2 \chapter{Zermelo-Fraenkel Set Theory}
5 The theory~\thydx{ZF} implements Zermelo-Fraenkel set
6 theory~\cite{halmos60,suppes72} as an extension of~\texttt{FOL}, classical
7 first-order logic. The theory includes a collection of derived natural
8 deduction rules, for use with Isabelle's classical reasoner. Much
9 of it is based on the work of No\"el~\cite{noel}.
11 A tremendous amount of set theory has been formally developed, including the
12 basic properties of relations, functions, ordinals and cardinals. Significant
13 results have been proved, such as the Schr\"oder-Bernstein Theorem, the
14 Wellordering Theorem and a version of Ramsey's Theorem. \texttt{ZF} provides
15 both the integers and the natural numbers. General methods have been
16 developed for solving recursion equations over monotonic functors; these have
17 been applied to yield constructions of lists, trees, infinite lists, etc.
19 \texttt{ZF} has a flexible package for handling inductive definitions,
20 such as inference systems, and datatype definitions, such as lists and
21 trees. Moreover it handles coinductive definitions, such as
22 bisimulation relations, and codatatype definitions, such as streams. It
23 provides a streamlined syntax for defining primitive recursive functions over
26 Because {\ZF} is an extension of {\FOL}, it provides the same
27 packages, namely \texttt{hyp_subst_tac}, the simplifier, and the
28 classical reasoner. The default simpset and claset are usually
31 Published articles~\cite{paulson-set-I,paulson-set-II} describe \texttt{ZF}
32 less formally than this chapter. Isabelle employs a novel treatment of
33 non-well-founded data structures within the standard {\sc zf} axioms including
34 the Axiom of Foundation~\cite{paulson-final}.
37 \section{Which version of axiomatic set theory?}
38 The two main axiom systems for set theory are Bernays-G\"odel~({\sc bg})
39 and Zermelo-Fraenkel~({\sc zf}). Resolution theorem provers can use {\sc
40 bg} because it is finite~\cite{boyer86,quaife92}. {\sc zf} does not
41 have a finite axiom system because of its Axiom Scheme of Replacement.
42 This makes it awkward to use with many theorem provers, since instances
43 of the axiom scheme have to be invoked explicitly. Since Isabelle has no
44 difficulty with axiom schemes, we may adopt either axiom system.
46 These two theories differ in their treatment of {\bf classes}, which are
47 collections that are `too big' to be sets. The class of all sets,~$V$,
48 cannot be a set without admitting Russell's Paradox. In {\sc bg}, both
49 classes and sets are individuals; $x\in V$ expresses that $x$ is a set. In
50 {\sc zf}, all variables denote sets; classes are identified with unary
51 predicates. The two systems define essentially the same sets and classes,
52 with similar properties. In particular, a class cannot belong to another
53 class (let alone a set).
55 Modern set theorists tend to prefer {\sc zf} because they are mainly concerned
56 with sets, rather than classes. {\sc bg} requires tiresome proofs that various
57 collections are sets; for instance, showing $x\in\{x\}$ requires showing that
64 \it name &\it meta-type & \it description \\
65 \cdx{Let} & $[\alpha,\alpha\To\beta]\To\beta$ & let binder\\
66 \cdx{0} & $i$ & empty set\\
67 \cdx{cons} & $[i,i]\To i$ & finite set constructor\\
68 \cdx{Upair} & $[i,i]\To i$ & unordered pairing\\
69 \cdx{Pair} & $[i,i]\To i$ & ordered pairing\\
70 \cdx{Inf} & $i$ & infinite set\\
71 \cdx{Pow} & $i\To i$ & powerset\\
72 \cdx{Union} \cdx{Inter} & $i\To i$ & set union/intersection \\
73 \cdx{split} & $[[i,i]\To i, i] \To i$ & generalized projection\\
74 \cdx{fst} \cdx{snd} & $i\To i$ & projections\\
75 \cdx{converse}& $i\To i$ & converse of a relation\\
76 \cdx{succ} & $i\To i$ & successor\\
77 \cdx{Collect} & $[i,i\To o]\To i$ & separation\\
78 \cdx{Replace} & $[i, [i,i]\To o] \To i$ & replacement\\
79 \cdx{PrimReplace} & $[i, [i,i]\To o] \To i$ & primitive replacement\\
80 \cdx{RepFun} & $[i, i\To i] \To i$ & functional replacement\\
81 \cdx{Pi} \cdx{Sigma} & $[i,i\To i]\To i$ & general product/sum\\
82 \cdx{domain} & $i\To i$ & domain of a relation\\
83 \cdx{range} & $i\To i$ & range of a relation\\
84 \cdx{field} & $i\To i$ & field of a relation\\
85 \cdx{Lambda} & $[i, i\To i]\To i$ & $\lambda$-abstraction\\
86 \cdx{restrict}& $[i, i] \To i$ & restriction of a function\\
87 \cdx{The} & $[i\To o]\To i$ & definite description\\
88 \cdx{if} & $[o,i,i]\To i$ & conditional\\
89 \cdx{Ball} \cdx{Bex} & $[i, i\To o]\To o$ & bounded quantifiers
92 \subcaption{Constants}
96 \index{*"-"`"` symbol}
97 \index{*"` symbol}\index{function applications!in \ZF}
101 \begin{tabular}{rrrr}
102 \it symbol & \it meta-type & \it priority & \it description \\
103 \tt `` & $[i,i]\To i$ & Left 90 & image \\
104 \tt -`` & $[i,i]\To i$ & Left 90 & inverse image \\
105 \tt ` & $[i,i]\To i$ & Left 90 & application \\
106 \sdx{Int} & $[i,i]\To i$ & Left 70 & intersection ($\int$) \\
107 \sdx{Un} & $[i,i]\To i$ & Left 65 & union ($\un$) \\
108 \tt - & $[i,i]\To i$ & Left 65 & set difference ($-$) \\[1ex]
109 \tt: & $[i,i]\To o$ & Left 50 & membership ($\in$) \\
110 \tt <= & $[i,i]\To o$ & Left 50 & subset ($\subseteq$)
114 \caption{Constants of {\ZF}} \label{zf-constants}
118 \section{The syntax of set theory}
119 The language of set theory, as studied by logicians, has no constants. The
120 traditional axioms merely assert the existence of empty sets, unions,
121 powersets, etc.; this would be intolerable for practical reasoning. The
122 Isabelle theory declares constants for primitive sets. It also extends
123 \texttt{FOL} with additional syntax for finite sets, ordered pairs,
124 comprehension, general union/intersection, general sums/products, and
125 bounded quantifiers. In most other respects, Isabelle implements precisely
126 Zermelo-Fraenkel set theory.
128 Figure~\ref{zf-constants} lists the constants and infixes of~\ZF, while
129 Figure~\ref{zf-trans} presents the syntax translations. Finally,
130 Figure~\ref{zf-syntax} presents the full grammar for set theory, including
131 the constructs of \FOL.
133 Local abbreviations can be introduced by a \texttt{let} construct whose
134 syntax appears in Fig.\ts\ref{zf-syntax}. Internally it is translated into
135 the constant~\cdx{Let}. It can be expanded by rewriting with its
136 definition, \tdx{Let_def}.
138 Apart from \texttt{let}, set theory does not use polymorphism. All terms in
139 {\ZF} have type~\tydx{i}, which is the type of individuals and has class~{\tt
140 term}. The type of first-order formulae, remember, is~\textit{o}.
142 Infix operators include binary union and intersection ($A\un B$ and
143 $A\int B$), set difference ($A-B$), and the subset and membership
144 relations. Note that $a$\verb|~:|$b$ is translated to $\neg(a\in b)$. The
145 union and intersection operators ($\bigcup A$ and $\bigcap A$) form the
146 union or intersection of a set of sets; $\bigcup A$ means the same as
147 $\bigcup@{x\in A}x$. Of these operators, only $\bigcup A$ is primitive.
149 The constant \cdx{Upair} constructs unordered pairs; thus {\tt
150 Upair($A$,$B$)} denotes the set~$\{A,B\}$ and \texttt{Upair($A$,$A$)}
151 denotes the singleton~$\{A\}$. General union is used to define binary
152 union. The Isabelle version goes on to define the constant
155 A\cup B & \equiv & \bigcup(\texttt{Upair}(A,B)) \\
156 \texttt{cons}(a,B) & \equiv & \texttt{Upair}(a,a) \un B
158 The $\{a@1, \ldots\}$ notation abbreviates finite sets constructed in the
159 obvious manner using~\texttt{cons} and~$\emptyset$ (the empty set):
161 \{a,b,c\} & \equiv & \texttt{cons}(a,\texttt{cons}(b,\texttt{cons}(c,\emptyset)))
164 The constant \cdx{Pair} constructs ordered pairs, as in {\tt
165 Pair($a$,$b$)}. Ordered pairs may also be written within angle brackets,
166 as {\tt<$a$,$b$>}. The $n$-tuple {\tt<$a@1$,\ldots,$a@{n-1}$,$a@n$>}
167 abbreviates the nest of pairs\par\nobreak
168 \centerline{\texttt{Pair($a@1$,\ldots,Pair($a@{n-1}$,$a@n$)\ldots).}}
170 In {\ZF}, a function is a set of pairs. A {\ZF} function~$f$ is simply an
171 individual as far as Isabelle is concerned: its Isabelle type is~$i$, not
172 say $i\To i$. The infix operator~{\tt`} denotes the application of a
173 function set to its argument; we must write~$f{\tt`}x$, not~$f(x)$. The
174 syntax for image is~$f{\tt``}A$ and that for inverse image is~$f{\tt-``}A$.
178 \index{lambda abs@$\lambda$-abstractions!in \ZF}
181 \begin{center} \footnotesize\tt\frenchspacing
183 \it external & \it internal & \it description \\
184 $a$ \ttilde: $b$ & \ttilde($a$ : $b$) & \rm negated membership\\
185 \ttlbrace$a@1$, $\ldots$, $a@n$\ttrbrace & cons($a@1$,$\ldots$,cons($a@n$,0)) &
187 <$a@1$, $\ldots$, $a@{n-1}$, $a@n$> &
188 Pair($a@1$,\ldots,Pair($a@{n-1}$,$a@n$)\ldots) &
189 \rm ordered $n$-tuple \\
190 \ttlbrace$x$:$A . P[x]$\ttrbrace & Collect($A$,$\lambda x. P[x]$) &
192 \ttlbrace$y . x$:$A$, $Q[x,y]$\ttrbrace & Replace($A$,$\lambda x\,y. Q[x,y]$) &
194 \ttlbrace$b[x] . x$:$A$\ttrbrace & RepFun($A$,$\lambda x. b[x]$) &
195 \rm functional replacement \\
196 \sdx{INT} $x$:$A . B[x]$ & Inter(\ttlbrace$B[x] . x$:$A$\ttrbrace) &
197 \rm general intersection \\
198 \sdx{UN} $x$:$A . B[x]$ & Union(\ttlbrace$B[x] . x$:$A$\ttrbrace) &
200 \sdx{PROD} $x$:$A . B[x]$ & Pi($A$,$\lambda x. B[x]$) &
201 \rm general product \\
202 \sdx{SUM} $x$:$A . B[x]$ & Sigma($A$,$\lambda x. B[x]$) &
204 $A$ -> $B$ & Pi($A$,$\lambda x. B$) &
205 \rm function space \\
206 $A$ * $B$ & Sigma($A$,$\lambda x. B$) &
207 \rm binary product \\
208 \sdx{THE} $x . P[x]$ & The($\lambda x. P[x]$) &
209 \rm definite description \\
210 \sdx{lam} $x$:$A . b[x]$ & Lambda($A$,$\lambda x. b[x]$) &
211 \rm $\lambda$-abstraction\\[1ex]
212 \sdx{ALL} $x$:$A . P[x]$ & Ball($A$,$\lambda x. P[x]$) &
213 \rm bounded $\forall$ \\
214 \sdx{EX} $x$:$A . P[x]$ & Bex($A$,$\lambda x. P[x]$) &
215 \rm bounded $\exists$
218 \caption{Translations for {\ZF}} \label{zf-trans}
227 term & = & \hbox{expression of type~$i$} \\
228 & | & "let"~id~"="~term";"\dots";"~id~"="~term~"in"~term \\
229 & | & "if"~term~"then"~term~"else"~term \\
230 & | & "{\ttlbrace} " term\; ("," term)^* " {\ttrbrace}" \\
231 & | & "< " term\; ("," term)^* " >" \\
232 & | & "{\ttlbrace} " id ":" term " . " formula " {\ttrbrace}" \\
233 & | & "{\ttlbrace} " id " . " id ":" term ", " formula " {\ttrbrace}" \\
234 & | & "{\ttlbrace} " term " . " id ":" term " {\ttrbrace}" \\
235 & | & term " `` " term \\
236 & | & term " -`` " term \\
237 & | & term " ` " term \\
238 & | & term " * " term \\
239 & | & term " Int " term \\
240 & | & term " Un " term \\
241 & | & term " - " term \\
242 & | & term " -> " term \\
243 & | & "THE~~" id " . " formula\\
244 & | & "lam~~" id ":" term " . " term \\
245 & | & "INT~~" id ":" term " . " term \\
246 & | & "UN~~~" id ":" term " . " term \\
247 & | & "PROD~" id ":" term " . " term \\
248 & | & "SUM~~" id ":" term " . " term \\[2ex]
249 formula & = & \hbox{expression of type~$o$} \\
250 & | & term " : " term \\
251 & | & term " \ttilde: " term \\
252 & | & term " <= " term \\
253 & | & term " = " term \\
254 & | & term " \ttilde= " term \\
255 & | & "\ttilde\ " formula \\
256 & | & formula " \& " formula \\
257 & | & formula " | " formula \\
258 & | & formula " --> " formula \\
259 & | & formula " <-> " formula \\
260 & | & "ALL " id ":" term " . " formula \\
261 & | & "EX~~" id ":" term " . " formula \\
262 & | & "ALL~" id~id^* " . " formula \\
263 & | & "EX~~" id~id^* " . " formula \\
264 & | & "EX!~" id~id^* " . " formula
267 \caption{Full grammar for {\ZF}} \label{zf-syntax}
271 \section{Binding operators}
272 The constant \cdx{Collect} constructs sets by the principle of {\bf
273 separation}. The syntax for separation is
274 \hbox{\tt\ttlbrace$x$:$A$.\ $P[x]$\ttrbrace}, where $P[x]$ is a formula
275 that may contain free occurrences of~$x$. It abbreviates the set {\tt
276 Collect($A$,$\lambda x. P[x]$)}, which consists of all $x\in A$ that
277 satisfy~$P[x]$. Note that \texttt{Collect} is an unfortunate choice of
278 name: some set theories adopt a set-formation principle, related to
279 replacement, called collection.
281 The constant \cdx{Replace} constructs sets by the principle of {\bf
282 replacement}. The syntax
283 \hbox{\tt\ttlbrace$y$.\ $x$:$A$,$Q[x,y]$\ttrbrace} denotes the set {\tt
284 Replace($A$,$\lambda x\,y. Q[x,y]$)}, which consists of all~$y$ such
285 that there exists $x\in A$ satisfying~$Q[x,y]$. The Replacement Axiom
286 has the condition that $Q$ must be single-valued over~$A$: for
287 all~$x\in A$ there exists at most one $y$ satisfying~$Q[x,y]$. A
288 single-valued binary predicate is also called a {\bf class function}.
290 The constant \cdx{RepFun} expresses a special case of replacement,
291 where $Q[x,y]$ has the form $y=b[x]$. Such a $Q$ is trivially
292 single-valued, since it is just the graph of the meta-level
293 function~$\lambda x. b[x]$. The resulting set consists of all $b[x]$
294 for~$x\in A$. This is analogous to the \ML{} functional \texttt{map},
295 since it applies a function to every element of a set. The syntax is
296 \hbox{\tt\ttlbrace$b[x]$.\ $x$:$A$\ttrbrace}, which expands to {\tt
297 RepFun($A$,$\lambda x. b[x]$)}.
299 \index{*INT symbol}\index{*UN symbol}
300 General unions and intersections of indexed
301 families of sets, namely $\bigcup@{x\in A}B[x]$ and $\bigcap@{x\in A}B[x]$,
302 are written \hbox{\tt UN $x$:$A$.\ $B[x]$} and \hbox{\tt INT $x$:$A$.\ $B[x]$}.
303 Their meaning is expressed using \texttt{RepFun} as
305 \bigcup(\{B[x]. x\in A\}) \qquad\hbox{and}\qquad
306 \bigcap(\{B[x]. x\in A\}).
308 General sums $\sum@{x\in A}B[x]$ and products $\prod@{x\in A}B[x]$ can be
309 constructed in set theory, where $B[x]$ is a family of sets over~$A$. They
310 have as special cases $A\times B$ and $A\to B$, where $B$ is simply a set.
311 This is similar to the situation in Constructive Type Theory (set theory
312 has `dependent sets') and calls for similar syntactic conventions. The
313 constants~\cdx{Sigma} and~\cdx{Pi} construct general sums and
314 products. Instead of \texttt{Sigma($A$,$B$)} and \texttt{Pi($A$,$B$)} we may
316 \hbox{\tt SUM $x$:$A$.\ $B[x]$} and \hbox{\tt PROD $x$:$A$.\ $B[x]$}.
317 \index{*SUM symbol}\index{*PROD symbol}%
318 The special cases as \hbox{\tt$A$*$B$} and \hbox{\tt$A$->$B$} abbreviate
319 general sums and products over a constant family.\footnote{Unlike normal
320 infix operators, {\tt*} and {\tt->} merely define abbreviations; there are
321 no constants~\texttt{op~*} and~\hbox{\tt op~->}.} Isabelle accepts these
322 abbreviations in parsing and uses them whenever possible for printing.
325 As mentioned above, whenever the axioms assert the existence and uniqueness
326 of a set, Isabelle's set theory declares a constant for that set. These
327 constants can express the {\bf definite description} operator~$\iota
328 x. P[x]$, which stands for the unique~$a$ satisfying~$P[a]$, if such exists.
329 Since all terms in {\ZF} denote something, a description is always
330 meaningful, but we do not know its value unless $P[x]$ defines it uniquely.
331 Using the constant~\cdx{The}, we may write descriptions as {\tt
332 The($\lambda x. P[x]$)} or use the syntax \hbox{\tt THE $x$.\ $P[x]$}.
335 Function sets may be written in $\lambda$-notation; $\lambda x\in A. b[x]$
336 stands for the set of all pairs $\pair{x,b[x]}$ for $x\in A$. In order for
337 this to be a set, the function's domain~$A$ must be given. Using the
338 constant~\cdx{Lambda}, we may express function sets as {\tt
339 Lambda($A$,$\lambda x. b[x]$)} or use the syntax \hbox{\tt lam $x$:$A$.\ $b[x]$}.
341 Isabelle's set theory defines two {\bf bounded quantifiers}:
343 \forall x\in A. P[x] &\hbox{abbreviates}& \forall x. x\in A\imp P[x] \\
344 \exists x\in A. P[x] &\hbox{abbreviates}& \exists x. x\in A\conj P[x]
346 The constants~\cdx{Ball} and~\cdx{Bex} are defined
347 accordingly. Instead of \texttt{Ball($A$,$P$)} and \texttt{Bex($A$,$P$)} we may
349 \hbox{\tt ALL $x$:$A$.\ $P[x]$} and \hbox{\tt EX $x$:$A$.\ $P[x]$}.
356 \tdx{Let_def} Let(s, f) == f(s)
358 \tdx{Ball_def} Ball(A,P) == ALL x. x:A --> P(x)
359 \tdx{Bex_def} Bex(A,P) == EX x. x:A & P(x)
361 \tdx{subset_def} A <= B == ALL x:A. x:B
362 \tdx{extension} A = B <-> A <= B & B <= A
364 \tdx{Union_iff} A : Union(C) <-> (EX B:C. A:B)
365 \tdx{Pow_iff} A : Pow(B) <-> A <= B
366 \tdx{foundation} A=0 | (EX x:A. ALL y:x. ~ y:A)
368 \tdx{replacement} (ALL x:A. ALL y z. P(x,y) & P(x,z) --> y=z) ==>
369 b : PrimReplace(A,P) <-> (EX x:A. P(x,b))
370 \subcaption{The Zermelo-Fraenkel Axioms}
372 \tdx{Replace_def} Replace(A,P) ==
373 PrimReplace(A, \%x y. (EX!z. P(x,z)) & P(x,y))
374 \tdx{RepFun_def} RepFun(A,f) == {\ttlbrace}y . x:A, y=f(x)\ttrbrace
375 \tdx{the_def} The(P) == Union({\ttlbrace}y . x:{\ttlbrace}0{\ttrbrace}, P(y){\ttrbrace})
376 \tdx{if_def} if(P,a,b) == THE z. P & z=a | ~P & z=b
377 \tdx{Collect_def} Collect(A,P) == {\ttlbrace}y . x:A, x=y & P(x){\ttrbrace}
378 \tdx{Upair_def} Upair(a,b) ==
379 {\ttlbrace}y. x:Pow(Pow(0)), (x=0 & y=a) | (x=Pow(0) & y=b){\ttrbrace}
380 \subcaption{Consequences of replacement}
382 \tdx{Inter_def} Inter(A) == {\ttlbrace}x:Union(A) . ALL y:A. x:y{\ttrbrace}
383 \tdx{Un_def} A Un B == Union(Upair(A,B))
384 \tdx{Int_def} A Int B == Inter(Upair(A,B))
385 \tdx{Diff_def} A - B == {\ttlbrace}x:A . x~:B{\ttrbrace}
386 \subcaption{Union, intersection, difference}
388 \caption{Rules and axioms of {\ZF}} \label{zf-rules}
394 \tdx{cons_def} cons(a,A) == Upair(a,a) Un A
395 \tdx{succ_def} succ(i) == cons(i,i)
396 \tdx{infinity} 0:Inf & (ALL y:Inf. succ(y): Inf)
397 \subcaption{Finite and infinite sets}
399 \tdx{Pair_def} <a,b> == {\ttlbrace}{\ttlbrace}a,a{\ttrbrace}, {\ttlbrace}a,b{\ttrbrace}{\ttrbrace}
400 \tdx{split_def} split(c,p) == THE y. EX a b. p=<a,b> & y=c(a,b)
401 \tdx{fst_def} fst(A) == split(\%x y. x, p)
402 \tdx{snd_def} snd(A) == split(\%x y. y, p)
403 \tdx{Sigma_def} Sigma(A,B) == UN x:A. UN y:B(x). {\ttlbrace}<x,y>{\ttrbrace}
404 \subcaption{Ordered pairs and Cartesian products}
406 \tdx{converse_def} converse(r) == {\ttlbrace}z. w:r, EX x y. w=<x,y> & z=<y,x>{\ttrbrace}
407 \tdx{domain_def} domain(r) == {\ttlbrace}x. w:r, EX y. w=<x,y>{\ttrbrace}
408 \tdx{range_def} range(r) == domain(converse(r))
409 \tdx{field_def} field(r) == domain(r) Un range(r)
410 \tdx{image_def} r `` A == {\ttlbrace}y : range(r) . EX x:A. <x,y> : r{\ttrbrace}
411 \tdx{vimage_def} r -`` A == converse(r)``A
412 \subcaption{Operations on relations}
414 \tdx{lam_def} Lambda(A,b) == {\ttlbrace}<x,b(x)> . x:A{\ttrbrace}
415 \tdx{apply_def} f`a == THE y. <a,y> : f
416 \tdx{Pi_def} Pi(A,B) == {\ttlbrace}f: Pow(Sigma(A,B)). ALL x:A. EX! y. <x,y>: f{\ttrbrace}
417 \tdx{restrict_def} restrict(f,A) == lam x:A. f`x
418 \subcaption{Functions and general product}
420 \caption{Further definitions of {\ZF}} \label{zf-defs}
425 \section{The Zermelo-Fraenkel axioms}
426 The axioms appear in Fig.\ts \ref{zf-rules}. They resemble those
427 presented by Suppes~\cite{suppes72}. Most of the theory consists of
428 definitions. In particular, bounded quantifiers and the subset relation
429 appear in other axioms. Object-level quantifiers and implications have
430 been replaced by meta-level ones wherever possible, to simplify use of the
431 axioms. See the file \texttt{ZF/ZF.thy} for details.
433 The traditional replacement axiom asserts
434 \[ y \in \texttt{PrimReplace}(A,P) \bimp (\exists x\in A. P(x,y)) \]
435 subject to the condition that $P(x,y)$ is single-valued for all~$x\in A$.
436 The Isabelle theory defines \cdx{Replace} to apply
437 \cdx{PrimReplace} to the single-valued part of~$P$, namely
438 \[ (\exists!z. P(x,z)) \conj P(x,y). \]
439 Thus $y\in \texttt{Replace}(A,P)$ if and only if there is some~$x$ such that
440 $P(x,-)$ holds uniquely for~$y$. Because the equivalence is unconditional,
441 \texttt{Replace} is much easier to use than \texttt{PrimReplace}; it defines the
442 same set, if $P(x,y)$ is single-valued. The nice syntax for replacement
443 expands to \texttt{Replace}.
445 Other consequences of replacement include functional replacement
446 (\cdx{RepFun}) and definite descriptions (\cdx{The}).
447 Axioms for separation (\cdx{Collect}) and unordered pairs
448 (\cdx{Upair}) are traditionally assumed, but they actually follow
449 from replacement~\cite[pages 237--8]{suppes72}.
451 The definitions of general intersection, etc., are straightforward. Note
452 the definition of \texttt{cons}, which underlies the finite set notation.
453 The axiom of infinity gives us a set that contains~0 and is closed under
454 successor (\cdx{succ}). Although this set is not uniquely defined,
455 the theory names it (\cdx{Inf}) in order to simplify the
456 construction of the natural numbers.
458 Further definitions appear in Fig.\ts\ref{zf-defs}. Ordered pairs are
459 defined in the standard way, $\pair{a,b}\equiv\{\{a\},\{a,b\}\}$. Recall
460 that \cdx{Sigma}$(A,B)$ generalizes the Cartesian product of two
461 sets. It is defined to be the union of all singleton sets
462 $\{\pair{x,y}\}$, for $x\in A$ and $y\in B(x)$. This is a typical usage of
465 The projections \cdx{fst} and~\cdx{snd} are defined in terms of the
466 generalized projection \cdx{split}. The latter has been borrowed from
467 Martin-L\"of's Type Theory, and is often easier to use than \cdx{fst}
470 Operations on relations include converse, domain, range, and image. The
471 set ${\tt Pi}(A,B)$ generalizes the space of functions between two sets.
472 Note the simple definitions of $\lambda$-abstraction (using
473 \cdx{RepFun}) and application (using a definite description). The
474 function \cdx{restrict}$(f,A)$ has the same values as~$f$, but only
482 \tdx{ballI} [| !!x. x:A ==> P(x) |] ==> ALL x:A. P(x)
483 \tdx{bspec} [| ALL x:A. P(x); x: A |] ==> P(x)
484 \tdx{ballE} [| ALL x:A. P(x); P(x) ==> Q; ~ x:A ==> Q |] ==> Q
486 \tdx{ball_cong} [| A=A'; !!x. x:A' ==> P(x) <-> P'(x) |] ==>
487 (ALL x:A. P(x)) <-> (ALL x:A'. P'(x))
489 \tdx{bexI} [| P(x); x: A |] ==> EX x:A. P(x)
490 \tdx{bexCI} [| ALL x:A. ~P(x) ==> P(a); a: A |] ==> EX x:A. P(x)
491 \tdx{bexE} [| EX x:A. P(x); !!x. [| x:A; P(x) |] ==> Q |] ==> Q
493 \tdx{bex_cong} [| A=A'; !!x. x:A' ==> P(x) <-> P'(x) |] ==>
494 (EX x:A. P(x)) <-> (EX x:A'. P'(x))
495 \subcaption{Bounded quantifiers}
497 \tdx{subsetI} (!!x. x:A ==> x:B) ==> A <= B
498 \tdx{subsetD} [| A <= B; c:A |] ==> c:B
499 \tdx{subsetCE} [| A <= B; ~(c:A) ==> P; c:B ==> P |] ==> P
500 \tdx{subset_refl} A <= A
501 \tdx{subset_trans} [| A<=B; B<=C |] ==> A<=C
503 \tdx{equalityI} [| A <= B; B <= A |] ==> A = B
504 \tdx{equalityD1} A = B ==> A<=B
505 \tdx{equalityD2} A = B ==> B<=A
506 \tdx{equalityE} [| A = B; [| A<=B; B<=A |] ==> P |] ==> P
507 \subcaption{Subsets and extensionality}
509 \tdx{emptyE} a:0 ==> P
510 \tdx{empty_subsetI} 0 <= A
511 \tdx{equals0I} [| !!y. y:A ==> False |] ==> A=0
512 \tdx{equals0D} [| A=0; a:A |] ==> P
514 \tdx{PowI} A <= B ==> A : Pow(B)
515 \tdx{PowD} A : Pow(B) ==> A<=B
516 \subcaption{The empty set; power sets}
518 \caption{Basic derived rules for {\ZF}} \label{zf-lemmas1}
522 \section{From basic lemmas to function spaces}
523 Faced with so many definitions, it is essential to prove lemmas. Even
524 trivial theorems like $A \int B = B \int A$ would be difficult to
525 prove from the definitions alone. Isabelle's set theory derives many
526 rules using a natural deduction style. Ideally, a natural deduction
527 rule should introduce or eliminate just one operator, but this is not
528 always practical. For most operators, we may forget its definition
529 and use its derived rules instead.
531 \subsection{Fundamental lemmas}
532 Figure~\ref{zf-lemmas1} presents the derived rules for the most basic
533 operators. The rules for the bounded quantifiers resemble those for the
534 ordinary quantifiers, but note that \tdx{ballE} uses a negated assumption
535 in the style of Isabelle's classical reasoner. The \rmindex{congruence
536 rules} \tdx{ball_cong} and \tdx{bex_cong} are required by Isabelle's
537 simplifier, but have few other uses. Congruence rules must be specially
538 derived for all binding operators, and henceforth will not be shown.
540 Figure~\ref{zf-lemmas1} also shows rules for the subset and equality
541 relations (proof by extensionality), and rules about the empty set and the
544 Figure~\ref{zf-lemmas2} presents rules for replacement and separation.
545 The rules for \cdx{Replace} and \cdx{RepFun} are much simpler than
546 comparable rules for \texttt{PrimReplace} would be. The principle of
547 separation is proved explicitly, although most proofs should use the
548 natural deduction rules for \texttt{Collect}. The elimination rule
549 \tdx{CollectE} is equivalent to the two destruction rules
550 \tdx{CollectD1} and \tdx{CollectD2}, but each rule is suited to
551 particular circumstances. Although too many rules can be confusing, there
552 is no reason to aim for a minimal set of rules. See the file
553 \texttt{ZF/ZF.ML} for a complete listing.
555 Figure~\ref{zf-lemmas3} presents rules for general union and intersection.
556 The empty intersection should be undefined. We cannot have
557 $\bigcap(\emptyset)=V$ because $V$, the universal class, is not a set. All
558 expressions denote something in {\ZF} set theory; the definition of
559 intersection implies $\bigcap(\emptyset)=\emptyset$, but this value is
560 arbitrary. The rule \tdx{InterI} must have a premise to exclude
561 the empty intersection. Some of the laws governing intersections require
565 %the [p] gives better page breaking for the book
568 \tdx{ReplaceI} [| x: A; P(x,b); !!y. P(x,y) ==> y=b |] ==>
569 b : {\ttlbrace}y. x:A, P(x,y){\ttrbrace}
571 \tdx{ReplaceE} [| b : {\ttlbrace}y. x:A, P(x,y){\ttrbrace};
572 !!x. [| x: A; P(x,b); ALL y. P(x,y)-->y=b |] ==> R
575 \tdx{RepFunI} [| a : A |] ==> f(a) : {\ttlbrace}f(x). x:A{\ttrbrace}
576 \tdx{RepFunE} [| b : {\ttlbrace}f(x). x:A{\ttrbrace};
577 !!x.[| x:A; b=f(x) |] ==> P |] ==> P
579 \tdx{separation} a : {\ttlbrace}x:A. P(x){\ttrbrace} <-> a:A & P(a)
580 \tdx{CollectI} [| a:A; P(a) |] ==> a : {\ttlbrace}x:A. P(x){\ttrbrace}
581 \tdx{CollectE} [| a : {\ttlbrace}x:A. P(x){\ttrbrace}; [| a:A; P(a) |] ==> R |] ==> R
582 \tdx{CollectD1} a : {\ttlbrace}x:A. P(x){\ttrbrace} ==> a:A
583 \tdx{CollectD2} a : {\ttlbrace}x:A. P(x){\ttrbrace} ==> P(a)
585 \caption{Replacement and separation} \label{zf-lemmas2}
591 \tdx{UnionI} [| B: C; A: B |] ==> A: Union(C)
592 \tdx{UnionE} [| A : Union(C); !!B.[| A: B; B: C |] ==> R |] ==> R
594 \tdx{InterI} [| !!x. x: C ==> A: x; c:C |] ==> A : Inter(C)
595 \tdx{InterD} [| A : Inter(C); B : C |] ==> A : B
596 \tdx{InterE} [| A : Inter(C); A:B ==> R; ~ B:C ==> R |] ==> R
598 \tdx{UN_I} [| a: A; b: B(a) |] ==> b: (UN x:A. B(x))
599 \tdx{UN_E} [| b : (UN x:A. B(x)); !!x.[| x: A; b: B(x) |] ==> R
602 \tdx{INT_I} [| !!x. x: A ==> b: B(x); a: A |] ==> b: (INT x:A. B(x))
603 \tdx{INT_E} [| b : (INT x:A. B(x)); a: A |] ==> b : B(a)
605 \caption{General union and intersection} \label{zf-lemmas3}
613 \tdx{pairing} a:Upair(b,c) <-> (a=b | a=c)
614 \tdx{UpairI1} a : Upair(a,b)
615 \tdx{UpairI2} b : Upair(a,b)
616 \tdx{UpairE} [| a : Upair(b,c); a = b ==> P; a = c ==> P |] ==> P
618 \caption{Unordered pairs} \label{zf-upair1}
624 \tdx{UnI1} c : A ==> c : A Un B
625 \tdx{UnI2} c : B ==> c : A Un B
626 \tdx{UnCI} (~c : B ==> c : A) ==> c : A Un B
627 \tdx{UnE} [| c : A Un B; c:A ==> P; c:B ==> P |] ==> P
629 \tdx{IntI} [| c : A; c : B |] ==> c : A Int B
630 \tdx{IntD1} c : A Int B ==> c : A
631 \tdx{IntD2} c : A Int B ==> c : B
632 \tdx{IntE} [| c : A Int B; [| c:A; c:B |] ==> P |] ==> P
634 \tdx{DiffI} [| c : A; ~ c : B |] ==> c : A - B
635 \tdx{DiffD1} c : A - B ==> c : A
636 \tdx{DiffD2} c : A - B ==> c ~: B
637 \tdx{DiffE} [| c : A - B; [| c:A; ~ c:B |] ==> P |] ==> P
639 \caption{Union, intersection, difference} \label{zf-Un}
645 \tdx{consI1} a : cons(a,B)
646 \tdx{consI2} a : B ==> a : cons(b,B)
647 \tdx{consCI} (~ a:B ==> a=b) ==> a: cons(b,B)
648 \tdx{consE} [| a : cons(b,A); a=b ==> P; a:A ==> P |] ==> P
650 \tdx{singletonI} a : {\ttlbrace}a{\ttrbrace}
651 \tdx{singletonE} [| a : {\ttlbrace}b{\ttrbrace}; a=b ==> P |] ==> P
653 \caption{Finite and singleton sets} \label{zf-upair2}
659 \tdx{succI1} i : succ(i)
660 \tdx{succI2} i : j ==> i : succ(j)
661 \tdx{succCI} (~ i:j ==> i=j) ==> i: succ(j)
662 \tdx{succE} [| i : succ(j); i=j ==> P; i:j ==> P |] ==> P
663 \tdx{succ_neq_0} [| succ(n)=0 |] ==> P
664 \tdx{succ_inject} succ(m) = succ(n) ==> m=n
666 \caption{The successor function} \label{zf-succ}
672 \tdx{the_equality} [| P(a); !!x. P(x) ==> x=a |] ==> (THE x. P(x)) = a
673 \tdx{theI} EX! x. P(x) ==> P(THE x. P(x))
675 \tdx{if_P} P ==> (if P then a else b) = a
676 \tdx{if_not_P} ~P ==> (if P then a else b) = b
678 \tdx{mem_asym} [| a:b; b:a |] ==> P
679 \tdx{mem_irrefl} a:a ==> P
681 \caption{Descriptions; non-circularity} \label{zf-the}
685 \subsection{Unordered pairs and finite sets}
686 Figure~\ref{zf-upair1} presents the principle of unordered pairing, along
687 with its derived rules. Binary union and intersection are defined in terms
688 of ordered pairs (Fig.\ts\ref{zf-Un}). Set difference is also included. The
689 rule \tdx{UnCI} is useful for classical reasoning about unions,
690 like \texttt{disjCI}\@; it supersedes \tdx{UnI1} and
691 \tdx{UnI2}, but these rules are often easier to work with. For
692 intersection and difference we have both elimination and destruction rules.
693 Again, there is no reason to provide a minimal rule set.
695 Figure~\ref{zf-upair2} is concerned with finite sets: it presents rules
696 for~\texttt{cons}, the finite set constructor, and rules for singleton
697 sets. Figure~\ref{zf-succ} presents derived rules for the successor
698 function, which is defined in terms of~\texttt{cons}. The proof that {\tt
699 succ} is injective appears to require the Axiom of Foundation.
701 Definite descriptions (\sdx{THE}) are defined in terms of the singleton
702 set~$\{0\}$, but their derived rules fortunately hide this
703 (Fig.\ts\ref{zf-the}). The rule~\tdx{theI} is difficult to apply
704 because of the two occurrences of~$\Var{P}$. However,
705 \tdx{the_equality} does not have this problem and the files contain
706 many examples of its use.
708 Finally, the impossibility of having both $a\in b$ and $b\in a$
709 (\tdx{mem_asym}) is proved by applying the Axiom of Foundation to
710 the set $\{a,b\}$. The impossibility of $a\in a$ is a trivial consequence.
712 See the file \texttt{ZF/upair.ML} for full proofs of the rules discussed in
720 \tdx{Union_upper} B:A ==> B <= Union(A)
721 \tdx{Union_least} [| !!x. x:A ==> x<=C |] ==> Union(A) <= C
723 \tdx{Inter_lower} B:A ==> Inter(A) <= B
724 \tdx{Inter_greatest} [| a:A; !!x. x:A ==> C<=x |] ==> C <= Inter(A)
726 \tdx{Un_upper1} A <= A Un B
727 \tdx{Un_upper2} B <= A Un B
728 \tdx{Un_least} [| A<=C; B<=C |] ==> A Un B <= C
730 \tdx{Int_lower1} A Int B <= A
731 \tdx{Int_lower2} A Int B <= B
732 \tdx{Int_greatest} [| C<=A; C<=B |] ==> C <= A Int B
734 \tdx{Diff_subset} A-B <= A
735 \tdx{Diff_contains} [| C<=A; C Int B = 0 |] ==> C <= A-B
737 \tdx{Collect_subset} Collect(A,P) <= A
739 \caption{Subset and lattice properties} \label{zf-subset}
743 \subsection{Subset and lattice properties}
744 The subset relation is a complete lattice. Unions form least upper bounds;
745 non-empty intersections form greatest lower bounds. Figure~\ref{zf-subset}
746 shows the corresponding rules. A few other laws involving subsets are
747 included. Proofs are in the file \texttt{ZF/subset.ML}.
749 Reasoning directly about subsets often yields clearer proofs than
750 reasoning about the membership relation. Section~\ref{sec:ZF-pow-example}
751 below presents an example of this, proving the equation ${{\tt Pow}(A)\cap
752 {\tt Pow}(B)}= {\tt Pow}(A\cap B)$.
758 \tdx{Pair_inject1} <a,b> = <c,d> ==> a=c
759 \tdx{Pair_inject2} <a,b> = <c,d> ==> b=d
760 \tdx{Pair_inject} [| <a,b> = <c,d>; [| a=c; b=d |] ==> P |] ==> P
761 \tdx{Pair_neq_0} <a,b>=0 ==> P
763 \tdx{fst_conv} fst(<a,b>) = a
764 \tdx{snd_conv} snd(<a,b>) = b
765 \tdx{split} split(\%x y. c(x,y), <a,b>) = c(a,b)
767 \tdx{SigmaI} [| a:A; b:B(a) |] ==> <a,b> : Sigma(A,B)
769 \tdx{SigmaE} [| c: Sigma(A,B);
770 !!x y.[| x:A; y:B(x); c=<x,y> |] ==> P |] ==> P
772 \tdx{SigmaE2} [| <a,b> : Sigma(A,B);
773 [| a:A; b:B(a) |] ==> P |] ==> P
775 \caption{Ordered pairs; projections; general sums} \label{zf-pair}
779 \subsection{Ordered pairs} \label{sec:pairs}
781 Figure~\ref{zf-pair} presents the rules governing ordered pairs,
782 projections and general sums. File \texttt{ZF/pair.ML} contains the
783 full (and tedious) proof that $\{\{a\},\{a,b\}\}$ functions as an ordered
784 pair. This property is expressed as two destruction rules,
785 \tdx{Pair_inject1} and \tdx{Pair_inject2}, and equivalently
786 as the elimination rule \tdx{Pair_inject}.
788 The rule \tdx{Pair_neq_0} asserts $\pair{a,b}\neq\emptyset$. This
789 is a property of $\{\{a\},\{a,b\}\}$, and need not hold for other
790 encodings of ordered pairs. The non-standard ordered pairs mentioned below
791 satisfy $\pair{\emptyset;\emptyset}=\emptyset$.
793 The natural deduction rules \tdx{SigmaI} and \tdx{SigmaE}
794 assert that \cdx{Sigma}$(A,B)$ consists of all pairs of the form
795 $\pair{x,y}$, for $x\in A$ and $y\in B(x)$. The rule \tdx{SigmaE2}
796 merely states that $\pair{a,b}\in \texttt{Sigma}(A,B)$ implies $a\in A$ and
799 In addition, it is possible to use tuples as patterns in abstractions:
801 {\tt\%<$x$,$y$>. $t$} \quad stands for\quad \texttt{split(\%$x$ $y$.\ $t$)}
803 Nested patterns are translated recursively:
804 {\tt\%<$x$,$y$,$z$>. $t$} $\leadsto$ {\tt\%<$x$,<$y$,$z$>>. $t$} $\leadsto$
805 \texttt{split(\%$x$.\%<$y$,$z$>. $t$)} $\leadsto$ \texttt{split(\%$x$. split(\%$y$
806 $z$.\ $t$))}. The reverse translation is performed upon printing.
808 The translation between patterns and \texttt{split} is performed automatically
809 by the parser and printer. Thus the internal and external form of a term
810 may differ, which affects proofs. For example the term {\tt
811 (\%<x,y>.<y,x>)<a,b>} requires the theorem \texttt{split} to rewrite to
814 In addition to explicit $\lambda$-abstractions, patterns can be used in any
815 variable binding construct which is internally described by a
816 $\lambda$-abstraction. Here are some important examples:
818 \item[Let:] \texttt{let {\it pattern} = $t$ in $u$}
819 \item[Choice:] \texttt{THE~{\it pattern}~.~$P$}
820 \item[Set operations:] \texttt{UN~{\it pattern}:$A$.~$B$}
821 \item[Comprehension:] \texttt{{\ttlbrace}~{\it pattern}:$A$~.~$P$~{\ttrbrace}}
829 \tdx{domainI} <a,b>: r ==> a : domain(r)
830 \tdx{domainE} [| a : domain(r); !!y. <a,y>: r ==> P |] ==> P
831 \tdx{domain_subset} domain(Sigma(A,B)) <= A
833 \tdx{rangeI} <a,b>: r ==> b : range(r)
834 \tdx{rangeE} [| b : range(r); !!x. <x,b>: r ==> P |] ==> P
835 \tdx{range_subset} range(A*B) <= B
837 \tdx{fieldI1} <a,b>: r ==> a : field(r)
838 \tdx{fieldI2} <a,b>: r ==> b : field(r)
839 \tdx{fieldCI} (~ <c,a>:r ==> <a,b>: r) ==> a : field(r)
841 \tdx{fieldE} [| a : field(r);
846 \tdx{field_subset} field(A*A) <= A
848 \caption{Domain, range and field of a relation} \label{zf-domrange}
853 \tdx{imageI} [| <a,b>: r; a:A |] ==> b : r``A
854 \tdx{imageE} [| b: r``A; !!x.[| <x,b>: r; x:A |] ==> P |] ==> P
856 \tdx{vimageI} [| <a,b>: r; b:B |] ==> a : r-``B
857 \tdx{vimageE} [| a: r-``B; !!x.[| <a,x>: r; x:B |] ==> P |] ==> P
859 \caption{Image and inverse image} \label{zf-domrange2}
863 \subsection{Relations}
864 Figure~\ref{zf-domrange} presents rules involving relations, which are sets
865 of ordered pairs. The converse of a relation~$r$ is the set of all pairs
866 $\pair{y,x}$ such that $\pair{x,y}\in r$; if $r$ is a function, then
867 {\cdx{converse}$(r)$} is its inverse. The rules for the domain
868 operation, namely \tdx{domainI} and~\tdx{domainE}, assert that
869 \cdx{domain}$(r)$ consists of all~$x$ such that $r$ contains
870 some pair of the form~$\pair{x,y}$. The range operation is similar, and
871 the field of a relation is merely the union of its domain and range.
873 Figure~\ref{zf-domrange2} presents rules for images and inverse images.
874 Note that these operations are generalisations of range and domain,
875 respectively. See the file \texttt{ZF/domrange.ML} for derivations of the
883 \tdx{fun_is_rel} f: Pi(A,B) ==> f <= Sigma(A,B)
885 \tdx{apply_equality} [| <a,b>: f; f: Pi(A,B) |] ==> f`a = b
886 \tdx{apply_equality2} [| <a,b>: f; <a,c>: f; f: Pi(A,B) |] ==> b=c
888 \tdx{apply_type} [| f: Pi(A,B); a:A |] ==> f`a : B(a)
889 \tdx{apply_Pair} [| f: Pi(A,B); a:A |] ==> <a,f`a>: f
890 \tdx{apply_iff} f: Pi(A,B) ==> <a,b>: f <-> a:A & f`a = b
892 \tdx{fun_extension} [| f : Pi(A,B); g: Pi(A,D);
893 !!x. x:A ==> f`x = g`x |] ==> f=g
895 \tdx{domain_type} [| <a,b> : f; f: Pi(A,B) |] ==> a : A
896 \tdx{range_type} [| <a,b> : f; f: Pi(A,B) |] ==> b : B(a)
898 \tdx{Pi_type} [| f: A->C; !!x. x:A ==> f`x: B(x) |] ==> f: Pi(A,B)
899 \tdx{domain_of_fun} f: Pi(A,B) ==> domain(f)=A
900 \tdx{range_of_fun} f: Pi(A,B) ==> f: A->range(f)
902 \tdx{restrict} a : A ==> restrict(f,A) ` a = f`a
903 \tdx{restrict_type} [| !!x. x:A ==> f`x: B(x) |] ==>
904 restrict(f,A) : Pi(A,B)
906 \caption{Functions} \label{zf-func1}
912 \tdx{lamI} a:A ==> <a,b(a)> : (lam x:A. b(x))
913 \tdx{lamE} [| p: (lam x:A. b(x)); !!x.[| x:A; p=<x,b(x)> |] ==> P
916 \tdx{lam_type} [| !!x. x:A ==> b(x): B(x) |] ==> (lam x:A. b(x)) : Pi(A,B)
918 \tdx{beta} a : A ==> (lam x:A. b(x)) ` a = b(a)
919 \tdx{eta} f : Pi(A,B) ==> (lam x:A. f`x) = f
921 \caption{$\lambda$-abstraction} \label{zf-lam}
927 \tdx{fun_empty} 0: 0->0
928 \tdx{fun_single} {\ttlbrace}<a,b>{\ttrbrace} : {\ttlbrace}a{\ttrbrace} -> {\ttlbrace}b{\ttrbrace}
930 \tdx{fun_disjoint_Un} [| f: A->B; g: C->D; A Int C = 0 |] ==>
931 (f Un g) : (A Un C) -> (B Un D)
933 \tdx{fun_disjoint_apply1} [| a:A; f: A->B; g: C->D; A Int C = 0 |] ==>
936 \tdx{fun_disjoint_apply2} [| c:C; f: A->B; g: C->D; A Int C = 0 |] ==>
939 \caption{Constructing functions from smaller sets} \label{zf-func2}
943 \subsection{Functions}
944 Functions, represented by graphs, are notoriously difficult to reason
945 about. The file \texttt{ZF/func.ML} derives many rules, which overlap more
946 than they ought. This section presents the more important rules.
948 Figure~\ref{zf-func1} presents the basic properties of \cdx{Pi}$(A,B)$,
949 the generalized function space. For example, if $f$ is a function and
950 $\pair{a,b}\in f$, then $f`a=b$ (\tdx{apply_equality}). Two functions
951 are equal provided they have equal domains and deliver equals results
952 (\tdx{fun_extension}).
954 By \tdx{Pi_type}, a function typing of the form $f\in A\to C$ can be
955 refined to the dependent typing $f\in\prod@{x\in A}B(x)$, given a suitable
956 family of sets $\{B(x)\}@{x\in A}$. Conversely, by \tdx{range_of_fun},
957 any dependent typing can be flattened to yield a function type of the form
958 $A\to C$; here, $C={\tt range}(f)$.
960 Among the laws for $\lambda$-abstraction, \tdx{lamI} and \tdx{lamE}
961 describe the graph of the generated function, while \tdx{beta} and
962 \tdx{eta} are the standard conversions. We essentially have a
963 dependently-typed $\lambda$-calculus (Fig.\ts\ref{zf-lam}).
965 Figure~\ref{zf-func2} presents some rules that can be used to construct
966 functions explicitly. We start with functions consisting of at most one
967 pair, and may form the union of two functions provided their domains are
973 \tdx{Int_absorb} A Int A = A
974 \tdx{Int_commute} A Int B = B Int A
975 \tdx{Int_assoc} (A Int B) Int C = A Int (B Int C)
976 \tdx{Int_Un_distrib} (A Un B) Int C = (A Int C) Un (B Int C)
978 \tdx{Un_absorb} A Un A = A
979 \tdx{Un_commute} A Un B = B Un A
980 \tdx{Un_assoc} (A Un B) Un C = A Un (B Un C)
981 \tdx{Un_Int_distrib} (A Int B) Un C = (A Un C) Int (B Un C)
983 \tdx{Diff_cancel} A-A = 0
984 \tdx{Diff_disjoint} A Int (B-A) = 0
985 \tdx{Diff_partition} A<=B ==> A Un (B-A) = B
986 \tdx{double_complement} [| A<=B; B<= C |] ==> (B - (C-A)) = A
987 \tdx{Diff_Un} A - (B Un C) = (A-B) Int (A-C)
988 \tdx{Diff_Int} A - (B Int C) = (A-B) Un (A-C)
990 \tdx{Union_Un_distrib} Union(A Un B) = Union(A) Un Union(B)
991 \tdx{Inter_Un_distrib} [| a:A; b:B |] ==>
992 Inter(A Un B) = Inter(A) Int Inter(B)
994 \tdx{Int_Union_RepFun} A Int Union(B) = (UN C:B. A Int C)
996 \tdx{Un_Inter_RepFun} b:B ==>
997 A Un Inter(B) = (INT C:B. A Un C)
999 \tdx{SUM_Un_distrib1} (SUM x:A Un B. C(x)) =
1000 (SUM x:A. C(x)) Un (SUM x:B. C(x))
1002 \tdx{SUM_Un_distrib2} (SUM x:C. A(x) Un B(x)) =
1003 (SUM x:C. A(x)) Un (SUM x:C. B(x))
1005 \tdx{SUM_Int_distrib1} (SUM x:A Int B. C(x)) =
1006 (SUM x:A. C(x)) Int (SUM x:B. C(x))
1008 \tdx{SUM_Int_distrib2} (SUM x:C. A(x) Int B(x)) =
1009 (SUM x:C. A(x)) Int (SUM x:C. B(x))
1011 \caption{Equalities} \label{zf-equalities}
1017 % \cdx{1} & $i$ & & $\{\emptyset\}$ \\
1018 % \cdx{bool} & $i$ & & the set $\{\emptyset,1\}$ \\
1019 % \cdx{cond} & $[i,i,i]\To i$ & & conditional for \texttt{bool} \\
1020 % \cdx{not} & $i\To i$ & & negation for \texttt{bool} \\
1021 % \sdx{and} & $[i,i]\To i$ & Left 70 & conjunction for \texttt{bool} \\
1022 % \sdx{or} & $[i,i]\To i$ & Left 65 & disjunction for \texttt{bool} \\
1023 % \sdx{xor} & $[i,i]\To i$ & Left 65 & exclusive-or for \texttt{bool}
1027 \tdx{bool_def} bool == {\ttlbrace}0,1{\ttrbrace}
1028 \tdx{cond_def} cond(b,c,d) == if b=1 then c else d
1029 \tdx{not_def} not(b) == cond(b,0,1)
1030 \tdx{and_def} a and b == cond(a,b,0)
1031 \tdx{or_def} a or b == cond(a,1,b)
1032 \tdx{xor_def} a xor b == cond(a,not(b),b)
1034 \tdx{bool_1I} 1 : bool
1035 \tdx{bool_0I} 0 : bool
1036 \tdx{boolE} [| c: bool; c=1 ==> P; c=0 ==> P |] ==> P
1037 \tdx{cond_1} cond(1,c,d) = c
1038 \tdx{cond_0} cond(0,c,d) = d
1040 \caption{The booleans} \label{zf-bool}
1044 \section{Further developments}
1045 The next group of developments is complex and extensive, and only
1046 highlights can be covered here. It involves many theories and ML files of
1049 Figure~\ref{zf-equalities} presents commutative, associative, distributive,
1050 and idempotency laws of union and intersection, along with other equations.
1051 See file \texttt{ZF/equalities.ML}.
1053 Theory \thydx{Bool} defines $\{0,1\}$ as a set of booleans, with the usual
1054 operators including a conditional (Fig.\ts\ref{zf-bool}). Although {\ZF} is a
1055 first-order theory, you can obtain the effect of higher-order logic using
1056 \texttt{bool}-valued functions, for example. The constant~\texttt{1} is
1057 translated to \texttt{succ(0)}.
1062 \it symbol & \it meta-type & \it priority & \it description \\
1063 \tt + & $[i,i]\To i$ & Right 65 & disjoint union operator\\
1064 \cdx{Inl}~~\cdx{Inr} & $i\To i$ & & injections\\
1065 \cdx{case} & $[i\To i,i\To i, i]\To i$ & & conditional for $A+B$
1068 \tdx{sum_def} A+B == {\ttlbrace}0{\ttrbrace}*A Un {\ttlbrace}1{\ttrbrace}*B
1069 \tdx{Inl_def} Inl(a) == <0,a>
1070 \tdx{Inr_def} Inr(b) == <1,b>
1071 \tdx{case_def} case(c,d,u) == split(\%y z. cond(y, d(z), c(z)), u)
1073 \tdx{sum_InlI} a : A ==> Inl(a) : A+B
1074 \tdx{sum_InrI} b : B ==> Inr(b) : A+B
1076 \tdx{Inl_inject} Inl(a)=Inl(b) ==> a=b
1077 \tdx{Inr_inject} Inr(a)=Inr(b) ==> a=b
1078 \tdx{Inl_neq_Inr} Inl(a)=Inr(b) ==> P
1080 \tdx{sumE2} u: A+B ==> (EX x. x:A & u=Inl(x)) | (EX y. y:B & u=Inr(y))
1082 \tdx{case_Inl} case(c,d,Inl(a)) = c(a)
1083 \tdx{case_Inr} case(c,d,Inr(b)) = d(b)
1085 \caption{Disjoint unions} \label{zf-sum}
1089 Theory \thydx{Sum} defines the disjoint union of two sets, with
1090 injections and a case analysis operator (Fig.\ts\ref{zf-sum}). Disjoint
1091 unions play a role in datatype definitions, particularly when there is
1092 mutual recursion~\cite{paulson-set-II}.
1096 \tdx{QPair_def} <a;b> == a+b
1097 \tdx{qsplit_def} qsplit(c,p) == THE y. EX a b. p=<a;b> & y=c(a,b)
1098 \tdx{qfsplit_def} qfsplit(R,z) == EX x y. z=<x;y> & R(x,y)
1099 \tdx{qconverse_def} qconverse(r) == {\ttlbrace}z. w:r, EX x y. w=<x;y> & z=<y;x>{\ttrbrace}
1100 \tdx{QSigma_def} QSigma(A,B) == UN x:A. UN y:B(x). {\ttlbrace}<x;y>{\ttrbrace}
1102 \tdx{qsum_def} A <+> B == ({\ttlbrace}0{\ttrbrace} <*> A) Un ({\ttlbrace}1{\ttrbrace} <*> B)
1103 \tdx{QInl_def} QInl(a) == <0;a>
1104 \tdx{QInr_def} QInr(b) == <1;b>
1105 \tdx{qcase_def} qcase(c,d) == qsplit(\%y z. cond(y, d(z), c(z)))
1107 \caption{Non-standard pairs, products and sums} \label{zf-qpair}
1110 Theory \thydx{QPair} defines a notion of ordered pair that admits
1111 non-well-founded tupling (Fig.\ts\ref{zf-qpair}). Such pairs are written
1112 {\tt<$a$;$b$>}. It also defines the eliminator \cdx{qsplit}, the
1113 converse operator \cdx{qconverse}, and the summation operator
1114 \cdx{QSigma}. These are completely analogous to the corresponding
1115 versions for standard ordered pairs. The theory goes on to define a
1116 non-standard notion of disjoint sum using non-standard pairs. All of these
1117 concepts satisfy the same properties as their standard counterparts; in
1118 addition, {\tt<$a$;$b$>} is continuous. The theory supports coinductive
1119 definitions, for example of infinite lists~\cite{paulson-final}.
1123 \tdx{bnd_mono_def} bnd_mono(D,h) ==
1124 h(D)<=D & (ALL W X. W<=X --> X<=D --> h(W) <= h(X))
1126 \tdx{lfp_def} lfp(D,h) == Inter({\ttlbrace}X: Pow(D). h(X) <= X{\ttrbrace})
1127 \tdx{gfp_def} gfp(D,h) == Union({\ttlbrace}X: Pow(D). X <= h(X){\ttrbrace})
1130 \tdx{lfp_lowerbound} [| h(A) <= A; A<=D |] ==> lfp(D,h) <= A
1132 \tdx{lfp_subset} lfp(D,h) <= D
1134 \tdx{lfp_greatest} [| bnd_mono(D,h);
1135 !!X. [| h(X) <= X; X<=D |] ==> A<=X
1136 |] ==> A <= lfp(D,h)
1138 \tdx{lfp_Tarski} bnd_mono(D,h) ==> lfp(D,h) = h(lfp(D,h))
1140 \tdx{induct} [| a : lfp(D,h); bnd_mono(D,h);
1141 !!x. x : h(Collect(lfp(D,h),P)) ==> P(x)
1144 \tdx{lfp_mono} [| bnd_mono(D,h); bnd_mono(E,i);
1145 !!X. X<=D ==> h(X) <= i(X)
1146 |] ==> lfp(D,h) <= lfp(E,i)
1148 \tdx{gfp_upperbound} [| A <= h(A); A<=D |] ==> A <= gfp(D,h)
1150 \tdx{gfp_subset} gfp(D,h) <= D
1152 \tdx{gfp_least} [| bnd_mono(D,h);
1153 !!X. [| X <= h(X); X<=D |] ==> X<=A
1154 |] ==> gfp(D,h) <= A
1156 \tdx{gfp_Tarski} bnd_mono(D,h) ==> gfp(D,h) = h(gfp(D,h))
1158 \tdx{coinduct} [| bnd_mono(D,h); a: X; X <= h(X Un gfp(D,h)); X <= D
1161 \tdx{gfp_mono} [| bnd_mono(D,h); D <= E;
1162 !!X. X<=D ==> h(X) <= i(X)
1163 |] ==> gfp(D,h) <= gfp(E,i)
1165 \caption{Least and greatest fixedpoints} \label{zf-fixedpt}
1168 The Knaster-Tarski Theorem states that every monotone function over a
1169 complete lattice has a fixedpoint. Theory \thydx{Fixedpt} proves the
1170 Theorem only for a particular lattice, namely the lattice of subsets of a
1171 set (Fig.\ts\ref{zf-fixedpt}). The theory defines least and greatest
1172 fixedpoint operators with corresponding induction and coinduction rules.
1173 These are essential to many definitions that follow, including the natural
1174 numbers and the transitive closure operator. The (co)inductive definition
1175 package also uses the fixedpoint operators~\cite{paulson-CADE}. See
1176 Davey and Priestley~\cite{davey&priestley} for more on the Knaster-Tarski
1177 Theorem and my paper~\cite{paulson-set-II} for discussion of the Isabelle
1180 Monotonicity properties are proved for most of the set-forming operations:
1181 union, intersection, Cartesian product, image, domain, range, etc. These
1182 are useful for applying the Knaster-Tarski Fixedpoint Theorem. The proofs
1183 themselves are trivial applications of Isabelle's classical reasoner. See
1184 file \texttt{ZF/mono.ML}.
1189 \it symbol & \it meta-type & \it priority & \it description \\
1190 \sdx{O} & $[i,i]\To i$ & Right 60 & composition ($\circ$) \\
1191 \cdx{id} & $i\To i$ & & identity function \\
1192 \cdx{inj} & $[i,i]\To i$ & & injective function space\\
1193 \cdx{surj} & $[i,i]\To i$ & & surjective function space\\
1194 \cdx{bij} & $[i,i]\To i$ & & bijective function space
1198 \tdx{comp_def} r O s == {\ttlbrace}xz : domain(s)*range(r) .
1199 EX x y z. xz=<x,z> & <x,y>:s & <y,z>:r{\ttrbrace}
1200 \tdx{id_def} id(A) == (lam x:A. x)
1201 \tdx{inj_def} inj(A,B) == {\ttlbrace} f: A->B. ALL w:A. ALL x:A. f`w=f`x --> w=x {\ttrbrace}
1202 \tdx{surj_def} surj(A,B) == {\ttlbrace} f: A->B . ALL y:B. EX x:A. f`x=y {\ttrbrace}
1203 \tdx{bij_def} bij(A,B) == inj(A,B) Int surj(A,B)
1206 \tdx{left_inverse} [| f: inj(A,B); a: A |] ==> converse(f)`(f`a) = a
1207 \tdx{right_inverse} [| f: inj(A,B); b: range(f) |] ==>
1208 f`(converse(f)`b) = b
1210 \tdx{inj_converse_inj} f: inj(A,B) ==> converse(f): inj(range(f), A)
1211 \tdx{bij_converse_bij} f: bij(A,B) ==> converse(f): bij(B,A)
1213 \tdx{comp_type} [| s<=A*B; r<=B*C |] ==> (r O s) <= A*C
1214 \tdx{comp_assoc} (r O s) O t = r O (s O t)
1216 \tdx{left_comp_id} r<=A*B ==> id(B) O r = r
1217 \tdx{right_comp_id} r<=A*B ==> r O id(A) = r
1219 \tdx{comp_func} [| g:A->B; f:B->C |] ==> (f O g):A->C
1220 \tdx{comp_func_apply} [| g:A->B; f:B->C; a:A |] ==> (f O g)`a = f`(g`a)
1222 \tdx{comp_inj} [| g:inj(A,B); f:inj(B,C) |] ==> (f O g):inj(A,C)
1223 \tdx{comp_surj} [| g:surj(A,B); f:surj(B,C) |] ==> (f O g):surj(A,C)
1224 \tdx{comp_bij} [| g:bij(A,B); f:bij(B,C) |] ==> (f O g):bij(A,C)
1226 \tdx{left_comp_inverse} f: inj(A,B) ==> converse(f) O f = id(A)
1227 \tdx{right_comp_inverse} f: surj(A,B) ==> f O converse(f) = id(B)
1229 \tdx{bij_disjoint_Un}
1230 [| f: bij(A,B); g: bij(C,D); A Int C = 0; B Int D = 0 |] ==>
1231 (f Un g) : bij(A Un C, B Un D)
1233 \tdx{restrict_bij} [| f:inj(A,B); C<=A |] ==> restrict(f,C): bij(C, f``C)
1235 \caption{Permutations} \label{zf-perm}
1238 The theory \thydx{Perm} is concerned with permutations (bijections) and
1239 related concepts. These include composition of relations, the identity
1240 relation, and three specialized function spaces: injective, surjective and
1241 bijective. Figure~\ref{zf-perm} displays many of their properties that
1242 have been proved. These results are fundamental to a treatment of
1243 equipollence and cardinality.
1245 \begin{figure}\small
1246 \index{#*@{\tt\#*} symbol}
1249 \index{#+@{\tt\#+} symbol}
1250 \index{#-@{\tt\#-} symbol}
1252 \it symbol & \it meta-type & \it priority & \it description \\
1253 \cdx{nat} & $i$ & & set of natural numbers \\
1254 \cdx{nat_case}& $[i,i\To i,i]\To i$ & & conditional for $nat$\\
1255 \tt \#* & $[i,i]\To i$ & Left 70 & multiplication \\
1256 \tt div & $[i,i]\To i$ & Left 70 & division\\
1257 \tt mod & $[i,i]\To i$ & Left 70 & modulus\\
1258 \tt \#+ & $[i,i]\To i$ & Left 65 & addition\\
1259 \tt \#- & $[i,i]\To i$ & Left 65 & subtraction
1263 \tdx{nat_def} nat == lfp(lam r: Pow(Inf). {\ttlbrace}0{\ttrbrace} Un {\ttlbrace}succ(x). x:r{\ttrbrace}
1265 \tdx{mod_def} m mod n == transrec(m, \%j f. if j:n then j else f`(j#-n))
1266 \tdx{div_def} m div n == transrec(m, \%j f. if j:n then 0 else succ(f`(j#-n)))
1268 \tdx{nat_case_def} nat_case(a,b,k) ==
1269 THE y. k=0 & y=a | (EX x. k=succ(x) & y=b(x))
1271 \tdx{nat_0I} 0 : nat
1272 \tdx{nat_succI} n : nat ==> succ(n) : nat
1275 [| n: nat; P(0); !!x. [| x: nat; P(x) |] ==> P(succ(x))
1278 \tdx{nat_case_0} nat_case(a,b,0) = a
1279 \tdx{nat_case_succ} nat_case(a,b,succ(m)) = b(m)
1281 \tdx{add_0} 0 #+ n = n
1282 \tdx{add_succ} succ(m) #+ n = succ(m #+ n)
1284 \tdx{mult_type} [| m:nat; n:nat |] ==> m #* n : nat
1285 \tdx{mult_0} 0 #* n = 0
1286 \tdx{mult_succ} succ(m) #* n = n #+ (m #* n)
1287 \tdx{mult_commute} [| m:nat; n:nat |] ==> m #* n = n #* m
1288 \tdx{add_mult_dist} [| m:nat; k:nat |] ==> (m #+ n) #* k = (m #* k){\thinspace}#+{\thinspace}(n #* k)
1290 [| m:nat; n:nat; k:nat |] ==> (m #* n) #* k = m #* (n #* k)
1291 \tdx{mod_quo_equality}
1292 [| 0:n; m:nat; n:nat |] ==> (m div n)#*n #+ m mod n = m
1294 \caption{The natural numbers} \label{zf-nat}
1297 Theory \thydx{Nat} defines the natural numbers and mathematical
1298 induction, along with a case analysis operator. The set of natural
1299 numbers, here called \texttt{nat}, is known in set theory as the ordinal~$\omega$.
1301 Theory \thydx{Arith} develops arithmetic on the natural numbers
1302 (Fig.\ts\ref{zf-nat}). Addition, multiplication and subtraction are defined
1303 by primitive recursion. Division and remainder are defined by repeated
1304 subtraction, which requires well-founded recursion; the termination argument
1305 relies on the divisor's being non-zero. Many properties are proved:
1306 commutative, associative and distributive laws, identity and cancellation
1307 laws, etc. The most interesting result is perhaps the theorem $a \bmod b +
1310 Theory \thydx{Univ} defines a `universe' $\texttt{univ}(A)$, which is used by
1311 the datatype package. This set contains $A$ and the
1312 natural numbers. Vitally, it is closed under finite products: ${\tt
1313 univ}(A)\times{\tt univ}(A)\subseteq{\tt univ}(A)$. This theory also
1314 defines the cumulative hierarchy of axiomatic set theory, which
1315 traditionally is written $V@\alpha$ for an ordinal~$\alpha$. The
1316 `universe' is a simple generalization of~$V@\omega$.
1318 Theory \thydx{QUniv} defines a `universe' ${\tt quniv}(A)$, which is used by
1319 the datatype package to construct codatatypes such as streams. It is
1320 analogous to ${\tt univ}(A)$ (and is defined in terms of it) but is closed
1321 under the non-standard product and sum.
1323 Theory \texttt{Finite} (Figure~\ref{zf-fin}) defines the finite set operator;
1324 ${\tt Fin}(A)$ is the set of all finite sets over~$A$. The theory employs
1325 Isabelle's inductive definition package, which proves various rules
1326 automatically. The induction rule shown is stronger than the one proved by
1327 the package. The theory also defines the set of all finite functions
1328 between two given sets.
1332 \tdx{Fin.emptyI} 0 : Fin(A)
1333 \tdx{Fin.consI} [| a: A; b: Fin(A) |] ==> cons(a,b) : Fin(A)
1338 !!x y. [| x: A; y: Fin(A); x~:y; P(y) |] ==> P(cons(x,y))
1341 \tdx{Fin_mono} A<=B ==> Fin(A) <= Fin(B)
1342 \tdx{Fin_UnI} [| b: Fin(A); c: Fin(A) |] ==> b Un c : Fin(A)
1343 \tdx{Fin_UnionI} C : Fin(Fin(A)) ==> Union(C) : Fin(A)
1344 \tdx{Fin_subset} [| c<=b; b: Fin(A) |] ==> c: Fin(A)
1346 \caption{The finite set operator} \label{zf-fin}
1351 \it symbol & \it meta-type & \it priority & \it description \\
1352 \cdx{list} & $i\To i$ && lists over some set\\
1353 \cdx{list_case} & $[i, [i,i]\To i, i] \To i$ && conditional for $list(A)$ \\
1354 \cdx{map} & $[i\To i, i] \To i$ & & mapping functional\\
1355 \cdx{length} & $i\To i$ & & length of a list\\
1356 \cdx{rev} & $i\To i$ & & reverse of a list\\
1357 \tt \at & $[i,i]\To i$ & Right 60 & append for lists\\
1358 \cdx{flat} & $i\To i$ & & append of list of lists
1361 \underscoreon %%because @ is used here
1363 \tdx{NilI} Nil : list(A)
1364 \tdx{ConsI} [| a: A; l: list(A) |] ==> Cons(a,l) : list(A)
1369 !!x y. [| x: A; y: list(A); P(y) |] ==> P(Cons(x,y))
1372 \tdx{Cons_iff} Cons(a,l)=Cons(a',l') <-> a=a' & l=l'
1373 \tdx{Nil_Cons_iff} ~ Nil=Cons(a,l)
1375 \tdx{list_mono} A<=B ==> list(A) <= list(B)
1377 \tdx{map_ident} l: list(A) ==> map(\%u. u, l) = l
1378 \tdx{map_compose} l: list(A) ==> map(h, map(j,l)) = map(\%u. h(j(u)), l)
1379 \tdx{map_app_distrib} xs: list(A) ==> map(h, xs@ys) = map(h,xs) @ map(h,ys)
1381 [| l: list(A); !!x. x: A ==> h(x): B |] ==> map(h,l) : list(B)
1383 ls: list(list(A)) ==> map(h, flat(ls)) = flat(map(map(h),ls))
1385 \caption{Lists} \label{zf-list}
1389 Figure~\ref{zf-list} presents the set of lists over~$A$, ${\tt list}(A)$. The
1390 definition employs Isabelle's datatype package, which defines the introduction
1391 and induction rules automatically, as well as the constructors, case operator
1392 (\verb|list_case|) and recursion operator. The theory then defines the usual
1393 list functions by primitive recursion. See theory \texttt{List}.
1396 \section{Simplification and classical reasoning}
1398 {\ZF} inherits simplification from {\FOL} but adopts it for set theory. The
1399 extraction of rewrite rules takes the {\ZF} primitives into account. It can
1400 strip bounded universal quantifiers from a formula; for example, ${\forall
1401 x\in A. f(x)=g(x)}$ yields the conditional rewrite rule $x\in A \Imp
1402 f(x)=g(x)$. Given $a\in\{x\in A. P(x)\}$ it extracts rewrite rules from $a\in
1403 A$ and~$P(a)$. It can also break down $a\in A\int B$ and $a\in A-B$.
1405 Simplification tactics tactics such as \texttt{Asm_simp_tac} and
1406 \texttt{Full_simp_tac} use the default simpset (\texttt{simpset()}), which
1407 works for most purposes. A small simplification set for set theory is
1408 called~\ttindexbold{ZF_ss}, and you can even use \ttindex{FOL_ss} as a minimal
1409 starting point. \texttt{ZF_ss} contains congruence rules for all the binding
1410 operators of {\ZF}\@. It contains all the conversion rules, such as
1411 \texttt{fst} and \texttt{snd}, as well as the rewrites shown in
1412 Fig.\ts\ref{zf-simpdata}. See the file \texttt{ZF/simpdata.ML} for a fuller
1415 As for the classical reasoner, tactics such as \texttt{Blast_tac} and {\tt
1416 Best_tac} refer to the default claset (\texttt{claset()}). This works for
1417 most purposes. Named clasets include \ttindexbold{ZF_cs} (basic set theory)
1418 and \ttindexbold{le_cs} (useful for reasoning about the relations $<$ and
1419 $\le$). You can use \ttindex{FOL_cs} as a minimal basis for building your own
1420 clasets. See \iflabelundefined{chap:classical}{the {\em Reference Manual\/}}%
1421 {Chap.\ts\ref{chap:classical}} for more discussion of classical proof methods.
1426 a\in \emptyset & \bimp & \bot\\
1427 a \in A \un B & \bimp & a\in A \disj a\in B\\
1428 a \in A \int B & \bimp & a\in A \conj a\in B\\
1429 a \in A-B & \bimp & a\in A \conj \neg (a\in B)\\
1430 \pair{a,b}\in {\tt Sigma}(A,B)
1431 & \bimp & a\in A \conj b\in B(a)\\
1432 a \in {\tt Collect}(A,P) & \bimp & a\in A \conj P(a)\\
1433 (\forall x \in \emptyset. P(x)) & \bimp & \top\\
1434 (\forall x \in A. \top) & \bimp & \top
1436 \caption{Some rewrite rules for set theory} \label{zf-simpdata}
1440 \section{Datatype definitions}
1441 \label{sec:ZF:datatype}
1444 The \ttindex{datatype} definition package of \ZF\ constructs inductive
1445 datatypes similar to those of \ML. It can also construct coinductive
1446 datatypes (codatatypes), which are non-well-founded structures such as
1447 streams. It defines the set using a fixed-point construction and proves
1448 induction rules, as well as theorems for recursion and case combinators. It
1449 supplies mechanisms for reasoning about freeness. The datatype package can
1450 handle both mutual and indirect recursion.
1454 \label{subsec:datatype:basics}
1456 A \texttt{datatype} definition has the following form:
1459 \mathtt{datatype} & t@1(A@1,\ldots,A@h) & = &
1460 constructor^1@1 ~\mid~ \ldots ~\mid~ constructor^1@{k@1} \\
1462 \mathtt{and} & t@n(A@1,\ldots,A@h) & = &
1463 constructor^n@1~ ~\mid~ \ldots ~\mid~ constructor^n@{k@n}
1466 Here $t@1$, \ldots,~$t@n$ are identifiers and $A@1$, \ldots,~$A@h$ are
1467 variables: the datatype's parameters. Each constructor specification has the
1469 \[ C \hbox{\tt~( } \hbox{\tt"} x@1 \hbox{\tt:} T@1 \hbox{\tt"},\;
1471 \hbox{\tt"} x@m \hbox{\tt:} T@m \hbox{\tt"}
1474 Here $C$ is the constructor name, and variables $x@1$, \ldots,~$x@m$ are the
1475 constructor arguments, belonging to the sets $T@1$, \ldots, $T@m$,
1476 respectively. Typically each $T@j$ is either a constant set, a datatype
1477 parameter (one of $A@1$, \ldots, $A@h$) or a recursive occurrence of one of
1478 the datatypes, say $t@i(A@1,\ldots,A@h)$. More complex possibilities exist,
1479 but they are much harder to realize. Often, additional information must be
1480 supplied in the form of theorems.
1482 A datatype can occur recursively as the argument of some function~$F$. This
1483 is called a {\em nested} (or \emph{indirect}) occurrence. It is only allowed
1484 if the datatype package is given a theorem asserting that $F$ is monotonic.
1485 If the datatype has indirect occurrences, then Isabelle/ZF does not support
1486 recursive function definitions.
1488 A simple example of a datatype is \texttt{list}, which is built-in, and is
1492 datatype "list(A)" = Nil | Cons ("a:A", "l: list(A)")
1494 Note that the datatype operator must be declared as a constant first.
1495 However, the package declares the constructors. Here, \texttt{Nil} gets type
1496 $i$ and \texttt{Cons} gets type $[i,i]\To i$.
1498 Trees and forests can be modelled by the mutually recursive datatype
1501 consts tree, forest, tree_forest :: i=>i
1502 datatype "tree(A)" = Tcons ("a: A", "f: forest(A)")
1503 and "forest(A)" = Fnil | Fcons ("t: tree(A)", "f: forest(A)")
1505 Here $\texttt{tree}(A)$ is the set of trees over $A$, $\texttt{forest}(A)$ is
1506 the set of forests over $A$, and $\texttt{tree_forest}(A)$ is the union of
1507 the previous two sets. All three operators must be declared first.
1509 The datatype \texttt{term}, which is defined by
1512 datatype "term(A)" = Apply ("a: A", "l: list(term(A))")
1515 is an example of nested recursion. (The theorem \texttt{list_mono} is proved
1516 in file \texttt{List.ML}, and the \texttt{term} example is devaloped in theory
1519 \subsubsection{Freeness of the constructors}
1521 Constructors satisfy {\em freeness} properties. Constructions are distinct,
1522 for example $\texttt{Nil}\not=\texttt{Cons}(a,l)$, and they are injective, for
1523 example $\texttt{Cons}(a,l)=\texttt{Cons}(a',l') \bimp a=a' \conj l=l'$.
1524 Because the number of freeness is quadratic in the number of constructors, the
1525 datatype package does not prove them, but instead provides several means of
1526 proving them dynamically. For the \texttt{list} datatype, freeness reasoning
1527 can be done in two ways: by simplifying with the theorems
1528 \texttt{list.free_iffs} or by invoking the classical reasoner with
1529 \texttt{list.free_SEs} as safe elimination rules. Occasionally this exposes
1530 the underlying representation of some constructor, which can be rectified
1531 using the command \hbox{\tt fold_tac list.con_defs}.
1533 \subsubsection{Structural induction}
1535 The datatype package also provides structural induction rules. For datatypes
1536 without mutual or nested recursion, the rule has the form exemplified by
1537 \texttt{list.induct} in Fig.\ts\ref{zf-list}. For mutually recursive
1538 datatypes, the induction rule is supplied in two forms. Consider datatype
1539 \texttt{TF}. The rule \texttt{tree_forest.induct} performs induction over a
1540 single predicate~\texttt{P}, which is presumed to be defined for both trees
1543 [| x : tree_forest(A);
1544 !!a f. [| a : A; f : forest(A); P(f) |] ==> P(Tcons(a, f)); P(Fnil);
1545 !!f t. [| t : tree(A); P(t); f : forest(A); P(f) |]
1549 The rule \texttt{tree_forest.mutual_induct} performs induction over two
1550 distinct predicates, \texttt{P_tree} and \texttt{P_forest}.
1553 [| a : A; f : forest(A); P_forest(f) |] ==> P_tree(Tcons(a, f));
1555 !!f t. [| t : tree(A); P_tree(t); f : forest(A); P_forest(f) |]
1556 ==> P_forest(Fcons(t, f))
1557 |] ==> (ALL za. za : tree(A) --> P_tree(za)) &
1558 (ALL za. za : forest(A) --> P_forest(za))
1561 For datatypes with nested recursion, such as the \texttt{term} example from
1562 above, things are a bit more complicated. The rule \texttt{term.induct}
1563 refers to the monotonic operator, \texttt{list}:
1566 !!a l. [| a : A; l : list(Collect(term(A), P)) |] ==> P(Apply(a, l))
1569 The file \texttt{ex/Term.ML} derives two higher-level induction rules, one of
1570 which is particularly useful for proving equations:
1573 !!x zs. [| x : A; zs : list(term(A)); map(f, zs) = map(g, zs) |]
1574 ==> f(Apply(x, zs)) = g(Apply(x, zs))
1577 How this can be generalized to other nested datatypes is a matter for future
1581 \subsubsection{The \texttt{case} operator}
1583 The package defines an operator for performing case analysis over the
1584 datatype. For \texttt{list}, it is called \texttt{list_case} and satisfies
1587 list_case(f_Nil, f_Cons, []) = f_Nil
1588 list_case(f_Nil, f_Cons, Cons(a, l)) = f_Cons(a, l)
1590 Here \texttt{f_Nil} is the value to return if the argument is \texttt{Nil} and
1591 \texttt{f_Cons} is a function that computes the value to return if the
1592 argument has the form $\texttt{Cons}(a,l)$. The function can be expressed as
1593 an abstraction, over patterns if desired (\S\ref{sec:pairs}).
1595 For mutually recursive datatypes, there is a single \texttt{case} operator.
1596 In the tree/forest example, the constant \texttt{tree_forest_case} handles all
1597 of the constructors of the two datatypes.
1602 \subsection{Defining datatypes}
1604 The theory syntax for datatype definitions is shown in
1605 Fig.~\ref{datatype-grammar}. In order to be well-formed, a datatype
1606 definition has to obey the rules stated in the previous section. As a result
1607 the theory is extended with the new types, the constructors, and the theorems
1608 listed in the previous section. The quotation marks are necessary because
1609 they enclose general Isabelle formul\ae.
1613 datatype : ( 'datatype' | 'codatatype' ) datadecls;
1615 datadecls: ( '"' id arglist '"' '=' (constructor + '|') ) + 'and'
1617 constructor : name ( () | consargs ) ( () | ( '(' mixfix ')' ) )
1619 consargs : '(' ('"' var ':' term '"' + ',') ')'
1622 \caption{Syntax of datatype declarations}
1623 \label{datatype-grammar}
1626 Codatatypes are declared like datatypes and are identical to them in every
1627 respect except that they have a coinduction rule instead of an induction rule.
1628 Note that while an induction rule has the effect of limiting the values
1629 contained in the set, a coinduction rule gives a way of constructing new
1632 Most of the theorems about datatypes become part of the default simpset. You
1633 never need to see them again because the simplifier applies them
1634 automatically. Add freeness properties (\texttt{free_iffs}) to the simpset
1635 when you want them. Induction or exhaustion are usually invoked by hand,
1636 usually via these special-purpose tactics:
1637 \begin{ttdescription}
1638 \item[\ttindexbold{induct_tac} {\tt"}$x${\tt"} $i$] applies structural
1639 induction on variable $x$ to subgoal $i$, provided the type of $x$ is a
1640 datatype. The induction variable should not occur among other assumptions
1643 In some cases, induction is overkill and a case distinction over all
1644 constructors of the datatype suffices.
1645 \begin{ttdescription}
1646 \item[\ttindexbold{exhaust_tac} {\tt"}$x${\tt"} $i$]
1647 performs an exhaustive case analysis for the variable~$x$.
1650 Both tactics can only be applied to a variable, whose typing must be given in
1651 some assumption, for example the assumption \texttt{x:\ list(A)}. The tactics
1652 also work for the natural numbers (\texttt{nat}) and disjoint sums, although
1653 these sets were not defined using the datatype package. (Disjoint sums are
1654 not recursive, so only \texttt{exhaust_tac} is available.)
1657 Here are some more details for the technically minded. Processing the
1658 theory file produces an \ML\ structure which, in addition to the usual
1659 components, contains a structure named $t$ for each datatype $t$ defined in
1660 the file. Each structure $t$ contains the following elements:
1662 val intrs : thm list \textrm{the introduction rules}
1663 val elim : thm \textrm{the elimination (case analysis) rule}
1664 val induct : thm \textrm{the standard induction rule}
1665 val mutual_induct : thm \textrm{the mutual induction rule, or \texttt{True}}
1666 val case_eqns : thm list \textrm{equations for the case operator}
1667 val recursor_eqns : thm list \textrm{equations for the recursor}
1668 val con_defs : thm list \textrm{definitions of the case operator and constructors}
1669 val free_iffs : thm list \textrm{logical equivalences for proving freeness}
1670 val free_SEs : thm list \textrm{elimination rules for proving freeness}
1671 val mk_free : string -> thm \textrm{A function for proving freeness theorems}
1672 val mk_cases : string -> thm \textrm{case analysis, see below}
1673 val defs : thm list \textrm{definitions of operators}
1674 val bnd_mono : thm list \textrm{monotonicity property}
1675 val dom_subset : thm list \textrm{inclusion in `bounding set'}
1677 Furthermore there is the theorem $C$\texttt{_I} for every constructor~$C$; for
1678 example, the \texttt{list} datatype's introduction rules are bound to the
1679 identifiers \texttt{Nil_I} and \texttt{Cons_I}.
1681 For a codatatype, the component \texttt{coinduct} is the coinduction rule,
1682 replacing the \texttt{induct} component.
1684 See the theories \texttt{ex/Ntree} and \texttt{ex/Brouwer} for examples of
1685 infinitely branching datatypes. See theory \texttt{ex/LList} for an example
1686 of a codatatype. Some of these theories illustrate the use of additional,
1687 undocumented features of the datatype package. Datatype definitions are
1688 reduced to inductive definitions, and the advanced features should be
1689 understood in that light.
1692 \subsection{Examples}
1694 \subsubsection{The datatype of binary trees}
1696 Let us define the set $\texttt{bt}(A)$ of binary trees over~$A$. The theory
1697 must contain these lines:
1700 datatype "bt(A)" = Lf | Br ("a: A", "t1: bt(A)", "t2: bt(A)")
1702 After loading the theory, we can prove, for example, that no tree equals its
1703 left branch. To ease the induction, we state the goal using quantifiers.
1705 Goal "l : bt(A) ==> ALL x r. Br(x,l,r) ~= l";
1707 {\out l : bt(A) ==> ALL x r. Br(x, l, r) ~= l}
1708 {\out 1. l : bt(A) ==> ALL x r. Br(x, l, r) ~= l}
1710 This can be proved by the structural induction tactic:
1712 by (induct_tac "l" 1);
1714 {\out l : bt(A) ==> ALL x r. Br(x, l, r) ~= l}
1715 {\out 1. ALL x r. Br(x, Lf, r) ~= Lf}
1716 {\out 2. !!a t1 t2.}
1717 {\out [| a : A; t1 : bt(A); ALL x r. Br(x, t1, r) ~= t1; t2 : bt(A);}
1718 {\out ALL x r. Br(x, t2, r) ~= t2 |]}
1719 {\out ==> ALL x r. Br(x, Br(a, t1, t2), r) ~= Br(a, t1, t2)}
1721 Both subgoals are proved using the simplifier. Tactic
1722 \texttt{asm_full_simp_tac} is used, rewriting the assumptions.
1723 This is because simplification using the freeness properties can unfold the
1724 definition of constructor~\texttt{Br}, so we arrange that all occurrences are
1727 by (ALLGOALS (asm_full_simp_tac (simpset() addsimps bt.free_iffs)));
1729 {\out l : bt(A) ==> ALL x r. Br(x, l, r) ~= l}
1732 To remove the quantifiers from the induction formula, we save the theorem using
1733 \ttindex{qed_spec_mp}.
1735 qed_spec_mp "Br_neq_left";
1736 {\out val Br_neq_left = "?l : bt(?A) ==> Br(?x, ?l, ?r) ~= ?l" : thm}
1739 When there are only a few constructors, we might prefer to prove the freenness
1740 theorems for each constructor. This is trivial, using the function given us
1743 val Br_iff = bt.mk_free "Br(a,l,r)=Br(a',l',r') <-> a=a' & l=l' & r=r'";
1745 {\out "Br(?a, ?l, ?r) = Br(?a', ?l', ?r') <->}
1746 {\out ?a = ?a' & ?l = ?l' & ?r = ?r'" : thm}
1749 The purpose of \ttindex{mk_cases} is to generate instances of the elimination
1750 (case analysis) rule that have been simplified using freeness reasoning. For
1751 example, this instance of the elimination rule propagates type-checking
1752 information from the premise $\texttt{Br}(a,l,r)\in\texttt{bt}(A)$:
1754 val BrE = bt.mk_cases "Br(a,l,r) : bt(A)";
1756 {\out "[| Br(?a, ?l, ?r) : bt(?A);}
1757 {\out [| ?a : ?A; ?l : bt(?A); ?r : bt(?A) |] ==> ?Q |] ==> ?Q" : thm}
1761 \subsubsection{Mixfix syntax in datatypes}
1763 Mixfix syntax is sometimes convenient. The theory \texttt{ex/PropLog} makes a
1764 deep embedding of propositional logic:
1767 datatype "prop" = Fls
1768 | Var ("n: nat") ("#_" [100] 100)
1769 | "=>" ("p: prop", "q: prop") (infixr 90)
1771 The second constructor has a special $\#n$ syntax, while the third constructor
1772 is an infixed arrow.
1775 \subsubsection{A giant enumeration type}
1777 This example shows a datatype that consists of 60 constructors:
1781 "enum" = C00 | C01 | C02 | C03 | C04 | C05 | C06 | C07 | C08 | C09
1782 | C10 | C11 | C12 | C13 | C14 | C15 | C16 | C17 | C18 | C19
1783 | C20 | C21 | C22 | C23 | C24 | C25 | C26 | C27 | C28 | C29
1784 | C30 | C31 | C32 | C33 | C34 | C35 | C36 | C37 | C38 | C39
1785 | C40 | C41 | C42 | C43 | C44 | C45 | C46 | C47 | C48 | C49
1786 | C50 | C51 | C52 | C53 | C54 | C55 | C56 | C57 | C58 | C59
1789 The datatype package scales well. Even though all properties are proved
1790 rather than assumed, full processing of this definition takes under 15 seconds
1791 (on a 300 MHz Pentium). The constructors have a balanced representation,
1792 essentially binary notation, so freeness properties can be proved fast.
1795 by (simp_tac (simpset() addsimps enum.free_iffs) 1);
1797 You need not derive such inequalities explicitly. The simplifier will dispose
1798 of them automatically, given the theorem list \texttt{free_iffs}.
1803 \subsection{Recursive function definitions}\label{sec:ZF:recursive}
1804 \index{recursive functions|see{recursion}}
1807 Datatypes come with a uniform way of defining functions, {\bf primitive
1808 recursion}. Such definitions rely on the recursion operator defined by the
1809 datatype package. Isabelle proves the desired recursion equations as
1812 In principle, one could introduce primitive recursive functions by asserting
1813 their reduction rules as new axioms. Here is a dangerous way of defining the
1814 append function for lists:
1815 \begin{ttbox}\slshape
1816 consts "\at" :: [i,i]=>i (infixr 60)
1818 app_Nil "[] \at ys = ys"
1819 app_Cons "(Cons(a,l)) \at ys = Cons(a, l \at ys)"
1821 Asserting axioms brings the danger of accidentally asserting nonsense. It
1822 should be avoided at all costs!
1824 The \ttindex{primrec} declaration is a safe means of defining primitive
1825 recursive functions on datatypes:
1827 consts "\at" :: [i,i]=>i (infixr 60)
1830 "(Cons(a,l)) \at ys = Cons(a, l \at ys)"
1832 Isabelle will now check that the two rules do indeed form a primitive
1833 recursive definition. For example, the declaration
1838 is rejected with an error message ``\texttt{Extra variables on rhs}''.
1841 \subsubsection{Syntax of recursive definitions}
1843 The general form of a primitive recursive definition is
1846 {\it reduction rules}
1848 where \textit{reduction rules} specify one or more equations of the form
1849 \[ f \, x@1 \, \dots \, x@m \, (C \, y@1 \, \dots \, y@k) \, z@1 \,
1850 \dots \, z@n = r \] such that $C$ is a constructor of the datatype, $r$
1851 contains only the free variables on the left-hand side, and all recursive
1852 calls in $r$ are of the form $f \, \dots \, y@i \, \dots$ for some $i$.
1853 There must be at most one reduction rule for each constructor. The order is
1854 immaterial. For missing constructors, the function is defined to return zero.
1856 All reduction rules are added to the default simpset.
1857 If you would like to refer to some rule by name, then you must prefix
1858 the rule with an identifier. These identifiers, like those in the
1859 \texttt{rules} section of a theory, will be visible at the \ML\ level.
1861 The reduction rules for {\tt\at} become part of the default simpset, which
1862 leads to short proof scripts:
1863 \begin{ttbox}\underscoreon
1864 Goal "xs: list(A) ==> (xs @ ys) @ zs = xs @ (ys @ zs)";
1865 by (induct\_tac "xs" 1);
1866 by (ALLGOALS Asm\_simp\_tac);
1869 You can even use the \texttt{primrec} form with non-recursive datatypes and
1870 with codatatypes. Recursion is not allowed, but it provides a convenient
1871 syntax for defining functions by cases.
1874 \subsubsection{Example: varying arguments}
1876 All arguments, other than the recursive one, must be the same in each equation
1877 and in each recursive call. To get around this restriction, use explict
1878 $\lambda$-abstraction and function application. Here is an example, drawn
1879 from the theory \texttt{Resid/Substitution}. The type of redexes is declared
1884 "redexes" = Var ("n: nat")
1885 | Fun ("t: redexes")
1886 | App ("b:bool" ,"f:redexes" , "a:redexes")
1889 The function \texttt{lift} takes a second argument, $k$, which varies in
1893 "lift(Var(i)) = (lam k:nat. if i<k then Var(i) else Var(succ(i)))"
1894 "lift(Fun(t)) = (lam k:nat. Fun(lift(t) ` succ(k)))"
1895 "lift(App(b,f,a)) = (lam k:nat. App(b, lift(f)`k, lift(a)`k))"
1897 Now \texttt{lift(r)`k} satisfies the required recursion equations.
1899 \index{recursion!primitive|)}
1903 \section{Inductive and coinductive definitions}
1904 \index{*inductive|(}
1905 \index{*coinductive|(}
1907 An {\bf inductive definition} specifies the least set~$R$ closed under given
1908 rules. (Applying a rule to elements of~$R$ yields a result within~$R$.) For
1909 example, a structural operational semantics is an inductive definition of an
1910 evaluation relation. Dually, a {\bf coinductive definition} specifies the
1911 greatest set~$R$ consistent with given rules. (Every element of~$R$ can be
1912 seen as arising by applying a rule to elements of~$R$.) An important example
1913 is using bisimulation relations to formalise equivalence of processes and
1914 infinite data structures.
1916 A theory file may contain any number of inductive and coinductive
1917 definitions. They may be intermixed with other declarations; in
1918 particular, the (co)inductive sets {\bf must} be declared separately as
1919 constants, and may have mixfix syntax or be subject to syntax translations.
1921 Each (co)inductive definition adds definitions to the theory and also
1922 proves some theorems. Each definition creates an \ML\ structure, which is a
1923 substructure of the main theory structure.
1924 This package is described in detail in a separate paper,%
1925 \footnote{It appeared in CADE~\cite{paulson-CADE}; a longer version is
1926 distributed with Isabelle as \emph{A Fixedpoint Approach to
1927 (Co)Inductive and (Co)Datatype Definitions}.} %
1928 which you might refer to for background information.
1931 \subsection{The syntax of a (co)inductive definition}
1932 An inductive definition has the form
1935 domains {\it domain declarations}
1936 intrs {\it introduction rules}
1937 monos {\it monotonicity theorems}
1938 con_defs {\it constructor definitions}
1939 type_intrs {\it introduction rules for type-checking}
1940 type_elims {\it elimination rules for type-checking}
1942 A coinductive definition is identical, but starts with the keyword
1945 The {\tt monos}, {\tt con\_defs}, {\tt type\_intrs} and {\tt type\_elims}
1946 sections are optional. If present, each is specified either as a list of
1947 identifiers or as a string. If the latter, then the string must be a valid
1948 \textsc{ml} expression of type {\tt thm list}. The string is simply inserted
1949 into the {\tt _thy.ML} file; if it is ill-formed, it will trigger \textsc{ml}
1950 error messages. You can then inspect the file on the temporary directory.
1953 \item[\it domain declarations] consist of one or more items of the form
1954 {\it string\/}~{\tt <=}~{\it string}, associating each recursive set with
1955 its domain. (The domain is some existing set that is large enough to
1956 hold the new set being defined.)
1958 \item[\it introduction rules] specify one or more introduction rules in
1959 the form {\it ident\/}~{\it string}, where the identifier gives the name of
1960 the rule in the result structure.
1962 \item[\it monotonicity theorems] are required for each operator applied to
1963 a recursive set in the introduction rules. There \textbf{must} be a theorem
1964 of the form $A\subseteq B\Imp M(A)\subseteq M(B)$, for each premise $t\in M(R_i)$
1965 in an introduction rule!
1967 \item[\it constructor definitions] contain definitions of constants
1968 appearing in the introduction rules. The (co)datatype package supplies
1969 the constructors' definitions here. Most (co)inductive definitions omit
1970 this section; one exception is the primitive recursive functions example;
1971 see theory \texttt{ex/Primrec}.
1973 \item[\it type\_intrs] consists of introduction rules for type-checking the
1974 definition: for demonstrating that the new set is included in its domain.
1975 (The proof uses depth-first search.)
1977 \item[\it type\_elims] consists of elimination rules for type-checking the
1978 definition. They are presumed to be safe and are applied as often as
1979 possible prior to the {\tt type\_intrs} search.
1982 The package has a few restrictions:
1984 \item The theory must separately declare the recursive sets as
1987 \item The names of the recursive sets must be identifiers, not infix
1990 \item Side-conditions must not be conjunctions. However, an introduction rule
1991 may contain any number of side-conditions.
1993 \item Side-conditions of the form $x=t$, where the variable~$x$ does not
1994 occur in~$t$, will be substituted through the rule \verb|mutual_induct|.
1998 \subsection{Example of an inductive definition}
2000 Two declarations, included in a theory file, define the finite powerset
2001 operator. First we declare the constant~\texttt{Fin}. Then we declare it
2002 inductively, with two introduction rules:
2007 domains "Fin(A)" <= "Pow(A)"
2010 consI "[| a: A; b: Fin(A) |] ==> cons(a,b) : Fin(A)"
2011 type_intrs empty_subsetI, cons_subsetI, PowI
2012 type_elims "[make_elim PowD]"
2014 The resulting theory structure contains a substructure, called~\texttt{Fin}.
2015 It contains the \texttt{Fin}$~A$ introduction rules as the list
2016 \texttt{Fin.intrs}, and also individually as \texttt{Fin.emptyI} and
2017 \texttt{Fin.consI}. The induction rule is \texttt{Fin.induct}.
2019 The chief problem with making (co)inductive definitions involves type-checking
2020 the rules. Sometimes, additional theorems need to be supplied under
2021 \texttt{type_intrs} or \texttt{type_elims}. If the package fails when trying
2022 to prove your introduction rules, then set the flag \ttindexbold{trace_induct}
2023 to \texttt{true} and try again. (See the manual \emph{A Fixedpoint Approach
2024 \ldots} for more discussion of type-checking.)
2026 In the example above, $\texttt{Pow}(A)$ is given as the domain of
2027 $\texttt{Fin}(A)$, for obviously every finite subset of~$A$ is a subset
2028 of~$A$. However, the inductive definition package can only prove that given a
2030 Here is the output that results (with the flag set) when the
2031 \texttt{type_intrs} and \texttt{type_elims} are omitted from the inductive
2034 Inductive definition Finite.Fin
2037 \%X. {z: Pow(A) . z = 0 | (EX a b. z = cons(a, b) & a : A & b : X)})
2038 Proving monotonicity...
2040 Proving the introduction rules...
2041 The typechecking subgoal:
2045 The subgoal after monos, type_elims:
2048 *** prove_goal: tactic failed
2050 We see the need to supply theorems to let the package prove
2051 $\emptyset\in\texttt{Pow}(A)$. Restoring the \texttt{type_intrs} but not the
2052 \texttt{type_elims}, we again get an error message:
2054 The typechecking subgoal:
2058 The subgoal after monos, type_elims:
2062 The typechecking subgoal:
2064 1. [| a : A; b : Fin(A) |] ==> cons(a, b) : Pow(A)
2066 The subgoal after monos, type_elims:
2068 1. [| a : A; b : Pow(A) |] ==> cons(a, b) : Pow(A)
2069 *** prove_goal: tactic failed
2071 The first rule has been type-checked, but the second one has failed. The
2072 simplest solution to such problems is to prove the failed subgoal separately
2073 and to supply it under \texttt{type_intrs}. The solution actually used is
2074 to supply, under \texttt{type_elims}, a rule that changes
2075 $b\in\texttt{Pow}(A)$ to $b\subseteq A$; together with \texttt{cons_subsetI}
2076 and \texttt{PowI}, it is enough to complete the type-checking.
2080 \subsection{Further examples}
2082 An inductive definition may involve arbitrary monotonic operators. Here is a
2083 standard example: the accessible part of a relation. Note the use
2084 of~\texttt{Pow} in the introduction rule and the corresponding mention of the
2085 rule \verb|Pow_mono| in the \texttt{monos} list. If the desired rule has a
2086 universally quantified premise, usually the effect can be obtained using
2091 domains "acc(r)" <= "field(r)"
2093 vimage "[| r-``{a}: Pow(acc(r)); a: field(r) |] ==> a: acc(r)"
2097 Finally, here is a coinductive definition. It captures (as a bisimulation)
2098 the notion of equality on lazy lists, which are first defined as a codatatype:
2100 consts llist :: i=>i
2101 codatatype "llist(A)" = LNil | LCons ("a: A", "l: llist(A)")
2106 domains "lleq(A)" <= "llist(A) * llist(A)"
2108 LNil "<LNil, LNil> : lleq(A)"
2109 LCons "[| a:A; <l,l'>: lleq(A) |]
2110 ==> <LCons(a,l), LCons(a,l')>: lleq(A)"
2111 type_intrs "llist.intrs"
2113 This use of \texttt{type_intrs} is typical: the relation concerns the
2114 codatatype \texttt{llist}, so naturally the introduction rules for that
2115 codatatype will be required for type-checking the rules.
2117 The Isabelle distribution contains many other inductive definitions. Simple
2118 examples are collected on subdirectory \texttt{ZF/ex}. The directory
2119 \texttt{Coind} and the theory \texttt{ZF/ex/LList} contain coinductive
2120 definitions. Larger examples may be found on other subdirectories of
2121 \texttt{ZF}, such as \texttt{IMP}, and \texttt{Resid}.
2124 \subsection{The result structure}
2126 Each (co)inductive set defined in a theory file generates an \ML\ substructure
2127 having the same name. The the substructure contains the following elements:
2130 val intrs : thm list \textrm{the introduction rules}
2131 val elim : thm \textrm{the elimination (case analysis) rule}
2132 val mk_cases : string -> thm \textrm{case analysis, see below}
2133 val induct : thm \textrm{the standard induction rule}
2134 val mutual_induct : thm \textrm{the mutual induction rule, or \texttt{True}}
2135 val defs : thm list \textrm{definitions of operators}
2136 val bnd_mono : thm list \textrm{monotonicity property}
2137 val dom_subset : thm list \textrm{inclusion in `bounding set'}
2139 Furthermore there is the theorem $C$\texttt{_I} for every constructor~$C$; for
2140 example, the \texttt{list} datatype's introduction rules are bound to the
2141 identifiers \texttt{Nil_I} and \texttt{Cons_I}.
2143 For a codatatype, the component \texttt{coinduct} is the coinduction rule,
2144 replacing the \texttt{induct} component.
2146 Recall that \ttindex{mk_cases} generates simplified instances of the
2147 elimination (case analysis) rule. It is as useful for inductive definitions
2148 as it is for datatypes. There are many examples in the theory
2149 \texttt{ex/Comb}, which is discussed at length
2150 elsewhere~\cite{paulson-generic}. The theory first defines the datatype
2151 \texttt{comb} of combinators:
2156 | "#" ("p: comb", "q: comb") (infixl 90)
2158 The theory goes on to define contraction and parallel contraction
2159 inductively. Then the file \texttt{ex/Comb.ML} defines special cases of
2160 contraction using \texttt{mk_cases}:
2162 val K_contractE = contract.mk_cases "K -1-> r";
2163 {\out val K_contractE = "K -1-> ?r ==> ?Q" : thm}
2165 We can read this as saying that the combinator \texttt{K} cannot reduce to
2166 anything. Similar elimination rules for \texttt{S} and application are also
2167 generated and are supplied to the classical reasoner. Note that
2168 \texttt{comb.con_defs} is given to \texttt{mk_cases} to allow freeness
2169 reasoning on datatype \texttt{comb}.
2171 \index{*coinductive|)} \index{*inductive|)}
2176 \section{The outer reaches of set theory}
2178 The constructions of the natural numbers and lists use a suite of
2179 operators for handling recursive function definitions. I have described
2180 the developments in detail elsewhere~\cite{paulson-set-II}. Here is a brief
2183 \item Theory \texttt{Trancl} defines the transitive closure of a relation
2184 (as a least fixedpoint).
2186 \item Theory \texttt{WF} proves the Well-Founded Recursion Theorem, using an
2187 elegant approach of Tobias Nipkow. This theorem permits general
2188 recursive definitions within set theory.
2190 \item Theory \texttt{Ord} defines the notions of transitive set and ordinal
2191 number. It derives transfinite induction. A key definition is {\bf
2192 less than}: $i<j$ if and only if $i$ and $j$ are both ordinals and
2193 $i\in j$. As a special case, it includes less than on the natural
2196 \item Theory \texttt{Epsilon} derives $\varepsilon$-induction and
2197 $\varepsilon$-recursion, which are generalisations of transfinite
2198 induction and recursion. It also defines \cdx{rank}$(x)$, which
2199 is the least ordinal $\alpha$ such that $x$ is constructed at
2200 stage $\alpha$ of the cumulative hierarchy (thus $x\in
2204 Other important theories lead to a theory of cardinal numbers. They have
2205 not yet been written up anywhere. Here is a summary:
2207 \item Theory \texttt{Rel} defines the basic properties of relations, such as
2208 (ir)reflexivity, (a)symmetry, and transitivity.
2210 \item Theory \texttt{EquivClass} develops a theory of equivalence
2211 classes, not using the Axiom of Choice.
2213 \item Theory \texttt{Order} defines partial orderings, total orderings and
2216 \item Theory \texttt{OrderArith} defines orderings on sum and product sets.
2217 These can be used to define ordinal arithmetic and have applications to
2218 cardinal arithmetic.
2220 \item Theory \texttt{OrderType} defines order types. Every wellordering is
2221 equivalent to a unique ordinal, which is its order type.
2223 \item Theory \texttt{Cardinal} defines equipollence and cardinal numbers.
2225 \item Theory \texttt{CardinalArith} defines cardinal addition and
2226 multiplication, and proves their elementary laws. It proves that there
2227 is no greatest cardinal. It also proves a deep result, namely
2228 $\kappa\otimes\kappa=\kappa$ for every infinite cardinal~$\kappa$; see
2229 Kunen~\cite[page 29]{kunen80}. None of these results assume the Axiom of
2230 Choice, which complicates their proofs considerably.
2233 The following developments involve the Axiom of Choice (AC):
2235 \item Theory \texttt{AC} asserts the Axiom of Choice and proves some simple
2238 \item Theory \texttt{Zorn} proves Hausdorff's Maximal Principle, Zorn's Lemma
2239 and the Wellordering Theorem, following Abrial and
2240 Laffitte~\cite{abrial93}.
2242 \item Theory \verb|Cardinal_AC| uses AC to prove simplified theorems about
2243 the cardinals. It also proves a theorem needed to justify
2244 infinitely branching datatype declarations: if $\kappa$ is an infinite
2245 cardinal and $|X(\alpha)| \le \kappa$ for all $\alpha<\kappa$ then
2246 $|\union\sb{\alpha<\kappa} X(\alpha)| \le \kappa$.
2248 \item Theory \texttt{InfDatatype} proves theorems to justify infinitely
2249 branching datatypes. Arbitrary index sets are allowed, provided their
2250 cardinalities have an upper bound. The theory also justifies some
2251 unusual cases of finite branching, involving the finite powerset operator
2252 and the finite function space operator.
2257 \section{The examples directories}
2258 Directory \texttt{HOL/IMP} contains a mechanised version of a semantic
2259 equivalence proof taken from Winskel~\cite{winskel93}. It formalises the
2260 denotational and operational semantics of a simple while-language, then
2261 proves the two equivalent. It contains several datatype and inductive
2262 definitions, and demonstrates their use.
2264 The directory \texttt{ZF/ex} contains further developments in {\ZF} set
2265 theory. Here is an overview; see the files themselves for more details. I
2266 describe much of this material in other
2267 publications~\cite{paulson-set-I,paulson-set-II,paulson-CADE}.
2269 \item File \texttt{misc.ML} contains miscellaneous examples such as
2270 Cantor's Theorem, the Schr\"oder-Bernstein Theorem and the `Composition
2271 of homomorphisms' challenge~\cite{boyer86}.
2273 \item Theory \texttt{Ramsey} proves the finite exponent 2 version of
2274 Ramsey's Theorem, following Basin and Kaufmann's
2275 presentation~\cite{basin91}.
2277 \item Theory \texttt{Integ} develops a theory of the integers as
2278 equivalence classes of pairs of natural numbers.
2280 \item Theory \texttt{Primrec} develops some computation theory. It
2281 inductively defines the set of primitive recursive functions and presents a
2282 proof that Ackermann's function is not primitive recursive.
2284 \item Theory \texttt{Primes} defines the Greatest Common Divisor of two
2285 natural numbers and and the ``divides'' relation.
2287 \item Theory \texttt{Bin} defines a datatype for two's complement binary
2288 integers, then proves rewrite rules to perform binary arithmetic. For
2289 instance, $1359\times {-}2468 = {-}3354012$ takes under 14 seconds.
2291 \item Theory \texttt{BT} defines the recursive data structure ${\tt
2292 bt}(A)$, labelled binary trees.
2294 \item Theory \texttt{Term} defines a recursive data structure for terms
2295 and term lists. These are simply finite branching trees.
2297 \item Theory \texttt{TF} defines primitives for solving mutually
2298 recursive equations over sets. It constructs sets of trees and forests
2299 as an example, including induction and recursion rules that handle the
2302 \item Theory \texttt{Prop} proves soundness and completeness of
2303 propositional logic~\cite{paulson-set-II}. This illustrates datatype
2304 definitions, inductive definitions, structural induction and rule
2307 \item Theory \texttt{ListN} inductively defines the lists of $n$
2308 elements~\cite{paulin92}.
2310 \item Theory \texttt{Acc} inductively defines the accessible part of a
2311 relation~\cite{paulin92}.
2313 \item Theory \texttt{Comb} defines the datatype of combinators and
2314 inductively defines contraction and parallel contraction. It goes on to
2315 prove the Church-Rosser Theorem. This case study follows Camilleri and
2316 Melham~\cite{camilleri92}.
2318 \item Theory \texttt{LList} defines lazy lists and a coinduction
2319 principle for proving equations between them.
2323 \section{A proof about powersets}\label{sec:ZF-pow-example}
2324 To demonstrate high-level reasoning about subsets, let us prove the
2325 equation ${{\tt Pow}(A)\cap {\tt Pow}(B)}= {\tt Pow}(A\cap B)$. Compared
2326 with first-order logic, set theory involves a maze of rules, and theorems
2327 have many different proofs. Attempting other proofs of the theorem might
2328 be instructive. This proof exploits the lattice properties of
2329 intersection. It also uses the monotonicity of the powerset operation,
2330 from \texttt{ZF/mono.ML}:
2332 \tdx{Pow_mono} A<=B ==> Pow(A) <= Pow(B)
2334 We enter the goal and make the first step, which breaks the equation into
2335 two inclusions by extensionality:\index{*equalityI theorem}
2337 Goal "Pow(A Int B) = Pow(A) Int Pow(B)";
2339 {\out Pow(A Int B) = Pow(A) Int Pow(B)}
2340 {\out 1. Pow(A Int B) = Pow(A) Int Pow(B)}
2342 by (resolve_tac [equalityI] 1);
2344 {\out Pow(A Int B) = Pow(A) Int Pow(B)}
2345 {\out 1. Pow(A Int B) <= Pow(A) Int Pow(B)}
2346 {\out 2. Pow(A) Int Pow(B) <= Pow(A Int B)}
2348 Both inclusions could be tackled straightforwardly using \texttt{subsetI}.
2349 A shorter proof results from noting that intersection forms the greatest
2350 lower bound:\index{*Int_greatest theorem}
2352 by (resolve_tac [Int_greatest] 1);
2354 {\out Pow(A Int B) = Pow(A) Int Pow(B)}
2355 {\out 1. Pow(A Int B) <= Pow(A)}
2356 {\out 2. Pow(A Int B) <= Pow(B)}
2357 {\out 3. Pow(A) Int Pow(B) <= Pow(A Int B)}
2359 Subgoal~1 follows by applying the monotonicity of \texttt{Pow} to $A\int
2360 B\subseteq A$; subgoal~2 follows similarly:
2361 \index{*Int_lower1 theorem}\index{*Int_lower2 theorem}
2363 by (resolve_tac [Int_lower1 RS Pow_mono] 1);
2365 {\out Pow(A Int B) = Pow(A) Int Pow(B)}
2366 {\out 1. Pow(A Int B) <= Pow(B)}
2367 {\out 2. Pow(A) Int Pow(B) <= Pow(A Int B)}
2369 by (resolve_tac [Int_lower2 RS Pow_mono] 1);
2371 {\out Pow(A Int B) = Pow(A) Int Pow(B)}
2372 {\out 1. Pow(A) Int Pow(B) <= Pow(A Int B)}
2374 We are left with the opposite inclusion, which we tackle in the
2375 straightforward way:\index{*subsetI theorem}
2377 by (resolve_tac [subsetI] 1);
2379 {\out Pow(A Int B) = Pow(A) Int Pow(B)}
2380 {\out 1. !!x. x : Pow(A) Int Pow(B) ==> x : Pow(A Int B)}
2382 The subgoal is to show $x\in {\tt Pow}(A\cap B)$ assuming $x\in{\tt
2383 Pow}(A)\cap {\tt Pow}(B)$; eliminating this assumption produces two
2384 subgoals. The rule \tdx{IntE} treats the intersection like a conjunction
2385 instead of unfolding its definition.
2387 by (eresolve_tac [IntE] 1);
2389 {\out Pow(A Int B) = Pow(A) Int Pow(B)}
2390 {\out 1. !!x. [| x : Pow(A); x : Pow(B) |] ==> x : Pow(A Int B)}
2392 The next step replaces the \texttt{Pow} by the subset
2393 relation~($\subseteq$).\index{*PowI theorem}
2395 by (resolve_tac [PowI] 1);
2397 {\out Pow(A Int B) = Pow(A) Int Pow(B)}
2398 {\out 1. !!x. [| x : Pow(A); x : Pow(B) |] ==> x <= A Int B}
2400 We perform the same replacement in the assumptions. This is a good
2401 demonstration of the tactic \ttindex{dresolve_tac}:\index{*PowD theorem}
2403 by (REPEAT (dresolve_tac [PowD] 1));
2405 {\out Pow(A Int B) = Pow(A) Int Pow(B)}
2406 {\out 1. !!x. [| x <= A; x <= B |] ==> x <= A Int B}
2408 The assumptions are that $x$ is a lower bound of both $A$ and~$B$, but
2409 $A\int B$ is the greatest lower bound:\index{*Int_greatest theorem}
2411 by (resolve_tac [Int_greatest] 1);
2413 {\out Pow(A Int B) = Pow(A) Int Pow(B)}
2414 {\out 1. !!x. [| x <= A; x <= B |] ==> x <= A}
2415 {\out 2. !!x. [| x <= A; x <= B |] ==> x <= B}
2417 To conclude the proof, we clear up the trivial subgoals:
2419 by (REPEAT (assume_tac 1));
2421 {\out Pow(A Int B) = Pow(A) Int Pow(B)}
2425 We could have performed this proof in one step by applying
2426 \ttindex{Blast_tac}. Let us
2427 go back to the start:
2431 {\out Pow(A Int B) = Pow(A) Int Pow(B)}
2432 {\out 1. Pow(A Int B) = Pow(A) Int Pow(B)}
2439 {\out Pow(A Int B) = Pow(A) Int Pow(B)}
2442 Past researchers regarded this as a difficult proof, as indeed it is if all
2443 the symbols are replaced by their definitions.
2446 \section{Monotonicity of the union operator}
2447 For another example, we prove that general union is monotonic:
2448 ${C\subseteq D}$ implies $\bigcup(C)\subseteq \bigcup(D)$. To begin, we
2449 tackle the inclusion using \tdx{subsetI}:
2451 Goal "C<=D ==> Union(C) <= Union(D)";
2453 {\out C <= D ==> Union(C) <= Union(D)}
2454 {\out 1. C <= D ==> Union(C) <= Union(D)}
2456 by (resolve_tac [subsetI] 1);
2458 {\out C <= D ==> Union(C) <= Union(D)}
2459 {\out 1. !!x. [| C <= D; x : Union(C) |] ==> x : Union(D)}
2461 Big union is like an existential quantifier --- the occurrence in the
2462 assumptions must be eliminated early, since it creates parameters.
2463 \index{*UnionE theorem}
2465 by (eresolve_tac [UnionE] 1);
2467 {\out C <= D ==> Union(C) <= Union(D)}
2468 {\out 1. !!x B. [| C <= D; x : B; B : C |] ==> x : Union(D)}
2470 Now we may apply \tdx{UnionI}, which creates an unknown involving the
2471 parameters. To show $x\in \bigcup(D)$ it suffices to show that $x$ belongs
2472 to some element, say~$\Var{B2}(x,B)$, of~$D$.
2474 by (resolve_tac [UnionI] 1);
2476 {\out C <= D ==> Union(C) <= Union(D)}
2477 {\out 1. !!x B. [| C <= D; x : B; B : C |] ==> ?B2(x,B) : D}
2478 {\out 2. !!x B. [| C <= D; x : B; B : C |] ==> x : ?B2(x,B)}
2480 Combining \tdx{subsetD} with the assumption $C\subseteq D$ yields
2481 $\Var{a}\in C \Imp \Var{a}\in D$, which reduces subgoal~1. Note that
2482 \texttt{eresolve_tac} has removed that assumption.
2484 by (eresolve_tac [subsetD] 1);
2486 {\out C <= D ==> Union(C) <= Union(D)}
2487 {\out 1. !!x B. [| x : B; B : C |] ==> ?B2(x,B) : C}
2488 {\out 2. !!x B. [| C <= D; x : B; B : C |] ==> x : ?B2(x,B)}
2490 The rest is routine. Observe how~$\Var{B2}(x,B)$ is instantiated.
2494 {\out C <= D ==> Union(C) <= Union(D)}
2495 {\out 1. !!x B. [| C <= D; x : B; B : C |] ==> x : B}
2498 {\out C <= D ==> Union(C) <= Union(D)}
2501 Again, \ttindex{Blast_tac} can prove the theorem in one step.
2508 {\out C <= D ==> Union(C) <= Union(D)}
2512 The file \texttt{ZF/equalities.ML} has many similar proofs. Reasoning about
2513 general intersection can be difficult because of its anomalous behaviour on
2514 the empty set. However, \ttindex{Blast_tac} copes well with these. Here is
2515 a typical example, borrowed from Devlin~\cite[page 12]{devlin79}:
2517 a:C ==> (INT x:C. A(x) Int B(x)) = (INT x:C. A(x)) Int (INT x:C. B(x))
2519 In traditional notation this is
2520 \[ a\in C \,\Imp\, \inter@{x\in C} \Bigl(A(x) \int B(x)\Bigr) =
2521 \Bigl(\inter@{x\in C} A(x)\Bigr) \int
2522 \Bigl(\inter@{x\in C} B(x)\Bigr) \]
2524 \section{Low-level reasoning about functions}
2525 The derived rules \texttt{lamI}, \texttt{lamE}, \texttt{lam_type}, \texttt{beta}
2526 and \texttt{eta} support reasoning about functions in a
2527 $\lambda$-calculus style. This is generally easier than regarding
2528 functions as sets of ordered pairs. But sometimes we must look at the
2529 underlying representation, as in the following proof
2530 of~\tdx{fun_disjoint_apply1}. This states that if $f$ and~$g$ are
2531 functions with disjoint domains~$A$ and~$C$, and if $a\in A$, then
2534 Goal "[| a:A; f: A->B; g: C->D; A Int C = 0 |] ==> \ttback
2535 \ttback (f Un g)`a = f`a";
2537 {\out [| a : A; f : A -> B; g : C -> D; A Int C = 0 |]}
2538 {\out ==> (f Un g) ` a = f ` a}
2539 {\out 1. [| a : A; f : A -> B; g : C -> D; A Int C = 0 |]}
2540 {\out ==> (f Un g) ` a = f ` a}
2542 Using \tdx{apply_equality}, we reduce the equality to reasoning about
2543 ordered pairs. The second subgoal is to verify that $f\un g$ is a function.
2544 To save space, the assumptions will be abbreviated below.
2546 by (resolve_tac [apply_equality] 1);
2548 {\out [| \ldots |] ==> (f Un g) ` a = f ` a}
2549 {\out 1. [| \ldots |] ==> <a,f ` a> : f Un g}
2550 {\out 2. [| \ldots |] ==> f Un g : (PROD x:?A. ?B(x))}
2552 We must show that the pair belongs to~$f$ or~$g$; by~\tdx{UnI1} we
2555 by (resolve_tac [UnI1] 1);
2557 {\out [| \ldots |] ==> (f Un g) ` a = f ` a}
2558 {\out 1. [| \ldots |] ==> <a,f ` a> : f}
2559 {\out 2. [| \ldots |] ==> f Un g : (PROD x:?A. ?B(x))}
2561 To show $\pair{a,f`a}\in f$ we use \tdx{apply_Pair}, which is
2562 essentially the converse of \tdx{apply_equality}:
2564 by (resolve_tac [apply_Pair] 1);
2566 {\out [| \ldots |] ==> (f Un g) ` a = f ` a}
2567 {\out 1. [| \ldots |] ==> f : (PROD x:?A2. ?B2(x))}
2568 {\out 2. [| \ldots |] ==> a : ?A2}
2569 {\out 3. [| \ldots |] ==> f Un g : (PROD x:?A. ?B(x))}
2571 Using the assumptions $f\in A\to B$ and $a\in A$, we solve the two subgoals
2572 from \tdx{apply_Pair}. Recall that a $\Pi$-set is merely a generalized
2573 function space, and observe that~{\tt?A2} is instantiated to~\texttt{A}.
2577 {\out [| \ldots |] ==> (f Un g) ` a = f ` a}
2578 {\out 1. [| \ldots |] ==> a : A}
2579 {\out 2. [| \ldots |] ==> f Un g : (PROD x:?A. ?B(x))}
2582 {\out [| \ldots |] ==> (f Un g) ` a = f ` a}
2583 {\out 1. [| \ldots |] ==> f Un g : (PROD x:?A. ?B(x))}
2585 To construct functions of the form $f\un g$, we apply
2586 \tdx{fun_disjoint_Un}:
2588 by (resolve_tac [fun_disjoint_Un] 1);
2590 {\out [| \ldots |] ==> (f Un g) ` a = f ` a}
2591 {\out 1. [| \ldots |] ==> f : ?A3 -> ?B3}
2592 {\out 2. [| \ldots |] ==> g : ?C3 -> ?D3}
2593 {\out 3. [| \ldots |] ==> ?A3 Int ?C3 = 0}
2595 The remaining subgoals are instances of the assumptions. Again, observe how
2596 unknowns are instantiated:
2600 {\out [| \ldots |] ==> (f Un g) ` a = f ` a}
2601 {\out 1. [| \ldots |] ==> g : ?C3 -> ?D3}
2602 {\out 2. [| \ldots |] ==> A Int ?C3 = 0}
2605 {\out [| \ldots |] ==> (f Un g) ` a = f ` a}
2606 {\out 1. [| \ldots |] ==> A Int C = 0}
2609 {\out [| \ldots |] ==> (f Un g) ` a = f ` a}
2612 See the files \texttt{ZF/func.ML} and \texttt{ZF/WF.ML} for more
2613 examples of reasoning about functions.
2615 \index{set theory|)}