>:s & <y,z>:r{\ttrbrace} \tdx{id_def} id(A) == (lam x:A. x) \tdx{inj_def} inj(A,B) == {\ttlbrace} f: A->B. ALL w:A. ALL x:A. f`w=f`x --> w=x {\ttrbrace} \tdx{surj_def} surj(A,B) == {\ttlbrace} f: A->B . ALL y:B. EX x:A. f`x=y {\ttrbrace} \tdx{bij_def} bij(A,B) == inj(A,B) Int surj(A,B) \tdx{left_inverse} [| f: inj(A,B); a: A |] ==> converse(f)`(f`a) = a \tdx{right_inverse} [| f: inj(A,B); b: range(f) |] ==> f`(converse(f)`b) = b \tdx{inj_converse_inj} f: inj(A,B) ==> converse(f): inj(range(f), A) \tdx{bij_converse_bij} f: bij(A,B) ==> converse(f): bij(B,A) \tdx{comp_type} [| s<=A*B; r<=B*C |] ==> (r O s) <= A*C \tdx{comp_assoc} (r O s) O t = r O (s O t) \tdx{left_comp_id} r<=A*B ==> id(B) O r = r \tdx{right_comp_id} r<=A*B ==> r O id(A) = r \tdx{comp_func} [| g:A->B; f:B->C |] ==> (f O g):A->C \tdx{comp_func_apply} [| g:A->B; f:B->C; a:A |] ==> (f O g)`a = f`(g`a) \tdx{comp_inj} [| g:inj(A,B); f:inj(B,C) |] ==> (f O g):inj(A,C) \tdx{comp_surj} [| g:surj(A,B); f:surj(B,C) |] ==> (f O g):surj(A,C) \tdx{comp_bij} [| g:bij(A,B); f:bij(B,C) |] ==> (f O g):bij(A,C) \tdx{left_comp_inverse} f: inj(A,B) ==> converse(f) O f = id(A) \tdx{right_comp_inverse} f: surj(A,B) ==> f O converse(f) = id(B) \tdx{bij_disjoint_Un} [| f: bij(A,B); g: bij(C,D); A Int C = 0; B Int D = 0 |] ==> (f Un g) : bij(A Un C, B Un D) \tdx{restrict_bij} [| f:inj(A,B); C<=A |] ==> restrict(f,C): bij(C, f``C) \end{ttbox} \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' $\texttt{univ}(A)$, which is used by the datatype package. This set contains $A$ and the natural numbers. Vitally, it is closed under finite products: ${\tt univ}(A)\times{\tt univ}(A)\subseteq{\tt univ}(A)$. This 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' ${\tt quniv}(A)$, which is used by the datatype package to construct codatatypes such as streams. It is analogous to ${\tt univ}(A)$ (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} 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 $a\in A-B$. Simplification tactics tactics such as \texttt{Asm_simp_tac} and \texttt{Full_simp_tac} use the default simpset (\texttt{simpset()}), which works for most purposes. A small simplification set for set theory is called~\ttindexbold{ZF_ss}, and you can even use \ttindex{FOL_ss} as a minimal starting point. \texttt{ZF_ss} contains congruence rules for all the binding operators of ZF. It contains all the conversion rules, such as \texttt{fst} and \texttt{snd}, as well as the rewrites shown in Fig.\ts\ref{zf-simpdata}. See the file \texttt{ZF/simpdata.ML} for a fuller list. \subsection{Classical Reasoning} As for the classical reasoner, tactics such as \texttt{Blast_tac} and {\tt Best_tac} refer to the default claset (\texttt{claset()}). This works for most purposes. Named clasets include \ttindexbold{ZF_cs} (basic set theory) and \ttindexbold{le_cs} (useful for reasoning about the relations $<$ and $\le$). You can use \ttindex{FOL_cs} as a minimal basis for building your own clasets. See \iflabelundefined{chap:classical}{the {\em Reference Manual\/}}% {Chap.\ts\ref{chap:classical}} for more discussion of classical proof methods. \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 \neg (a\in B)\\ \pair{a,b}\in {\tt Sigma}(A,B) & \bimp & a\in A \conj b\in B(a)\\ a \in {\tt Collect}(A,P) & \bimp & a\in A \conj P(a)\\ (\forall x \in \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} [| ?P ==> ?a: ?A; ~?P ==> ?b: ?A |] ==> (if ?P then ?a else ?b): ?A [| ?m : nat; ?n : nat |] ==> ?m #+ ?n : nat ?a : ?A ==> Inl(?a) : ?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 \texttt{i\#+j :\ ?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 insert the rules using \texttt{AddTCs} and the rest should be automatic. In particular, the simplifier will use type-checking to help satisfy conditional rewrite rules. Call the tactic \ttindex{Typecheck_tac} to break down all subgoals using type-checking rules. 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 \texttt{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 \texttt{typecheck_tac} using the default tcset. \end{ttdescription} To supply some type-checking rules temporarily, using \texttt{Addrules} and later \texttt{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} 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{ttbox} \tdx{nat_def} nat == lfp(lam r: Pow(Inf). {\ttlbrace}0{\ttrbrace} Un {\ttlbrace}succ(x). x:r{\ttrbrace} \tdx{nat_case_def} nat_case(a,b,k) == THE y. k=0 & y=a | (EX x. k=succ(x) & y=b(x)) \tdx{nat_0I} 0 : nat \tdx{nat_succI} n : nat ==> succ(n) : nat \tdx{nat_induct} [| n: nat; P(0); !!x. [| x: 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 : 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: nat ==> (m div n)#*n #+ m mod n = m \end{ttbox} \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 \texttt{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 $t\in\texttt{nat}$, the arithmetic operators coerce their arguments to be natural numbers. The function \cdx{natify} is defined such that $\texttt{natify}(n) = n$ if $n$ is a natural number, $\texttt{natify}(\texttt{succ}(x)) = \texttt{succ}(\texttt{natify}(x))$ for all $x$, and finally $\texttt{natify}(x)=0$ 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 \texttt{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{ttbox} 1. i #+ j #+ k #- j < k #+ l > by (Simp_tac 1); 1. natify(i) < natify(l) \end{ttbox} Given the assumptions \texttt{i:nat} and \texttt{l: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{ttbox} \tdx{zadd_0_intify} 0 $+ n = intify(n) \tdx{zmult_type} m $* n : 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 $* k) \end{ttbox} \caption{The integers} \label{zf-int} \end{figure} \index{integers} Theory \thydx{Int} defines the integers, as equivalence classes of natural numbers. Figure~\ref{zf-int} presents a tidy collection of laws. In fact, a large library of facts is proved, including monotonicity laws for addition and multiplication, covering both positive and negative operands. As with the natural numbers, the need for typing proofs is minimized. All the operators defined in Fig.\ts\ref{zf-int} coerce their operands to integers by applying the function \cdx{intify}. This function is the identity on integers and maps other operands to zero. Decimal notation is provided for the integers. Numbers, written as \texttt{\#$nnn$} or \texttt{\#-$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{ttbox} 1. y $+ z $+ #-3 $* x $+ y $<= x $* #2 $+ z > by (Simp_tac 1); 1. #2 $* y $<= #5 $* x \end{ttbox} 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 those of \ML. 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 \texttt{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 \texttt{list}, which is built-in, and is defined by \begin{ttbox} consts list :: i=>i datatype "list(A)" = Nil | Cons ("a:A", "l: list(A)") \end{ttbox} Note that the datatype operator must be declared as a constant first. However, the package declares the constructors. Here, \texttt{Nil} gets type $i$ and \texttt{Cons} gets type $[i,i]\To i$. Trees and forests can be modelled by the mutually recursive datatype definition \begin{ttbox} consts tree, forest, tree_forest :: i=>i datatype "tree(A)" = Tcons ("a: A", "f: forest(A)") and "forest(A)" = Fnil | Fcons ("t: tree(A)", "f: forest(A)") \end{ttbox} Here $\texttt{tree}(A)$ is the set of trees over $A$, $\texttt{forest}(A)$ is the set of forests over $A$, and $\texttt{tree_forest}(A)$ is the union of the previous two sets. All three operators must be declared first. The datatype \texttt{term}, which is defined by \begin{ttbox} consts term :: i=>i datatype "term(A)" = Apply ("a: A", "l: list(term(A))") monos "[list_mono]" \end{ttbox} is an example of nested recursion. (The theorem \texttt{list_mono} is proved in file \texttt{List.ML}, and the \texttt{term} example is devaloped in theory \thydx{ex/Term}.) \subsubsection{Freeness of the constructors} Constructors satisfy {\em freeness} properties. Constructions are distinct, for example $\texttt{Nil}\not=\texttt{Cons}(a,l)$, and they are injective, for example $\texttt{Cons}(a,l)=\texttt{Cons}(a',l') \bimp a=a' \conj l=l'$. 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 \texttt{list} datatype, they are called \texttt{list.free_SEs}. Occasionally this exposes the underlying representation of some constructor, which can be rectified using the command \hbox{\tt fold_tac list.con_defs}. \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 \texttt{list.induct} in Fig.\ts\ref{zf-list}. For mutually recursive datatypes, the induction rule is supplied in two forms. Consider datatype \texttt{TF}. The rule \texttt{tree_forest.induct} performs induction over a single predicate~\texttt{P}, which is presumed to be defined for both trees and forests: \begin{ttbox} [| x : tree_forest(A); !!a f. [| a : A; f : forest(A); P(f) |] ==> P(Tcons(a, f)); P(Fnil); !!f t. [| t : tree(A); P(t); f : forest(A); P(f) |] ==> P(Fcons(t, f)) |] ==> P(x) \end{ttbox} The rule \texttt{tree_forest.mutual_induct} performs induction over two distinct predicates, \texttt{P_tree} and \texttt{P_forest}. \begin{ttbox} [| !!a f. [| a : A; f : forest(A); P_forest(f) |] ==> P_tree(Tcons(a, f)); P_forest(Fnil); !!f t. [| t : tree(A); P_tree(t); f : forest(A); P_forest(f) |] ==> P_forest(Fcons(t, f)) |] ==> (ALL za. za : tree(A) --> P_tree(za)) & (ALL za. za : forest(A) --> P_forest(za)) \end{ttbox} For datatypes with nested recursion, such as the \texttt{term} example from above, things are a bit more complicated. The rule \texttt{term.induct} refers to the monotonic operator, \texttt{list}: \begin{ttbox} [| x : term(A); !!a l. [| a: A; l: list(Collect(term(A), P)) |] ==> P(Apply(a, l)) |] ==> P(x) \end{ttbox} The file \texttt{ex/Term.ML} derives two higher-level induction rules, one of which is particularly useful for proving equations: \begin{ttbox} [| t : term(A); !!x zs. [| x : A; zs : list(term(A)); map(f, zs) = map(g, zs) |] ==> f(Apply(x, zs)) = g(Apply(x, zs)) |] ==> f(t) = g(t) \end{ttbox} How this can be generalized to other nested datatypes is a matter for future research. \subsubsection{The \texttt{case} operator} The package defines an operator for performing case analysis over the datatype. For \texttt{list}, it is called \texttt{list_case} and satisfies the equations \begin{ttbox} list_case(f_Nil, f_Cons, []) = f_Nil list_case(f_Nil, f_Cons, Cons(a, l)) = f_Cons(a, l) \end{ttbox} Here \texttt{f_Nil} is the value to return if the argument is \texttt{Nil} and \texttt{f_Cons} is a function that computes the value to return if the argument has the form $\texttt{Cons}(a,l)$. The function can be expressed as an abstraction, over patterns if desired (\S\ref{sec:pairs}). For mutually recursive datatypes, there is a single \texttt{case} operator. In the tree/forest example, the constant \texttt{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. Induction or exhaustion are usually invoked by hand, usually via these special-purpose tactics: \begin{ttdescription} \item[\ttindexbold{induct_tac} {\tt"}$x${\tt"} $i$] applies structural induction on variable $x$ to subgoal $i$, provided the type of $x$ is a datatype. The induction variable should not occur among other assumptions of the subgoal. \end{ttdescription} In some cases, induction is overkill and a case distinction over all constructors of the datatype suffices. \begin{ttdescription} \item[\ttindexbold{exhaust_tac} {\tt"}$x${\tt"} $i$] performs an exhaustive 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 \texttt{x:\ list(A)}. The tactics also work for the natural numbers (\texttt{nat}) and disjoint sums, although these sets were not defined using the datatype package. (Disjoint sums are not recursive, so only \texttt{exhaust_tac} is available.) \bigskip Here are some more details for the technically minded. Processing the theory file produces an \ML\ structure which, in addition to the usual components, contains a structure named $t$ for each datatype $t$ defined in the file. Each structure $t$ contains the following elements: \begin{ttbox} val intrs : thm list \textrm{the introduction rules} val elim : thm \textrm{the elimination (case analysis) rule} val induct : thm \textrm{the standard induction rule} val mutual_induct : thm \textrm{the mutual induction rule, or \texttt{True}} val case_eqns : thm list \textrm{equations for the case operator} val recursor_eqns : thm list \textrm{equations for the recursor} val con_defs : thm list \textrm{definitions of the case operator and constructors} val free_iffs : thm list \textrm{logical equivalences for proving freeness} val free_SEs : thm list \textrm{elimination rules for proving freeness} val mk_free : string -> thm \textrm{A function for proving freeness theorems} val mk_cases : string -> thm \textrm{case analysis, see below} val defs : thm list \textrm{definitions of operators} val bnd_mono : thm list \textrm{monotonicity property} val dom_subset : thm list \textrm{inclusion in `bounding set'} \end{ttbox} Furthermore there is the theorem $C$\texttt{_I} for every constructor~$C$; for example, the \texttt{list} datatype's introduction rules are bound to the identifiers \texttt{Nil_I} and \texttt{Cons_I}. For a codatatype, the component \texttt{coinduct} is the coinduction rule, replacing the \texttt{induct} component. See the theories \texttt{ex/Ntree} and \texttt{ex/Brouwer} for examples of infinitely branching datatypes. See theory \texttt{ex/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 $\texttt{bt}(A)$ of binary trees over~$A$. The theory must contain these lines: \begin{ttbox} consts bt :: i=>i datatype "bt(A)" = Lf | Br ("a: A", "t1: bt(A)", "t2: bt(A)") \end{ttbox} After loading the theory, we can prove, for example, that no tree equals its left branch. To ease the induction, we state the goal using quantifiers. \begin{ttbox} Goal "l : bt(A) ==> ALL x r. Br(x,l,r) ~= l"; {\out Level 0} {\out l : bt(A) ==> ALL x r. Br(x, l, r) ~= l} {\out 1. l : bt(A) ==> ALL x r. Br(x, l, r) ~= l} \end{ttbox} This can be proved by the structural induction tactic: \begin{ttbox} by (induct_tac "l" 1); {\out Level 1} {\out l : bt(A) ==> ALL x r. Br(x, l, r) ~= l} {\out 1. ALL x r. Br(x, Lf, r) ~= Lf} {\out 2. !!a t1 t2.} {\out [| a : A; t1 : bt(A);} {\out ALL x r. Br(x, t1, r) ~= t1; t2 : bt(A);} {\out ALL x r. Br(x, t2, r) ~= t2 |]} {\out ==> ALL x r. Br(x, Br(a, t1, t2), r) ~= Br(a, t1, t2)} \end{ttbox} Both subgoals are proved using \texttt{Auto_tac}, which performs the necessary freeness reasoning. \begin{ttbox} by Auto_tac; {\out Level 2} {\out l : bt(A) ==> ALL x r. Br(x, l, r) ~= l} {\out No subgoals!} \end{ttbox} To remove the quantifiers from the induction formula, we save the theorem using \ttindex{qed_spec_mp}. \begin{ttbox} qed_spec_mp "Br_neq_left"; {\out val Br_neq_left = "?l : bt(?A) ==> Br(?x, ?l, ?r) ~= ?l" : thm} \end{ttbox} When there are only a few constructors, we might prefer to prove the freenness theorems for each constructor. This is trivial, using the function given us for that purpose: \begin{ttbox} val Br_iff = bt.mk_free "Br(a,l,r)=Br(a',l',r') <-> a=a' & l=l' & r=r'"; {\out val Br_iff =} {\out "Br(?a, ?l, ?r) = Br(?a', ?l', ?r') <->} {\out ?a = ?a' & ?l = ?l' & ?r = ?r'" : thm} \end{ttbox} The purpose of \ttindex{mk_cases} is to generate instances of the elimination (case analysis) rule that have been simplified using freeness reasoning. For example, this instance of the elimination rule propagates type-checking information from the premise $\texttt{Br}(a,l,r)\in\texttt{bt}(A)$: \begin{ttbox} val BrE = bt.mk_cases "Br(a,l,r) : bt(A)"; {\out val BrE =} {\out "[| Br(?a, ?l, ?r) : bt(?A);} {\out [| ?a : ?A; ?l : bt(?A); ?r : bt(?A) |] ==> ?Q |]} {\out ==> ?Q" : thm} \end{ttbox} \subsubsection{Mixfix syntax in datatypes} Mixfix syntax is sometimes convenient. The theory \texttt{ex/PropLog} makes a deep embedding of propositional logic: \begin{ttbox} consts prop :: i datatype "prop" = Fls | Var ("n: nat") ("#_" [100] 100) | "=>" ("p: prop", "q: prop") (infixr 90) \end{ttbox} 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{ttbox} 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{ttbox} The datatype package scales well. Even though all properties are proved rather than assumed, full processing of this definition takes under 15 seconds (on a 300 MHz Pentium). The constructors have a balanced representation, essentially binary notation, so freeness properties can be proved fast. \begin{ttbox} Goal "C00 ~= C01"; by (Simp_tac 1); \end{ttbox} 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 new axioms. Here is a dangerous way of defining the append function for lists: \begin{ttbox}\slshape consts "\at" :: [i,i]=>i (infixr 60) rules app_Nil "[] \at ys = ys" app_Cons "(Cons(a,l)) \at ys = Cons(a, l \at ys)" \end{ttbox} Asserting axioms brings the danger of accidentally asserting nonsense. It should be avoided at all costs! The \ttindex{primrec} declaration is a safe means of defining primitive recursive functions on datatypes: \begin{ttbox} consts "\at" :: [i,i]=>i (infixr 60) primrec "[] \at ys = ys" "(Cons(a,l)) \at ys = Cons(a, l \at ys)" \end{ttbox} Isabelle will now check that the two rules do indeed form a primitive recursive definition. For example, the declaration \begin{ttbox} primrec "[] \at ys = us" \end{ttbox} is rejected with an error message ``\texttt{Extra variables on rhs}''. \subsubsection{Syntax of recursive definitions} The general form of a primitive recursive definition is \begin{ttbox} 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 \texttt{rules} section of a theory, will be visible at the \ML\ level. The reduction rules for {\tt\at} become part of the default simpset, which leads to short proof scripts: \begin{ttbox}\underscoreon Goal "xs: list(A) ==> (xs @ ys) @ zs = xs @ (ys @ zs)"; by (induct\_tac "xs" 1); by (ALLGOALS Asm\_simp\_tac); \end{ttbox} You can even use the \texttt{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. Here is an example, drawn from the theory \texttt{Resid/Substitution}. The type of redexes is declared as follows: \begin{ttbox} consts redexes :: i datatype "redexes" = Var ("n: nat") | Fun ("t: redexes") | App ("b:bool" ,"f:redexes" , "a:redexes") \end{ttbox} The function \texttt{lift} takes a second argument, $k$, which varies in recursive calls. \begin{ttbox} primrec "lift(Var(i)) = (lam k:nat. if i<k then Var(i) else Var(succ(i)))" "lift(Fun(t)) = (lam k:nat. Fun(lift(t) ` succ(k)))" "lift(App(b,f,a)) = (lam k:nat. App(b, lift(f)`k, lift(a)`k))" \end{ttbox} Now \texttt{lift(r)`k} satisfies the required recursion equations. \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. Each definition creates an \ML\ structure, which is a substructure of the main theory structure. This package 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} inductive domains {\it domain declarations} intrs {\it introduction rules} monos {\it monotonicity theorems} con_defs {\it constructor definitions} type_intrs {\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 {\tt co\-inductive}. The {\tt monos}, {\tt con\_defs}, {\tt type\_intrs} and {\tt type\_elims} sections are optional. If present, each is specified either as a list of identifiers or as a string. If the latter, then the string must be a valid \textsc{ml} expression of type {\tt thm list}. The string is simply inserted into the {\tt _thy.ML} file; if it is ill-formed, it will trigger \textsc{ml} error messages. You can then inspect the file on the temporary directory. \begin{description} \item[\it domain declarations] are items of the form {\it string\/}~{\tt <=}~{\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 \texttt{ex/Primrec}. \item[\it type\_intrs] 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 {\tt type\_intrs} 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 \verb|mutual_induct|. \end{itemize} \subsection{Example of an inductive definition} Two declarations, included in a theory file, define the finite powerset operator. First we declare the constant~\texttt{Fin}. Then we declare it inductively, with two introduction rules: \begin{ttbox} consts Fin :: i=>i inductive domains "Fin(A)" <= "Pow(A)" intrs emptyI "0 : Fin(A)" consI "[| a: A; b: Fin(A) |] ==> cons(a,b) : Fin(A)" type_intrs empty_subsetI, cons_subsetI, PowI type_elims "[make_elim PowD]" \end{ttbox} The resulting theory structure contains a substructure, called~\texttt{Fin}. It contains the \texttt{Fin}$~A$ introduction rules as the list \texttt{Fin.intrs}, and also individually as \texttt{Fin.emptyI} and \texttt{Fin.consI}. The induction rule is \texttt{Fin.induct}. The chief problem with making (co)inductive definitions involves type-checking the rules. Sometimes, additional theorems need to be supplied under \texttt{type_intrs} or \texttt{type_elims}. If the package fails when trying to prove your introduction rules, then set the flag \ttindexbold{trace_induct} to \texttt{true} and try again. (See the manual \emph{A Fixedpoint Approach \ldots} for more discussion of type-checking.) In the example above, $\texttt{Pow}(A)$ is given as the domain of $\texttt{Fin}(A)$, 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 \texttt{type_intrs} and \texttt{type_elims} are omitted from the inductive definition above: \begin{ttbox} Inductive definition Finite.Fin Fin(A) == lfp(Pow(A), \%X. {z: Pow(A) . z = 0 | (EX a b. z = cons(a, b) & a : A & b : X)}) Proving monotonicity... \ttbreak Proving the introduction rules... The type-checking subgoal: 0 : Fin(A) 1. 0 : Pow(A) \ttbreak The subgoal after monos, type_elims: 0 : Fin(A) 1. 0 : Pow(A) *** prove_goal: tactic failed \end{ttbox} We see the need to supply theorems to let the package prove $\emptyset\in\texttt{Pow}(A)$. Restoring the \texttt{type_intrs} but not the \texttt{type_elims}, we again get an error message: \begin{ttbox} The type-checking subgoal: 0 : Fin(A) 1. 0 : Pow(A) \ttbreak The subgoal after monos, type_elims: 0 : Fin(A) 1. 0 : Pow(A) \ttbreak The type-checking subgoal: cons(a, b) : Fin(A) 1. [| a : A; b : Fin(A) |] ==> cons(a, b) : Pow(A) \ttbreak The subgoal after monos, type_elims: cons(a, b) : Fin(A) 1. [| a : A; b : Pow(A) |] ==> cons(a, b) : Pow(A) *** prove_goal: tactic failed \end{ttbox} 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 \texttt{type_intrs}. The solution actually used is to supply, under \texttt{type_elims}, a rule that changes $b\in\texttt{Pow}(A)$ to $b\subseteq A$; together with \texttt{cons_subsetI} and \texttt{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~\texttt{Pow} in the introduction rule and the corresponding mention of the rule \verb|Pow_mono| in the \texttt{monos} list. If the desired rule has a universally quantified premise, usually the effect can be obtained using \texttt{Pow}. \begin{ttbox} consts acc :: i=>i inductive domains "acc(r)" <= "field(r)" intrs vimage "[| r-``{a}: Pow(acc(r)); a: field(r) |] ==> a: acc(r)" monos Pow_mono \end{ttbox} Finally, here is a coinductive definition. It captures (as a bisimulation) the notion of equality on lazy lists, which are first defined as a codatatype: \begin{ttbox} consts llist :: i=>i codatatype "llist(A)" = LNil | LCons ("a: A", "l: llist(A)") \ttbreak consts lleq :: i=>i coinductive domains "lleq(A)" <= "llist(A) * llist(A)" intrs LNil "<LNil, LNil> : lleq(A)" LCons "[| a:A; <l,l'>: lleq(A) |] ==> <LCons(a,l), LCons(a,l')>: lleq(A)" type_intrs "llist.intrs" \end{ttbox} This use of \texttt{type_intrs} is typical: the relation concerns the codatatype \texttt{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 \texttt{ZF/ex}. The directory \texttt{Coind} and the theory \texttt{ZF/ex/LList} contain coinductive definitions. Larger examples may be found on other subdirectories of \texttt{ZF}, such as \texttt{IMP}, and \texttt{Resid}. \subsection{The result structure} Each (co)inductive set defined in a theory file generates an \ML\ substructure having the same name. The the substructure contains the following elements: \begin{ttbox} val intrs : thm list \textrm{the introduction rules} val elim : thm \textrm{the elimination (case analysis) rule} val mk_cases : string -> thm \textrm{case analysis, see below} val induct : thm \textrm{the standard induction rule} val mutual_induct : thm \textrm{the mutual induction rule, or \texttt{True}} val defs : thm list \textrm{definitions of operators} val bnd_mono : thm list \textrm{monotonicity property} val dom_subset : thm list \textrm{inclusion in `bounding set'} \end{ttbox} Furthermore there is the theorem $C$\texttt{_I} for every constructor~$C$; for example, the \texttt{list} datatype's introduction rules are bound to the identifiers \texttt{Nil_I} and \texttt{Cons_I}. For a codatatype, the component \texttt{coinduct} is the coinduction rule, replacing the \texttt{induct} component. Recall that \ttindex{mk_cases} generates simplified instances of the elimination (case analysis) rule. It is as useful for inductive definitions as it is for datatypes. There are many examples in the theory \texttt{ex/Comb}, which is discussed at length elsewhere~\cite{paulson-generic}. The theory first defines the datatype \texttt{comb} of combinators: \begin{ttbox} consts comb :: i datatype "comb" = K | S | "#" ("p: comb", "q: comb") (infixl 90) \end{ttbox} The theory goes on to define contraction and parallel contraction inductively. Then the file \texttt{ex/Comb.ML} defines special cases of contraction using \texttt{mk_cases}: \begin{ttbox} val K_contractE = contract.mk_cases "K -1-> r"; {\out val K_contractE = "K -1-> ?r ==> ?Q" : thm} \end{ttbox} We can read this as saying that the combinator \texttt{K} cannot reduce to anything. Similar elimination rules for \texttt{S} and application are also generated and are supplied to the classical reasoner. Note that \texttt{comb.con_defs} is given to \texttt{mk_cases} to allow freeness reasoning on datatype \texttt{comb}. \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 \texttt{Trancl} defines the transitive closure of a relation (as a least fixedpoint). \item Theory \texttt{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 \texttt{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 \texttt{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 \texttt{Rel} defines the basic properties of relations, such as (ir)reflexivity, (a)symmetry, and transitivity. \item Theory \texttt{EquivClass} develops a theory of equivalence classes, not using the Axiom of Choice. \item Theory \texttt{Order} defines partial orderings, total orderings and wellorderings. \item Theory \texttt{OrderArith} defines orderings on sum and product sets. These can be used to define ordinal arithmetic and have applications to cardinal arithmetic. \item Theory \texttt{OrderType} defines order types. Every wellordering is equivalent to a unique ordinal, which is its order type. \item Theory \texttt{Cardinal} defines equipollence and cardinal numbers. \item Theory \texttt{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 \texttt{AC} asserts the Axiom of Choice and proves some simple equivalent forms. \item Theory \texttt{Zorn} proves Hausdorff's Maximal Principle, Zorn's Lemma and the Wellordering Theorem, following Abrial and Laffitte~\cite{abrial93}. \item Theory \verb|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 \texttt{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 \texttt{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 \texttt{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-CADE}. \begin{itemize} \item File \texttt{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 \texttt{Ramsey} proves the finite exponent 2 version of Ramsey's Theorem, following Basin and Kaufmann's presentation~\cite{basin91}. \item Theory \texttt{Integ} develops a theory of the integers as equivalence classes of pairs of natural numbers. \item Theory \texttt{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 \texttt{Primes} defines the Greatest Common Divisor of two natural numbers and and the ``divides'' relation. \item Theory \texttt{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 under 14 seconds. \item Theory \texttt{BT} defines the recursive data structure ${\tt bt}(A)$, labelled binary trees. \item Theory \texttt{Term} defines a recursive data structure for terms and term lists. These are simply finite branching trees. \item Theory \texttt{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 \texttt{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 \texttt{ListN} inductively defines the lists of $n$ elements~\cite{paulin-tlca}. \item Theory \texttt{Acc} inductively defines the accessible part of a relation~\cite{paulin-tlca}. \item Theory \texttt{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 \texttt{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 ${{\tt Pow}(A)\cap {\tt Pow}(B)}= {\tt Pow}(A\cap B)$. 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 \texttt{ZF/mono.ML}: \begin{ttbox} \tdx{Pow_mono} A<=B ==> Pow(A) <= Pow(B) \end{ttbox} We enter the goal and make the first step, which breaks the equation into two inclusions by extensionality:\index{*equalityI theorem} \begin{ttbox} Goal "Pow(A Int B) = Pow(A) Int Pow(B)"; {\out Level 0} {\out Pow(A Int B) = Pow(A) Int Pow(B)} {\out 1. Pow(A Int B) = Pow(A) Int Pow(B)} \ttbreak by (resolve_tac [equalityI] 1); {\out Level 1} {\out Pow(A Int B) = Pow(A) Int Pow(B)} {\out 1. Pow(A Int B) <= Pow(A) Int Pow(B)} {\out 2. Pow(A) Int Pow(B) <= Pow(A Int B)} \end{ttbox} Both inclusions could be tackled straightforwardly using \texttt{subsetI}. A shorter proof results from noting that intersection forms the greatest lower bound:\index{*Int_greatest theorem} \begin{ttbox} by (resolve_tac [Int_greatest] 1); {\out Level 2} {\out Pow(A Int B) = Pow(A) Int Pow(B)} {\out 1. Pow(A Int B) <= Pow(A)} {\out 2. Pow(A Int B) <= Pow(B)} {\out 3. Pow(A) Int Pow(B) <= Pow(A Int B)} \end{ttbox} Subgoal~1 follows by applying the monotonicity of \texttt{Pow} to $A\int B\subseteq A$; subgoal~2 follows similarly: \index{*Int_lower1 theorem}\index{*Int_lower2 theorem} \begin{ttbox} by (resolve_tac [Int_lower1 RS Pow_mono] 1); {\out Level 3} {\out Pow(A Int B) = Pow(A) Int Pow(B)} {\out 1. Pow(A Int B) <= Pow(B)} {\out 2. Pow(A) Int Pow(B) <= Pow(A Int B)} \ttbreak by (resolve_tac [Int_lower2 RS Pow_mono] 1); {\out Level 4} {\out Pow(A Int B) = Pow(A) Int Pow(B)} {\out 1. Pow(A) Int Pow(B) <= Pow(A Int B)} \end{ttbox} We are left with the opposite inclusion, which we tackle in the straightforward way:\index{*subsetI theorem} \begin{ttbox} by (resolve_tac [subsetI] 1); {\out Level 5} {\out Pow(A Int B) = Pow(A) Int Pow(B)} {\out 1. !!x. x : Pow(A) Int Pow(B) ==> x : Pow(A Int B)} \end{ttbox} The subgoal is to show $x\in {\tt Pow}(A\cap B)$ assuming $x\in{\tt Pow}(A)\cap {\tt Pow}(B)$; eliminating this assumption produces two subgoals. The rule \tdx{IntE} treats the intersection like a conjunction instead of unfolding its definition. \begin{ttbox} by (eresolve_tac [IntE] 1); {\out Level 6} {\out Pow(A Int B) = Pow(A) Int Pow(B)} {\out 1. !!x. [| x : Pow(A); x : Pow(B) |] ==> x : Pow(A Int B)} \end{ttbox} The next step replaces the \texttt{Pow} by the subset relation~($\subseteq$).\index{*PowI theorem} \begin{ttbox} by (resolve_tac [PowI] 1); {\out Level 7} {\out Pow(A Int B) = Pow(A) Int Pow(B)} {\out 1. !!x. [| x : Pow(A); x : Pow(B) |] ==> x <= A Int B} \end{ttbox} We perform the same replacement in the assumptions. This is a good demonstration of the tactic \ttindex{dresolve_tac}:\index{*PowD theorem} \begin{ttbox} by (REPEAT (dresolve_tac [PowD] 1)); {\out Level 8} {\out Pow(A Int B) = Pow(A) Int Pow(B)} {\out 1. !!x. [| x <= A; x <= B |] ==> x <= A Int B} \end{ttbox} 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{ttbox} by (resolve_tac [Int_greatest] 1); {\out Level 9} {\out Pow(A Int B) = Pow(A) Int Pow(B)} {\out 1. !!x. [| x <= A; x <= B |] ==> x <= A} {\out 2. !!x. [| x <= A; x <= B |] ==> x <= B} \end{ttbox} To conclude the proof, we clear up the trivial subgoals: \begin{ttbox} by (REPEAT (assume_tac 1)); {\out Level 10} {\out Pow(A Int B) = Pow(A) Int Pow(B)} {\out No subgoals!} \end{ttbox} \medskip We could have performed this proof in one step by applying \ttindex{Blast_tac}. Let us go back to the start: \begin{ttbox} choplev 0; {\out Level 0} {\out Pow(A Int B) = Pow(A) Int Pow(B)} {\out 1. Pow(A Int B) = Pow(A) Int Pow(B)} by (Blast_tac 1); {\out Depth = 0} {\out Depth = 1} {\out Depth = 2} {\out Depth = 3} {\out Level 1} {\out Pow(A Int B) = Pow(A) Int Pow(B)} {\out No subgoals!} \end{ttbox} 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{ttbox} Goal "C<=D ==> Union(C) <= Union(D)"; {\out Level 0} {\out C <= D ==> Union(C) <= Union(D)} {\out 1. C <= D ==> Union(C) <= Union(D)} \ttbreak by (resolve_tac [subsetI] 1); {\out Level 1} {\out C <= D ==> Union(C) <= Union(D)} {\out 1. !!x. [| C <= D; x : Union(C) |] ==> x : Union(D)} \end{ttbox} Big union is like an existential quantifier --- the occurrence in the assumptions must be eliminated early, since it creates parameters. \index{*UnionE theorem} \begin{ttbox} by (eresolve_tac [UnionE] 1); {\out Level 2} {\out C <= D ==> Union(C) <= Union(D)} {\out 1. !!x B. [| C <= D; x : B; B : C |] ==> x : Union(D)} \end{ttbox} Now we may apply \tdx{UnionI}, which creates an unknown involving the parameters. To show $x\in \bigcup(D)$ it suffices to show that $x$ belongs to some element, say~$\Var{B2}(x,B)$, of~$D$. \begin{ttbox} by (resolve_tac [UnionI] 1); {\out Level 3} {\out C <= D ==> Union(C) <= Union(D)} {\out 1. !!x B. [| C <= D; x : B; B : C |] ==> ?B2(x,B) : D} {\out 2. !!x B. [| C <= D; x : B; B : C |] ==> x : ?B2(x,B)} \end{ttbox} Combining \tdx{subsetD} with the assumption $C\subseteq D$ yields $\Var{a}\in C \Imp \Var{a}\in D$, which reduces subgoal~1. Note that \texttt{eresolve_tac} has removed that assumption. \begin{ttbox} by (eresolve_tac [subsetD] 1); {\out Level 4} {\out C <= D ==> Union(C) <= Union(D)} {\out 1. !!x B. [| x : B; B : C |] ==> ?B2(x,B) : C} {\out 2. !!x B. [| C <= D; x : B; B : C |] ==> x : ?B2(x,B)} \end{ttbox} The rest is routine. Observe how~$\Var{B2}(x,B)$ is instantiated. \begin{ttbox} by (assume_tac 1); {\out Level 5} {\out C <= D ==> Union(C) <= Union(D)} {\out 1. !!x B. [| C <= D; x : B; B : C |] ==> x : B} by (assume_tac 1); {\out Level 6} {\out C <= D ==> Union(C) <= Union(D)} {\out No subgoals!} \end{ttbox} Again, \ttindex{Blast_tac} can prove the theorem in one step. \begin{ttbox} by (Blast_tac 1); {\out Depth = 0} {\out Depth = 1} {\out Depth = 2} {\out Level 1} {\out C <= D ==> Union(C) <= Union(D)} {\out No subgoals!} \end{ttbox} The file \texttt{ZF/equalities.ML} has many similar proofs. Reasoning about general intersection can be difficult because of its anomalous behaviour on the empty set. However, \ttindex{Blast_tac} copes well with these. Here is a typical example, borrowed from Devlin~\cite[page 12]{devlin79}: \begin{ttbox} a:C ==> (INT x:C. A(x) Int B(x)) = (INT x:C. A(x)) Int (INT x:C. B(x)) \end{ttbox} In traditional notation this is \[ a\in C \,\Imp\, \inter@{x\in C} \Bigl(A(x) \int B(x)\Bigr) = \Bigl(\inter@{x\in C} A(x)\Bigr) \int \Bigl(\inter@{x\in C} B(x)\Bigr) \] \section{Low-level reasoning about functions} The derived rules \texttt{lamI}, \texttt{lamE}, \texttt{lam_type}, \texttt{beta} and \texttt{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{ttbox} Goal "[| a:A; f: A->B; g: C->D; A Int C = 0 |] ==> \ttback \ttback (f Un g)`a = f`a"; {\out Level 0} {\out [| a : A; f : A -> B; g : C -> D; A Int C = 0 |]} {\out ==> (f Un g) ` a = f ` a} {\out 1. [| a : A; f : A -> B; g : C -> D; A Int C = 0 |]} {\out ==> (f Un g) ` a = f ` a} \end{ttbox} Using \tdx{apply_equality}, we reduce the equality to reasoning about ordered pairs. The second subgoal is to verify that $f\un g$ is a function. To save space, the assumptions will be abbreviated below. \begin{ttbox} by (resolve_tac [apply_equality] 1); {\out Level 1} {\out [| \ldots |] ==> (f Un g) ` a = f ` a} {\out 1. [| \ldots |] ==> <a,f ` a> : f Un g} {\out 2. [| \ldots |] ==> f Un g : (PROD x:?A. ?B(x))} \end{ttbox} We must show that the pair belongs to~$f$ or~$g$; by~\tdx{UnI1} we choose~$f$: \begin{ttbox} by (resolve_tac [UnI1] 1); {\out Level 2} {\out [| \ldots |] ==> (f Un g) ` a = f ` a} {\out 1. [| \ldots |] ==> <a,f ` a> : f} {\out 2. [| \ldots |] ==> f Un g : (PROD x:?A. ?B(x))} \end{ttbox} To show $\pair{a,f`a}\in f$ we use \tdx{apply_Pair}, which is essentially the converse of \tdx{apply_equality}: \begin{ttbox} by (resolve_tac [apply_Pair] 1); {\out Level 3} {\out [| \ldots |] ==> (f Un g) ` a = f ` a} {\out 1. [| \ldots |] ==> f : (PROD x:?A2. ?B2(x))} {\out 2. [| \ldots |] ==> a : ?A2} {\out 3. [| \ldots |] ==> f Un g : (PROD x:?A. ?B(x))} \end{ttbox} 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} is instantiated to~\texttt{A}. \begin{ttbox} by (assume_tac 1); {\out Level 4} {\out [| \ldots |] ==> (f Un g) ` a = f ` a} {\out 1. [| \ldots |] ==> a : A} {\out 2. [| \ldots |] ==> f Un g : (PROD x:?A. ?B(x))} by (assume_tac 1); {\out Level 5} {\out [| \ldots |] ==> (f Un g) ` a = f ` a} {\out 1. [| \ldots |] ==> f Un g : (PROD x:?A. ?B(x))} \end{ttbox} To construct functions of the form $f\un g$, we apply \tdx{fun_disjoint_Un}: \begin{ttbox} by (resolve_tac [fun_disjoint_Un] 1); {\out Level 6} {\out [| \ldots |] ==> (f Un g) ` a = f ` a} {\out 1. [| \ldots |] ==> f : ?A3 -> ?B3} {\out 2. [| \ldots |] ==> g : ?C3 -> ?D3} {\out 3. [| \ldots |] ==> ?A3 Int ?C3 = 0} \end{ttbox} The remaining subgoals are instances of the assumptions. Again, observe how unknowns are instantiated: \begin{ttbox} by (assume_tac 1); {\out Level 7} {\out [| \ldots |] ==> (f Un g) ` a = f ` a} {\out 1. [| \ldots |] ==> g : ?C3 -> ?D3} {\out 2. [| \ldots |] ==> A Int ?C3 = 0} by (assume_tac 1); {\out Level 8} {\out [| \ldots |] ==> (f Un g) ` a = f ` a} {\out 1. [| \ldots |] ==> A Int C = 0} by (assume_tac 1); {\out Level 9} {\out [| \ldots |] ==> (f Un g) ` a = f ` a} {\out No subgoals!} \end{ttbox} See the files \texttt{ZF/func.ML} and \texttt{ZF/WF.ML} for more examples of reasoning about functions. \index{set theory|)}