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}.
905 \section{Generic packages}
906 \label{sec:HOL:generic-packages}
908 \HOL\ instantiates most of Isabelle's generic packages, making available the
909 simplifier and the classical reasoner.
911 \subsection{Simplification and substitution}
913 Simplification tactics tactics such as \texttt{Asm_simp_tac} and \texttt{Full_simp_tac} use the default simpset
914 (\texttt{simpset()}), which works for most purposes. A quite minimal
915 simplification set for higher-order logic is~\ttindexbold{HOL_ss};
916 even more frugal is \ttindexbold{HOL_basic_ss}. Equality~($=$), which
917 also expresses logical equivalence, may be used for rewriting. See
918 the file \texttt{HOL/simpdata.ML} for a complete listing of the basic
919 simplification rules.
921 See \iflabelundefined{chap:classical}{the {\em Reference Manual\/}}%
922 {Chaps.\ts\ref{substitution} and~\ref{simp-chap}} for details of substitution
925 \begin{warn}\index{simplification!of conjunctions}%
926 Reducing $a=b\conj P(a)$ to $a=b\conj P(b)$ is sometimes advantageous. The
927 left part of a conjunction helps in simplifying the right part. This effect
928 is not available by default: it can be slow. It can be obtained by
929 including \ttindex{conj_cong} in a simpset, \verb$addcongs [conj_cong]$.
932 If the simplifier cannot use a certain rewrite rule --- either because
933 of nontermination or because its left-hand side is too flexible ---
934 then you might try \texttt{stac}:
935 \begin{ttdescription}
936 \item[\ttindexbold{stac} $thm$ $i,$] where $thm$ is of the form $lhs = rhs$,
937 replaces in subgoal $i$ instances of $lhs$ by corresponding instances of
938 $rhs$. In case of multiple instances of $lhs$ in subgoal $i$, backtracking
939 may be necessary to select the desired ones.
941 If $thm$ is a conditional equality, the instantiated condition becomes an
942 additional (first) subgoal.
945 \HOL{} provides the tactic \ttindex{hyp_subst_tac}, which substitutes
946 for an equality throughout a subgoal and its hypotheses. This tactic uses
947 \HOL's general substitution rule.
949 \subsubsection{Case splitting}
950 \label{subsec:HOL:case:splitting}
952 \HOL{} also provides convenient means for case splitting during
953 rewriting. Goals containing a subterm of the form \texttt{if}~$b$~{\tt
954 then\dots else\dots} often require a case distinction on $b$. This is
955 expressed by the theorem \tdx{split_if}:
957 \Var{P}(\mbox{\tt if}~\Var{b}~{\tt then}~\Var{x}~\mbox{\tt else}~\Var{y})~=~
958 ((\Var{b} \to \Var{P}(\Var{x})) \land (\neg \Var{b} \to \Var{P}(\Var{y})))
961 For example, a simple instance of $(*)$ is
963 x \in (\mbox{\tt if}~x \in A~{\tt then}~A~\mbox{\tt else}~\{x\})~=~
964 ((x \in A \to x \in A) \land (x \notin A \to x \in \{x\}))
966 Because $(*)$ is too general as a rewrite rule for the simplifier (the
967 left-hand side is not a higher-order pattern in the sense of
968 \iflabelundefined{chap:simplification}{the {\em Reference Manual\/}}%
969 {Chap.\ts\ref{chap:simplification}}), there is a special infix function
970 \ttindexbold{addsplits} of type \texttt{simpset * thm list -> simpset}
971 (analogous to \texttt{addsimps}) that adds rules such as $(*)$ to a
974 by(simp_tac (simpset() addsplits [split_if]) 1);
976 The effect is that after each round of simplification, one occurrence of
977 \texttt{if} is split acording to \texttt{split_if}, until all occurences of
978 \texttt{if} have been eliminated.
980 It turns out that using \texttt{split_if} is almost always the right thing to
981 do. Hence \texttt{split_if} is already included in the default simpset. If
982 you want to delete it from a simpset, use \ttindexbold{delsplits}, which is
983 the inverse of \texttt{addsplits}:
985 by(simp_tac (simpset() delsplits [split_if]) 1);
988 In general, \texttt{addsplits} accepts rules of the form
990 \Var{P}(c~\Var{x@1}~\dots~\Var{x@n})~=~ rhs
992 where $c$ is a constant and $rhs$ is arbitrary. Note that $(*)$ is of the
993 right form because internally the left-hand side is
994 $\Var{P}(\mathtt{If}~\Var{b}~\Var{x}~~\Var{y})$. Important further examples
995 are splitting rules for \texttt{case} expressions (see~\S\ref{subsec:list}
996 and~\S\ref{subsec:datatype:basics}).
998 Analogous to \texttt{Addsimps} and \texttt{Delsimps}, there are also
999 imperative versions of \texttt{addsplits} and \texttt{delsplits}
1001 \ttindexbold{Addsplits}: thm list -> unit
1002 \ttindexbold{Delsplits}: thm list -> unit
1004 for adding splitting rules to, and deleting them from the current simpset.
1006 \subsection{Classical reasoning}
1008 \HOL\ derives classical introduction rules for $\disj$ and~$\exists$, as
1009 well as classical elimination rules for~$\imp$ and~$\bimp$, and the swap
1010 rule; recall Fig.\ts\ref{hol-lemmas2} above.
1012 The classical reasoner is installed. Tactics such as \texttt{Blast_tac} and
1013 {\tt Best_tac} refer to the default claset (\texttt{claset()}), which works
1014 for most purposes. Named clasets include \ttindexbold{prop_cs}, which
1015 includes the propositional rules, and \ttindexbold{HOL_cs}, which also
1016 includes quantifier rules. See the file \texttt{HOL/cladata.ML} for lists of
1017 the classical rules,
1018 and \iflabelundefined{chap:classical}{the {\em Reference Manual\/}}%
1019 {Chap.\ts\ref{chap:classical}} for more discussion of classical proof methods.
1022 \section{Calling the decision procedure SVC}\label{sec:HOL:SVC}
1024 \index{SVC decision procedure|(}
1026 The Stanford Validity Checker (SVC) is a tool that can check the validity of
1027 certain types of formulae. If it is installed on your machine, then
1028 Isabelle/HOL can be configured to call it through the tactic
1029 \ttindex{svc_tac}. It is ideal for large tautologies and complex problems in
1030 linear arithmetic. Subexpressions that SVC cannot handle are automatically
1031 replaced by variables, so you can call the tactic on any subgoal. See the
1032 file \texttt{HOL/ex/svc_test.ML} for examples.
1034 svc_tac : int -> tactic
1035 Svc.trace : bool ref \hfill{\bf initially false}
1038 \begin{ttdescription}
1039 \item[\ttindexbold{svc_tac} $i$] attempts to prove subgoal~$i$ by translating
1040 it into a formula recognized by~SVC\@. If it succeeds then the subgoal is
1041 removed. It fails if SVC is unable to prove the subgoal. It crashes with
1042 an error message if SVC appears not to be installed. Numeric variables may
1043 have types \texttt{nat}, \texttt{int} or \texttt{real}.
1045 \item[\ttindexbold{Svc.trace}] is a flag that, if set, causes \texttt{svc_tac}
1046 to trace its operations: abstraction of the subgoal, translation to SVC
1047 syntax, SVC's response.
1050 Here is an example, with tracing turned on:
1053 {\out val it : bool = true}
1054 Goal "(#3::nat)*a <= #2 + #4*b + #6*c & #11 <= #2*a + b + #2*c & \ttback
1055 \ttback a + #3*b <= #5 + #2*c --> #2 + #3*b <= #2*a + #6*c";
1058 {\out Subgoal abstracted to}
1059 {\out #3 * a <= #2 + #4 * b + #6 * c &}
1060 {\out #11 <= #2 * a + b + #2 * c & a + #3 * b <= #5 + #2 * c -->}
1061 {\out #2 + #3 * b <= #2 * a + #6 * c}
1063 {\out (=> (<= 0 (F_c) ) (=> (<= 0 (F_b) ) (=> (<= 0 (F_a) )}
1064 {\out (=> (AND (<= {* 3 (F_a) } {+ {+ 2 {* 4 (F_b) } } }
1065 {\out {* 6 (F_c) } } ) (AND (<= 11 {+ {+ {* 2 (F_a) } (F_b) }}
1066 {\out {* 2 (F_c) } } ) (<= {+ (F_a) {* 3 (F_b) } } {+ 5 }
1067 {\out {* 2 (F_c) } } ) ) ) (< {+ 2 {* 3 (F_b) } } {+ 1 {+ }
1068 {\out {* 2 (F_a) } {* 6 (F_c) } } } ) ) ) ) ) }
1072 {\out #3 * a <= #2 + #4 * b + #6 * c &}
1073 {\out #11 <= #2 * a + b + #2 * c & a + #3 * b <= #5 + #2 * c -->}
1074 {\out #2 + #3 * b <= #2 * a + #6 * c}
1080 Calling \ttindex{svc_tac} entails an above-average risk of
1081 unsoundness. Isabelle does not check SVC's result independently. Moreover,
1082 the tactic translates the submitted formula using code that lies outside
1083 Isabelle's inference core. Theorems that depend upon results proved using SVC
1084 (and other oracles) are displayed with the annotation \texttt{[!]} attached.
1085 You can also use \texttt{\#der (rep_thm $th$)} to examine the proof object of
1086 theorem~$th$, as described in the \emph{Reference Manual}.
1089 To start, first download SVC from the Internet at URL
1091 http://agamemnon.stanford.edu/~levitt/vc/index.html
1093 and install it using the instructions supplied. SVC requires two environment
1095 \begin{ttdescription}
1096 \item[\ttindexbold{SVC_HOME}] is an absolute pathname to the SVC
1097 distribution directory.
1099 \item[\ttindexbold{SVC_MACHINE}] identifies the type of computer and
1102 You can set these environment variables either using the Unix shell or through
1103 an Isabelle \texttt{settings} file. Isabelle assumes SVC to be installed if
1104 \texttt{SVC_HOME} is defined.
1106 \paragraph*{Acknowledgement.} This interface uses code supplied by S{\o}ren
1110 \index{SVC decision procedure|)}
1115 \section{Types}\label{sec:HOL:Types}
1116 This section describes \HOL's basic predefined types ($\alpha \times
1117 \beta$, $\alpha + \beta$, $nat$ and $\alpha \; list$) and ways for
1118 introducing new types in general. The most important type
1119 construction, the \texttt{datatype}, is treated separately in
1120 \S\ref{sec:HOL:datatype}.
1123 \subsection{Product and sum types}\index{*"* type}\index{*"+ type}
1124 \label{subsec:prod-sum}
1126 \begin{figure}[htbp]
1128 \it symbol & \it meta-type & & \it description \\
1129 \cdx{Pair} & $[\alpha,\beta]\To \alpha\times\beta$
1130 & & ordered pairs $(a,b)$ \\
1131 \cdx{fst} & $\alpha\times\beta \To \alpha$ & & first projection\\
1132 \cdx{snd} & $\alpha\times\beta \To \beta$ & & second projection\\
1133 \cdx{split} & $[[\alpha,\beta]\To\gamma, \alpha\times\beta] \To \gamma$
1134 & & generalized projection\\
1136 $[\alpha\,set, \alpha\To\beta\,set]\To(\alpha\times\beta)set$ &
1137 & general sum of sets
1139 \begin{ttbox}\makeatletter
1140 %\tdx{fst_def} fst p == @a. ? b. p = (a,b)
1141 %\tdx{snd_def} snd p == @b. ? a. p = (a,b)
1142 %\tdx{split_def} split c p == c (fst p) (snd p)
1143 \tdx{Sigma_def} Sigma A B == UN x:A. UN y:B x. {\ttlbrace}(x,y){\ttrbrace}
1145 \tdx{Pair_eq} ((a,b) = (a',b')) = (a=a' & b=b')
1146 \tdx{Pair_inject} [| (a, b) = (a',b'); [| a=a'; b=b' |] ==> R |] ==> R
1147 \tdx{PairE} [| !!x y. p = (x,y) ==> Q |] ==> Q
1149 \tdx{fst_conv} fst (a,b) = a
1150 \tdx{snd_conv} snd (a,b) = b
1151 \tdx{surjective_pairing} p = (fst p,snd p)
1153 \tdx{split} split c (a,b) = c a b
1154 \tdx{split_split} R(split c p) = (! x y. p = (x,y) --> R(c x y))
1156 \tdx{SigmaI} [| a:A; b:B a |] ==> (a,b) : Sigma A B
1157 \tdx{SigmaE} [| c:Sigma A B; !!x y.[| x:A; y:B x; c=(x,y) |] ==> P |] ==> P
1159 \caption{Type $\alpha\times\beta$}\label{hol-prod}
1162 Theory \thydx{Prod} (Fig.\ts\ref{hol-prod}) defines the product type
1163 $\alpha\times\beta$, with the ordered pair syntax $(a, b)$. General
1164 tuples are simulated by pairs nested to the right:
1166 \begin{tabular}{c|c}
1167 external & internal \\
1169 $\tau@1 \times \dots \times \tau@n$ & $\tau@1 \times (\dots (\tau@{n-1} \times \tau@n)\dots)$ \\
1171 $(t@1,\dots,t@n)$ & $(t@1,(\dots,(t@{n-1},t@n)\dots)$ \\
1174 In addition, it is possible to use tuples
1175 as patterns in abstractions:
1177 {\tt\%($x$,$y$). $t$} \quad stands for\quad \texttt{split(\%$x$\thinspace$y$.\ $t$)}
1179 Nested patterns are also supported. They are translated stepwise:
1180 {\tt\%($x$,$y$,$z$). $t$} $\leadsto$ {\tt\%($x$,($y$,$z$)). $t$} $\leadsto$
1181 {\tt split(\%$x$.\%($y$,$z$). $t$)} $\leadsto$ \texttt{split(\%$x$. split(\%$y$
1182 $z$.\ $t$))}. The reverse translation is performed upon printing.
1184 The translation between patterns and \texttt{split} is performed automatically
1185 by the parser and printer. Thus the internal and external form of a term
1186 may differ, which can affects proofs. For example the term {\tt
1187 (\%(x,y).(y,x))(a,b)} requires the theorem \texttt{split} (which is in the
1188 default simpset) to rewrite to {\tt(b,a)}.
1190 In addition to explicit $\lambda$-abstractions, patterns can be used in any
1191 variable binding construct which is internally described by a
1192 $\lambda$-abstraction. Some important examples are
1194 \item[Let:] \texttt{let {\it pattern} = $t$ in $u$}
1195 \item[Quantifiers:] \texttt{!~{\it pattern}:$A$.~$P$}
1196 \item[Choice:] {\underscoreon \tt @~{\it pattern}~.~$P$}
1197 \item[Set operations:] \texttt{UN~{\it pattern}:$A$.~$B$}
1198 \item[Sets:] \texttt{{\ttlbrace}~{\it pattern}~.~$P$~{\ttrbrace}}
1201 There is a simple tactic which supports reasoning about patterns:
1202 \begin{ttdescription}
1203 \item[\ttindexbold{split_all_tac} $i$] replaces in subgoal $i$ all
1204 {\tt!!}-quantified variables of product type by individual variables for
1205 each component. A simple example:
1207 {\out 1. !!p. (\%(x,y,z). (x, y, z)) p = p}
1208 by(split_all_tac 1);
1209 {\out 1. !!x xa ya. (\%(x,y,z). (x, y, z)) (x, xa, ya) = (x, xa, ya)}
1213 Theory \texttt{Prod} also introduces the degenerate product type \texttt{unit}
1214 which contains only a single element named {\tt()} with the property
1216 \tdx{unit_eq} u = ()
1220 Theory \thydx{Sum} (Fig.~\ref{hol-sum}) defines the sum type $\alpha+\beta$
1221 which associates to the right and has a lower priority than $*$: $\tau@1 +
1222 \tau@2 + \tau@3*\tau@4$ means $\tau@1 + (\tau@2 + (\tau@3*\tau@4))$.
1224 The definition of products and sums in terms of existing types is not
1225 shown. The constructions are fairly standard and can be found in the
1226 respective theory files. Although the sum and product types are
1227 constructed manually for foundational reasons, they are represented as
1228 actual datatypes later (see \S\ref{subsec:datatype:rep_datatype}).
1229 Therefore, the theory \texttt{Datatype} should be used instead of
1230 \texttt{Sum} or \texttt{Prod}.
1234 \it symbol & \it meta-type & & \it description \\
1235 \cdx{Inl} & $\alpha \To \alpha+\beta$ & & first injection\\
1236 \cdx{Inr} & $\beta \To \alpha+\beta$ & & second injection\\
1237 \cdx{sum_case} & $[\alpha\To\gamma, \beta\To\gamma, \alpha+\beta] \To\gamma$
1240 \begin{ttbox}\makeatletter
1241 \tdx{Inl_not_Inr} Inl a ~= Inr b
1243 \tdx{inj_Inl} inj Inl
1244 \tdx{inj_Inr} inj Inr
1246 \tdx{sumE} [| !!x. P(Inl x); !!y. P(Inr y) |] ==> P s
1248 \tdx{sum_case_Inl} sum_case f g (Inl x) = f x
1249 \tdx{sum_case_Inr} sum_case f g (Inr x) = g x
1251 \tdx{surjective_sum} sum_case (\%x. f(Inl x)) (\%y. f(Inr y)) s = f s
1252 \tdx{sum.split_case} R(sum_case f g s) = ((! x. s = Inl(x) --> R(f(x))) &
1253 (! y. s = Inr(y) --> R(g(y))))
1255 \caption{Type $\alpha+\beta$}\label{hol-sum}
1266 \it symbol & \it meta-type & \it priority & \it description \\
1267 \cdx{0} & $nat$ & & zero \\
1268 \cdx{Suc} & $nat \To nat$ & & successor function\\
1269 % \cdx{nat_case} & $[\alpha, nat\To\alpha, nat] \To\alpha$ & & conditional\\
1270 % \cdx{nat_rec} & $[nat, \alpha, [nat, \alpha]\To\alpha] \To \alpha$
1271 % & & primitive recursor\\
1272 \tt * & $[nat,nat]\To nat$ & Left 70 & multiplication \\
1273 \tt div & $[nat,nat]\To nat$ & Left 70 & division\\
1274 \tt mod & $[nat,nat]\To nat$ & Left 70 & modulus\\
1275 \tt + & $[nat,nat]\To nat$ & Left 65 & addition\\
1276 \tt - & $[nat,nat]\To nat$ & Left 65 & subtraction
1278 \subcaption{Constants and infixes}
1280 \begin{ttbox}\makeatother
1281 \tdx{nat_induct} [| P 0; !!n. P n ==> P(Suc n) |] ==> P n
1283 \tdx{Suc_not_Zero} Suc m ~= 0
1284 \tdx{inj_Suc} inj Suc
1285 \tdx{n_not_Suc_n} n~=Suc n
1286 \subcaption{Basic properties}
1288 \caption{The type of natural numbers, \tydx{nat}} \label{hol-nat1}
1293 \begin{ttbox}\makeatother
1295 (Suc m)+n = Suc(m+n)
1304 \tdx{mod_less} m<n ==> m mod n = m
1305 \tdx{mod_geq} [| 0<n; ~m<n |] ==> m mod n = (m-n) mod n
1307 \tdx{div_less} m<n ==> m div n = 0
1308 \tdx{div_geq} [| 0<n; ~m<n |] ==> m div n = Suc((m-n) div n)
1310 \caption{Recursion equations for the arithmetic operators} \label{hol-nat2}
1313 \subsection{The type of natural numbers, \textit{nat}}
1314 \index{nat@{\textit{nat}} type|(}
1316 The theory \thydx{NatDef} defines the natural numbers in a roundabout but
1317 traditional way. The axiom of infinity postulates a type~\tydx{ind} of
1318 individuals, which is non-empty and closed under an injective operation. The
1319 natural numbers are inductively generated by choosing an arbitrary individual
1320 for~0 and using the injective operation to take successors. This is a least
1321 fixedpoint construction. For details see the file \texttt{NatDef.thy}.
1323 Type~\tydx{nat} is an instance of class~\cldx{ord}, which makes the
1324 overloaded functions of this class (esp.\ \cdx{<} and \cdx{<=}, but also
1325 \cdx{min}, \cdx{max} and \cdx{LEAST}) available on \tydx{nat}. Theory
1326 \thydx{Nat} builds on \texttt{NatDef} and shows that {\tt<=} is a partial order,
1327 so \tydx{nat} is also an instance of class \cldx{order}.
1329 Theory \thydx{Arith} develops arithmetic on the natural numbers. It defines
1330 addition, multiplication and subtraction. Theory \thydx{Divides} defines
1331 division, remainder and the ``divides'' relation. The numerous theorems
1332 proved include commutative, associative, distributive, identity and
1333 cancellation laws. See Figs.\ts\ref{hol-nat1} and~\ref{hol-nat2}. The
1334 recursion equations for the operators \texttt{+}, \texttt{-} and \texttt{*} on
1335 \texttt{nat} are part of the default simpset.
1337 Functions on \tydx{nat} can be defined by primitive or well-founded recursion;
1338 see \S\ref{sec:HOL:recursive}. A simple example is addition.
1339 Here, \texttt{op +} is the name of the infix operator~\texttt{+}, following
1340 the standard convention.
1344 "Suc m + n = Suc (m + n)"
1346 There is also a \sdx{case}-construct
1349 case \(e\) of 0 => \(a\) | Suc \(m\) => \(b\)
1351 Note that Isabelle insists on precisely this format; you may not even change
1352 the order of the two cases.
1353 Both \texttt{primrec} and \texttt{case} are realized by a recursion operator
1354 \cdx{nat_rec}, which is available because \textit{nat} is represented as
1355 a datatype (see \S\ref{subsec:datatype:rep_datatype}).
1357 %The predecessor relation, \cdx{pred_nat}, is shown to be well-founded.
1358 %Recursion along this relation resembles primitive recursion, but is
1359 %stronger because we are in higher-order logic; using primitive recursion to
1360 %define a higher-order function, we can easily Ackermann's function, which
1361 %is not primitive recursive \cite[page~104]{thompson91}.
1362 %The transitive closure of \cdx{pred_nat} is~$<$. Many functions on the
1363 %natural numbers are most easily expressed using recursion along~$<$.
1365 Tactic {\tt\ttindex{induct_tac} "$n$" $i$} performs induction on variable~$n$
1366 in subgoal~$i$ using theorem \texttt{nat_induct}. There is also the derived
1367 theorem \tdx{less_induct}:
1369 [| !!n. [| ! m. m<n --> P m |] ==> P n |] ==> P n
1373 Reasoning about arithmetic inequalities can be tedious. Fortunately HOL
1374 provides a decision procedure for quantifier-free linear arithmetic (i.e.\
1375 only addition and subtraction). The simplifier invokes a weak version of this
1376 decision procedure automatically. If this is not sufficent, you can invoke
1377 the full procedure \ttindex{arith_tac} explicitly. It copes with arbitrary
1378 formulae involving {\tt=}, {\tt<}, {\tt<=}, {\tt+}, {\tt-}, {\tt Suc}, {\tt
1379 min}, {\tt max} and numerical constants; other subterms are treated as
1380 atomic; subformulae not involving type $nat$ are ignored; quantified
1381 subformulae are ignored unless they are positive universal or negative
1382 existential. Note that the running time is exponential in the number of
1383 occurrences of {\tt min}, {\tt max}, and {\tt-} because they require case
1384 distinctions. Note also that \texttt{arith_tac} is not complete: if
1385 divisibility plays a role, it may fail to prove a valid formula, for example
1386 $m+m \neq n+n+1$. Fortunately such examples are rare in practice.
1388 If \texttt{arith_tac} fails you, try to find relevant arithmetic results in
1389 the library. The theory \texttt{NatDef} contains theorems about {\tt<} and
1390 {\tt<=}, the theory \texttt{Arith} contains theorems about \texttt{+},
1391 \texttt{-} and \texttt{*}, and theory \texttt{Divides} contains theorems about
1392 \texttt{div} and \texttt{mod}. Use the \texttt{find}-functions to locate them
1393 (see the {\em Reference Manual\/}).
1396 \index{#@{\tt[]} symbol}
1397 \index{#@{\tt\#} symbol}
1398 \index{"@@{\tt\at} symbol}
1401 \it symbol & \it meta-type & \it priority & \it description \\
1402 \tt[] & $\alpha\,list$ & & empty list\\
1403 \tt \# & $[\alpha,\alpha\,list]\To \alpha\,list$ & Right 65 &
1405 \cdx{null} & $\alpha\,list \To bool$ & & emptiness test\\
1406 \cdx{hd} & $\alpha\,list \To \alpha$ & & head \\
1407 \cdx{tl} & $\alpha\,list \To \alpha\,list$ & & tail \\
1408 \cdx{last} & $\alpha\,list \To \alpha$ & & last element \\
1409 \cdx{butlast} & $\alpha\,list \To \alpha\,list$ & & drop last element \\
1410 \tt\at & $[\alpha\,list,\alpha\,list]\To \alpha\,list$ & Left 65 & append \\
1411 \cdx{map} & $(\alpha\To\beta) \To (\alpha\,list \To \beta\,list)$
1413 \cdx{filter} & $(\alpha \To bool) \To (\alpha\,list \To \alpha\,list)$
1414 & & filter functional\\
1415 \cdx{set}& $\alpha\,list \To \alpha\,set$ & & elements\\
1416 \sdx{mem} & $\alpha \To \alpha\,list \To bool$ & Left 55 & membership\\
1417 \cdx{foldl} & $(\beta\To\alpha\To\beta) \To \beta \To \alpha\,list \To \beta$ &
1419 \cdx{concat} & $(\alpha\,list)list\To \alpha\,list$ & & concatenation \\
1420 \cdx{rev} & $\alpha\,list \To \alpha\,list$ & & reverse \\
1421 \cdx{length} & $\alpha\,list \To nat$ & & length \\
1422 \tt! & $\alpha\,list \To nat \To \alpha$ & Left 100 & indexing \\
1423 \cdx{take}, \cdx{drop} & $nat \To \alpha\,list \To \alpha\,list$ &&
1424 take or drop a prefix \\
1427 $(\alpha \To bool) \To \alpha\,list \To \alpha\,list$ &&
1428 take or drop a prefix
1430 \subcaption{Constants and infixes}
1432 \begin{center} \tt\frenchspacing
1433 \begin{tabular}{rrr}
1434 \it external & \it internal & \it description \\{}
1435 [$x@1$, $\dots$, $x@n$] & $x@1$ \# $\cdots$ \# $x@n$ \# [] &
1436 \rm finite list \\{}
1437 [$x$:$l$. $P$] & filter ($\lambda x{.}P$) $l$ &
1438 \rm list comprehension
1441 \subcaption{Translations}
1442 \caption{The theory \thydx{List}} \label{hol-list}
1447 \begin{ttbox}\makeatother
1456 (x#xs) @ ys = x # xs @ ys
1459 map f (x#xs) = f x # map f xs
1462 filter P (x#xs) = (if P x then x#filter P xs else filter P xs)
1464 set [] = \ttlbrace\ttrbrace
1465 set (x#xs) = insert x (set xs)
1468 x mem (y#ys) = (if y=x then True else x mem ys)
1471 foldl f a (x#xs) = foldl f (f a x) xs
1474 concat(x#xs) = x @ concat(xs)
1477 rev(x#xs) = rev(xs) @ [x]
1480 length(x#xs) = Suc(length(xs))
1483 xs!(Suc n) = (tl xs)!n
1486 take n (x#xs) = (case n of 0 => [] | Suc(m) => x # take m xs)
1489 drop n (x#xs) = (case n of 0 => x#xs | Suc(m) => drop m xs)
1492 takeWhile P (x#xs) = (if P x then x#takeWhile P xs else [])
1495 dropWhile P (x#xs) = (if P x then dropWhile P xs else xs)
1497 \caption{Recursions equations for list processing functions}
1498 \label{fig:HOL:list-simps}
1500 \index{nat@{\textit{nat}} type|)}
1503 \subsection{The type constructor for lists, \textit{list}}
1505 \index{list@{\textit{list}} type|(}
1507 Figure~\ref{hol-list} presents the theory \thydx{List}: the basic list
1508 operations with their types and syntax. Type $\alpha \; list$ is
1509 defined as a \texttt{datatype} with the constructors {\tt[]} and {\tt\#}.
1510 As a result the generic structural induction and case analysis tactics
1511 \texttt{induct\_tac} and \texttt{exhaust\_tac} also become available for
1512 lists. A \sdx{case} construct of the form
1514 case $e$ of [] => $a$ | \(x\)\#\(xs\) => b
1516 is defined by translation. For details see~\S\ref{sec:HOL:datatype}. There
1517 is also a case splitting rule \tdx{split_list_case}
1520 P(\mathtt{case}~e~\mathtt{of}~\texttt{[] =>}~a ~\texttt{|}~
1521 x\texttt{\#}xs~\texttt{=>}~f~x~xs) ~= \\
1522 ((e = \texttt{[]} \to P(a)) \land
1523 (\forall x~ xs. e = x\texttt{\#}xs \to P(f~x~xs)))
1526 which can be fed to \ttindex{addsplits} just like
1527 \texttt{split_if} (see~\S\ref{subsec:HOL:case:splitting}).
1529 \texttt{List} provides a basic library of list processing functions defined by
1530 primitive recursion (see~\S\ref{sec:HOL:primrec}). The recursion equations
1531 are shown in Fig.\ts\ref{fig:HOL:list-simps}.
1533 \index{list@{\textit{list}} type|)}
1536 \subsection{Introducing new types} \label{sec:typedef}
1538 The \HOL-methodology dictates that all extensions to a theory should
1539 be \textbf{definitional}. The type definition mechanism that
1540 meets this criterion is \ttindex{typedef}. Note that \emph{type synonyms},
1541 which are inherited from {\Pure} and described elsewhere, are just
1542 syntactic abbreviations that have no logical meaning.
1545 Types in \HOL\ must be non-empty; otherwise the quantifier rules would be
1546 unsound, because $\exists x. x=x$ is a theorem \cite[\S7]{paulson-COLOG}.
1548 A \bfindex{type definition} identifies the new type with a subset of
1549 an existing type. More precisely, the new type is defined by
1550 exhibiting an existing type~$\tau$, a set~$A::\tau\,set$, and a
1551 theorem of the form $x:A$. Thus~$A$ is a non-empty subset of~$\tau$,
1552 and the new type denotes this subset. New functions are defined that
1553 establish an isomorphism between the new type and the subset. If
1554 type~$\tau$ involves type variables $\alpha@1$, \ldots, $\alpha@n$,
1555 then the type definition creates a type constructor
1556 $(\alpha@1,\ldots,\alpha@n)ty$ rather than a particular type.
1558 \begin{figure}[htbp]
1560 typedef : 'typedef' ( () | '(' name ')') type '=' set witness;
1562 type : typevarlist name ( () | '(' infix ')' );
1564 witness : () | '(' id ')';
1566 \caption{Syntax of type definitions}
1567 \label{fig:HOL:typedef}
1570 The syntax for type definitions is shown in Fig.~\ref{fig:HOL:typedef}. For
1571 the definition of `typevarlist' and `infix' see
1572 \iflabelundefined{chap:classical}
1573 {the appendix of the {\em Reference Manual\/}}%
1574 {Appendix~\ref{app:TheorySyntax}}. The remaining nonterminals have the
1577 \item[\it type:] the new type constructor $(\alpha@1,\dots,\alpha@n)ty$ with
1578 optional infix annotation.
1579 \item[\it name:] an alphanumeric name $T$ for the type constructor
1580 $ty$, in case $ty$ is a symbolic name. Defaults to $ty$.
1581 \item[\it set:] the representing subset $A$.
1582 \item[\it witness:] name of a theorem of the form $a:A$ proving
1583 non-emptiness. It can be omitted in case Isabelle manages to prove
1584 non-emptiness automatically.
1586 If all context conditions are met (no duplicate type variables in
1587 `typevarlist', no extra type variables in `set', and no free term variables
1588 in `set'), the following components are added to the theory:
1590 \item a type $ty :: (term,\dots,term)term$
1594 Rep_T &::& (\alpha@1,\dots,\alpha@n)ty \To \tau \\
1595 Abs_T &::& \tau \To (\alpha@1,\dots,\alpha@n)ty
1597 \item a definition and three axioms
1600 T{\tt_def} & T \equiv A \\
1601 {\tt Rep_}T & Rep_T\,x \in T \\
1602 {\tt Rep_}T{\tt_inverse} & Abs_T\,(Rep_T\,x) = x \\
1603 {\tt Abs_}T{\tt_inverse} & y \in T \Imp Rep_T\,(Abs_T\,y) = y
1606 stating that $(\alpha@1,\dots,\alpha@n)ty$ is isomorphic to $A$ by $Rep_T$
1607 and its inverse $Abs_T$.
1609 Below are two simple examples of \HOL\ type definitions. Non-emptiness
1610 is proved automatically here.
1612 typedef unit = "{\ttlbrace}True{\ttrbrace}"
1615 ('a, 'b) "*" (infixr 20)
1616 = "{\ttlbrace}f . EX (a::'a) (b::'b). f = (\%x y. x = a & y = b){\ttrbrace}"
1619 Type definitions permit the introduction of abstract data types in a safe
1620 way, namely by providing models based on already existing types. Given some
1621 abstract axiomatic description $P$ of a type, this involves two steps:
1623 \item Find an appropriate type $\tau$ and subset $A$ which has the desired
1624 properties $P$, and make a type definition based on this representation.
1625 \item Prove that $P$ holds for $ty$ by lifting $P$ from the representation.
1627 You can now forget about the representation and work solely in terms of the
1628 abstract properties $P$.
1631 If you introduce a new type (constructor) $ty$ axiomatically, i.e.\ by
1632 declaring the type and its operations and by stating the desired axioms, you
1633 should make sure the type has a non-empty model. You must also have a clause
1636 arities \(ty\) :: (term,\thinspace\(\dots\),{\thinspace}term){\thinspace}term
1638 in your theory file to tell Isabelle that $ty$ is in class \texttt{term}, the
1639 class of all \HOL\ types.
1645 At a first approximation, records are just a minor generalisation of tuples,
1646 where components may be addressed by labels instead of just position (think of
1647 {\ML}, for example). The version of records offered by Isabelle/HOL is
1648 slightly more advanced, though, supporting \emph{extensible record schemes}.
1649 This admits operations that are polymorphic with respect to record extension,
1650 yielding ``object-oriented'' effects like (single) inheritance. See also
1651 \cite{NaraschewskiW-TPHOLs98} for more details on object-oriented
1652 verification and record subtyping in HOL.
1657 Isabelle/HOL supports fixed and schematic records both at the level of terms
1658 and types. The concrete syntax is as follows:
1661 \begin{tabular}{l|l|l}
1662 & record terms & record types \\ \hline
1663 fixed & $\record{x = a\fs y = b}$ & $\record{x \ty A\fs y \ty B}$ \\
1664 schematic & $\record{x = a\fs y = b\fs \more = m}$ &
1665 $\record{x \ty A\fs y \ty B\fs \more \ty M}$ \\
1669 \noindent The \textsc{ascii} representation of $\record{x = a}$ is \texttt{(| x = a |)}.
1671 A fixed record $\record{x = a\fs y = b}$ has field $x$ of value $a$ and field
1672 $y$ of value $b$. The corresponding type is $\record{x \ty A\fs y \ty B}$,
1673 assuming that $a \ty A$ and $b \ty B$.
1675 A record scheme like $\record{x = a\fs y = b\fs \more = m}$ contains fields
1676 $x$ and $y$ as before, but also possibly further fields as indicated by the
1677 ``$\more$'' notation (which is actually part of the syntax). The improper
1678 field ``$\more$'' of a record scheme is called the \emph{more part}.
1679 Logically it is just a free variable, which is occasionally referred to as
1680 \emph{row variable} in the literature. The more part of a record scheme may
1681 be instantiated by zero or more further components. For example, above scheme
1682 might get instantiated to $\record{x = a\fs y = b\fs z = c\fs \more = m'}$,
1683 where $m'$ refers to a different more part. Fixed records are special
1684 instances of record schemes, where ``$\more$'' is properly terminated by the
1685 $() :: unit$ element. Actually, $\record{x = a\fs y = b}$ is just an
1686 abbreviation for $\record{x = a\fs y = b\fs \more = ()}$.
1690 There are two key features that make extensible records in a simply typed
1691 language like HOL feasible:
1693 \item the more part is internalised, as a free term or type variable,
1694 \item field names are externalised, they cannot be accessed within the logic
1695 as first-class values.
1700 In Isabelle/HOL record types have to be defined explicitly, fixing their field
1701 names and types, and their (optional) parent record (see
1702 \S\ref{sec:HOL:record-def}). Afterwards, records may be formed using above
1703 syntax, while obeying the canonical order of fields as given by their
1704 declaration. The record package also provides several operations like
1705 selectors and updates (see \S\ref{sec:HOL:record-ops}), together with
1706 characteristic properties (see \S\ref{sec:HOL:record-thms}).
1708 There is an example theory demonstrating most basic aspects of extensible
1709 records (see theory \texttt{HOL/ex/Points} in the Isabelle sources).
1712 \subsection{Defining records}\label{sec:HOL:record-def}
1714 The theory syntax for record type definitions is shown in
1715 Fig.~\ref{fig:HOL:record}. For the definition of `typevarlist' and `type' see
1716 \iflabelundefined{chap:classical}
1717 {the appendix of the {\em Reference Manual\/}}%
1718 {Appendix~\ref{app:TheorySyntax}}.
1720 \begin{figure}[htbp]
1722 record : 'record' typevarlist name '=' parent (field +);
1724 parent : ( () | type '+');
1725 field : name '::' type;
1727 \caption{Syntax of record type definitions}
1728 \label{fig:HOL:record}
1731 A general \ttindex{record} specification is of the following form:
1733 \mathtt{record}~(\alpha@1, \dots, \alpha@n) \, t ~=~
1734 (\tau@1, \dots, \tau@m) \, s ~+~ c@1 :: \sigma@1 ~ \dots ~ c@l :: \sigma@l
1736 where $\vec\alpha@n$ are distinct type variables, and $\vec\tau@m$,
1737 $\vec\sigma@l$ are types containing at most variables from $\vec\alpha@n$.
1738 Type constructor $t$ has to be new, while $s$ has to specify an existing
1739 record type. Furthermore, the $\vec c@l$ have to be distinct field names.
1740 There has to be at least one field.
1742 In principle, field names may never be shared with other records. This is no
1743 actual restriction in practice, since $\vec c@l$ are internally declared
1744 within a separate name space qualified by the name $t$ of the record.
1748 Above definition introduces a new record type $(\vec\alpha@n) \, t$ by
1749 extending an existing one $(\vec\tau@m) \, s$ by new fields $\vec c@l \ty
1750 \vec\sigma@l$. The parent record specification is optional, by omitting it
1751 $t$ becomes a \emph{root record}. The hierarchy of all records declared
1752 within a theory forms a forest structure, i.e.\ a set of trees, where any of
1753 these is rooted by some root record.
1755 For convenience, $(\vec\alpha@n) \, t$ is made a type abbreviation for the
1756 fixed record type $\record{\vec c@l \ty \vec\sigma@l}$, and $(\vec\alpha@n,
1757 \zeta) \, t_scheme$ is made an abbreviation for $\record{\vec c@l \ty
1758 \vec\sigma@l\fs \more \ty \zeta}$.
1762 The following simple example defines a root record type $point$ with fields $x
1763 \ty nat$ and $y \ty nat$, and record type $cpoint$ by extending $point$ with
1764 an additional $colour$ component.
1771 record cpoint = point +
1776 \subsection{Record operations}\label{sec:HOL:record-ops}
1778 Any record definition of the form presented above produces certain standard
1779 operations. Selectors and updates are provided for any field, including the
1780 improper one ``$more$''. There are also cumulative record constructor
1783 To simplify the presentation below, we first assume that $(\vec\alpha@n) \, t$
1784 is a root record with fields $\vec c@l \ty \vec\sigma@l$.
1788 \textbf{Selectors} and \textbf{updates} are available for any field (including
1789 ``$more$'') as follows:
1790 \begin{matharray}{lll}
1791 c@i & \ty & \record{\vec c@l \ty \vec \sigma@l, \more \ty \zeta} \To \sigma@i \\
1792 c@i_update & \ty & \sigma@i \To \record{\vec c@l \ty \vec \sigma@l, \more \ty \zeta} \To
1793 \record{\vec c@l \ty \vec \sigma@l, \more \ty \zeta}
1796 There is some special syntax for updates: $r \, \record{x \asn a}$ abbreviates
1797 term $x_update \, a \, r$. Repeated updates are also supported: $r \,
1798 \record{x \asn a} \, \record{y \asn b} \, \record{z \asn c}$ may be written as
1799 $r \, \record{x \asn a\fs y \asn b\fs z \asn c}$. Note that because of
1800 postfix notation the order of fields shown here is reverse than in the actual
1801 term. This might lead to confusion in conjunction with proof tools like
1804 Since repeated updates are just function applications, fields may be freely
1805 permuted in $\record{x \asn a\fs y \asn b\fs z \asn c}$, as far as the logic
1806 is concerned. Thus commutativity of updates can be proven within the logic
1807 for any two fields, but not as a general theorem: fields are not first-class
1812 \textbf{Make} operations provide cumulative record constructor functions:
1813 \begin{matharray}{lll}
1814 make & \ty & \vec\sigma@l \To \record{\vec c@l \ty \vec \sigma@l} \\
1815 make_scheme & \ty & \vec\sigma@l \To
1816 \zeta \To \record{\vec c@l \ty \vec \sigma@l, \more \ty \zeta} \\
1819 These functions are curried. The corresponding definitions in terms of actual
1820 record terms are part of the standard simpset. Thus $point\dtt make\,a\,b$
1821 rewrites to $\record{x = a\fs y = b}$.
1825 Any of above selector, update and make operations are declared within a local
1826 name space prefixed by the name $t$ of the record. In case that different
1827 records share base names of fields, one has to qualify names explicitly (e.g.\
1828 $t\dtt c@i_update$). This is recommended especially for operations like
1829 $make$ or $update_more$ that always have the same base name. Just use $t\dtt
1830 make$ etc.\ to avoid confusion.
1834 We reconsider the case of non-root records, which are derived of some parent
1835 record. In general, the latter may depend on another parent as well,
1836 resulting in a list of \emph{ancestor records}. Appending the lists of fields
1837 of all ancestors results in a certain field prefix. The record package
1838 automatically takes care of this by lifting operations over this context of
1839 ancestor fields. Assuming that $(\vec\alpha@n) \, t$ has ancestor fields
1840 $\vec d@k \ty \vec\rho@k$, selectors will get the following types:
1841 \begin{matharray}{lll}
1842 c@i & \ty & \record{\vec d@k \ty \vec\rho@k, \vec c@l \ty \vec \sigma@l, \more \ty \zeta}
1846 Update and make operations are analogous.
1849 \subsection{Proof tools}\label{sec:HOL:record-thms}
1851 The record package provides the following proof rules for any record type $t$.
1854 \item Standard conversions (selectors or updates applied to record constructor
1855 terms, make function definitions) are part of the standard simpset (via
1858 \item Selectors applied to updated records are automatically reduced by
1859 simplification procedure \ttindex{record_simproc}, which is part of the
1862 \item Inject equations of a form analogous to $((x, y) = (x', y')) \equiv x=x'
1863 \conj y=y'$ are made part of the standard simpset and claset (via
1866 \item A tactic for record field splitting (\ttindex{record_split_tac}) may be
1867 made part of the claset (via \texttt{addSWrapper}). This tactic is based on
1868 rules analogous to $(\All x PROP~P~x) \equiv (\All{a~b} PROP~P(a, b))$ for
1872 The first two kinds of rules are stored within the theory as $t\dtt simps$ and
1873 $t\dtt iffs$, respectively. In some situations it might be appropriate to
1874 expand the definitions of updates: $t\dtt update_defs$. Note that these names
1875 are \emph{not} bound at the {\ML} level.
1879 The example theory \texttt{HOL/ex/Points} demonstrates typical proofs
1880 concerning records. The basic idea is to make \ttindex{record_split_tac}
1881 expand quantified record variables and then simplify by the conversion rules.
1882 By using a combination of the simplifier and classical prover together with
1883 the default simpset and claset, record problems should be solved with a single
1884 stroke of \texttt{Auto_tac} or \texttt{Force_tac}. Most of the time, plain
1885 \texttt{Simp_tac} should be sufficient, though.
1888 \section{Datatype definitions}
1889 \label{sec:HOL:datatype}
1892 Inductive datatypes, similar to those of \ML, frequently appear in
1893 applications of Isabelle/HOL. In principle, such types could be defined by
1894 hand via \texttt{typedef} (see \S\ref{sec:typedef}), but this would be far too
1895 tedious. The \ttindex{datatype} definition package of Isabelle/HOL (cf.\
1896 \cite{Berghofer-Wenzel:1999:TPHOL}) automates such chores. It generates an
1897 appropriate \texttt{typedef} based on a least fixed-point construction, and
1898 proves freeness theorems and induction rules, as well as theorems for
1899 recursion and case combinators. The user just has to give a simple
1900 specification of new inductive types using a notation similar to {\ML} or
1903 The current datatype package can handle both mutual and indirect recursion.
1904 It also offers to represent existing types as datatypes giving the advantage
1905 of a more uniform view on standard theories.
1909 \label{subsec:datatype:basics}
1911 A general \texttt{datatype} definition is of the following form:
1914 \mathtt{datatype} & (\alpha@1,\ldots,\alpha@h)t@1 & = &
1915 C^1@1~\tau^1@{1,1}~\ldots~\tau^1@{1,m^1@1} ~\mid~ \ldots ~\mid~
1916 C^1@{k@1}~\tau^1@{k@1,1}~\ldots~\tau^1@{k@1,m^1@{k@1}} \\
1918 \mathtt{and} & (\alpha@1,\ldots,\alpha@h)t@n & = &
1919 C^n@1~\tau^n@{1,1}~\ldots~\tau^n@{1,m^n@1} ~\mid~ \ldots ~\mid~
1920 C^n@{k@n}~\tau^n@{k@n,1}~\ldots~\tau^n@{k@n,m^n@{k@n}}
1923 where $\alpha@i$ are type variables, $C^j@i$ are distinct constructor
1924 names and $\tau^j@{i,i'}$ are {\em admissible} types containing at
1925 most the type variables $\alpha@1, \ldots, \alpha@h$. A type $\tau$
1926 occurring in a \texttt{datatype} definition is {\em admissible} iff
1928 \item $\tau$ is non-recursive, i.e.\ $\tau$ does not contain any of the
1929 newly defined type constructors $t@1,\ldots,t@n$, or
1930 \item $\tau = (\alpha@1,\ldots,\alpha@h)t@{j'}$ where $1 \leq j' \leq n$, or
1931 \item $\tau = (\tau'@1,\ldots,\tau'@{h'})t'$, where $t'$ is
1932 the type constructor of an already existing datatype and $\tau'@1,\ldots,\tau'@{h'}$
1933 are admissible types.
1934 \item $\tau = \sigma \rightarrow \tau'$, where $\tau'$ is an admissible
1935 type and $\sigma$ is non-recursive (i.e. the occurrences of the newly defined
1936 types are {\em strictly positive})
1938 If some $(\alpha@1,\ldots,\alpha@h)t@{j'}$ occurs in a type $\tau^j@{i,i'}$
1941 (\ldots,\ldots ~ (\alpha@1,\ldots,\alpha@h)t@{j'} ~ \ldots,\ldots)t'
1943 this is called a {\em nested} (or \emph{indirect}) occurrence. A very simple
1944 example of a datatype is the type \texttt{list}, which can be defined by
1946 datatype 'a list = Nil
1949 Arithmetic expressions \texttt{aexp} and boolean expressions \texttt{bexp} can be modelled
1950 by the mutually recursive datatype definition
1952 datatype 'a aexp = If_then_else ('a bexp) ('a aexp) ('a aexp)
1953 | Sum ('a aexp) ('a aexp)
1954 | Diff ('a aexp) ('a aexp)
1957 and 'a bexp = Less ('a aexp) ('a aexp)
1958 | And ('a bexp) ('a bexp)
1959 | Or ('a bexp) ('a bexp)
1961 The datatype \texttt{term}, which is defined by
1963 datatype ('a, 'b) term = Var 'a
1964 | App 'b ((('a, 'b) term) list)
1966 is an example for a datatype with nested recursion. Using nested recursion
1967 involving function spaces, we may also define infinitely branching datatypes, e.g.
1969 datatype 'a tree = Atom 'a | Branch "nat => 'a tree"
1974 Types in HOL must be non-empty. Each of the new datatypes
1975 $(\alpha@1,\ldots,\alpha@h)t@j$ with $1 \le j \le n$ is non-empty iff it has a
1976 constructor $C^j@i$ with the following property: for all argument types
1977 $\tau^j@{i,i'}$ of the form $(\alpha@1,\ldots,\alpha@h)t@{j'}$ the datatype
1978 $(\alpha@1,\ldots,\alpha@h)t@{j'}$ is non-empty.
1980 If there are no nested occurrences of the newly defined datatypes, obviously
1981 at least one of the newly defined datatypes $(\alpha@1,\ldots,\alpha@h)t@j$
1982 must have a constructor $C^j@i$ without recursive arguments, a \emph{base
1983 case}, to ensure that the new types are non-empty. If there are nested
1984 occurrences, a datatype can even be non-empty without having a base case
1985 itself. Since \texttt{list} is a non-empty datatype, \texttt{datatype t = C (t
1986 list)} is non-empty as well.
1989 \subsubsection{Freeness of the constructors}
1991 The datatype constructors are automatically defined as functions of their
1993 \[ C^j@i :: [\tau^j@{i,1},\dots,\tau^j@{i,m^j@i}] \To (\alpha@1,\dots,\alpha@h)t@j \]
1994 These functions have certain {\em freeness} properties. They construct
1997 C^j@i~x@1~\dots~x@{m^j@i} \neq C^j@{i'}~y@1~\dots~y@{m^j@{i'}} \qquad
1998 \mbox{for all}~ i \neq i'.
2000 The constructor functions are injective:
2002 (C^j@i~x@1~\dots~x@{m^j@i} = C^j@i~y@1~\dots~y@{m^j@i}) =
2003 (x@1 = y@1 \land \dots \land x@{m^j@i} = y@{m^j@i})
2005 Since the number of distinctness inequalities is quadratic in the number of
2006 constructors, the datatype package avoids proving them separately if there are
2007 too many constructors. Instead, specific inequalities are proved by a suitable
2008 simplification procedure on demand.\footnote{This procedure, which is already part
2009 of the default simpset, may be referred to by the ML identifier
2010 \texttt{DatatypePackage.distinct_simproc}.}
2012 \subsubsection{Structural induction}
2014 The datatype package also provides structural induction rules. For
2015 datatypes without nested recursion, this is of the following form:
2017 \infer{P@1~x@1 \wedge \dots \wedge P@n~x@n}
2019 \Forall x@1 \dots x@{m^1@1}.
2020 \List{P@{s^1@{1,1}}~x@{r^1@{1,1}}; \dots;
2021 P@{s^1@{1,l^1@1}}~x@{r^1@{1,l^1@1}}} & \Imp &
2022 P@1~\left(C^1@1~x@1~\dots~x@{m^1@1}\right) \\
2024 \Forall x@1 \dots x@{m^1@{k@1}}.
2025 \List{P@{s^1@{k@1,1}}~x@{r^1@{k@1,1}}; \dots;
2026 P@{s^1@{k@1,l^1@{k@1}}}~x@{r^1@{k@1,l^1@{k@1}}}} & \Imp &
2027 P@1~\left(C^1@{k@1}~x@1~\ldots~x@{m^1@{k@1}}\right) \\
2029 \Forall x@1 \dots x@{m^n@1}.
2030 \List{P@{s^n@{1,1}}~x@{r^n@{1,1}}; \dots;
2031 P@{s^n@{1,l^n@1}}~x@{r^n@{1,l^n@1}}} & \Imp &
2032 P@n~\left(C^n@1~x@1~\ldots~x@{m^n@1}\right) \\
2034 \Forall x@1 \dots x@{m^n@{k@n}}.
2035 \List{P@{s^n@{k@n,1}}~x@{r^n@{k@n,1}}; \dots
2036 P@{s^n@{k@n,l^n@{k@n}}}~x@{r^n@{k@n,l^n@{k@n}}}} & \Imp &
2037 P@n~\left(C^n@{k@n}~x@1~\ldots~x@{m^n@{k@n}}\right)
2044 \left\{\left(r^j@{i,1},s^j@{i,1}\right),\ldots,
2045 \left(r^j@{i,l^j@i},s^j@{i,l^j@i}\right)\right\} = \\[2ex]
2046 && \left\{(i',i'')~\left|~
2047 1\leq i' \leq m^j@i \wedge 1 \leq i'' \leq n \wedge
2048 \tau^j@{i,i'} = (\alpha@1,\ldots,\alpha@h)t@{i''}\right.\right\}
2051 i.e.\ the properties $P@j$ can be assumed for all recursive arguments.
2053 For datatypes with nested recursion, such as the \texttt{term} example from
2054 above, things are a bit more complicated. Conceptually, Isabelle/HOL unfolds
2057 datatype ('a, 'b) term = Var 'a
2058 | App 'b ((('a, 'b) term) list)
2060 to an equivalent definition without nesting:
2062 datatype ('a, 'b) term = Var
2063 | App 'b (('a, 'b) term_list)
2064 and ('a, 'b) term_list = Nil'
2065 | Cons' (('a,'b) term) (('a,'b) term_list)
2067 Note however, that the type \texttt{('a,'b) term_list} and the constructors {\tt
2068 Nil'} and \texttt{Cons'} are not really introduced. One can directly work with
2069 the original (isomorphic) type \texttt{(('a, 'b) term) list} and its existing
2070 constructors \texttt{Nil} and \texttt{Cons}. Thus, the structural induction rule for
2071 \texttt{term} gets the form
2073 \infer{P@1~x@1 \wedge P@2~x@2}
2075 \Forall x.~P@1~(\mathtt{Var}~x) \\
2076 \Forall x@1~x@2.~P@2~x@2 \Imp P@1~(\mathtt{App}~x@1~x@2) \\
2078 \Forall x@1~x@2. \List{P@1~x@1; P@2~x@2} \Imp P@2~(\mathtt{Cons}~x@1~x@2)
2081 Note that there are two predicates $P@1$ and $P@2$, one for the type \texttt{('a,'b) term}
2082 and one for the type \texttt{(('a, 'b) term) list}.
2084 For a datatype with function types such as \texttt{'a tree}, the induction rule
2088 {\Forall a.~P~(\mathtt{Atom}~a) &
2089 \Forall ts.~(\forall x.~P~(ts~x)) \Imp P~(\mathtt{Branch}~ts)}
2092 \medskip In principle, inductive types are already fully determined by
2093 freeness and structural induction. For convenience in applications,
2094 the following derived constructions are automatically provided for any
2097 \subsubsection{The \sdx{case} construct}
2099 The type comes with an \ML-like \texttt{case}-construct:
2102 \mbox{\tt case}~e~\mbox{\tt of} & C^j@1~x@{1,1}~\dots~x@{1,m^j@1} & \To & e@1 \\
2104 \mid & C^j@{k@j}~x@{k@j,1}~\dots~x@{k@j,m^j@{k@j}} & \To & e@{k@j}
2107 where the $x@{i,j}$ are either identifiers or nested tuple patterns as in
2108 \S\ref{subsec:prod-sum}.
2110 All constructors must be present, their order is fixed, and nested patterns
2111 are not supported (with the exception of tuples). Violating this
2112 restriction results in strange error messages.
2115 To perform case distinction on a goal containing a \texttt{case}-construct,
2116 the theorem $t@j.$\texttt{split} is provided:
2118 \begin{array}{@{}rcl@{}}
2119 P(t@j_\mathtt{case}~f@1~\dots~f@{k@j}~e) &\!\!\!=&
2120 \!\!\! ((\forall x@1 \dots x@{m^j@1}. e = C^j@1~x@1\dots x@{m^j@1} \to
2121 P(f@1~x@1\dots x@{m^j@1})) \\
2122 &&\!\!\! ~\land~ \dots ~\land \\
2123 &&~\!\!\! (\forall x@1 \dots x@{m^j@{k@j}}. e = C^j@{k@j}~x@1\dots x@{m^j@{k@j}} \to
2124 P(f@{k@j}~x@1\dots x@{m^j@{k@j}})))
2127 where $t@j$\texttt{_case} is the internal name of the \texttt{case}-construct.
2128 This theorem can be added to a simpset via \ttindex{addsplits}
2129 (see~\S\ref{subsec:HOL:case:splitting}).
2131 \subsubsection{The function \cdx{size}}\label{sec:HOL:size}
2133 Theory \texttt{Arith} declares a generic function \texttt{size} of type
2134 $\alpha\To nat$. Each datatype defines a particular instance of \texttt{size}
2135 by overloading according to the following scheme:
2136 %%% FIXME: This formula is too big and is completely unreadable
2138 size(C^j@i~x@1~\dots~x@{m^j@i}) = \!
2141 0 & \!\mbox{if $Rec^j@i = \emptyset$} \\
2142 1+\sum\limits@{h=1}^{l^j@i}size~x@{r^j@{i,h}} &
2143 \!\mbox{if $Rec^j@i = \left\{\left(r^j@{i,1},s^j@{i,1}\right),\ldots,
2144 \left(r^j@{i,l^j@i},s^j@{i,l^j@i}\right)\right\}$}
2148 where $Rec^j@i$ is defined above. Viewing datatypes as generalised trees, the
2149 size of a leaf is 0 and the size of a node is the sum of the sizes of its
2152 \subsection{Defining datatypes}
2154 The theory syntax for datatype definitions is shown in
2155 Fig.~\ref{datatype-grammar}. In order to be well-formed, a datatype
2156 definition has to obey the rules stated in the previous section. As a result
2157 the theory is extended with the new types, the constructors, and the theorems
2158 listed in the previous section.
2162 datatype : 'datatype' typedecls;
2164 typedecls: ( newtype '=' (cons + '|') ) + 'and'
2166 newtype : typevarlist id ( () | '(' infix ')' )
2168 cons : name (argtype *) ( () | ( '(' mixfix ')' ) )
2170 argtype : id | tid | ('(' typevarlist id ')')
2173 \caption{Syntax of datatype declarations}
2174 \label{datatype-grammar}
2177 Most of the theorems about datatypes become part of the default simpset and
2178 you never need to see them again because the simplifier applies them
2179 automatically. Only induction or exhaustion are usually invoked by hand.
2180 \begin{ttdescription}
2181 \item[\ttindexbold{induct_tac} {\tt"}$x${\tt"} $i$]
2182 applies structural induction on variable $x$ to subgoal $i$, provided the
2183 type of $x$ is a datatype.
2184 \item[\ttindexbold{mutual_induct_tac}
2185 {\tt["}$x@1${\tt",}$\ldots${\tt,"}$x@n${\tt"]} $i$] applies simultaneous
2186 structural induction on the variables $x@1,\ldots,x@n$ to subgoal $i$. This
2187 is the canonical way to prove properties of mutually recursive datatypes
2188 such as \texttt{aexp} and \texttt{bexp}, or datatypes with nested recursion such as
2191 In some cases, induction is overkill and a case distinction over all
2192 constructors of the datatype suffices.
2193 \begin{ttdescription}
2194 \item[\ttindexbold{exhaust_tac} {\tt"}$u${\tt"} $i$]
2195 performs an exhaustive case analysis for the term $u$ whose type
2196 must be a datatype. If the datatype has $k@j$ constructors
2197 $C^j@1$, \dots $C^j@{k@j}$, subgoal $i$ is replaced by $k@j$ new subgoals which
2198 contain the additional assumption $u = C^j@{i'}~x@1~\dots~x@{m^j@{i'}}$ for
2199 $i'=1$, $\dots$,~$k@j$.
2202 Note that induction is only allowed on free variables that should not occur
2203 among the premises of the subgoal. Exhaustion applies to arbitrary terms.
2208 For the technically minded, we exhibit some more details. Processing the
2209 theory file produces an \ML\ structure which, in addition to the usual
2210 components, contains a structure named $t$ for each datatype $t$ defined in
2211 the file. Each structure $t$ contains the following elements:
2213 val distinct : thm list
2214 val inject : thm list
2217 val cases : thm list
2222 val simps : thm list
2224 \texttt{distinct}, \texttt{inject}, \texttt{induct}, \texttt{size}
2225 and \texttt{split} contain the theorems
2226 described above. For user convenience, \texttt{distinct} contains
2227 inequalities in both directions. The reduction rules of the {\tt
2228 case}-construct are in \texttt{cases}. All theorems from {\tt
2229 distinct}, \texttt{inject} and \texttt{cases} are combined in \texttt{simps}.
2230 In case of mutually recursive datatypes, \texttt{recs}, \texttt{size}, \texttt{induct}
2231 and \texttt{simps} are contained in a separate structure named $t@1_\ldots_t@n$.
2234 \subsection{Representing existing types as datatypes}\label{subsec:datatype:rep_datatype}
2236 For foundational reasons, some basic types such as \texttt{nat}, \texttt{*}, {\tt
2237 +}, \texttt{bool} and \texttt{unit} are not defined in a \texttt{datatype} section,
2238 but by more primitive means using \texttt{typedef}. To be able to use the
2239 tactics \texttt{induct_tac} and \texttt{exhaust_tac} and to define functions by
2240 primitive recursion on these types, such types may be represented as actual
2241 datatypes. This is done by specifying an induction rule, as well as theorems
2242 stating the distinctness and injectivity of constructors in a {\tt
2243 rep_datatype} section. For type \texttt{nat} this works as follows:
2246 distinct Suc_not_Zero, Zero_not_Suc
2250 The datatype package automatically derives additional theorems for recursion
2251 and case combinators from these rules. Any of the basic HOL types mentioned
2252 above are represented as datatypes. Try an induction on \texttt{bool}
2256 \subsection{Examples}
2258 \subsubsection{The datatype $\alpha~mylist$}
2260 We want to define a type $\alpha~mylist$. To do this we have to build a new
2261 theory that contains the type definition. We start from the theory
2262 \texttt{Datatype} instead of \texttt{Main} in order to avoid clashes with the
2263 \texttt{List} theory of Isabelle/HOL.
2266 datatype 'a mylist = Nil | Cons 'a ('a mylist)
2269 After loading the theory, we can prove $Cons~x~xs\neq xs$, for example. To
2270 ease the induction applied below, we state the goal with $x$ quantified at the
2271 object-level. This will be stripped later using \ttindex{qed_spec_mp}.
2273 Goal "!x. Cons x xs ~= xs";
2275 {\out ! x. Cons x xs ~= xs}
2276 {\out 1. ! x. Cons x xs ~= xs}
2278 This can be proved by the structural induction tactic:
2280 by (induct_tac "xs" 1);
2282 {\out ! x. Cons x xs ~= xs}
2283 {\out 1. ! x. Cons x Nil ~= Nil}
2284 {\out 2. !!a mylist.}
2285 {\out ! x. Cons x mylist ~= mylist ==>}
2286 {\out ! x. Cons x (Cons a mylist) ~= Cons a mylist}
2288 The first subgoal can be proved using the simplifier. Isabelle/HOL has
2289 already added the freeness properties of lists to the default simplification
2294 {\out ! x. Cons x xs ~= xs}
2295 {\out 1. !!a mylist.}
2296 {\out ! x. Cons x mylist ~= mylist ==>}
2297 {\out ! x. Cons x (Cons a mylist) ~= Cons a mylist}
2299 Similarly, we prove the remaining goal.
2301 by (Asm_simp_tac 1);
2303 {\out ! x. Cons x xs ~= xs}
2306 qed_spec_mp "not_Cons_self";
2307 {\out val not_Cons_self = "Cons x xs ~= xs" : thm}
2309 Because both subgoals could have been proved by \texttt{Asm_simp_tac}
2310 we could have done that in one step:
2312 by (ALLGOALS Asm_simp_tac);
2316 \subsubsection{The datatype $\alpha~mylist$ with mixfix syntax}
2318 In this example we define the type $\alpha~mylist$ again but this time
2319 we want to write \texttt{[]} for \texttt{Nil} and we want to use infix
2320 notation \verb|#| for \texttt{Cons}. To do this we simply add mixfix
2321 annotations after the constructor declarations as follows:
2324 datatype 'a mylist =
2326 Cons 'a ('a mylist) (infixr "#" 70)
2329 Now the theorem in the previous example can be written \verb|x#xs ~= xs|.
2332 \subsubsection{A datatype for weekdays}
2334 This example shows a datatype that consists of 7 constructors:
2337 datatype days = Mon | Tue | Wed | Thu | Fri | Sat | Sun
2340 Because there are more than 6 constructors, inequality is expressed via a function
2341 \verb|days_ord|. The theorem \verb|Mon ~= Tue| is not directly
2342 contained among the distinctness theorems, but the simplifier can
2343 prove it thanks to rewrite rules inherited from theory \texttt{Arith}:
2348 You need not derive such inequalities explicitly: the simplifier will dispose
2349 of them automatically.
2353 \section{Recursive function definitions}\label{sec:HOL:recursive}
2354 \index{recursive functions|see{recursion}}
2356 Isabelle/HOL provides two main mechanisms of defining recursive functions.
2358 \item \textbf{Primitive recursion} is available only for datatypes, and it is
2359 somewhat restrictive. Recursive calls are only allowed on the argument's
2360 immediate constituents. On the other hand, it is the form of recursion most
2361 often wanted, and it is easy to use.
2363 \item \textbf{Well-founded recursion} requires that you supply a well-founded
2364 relation that governs the recursion. Recursive calls are only allowed if
2365 they make the argument decrease under the relation. Complicated recursion
2366 forms, such as nested recursion, can be dealt with. Termination can even be
2367 proved at a later time, though having unsolved termination conditions around
2368 can make work difficult.%
2369 \footnote{This facility is based on Konrad Slind's TFL
2370 package~\cite{slind-tfl}. Thanks are due to Konrad for implementing TFL
2371 and assisting with its installation.}
2374 Following good HOL tradition, these declarations do not assert arbitrary
2375 axioms. Instead, they define the function using a recursion operator. Both
2376 HOL and ZF derive the theory of well-founded recursion from first
2377 principles~\cite{paulson-set-II}. Primitive recursion over some datatype
2378 relies on the recursion operator provided by the datatype package. With
2379 either form of function definition, Isabelle proves the desired recursion
2380 equations as theorems.
2383 \subsection{Primitive recursive functions}
2384 \label{sec:HOL:primrec}
2385 \index{recursion!primitive|(}
2388 Datatypes come with a uniform way of defining functions, {\bf primitive
2389 recursion}. In principle, one could introduce primitive recursive functions
2390 by asserting their reduction rules as new axioms, but this is not recommended:
2391 \begin{ttbox}\slshape
2393 consts app :: ['a list, 'a list] => 'a list
2395 app_Nil "app [] ys = ys"
2396 app_Cons "app (x#xs) ys = x#app xs ys"
2399 Asserting axioms brings the danger of accidentally asserting nonsense, as
2400 in \verb$app [] ys = us$.
2402 The \ttindex{primrec} declaration is a safe means of defining primitive
2403 recursive functions on datatypes:
2406 consts app :: ['a list, 'a list] => 'a list
2409 "app (x#xs) ys = x#app xs ys"
2412 Isabelle will now check that the two rules do indeed form a primitive
2413 recursive definition. For example
2418 is rejected with an error message ``\texttt{Extra variables on rhs}''.
2422 The general form of a primitive recursive definition is
2425 {\it reduction rules}
2427 where \textit{reduction rules} specify one or more equations of the form
2428 \[ f \, x@1 \, \dots \, x@m \, (C \, y@1 \, \dots \, y@k) \, z@1 \,
2429 \dots \, z@n = r \] such that $C$ is a constructor of the datatype, $r$
2430 contains only the free variables on the left-hand side, and all recursive
2431 calls in $r$ are of the form $f \, \dots \, y@i \, \dots$ for some $i$. There
2432 must be at most one reduction rule for each constructor. The order is
2433 immaterial. For missing constructors, the function is defined to return a
2436 If you would like to refer to some rule by name, then you must prefix
2437 the rule with an identifier. These identifiers, like those in the
2438 \texttt{rules} section of a theory, will be visible at the \ML\ level.
2440 The primitive recursive function can have infix or mixfix syntax:
2441 \begin{ttbox}\underscoreon
2442 consts "@" :: ['a list, 'a list] => 'a list (infixr 60)
2445 "(x#xs) @ ys = x#(xs @ ys)"
2448 The reduction rules become part of the default simpset, which
2449 leads to short proof scripts:
2450 \begin{ttbox}\underscoreon
2451 Goal "(xs @ ys) @ zs = xs @ (ys @ zs)";
2452 by (induct\_tac "xs" 1);
2453 by (ALLGOALS Asm\_simp\_tac);
2456 \subsubsection{Example: Evaluation of expressions}
2457 Using mutual primitive recursion, we can define evaluation functions \texttt{evala}
2458 and \texttt{eval_bexp} for the datatypes of arithmetic and boolean expressions mentioned in
2459 \S\ref{subsec:datatype:basics}:
2462 evala :: "['a => nat, 'a aexp] => nat"
2463 evalb :: "['a => nat, 'a bexp] => bool"
2466 "evala env (If_then_else b a1 a2) =
2467 (if evalb env b then evala env a1 else evala env a2)"
2468 "evala env (Sum a1 a2) = evala env a1 + evala env a2"
2469 "evala env (Diff a1 a2) = evala env a1 - evala env a2"
2470 "evala env (Var v) = env v"
2471 "evala env (Num n) = n"
2473 "evalb env (Less a1 a2) = (evala env a1 < evala env a2)"
2474 "evalb env (And b1 b2) = (evalb env b1 & evalb env b2)"
2475 "evalb env (Or b1 b2) = (evalb env b1 & evalb env b2)"
2477 Since the value of an expression depends on the value of its variables,
2478 the functions \texttt{evala} and \texttt{evalb} take an additional
2479 parameter, an {\em environment} of type \texttt{'a => nat}, which maps
2480 variables to their values.
2482 Similarly, we may define substitution functions \texttt{substa}
2483 and \texttt{substb} for expressions: The mapping \texttt{f} of type
2484 \texttt{'a => 'a aexp} given as a parameter is lifted canonically
2485 on the types \texttt{'a aexp} and \texttt{'a bexp}:
2488 substa :: "['a => 'b aexp, 'a aexp] => 'b aexp"
2489 substb :: "['a => 'b aexp, 'a bexp] => 'b bexp"
2492 "substa f (If_then_else b a1 a2) =
2493 If_then_else (substb f b) (substa f a1) (substa f a2)"
2494 "substa f (Sum a1 a2) = Sum (substa f a1) (substa f a2)"
2495 "substa f (Diff a1 a2) = Diff (substa f a1) (substa f a2)"
2496 "substa f (Var v) = f v"
2497 "substa f (Num n) = Num n"
2499 "substb f (Less a1 a2) = Less (substa f a1) (substa f a2)"
2500 "substb f (And b1 b2) = And (substb f b1) (substb f b2)"
2501 "substb f (Or b1 b2) = Or (substb f b1) (substb f b2)"
2503 In textbooks about semantics one often finds {\em substitution theorems},
2504 which express the relationship between substitution and evaluation. For
2505 \texttt{'a aexp} and \texttt{'a bexp}, we can prove such a theorem by mutual
2506 induction, followed by simplification:
2509 "evala env (substa (Var(v := a')) a) =
2510 evala (env(v := evala env a')) a &
2511 evalb env (substb (Var(v := a')) b) =
2512 evalb (env(v := evala env a')) b";
2513 by (mutual_induct_tac ["a","b"] 1);
2514 by (ALLGOALS Asm_full_simp_tac);
2517 \subsubsection{Example: A substitution function for terms}
2518 Functions on datatypes with nested recursion, such as the type
2519 \texttt{term} mentioned in \S\ref{subsec:datatype:basics}, are
2520 also defined by mutual primitive recursion. A substitution
2521 function \texttt{subst_term} on type \texttt{term}, similar to the functions
2522 \texttt{substa} and \texttt{substb} described above, can
2523 be defined as follows:
2526 subst_term :: "['a => ('a, 'b) term, ('a, 'b) term] => ('a, 'b) term"
2528 "['a => ('a, 'b) term, ('a, 'b) term list] => ('a, 'b) term list"
2531 "subst_term f (Var a) = f a"
2532 "subst_term f (App b ts) = App b (subst_term_list f ts)"
2534 "subst_term_list f [] = []"
2535 "subst_term_list f (t # ts) =
2536 subst_term f t # subst_term_list f ts"
2538 The recursion scheme follows the structure of the unfolded definition of type
2539 \texttt{term} shown in \S\ref{subsec:datatype:basics}. To prove properties of
2540 this substitution function, mutual induction is needed:
2543 "(subst_term ((subst_term f1) o f2) t) =
2544 (subst_term f1 (subst_term f2 t)) &
2545 (subst_term_list ((subst_term f1) o f2) ts) =
2546 (subst_term_list f1 (subst_term_list f2 ts))";
2547 by (mutual_induct_tac ["t", "ts"] 1);
2548 by (ALLGOALS Asm_full_simp_tac);
2551 \subsubsection{Example: A map function for infinitely branching trees}
2552 Defining functions on infinitely branching datatypes by primitive
2553 recursion is just as easy. For example, we can define a function
2554 \texttt{map_tree} on \texttt{'a tree} as follows:
2557 map_tree :: "('a => 'b) => 'a tree => 'b tree"
2560 "map_tree f (Atom a) = Atom (f a)"
2561 "map_tree f (Branch ts) = Branch (\%x. map_tree f (ts x))"
2563 Note that all occurrences of functions such as \texttt{ts} in the
2564 \texttt{primrec} clauses must be applied to an argument. In particular,
2565 \texttt{map_tree f o ts} is not allowed.
2567 \index{recursion!primitive|)}
2571 \subsection{General recursive functions}
2572 \label{sec:HOL:recdef}
2573 \index{recursion!general|(}
2576 Using \texttt{recdef}, you can declare functions involving nested recursion
2577 and pattern-matching. Recursion need not involve datatypes and there are few
2578 syntactic restrictions. Termination is proved by showing that each recursive
2579 call makes the argument smaller in a suitable sense, which you specify by
2580 supplying a well-founded relation.
2582 Here is a simple example, the Fibonacci function. The first line declares
2583 \texttt{fib} to be a constant. The well-founded relation is simply~$<$ (on
2584 the natural numbers). Pattern-matching is used here: \texttt{1} is a
2585 macro for \texttt{Suc~0}.
2587 consts fib :: "nat => nat"
2588 recdef fib "less_than"
2591 "fib (Suc(Suc x)) = (fib x + fib (Suc x))"
2594 With \texttt{recdef}, function definitions may be incomplete, and patterns may
2595 overlap, as in functional programming. The \texttt{recdef} package
2596 disambiguates overlapping patterns by taking the order of rules into account.
2597 For missing patterns, the function is defined to return a default value.
2599 %For example, here is a declaration of the list function \cdx{hd}:
2601 %consts hd :: 'a list => 'a
2605 %Because this function is not recursive, we may supply the empty well-founded
2608 The well-founded relation defines a notion of ``smaller'' for the function's
2609 argument type. The relation $\prec$ is \textbf{well-founded} provided it
2610 admits no infinitely decreasing chains
2611 \[ \cdots\prec x@n\prec\cdots\prec x@1. \]
2612 If the function's argument has type~$\tau$, then $\prec$ has to be a relation
2613 over~$\tau$: it must have type $(\tau\times\tau)set$.
2615 Proving well-foundedness can be tricky, so Isabelle/HOL provides a collection
2616 of operators for building well-founded relations. The package recognises
2617 these operators and automatically proves that the constructed relation is
2618 well-founded. Here are those operators, in order of importance:
2620 \item \texttt{less_than} is ``less than'' on the natural numbers.
2621 (It has type $(nat\times nat)set$, while $<$ has type $[nat,nat]\To bool$.
2623 \item $\mathop{\mathtt{measure}} f$, where $f$ has type $\tau\To nat$, is the
2624 relation~$\prec$ on type~$\tau$ such that $x\prec y$ iff $f(x)<f(y)$.
2625 Typically, $f$ takes the recursive function's arguments (as a tuple) and
2626 returns a result expressed in terms of the function \texttt{size}. It is
2627 called a \textbf{measure function}. Recall that \texttt{size} is overloaded
2628 and is defined on all datatypes (see \S\ref{sec:HOL:size}).
2630 \item $\mathop{\mathtt{inv_image}} f\;R$ is a generalisation of
2631 \texttt{measure}. It specifies a relation such that $x\prec y$ iff $f(x)$
2632 is less than $f(y)$ according to~$R$, which must itself be a well-founded
2635 \item $R@1\texttt{**}R@2$ is the lexicographic product of two relations. It
2636 is a relation on pairs and satisfies $(x@1,x@2)\prec(y@1,y@2)$ iff $x@1$
2637 is less than $y@1$ according to~$R@1$ or $x@1=y@1$ and $x@2$
2638 is less than $y@2$ according to~$R@2$.
2640 \item \texttt{finite_psubset} is the proper subset relation on finite sets.
2643 We can use \texttt{measure} to declare Euclid's algorithm for the greatest
2644 common divisor. The measure function, $\lambda(m,n). n$, specifies that the
2645 recursion terminates because argument~$n$ decreases.
2647 recdef gcd "measure ((\%(m,n). n) ::nat*nat=>nat)"
2648 "gcd (m, n) = (if n=0 then m else gcd(n, m mod n))"
2651 The general form of a well-founded recursive definition is
2653 recdef {\it function} {\it rel}
2654 congs {\it congruence rules} {\bf(optional)}
2655 simpset {\it simplification set} {\bf(optional)}
2656 {\it reduction rules}
2660 \item \textit{function} is the name of the function, either as an \textit{id}
2661 or a \textit{string}.
2663 \item \textit{rel} is a {\HOL} expression for the well-founded termination
2666 \item \textit{congruence rules} are required only in highly exceptional
2669 \item The \textit{simplification set} is used to prove that the supplied
2670 relation is well-founded. It is also used to prove the \textbf{termination
2671 conditions}: assertions that arguments of recursive calls decrease under
2672 \textit{rel}. By default, simplification uses \texttt{simpset()}, which
2673 is sufficient to prove well-foundedness for the built-in relations listed
2676 \item \textit{reduction rules} specify one or more recursion equations. Each
2677 left-hand side must have the form $f\,t$, where $f$ is the function and $t$
2678 is a tuple of distinct variables. If more than one equation is present then
2679 $f$ is defined by pattern-matching on components of its argument whose type
2680 is a \texttt{datatype}.
2682 Unlike with \texttt{primrec}, the reduction rules are not added to the
2683 default simpset, and individual rules may not be labelled with identifiers.
2684 However, the identifier $f$\texttt{.rules} is visible at the \ML\ level
2685 as a list of theorems.
2688 With the definition of \texttt{gcd} shown above, Isabelle/HOL is unable to
2689 prove one termination condition. It remains as a precondition of the
2693 {\out ["! m n. n ~= 0 --> m mod n < n}
2694 {\out ==> gcd (?m, ?n) = (if ?n = 0 then ?m else gcd (?n, ?m mod ?n))"] }
2697 The theory \texttt{HOL/ex/Primes} illustrates how to prove termination
2698 conditions afterwards. The function \texttt{Tfl.tgoalw} is like the standard
2699 function \texttt{goalw}, which sets up a goal to prove, but its argument
2700 should be the identifier $f$\texttt{.rules} and its effect is to set up a
2701 proof of the termination conditions:
2703 Tfl.tgoalw thy [] gcd.rules;
2705 {\out ! m n. n ~= 0 --> m mod n < n}
2706 {\out 1. ! m n. n ~= 0 --> m mod n < n}
2708 This subgoal has a one-step proof using \texttt{simp_tac}. Once the theorem
2709 is proved, it can be used to eliminate the termination conditions from
2710 elements of \texttt{gcd.rules}. Theory \texttt{HOL/Subst/Unify} is a much
2711 more complicated example of this process, where the termination conditions can
2712 only be proved by complicated reasoning involving the recursive function
2715 Isabelle/HOL can prove the \texttt{gcd} function's termination condition
2716 automatically if supplied with the right simpset.
2718 recdef gcd "measure ((\%(m,n). n) ::nat*nat=>nat)"
2719 simpset "simpset() addsimps [mod_less_divisor, zero_less_eq]"
2720 "gcd (m, n) = (if n=0 then m else gcd(n, m mod n))"
2723 A \texttt{recdef} definition also returns an induction rule specialised for
2724 the recursive function. For the \texttt{gcd} function above, the induction
2728 {\out "(!!m n. n ~= 0 --> ?P n (m mod n) ==> ?P m n) ==> ?P ?u ?v" : thm}
2730 This rule should be used to reason inductively about the \texttt{gcd}
2731 function. It usually makes the induction hypothesis available at all
2732 recursive calls, leading to very direct proofs. If any termination conditions
2733 remain unproved, they will become additional premises of this rule.
2735 \index{recursion!general|)}
2739 \section{Inductive and coinductive definitions}
2740 \index{*inductive|(}
2741 \index{*coinductive|(}
2743 An {\bf inductive definition} specifies the least set~$R$ closed under given
2744 rules. (Applying a rule to elements of~$R$ yields a result within~$R$.) For
2745 example, a structural operational semantics is an inductive definition of an
2746 evaluation relation. Dually, a {\bf coinductive definition} specifies the
2747 greatest set~$R$ consistent with given rules. (Every element of~$R$ can be
2748 seen as arising by applying a rule to elements of~$R$.) An important example
2749 is using bisimulation relations to formalise equivalence of processes and
2750 infinite data structures.
2752 A theory file may contain any number of inductive and coinductive
2753 definitions. They may be intermixed with other declarations; in
2754 particular, the (co)inductive sets {\bf must} be declared separately as
2755 constants, and may have mixfix syntax or be subject to syntax translations.
2757 Each (co)inductive definition adds definitions to the theory and also
2758 proves some theorems. Each definition creates an \ML\ structure, which is a
2759 substructure of the main theory structure.
2761 This package is related to the \ZF\ one, described in a separate
2763 \footnote{It appeared in CADE~\cite{paulson-CADE}; a longer version is
2764 distributed with Isabelle.} %
2765 which you should refer to in case of difficulties. The package is simpler
2766 than \ZF's thanks to \HOL's extra-logical automatic type-checking. The types
2767 of the (co)inductive sets determine the domain of the fixedpoint definition,
2768 and the package does not have to use inference rules for type-checking.
2771 \subsection{The result structure}
2772 Many of the result structure's components have been discussed in the paper;
2773 others are self-explanatory.
2775 \item[\tt defs] is the list of definitions of the recursive sets.
2777 \item[\tt mono] is a monotonicity theorem for the fixedpoint operator.
2779 \item[\tt unfold] is a fixedpoint equation for the recursive set (the union of
2780 the recursive sets, in the case of mutual recursion).
2782 \item[\tt intrs] is the list of introduction rules, now proved as theorems, for
2783 the recursive sets. The rules are also available individually, using the
2784 names given them in the theory file.
2786 \item[\tt elims] is the list of elimination rule.
2788 \item[\tt elim] is the head of the list \texttt{elims}.
2790 \item[\tt mk_cases] is a function to create simplified instances of {\tt
2791 elim} using freeness reasoning on underlying datatypes.
2794 For an inductive definition, the result structure contains the
2795 rule \texttt{induct}. For a
2796 coinductive definition, it contains the rule \verb|coinduct|.
2798 Figure~\ref{def-result-fig} summarises the two result signatures,
2799 specifying the types of all these components.
2807 val intrs : thm list
2808 val elims : thm list
2810 val mk_cases : string -> thm
2811 {\it(Inductive definitions only)}
2813 {\it(coinductive definitions only)}
2818 \caption{The {\ML} result of a (co)inductive definition} \label{def-result-fig}
2821 \subsection{The syntax of a (co)inductive definition}
2822 An inductive definition has the form
2824 inductive {\it inductive sets}
2825 intrs {\it introduction rules}
2826 monos {\it monotonicity theorems}
2827 con_defs {\it constructor definitions}
2829 A coinductive definition is identical, except that it starts with the keyword
2830 \texttt{coinductive}.
2832 The \texttt{monos} and \texttt{con_defs} sections are optional. If present,
2833 each is specified by a list of identifiers.
2836 \item The \textit{inductive sets} are specified by one or more strings.
2838 \item The \textit{introduction rules} specify one or more introduction rules in
2839 the form \textit{ident\/}~\textit{string}, where the identifier gives the name of
2840 the rule in the result structure.
2842 \item The \textit{monotonicity theorems} are required for each operator
2843 applied to a recursive set in the introduction rules. There {\bf must}
2844 be a theorem of the form $A\subseteq B\Imp M(A)\subseteq M(B)$, for each
2845 premise $t\in M(R@i)$ in an introduction rule!
2847 \item The \textit{constructor definitions} contain definitions of constants
2848 appearing in the introduction rules. In most cases it can be omitted.
2852 \subsection{Example of an inductive definition}
2853 Two declarations, included in a theory file, define the finite powerset
2854 operator. First we declare the constant~\texttt{Fin}. Then we declare it
2855 inductively, with two introduction rules:
2857 consts Fin :: 'a set => 'a set set
2860 emptyI "{\ttlbrace}{\ttrbrace} : Fin A"
2861 insertI "[| a: A; b: Fin A |] ==> insert a b : Fin A"
2863 The resulting theory structure contains a substructure, called~\texttt{Fin}.
2864 It contains the \texttt{Fin}$~A$ introduction rules as the list \texttt{Fin.intrs},
2865 and also individually as \texttt{Fin.emptyI} and \texttt{Fin.consI}. The induction
2866 rule is \texttt{Fin.induct}.
2868 For another example, here is a theory file defining the accessible
2869 part of a relation. The main thing to note is the use of~\texttt{Pow} in
2870 the sole introduction rule, and the corresponding mention of the rule
2871 \verb|Pow_mono| in the \texttt{monos} list. The paper
2872 \cite{paulson-CADE} discusses a \ZF\ version of this example in more
2876 consts pred :: "['b, ('a * 'b)set] => 'a set" (*Set of predecessors*)
2877 acc :: "('a * 'a)set => 'a set" (*Accessible part*)
2878 defs pred_def "pred x r == {y. (y,x):r}"
2881 pred "pred a r: Pow(acc r) ==> a: acc r"
2885 The Isabelle distribution contains many other inductive definitions. Simple
2886 examples are collected on subdirectory \texttt{HOL/Induct}. The theory
2887 \texttt{HOL/Induct/LList} contains coinductive definitions. Larger examples
2888 may be found on other subdirectories of \texttt{HOL}, such as \texttt{IMP},
2889 \texttt{Lambda} and \texttt{Auth}.
2891 \index{*coinductive|)} \index{*inductive|)}
2894 \section{The examples directories}
2896 Directory \texttt{HOL/Auth} contains theories for proving the correctness of
2897 cryptographic protocols~\cite{paulson-jcs}. The approach is based upon
2898 operational semantics rather than the more usual belief logics. On the same
2899 directory are proofs for some standard examples, such as the Needham-Schroeder
2900 public-key authentication protocol and the Otway-Rees
2903 Directory \texttt{HOL/IMP} contains a formalization of various denotational,
2904 operational and axiomatic semantics of a simple while-language, the necessary
2905 equivalence proofs, soundness and completeness of the Hoare rules with
2906 respect to the denotational semantics, and soundness and completeness of a
2907 verification condition generator. Much of development is taken from
2908 Winskel~\cite{winskel93}. For details see~\cite{nipkow-IMP}.
2910 Directory \texttt{HOL/Hoare} contains a user friendly surface syntax for Hoare
2911 logic, including a tactic for generating verification-conditions.
2913 Directory \texttt{HOL/MiniML} contains a formalization of the type system of
2914 the core functional language Mini-ML and a correctness proof for its type
2915 inference algorithm $\cal W$~\cite{milner78,nipkow-W}.
2917 Directory \texttt{HOL/Lambda} contains a formalization of untyped
2918 $\lambda$-calculus in de~Bruijn notation and Church-Rosser proofs for $\beta$
2919 and $\eta$ reduction~\cite{Nipkow-CR}.
2921 Directory \texttt{HOL/Subst} contains Martin Coen's mechanization of a theory of
2922 substitutions and unifiers. It is based on Paulson's previous
2923 mechanisation in {\LCF}~\cite{paulson85} of Manna and Waldinger's
2924 theory~\cite{mw81}. It demonstrates a complicated use of \texttt{recdef},
2925 with nested recursion.
2927 Directory \texttt{HOL/Induct} presents simple examples of (co)inductive
2928 definitions and datatypes.
2930 \item Theory \texttt{PropLog} proves the soundness and completeness of
2931 classical propositional logic, given a truth table semantics. The only
2932 connective is $\imp$. A Hilbert-style axiom system is specified, and its
2933 set of theorems defined inductively. A similar proof in \ZF{} is
2934 described elsewhere~\cite{paulson-set-II}.
2936 \item Theory \texttt{Term} defines the datatype \texttt{term}.
2938 \item Theory \texttt{ABexp} defines arithmetic and boolean expressions
2939 as mutually recursive datatypes.
2941 \item The definition of lazy lists demonstrates methods for handling
2942 infinite data structures and coinduction in higher-order
2943 logic~\cite{paulson-coind}.%
2944 \footnote{To be precise, these lists are \emph{potentially infinite} rather
2945 than lazy. Lazy implies a particular operational semantics.}
2946 Theory \thydx{LList} defines an operator for
2947 corecursion on lazy lists, which is used to define a few simple functions
2948 such as map and append. A coinduction principle is defined
2949 for proving equations on lazy lists.
2951 \item Theory \thydx{LFilter} defines the filter functional for lazy lists.
2952 This functional is notoriously difficult to define because finding the next
2953 element meeting the predicate requires possibly unlimited search. It is not
2954 computable, but can be expressed using a combination of induction and
2957 \item Theory \thydx{Exp} illustrates the use of iterated inductive definitions
2958 to express a programming language semantics that appears to require mutual
2959 induction. Iterated induction allows greater modularity.
2962 Directory \texttt{HOL/ex} contains other examples and experimental proofs in
2965 \item Theory \texttt{Recdef} presents many examples of using \texttt{recdef}
2966 to define recursive functions. Another example is \texttt{Fib}, which
2967 defines the Fibonacci function.
2969 \item Theory \texttt{Primes} defines the Greatest Common Divisor of two
2970 natural numbers and proves a key lemma of the Fundamental Theorem of
2971 Arithmetic: if $p$ is prime and $p$ divides $m\times n$ then $p$ divides~$m$
2974 \item Theory \texttt{Primrec} develops some computation theory. It
2975 inductively defines the set of primitive recursive functions and presents a
2976 proof that Ackermann's function is not primitive recursive.
2978 \item File \texttt{cla.ML} demonstrates the classical reasoner on over sixty
2979 predicate calculus theorems, ranging from simple tautologies to
2980 moderately difficult problems involving equality and quantifiers.
2982 \item File \texttt{meson.ML} contains an experimental implementation of the {\sc
2983 meson} proof procedure, inspired by Plaisted~\cite{plaisted90}. It is
2984 much more powerful than Isabelle's classical reasoner. But it is less
2985 useful in practice because it works only for pure logic; it does not
2986 accept derived rules for the set theory primitives, for example.
2988 \item File \texttt{mesontest.ML} contains test data for the {\sc meson} proof
2989 procedure. These are mostly taken from Pelletier \cite{pelletier86}.
2991 \item File \texttt{set.ML} proves Cantor's Theorem, which is presented in
2992 \S\ref{sec:hol-cantor} below, and the Schr\"oder-Bernstein Theorem.
2994 \item Theory \texttt{MT} contains Jacob Frost's formalization~\cite{frost93} of
2995 Milner and Tofte's coinduction example~\cite{milner-coind}. This
2996 substantial proof concerns the soundness of a type system for a simple
2997 functional language. The semantics of recursion is given by a cyclic
2998 environment, which makes a coinductive argument appropriate.
3003 \section{Example: Cantor's Theorem}\label{sec:hol-cantor}
3004 Cantor's Theorem states that every set has more subsets than it has
3005 elements. It has become a favourite example in higher-order logic since
3006 it is so easily expressed:
3007 \[ \forall f::\alpha \To \alpha \To bool. \exists S::\alpha\To bool.
3008 \forall x::\alpha. f~x \not= S
3011 Viewing types as sets, $\alpha\To bool$ represents the powerset
3012 of~$\alpha$. This version states that for every function from $\alpha$ to
3013 its powerset, some subset is outside its range.
3015 The Isabelle proof uses \HOL's set theory, with the type $\alpha\,set$ and
3016 the operator \cdx{range}.
3020 The set~$S$ is given as an unknown instead of a
3021 quantified variable so that we may inspect the subset found by the proof.
3023 Goal "?S ~: range\thinspace(f :: 'a=>'a set)";
3025 {\out ?S ~: range f}
3026 {\out 1. ?S ~: range f}
3028 The first two steps are routine. The rule \tdx{rangeE} replaces
3029 $\Var{S}\in \texttt{range} \, f$ by $\Var{S}=f~x$ for some~$x$.
3031 by (resolve_tac [notI] 1);
3033 {\out ?S ~: range f}
3034 {\out 1. ?S : range f ==> False}
3036 by (eresolve_tac [rangeE] 1);
3038 {\out ?S ~: range f}
3039 {\out 1. !!x. ?S = f x ==> False}
3041 Next, we apply \tdx{equalityCE}, reasoning that since $\Var{S}=f~x$,
3042 we have $\Var{c}\in \Var{S}$ if and only if $\Var{c}\in f~x$ for
3045 by (eresolve_tac [equalityCE] 1);
3047 {\out ?S ~: range f}
3048 {\out 1. !!x. [| ?c3 x : ?S; ?c3 x : f x |] ==> False}
3049 {\out 2. !!x. [| ?c3 x ~: ?S; ?c3 x ~: f x |] ==> False}
3051 Now we use a bit of creativity. Suppose that~$\Var{S}$ has the form of a
3052 comprehension. Then $\Var{c}\in\{x.\Var{P}~x\}$ implies
3053 $\Var{P}~\Var{c}$. Destruct-resolution using \tdx{CollectD}
3054 instantiates~$\Var{S}$ and creates the new assumption.
3056 by (dresolve_tac [CollectD] 1);
3058 {\out {\ttlbrace}x. ?P7 x{\ttrbrace} ~: range f}
3059 {\out 1. !!x. [| ?c3 x : f x; ?P7(?c3 x) |] ==> False}
3060 {\out 2. !!x. [| ?c3 x ~: {\ttlbrace}x. ?P7 x{\ttrbrace}; ?c3 x ~: f x |] ==> False}
3062 Forcing a contradiction between the two assumptions of subgoal~1
3063 completes the instantiation of~$S$. It is now the set $\{x. x\not\in
3064 f~x\}$, which is the standard diagonal construction.
3068 {\out {\ttlbrace}x. x ~: f x{\ttrbrace} ~: range f}
3069 {\out 1. !!x. [| x ~: {\ttlbrace}x. x ~: f x{\ttrbrace}; x ~: f x |] ==> False}
3071 The rest should be easy. To apply \tdx{CollectI} to the negated
3072 assumption, we employ \ttindex{swap_res_tac}:
3074 by (swap_res_tac [CollectI] 1);
3076 {\out {\ttlbrace}x. x ~: f x{\ttrbrace} ~: range f}
3077 {\out 1. !!x. [| x ~: f x; ~ False |] ==> x ~: f x}
3081 {\out {\ttlbrace}x. x ~: f x{\ttrbrace} ~: range f}
3084 How much creativity is required? As it happens, Isabelle can prove this
3085 theorem automatically. The default classical set \texttt{claset()} contains rules
3086 for most of the constructs of \HOL's set theory. We must augment it with
3087 \tdx{equalityCE} to break up set equalities, and then apply best-first
3088 search. Depth-first search would diverge, but best-first search
3089 successfully navigates through the large search space.
3090 \index{search!best-first}
3094 {\out ?S ~: range f}
3095 {\out 1. ?S ~: range f}
3097 by (best_tac (claset() addSEs [equalityCE]) 1);
3099 {\out {\ttlbrace}x. x ~: f x{\ttrbrace} ~: range f}
3102 If you run this example interactively, make sure your current theory contains
3103 theory \texttt{Set}, for example by executing \ttindex{context}~{\tt Set.thy}.
3104 Otherwise the default claset may not contain the rules for set theory.
3105 \index{higher-order logic|)}
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