2 \chapter{Higher-Order Logic}
3 \index{higher-order logic|(}
4 \index{HOL system@{\sc hol} system}
6 The theory~\thydx{HOL} implements higher-order logic. It is based on
7 Gordon's~{\sc hol} system~\cite{mgordon-hol}, which itself is based on
8 Church's original paper~\cite{church40}. Andrews's
9 book~\cite{andrews86} is a full description of the original
10 Church-style higher-order logic. Experience with the {\sc hol} system
11 has demonstrated that higher-order logic is widely applicable in many
12 areas of mathematics and computer science, not just hardware
13 verification, {\sc hol}'s original \textit{raison d'\^etre\/}. It is
14 weaker than {\ZF} set theory but for most applications this does not
15 matter. If you prefer {\ML} to Lisp, you will probably prefer \HOL\
18 The syntax of \HOL\footnote{Earlier versions of Isabelle's \HOL\ used a
19 different syntax. Ancient releases of Isabelle included still another version
20 of~\HOL, with explicit type inference rules~\cite{paulson-COLOG}. This
21 version no longer exists, but \thydx{ZF} supports a similar style of
22 reasoning.} follows $\lambda$-calculus and functional programming. Function
23 application is curried. To apply the function~$f$ of type
24 $\tau@1\To\tau@2\To\tau@3$ to the arguments~$a$ and~$b$ in \HOL, you simply
25 write $f\,a\,b$. There is no `apply' operator as in \thydx{ZF}. Note that
26 $f(a,b)$ means ``$f$ applied to the pair $(a,b)$'' in \HOL. We write ordered
27 pairs as $(a,b)$, not $\langle a,b\rangle$ as in {\ZF}.
29 \HOL\ has a distinct feel, compared with {\ZF} and {\CTT}. It
30 identifies object-level types with meta-level types, taking advantage of
31 Isabelle's built-in type-checker. It identifies object-level functions
32 with meta-level functions, so it uses Isabelle's operations for abstraction
35 These identifications allow Isabelle to support \HOL\ particularly
36 nicely, but they also mean that \HOL\ requires more sophistication
37 from the user --- in particular, an understanding of Isabelle's type
38 system. Beginners should work with \texttt{show_types} (or even
39 \texttt{show_sorts}) set to \texttt{true}.
41 %working in first-order logic before attempting to use higher-order logic.
42 %This chapter assumes familiarity with~{\FOL{}}.
47 \it name &\it meta-type & \it description \\
48 \cdx{Trueprop}& $bool\To prop$ & coercion to $prop$\\
49 \cdx{Not} & $bool\To bool$ & negation ($\neg$) \\
50 \cdx{True} & $bool$ & tautology ($\top$) \\
51 \cdx{False} & $bool$ & absurdity ($\bot$) \\
52 \cdx{If} & $[bool,\alpha,\alpha]\To\alpha$ & conditional \\
53 \cdx{Let} & $[\alpha,\alpha\To\beta]\To\beta$ & let binder
55 \subcaption{Constants}
58 \index{"@@{\tt\at} symbol}
59 \index{*"! symbol}\index{*"? symbol}
60 \index{*"?"! symbol}\index{*"E"X"! symbol}
61 \it symbol &\it name &\it meta-type & \it description \\
62 \sdx{SOME} or \tt\at & \cdx{Eps} & $(\alpha\To bool)\To\alpha$ &
63 Hilbert description ($\varepsilon$) \\
64 \sdx{ALL} or {\tt!~} & \cdx{All} & $(\alpha\To bool)\To bool$ &
65 universal quantifier ($\forall$) \\
66 \sdx{EX} or {\tt?~} & \cdx{Ex} & $(\alpha\To bool)\To bool$ &
67 existential quantifier ($\exists$) \\
68 \texttt{EX!} or {\tt?!} & \cdx{Ex1} & $(\alpha\To bool)\To bool$ &
69 unique existence ($\exists!$)\\
70 \texttt{LEAST} & \cdx{Least} & $(\alpha::ord \To bool)\To\alpha$ &
77 \index{&@{\tt\&} symbol}
79 \index{*"-"-"> symbol}
80 \it symbol & \it meta-type & \it priority & \it description \\
81 \sdx{o} & $[\beta\To\gamma,\alpha\To\beta]\To (\alpha\To\gamma)$ &
82 Left 55 & composition ($\circ$) \\
83 \tt = & $[\alpha,\alpha]\To bool$ & Left 50 & equality ($=$) \\
84 \tt < & $[\alpha::ord,\alpha]\To bool$ & Left 50 & less than ($<$) \\
85 \tt <= & $[\alpha::ord,\alpha]\To bool$ & Left 50 &
86 less than or equals ($\leq$)\\
87 \tt \& & $[bool,bool]\To bool$ & Right 35 & conjunction ($\conj$) \\
88 \tt | & $[bool,bool]\To bool$ & Right 30 & disjunction ($\disj$) \\
89 \tt --> & $[bool,bool]\To bool$ & Right 25 & implication ($\imp$)
92 \caption{Syntax of \texttt{HOL}} \label{hol-constants}
100 \[\begin{array}{rclcl}
101 term & = & \hbox{expression of class~$term$} \\
102 & | & "SOME~" id " . " formula
103 & | & "\at~" id " . " formula \\
105 \multicolumn{3}{l}{"let"~id~"="~term";"\dots";"~id~"="~term~"in"~term} \\
107 \multicolumn{3}{l}{"if"~formula~"then"~term~"else"~term} \\
108 & | & "LEAST"~ id " . " formula \\[2ex]
109 formula & = & \hbox{expression of type~$bool$} \\
110 & | & term " = " term \\
111 & | & term " \ttilde= " term \\
112 & | & term " < " term \\
113 & | & term " <= " term \\
114 & | & "\ttilde\ " formula \\
115 & | & formula " \& " formula \\
116 & | & formula " | " formula \\
117 & | & formula " --> " formula \\
118 & | & "ALL~" id~id^* " . " formula
119 & | & "!~~~" id~id^* " . " formula \\
120 & | & "EX~~" id~id^* " . " formula
121 & | & "?~~~" id~id^* " . " formula \\
122 & | & "EX!~" id~id^* " . " formula
123 & | & "?!~~" id~id^* " . " formula \\
126 \caption{Full grammar for \HOL} \label{hol-grammar}
132 Figure~\ref{hol-constants} lists the constants (including infixes and
133 binders), while Fig.\ts\ref{hol-grammar} presents the grammar of
134 higher-order logic. Note that $a$\verb|~=|$b$ is translated to
138 \HOL\ has no if-and-only-if connective; logical equivalence is expressed
139 using equality. But equality has a high priority, as befitting a
140 relation, while if-and-only-if typically has the lowest priority. Thus,
141 $\neg\neg P=P$ abbreviates $\neg\neg (P=P)$ and not $(\neg\neg P)=P$.
142 When using $=$ to mean logical equivalence, enclose both operands in
146 \subsection{Types and classes}
147 The universal type class of higher-order terms is called~\cldx{term}.
148 By default, explicit type variables have class \cldx{term}. In
149 particular the equality symbol and quantifiers are polymorphic over
152 The type of formulae, \tydx{bool}, belongs to class \cldx{term}; thus,
153 formulae are terms. The built-in type~\tydx{fun}, which constructs
154 function types, is overloaded with arity {\tt(term,\thinspace
155 term)\thinspace term}. Thus, $\sigma\To\tau$ belongs to class~{\tt
156 term} if $\sigma$ and~$\tau$ do, allowing quantification over
159 \HOL\ offers various methods for introducing new types.
160 See~\S\ref{sec:HOL:Types} and~\S\ref{sec:HOL:datatype}.
162 Theory \thydx{Ord} defines the syntactic class \cldx{ord} of order
163 signatures; the relations $<$ and $\leq$ are polymorphic over this
164 class, as are the functions \cdx{mono}, \cdx{min} and \cdx{max}, and
165 the \cdx{LEAST} operator. \thydx{Ord} also defines a subclass
166 \cldx{order} of \cldx{ord} which axiomatizes partially ordered types
169 Three other syntactic type classes --- \cldx{plus}, \cldx{minus} and
170 \cldx{times} --- permit overloading of the operators {\tt+},\index{*"+
171 symbol} {\tt-}\index{*"- symbol} and {\tt*}.\index{*"* symbol} In
172 particular, {\tt-} is instantiated for set difference and subtraction
175 If you state a goal containing overloaded functions, you may need to include
176 type constraints. Type inference may otherwise make the goal more
177 polymorphic than you intended, with confusing results. For example, the
178 variables $i$, $j$ and $k$ in the goal $i \le j \Imp i \le j+k$ have type
179 $\alpha::\{ord,plus\}$, although you may have expected them to have some
180 numeric type, e.g. $nat$. Instead you should have stated the goal as
181 $(i::nat) \le j \Imp i \le j+k$, which causes all three variables to have
185 If resolution fails for no obvious reason, try setting
186 \ttindex{show_types} to \texttt{true}, causing Isabelle to display
187 types of terms. Possibly set \ttindex{show_sorts} to \texttt{true} as
188 well, causing Isabelle to display type classes and sorts.
190 \index{unification!incompleteness of}
191 Where function types are involved, Isabelle's unification code does not
192 guarantee to find instantiations for type variables automatically. Be
193 prepared to use \ttindex{res_inst_tac} instead of \texttt{resolve_tac},
194 possibly instantiating type variables. Setting
195 \ttindex{Unify.trace_types} to \texttt{true} causes Isabelle to report
196 omitted search paths during unification.\index{tracing!of unification}
202 Hilbert's {\bf description} operator~$\varepsilon x. P[x]$ stands for
203 some~$x$ satisfying~$P$, if such exists. Since all terms in \HOL\
204 denote something, a description is always meaningful, but we do not
205 know its value unless $P$ defines it uniquely. We may write
206 descriptions as \cdx{Eps}($\lambda x. P[x]$) or use the syntax
207 \hbox{\tt SOME~$x$.~$P[x]$}.
209 Existential quantification is defined by
210 \[ \exists x. P~x \;\equiv\; P(\varepsilon x. P~x). \]
211 The unique existence quantifier, $\exists!x. P$, is defined in terms
212 of~$\exists$ and~$\forall$. An Isabelle binder, it admits nested
213 quantifications. For instance, $\exists!x\,y. P\,x\,y$ abbreviates
214 $\exists!x. \exists!y. P\,x\,y$; note that this does not mean that there
215 exists a unique pair $(x,y)$ satisfying~$P\,x\,y$.
219 \index{*"! symbol}\index{*"? symbol}\index{HOL system@{\sc hol} system} The
220 basic Isabelle/HOL binders have two notations. Apart from the usual
221 \texttt{ALL} and \texttt{EX} for $\forall$ and $\exists$, Isabelle/HOL also
222 supports the original notation of Gordon's {\sc hol} system: \texttt{!}\
223 and~\texttt{?}. In the latter case, the existential quantifier \emph{must} be
224 followed by a space; thus {\tt?x} is an unknown, while \verb'? x. f x=y' is a
225 quantification. Both notations are accepted for input. The print mode
226 ``\ttindexbold{HOL}'' governs the output notation. If enabled (e.g.\ by
227 passing option \texttt{-m HOL} to the \texttt{isabelle} executable),
228 then~{\tt!}\ and~{\tt?}\ are displayed.
232 If $\tau$ is a type of class \cldx{ord}, $P$ a formula and $x$ a
233 variable of type $\tau$, then the term \cdx{LEAST}~$x. P[x]$ is defined
234 to be the least (w.r.t.\ $\le$) $x$ such that $P~x$ holds (see
235 Fig.~\ref{hol-defs}). The definition uses Hilbert's $\varepsilon$
236 choice operator, so \texttt{Least} is always meaningful, but may yield
237 nothing useful in case there is not a unique least element satisfying
238 $P$.\footnote{Class $ord$ does not require much of its instances, so
239 $\le$ need not be a well-ordering, not even an order at all!}
241 \medskip All these binders have priority 10.
244 The low priority of binders means that they need to be enclosed in
245 parenthesis when they occur in the context of other operations. For example,
246 instead of $P \land \forall x. Q$ you need to write $P \land (\forall x. Q)$.
250 \subsection{The let and case constructions}
251 Local abbreviations can be introduced by a \texttt{let} construct whose
252 syntax appears in Fig.\ts\ref{hol-grammar}. Internally it is translated into
253 the constant~\cdx{Let}. It can be expanded by rewriting with its
254 definition, \tdx{Let_def}.
256 \HOL\ also defines the basic syntax
257 \[\dquotes"case"~e~"of"~c@1~"=>"~e@1~"|" \dots "|"~c@n~"=>"~e@n\]
258 as a uniform means of expressing \texttt{case} constructs. Therefore \texttt{case}
259 and \sdx{of} are reserved words. Initially, this is mere syntax and has no
260 logical meaning. By declaring translations, you can cause instances of the
261 \texttt{case} construct to denote applications of particular case operators.
262 This is what happens automatically for each \texttt{datatype} definition
263 (see~\S\ref{sec:HOL:datatype}).
266 Both \texttt{if} and \texttt{case} constructs have as low a priority as
267 quantifiers, which requires additional enclosing parentheses in the context
268 of most other operations. For example, instead of $f~x = {\tt if\dots
269 then\dots else}\dots$ you need to write $f~x = ({\tt if\dots then\dots
273 \section{Rules of inference}
276 \begin{ttbox}\makeatother
277 \tdx{refl} t = (t::'a)
278 \tdx{subst} [| s = t; P s |] ==> P (t::'a)
279 \tdx{ext} (!!x::'a. (f x :: 'b) = g x) ==> (\%x. f x) = (\%x. g x)
280 \tdx{impI} (P ==> Q) ==> P-->Q
281 \tdx{mp} [| P-->Q; P |] ==> Q
282 \tdx{iff} (P-->Q) --> (Q-->P) --> (P=Q)
283 \tdx{selectI} P(x::'a) ==> P(@x. P x)
284 \tdx{True_or_False} (P=True) | (P=False)
286 \caption{The \texttt{HOL} rules} \label{hol-rules}
289 Figure~\ref{hol-rules} shows the primitive inference rules of~\HOL{},
290 with their~{\ML} names. Some of the rules deserve additional
292 \begin{ttdescription}
293 \item[\tdx{ext}] expresses extensionality of functions.
294 \item[\tdx{iff}] asserts that logically equivalent formulae are
296 \item[\tdx{selectI}] gives the defining property of the Hilbert
297 $\varepsilon$-operator. It is a form of the Axiom of Choice. The derived rule
298 \tdx{select_equality} (see below) is often easier to use.
299 \item[\tdx{True_or_False}] makes the logic classical.\footnote{In
300 fact, the $\varepsilon$-operator already makes the logic classical, as
301 shown by Diaconescu; see Paulson~\cite{paulson-COLOG} for details.}
305 \begin{figure}\hfuzz=4pt%suppress "Overfull \hbox" message
306 \begin{ttbox}\makeatother
307 \tdx{True_def} True == ((\%x::bool. x)=(\%x. x))
308 \tdx{All_def} All == (\%P. P = (\%x. True))
309 \tdx{Ex_def} Ex == (\%P. P(@x. P x))
310 \tdx{False_def} False == (!P. P)
311 \tdx{not_def} not == (\%P. P-->False)
312 \tdx{and_def} op & == (\%P Q. !R. (P-->Q-->R) --> R)
313 \tdx{or_def} op | == (\%P Q. !R. (P-->R) --> (Q-->R) --> R)
314 \tdx{Ex1_def} Ex1 == (\%P. ? x. P x & (! y. P y --> y=x))
316 \tdx{o_def} op o == (\%(f::'b=>'c) g x::'a. f(g x))
317 \tdx{if_def} If P x y ==
318 (\%P x y. @z::'a.(P=True --> z=x) & (P=False --> z=y))
319 \tdx{Let_def} Let s f == f s
320 \tdx{Least_def} Least P == @x. P(x) & (ALL y. P(y) --> x <= y)"
322 \caption{The \texttt{HOL} definitions} \label{hol-defs}
326 \HOL{} follows standard practice in higher-order logic: only a few
327 connectives are taken as primitive, with the remainder defined obscurely
328 (Fig.\ts\ref{hol-defs}). Gordon's {\sc hol} system expresses the
329 corresponding definitions \cite[page~270]{mgordon-hol} using
330 object-equality~({\tt=}), which is possible because equality in
331 higher-order logic may equate formulae and even functions over formulae.
332 But theory~\HOL{}, like all other Isabelle theories, uses
333 meta-equality~({\tt==}) for definitions.
335 The definitions above should never be expanded and are shown for completeness
336 only. Instead users should reason in terms of the derived rules shown below
337 or, better still, using high-level tactics
338 (see~\S\ref{sec:HOL:generic-packages}).
341 Some of the rules mention type variables; for example, \texttt{refl}
342 mentions the type variable~{\tt'a}. This allows you to instantiate
343 type variables explicitly by calling \texttt{res_inst_tac}.
348 \tdx{sym} s=t ==> t=s
349 \tdx{trans} [| r=s; s=t |] ==> r=t
350 \tdx{ssubst} [| t=s; P s |] ==> P t
351 \tdx{box_equals} [| a=b; a=c; b=d |] ==> c=d
352 \tdx{arg_cong} x = y ==> f x = f y
353 \tdx{fun_cong} f = g ==> f x = g x
354 \tdx{cong} [| f = g; x = y |] ==> f x = g y
355 \tdx{not_sym} t ~= s ==> s ~= t
356 \subcaption{Equality}
359 \tdx{FalseE} False ==> P
361 \tdx{conjI} [| P; Q |] ==> P&Q
362 \tdx{conjunct1} [| P&Q |] ==> P
363 \tdx{conjunct2} [| P&Q |] ==> Q
364 \tdx{conjE} [| P&Q; [| P; Q |] ==> R |] ==> R
366 \tdx{disjI1} P ==> P|Q
367 \tdx{disjI2} Q ==> P|Q
368 \tdx{disjE} [| P | Q; P ==> R; Q ==> R |] ==> R
370 \tdx{notI} (P ==> False) ==> ~ P
371 \tdx{notE} [| ~ P; P |] ==> R
372 \tdx{impE} [| P-->Q; P; Q ==> R |] ==> R
373 \subcaption{Propositional logic}
375 \tdx{iffI} [| P ==> Q; Q ==> P |] ==> P=Q
376 \tdx{iffD1} [| P=Q; P |] ==> Q
377 \tdx{iffD2} [| P=Q; Q |] ==> P
378 \tdx{iffE} [| P=Q; [| P --> Q; Q --> P |] ==> R |] ==> R
380 %\tdx{eqTrueI} P ==> P=True
381 %\tdx{eqTrueE} P=True ==> P
382 \subcaption{Logical equivalence}
385 \caption{Derived rules for \HOL} \label{hol-lemmas1}
390 \begin{ttbox}\makeatother
391 \tdx{allI} (!!x. P x) ==> !x. P x
392 \tdx{spec} !x. P x ==> P x
393 \tdx{allE} [| !x. P x; P x ==> R |] ==> R
394 \tdx{all_dupE} [| !x. P x; [| P x; !x. P x |] ==> R |] ==> R
396 \tdx{exI} P x ==> ? x. P x
397 \tdx{exE} [| ? x. P x; !!x. P x ==> Q |] ==> Q
399 \tdx{ex1I} [| P a; !!x. P x ==> x=a |] ==> ?! x. P x
400 \tdx{ex1E} [| ?! x. P x; !!x. [| P x; ! y. P y --> y=x |] ==> R
403 \tdx{select_equality} [| P a; !!x. P x ==> x=a |] ==> (@x. P x) = a
404 \subcaption{Quantifiers and descriptions}
406 \tdx{ccontr} (~P ==> False) ==> P
407 \tdx{classical} (~P ==> P) ==> P
408 \tdx{excluded_middle} ~P | P
410 \tdx{disjCI} (~Q ==> P) ==> P|Q
411 \tdx{exCI} (! x. ~ P x ==> P a) ==> ? x. P x
412 \tdx{impCE} [| P-->Q; ~ P ==> R; Q ==> R |] ==> R
413 \tdx{iffCE} [| P=Q; [| P;Q |] ==> R; [| ~P; ~Q |] ==> R |] ==> R
414 \tdx{notnotD} ~~P ==> P
415 \tdx{swap} ~P ==> (~Q ==> P) ==> Q
416 \subcaption{Classical logic}
418 %\tdx{if_True} (if True then x else y) = x
419 %\tdx{if_False} (if False then x else y) = y
420 \tdx{if_P} P ==> (if P then x else y) = x
421 \tdx{if_not_P} ~ P ==> (if P then x else y) = y
422 \tdx{split_if} P(if Q then x else y) = ((Q --> P x) & (~Q --> P y))
423 \subcaption{Conditionals}
425 \caption{More derived rules} \label{hol-lemmas2}
428 Some derived rules are shown in Figures~\ref{hol-lemmas1}
429 and~\ref{hol-lemmas2}, with their {\ML} names. These include natural rules
430 for the logical connectives, as well as sequent-style elimination rules for
431 conjunctions, implications, and universal quantifiers.
433 Note the equality rules: \tdx{ssubst} performs substitution in
434 backward proofs, while \tdx{box_equals} supports reasoning by
435 simplifying both sides of an equation.
437 The following simple tactics are occasionally useful:
438 \begin{ttdescription}
439 \item[\ttindexbold{strip_tac} $i$] applies \texttt{allI} and \texttt{impI}
440 repeatedly to remove all outermost universal quantifiers and implications
442 \item[\ttindexbold{case_tac} {\tt"}$P${\tt"} $i$] performs case distinction
443 on $P$ for subgoal $i$: the latter is replaced by two identical subgoals
444 with the added assumptions $P$ and $\neg P$, respectively.
451 \it name &\it meta-type & \it description \\
452 \index{{}@\verb'{}' symbol}
453 \verb|{}| & $\alpha\,set$ & the empty set \\
454 \cdx{insert} & $[\alpha,\alpha\,set]\To \alpha\,set$
455 & insertion of element \\
456 \cdx{Collect} & $(\alpha\To bool)\To\alpha\,set$
458 \cdx{Compl} & $\alpha\,set\To\alpha\,set$
460 \cdx{INTER} & $[\alpha\,set,\alpha\To\beta\,set]\To\beta\,set$
461 & intersection over a set\\
462 \cdx{UNION} & $[\alpha\,set,\alpha\To\beta\,set]\To\beta\,set$
464 \cdx{Inter} & $(\alpha\,set)set\To\alpha\,set$
465 &set of sets intersection \\
466 \cdx{Union} & $(\alpha\,set)set\To\alpha\,set$
467 &set of sets union \\
468 \cdx{Pow} & $\alpha\,set \To (\alpha\,set)set$
470 \cdx{range} & $(\alpha\To\beta )\To\beta\,set$
471 & range of a function \\[1ex]
472 \cdx{Ball}~~\cdx{Bex} & $[\alpha\,set,\alpha\To bool]\To bool$
473 & bounded quantifiers
476 \subcaption{Constants}
479 \begin{tabular}{llrrr}
480 \it symbol &\it name &\it meta-type & \it priority & \it description \\
481 \sdx{INT} & \cdx{INTER1} & $(\alpha\To\beta\,set)\To\beta\,set$ & 10 &
482 intersection over a type\\
483 \sdx{UN} & \cdx{UNION1} & $(\alpha\To\beta\,set)\To\beta\,set$ & 10 &
493 \begin{tabular}{rrrr}
494 \it symbol & \it meta-type & \it priority & \it description \\
495 \tt `` & $[\alpha\To\beta ,\alpha\,set]\To \beta\,set$
497 \sdx{Int} & $[\alpha\,set,\alpha\,set]\To\alpha\,set$
498 & Left 70 & intersection ($\int$) \\
499 \sdx{Un} & $[\alpha\,set,\alpha\,set]\To\alpha\,set$
500 & Left 65 & union ($\un$) \\
501 \tt: & $[\alpha ,\alpha\,set]\To bool$
502 & Left 50 & membership ($\in$) \\
503 \tt <= & $[\alpha\,set,\alpha\,set]\To bool$
504 & Left 50 & subset ($\subseteq$)
508 \caption{Syntax of the theory \texttt{Set}} \label{hol-set-syntax}
513 \begin{center} \tt\frenchspacing
516 \it external & \it internal & \it description \\
517 $a$ \ttilde: $b$ & \ttilde($a$ : $b$) & \rm non-membership\\
518 {\ttlbrace}$a@1$, $\ldots${\ttrbrace} & insert $a@1$ $\ldots$ {\ttlbrace}{\ttrbrace} & \rm finite set \\
519 {\ttlbrace}$x$. $P[x]${\ttrbrace} & Collect($\lambda x. P[x]$) &
521 \sdx{INT} $x$:$A$. $B[x]$ & INTER $A$ $\lambda x. B[x]$ &
523 \sdx{UN}{\tt\ } $x$:$A$. $B[x]$ & UNION $A$ $\lambda x. B[x]$ &
525 \sdx{ALL} $x$:$A$. $P[x]$ or \sdx{!} $x$:$A$. $P[x]$ &
526 Ball $A$ $\lambda x. P[x]$ &
527 \rm bounded $\forall$ \\
528 \sdx{EX}{\tt\ } $x$:$A$. $P[x]$ or \sdx{?} $x$:$A$. $P[x]$ &
529 Bex $A$ $\lambda x. P[x]$ & \rm bounded $\exists$
532 \subcaption{Translations}
535 \[\begin{array}{rclcl}
536 term & = & \hbox{other terms\ldots} \\
537 & | & "{\ttlbrace}{\ttrbrace}" \\
538 & | & "{\ttlbrace} " term\; ("," term)^* " {\ttrbrace}" \\
539 & | & "{\ttlbrace} " id " . " formula " {\ttrbrace}" \\
540 & | & term " `` " term \\
541 & | & term " Int " term \\
542 & | & term " Un " term \\
543 & | & "INT~~" id ":" term " . " term \\
544 & | & "UN~~~" id ":" term " . " term \\
545 & | & "INT~~" id~id^* " . " term \\
546 & | & "UN~~~" id~id^* " . " term \\[2ex]
547 formula & = & \hbox{other formulae\ldots} \\
548 & | & term " : " term \\
549 & | & term " \ttilde: " term \\
550 & | & term " <= " term \\
551 & | & "ALL " id ":" term " . " formula
552 & | & "!~" id ":" term " . " formula \\
553 & | & "EX~~" id ":" term " . " formula
554 & | & "?~" id ":" term " . " formula \\
557 \subcaption{Full Grammar}
558 \caption{Syntax of the theory \texttt{Set} (continued)} \label{hol-set-syntax2}
562 \section{A formulation of set theory}
563 Historically, higher-order logic gives a foundation for Russell and
564 Whitehead's theory of classes. Let us use modern terminology and call them
565 {\bf sets}, but note that these sets are distinct from those of {\ZF} set
566 theory, and behave more like {\ZF} classes.
569 Sets are given by predicates over some type~$\sigma$. Types serve to
570 define universes for sets, but type-checking is still significant.
572 There is a universal set (for each type). Thus, sets have complements, and
573 may be defined by absolute comprehension.
575 Although sets may contain other sets as elements, the containing set must
576 have a more complex type.
578 Finite unions and intersections have the same behaviour in \HOL\ as they
579 do in~{\ZF}. In \HOL\ the intersection of the empty set is well-defined,
580 denoting the universal set for the given type.
582 \subsection{Syntax of set theory}\index{*set type}
583 \HOL's set theory is called \thydx{Set}. The type $\alpha\,set$ is
584 essentially the same as $\alpha\To bool$. The new type is defined for
585 clarity and to avoid complications involving function types in unification.
586 The isomorphisms between the two types are declared explicitly. They are
587 very natural: \texttt{Collect} maps $\alpha\To bool$ to $\alpha\,set$, while
588 \hbox{\tt op :} maps in the other direction (ignoring argument order).
590 Figure~\ref{hol-set-syntax} lists the constants, infixes, and syntax
591 translations. Figure~\ref{hol-set-syntax2} presents the grammar of the new
592 constructs. Infix operators include union and intersection ($A\un B$
593 and $A\int B$), the subset and membership relations, and the image
594 operator~{\tt``}\@. Note that $a$\verb|~:|$b$ is translated to
597 The $\{a@1,\ldots\}$ notation abbreviates finite sets constructed in
598 the obvious manner using~\texttt{insert} and~$\{\}$:
600 \{a, b, c\} & \equiv &
601 \texttt{insert} \, a \, ({\tt insert} \, b \, ({\tt insert} \, c \, \{\}))
604 The set \hbox{\tt{\ttlbrace}$x$.\ $P[x]${\ttrbrace}} consists of all $x$ (of suitable type)
605 that satisfy~$P[x]$, where $P[x]$ is a formula that may contain free
606 occurrences of~$x$. This syntax expands to \cdx{Collect}$(\lambda
607 x. P[x])$. It defines sets by absolute comprehension, which is impossible
608 in~{\ZF}; the type of~$x$ implicitly restricts the comprehension.
610 The set theory defines two {\bf bounded quantifiers}:
612 \forall x\in A. P[x] &\hbox{abbreviates}& \forall x. x\in A\imp P[x] \\
613 \exists x\in A. P[x] &\hbox{abbreviates}& \exists x. x\in A\conj P[x]
615 The constants~\cdx{Ball} and~\cdx{Bex} are defined
616 accordingly. Instead of \texttt{Ball $A$ $P$} and \texttt{Bex $A$ $P$} we may
617 write\index{*"! symbol}\index{*"? symbol}
618 \index{*ALL symbol}\index{*EX symbol}
620 \hbox{\tt ALL~$x$:$A$.\ $P[x]$} and \hbox{\tt EX~$x$:$A$.\ $P[x]$}. The
621 original notation of Gordon's {\sc hol} system is supported as well: \sdx{!}\
624 Unions and intersections over sets, namely $\bigcup@{x\in A}B[x]$ and
625 $\bigcap@{x\in A}B[x]$, are written
626 \sdx{UN}~\hbox{\tt$x$:$A$.\ $B[x]$} and
627 \sdx{INT}~\hbox{\tt$x$:$A$.\ $B[x]$}.
629 Unions and intersections over types, namely $\bigcup@x B[x]$ and $\bigcap@x
630 B[x]$, are written \sdx{UN}~\hbox{\tt$x$.\ $B[x]$} and
631 \sdx{INT}~\hbox{\tt$x$.\ $B[x]$}. They are equivalent to the previous
632 union and intersection operators when $A$ is the universal set.
634 The operators $\bigcup A$ and $\bigcap A$ act upon sets of sets. They are
635 not binders, but are equal to $\bigcup@{x\in A}x$ and $\bigcap@{x\in A}x$,
640 \begin{figure} \underscoreon
642 \tdx{mem_Collect_eq} (a : {\ttlbrace}x. P x{\ttrbrace}) = P a
643 \tdx{Collect_mem_eq} {\ttlbrace}x. x:A{\ttrbrace} = A
645 \tdx{empty_def} {\ttlbrace}{\ttrbrace} == {\ttlbrace}x. False{\ttrbrace}
646 \tdx{insert_def} insert a B == {\ttlbrace}x. x=a{\ttrbrace} Un B
647 \tdx{Ball_def} Ball A P == ! x. x:A --> P x
648 \tdx{Bex_def} Bex A P == ? x. x:A & P x
649 \tdx{subset_def} A <= B == ! x:A. x:B
650 \tdx{Un_def} A Un B == {\ttlbrace}x. x:A | x:B{\ttrbrace}
651 \tdx{Int_def} A Int B == {\ttlbrace}x. x:A & x:B{\ttrbrace}
652 \tdx{set_diff_def} A - B == {\ttlbrace}x. x:A & x~:B{\ttrbrace}
653 \tdx{Compl_def} Compl A == {\ttlbrace}x. ~ x:A{\ttrbrace}
654 \tdx{INTER_def} INTER A B == {\ttlbrace}y. ! x:A. y: B x{\ttrbrace}
655 \tdx{UNION_def} UNION A B == {\ttlbrace}y. ? x:A. y: B x{\ttrbrace}
656 \tdx{INTER1_def} INTER1 B == INTER {\ttlbrace}x. True{\ttrbrace} B
657 \tdx{UNION1_def} UNION1 B == UNION {\ttlbrace}x. True{\ttrbrace} B
658 \tdx{Inter_def} Inter S == (INT x:S. x)
659 \tdx{Union_def} Union S == (UN x:S. x)
660 \tdx{Pow_def} Pow A == {\ttlbrace}B. B <= A{\ttrbrace}
661 \tdx{image_def} f``A == {\ttlbrace}y. ? x:A. y=f x{\ttrbrace}
662 \tdx{range_def} range f == {\ttlbrace}y. ? x. y=f x{\ttrbrace}
664 \caption{Rules of the theory \texttt{Set}} \label{hol-set-rules}
668 \begin{figure} \underscoreon
670 \tdx{CollectI} [| P a |] ==> a : {\ttlbrace}x. P x{\ttrbrace}
671 \tdx{CollectD} [| a : {\ttlbrace}x. P x{\ttrbrace} |] ==> P a
672 \tdx{CollectE} [| a : {\ttlbrace}x. P x{\ttrbrace}; P a ==> W |] ==> W
674 \tdx{ballI} [| !!x. x:A ==> P x |] ==> ! x:A. P x
675 \tdx{bspec} [| ! x:A. P x; x:A |] ==> P x
676 \tdx{ballE} [| ! x:A. P x; P x ==> Q; ~ x:A ==> Q |] ==> Q
678 \tdx{bexI} [| P x; x:A |] ==> ? x:A. P x
679 \tdx{bexCI} [| ! x:A. ~ P x ==> P a; a:A |] ==> ? x:A. P x
680 \tdx{bexE} [| ? x:A. P x; !!x. [| x:A; P x |] ==> Q |] ==> Q
681 \subcaption{Comprehension and Bounded quantifiers}
683 \tdx{subsetI} (!!x. x:A ==> x:B) ==> A <= B
684 \tdx{subsetD} [| A <= B; c:A |] ==> c:B
685 \tdx{subsetCE} [| A <= B; ~ (c:A) ==> P; c:B ==> P |] ==> P
687 \tdx{subset_refl} A <= A
688 \tdx{subset_trans} [| A<=B; B<=C |] ==> A<=C
690 \tdx{equalityI} [| A <= B; B <= A |] ==> A = B
691 \tdx{equalityD1} A = B ==> A<=B
692 \tdx{equalityD2} A = B ==> B<=A
693 \tdx{equalityE} [| A = B; [| A<=B; B<=A |] ==> P |] ==> P
695 \tdx{equalityCE} [| A = B; [| c:A; c:B |] ==> P;
696 [| ~ c:A; ~ c:B |] ==> P
698 \subcaption{The subset and equality relations}
700 \caption{Derived rules for set theory} \label{hol-set1}
704 \begin{figure} \underscoreon
706 \tdx{emptyE} a : {\ttlbrace}{\ttrbrace} ==> P
708 \tdx{insertI1} a : insert a B
709 \tdx{insertI2} a : B ==> a : insert b B
710 \tdx{insertE} [| a : insert b A; a=b ==> P; a:A ==> P |] ==> P
712 \tdx{ComplI} [| c:A ==> False |] ==> c : Compl A
713 \tdx{ComplD} [| c : Compl A |] ==> ~ c:A
715 \tdx{UnI1} c:A ==> c : A Un B
716 \tdx{UnI2} c:B ==> c : A Un B
717 \tdx{UnCI} (~c:B ==> c:A) ==> c : A Un B
718 \tdx{UnE} [| c : A Un B; c:A ==> P; c:B ==> P |] ==> P
720 \tdx{IntI} [| c:A; c:B |] ==> c : A Int B
721 \tdx{IntD1} c : A Int B ==> c:A
722 \tdx{IntD2} c : A Int B ==> c:B
723 \tdx{IntE} [| c : A Int B; [| c:A; c:B |] ==> P |] ==> P
725 \tdx{UN_I} [| a:A; b: B a |] ==> b: (UN x:A. B x)
726 \tdx{UN_E} [| b: (UN x:A. B x); !!x.[| x:A; b:B x |] ==> R |] ==> R
728 \tdx{INT_I} (!!x. x:A ==> b: B x) ==> b : (INT x:A. B x)
729 \tdx{INT_D} [| b: (INT x:A. B x); a:A |] ==> b: B a
730 \tdx{INT_E} [| b: (INT x:A. B x); b: B a ==> R; ~ a:A ==> R |] ==> R
732 \tdx{UnionI} [| X:C; A:X |] ==> A : Union C
733 \tdx{UnionE} [| A : Union C; !!X.[| A:X; X:C |] ==> R |] ==> R
735 \tdx{InterI} [| !!X. X:C ==> A:X |] ==> A : Inter C
736 \tdx{InterD} [| A : Inter C; X:C |] ==> A:X
737 \tdx{InterE} [| A : Inter C; A:X ==> R; ~ X:C ==> R |] ==> R
739 \tdx{PowI} A<=B ==> A: Pow B
740 \tdx{PowD} A: Pow B ==> A<=B
742 \tdx{imageI} [| x:A |] ==> f x : f``A
743 \tdx{imageE} [| b : f``A; !!x.[| b=f x; x:A |] ==> P |] ==> P
745 \tdx{rangeI} f x : range f
746 \tdx{rangeE} [| b : range f; !!x.[| b=f x |] ==> P |] ==> P
748 \caption{Further derived rules for set theory} \label{hol-set2}
752 \subsection{Axioms and rules of set theory}
753 Figure~\ref{hol-set-rules} presents the rules of theory \thydx{Set}. The
754 axioms \tdx{mem_Collect_eq} and \tdx{Collect_mem_eq} assert
755 that the functions \texttt{Collect} and \hbox{\tt op :} are isomorphisms. Of
756 course, \hbox{\tt op :} also serves as the membership relation.
758 All the other axioms are definitions. They include the empty set, bounded
759 quantifiers, unions, intersections, complements and the subset relation.
760 They also include straightforward constructions on functions: image~({\tt``})
763 %The predicate \cdx{inj_on} is used for simulating type definitions.
764 %The statement ${\tt inj_on}~f~A$ asserts that $f$ is injective on the
765 %set~$A$, which specifies a subset of its domain type. In a type
766 %definition, $f$ is the abstraction function and $A$ is the set of valid
767 %representations; we should not expect $f$ to be injective outside of~$A$.
769 %\begin{figure} \underscoreon
771 %\tdx{Inv_f_f} inj f ==> Inv f (f x) = x
772 %\tdx{f_Inv_f} y : range f ==> f(Inv f y) = y
775 % [| Inv f x=Inv f y; x: range f; y: range f |] ==> x=y
778 %\tdx{monoI} [| !!A B. A <= B ==> f A <= f B |] ==> mono f
779 %\tdx{monoD} [| mono f; A <= B |] ==> f A <= f B
781 %\tdx{injI} [| !! x y. f x = f y ==> x=y |] ==> inj f
782 %\tdx{inj_inverseI} (!!x. g(f x) = x) ==> inj f
783 %\tdx{injD} [| inj f; f x = f y |] ==> x=y
785 %\tdx{inj_onI} (!!x y. [| f x=f y; x:A; y:A |] ==> x=y) ==> inj_on f A
786 %\tdx{inj_onD} [| inj_on f A; f x=f y; x:A; y:A |] ==> x=y
788 %\tdx{inj_on_inverseI}
789 % (!!x. x:A ==> g(f x) = x) ==> inj_on f A
790 %\tdx{inj_on_contraD}
791 % [| inj_on f A; x~=y; x:A; y:A |] ==> ~ f x=f y
793 %\caption{Derived rules involving functions} \label{hol-fun}
797 \begin{figure} \underscoreon
799 \tdx{Union_upper} B:A ==> B <= Union A
800 \tdx{Union_least} [| !!X. X:A ==> X<=C |] ==> Union A <= C
802 \tdx{Inter_lower} B:A ==> Inter A <= B
803 \tdx{Inter_greatest} [| !!X. X:A ==> C<=X |] ==> C <= Inter A
805 \tdx{Un_upper1} A <= A Un B
806 \tdx{Un_upper2} B <= A Un B
807 \tdx{Un_least} [| A<=C; B<=C |] ==> A Un B <= C
809 \tdx{Int_lower1} A Int B <= A
810 \tdx{Int_lower2} A Int B <= B
811 \tdx{Int_greatest} [| C<=A; C<=B |] ==> C <= A Int B
813 \caption{Derived rules involving subsets} \label{hol-subset}
817 \begin{figure} \underscoreon \hfuzz=4pt%suppress "Overfull \hbox" message
819 \tdx{Int_absorb} A Int A = A
820 \tdx{Int_commute} A Int B = B Int A
821 \tdx{Int_assoc} (A Int B) Int C = A Int (B Int C)
822 \tdx{Int_Un_distrib} (A Un B) Int C = (A Int C) Un (B Int C)
824 \tdx{Un_absorb} A Un A = A
825 \tdx{Un_commute} A Un B = B Un A
826 \tdx{Un_assoc} (A Un B) Un C = A Un (B Un C)
827 \tdx{Un_Int_distrib} (A Int B) Un C = (A Un C) Int (B Un C)
829 \tdx{Compl_disjoint} A Int (Compl A) = {\ttlbrace}x. False{\ttrbrace}
830 \tdx{Compl_partition} A Un (Compl A) = {\ttlbrace}x. True{\ttrbrace}
831 \tdx{double_complement} Compl(Compl A) = A
832 \tdx{Compl_Un} Compl(A Un B) = (Compl A) Int (Compl B)
833 \tdx{Compl_Int} Compl(A Int B) = (Compl A) Un (Compl B)
835 \tdx{Union_Un_distrib} Union(A Un B) = (Union A) Un (Union B)
836 \tdx{Int_Union} A Int (Union B) = (UN C:B. A Int C)
837 \tdx{Un_Union_image} (UN x:C.(A x) Un (B x)) = Union(A``C) Un Union(B``C)
839 \tdx{Inter_Un_distrib} Inter(A Un B) = (Inter A) Int (Inter B)
840 \tdx{Un_Inter} A Un (Inter B) = (INT C:B. A Un C)
841 \tdx{Int_Inter_image} (INT x:C.(A x) Int (B x)) = Inter(A``C) Int Inter(B``C)
843 \caption{Set equalities} \label{hol-equalities}
847 Figures~\ref{hol-set1} and~\ref{hol-set2} present derived rules. Most are
848 obvious and resemble rules of Isabelle's {\ZF} set theory. Certain rules,
849 such as \tdx{subsetCE}, \tdx{bexCI} and \tdx{UnCI},
850 are designed for classical reasoning; the rules \tdx{subsetD},
851 \tdx{bexI}, \tdx{Un1} and~\tdx{Un2} are not
852 strictly necessary but yield more natural proofs. Similarly,
853 \tdx{equalityCE} supports classical reasoning about extensionality,
854 after the fashion of \tdx{iffCE}. See the file \texttt{HOL/Set.ML} for
855 proofs pertaining to set theory.
857 Figure~\ref{hol-subset} presents lattice properties of the subset relation.
858 Unions form least upper bounds; non-empty intersections form greatest lower
859 bounds. Reasoning directly about subsets often yields clearer proofs than
860 reasoning about the membership relation. See the file \texttt{HOL/subset.ML}.
862 Figure~\ref{hol-equalities} presents many common set equalities. They
863 include commutative, associative and distributive laws involving unions,
864 intersections and complements. For a complete listing see the file {\tt
868 \texttt{Blast_tac} proves many set-theoretic theorems automatically.
869 Hence you seldom need to refer to the theorems above.
875 \it name &\it meta-type & \it description \\
876 \cdx{inj}~~\cdx{surj}& $(\alpha\To\beta )\To bool$
877 & injective/surjective \\
878 \cdx{inj_on} & $[\alpha\To\beta ,\alpha\,set]\To bool$
879 & injective over subset\\
880 \cdx{inv} & $(\alpha\To\beta)\To(\beta\To\alpha)$ & inverse function
886 \tdx{inj_def} inj f == ! x y. f x=f y --> x=y
887 \tdx{surj_def} surj f == ! y. ? x. y=f x
888 \tdx{inj_on_def} inj_on f A == !x:A. !y:A. f x=f y --> x=y
889 \tdx{inv_def} inv f == (\%y. @x. f(x)=y)
891 \caption{Theory \thydx{Fun}} \label{fig:HOL:Fun}
894 \subsection{Properties of functions}\nopagebreak
895 Figure~\ref{fig:HOL:Fun} presents a theory of simple properties of functions.
896 Note that ${\tt inv}~f$ uses Hilbert's $\varepsilon$ to yield an inverse
897 of~$f$. See the file \texttt{HOL/Fun.ML} for a complete listing of the derived
898 rules. Reasoning about function composition (the operator~\sdx{o}) and the
899 predicate~\cdx{surj} is done simply by expanding the definitions.
901 There is also a large collection of monotonicity theorems for constructions
902 on sets in the file \texttt{HOL/mono.ML}.
904 \section{Generic packages}
905 \label{sec:HOL:generic-packages}
907 \HOL\ instantiates most of Isabelle's generic packages, making available the
908 simplifier and the classical reasoner.
910 \subsection{Simplification and substitution}
912 Simplification tactics tactics such as \texttt{Asm_simp_tac} and \texttt{Full_simp_tac} use the default simpset
913 (\texttt{simpset()}), which works for most purposes. A quite minimal
914 simplification set for higher-order logic is~\ttindexbold{HOL_ss};
915 even more frugal is \ttindexbold{HOL_basic_ss}. Equality~($=$), which
916 also expresses logical equivalence, may be used for rewriting. See
917 the file \texttt{HOL/simpdata.ML} for a complete listing of the basic
918 simplification rules.
920 See \iflabelundefined{chap:classical}{the {\em Reference Manual\/}}%
921 {Chaps.\ts\ref{substitution} and~\ref{simp-chap}} for details of substitution
924 \begin{warn}\index{simplification!of conjunctions}%
925 Reducing $a=b\conj P(a)$ to $a=b\conj P(b)$ is sometimes advantageous. The
926 left part of a conjunction helps in simplifying the right part. This effect
927 is not available by default: it can be slow. It can be obtained by
928 including \ttindex{conj_cong} in a simpset, \verb$addcongs [conj_cong]$.
931 If the simplifier cannot use a certain rewrite rule --- either because
932 of nontermination or because its left-hand side is too flexible ---
933 then you might try \texttt{stac}:
934 \begin{ttdescription}
935 \item[\ttindexbold{stac} $thm$ $i,$] where $thm$ is of the form $lhs = rhs$,
936 replaces in subgoal $i$ instances of $lhs$ by corresponding instances of
937 $rhs$. In case of multiple instances of $lhs$ in subgoal $i$, backtracking
938 may be necessary to select the desired ones.
940 If $thm$ is a conditional equality, the instantiated condition becomes an
941 additional (first) subgoal.
944 \HOL{} provides the tactic \ttindex{hyp_subst_tac}, which substitutes
945 for an equality throughout a subgoal and its hypotheses. This tactic uses
946 \HOL's general substitution rule.
948 \subsubsection{Case splitting}
949 \label{subsec:HOL:case:splitting}
951 \HOL{} also provides convenient means for case splitting during
952 rewriting. Goals containing a subterm of the form \texttt{if}~$b$~{\tt
953 then\dots else\dots} often require a case distinction on $b$. This is
954 expressed by the theorem \tdx{split_if}:
956 \Var{P}(\mbox{\tt if}~\Var{b}~{\tt then}~\Var{x}~\mbox{\tt else}~\Var{y})~=~
957 ((\Var{b} \to \Var{P}(\Var{x})) \land (\neg \Var{b} \to \Var{P}(\Var{y})))
960 For example, a simple instance of $(*)$ is
962 x \in (\mbox{\tt if}~x \in A~{\tt then}~A~\mbox{\tt else}~\{x\})~=~
963 ((x \in A \to x \in A) \land (x \notin A \to x \in \{x\}))
965 Because $(*)$ is too general as a rewrite rule for the simplifier (the
966 left-hand side is not a higher-order pattern in the sense of
967 \iflabelundefined{chap:simplification}{the {\em Reference Manual\/}}%
968 {Chap.\ts\ref{chap:simplification}}), there is a special infix function
969 \ttindexbold{addsplits} of type \texttt{simpset * thm list -> simpset}
970 (analogous to \texttt{addsimps}) that adds rules such as $(*)$ to a
973 by(simp_tac (simpset() addsplits [split_if]) 1);
975 The effect is that after each round of simplification, one occurrence of
976 \texttt{if} is split acording to \texttt{split_if}, until all occurences of
977 \texttt{if} have been eliminated.
979 It turns out that using \texttt{split_if} is almost always the right thing to
980 do. Hence \texttt{split_if} is already included in the default simpset. If
981 you want to delete it from a simpset, use \ttindexbold{delsplits}, which is
982 the inverse of \texttt{addsplits}:
984 by(simp_tac (simpset() delsplits [split_if]) 1);
987 In general, \texttt{addsplits} accepts rules of the form
989 \Var{P}(c~\Var{x@1}~\dots~\Var{x@n})~=~ rhs
991 where $c$ is a constant and $rhs$ is arbitrary. Note that $(*)$ is of the
992 right form because internally the left-hand side is
993 $\Var{P}(\mathtt{If}~\Var{b}~\Var{x}~~\Var{y})$. Important further examples
994 are splitting rules for \texttt{case} expressions (see~\S\ref{subsec:list}
995 and~\S\ref{subsec:datatype:basics}).
997 Analogous to \texttt{Addsimps} and \texttt{Delsimps}, there are also
998 imperative versions of \texttt{addsplits} and \texttt{delsplits}
1000 \ttindexbold{Addsplits}: thm list -> unit
1001 \ttindexbold{Delsplits}: thm list -> unit
1003 for adding splitting rules to, and deleting them from the current simpset.
1005 \subsection{Classical reasoning}
1007 \HOL\ derives classical introduction rules for $\disj$ and~$\exists$, as
1008 well as classical elimination rules for~$\imp$ and~$\bimp$, and the swap
1009 rule; recall Fig.\ts\ref{hol-lemmas2} above.
1011 The classical reasoner is installed. Tactics such as \texttt{Blast_tac} and {\tt
1012 Best_tac} refer to the default claset (\texttt{claset()}), which works for most
1013 purposes. Named clasets include \ttindexbold{prop_cs}, which includes the
1014 propositional rules, and \ttindexbold{HOL_cs}, which also includes quantifier
1015 rules. See the file \texttt{HOL/cladata.ML} for lists of the classical rules,
1016 and \iflabelundefined{chap:classical}{the {\em Reference Manual\/}}%
1017 {Chap.\ts\ref{chap:classical}} for more discussion of classical proof methods.
1020 \section{Types}\label{sec:HOL:Types}
1021 This section describes \HOL's basic predefined types ($\alpha \times
1022 \beta$, $\alpha + \beta$, $nat$ and $\alpha \; list$) and ways for
1023 introducing new types in general. The most important type
1024 construction, the \texttt{datatype}, is treated separately in
1025 \S\ref{sec:HOL:datatype}.
1028 \subsection{Product and sum types}\index{*"* type}\index{*"+ type}
1029 \label{subsec:prod-sum}
1031 \begin{figure}[htbp]
1033 \it symbol & \it meta-type & & \it description \\
1034 \cdx{Pair} & $[\alpha,\beta]\To \alpha\times\beta$
1035 & & ordered pairs $(a,b)$ \\
1036 \cdx{fst} & $\alpha\times\beta \To \alpha$ & & first projection\\
1037 \cdx{snd} & $\alpha\times\beta \To \beta$ & & second projection\\
1038 \cdx{split} & $[[\alpha,\beta]\To\gamma, \alpha\times\beta] \To \gamma$
1039 & & generalized projection\\
1041 $[\alpha\,set, \alpha\To\beta\,set]\To(\alpha\times\beta)set$ &
1042 & general sum of sets
1044 \begin{ttbox}\makeatletter
1045 %\tdx{fst_def} fst p == @a. ? b. p = (a,b)
1046 %\tdx{snd_def} snd p == @b. ? a. p = (a,b)
1047 %\tdx{split_def} split c p == c (fst p) (snd p)
1048 \tdx{Sigma_def} Sigma A B == UN x:A. UN y:B x. {\ttlbrace}(x,y){\ttrbrace}
1050 \tdx{Pair_eq} ((a,b) = (a',b')) = (a=a' & b=b')
1051 \tdx{Pair_inject} [| (a, b) = (a',b'); [| a=a'; b=b' |] ==> R |] ==> R
1052 \tdx{PairE} [| !!x y. p = (x,y) ==> Q |] ==> Q
1054 \tdx{fst_conv} fst (a,b) = a
1055 \tdx{snd_conv} snd (a,b) = b
1056 \tdx{surjective_pairing} p = (fst p,snd p)
1058 \tdx{split} split c (a,b) = c a b
1059 \tdx{split_split} R(split c p) = (! x y. p = (x,y) --> R(c x y))
1061 \tdx{SigmaI} [| a:A; b:B a |] ==> (a,b) : Sigma A B
1062 \tdx{SigmaE} [| c:Sigma A B; !!x y.[| x:A; y:B x; c=(x,y) |] ==> P |] ==> P
1064 \caption{Type $\alpha\times\beta$}\label{hol-prod}
1067 Theory \thydx{Prod} (Fig.\ts\ref{hol-prod}) defines the product type
1068 $\alpha\times\beta$, with the ordered pair syntax $(a, b)$. General
1069 tuples are simulated by pairs nested to the right:
1071 \begin{tabular}{c|c}
1072 external & internal \\
1074 $\tau@1 \times \dots \times \tau@n$ & $\tau@1 \times (\dots (\tau@{n-1} \times \tau@n)\dots)$ \\
1076 $(t@1,\dots,t@n)$ & $(t@1,(\dots,(t@{n-1},t@n)\dots)$ \\
1079 In addition, it is possible to use tuples
1080 as patterns in abstractions:
1082 {\tt\%($x$,$y$). $t$} \quad stands for\quad \texttt{split(\%$x$\thinspace$y$.\ $t$)}
1084 Nested patterns are also supported. They are translated stepwise:
1085 {\tt\%($x$,$y$,$z$). $t$} $\leadsto$ {\tt\%($x$,($y$,$z$)). $t$} $\leadsto$
1086 {\tt split(\%$x$.\%($y$,$z$). $t$)} $\leadsto$ \texttt{split(\%$x$. split(\%$y$
1087 $z$.\ $t$))}. The reverse translation is performed upon printing.
1089 The translation between patterns and \texttt{split} is performed automatically
1090 by the parser and printer. Thus the internal and external form of a term
1091 may differ, which can affects proofs. For example the term {\tt
1092 (\%(x,y).(y,x))(a,b)} requires the theorem \texttt{split} (which is in the
1093 default simpset) to rewrite to {\tt(b,a)}.
1095 In addition to explicit $\lambda$-abstractions, patterns can be used in any
1096 variable binding construct which is internally described by a
1097 $\lambda$-abstraction. Some important examples are
1099 \item[Let:] \texttt{let {\it pattern} = $t$ in $u$}
1100 \item[Quantifiers:] \texttt{!~{\it pattern}:$A$.~$P$}
1101 \item[Choice:] {\underscoreon \tt @~{\it pattern}~.~$P$}
1102 \item[Set operations:] \texttt{UN~{\it pattern}:$A$.~$B$}
1103 \item[Sets:] \texttt{{\ttlbrace}~{\it pattern}~.~$P$~{\ttrbrace}}
1106 There is a simple tactic which supports reasoning about patterns:
1107 \begin{ttdescription}
1108 \item[\ttindexbold{split_all_tac} $i$] replaces in subgoal $i$ all
1109 {\tt!!}-quantified variables of product type by individual variables for
1110 each component. A simple example:
1112 {\out 1. !!p. (\%(x,y,z). (x, y, z)) p = p}
1113 by(split_all_tac 1);
1114 {\out 1. !!x xa ya. (\%(x,y,z). (x, y, z)) (x, xa, ya) = (x, xa, ya)}
1118 Theory \texttt{Prod} also introduces the degenerate product type \texttt{unit}
1119 which contains only a single element named {\tt()} with the property
1121 \tdx{unit_eq} u = ()
1125 Theory \thydx{Sum} (Fig.~\ref{hol-sum}) defines the sum type $\alpha+\beta$
1126 which associates to the right and has a lower priority than $*$: $\tau@1 +
1127 \tau@2 + \tau@3*\tau@4$ means $\tau@1 + (\tau@2 + (\tau@3*\tau@4))$.
1129 The definition of products and sums in terms of existing types is not
1130 shown. The constructions are fairly standard and can be found in the
1131 respective theory files.
1135 \it symbol & \it meta-type & & \it description \\
1136 \cdx{Inl} & $\alpha \To \alpha+\beta$ & & first injection\\
1137 \cdx{Inr} & $\beta \To \alpha+\beta$ & & second injection\\
1138 \cdx{sum_case} & $[\alpha\To\gamma, \beta\To\gamma, \alpha+\beta] \To\gamma$
1141 \begin{ttbox}\makeatletter
1142 %\tdx{sum_case_def} sum_case == (\%f g p. @z. (!x. p=Inl x --> z=f x) &
1143 % (!y. p=Inr y --> z=g y))
1145 \tdx{Inl_not_Inr} Inl a ~= Inr b
1147 \tdx{inj_Inl} inj Inl
1148 \tdx{inj_Inr} inj Inr
1150 \tdx{sumE} [| !!x. P(Inl x); !!y. P(Inr y) |] ==> P s
1152 \tdx{sum_case_Inl} sum_case f g (Inl x) = f x
1153 \tdx{sum_case_Inr} sum_case f g (Inr x) = g x
1155 \tdx{surjective_sum} sum_case (\%x. f(Inl x)) (\%y. f(Inr y)) s = f s
1156 \tdx{split_sum_case} R(sum_case f g s) = ((! x. s = Inl(x) --> R(f(x))) &
1157 (! y. s = Inr(y) --> R(g(y))))
1159 \caption{Type $\alpha+\beta$}\label{hol-sum}
1170 \it symbol & \it meta-type & \it priority & \it description \\
1171 \cdx{0} & $nat$ & & zero \\
1172 \cdx{Suc} & $nat \To nat$ & & successor function\\
1173 % \cdx{nat_case} & $[\alpha, nat\To\alpha, nat] \To\alpha$ & & conditional\\
1174 % \cdx{nat_rec} & $[nat, \alpha, [nat, \alpha]\To\alpha] \To \alpha$
1175 % & & primitive recursor\\
1176 \tt * & $[nat,nat]\To nat$ & Left 70 & multiplication \\
1177 \tt div & $[nat,nat]\To nat$ & Left 70 & division\\
1178 \tt mod & $[nat,nat]\To nat$ & Left 70 & modulus\\
1179 \tt + & $[nat,nat]\To nat$ & Left 65 & addition\\
1180 \tt - & $[nat,nat]\To nat$ & Left 65 & subtraction
1182 \subcaption{Constants and infixes}
1184 \begin{ttbox}\makeatother
1185 \tdx{nat_induct} [| P 0; !!n. P n ==> P(Suc n) |] ==> P n
1187 \tdx{Suc_not_Zero} Suc m ~= 0
1188 \tdx{inj_Suc} inj Suc
1189 \tdx{n_not_Suc_n} n~=Suc n
1190 \subcaption{Basic properties}
1192 \caption{The type of natural numbers, \tydx{nat}} \label{hol-nat1}
1197 \begin{ttbox}\makeatother
1199 (Suc m)+n = Suc(m+n)
1208 \tdx{mod_less} m<n ==> m mod n = m
1209 \tdx{mod_geq} [| 0<n; ~m<n |] ==> m mod n = (m-n) mod n
1211 \tdx{div_less} m<n ==> m div n = 0
1212 \tdx{div_geq} [| 0<n; ~m<n |] ==> m div n = Suc((m-n) div n)
1214 \caption{Recursion equations for the arithmetic operators} \label{hol-nat2}
1217 \subsection{The type of natural numbers, \textit{nat}}
1218 \index{nat@{\textit{nat}} type|(}
1220 The theory \thydx{NatDef} defines the natural numbers in a roundabout but
1221 traditional way. The axiom of infinity postulates a type~\tydx{ind} of
1222 individuals, which is non-empty and closed under an injective operation. The
1223 natural numbers are inductively generated by choosing an arbitrary individual
1224 for~0 and using the injective operation to take successors. This is a least
1225 fixedpoint construction. For details see the file \texttt{NatDef.thy}.
1227 Type~\tydx{nat} is an instance of class~\cldx{ord}, which makes the
1228 overloaded functions of this class (esp.\ \cdx{<} and \cdx{<=}, but also
1229 \cdx{min}, \cdx{max} and \cdx{LEAST}) available on \tydx{nat}. Theory
1230 \thydx{Nat} builds on \texttt{NatDef} and shows that {\tt<=} is a partial order,
1231 so \tydx{nat} is also an instance of class \cldx{order}.
1233 Theory \thydx{Arith} develops arithmetic on the natural numbers. It defines
1234 addition, multiplication and subtraction. Theory \thydx{Divides} defines
1235 division, remainder and the ``divides'' relation. The numerous theorems
1236 proved include commutative, associative, distributive, identity and
1237 cancellation laws. See Figs.\ts\ref{hol-nat1} and~\ref{hol-nat2}. The
1238 recursion equations for the operators \texttt{+}, \texttt{-} and \texttt{*} on
1239 \texttt{nat} are part of the default simpset.
1241 Functions on \tydx{nat} can be defined by primitive or well-founded recursion;
1242 see \S\ref{sec:HOL:recursive}. A simple example is addition.
1243 Here, \texttt{op +} is the name of the infix operator~\texttt{+}, following
1244 the standard convention.
1248 "Suc m + n = Suc (m + n)"
1250 There is also a \sdx{case}-construct
1253 case \(e\) of 0 => \(a\) | Suc \(m\) => \(b\)
1255 Note that Isabelle insists on precisely this format; you may not even change
1256 the order of the two cases.
1257 Both \texttt{primrec} and \texttt{case} are realized by a recursion operator
1258 \cdx{nat_rec}, the details of which can be found in theory \texttt{NatDef}.
1260 %The predecessor relation, \cdx{pred_nat}, is shown to be well-founded.
1261 %Recursion along this relation resembles primitive recursion, but is
1262 %stronger because we are in higher-order logic; using primitive recursion to
1263 %define a higher-order function, we can easily Ackermann's function, which
1264 %is not primitive recursive \cite[page~104]{thompson91}.
1265 %The transitive closure of \cdx{pred_nat} is~$<$. Many functions on the
1266 %natural numbers are most easily expressed using recursion along~$<$.
1268 Tactic {\tt\ttindex{induct_tac} "$n$" $i$} performs induction on variable~$n$
1269 in subgoal~$i$ using theorem \texttt{nat_induct}. There is also the derived
1270 theorem \tdx{less_induct}:
1272 [| !!n. [| ! m. m<n --> P m |] ==> P n |] ==> P n
1276 Reasoning about arithmetic inequalities can be tedious. Fortunately HOL
1277 provides a decision procedure for quantifier-free linear arithmetic (i.e.\
1278 only addition and subtraction). The simplifier invokes a weak version of this
1279 decision procedure automatically. If this is not sufficent, you can invoke
1280 the full procedure \ttindex{arith_tac} explicitly. It copes with arbitrary
1281 formulae involving {\tt=}, {\tt<}, {\tt<=}, {\tt+}, {\tt-}, {\tt Suc}, {\tt
1282 min}, {\tt max} and numerical constants; other subterms are treated as
1283 atomic; subformulae not involving type $nat$ are ignored; quantified
1284 subformulae are ignored unless they are positive universal or negative
1285 existential. Note that the running time is exponential in the number of
1286 occurrences of {\tt min}, {\tt max}, and {\tt-} because they require case
1287 distinctions. Note also that \texttt{arith_tac} is not complete: if
1288 divisibility plays a role, it may fail to prove a valid formula, for example
1289 $m+m \neq n+n+1$. Fortunately such examples are rare in practice.
1291 If \texttt{arith_tac} fails you, try to find relevant arithmetic results in
1292 the library. The theory \texttt{NatDef} contains theorems about {\tt<} and
1293 {\tt<=}, the theory \texttt{Arith} contains theorems about \texttt{+},
1294 \texttt{-} and \texttt{*}, and theory \texttt{Divides} contains theorems about
1295 \texttt{div} and \texttt{mod}. Use the \texttt{find}-functions to locate them
1296 (see the {\em Reference Manual\/}).
1299 \index{#@{\tt[]} symbol}
1300 \index{#@{\tt\#} symbol}
1301 \index{"@@{\tt\at} symbol}
1304 \it symbol & \it meta-type & \it priority & \it description \\
1305 \tt[] & $\alpha\,list$ & & empty list\\
1306 \tt \# & $[\alpha,\alpha\,list]\To \alpha\,list$ & Right 65 &
1308 \cdx{null} & $\alpha\,list \To bool$ & & emptiness test\\
1309 \cdx{hd} & $\alpha\,list \To \alpha$ & & head \\
1310 \cdx{tl} & $\alpha\,list \To \alpha\,list$ & & tail \\
1311 \cdx{last} & $\alpha\,list \To \alpha$ & & last element \\
1312 \cdx{butlast} & $\alpha\,list \To \alpha\,list$ & & drop last element \\
1313 \tt\at & $[\alpha\,list,\alpha\,list]\To \alpha\,list$ & Left 65 & append \\
1314 \cdx{map} & $(\alpha\To\beta) \To (\alpha\,list \To \beta\,list)$
1316 \cdx{filter} & $(\alpha \To bool) \To (\alpha\,list \To \alpha\,list)$
1317 & & filter functional\\
1318 \cdx{set}& $\alpha\,list \To \alpha\,set$ & & elements\\
1319 \sdx{mem} & $\alpha \To \alpha\,list \To bool$ & Left 55 & membership\\
1320 \cdx{foldl} & $(\beta\To\alpha\To\beta) \To \beta \To \alpha\,list \To \beta$ &
1322 \cdx{concat} & $(\alpha\,list)list\To \alpha\,list$ & & concatenation \\
1323 \cdx{rev} & $\alpha\,list \To \alpha\,list$ & & reverse \\
1324 \cdx{length} & $\alpha\,list \To nat$ & & length \\
1325 \tt! & $\alpha\,list \To nat \To \alpha$ & Left 100 & indexing \\
1326 \cdx{take}, \cdx{drop} & $nat \To \alpha\,list \To \alpha\,list$ &&
1327 take or drop a prefix \\
1330 $(\alpha \To bool) \To \alpha\,list \To \alpha\,list$ &&
1331 take or drop a prefix
1333 \subcaption{Constants and infixes}
1335 \begin{center} \tt\frenchspacing
1336 \begin{tabular}{rrr}
1337 \it external & \it internal & \it description \\{}
1338 [$x@1$, $\dots$, $x@n$] & $x@1$ \# $\cdots$ \# $x@n$ \# [] &
1339 \rm finite list \\{}
1340 [$x$:$l$. $P$] & filter ($\lambda x{.}P$) $l$ &
1341 \rm list comprehension
1344 \subcaption{Translations}
1345 \caption{The theory \thydx{List}} \label{hol-list}
1350 \begin{ttbox}\makeatother
1359 (x#xs) @ ys = x # xs @ ys
1362 map f (x#xs) = f x # map f xs
1365 filter P (x#xs) = (if P x then x#filter P xs else filter P xs)
1367 set [] = \ttlbrace\ttrbrace
1368 set (x#xs) = insert x (set xs)
1371 x mem (y#ys) = (if y=x then True else x mem ys)
1374 foldl f a (x#xs) = foldl f (f a x) xs
1377 concat(x#xs) = x @ concat(xs)
1380 rev(x#xs) = rev(xs) @ [x]
1383 length(x#xs) = Suc(length(xs))
1386 xs!(Suc n) = (tl xs)!n
1389 take n (x#xs) = (case n of 0 => [] | Suc(m) => x # take m xs)
1392 drop n (x#xs) = (case n of 0 => x#xs | Suc(m) => drop m xs)
1395 takeWhile P (x#xs) = (if P x then x#takeWhile P xs else [])
1398 dropWhile P (x#xs) = (if P x then dropWhile P xs else xs)
1400 \caption{Recursions equations for list processing functions}
1401 \label{fig:HOL:list-simps}
1403 \index{nat@{\textit{nat}} type|)}
1406 \subsection{The type constructor for lists, \textit{list}}
1408 \index{list@{\textit{list}} type|(}
1410 Figure~\ref{hol-list} presents the theory \thydx{List}: the basic list
1411 operations with their types and syntax. Type $\alpha \; list$ is
1412 defined as a \texttt{datatype} with the constructors {\tt[]} and {\tt\#}.
1413 As a result the generic structural induction and case analysis tactics
1414 \texttt{induct\_tac} and \texttt{exhaust\_tac} also become available for
1415 lists. A \sdx{case} construct of the form
1417 case $e$ of [] => $a$ | \(x\)\#\(xs\) => b
1419 is defined by translation. For details see~\S\ref{sec:HOL:datatype}. There
1420 is also a case splitting rule \tdx{split_list_case}
1423 P(\mathtt{case}~e~\mathtt{of}~\texttt{[] =>}~a ~\texttt{|}~
1424 x\texttt{\#}xs~\texttt{=>}~f~x~xs) ~= \\
1425 ((e = \texttt{[]} \to P(a)) \land
1426 (\forall x~ xs. e = x\texttt{\#}xs \to P(f~x~xs)))
1429 which can be fed to \ttindex{addsplits} just like
1430 \texttt{split_if} (see~\S\ref{subsec:HOL:case:splitting}).
1432 \texttt{List} provides a basic library of list processing functions defined by
1433 primitive recursion (see~\S\ref{sec:HOL:primrec}). The recursion equations
1434 are shown in Fig.\ts\ref{fig:HOL:list-simps}.
1436 \index{list@{\textit{list}} type|)}
1439 \subsection{Introducing new types} \label{sec:typedef}
1441 The \HOL-methodology dictates that all extensions to a theory should
1442 be \textbf{definitional}. The type definition mechanism that
1443 meets this criterion is \ttindex{typedef}. Note that \emph{type synonyms},
1444 which are inherited from {\Pure} and described elsewhere, are just
1445 syntactic abbreviations that have no logical meaning.
1448 Types in \HOL\ must be non-empty; otherwise the quantifier rules would be
1449 unsound, because $\exists x. x=x$ is a theorem \cite[\S7]{paulson-COLOG}.
1451 A \bfindex{type definition} identifies the new type with a subset of
1452 an existing type. More precisely, the new type is defined by
1453 exhibiting an existing type~$\tau$, a set~$A::\tau\,set$, and a
1454 theorem of the form $x:A$. Thus~$A$ is a non-empty subset of~$\tau$,
1455 and the new type denotes this subset. New functions are defined that
1456 establish an isomorphism between the new type and the subset. If
1457 type~$\tau$ involves type variables $\alpha@1$, \ldots, $\alpha@n$,
1458 then the type definition creates a type constructor
1459 $(\alpha@1,\ldots,\alpha@n)ty$ rather than a particular type.
1461 \begin{figure}[htbp]
1463 typedef : 'typedef' ( () | '(' name ')') type '=' set witness;
1465 type : typevarlist name ( () | '(' infix ')' );
1467 witness : () | '(' id ')';
1469 \caption{Syntax of type definitions}
1470 \label{fig:HOL:typedef}
1473 The syntax for type definitions is shown in Fig.~\ref{fig:HOL:typedef}. For
1474 the definition of `typevarlist' and `infix' see
1475 \iflabelundefined{chap:classical}
1476 {the appendix of the {\em Reference Manual\/}}%
1477 {Appendix~\ref{app:TheorySyntax}}. The remaining nonterminals have the
1480 \item[\it type:] the new type constructor $(\alpha@1,\dots,\alpha@n)ty$ with
1481 optional infix annotation.
1482 \item[\it name:] an alphanumeric name $T$ for the type constructor
1483 $ty$, in case $ty$ is a symbolic name. Defaults to $ty$.
1484 \item[\it set:] the representing subset $A$.
1485 \item[\it witness:] name of a theorem of the form $a:A$ proving
1486 non-emptiness. It can be omitted in case Isabelle manages to prove
1487 non-emptiness automatically.
1489 If all context conditions are met (no duplicate type variables in
1490 `typevarlist', no extra type variables in `set', and no free term variables
1491 in `set'), the following components are added to the theory:
1493 \item a type $ty :: (term,\dots,term)term$
1497 Rep_T &::& (\alpha@1,\dots,\alpha@n)ty \To \tau \\
1498 Abs_T &::& \tau \To (\alpha@1,\dots,\alpha@n)ty
1500 \item a definition and three axioms
1503 T{\tt_def} & T \equiv A \\
1504 {\tt Rep_}T & Rep_T\,x \in T \\
1505 {\tt Rep_}T{\tt_inverse} & Abs_T\,(Rep_T\,x) = x \\
1506 {\tt Abs_}T{\tt_inverse} & y \in T \Imp Rep_T\,(Abs_T\,y) = y
1509 stating that $(\alpha@1,\dots,\alpha@n)ty$ is isomorphic to $A$ by $Rep_T$
1510 and its inverse $Abs_T$.
1512 Below are two simple examples of \HOL\ type definitions. Non-emptiness
1513 is proved automatically here.
1515 typedef unit = "{\ttlbrace}True{\ttrbrace}"
1518 ('a, 'b) "*" (infixr 20)
1519 = "{\ttlbrace}f . EX (a::'a) (b::'b). f = (\%x y. x = a & y = b){\ttrbrace}"
1522 Type definitions permit the introduction of abstract data types in a safe
1523 way, namely by providing models based on already existing types. Given some
1524 abstract axiomatic description $P$ of a type, this involves two steps:
1526 \item Find an appropriate type $\tau$ and subset $A$ which has the desired
1527 properties $P$, and make a type definition based on this representation.
1528 \item Prove that $P$ holds for $ty$ by lifting $P$ from the representation.
1530 You can now forget about the representation and work solely in terms of the
1531 abstract properties $P$.
1534 If you introduce a new type (constructor) $ty$ axiomatically, i.e.\ by
1535 declaring the type and its operations and by stating the desired axioms, you
1536 should make sure the type has a non-empty model. You must also have a clause
1539 arities \(ty\) :: (term,\thinspace\(\dots\),{\thinspace}term){\thinspace}term
1541 in your theory file to tell Isabelle that $ty$ is in class \texttt{term}, the
1542 class of all \HOL\ types.
1548 At a first approximation, records are just a minor generalisation of tuples,
1549 where components may be addressed by labels instead of just position (think of
1550 {\ML}, for example). The version of records offered by Isabelle/HOL is
1551 slightly more advanced, though, supporting \emph{extensible record schemes}.
1552 This admits operations that are polymorphic with respect to record extension,
1553 yielding ``object-oriented'' effects like (single) inheritance. See also
1554 \cite{NaraschewskiW-TPHOLs98} for more details on object-oriented
1555 verification and record subtyping in HOL.
1560 Isabelle/HOL supports fixed and schematic records both at the level of terms
1561 and types. The concrete syntax is as follows:
1564 \begin{tabular}{l|l|l}
1565 & record terms & record types \\ \hline
1566 fixed & $\record{x = a\fs y = b}$ & $\record{x \ty A\fs y \ty B}$ \\
1567 schematic & $\record{x = a\fs y = b\fs \more = m}$ &
1568 $\record{x \ty A\fs y \ty B\fs \more \ty M}$ \\
1572 \noindent The \textsc{ascii} representation of $\record{x = a}$ is \texttt{(| x = a |)}.
1574 A fixed record $\record{x = a\fs y = b}$ has field $x$ of value $a$ and field
1575 $y$ of value $b$. The corresponding type is $\record{x \ty A\fs y \ty B}$,
1576 assuming that $a \ty A$ and $b \ty B$.
1578 A record scheme like $\record{x = a\fs y = b\fs \more = m}$ contains fields
1579 $x$ and $y$ as before, but also possibly further fields as indicated by the
1580 ``$\more$'' notation (which is actually part of the syntax). The improper
1581 field ``$\more$'' of a record scheme is called the \emph{more part}.
1582 Logically it is just a free variable, which is occasionally referred to as
1583 \emph{row variable} in the literature. The more part of a record scheme may
1584 be instantiated by zero or more further components. For example, above scheme
1585 might get instantiated to $\record{x = a\fs y = b\fs z = c\fs \more = m'}$,
1586 where $m'$ refers to a different more part. Fixed records are special
1587 instances of record schemes, where ``$\more$'' is properly terminated by the
1588 $() :: unit$ element. Actually, $\record{x = a\fs y = b}$ is just an
1589 abbreviation for $\record{x = a\fs y = b\fs \more = ()}$.
1593 There are two key features that make extensible records in a simply typed
1594 language like HOL feasible:
1596 \item the more part is internalised, as a free term or type variable,
1597 \item field names are externalised, they cannot be accessed within the logic
1598 as first-class values.
1603 In Isabelle/HOL record types have to be defined explicitly, fixing their field
1604 names and types, and their (optional) parent record (see
1605 \S\ref{sec:HOL:record-def}). Afterwards, records may be formed using above
1606 syntax, while obeying the canonical order of fields as given by their
1607 declaration. The record package also provides several operations like
1608 selectors and updates (see \S\ref{sec:HOL:record-ops}), together with
1609 characteristic properties (see \S\ref{sec:HOL:record-thms}).
1611 There is an example theory demonstrating most basic aspects of extensible
1612 records (see theory \texttt{HOL/ex/Points} in the Isabelle sources).
1615 \subsection{Defining records}\label{sec:HOL:record-def}
1617 The theory syntax for record type definitions is shown in
1618 Fig.~\ref{fig:HOL:record}. For the definition of `typevarlist' and `type' see
1619 \iflabelundefined{chap:classical}
1620 {the appendix of the {\em Reference Manual\/}}%
1621 {Appendix~\ref{app:TheorySyntax}}.
1623 \begin{figure}[htbp]
1625 record : 'record' typevarlist name '=' parent (field +);
1627 parent : ( () | type '+');
1628 field : name '::' type;
1630 \caption{Syntax of record type definitions}
1631 \label{fig:HOL:record}
1634 A general \ttindex{record} specification is of the following form:
1636 \mathtt{record}~(\alpha@1, \dots, \alpha@n) \, t ~=~
1637 (\tau@1, \dots, \tau@m) \, s ~+~ c@1 :: \sigma@1 ~ \dots ~ c@l :: \sigma@l
1639 where $\vec\alpha@n$ are distinct type variables, and $\vec\tau@m$,
1640 $\vec\sigma@l$ are types containing at most variables from $\vec\alpha@n$.
1641 Type constructor $t$ has to be new, while $s$ has to specify an existing
1642 record type. Furthermore, the $\vec c@l$ have to be distinct field names.
1643 There has to be at least one field.
1645 In principle, field names may never be shared with other records. This is no
1646 actual restriction in practice, since $\vec c@l$ are internally declared
1647 within a separate name space qualified by the name $t$ of the record.
1651 Above definition introduces a new record type $(\vec\alpha@n) \, t$ by
1652 extending an existing one $(\vec\tau@m) \, s$ by new fields $\vec c@l \ty
1653 \vec\sigma@l$. The parent record specification is optional, by omitting it
1654 $t$ becomes a \emph{root record}. The hierarchy of all records declared
1655 within a theory forms a forest structure, i.e.\ a set of trees, where any of
1656 these is rooted by some root record.
1658 For convenience, $(\vec\alpha@n) \, t$ is made a type abbreviation for the
1659 fixed record type $\record{\vec c@l \ty \vec\sigma@l}$, and $(\vec\alpha@n,
1660 \zeta) \, t_scheme$ is made an abbreviation for $\record{\vec c@l \ty
1661 \vec\sigma@l\fs \more \ty \zeta}$.
1665 The following simple example defines a root record type $point$ with fields $x
1666 \ty nat$ and $y \ty nat$, and record type $cpoint$ by extending $point$ with
1667 an additional $colour$ component.
1674 record cpoint = point +
1679 \subsection{Record operations}\label{sec:HOL:record-ops}
1681 Any record definition of the form presented above produces certain standard
1682 operations. Selectors and updates are provided for any field, including the
1683 improper one ``$more$''. There are also cumulative record constructor
1686 To simplify the presentation below, we first assume that $(\vec\alpha@n) \, t$
1687 is a root record with fields $\vec c@l \ty \vec\sigma@l$.
1691 \textbf{Selectors} and \textbf{updates} are available for any field (including
1692 ``$more$'') as follows:
1693 \begin{matharray}{lll}
1694 c@i & \ty & \record{\vec c@l \ty \vec \sigma@l, \more \ty \zeta} \To \sigma@i \\
1695 c@i_update & \ty & \sigma@i \To \record{\vec c@l \ty \vec \sigma@l, \more \ty \zeta} \To
1696 \record{\vec c@l \ty \vec \sigma@l, \more \ty \zeta}
1699 There is some special syntax for updates: $r \, \record{x \asn a}$ abbreviates
1700 term $x_update \, a \, r$. Repeated updates are also supported: $r \,
1701 \record{x \asn a} \, \record{y \asn b} \, \record{z \asn c}$ may be written as
1702 $r \, \record{x \asn a\fs y \asn b\fs z \asn c}$. Note that because of
1703 postfix notation the order of fields shown here is reverse than in the actual
1704 term. This might lead to confusion in conjunction with proof tools like
1707 Since repeated updates are just function applications, fields may be freely
1708 permuted in $\record{x \asn a\fs y \asn b\fs z \asn c}$, as far as the logic
1709 is concerned. Thus commutativity of updates can be proven within the logic
1710 for any two fields, but not as a general theorem: fields are not first-class
1715 \textbf{Make} operations provide cumulative record constructor functions:
1716 \begin{matharray}{lll}
1717 make & \ty & \vec\sigma@l \To \record{\vec c@l \ty \vec \sigma@l} \\
1718 make_scheme & \ty & \vec\sigma@l \To
1719 \zeta \To \record{\vec c@l \ty \vec \sigma@l, \more \ty \zeta} \\
1722 These functions are curried. The corresponding definitions in terms of actual
1723 record terms are part of the standard simpset. Thus $point\dtt make\,a\,b$
1724 rewrites to $\record{x = a\fs y = b}$.
1728 Any of above selector, update and make operations are declared within a local
1729 name space prefixed by the name $t$ of the record. In case that different
1730 records share base names of fields, one has to qualify names explicitly (e.g.\
1731 $t\dtt c@i_update$). This is recommended especially for operations like
1732 $make$ or $update_more$ that always have the same base name. Just use $t\dtt
1733 make$ etc.\ to avoid confusion.
1737 We reconsider the case of non-root records, which are derived of some parent
1738 record. In general, the latter may depend on another parent as well,
1739 resulting in a list of \emph{ancestor records}. Appending the lists of fields
1740 of all ancestors results in a certain field prefix. The record package
1741 automatically takes care of this by lifting operations over this context of
1742 ancestor fields. Assuming that $(\vec\alpha@n) \, t$ has ancestor fields
1743 $\vec d@k \ty \vec\rho@k$, selectors will get the following types:
1744 \begin{matharray}{lll}
1745 c@i & \ty & \record{\vec d@k \ty \vec\rho@k, \vec c@l \ty \vec \sigma@l, \more \ty \zeta}
1749 Update and make operations are analogous.
1752 \subsection{Proof tools}\label{sec:HOL:record-thms}
1754 The record package provides the following proof rules for any record type $t$.
1757 \item Standard conversions (selectors or updates applied to record constructor
1758 terms, make function definitions) are part of the standard simpset (via
1761 \item Inject equations of a form analogous to $((x, y) = (x', y')) \equiv x=x'
1762 \conj y=y'$ are made part of the standard simpset and claset (via
1765 \item A tactic for record field splitting (\ttindex{record_split_tac}) is made
1766 part of the standard claset (via \texttt{addSWrapper}). This tactic is
1767 based on rules analogous to $(\All x PROP~P~x) \equiv (\All{a~b} PROP~P(a,
1771 The first two kinds of rules are stored within the theory as $t\dtt simps$ and
1772 $t\dtt iffs$, respectively. In some situations it might be appropriate to
1773 expand the definitions of updates: $t\dtt updates$. Following a new trend in
1774 Isabelle system architecture, these names are \emph{not} bound at the {\ML}
1779 The example theory \texttt{HOL/ex/Points} demonstrates typical proofs
1780 concerning records. The basic idea is to make \ttindex{record_split_tac}
1781 expand quantified record variables and then simplify by the conversion rules.
1782 By using a combination of the simplifier and classical prover together with
1783 the default simpset and claset, record problems should be solved with a single
1784 stroke of \texttt{Auto_tac} or \texttt{Force_tac}.
1787 \section{Datatype definitions}
1788 \label{sec:HOL:datatype}
1791 Inductive datatypes, similar to those of \ML, frequently appear in
1792 applications of Isabelle/HOL. In principle, such types could be defined by
1793 hand via \texttt{typedef} (see \S\ref{sec:typedef}), but this would be far too
1794 tedious. The \ttindex{datatype} definition package of Isabelle/HOL (cf.\
1795 \cite{Berghofer-Wenzel:1999:TPHOL}) automates such chores. It generates an
1796 appropriate \texttt{typedef} based on a least fixed-point construction, and
1797 proves freeness theorems and induction rules, as well as theorems for
1798 recursion and case combinators. The user just has to give a simple
1799 specification of new inductive types using a notation similar to {\ML} or
1802 The current datatype package can handle both mutual and indirect recursion.
1803 It also offers to represent existing types as datatypes giving the advantage
1804 of a more uniform view on standard theories.
1808 \label{subsec:datatype:basics}
1810 A general \texttt{datatype} definition is of the following form:
1813 \mathtt{datatype} & (\alpha@1,\ldots,\alpha@h)t@1 & = &
1814 C^1@1~\tau^1@{1,1}~\ldots~\tau^1@{1,m^1@1} ~\mid~ \ldots ~\mid~
1815 C^1@{k@1}~\tau^1@{k@1,1}~\ldots~\tau^1@{k@1,m^1@{k@1}} \\
1817 \mathtt{and} & (\alpha@1,\ldots,\alpha@h)t@n & = &
1818 C^n@1~\tau^n@{1,1}~\ldots~\tau^n@{1,m^n@1} ~\mid~ \ldots ~\mid~
1819 C^n@{k@n}~\tau^n@{k@n,1}~\ldots~\tau^n@{k@n,m^n@{k@n}}
1822 where $\alpha@i$ are type variables, $C^j@i$ are distinct constructor
1823 names and $\tau^j@{i,i'}$ are {\em admissible} types containing at
1824 most the type variables $\alpha@1, \ldots, \alpha@h$. A type $\tau$
1825 occurring in a \texttt{datatype} definition is {\em admissible} iff
1827 \item $\tau$ is non-recursive, i.e.\ $\tau$ does not contain any of the
1828 newly defined type constructors $t@1,\ldots,t@n$, or
1829 \item $\tau = (\alpha@1,\ldots,\alpha@h)t@{j'}$ where $1 \leq j' \leq n$, or
1830 \item $\tau = (\tau'@1,\ldots,\tau'@{h'})t'$, where $t'$ is
1831 the type constructor of an already existing datatype and $\tau'@1,\ldots,\tau'@{h'}$
1832 are admissible types.
1833 \item $\tau = \sigma \rightarrow \tau'$, where $\tau'$ is an admissible
1834 type and $\sigma$ is non-recursive (i.e. the occurrences of the newly defined
1835 types are {\em strictly positive})
1837 If some $(\alpha@1,\ldots,\alpha@h)t@{j'}$ occurs in a type $\tau^j@{i,i'}$
1840 (\ldots,\ldots ~ (\alpha@1,\ldots,\alpha@h)t@{j'} ~ \ldots,\ldots)t'
1842 this is called a {\em nested} (or \emph{indirect}) occurrence. A very simple
1843 example of a datatype is the type \texttt{list}, which can be defined by
1845 datatype 'a list = Nil
1848 Arithmetic expressions \texttt{aexp} and boolean expressions \texttt{bexp} can be modelled
1849 by the mutually recursive datatype definition
1851 datatype 'a aexp = If_then_else ('a bexp) ('a aexp) ('a aexp)
1852 | Sum ('a aexp) ('a aexp)
1853 | Diff ('a aexp) ('a aexp)
1856 and 'a bexp = Less ('a aexp) ('a aexp)
1857 | And ('a bexp) ('a bexp)
1858 | Or ('a bexp) ('a bexp)
1860 The datatype \texttt{term}, which is defined by
1862 datatype ('a, 'b) term = Var 'a
1863 | App 'b ((('a, 'b) term) list)
1865 is an example for a datatype with nested recursion. Using nested recursion
1866 involving function spaces, we may also define infinitely branching datatypes, e.g.
1868 datatype 'a tree = Atom 'a | Branch "nat => 'a tree"
1873 Types in HOL must be non-empty. Each of the new datatypes
1874 $(\alpha@1,\ldots,\alpha@h)t@j$ with $1 \le j \le n$ is non-empty iff it has a
1875 constructor $C^j@i$ with the following property: for all argument types
1876 $\tau^j@{i,i'}$ of the form $(\alpha@1,\ldots,\alpha@h)t@{j'}$ the datatype
1877 $(\alpha@1,\ldots,\alpha@h)t@{j'}$ is non-empty.
1879 If there are no nested occurrences of the newly defined datatypes, obviously
1880 at least one of the newly defined datatypes $(\alpha@1,\ldots,\alpha@h)t@j$
1881 must have a constructor $C^j@i$ without recursive arguments, a \emph{base
1882 case}, to ensure that the new types are non-empty. If there are nested
1883 occurrences, a datatype can even be non-empty without having a base case
1884 itself. Since \texttt{list} is a non-empty datatype, \texttt{datatype t = C (t
1885 list)} is non-empty as well.
1888 \subsubsection{Freeness of the constructors}
1890 The datatype constructors are automatically defined as functions of their
1892 \[ C^j@i :: [\tau^j@{i,1},\dots,\tau^j@{i,m^j@i}] \To (\alpha@1,\dots,\alpha@h)t@j \]
1893 These functions have certain {\em freeness} properties. They construct
1896 C^j@i~x@1~\dots~x@{m^j@i} \neq C^j@{i'}~y@1~\dots~y@{m^j@{i'}} \qquad
1897 \mbox{for all}~ i \neq i'.
1899 The constructor functions are injective:
1901 (C^j@i~x@1~\dots~x@{m^j@i} = C^j@i~y@1~\dots~y@{m^j@i}) =
1902 (x@1 = y@1 \land \dots \land x@{m^j@i} = y@{m^j@i})
1904 Since the number of distinctness inequalities is quadratic in the number of
1905 constructors, the datatype package avoids proving them separately if there are
1906 too many constructors. Instead, specific inequalities are proved by a suitable
1907 simplification procedure on demand.\footnote{This procedure, which is already part
1908 of the default simpset, may be referred to by the ML identifier
1909 \texttt{DatatypePackage.distinct_simproc}.}
1911 \subsubsection{Structural induction}
1913 The datatype package also provides structural induction rules. For
1914 datatypes without nested recursion, this is of the following form:
1916 \infer{P@1~x@1 \wedge \dots \wedge P@n~x@n}
1918 \Forall x@1 \dots x@{m^1@1}.
1919 \List{P@{s^1@{1,1}}~x@{r^1@{1,1}}; \dots;
1920 P@{s^1@{1,l^1@1}}~x@{r^1@{1,l^1@1}}} & \Imp &
1921 P@1~\left(C^1@1~x@1~\dots~x@{m^1@1}\right) \\
1923 \Forall x@1 \dots x@{m^1@{k@1}}.
1924 \List{P@{s^1@{k@1,1}}~x@{r^1@{k@1,1}}; \dots;
1925 P@{s^1@{k@1,l^1@{k@1}}}~x@{r^1@{k@1,l^1@{k@1}}}} & \Imp &
1926 P@1~\left(C^1@{k@1}~x@1~\ldots~x@{m^1@{k@1}}\right) \\
1928 \Forall x@1 \dots x@{m^n@1}.
1929 \List{P@{s^n@{1,1}}~x@{r^n@{1,1}}; \dots;
1930 P@{s^n@{1,l^n@1}}~x@{r^n@{1,l^n@1}}} & \Imp &
1931 P@n~\left(C^n@1~x@1~\ldots~x@{m^n@1}\right) \\
1933 \Forall x@1 \dots x@{m^n@{k@n}}.
1934 \List{P@{s^n@{k@n,1}}~x@{r^n@{k@n,1}}; \dots
1935 P@{s^n@{k@n,l^n@{k@n}}}~x@{r^n@{k@n,l^n@{k@n}}}} & \Imp &
1936 P@n~\left(C^n@{k@n}~x@1~\ldots~x@{m^n@{k@n}}\right)
1943 \left\{\left(r^j@{i,1},s^j@{i,1}\right),\ldots,
1944 \left(r^j@{i,l^j@i},s^j@{i,l^j@i}\right)\right\} = \\[2ex]
1945 && \left\{(i',i'')~\left|~
1946 1\leq i' \leq m^j@i \wedge 1 \leq i'' \leq n \wedge
1947 \tau^j@{i,i'} = (\alpha@1,\ldots,\alpha@h)t@{i''}\right.\right\}
1950 i.e.\ the properties $P@j$ can be assumed for all recursive arguments.
1952 For datatypes with nested recursion, such as the \texttt{term} example from
1953 above, things are a bit more complicated. Conceptually, Isabelle/HOL unfolds
1956 datatype ('a, 'b) term = Var 'a
1957 | App 'b ((('a, 'b) term) list)
1959 to an equivalent definition without nesting:
1961 datatype ('a, 'b) term = Var
1962 | App 'b (('a, 'b) term_list)
1963 and ('a, 'b) term_list = Nil'
1964 | Cons' (('a,'b) term) (('a,'b) term_list)
1966 Note however, that the type \texttt{('a,'b) term_list} and the constructors {\tt
1967 Nil'} and \texttt{Cons'} are not really introduced. One can directly work with
1968 the original (isomorphic) type \texttt{(('a, 'b) term) list} and its existing
1969 constructors \texttt{Nil} and \texttt{Cons}. Thus, the structural induction rule for
1970 \texttt{term} gets the form
1972 \infer{P@1~x@1 \wedge P@2~x@2}
1974 \Forall x.~P@1~(\mathtt{Var}~x) \\
1975 \Forall x@1~x@2.~P@2~x@2 \Imp P@1~(\mathtt{App}~x@1~x@2) \\
1977 \Forall x@1~x@2. \List{P@1~x@1; P@2~x@2} \Imp P@2~(\mathtt{Cons}~x@1~x@2)
1980 Note that there are two predicates $P@1$ and $P@2$, one for the type \texttt{('a,'b) term}
1981 and one for the type \texttt{(('a, 'b) term) list}.
1983 For a datatype with function types such as \texttt{'a tree}, the induction rule
1987 {\Forall a.~P~(\mathtt{Atom}~a) &
1988 \Forall ts.~(\forall x.~P~(ts~x)) \Imp P~(\mathtt{Branch}~ts)}
1991 \medskip In principle, inductive types are already fully determined by
1992 freeness and structural induction. For convenience in applications,
1993 the following derived constructions are automatically provided for any
1996 \subsubsection{The \sdx{case} construct}
1998 The type comes with an \ML-like \texttt{case}-construct:
2001 \mbox{\tt case}~e~\mbox{\tt of} & C^j@1~x@{1,1}~\dots~x@{1,m^j@1} & \To & e@1 \\
2003 \mid & C^j@{k@j}~x@{k@j,1}~\dots~x@{k@j,m^j@{k@j}} & \To & e@{k@j}
2006 where the $x@{i,j}$ are either identifiers or nested tuple patterns as in
2007 \S\ref{subsec:prod-sum}.
2009 All constructors must be present, their order is fixed, and nested patterns
2010 are not supported (with the exception of tuples). Violating this
2011 restriction results in strange error messages.
2014 To perform case distinction on a goal containing a \texttt{case}-construct,
2015 the theorem $t@j.$\texttt{split} is provided:
2017 \begin{array}{@{}rcl@{}}
2018 P(t@j_\mathtt{case}~f@1~\dots~f@{k@j}~e) &\!\!\!=&
2019 \!\!\! ((\forall x@1 \dots x@{m^j@1}. e = C^j@1~x@1\dots x@{m^j@1} \to
2020 P(f@1~x@1\dots x@{m^j@1})) \\
2021 &&\!\!\! ~\land~ \dots ~\land \\
2022 &&~\!\!\! (\forall x@1 \dots x@{m^j@{k@j}}. e = C^j@{k@j}~x@1\dots x@{m^j@{k@j}} \to
2023 P(f@{k@j}~x@1\dots x@{m^j@{k@j}})))
2026 where $t@j$\texttt{_case} is the internal name of the \texttt{case}-construct.
2027 This theorem can be added to a simpset via \ttindex{addsplits}
2028 (see~\S\ref{subsec:HOL:case:splitting}).
2030 \subsubsection{The function \cdx{size}}\label{sec:HOL:size}
2032 Theory \texttt{Arith} declares a generic function \texttt{size} of type
2033 $\alpha\To nat$. Each datatype defines a particular instance of \texttt{size}
2034 by overloading according to the following scheme:
2035 %%% FIXME: This formula is too big and is completely unreadable
2037 size(C^j@i~x@1~\dots~x@{m^j@i}) = \!
2040 0 & \!\mbox{if $Rec^j@i = \emptyset$} \\
2041 1+\sum\limits@{h=1}^{l^j@i}size~x@{r^j@{i,h}} &
2042 \!\mbox{if $Rec^j@i = \left\{\left(r^j@{i,1},s^j@{i,1}\right),\ldots,
2043 \left(r^j@{i,l^j@i},s^j@{i,l^j@i}\right)\right\}$}
2047 where $Rec^j@i$ is defined above. Viewing datatypes as generalised trees, the
2048 size of a leaf is 0 and the size of a node is the sum of the sizes of its
2051 \subsection{Defining datatypes}
2053 The theory syntax for datatype definitions is shown in
2054 Fig.~\ref{datatype-grammar}. In order to be well-formed, a datatype
2055 definition has to obey the rules stated in the previous section. As a result
2056 the theory is extended with the new types, the constructors, and the theorems
2057 listed in the previous section.
2061 datatype : 'datatype' typedecls;
2063 typedecls: ( newtype '=' (cons + '|') ) + 'and'
2065 newtype : typevarlist id ( () | '(' infix ')' )
2067 cons : name (argtype *) ( () | ( '(' mixfix ')' ) )
2069 argtype : id | tid | ('(' typevarlist id ')')
2072 \caption{Syntax of datatype declarations}
2073 \label{datatype-grammar}
2076 Most of the theorems about datatypes become part of the default simpset and
2077 you never need to see them again because the simplifier applies them
2078 automatically. Only induction or exhaustion are usually invoked by hand.
2079 \begin{ttdescription}
2080 \item[\ttindexbold{induct_tac} {\tt"}$x${\tt"} $i$]
2081 applies structural induction on variable $x$ to subgoal $i$, provided the
2082 type of $x$ is a datatype.
2083 \item[\ttindexbold{mutual_induct_tac}
2084 {\tt["}$x@1${\tt",}$\ldots${\tt,"}$x@n${\tt"]} $i$] applies simultaneous
2085 structural induction on the variables $x@1,\ldots,x@n$ to subgoal $i$. This
2086 is the canonical way to prove properties of mutually recursive datatypes
2087 such as \texttt{aexp} and \texttt{bexp}, or datatypes with nested recursion such as
2090 In some cases, induction is overkill and a case distinction over all
2091 constructors of the datatype suffices.
2092 \begin{ttdescription}
2093 \item[\ttindexbold{exhaust_tac} {\tt"}$u${\tt"} $i$]
2094 performs an exhaustive case analysis for the term $u$ whose type
2095 must be a datatype. If the datatype has $k@j$ constructors
2096 $C^j@1$, \dots $C^j@{k@j}$, subgoal $i$ is replaced by $k@j$ new subgoals which
2097 contain the additional assumption $u = C^j@{i'}~x@1~\dots~x@{m^j@{i'}}$ for
2098 $i'=1$, $\dots$,~$k@j$.
2101 Note that induction is only allowed on free variables that should not occur
2102 among the premises of the subgoal. Exhaustion applies to arbitrary terms.
2107 For the technically minded, we exhibit some more details. Processing the
2108 theory file produces an \ML\ structure which, in addition to the usual
2109 components, contains a structure named $t$ for each datatype $t$ defined in
2110 the file. Each structure $t$ contains the following elements:
2112 val distinct : thm list
2113 val inject : thm list
2116 val cases : thm list
2121 val simps : thm list
2123 \texttt{distinct}, \texttt{inject}, \texttt{induct}, \texttt{size}
2124 and \texttt{split} contain the theorems
2125 described above. For user convenience, \texttt{distinct} contains
2126 inequalities in both directions. The reduction rules of the {\tt
2127 case}-construct are in \texttt{cases}. All theorems from {\tt
2128 distinct}, \texttt{inject} and \texttt{cases} are combined in \texttt{simps}.
2129 In case of mutually recursive datatypes, \texttt{recs}, \texttt{size}, \texttt{induct}
2130 and \texttt{simps} are contained in a separate structure named $t@1_\ldots_t@n$.
2133 \subsection{Representing existing types as datatypes}
2135 For foundational reasons, some basic types such as \texttt{nat}, \texttt{*}, {\tt
2136 +}, \texttt{bool} and \texttt{unit} are not defined in a \texttt{datatype} section,
2137 but by more primitive means using \texttt{typedef}. To be able to use the
2138 tactics \texttt{induct_tac} and \texttt{exhaust_tac} and to define functions by
2139 primitive recursion on these types, such types may be represented as actual
2140 datatypes. This is done by specifying an induction rule, as well as theorems
2141 stating the distinctness and injectivity of constructors in a {\tt
2142 rep_datatype} section. For type \texttt{nat} this works as follows:
2145 distinct Suc_not_Zero, Zero_not_Suc
2149 The datatype package automatically derives additional theorems for recursion
2150 and case combinators from these rules. Any of the basic HOL types mentioned
2151 above are represented as datatypes. Try an induction on \texttt{bool}
2155 \subsection{Examples}
2157 \subsubsection{The datatype $\alpha~mylist$}
2159 We want to define a type $\alpha~mylist$. To do this we have to build a new
2160 theory that contains the type definition. We start from the theory
2161 \texttt{Datatype} instead of \texttt{Main} in order to avoid clashes with the
2162 \texttt{List} theory of Isabelle/HOL.
2165 datatype 'a mylist = Nil | Cons 'a ('a mylist)
2168 After loading the theory, we can prove $Cons~x~xs\neq xs$, for example. To
2169 ease the induction applied below, we state the goal with $x$ quantified at the
2170 object-level. This will be stripped later using \ttindex{qed_spec_mp}.
2172 Goal "!x. Cons x xs ~= xs";
2174 {\out ! x. Cons x xs ~= xs}
2175 {\out 1. ! x. Cons x xs ~= xs}
2177 This can be proved by the structural induction tactic:
2179 by (induct_tac "xs" 1);
2181 {\out ! x. Cons x xs ~= xs}
2182 {\out 1. ! x. Cons x Nil ~= Nil}
2183 {\out 2. !!a mylist.}
2184 {\out ! x. Cons x mylist ~= mylist ==>}
2185 {\out ! x. Cons x (Cons a mylist) ~= Cons a mylist}
2187 The first subgoal can be proved using the simplifier. Isabelle/HOL has
2188 already added the freeness properties of lists to the default simplification
2193 {\out ! x. Cons x xs ~= xs}
2194 {\out 1. !!a mylist.}
2195 {\out ! x. Cons x mylist ~= mylist ==>}
2196 {\out ! x. Cons x (Cons a mylist) ~= Cons a mylist}
2198 Similarly, we prove the remaining goal.
2200 by (Asm_simp_tac 1);
2202 {\out ! x. Cons x xs ~= xs}
2205 qed_spec_mp "not_Cons_self";
2206 {\out val not_Cons_self = "Cons x xs ~= xs" : thm}
2208 Because both subgoals could have been proved by \texttt{Asm_simp_tac}
2209 we could have done that in one step:
2211 by (ALLGOALS Asm_simp_tac);
2215 \subsubsection{The datatype $\alpha~mylist$ with mixfix syntax}
2217 In this example we define the type $\alpha~mylist$ again but this time
2218 we want to write \texttt{[]} for \texttt{Nil} and we want to use infix
2219 notation \verb|#| for \texttt{Cons}. To do this we simply add mixfix
2220 annotations after the constructor declarations as follows:
2223 datatype 'a mylist =
2225 Cons 'a ('a mylist) (infixr "#" 70)
2228 Now the theorem in the previous example can be written \verb|x#xs ~= xs|.
2231 \subsubsection{A datatype for weekdays}
2233 This example shows a datatype that consists of 7 constructors:
2236 datatype days = Mon | Tue | Wed | Thu | Fri | Sat | Sun
2239 Because there are more than 6 constructors, inequality is expressed via a function
2240 \verb|days_ord|. The theorem \verb|Mon ~= Tue| is not directly
2241 contained among the distinctness theorems, but the simplifier can
2242 prove it thanks to rewrite rules inherited from theory \texttt{Arith}:
2247 You need not derive such inequalities explicitly: the simplifier will dispose
2248 of them automatically.
2252 \section{Recursive function definitions}\label{sec:HOL:recursive}
2253 \index{recursive functions|see{recursion}}
2255 Isabelle/HOL provides two main mechanisms of defining recursive functions.
2257 \item \textbf{Primitive recursion} is available only for datatypes, and it is
2258 somewhat restrictive. Recursive calls are only allowed on the argument's
2259 immediate constituents. On the other hand, it is the form of recursion most
2260 often wanted, and it is easy to use.
2262 \item \textbf{Well-founded recursion} requires that you supply a well-founded
2263 relation that governs the recursion. Recursive calls are only allowed if
2264 they make the argument decrease under the relation. Complicated recursion
2265 forms, such as nested recursion, can be dealt with. Termination can even be
2266 proved at a later time, though having unsolved termination conditions around
2267 can make work difficult.%
2268 \footnote{This facility is based on Konrad Slind's TFL
2269 package~\cite{slind-tfl}. Thanks are due to Konrad for implementing TFL
2270 and assisting with its installation.}
2273 Following good HOL tradition, these declarations do not assert arbitrary
2274 axioms. Instead, they define the function using a recursion operator. Both
2275 HOL and ZF derive the theory of well-founded recursion from first
2276 principles~\cite{paulson-set-II}. Primitive recursion over some datatype
2277 relies on the recursion operator provided by the datatype package. With
2278 either form of function definition, Isabelle proves the desired recursion
2279 equations as theorems.
2282 \subsection{Primitive recursive functions}
2283 \label{sec:HOL:primrec}
2284 \index{recursion!primitive|(}
2287 Datatypes come with a uniform way of defining functions, {\bf primitive
2288 recursion}. In principle, one could introduce primitive recursive functions
2289 by asserting their reduction rules as new axioms, but this is not recommended:
2290 \begin{ttbox}\slshape
2292 consts app :: ['a list, 'a list] => 'a list
2294 app_Nil "app [] ys = ys"
2295 app_Cons "app (x#xs) ys = x#app xs ys"
2298 Asserting axioms brings the danger of accidentally asserting nonsense, as
2299 in \verb$app [] ys = us$.
2301 The \ttindex{primrec} declaration is a safe means of defining primitive
2302 recursive functions on datatypes:
2305 consts app :: ['a list, 'a list] => 'a list
2308 "app (x#xs) ys = x#app xs ys"
2311 Isabelle will now check that the two rules do indeed form a primitive
2312 recursive definition. For example
2317 is rejected with an error message ``\texttt{Extra variables on rhs}''.
2321 The general form of a primitive recursive definition is
2324 {\it reduction rules}
2326 where \textit{reduction rules} specify one or more equations of the form
2327 \[ f \, x@1 \, \dots \, x@m \, (C \, y@1 \, \dots \, y@k) \, z@1 \,
2328 \dots \, z@n = r \] such that $C$ is a constructor of the datatype, $r$
2329 contains only the free variables on the left-hand side, and all recursive
2330 calls in $r$ are of the form $f \, \dots \, y@i \, \dots$ for some $i$. There
2331 must be at most one reduction rule for each constructor. The order is
2332 immaterial. For missing constructors, the function is defined to return a
2335 If you would like to refer to some rule by name, then you must prefix
2336 the rule with an identifier. These identifiers, like those in the
2337 \texttt{rules} section of a theory, will be visible at the \ML\ level.
2339 The primitive recursive function can have infix or mixfix syntax:
2340 \begin{ttbox}\underscoreon
2341 consts "@" :: ['a list, 'a list] => 'a list (infixr 60)
2344 "(x#xs) @ ys = x#(xs @ ys)"
2347 The reduction rules become part of the default simpset, which
2348 leads to short proof scripts:
2349 \begin{ttbox}\underscoreon
2350 Goal "(xs @ ys) @ zs = xs @ (ys @ zs)";
2351 by (induct\_tac "xs" 1);
2352 by (ALLGOALS Asm\_simp\_tac);
2355 \subsubsection{Example: Evaluation of expressions}
2356 Using mutual primitive recursion, we can define evaluation functions \texttt{evala}
2357 and \texttt{eval_bexp} for the datatypes of arithmetic and boolean expressions mentioned in
2358 \S\ref{subsec:datatype:basics}:
2361 evala :: "['a => nat, 'a aexp] => nat"
2362 evalb :: "['a => nat, 'a bexp] => bool"
2365 "evala env (If_then_else b a1 a2) =
2366 (if evalb env b then evala env a1 else evala env a2)"
2367 "evala env (Sum a1 a2) = evala env a1 + evala env a2"
2368 "evala env (Diff a1 a2) = evala env a1 - evala env a2"
2369 "evala env (Var v) = env v"
2370 "evala env (Num n) = n"
2372 "evalb env (Less a1 a2) = (evala env a1 < evala env a2)"
2373 "evalb env (And b1 b2) = (evalb env b1 & evalb env b2)"
2374 "evalb env (Or b1 b2) = (evalb env b1 & evalb env b2)"
2376 Since the value of an expression depends on the value of its variables,
2377 the functions \texttt{evala} and \texttt{evalb} take an additional
2378 parameter, an {\em environment} of type \texttt{'a => nat}, which maps
2379 variables to their values.
2381 Similarly, we may define substitution functions \texttt{substa}
2382 and \texttt{substb} for expressions: The mapping \texttt{f} of type
2383 \texttt{'a => 'a aexp} given as a parameter is lifted canonically
2384 on the types \texttt{'a aexp} and \texttt{'a bexp}:
2387 substa :: "['a => 'b aexp, 'a aexp] => 'b aexp"
2388 substb :: "['a => 'b aexp, 'a bexp] => 'b bexp"
2391 "substa f (If_then_else b a1 a2) =
2392 If_then_else (substb f b) (substa f a1) (substa f a2)"
2393 "substa f (Sum a1 a2) = Sum (substa f a1) (substa f a2)"
2394 "substa f (Diff a1 a2) = Diff (substa f a1) (substa f a2)"
2395 "substa f (Var v) = f v"
2396 "substa f (Num n) = Num n"
2398 "substb f (Less a1 a2) = Less (substa f a1) (substa f a2)"
2399 "substb f (And b1 b2) = And (substb f b1) (substb f b2)"
2400 "substb f (Or b1 b2) = Or (substb f b1) (substb f b2)"
2402 In textbooks about semantics one often finds {\em substitution theorems},
2403 which express the relationship between substitution and evaluation. For
2404 \texttt{'a aexp} and \texttt{'a bexp}, we can prove such a theorem by mutual
2405 induction, followed by simplification:
2408 "evala env (substa (Var(v := a')) a) =
2409 evala (env(v := evala env a')) a &
2410 evalb env (substb (Var(v := a')) b) =
2411 evalb (env(v := evala env a')) b";
2412 by (mutual_induct_tac ["a","b"] 1);
2413 by (ALLGOALS Asm_full_simp_tac);
2416 \subsubsection{Example: A substitution function for terms}
2417 Functions on datatypes with nested recursion, such as the type
2418 \texttt{term} mentioned in \S\ref{subsec:datatype:basics}, are
2419 also defined by mutual primitive recursion. A substitution
2420 function \texttt{subst_term} on type \texttt{term}, similar to the functions
2421 \texttt{substa} and \texttt{substb} described above, can
2422 be defined as follows:
2425 subst_term :: "['a => ('a, 'b) term, ('a, 'b) term] => ('a, 'b) term"
2427 "['a => ('a, 'b) term, ('a, 'b) term list] => ('a, 'b) term list"
2430 "subst_term f (Var a) = f a"
2431 "subst_term f (App b ts) = App b (subst_term_list f ts)"
2433 "subst_term_list f [] = []"
2434 "subst_term_list f (t # ts) =
2435 subst_term f t # subst_term_list f ts"
2437 The recursion scheme follows the structure of the unfolded definition of type
2438 \texttt{term} shown in \S\ref{subsec:datatype:basics}. To prove properties of
2439 this substitution function, mutual induction is needed:
2442 "(subst_term ((subst_term f1) o f2) t) =
2443 (subst_term f1 (subst_term f2 t)) &
2444 (subst_term_list ((subst_term f1) o f2) ts) =
2445 (subst_term_list f1 (subst_term_list f2 ts))";
2446 by (mutual_induct_tac ["t", "ts"] 1);
2447 by (ALLGOALS Asm_full_simp_tac);
2450 \subsubsection{Example: A map function for infinitely branching trees}
2451 Defining functions on infinitely branching datatypes by primitive
2452 recursion is just as easy. For example, we can define a function
2453 \texttt{map_tree} on \texttt{'a tree} as follows:
2456 map_tree :: "('a => 'b) => 'a tree => 'b tree"
2459 "map_tree f (Atom a) = Atom (f a)"
2460 "map_tree f (Branch ts) = Branch (\%x. map_tree f (ts x))"
2462 Note that all occurrences of functions such as \texttt{ts} in the
2463 \texttt{primrec} clauses must be applied to an argument. In particular,
2464 \texttt{map_tree f o ts} is not allowed.
2466 \index{recursion!primitive|)}
2470 \subsection{General recursive functions}
2471 \label{sec:HOL:recdef}
2472 \index{recursion!general|(}
2475 Using \texttt{recdef}, you can declare functions involving nested recursion
2476 and pattern-matching. Recursion need not involve datatypes and there are few
2477 syntactic restrictions. Termination is proved by showing that each recursive
2478 call makes the argument smaller in a suitable sense, which you specify by
2479 supplying a well-founded relation.
2481 Here is a simple example, the Fibonacci function. The first line declares
2482 \texttt{fib} to be a constant. The well-founded relation is simply~$<$ (on
2483 the natural numbers). Pattern-matching is used here: \texttt{1} is a
2484 macro for \texttt{Suc~0}.
2486 consts fib :: "nat => nat"
2487 recdef fib "less_than"
2490 "fib (Suc(Suc x)) = (fib x + fib (Suc x))"
2493 With \texttt{recdef}, function definitions may be incomplete, and patterns may
2494 overlap, as in functional programming. The \texttt{recdef} package
2495 disambiguates overlapping patterns by taking the order of rules into account.
2496 For missing patterns, the function is defined to return a default value.
2498 %For example, here is a declaration of the list function \cdx{hd}:
2500 %consts hd :: 'a list => 'a
2504 %Because this function is not recursive, we may supply the empty well-founded
2507 The well-founded relation defines a notion of ``smaller'' for the function's
2508 argument type. The relation $\prec$ is \textbf{well-founded} provided it
2509 admits no infinitely decreasing chains
2510 \[ \cdots\prec x@n\prec\cdots\prec x@1. \]
2511 If the function's argument has type~$\tau$, then $\prec$ has to be a relation
2512 over~$\tau$: it must have type $(\tau\times\tau)set$.
2514 Proving well-foundedness can be tricky, so Isabelle/HOL provides a collection
2515 of operators for building well-founded relations. The package recognises
2516 these operators and automatically proves that the constructed relation is
2517 well-founded. Here are those operators, in order of importance:
2519 \item \texttt{less_than} is ``less than'' on the natural numbers.
2520 (It has type $(nat\times nat)set$, while $<$ has type $[nat,nat]\To bool$.
2522 \item $\mathop{\mathtt{measure}} f$, where $f$ has type $\tau\To nat$, is the
2523 relation~$\prec$ on type~$\tau$ such that $x\prec y$ iff $f(x)<f(y)$.
2524 Typically, $f$ takes the recursive function's arguments (as a tuple) and
2525 returns a result expressed in terms of the function \texttt{size}. It is
2526 called a \textbf{measure function}. Recall that \texttt{size} is overloaded
2527 and is defined on all datatypes (see \S\ref{sec:HOL:size}).
2529 \item $\mathop{\mathtt{inv_image}} f\;R$ is a generalisation of
2530 \texttt{measure}. It specifies a relation such that $x\prec y$ iff $f(x)$
2531 is less than $f(y)$ according to~$R$, which must itself be a well-founded
2534 \item $R@1\texttt{**}R@2$ is the lexicographic product of two relations. It
2535 is a relation on pairs and satisfies $(x@1,x@2)\prec(y@1,y@2)$ iff $x@1$
2536 is less than $y@1$ according to~$R@1$ or $x@1=y@1$ and $x@2$
2537 is less than $y@2$ according to~$R@2$.
2539 \item \texttt{finite_psubset} is the proper subset relation on finite sets.
2542 We can use \texttt{measure} to declare Euclid's algorithm for the greatest
2543 common divisor. The measure function, $\lambda(m,n). n$, specifies that the
2544 recursion terminates because argument~$n$ decreases.
2546 recdef gcd "measure ((\%(m,n). n) ::nat*nat=>nat)"
2547 "gcd (m, n) = (if n=0 then m else gcd(n, m mod n))"
2550 The general form of a well-founded recursive definition is
2552 recdef {\it function} {\it rel}
2553 congs {\it congruence rules} {\bf(optional)}
2554 simpset {\it simplification set} {\bf(optional)}
2555 {\it reduction rules}
2559 \item \textit{function} is the name of the function, either as an \textit{id}
2560 or a \textit{string}.
2562 \item \textit{rel} is a {\HOL} expression for the well-founded termination
2565 \item \textit{congruence rules} are required only in highly exceptional
2568 \item The \textit{simplification set} is used to prove that the supplied
2569 relation is well-founded. It is also used to prove the \textbf{termination
2570 conditions}: assertions that arguments of recursive calls decrease under
2571 \textit{rel}. By default, simplification uses \texttt{simpset()}, which
2572 is sufficient to prove well-foundedness for the built-in relations listed
2575 \item \textit{reduction rules} specify one or more recursion equations. Each
2576 left-hand side must have the form $f\,t$, where $f$ is the function and $t$
2577 is a tuple of distinct variables. If more than one equation is present then
2578 $f$ is defined by pattern-matching on components of its argument whose type
2579 is a \texttt{datatype}.
2581 Unlike with \texttt{primrec}, the reduction rules are not added to the
2582 default simpset, and individual rules may not be labelled with identifiers.
2583 However, the identifier $f$\texttt{.rules} is visible at the \ML\ level
2584 as a list of theorems.
2587 With the definition of \texttt{gcd} shown above, Isabelle/HOL is unable to
2588 prove one termination condition. It remains as a precondition of the
2592 {\out ["! m n. n ~= 0 --> m mod n < n}
2593 {\out ==> gcd (?m, ?n) = (if ?n = 0 then ?m else gcd (?n, ?m mod ?n))"] }
2596 The theory \texttt{HOL/ex/Primes} illustrates how to prove termination
2597 conditions afterwards. The function \texttt{Tfl.tgoalw} is like the standard
2598 function \texttt{goalw}, which sets up a goal to prove, but its argument
2599 should be the identifier $f$\texttt{.rules} and its effect is to set up a
2600 proof of the termination conditions:
2602 Tfl.tgoalw thy [] gcd.rules;
2604 {\out ! m n. n ~= 0 --> m mod n < n}
2605 {\out 1. ! m n. n ~= 0 --> m mod n < n}
2607 This subgoal has a one-step proof using \texttt{simp_tac}. Once the theorem
2608 is proved, it can be used to eliminate the termination conditions from
2609 elements of \texttt{gcd.rules}. Theory \texttt{HOL/Subst/Unify} is a much
2610 more complicated example of this process, where the termination conditions can
2611 only be proved by complicated reasoning involving the recursive function
2614 Isabelle/HOL can prove the \texttt{gcd} function's termination condition
2615 automatically if supplied with the right simpset.
2617 recdef gcd "measure ((\%(m,n). n) ::nat*nat=>nat)"
2618 simpset "simpset() addsimps [mod_less_divisor, zero_less_eq]"
2619 "gcd (m, n) = (if n=0 then m else gcd(n, m mod n))"
2622 A \texttt{recdef} definition also returns an induction rule specialised for
2623 the recursive function. For the \texttt{gcd} function above, the induction
2627 {\out "(!!m n. n ~= 0 --> ?P n (m mod n) ==> ?P m n) ==> ?P ?u ?v" : thm}
2629 This rule should be used to reason inductively about the \texttt{gcd}
2630 function. It usually makes the induction hypothesis available at all
2631 recursive calls, leading to very direct proofs. If any termination conditions
2632 remain unproved, they will become additional premises of this rule.
2634 \index{recursion!general|)}
2638 \section{Inductive and coinductive definitions}
2639 \index{*inductive|(}
2640 \index{*coinductive|(}
2642 An {\bf inductive definition} specifies the least set~$R$ closed under given
2643 rules. (Applying a rule to elements of~$R$ yields a result within~$R$.) For
2644 example, a structural operational semantics is an inductive definition of an
2645 evaluation relation. Dually, a {\bf coinductive definition} specifies the
2646 greatest set~$R$ consistent with given rules. (Every element of~$R$ can be
2647 seen as arising by applying a rule to elements of~$R$.) An important example
2648 is using bisimulation relations to formalise equivalence of processes and
2649 infinite data structures.
2651 A theory file may contain any number of inductive and coinductive
2652 definitions. They may be intermixed with other declarations; in
2653 particular, the (co)inductive sets {\bf must} be declared separately as
2654 constants, and may have mixfix syntax or be subject to syntax translations.
2656 Each (co)inductive definition adds definitions to the theory and also
2657 proves some theorems. Each definition creates an \ML\ structure, which is a
2658 substructure of the main theory structure.
2660 This package is related to the \ZF\ one, described in a separate
2662 \footnote{It appeared in CADE~\cite{paulson-CADE}; a longer version is
2663 distributed with Isabelle.} %
2664 which you should refer to in case of difficulties. The package is simpler
2665 than \ZF's thanks to \HOL's extra-logical automatic type-checking. The types
2666 of the (co)inductive sets determine the domain of the fixedpoint definition,
2667 and the package does not have to use inference rules for type-checking.
2670 \subsection{The result structure}
2671 Many of the result structure's components have been discussed in the paper;
2672 others are self-explanatory.
2674 \item[\tt defs] is the list of definitions of the recursive sets.
2676 \item[\tt mono] is a monotonicity theorem for the fixedpoint operator.
2678 \item[\tt unfold] is a fixedpoint equation for the recursive set (the union of
2679 the recursive sets, in the case of mutual recursion).
2681 \item[\tt intrs] is the list of introduction rules, now proved as theorems, for
2682 the recursive sets. The rules are also available individually, using the
2683 names given them in the theory file.
2685 \item[\tt elims] is the list of elimination rule.
2687 \item[\tt elim] is the head of the list \texttt{elims}.
2689 \item[\tt mk_cases] is a function to create simplified instances of {\tt
2690 elim} using freeness reasoning on underlying datatypes.
2693 For an inductive definition, the result structure contains the
2694 rule \texttt{induct}. For a
2695 coinductive definition, it contains the rule \verb|coinduct|.
2697 Figure~\ref{def-result-fig} summarises the two result signatures,
2698 specifying the types of all these components.
2706 val intrs : thm list
2707 val elims : thm list
2709 val mk_cases : string -> thm
2710 {\it(Inductive definitions only)}
2712 {\it(coinductive definitions only)}
2717 \caption{The {\ML} result of a (co)inductive definition} \label{def-result-fig}
2720 \subsection{The syntax of a (co)inductive definition}
2721 An inductive definition has the form
2723 inductive {\it inductive sets}
2724 intrs {\it introduction rules}
2725 monos {\it monotonicity theorems}
2726 con_defs {\it constructor definitions}
2728 A coinductive definition is identical, except that it starts with the keyword
2729 \texttt{coinductive}.
2731 The \texttt{monos} and \texttt{con_defs} sections are optional. If present,
2732 each is specified by a list of identifiers.
2735 \item The \textit{inductive sets} are specified by one or more strings.
2737 \item The \textit{introduction rules} specify one or more introduction rules in
2738 the form \textit{ident\/}~\textit{string}, where the identifier gives the name of
2739 the rule in the result structure.
2741 \item The \textit{monotonicity theorems} are required for each operator
2742 applied to a recursive set in the introduction rules. There {\bf must}
2743 be a theorem of the form $A\subseteq B\Imp M(A)\subseteq M(B)$, for each
2744 premise $t\in M(R@i)$ in an introduction rule!
2746 \item The \textit{constructor definitions} contain definitions of constants
2747 appearing in the introduction rules. In most cases it can be omitted.
2751 \subsection{Example of an inductive definition}
2752 Two declarations, included in a theory file, define the finite powerset
2753 operator. First we declare the constant~\texttt{Fin}. Then we declare it
2754 inductively, with two introduction rules:
2756 consts Fin :: 'a set => 'a set set
2759 emptyI "{\ttlbrace}{\ttrbrace} : Fin A"
2760 insertI "[| a: A; b: Fin A |] ==> insert a b : Fin A"
2762 The resulting theory structure contains a substructure, called~\texttt{Fin}.
2763 It contains the \texttt{Fin}$~A$ introduction rules as the list \texttt{Fin.intrs},
2764 and also individually as \texttt{Fin.emptyI} and \texttt{Fin.consI}. The induction
2765 rule is \texttt{Fin.induct}.
2767 For another example, here is a theory file defining the accessible
2768 part of a relation. The main thing to note is the use of~\texttt{Pow} in
2769 the sole introduction rule, and the corresponding mention of the rule
2770 \verb|Pow_mono| in the \texttt{monos} list. The paper
2771 \cite{paulson-CADE} discusses a \ZF\ version of this example in more
2775 consts pred :: "['b, ('a * 'b)set] => 'a set" (*Set of predecessors*)
2776 acc :: "('a * 'a)set => 'a set" (*Accessible part*)
2777 defs pred_def "pred x r == {y. (y,x):r}"
2780 pred "pred a r: Pow(acc r) ==> a: acc r"
2784 The Isabelle distribution contains many other inductive definitions. Simple
2785 examples are collected on subdirectory \texttt{HOL/Induct}. The theory
2786 \texttt{HOL/Induct/LList} contains coinductive definitions. Larger examples
2787 may be found on other subdirectories of \texttt{HOL}, such as \texttt{IMP},
2788 \texttt{Lambda} and \texttt{Auth}.
2790 \index{*coinductive|)} \index{*inductive|)}
2793 \section{The examples directories}
2795 Directory \texttt{HOL/Auth} contains theories for proving the correctness of
2796 cryptographic protocols~\cite{paulson-jcs}. The approach is based upon
2797 operational semantics rather than the more usual belief logics. On the same
2798 directory are proofs for some standard examples, such as the Needham-Schroeder
2799 public-key authentication protocol and the Otway-Rees
2802 Directory \texttt{HOL/IMP} contains a formalization of various denotational,
2803 operational and axiomatic semantics of a simple while-language, the necessary
2804 equivalence proofs, soundness and completeness of the Hoare rules with
2805 respect to the denotational semantics, and soundness and completeness of a
2806 verification condition generator. Much of development is taken from
2807 Winskel~\cite{winskel93}. For details see~\cite{nipkow-IMP}.
2809 Directory \texttt{HOL/Hoare} contains a user friendly surface syntax for Hoare
2810 logic, including a tactic for generating verification-conditions.
2812 Directory \texttt{HOL/MiniML} contains a formalization of the type system of
2813 the core functional language Mini-ML and a correctness proof for its type
2814 inference algorithm $\cal W$~\cite{milner78,nipkow-W}.
2816 Directory \texttt{HOL/Lambda} contains a formalization of untyped
2817 $\lambda$-calculus in de~Bruijn notation and Church-Rosser proofs for $\beta$
2818 and $\eta$ reduction~\cite{Nipkow-CR}.
2820 Directory \texttt{HOL/Subst} contains Martin Coen's mechanization of a theory of
2821 substitutions and unifiers. It is based on Paulson's previous
2822 mechanisation in {\LCF}~\cite{paulson85} of Manna and Waldinger's
2823 theory~\cite{mw81}. It demonstrates a complicated use of \texttt{recdef},
2824 with nested recursion.
2826 Directory \texttt{HOL/Induct} presents simple examples of (co)inductive
2827 definitions and datatypes.
2829 \item Theory \texttt{PropLog} proves the soundness and completeness of
2830 classical propositional logic, given a truth table semantics. The only
2831 connective is $\imp$. A Hilbert-style axiom system is specified, and its
2832 set of theorems defined inductively. A similar proof in \ZF{} is
2833 described elsewhere~\cite{paulson-set-II}.
2835 \item Theory \texttt{Term} defines the datatype \texttt{term}.
2837 \item Theory \texttt{ABexp} defines arithmetic and boolean expressions
2838 as mutually recursive datatypes.
2840 \item The definition of lazy lists demonstrates methods for handling
2841 infinite data structures and coinduction in higher-order
2842 logic~\cite{paulson-coind}.%
2843 \footnote{To be precise, these lists are \emph{potentially infinite} rather
2844 than lazy. Lazy implies a particular operational semantics.}
2845 Theory \thydx{LList} defines an operator for
2846 corecursion on lazy lists, which is used to define a few simple functions
2847 such as map and append. A coinduction principle is defined
2848 for proving equations on lazy lists.
2850 \item Theory \thydx{LFilter} defines the filter functional for lazy lists.
2851 This functional is notoriously difficult to define because finding the next
2852 element meeting the predicate requires possibly unlimited search. It is not
2853 computable, but can be expressed using a combination of induction and
2856 \item Theory \thydx{Exp} illustrates the use of iterated inductive definitions
2857 to express a programming language semantics that appears to require mutual
2858 induction. Iterated induction allows greater modularity.
2861 Directory \texttt{HOL/ex} contains other examples and experimental proofs in
2864 \item Theory \texttt{Recdef} presents many examples of using \texttt{recdef}
2865 to define recursive functions. Another example is \texttt{Fib}, which
2866 defines the Fibonacci function.
2868 \item Theory \texttt{Primes} defines the Greatest Common Divisor of two
2869 natural numbers and proves a key lemma of the Fundamental Theorem of
2870 Arithmetic: if $p$ is prime and $p$ divides $m\times n$ then $p$ divides~$m$
2873 \item Theory \texttt{Primrec} develops some computation theory. It
2874 inductively defines the set of primitive recursive functions and presents a
2875 proof that Ackermann's function is not primitive recursive.
2877 \item File \texttt{cla.ML} demonstrates the classical reasoner on over sixty
2878 predicate calculus theorems, ranging from simple tautologies to
2879 moderately difficult problems involving equality and quantifiers.
2881 \item File \texttt{meson.ML} contains an experimental implementation of the {\sc
2882 meson} proof procedure, inspired by Plaisted~\cite{plaisted90}. It is
2883 much more powerful than Isabelle's classical reasoner. But it is less
2884 useful in practice because it works only for pure logic; it does not
2885 accept derived rules for the set theory primitives, for example.
2887 \item File \texttt{mesontest.ML} contains test data for the {\sc meson} proof
2888 procedure. These are mostly taken from Pelletier \cite{pelletier86}.
2890 \item File \texttt{set.ML} proves Cantor's Theorem, which is presented in
2891 \S\ref{sec:hol-cantor} below, and the Schr\"oder-Bernstein Theorem.
2893 \item Theory \texttt{MT} contains Jacob Frost's formalization~\cite{frost93} of
2894 Milner and Tofte's coinduction example~\cite{milner-coind}. This
2895 substantial proof concerns the soundness of a type system for a simple
2896 functional language. The semantics of recursion is given by a cyclic
2897 environment, which makes a coinductive argument appropriate.
2902 \section{Example: Cantor's Theorem}\label{sec:hol-cantor}
2903 Cantor's Theorem states that every set has more subsets than it has
2904 elements. It has become a favourite example in higher-order logic since
2905 it is so easily expressed:
2906 \[ \forall f::\alpha \To \alpha \To bool. \exists S::\alpha\To bool.
2907 \forall x::\alpha. f~x \not= S
2910 Viewing types as sets, $\alpha\To bool$ represents the powerset
2911 of~$\alpha$. This version states that for every function from $\alpha$ to
2912 its powerset, some subset is outside its range.
2914 The Isabelle proof uses \HOL's set theory, with the type $\alpha\,set$ and
2915 the operator \cdx{range}.
2919 The set~$S$ is given as an unknown instead of a
2920 quantified variable so that we may inspect the subset found by the proof.
2922 Goal "?S ~: range\thinspace(f :: 'a=>'a set)";
2924 {\out ?S ~: range f}
2925 {\out 1. ?S ~: range f}
2927 The first two steps are routine. The rule \tdx{rangeE} replaces
2928 $\Var{S}\in \texttt{range} \, f$ by $\Var{S}=f~x$ for some~$x$.
2930 by (resolve_tac [notI] 1);
2932 {\out ?S ~: range f}
2933 {\out 1. ?S : range f ==> False}
2935 by (eresolve_tac [rangeE] 1);
2937 {\out ?S ~: range f}
2938 {\out 1. !!x. ?S = f x ==> False}
2940 Next, we apply \tdx{equalityCE}, reasoning that since $\Var{S}=f~x$,
2941 we have $\Var{c}\in \Var{S}$ if and only if $\Var{c}\in f~x$ for
2944 by (eresolve_tac [equalityCE] 1);
2946 {\out ?S ~: range f}
2947 {\out 1. !!x. [| ?c3 x : ?S; ?c3 x : f x |] ==> False}
2948 {\out 2. !!x. [| ?c3 x ~: ?S; ?c3 x ~: f x |] ==> False}
2950 Now we use a bit of creativity. Suppose that~$\Var{S}$ has the form of a
2951 comprehension. Then $\Var{c}\in\{x.\Var{P}~x\}$ implies
2952 $\Var{P}~\Var{c}$. Destruct-resolution using \tdx{CollectD}
2953 instantiates~$\Var{S}$ and creates the new assumption.
2955 by (dresolve_tac [CollectD] 1);
2957 {\out {\ttlbrace}x. ?P7 x{\ttrbrace} ~: range f}
2958 {\out 1. !!x. [| ?c3 x : f x; ?P7(?c3 x) |] ==> False}
2959 {\out 2. !!x. [| ?c3 x ~: {\ttlbrace}x. ?P7 x{\ttrbrace}; ?c3 x ~: f x |] ==> False}
2961 Forcing a contradiction between the two assumptions of subgoal~1
2962 completes the instantiation of~$S$. It is now the set $\{x. x\not\in
2963 f~x\}$, which is the standard diagonal construction.
2967 {\out {\ttlbrace}x. x ~: f x{\ttrbrace} ~: range f}
2968 {\out 1. !!x. [| x ~: {\ttlbrace}x. x ~: f x{\ttrbrace}; x ~: f x |] ==> False}
2970 The rest should be easy. To apply \tdx{CollectI} to the negated
2971 assumption, we employ \ttindex{swap_res_tac}:
2973 by (swap_res_tac [CollectI] 1);
2975 {\out {\ttlbrace}x. x ~: f x{\ttrbrace} ~: range f}
2976 {\out 1. !!x. [| x ~: f x; ~ False |] ==> x ~: f x}
2980 {\out {\ttlbrace}x. x ~: f x{\ttrbrace} ~: range f}
2983 How much creativity is required? As it happens, Isabelle can prove this
2984 theorem automatically. The default classical set \texttt{claset()} contains rules
2985 for most of the constructs of \HOL's set theory. We must augment it with
2986 \tdx{equalityCE} to break up set equalities, and then apply best-first
2987 search. Depth-first search would diverge, but best-first search
2988 successfully navigates through the large search space.
2989 \index{search!best-first}
2993 {\out ?S ~: range f}
2994 {\out 1. ?S ~: range f}
2996 by (best_tac (claset() addSEs [equalityCE]) 1);
2998 {\out {\ttlbrace}x. x ~: f x{\ttrbrace} ~: range f}
3001 If you run this example interactively, make sure your current theory contains
3002 theory \texttt{Set}, for example by executing \ttindex{context}~{\tt Set.thy}.
3003 Otherwise the default claset may not contain the rules for set theory.
3004 \index{higher-order logic|)}
3006 %%% Local Variables:
3008 %%% TeX-master: "logics"