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     1 %% $Id$

     2 \chapter{Zermelo-Fraenkel Set Theory}

     3 \index{set theory|(}

     4

     5 The theory~\thydx{ZF} implements Zermelo-Fraenkel set

     6 theory~\cite{halmos60,suppes72} as an extension of~\texttt{FOL}, classical

     7 first-order logic.  The theory includes a collection of derived natural

     8 deduction rules, for use with Isabelle's classical reasoner.  Much

     9 of it is based on the work of No\"el~\cite{noel}.

    10

    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.

    18

    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

    24 datatypes.

    25

    26 Because ZF is an extension of FOL, it provides the same packages, namely

    27 \texttt{hyp_subst_tac}, the simplifier, and the classical reasoner.  The

    28 default simpset and claset are usually satisfactory.

    29

    30 Published articles~\cite{paulson-set-I,paulson-set-II} describe \texttt{ZF}

    31 less formally than this chapter.  Isabelle employs a novel treatment of

    32 non-well-founded data structures within the standard {\sc zf} axioms including

    33 the Axiom of Foundation~\cite{paulson-mscs}.

    34

    35

    36 \section{Which version of axiomatic set theory?}

    37 The two main axiom systems for set theory are Bernays-G\"odel~({\sc bg})

    38 and Zermelo-Fraenkel~({\sc zf}).  Resolution theorem provers can use {\sc

    39   bg} because it is finite~\cite{boyer86,quaife92}.  {\sc zf} does not

    40 have a finite axiom system because of its Axiom Scheme of Replacement.

    41 This makes it awkward to use with many theorem provers, since instances

    42 of the axiom scheme have to be invoked explicitly.  Since Isabelle has no

    43 difficulty with axiom schemes, we may adopt either axiom system.

    44

    45 These two theories differ in their treatment of {\bf classes}, which are

    46 collections that are too big' to be sets.  The class of all sets,~$V$,

    47 cannot be a set without admitting Russell's Paradox.  In {\sc bg}, both

    48 classes and sets are individuals; $x\in V$ expresses that $x$ is a set.  In

    49 {\sc zf}, all variables denote sets; classes are identified with unary

    50 predicates.  The two systems define essentially the same sets and classes,

    51 with similar properties.  In particular, a class cannot belong to another

    52 class (let alone a set).

    53

    54 Modern set theorists tend to prefer {\sc zf} because they are mainly concerned

    55 with sets, rather than classes.  {\sc bg} requires tiresome proofs that various

    56 collections are sets; for instance, showing $x\in\{x\}$ requires showing that

    57 $x$ is a set.

    58

    59

    60 \begin{figure} \small

    61 \begin{center}

    62 \begin{tabular}{rrr}

    63   \it name      &\it meta-type  & \it description \\

    64   \cdx{Let}     & $[\alpha,\alpha\To\beta]\To\beta$ & let binder\\

    65   \cdx{0}       & $i$           & empty set\\

    66   \cdx{cons}    & $[i,i]\To i$  & finite set constructor\\

    67   \cdx{Upair}   & $[i,i]\To i$  & unordered pairing\\

    68   \cdx{Pair}    & $[i,i]\To i$  & ordered pairing\\

    69   \cdx{Inf}     & $i$   & infinite set\\

    70   \cdx{Pow}     & $i\To i$      & powerset\\

    71   \cdx{Union} \cdx{Inter} & $i\To i$    & set union/intersection \\

    72   \cdx{split}   & $[[i,i]\To i, i] \To i$ & generalized projection\\

    73   \cdx{fst} \cdx{snd}   & $i\To i$      & projections\\

    74   \cdx{converse}& $i\To i$      & converse of a relation\\

    75   \cdx{succ}    & $i\To i$      & successor\\

    76   \cdx{Collect} & $[i,i\To o]\To i$     & separation\\

    77   \cdx{Replace} & $[i, [i,i]\To o] \To i$       & replacement\\

    78   \cdx{PrimReplace} & $[i, [i,i]\To o] \To i$   & primitive replacement\\

    79   \cdx{RepFun}  & $[i, i\To i] \To i$   & functional replacement\\

    80   \cdx{Pi} \cdx{Sigma}  & $[i,i\To i]\To i$     & general product/sum\\

    81   \cdx{domain}  & $i\To i$      & domain of a relation\\

    82   \cdx{range}   & $i\To i$      & range of a relation\\

    83   \cdx{field}   & $i\To i$      & field of a relation\\

    84   \cdx{Lambda}  & $[i, i\To i]\To i$    & $\lambda$-abstraction\\

    85   \cdx{restrict}& $[i, i] \To i$        & restriction of a function\\

    86   \cdx{The}     & $[i\To o]\To i$       & definite description\\

    87   \cdx{if}      & $[o,i,i]\To i$        & conditional\\

    88   \cdx{Ball} \cdx{Bex}  & $[i, i\To o]\To o$    & bounded quantifiers

    89 \end{tabular}

    90 \end{center}

    91 \subcaption{Constants}

    92

    93 \begin{center}

    94 \index{*"" symbol}

    95 \index{*"-"" symbol}

    96 \index{*" symbol}\index{function applications!in ZF}

    97 \index{*"- symbol}

    98 \index{*": symbol}

    99 \index{*"<"= symbol}

   100 \begin{tabular}{rrrr}

   101   \it symbol  & \it meta-type & \it priority & \it description \\

   102   \tt         & $[i,i]\To i$  &  Left 90      & image \\

   103   \tt -       & $[i,i]\To i$  &  Left 90      & inverse image \\

   104   \tt          & $[i,i]\To i$  &  Left 90      & application \\

   105   \sdx{Int}     & $[i,i]\To i$  &  Left 70      & intersection ($\int$) \\

   106   \sdx{Un}      & $[i,i]\To i$  &  Left 65      & union ($\un$) \\

   107   \tt -         & $[i,i]\To i$  &  Left 65      & set difference ($-$) \$1ex]   108 \tt: & [i,i]\To o & Left 50 & membership (\in) \\   109 \tt <= & [i,i]\To o & Left 50 & subset (\subseteq)   110 \end{tabular}   111 \end{center}   112 \subcaption{Infixes}   113 \caption{Constants of ZF} \label{zf-constants}   114 \end{figure}   115   116   117 \section{The syntax of set theory}   118 The language of set theory, as studied by logicians, has no constants. The   119 traditional axioms merely assert the existence of empty sets, unions,   120 powersets, etc.; this would be intolerable for practical reasoning. The   121 Isabelle theory declares constants for primitive sets. It also extends   122 \texttt{FOL} with additional syntax for finite sets, ordered pairs,   123 comprehension, general union/intersection, general sums/products, and   124 bounded quantifiers. In most other respects, Isabelle implements precisely   125 Zermelo-Fraenkel set theory.   126   127 Figure~\ref{zf-constants} lists the constants and infixes of~ZF, while   128 Figure~\ref{zf-trans} presents the syntax translations. Finally,   129 Figure~\ref{zf-syntax} presents the full grammar for set theory, including the   130 constructs of FOL.   131   132 Local abbreviations can be introduced by a \texttt{let} construct whose   133 syntax appears in Fig.\ts\ref{zf-syntax}. Internally it is translated into   134 the constant~\cdx{Let}. It can be expanded by rewriting with its   135 definition, \tdx{Let_def}.   136   137 Apart from \texttt{let}, set theory does not use polymorphism. All terms in   138 ZF have type~\tydx{i}, which is the type of individuals and has class~{\tt   139 term}. The type of first-order formulae, remember, is~\textit{o}.   140   141 Infix operators include binary union and intersection (A\un B and   142 A\int B), set difference (A-B), and the subset and membership   143 relations. Note that a\verb|~:|b is translated to \neg(a\in b). The   144 union and intersection operators (\bigcup A and \bigcap A) form the   145 union or intersection of a set of sets; \bigcup A means the same as   146 \bigcup@{x\in A}x. Of these operators, only \bigcup A is primitive.   147   148 The constant \cdx{Upair} constructs unordered pairs; thus {\tt   149 Upair(A,B)} denotes the set~\{A,B\} and \texttt{Upair(A,A)}   150 denotes the singleton~\{A\}. General union is used to define binary   151 union. The Isabelle version goes on to define the constant   152 \cdx{cons}:   153 \begin{eqnarray*}   154 A\cup B & \equiv & \bigcup(\texttt{Upair}(A,B)) \\   155 \texttt{cons}(a,B) & \equiv & \texttt{Upair}(a,a) \un B   156 \end{eqnarray*}   157 The \{a@1, \ldots\} notation abbreviates finite sets constructed in the   158 obvious manner using~\texttt{cons} and~\emptyset (the empty set):   159 \begin{eqnarray*}   160 \{a,b,c\} & \equiv & \texttt{cons}(a,\texttt{cons}(b,\texttt{cons}(c,\emptyset)))   161 \end{eqnarray*}   162   163 The constant \cdx{Pair} constructs ordered pairs, as in {\tt   164 Pair(a,b)}. Ordered pairs may also be written within angle brackets,   165 as {\tt<a,b>}. The n-tuple {\tt<a@1,\ldots,a@{n-1},a@n>}   166 abbreviates the nest of pairs\par\nobreak   167 \centerline{\texttt{Pair(a@1,\ldots,Pair(a@{n-1},a@n)\ldots).}}   168   169 In ZF, a function is a set of pairs. A ZF function~f is simply an   170 individual as far as Isabelle is concerned: its Isabelle type is~i, not say   171 i\To i. The infix operator~{\tt} denotes the application of a function set   172 to its argument; we must write~f{\tt}x, not~f(x). The syntax for image   173 is~f{\tt}A and that for inverse image is~f{\tt-}A.   174   175   176 \begin{figure}   177 \index{lambda abs@\lambda-abstractions!in ZF}   178 \index{*"-"> symbol}   179 \index{*"* symbol}   180 \begin{center} \footnotesize\tt\frenchspacing   181 \begin{tabular}{rrr}   182 \it external & \it internal & \it description \\   183 a \ttilde: b & \ttilde(a : b) & \rm negated membership\\   184 \ttlbracea@1, \ldots, a@n\ttrbrace & cons(a@1,\ldots,cons(a@n,0)) &   185 \rm finite set \\   186 <a@1, \ldots, a@{n-1}, a@n> &   187 Pair(a@1,\ldots,Pair(a@{n-1},a@n)\ldots) &   188 \rm ordered n-tuple \\   189 \ttlbracex:A . P[x]\ttrbrace & Collect(A,\lambda x. P[x]) &   190 \rm separation \\   191 \ttlbracey . x:A, Q[x,y]\ttrbrace & Replace(A,\lambda x\,y. Q[x,y]) &   192 \rm replacement \\   193 \ttlbraceb[x] . x:A\ttrbrace & RepFun(A,\lambda x. b[x]) &   194 \rm functional replacement \\   195 \sdx{INT} x:A . B[x] & Inter(\ttlbraceB[x] . x:A\ttrbrace) &   196 \rm general intersection \\   197 \sdx{UN} x:A . B[x] & Union(\ttlbraceB[x] . x:A\ttrbrace) &   198 \rm general union \\   199 \sdx{PROD} x:A . B[x] & Pi(A,\lambda x. B[x]) &   200 \rm general product \\   201 \sdx{SUM} x:A . B[x] & Sigma(A,\lambda x. B[x]) &   202 \rm general sum \\   203 A -> B & Pi(A,\lambda x. B) &   204 \rm function space \\   205 A * B & Sigma(A,\lambda x. B) &   206 \rm binary product \\   207 \sdx{THE} x . P[x] & The(\lambda x. P[x]) &   208 \rm definite description \\   209 \sdx{lam} x:A . b[x] & Lambda(A,\lambda x. b[x]) &   210 \rm \lambda-abstraction\\[1ex]   211 \sdx{ALL} x:A . P[x] & Ball(A,\lambda x. P[x]) &   212 \rm bounded \forall \\   213 \sdx{EX} x:A . P[x] & Bex(A,\lambda x. P[x]) &   214 \rm bounded \exists   215 \end{tabular}   216 \end{center}   217 \caption{Translations for ZF} \label{zf-trans}   218 \end{figure}   219   220   221 \begin{figure}   222 \index{*let symbol}   223 \index{*in symbol}   224 \dquotes   225 \[\begin{array}{rcl}   226 term & = & \hbox{expression of type~i} \\   227 & | & "let"~id~"="~term";"\dots";"~id~"="~term~"in"~term \\   228 & | & "if"~term~"then"~term~"else"~term \\   229 & | & "{\ttlbrace} " term\; ("," term)^* " {\ttrbrace}" \\   230 & | & "< " term\; ("," term)^* " >" \\   231 & | & "{\ttlbrace} " id ":" term " . " formula " {\ttrbrace}" \\   232 & | & "{\ttlbrace} " id " . " id ":" term ", " formula " {\ttrbrace}" \\   233 & | & "{\ttlbrace} " term " . " id ":" term " {\ttrbrace}" \\   234 & | & term "  " term \\   235 & | & term " - " term \\   236 & | & term "  " term \\   237 & | & term " * " term \\   238 & | & term " Int " term \\   239 & | & term " Un " term \\   240 & | & term " - " term \\   241 & | & term " -> " term \\   242 & | & "THE~~" id " . " formula\\   243 & | & "lam~~" id ":" term " . " term \\   244 & | & "INT~~" id ":" term " . " term \\   245 & | & "UN~~~" id ":" term " . " term \\   246 & | & "PROD~" id ":" term " . " term \\   247 & | & "SUM~~" id ":" term " . " term \\[2ex]   248 formula & = & \hbox{expression of type~o} \\   249 & | & term " : " term \\   250 & | & term " \ttilde: " term \\   251 & | & term " <= " term \\   252 & | & term " = " term \\   253 & | & term " \ttilde= " term \\   254 & | & "\ttilde\ " formula \\   255 & | & formula " \& " formula \\   256 & | & formula " | " formula \\   257 & | & formula " --> " formula \\   258 & | & formula " <-> " formula \\   259 & | & "ALL " id ":" term " . " formula \\   260 & | & "EX~~" id ":" term " . " formula \\   261 & | & "ALL~" id~id^* " . " formula \\   262 & | & "EX~~" id~id^* " . " formula \\   263 & | & "EX!~" id~id^* " . " formula   264 \end{array}   265$

   266 \caption{Full grammar for ZF} \label{zf-syntax}

   267 \end{figure}

   268

   269

   270 \section{Binding operators}

   271 The constant \cdx{Collect} constructs sets by the principle of {\bf

   272   separation}.  The syntax for separation is

   273 \hbox{\tt\ttlbrace$x$:$A$.\ $P[x]$\ttrbrace}, where $P[x]$ is a formula

   274 that may contain free occurrences of~$x$.  It abbreviates the set {\tt

   275   Collect($A$,$\lambda x. P[x]$)}, which consists of all $x\in A$ that

   276 satisfy~$P[x]$.  Note that \texttt{Collect} is an unfortunate choice of

   277 name: some set theories adopt a set-formation principle, related to

   278 replacement, called collection.

   279

   280 The constant \cdx{Replace} constructs sets by the principle of {\bf

   281   replacement}.  The syntax

   282 \hbox{\tt\ttlbrace$y$.\ $x$:$A$,$Q[x,y]$\ttrbrace} denotes the set {\tt

   283   Replace($A$,$\lambda x\,y. Q[x,y]$)}, which consists of all~$y$ such

   284 that there exists $x\in A$ satisfying~$Q[x,y]$.  The Replacement Axiom

   285 has the condition that $Q$ must be single-valued over~$A$: for

   286 all~$x\in A$ there exists at most one $y$ satisfying~$Q[x,y]$.  A

   287 single-valued binary predicate is also called a {\bf class function}.

   288

   289 The constant \cdx{RepFun} expresses a special case of replacement,

   290 where $Q[x,y]$ has the form $y=b[x]$.  Such a $Q$ is trivially

   291 single-valued, since it is just the graph of the meta-level

   292 function~$\lambda x. b[x]$.  The resulting set consists of all $b[x]$

   293 for~$x\in A$.  This is analogous to the \ML{} functional \texttt{map},

   294 since it applies a function to every element of a set.  The syntax is

   295 \hbox{\tt\ttlbrace$b[x]$.\ $x$:$A$\ttrbrace}, which expands to {\tt

   296   RepFun($A$,$\lambda x. b[x]$)}.

   297

   298 \index{*INT symbol}\index{*UN symbol}

   299 General unions and intersections of indexed

   300 families of sets, namely $\bigcup@{x\in A}B[x]$ and $\bigcap@{x\in A}B[x]$,

   301 are written \hbox{\tt UN $x$:$A$.\ $B[x]$} and \hbox{\tt INT $x$:$A$.\ $B[x]$}.

   302 Their meaning is expressed using \texttt{RepFun} as

   303 $  304 \bigcup(\{B[x]. x\in A\}) \qquad\hbox{and}\qquad   305 \bigcap(\{B[x]. x\in A\}).   306$

   307 General sums $\sum@{x\in A}B[x]$ and products $\prod@{x\in A}B[x]$ can be

   308 constructed in set theory, where $B[x]$ is a family of sets over~$A$.  They

   309 have as special cases $A\times B$ and $A\to B$, where $B$ is simply a set.

   310 This is similar to the situation in Constructive Type Theory (set theory

   311 has dependent sets') and calls for similar syntactic conventions.  The

   312 constants~\cdx{Sigma} and~\cdx{Pi} construct general sums and

   313 products.  Instead of \texttt{Sigma($A$,$B$)} and \texttt{Pi($A$,$B$)} we may

   314 write

   315 \hbox{\tt SUM $x$:$A$.\ $B[x]$} and \hbox{\tt PROD $x$:$A$.\ $B[x]$}.

   316 \index{*SUM symbol}\index{*PROD symbol}%

   317 The special cases as \hbox{\tt$A$*$B$} and \hbox{\tt$A$->$B$} abbreviate

   318 general sums and products over a constant family.\footnote{Unlike normal

   319 infix operators, {\tt*} and {\tt->} merely define abbreviations; there are

   320 no constants~\texttt{op~*} and~\hbox{\tt op~->}.} Isabelle accepts these

   321 abbreviations in parsing and uses them whenever possible for printing.

   322

   323 \index{*THE symbol} As mentioned above, whenever the axioms assert the

   324 existence and uniqueness of a set, Isabelle's set theory declares a constant

   325 for that set.  These constants can express the {\bf definite description}

   326 operator~$\iota x. P[x]$, which stands for the unique~$a$ satisfying~$P[a]$,

   327 if such exists.  Since all terms in ZF denote something, a description is

   328 always meaningful, but we do not know its value unless $P[x]$ defines it

   329 uniquely.  Using the constant~\cdx{The}, we may write descriptions as {\tt

   330   The($\lambda x. P[x]$)} or use the syntax \hbox{\tt THE $x$.\ $P[x]$}.

   331

   332 \index{*lam symbol}

   333 Function sets may be written in $\lambda$-notation; $\lambda x\in A. b[x]$

   334 stands for the set of all pairs $\pair{x,b[x]}$ for $x\in A$.  In order for

   335 this to be a set, the function's domain~$A$ must be given.  Using the

   336 constant~\cdx{Lambda}, we may express function sets as {\tt

   337 Lambda($A$,$\lambda x. b[x]$)} or use the syntax \hbox{\tt lam $x$:$A$.\ $b[x]$}.

   338

   339 Isabelle's set theory defines two {\bf bounded quantifiers}:

   340 \begin{eqnarray*}

   341    \forall x\in A. P[x] &\hbox{abbreviates}& \forall x. x\in A\imp P[x] \\

   342    \exists x\in A. P[x] &\hbox{abbreviates}& \exists x. x\in A\conj P[x]

   343 \end{eqnarray*}

   344 The constants~\cdx{Ball} and~\cdx{Bex} are defined

   345 accordingly.  Instead of \texttt{Ball($A$,$P$)} and \texttt{Bex($A$,$P$)} we may

   346 write

   347 \hbox{\tt ALL $x$:$A$.\ $P[x]$} and \hbox{\tt EX $x$:$A$.\ $P[x]$}.

   348

   349

   350 %%%% ZF.thy

   351

   352 \begin{figure}

   353 \begin{ttbox}

   354 \tdx{Let_def}            Let(s, f) == f(s)

   355

   356 \tdx{Ball_def}           Ball(A,P) == ALL x. x:A --> P(x)

   357 \tdx{Bex_def}            Bex(A,P)  == EX x. x:A & P(x)

   358

   359 \tdx{subset_def}         A <= B  == ALL x:A. x:B

   360 \tdx{extension}          A = B  <->  A <= B & B <= A

   361

   362 \tdx{Union_iff}          A : Union(C) <-> (EX B:C. A:B)

   363 \tdx{Pow_iff}            A : Pow(B) <-> A <= B

   364 \tdx{foundation}         A=0 | (EX x:A. ALL y:x. ~ y:A)

   365

   366 \tdx{replacement}        (ALL x:A. ALL y z. P(x,y) & P(x,z) --> y=z) ==>

   367                    b : PrimReplace(A,P) <-> (EX x:A. P(x,b))

   368 \subcaption{The Zermelo-Fraenkel Axioms}

   369

   370 \tdx{Replace_def}  Replace(A,P) ==

   371                    PrimReplace(A, \%x y. (EX!z. P(x,z)) & P(x,y))

   372 \tdx{RepFun_def}   RepFun(A,f)  == {\ttlbrace}y . x:A, y=f(x)\ttrbrace

   373 \tdx{the_def}      The(P)       == Union({\ttlbrace}y . x:{\ttlbrace}0{\ttrbrace}, P(y){\ttrbrace})

   374 \tdx{if_def}       if(P,a,b)    == THE z. P & z=a | ~P & z=b

   375 \tdx{Collect_def}  Collect(A,P) == {\ttlbrace}y . x:A, x=y & P(x){\ttrbrace}

   376 \tdx{Upair_def}    Upair(a,b)   ==

   377                  {\ttlbrace}y. x:Pow(Pow(0)), (x=0 & y=a) | (x=Pow(0) & y=b){\ttrbrace}

   378 \subcaption{Consequences of replacement}

   379

   380 \tdx{Inter_def}    Inter(A) == {\ttlbrace}x:Union(A) . ALL y:A. x:y{\ttrbrace}

   381 \tdx{Un_def}       A Un  B  == Union(Upair(A,B))

   382 \tdx{Int_def}      A Int B  == Inter(Upair(A,B))

   383 \tdx{Diff_def}     A - B    == {\ttlbrace}x:A . x~:B{\ttrbrace}

   384 \subcaption{Union, intersection, difference}

   385 \end{ttbox}

   386 \caption{Rules and axioms of ZF} \label{zf-rules}

   387 \end{figure}

   388

   389

   390 \begin{figure}

   391 \begin{ttbox}

   392 \tdx{cons_def}     cons(a,A) == Upair(a,a) Un A

   393 \tdx{succ_def}     succ(i) == cons(i,i)

   394 \tdx{infinity}     0:Inf & (ALL y:Inf. succ(y): Inf)

   395 \subcaption{Finite and infinite sets}

   396

   397 \tdx{Pair_def}       <a,b>      == {\ttlbrace}{\ttlbrace}a,a{\ttrbrace}, {\ttlbrace}a,b{\ttrbrace}{\ttrbrace}

   398 \tdx{split_def}      split(c,p) == THE y. EX a b. p=<a,b> & y=c(a,b)

   399 \tdx{fst_def}        fst(A)     == split(\%x y. x, p)

   400 \tdx{snd_def}        snd(A)     == split(\%x y. y, p)

   401 \tdx{Sigma_def}      Sigma(A,B) == UN x:A. UN y:B(x). {\ttlbrace}<x,y>{\ttrbrace}

   402 \subcaption{Ordered pairs and Cartesian products}

   403

   404 \tdx{converse_def}   converse(r) == {\ttlbrace}z. w:r, EX x y. w=<x,y> & z=<y,x>{\ttrbrace}

   405 \tdx{domain_def}     domain(r)   == {\ttlbrace}x. w:r, EX y. w=<x,y>{\ttrbrace}

   406 \tdx{range_def}      range(r)    == domain(converse(r))

   407 \tdx{field_def}      field(r)    == domain(r) Un range(r)

   408 \tdx{image_def}      r  A      == {\ttlbrace}y : range(r) . EX x:A. <x,y> : r{\ttrbrace}

   409 \tdx{vimage_def}     r - A     == converse(r)A

   410 \subcaption{Operations on relations}

   411

   412 \tdx{lam_def}    Lambda(A,b) == {\ttlbrace}<x,b(x)> . x:A{\ttrbrace}

   413 \tdx{apply_def}  fa         == THE y. <a,y> : f

   414 \tdx{Pi_def}     Pi(A,B) == {\ttlbrace}f: Pow(Sigma(A,B)). ALL x:A. EX! y. <x,y>: f{\ttrbrace}

   415 \tdx{restrict_def}   restrict(f,A) == lam x:A. fx

   416 \subcaption{Functions and general product}

   417 \end{ttbox}

   418 \caption{Further definitions of ZF} \label{zf-defs}

   419 \end{figure}

   420

   421

   422

   423 \section{The Zermelo-Fraenkel axioms}

   424 The axioms appear in Fig.\ts \ref{zf-rules}.  They resemble those

   425 presented by Suppes~\cite{suppes72}.  Most of the theory consists of

   426 definitions.  In particular, bounded quantifiers and the subset relation

   427 appear in other axioms.  Object-level quantifiers and implications have

   428 been replaced by meta-level ones wherever possible, to simplify use of the

   429 axioms.  See the file \texttt{ZF/ZF.thy} for details.

   430

   431 The traditional replacement axiom asserts

   432 $y \in \texttt{PrimReplace}(A,P) \bimp (\exists x\in A. P(x,y))$

   433 subject to the condition that $P(x,y)$ is single-valued for all~$x\in A$.

   434 The Isabelle theory defines \cdx{Replace} to apply

   435 \cdx{PrimReplace} to the single-valued part of~$P$, namely

   436 $(\exists!z. P(x,z)) \conj P(x,y).$

   437 Thus $y\in \texttt{Replace}(A,P)$ if and only if there is some~$x$ such that

   438 $P(x,-)$ holds uniquely for~$y$.  Because the equivalence is unconditional,

   439 \texttt{Replace} is much easier to use than \texttt{PrimReplace}; it defines the

   440 same set, if $P(x,y)$ is single-valued.  The nice syntax for replacement

   441 expands to \texttt{Replace}.

   442

   443 Other consequences of replacement include functional replacement

   444 (\cdx{RepFun}) and definite descriptions (\cdx{The}).

   445 Axioms for separation (\cdx{Collect}) and unordered pairs

   446 (\cdx{Upair}) are traditionally assumed, but they actually follow

   447 from replacement~\cite[pages 237--8]{suppes72}.

   448

   449 The definitions of general intersection, etc., are straightforward.  Note

   450 the definition of \texttt{cons}, which underlies the finite set notation.

   451 The axiom of infinity gives us a set that contains~0 and is closed under

   452 successor (\cdx{succ}).  Although this set is not uniquely defined,

   453 the theory names it (\cdx{Inf}) in order to simplify the

   454 construction of the natural numbers.

   455

   456 Further definitions appear in Fig.\ts\ref{zf-defs}.  Ordered pairs are

   457 defined in the standard way, $\pair{a,b}\equiv\{\{a\},\{a,b\}\}$.  Recall

   458 that \cdx{Sigma}$(A,B)$ generalizes the Cartesian product of two

   459 sets.  It is defined to be the union of all singleton sets

   460 $\{\pair{x,y}\}$, for $x\in A$ and $y\in B(x)$.  This is a typical usage of

   461 general union.

   462

   463 The projections \cdx{fst} and~\cdx{snd} are defined in terms of the

   464 generalized projection \cdx{split}.  The latter has been borrowed from

   465 Martin-L\"of's Type Theory, and is often easier to use than \cdx{fst}

   466 and~\cdx{snd}.

   467

   468 Operations on relations include converse, domain, range, and image.  The

   469 set ${\tt Pi}(A,B)$ generalizes the space of functions between two sets.

   470 Note the simple definitions of $\lambda$-abstraction (using

   471 \cdx{RepFun}) and application (using a definite description).  The

   472 function \cdx{restrict}$(f,A)$ has the same values as~$f$, but only

   473 over the domain~$A$.

   474

   475

   476 %%%% zf.ML

   477

   478 \begin{figure}

   479 \begin{ttbox}

   480 \tdx{ballI}       [| !!x. x:A ==> P(x) |] ==> ALL x:A. P(x)

   481 \tdx{bspec}       [| ALL x:A. P(x);  x: A |] ==> P(x)

   482 \tdx{ballE}       [| ALL x:A. P(x);  P(x) ==> Q;  ~ x:A ==> Q |] ==> Q

   483

   484 \tdx{ball_cong}   [| A=A';  !!x. x:A' ==> P(x) <-> P'(x) |] ==>

   485             (ALL x:A. P(x)) <-> (ALL x:A'. P'(x))

   486

   487 \tdx{bexI}        [| P(x);  x: A |] ==> EX x:A. P(x)

   488 \tdx{bexCI}       [| ALL x:A. ~P(x) ==> P(a);  a: A |] ==> EX x:A. P(x)

   489 \tdx{bexE}        [| EX x:A. P(x);  !!x. [| x:A; P(x) |] ==> Q |] ==> Q

   490

   491 \tdx{bex_cong}    [| A=A';  !!x. x:A' ==> P(x) <-> P'(x) |] ==>

   492             (EX x:A. P(x)) <-> (EX x:A'. P'(x))

   493 \subcaption{Bounded quantifiers}

   494

   495 \tdx{subsetI}       (!!x. x:A ==> x:B) ==> A <= B

   496 \tdx{subsetD}       [| A <= B;  c:A |] ==> c:B

   497 \tdx{subsetCE}      [| A <= B;  ~(c:A) ==> P;  c:B ==> P |] ==> P

   498 \tdx{subset_refl}   A <= A

   499 \tdx{subset_trans}  [| A<=B;  B<=C |] ==> A<=C

   500

   501 \tdx{equalityI}     [| A <= B;  B <= A |] ==> A = B

   502 \tdx{equalityD1}    A = B ==> A<=B

   503 \tdx{equalityD2}    A = B ==> B<=A

   504 \tdx{equalityE}     [| A = B;  [| A<=B; B<=A |] ==> P |]  ==>  P

   505 \subcaption{Subsets and extensionality}

   506

   507 \tdx{emptyE}          a:0 ==> P

   508 \tdx{empty_subsetI}   0 <= A

   509 \tdx{equals0I}        [| !!y. y:A ==> False |] ==> A=0

   510 \tdx{equals0D}        [| A=0;  a:A |] ==> P

   511

   512 \tdx{PowI}            A <= B ==> A : Pow(B)

   513 \tdx{PowD}            A : Pow(B)  ==>  A<=B

   514 \subcaption{The empty set; power sets}

   515 \end{ttbox}

   516 \caption{Basic derived rules for ZF} \label{zf-lemmas1}

   517 \end{figure}

   518

   519

   520 \section{From basic lemmas to function spaces}

   521 Faced with so many definitions, it is essential to prove lemmas.  Even

   522 trivial theorems like $A \int B = B \int A$ would be difficult to

   523 prove from the definitions alone.  Isabelle's set theory derives many

   524 rules using a natural deduction style.  Ideally, a natural deduction

   525 rule should introduce or eliminate just one operator, but this is not

   526 always practical.  For most operators, we may forget its definition

   527 and use its derived rules instead.

   528

   529 \subsection{Fundamental lemmas}

   530 Figure~\ref{zf-lemmas1} presents the derived rules for the most basic

   531 operators.  The rules for the bounded quantifiers resemble those for the

   532 ordinary quantifiers, but note that \tdx{ballE} uses a negated assumption

   533 in the style of Isabelle's classical reasoner.  The \rmindex{congruence

   534   rules} \tdx{ball_cong} and \tdx{bex_cong} are required by Isabelle's

   535 simplifier, but have few other uses.  Congruence rules must be specially

   536 derived for all binding operators, and henceforth will not be shown.

   537

   538 Figure~\ref{zf-lemmas1} also shows rules for the subset and equality

   539 relations (proof by extensionality), and rules about the empty set and the

   540 power set operator.

   541

   542 Figure~\ref{zf-lemmas2} presents rules for replacement and separation.

   543 The rules for \cdx{Replace} and \cdx{RepFun} are much simpler than

   544 comparable rules for \texttt{PrimReplace} would be.  The principle of

   545 separation is proved explicitly, although most proofs should use the

   546 natural deduction rules for \texttt{Collect}.  The elimination rule

   547 \tdx{CollectE} is equivalent to the two destruction rules

   548 \tdx{CollectD1} and \tdx{CollectD2}, but each rule is suited to

   549 particular circumstances.  Although too many rules can be confusing, there

   550 is no reason to aim for a minimal set of rules.  See the file

   551 \texttt{ZF/ZF.ML} for a complete listing.

   552

   553 Figure~\ref{zf-lemmas3} presents rules for general union and intersection.

   554 The empty intersection should be undefined.  We cannot have

   555 $\bigcap(\emptyset)=V$ because $V$, the universal class, is not a set.  All

   556 expressions denote something in ZF set theory; the definition of

   557 intersection implies $\bigcap(\emptyset)=\emptyset$, but this value is

   558 arbitrary.  The rule \tdx{InterI} must have a premise to exclude

   559 the empty intersection.  Some of the laws governing intersections require

   560 similar premises.

   561

   562

   563 %the [p] gives better page breaking for the book

   564 \begin{figure}[p]

   565 \begin{ttbox}

   566 \tdx{ReplaceI}      [| x: A;  P(x,b);  !!y. P(x,y) ==> y=b |] ==>

   567               b : {\ttlbrace}y. x:A, P(x,y){\ttrbrace}

   568

   569 \tdx{ReplaceE}      [| b : {\ttlbrace}y. x:A, P(x,y){\ttrbrace};

   570                  !!x. [| x: A;  P(x,b);  ALL y. P(x,y)-->y=b |] ==> R

   571               |] ==> R

   572

   573 \tdx{RepFunI}       [| a : A |] ==> f(a) : {\ttlbrace}f(x). x:A{\ttrbrace}

   574 \tdx{RepFunE}       [| b : {\ttlbrace}f(x). x:A{\ttrbrace};

   575                  !!x.[| x:A;  b=f(x) |] ==> P |] ==> P

   576

   577 \tdx{separation}     a : {\ttlbrace}x:A. P(x){\ttrbrace} <-> a:A & P(a)

   578 \tdx{CollectI}       [| a:A;  P(a) |] ==> a : {\ttlbrace}x:A. P(x){\ttrbrace}

   579 \tdx{CollectE}       [| a : {\ttlbrace}x:A. P(x){\ttrbrace};  [| a:A; P(a) |] ==> R |] ==> R

   580 \tdx{CollectD1}      a : {\ttlbrace}x:A. P(x){\ttrbrace} ==> a:A

   581 \tdx{CollectD2}      a : {\ttlbrace}x:A. P(x){\ttrbrace} ==> P(a)

   582 \end{ttbox}

   583 \caption{Replacement and separation} \label{zf-lemmas2}

   584 \end{figure}

   585

   586

   587 \begin{figure}

   588 \begin{ttbox}

   589 \tdx{UnionI}    [| B: C;  A: B |] ==> A: Union(C)

   590 \tdx{UnionE}    [| A : Union(C);  !!B.[| A: B;  B: C |] ==> R |] ==> R

   591

   592 \tdx{InterI}    [| !!x. x: C ==> A: x;  c:C |] ==> A : Inter(C)

   593 \tdx{InterD}    [| A : Inter(C);  B : C |] ==> A : B

   594 \tdx{InterE}    [| A : Inter(C);  A:B ==> R;  ~ B:C ==> R |] ==> R

   595

   596 \tdx{UN_I}      [| a: A;  b: B(a) |] ==> b: (UN x:A. B(x))

   597 \tdx{UN_E}      [| b : (UN x:A. B(x));  !!x.[| x: A;  b: B(x) |] ==> R

   598           |] ==> R

   599

   600 \tdx{INT_I}     [| !!x. x: A ==> b: B(x);  a: A |] ==> b: (INT x:A. B(x))

   601 \tdx{INT_E}     [| b : (INT x:A. B(x));  a: A |] ==> b : B(a)

   602 \end{ttbox}

   603 \caption{General union and intersection} \label{zf-lemmas3}

   604 \end{figure}

   605

   606

   607 %%% upair.ML

   608

   609 \begin{figure}

   610 \begin{ttbox}

   611 \tdx{pairing}      a:Upair(b,c) <-> (a=b | a=c)

   612 \tdx{UpairI1}      a : Upair(a,b)

   613 \tdx{UpairI2}      b : Upair(a,b)

   614 \tdx{UpairE}       [| a : Upair(b,c);  a = b ==> P;  a = c ==> P |] ==> P

   615 \end{ttbox}

   616 \caption{Unordered pairs} \label{zf-upair1}

   617 \end{figure}

   618

   619

   620 \begin{figure}

   621 \begin{ttbox}

   622 \tdx{UnI1}         c : A ==> c : A Un B

   623 \tdx{UnI2}         c : B ==> c : A Un B

   624 \tdx{UnCI}         (~c : B ==> c : A) ==> c : A Un B

   625 \tdx{UnE}          [| c : A Un B;  c:A ==> P;  c:B ==> P |] ==> P

   626

   627 \tdx{IntI}         [| c : A;  c : B |] ==> c : A Int B

   628 \tdx{IntD1}        c : A Int B ==> c : A

   629 \tdx{IntD2}        c : A Int B ==> c : B

   630 \tdx{IntE}         [| c : A Int B;  [| c:A; c:B |] ==> P |] ==> P

   631

   632 \tdx{DiffI}        [| c : A;  ~ c : B |] ==> c : A - B

   633 \tdx{DiffD1}       c : A - B ==> c : A

   634 \tdx{DiffD2}       c : A - B ==> c ~: B

   635 \tdx{DiffE}        [| c : A - B;  [| c:A; ~ c:B |] ==> P |] ==> P

   636 \end{ttbox}

   637 \caption{Union, intersection, difference} \label{zf-Un}

   638 \end{figure}

   639

   640

   641 \begin{figure}

   642 \begin{ttbox}

   643 \tdx{consI1}       a : cons(a,B)

   644 \tdx{consI2}       a : B ==> a : cons(b,B)

   645 \tdx{consCI}       (~ a:B ==> a=b) ==> a: cons(b,B)

   646 \tdx{consE}        [| a : cons(b,A);  a=b ==> P;  a:A ==> P |] ==> P

   647

   648 \tdx{singletonI}   a : {\ttlbrace}a{\ttrbrace}

   649 \tdx{singletonE}   [| a : {\ttlbrace}b{\ttrbrace}; a=b ==> P |] ==> P

   650 \end{ttbox}

   651 \caption{Finite and singleton sets} \label{zf-upair2}

   652 \end{figure}

   653

   654

   655 \begin{figure}

   656 \begin{ttbox}

   657 \tdx{succI1}       i : succ(i)

   658 \tdx{succI2}       i : j ==> i : succ(j)

   659 \tdx{succCI}       (~ i:j ==> i=j) ==> i: succ(j)

   660 \tdx{succE}        [| i : succ(j);  i=j ==> P;  i:j ==> P |] ==> P

   661 \tdx{succ_neq_0}   [| succ(n)=0 |] ==> P

   662 \tdx{succ_inject}  succ(m) = succ(n) ==> m=n

   663 \end{ttbox}

   664 \caption{The successor function} \label{zf-succ}

   665 \end{figure}

   666

   667

   668 \begin{figure}

   669 \begin{ttbox}

   670 \tdx{the_equality}     [| P(a);  !!x. P(x) ==> x=a |] ==> (THE x. P(x)) = a

   671 \tdx{theI}             EX! x. P(x) ==> P(THE x. P(x))

   672

   673 \tdx{if_P}              P ==> (if P then a else b) = a

   674 \tdx{if_not_P}         ~P ==> (if P then a else b) = b

   675

   676 \tdx{mem_asym}         [| a:b;  b:a |] ==> P

   677 \tdx{mem_irrefl}       a:a ==> P

   678 \end{ttbox}

   679 \caption{Descriptions; non-circularity} \label{zf-the}

   680 \end{figure}

   681

   682

   683 \subsection{Unordered pairs and finite sets}

   684 Figure~\ref{zf-upair1} presents the principle of unordered pairing, along

   685 with its derived rules.  Binary union and intersection are defined in terms

   686 of ordered pairs (Fig.\ts\ref{zf-Un}).  Set difference is also included.  The

   687 rule \tdx{UnCI} is useful for classical reasoning about unions,

   688 like \texttt{disjCI}\@; it supersedes \tdx{UnI1} and

   689 \tdx{UnI2}, but these rules are often easier to work with.  For

   690 intersection and difference we have both elimination and destruction rules.

   691 Again, there is no reason to provide a minimal rule set.

   692

   693 Figure~\ref{zf-upair2} is concerned with finite sets: it presents rules

   694 for~\texttt{cons}, the finite set constructor, and rules for singleton

   695 sets.  Figure~\ref{zf-succ} presents derived rules for the successor

   696 function, which is defined in terms of~\texttt{cons}.  The proof that {\tt

   697   succ} is injective appears to require the Axiom of Foundation.

   698

   699 Definite descriptions (\sdx{THE}) are defined in terms of the singleton

   700 set~$\{0\}$, but their derived rules fortunately hide this

   701 (Fig.\ts\ref{zf-the}).  The rule~\tdx{theI} is difficult to apply

   702 because of the two occurrences of~$\Var{P}$.  However,

   703 \tdx{the_equality} does not have this problem and the files contain

   704 many examples of its use.

   705

   706 Finally, the impossibility of having both $a\in b$ and $b\in a$

   707 (\tdx{mem_asym}) is proved by applying the Axiom of Foundation to

   708 the set $\{a,b\}$.  The impossibility of $a\in a$ is a trivial consequence.

   709

   710 See the file \texttt{ZF/upair.ML} for full proofs of the rules discussed in

   711 this section.

   712

   713

   714 %%% subset.ML

   715

   716 \begin{figure}

   717 \begin{ttbox}

   718 \tdx{Union_upper}       B:A ==> B <= Union(A)

   719 \tdx{Union_least}       [| !!x. x:A ==> x<=C |] ==> Union(A) <= C

   720

   721 \tdx{Inter_lower}       B:A ==> Inter(A) <= B

   722 \tdx{Inter_greatest}    [| a:A;  !!x. x:A ==> C<=x |] ==> C <= Inter(A)

   723

   724 \tdx{Un_upper1}         A <= A Un B

   725 \tdx{Un_upper2}         B <= A Un B

   726 \tdx{Un_least}          [| A<=C;  B<=C |] ==> A Un B <= C

   727

   728 \tdx{Int_lower1}        A Int B <= A

   729 \tdx{Int_lower2}        A Int B <= B

   730 \tdx{Int_greatest}      [| C<=A;  C<=B |] ==> C <= A Int B

   731

   732 \tdx{Diff_subset}       A-B <= A

   733 \tdx{Diff_contains}     [| C<=A;  C Int B = 0 |] ==> C <= A-B

   734

   735 \tdx{Collect_subset}    Collect(A,P) <= A

   736 \end{ttbox}

   737 \caption{Subset and lattice properties} \label{zf-subset}

   738 \end{figure}

   739

   740

   741 \subsection{Subset and lattice properties}

   742 The subset relation is a complete lattice.  Unions form least upper bounds;

   743 non-empty intersections form greatest lower bounds.  Figure~\ref{zf-subset}

   744 shows the corresponding rules.  A few other laws involving subsets are

   745 included.  Proofs are in the file \texttt{ZF/subset.ML}.

   746

   747 Reasoning directly about subsets often yields clearer proofs than

   748 reasoning about the membership relation.  Section~\ref{sec:ZF-pow-example}

   749 below presents an example of this, proving the equation ${{\tt Pow}(A)\cap   750 {\tt Pow}(B)}= {\tt Pow}(A\cap B)$.

   751

   752 %%% pair.ML

   753

   754 \begin{figure}

   755 \begin{ttbox}

   756 \tdx{Pair_inject1}    <a,b> = <c,d> ==> a=c

   757 \tdx{Pair_inject2}    <a,b> = <c,d> ==> b=d

   758 \tdx{Pair_inject}     [| <a,b> = <c,d>;  [| a=c; b=d |] ==> P |] ==> P

   759 \tdx{Pair_neq_0}      <a,b>=0 ==> P

   760

   761 \tdx{fst_conv}        fst(<a,b>) = a

   762 \tdx{snd_conv}        snd(<a,b>) = b

   763 \tdx{split}           split(\%x y. c(x,y), <a,b>) = c(a,b)

   764

   765 \tdx{SigmaI}          [| a:A;  b:B(a) |] ==> <a,b> : Sigma(A,B)

   766

   767 \tdx{SigmaE}          [| c: Sigma(A,B);

   768                    !!x y.[| x:A; y:B(x); c=<x,y> |] ==> P |] ==> P

   769

   770 \tdx{SigmaE2}         [| <a,b> : Sigma(A,B);

   771                    [| a:A;  b:B(a) |] ==> P   |] ==> P

   772 \end{ttbox}

   773 \caption{Ordered pairs; projections; general sums} \label{zf-pair}

   774 \end{figure}

   775

   776

   777 \subsection{Ordered pairs} \label{sec:pairs}

   778

   779 Figure~\ref{zf-pair} presents the rules governing ordered pairs,

   780 projections and general sums.  File \texttt{ZF/pair.ML} contains the

   781 full (and tedious) proof that $\{\{a\},\{a,b\}\}$ functions as an ordered

   782 pair.  This property is expressed as two destruction rules,

   783 \tdx{Pair_inject1} and \tdx{Pair_inject2}, and equivalently

   784 as the elimination rule \tdx{Pair_inject}.

   785

   786 The rule \tdx{Pair_neq_0} asserts $\pair{a,b}\neq\emptyset$.  This

   787 is a property of $\{\{a\},\{a,b\}\}$, and need not hold for other

   788 encodings of ordered pairs.  The non-standard ordered pairs mentioned below

   789 satisfy $\pair{\emptyset;\emptyset}=\emptyset$.

   790

   791 The natural deduction rules \tdx{SigmaI} and \tdx{SigmaE}

   792 assert that \cdx{Sigma}$(A,B)$ consists of all pairs of the form

   793 $\pair{x,y}$, for $x\in A$ and $y\in B(x)$.  The rule \tdx{SigmaE2}

   794 merely states that $\pair{a,b}\in \texttt{Sigma}(A,B)$ implies $a\in A$ and

   795 $b\in B(a)$.

   796

   797 In addition, it is possible to use tuples as patterns in abstractions:

   798 \begin{center}

   799 {\tt\%<$x$,$y$>. $t$} \quad stands for\quad \texttt{split(\%$x$ $y$.\ $t$)}

   800 \end{center}

   801 Nested patterns are translated recursively:

   802 {\tt\%<$x$,$y$,$z$>. $t$} $\leadsto$ {\tt\%<$x$,<$y$,$z$>>. $t$} $\leadsto$

   803 \texttt{split(\%$x$.\%<$y$,$z$>. $t$)} $\leadsto$ \texttt{split(\%$x$. split(\%$y$

   804   $z$.\ $t$))}.  The reverse translation is performed upon printing.

   805 \begin{warn}

   806   The translation between patterns and \texttt{split} is performed automatically

   807   by the parser and printer.  Thus the internal and external form of a term

   808   may differ, which affects proofs.  For example the term {\tt

   809     (\%<x,y>.<y,x>)<a,b>} requires the theorem \texttt{split} to rewrite to

   810   {\tt<b,a>}.

   811 \end{warn}

   812 In addition to explicit $\lambda$-abstractions, patterns can be used in any

   813 variable binding construct which is internally described by a

   814 $\lambda$-abstraction.  Here are some important examples:

   815 \begin{description}

   816 \item[Let:] \texttt{let {\it pattern} = $t$ in $u$}

   817 \item[Choice:] \texttt{THE~{\it pattern}~.~$P$}

   818 \item[Set operations:] \texttt{UN~{\it pattern}:$A$.~$B$}

   819 \item[Comprehension:] \texttt{{\ttlbrace}~{\it pattern}:$A$~.~$P$~{\ttrbrace}}

   820 \end{description}

   821

   822

   823 %%% domrange.ML

   824

   825 \begin{figure}

   826 \begin{ttbox}

   827 \tdx{domainI}        <a,b>: r ==> a : domain(r)

   828 \tdx{domainE}        [| a : domain(r);  !!y. <a,y>: r ==> P |] ==> P

   829 \tdx{domain_subset}  domain(Sigma(A,B)) <= A

   830

   831 \tdx{rangeI}         <a,b>: r ==> b : range(r)

   832 \tdx{rangeE}         [| b : range(r);  !!x. <x,b>: r ==> P |] ==> P

   833 \tdx{range_subset}   range(A*B) <= B

   834

   835 \tdx{fieldI1}        <a,b>: r ==> a : field(r)

   836 \tdx{fieldI2}        <a,b>: r ==> b : field(r)

   837 \tdx{fieldCI}        (~ <c,a>:r ==> <a,b>: r) ==> a : field(r)

   838

   839 \tdx{fieldE}         [| a : field(r);

   840                   !!x. <a,x>: r ==> P;

   841                   !!x. <x,a>: r ==> P

   842                |] ==> P

   843

   844 \tdx{field_subset}   field(A*A) <= A

   845 \end{ttbox}

   846 \caption{Domain, range and field of a relation} \label{zf-domrange}

   847 \end{figure}

   848

   849 \begin{figure}

   850 \begin{ttbox}

   851 \tdx{imageI}         [| <a,b>: r;  a:A |] ==> b : rA

   852 \tdx{imageE}         [| b: rA;  !!x.[| <x,b>: r;  x:A |] ==> P |] ==> P

   853

   854 \tdx{vimageI}        [| <a,b>: r;  b:B |] ==> a : r-B

   855 \tdx{vimageE}        [| a: r-B;  !!x.[| <a,x>: r;  x:B |] ==> P |] ==> P

   856 \end{ttbox}

   857 \caption{Image and inverse image} \label{zf-domrange2}

   858 \end{figure}

   859

   860

   861 \subsection{Relations}

   862 Figure~\ref{zf-domrange} presents rules involving relations, which are sets

   863 of ordered pairs.  The converse of a relation~$r$ is the set of all pairs

   864 $\pair{y,x}$ such that $\pair{x,y}\in r$; if $r$ is a function, then

   865 {\cdx{converse}$(r)$} is its inverse.  The rules for the domain

   866 operation, namely \tdx{domainI} and~\tdx{domainE}, assert that

   867 \cdx{domain}$(r)$ consists of all~$x$ such that $r$ contains

   868 some pair of the form~$\pair{x,y}$.  The range operation is similar, and

   869 the field of a relation is merely the union of its domain and range.

   870

   871 Figure~\ref{zf-domrange2} presents rules for images and inverse images.

   872 Note that these operations are generalisations of range and domain,

   873 respectively.  See the file \texttt{ZF/domrange.ML} for derivations of the

   874 rules.

   875

   876

   877 %%% func.ML

   878

   879 \begin{figure}

   880 \begin{ttbox}

   881 \tdx{fun_is_rel}      f: Pi(A,B) ==> f <= Sigma(A,B)

   882

   883 \tdx{apply_equality}  [| <a,b>: f;  f: Pi(A,B) |] ==> fa = b

   884 \tdx{apply_equality2} [| <a,b>: f;  <a,c>: f;  f: Pi(A,B) |] ==> b=c

   885

   886 \tdx{apply_type}      [| f: Pi(A,B);  a:A |] ==> fa : B(a)

   887 \tdx{apply_Pair}      [| f: Pi(A,B);  a:A |] ==> <a,fa>: f

   888 \tdx{apply_iff}       f: Pi(A,B) ==> <a,b>: f <-> a:A & fa = b

   889

   890 \tdx{fun_extension}   [| f : Pi(A,B);  g: Pi(A,D);

   891                    !!x. x:A ==> fx = gx     |] ==> f=g

   892

   893 \tdx{domain_type}     [| <a,b> : f;  f: Pi(A,B) |] ==> a : A

   894 \tdx{range_type}      [| <a,b> : f;  f: Pi(A,B) |] ==> b : B(a)

   895

   896 \tdx{Pi_type}         [| f: A->C;  !!x. x:A ==> fx: B(x) |] ==> f: Pi(A,B)

   897 \tdx{domain_of_fun}   f: Pi(A,B) ==> domain(f)=A

   898 \tdx{range_of_fun}    f: Pi(A,B) ==> f: A->range(f)

   899

   900 \tdx{restrict}        a : A ==> restrict(f,A)  a = fa

   901 \tdx{restrict_type}   [| !!x. x:A ==> fx: B(x) |] ==>

   902                 restrict(f,A) : Pi(A,B)

   903 \end{ttbox}

   904 \caption{Functions} \label{zf-func1}

   905 \end{figure}

   906

   907

   908 \begin{figure}

   909 \begin{ttbox}

   910 \tdx{lamI}      a:A ==> <a,b(a)> : (lam x:A. b(x))

   911 \tdx{lamE}      [| p: (lam x:A. b(x));  !!x.[| x:A; p=<x,b(x)> |] ==> P

   912           |] ==>  P

   913

   914 \tdx{lam_type}  [| !!x. x:A ==> b(x): B(x) |] ==> (lam x:A. b(x)) : Pi(A,B)

   915

   916 \tdx{beta}      a : A ==> (lam x:A. b(x))  a = b(a)

   917 \tdx{eta}       f : Pi(A,B) ==> (lam x:A. fx) = f

   918 \end{ttbox}

   919 \caption{$\lambda$-abstraction} \label{zf-lam}

   920 \end{figure}

   921

   922

   923 \begin{figure}

   924 \begin{ttbox}

   925 \tdx{fun_empty}            0: 0->0

   926 \tdx{fun_single}           {\ttlbrace}<a,b>{\ttrbrace} : {\ttlbrace}a{\ttrbrace} -> {\ttlbrace}b{\ttrbrace}

   927

   928 \tdx{fun_disjoint_Un}      [| f: A->B;  g: C->D;  A Int C = 0  |] ==>

   929                      (f Un g) : (A Un C) -> (B Un D)

   930

   931 \tdx{fun_disjoint_apply1}  [| a:A;  f: A->B;  g: C->D;  A Int C = 0 |] ==>

   932                      (f Un g)a = fa

   933

   934 \tdx{fun_disjoint_apply2}  [| c:C;  f: A->B;  g: C->D;  A Int C = 0 |] ==>

   935                      (f Un g)c = gc

   936 \end{ttbox}

   937 \caption{Constructing functions from smaller sets} \label{zf-func2}

   938 \end{figure}

   939

   940

   941 \subsection{Functions}

   942 Functions, represented by graphs, are notoriously difficult to reason

   943 about.  The file \texttt{ZF/func.ML} derives many rules, which overlap more

   944 than they ought.  This section presents the more important rules.

   945

   946 Figure~\ref{zf-func1} presents the basic properties of \cdx{Pi}$(A,B)$,

   947 the generalized function space.  For example, if $f$ is a function and

   948 $\pair{a,b}\in f$, then $fa=b$ (\tdx{apply_equality}).  Two functions

   949 are equal provided they have equal domains and deliver equals results

   950 (\tdx{fun_extension}).

   951

   952 By \tdx{Pi_type}, a function typing of the form $f\in A\to C$ can be

   953 refined to the dependent typing $f\in\prod@{x\in A}B(x)$, given a suitable

   954 family of sets $\{B(x)\}@{x\in A}$.  Conversely, by \tdx{range_of_fun},

   955 any dependent typing can be flattened to yield a function type of the form

   956 $A\to C$; here, $C={\tt range}(f)$.

   957

   958 Among the laws for $\lambda$-abstraction, \tdx{lamI} and \tdx{lamE}

   959 describe the graph of the generated function, while \tdx{beta} and

   960 \tdx{eta} are the standard conversions.  We essentially have a

   961 dependently-typed $\lambda$-calculus (Fig.\ts\ref{zf-lam}).

   962

   963 Figure~\ref{zf-func2} presents some rules that can be used to construct

   964 functions explicitly.  We start with functions consisting of at most one

   965 pair, and may form the union of two functions provided their domains are

   966 disjoint.

   967

   968

   969 \begin{figure}

   970 \begin{ttbox}

   971 \tdx{Int_absorb}         A Int A = A

   972 \tdx{Int_commute}        A Int B = B Int A

   973 \tdx{Int_assoc}          (A Int B) Int C  =  A Int (B Int C)

   974 \tdx{Int_Un_distrib}     (A Un B) Int C  =  (A Int C) Un (B Int C)

   975

   976 \tdx{Un_absorb}          A Un A = A

   977 \tdx{Un_commute}         A Un B = B Un A

   978 \tdx{Un_assoc}           (A Un B) Un C  =  A Un (B Un C)

   979 \tdx{Un_Int_distrib}     (A Int B) Un C  =  (A Un C) Int (B Un C)

   980

   981 \tdx{Diff_cancel}        A-A = 0

   982 \tdx{Diff_disjoint}      A Int (B-A) = 0

   983 \tdx{Diff_partition}     A<=B ==> A Un (B-A) = B

   984 \tdx{double_complement}  [| A<=B; B<= C |] ==> (B - (C-A)) = A

   985 \tdx{Diff_Un}            A - (B Un C) = (A-B) Int (A-C)

   986 \tdx{Diff_Int}           A - (B Int C) = (A-B) Un (A-C)

   987

   988 \tdx{Union_Un_distrib}   Union(A Un B) = Union(A) Un Union(B)

   989 \tdx{Inter_Un_distrib}   [| a:A;  b:B |] ==>

   990                    Inter(A Un B) = Inter(A) Int Inter(B)

   991

   992 \tdx{Int_Union_RepFun}   A Int Union(B) = (UN C:B. A Int C)

   993

   994 \tdx{Un_Inter_RepFun}    b:B ==>

   995                    A Un Inter(B) = (INT C:B. A Un C)

   996

   997 \tdx{SUM_Un_distrib1}    (SUM x:A Un B. C(x)) =

   998                    (SUM x:A. C(x)) Un (SUM x:B. C(x))

   999

  1000 \tdx{SUM_Un_distrib2}    (SUM x:C. A(x) Un B(x)) =

  1001                    (SUM x:C. A(x))  Un  (SUM x:C. B(x))

  1002

  1003 \tdx{SUM_Int_distrib1}   (SUM x:A Int B. C(x)) =

  1004                    (SUM x:A. C(x)) Int (SUM x:B. C(x))

  1005

  1006 \tdx{SUM_Int_distrib2}   (SUM x:C. A(x) Int B(x)) =

  1007                    (SUM x:C. A(x)) Int (SUM x:C. B(x))

  1008 \end{ttbox}

  1009 \caption{Equalities} \label{zf-equalities}

  1010 \end{figure}

  1011

  1012

  1013 \begin{figure}

  1014 %\begin{constants}

  1015 %  \cdx{1}       & $i$           &       & $\{\emptyset\}$       \\

  1016 %  \cdx{bool}    & $i$           &       & the set $\{\emptyset,1\}$     \\

  1017 %  \cdx{cond}   & $[i,i,i]\To i$ &       & conditional for \texttt{bool}    \\

  1018 %  \cdx{not}    & $i\To i$       &       & negation for \texttt{bool}       \\

  1019 %  \sdx{and}    & $[i,i]\To i$   & Left 70 & conjunction for \texttt{bool}  \\

  1020 %  \sdx{or}     & $[i,i]\To i$   & Left 65 & disjunction for \texttt{bool}  \\

  1021 %  \sdx{xor}    & $[i,i]\To i$   & Left 65 & exclusive-or for \texttt{bool}

  1022 %\end{constants}

  1023 %

  1024 \begin{ttbox}

  1025 \tdx{bool_def}       bool == {\ttlbrace}0,1{\ttrbrace}

  1026 \tdx{cond_def}       cond(b,c,d) == if b=1 then c else d

  1027 \tdx{not_def}        not(b)  == cond(b,0,1)

  1028 \tdx{and_def}        a and b == cond(a,b,0)

  1029 \tdx{or_def}         a or b  == cond(a,1,b)

  1030 \tdx{xor_def}        a xor b == cond(a,not(b),b)

  1031

  1032 \tdx{bool_1I}        1 : bool

  1033 \tdx{bool_0I}        0 : bool

  1034 \tdx{boolE}          [| c: bool;  c=1 ==> P;  c=0 ==> P |] ==> P

  1035 \tdx{cond_1}         cond(1,c,d) = c

  1036 \tdx{cond_0}         cond(0,c,d) = d

  1037 \end{ttbox}

  1038 \caption{The booleans} \label{zf-bool}

  1039 \end{figure}

  1040

  1041

  1042 \section{Further developments}

  1043 The next group of developments is complex and extensive, and only

  1044 highlights can be covered here.  It involves many theories and ML files of

  1045 proofs.

  1046

  1047 Figure~\ref{zf-equalities} presents commutative, associative, distributive,

  1048 and idempotency laws of union and intersection, along with other equations.

  1049 See file \texttt{ZF/equalities.ML}.

  1050

  1051 Theory \thydx{Bool} defines $\{0,1\}$ as a set of booleans, with the usual

  1052 operators including a conditional (Fig.\ts\ref{zf-bool}).  Although ZF is a

  1053 first-order theory, you can obtain the effect of higher-order logic using

  1054 \texttt{bool}-valued functions, for example.  The constant~\texttt{1} is

  1055 translated to \texttt{succ(0)}.

  1056

  1057 \begin{figure}

  1058 \index{*"+ symbol}

  1059 \begin{constants}

  1060   \it symbol    & \it meta-type & \it priority & \it description \\

  1061   \tt +         & $[i,i]\To i$  &  Right 65     & disjoint union operator\\

  1062   \cdx{Inl}~~\cdx{Inr}  & $i\To i$      &       & injections\\

  1063   \cdx{case}    & $[i\To i,i\To i, i]\To i$ &   & conditional for $A+B$

  1064 \end{constants}

  1065 \begin{ttbox}

  1066 \tdx{sum_def}        A+B == {\ttlbrace}0{\ttrbrace}*A Un {\ttlbrace}1{\ttrbrace}*B

  1067 \tdx{Inl_def}        Inl(a) == <0,a>

  1068 \tdx{Inr_def}        Inr(b) == <1,b>

  1069 \tdx{case_def}       case(c,d,u) == split(\%y z. cond(y, d(z), c(z)), u)

  1070

  1071 \tdx{sum_InlI}       a : A ==> Inl(a) : A+B

  1072 \tdx{sum_InrI}       b : B ==> Inr(b) : A+B

  1073

  1074 \tdx{Inl_inject}     Inl(a)=Inl(b) ==> a=b

  1075 \tdx{Inr_inject}     Inr(a)=Inr(b) ==> a=b

  1076 \tdx{Inl_neq_Inr}    Inl(a)=Inr(b) ==> P

  1077

  1078 \tdx{sumE2}   u: A+B ==> (EX x. x:A & u=Inl(x)) | (EX y. y:B & u=Inr(y))

  1079

  1080 \tdx{case_Inl}       case(c,d,Inl(a)) = c(a)

  1081 \tdx{case_Inr}       case(c,d,Inr(b)) = d(b)

  1082 \end{ttbox}

  1083 \caption{Disjoint unions} \label{zf-sum}

  1084 \end{figure}

  1085

  1086

  1087 \subsection{Disjoint unions}

  1088

  1089 Theory \thydx{Sum} defines the disjoint union of two sets, with

  1090 injections and a case analysis operator (Fig.\ts\ref{zf-sum}).  Disjoint

  1091 unions play a role in datatype definitions, particularly when there is

  1092 mutual recursion~\cite{paulson-set-II}.

  1093

  1094 \begin{figure}

  1095 \begin{ttbox}

  1096 \tdx{QPair_def}       <a;b> == a+b

  1097 \tdx{qsplit_def}      qsplit(c,p)  == THE y. EX a b. p=<a;b> & y=c(a,b)

  1098 \tdx{qfsplit_def}     qfsplit(R,z) == EX x y. z=<x;y> & R(x,y)

  1099 \tdx{qconverse_def}   qconverse(r) == {\ttlbrace}z. w:r, EX x y. w=<x;y> & z=<y;x>{\ttrbrace}

  1100 \tdx{QSigma_def}      QSigma(A,B)  == UN x:A. UN y:B(x). {\ttlbrace}<x;y>{\ttrbrace}

  1101

  1102 \tdx{qsum_def}        A <+> B      == ({\ttlbrace}0{\ttrbrace} <*> A) Un ({\ttlbrace}1{\ttrbrace} <*> B)

  1103 \tdx{QInl_def}        QInl(a)      == <0;a>

  1104 \tdx{QInr_def}        QInr(b)      == <1;b>

  1105 \tdx{qcase_def}       qcase(c,d)   == qsplit(\%y z. cond(y, d(z), c(z)))

  1106 \end{ttbox}

  1107 \caption{Non-standard pairs, products and sums} \label{zf-qpair}

  1108 \end{figure}

  1109

  1110

  1111 \subsection{Non-standard ordered pairs}

  1112

  1113 Theory \thydx{QPair} defines a notion of ordered pair that admits

  1114 non-well-founded tupling (Fig.\ts\ref{zf-qpair}).  Such pairs are written

  1115 {\tt<$a$;$b$>}.  It also defines the eliminator \cdx{qsplit}, the

  1116 converse operator \cdx{qconverse}, and the summation operator

  1117 \cdx{QSigma}.  These are completely analogous to the corresponding

  1118 versions for standard ordered pairs.  The theory goes on to define a

  1119 non-standard notion of disjoint sum using non-standard pairs.  All of these

  1120 concepts satisfy the same properties as their standard counterparts; in

  1121 addition, {\tt<$a$;$b$>} is continuous.  The theory supports coinductive

  1122 definitions, for example of infinite lists~\cite{paulson-mscs}.

  1123

  1124 \begin{figure}

  1125 \begin{ttbox}

  1126 \tdx{bnd_mono_def}   bnd_mono(D,h) ==

  1127                  h(D)<=D & (ALL W X. W<=X --> X<=D --> h(W) <= h(X))

  1128

  1129 \tdx{lfp_def}        lfp(D,h) == Inter({\ttlbrace}X: Pow(D). h(X) <= X{\ttrbrace})

  1130 \tdx{gfp_def}        gfp(D,h) == Union({\ttlbrace}X: Pow(D). X <= h(X){\ttrbrace})

  1131

  1132

  1133 \tdx{lfp_lowerbound} [| h(A) <= A;  A<=D |] ==> lfp(D,h) <= A

  1134

  1135 \tdx{lfp_subset}     lfp(D,h) <= D

  1136

  1137 \tdx{lfp_greatest}   [| bnd_mono(D,h);

  1138                   !!X. [| h(X) <= X;  X<=D |] ==> A<=X

  1139                |] ==> A <= lfp(D,h)

  1140

  1141 \tdx{lfp_Tarski}     bnd_mono(D,h) ==> lfp(D,h) = h(lfp(D,h))

  1142

  1143 \tdx{induct}         [| a : lfp(D,h);  bnd_mono(D,h);

  1144                   !!x. x : h(Collect(lfp(D,h),P)) ==> P(x)

  1145                |] ==> P(a)

  1146

  1147 \tdx{lfp_mono}       [| bnd_mono(D,h);  bnd_mono(E,i);

  1148                   !!X. X<=D ==> h(X) <= i(X)

  1149                |] ==> lfp(D,h) <= lfp(E,i)

  1150

  1151 \tdx{gfp_upperbound} [| A <= h(A);  A<=D |] ==> A <= gfp(D,h)

  1152

  1153 \tdx{gfp_subset}     gfp(D,h) <= D

  1154

  1155 \tdx{gfp_least}      [| bnd_mono(D,h);

  1156                   !!X. [| X <= h(X);  X<=D |] ==> X<=A

  1157                |] ==> gfp(D,h) <= A

  1158

  1159 \tdx{gfp_Tarski}     bnd_mono(D,h) ==> gfp(D,h) = h(gfp(D,h))

  1160

  1161 \tdx{coinduct}       [| bnd_mono(D,h); a: X; X <= h(X Un gfp(D,h)); X <= D

  1162                |] ==> a : gfp(D,h)

  1163

  1164 \tdx{gfp_mono}       [| bnd_mono(D,h);  D <= E;

  1165                   !!X. X<=D ==> h(X) <= i(X)

  1166                |] ==> gfp(D,h) <= gfp(E,i)

  1167 \end{ttbox}

  1168 \caption{Least and greatest fixedpoints} \label{zf-fixedpt}

  1169 \end{figure}

  1170

  1171

  1172 \subsection{Least and greatest fixedpoints}

  1173

  1174 The Knaster-Tarski Theorem states that every monotone function over a

  1175 complete lattice has a fixedpoint.  Theory \thydx{Fixedpt} proves the

  1176 Theorem only for a particular lattice, namely the lattice of subsets of a

  1177 set (Fig.\ts\ref{zf-fixedpt}).  The theory defines least and greatest

  1178 fixedpoint operators with corresponding induction and coinduction rules.

  1179 These are essential to many definitions that follow, including the natural

  1180 numbers and the transitive closure operator.  The (co)inductive definition

  1181 package also uses the fixedpoint operators~\cite{paulson-CADE}.  See

  1182 Davey and Priestley~\cite{davey-priestley} for more on the Knaster-Tarski

  1183 Theorem and my paper~\cite{paulson-set-II} for discussion of the Isabelle

  1184 proofs.

  1185

  1186 Monotonicity properties are proved for most of the set-forming operations:

  1187 union, intersection, Cartesian product, image, domain, range, etc.  These

  1188 are useful for applying the Knaster-Tarski Fixedpoint Theorem.  The proofs

  1189 themselves are trivial applications of Isabelle's classical reasoner.  See

  1190 file \texttt{ZF/mono.ML}.

  1191

  1192

  1193 \subsection{Finite sets and lists}

  1194

  1195 Theory \texttt{Finite} (Figure~\ref{zf-fin}) defines the finite set operator;

  1196 ${\tt Fin}(A)$ is the set of all finite sets over~$A$.  The theory employs

  1197 Isabelle's inductive definition package, which proves various rules

  1198 automatically.  The induction rule shown is stronger than the one proved by

  1199 the package.  The theory also defines the set of all finite functions

  1200 between two given sets.

  1201

  1202 \begin{figure}

  1203 \begin{ttbox}

  1204 \tdx{Fin.emptyI}      0 : Fin(A)

  1205 \tdx{Fin.consI}       [| a: A;  b: Fin(A) |] ==> cons(a,b) : Fin(A)

  1206

  1207 \tdx{Fin_induct}

  1208     [| b: Fin(A);

  1209        P(0);

  1210        !!x y. [| x: A;  y: Fin(A);  x~:y;  P(y) |] ==> P(cons(x,y))

  1211     |] ==> P(b)

  1212

  1213 \tdx{Fin_mono}        A<=B ==> Fin(A) <= Fin(B)

  1214 \tdx{Fin_UnI}         [| b: Fin(A);  c: Fin(A) |] ==> b Un c : Fin(A)

  1215 \tdx{Fin_UnionI}      C : Fin(Fin(A)) ==> Union(C) : Fin(A)

  1216 \tdx{Fin_subset}      [| c<=b;  b: Fin(A) |] ==> c: Fin(A)

  1217 \end{ttbox}

  1218 \caption{The finite set operator} \label{zf-fin}

  1219 \end{figure}

  1220

  1221 \begin{figure}

  1222 \begin{constants}

  1223   \it symbol  & \it meta-type & \it priority & \it description \\

  1224   \cdx{list}    & $i\To i$      && lists over some set\\

  1225   \cdx{list_case} & $[i, [i,i]\To i, i] \To i$  && conditional for $list(A)$ \\

  1226   \cdx{map}     & $[i\To i, i] \To i$   &       & mapping functional\\

  1227   \cdx{length}  & $i\To i$              &       & length of a list\\

  1228   \cdx{rev}     & $i\To i$              &       & reverse of a list\\

  1229   \tt \at       & $[i,i]\To i$  &  Right 60     & append for lists\\

  1230   \cdx{flat}    & $i\To i$   &                  & append of list of lists

  1231 \end{constants}

  1232

  1233 \underscoreon %%because @ is used here

  1234 \begin{ttbox}

  1235 \tdx{NilI}            Nil : list(A)

  1236 \tdx{ConsI}           [| a: A;  l: list(A) |] ==> Cons(a,l) : list(A)

  1237

  1238 \tdx{List.induct}

  1239     [| l: list(A);

  1240        P(Nil);

  1241        !!x y. [| x: A;  y: list(A);  P(y) |] ==> P(Cons(x,y))

  1242     |] ==> P(l)

  1243

  1244 \tdx{Cons_iff}        Cons(a,l)=Cons(a',l') <-> a=a' & l=l'

  1245 \tdx{Nil_Cons_iff}    ~ Nil=Cons(a,l)

  1246

  1247 \tdx{list_mono}       A<=B ==> list(A) <= list(B)

  1248

  1249 \tdx{map_ident}       l: list(A) ==> map(\%u. u, l) = l

  1250 \tdx{map_compose}     l: list(A) ==> map(h, map(j,l)) = map(\%u. h(j(u)), l)

  1251 \tdx{map_app_distrib} xs: list(A) ==> map(h, xs@ys) = map(h,xs) @ map(h,ys)

  1252 \tdx{map_type}

  1253     [| l: list(A);  !!x. x: A ==> h(x): B |] ==> map(h,l) : list(B)

  1254 \tdx{map_flat}

  1255     ls: list(list(A)) ==> map(h, flat(ls)) = flat(map(map(h),ls))

  1256 \end{ttbox}

  1257 \caption{Lists} \label{zf-list}

  1258 \end{figure}

  1259

  1260

  1261 Figure~\ref{zf-list} presents the set of lists over~$A$, ${\tt list}(A)$.  The

  1262 definition employs Isabelle's datatype package, which defines the introduction

  1263 and induction rules automatically, as well as the constructors, case operator

  1264 (\verb|list_case|) and recursion operator.  The theory then defines the usual

  1265 list functions by primitive recursion.  See theory \texttt{List}.

  1266

  1267

  1268 \subsection{Miscellaneous}

  1269

  1270 \begin{figure}

  1271 \begin{constants}

  1272   \it symbol  & \it meta-type & \it priority & \it description \\

  1273   \sdx{O}       & $[i,i]\To i$  &  Right 60     & composition ($\circ$) \\

  1274   \cdx{id}      & $i\To i$      &       & identity function \\

  1275   \cdx{inj}     & $[i,i]\To i$  &       & injective function space\\

  1276   \cdx{surj}    & $[i,i]\To i$  &       & surjective function space\\

  1277   \cdx{bij}     & $[i,i]\To i$  &       & bijective function space

  1278 \end{constants}

  1279

  1280 \begin{ttbox}

  1281 \tdx{comp_def}  r O s     == {\ttlbrace}xz : domain(s)*range(r) .

  1282                         EX x y z. xz=<x,z> & <x,y>:s & <y,z>:r{\ttrbrace}

  1283 \tdx{id_def}    id(A)     == (lam x:A. x)

  1284 \tdx{inj_def}   inj(A,B)  == {\ttlbrace} f: A->B. ALL w:A. ALL x:A. fw=fx --> w=x {\ttrbrace}

  1285 \tdx{surj_def}  surj(A,B) == {\ttlbrace} f: A->B . ALL y:B. EX x:A. fx=y {\ttrbrace}

  1286 \tdx{bij_def}   bij(A,B)  == inj(A,B) Int surj(A,B)

  1287

  1288

  1289 \tdx{left_inverse}     [| f: inj(A,B);  a: A |] ==> converse(f)(fa) = a

  1290 \tdx{right_inverse}    [| f: inj(A,B);  b: range(f) |] ==>

  1291                  f(converse(f)b) = b

  1292

  1293 \tdx{inj_converse_inj} f: inj(A,B) ==> converse(f): inj(range(f), A)

  1294 \tdx{bij_converse_bij} f: bij(A,B) ==> converse(f): bij(B,A)

  1295

  1296 \tdx{comp_type}        [| s<=A*B;  r<=B*C |] ==> (r O s) <= A*C

  1297 \tdx{comp_assoc}       (r O s) O t = r O (s O t)

  1298

  1299 \tdx{left_comp_id}     r<=A*B ==> id(B) O r = r

  1300 \tdx{right_comp_id}    r<=A*B ==> r O id(A) = r

  1301

  1302 \tdx{comp_func}        [| g:A->B; f:B->C |] ==> (f O g):A->C

  1303 \tdx{comp_func_apply}  [| g:A->B; f:B->C; a:A |] ==> (f O g)a = f(ga)

  1304

  1305 \tdx{comp_inj}         [| g:inj(A,B);  f:inj(B,C)  |] ==> (f O g):inj(A,C)

  1306 \tdx{comp_surj}        [| g:surj(A,B); f:surj(B,C) |] ==> (f O g):surj(A,C)

  1307 \tdx{comp_bij}         [| g:bij(A,B); f:bij(B,C) |] ==> (f O g):bij(A,C)

  1308

  1309 \tdx{left_comp_inverse}     f: inj(A,B) ==> converse(f) O f = id(A)

  1310 \tdx{right_comp_inverse}    f: surj(A,B) ==> f O converse(f) = id(B)

  1311

  1312 \tdx{bij_disjoint_Un}

  1313     [| f: bij(A,B);  g: bij(C,D);  A Int C = 0;  B Int D = 0 |] ==>

  1314     (f Un g) : bij(A Un C, B Un D)

  1315

  1316 \tdx{restrict_bij}  [| f:inj(A,B);  C<=A |] ==> restrict(f,C): bij(C, fC)

  1317 \end{ttbox}

  1318 \caption{Permutations} \label{zf-perm}

  1319 \end{figure}

  1320

  1321 The theory \thydx{Perm} is concerned with permutations (bijections) and

  1322 related concepts.  These include composition of relations, the identity

  1323 relation, and three specialized function spaces: injective, surjective and

  1324 bijective.  Figure~\ref{zf-perm} displays many of their properties that

  1325 have been proved.  These results are fundamental to a treatment of

  1326 equipollence and cardinality.

  1327

  1328 Theory \thydx{Univ} defines a universe' $\texttt{univ}(A)$, which is used by

  1329 the datatype package.  This set contains $A$ and the

  1330 natural numbers.  Vitally, it is closed under finite products: ${\tt   1331 univ}(A)\times{\tt univ}(A)\subseteq{\tt univ}(A)$.  This theory also

  1332 defines the cumulative hierarchy of axiomatic set theory, which

  1333 traditionally is written $V@\alpha$ for an ordinal~$\alpha$.  The

  1334 universe' is a simple generalization of~$V@\omega$.

  1335

  1336 Theory \thydx{QUniv} defines a universe' ${\tt quniv}(A)$, which is used by

  1337 the datatype package to construct codatatypes such as streams.  It is

  1338 analogous to ${\tt univ}(A)$ (and is defined in terms of it) but is closed

  1339 under the non-standard product and sum.

  1340

  1341

  1342 \section{Automatic Tools}

  1343

  1344 ZF provides the simplifier and the classical reasoner.  Moreover it supplies a

  1345 specialized tool to infer types' of terms.

  1346

  1347 \subsection{Simplification}

  1348

  1349 ZF inherits simplification from FOL but adopts it for set theory.  The

  1350 extraction of rewrite rules takes the ZF primitives into account.  It can

  1351 strip bounded universal quantifiers from a formula; for example, ${\forall   1352 x\in A. f(x)=g(x)}$ yields the conditional rewrite rule $x\in A \Imp   1353 f(x)=g(x)$.  Given $a\in\{x\in A. P(x)\}$ it extracts rewrite rules from $a\in   1354 A$ and~$P(a)$.  It can also break down $a\in A\int B$ and $a\in A-B$.

  1355

  1356 Simplification tactics tactics such as \texttt{Asm_simp_tac} and

  1357 \texttt{Full_simp_tac} use the default simpset (\texttt{simpset()}), which

  1358 works for most purposes.  A small simplification set for set theory is

  1359 called~\ttindexbold{ZF_ss}, and you can even use \ttindex{FOL_ss} as a minimal

  1360 starting point.  \texttt{ZF_ss} contains congruence rules for all the binding

  1361 operators of ZF.  It contains all the conversion rules, such as \texttt{fst}

  1362 and \texttt{snd}, as well as the rewrites shown in Fig.\ts\ref{zf-simpdata}.

  1363 See the file \texttt{ZF/simpdata.ML} for a fuller list.

  1364

  1365

  1366 \subsection{Classical Reasoning}

  1367

  1368 As for the classical reasoner, tactics such as \texttt{Blast_tac} and {\tt

  1369   Best_tac} refer to the default claset (\texttt{claset()}).  This works for

  1370 most purposes.  Named clasets include \ttindexbold{ZF_cs} (basic set theory)

  1371 and \ttindexbold{le_cs} (useful for reasoning about the relations $<$ and

  1372 $\le$).  You can use \ttindex{FOL_cs} as a minimal basis for building your own

  1373 clasets.  See \iflabelundefined{chap:classical}{the {\em Reference Manual\/}}%

  1374 {Chap.\ts\ref{chap:classical}} for more discussion of classical proof methods.

  1375

  1376

  1377 \begin{figure}

  1378 \begin{eqnarray*}

  1379   a\in \emptyset        & \bimp &  \bot\\

  1380   a \in A \un B      & \bimp &  a\in A \disj a\in B\\

  1381   a \in A \int B      & \bimp &  a\in A \conj a\in B\\

  1382   a \in A-B             & \bimp &  a\in A \conj \neg (a\in B)\\

  1383   \pair{a,b}\in {\tt Sigma}(A,B)

  1384                         & \bimp &  a\in A \conj b\in B(a)\\

  1385   a \in {\tt Collect}(A,P)      & \bimp &  a\in A \conj P(a)\\

  1386   (\forall x \in \emptyset. P(x)) & \bimp &  \top\\

  1387   (\forall x \in A. \top)       & \bimp &  \top

  1388 \end{eqnarray*}

  1389 \caption{Some rewrite rules for set theory} \label{zf-simpdata}

  1390 \end{figure}

  1391

  1392

  1393 \subsection{Type-Checking Tactics}

  1394 \index{type-checking tactics}

  1395

  1396 Isabelle/ZF provides simple tactics to help automate those proofs that are

  1397 essentially type-checking.  Such proofs are built by applying rules such as

  1398 these:

  1399 \begin{ttbox}

  1400 [| ?P ==> ?a: ?A; ~?P ==> ?b: ?A |] ==> (if ?P then ?a else ?b): ?A

  1401

  1402 [| ?m : nat; ?n : nat |] ==> ?m #+ ?n : nat

  1403

  1404 ?a : ?A ==> Inl(?a) : ?A + ?B

  1405 \end{ttbox}

  1406 In typical applications, the goal has the form $t\in\Var{A}$: in other words,

  1407 we have a specific term~$t$ and need to infer its type' by instantiating the

  1408 set variable~$\Var{A}$.  Neither the simplifier nor the classical reasoner

  1409 does this job well.  The if-then-else rule, and many similar ones, can make

  1410 the classical reasoner loop.  The simplifier refuses (on principle) to

  1411 instantiate variables during rewriting, so goals such as \texttt{i\#+j :\ ?A}

  1412 are left unsolved.

  1413

  1414 The simplifier calls the type-checker to solve rewritten subgoals: this stage

  1415 can indeed instantiate variables.  If you have defined new constants and

  1416 proved type-checking rules for them, then insert the rules using

  1417 \texttt{AddTCs} and the rest should be automatic.  In particular, the

  1418 simplifier will use type-checking to help satisfy conditional rewrite rules.

  1419 Call the tactic \ttindex{Typecheck_tac} to break down all subgoals using

  1420 type-checking rules.

  1421

  1422 Though the easiest way to invoke the type-checker is via the simplifier,

  1423 specialized applications may require more detailed knowledge of

  1424 the type-checking primitives.  They are modelled on the simplifier's:

  1425 \begin{ttdescription}

  1426 \item[\ttindexbold{tcset}] is the type of tcsets: sets of type-checking rules.

  1427

  1428 \item[\ttindexbold{addTCs}] is an infix operator to add type-checking rules to

  1429   a tcset.

  1430

  1431 \item[\ttindexbold{delTCs}] is an infix operator to remove type-checking rules

  1432   from a tcset.

  1433

  1434 \item[\ttindexbold{typecheck_tac}] is a tactic for attempting to prove all

  1435   subgoals using the rules given in its argument, a tcset.

  1436 \end{ttdescription}

  1437

  1438 Tcsets, like simpsets, are associated with theories and are merged when

  1439 theories are merged.  There are further primitives that use the default tcset.

  1440 \begin{ttdescription}

  1441 \item[\ttindexbold{tcset}] is a function to return the default tcset; use the

  1442   expression \texttt{tcset()}.

  1443

  1444 \item[\ttindexbold{AddTCs}] adds type-checking rules to the default tcset.

  1445

  1446 \item[\ttindexbold{DelTCs}] removes type-checking rules from the default

  1447   tcset.

  1448

  1449 \item[\ttindexbold{Typecheck_tac}] calls \texttt{typecheck_tac} using the

  1450   default tcset.

  1451 \end{ttdescription}

  1452

  1453 To supply some type-checking rules temporarily, using \texttt{Addrules} and

  1454 later \texttt{Delrules} is the simplest way.  There is also a high-tech

  1455 approach.  Call the simplifier with a new solver expressed using

  1456 \ttindexbold{type_solver_tac} and your temporary type-checking rules.

  1457 \begin{ttbox}

  1458 by (asm_simp_tac

  1459      (simpset() setSolver type_solver_tac (tcset() addTCs prems)) 2);

  1460 \end{ttbox}

  1461

  1462

  1463 \section{Natural number and integer arithmetic}

  1464

  1465 \index{arithmetic|(}

  1466

  1467 \begin{figure}\small

  1468 \index{#*@{\tt\#*} symbol}

  1469 \index{*div symbol}

  1470 \index{*mod symbol}

  1471 \index{#+@{\tt\#+} symbol}

  1472 \index{#-@{\tt\#-} symbol}

  1473 \begin{constants}

  1474   \it symbol  & \it meta-type & \it priority & \it description \\

  1475   \cdx{nat}     & $i$                   &       & set of natural numbers \\

  1476   \cdx{nat_case}& $[i,i\To i,i]\To i$     &     & conditional for $nat$\\

  1477   \tt \#*       & $[i,i]\To i$  &  Left 70      & multiplication \\

  1478   \tt div       & $[i,i]\To i$  &  Left 70      & division\\

  1479   \tt mod       & $[i,i]\To i$  &  Left 70      & modulus\\

  1480   \tt \#+       & $[i,i]\To i$  &  Left 65      & addition\\

  1481   \tt \#-       & $[i,i]\To i$  &  Left 65      & subtraction

  1482 \end{constants}

  1483

  1484 \begin{ttbox}

  1485 \tdx{nat_def}  nat == lfp(lam r: Pow(Inf). {\ttlbrace}0{\ttrbrace} Un {\ttlbrace}succ(x). x:r{\ttrbrace}

  1486

  1487 \tdx{nat_case_def}  nat_case(a,b,k) ==

  1488               THE y. k=0 & y=a | (EX x. k=succ(x) & y=b(x))

  1489

  1490 \tdx{nat_0I}           0 : nat

  1491 \tdx{nat_succI}        n : nat ==> succ(n) : nat

  1492

  1493 \tdx{nat_induct}

  1494     [| n: nat;  P(0);  !!x. [| x: nat;  P(x) |] ==> P(succ(x))

  1495     |] ==> P(n)

  1496

  1497 \tdx{nat_case_0}      nat_case(a,b,0) = a

  1498 \tdx{nat_case_succ}   nat_case(a,b,succ(m)) = b(m)

  1499

  1500 \tdx{add_0_natify}     0 #+ n = natify(n)

  1501 \tdx{add_succ}         succ(m) #+ n = succ(m #+ n)

  1502

  1503 \tdx{mult_type}        m #* n : nat

  1504 \tdx{mult_0}           0 #* n = 0

  1505 \tdx{mult_succ}        succ(m) #* n = n #+ (m #* n)

  1506 \tdx{mult_commute}     m #* n = n #* m

  1507 \tdx{add_mult_dist}    (m #+ n) #* k = (m #* k) #+ (n #* k)

  1508 \tdx{mult_assoc}       (m #* n) #* k = m #* (n #* k)

  1509 \tdx{mod_div_equality} m: nat ==> (m div n)#*n #+ m mod n = m

  1510 \end{ttbox}

  1511 \caption{The natural numbers} \label{zf-nat}

  1512 \end{figure}

  1513

  1514 \index{natural numbers}

  1515

  1516 Theory \thydx{Nat} defines the natural numbers and mathematical

  1517 induction, along with a case analysis operator.  The set of natural

  1518 numbers, here called \texttt{nat}, is known in set theory as the ordinal~$\omega$.

  1519

  1520 Theory \thydx{Arith} develops arithmetic on the natural numbers

  1521 (Fig.\ts\ref{zf-nat}).  Addition, multiplication and subtraction are defined

  1522 by primitive recursion.  Division and remainder are defined by repeated

  1523 subtraction, which requires well-founded recursion; the termination argument

  1524 relies on the divisor's being non-zero.  Many properties are proved:

  1525 commutative, associative and distributive laws, identity and cancellation

  1526 laws, etc.  The most interesting result is perhaps the theorem $a \bmod b +   1527 (a/b)\times b = a$.

  1528

  1529 To minimize the need for tedious proofs of $t\in\texttt{nat}$, the arithmetic

  1530 operators coerce their arguments to be natural numbers.  The function

  1531 \cdx{natify} is defined such that $\texttt{natify}(n) = n$ if $n$ is a natural

  1532 number, $\texttt{natify}(\texttt{succ}(x)) =   1533 \texttt{succ}(\texttt{natify}(x))$ for all $x$, and finally

  1534 $\texttt{natify}(x)=0$ in all other cases.  The benefit is that the addition,

  1535 subtraction, multiplication, division and remainder operators always return

  1536 natural numbers, regardless of their arguments.  Algebraic laws (commutative,

  1537 associative, distributive) are unconditional.  Occurrences of \texttt{natify}

  1538 as operands of those operators are simplified away.  Any remaining occurrences

  1539 can either be tolerated or else eliminated by proving that the argument is a

  1540 natural number.

  1541

  1542 The simplifier automatically cancels common terms on the opposite sides of

  1543 subtraction and of relations ($=$, $<$ and $\le$).  Here is an example:

  1544 \begin{ttbox}

  1545  1. i #+ j #+ k #- j < k #+ l

  1546 > by (Simp_tac 1);

  1547  1. natify(i) < natify(l)

  1548 \end{ttbox}

  1549 Given the assumptions \texttt{i:nat} and \texttt{l:nat}, both occurrences of

  1550 \cdx{natify} would be simplified away.

  1551

  1552

  1553 \begin{figure}\small

  1554 \index{$*@{\tt\$*} symbol}

  1555 \index{$+@{\tt\$+} symbol}

  1556 \index{$-@{\tt\$-} symbol}

  1557 \begin{constants}

  1558   \it symbol  & \it meta-type & \it priority & \it description \\

  1559   \cdx{int}     & $i$                   &       & set of integers \\

  1560   \tt \$* &$[i,i]\To i$& Left 70 & multiplication \\   1561 \tt \$+       & $[i,i]\To i$  &  Left 65      & addition\\

  1562   \tt \$- &$[i,i]\To i$& Left 65 & subtraction\\   1563 \tt \$<       & $[i,i]\To o$  &  Left 50      & $<$ on integers\\

  1564   \tt \$<= &$[i,i]\To o$& Left 50 &$\le$on integers   1565 \end{constants}   1566   1567 \begin{ttbox}   1568 \tdx{zadd_0_intify} 0$+ n = intify(n)

  1569

  1570 \tdx{zmult_type}        m $* n : int   1571 \tdx{zmult_0} 0$* n = 0

  1572 \tdx{zmult_commute}     m $* n = n$* m

  1573 \tdx{zadd_zmult_dist}    (m $+ n)$* k = (m $* k)$+ (n $* k)   1574 \tdx{zmult_assoc} (m$* n) $* k = m$* (n $* k)   1575 \end{ttbox}   1576 \caption{The integers} \label{zf-int}   1577 \end{figure}   1578   1579   1580 \index{integers}   1581   1582 Theory \thydx{Int} defines the integers, as equivalence classes of natural   1583 numbers. Figure~\ref{zf-int} presents a tidy collection of laws. In   1584 fact, a large library of facts is proved, including monotonicity laws for   1585 addition and multiplication, covering both positive and negative operands.   1586   1587 As with the natural numbers, the need for typing proofs is minimized. All the   1588 operators defined in Fig.\ts\ref{zf-int} coerce their operands to integers by   1589 applying the function \cdx{intify}. This function is the identity on integers   1590 and maps other operands to zero.   1591   1592 Decimal notation is provided for the integers. Numbers, written as   1593 \texttt{\#$nnn$} or \texttt{\#-$nnn$}, are represented internally in   1594 two's-complement binary. Expressions involving addition, subtraction and   1595 multiplication of numeral constants are evaluated (with acceptable efficiency)   1596 by simplification. The simplifier also collects similar terms, multiplying   1597 them by a numerical coefficient. It also cancels occurrences of the same   1598 terms on the other side of the relational operators. Example:   1599 \begin{ttbox}   1600 1. y$+ z $+ #-3$* x $+ y$<= x $* #2$+ z

  1601 > by (Simp_tac 1);

  1602  1. #2 $* y$<= #5 $* x   1603 \end{ttbox}   1604 For more information on the integers, please see the theories on directory   1605 \texttt{ZF/Integ}.   1606   1607 \index{arithmetic|)}   1608   1609   1610 \section{Datatype definitions}   1611 \label{sec:ZF:datatype}   1612 \index{*datatype|(}   1613   1614 The \ttindex{datatype} definition package of ZF constructs inductive datatypes   1615 similar to those of \ML. It can also construct coinductive datatypes   1616 (codatatypes), which are non-well-founded structures such as streams. It   1617 defines the set using a fixed-point construction and proves induction rules,   1618 as well as theorems for recursion and case combinators. It supplies   1619 mechanisms for reasoning about freeness. The datatype package can handle both   1620 mutual and indirect recursion.   1621   1622   1623 \subsection{Basics}   1624 \label{subsec:datatype:basics}   1625   1626 A \texttt{datatype} definition has the following form:   1627 $  1628 \begin{array}{llcl}   1629 \mathtt{datatype} & t@1(A@1,\ldots,A@h) & = &   1630 constructor^1@1 ~\mid~ \ldots ~\mid~ constructor^1@{k@1} \\   1631 & & \vdots \\   1632 \mathtt{and} & t@n(A@1,\ldots,A@h) & = &   1633 constructor^n@1~ ~\mid~ \ldots ~\mid~ constructor^n@{k@n}   1634 \end{array}   1635$   1636 Here$t@1$, \ldots,~$t@n$are identifiers and$A@1$, \ldots,~$A@h$are   1637 variables: the datatype's parameters. Each constructor specification has the   1638 form \dquotesoff   1639 $C \hbox{\tt~( } \hbox{\tt"} x@1 \hbox{\tt:} T@1 \hbox{\tt"},\;   1640 \ldots,\;   1641 \hbox{\tt"} x@m \hbox{\tt:} T@m \hbox{\tt"}   1642 \hbox{\tt~)}   1643$   1644 Here$C$is the constructor name, and variables$x@1$, \ldots,~$x@m$are the   1645 constructor arguments, belonging to the sets$T@1$, \ldots,$T@m$,   1646 respectively. Typically each$T@j$is either a constant set, a datatype   1647 parameter (one of$A@1$, \ldots,$A@h$) or a recursive occurrence of one of   1648 the datatypes, say$t@i(A@1,\ldots,A@h)$. More complex possibilities exist,   1649 but they are much harder to realize. Often, additional information must be   1650 supplied in the form of theorems.   1651   1652 A datatype can occur recursively as the argument of some function~$F$. This   1653 is called a {\em nested} (or \emph{indirect}) occurrence. It is only allowed   1654 if the datatype package is given a theorem asserting that$F$is monotonic.   1655 If the datatype has indirect occurrences, then Isabelle/ZF does not support   1656 recursive function definitions.   1657   1658 A simple example of a datatype is \texttt{list}, which is built-in, and is   1659 defined by   1660 \begin{ttbox}   1661 consts list :: i=>i   1662 datatype "list(A)" = Nil | Cons ("a:A", "l: list(A)")   1663 \end{ttbox}   1664 Note that the datatype operator must be declared as a constant first.   1665 However, the package declares the constructors. Here, \texttt{Nil} gets type   1666$i$and \texttt{Cons} gets type$[i,i]\To i$.   1667   1668 Trees and forests can be modelled by the mutually recursive datatype   1669 definition   1670 \begin{ttbox}   1671 consts tree, forest, tree_forest :: i=>i   1672 datatype "tree(A)" = Tcons ("a: A", "f: forest(A)")   1673 and "forest(A)" = Fnil | Fcons ("t: tree(A)", "f: forest(A)")   1674 \end{ttbox}   1675 Here$\texttt{tree}(A)$is the set of trees over$A$,$\texttt{forest}(A)$is   1676 the set of forests over$A$, and$\texttt{tree_forest}(A)$is the union of   1677 the previous two sets. All three operators must be declared first.   1678   1679 The datatype \texttt{term}, which is defined by   1680 \begin{ttbox}   1681 consts term :: i=>i   1682 datatype "term(A)" = Apply ("a: A", "l: list(term(A))")   1683 monos "[list_mono]"   1684 \end{ttbox}   1685 is an example of nested recursion. (The theorem \texttt{list_mono} is proved   1686 in file \texttt{List.ML}, and the \texttt{term} example is devaloped in theory   1687 \thydx{ex/Term}.)   1688   1689 \subsubsection{Freeness of the constructors}   1690   1691 Constructors satisfy {\em freeness} properties. Constructions are distinct,   1692 for example$\texttt{Nil}\not=\texttt{Cons}(a,l)$, and they are injective, for   1693 example$\texttt{Cons}(a,l)=\texttt{Cons}(a',l') \bimp a=a' \conj l=l'$.   1694 Because the number of freeness is quadratic in the number of constructors, the   1695 datatype package does not prove them. Instead, it ensures that simplification   1696 will prove them dynamically: when the simplifier encounters a formula   1697 asserting the equality of two datatype constructors, it performs freeness   1698 reasoning.   1699   1700 Freeness reasoning can also be done using the classical reasoner, but it is   1701 more complicated. You have to add some safe elimination rules rules to the   1702 claset. For the \texttt{list} datatype, they are called   1703 \texttt{list.free_SEs}. Occasionally this exposes the underlying   1704 representation of some constructor, which can be rectified using the command   1705 \hbox{\tt fold_tac list.con_defs}.   1706   1707   1708 \subsubsection{Structural induction}   1709   1710 The datatype package also provides structural induction rules. For datatypes   1711 without mutual or nested recursion, the rule has the form exemplified by   1712 \texttt{list.induct} in Fig.\ts\ref{zf-list}. For mutually recursive   1713 datatypes, the induction rule is supplied in two forms. Consider datatype   1714 \texttt{TF}. The rule \texttt{tree_forest.induct} performs induction over a   1715 single predicate~\texttt{P}, which is presumed to be defined for both trees   1716 and forests:   1717 \begin{ttbox}   1718 [| x : tree_forest(A);   1719 !!a f. [| a : A; f : forest(A); P(f) |] ==> P(Tcons(a, f));   1720 P(Fnil);   1721 !!f t. [| t : tree(A); P(t); f : forest(A); P(f) |]   1722 ==> P(Fcons(t, f))   1723 |] ==> P(x)   1724 \end{ttbox}   1725 The rule \texttt{tree_forest.mutual_induct} performs induction over two   1726 distinct predicates, \texttt{P_tree} and \texttt{P_forest}.   1727 \begin{ttbox}   1728 [| !!a f.   1729 [| a : A; f : forest(A); P_forest(f) |] ==> P_tree(Tcons(a, f));   1730 P_forest(Fnil);   1731 !!f t. [| t : tree(A); P_tree(t); f : forest(A); P_forest(f) |]   1732 ==> P_forest(Fcons(t, f))   1733 |] ==> (ALL za. za : tree(A) --> P_tree(za)) &   1734 (ALL za. za : forest(A) --> P_forest(za))   1735 \end{ttbox}   1736   1737 For datatypes with nested recursion, such as the \texttt{term} example from   1738 above, things are a bit more complicated. The rule \texttt{term.induct}   1739 refers to the monotonic operator, \texttt{list}:   1740 \begin{ttbox}   1741 [| x : term(A);   1742 !!a l. [| a: A; l: list(Collect(term(A), P)) |] ==> P(Apply(a, l))   1743 |] ==> P(x)   1744 \end{ttbox}   1745 The file \texttt{ex/Term.ML} derives two higher-level induction rules, one of   1746 which is particularly useful for proving equations:   1747 \begin{ttbox}   1748 [| t : term(A);   1749 !!x zs. [| x : A; zs : list(term(A)); map(f, zs) = map(g, zs) |]   1750 ==> f(Apply(x, zs)) = g(Apply(x, zs))   1751 |] ==> f(t) = g(t)   1752 \end{ttbox}   1753 How this can be generalized to other nested datatypes is a matter for future   1754 research.   1755   1756   1757 \subsubsection{The \texttt{case} operator}   1758   1759 The package defines an operator for performing case analysis over the   1760 datatype. For \texttt{list}, it is called \texttt{list_case} and satisfies   1761 the equations   1762 \begin{ttbox}   1763 list_case(f_Nil, f_Cons, []) = f_Nil   1764 list_case(f_Nil, f_Cons, Cons(a, l)) = f_Cons(a, l)   1765 \end{ttbox}   1766 Here \texttt{f_Nil} is the value to return if the argument is \texttt{Nil} and   1767 \texttt{f_Cons} is a function that computes the value to return if the   1768 argument has the form$\texttt{Cons}(a,l)$. The function can be expressed as   1769 an abstraction, over patterns if desired (\S\ref{sec:pairs}).   1770   1771 For mutually recursive datatypes, there is a single \texttt{case} operator.   1772 In the tree/forest example, the constant \texttt{tree_forest_case} handles all   1773 of the constructors of the two datatypes.   1774   1775   1776   1777   1778 \subsection{Defining datatypes}   1779   1780 The theory syntax for datatype definitions is shown in   1781 Fig.~\ref{datatype-grammar}. In order to be well-formed, a datatype   1782 definition has to obey the rules stated in the previous section. As a result   1783 the theory is extended with the new types, the constructors, and the theorems   1784 listed in the previous section. The quotation marks are necessary because   1785 they enclose general Isabelle formul\ae.   1786   1787 \begin{figure}   1788 \begin{rail}   1789 datatype : ( 'datatype' | 'codatatype' ) datadecls;   1790   1791 datadecls: ( '"' id arglist '"' '=' (constructor + '|') ) + 'and'   1792 ;   1793 constructor : name ( () | consargs ) ( () | ( '(' mixfix ')' ) )   1794 ;   1795 consargs : '(' ('"' var ':' term '"' + ',') ')'   1796 ;   1797 \end{rail}   1798 \caption{Syntax of datatype declarations}   1799 \label{datatype-grammar}   1800 \end{figure}   1801   1802 Codatatypes are declared like datatypes and are identical to them in every   1803 respect except that they have a coinduction rule instead of an induction rule.   1804 Note that while an induction rule has the effect of limiting the values   1805 contained in the set, a coinduction rule gives a way of constructing new   1806 values of the set.   1807   1808 Most of the theorems about datatypes become part of the default simpset. You   1809 never need to see them again because the simplifier applies them   1810 automatically. Induction or exhaustion are usually invoked by hand,   1811 usually via these special-purpose tactics:   1812 \begin{ttdescription}   1813 \item[\ttindexbold{induct_tac} {\tt"}$x${\tt"}$i$] applies structural   1814 induction on variable$x$to subgoal$i$, provided the type of$x$is a   1815 datatype. The induction variable should not occur among other assumptions   1816 of the subgoal.   1817 \end{ttdescription}   1818 In some cases, induction is overkill and a case distinction over all   1819 constructors of the datatype suffices.   1820 \begin{ttdescription}   1821 \item[\ttindexbold{exhaust_tac} {\tt"}$x${\tt"}$i$]   1822 performs an exhaustive case analysis for the variable~$x$.   1823 \end{ttdescription}   1824   1825 Both tactics can only be applied to a variable, whose typing must be given in   1826 some assumption, for example the assumption \texttt{x:\ list(A)}. The tactics   1827 also work for the natural numbers (\texttt{nat}) and disjoint sums, although   1828 these sets were not defined using the datatype package. (Disjoint sums are   1829 not recursive, so only \texttt{exhaust_tac} is available.)   1830   1831 \bigskip   1832 Here are some more details for the technically minded. Processing the   1833 theory file produces an \ML\ structure which, in addition to the usual   1834 components, contains a structure named$t$for each datatype$t$defined in   1835 the file. Each structure$t$contains the following elements:   1836 \begin{ttbox}   1837 val intrs : thm list \textrm{the introduction rules}   1838 val elim : thm \textrm{the elimination (case analysis) rule}   1839 val induct : thm \textrm{the standard induction rule}   1840 val mutual_induct : thm \textrm{the mutual induction rule, or \texttt{True}}   1841 val case_eqns : thm list \textrm{equations for the case operator}   1842 val recursor_eqns : thm list \textrm{equations for the recursor}   1843 val con_defs : thm list \textrm{definitions of the case operator and constructors}   1844 val free_iffs : thm list \textrm{logical equivalences for proving freeness}   1845 val free_SEs : thm list \textrm{elimination rules for proving freeness}   1846 val mk_free : string -> thm \textrm{A function for proving freeness theorems}   1847 val mk_cases : string -> thm \textrm{case analysis, see below}   1848 val defs : thm list \textrm{definitions of operators}   1849 val bnd_mono : thm list \textrm{monotonicity property}   1850 val dom_subset : thm list \textrm{inclusion in bounding set'}   1851 \end{ttbox}   1852 Furthermore there is the theorem$C$\texttt{_I} for every constructor~$C$; for   1853 example, the \texttt{list} datatype's introduction rules are bound to the   1854 identifiers \texttt{Nil_I} and \texttt{Cons_I}.   1855   1856 For a codatatype, the component \texttt{coinduct} is the coinduction rule,   1857 replacing the \texttt{induct} component.   1858   1859 See the theories \texttt{ex/Ntree} and \texttt{ex/Brouwer} for examples of   1860 infinitely branching datatypes. See theory \texttt{ex/LList} for an example   1861 of a codatatype. Some of these theories illustrate the use of additional,   1862 undocumented features of the datatype package. Datatype definitions are   1863 reduced to inductive definitions, and the advanced features should be   1864 understood in that light.   1865   1866   1867 \subsection{Examples}   1868   1869 \subsubsection{The datatype of binary trees}   1870   1871 Let us define the set$\texttt{bt}(A)$of binary trees over~$A$. The theory   1872 must contain these lines:   1873 \begin{ttbox}   1874 consts bt :: i=>i   1875 datatype "bt(A)" = Lf | Br ("a: A", "t1: bt(A)", "t2: bt(A)")   1876 \end{ttbox}   1877 After loading the theory, we can prove, for example, that no tree equals its   1878 left branch. To ease the induction, we state the goal using quantifiers.   1879 \begin{ttbox}   1880 Goal "l : bt(A) ==> ALL x r. Br(x,l,r) ~= l";   1881 {\out Level 0}   1882 {\out l : bt(A) ==> ALL x r. Br(x, l, r) ~= l}   1883 {\out 1. l : bt(A) ==> ALL x r. Br(x, l, r) ~= l}   1884 \end{ttbox}   1885 This can be proved by the structural induction tactic:   1886 \begin{ttbox}   1887 by (induct_tac "l" 1);   1888 {\out Level 1}   1889 {\out l : bt(A) ==> ALL x r. Br(x, l, r) ~= l}   1890 {\out 1. ALL x r. Br(x, Lf, r) ~= Lf}   1891 {\out 2. !!a t1 t2.}   1892 {\out [| a : A; t1 : bt(A);}   1893 {\out ALL x r. Br(x, t1, r) ~= t1; t2 : bt(A);}   1894 {\out ALL x r. Br(x, t2, r) ~= t2 |]}   1895 {\out ==> ALL x r. Br(x, Br(a, t1, t2), r) ~= Br(a, t1, t2)}   1896 \end{ttbox}   1897 Both subgoals are proved using \texttt{Auto_tac}, which performs the necessary   1898 freeness reasoning.   1899 \begin{ttbox}   1900 by Auto_tac;   1901 {\out Level 2}   1902 {\out l : bt(A) ==> ALL x r. Br(x, l, r) ~= l}   1903 {\out No subgoals!}   1904 \end{ttbox}   1905 To remove the quantifiers from the induction formula, we save the theorem using   1906 \ttindex{qed_spec_mp}.   1907 \begin{ttbox}   1908 qed_spec_mp "Br_neq_left";   1909 {\out val Br_neq_left = "?l : bt(?A) ==> Br(?x, ?l, ?r) ~= ?l" : thm}   1910 \end{ttbox}   1911   1912 When there are only a few constructors, we might prefer to prove the freenness   1913 theorems for each constructor. This is trivial, using the function given us   1914 for that purpose:   1915 \begin{ttbox}   1916 val Br_iff =   1917 bt.mk_free "Br(a,l,r)=Br(a',l',r') <-> a=a' & l=l' & r=r'";   1918 {\out val Br_iff =}   1919 {\out "Br(?a, ?l, ?r) = Br(?a', ?l', ?r') <->}   1920 {\out ?a = ?a' & ?l = ?l' & ?r = ?r'" : thm}   1921 \end{ttbox}   1922   1923 The purpose of \ttindex{mk_cases} is to generate instances of the elimination   1924 (case analysis) rule that have been simplified using freeness reasoning. For   1925 example, this instance of the elimination rule propagates type-checking   1926 information from the premise$\texttt{Br}(a,l,r)\in\texttt{bt}(A)$:   1927 \begin{ttbox}   1928 val BrE = bt.mk_cases "Br(a,l,r) : bt(A)";   1929 {\out val BrE =}   1930 {\out "[| Br(?a, ?l, ?r) : bt(?A);}   1931 {\out [| ?a : ?A; ?l : bt(?A); ?r : bt(?A) |] ==> ?Q |]}   1932 {\out ==> ?Q" : thm}   1933 \end{ttbox}   1934   1935   1936 \subsubsection{Mixfix syntax in datatypes}   1937   1938 Mixfix syntax is sometimes convenient. The theory \texttt{ex/PropLog} makes a   1939 deep embedding of propositional logic:   1940 \begin{ttbox}   1941 consts prop :: i   1942 datatype "prop" = Fls   1943 | Var ("n: nat") ("#_" [100] 100)   1944 | "=>" ("p: prop", "q: prop") (infixr 90)   1945 \end{ttbox}   1946 The second constructor has a special$\#n$syntax, while the third constructor   1947 is an infixed arrow.   1948   1949   1950 \subsubsection{A giant enumeration type}   1951   1952 This example shows a datatype that consists of 60 constructors:   1953 \begin{ttbox}   1954 consts enum :: i   1955 datatype   1956 "enum" = C00 | C01 | C02 | C03 | C04 | C05 | C06 | C07 | C08 | C09   1957 | C10 | C11 | C12 | C13 | C14 | C15 | C16 | C17 | C18 | C19   1958 | C20 | C21 | C22 | C23 | C24 | C25 | C26 | C27 | C28 | C29   1959 | C30 | C31 | C32 | C33 | C34 | C35 | C36 | C37 | C38 | C39   1960 | C40 | C41 | C42 | C43 | C44 | C45 | C46 | C47 | C48 | C49   1961 | C50 | C51 | C52 | C53 | C54 | C55 | C56 | C57 | C58 | C59   1962 end   1963 \end{ttbox}   1964 The datatype package scales well. Even though all properties are proved   1965 rather than assumed, full processing of this definition takes under 15 seconds   1966 (on a 300 MHz Pentium). The constructors have a balanced representation,   1967 essentially binary notation, so freeness properties can be proved fast.   1968 \begin{ttbox}   1969 Goal "C00 ~= C01";   1970 by (Simp_tac 1);   1971 \end{ttbox}   1972 You need not derive such inequalities explicitly. The simplifier will dispose   1973 of them automatically.   1974   1975 \index{*datatype|)}   1976   1977   1978 \subsection{Recursive function definitions}\label{sec:ZF:recursive}   1979 \index{recursive functions|see{recursion}}   1980 \index{*primrec|(}   1981 \index{recursion!primitive|(}   1982   1983 Datatypes come with a uniform way of defining functions, {\bf primitive   1984 recursion}. Such definitions rely on the recursion operator defined by the   1985 datatype package. Isabelle proves the desired recursion equations as   1986 theorems.   1987   1988 In principle, one could introduce primitive recursive functions by asserting   1989 their reduction rules as new axioms. Here is a dangerous way of defining the   1990 append function for lists:   1991 \begin{ttbox}\slshape   1992 consts "\at" :: [i,i]=>i (infixr 60)   1993 rules   1994 app_Nil "[] \at ys = ys"   1995 app_Cons "(Cons(a,l)) \at ys = Cons(a, l \at ys)"   1996 \end{ttbox}   1997 Asserting axioms brings the danger of accidentally asserting nonsense. It   1998 should be avoided at all costs!   1999   2000 The \ttindex{primrec} declaration is a safe means of defining primitive   2001 recursive functions on datatypes:   2002 \begin{ttbox}   2003 consts "\at" :: [i,i]=>i (infixr 60)   2004 primrec   2005 "[] \at ys = ys"   2006 "(Cons(a,l)) \at ys = Cons(a, l \at ys)"   2007 \end{ttbox}   2008 Isabelle will now check that the two rules do indeed form a primitive   2009 recursive definition. For example, the declaration   2010 \begin{ttbox}   2011 primrec   2012 "[] \at ys = us"   2013 \end{ttbox}   2014 is rejected with an error message \texttt{Extra variables on rhs}''.   2015   2016   2017 \subsubsection{Syntax of recursive definitions}   2018   2019 The general form of a primitive recursive definition is   2020 \begin{ttbox}   2021 primrec   2022 {\it reduction rules}   2023 \end{ttbox}   2024 where \textit{reduction rules} specify one or more equations of the form   2025 $f \, x@1 \, \dots \, x@m \, (C \, y@1 \, \dots \, y@k) \, z@1 \,   2026 \dots \, z@n = r$ such that$C$is a constructor of the datatype,$r$  2027 contains only the free variables on the left-hand side, and all recursive   2028 calls in$r$are of the form$f \, \dots \, y@i \, \dots$for some$i$.   2029 There must be at most one reduction rule for each constructor. The order is   2030 immaterial. For missing constructors, the function is defined to return zero.   2031   2032 All reduction rules are added to the default simpset.   2033 If you would like to refer to some rule by name, then you must prefix   2034 the rule with an identifier. These identifiers, like those in the   2035 \texttt{rules} section of a theory, will be visible at the \ML\ level.   2036   2037 The reduction rules for {\tt\at} become part of the default simpset, which   2038 leads to short proof scripts:   2039 \begin{ttbox}\underscoreon   2040 Goal "xs: list(A) ==> (xs @ ys) @ zs = xs @ (ys @ zs)";   2041 by (induct\_tac "xs" 1);   2042 by (ALLGOALS Asm\_simp\_tac);   2043 \end{ttbox}   2044   2045 You can even use the \texttt{primrec} form with non-recursive datatypes and   2046 with codatatypes. Recursion is not allowed, but it provides a convenient   2047 syntax for defining functions by cases.   2048   2049   2050 \subsubsection{Example: varying arguments}   2051   2052 All arguments, other than the recursive one, must be the same in each equation   2053 and in each recursive call. To get around this restriction, use explict   2054$\lambda$-abstraction and function application. Here is an example, drawn   2055 from the theory \texttt{Resid/Substitution}. The type of redexes is declared   2056 as follows:   2057 \begin{ttbox}   2058 consts redexes :: i   2059 datatype   2060 "redexes" = Var ("n: nat")   2061 | Fun ("t: redexes")   2062 | App ("b:bool" ,"f:redexes" , "a:redexes")   2063 \end{ttbox}   2064   2065 The function \texttt{lift} takes a second argument,$k$, which varies in   2066 recursive calls.   2067 \begin{ttbox}   2068 primrec   2069 "lift(Var(i)) = (lam k:nat. if i<k then Var(i) else Var(succ(i)))"   2070 "lift(Fun(t)) = (lam k:nat. Fun(lift(t)  succ(k)))"   2071 "lift(App(b,f,a)) = (lam k:nat. App(b, lift(f)k, lift(a)k))"   2072 \end{ttbox}   2073 Now \texttt{lift(r)k} satisfies the required recursion equations.   2074   2075 \index{recursion!primitive|)}   2076 \index{*primrec|)}   2077   2078   2079 \section{Inductive and coinductive definitions}   2080 \index{*inductive|(}   2081 \index{*coinductive|(}   2082   2083 An {\bf inductive definition} specifies the least set~$R$closed under given   2084 rules. (Applying a rule to elements of~$R$yields a result within~$R$.) For   2085 example, a structural operational semantics is an inductive definition of an   2086 evaluation relation. Dually, a {\bf coinductive definition} specifies the   2087 greatest set~$R$consistent with given rules. (Every element of~$R$can be   2088 seen as arising by applying a rule to elements of~$R$.) An important example   2089 is using bisimulation relations to formalise equivalence of processes and   2090 infinite data structures.   2091   2092 A theory file may contain any number of inductive and coinductive   2093 definitions. They may be intermixed with other declarations; in   2094 particular, the (co)inductive sets {\bf must} be declared separately as   2095 constants, and may have mixfix syntax or be subject to syntax translations.   2096   2097 Each (co)inductive definition adds definitions to the theory and also   2098 proves some theorems. Each definition creates an \ML\ structure, which is a   2099 substructure of the main theory structure.   2100 This package is described in detail in a separate paper,%   2101 \footnote{It appeared in CADE~\cite{paulson-CADE}; a longer version is   2102 distributed with Isabelle as \emph{A Fixedpoint Approach to   2103 (Co)Inductive and (Co)Datatype Definitions}.} %   2104 which you might refer to for background information.   2105   2106   2107 \subsection{The syntax of a (co)inductive definition}   2108 An inductive definition has the form   2109 \begin{ttbox}   2110 inductive   2111 domains {\it domain declarations}   2112 intrs {\it introduction rules}   2113 monos {\it monotonicity theorems}   2114 con_defs {\it constructor definitions}   2115 type_intrs {\it introduction rules for type-checking}   2116 type_elims {\it elimination rules for type-checking}   2117 \end{ttbox}   2118 A coinductive definition is identical, but starts with the keyword   2119 {\tt co\-inductive}.   2120   2121 The {\tt monos}, {\tt con\_defs}, {\tt type\_intrs} and {\tt type\_elims}   2122 sections are optional. If present, each is specified either as a list of   2123 identifiers or as a string. If the latter, then the string must be a valid   2124 \textsc{ml} expression of type {\tt thm list}. The string is simply inserted   2125 into the {\tt _thy.ML} file; if it is ill-formed, it will trigger \textsc{ml}   2126 error messages. You can then inspect the file on the temporary directory.   2127   2128 \begin{description}   2129 \item[\it domain declarations] are items of the form   2130 {\it string\/}~{\tt <=}~{\it string}, associating each recursive set with   2131 its domain. (The domain is some existing set that is large enough to   2132 hold the new set being defined.)   2133   2134 \item[\it introduction rules] specify one or more introduction rules in   2135 the form {\it ident\/}~{\it string}, where the identifier gives the name of   2136 the rule in the result structure.   2137   2138 \item[\it monotonicity theorems] are required for each operator applied to   2139 a recursive set in the introduction rules. There \textbf{must} be a theorem   2140 of the form$A\subseteq B\Imp M(A)\subseteq M(B)$, for each premise$t\in M(R_i)$  2141 in an introduction rule!   2142   2143 \item[\it constructor definitions] contain definitions of constants   2144 appearing in the introduction rules. The (co)datatype package supplies   2145 the constructors' definitions here. Most (co)inductive definitions omit   2146 this section; one exception is the primitive recursive functions example;   2147 see theory \texttt{ex/Primrec}.   2148   2149 \item[\it type\_intrs] consists of introduction rules for type-checking the   2150 definition: for demonstrating that the new set is included in its domain.   2151 (The proof uses depth-first search.)   2152   2153 \item[\it type\_elims] consists of elimination rules for type-checking the   2154 definition. They are presumed to be safe and are applied as often as   2155 possible prior to the {\tt type\_intrs} search.   2156 \end{description}   2157   2158 The package has a few restrictions:   2159 \begin{itemize}   2160 \item The theory must separately declare the recursive sets as   2161 constants.   2162   2163 \item The names of the recursive sets must be identifiers, not infix   2164 operators.   2165   2166 \item Side-conditions must not be conjunctions. However, an introduction rule   2167 may contain any number of side-conditions.   2168   2169 \item Side-conditions of the form$x=t$, where the variable~$x$does not   2170 occur in~$t$, will be substituted through the rule \verb|mutual_induct|.   2171 \end{itemize}   2172   2173   2174 \subsection{Example of an inductive definition}   2175   2176 Two declarations, included in a theory file, define the finite powerset   2177 operator. First we declare the constant~\texttt{Fin}. Then we declare it   2178 inductively, with two introduction rules:   2179 \begin{ttbox}   2180 consts Fin :: i=>i   2181   2182 inductive   2183 domains "Fin(A)" <= "Pow(A)"   2184 intrs   2185 emptyI "0 : Fin(A)"   2186 consI "[| a: A; b: Fin(A) |] ==> cons(a,b) : Fin(A)"   2187 type_intrs empty_subsetI, cons_subsetI, PowI   2188 type_elims "[make_elim PowD]"   2189 \end{ttbox}   2190 The resulting theory structure contains a substructure, called~\texttt{Fin}.   2191 It contains the \texttt{Fin}$~A$introduction rules as the list   2192 \texttt{Fin.intrs}, and also individually as \texttt{Fin.emptyI} and   2193 \texttt{Fin.consI}. The induction rule is \texttt{Fin.induct}.   2194   2195 The chief problem with making (co)inductive definitions involves type-checking   2196 the rules. Sometimes, additional theorems need to be supplied under   2197 \texttt{type_intrs} or \texttt{type_elims}. If the package fails when trying   2198 to prove your introduction rules, then set the flag \ttindexbold{trace_induct}   2199 to \texttt{true} and try again. (See the manual \emph{A Fixedpoint Approach   2200 \ldots} for more discussion of type-checking.)   2201   2202 In the example above,$\texttt{Pow}(A)$is given as the domain of   2203$\texttt{Fin}(A)$, for obviously every finite subset of~$A$is a subset   2204 of~$A$. However, the inductive definition package can only prove that given a   2205 few hints.   2206 Here is the output that results (with the flag set) when the   2207 \texttt{type_intrs} and \texttt{type_elims} are omitted from the inductive   2208 definition above:   2209 \begin{ttbox}   2210 Inductive definition Finite.Fin   2211 Fin(A) ==   2212 lfp(Pow(A),   2213 \%X. {z: Pow(A) . z = 0 | (EX a b. z = cons(a, b) & a : A & b : X)})   2214 Proving monotonicity...   2215 \ttbreak   2216 Proving the introduction rules...   2217 The type-checking subgoal:   2218 0 : Fin(A)   2219 1. 0 : Pow(A)   2220 \ttbreak   2221 The subgoal after monos, type_elims:   2222 0 : Fin(A)   2223 1. 0 : Pow(A)   2224 *** prove_goal: tactic failed   2225 \end{ttbox}   2226 We see the need to supply theorems to let the package prove   2227$\emptyset\in\texttt{Pow}(A)$. Restoring the \texttt{type_intrs} but not the   2228 \texttt{type_elims}, we again get an error message:   2229 \begin{ttbox}   2230 The type-checking subgoal:   2231 0 : Fin(A)   2232 1. 0 : Pow(A)   2233 \ttbreak   2234 The subgoal after monos, type_elims:   2235 0 : Fin(A)   2236 1. 0 : Pow(A)   2237 \ttbreak   2238 The type-checking subgoal:   2239 cons(a, b) : Fin(A)   2240 1. [| a : A; b : Fin(A) |] ==> cons(a, b) : Pow(A)   2241 \ttbreak   2242 The subgoal after monos, type_elims:   2243 cons(a, b) : Fin(A)   2244 1. [| a : A; b : Pow(A) |] ==> cons(a, b) : Pow(A)   2245 *** prove_goal: tactic failed   2246 \end{ttbox}   2247 The first rule has been type-checked, but the second one has failed. The   2248 simplest solution to such problems is to prove the failed subgoal separately   2249 and to supply it under \texttt{type_intrs}. The solution actually used is   2250 to supply, under \texttt{type_elims}, a rule that changes   2251$b\in\texttt{Pow}(A)$to$b\subseteq A$; together with \texttt{cons_subsetI}   2252 and \texttt{PowI}, it is enough to complete the type-checking.   2253   2254   2255   2256 \subsection{Further examples}   2257   2258 An inductive definition may involve arbitrary monotonic operators. Here is a   2259 standard example: the accessible part of a relation. Note the use   2260 of~\texttt{Pow} in the introduction rule and the corresponding mention of the   2261 rule \verb|Pow_mono| in the \texttt{monos} list. If the desired rule has a   2262 universally quantified premise, usually the effect can be obtained using   2263 \texttt{Pow}.   2264 \begin{ttbox}   2265 consts acc :: i=>i   2266 inductive   2267 domains "acc(r)" <= "field(r)"   2268 intrs   2269 vimage "[| r-{a}: Pow(acc(r)); a: field(r) |] ==> a: acc(r)"   2270 monos Pow_mono   2271 \end{ttbox}   2272   2273 Finally, here is a coinductive definition. It captures (as a bisimulation)   2274 the notion of equality on lazy lists, which are first defined as a codatatype:   2275 \begin{ttbox}   2276 consts llist :: i=>i   2277 codatatype "llist(A)" = LNil | LCons ("a: A", "l: llist(A)")   2278 \ttbreak   2279   2280 consts lleq :: i=>i   2281 coinductive   2282 domains "lleq(A)" <= "llist(A) * llist(A)"   2283 intrs   2284 LNil "<LNil, LNil> : lleq(A)"   2285 LCons "[| a:A; <l,l'>: lleq(A) |]   2286 ==> <LCons(a,l), LCons(a,l')>: lleq(A)"   2287 type_intrs "llist.intrs"   2288 \end{ttbox}   2289 This use of \texttt{type_intrs} is typical: the relation concerns the   2290 codatatype \texttt{llist}, so naturally the introduction rules for that   2291 codatatype will be required for type-checking the rules.   2292   2293 The Isabelle distribution contains many other inductive definitions. Simple   2294 examples are collected on subdirectory \texttt{ZF/ex}. The directory   2295 \texttt{Coind} and the theory \texttt{ZF/ex/LList} contain coinductive   2296 definitions. Larger examples may be found on other subdirectories of   2297 \texttt{ZF}, such as \texttt{IMP}, and \texttt{Resid}.   2298   2299   2300 \subsection{The result structure}   2301   2302 Each (co)inductive set defined in a theory file generates an \ML\ substructure   2303 having the same name. The the substructure contains the following elements:   2304   2305 \begin{ttbox}   2306 val intrs : thm list \textrm{the introduction rules}   2307 val elim : thm \textrm{the elimination (case analysis) rule}   2308 val mk_cases : string -> thm \textrm{case analysis, see below}   2309 val induct : thm \textrm{the standard induction rule}   2310 val mutual_induct : thm \textrm{the mutual induction rule, or \texttt{True}}   2311 val defs : thm list \textrm{definitions of operators}   2312 val bnd_mono : thm list \textrm{monotonicity property}   2313 val dom_subset : thm list \textrm{inclusion in bounding set'}   2314 \end{ttbox}   2315 Furthermore there is the theorem$C$\texttt{_I} for every constructor~$C$; for   2316 example, the \texttt{list} datatype's introduction rules are bound to the   2317 identifiers \texttt{Nil_I} and \texttt{Cons_I}.   2318   2319 For a codatatype, the component \texttt{coinduct} is the coinduction rule,   2320 replacing the \texttt{induct} component.   2321   2322 Recall that \ttindex{mk_cases} generates simplified instances of the   2323 elimination (case analysis) rule. It is as useful for inductive definitions   2324 as it is for datatypes. There are many examples in the theory   2325 \texttt{ex/Comb}, which is discussed at length   2326 elsewhere~\cite{paulson-generic}. The theory first defines the datatype   2327 \texttt{comb} of combinators:   2328 \begin{ttbox}   2329 consts comb :: i   2330 datatype "comb" = K   2331 | S   2332 | "#" ("p: comb", "q: comb") (infixl 90)   2333 \end{ttbox}   2334 The theory goes on to define contraction and parallel contraction   2335 inductively. Then the file \texttt{ex/Comb.ML} defines special cases of   2336 contraction using \texttt{mk_cases}:   2337 \begin{ttbox}   2338 val K_contractE = contract.mk_cases "K -1-> r";   2339 {\out val K_contractE = "K -1-> ?r ==> ?Q" : thm}   2340 \end{ttbox}   2341 We can read this as saying that the combinator \texttt{K} cannot reduce to   2342 anything. Similar elimination rules for \texttt{S} and application are also   2343 generated and are supplied to the classical reasoner. Note that   2344 \texttt{comb.con_defs} is given to \texttt{mk_cases} to allow freeness   2345 reasoning on datatype \texttt{comb}.   2346   2347 \index{*coinductive|)} \index{*inductive|)}   2348   2349   2350   2351   2352 \section{The outer reaches of set theory}   2353   2354 The constructions of the natural numbers and lists use a suite of   2355 operators for handling recursive function definitions. I have described   2356 the developments in detail elsewhere~\cite{paulson-set-II}. Here is a brief   2357 summary:   2358 \begin{itemize}   2359 \item Theory \texttt{Trancl} defines the transitive closure of a relation   2360 (as a least fixedpoint).   2361   2362 \item Theory \texttt{WF} proves the Well-Founded Recursion Theorem, using an   2363 elegant approach of Tobias Nipkow. This theorem permits general   2364 recursive definitions within set theory.   2365   2366 \item Theory \texttt{Ord} defines the notions of transitive set and ordinal   2367 number. It derives transfinite induction. A key definition is {\bf   2368 less than}:$i<j$if and only if$i$and$j$are both ordinals and   2369$i\in j$. As a special case, it includes less than on the natural   2370 numbers.   2371   2372 \item Theory \texttt{Epsilon} derives$\varepsilon$-induction and   2373$\varepsilon$-recursion, which are generalisations of transfinite   2374 induction and recursion. It also defines \cdx{rank}$(x)$, which   2375 is the least ordinal$\alpha$such that$x$is constructed at   2376 stage$\alpha$of the cumulative hierarchy (thus$x\in

  2377     V@{\alpha+1}$).   2378 \end{itemize}   2379   2380 Other important theories lead to a theory of cardinal numbers. They have   2381 not yet been written up anywhere. Here is a summary:   2382 \begin{itemize}   2383 \item Theory \texttt{Rel} defines the basic properties of relations, such as   2384 (ir)reflexivity, (a)symmetry, and transitivity.   2385   2386 \item Theory \texttt{EquivClass} develops a theory of equivalence   2387 classes, not using the Axiom of Choice.   2388   2389 \item Theory \texttt{Order} defines partial orderings, total orderings and   2390 wellorderings.   2391   2392 \item Theory \texttt{OrderArith} defines orderings on sum and product sets.   2393 These can be used to define ordinal arithmetic and have applications to   2394 cardinal arithmetic.   2395   2396 \item Theory \texttt{OrderType} defines order types. Every wellordering is   2397 equivalent to a unique ordinal, which is its order type.   2398   2399 \item Theory \texttt{Cardinal} defines equipollence and cardinal numbers.   2400   2401 \item Theory \texttt{CardinalArith} defines cardinal addition and   2402 multiplication, and proves their elementary laws. It proves that there   2403 is no greatest cardinal. It also proves a deep result, namely   2404$\kappa\otimes\kappa=\kappa$for every infinite cardinal~$\kappa$; see   2405 Kunen~\cite[page 29]{kunen80}. None of these results assume the Axiom of   2406 Choice, which complicates their proofs considerably.   2407 \end{itemize}   2408   2409 The following developments involve the Axiom of Choice (AC):   2410 \begin{itemize}   2411 \item Theory \texttt{AC} asserts the Axiom of Choice and proves some simple   2412 equivalent forms.   2413   2414 \item Theory \texttt{Zorn} proves Hausdorff's Maximal Principle, Zorn's Lemma   2415 and the Wellordering Theorem, following Abrial and   2416 Laffitte~\cite{abrial93}.   2417   2418 \item Theory \verb|Cardinal_AC| uses AC to prove simplified theorems about   2419 the cardinals. It also proves a theorem needed to justify   2420 infinitely branching datatype declarations: if$\kappa$is an infinite   2421 cardinal and$|X(\alpha)| \le \kappa$for all$\alpha<\kappa$then   2422$|\union\sb{\alpha<\kappa} X(\alpha)| \le \kappa$.   2423   2424 \item Theory \texttt{InfDatatype} proves theorems to justify infinitely   2425 branching datatypes. Arbitrary index sets are allowed, provided their   2426 cardinalities have an upper bound. The theory also justifies some   2427 unusual cases of finite branching, involving the finite powerset operator   2428 and the finite function space operator.   2429 \end{itemize}   2430   2431   2432   2433 \section{The examples directories}   2434 Directory \texttt{HOL/IMP} contains a mechanised version of a semantic   2435 equivalence proof taken from Winskel~\cite{winskel93}. It formalises the   2436 denotational and operational semantics of a simple while-language, then   2437 proves the two equivalent. It contains several datatype and inductive   2438 definitions, and demonstrates their use.   2439   2440 The directory \texttt{ZF/ex} contains further developments in ZF set theory.   2441 Here is an overview; see the files themselves for more details. I describe   2442 much of this material in other   2443 publications~\cite{paulson-set-I,paulson-set-II,paulson-CADE}.   2444 \begin{itemize}   2445 \item File \texttt{misc.ML} contains miscellaneous examples such as   2446 Cantor's Theorem, the Schr\"oder-Bernstein Theorem and the Composition   2447 of homomorphisms' challenge~\cite{boyer86}.   2448   2449 \item Theory \texttt{Ramsey} proves the finite exponent 2 version of   2450 Ramsey's Theorem, following Basin and Kaufmann's   2451 presentation~\cite{basin91}.   2452   2453 \item Theory \texttt{Integ} develops a theory of the integers as   2454 equivalence classes of pairs of natural numbers.   2455   2456 \item Theory \texttt{Primrec} develops some computation theory. It   2457 inductively defines the set of primitive recursive functions and presents a   2458 proof that Ackermann's function is not primitive recursive.   2459   2460 \item Theory \texttt{Primes} defines the Greatest Common Divisor of two   2461 natural numbers and and the divides'' relation.   2462   2463 \item Theory \texttt{Bin} defines a datatype for two's complement binary   2464 integers, then proves rewrite rules to perform binary arithmetic. For   2465 instance,$1359\times {-}2468 = {-}3354012$takes under 14 seconds.   2466   2467 \item Theory \texttt{BT} defines the recursive data structure${\tt

  2468     bt}(A)$, labelled binary trees.   2469   2470 \item Theory \texttt{Term} defines a recursive data structure for terms   2471 and term lists. These are simply finite branching trees.   2472   2473 \item Theory \texttt{TF} defines primitives for solving mutually   2474 recursive equations over sets. It constructs sets of trees and forests   2475 as an example, including induction and recursion rules that handle the   2476 mutual recursion.   2477   2478 \item Theory \texttt{Prop} proves soundness and completeness of   2479 propositional logic~\cite{paulson-set-II}. This illustrates datatype   2480 definitions, inductive definitions, structural induction and rule   2481 induction.   2482   2483 \item Theory \texttt{ListN} inductively defines the lists of$n$  2484 elements~\cite{paulin-tlca}.   2485   2486 \item Theory \texttt{Acc} inductively defines the accessible part of a   2487 relation~\cite{paulin-tlca}.   2488   2489 \item Theory \texttt{Comb} defines the datatype of combinators and   2490 inductively defines contraction and parallel contraction. It goes on to   2491 prove the Church-Rosser Theorem. This case study follows Camilleri and   2492 Melham~\cite{camilleri92}.   2493   2494 \item Theory \texttt{LList} defines lazy lists and a coinduction   2495 principle for proving equations between them.   2496 \end{itemize}   2497   2498   2499 \section{A proof about powersets}\label{sec:ZF-pow-example}   2500 To demonstrate high-level reasoning about subsets, let us prove the   2501 equation${{\tt Pow}(A)\cap {\tt Pow}(B)}= {\tt Pow}(A\cap B)$. Compared   2502 with first-order logic, set theory involves a maze of rules, and theorems   2503 have many different proofs. Attempting other proofs of the theorem might   2504 be instructive. This proof exploits the lattice properties of   2505 intersection. It also uses the monotonicity of the powerset operation,   2506 from \texttt{ZF/mono.ML}:   2507 \begin{ttbox}   2508 \tdx{Pow_mono} A<=B ==> Pow(A) <= Pow(B)   2509 \end{ttbox}   2510 We enter the goal and make the first step, which breaks the equation into   2511 two inclusions by extensionality:\index{*equalityI theorem}   2512 \begin{ttbox}   2513 Goal "Pow(A Int B) = Pow(A) Int Pow(B)";   2514 {\out Level 0}   2515 {\out Pow(A Int B) = Pow(A) Int Pow(B)}   2516 {\out 1. Pow(A Int B) = Pow(A) Int Pow(B)}   2517 \ttbreak   2518 by (resolve_tac [equalityI] 1);   2519 {\out Level 1}   2520 {\out Pow(A Int B) = Pow(A) Int Pow(B)}   2521 {\out 1. Pow(A Int B) <= Pow(A) Int Pow(B)}   2522 {\out 2. Pow(A) Int Pow(B) <= Pow(A Int B)}   2523 \end{ttbox}   2524 Both inclusions could be tackled straightforwardly using \texttt{subsetI}.   2525 A shorter proof results from noting that intersection forms the greatest   2526 lower bound:\index{*Int_greatest theorem}   2527 \begin{ttbox}   2528 by (resolve_tac [Int_greatest] 1);   2529 {\out Level 2}   2530 {\out Pow(A Int B) = Pow(A) Int Pow(B)}   2531 {\out 1. Pow(A Int B) <= Pow(A)}   2532 {\out 2. Pow(A Int B) <= Pow(B)}   2533 {\out 3. Pow(A) Int Pow(B) <= Pow(A Int B)}   2534 \end{ttbox}   2535 Subgoal~1 follows by applying the monotonicity of \texttt{Pow} to$A\int

  2536 B\subseteq A$; subgoal~2 follows similarly:   2537 \index{*Int_lower1 theorem}\index{*Int_lower2 theorem}   2538 \begin{ttbox}   2539 by (resolve_tac [Int_lower1 RS Pow_mono] 1);   2540 {\out Level 3}   2541 {\out Pow(A Int B) = Pow(A) Int Pow(B)}   2542 {\out 1. Pow(A Int B) <= Pow(B)}   2543 {\out 2. Pow(A) Int Pow(B) <= Pow(A Int B)}   2544 \ttbreak   2545 by (resolve_tac [Int_lower2 RS Pow_mono] 1);   2546 {\out Level 4}   2547 {\out Pow(A Int B) = Pow(A) Int Pow(B)}   2548 {\out 1. Pow(A) Int Pow(B) <= Pow(A Int B)}   2549 \end{ttbox}   2550 We are left with the opposite inclusion, which we tackle in the   2551 straightforward way:\index{*subsetI theorem}   2552 \begin{ttbox}   2553 by (resolve_tac [subsetI] 1);   2554 {\out Level 5}   2555 {\out Pow(A Int B) = Pow(A) Int Pow(B)}   2556 {\out 1. !!x. x : Pow(A) Int Pow(B) ==> x : Pow(A Int B)}   2557 \end{ttbox}   2558 The subgoal is to show$x\in {\tt Pow}(A\cap B)$assuming$x\in{\tt

  2559 Pow}(A)\cap {\tt Pow}(B)$; eliminating this assumption produces two   2560 subgoals. The rule \tdx{IntE} treats the intersection like a conjunction   2561 instead of unfolding its definition.   2562 \begin{ttbox}   2563 by (eresolve_tac [IntE] 1);   2564 {\out Level 6}   2565 {\out Pow(A Int B) = Pow(A) Int Pow(B)}   2566 {\out 1. !!x. [| x : Pow(A); x : Pow(B) |] ==> x : Pow(A Int B)}   2567 \end{ttbox}   2568 The next step replaces the \texttt{Pow} by the subset   2569 relation~($\subseteq$).\index{*PowI theorem}   2570 \begin{ttbox}   2571 by (resolve_tac [PowI] 1);   2572 {\out Level 7}   2573 {\out Pow(A Int B) = Pow(A) Int Pow(B)}   2574 {\out 1. !!x. [| x : Pow(A); x : Pow(B) |] ==> x <= A Int B}   2575 \end{ttbox}   2576 We perform the same replacement in the assumptions. This is a good   2577 demonstration of the tactic \ttindex{dresolve_tac}:\index{*PowD theorem}   2578 \begin{ttbox}   2579 by (REPEAT (dresolve_tac [PowD] 1));   2580 {\out Level 8}   2581 {\out Pow(A Int B) = Pow(A) Int Pow(B)}   2582 {\out 1. !!x. [| x <= A; x <= B |] ==> x <= A Int B}   2583 \end{ttbox}   2584 The assumptions are that$x$is a lower bound of both$A$and~$B$, but   2585$A\int B$is the greatest lower bound:\index{*Int_greatest theorem}   2586 \begin{ttbox}   2587 by (resolve_tac [Int_greatest] 1);   2588 {\out Level 9}   2589 {\out Pow(A Int B) = Pow(A) Int Pow(B)}   2590 {\out 1. !!x. [| x <= A; x <= B |] ==> x <= A}   2591 {\out 2. !!x. [| x <= A; x <= B |] ==> x <= B}   2592 \end{ttbox}   2593 To conclude the proof, we clear up the trivial subgoals:   2594 \begin{ttbox}   2595 by (REPEAT (assume_tac 1));   2596 {\out Level 10}   2597 {\out Pow(A Int B) = Pow(A) Int Pow(B)}   2598 {\out No subgoals!}   2599 \end{ttbox}   2600 \medskip   2601 We could have performed this proof in one step by applying   2602 \ttindex{Blast_tac}. Let us   2603 go back to the start:   2604 \begin{ttbox}   2605 choplev 0;   2606 {\out Level 0}   2607 {\out Pow(A Int B) = Pow(A) Int Pow(B)}   2608 {\out 1. Pow(A Int B) = Pow(A) Int Pow(B)}   2609 by (Blast_tac 1);   2610 {\out Depth = 0}   2611 {\out Depth = 1}   2612 {\out Depth = 2}   2613 {\out Depth = 3}   2614 {\out Level 1}   2615 {\out Pow(A Int B) = Pow(A) Int Pow(B)}   2616 {\out No subgoals!}   2617 \end{ttbox}   2618 Past researchers regarded this as a difficult proof, as indeed it is if all   2619 the symbols are replaced by their definitions.   2620 \goodbreak   2621   2622 \section{Monotonicity of the union operator}   2623 For another example, we prove that general union is monotonic:   2624${C\subseteq D}$implies$\bigcup(C)\subseteq \bigcup(D)$. To begin, we   2625 tackle the inclusion using \tdx{subsetI}:   2626 \begin{ttbox}   2627 Goal "C<=D ==> Union(C) <= Union(D)";   2628 {\out Level 0}   2629 {\out C <= D ==> Union(C) <= Union(D)}   2630 {\out 1. C <= D ==> Union(C) <= Union(D)}   2631 \ttbreak   2632 by (resolve_tac [subsetI] 1);   2633 {\out Level 1}   2634 {\out C <= D ==> Union(C) <= Union(D)}   2635 {\out 1. !!x. [| C <= D; x : Union(C) |] ==> x : Union(D)}   2636 \end{ttbox}   2637 Big union is like an existential quantifier --- the occurrence in the   2638 assumptions must be eliminated early, since it creates parameters.   2639 \index{*UnionE theorem}   2640 \begin{ttbox}   2641 by (eresolve_tac [UnionE] 1);   2642 {\out Level 2}   2643 {\out C <= D ==> Union(C) <= Union(D)}   2644 {\out 1. !!x B. [| C <= D; x : B; B : C |] ==> x : Union(D)}   2645 \end{ttbox}   2646 Now we may apply \tdx{UnionI}, which creates an unknown involving the   2647 parameters. To show$x\in \bigcup(D)$it suffices to show that$x$belongs   2648 to some element, say~$\Var{B2}(x,B)$, of~$D$.   2649 \begin{ttbox}   2650 by (resolve_tac [UnionI] 1);   2651 {\out Level 3}   2652 {\out C <= D ==> Union(C) <= Union(D)}   2653 {\out 1. !!x B. [| C <= D; x : B; B : C |] ==> ?B2(x,B) : D}   2654 {\out 2. !!x B. [| C <= D; x : B; B : C |] ==> x : ?B2(x,B)}   2655 \end{ttbox}   2656 Combining \tdx{subsetD} with the assumption$C\subseteq D$yields   2657$\Var{a}\in C \Imp \Var{a}\in D$, which reduces subgoal~1. Note that   2658 \texttt{eresolve_tac} has removed that assumption.   2659 \begin{ttbox}   2660 by (eresolve_tac [subsetD] 1);   2661 {\out Level 4}   2662 {\out C <= D ==> Union(C) <= Union(D)}   2663 {\out 1. !!x B. [| x : B; B : C |] ==> ?B2(x,B) : C}   2664 {\out 2. !!x B. [| C <= D; x : B; B : C |] ==> x : ?B2(x,B)}   2665 \end{ttbox}   2666 The rest is routine. Observe how~$\Var{B2}(x,B)$is instantiated.   2667 \begin{ttbox}   2668 by (assume_tac 1);   2669 {\out Level 5}   2670 {\out C <= D ==> Union(C) <= Union(D)}   2671 {\out 1. !!x B. [| C <= D; x : B; B : C |] ==> x : B}   2672 by (assume_tac 1);   2673 {\out Level 6}   2674 {\out C <= D ==> Union(C) <= Union(D)}   2675 {\out No subgoals!}   2676 \end{ttbox}   2677 Again, \ttindex{Blast_tac} can prove the theorem in one step.   2678 \begin{ttbox}   2679 by (Blast_tac 1);   2680 {\out Depth = 0}   2681 {\out Depth = 1}   2682 {\out Depth = 2}   2683 {\out Level 1}   2684 {\out C <= D ==> Union(C) <= Union(D)}   2685 {\out No subgoals!}   2686 \end{ttbox}   2687   2688 The file \texttt{ZF/equalities.ML} has many similar proofs. Reasoning about   2689 general intersection can be difficult because of its anomalous behaviour on   2690 the empty set. However, \ttindex{Blast_tac} copes well with these. Here is   2691 a typical example, borrowed from Devlin~\cite[page 12]{devlin79}:   2692 \begin{ttbox}   2693 a:C ==> (INT x:C. A(x) Int B(x)) = (INT x:C. A(x)) Int (INT x:C. B(x))   2694 \end{ttbox}   2695 In traditional notation this is   2696 $a\in C \,\Imp\, \inter@{x\in C} \Bigl(A(x) \int B(x)\Bigr) =   2697 \Bigl(\inter@{x\in C} A(x)\Bigr) \int   2698 \Bigl(\inter@{x\in C} B(x)\Bigr)$   2699   2700 \section{Low-level reasoning about functions}   2701 The derived rules \texttt{lamI}, \texttt{lamE}, \texttt{lam_type}, \texttt{beta}   2702 and \texttt{eta} support reasoning about functions in a   2703$\lambda$-calculus style. This is generally easier than regarding   2704 functions as sets of ordered pairs. But sometimes we must look at the   2705 underlying representation, as in the following proof   2706 of~\tdx{fun_disjoint_apply1}. This states that if$f$and~$g$are   2707 functions with disjoint domains~$A$and~$C$, and if$a\in A$, then   2708$(f\un g)a = fa$:   2709 \begin{ttbox}   2710 Goal "[| a:A; f: A->B; g: C->D; A Int C = 0 |] ==> \ttback   2711 \ttback (f Un g)a = fa";   2712 {\out Level 0}   2713 {\out [| a : A; f : A -> B; g : C -> D; A Int C = 0 |]}   2714 {\out ==> (f Un g)  a = f  a}   2715 {\out 1. [| a : A; f : A -> B; g : C -> D; A Int C = 0 |]}   2716 {\out ==> (f Un g)  a = f  a}   2717 \end{ttbox}   2718 Using \tdx{apply_equality}, we reduce the equality to reasoning about   2719 ordered pairs. The second subgoal is to verify that$f\un g$is a function.   2720 To save space, the assumptions will be abbreviated below.   2721 \begin{ttbox}   2722 by (resolve_tac [apply_equality] 1);   2723 {\out Level 1}   2724 {\out [| \ldots |] ==> (f Un g)  a = f  a}   2725 {\out 1. [| \ldots |] ==> <a,f  a> : f Un g}   2726 {\out 2. [| \ldots |] ==> f Un g : (PROD x:?A. ?B(x))}   2727 \end{ttbox}   2728 We must show that the pair belongs to~$f$or~$g$; by~\tdx{UnI1} we   2729 choose~$f$:   2730 \begin{ttbox}   2731 by (resolve_tac [UnI1] 1);   2732 {\out Level 2}   2733 {\out [| \ldots |] ==> (f Un g)  a = f  a}   2734 {\out 1. [| \ldots |] ==> <a,f  a> : f}   2735 {\out 2. [| \ldots |] ==> f Un g : (PROD x:?A. ?B(x))}   2736 \end{ttbox}   2737 To show$\pair{a,fa}\in f$we use \tdx{apply_Pair}, which is   2738 essentially the converse of \tdx{apply_equality}:   2739 \begin{ttbox}   2740 by (resolve_tac [apply_Pair] 1);   2741 {\out Level 3}   2742 {\out [| \ldots |] ==> (f Un g)  a = f  a}   2743 {\out 1. [| \ldots |] ==> f : (PROD x:?A2. ?B2(x))}   2744 {\out 2. [| \ldots |] ==> a : ?A2}   2745 {\out 3. [| \ldots |] ==> f Un g : (PROD x:?A. ?B(x))}   2746 \end{ttbox}   2747 Using the assumptions$f\in A\to B$and$a\in A$, we solve the two subgoals   2748 from \tdx{apply_Pair}. Recall that a$\Pi$-set is merely a generalized   2749 function space, and observe that~{\tt?A2} is instantiated to~\texttt{A}.   2750 \begin{ttbox}   2751 by (assume_tac 1);   2752 {\out Level 4}   2753 {\out [| \ldots |] ==> (f Un g)  a = f  a}   2754 {\out 1. [| \ldots |] ==> a : A}   2755 {\out 2. [| \ldots |] ==> f Un g : (PROD x:?A. ?B(x))}   2756 by (assume_tac 1);   2757 {\out Level 5}   2758 {\out [| \ldots |] ==> (f Un g)  a = f  a}   2759 {\out 1. [| \ldots |] ==> f Un g : (PROD x:?A. ?B(x))}   2760 \end{ttbox}   2761 To construct functions of the form$f\un g\$, we apply

  2762 \tdx{fun_disjoint_Un}:

  2763 \begin{ttbox}

  2764 by (resolve_tac [fun_disjoint_Un] 1);

  2765 {\out Level 6}

  2766 {\out [| \ldots |] ==> (f Un g)  a = f  a}

  2767 {\out  1. [| \ldots |] ==> f : ?A3 -> ?B3}

  2768 {\out  2. [| \ldots |] ==> g : ?C3 -> ?D3}

  2769 {\out  3. [| \ldots |] ==> ?A3 Int ?C3 = 0}

  2770 \end{ttbox}

  2771 The remaining subgoals are instances of the assumptions.  Again, observe how

  2772 unknowns are instantiated:

  2773 \begin{ttbox}

  2774 by (assume_tac 1);

  2775 {\out Level 7}

  2776 {\out [| \ldots |] ==> (f Un g)  a = f  a}

  2777 {\out  1. [| \ldots |] ==> g : ?C3 -> ?D3}

  2778 {\out  2. [| \ldots |] ==> A Int ?C3 = 0}

  2779 by (assume_tac 1);

  2780 {\out Level 8}

  2781 {\out [| \ldots |] ==> (f Un g)  a = f  a}

  2782 {\out  1. [| \ldots |] ==> A Int C = 0}

  2783 by (assume_tac 1);

  2784 {\out Level 9}

  2785 {\out [| \ldots |] ==> (f Un g)  a = f  a}

  2786 {\out No subgoals!}

  2787 \end{ttbox}

  2788 See the files \texttt{ZF/func.ML} and \texttt{ZF/WF.ML} for more

  2789 examples of reasoning about functions.

  2790

  2791 \index{set theory|)}
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