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. Some
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 Published articles~\cite{paulson-set-I,paulson-set-II} describe \texttt{ZF}
27 less formally than this chapter. Isabelle employs a novel treatment of
28 non-well-founded data structures within the standard {\sc zf} axioms including
29 the Axiom of Foundation~\cite{paulson-mscs}.
32 \section{Which version of axiomatic set theory?}
33 The two main axiom systems for set theory are Bernays-G\"odel~({\sc bg})
34 and Zermelo-Fraenkel~({\sc zf}). Resolution theorem provers can use {\sc
35 bg} because it is finite~\cite{boyer86,quaife92}. {\sc zf} does not
36 have a finite axiom system because of its Axiom Scheme of Replacement.
37 This makes it awkward to use with many theorem provers, since instances
38 of the axiom scheme have to be invoked explicitly. Since Isabelle has no
39 difficulty with axiom schemes, we may adopt either axiom system.
41 These two theories differ in their treatment of {\bf classes}, which are
42 collections that are `too big' to be sets. The class of all sets,~$V$,
43 cannot be a set without admitting Russell's Paradox. In {\sc bg}, both
44 classes and sets are individuals; $x\in V$ expresses that $x$ is a set. In
45 {\sc zf}, all variables denote sets; classes are identified with unary
46 predicates. The two systems define essentially the same sets and classes,
47 with similar properties. In particular, a class cannot belong to another
48 class (let alone a set).
50 Modern set theorists tend to prefer {\sc zf} because they are mainly concerned
51 with sets, rather than classes. {\sc bg} requires tiresome proofs that various
52 collections are sets; for instance, showing $x\in\{x\}$ requires showing that
59 \it name &\it meta-type & \it description \\
60 \cdx{Let} & $[\alpha,\alpha\To\beta]\To\beta$ & let binder\\
61 \cdx{0} & $i$ & empty set\\
62 \cdx{cons} & $[i,i]\To i$ & finite set constructor\\
63 \cdx{Upair} & $[i,i]\To i$ & unordered pairing\\
64 \cdx{Pair} & $[i,i]\To i$ & ordered pairing\\
65 \cdx{Inf} & $i$ & infinite set\\
66 \cdx{Pow} & $i\To i$ & powerset\\
67 \cdx{Union} \cdx{Inter} & $i\To i$ & set union/intersection \\
68 \cdx{split} & $[[i,i]\To i, i] \To i$ & generalized projection\\
69 \cdx{fst} \cdx{snd} & $i\To i$ & projections\\
70 \cdx{converse}& $i\To i$ & converse of a relation\\
71 \cdx{succ} & $i\To i$ & successor\\
72 \cdx{Collect} & $[i,i\To o]\To i$ & separation\\
73 \cdx{Replace} & $[i, [i,i]\To o] \To i$ & replacement\\
74 \cdx{PrimReplace} & $[i, [i,i]\To o] \To i$ & primitive replacement\\
75 \cdx{RepFun} & $[i, i\To i] \To i$ & functional replacement\\
76 \cdx{Pi} \cdx{Sigma} & $[i,i\To i]\To i$ & general product/sum\\
77 \cdx{domain} & $i\To i$ & domain of a relation\\
78 \cdx{range} & $i\To i$ & range of a relation\\
79 \cdx{field} & $i\To i$ & field of a relation\\
80 \cdx{Lambda} & $[i, i\To i]\To i$ & $\lambda$-abstraction\\
81 \cdx{restrict}& $[i, i] \To i$ & restriction of a function\\
82 \cdx{The} & $[i\To o]\To i$ & definite description\\
83 \cdx{if} & $[o,i,i]\To i$ & conditional\\
84 \cdx{Ball} \cdx{Bex} & $[i, i\To o]\To o$ & bounded quantifiers
87 \subcaption{Constants}
91 \index{*"-"`"` symbol}
92 \index{*"` symbol}\index{function applications}
97 \it symbol & \it meta-type & \it priority & \it description \\
98 \tt `` & $[i,i]\To i$ & Left 90 & image \\
99 \tt -`` & $[i,i]\To i$ & Left 90 & inverse image \\
100 \tt ` & $[i,i]\To i$ & Left 90 & application \\
101 \sdx{Int} & $[i,i]\To i$ & Left 70 & intersection ($\int$) \\
102 \sdx{Un} & $[i,i]\To i$ & Left 65 & union ($\un$) \\
103 \tt - & $[i,i]\To i$ & Left 65 & set difference ($-$) \\[1ex]
104 \tt: & $[i,i]\To o$ & Left 50 & membership ($\in$) \\
105 \tt <= & $[i,i]\To o$ & Left 50 & subset ($\subseteq$)
109 \caption{Constants of ZF} \label{zf-constants}
113 \section{The syntax of set theory}
114 The language of set theory, as studied by logicians, has no constants. The
115 traditional axioms merely assert the existence of empty sets, unions,
116 powersets, etc.; this would be intolerable for practical reasoning. The
117 Isabelle theory declares constants for primitive sets. It also extends
118 \texttt{FOL} with additional syntax for finite sets, ordered pairs,
119 comprehension, general union/intersection, general sums/products, and
120 bounded quantifiers. In most other respects, Isabelle implements precisely
121 Zermelo-Fraenkel set theory.
123 Figure~\ref{zf-constants} lists the constants and infixes of~ZF, while
124 Figure~\ref{zf-trans} presents the syntax translations. Finally,
125 Figure~\ref{zf-syntax} presents the full grammar for set theory, including the
128 Local abbreviations can be introduced by a \isa{let} construct whose
129 syntax appears in Fig.\ts\ref{zf-syntax}. Internally it is translated into
130 the constant~\cdx{Let}. It can be expanded by rewriting with its
131 definition, \tdx{Let_def}.
133 Apart from \isa{let}, set theory does not use polymorphism. All terms in
134 ZF have type~\tydx{i}, which is the type of individuals and has
135 class~\cldx{term}. The type of first-order formulae, remember,
138 Infix operators include binary union and intersection ($A\un B$ and
139 $A\int B$), set difference ($A-B$), and the subset and membership
140 relations. Note that $a$\verb|~:|$b$ is translated to $\lnot(a\in b)$,
141 which is equivalent to $a\notin b$. The
142 union and intersection operators ($\bigcup A$ and $\bigcap A$) form the
143 union or intersection of a set of sets; $\bigcup A$ means the same as
144 $\bigcup@{x\in A}x$. Of these operators, only $\bigcup A$ is primitive.
146 The constant \cdx{Upair} constructs unordered pairs; thus \isa{Upair($A$,$B$)} denotes the set~$\{A,B\}$ and
147 \isa{Upair($A$,$A$)} denotes the singleton~$\{A\}$. General union is
148 used to define binary union. The Isabelle version goes on to define
152 A\cup B & \equiv & \bigcup(\isa{Upair}(A,B)) \\
153 \isa{cons}(a,B) & \equiv & \isa{Upair}(a,a) \un B
155 The $\{a@1, \ldots\}$ notation abbreviates finite sets constructed in the
156 obvious manner using~\isa{cons} and~$\emptyset$ (the empty set) \isasymin \begin{eqnarray*}
157 \{a,b,c\} & \equiv & \isa{cons}(a,\isa{cons}(b,\isa{cons}(c,\emptyset)))
160 The constant \cdx{Pair} constructs ordered pairs, as in \isa{Pair($a$,$b$)}. Ordered pairs may also be written within angle brackets,
161 as {\tt<$a$,$b$>}. The $n$-tuple {\tt<$a@1$,\ldots,$a@{n-1}$,$a@n$>}
162 abbreviates the nest of pairs\par\nobreak
163 \centerline{\isa{Pair($a@1$,\ldots,Pair($a@{n-1}$,$a@n$)\ldots).}}
165 In ZF, a function is a set of pairs. A ZF function~$f$ is simply an
166 individual as far as Isabelle is concerned: its Isabelle type is~$i$, not say
167 $i\To i$. The infix operator~{\tt`} denotes the application of a function set
168 to its argument; we must write~$f{\tt`}x$, not~$f(x)$. The syntax for image
169 is~$f{\tt``}A$ and that for inverse image is~$f{\tt-``}A$.
173 \index{lambda abs@$\lambda$-abstractions}
176 \begin{center} \footnotesize\tt\frenchspacing
178 \it external & \it internal & \it description \\
179 $a$ \ttilde: $b$ & \ttilde($a$ : $b$) & \rm negated membership\\
180 \ttlbrace$a@1$, $\ldots$, $a@n$\ttrbrace & cons($a@1$,$\ldots$,cons($a@n$,0)) &
182 <$a@1$, $\ldots$, $a@{n-1}$, $a@n$> &
183 Pair($a@1$,\ldots,Pair($a@{n-1}$,$a@n$)\ldots) &
184 \rm ordered $n$-tuple \\
185 \ttlbrace$x$:$A . P[x]$\ttrbrace & Collect($A$,$\lambda x. P[x]$) &
187 \ttlbrace$y . x$:$A$, $Q[x,y]$\ttrbrace & Replace($A$,$\lambda x\,y. Q[x,y]$) &
189 \ttlbrace$b[x] . x$:$A$\ttrbrace & RepFun($A$,$\lambda x. b[x]$) &
190 \rm functional replacement \\
191 \sdx{INT} $x$:$A . B[x]$ & Inter(\ttlbrace$B[x] . x$:$A$\ttrbrace) &
192 \rm general intersection \\
193 \sdx{UN} $x$:$A . B[x]$ & Union(\ttlbrace$B[x] . x$:$A$\ttrbrace) &
195 \sdx{PROD} $x$:$A . B[x]$ & Pi($A$,$\lambda x. B[x]$) &
196 \rm general product \\
197 \sdx{SUM} $x$:$A . B[x]$ & Sigma($A$,$\lambda x. B[x]$) &
199 $A$ -> $B$ & Pi($A$,$\lambda x. B$) &
200 \rm function space \\
201 $A$ * $B$ & Sigma($A$,$\lambda x. B$) &
202 \rm binary product \\
203 \sdx{THE} $x . P[x]$ & The($\lambda x. P[x]$) &
204 \rm definite description \\
205 \sdx{lam} $x$:$A . b[x]$ & Lambda($A$,$\lambda x. b[x]$) &
206 \rm $\lambda$-abstraction\\[1ex]
207 \sdx{ALL} $x$:$A . P[x]$ & Ball($A$,$\lambda x. P[x]$) &
208 \rm bounded $\forall$ \\
209 \sdx{EX} $x$:$A . P[x]$ & Bex($A$,$\lambda x. P[x]$) &
210 \rm bounded $\exists$
213 \caption{Translations for ZF} \label{zf-trans}
222 term & = & \hbox{expression of type~$i$} \\
223 & | & "let"~id~"="~term";"\dots";"~id~"="~term~"in"~term \\
224 & | & "if"~term~"then"~term~"else"~term \\
225 & | & "{\ttlbrace} " term\; ("," term)^* " {\ttrbrace}" \\
226 & | & "< " term\; ("," term)^* " >" \\
227 & | & "{\ttlbrace} " id ":" term " . " formula " {\ttrbrace}" \\
228 & | & "{\ttlbrace} " id " . " id ":" term ", " formula " {\ttrbrace}" \\
229 & | & "{\ttlbrace} " term " . " id ":" term " {\ttrbrace}" \\
230 & | & term " `` " term \\
231 & | & term " -`` " term \\
232 & | & term " ` " term \\
233 & | & term " * " term \\
234 & | & term " \isasyminter " term \\
235 & | & term " \isasymunion " term \\
236 & | & term " - " term \\
237 & | & term " -> " term \\
238 & | & "THE~~" id " . " formula\\
239 & | & "lam~~" id ":" term " . " term \\
240 & | & "INT~~" id ":" term " . " term \\
241 & | & "UN~~~" id ":" term " . " term \\
242 & | & "PROD~" id ":" term " . " term \\
243 & | & "SUM~~" id ":" term " . " term \\[2ex]
244 formula & = & \hbox{expression of type~$o$} \\
245 & | & term " : " term \\
246 & | & term " \ttilde: " term \\
247 & | & term " <= " term \\
248 & | & term " = " term \\
249 & | & term " \ttilde= " term \\
250 & | & "\ttilde\ " formula \\
251 & | & formula " \& " formula \\
252 & | & formula " | " formula \\
253 & | & formula " --> " formula \\
254 & | & formula " <-> " formula \\
255 & | & "ALL " id ":" term " . " formula \\
256 & | & "EX~~" id ":" term " . " formula \\
257 & | & "ALL~" id~id^* " . " formula \\
258 & | & "EX~~" id~id^* " . " formula \\
259 & | & "EX!~" id~id^* " . " formula
262 \caption{Full grammar for ZF} \label{zf-syntax}
266 \section{Binding operators}
267 The constant \cdx{Collect} constructs sets by the principle of {\bf
268 separation}. The syntax for separation is
269 \hbox{\tt\ttlbrace$x$:$A$.\ $P[x]$\ttrbrace}, where $P[x]$ is a formula
270 that may contain free occurrences of~$x$. It abbreviates the set \isa{Collect($A$,$\lambda x. P[x]$)}, which consists of all $x\in A$ that
271 satisfy~$P[x]$. Note that \isa{Collect} is an unfortunate choice of
272 name: some set theories adopt a set-formation principle, related to
273 replacement, called collection.
275 The constant \cdx{Replace} constructs sets by the principle of {\bf
276 replacement}. The syntax
277 \hbox{\tt\ttlbrace$y$.\ $x$:$A$,$Q[x,y]$\ttrbrace} denotes the set
278 \isa{Replace($A$,$\lambda x\,y. Q[x,y]$)}, which consists of all~$y$ such
279 that there exists $x\in A$ satisfying~$Q[x,y]$. The Replacement Axiom
280 has the condition that $Q$ must be single-valued over~$A$: for
281 all~$x\in A$ there exists at most one $y$ satisfying~$Q[x,y]$. A
282 single-valued binary predicate is also called a {\bf class function}.
284 The constant \cdx{RepFun} expresses a special case of replacement,
285 where $Q[x,y]$ has the form $y=b[x]$. Such a $Q$ is trivially
286 single-valued, since it is just the graph of the meta-level
287 function~$\lambda x. b[x]$. The resulting set consists of all $b[x]$
288 for~$x\in A$. This is analogous to the \ML{} functional \isa{map},
289 since it applies a function to every element of a set. The syntax is
290 \isa{\ttlbrace$b[x]$.\ $x$:$A$\ttrbrace}, which expands to
291 \isa{RepFun($A$,$\lambda x. b[x]$)}.
293 \index{*INT symbol}\index{*UN symbol}
294 General unions and intersections of indexed
295 families of sets, namely $\bigcup@{x\in A}B[x]$ and $\bigcap@{x\in A}B[x]$,
296 are written \isa{UN $x$:$A$.\ $B[x]$} and \isa{INT $x$:$A$.\ $B[x]$}.
297 Their meaning is expressed using \isa{RepFun} as
299 \bigcup(\{B[x]. x\in A\}) \qquad\hbox{and}\qquad
300 \bigcap(\{B[x]. x\in A\}).
302 General sums $\sum@{x\in A}B[x]$ and products $\prod@{x\in A}B[x]$ can be
303 constructed in set theory, where $B[x]$ is a family of sets over~$A$. They
304 have as special cases $A\times B$ and $A\to B$, where $B$ is simply a set.
305 This is similar to the situation in Constructive Type Theory (set theory
306 has `dependent sets') and calls for similar syntactic conventions. The
307 constants~\cdx{Sigma} and~\cdx{Pi} construct general sums and
308 products. Instead of \isa{Sigma($A$,$B$)} and \isa{Pi($A$,$B$)} we may
310 \isa{SUM $x$:$A$.\ $B[x]$} and \isa{PROD $x$:$A$.\ $B[x]$}.
311 \index{*SUM symbol}\index{*PROD symbol}%
312 The special cases as \hbox{\tt$A$*$B$} and \hbox{\tt$A$->$B$} abbreviate
313 general sums and products over a constant family.\footnote{Unlike normal
314 infix operators, {\tt*} and {\tt->} merely define abbreviations; there are
315 no constants~\isa{op~*} and~\isa{op~->}.} Isabelle accepts these
316 abbreviations in parsing and uses them whenever possible for printing.
318 \index{*THE symbol} As mentioned above, whenever the axioms assert the
319 existence and uniqueness of a set, Isabelle's set theory declares a constant
320 for that set. These constants can express the {\bf definite description}
321 operator~$\iota x. P[x]$, which stands for the unique~$a$ satisfying~$P[a]$,
322 if such exists. Since all terms in ZF denote something, a description is
323 always meaningful, but we do not know its value unless $P[x]$ defines it
324 uniquely. Using the constant~\cdx{The}, we may write descriptions as
325 \isa{The($\lambda x. P[x]$)} or use the syntax \isa{THE $x$.\ $P[x]$}.
328 Function sets may be written in $\lambda$-notation; $\lambda x\in A. b[x]$
329 stands for the set of all pairs $\pair{x,b[x]}$ for $x\in A$. In order for
330 this to be a set, the function's domain~$A$ must be given. Using the
331 constant~\cdx{Lambda}, we may express function sets as \isa{Lambda($A$,$\lambda x. b[x]$)} or use the syntax \isa{lam $x$:$A$.\ $b[x]$}.
333 Isabelle's set theory defines two {\bf bounded quantifiers}:
335 \forall x\in A. P[x] &\hbox{abbreviates}& \forall x. x\in A\imp P[x] \\
336 \exists x\in A. P[x] &\hbox{abbreviates}& \exists x. x\in A\conj P[x]
338 The constants~\cdx{Ball} and~\cdx{Bex} are defined
339 accordingly. Instead of \isa{Ball($A$,$P$)} and \isa{Bex($A$,$P$)} we may
341 \isa{ALL $x$:$A$.\ $P[x]$} and \isa{EX $x$:$A$.\ $P[x]$}.
347 \begin{alltt*}\isastyleminor
348 \tdx{Let_def}: Let(s, f) == f(s)
350 \tdx{Ball_def}: Ball(A,P) == {\isasymforall}x. x \isasymin A --> P(x)
351 \tdx{Bex_def}: Bex(A,P) == {\isasymexists}x. x \isasymin A & P(x)
353 \tdx{subset_def}: A \isasymsubseteq B == {\isasymforall}x \isasymin A. x \isasymin B
354 \tdx{extension}: A = B <-> A \isasymsubseteq B & B \isasymsubseteq A
356 \tdx{Union_iff}: A \isasymin Union(C) <-> ({\isasymexists}B \isasymin C. A \isasymin B)
357 \tdx{Pow_iff}: A \isasymin Pow(B) <-> A \isasymsubseteq B
358 \tdx{foundation}: A=0 | ({\isasymexists}x \isasymin A. {\isasymforall}y \isasymin x. y \isasymnotin A)
360 \tdx{replacement}: ({\isasymforall}x \isasymin A. {\isasymforall}y z. P(x,y) & P(x,z) --> y=z) ==>
361 b \isasymin PrimReplace(A,P) <-> ({\isasymexists}x{\isasymin}A. P(x,b))
362 \subcaption{The Zermelo-Fraenkel Axioms}
364 \tdx{Replace_def}: Replace(A,P) ==
365 PrimReplace(A, \%x y. (\isasymexists!z. P(x,z)) & P(x,y))
366 \tdx{RepFun_def}: RepFun(A,f) == {\ttlbrace}y . x \isasymin A, y=f(x)\ttrbrace
367 \tdx{the_def}: The(P) == Union({\ttlbrace}y . x \isasymin {\ttlbrace}0{\ttrbrace}, P(y){\ttrbrace})
368 \tdx{if_def}: if(P,a,b) == THE z. P & z=a | ~P & z=b
369 \tdx{Collect_def}: Collect(A,P) == {\ttlbrace}y . x \isasymin A, x=y & P(x){\ttrbrace}
370 \tdx{Upair_def}: Upair(a,b) ==
371 {\ttlbrace}y. x\isasymin{}Pow(Pow(0)), x=0 & y=a | x=Pow(0) & y=b{\ttrbrace}
372 \subcaption{Consequences of replacement}
374 \tdx{Inter_def}: Inter(A) == {\ttlbrace}x \isasymin Union(A) . {\isasymforall}y \isasymin A. x \isasymin y{\ttrbrace}
375 \tdx{Un_def}: A \isasymunion B == Union(Upair(A,B))
376 \tdx{Int_def}: A \isasyminter B == Inter(Upair(A,B))
377 \tdx{Diff_def}: A - B == {\ttlbrace}x \isasymin A . x \isasymnotin B{\ttrbrace}
378 \subcaption{Union, intersection, difference}
380 \caption{Rules and axioms of ZF} \label{zf-rules}
385 \begin{alltt*}\isastyleminor
386 \tdx{cons_def}: cons(a,A) == Upair(a,a) \isasymunion A
387 \tdx{succ_def}: succ(i) == cons(i,i)
388 \tdx{infinity}: 0 \isasymin Inf & ({\isasymforall}y \isasymin Inf. succ(y) \isasymin Inf)
389 \subcaption{Finite and infinite sets}
391 \tdx{Pair_def}: <a,b> == {\ttlbrace}{\ttlbrace}a,a{\ttrbrace}, {\ttlbrace}a,b{\ttrbrace}{\ttrbrace}
392 \tdx{split_def}: split(c,p) == THE y. {\isasymexists}a b. p=<a,b> & y=c(a,b)
393 \tdx{fst_def}: fst(A) == split(\%x y. x, p)
394 \tdx{snd_def}: snd(A) == split(\%x y. y, p)
395 \tdx{Sigma_def}: Sigma(A,B) == {\isasymUnion}x \isasymin A. {\isasymUnion}y \isasymin B(x). {\ttlbrace}<x,y>{\ttrbrace}
396 \subcaption{Ordered pairs and Cartesian products}
398 \tdx{converse_def}: converse(r) == {\ttlbrace}z. w\isasymin{}r, {\isasymexists}x y. w=<x,y> & z=<y,x>{\ttrbrace}
399 \tdx{domain_def}: domain(r) == {\ttlbrace}x. w \isasymin r, {\isasymexists}y. w=<x,y>{\ttrbrace}
400 \tdx{range_def}: range(r) == domain(converse(r))
401 \tdx{field_def}: field(r) == domain(r) \isasymunion range(r)
402 \tdx{image_def}: r `` A == {\ttlbrace}y\isasymin{}range(r) . {\isasymexists}x \isasymin A. <x,y> \isasymin r{\ttrbrace}
403 \tdx{vimage_def}: r -`` A == converse(r)``A
404 \subcaption{Operations on relations}
406 \tdx{lam_def}: Lambda(A,b) == {\ttlbrace}<x,b(x)> . x \isasymin A{\ttrbrace}
407 \tdx{apply_def}: f`a == THE y. <a,y> \isasymin f
408 \tdx{Pi_def}: Pi(A,B) == {\ttlbrace}f\isasymin{}Pow(Sigma(A,B)). {\isasymforall}x\isasymin{}A. \isasymexists!y. <x,y>\isasymin{}f{\ttrbrace}
409 \tdx{restrict_def}: restrict(f,A) == lam x \isasymin A. f`x
410 \subcaption{Functions and general product}
412 \caption{Further definitions of ZF} \label{zf-defs}
417 \section{The Zermelo-Fraenkel axioms}
418 The axioms appear in Fig.\ts \ref{zf-rules}. They resemble those
419 presented by Suppes~\cite{suppes72}. Most of the theory consists of
420 definitions. In particular, bounded quantifiers and the subset relation
421 appear in other axioms. Object-level quantifiers and implications have
422 been replaced by meta-level ones wherever possible, to simplify use of the
425 The traditional replacement axiom asserts
426 \[ y \in \isa{PrimReplace}(A,P) \bimp (\exists x\in A. P(x,y)) \]
427 subject to the condition that $P(x,y)$ is single-valued for all~$x\in A$.
428 The Isabelle theory defines \cdx{Replace} to apply
429 \cdx{PrimReplace} to the single-valued part of~$P$, namely
430 \[ (\exists!z. P(x,z)) \conj P(x,y). \]
431 Thus $y\in \isa{Replace}(A,P)$ if and only if there is some~$x$ such that
432 $P(x,-)$ holds uniquely for~$y$. Because the equivalence is unconditional,
433 \isa{Replace} is much easier to use than \isa{PrimReplace}; it defines the
434 same set, if $P(x,y)$ is single-valued. The nice syntax for replacement
435 expands to \isa{Replace}.
437 Other consequences of replacement include replacement for
439 (\cdx{RepFun}) and definite descriptions (\cdx{The}).
440 Axioms for separation (\cdx{Collect}) and unordered pairs
441 (\cdx{Upair}) are traditionally assumed, but they actually follow
442 from replacement~\cite[pages 237--8]{suppes72}.
444 The definitions of general intersection, etc., are straightforward. Note
445 the definition of \isa{cons}, which underlies the finite set notation.
446 The axiom of infinity gives us a set that contains~0 and is closed under
447 successor (\cdx{succ}). Although this set is not uniquely defined,
448 the theory names it (\cdx{Inf}) in order to simplify the
449 construction of the natural numbers.
451 Further definitions appear in Fig.\ts\ref{zf-defs}. Ordered pairs are
452 defined in the standard way, $\pair{a,b}\equiv\{\{a\},\{a,b\}\}$. Recall
453 that \cdx{Sigma}$(A,B)$ generalizes the Cartesian product of two
454 sets. It is defined to be the union of all singleton sets
455 $\{\pair{x,y}\}$, for $x\in A$ and $y\in B(x)$. This is a typical usage of
458 The projections \cdx{fst} and~\cdx{snd} are defined in terms of the
459 generalized projection \cdx{split}. The latter has been borrowed from
460 Martin-L\"of's Type Theory, and is often easier to use than \cdx{fst}
463 Operations on relations include converse, domain, range, and image. The
464 set $\isa{Pi}(A,B)$ generalizes the space of functions between two sets.
465 Note the simple definitions of $\lambda$-abstraction (using
466 \cdx{RepFun}) and application (using a definite description). The
467 function \cdx{restrict}$(f,A)$ has the same values as~$f$, but only
474 \begin{alltt*}\isastyleminor
475 \tdx{ballI}: [| !!x. x\isasymin{}A ==> P(x) |] ==> {\isasymforall}x\isasymin{}A. P(x)
476 \tdx{bspec}: [| {\isasymforall}x\isasymin{}A. P(x); x\isasymin{}A |] ==> P(x)
477 \tdx{ballE}: [| {\isasymforall}x\isasymin{}A. P(x); P(x) ==> Q; x \isasymnotin A ==> Q |] ==> Q
479 \tdx{ball_cong}: [| A=A'; !!x. x\isasymin{}A' ==> P(x) <-> P'(x) |] ==>
480 ({\isasymforall}x\isasymin{}A. P(x)) <-> ({\isasymforall}x\isasymin{}A'. P'(x))
482 \tdx{bexI}: [| P(x); x\isasymin{}A |] ==> {\isasymexists}x\isasymin{}A. P(x)
483 \tdx{bexCI}: [| {\isasymforall}x\isasymin{}A. ~P(x) ==> P(a); a\isasymin{}A |] ==> {\isasymexists}x\isasymin{}A. P(x)
484 \tdx{bexE}: [| {\isasymexists}x\isasymin{}A. P(x); !!x. [| x\isasymin{}A; P(x) |] ==> Q |] ==> Q
486 \tdx{bex_cong}: [| A=A'; !!x. x\isasymin{}A' ==> P(x) <-> P'(x) |] ==>
487 ({\isasymexists}x\isasymin{}A. P(x)) <-> ({\isasymexists}x\isasymin{}A'. P'(x))
488 \subcaption{Bounded quantifiers}
490 \tdx{subsetI}: (!!x. x \isasymin A ==> x \isasymin B) ==> A \isasymsubseteq B
491 \tdx{subsetD}: [| A \isasymsubseteq B; c \isasymin A |] ==> c \isasymin B
492 \tdx{subsetCE}: [| A \isasymsubseteq B; c \isasymnotin A ==> P; c \isasymin B ==> P |] ==> P
493 \tdx{subset_refl}: A \isasymsubseteq A
494 \tdx{subset_trans}: [| A \isasymsubseteq B; B \isasymsubseteq C |] ==> A \isasymsubseteq C
496 \tdx{equalityI}: [| A \isasymsubseteq B; B \isasymsubseteq A |] ==> A = B
497 \tdx{equalityD1}: A = B ==> A \isasymsubseteq B
498 \tdx{equalityD2}: A = B ==> B \isasymsubseteq A
499 \tdx{equalityE}: [| A = B; [| A \isasymsubseteq B; B \isasymsubseteq A |] ==> P |] ==> P
500 \subcaption{Subsets and extensionality}
502 \tdx{emptyE}: a \isasymin 0 ==> P
503 \tdx{empty_subsetI}: 0 \isasymsubseteq A
504 \tdx{equals0I}: [| !!y. y \isasymin A ==> False |] ==> A=0
505 \tdx{equals0D}: [| A=0; a \isasymin A |] ==> P
507 \tdx{PowI}: A \isasymsubseteq B ==> A \isasymin Pow(B)
508 \tdx{PowD}: A \isasymin Pow(B) ==> A \isasymsubseteq B
509 \subcaption{The empty set; power sets}
511 \caption{Basic derived rules for ZF} \label{zf-lemmas1}
515 \section{From basic lemmas to function spaces}
516 Faced with so many definitions, it is essential to prove lemmas. Even
517 trivial theorems like $A \int B = B \int A$ would be difficult to
518 prove from the definitions alone. Isabelle's set theory derives many
519 rules using a natural deduction style. Ideally, a natural deduction
520 rule should introduce or eliminate just one operator, but this is not
521 always practical. For most operators, we may forget its definition
522 and use its derived rules instead.
524 \subsection{Fundamental lemmas}
525 Figure~\ref{zf-lemmas1} presents the derived rules for the most basic
526 operators. The rules for the bounded quantifiers resemble those for the
527 ordinary quantifiers, but note that \tdx{ballE} uses a negated assumption
528 in the style of Isabelle's classical reasoner. The \rmindex{congruence
529 rules} \tdx{ball_cong} and \tdx{bex_cong} are required by Isabelle's
530 simplifier, but have few other uses. Congruence rules must be specially
531 derived for all binding operators, and henceforth will not be shown.
533 Figure~\ref{zf-lemmas1} also shows rules for the subset and equality
534 relations (proof by extensionality), and rules about the empty set and the
537 Figure~\ref{zf-lemmas2} presents rules for replacement and separation.
538 The rules for \cdx{Replace} and \cdx{RepFun} are much simpler than
539 comparable rules for \isa{PrimReplace} would be. The principle of
540 separation is proved explicitly, although most proofs should use the
541 natural deduction rules for \isa{Collect}. The elimination rule
542 \tdx{CollectE} is equivalent to the two destruction rules
543 \tdx{CollectD1} and \tdx{CollectD2}, but each rule is suited to
544 particular circumstances. Although too many rules can be confusing, there
545 is no reason to aim for a minimal set of rules.
547 Figure~\ref{zf-lemmas3} presents rules for general union and intersection.
548 The empty intersection should be undefined. We cannot have
549 $\bigcap(\emptyset)=V$ because $V$, the universal class, is not a set. All
550 expressions denote something in ZF set theory; the definition of
551 intersection implies $\bigcap(\emptyset)=\emptyset$, but this value is
552 arbitrary. The rule \tdx{InterI} must have a premise to exclude
553 the empty intersection. Some of the laws governing intersections require
557 %the [p] gives better page breaking for the book
559 \begin{alltt*}\isastyleminor
560 \tdx{ReplaceI}: [| x\isasymin{}A; P(x,b); !!y. P(x,y) ==> y=b |] ==>
561 b\isasymin{}{\ttlbrace}y. x\isasymin{}A, P(x,y){\ttrbrace}
563 \tdx{ReplaceE}: [| b\isasymin{}{\ttlbrace}y. x\isasymin{}A, P(x,y){\ttrbrace};
564 !!x. [| x\isasymin{}A; P(x,b); {\isasymforall}y. P(x,y)-->y=b |] ==> R
567 \tdx{RepFunI}: [| a\isasymin{}A |] ==> f(a)\isasymin{}{\ttlbrace}f(x). x\isasymin{}A{\ttrbrace}
568 \tdx{RepFunE}: [| b\isasymin{}{\ttlbrace}f(x). x\isasymin{}A{\ttrbrace};
569 !!x.[| x\isasymin{}A; b=f(x) |] ==> P |] ==> P
571 \tdx{separation}: a\isasymin{}{\ttlbrace}x\isasymin{}A. P(x){\ttrbrace} <-> a\isasymin{}A & P(a)
572 \tdx{CollectI}: [| a\isasymin{}A; P(a) |] ==> a\isasymin{}{\ttlbrace}x\isasymin{}A. P(x){\ttrbrace}
573 \tdx{CollectE}: [| a\isasymin{}{\ttlbrace}x\isasymin{}A. P(x){\ttrbrace}; [| a\isasymin{}A; P(a) |] ==> R |] ==> R
574 \tdx{CollectD1}: a\isasymin{}{\ttlbrace}x\isasymin{}A. P(x){\ttrbrace} ==> a\isasymin{}A
575 \tdx{CollectD2}: a\isasymin{}{\ttlbrace}x\isasymin{}A. P(x){\ttrbrace} ==> P(a)
577 \caption{Replacement and separation} \label{zf-lemmas2}
582 \begin{alltt*}\isastyleminor
583 \tdx{UnionI}: [| B\isasymin{}C; A\isasymin{}B |] ==> A\isasymin{}Union(C)
584 \tdx{UnionE}: [| A\isasymin{}Union(C); !!B.[| A\isasymin{}B; B\isasymin{}C |] ==> R |] ==> R
586 \tdx{InterI}: [| !!x. x\isasymin{}C ==> A\isasymin{}x; c\isasymin{}C |] ==> A\isasymin{}Inter(C)
587 \tdx{InterD}: [| A\isasymin{}Inter(C); B\isasymin{}C |] ==> A\isasymin{}B
588 \tdx{InterE}: [| A\isasymin{}Inter(C); A\isasymin{}B ==> R; B \isasymnotin C ==> R |] ==> R
590 \tdx{UN_I}: [| a\isasymin{}A; b\isasymin{}B(a) |] ==> b\isasymin{}({\isasymUnion}x\isasymin{}A. B(x))
591 \tdx{UN_E}: [| b\isasymin{}({\isasymUnion}x\isasymin{}A. B(x)); !!x.[| x\isasymin{}A; b\isasymin{}B(x) |] ==> R
594 \tdx{INT_I}: [| !!x. x\isasymin{}A ==> b\isasymin{}B(x); a\isasymin{}A |] ==> b\isasymin{}({\isasymInter}x\isasymin{}A. B(x))
595 \tdx{INT_E}: [| b\isasymin{}({\isasymInter}x\isasymin{}A. B(x)); a\isasymin{}A |] ==> b\isasymin{}B(a)
597 \caption{General union and intersection} \label{zf-lemmas3}
604 \begin{alltt*}\isastyleminor
605 \tdx{pairing}: a\isasymin{}Upair(b,c) <-> (a=b | a=c)
606 \tdx{UpairI1}: a\isasymin{}Upair(a,b)
607 \tdx{UpairI2}: b\isasymin{}Upair(a,b)
608 \tdx{UpairE}: [| a\isasymin{}Upair(b,c); a=b ==> P; a=c ==> P |] ==> P
610 \caption{Unordered pairs} \label{zf-upair1}
615 \begin{alltt*}\isastyleminor
616 \tdx{UnI1}: c\isasymin{}A ==> c\isasymin{}A \isasymunion B
617 \tdx{UnI2}: c\isasymin{}B ==> c\isasymin{}A \isasymunion B
618 \tdx{UnCI}: (c \isasymnotin B ==> c\isasymin{}A) ==> c\isasymin{}A \isasymunion B
619 \tdx{UnE}: [| c\isasymin{}A \isasymunion B; c\isasymin{}A ==> P; c\isasymin{}B ==> P |] ==> P
621 \tdx{IntI}: [| c\isasymin{}A; c\isasymin{}B |] ==> c\isasymin{}A \isasyminter B
622 \tdx{IntD1}: c\isasymin{}A \isasyminter B ==> c\isasymin{}A
623 \tdx{IntD2}: c\isasymin{}A \isasyminter B ==> c\isasymin{}B
624 \tdx{IntE}: [| c\isasymin{}A \isasyminter B; [| c\isasymin{}A; c\isasymin{}B |] ==> P |] ==> P
626 \tdx{DiffI}: [| c\isasymin{}A; c \isasymnotin B |] ==> c\isasymin{}A - B
627 \tdx{DiffD1}: c\isasymin{}A - B ==> c\isasymin{}A
628 \tdx{DiffD2}: c\isasymin{}A - B ==> c \isasymnotin B
629 \tdx{DiffE}: [| c\isasymin{}A - B; [| c\isasymin{}A; c \isasymnotin B |] ==> P |] ==> P
631 \caption{Union, intersection, difference} \label{zf-Un}
636 \begin{alltt*}\isastyleminor
637 \tdx{consI1}: a\isasymin{}cons(a,B)
638 \tdx{consI2}: a\isasymin{}B ==> a\isasymin{}cons(b,B)
639 \tdx{consCI}: (a \isasymnotin B ==> a=b) ==> a\isasymin{}cons(b,B)
640 \tdx{consE}: [| a\isasymin{}cons(b,A); a=b ==> P; a\isasymin{}A ==> P |] ==> P
642 \tdx{singletonI}: a\isasymin{}{\ttlbrace}a{\ttrbrace}
643 \tdx{singletonE}: [| a\isasymin{}{\ttlbrace}b{\ttrbrace}; a=b ==> P |] ==> P
645 \caption{Finite and singleton sets} \label{zf-upair2}
650 \begin{alltt*}\isastyleminor
651 \tdx{succI1}: i\isasymin{}succ(i)
652 \tdx{succI2}: i\isasymin{}j ==> i\isasymin{}succ(j)
653 \tdx{succCI}: (i \isasymnotin j ==> i=j) ==> i\isasymin{}succ(j)
654 \tdx{succE}: [| i\isasymin{}succ(j); i=j ==> P; i\isasymin{}j ==> P |] ==> P
655 \tdx{succ_neq_0}: [| succ(n)=0 |] ==> P
656 \tdx{succ_inject}: succ(m) = succ(n) ==> m=n
658 \caption{The successor function} \label{zf-succ}
663 \begin{alltt*}\isastyleminor
664 \tdx{the_equality}: [| P(a); !!x. P(x) ==> x=a |] ==> (THE x. P(x))=a
665 \tdx{theI}: \isasymexists! x. P(x) ==> P(THE x. P(x))
667 \tdx{if_P}: P ==> (if P then a else b) = a
668 \tdx{if_not_P}: ~P ==> (if P then a else b) = b
670 \tdx{mem_asym}: [| a\isasymin{}b; b\isasymin{}a |] ==> P
671 \tdx{mem_irrefl}: a\isasymin{}a ==> P
673 \caption{Descriptions; non-circularity} \label{zf-the}
677 \subsection{Unordered pairs and finite sets}
678 Figure~\ref{zf-upair1} presents the principle of unordered pairing, along
679 with its derived rules. Binary union and intersection are defined in terms
680 of ordered pairs (Fig.\ts\ref{zf-Un}). Set difference is also included. The
681 rule \tdx{UnCI} is useful for classical reasoning about unions,
682 like \isa{disjCI}\@; it supersedes \tdx{UnI1} and
683 \tdx{UnI2}, but these rules are often easier to work with. For
684 intersection and difference we have both elimination and destruction rules.
685 Again, there is no reason to provide a minimal rule set.
687 Figure~\ref{zf-upair2} is concerned with finite sets: it presents rules
688 for~\isa{cons}, the finite set constructor, and rules for singleton
689 sets. Figure~\ref{zf-succ} presents derived rules for the successor
690 function, which is defined in terms of~\isa{cons}. The proof that
691 \isa{succ} is injective appears to require the Axiom of Foundation.
693 Definite descriptions (\sdx{THE}) are defined in terms of the singleton
694 set~$\{0\}$, but their derived rules fortunately hide this
695 (Fig.\ts\ref{zf-the}). The rule~\tdx{theI} is difficult to apply
696 because of the two occurrences of~$\Var{P}$. However,
697 \tdx{the_equality} does not have this problem and the files contain
698 many examples of its use.
700 Finally, the impossibility of having both $a\in b$ and $b\in a$
701 (\tdx{mem_asym}) is proved by applying the Axiom of Foundation to
702 the set $\{a,b\}$. The impossibility of $a\in a$ is a trivial consequence.
708 \begin{alltt*}\isastyleminor
709 \tdx{Union_upper}: B\isasymin{}A ==> B \isasymsubseteq Union(A)
710 \tdx{Union_least}: [| !!x. x\isasymin{}A ==> x \isasymsubseteq C |] ==> Union(A) \isasymsubseteq C
712 \tdx{Inter_lower}: B\isasymin{}A ==> Inter(A) \isasymsubseteq B
713 \tdx{Inter_greatest}: [| a\isasymin{}A; !!x. x\isasymin{}A ==> C \isasymsubseteq x |] ==> C\isasymsubseteq{}Inter(A)
715 \tdx{Un_upper1}: A \isasymsubseteq A \isasymunion B
716 \tdx{Un_upper2}: B \isasymsubseteq A \isasymunion B
717 \tdx{Un_least}: [| A \isasymsubseteq C; B \isasymsubseteq C |] ==> A \isasymunion B \isasymsubseteq C
719 \tdx{Int_lower1}: A \isasyminter B \isasymsubseteq A
720 \tdx{Int_lower2}: A \isasyminter B \isasymsubseteq B
721 \tdx{Int_greatest}: [| C \isasymsubseteq A; C \isasymsubseteq B |] ==> C \isasymsubseteq A \isasyminter B
723 \tdx{Diff_subset}: A-B \isasymsubseteq A
724 \tdx{Diff_contains}: [| C \isasymsubseteq A; C \isasyminter B = 0 |] ==> C \isasymsubseteq A-B
726 \tdx{Collect_subset}: Collect(A,P) \isasymsubseteq A
728 \caption{Subset and lattice properties} \label{zf-subset}
732 \subsection{Subset and lattice properties}
733 The subset relation is a complete lattice. Unions form least upper bounds;
734 non-empty intersections form greatest lower bounds. Figure~\ref{zf-subset}
735 shows the corresponding rules. A few other laws involving subsets are
737 Reasoning directly about subsets often yields clearer proofs than
738 reasoning about the membership relation. Section~\ref{sec:ZF-pow-example}
739 below presents an example of this, proving the equation
740 ${\isa{Pow}(A)\cap \isa{Pow}(B)}= \isa{Pow}(A\cap B)$.
745 \begin{alltt*}\isastyleminor
746 \tdx{Pair_inject1}: <a,b> = <c,d> ==> a=c
747 \tdx{Pair_inject2}: <a,b> = <c,d> ==> b=d
748 \tdx{Pair_inject}: [| <a,b> = <c,d>; [| a=c; b=d |] ==> P |] ==> P
749 \tdx{Pair_neq_0}: <a,b>=0 ==> P
751 \tdx{fst_conv}: fst(<a,b>) = a
752 \tdx{snd_conv}: snd(<a,b>) = b
753 \tdx{split}: split(\%x y. c(x,y), <a,b>) = c(a,b)
755 \tdx{SigmaI}: [| a\isasymin{}A; b\isasymin{}B(a) |] ==> <a,b>\isasymin{}Sigma(A,B)
757 \tdx{SigmaE}: [| c\isasymin{}Sigma(A,B);
758 !!x y.[| x\isasymin{}A; y\isasymin{}B(x); c=<x,y> |] ==> P |] ==> P
760 \tdx{SigmaE2}: [| <a,b>\isasymin{}Sigma(A,B);
761 [| a\isasymin{}A; b\isasymin{}B(a) |] ==> P |] ==> P
763 \caption{Ordered pairs; projections; general sums} \label{zf-pair}
767 \subsection{Ordered pairs} \label{sec:pairs}
769 Figure~\ref{zf-pair} presents the rules governing ordered pairs,
770 projections and general sums --- in particular, that
771 $\{\{a\},\{a,b\}\}$ functions as an ordered pair. This property is
772 expressed as two destruction rules,
773 \tdx{Pair_inject1} and \tdx{Pair_inject2}, and equivalently
774 as the elimination rule \tdx{Pair_inject}.
776 The rule \tdx{Pair_neq_0} asserts $\pair{a,b}\neq\emptyset$. This
777 is a property of $\{\{a\},\{a,b\}\}$, and need not hold for other
778 encodings of ordered pairs. The non-standard ordered pairs mentioned below
779 satisfy $\pair{\emptyset;\emptyset}=\emptyset$.
781 The natural deduction rules \tdx{SigmaI} and \tdx{SigmaE}
782 assert that \cdx{Sigma}$(A,B)$ consists of all pairs of the form
783 $\pair{x,y}$, for $x\in A$ and $y\in B(x)$. The rule \tdx{SigmaE2}
784 merely states that $\pair{a,b}\in \isa{Sigma}(A,B)$ implies $a\in A$ and
787 In addition, it is possible to use tuples as patterns in abstractions:
789 {\tt\%<$x$,$y$>. $t$} \quad stands for\quad \isa{split(\%$x$ $y$.\ $t$)}
791 Nested patterns are translated recursively:
792 {\tt\%<$x$,$y$,$z$>. $t$} $\leadsto$ {\tt\%<$x$,<$y$,$z$>>. $t$} $\leadsto$
793 \isa{split(\%$x$.\%<$y$,$z$>. $t$)} $\leadsto$ \isa{split(\%$x$. split(\%$y$
794 $z$.\ $t$))}. The reverse translation is performed upon printing.
796 The translation between patterns and \isa{split} is performed automatically
797 by the parser and printer. Thus the internal and external form of a term
798 may differ, which affects proofs. For example the term \isa{(\%<x,y>.<y,x>)<a,b>} requires the theorem \isa{split} to rewrite to
801 In addition to explicit $\lambda$-abstractions, patterns can be used in any
802 variable binding construct which is internally described by a
803 $\lambda$-abstraction. Here are some important examples:
805 \item[Let:] \isa{let {\it pattern} = $t$ in $u$}
806 \item[Choice:] \isa{THE~{\it pattern}~.~$P$}
807 \item[Set operations:] \isa{\isasymUnion~{\it pattern}:$A$.~$B$}
808 \item[Comprehension:] \isa{{\ttlbrace}~{\it pattern}:$A$~.~$P$~{\ttrbrace}}
815 \begin{alltt*}\isastyleminor
816 \tdx{domainI}: <a,b>\isasymin{}r ==> a\isasymin{}domain(r)
817 \tdx{domainE}: [| a\isasymin{}domain(r); !!y. <a,y>\isasymin{}r ==> P |] ==> P
818 \tdx{domain_subset}: domain(Sigma(A,B)) \isasymsubseteq A
820 \tdx{rangeI}: <a,b>\isasymin{}r ==> b\isasymin{}range(r)
821 \tdx{rangeE}: [| b\isasymin{}range(r); !!x. <x,b>\isasymin{}r ==> P |] ==> P
822 \tdx{range_subset}: range(A*B) \isasymsubseteq B
824 \tdx{fieldI1}: <a,b>\isasymin{}r ==> a\isasymin{}field(r)
825 \tdx{fieldI2}: <a,b>\isasymin{}r ==> b\isasymin{}field(r)
826 \tdx{fieldCI}: (<c,a> \isasymnotin r ==> <a,b>\isasymin{}r) ==> a\isasymin{}field(r)
828 \tdx{fieldE}: [| a\isasymin{}field(r);
829 !!x. <a,x>\isasymin{}r ==> P;
830 !!x. <x,a>\isasymin{}r ==> P
833 \tdx{field_subset}: field(A*A) \isasymsubseteq A
835 \caption{Domain, range and field of a relation} \label{zf-domrange}
839 \begin{alltt*}\isastyleminor
840 \tdx{imageI}: [| <a,b>\isasymin{}r; a\isasymin{}A |] ==> b\isasymin{}r``A
841 \tdx{imageE}: [| b\isasymin{}r``A; !!x.[| <x,b>\isasymin{}r; x\isasymin{}A |] ==> P |] ==> P
843 \tdx{vimageI}: [| <a,b>\isasymin{}r; b\isasymin{}B |] ==> a\isasymin{}r-``B
844 \tdx{vimageE}: [| a\isasymin{}r-``B; !!x.[| <a,x>\isasymin{}r; x\isasymin{}B |] ==> P |] ==> P
846 \caption{Image and inverse image} \label{zf-domrange2}
850 \subsection{Relations}
851 Figure~\ref{zf-domrange} presents rules involving relations, which are sets
852 of ordered pairs. The converse of a relation~$r$ is the set of all pairs
853 $\pair{y,x}$ such that $\pair{x,y}\in r$; if $r$ is a function, then
854 {\cdx{converse}$(r)$} is its inverse. The rules for the domain
855 operation, namely \tdx{domainI} and~\tdx{domainE}, assert that
856 \cdx{domain}$(r)$ consists of all~$x$ such that $r$ contains
857 some pair of the form~$\pair{x,y}$. The range operation is similar, and
858 the field of a relation is merely the union of its domain and range.
860 Figure~\ref{zf-domrange2} presents rules for images and inverse images.
861 Note that these operations are generalisations of range and domain,
868 \begin{alltt*}\isastyleminor
869 \tdx{fun_is_rel}: f\isasymin{}Pi(A,B) ==> f \isasymsubseteq Sigma(A,B)
871 \tdx{apply_equality}: [| <a,b>\isasymin{}f; f\isasymin{}Pi(A,B) |] ==> f`a = b
872 \tdx{apply_equality2}: [| <a,b>\isasymin{}f; <a,c>\isasymin{}f; f\isasymin{}Pi(A,B) |] ==> b=c
874 \tdx{apply_type}: [| f\isasymin{}Pi(A,B); a\isasymin{}A |] ==> f`a\isasymin{}B(a)
875 \tdx{apply_Pair}: [| f\isasymin{}Pi(A,B); a\isasymin{}A |] ==> <a,f`a>\isasymin{}f
876 \tdx{apply_iff}: f\isasymin{}Pi(A,B) ==> <a,b>\isasymin{}f <-> a\isasymin{}A & f`a = b
878 \tdx{fun_extension}: [| f\isasymin{}Pi(A,B); g\isasymin{}Pi(A,D);
879 !!x. x\isasymin{}A ==> f`x = g`x |] ==> f=g
881 \tdx{domain_type}: [| <a,b>\isasymin{}f; f\isasymin{}Pi(A,B) |] ==> a\isasymin{}A
882 \tdx{range_type}: [| <a,b>\isasymin{}f; f\isasymin{}Pi(A,B) |] ==> b\isasymin{}B(a)
884 \tdx{Pi_type}: [| f\isasymin{}A->C; !!x. x\isasymin{}A ==> f`x\isasymin{}B(x) |] ==> f\isasymin{}Pi(A,B)
885 \tdx{domain_of_fun}: f\isasymin{}Pi(A,B) ==> domain(f)=A
886 \tdx{range_of_fun}: f\isasymin{}Pi(A,B) ==> f\isasymin{}A->range(f)
888 \tdx{restrict}: a\isasymin{}A ==> restrict(f,A) ` a = f`a
889 \tdx{restrict_type}: [| !!x. x\isasymin{}A ==> f`x\isasymin{}B(x) |] ==>
890 restrict(f,A)\isasymin{}Pi(A,B)
892 \caption{Functions} \label{zf-func1}
897 \begin{alltt*}\isastyleminor
898 \tdx{lamI}: a\isasymin{}A ==> <a,b(a)>\isasymin{}(lam x\isasymin{}A. b(x))
899 \tdx{lamE}: [| p\isasymin{}(lam x\isasymin{}A. b(x)); !!x.[| x\isasymin{}A; p=<x,b(x)> |] ==> P
902 \tdx{lam_type}: [| !!x. x\isasymin{}A ==> b(x)\isasymin{}B(x) |] ==> (lam x\isasymin{}A. b(x))\isasymin{}Pi(A,B)
904 \tdx{beta}: a\isasymin{}A ==> (lam x\isasymin{}A. b(x)) ` a = b(a)
905 \tdx{eta}: f\isasymin{}Pi(A,B) ==> (lam x\isasymin{}A. f`x) = f
907 \caption{$\lambda$-abstraction} \label{zf-lam}
912 \begin{alltt*}\isastyleminor
913 \tdx{fun_empty}: 0\isasymin{}0->0
914 \tdx{fun_single}: {\ttlbrace}<a,b>{\ttrbrace}\isasymin{}{\ttlbrace}a{\ttrbrace} -> {\ttlbrace}b{\ttrbrace}
916 \tdx{fun_disjoint_Un}: [| f\isasymin{}A->B; g\isasymin{}C->D; A \isasyminter C = 0 |] ==>
917 (f \isasymunion g)\isasymin{}(A \isasymunion C) -> (B \isasymunion D)
919 \tdx{fun_disjoint_apply1}: [| a\isasymin{}A; f\isasymin{}A->B; g\isasymin{}C->D; A\isasyminter{}C = 0 |] ==>
920 (f \isasymunion g)`a = f`a
922 \tdx{fun_disjoint_apply2}: [| c\isasymin{}C; f\isasymin{}A->B; g\isasymin{}C->D; A\isasyminter{}C = 0 |] ==>
923 (f \isasymunion g)`c = g`c
925 \caption{Constructing functions from smaller sets} \label{zf-func2}
929 \subsection{Functions}
930 Functions, represented by graphs, are notoriously difficult to reason
931 about. The ZF theory provides many derived rules, which overlap more
932 than they ought. This section presents the more important rules.
934 Figure~\ref{zf-func1} presents the basic properties of \cdx{Pi}$(A,B)$,
935 the generalized function space. For example, if $f$ is a function and
936 $\pair{a,b}\in f$, then $f`a=b$ (\tdx{apply_equality}). Two functions
937 are equal provided they have equal domains and deliver equals results
938 (\tdx{fun_extension}).
940 By \tdx{Pi_type}, a function typing of the form $f\in A\to C$ can be
941 refined to the dependent typing $f\in\prod@{x\in A}B(x)$, given a suitable
942 family of sets $\{B(x)\}@{x\in A}$. Conversely, by \tdx{range_of_fun},
943 any dependent typing can be flattened to yield a function type of the form
944 $A\to C$; here, $C=\isa{range}(f)$.
946 Among the laws for $\lambda$-abstraction, \tdx{lamI} and \tdx{lamE}
947 describe the graph of the generated function, while \tdx{beta} and
948 \tdx{eta} are the standard conversions. We essentially have a
949 dependently-typed $\lambda$-calculus (Fig.\ts\ref{zf-lam}).
951 Figure~\ref{zf-func2} presents some rules that can be used to construct
952 functions explicitly. We start with functions consisting of at most one
953 pair, and may form the union of two functions provided their domains are
958 \begin{alltt*}\isastyleminor
959 \tdx{Int_absorb}: A \isasyminter A = A
960 \tdx{Int_commute}: A \isasyminter B = B \isasyminter A
961 \tdx{Int_assoc}: (A \isasyminter B) \isasyminter C = A \isasyminter (B \isasyminter C)
962 \tdx{Int_Un_distrib}: (A \isasymunion B) \isasyminter C = (A \isasyminter C) \isasymunion (B \isasyminter C)
964 \tdx{Un_absorb}: A \isasymunion A = A
965 \tdx{Un_commute}: A \isasymunion B = B \isasymunion A
966 \tdx{Un_assoc}: (A \isasymunion B) \isasymunion C = A \isasymunion (B \isasymunion C)
967 \tdx{Un_Int_distrib}: (A \isasyminter B) \isasymunion C = (A \isasymunion C) \isasyminter (B \isasymunion C)
969 \tdx{Diff_cancel}: A-A = 0
970 \tdx{Diff_disjoint}: A \isasyminter (B-A) = 0
971 \tdx{Diff_partition}: A \isasymsubseteq B ==> A \isasymunion (B-A) = B
972 \tdx{double_complement}: [| A \isasymsubseteq B; B \isasymsubseteq C |] ==> (B - (C-A)) = A
973 \tdx{Diff_Un}: A - (B \isasymunion C) = (A-B) \isasyminter (A-C)
974 \tdx{Diff_Int}: A - (B \isasyminter C) = (A-B) \isasymunion (A-C)
976 \tdx{Union_Un_distrib}: Union(A \isasymunion B) = Union(A) \isasymunion Union(B)
977 \tdx{Inter_Un_distrib}: [| a \isasymin A; b \isasymin B |] ==>
978 Inter(A \isasymunion B) = Inter(A) \isasyminter Inter(B)
980 \tdx{Int_Union_RepFun}: A \isasyminter Union(B) = ({\isasymUnion}C \isasymin B. A \isasyminter C)
982 \tdx{Un_Inter_RepFun}: b \isasymin B ==>
983 A \isasymunion Inter(B) = ({\isasymInter}C \isasymin B. A \isasymunion C)
985 \tdx{SUM_Un_distrib1}: (SUM x \isasymin A \isasymunion B. C(x)) =
986 (SUM x \isasymin A. C(x)) \isasymunion (SUM x \isasymin B. C(x))
988 \tdx{SUM_Un_distrib2}: (SUM x \isasymin C. A(x) \isasymunion B(x)) =
989 (SUM x \isasymin C. A(x)) \isasymunion (SUM x \isasymin C. B(x))
991 \tdx{SUM_Int_distrib1}: (SUM x \isasymin A \isasyminter B. C(x)) =
992 (SUM x \isasymin A. C(x)) \isasyminter (SUM x \isasymin B. C(x))
994 \tdx{SUM_Int_distrib2}: (SUM x \isasymin C. A(x) \isasyminter B(x)) =
995 (SUM x \isasymin C. A(x)) \isasyminter (SUM x \isasymin C. B(x))
997 \caption{Equalities} \label{zf-equalities}
1003 % \cdx{1} & $i$ & & $\{\emptyset\}$ \\
1004 % \cdx{bool} & $i$ & & the set $\{\emptyset,1\}$ \\
1005 % \cdx{cond} & $[i,i,i]\To i$ & & conditional for \isa{bool} \\
1006 % \cdx{not} & $i\To i$ & & negation for \isa{bool} \\
1007 % \sdx{and} & $[i,i]\To i$ & Left 70 & conjunction for \isa{bool} \\
1008 % \sdx{or} & $[i,i]\To i$ & Left 65 & disjunction for \isa{bool} \\
1009 % \sdx{xor} & $[i,i]\To i$ & Left 65 & exclusive-or for \isa{bool}
1012 \begin{alltt*}\isastyleminor
1013 \tdx{bool_def}: bool == {\ttlbrace}0,1{\ttrbrace}
1014 \tdx{cond_def}: cond(b,c,d) == if b=1 then c else d
1015 \tdx{not_def}: not(b) == cond(b,0,1)
1016 \tdx{and_def}: a and b == cond(a,b,0)
1017 \tdx{or_def}: a or b == cond(a,1,b)
1018 \tdx{xor_def}: a xor b == cond(a,not(b),b)
1020 \tdx{bool_1I}: 1 \isasymin bool
1021 \tdx{bool_0I}: 0 \isasymin bool
1022 \tdx{boolE}: [| c \isasymin bool; c=1 ==> P; c=0 ==> P |] ==> P
1023 \tdx{cond_1}: cond(1,c,d) = c
1024 \tdx{cond_0}: cond(0,c,d) = d
1026 \caption{The booleans} \label{zf-bool}
1030 \section{Further developments}
1031 The next group of developments is complex and extensive, and only
1032 highlights can be covered here. It involves many theories and proofs.
1034 Figure~\ref{zf-equalities} presents commutative, associative, distributive,
1035 and idempotency laws of union and intersection, along with other equations.
1037 Theory \thydx{Bool} defines $\{0,1\}$ as a set of booleans, with the usual
1038 operators including a conditional (Fig.\ts\ref{zf-bool}). Although ZF is a
1039 first-order theory, you can obtain the effect of higher-order logic using
1040 \isa{bool}-valued functions, for example. The constant~\isa{1} is
1041 translated to \isa{succ(0)}.
1046 \it symbol & \it meta-type & \it priority & \it description \\
1047 \tt + & $[i,i]\To i$ & Right 65 & disjoint union operator\\
1048 \cdx{Inl}~~\cdx{Inr} & $i\To i$ & & injections\\
1049 \cdx{case} & $[i\To i,i\To i, i]\To i$ & & conditional for $A+B$
1051 \begin{alltt*}\isastyleminor
1052 \tdx{sum_def}: A+B == {\ttlbrace}0{\ttrbrace}*A \isasymunion {\ttlbrace}1{\ttrbrace}*B
1053 \tdx{Inl_def}: Inl(a) == <0,a>
1054 \tdx{Inr_def}: Inr(b) == <1,b>
1055 \tdx{case_def}: case(c,d,u) == split(\%y z. cond(y, d(z), c(z)), u)
1057 \tdx{InlI}: a \isasymin A ==> Inl(a) \isasymin A+B
1058 \tdx{InrI}: b \isasymin B ==> Inr(b) \isasymin A+B
1060 \tdx{Inl_inject}: Inl(a)=Inl(b) ==> a=b
1061 \tdx{Inr_inject}: Inr(a)=Inr(b) ==> a=b
1062 \tdx{Inl_neq_Inr}: Inl(a)=Inr(b) ==> P
1064 \tdx{sum_iff}: u \isasymin A+B <-> ({\isasymexists}x\isasymin{}A. u=Inl(x)) | ({\isasymexists}y\isasymin{}B. u=Inr(y))
1066 \tdx{case_Inl}: case(c,d,Inl(a)) = c(a)
1067 \tdx{case_Inr}: case(c,d,Inr(b)) = d(b)
1069 \caption{Disjoint unions} \label{zf-sum}
1073 \subsection{Disjoint unions}
1075 Theory \thydx{Sum} defines the disjoint union of two sets, with
1076 injections and a case analysis operator (Fig.\ts\ref{zf-sum}). Disjoint
1077 unions play a role in datatype definitions, particularly when there is
1078 mutual recursion~\cite{paulson-set-II}.
1081 \begin{alltt*}\isastyleminor
1082 \tdx{QPair_def}: <a;b> == a+b
1083 \tdx{qsplit_def}: qsplit(c,p) == THE y. {\isasymexists}a b. p=<a;b> & y=c(a,b)
1084 \tdx{qfsplit_def}: qfsplit(R,z) == {\isasymexists}x y. z=<x;y> & R(x,y)
1085 \tdx{qconverse_def}: qconverse(r) == {\ttlbrace}z. w \isasymin r, {\isasymexists}x y. w=<x;y> & z=<y;x>{\ttrbrace}
1086 \tdx{QSigma_def}: QSigma(A,B) == {\isasymUnion}x \isasymin A. {\isasymUnion}y \isasymin B(x). {\ttlbrace}<x;y>{\ttrbrace}
1088 \tdx{qsum_def}: A <+> B == ({\ttlbrace}0{\ttrbrace} <*> A) \isasymunion ({\ttlbrace}1{\ttrbrace} <*> B)
1089 \tdx{QInl_def}: QInl(a) == <0;a>
1090 \tdx{QInr_def}: QInr(b) == <1;b>
1091 \tdx{qcase_def}: qcase(c,d) == qsplit(\%y z. cond(y, d(z), c(z)))
1093 \caption{Non-standard pairs, products and sums} \label{zf-qpair}
1097 \subsection{Non-standard ordered pairs}
1099 Theory \thydx{QPair} defines a notion of ordered pair that admits
1100 non-well-founded tupling (Fig.\ts\ref{zf-qpair}). Such pairs are written
1101 {\tt<$a$;$b$>}. It also defines the eliminator \cdx{qsplit}, the
1102 converse operator \cdx{qconverse}, and the summation operator
1103 \cdx{QSigma}. These are completely analogous to the corresponding
1104 versions for standard ordered pairs. The theory goes on to define a
1105 non-standard notion of disjoint sum using non-standard pairs. All of these
1106 concepts satisfy the same properties as their standard counterparts; in
1107 addition, {\tt<$a$;$b$>} is continuous. The theory supports coinductive
1108 definitions, for example of infinite lists~\cite{paulson-mscs}.
1111 \begin{alltt*}\isastyleminor
1112 \tdx{bnd_mono_def}: bnd_mono(D,h) ==
1113 h(D)\isasymsubseteq{}D & ({\isasymforall}W X. W\isasymsubseteq{}X --> X\isasymsubseteq{}D --> h(W)\isasymsubseteq{}h(X))
1115 \tdx{lfp_def}: lfp(D,h) == Inter({\ttlbrace}X \isasymin Pow(D). h(X) \isasymsubseteq X{\ttrbrace})
1116 \tdx{gfp_def}: gfp(D,h) == Union({\ttlbrace}X \isasymin Pow(D). X \isasymsubseteq h(X){\ttrbrace})
1119 \tdx{lfp_lowerbound}: [| h(A) \isasymsubseteq A; A \isasymsubseteq D |] ==> lfp(D,h) \isasymsubseteq A
1121 \tdx{lfp_subset}: lfp(D,h) \isasymsubseteq D
1123 \tdx{lfp_greatest}: [| bnd_mono(D,h);
1124 !!X. [| h(X) \isasymsubseteq X; X \isasymsubseteq D |] ==> A \isasymsubseteq X
1125 |] ==> A \isasymsubseteq lfp(D,h)
1127 \tdx{lfp_Tarski}: bnd_mono(D,h) ==> lfp(D,h) = h(lfp(D,h))
1129 \tdx{induct}: [| a \isasymin lfp(D,h); bnd_mono(D,h);
1130 !!x. x \isasymin h(Collect(lfp(D,h),P)) ==> P(x)
1133 \tdx{lfp_mono}: [| bnd_mono(D,h); bnd_mono(E,i);
1134 !!X. X \isasymsubseteq D ==> h(X) \isasymsubseteq i(X)
1135 |] ==> lfp(D,h) \isasymsubseteq lfp(E,i)
1137 \tdx{gfp_upperbound}: [| A \isasymsubseteq h(A); A \isasymsubseteq D |] ==> A \isasymsubseteq gfp(D,h)
1139 \tdx{gfp_subset}: gfp(D,h) \isasymsubseteq D
1141 \tdx{gfp_least}: [| bnd_mono(D,h);
1142 !!X. [| X \isasymsubseteq h(X); X \isasymsubseteq D |] ==> X \isasymsubseteq A
1143 |] ==> gfp(D,h) \isasymsubseteq A
1145 \tdx{gfp_Tarski}: bnd_mono(D,h) ==> gfp(D,h) = h(gfp(D,h))
1147 \tdx{coinduct}: [| bnd_mono(D,h); a \isasymin X; X \isasymsubseteq h(X \isasymunion gfp(D,h)); X \isasymsubseteq D
1148 |] ==> a \isasymin gfp(D,h)
1150 \tdx{gfp_mono}: [| bnd_mono(D,h); D \isasymsubseteq E;
1151 !!X. X \isasymsubseteq D ==> h(X) \isasymsubseteq i(X)
1152 |] ==> gfp(D,h) \isasymsubseteq gfp(E,i)
1154 \caption{Least and greatest fixedpoints} \label{zf-fixedpt}
1158 \subsection{Least and greatest fixedpoints}
1160 The Knaster-Tarski Theorem states that every monotone function over a
1161 complete lattice has a fixedpoint. Theory \thydx{Fixedpt} proves the
1162 Theorem only for a particular lattice, namely the lattice of subsets of a
1163 set (Fig.\ts\ref{zf-fixedpt}). The theory defines least and greatest
1164 fixedpoint operators with corresponding induction and coinduction rules.
1165 These are essential to many definitions that follow, including the natural
1166 numbers and the transitive closure operator. The (co)inductive definition
1167 package also uses the fixedpoint operators~\cite{paulson-CADE}. See
1168 Davey and Priestley~\cite{davey-priestley} for more on the Knaster-Tarski
1169 Theorem and my paper~\cite{paulson-set-II} for discussion of the Isabelle
1172 Monotonicity properties are proved for most of the set-forming operations:
1173 union, intersection, Cartesian product, image, domain, range, etc. These
1174 are useful for applying the Knaster-Tarski Fixedpoint Theorem. The proofs
1175 themselves are trivial applications of Isabelle's classical reasoner.
1178 \subsection{Finite sets and lists}
1180 Theory \texttt{Finite} (Figure~\ref{zf-fin}) defines the finite set operator;
1181 $\isa{Fin}(A)$ is the set of all finite sets over~$A$. The theory employs
1182 Isabelle's inductive definition package, which proves various rules
1183 automatically. The induction rule shown is stronger than the one proved by
1184 the package. The theory also defines the set of all finite functions
1185 between two given sets.
1188 \begin{alltt*}\isastyleminor
1189 \tdx{Fin.emptyI} 0 \isasymin Fin(A)
1190 \tdx{Fin.consI} [| a \isasymin A; b \isasymin Fin(A) |] ==> cons(a,b) \isasymin Fin(A)
1193 [| b \isasymin Fin(A);
1195 !!x y. [| x\isasymin{}A; y\isasymin{}Fin(A); x\isasymnotin{}y; P(y) |] ==> P(cons(x,y))
1198 \tdx{Fin_mono}: A \isasymsubseteq B ==> Fin(A) \isasymsubseteq Fin(B)
1199 \tdx{Fin_UnI}: [| b \isasymin Fin(A); c \isasymin Fin(A) |] ==> b \isasymunion c \isasymin Fin(A)
1200 \tdx{Fin_UnionI}: C \isasymin Fin(Fin(A)) ==> Union(C) \isasymin Fin(A)
1201 \tdx{Fin_subset}: [| c \isasymsubseteq b; b \isasymin Fin(A) |] ==> c \isasymin Fin(A)
1203 \caption{The finite set operator} \label{zf-fin}
1208 \it symbol & \it meta-type & \it priority & \it description \\
1209 \cdx{list} & $i\To i$ && lists over some set\\
1210 \cdx{list_case} & $[i, [i,i]\To i, i] \To i$ && conditional for $list(A)$ \\
1211 \cdx{map} & $[i\To i, i] \To i$ & & mapping functional\\
1212 \cdx{length} & $i\To i$ & & length of a list\\
1213 \cdx{rev} & $i\To i$ & & reverse of a list\\
1214 \tt \at & $[i,i]\To i$ & Right 60 & append for lists\\
1215 \cdx{flat} & $i\To i$ & & append of list of lists
1218 \underscoreon %%because @ is used here
1219 \begin{alltt*}\isastyleminor
1220 \tdx{NilI}: Nil \isasymin list(A)
1221 \tdx{ConsI}: [| a \isasymin A; l \isasymin list(A) |] ==> Cons(a,l) \isasymin list(A)
1224 [| l \isasymin list(A);
1226 !!x y. [| x \isasymin A; y \isasymin list(A); P(y) |] ==> P(Cons(x,y))
1229 \tdx{Cons_iff}: Cons(a,l)=Cons(a',l') <-> a=a' & l=l'
1230 \tdx{Nil_Cons_iff}: Nil \isasymnoteq Cons(a,l)
1232 \tdx{list_mono}: A \isasymsubseteq B ==> list(A) \isasymsubseteq list(B)
1234 \tdx{map_ident}: l\isasymin{}list(A) ==> map(\%u. u, l) = l
1235 \tdx{map_compose}: l\isasymin{}list(A) ==> map(h, map(j,l)) = map(\%u. h(j(u)), l)
1236 \tdx{map_app_distrib}: xs\isasymin{}list(A) ==> map(h, xs@ys) = map(h,xs)@map(h,ys)
1238 [| l\isasymin{}list(A); !!x. x\isasymin{}A ==> h(x)\isasymin{}B |] ==> map(h,l)\isasymin{}list(B)
1240 ls: list(list(A)) ==> map(h, flat(ls)) = flat(map(map(h),ls))
1242 \caption{Lists} \label{zf-list}
1246 Figure~\ref{zf-list} presents the set of lists over~$A$, $\isa{list}(A)$. The
1247 definition employs Isabelle's datatype package, which defines the introduction
1248 and induction rules automatically, as well as the constructors, case operator
1249 (\isa{list\_case}) and recursion operator. The theory then defines the usual
1250 list functions by primitive recursion. See theory \texttt{List}.
1253 \subsection{Miscellaneous}
1257 \it symbol & \it meta-type & \it priority & \it description \\
1258 \sdx{O} & $[i,i]\To i$ & Right 60 & composition ($\circ$) \\
1259 \cdx{id} & $i\To i$ & & identity function \\
1260 \cdx{inj} & $[i,i]\To i$ & & injective function space\\
1261 \cdx{surj} & $[i,i]\To i$ & & surjective function space\\
1262 \cdx{bij} & $[i,i]\To i$ & & bijective function space
1265 \begin{alltt*}\isastyleminor
1266 \tdx{comp_def}: r O s == {\ttlbrace}xz \isasymin domain(s)*range(r) .
1267 {\isasymexists}x y z. xz=<x,z> & <x,y> \isasymin s & <y,z> \isasymin r{\ttrbrace}
1268 \tdx{id_def}: id(A) == (lam x \isasymin A. x)
1269 \tdx{inj_def}: inj(A,B) == {\ttlbrace} f\isasymin{}A->B. {\isasymforall}w\isasymin{}A. {\isasymforall}x\isasymin{}A. f`w=f`x --> w=x {\ttrbrace}
1270 \tdx{surj_def}: surj(A,B) == {\ttlbrace} f\isasymin{}A->B . {\isasymforall}y\isasymin{}B. {\isasymexists}x\isasymin{}A. f`x=y {\ttrbrace}
1271 \tdx{bij_def}: bij(A,B) == inj(A,B) \isasyminter surj(A,B)
1274 \tdx{left_inverse}: [| f\isasymin{}inj(A,B); a\isasymin{}A |] ==> converse(f)`(f`a) = a
1275 \tdx{right_inverse}: [| f\isasymin{}inj(A,B); b\isasymin{}range(f) |] ==>
1276 f`(converse(f)`b) = b
1278 \tdx{inj_converse_inj}: f\isasymin{}inj(A,B) ==> converse(f) \isasymin inj(range(f),A)
1279 \tdx{bij_converse_bij}: f\isasymin{}bij(A,B) ==> converse(f) \isasymin bij(B,A)
1281 \tdx{comp_type}: [| s \isasymsubseteq A*B; r \isasymsubseteq B*C |] ==> (r O s) \isasymsubseteq A*C
1282 \tdx{comp_assoc}: (r O s) O t = r O (s O t)
1284 \tdx{left_comp_id}: r \isasymsubseteq A*B ==> id(B) O r = r
1285 \tdx{right_comp_id}: r \isasymsubseteq A*B ==> r O id(A) = r
1287 \tdx{comp_func}: [| g\isasymin{}A->B; f\isasymin{}B->C |] ==> (f O g) \isasymin A->C
1288 \tdx{comp_func_apply}: [| g\isasymin{}A->B; f\isasymin{}B->C; a\isasymin{}A |] ==> (f O g)`a = f`(g`a)
1290 \tdx{comp_inj}: [| g\isasymin{}inj(A,B); f\isasymin{}inj(B,C) |] ==> (f O g)\isasymin{}inj(A,C)
1291 \tdx{comp_surj}: [| g\isasymin{}surj(A,B); f\isasymin{}surj(B,C) |] ==> (f O g)\isasymin{}surj(A,C)
1292 \tdx{comp_bij}: [| g\isasymin{}bij(A,B); f\isasymin{}bij(B,C) |] ==> (f O g)\isasymin{}bij(A,C)
1294 \tdx{left_comp_inverse}: f\isasymin{}inj(A,B) ==> converse(f) O f = id(A)
1295 \tdx{right_comp_inverse}: f\isasymin{}surj(A,B) ==> f O converse(f) = id(B)
1297 \tdx{bij_disjoint_Un}:
1298 [| f\isasymin{}bij(A,B); g\isasymin{}bij(C,D); A \isasyminter C = 0; B \isasyminter D = 0 |] ==>
1299 (f \isasymunion g)\isasymin{}bij(A \isasymunion C, B \isasymunion D)
1301 \tdx{restrict_bij}: [| f\isasymin{}inj(A,B); C\isasymsubseteq{}A |] ==> restrict(f,C)\isasymin{}bij(C, f``C)
1303 \caption{Permutations} \label{zf-perm}
1306 The theory \thydx{Perm} is concerned with permutations (bijections) and
1307 related concepts. These include composition of relations, the identity
1308 relation, and three specialized function spaces: injective, surjective and
1309 bijective. Figure~\ref{zf-perm} displays many of their properties that
1310 have been proved. These results are fundamental to a treatment of
1311 equipollence and cardinality.
1313 Theory \thydx{Univ} defines a `universe' $\isa{univ}(A)$, which is used by
1314 the datatype package. This set contains $A$ and the
1315 natural numbers. Vitally, it is closed under finite products:
1316 $\isa{univ}(A)\times\isa{univ}(A)\subseteq\isa{univ}(A)$. This theory also
1317 defines the cumulative hierarchy of axiomatic set theory, which
1318 traditionally is written $V@\alpha$ for an ordinal~$\alpha$. The
1319 `universe' is a simple generalization of~$V@\omega$.
1321 Theory \thydx{QUniv} defines a `universe' $\isa{quniv}(A)$, which is used by
1322 the datatype package to construct codatatypes such as streams. It is
1323 analogous to $\isa{univ}(A)$ (and is defined in terms of it) but is closed
1324 under the non-standard product and sum.
1327 \section{Automatic Tools}
1329 ZF provides the simplifier and the classical reasoner. Moreover it supplies a
1330 specialized tool to infer `types' of terms.
1332 \subsection{Simplification and Classical Reasoning}
1334 ZF inherits simplification from FOL but adopts it for set theory. The
1335 extraction of rewrite rules takes the ZF primitives into account. It can
1336 strip bounded universal quantifiers from a formula; for example, ${\forall
1337 x\in A. f(x)=g(x)}$ yields the conditional rewrite rule $x\in A \Imp
1338 f(x)=g(x)$. Given $a\in\{x\in A. P(x)\}$ it extracts rewrite rules from $a\in
1339 A$ and~$P(a)$. It can also break down $a\in A\int B$ and $a\in A-B$.
1341 The default simpset used by \isa{simp} contains congruence rules for all of ZF's
1342 binding operators. It contains all the conversion rules, such as
1344 \isa{snd}, as well as the rewrites shown in Fig.\ts\ref{zf-simpdata}.
1346 Classical reasoner methods such as \isa{blast} and \isa{auto} refer to
1347 a rich collection of built-in axioms for all the set-theoretic
1353 a\in \emptyset & \bimp & \bot\\
1354 a \in A \un B & \bimp & a\in A \disj a\in B\\
1355 a \in A \int B & \bimp & a\in A \conj a\in B\\
1356 a \in A-B & \bimp & a\in A \conj \lnot (a\in B)\\
1357 \pair{a,b}\in \isa{Sigma}(A,B)
1358 & \bimp & a\in A \conj b\in B(a)\\
1359 a \in \isa{Collect}(A,P) & \bimp & a\in A \conj P(a)\\
1360 (\forall x \in \emptyset. P(x)) & \bimp & \top\\
1361 (\forall x \in A. \top) & \bimp & \top
1363 \caption{Some rewrite rules for set theory} \label{zf-simpdata}
1367 \subsection{Type-Checking Tactics}
1368 \index{type-checking tactics}
1370 Isabelle/ZF provides simple tactics to help automate those proofs that are
1371 essentially type-checking. Such proofs are built by applying rules such as
1373 \begin{ttbox}\isastyleminor
1374 [| ?P ==> ?a \isasymin ?A; ~?P ==> ?b \isasymin ?A |]
1375 ==> (if ?P then ?a else ?b) \isasymin ?A
1377 [| ?m \isasymin nat; ?n \isasymin nat |] ==> ?m #+ ?n \isasymin nat
1379 ?a \isasymin ?A ==> Inl(?a) \isasymin ?A + ?B
1381 In typical applications, the goal has the form $t\in\Var{A}$: in other words,
1382 we have a specific term~$t$ and need to infer its `type' by instantiating the
1383 set variable~$\Var{A}$. Neither the simplifier nor the classical reasoner
1384 does this job well. The if-then-else rule, and many similar ones, can make
1385 the classical reasoner loop. The simplifier refuses (on principle) to
1386 instantiate variables during rewriting, so goals such as \isa{i\#+j \isasymin \ ?A}
1389 The simplifier calls the type-checker to solve rewritten subgoals: this stage
1390 can indeed instantiate variables. If you have defined new constants and
1391 proved type-checking rules for them, then declare the rules using
1392 the attribute \isa{TC} and the rest should be automatic. In
1393 particular, the simplifier will use type-checking to help satisfy
1394 conditional rewrite rules. Call the method \ttindex{typecheck} to
1395 break down all subgoals using type-checking rules. You can add new
1396 type-checking rules temporarily like this:
1398 \isacommand{apply}\ (typecheck add:\ inj_is_fun)
1402 %Though the easiest way to invoke the type-checker is via the simplifier,
1403 %specialized applications may require more detailed knowledge of
1404 %the type-checking primitives. They are modelled on the simplifier's:
1405 %\begin{ttdescription}
1406 %\item[\ttindexbold{tcset}] is the type of tcsets: sets of type-checking rules.
1408 %\item[\ttindexbold{addTCs}] is an infix operator to add type-checking rules to
1411 %\item[\ttindexbold{delTCs}] is an infix operator to remove type-checking rules
1414 %\item[\ttindexbold{typecheck_tac}] is a tactic for attempting to prove all
1415 % subgoals using the rules given in its argument, a tcset.
1416 %\end{ttdescription}
1418 %Tcsets, like simpsets, are associated with theories and are merged when
1419 %theories are merged. There are further primitives that use the default tcset.
1420 %\begin{ttdescription}
1421 %\item[\ttindexbold{tcset}] is a function to return the default tcset; use the
1422 % expression \isa{tcset()}.
1424 %\item[\ttindexbold{AddTCs}] adds type-checking rules to the default tcset.
1426 %\item[\ttindexbold{DelTCs}] removes type-checking rules from the default
1429 %\item[\ttindexbold{Typecheck_tac}] calls \isa{typecheck_tac} using the
1431 %\end{ttdescription}
1433 %To supply some type-checking rules temporarily, using \isa{Addrules} and
1434 %later \isa{Delrules} is the simplest way. There is also a high-tech
1435 %approach. Call the simplifier with a new solver expressed using
1436 %\ttindexbold{type_solver_tac} and your temporary type-checking rules.
1437 %\begin{ttbox}\isastyleminor
1439 % (simpset() setSolver type_solver_tac (tcset() addTCs prems)) 2);
1443 \section{Natural number and integer arithmetic}
1445 \index{arithmetic|(}
1447 \begin{figure}\small
1448 \index{#*@{\tt\#*} symbol}
1451 \index{#+@{\tt\#+} symbol}
1452 \index{#-@{\tt\#-} symbol}
1454 \it symbol & \it meta-type & \it priority & \it description \\
1455 \cdx{nat} & $i$ & & set of natural numbers \\
1456 \cdx{nat_case}& $[i,i\To i,i]\To i$ & & conditional for $nat$\\
1457 \tt \#* & $[i,i]\To i$ & Left 70 & multiplication \\
1458 \tt div & $[i,i]\To i$ & Left 70 & division\\
1459 \tt mod & $[i,i]\To i$ & Left 70 & modulus\\
1460 \tt \#+ & $[i,i]\To i$ & Left 65 & addition\\
1461 \tt \#- & $[i,i]\To i$ & Left 65 & subtraction
1464 \begin{alltt*}\isastyleminor
1465 \tdx{nat_def}: nat == lfp(lam r \isasymin Pow(Inf). {\ttlbrace}0{\ttrbrace} \isasymunion {\ttlbrace}succ(x). x \isasymin r{\ttrbrace}
1467 \tdx{nat_case_def}: nat_case(a,b,k) ==
1468 THE y. k=0 & y=a | ({\isasymexists}x. k=succ(x) & y=b(x))
1470 \tdx{nat_0I}: 0 \isasymin nat
1471 \tdx{nat_succI}: n \isasymin nat ==> succ(n) \isasymin nat
1474 [| n \isasymin nat; P(0); !!x. [| x \isasymin nat; P(x) |] ==> P(succ(x))
1477 \tdx{nat_case_0}: nat_case(a,b,0) = a
1478 \tdx{nat_case_succ}: nat_case(a,b,succ(m)) = b(m)
1480 \tdx{add_0_natify}: 0 #+ n = natify(n)
1481 \tdx{add_succ}: succ(m) #+ n = succ(m #+ n)
1483 \tdx{mult_type}: m #* n \isasymin nat
1484 \tdx{mult_0}: 0 #* n = 0
1485 \tdx{mult_succ}: succ(m) #* n = n #+ (m #* n)
1486 \tdx{mult_commute}: m #* n = n #* m
1487 \tdx{add_mult_dist}: (m #+ n) #* k = (m #* k) #+ (n #* k)
1488 \tdx{mult_assoc}: (m #* n) #* k = m #* (n #* k)
1489 \tdx{mod_div_equality}: m \isasymin nat ==> (m div n)#*n #+ m mod n = m
1491 \caption{The natural numbers} \label{zf-nat}
1494 \index{natural numbers}
1496 Theory \thydx{Nat} defines the natural numbers and mathematical
1497 induction, along with a case analysis operator. The set of natural
1498 numbers, here called \isa{nat}, is known in set theory as the ordinal~$\omega$.
1500 Theory \thydx{Arith} develops arithmetic on the natural numbers
1501 (Fig.\ts\ref{zf-nat}). Addition, multiplication and subtraction are defined
1502 by primitive recursion. Division and remainder are defined by repeated
1503 subtraction, which requires well-founded recursion; the termination argument
1504 relies on the divisor's being non-zero. Many properties are proved:
1505 commutative, associative and distributive laws, identity and cancellation
1506 laws, etc. The most interesting result is perhaps the theorem $a \bmod b +
1509 To minimize the need for tedious proofs of $t\in\isa{nat}$, the arithmetic
1510 operators coerce their arguments to be natural numbers. The function
1511 \cdx{natify} is defined such that $\isa{natify}(n) = n$ if $n$ is a natural
1512 number, $\isa{natify}(\isa{succ}(x)) =
1513 \isa{succ}(\isa{natify}(x))$ for all $x$, and finally
1514 $\isa{natify}(x)=0$ in all other cases. The benefit is that the addition,
1515 subtraction, multiplication, division and remainder operators always return
1516 natural numbers, regardless of their arguments. Algebraic laws (commutative,
1517 associative, distributive) are unconditional. Occurrences of \isa{natify}
1518 as operands of those operators are simplified away. Any remaining occurrences
1519 can either be tolerated or else eliminated by proving that the argument is a
1522 The simplifier automatically cancels common terms on the opposite sides of
1523 subtraction and of relations ($=$, $<$ and $\le$). Here is an example:
1525 1. i \#+ j \#+ k \#- j < k \#+ l\isanewline
1526 \isacommand{apply}\ simp\isanewline
1527 1. natify(i) < natify(l)
1529 Given the assumptions \isa{i \isasymin nat} and \isa{l \isasymin nat}, both occurrences of
1530 \cdx{natify} would be simplified away.
1533 \begin{figure}\small
1534 \index{$*@{\tt\$*} symbol}
1535 \index{$+@{\tt\$+} symbol}
1536 \index{$-@{\tt\$-} symbol}
1538 \it symbol & \it meta-type & \it priority & \it description \\
1539 \cdx{int} & $i$ & & set of integers \\
1540 \tt \$* & $[i,i]\To i$ & Left 70 & multiplication \\
1541 \tt \$+ & $[i,i]\To i$ & Left 65 & addition\\
1542 \tt \$- & $[i,i]\To i$ & Left 65 & subtraction\\
1543 \tt \$< & $[i,i]\To o$ & Left 50 & $<$ on integers\\
1544 \tt \$<= & $[i,i]\To o$ & Left 50 & $\le$ on integers
1547 \begin{alltt*}\isastyleminor
1548 \tdx{zadd_0_intify}: 0 $+ n = intify(n)
1550 \tdx{zmult_type}: m $* n \isasymin int
1551 \tdx{zmult_0}: 0 $* n = 0
1552 \tdx{zmult_commute}: m $* n = n $* m
1553 \tdx{zadd_zmult_dist}: (m $+ n) $* k = (m $* k) $+ (n $* k)
1554 \tdx{zmult_assoc}: (m $* n) $* k = m $* (n $* k)
1556 \caption{The integers} \label{zf-int}
1562 Theory \thydx{Int} defines the integers, as equivalence classes of natural
1563 numbers. Figure~\ref{zf-int} presents a tidy collection of laws. In
1564 fact, a large library of facts is proved, including monotonicity laws for
1565 addition and multiplication, covering both positive and negative operands.
1567 As with the natural numbers, the need for typing proofs is minimized. All the
1568 operators defined in Fig.\ts\ref{zf-int} coerce their operands to integers by
1569 applying the function \cdx{intify}. This function is the identity on integers
1570 and maps other operands to zero.
1572 Decimal notation is provided for the integers. Numbers, written as
1573 \isa{\#$nnn$} or \isa{\#-$nnn$}, are represented internally in
1574 two's-complement binary. Expressions involving addition, subtraction and
1575 multiplication of numeral constants are evaluated (with acceptable efficiency)
1576 by simplification. The simplifier also collects similar terms, multiplying
1577 them by a numerical coefficient. It also cancels occurrences of the same
1578 terms on the other side of the relational operators. Example:
1580 1. y \$+ z \$+ \#-3 \$* x \$+ y \$<= x \$* \#2 \$+
1582 \isacommand{apply}\ simp\isanewline
1583 1. \#2 \$* y \$<= \#5 \$* x
1585 For more information on the integers, please see the theories on directory
1588 \index{arithmetic|)}
1591 \section{Datatype definitions}
1592 \label{sec:ZF:datatype}
1595 The \ttindex{datatype} definition package of ZF constructs inductive datatypes
1596 similar to \ML's. It can also construct coinductive datatypes
1597 (codatatypes), which are non-well-founded structures such as streams. It
1598 defines the set using a fixed-point construction and proves induction rules,
1599 as well as theorems for recursion and case combinators. It supplies
1600 mechanisms for reasoning about freeness. The datatype package can handle both
1601 mutual and indirect recursion.
1605 \label{subsec:datatype:basics}
1607 A \isa{datatype} definition has the following form:
1610 \mathtt{datatype} & t@1(A@1,\ldots,A@h) & = &
1611 constructor^1@1 ~\mid~ \ldots ~\mid~ constructor^1@{k@1} \\
1613 \mathtt{and} & t@n(A@1,\ldots,A@h) & = &
1614 constructor^n@1~ ~\mid~ \ldots ~\mid~ constructor^n@{k@n}
1617 Here $t@1$, \ldots,~$t@n$ are identifiers and $A@1$, \ldots,~$A@h$ are
1618 variables: the datatype's parameters. Each constructor specification has the
1620 \[ C \hbox{\tt~( } \hbox{\tt"} x@1 \hbox{\tt:} T@1 \hbox{\tt"},\;
1622 \hbox{\tt"} x@m \hbox{\tt:} T@m \hbox{\tt"}
1625 Here $C$ is the constructor name, and variables $x@1$, \ldots,~$x@m$ are the
1626 constructor arguments, belonging to the sets $T@1$, \ldots, $T@m$,
1627 respectively. Typically each $T@j$ is either a constant set, a datatype
1628 parameter (one of $A@1$, \ldots, $A@h$) or a recursive occurrence of one of
1629 the datatypes, say $t@i(A@1,\ldots,A@h)$. More complex possibilities exist,
1630 but they are much harder to realize. Often, additional information must be
1631 supplied in the form of theorems.
1633 A datatype can occur recursively as the argument of some function~$F$. This
1634 is called a {\em nested} (or \emph{indirect}) occurrence. It is only allowed
1635 if the datatype package is given a theorem asserting that $F$ is monotonic.
1636 If the datatype has indirect occurrences, then Isabelle/ZF does not support
1637 recursive function definitions.
1639 A simple example of a datatype is \isa{list}, which is built-in, and is
1641 \begin{alltt*}\isastyleminor
1642 consts list :: "i=>i"
1643 datatype "list(A)" = Nil | Cons ("a \isasymin A", "l \isasymin list(A)")
1645 Note that the datatype operator must be declared as a constant first.
1646 However, the package declares the constructors. Here, \isa{Nil} gets type
1647 $i$ and \isa{Cons} gets type $[i,i]\To i$.
1649 Trees and forests can be modelled by the mutually recursive datatype
1651 \begin{alltt*}\isastyleminor
1655 tree_forest :: "i=>i"
1656 datatype "tree(A)" = Tcons ("a{\isasymin}A", "f{\isasymin}forest(A)")
1657 and "forest(A)" = Fnil | Fcons ("t{\isasymin}tree(A)", "f{\isasymin}forest(A)")
1659 Here $\isa{tree}(A)$ is the set of trees over $A$, $\isa{forest}(A)$ is
1660 the set of forests over $A$, and $\isa{tree_forest}(A)$ is the union of
1661 the previous two sets. All three operators must be declared first.
1663 The datatype \isa{term}, which is defined by
1664 \begin{alltt*}\isastyleminor
1665 consts term :: "i=>i"
1666 datatype "term(A)" = Apply ("a \isasymin A", "l \isasymin list(term(A))")
1669 is an example of nested recursion. (The theorem \isa{list_mono} is proved
1670 in theory \isa{List}, and the \isa{term} example is developed in
1672 \thydx{Induct/Term}.)
1674 \subsubsection{Freeness of the constructors}
1676 Constructors satisfy {\em freeness} properties. Constructions are distinct,
1677 for example $\isa{Nil}\not=\isa{Cons}(a,l)$, and they are injective, for
1678 example $\isa{Cons}(a,l)=\isa{Cons}(a',l') \bimp a=a' \conj l=l'$.
1679 Because the number of freeness is quadratic in the number of constructors, the
1680 datatype package does not prove them. Instead, it ensures that simplification
1681 will prove them dynamically: when the simplifier encounters a formula
1682 asserting the equality of two datatype constructors, it performs freeness
1685 Freeness reasoning can also be done using the classical reasoner, but it is
1686 more complicated. You have to add some safe elimination rules rules to the
1687 claset. For the \isa{list} datatype, they are called
1688 \isa{list.free_elims}. Occasionally this exposes the underlying
1689 representation of some constructor, which can be rectified using the command
1690 \isa{unfold list.con_defs [symmetric]}.
1693 \subsubsection{Structural induction}
1695 The datatype package also provides structural induction rules. For datatypes
1696 without mutual or nested recursion, the rule has the form exemplified by
1697 \isa{list.induct} in Fig.\ts\ref{zf-list}. For mutually recursive
1698 datatypes, the induction rule is supplied in two forms. Consider datatype
1699 \isa{TF}. The rule \isa{tree_forest.induct} performs induction over a
1700 single predicate~\isa{P}, which is presumed to be defined for both trees
1702 \begin{alltt*}\isastyleminor
1703 [| x \isasymin tree_forest(A);
1704 !!a f. [| a \isasymin A; f \isasymin forest(A); P(f) |] ==> P(Tcons(a, f));
1706 !!f t. [| t \isasymin tree(A); P(t); f \isasymin forest(A); P(f) |]
1710 The rule \isa{tree_forest.mutual_induct} performs induction over two
1711 distinct predicates, \isa{P_tree} and \isa{P_forest}.
1712 \begin{alltt*}\isastyleminor
1714 [| a{\isasymin}A; f{\isasymin}forest(A); P_forest(f) |] ==> P_tree(Tcons(a,f));
1716 !!f t. [| t{\isasymin}tree(A); P_tree(t); f{\isasymin}forest(A); P_forest(f) |]
1717 ==> P_forest(Fcons(t, f))
1718 |] ==> ({\isasymforall}za. za \isasymin tree(A) --> P_tree(za)) &
1719 ({\isasymforall}za. za \isasymin forest(A) --> P_forest(za))
1722 For datatypes with nested recursion, such as the \isa{term} example from
1723 above, things are a bit more complicated. The rule \isa{term.induct}
1724 refers to the monotonic operator, \isa{list}:
1725 \begin{alltt*}\isastyleminor
1726 [| x \isasymin term(A);
1727 !!a l. [| a\isasymin{}A; l\isasymin{}list(Collect(term(A), P)) |] ==> P(Apply(a,l))
1730 The theory \isa{Induct/Term.thy} derives two higher-level induction rules,
1731 one of which is particularly useful for proving equations:
1732 \begin{alltt*}\isastyleminor
1733 [| t \isasymin term(A);
1734 !!x zs. [| x \isasymin A; zs \isasymin list(term(A)); map(f, zs) = map(g, zs) |]
1735 ==> f(Apply(x, zs)) = g(Apply(x, zs))
1738 How this can be generalized to other nested datatypes is a matter for future
1742 \subsubsection{The \isa{case} operator}
1744 The package defines an operator for performing case analysis over the
1745 datatype. For \isa{list}, it is called \isa{list_case} and satisfies
1747 \begin{ttbox}\isastyleminor
1748 list_case(f_Nil, f_Cons, []) = f_Nil
1749 list_case(f_Nil, f_Cons, Cons(a, l)) = f_Cons(a, l)
1751 Here \isa{f_Nil} is the value to return if the argument is \isa{Nil} and
1752 \isa{f_Cons} is a function that computes the value to return if the
1753 argument has the form $\isa{Cons}(a,l)$. The function can be expressed as
1754 an abstraction, over patterns if desired (\S\ref{sec:pairs}).
1756 For mutually recursive datatypes, there is a single \isa{case} operator.
1757 In the tree/forest example, the constant \isa{tree_forest_case} handles all
1758 of the constructors of the two datatypes.
1761 \subsection{Defining datatypes}
1763 The theory syntax for datatype definitions is shown in
1764 Fig.~\ref{datatype-grammar}. In order to be well-formed, a datatype
1765 definition has to obey the rules stated in the previous section. As a result
1766 the theory is extended with the new types, the constructors, and the theorems
1767 listed in the previous section. The quotation marks are necessary because
1768 they enclose general Isabelle formul\ae.
1772 datatype : ( 'datatype' | 'codatatype' ) datadecls;
1774 datadecls: ( '"' id arglist '"' '=' (constructor + '|') ) + 'and'
1776 constructor : name ( () | consargs ) ( () | ( '(' mixfix ')' ) )
1778 consargs : '(' ('"' var ' : ' term '"' + ',') ')'
1781 \caption{Syntax of datatype declarations}
1782 \label{datatype-grammar}
1785 Codatatypes are declared like datatypes and are identical to them in every
1786 respect except that they have a coinduction rule instead of an induction rule.
1787 Note that while an induction rule has the effect of limiting the values
1788 contained in the set, a coinduction rule gives a way of constructing new
1791 Most of the theorems about datatypes become part of the default simpset. You
1792 never need to see them again because the simplifier applies them
1795 \subsubsection{Specialized methods for datatypes}
1797 Induction and case-analysis can be invoked using these special-purpose
1799 \begin{ttdescription}
1800 \item[\methdx{induct_tac} $x$] applies structural
1801 induction on variable $x$ to subgoal~1, provided the type of $x$ is a
1802 datatype. The induction variable should not occur among other assumptions
1806 % we also have the ind_cases method, but what does it do?
1807 In some situations, induction is overkill and a case distinction over all
1808 constructors of the datatype suffices.
1809 \begin{ttdescription}
1810 \item[\methdx{case_tac} $x$]
1811 performs a case analysis for the variable~$x$.
1814 Both tactics can only be applied to a variable, whose typing must be given in
1815 some assumption, for example the assumption \isa{x \isasymin \ list(A)}. The tactics
1816 also work for the natural numbers (\isa{nat}) and disjoint sums, although
1817 these sets were not defined using the datatype package. (Disjoint sums are
1818 not recursive, so only \isa{case_tac} is available.)
1820 Structured Isar methods are also available. Below, $t$
1821 stands for the name of the datatype.
1822 \begin{ttdescription}
1823 \item[\methdx{induct} \isa{set:}\ $t$] is the Isar induction tactic.
1824 \item[\methdx{cases} \isa{set:}\ $t$] is the Isar case-analysis tactic.
1828 \subsubsection{The theorems proved by a datatype declaration}
1830 Here are some more details for the technically minded. Processing the
1831 datatype declaration of a set~$t$ produces a name space~$t$ containing
1832 the following theorems:
1833 \begin{ttbox}\isastyleminor
1834 intros \textrm{the introduction rules}
1835 cases \textrm{the case analysis rule}
1836 induct \textrm{the standard induction rule}
1837 mutual_induct \textrm{the mutual induction rule, if needed}
1838 case_eqns \textrm{equations for the case operator}
1839 recursor_eqns \textrm{equations for the recursor}
1840 simps \textrm{the union of} case_eqns \textrm{and} recursor_eqns
1841 con_defs \textrm{definitions of the case operator and constructors}
1842 free_iffs \textrm{logical equivalences for proving freeness}
1843 free_elims \textrm{elimination rules for proving freeness}
1844 defs \textrm{datatype definition(s)}
1846 Furthermore there is the theorem $C$ for every constructor~$C$; for
1847 example, the \isa{list} datatype's introduction rules are bound to the
1848 identifiers \isa{Nil} and \isa{Cons}.
1850 For a codatatype, the component \isa{coinduct} is the coinduction rule,
1851 replacing the \isa{induct} component.
1853 See the theories \isa{Induct/Ntree} and \isa{Induct/Brouwer} for examples of
1854 infinitely branching datatypes. See theory \isa{Induct/LList} for an example
1855 of a codatatype. Some of these theories illustrate the use of additional,
1856 undocumented features of the datatype package. Datatype definitions are
1857 reduced to inductive definitions, and the advanced features should be
1858 understood in that light.
1861 \subsection{Examples}
1863 \subsubsection{The datatype of binary trees}
1865 Let us define the set $\isa{bt}(A)$ of binary trees over~$A$. The theory
1866 must contain these lines:
1867 \begin{alltt*}\isastyleminor
1869 datatype "bt(A)" = Lf | Br ("a\isasymin{}A", "t1\isasymin{}bt(A)", "t2\isasymin{}bt(A)")
1871 After loading the theory, we can prove some theorem.
1872 We begin by declaring the constructor's typechecking rules
1873 as simplification rules:
1875 \isacommand{declare}\ bt.intros\ [simp]%
1878 Our first example is the theorem that no tree equals its
1879 left branch. To make the inductive hypothesis strong enough,
1880 the proof requires a quantified induction formula, but
1881 the \isa{rule\_format} attribute will remove the quantifiers
1882 before the theorem is stored.
1884 \isacommand{lemma}\ Br\_neq\_left\ [rule\_format]:\ "l\isasymin bt(A)\ ==>\ \isasymforall x\ r.\ Br(x,l,r)\isasymnoteq{}l"\isanewline
1885 \ 1.\ l\ \isasymin \ bt(A)\ \isasymLongrightarrow \ \isasymforall x\ r.\ Br(x,\ l,\ r)\ \isasymnoteq \ l%
1887 This can be proved by the structural induction tactic:
1889 \ \ \isacommand{apply}\ (induct\_tac\ l)\isanewline
1890 \ 1.\ \isasymforall x\ r.\ Br(x,\ Lf,\ r)\ \isasymnoteq \ Lf\isanewline
1891 \ 2.\ \isasymAnd a\ t1\ t2.\isanewline
1892 \isaindent{\ 2.\ \ \ \ }\isasymlbrakk a\ \isasymin \ A;\ t1\ \isasymin \ bt(A);\ \isasymforall x\ r.\ Br(x,\ t1,\ r)\ \isasymnoteq \ t1;\ t2\ \isasymin \ bt(A);\isanewline
1893 \isaindent{\ 2.\ \ \ \ \ \ \ }\isasymforall x\ r.\ Br(x,\ t2,\ r)\ \isasymnoteq \ t2\isasymrbrakk \isanewline
1894 \isaindent{\ 2.\ \ \ \ }\isasymLongrightarrow \ \isasymforall x\ r.\ Br(x,\ Br(a,\ t1,\ t2),\ r)\ \isasymnoteq \ Br(a,\ t1,\ t2)
1896 Both subgoals are proved using \isa{auto}, which performs the necessary
1899 \ \ \isacommand{apply}\ auto\isanewline
1900 No\ subgoals!\isanewline
1904 An alternative proof uses Isar's fancy \isa{induct} method, which
1905 automatically quantifies over all free variables:
1908 \isacommand{lemma}\ Br\_neq\_left':\ "l\ \isasymin \ bt(A)\ ==>\ (!!x\ r.\ Br(x,\ l,\ r)\ \isasymnoteq \ l)"\isanewline
1909 \ \ \isacommand{apply}\ (induct\ set:\ bt)\isanewline
1910 \ 1.\ \isasymAnd x\ r.\ Br(x,\ Lf,\ r)\ \isasymnoteq \ Lf\isanewline
1911 \ 2.\ \isasymAnd a\ t1\ t2\ x\ r.\isanewline
1912 \isaindent{\ 2.\ \ \ \ }\isasymlbrakk a\ \isasymin \ A;\ t1\ \isasymin \ bt(A);\ \isasymAnd x\ r.\ Br(x,\ t1,\ r)\ \isasymnoteq \ t1;\ t2\ \isasymin \ bt(A);\isanewline
1913 \isaindent{\ 2.\ \ \ \ \ \ \ }\isasymAnd x\ r.\ Br(x,\ t2,\ r)\ \isasymnoteq \ t2\isasymrbrakk \isanewline
1914 \isaindent{\ 2.\ \ \ \ }\isasymLongrightarrow \ Br(x,\ Br(a,\ t1,\ t2),\ r)\ \isasymnoteq \ Br(a,\ t1,\ t2)
1916 Compare the form of the induction hypotheses with the corresponding ones in
1917 the previous proof. As before, to conclude requires only \isa{auto}.
1919 When there are only a few constructors, we might prefer to prove the freenness
1920 theorems for each constructor. This is simple:
1922 \isacommand{lemma}\ Br\_iff:\ "Br(a,l,r)\ =\ Br(a',l',r')\ <->\ a=a'\ \&\ l=l'\ \&\ r=r'"\isanewline
1923 \ \ \isacommand{by}\ (blast\ elim!:\ bt.free\_elims)
1925 Here we see a demonstration of freeness reasoning using
1926 \isa{bt.free\_elims}, but simpler still is just to apply \isa{auto}.
1928 An \ttindex{inductive\_cases} declaration generates instances of the
1929 case analysis rule that have been simplified using freeness
1932 \isacommand{inductive\_cases}\ Br\_in\_bt:\ "Br(a,\ l,\ r)\ \isasymin \ bt(A)"
1934 The theorem just created is
1936 \isasymlbrakk Br(a,\ l,\ r)\ \isasymin \ bt(A);\ \isasymlbrakk a\ \isasymin \ A;\ l\ \isasymin \ bt(A);\ r\ \isasymin \ bt(A)\isasymrbrakk \ \isasymLongrightarrow \ Q\isasymrbrakk \ \isasymLongrightarrow \ Q.
1938 It is an elimination rule that from $\isa{Br}(a,l,r)\in\isa{bt}(A)$
1939 lets us infer $a\in A$, $l\in\isa{bt}(A)$ and
1943 \subsubsection{Mixfix syntax in datatypes}
1945 Mixfix syntax is sometimes convenient. The theory \isa{Induct/PropLog} makes a
1946 deep embedding of propositional logic:
1947 \begin{alltt*}\isastyleminor
1949 datatype "prop" = Fls
1950 | Var ("n \isasymin nat") ("#_" [100] 100)
1951 | "=>" ("p \isasymin prop", "q \isasymin prop") (infixr 90)
1953 The second constructor has a special $\#n$ syntax, while the third constructor
1954 is an infixed arrow.
1957 \subsubsection{A giant enumeration type}
1959 This example shows a datatype that consists of 60 constructors:
1960 \begin{alltt*}\isastyleminor
1963 "enum" = C00 | C01 | C02 | C03 | C04 | C05 | C06 | C07 | C08 | C09
1964 | C10 | C11 | C12 | C13 | C14 | C15 | C16 | C17 | C18 | C19
1965 | C20 | C21 | C22 | C23 | C24 | C25 | C26 | C27 | C28 | C29
1966 | C30 | C31 | C32 | C33 | C34 | C35 | C36 | C37 | C38 | C39
1967 | C40 | C41 | C42 | C43 | C44 | C45 | C46 | C47 | C48 | C49
1968 | C50 | C51 | C52 | C53 | C54 | C55 | C56 | C57 | C58 | C59
1971 The datatype package scales well. Even though all properties are proved
1972 rather than assumed, full processing of this definition takes around two seconds
1973 (on a 1.8GHz machine). The constructors have a balanced representation,
1974 related to binary notation, so freeness properties can be proved fast.
1976 \isacommand{lemma}\ "C00 \isasymnoteq\ C01"\isanewline
1977 \ \ \isacommand{by}\ simp
1979 You need not derive such inequalities explicitly. The simplifier will
1980 dispose of them automatically.
1985 \subsection{Recursive function definitions}\label{sec:ZF:recursive}
1986 \index{recursive functions|see{recursion}}
1988 \index{recursion!primitive|(}
1990 Datatypes come with a uniform way of defining functions, {\bf primitive
1991 recursion}. Such definitions rely on the recursion operator defined by the
1992 datatype package. Isabelle proves the desired recursion equations as
1995 In principle, one could introduce primitive recursive functions by asserting
1996 their reduction rules as axioms. Here is a dangerous way of defining a
1997 recursive function over binary trees:
1999 \isacommand{consts}\ \ n\_nodes\ ::\ "i\ =>\ i"\isanewline
2000 \isacommand{axioms}\isanewline
2001 \ \ n\_nodes\_Lf:\ "n\_nodes(Lf)\ =\ 0"\isanewline
2002 \ \ n\_nodes\_Br:\ "n\_nodes(Br(a,l,r))\ =\ succ(n\_nodes(l)\ \#+\ n\_nodes(r))"
2004 Asserting axioms brings the danger of accidentally introducing
2005 contradictions. It should be avoided whenever possible.
2007 The \ttindex{primrec} declaration is a safe means of defining primitive
2008 recursive functions on datatypes:
2010 \isacommand{consts}\ \ n\_nodes\ ::\ "i\ =>\ i"\isanewline
2011 \isacommand{primrec}\isanewline
2012 \ \ "n\_nodes(Lf)\ =\ 0"\isanewline
2013 \ \ "n\_nodes(Br(a,\ l,\ r))\ =\ succ(n\_nodes(l)\ \#+\ n\_nodes(r))"
2015 Isabelle will now derive the two equations from a low-level definition
2016 based upon well-founded recursion. If they do not define a legitimate
2017 recursion, then Isabelle will reject the declaration.
2020 \subsubsection{Syntax of recursive definitions}
2022 The general form of a primitive recursive definition is
2023 \begin{ttbox}\isastyleminor
2025 {\it reduction rules}
2027 where \textit{reduction rules} specify one or more equations of the form
2028 \[ f \, x@1 \, \dots \, x@m \, (C \, y@1 \, \dots \, y@k) \, z@1 \,
2029 \dots \, z@n = r \] such that $C$ is a constructor of the datatype, $r$
2030 contains only the free variables on the left-hand side, and all recursive
2031 calls in $r$ are of the form $f \, \dots \, y@i \, \dots$ for some $i$.
2032 There must be at most one reduction rule for each constructor. The order is
2033 immaterial. For missing constructors, the function is defined to return zero.
2035 All reduction rules are added to the default simpset.
2036 If you would like to refer to some rule by name, then you must prefix
2037 the rule with an identifier. These identifiers, like those in the
2038 \isa{rules} section of a theory, will be visible in proof scripts.
2040 The reduction rules become part of the default simpset, which
2041 leads to short proof scripts:
2043 \isacommand{lemma}\ n\_nodes\_type\ [simp]:\ "t\ \isasymin \ bt(A)\ ==>\ n\_nodes(t)\ \isasymin \ nat"\isanewline
2044 \ \ \isacommand{by}\ (induct\_tac\ t,\ auto)
2047 You can even use the \isa{primrec} form with non-recursive datatypes and
2048 with codatatypes. Recursion is not allowed, but it provides a convenient
2049 syntax for defining functions by cases.
2052 \subsubsection{Example: varying arguments}
2054 All arguments, other than the recursive one, must be the same in each equation
2055 and in each recursive call. To get around this restriction, use explict
2056 $\lambda$-abstraction and function application. For example, let us
2057 define the tail-recursive version of \isa{n\_nodes}, using an
2058 accumulating argument for the counter. The second argument, $k$, varies in
2061 \isacommand{consts}\ \ n\_nodes\_aux\ ::\ "i\ =>\ i"\isanewline
2062 \isacommand{primrec}\isanewline
2063 \ \ "n\_nodes\_aux(Lf)\ =\ (\isasymlambda k\ \isasymin \ nat.\ k)"\isanewline
2064 \ \ "n\_nodes\_aux(Br(a,l,r))\ =\ \isanewline
2065 \ \ \ \ \ \ (\isasymlambda k\ \isasymin \ nat.\ n\_nodes\_aux(r)\ `\ \ (n\_nodes\_aux(l)\ `\ succ(k)))"
2067 Now \isa{n\_nodes\_aux(t)\ `\ k} is our function in two arguments. We
2068 can prove a theorem relating it to \isa{n\_nodes}. Note the quantification
2069 over \isa{k\ \isasymin \ nat}:
2071 \isacommand{lemma}\ n\_nodes\_aux\_eq\ [rule\_format]:\isanewline
2072 \ \ \ \ \ "t\ \isasymin \ bt(A)\ ==>\ \isasymforall k\ \isasymin \ nat.\ n\_nodes\_aux(t)`k\ =\ n\_nodes(t)\ \#+\ k"\isanewline
2073 \ \ \isacommand{by}\ (induct\_tac\ t,\ simp\_all)
2076 Now, we can use \isa{n\_nodes\_aux} to define a tail-recursive version
2079 \isacommand{constdefs}\isanewline
2080 \ \ n\_nodes\_tail\ ::\ "i\ =>\ i"\isanewline
2081 \ \ \ "n\_nodes\_tail(t)\ ==\ n\_nodes\_aux(t)\ `\ 0"
2084 prove that \isa{n\_nodes\_tail} is equivalent to \isa{n\_nodes}:
2086 \isacommand{lemma}\ "t\ \isasymin \ bt(A)\ ==>\ n\_nodes\_tail(t)\ =\ n\_nodes(t)"\isanewline
2087 \ \isacommand{by}\ (simp\ add:\ n\_nodes\_tail\_def\ n\_nodes\_aux\_eq)
2093 \index{recursion!primitive|)}
2097 \section{Inductive and coinductive definitions}
2098 \index{*inductive|(}
2099 \index{*coinductive|(}
2101 An {\bf inductive definition} specifies the least set~$R$ closed under given
2102 rules. (Applying a rule to elements of~$R$ yields a result within~$R$.) For
2103 example, a structural operational semantics is an inductive definition of an
2104 evaluation relation. Dually, a {\bf coinductive definition} specifies the
2105 greatest set~$R$ consistent with given rules. (Every element of~$R$ can be
2106 seen as arising by applying a rule to elements of~$R$.) An important example
2107 is using bisimulation relations to formalise equivalence of processes and
2108 infinite data structures.
2110 A theory file may contain any number of inductive and coinductive
2111 definitions. They may be intermixed with other declarations; in
2112 particular, the (co)inductive sets {\bf must} be declared separately as
2113 constants, and may have mixfix syntax or be subject to syntax translations.
2115 Each (co)inductive definition adds definitions to the theory and also
2116 proves some theorems. It behaves identially to the analogous
2117 inductive definition except that instead of an induction rule there is
2118 a coinduction rule. Its treatment of coinduction is described in
2119 detail in a separate paper,%
2120 \footnote{It appeared in CADE~\cite{paulson-CADE}; a longer version is
2121 distributed with Isabelle as \emph{A Fixedpoint Approach to
2122 (Co)Inductive and (Co)Datatype Definitions}.} %
2123 which you might refer to for background information.
2126 \subsection{The syntax of a (co)inductive definition}
2127 An inductive definition has the form
2128 \begin{ttbox}\isastyleminor
2130 domains {\it domain declarations}
2131 intros {\it introduction rules}
2132 monos {\it monotonicity theorems}
2133 con_defs {\it constructor definitions}
2134 type_intros {\it introduction rules for type-checking}
2135 type_elims {\it elimination rules for type-checking}
2137 A coinductive definition is identical, but starts with the keyword
2138 \isa{co\-inductive}.
2140 The \isa{monos}, \isa{con\_defs}, \isa{type\_intros} and \isa{type\_elims}
2141 sections are optional. If present, each is specified as a list of
2142 theorems, which may contain Isar attributes as usual.
2145 \item[\it domain declarations] are items of the form
2146 {\it string\/}~\isa{\isasymsubseteq }~{\it string}, associating each recursive set with
2147 its domain. (The domain is some existing set that is large enough to
2148 hold the new set being defined.)
2150 \item[\it introduction rules] specify one or more introduction rules in
2151 the form {\it ident\/}~{\it string}, where the identifier gives the name of
2152 the rule in the result structure.
2154 \item[\it monotonicity theorems] are required for each operator applied to
2155 a recursive set in the introduction rules. There \textbf{must} be a theorem
2156 of the form $A\subseteq B\Imp M(A)\subseteq M(B)$, for each premise $t\in M(R_i)$
2157 in an introduction rule!
2159 \item[\it constructor definitions] contain definitions of constants
2160 appearing in the introduction rules. The (co)datatype package supplies
2161 the constructors' definitions here. Most (co)inductive definitions omit
2162 this section; one exception is the primitive recursive functions example;
2163 see theory \isa{Induct/Primrec}.
2165 \item[\it type\_intros] consists of introduction rules for type-checking the
2166 definition: for demonstrating that the new set is included in its domain.
2167 (The proof uses depth-first search.)
2169 \item[\it type\_elims] consists of elimination rules for type-checking the
2170 definition. They are presumed to be safe and are applied as often as
2171 possible prior to the \isa{type\_intros} search.
2174 The package has a few restrictions:
2176 \item The theory must separately declare the recursive sets as
2179 \item The names of the recursive sets must be identifiers, not infix
2182 \item Side-conditions must not be conjunctions. However, an introduction rule
2183 may contain any number of side-conditions.
2185 \item Side-conditions of the form $x=t$, where the variable~$x$ does not
2186 occur in~$t$, will be substituted through the rule \isa{mutual\_induct}.
2190 \subsection{Example of an inductive definition}
2192 Below, we shall see how Isabelle/ZF defines the finite powerset
2193 operator. The first step is to declare the constant~\isa{Fin}. Then we
2194 must declare it inductively, with two introduction rules:
2196 \isacommand{consts}\ \ Fin\ ::\ "i=>i"\isanewline
2197 \isacommand{inductive}\isanewline
2198 \ \ \isakeyword{domains}\ \ \ "Fin(A)"\ \isasymsubseteq\ "Pow(A)"\isanewline
2199 \ \ \isakeyword{intros}\isanewline
2200 \ \ \ \ emptyI:\ \ "0\ \isasymin\ Fin(A)"\isanewline
2201 \ \ \ \ consI:\ \ \ "[|\ a\ \isasymin\ A;\ \ b\ \isasymin\ Fin(A)\ |]\ ==>\ cons(a,b)\ \isasymin\ Fin(A)"\isanewline
2202 \ \ \isakeyword{type\_intros}\ \ empty\_subsetI\ cons\_subsetI\ PowI\isanewline
2203 \ \ \isakeyword{type\_elims}\ \ \ PowD\ [THEN\ revcut\_rl]\end{isabelle}
2204 The resulting theory contains a name space, called~\isa{Fin}.
2205 The \isa{Fin}$~A$ introduction rules can be referred to collectively as
2206 \isa{Fin.intros}, and also individually as \isa{Fin.emptyI} and
2207 \isa{Fin.consI}. The induction rule is \isa{Fin.induct}.
2209 The chief problem with making (co)inductive definitions involves type-checking
2210 the rules. Sometimes, additional theorems need to be supplied under
2211 \isa{type_intros} or \isa{type_elims}. If the package fails when trying
2212 to prove your introduction rules, then set the flag \ttindexbold{trace_induct}
2213 to \isa{true} and try again. (See the manual \emph{A Fixedpoint Approach
2214 \ldots} for more discussion of type-checking.)
2216 In the example above, $\isa{Pow}(A)$ is given as the domain of
2217 $\isa{Fin}(A)$, for obviously every finite subset of~$A$ is a subset
2218 of~$A$. However, the inductive definition package can only prove that given a
2220 Here is the output that results (with the flag set) when the
2221 \isa{type_intros} and \isa{type_elims} are omitted from the inductive
2223 \begin{alltt*}\isastyleminor
2224 Inductive definition Finite.Fin
2227 \%X. {z\isasymin{}Pow(A) . z = 0 | ({\isasymexists}a b. z = cons(a,b) & a\isasymin{}A & b\isasymin{}X)})
2228 Proving monotonicity...
2230 Proving the introduction rules...
2231 The type-checking subgoal:
2233 1. 0 \isasymin Pow(A)
2235 The subgoal after monos, type_elims:
2237 1. 0 \isasymin Pow(A)
2238 *** prove_goal: tactic failed
2240 We see the need to supply theorems to let the package prove
2241 $\emptyset\in\isa{Pow}(A)$. Restoring the \isa{type_intros} but not the
2242 \isa{type_elims}, we again get an error message:
2243 \begin{alltt*}\isastyleminor
2244 The type-checking subgoal:
2246 1. 0 \isasymin Pow(A)
2248 The subgoal after monos, type_elims:
2250 1. 0 \isasymin Pow(A)
2252 The type-checking subgoal:
2253 cons(a, b) \isasymin Fin(A)
2254 1. [| a \isasymin A; b \isasymin Fin(A) |] ==> cons(a, b) \isasymin Pow(A)
2256 The subgoal after monos, type_elims:
2257 cons(a, b) \isasymin Fin(A)
2258 1. [| a \isasymin A; b \isasymin Pow(A) |] ==> cons(a, b) \isasymin Pow(A)
2259 *** prove_goal: tactic failed
2261 The first rule has been type-checked, but the second one has failed. The
2262 simplest solution to such problems is to prove the failed subgoal separately
2263 and to supply it under \isa{type_intros}. The solution actually used is
2264 to supply, under \isa{type_elims}, a rule that changes
2265 $b\in\isa{Pow}(A)$ to $b\subseteq A$; together with \isa{cons_subsetI}
2266 and \isa{PowI}, it is enough to complete the type-checking.
2270 \subsection{Further examples}
2272 An inductive definition may involve arbitrary monotonic operators. Here is a
2273 standard example: the accessible part of a relation. Note the use
2274 of~\isa{Pow} in the introduction rule and the corresponding mention of the
2275 rule \isa{Pow\_mono} in the \isa{monos} list. If the desired rule has a
2276 universally quantified premise, usually the effect can be obtained using
2279 \isacommand{consts}\ \ acc\ ::\ "i\ =>\ i"\isanewline
2280 \isacommand{inductive}\isanewline
2281 \ \ \isakeyword{domains}\ "acc(r)"\ \isasymsubseteq \ "field(r)"\isanewline
2282 \ \ \isakeyword{intros}\isanewline
2283 \ \ \ \ vimage:\ \ "[|\ r-``\isacharbraceleft a\isacharbraceright\ \isasymin\ Pow(acc(r));\ a\ \isasymin \ field(r)\ |]
2285 \ \ \ \ \ \ \ \ \ \ \ \ \ \ ==>\ a\ \isasymin \ acc(r)"\isanewline
2286 \ \ \isakeyword{monos}\ \ Pow\_mono
2289 Finally, here are some coinductive definitions. We begin by defining
2290 lazy (potentially infinite) lists as a codatatype:
2292 \isacommand{consts}\ \ llist\ \ ::\ "i=>i"\isanewline
2293 \isacommand{codatatype}\isanewline
2294 \ \ "llist(A)"\ =\ LNil\ |\ LCons\ ("a\ \isasymin \ A",\ "l\ \isasymin \ llist(A)")\isanewline
2297 The notion of equality on such lists is modelled as a bisimulation:
2299 \isacommand{consts}\ \ lleq\ ::\ "i=>i"\isanewline
2300 \isacommand{coinductive}\isanewline
2301 \ \ \isakeyword{domains}\ "lleq(A)"\ <=\ "llist(A)\ *\ llist(A)"\isanewline
2302 \ \ \isakeyword{intros}\isanewline
2303 \ \ \ \ LNil:\ \ "<LNil,\ LNil>\ \isasymin \ lleq(A)"\isanewline
2304 \ \ \ \ LCons:\ "[|\ a\ \isasymin \ A;\ <l,l'>\ \isasymin \ lleq(A)\ |]\ \isanewline
2305 \ \ \ \ \ \ \ \ \ \ \ \ ==>\ <LCons(a,l),\ LCons(a,l')>\ \isasymin \ lleq(A)"\isanewline
2306 \ \ \isakeyword{type\_intros}\ \ llist.intros
2308 This use of \isa{type_intros} is typical: the relation concerns the
2309 codatatype \isa{llist}, so naturally the introduction rules for that
2310 codatatype will be required for type-checking the rules.
2312 The Isabelle distribution contains many other inductive definitions. Simple
2313 examples are collected on subdirectory \isa{ZF/Induct}. The directory
2314 \isa{Coind} and the theory \isa{ZF/Induct/LList} contain coinductive
2315 definitions. Larger examples may be found on other subdirectories of
2316 \isa{ZF}, such as \isa{IMP}, and \isa{Resid}.
2319 \subsection{Theorems generated}
2321 Each (co)inductive set defined in a theory file generates a name space
2322 containing the following elements:
2323 \begin{ttbox}\isastyleminor
2324 intros \textrm{the introduction rules}
2325 elim \textrm{the elimination (case analysis) rule}
2326 induct \textrm{the standard induction rule}
2327 mutual_induct \textrm{the mutual induction rule, if needed}
2328 defs \textrm{definitions of inductive sets}
2329 bnd_mono \textrm{monotonicity property}
2330 dom_subset \textrm{inclusion in `bounding set'}
2332 Furthermore, each introduction rule is available under its declared
2333 name. For a codatatype, the component \isa{coinduct} is the coinduction rule,
2334 replacing the \isa{induct} component.
2336 Recall that the \ttindex{inductive\_cases} declaration generates
2337 simplified instances of the case analysis rule. It is as useful for
2338 inductive definitions as it is for datatypes. There are many examples
2340 \isa{Induct/Comb}, which is discussed at length
2341 elsewhere~\cite{paulson-generic}. The theory first defines the
2343 \isa{comb} of combinators:
2344 \begin{alltt*}\isastyleminor
2348 | "#" ("p \isasymin comb", "q \isasymin comb") (infixl 90)
2350 The theory goes on to define contraction and parallel contraction
2351 inductively. Then the theory \isa{Induct/Comb.thy} defines special
2352 cases of contraction, such as this one:
2354 \isacommand{inductive\_cases}\ K\_contractE [elim!]:\ "K -1-> r"
2356 The theorem just created is \isa{K -1-> r \ \isasymLongrightarrow \ Q},
2357 which expresses that the combinator \isa{K} cannot reduce to
2358 anything. (From the assumption \isa{K-1->r}, we can conclude any desired
2359 formula \isa{Q}\@.) Similar elimination rules for \isa{S} and application are also
2360 generated. The attribute \isa{elim!}\ shown above supplies the generated
2361 theorem to the classical reasoner. This mode of working allows
2362 effective reasoniung about operational semantics.
2364 \index{*coinductive|)} \index{*inductive|)}
2368 \section{The outer reaches of set theory}
2370 The constructions of the natural numbers and lists use a suite of
2371 operators for handling recursive function definitions. I have described
2372 the developments in detail elsewhere~\cite{paulson-set-II}. Here is a brief
2375 \item Theory \isa{Trancl} defines the transitive closure of a relation
2376 (as a least fixedpoint).
2378 \item Theory \isa{WF} proves the well-founded recursion theorem, using an
2379 elegant approach of Tobias Nipkow. This theorem permits general
2380 recursive definitions within set theory.
2382 \item Theory \isa{Ord} defines the notions of transitive set and ordinal
2383 number. It derives transfinite induction. A key definition is {\bf
2384 less than}: $i<j$ if and only if $i$ and $j$ are both ordinals and
2385 $i\in j$. As a special case, it includes less than on the natural
2388 \item Theory \isa{Epsilon} derives $\varepsilon$-induction and
2389 $\varepsilon$-recursion, which are generalisations of transfinite
2390 induction and recursion. It also defines \cdx{rank}$(x)$, which is the
2391 least ordinal $\alpha$ such that $x$ is constructed at stage $\alpha$ of
2392 the cumulative hierarchy (thus $x\in V@{\alpha+1}$).
2395 Other important theories lead to a theory of cardinal numbers. They have
2396 not yet been written up anywhere. Here is a summary:
2398 \item Theory \isa{Rel} defines the basic properties of relations, such as
2399 reflexivity, symmetry and transitivity.
2401 \item Theory \isa{EquivClass} develops a theory of equivalence
2402 classes, not using the Axiom of Choice.
2404 \item Theory \isa{Order} defines partial orderings, total orderings and
2407 \item Theory \isa{OrderArith} defines orderings on sum and product sets.
2408 These can be used to define ordinal arithmetic and have applications to
2409 cardinal arithmetic.
2411 \item Theory \isa{OrderType} defines order types. Every wellordering is
2412 equivalent to a unique ordinal, which is its order type.
2414 \item Theory \isa{Cardinal} defines equipollence and cardinal numbers.
2416 \item Theory \isa{CardinalArith} defines cardinal addition and
2417 multiplication, and proves their elementary laws. It proves that there
2418 is no greatest cardinal. It also proves a deep result, namely
2419 $\kappa\otimes\kappa=\kappa$ for every infinite cardinal~$\kappa$; see
2420 Kunen~\cite[page 29]{kunen80}. None of these results assume the Axiom of
2421 Choice, which complicates their proofs considerably.
2424 The following developments involve the Axiom of Choice (AC):
2426 \item Theory \isa{AC} asserts the Axiom of Choice and proves some simple
2429 \item Theory \isa{Zorn} proves Hausdorff's Maximal Principle, Zorn's Lemma
2430 and the Wellordering Theorem, following Abrial and
2431 Laffitte~\cite{abrial93}.
2433 \item Theory \isa{Cardinal\_AC} uses AC to prove simplified theorems about
2434 the cardinals. It also proves a theorem needed to justify
2435 infinitely branching datatype declarations: if $\kappa$ is an infinite
2436 cardinal and $|X(\alpha)| \le \kappa$ for all $\alpha<\kappa$ then
2437 $|\union\sb{\alpha<\kappa} X(\alpha)| \le \kappa$.
2439 \item Theory \isa{InfDatatype} proves theorems to justify infinitely
2440 branching datatypes. Arbitrary index sets are allowed, provided their
2441 cardinalities have an upper bound. The theory also justifies some
2442 unusual cases of finite branching, involving the finite powerset operator
2443 and the finite function space operator.
2448 \section{The examples directories}
2449 Directory \isa{HOL/IMP} contains a mechanised version of a semantic
2450 equivalence proof taken from Winskel~\cite{winskel93}. It formalises the
2451 denotational and operational semantics of a simple while-language, then
2452 proves the two equivalent. It contains several datatype and inductive
2453 definitions, and demonstrates their use.
2455 The directory \isa{ZF/ex} contains further developments in ZF set theory.
2456 Here is an overview; see the files themselves for more details. I describe
2457 much of this material in other
2458 publications~\cite{paulson-set-I,paulson-set-II,paulson-fixedpt-milner}.
2460 \item File \isa{misc.ML} contains miscellaneous examples such as
2461 Cantor's Theorem, the Schr\"oder-Bernstein Theorem and the `Composition
2462 of homomorphisms' challenge~\cite{boyer86}.
2464 \item Theory \isa{Ramsey} proves the finite exponent 2 version of
2465 Ramsey's Theorem, following Basin and Kaufmann's
2466 presentation~\cite{basin91}.
2468 \item Theory \isa{Integ} develops a theory of the integers as
2469 equivalence classes of pairs of natural numbers.
2471 \item Theory \isa{Primrec} develops some computation theory. It
2472 inductively defines the set of primitive recursive functions and presents a
2473 proof that Ackermann's function is not primitive recursive.
2475 \item Theory \isa{Primes} defines the Greatest Common Divisor of two
2476 natural numbers and and the ``divides'' relation.
2478 \item Theory \isa{Bin} defines a datatype for two's complement binary
2479 integers, then proves rewrite rules to perform binary arithmetic. For
2480 instance, $1359\times {-}2468 = {-}3354012$ takes 0.3 seconds.
2482 \item Theory \isa{BT} defines the recursive data structure $\isa{bt}(A)$, labelled binary trees.
2484 \item Theory \isa{Term} defines a recursive data structure for terms
2485 and term lists. These are simply finite branching trees.
2487 \item Theory \isa{TF} defines primitives for solving mutually
2488 recursive equations over sets. It constructs sets of trees and forests
2489 as an example, including induction and recursion rules that handle the
2492 \item Theory \isa{Prop} proves soundness and completeness of
2493 propositional logic~\cite{paulson-set-II}. This illustrates datatype
2494 definitions, inductive definitions, structural induction and rule
2497 \item Theory \isa{ListN} inductively defines the lists of $n$
2498 elements~\cite{paulin-tlca}.
2500 \item Theory \isa{Acc} inductively defines the accessible part of a
2501 relation~\cite{paulin-tlca}.
2503 \item Theory \isa{Comb} defines the datatype of combinators and
2504 inductively defines contraction and parallel contraction. It goes on to
2505 prove the Church-Rosser Theorem. This case study follows Camilleri and
2506 Melham~\cite{camilleri92}.
2508 \item Theory \isa{LList} defines lazy lists and a coinduction
2509 principle for proving equations between them.
2513 \section{A proof about powersets}\label{sec:ZF-pow-example}
2514 To demonstrate high-level reasoning about subsets, let us prove the
2515 equation ${\isa{Pow}(A)\cap \isa{Pow}(B)}= \isa{Pow}(A\cap B)$. Compared
2516 with first-order logic, set theory involves a maze of rules, and theorems
2517 have many different proofs. Attempting other proofs of the theorem might
2518 be instructive. This proof exploits the lattice properties of
2519 intersection. It also uses the monotonicity of the powerset operation,
2520 from \isa{ZF/mono.ML}:
2522 \tdx{Pow_mono}: A \isasymsubseteq B ==> Pow(A) \isasymsubseteq Pow(B)
2524 We enter the goal and make the first step, which breaks the equation into
2525 two inclusions by extensionality:\index{*equalityI theorem}
2527 \isacommand{lemma}\ "Pow(A\ Int\ B)\ =\ Pow(A)\ Int\ Pow(B)"\isanewline
2528 \ 1.\ Pow(A\ \isasyminter \ B)\ =\ Pow(A)\ \isasyminter \ Pow(B)\isanewline
2529 \isacommand{apply}\ (rule\ equalityI)\isanewline
2530 \ 1.\ Pow(A\ \isasyminter \ B)\ \isasymsubseteq \ Pow(A)\ \isasyminter \ Pow(B)\isanewline
2531 \ 2.\ Pow(A)\ \isasyminter \ Pow(B)\ \isasymsubseteq \ Pow(A\ \isasyminter \ B)
2533 Both inclusions could be tackled straightforwardly using \isa{subsetI}.
2534 A shorter proof results from noting that intersection forms the greatest
2535 lower bound:\index{*Int_greatest theorem}
2537 \isacommand{apply}\ (rule\ Int\_greatest)\isanewline
2538 \ 1.\ Pow(A\ \isasyminter \ B)\ \isasymsubseteq \ Pow(A)\isanewline
2539 \ 2.\ Pow(A\ \isasyminter \ B)\ \isasymsubseteq \ Pow(B)\isanewline
2540 \ 3.\ Pow(A)\ \isasyminter \ Pow(B)\ \isasymsubseteq \ Pow(A\ \isasyminter \ B)
2542 Subgoal~1 follows by applying the monotonicity of \isa{Pow} to $A\int
2543 B\subseteq A$; subgoal~2 follows similarly:
2544 \index{*Int_lower1 theorem}\index{*Int_lower2 theorem}
2546 \isacommand{apply}\ (rule\ Int\_lower1\ [THEN\ Pow\_mono])\isanewline
2547 \ 1.\ Pow(A\ \isasyminter \ B)\ \isasymsubseteq \ Pow(B)\isanewline
2548 \ 2.\ Pow(A)\ \isasyminter \ Pow(B)\ \isasymsubseteq \ Pow(A\ \isasyminter \ B)
2550 \isacommand{apply}\ (rule\ Int\_lower2\ [THEN\ Pow\_mono])\isanewline
2551 \ 1.\ Pow(A)\ \isasyminter \ Pow(B)\ \isasymsubseteq \ Pow(A\ \isasyminter \ B)
2553 We are left with the opposite inclusion, which we tackle in the
2554 straightforward way:\index{*subsetI theorem}
2556 \isacommand{apply}\ (rule\ subsetI)\isanewline
2557 \ 1.\ \isasymAnd x.\ x\ \isasymin \ Pow(A)\ \isasyminter \ Pow(B)\ \isasymLongrightarrow \ x\ \isasymin \ Pow(A\ \isasyminter \ B)
2559 The subgoal is to show $x\in \isa{Pow}(A\cap B)$ assuming $x\in\isa{Pow}(A)\cap \isa{Pow}(B)$; eliminating this assumption produces two
2560 subgoals. The rule \tdx{IntE} treats the intersection like a conjunction
2561 instead of unfolding its definition.
2563 \isacommand{apply}\ (erule\ IntE)\isanewline
2564 \ 1.\ \isasymAnd x.\ \isasymlbrakk x\ \isasymin \ Pow(A);\ x\ \isasymin \ Pow(B)\isasymrbrakk \ \isasymLongrightarrow \ x\ \isasymin \ Pow(A\ \isasyminter \ B)
2566 The next step replaces the \isa{Pow} by the subset
2567 relation~($\subseteq$).\index{*PowI theorem}
2569 \isacommand{apply}\ (rule\ PowI)\isanewline
2570 \ 1.\ \isasymAnd x.\ \isasymlbrakk x\ \isasymin \ Pow(A);\ x\ \isasymin \ Pow(B)\isasymrbrakk \ \isasymLongrightarrow \ x\ \isasymsubseteq \ A\ \isasyminter \ B%
2572 We perform the same replacement in the assumptions. This is a good
2573 demonstration of the tactic \ttindex{drule}:\index{*PowD theorem}
2575 \isacommand{apply}\ (drule\ PowD)+\isanewline
2576 \ 1.\ \isasymAnd x.\ \isasymlbrakk x\ \isasymsubseteq \ A;\ x\ \isasymsubseteq \ B\isasymrbrakk \ \isasymLongrightarrow \ x\ \isasymsubseteq \ A\ \isasyminter \ B%
2578 The assumptions are that $x$ is a lower bound of both $A$ and~$B$, but
2579 $A\int B$ is the greatest lower bound:\index{*Int_greatest theorem}
2581 \isacommand{apply}\ (rule\ Int\_greatest)\isanewline
2582 \ 1.\ \isasymAnd x.\ \isasymlbrakk x\ \isasymsubseteq \ A;\ x\ \isasymsubseteq \ B\isasymrbrakk \ \isasymLongrightarrow \ x\ \isasymsubseteq \ A\isanewline
2583 \ 2.\ \isasymAnd x.\ \isasymlbrakk x\ \isasymsubseteq \ A;\ x\ \isasymsubseteq \ B\isasymrbrakk \ \isasymLongrightarrow \ x\ \isasymsubseteq \ B%
2585 To conclude the proof, we clear up the trivial subgoals:
2587 \isacommand{apply}\ (assumption+)\isanewline
2591 We could have performed this proof instantly by calling
2594 \isacommand{lemma}\ "Pow(A\ Int\ B)\ =\ Pow(A)\ Int\ Pow(B)"\isanewline
2597 Past researchers regarded this as a difficult proof, as indeed it is if all
2598 the symbols are replaced by their definitions.
2601 \section{Monotonicity of the union operator}
2602 For another example, we prove that general union is monotonic:
2603 ${C\subseteq D}$ implies $\bigcup(C)\subseteq \bigcup(D)$. To begin, we
2604 tackle the inclusion using \tdx{subsetI}:
2606 \isacommand{lemma}\ "C\isasymsubseteq D\ ==>\ Union(C)\
2607 \isasymsubseteq \ Union(D)"\isanewline
2608 \isacommand{apply}\ (rule\ subsetI)\isanewline
2609 \ 1.\ \isasymAnd x.\ \isasymlbrakk C\ \isasymsubseteq \ D;\ x\ \isasymin \ \isasymUnion C\isasymrbrakk \ \isasymLongrightarrow \ x\ \isasymin \ \isasymUnion D%
2611 Big union is like an existential quantifier --- the occurrence in the
2612 assumptions must be eliminated early, since it creates parameters.
2613 \index{*UnionE theorem}
2615 \isacommand{apply}\ (erule\ UnionE)\isanewline
2616 \ 1.\ \isasymAnd x\ B.\ \isasymlbrakk C\ \isasymsubseteq \ D;\ x\ \isasymin \ B;\ B\ \isasymin \ C\isasymrbrakk \ \isasymLongrightarrow \ x\ \isasymin \ \isasymUnion D%
2618 Now we may apply \tdx{UnionI}, which creates an unknown involving the
2619 parameters. To show \isa{x\ \isasymin \ \isasymUnion D} it suffices to show that~\isa{x} belongs
2620 to some element, say~\isa{?B2(x,B)}, of~\isa{D}\@.
2622 \isacommand{apply}\ (rule\ UnionI)\isanewline
2623 \ 1.\ \isasymAnd x\ B.\ \isasymlbrakk C\ \isasymsubseteq \ D;\ x\ \isasymin \ B;\ B\ \isasymin \ C\isasymrbrakk \ \isasymLongrightarrow \ ?B2(x,\ B)\ \isasymin \ D\isanewline
2624 \ 2.\ \isasymAnd x\ B.\ \isasymlbrakk C\ \isasymsubseteq \ D;\ x\ \isasymin \ B;\ B\ \isasymin \ C\isasymrbrakk \ \isasymLongrightarrow \ x\ \isasymin \ ?B2(x,\ B)
2626 Combining the rule \tdx{subsetD} with the assumption \isa{C\ \isasymsubseteq \ D} yields
2627 $\Var{a}\in C \Imp \Var{a}\in D$, which reduces subgoal~1. Note that
2628 \isa{erule} removes the subset assumption.
2630 \isacommand{apply}\ (erule\ subsetD)\isanewline
2631 \ 1.\ \isasymAnd x\ B.\ \isasymlbrakk x\ \isasymin \ B;\ B\ \isasymin \ C\isasymrbrakk \ \isasymLongrightarrow \ ?B2(x,\ B)\ \isasymin \ C\isanewline
2632 \ 2.\ \isasymAnd x\ B.\ \isasymlbrakk C\ \isasymsubseteq \ D;\ x\ \isasymin \ B;\ B\ \isasymin \ C\isasymrbrakk \ \isasymLongrightarrow \ x\ \isasymin \ ?B2(x,\ B)
2634 The rest is routine. Observe how the first call to \isa{assumption}
2635 instantiates \isa{?B2(x,B)} to~\isa{B}\@.
2637 \isacommand{apply}\ assumption\ \isanewline
2638 \ 1.\ \isasymAnd x\ B.\ \isasymlbrakk C\ \isasymsubseteq \ D;\ x\ \isasymin \ B;\ B\ \isasymin \ C\isasymrbrakk \ \isasymLongrightarrow \ x\ \isasymin \ B%
2640 \isacommand{apply}\ assumption\ \isanewline
2641 No\ subgoals!\isanewline
2644 Again, \isa{blast} can prove this theorem in one step.
2646 The theory \isa{ZF/equalities.thy} has many similar proofs. Reasoning about
2647 general intersection can be difficult because of its anomalous behaviour on
2648 the empty set. However, \isa{blast} copes well with these. Here is
2649 a typical example, borrowed from Devlin~\cite[page 12]{devlin79}:
2650 \[ a\in C \,\Imp\, \inter@{x\in C} \Bigl(A(x) \int B(x)\Bigr) =
2651 \Bigl(\inter@{x\in C} A(x)\Bigr) \int
2652 \Bigl(\inter@{x\in C} B(x)\Bigr) \]
2654 \section{Low-level reasoning about functions}
2655 The derived rules \isa{lamI}, \isa{lamE}, \isa{lam_type}, \isa{beta}
2656 and \isa{eta} support reasoning about functions in a
2657 $\lambda$-calculus style. This is generally easier than regarding
2658 functions as sets of ordered pairs. But sometimes we must look at the
2659 underlying representation, as in the following proof
2660 of~\tdx{fun_disjoint_apply1}. This states that if $f$ and~$g$ are
2661 functions with disjoint domains~$A$ and~$C$, and if $a\in A$, then
2664 \isacommand{lemma}\ "[|\ a\ \isasymin \ A;\ \ f\ \isasymin \ A->B;\ \ g\ \isasymin \ C->D;\ \ A\ \isasyminter \ C\ =\ 0\ |]
2666 \ \ \ \ \ \ \ \ ==>\ (f\ \isasymunion \ g)`a\ =\ f`a"
2668 Using \tdx{apply_equality}, we reduce the equality to reasoning about
2669 ordered pairs. The second subgoal is to verify that \isa{f\ \isasymunion \ g} is a function, since
2670 \isa{Pi(?A,?B)} denotes a dependent function space.
2672 \isacommand{apply}\ (rule\ apply\_equality)\isanewline
2673 \ 1.\ \isasymlbrakk a\ \isasymin \ A;\ f\ \isasymin \ A\ \isasymrightarrow \ B;\ g\ \isasymin \ C\ \isasymrightarrow \ D;\ A\ \isasyminter \ C\ =\ 0\isasymrbrakk \isanewline
2674 \isaindent{\ 1.\ }\isasymLongrightarrow \ \isasymlangle a,\ f\ `\ a\isasymrangle \ \isasymin \ f\ \isasymunion \ g\isanewline
2675 \ 2.\ \isasymlbrakk a\ \isasymin \ A;\ f\ \isasymin \ A\ \isasymrightarrow \ B;\ g\ \isasymin \ C\ \isasymrightarrow \ D;\ A\ \isasyminter \ C\ =\ 0\isasymrbrakk \isanewline
2676 \isaindent{\ 2.\ }\isasymLongrightarrow \ f\ \isasymunion \ g\ \isasymin \ Pi(?A,\ ?B)
2678 We must show that the pair belongs to~$f$ or~$g$; by~\tdx{UnI1} we
2681 \isacommand{apply}\ (rule\ UnI1)\isanewline
2682 \ 1.\ \isasymlbrakk a\ \isasymin \ A;\ f\ \isasymin \ A\ \isasymrightarrow \ B;\ g\ \isasymin \ C\ \isasymrightarrow \ D;\ A\ \isasyminter \ C\ =\ 0\isasymrbrakk \ \isasymLongrightarrow \ \isasymlangle a,\ f\ `\ a\isasymrangle \ \isasymin \ f\isanewline
2683 \ 2.\ \isasymlbrakk a\ \isasymin \ A;\ f\ \isasymin \ A\ \isasymrightarrow \ B;\ g\ \isasymin \ C\ \isasymrightarrow \ D;\ A\ \isasyminter \ C\ =\ 0\isasymrbrakk \isanewline
2684 \isaindent{\ 2.\ }\isasymLongrightarrow \ f\ \isasymunion \ g\ \isasymin \ Pi(?A,\ ?B)
2686 To show $\pair{a,f`a}\in f$ we use \tdx{apply_Pair}, which is
2687 essentially the converse of \tdx{apply_equality}:
2689 \isacommand{apply}\ (rule\ apply\_Pair)\isanewline
2690 \ 1.\ \isasymlbrakk a\ \isasymin \ A;\ f\ \isasymin \ A\ \isasymrightarrow \ B;\ g\ \isasymin \ C\ \isasymrightarrow \ D;\ A\ \isasyminter \ C\ =\ 0\isasymrbrakk \ \isasymLongrightarrow \ f\ \isasymin \ Pi(?A2,?B2)\isanewline
2691 \ 2.\ \isasymlbrakk a\ \isasymin \ A;\ f\ \isasymin \ A\ \isasymrightarrow \ B;\ g\ \isasymin \ C\ \isasymrightarrow \ D;\ A\ \isasyminter \ C\ =\ 0\isasymrbrakk \ \isasymLongrightarrow \ a\ \isasymin \ ?A2\isanewline
2692 \ 3.\ \isasymlbrakk a\ \isasymin \ A;\ f\ \isasymin \ A\ \isasymrightarrow \ B;\ g\ \isasymin \ C\ \isasymrightarrow \ D;\ A\ \isasyminter \ C\ =\ 0\isasymrbrakk \isanewline
2693 \isaindent{\ 3.\ }\isasymLongrightarrow \ f\ \isasymunion \ g\ \isasymin \ Pi(?A,\ ?B)
2695 Using the assumptions $f\in A\to B$ and $a\in A$, we solve the two subgoals
2696 from \tdx{apply_Pair}. Recall that a $\Pi$-set is merely a generalized
2697 function space, and observe that~{\tt?A2} gets instantiated to~\isa{A}.
2699 \isacommand{apply}\ assumption\ \isanewline
2700 \ 1.\ \isasymlbrakk a\ \isasymin \ A;\ f\ \isasymin \ A\ \isasymrightarrow \ B;\ g\ \isasymin \ C\ \isasymrightarrow \ D;\ A\ \isasyminter \ C\ =\ 0\isasymrbrakk \ \isasymLongrightarrow \ a\ \isasymin \ A\isanewline
2701 \ 2.\ \isasymlbrakk a\ \isasymin \ A;\ f\ \isasymin \ A\ \isasymrightarrow \ B;\ g\ \isasymin \ C\ \isasymrightarrow \ D;\ A\ \isasyminter \ C\ =\ 0\isasymrbrakk \isanewline
2702 \isaindent{\ 2.\ }\isasymLongrightarrow \ f\ \isasymunion \ g\ \isasymin \ Pi(?A,\ ?B)
2704 \isacommand{apply}\ assumption\ \isanewline
2705 \ 1.\ \isasymlbrakk a\ \isasymin \ A;\ f\ \isasymin \ A\ \isasymrightarrow \ B;\ g\ \isasymin \ C\ \isasymrightarrow \ D;\ A\ \isasyminter \ C\ =\ 0\isasymrbrakk \isanewline
2706 \isaindent{\ 1.\ }\isasymLongrightarrow \ f\ \isasymunion \ g\ \isasymin \ Pi(?A,\ ?B)
2708 To construct functions of the form $f\un g$, we apply
2709 \tdx{fun_disjoint_Un}:
2711 \isacommand{apply}\ (rule\ fun\_disjoint\_Un)\isanewline
2712 \ 1.\ \isasymlbrakk a\ \isasymin \ A;\ f\ \isasymin \ A\ \isasymrightarrow \ B;\ g\ \isasymin \ C\ \isasymrightarrow \ D;\ A\ \isasyminter \ C\ =\ 0\isasymrbrakk \ \isasymLongrightarrow \ f\ \isasymin \ ?A3\ \isasymrightarrow \ ?B3\isanewline
2713 \ 2.\ \isasymlbrakk a\ \isasymin \ A;\ f\ \isasymin \ A\ \isasymrightarrow \ B;\ g\ \isasymin \ C\ \isasymrightarrow \ D;\ A\ \isasyminter \ C\ =\ 0\isasymrbrakk \ \isasymLongrightarrow \ g\ \isasymin \ ?C3\ \isasymrightarrow \ ?D3\isanewline
2714 \ 3.\ \isasymlbrakk a\ \isasymin \ A;\ f\ \isasymin \ A\ \isasymrightarrow \ B;\ g\ \isasymin \ C\ \isasymrightarrow \ D;\ A\ \isasyminter \ C\ =\ 0\isasymrbrakk \ \isasymLongrightarrow \ ?A3\ \isasyminter \ ?C3\ =\ 0
2716 The remaining subgoals are instances of the assumptions. Again, observe how
2717 unknowns become instantiated:
2719 \isacommand{apply}\ assumption\ \isanewline
2720 \ 1.\ \isasymlbrakk a\ \isasymin \ A;\ f\ \isasymin \ A\ \isasymrightarrow \ B;\ g\ \isasymin \ C\ \isasymrightarrow \ D;\ A\ \isasyminter \ C\ =\ 0\isasymrbrakk \ \isasymLongrightarrow \ g\ \isasymin \ ?C3\ \isasymrightarrow \ ?D3\isanewline
2721 \ 2.\ \isasymlbrakk a\ \isasymin \ A;\ f\ \isasymin \ A\ \isasymrightarrow \ B;\ g\ \isasymin \ C\ \isasymrightarrow \ D;\ A\ \isasyminter \ C\ =\ 0\isasymrbrakk \ \isasymLongrightarrow \ A\ \isasyminter \ ?C3\ =\ 0
2723 \isacommand{apply}\ assumption\ \isanewline
2724 \ 1.\ \isasymlbrakk a\ \isasymin \ A;\ f\ \isasymin \ A\ \isasymrightarrow \ B;\ g\ \isasymin \ C\ \isasymrightarrow \ D;\ A\ \isasyminter \ C\ =\ 0\isasymrbrakk \ \isasymLongrightarrow \ A\ \isasyminter \ C\ =\ 0
2726 \isacommand{apply}\ assumption\ \isanewline
2727 No\ subgoals!\isanewline
2730 See the theories \isa{ZF/func.thy} and \isa{ZF/WF.thy} for more
2731 examples of reasoning about functions.
2733 \index{set theory|)}