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 book~\cite{andrews86} is a
9 full description of the original Church-style higher-order logic. Experience
10 with the {\sc hol} system has demonstrated that higher-order logic is widely
11 applicable in many areas of mathematics and computer science, not just
12 hardware verification, {\sc hol}'s original \textit{raison d'{\^e}tre\/}. It
13 is weaker than ZF set theory but for most applications this does not matter.
14 If you prefer {\ML} to Lisp, you will probably prefer HOL to~ZF.
16 The syntax of HOL\footnote{Earlier versions of Isabelle's HOL used a different
17 syntax. Ancient releases of Isabelle included still another version of~HOL,
18 with explicit type inference rules~\cite{paulson-COLOG}. This version no
19 longer exists, but \thydx{ZF} supports a similar style of reasoning.}
20 follows $\lambda$-calculus and functional programming. Function application
21 is curried. To apply the function~$f$ of type $\tau@1\To\tau@2\To\tau@3$ to
22 the arguments~$a$ and~$b$ in HOL, you simply write $f\,a\,b$. There is no
23 `apply' operator as in \thydx{ZF}. Note that $f(a,b)$ means ``$f$ applied to
24 the pair $(a,b)$'' in HOL. We write ordered pairs as $(a,b)$, not $\langle
27 HOL has a distinct feel, compared with ZF and CTT. It identifies object-level
28 types with meta-level types, taking advantage of Isabelle's built-in
29 type-checker. It identifies object-level functions with meta-level functions,
30 so it uses Isabelle's operations for abstraction and application.
32 These identifications allow Isabelle to support HOL particularly nicely, but
33 they also mean that HOL requires more sophistication from the user --- in
34 particular, an understanding of Isabelle's type system. Beginners should work
35 with \texttt{show_types} (or even \texttt{show_sorts}) set to \texttt{true}.
40 \it name &\it meta-type & \it description \\
41 \cdx{Trueprop}& $bool\To prop$ & coercion to $prop$\\
42 \cdx{Not} & $bool\To bool$ & negation ($\lnot$) \\
43 \cdx{True} & $bool$ & tautology ($\top$) \\
44 \cdx{False} & $bool$ & absurdity ($\bot$) \\
45 \cdx{If} & $[bool,\alpha,\alpha]\To\alpha$ & conditional \\
46 \cdx{Let} & $[\alpha,\alpha\To\beta]\To\beta$ & let binder
48 \subcaption{Constants}
51 \index{"@@{\tt\at} symbol}
52 \index{*"! symbol}\index{*"? symbol}
53 \index{*"?"! symbol}\index{*"E"X"! symbol}
54 \it symbol &\it name &\it meta-type & \it description \\
55 \sdx{SOME} or \tt\at & \cdx{Eps} & $(\alpha\To bool)\To\alpha$ &
56 Hilbert description ($\varepsilon$) \\
57 \sdx{ALL} or {\tt!~} & \cdx{All} & $(\alpha\To bool)\To bool$ &
58 universal quantifier ($\forall$) \\
59 \sdx{EX} or {\tt?~} & \cdx{Ex} & $(\alpha\To bool)\To bool$ &
60 existential quantifier ($\exists$) \\
61 \texttt{EX!} or {\tt?!} & \cdx{Ex1} & $(\alpha\To bool)\To bool$ &
62 unique existence ($\exists!$)\\
63 \texttt{LEAST} & \cdx{Least} & $(\alpha::ord \To bool)\To\alpha$ &
70 \index{&@{\tt\&} symbol}
72 \index{*"-"-"> symbol}
73 \it symbol & \it meta-type & \it priority & \it description \\
74 \sdx{o} & $[\beta\To\gamma,\alpha\To\beta]\To (\alpha\To\gamma)$ &
75 Left 55 & composition ($\circ$) \\
76 \tt = & $[\alpha,\alpha]\To bool$ & Left 50 & equality ($=$) \\
77 \tt < & $[\alpha::ord,\alpha]\To bool$ & Left 50 & less than ($<$) \\
78 \tt <= & $[\alpha::ord,\alpha]\To bool$ & Left 50 &
79 less than or equals ($\leq$)\\
80 \tt \& & $[bool,bool]\To bool$ & Right 35 & conjunction ($\conj$) \\
81 \tt | & $[bool,bool]\To bool$ & Right 30 & disjunction ($\disj$) \\
82 \tt --> & $[bool,bool]\To bool$ & Right 25 & implication ($\imp$)
85 \caption{Syntax of \texttt{HOL}} \label{hol-constants}
93 \[\begin{array}{rclcl}
94 term & = & \hbox{expression of class~$term$} \\
95 & | & "SOME~" id " . " formula
96 & | & "\at~" id " . " formula \\
98 \multicolumn{3}{l}{"let"~id~"="~term";"\dots";"~id~"="~term~"in"~term} \\
100 \multicolumn{3}{l}{"if"~formula~"then"~term~"else"~term} \\
101 & | & "LEAST"~ id " . " formula \\[2ex]
102 formula & = & \hbox{expression of type~$bool$} \\
103 & | & term " = " term \\
104 & | & term " \ttilde= " term \\
105 & | & term " < " term \\
106 & | & term " <= " term \\
107 & | & "\ttilde\ " formula \\
108 & | & formula " \& " formula \\
109 & | & formula " | " formula \\
110 & | & formula " --> " formula \\
111 & | & "ALL~" id~id^* " . " formula
112 & | & "!~~~" id~id^* " . " formula \\
113 & | & "EX~~" id~id^* " . " formula
114 & | & "?~~~" id~id^* " . " formula \\
115 & | & "EX!~" id~id^* " . " formula
116 & | & "?!~~" id~id^* " . " formula \\
119 \caption{Full grammar for HOL} \label{hol-grammar}
125 Figure~\ref{hol-constants} lists the constants (including infixes and
126 binders), while Fig.\ts\ref{hol-grammar} presents the grammar of
127 higher-order logic. Note that $a$\verb|~=|$b$ is translated to
131 HOL has no if-and-only-if connective; logical equivalence is expressed using
132 equality. But equality has a high priority, as befitting a relation, while
133 if-and-only-if typically has the lowest priority. Thus, $\lnot\lnot P=P$
134 abbreviates $\lnot\lnot (P=P)$ and not $(\lnot\lnot P)=P$. When using $=$
135 to mean logical equivalence, enclose both operands in parentheses.
138 \subsection{Types and overloading}
139 The universal type class of higher-order terms is called~\cldx{term}.
140 By default, explicit type variables have class \cldx{term}. In
141 particular the equality symbol and quantifiers are polymorphic over
144 The type of formulae, \tydx{bool}, belongs to class \cldx{term}; thus,
145 formulae are terms. The built-in type~\tydx{fun}, which constructs
146 function types, is overloaded with arity {\tt(term,\thinspace
147 term)\thinspace term}. Thus, $\sigma\To\tau$ belongs to class~{\tt
148 term} if $\sigma$ and~$\tau$ do, allowing quantification over
151 HOL allows new types to be declared as subsets of existing types;
152 see~{\S}\ref{sec:HOL:Types}. ML-like datatypes can also be declared; see
153 ~{\S}\ref{sec:HOL:datatype}.
155 Several syntactic type classes --- \cldx{plus}, \cldx{minus},
157 \cldx{power} --- permit overloading of the operators {\tt+},\index{*"+
158 symbol} {\tt-}\index{*"- symbol}, {\tt*}.\index{*"* symbol}
159 and \verb|^|.\index{^@\verb.^. symbol}
161 They are overloaded to denote the obvious arithmetic operations on types
162 \tdx{nat}, \tdx{int} and~\tdx{real}. (With the \verb|^| operator, the
163 exponent always has type~\tdx{nat}.) Non-arithmetic overloadings are also
164 done: the operator {\tt-} can denote set difference, while \verb|^| can
165 denote exponentiation of relations (iterated composition). Unary minus is
166 also written as~{\tt-} and is overloaded like its 2-place counterpart; it even
167 can stand for set complement.
169 The constant \cdx{0} is also overloaded. It serves as the zero element of
170 several types, of which the most important is \tdx{nat} (the natural
171 numbers). The type class \cldx{plus_ac0} comprises all types for which 0
172 and~+ satisfy the laws $x+y=y+x$, $(x+y)+z = x+(y+z)$ and $0+x = x$. These
173 types include the numeric ones \tdx{nat}, \tdx{int} and~\tdx{real} and also
174 multisets. The summation operator \cdx{setsum} is available for all types in
177 Theory \thydx{Ord} defines the syntactic class \cldx{ord} of order
178 signatures. The relations $<$ and $\leq$ are polymorphic over this
179 class, as are the functions \cdx{mono}, \cdx{min} and \cdx{max}, and
180 the \cdx{LEAST} operator. \thydx{Ord} also defines a subclass
181 \cldx{order} of \cldx{ord} which axiomatizes the types that are partially
182 ordered with respect to~$\leq$. A further subclass \cldx{linorder} of
183 \cldx{order} axiomatizes linear orderings.
184 For details, see the file \texttt{Ord.thy}.
186 If you state a goal containing overloaded functions, you may need to include
187 type constraints. Type inference may otherwise make the goal more
188 polymorphic than you intended, with confusing results. For example, the
189 variables $i$, $j$ and $k$ in the goal $i \leq j \Imp i \leq j+k$ have type
190 $\alpha::\{ord,plus\}$, although you may have expected them to have some
191 numeric type, e.g. $nat$. Instead you should have stated the goal as
192 $(i::nat) \leq j \Imp i \leq j+k$, which causes all three variables to have
196 If resolution fails for no obvious reason, try setting
197 \ttindex{show_types} to \texttt{true}, causing Isabelle to display
198 types of terms. Possibly set \ttindex{show_sorts} to \texttt{true} as
199 well, causing Isabelle to display type classes and sorts.
201 \index{unification!incompleteness of}
202 Where function types are involved, Isabelle's unification code does not
203 guarantee to find instantiations for type variables automatically. Be
204 prepared to use \ttindex{res_inst_tac} instead of \texttt{resolve_tac},
205 possibly instantiating type variables. Setting
206 \ttindex{Unify.trace_types} to \texttt{true} causes Isabelle to report
207 omitted search paths during unification.\index{tracing!of unification}
213 Hilbert's {\bf description} operator~$\varepsilon x. P[x]$ stands for some~$x$
214 satisfying~$P$, if such exists. Since all terms in HOL denote something, a
215 description is always meaningful, but we do not know its value unless $P$
216 defines it uniquely. We may write descriptions as \cdx{Eps}($\lambda x.
217 P[x]$) or use the syntax \hbox{\tt SOME~$x$.~$P[x]$}.
219 Existential quantification is defined by
220 \[ \exists x. P~x \;\equiv\; P(\varepsilon x. P~x). \]
221 The unique existence quantifier, $\exists!x. P$, is defined in terms
222 of~$\exists$ and~$\forall$. An Isabelle binder, it admits nested
223 quantifications. For instance, $\exists!x\,y. P\,x\,y$ abbreviates
224 $\exists!x. \exists!y. P\,x\,y$; note that this does not mean that there
225 exists a unique pair $(x,y)$ satisfying~$P\,x\,y$.
229 \index{*"! symbol}\index{*"? symbol}\index{HOL system@{\sc hol} system} The
230 basic Isabelle/HOL binders have two notations. Apart from the usual
231 \texttt{ALL} and \texttt{EX} for $\forall$ and $\exists$, Isabelle/HOL also
232 supports the original notation of Gordon's {\sc hol} system: \texttt{!}\
233 and~\texttt{?}. In the latter case, the existential quantifier \emph{must} be
234 followed by a space; thus {\tt?x} is an unknown, while \verb'? x. f x=y' is a
235 quantification. Both notations are accepted for input. The print mode
236 ``\ttindexbold{HOL}'' governs the output notation. If enabled (e.g.\ by
237 passing option \texttt{-m HOL} to the \texttt{isabelle} executable),
238 then~{\tt!}\ and~{\tt?}\ are displayed.
242 If $\tau$ is a type of class \cldx{ord}, $P$ a formula and $x$ a
243 variable of type $\tau$, then the term \cdx{LEAST}~$x. P[x]$ is defined
244 to be the least (w.r.t.\ $\leq$) $x$ such that $P~x$ holds (see
245 Fig.~\ref{hol-defs}). The definition uses Hilbert's $\varepsilon$
246 choice operator, so \texttt{Least} is always meaningful, but may yield
247 nothing useful in case there is not a unique least element satisfying
248 $P$.\footnote{Class $ord$ does not require much of its instances, so
249 $\leq$ need not be a well-ordering, not even an order at all!}
251 \medskip All these binders have priority 10.
254 The low priority of binders means that they need to be enclosed in
255 parenthesis when they occur in the context of other operations. For example,
256 instead of $P \land \forall x. Q$ you need to write $P \land (\forall x. Q)$.
260 \subsection{The let and case constructions}
261 Local abbreviations can be introduced by a \texttt{let} construct whose
262 syntax appears in Fig.\ts\ref{hol-grammar}. Internally it is translated into
263 the constant~\cdx{Let}. It can be expanded by rewriting with its
264 definition, \tdx{Let_def}.
266 HOL also defines the basic syntax
267 \[\dquotes"case"~e~"of"~c@1~"=>"~e@1~"|" \dots "|"~c@n~"=>"~e@n\]
268 as a uniform means of expressing \texttt{case} constructs. Therefore \texttt{case}
269 and \sdx{of} are reserved words. Initially, this is mere syntax and has no
270 logical meaning. By declaring translations, you can cause instances of the
271 \texttt{case} construct to denote applications of particular case operators.
272 This is what happens automatically for each \texttt{datatype} definition
273 (see~{\S}\ref{sec:HOL:datatype}).
276 Both \texttt{if} and \texttt{case} constructs have as low a priority as
277 quantifiers, which requires additional enclosing parentheses in the context
278 of most other operations. For example, instead of $f~x = {\tt if\dots
279 then\dots else}\dots$ you need to write $f~x = ({\tt if\dots then\dots
283 \section{Rules of inference}
286 \begin{ttbox}\makeatother
287 \tdx{refl} t = (t::'a)
288 \tdx{subst} [| s = t; P s |] ==> P (t::'a)
289 \tdx{ext} (!!x::'a. (f x :: 'b) = g x) ==> (\%x. f x) = (\%x. g x)
290 \tdx{impI} (P ==> Q) ==> P-->Q
291 \tdx{mp} [| P-->Q; P |] ==> Q
292 \tdx{iff} (P-->Q) --> (Q-->P) --> (P=Q)
293 \tdx{someI} P(x::'a) ==> P(@x. P x)
294 \tdx{True_or_False} (P=True) | (P=False)
296 \caption{The \texttt{HOL} rules} \label{hol-rules}
299 Figure~\ref{hol-rules} shows the primitive inference rules of~HOL, with
300 their~{\ML} names. Some of the rules deserve additional comments:
301 \begin{ttdescription}
302 \item[\tdx{ext}] expresses extensionality of functions.
303 \item[\tdx{iff}] asserts that logically equivalent formulae are
305 \item[\tdx{someI}] gives the defining property of the Hilbert
306 $\varepsilon$-operator. It is a form of the Axiom of Choice. The derived rule
307 \tdx{some_equality} (see below) is often easier to use.
308 \item[\tdx{True_or_False}] makes the logic classical.\footnote{In
309 fact, the $\varepsilon$-operator already makes the logic classical, as
310 shown by Diaconescu; see Paulson~\cite{paulson-COLOG} for details.}
314 \begin{figure}\hfuzz=4pt%suppress "Overfull \hbox" message
315 \begin{ttbox}\makeatother
316 \tdx{True_def} True == ((\%x::bool. x)=(\%x. x))
317 \tdx{All_def} All == (\%P. P = (\%x. True))
318 \tdx{Ex_def} Ex == (\%P. P(@x. P x))
319 \tdx{False_def} False == (!P. P)
320 \tdx{not_def} not == (\%P. P-->False)
321 \tdx{and_def} op & == (\%P Q. !R. (P-->Q-->R) --> R)
322 \tdx{or_def} op | == (\%P Q. !R. (P-->R) --> (Q-->R) --> R)
323 \tdx{Ex1_def} Ex1 == (\%P. ? x. P x & (! y. P y --> y=x))
325 \tdx{o_def} op o == (\%(f::'b=>'c) g x::'a. f(g x))
326 \tdx{if_def} If P x y ==
327 (\%P x y. @z::'a.(P=True --> z=x) & (P=False --> z=y))
328 \tdx{Let_def} Let s f == f s
329 \tdx{Least_def} Least P == @x. P(x) & (ALL y. P(y) --> x <= y)"
331 \caption{The \texttt{HOL} definitions} \label{hol-defs}
335 HOL follows standard practice in higher-order logic: only a few connectives
336 are taken as primitive, with the remainder defined obscurely
337 (Fig.\ts\ref{hol-defs}). Gordon's {\sc hol} system expresses the
338 corresponding definitions \cite[page~270]{mgordon-hol} using
339 object-equality~({\tt=}), which is possible because equality in higher-order
340 logic may equate formulae and even functions over formulae. But theory~HOL,
341 like all other Isabelle theories, uses meta-equality~({\tt==}) for
344 The definitions above should never be expanded and are shown for completeness
345 only. Instead users should reason in terms of the derived rules shown below
346 or, better still, using high-level tactics
347 (see~{\S}\ref{sec:HOL:generic-packages}).
350 Some of the rules mention type variables; for example, \texttt{refl}
351 mentions the type variable~{\tt'a}. This allows you to instantiate
352 type variables explicitly by calling \texttt{res_inst_tac}.
357 \tdx{sym} s=t ==> t=s
358 \tdx{trans} [| r=s; s=t |] ==> r=t
359 \tdx{ssubst} [| t=s; P s |] ==> P t
360 \tdx{box_equals} [| a=b; a=c; b=d |] ==> c=d
361 \tdx{arg_cong} x = y ==> f x = f y
362 \tdx{fun_cong} f = g ==> f x = g x
363 \tdx{cong} [| f = g; x = y |] ==> f x = g y
364 \tdx{not_sym} t ~= s ==> s ~= t
365 \subcaption{Equality}
368 \tdx{FalseE} False ==> P
370 \tdx{conjI} [| P; Q |] ==> P&Q
371 \tdx{conjunct1} [| P&Q |] ==> P
372 \tdx{conjunct2} [| P&Q |] ==> Q
373 \tdx{conjE} [| P&Q; [| P; Q |] ==> R |] ==> R
375 \tdx{disjI1} P ==> P|Q
376 \tdx{disjI2} Q ==> P|Q
377 \tdx{disjE} [| P | Q; P ==> R; Q ==> R |] ==> R
379 \tdx{notI} (P ==> False) ==> ~ P
380 \tdx{notE} [| ~ P; P |] ==> R
381 \tdx{impE} [| P-->Q; P; Q ==> R |] ==> R
382 \subcaption{Propositional logic}
384 \tdx{iffI} [| P ==> Q; Q ==> P |] ==> P=Q
385 \tdx{iffD1} [| P=Q; P |] ==> Q
386 \tdx{iffD2} [| P=Q; Q |] ==> P
387 \tdx{iffE} [| P=Q; [| P --> Q; Q --> P |] ==> R |] ==> R
388 \subcaption{Logical equivalence}
391 \caption{Derived rules for HOL} \label{hol-lemmas1}
394 %\tdx{eqTrueI} P ==> P=True
395 %\tdx{eqTrueE} P=True ==> P
399 \begin{ttbox}\makeatother
400 \tdx{allI} (!!x. P x) ==> !x. P x
401 \tdx{spec} !x. P x ==> P x
402 \tdx{allE} [| !x. P x; P x ==> R |] ==> R
403 \tdx{all_dupE} [| !x. P x; [| P x; !x. P x |] ==> R |] ==> R
405 \tdx{exI} P x ==> ? x. P x
406 \tdx{exE} [| ? x. P x; !!x. P x ==> Q |] ==> Q
408 \tdx{ex1I} [| P a; !!x. P x ==> x=a |] ==> ?! x. P x
409 \tdx{ex1E} [| ?! x. P x; !!x. [| P x; ! y. P y --> y=x |] ==> R
412 \tdx{some_equality} [| P a; !!x. P x ==> x=a |] ==> (@x. P x) = a
413 \subcaption{Quantifiers and descriptions}
415 \tdx{ccontr} (~P ==> False) ==> P
416 \tdx{classical} (~P ==> P) ==> P
417 \tdx{excluded_middle} ~P | P
419 \tdx{disjCI} (~Q ==> P) ==> P|Q
420 \tdx{exCI} (! x. ~ P x ==> P a) ==> ? x. P x
421 \tdx{impCE} [| P-->Q; ~ P ==> R; Q ==> R |] ==> R
422 \tdx{iffCE} [| P=Q; [| P;Q |] ==> R; [| ~P; ~Q |] ==> R |] ==> R
423 \tdx{notnotD} ~~P ==> P
424 \tdx{swap} ~P ==> (~Q ==> P) ==> Q
425 \subcaption{Classical logic}
427 \tdx{if_P} P ==> (if P then x else y) = x
428 \tdx{if_not_P} ~ P ==> (if P then x else y) = y
429 \tdx{split_if} P(if Q then x else y) = ((Q --> P x) & (~Q --> P y))
430 \subcaption{Conditionals}
432 \caption{More derived rules} \label{hol-lemmas2}
435 Some derived rules are shown in Figures~\ref{hol-lemmas1}
436 and~\ref{hol-lemmas2}, with their {\ML} names. These include natural rules
437 for the logical connectives, as well as sequent-style elimination rules for
438 conjunctions, implications, and universal quantifiers.
440 Note the equality rules: \tdx{ssubst} performs substitution in
441 backward proofs, while \tdx{box_equals} supports reasoning by
442 simplifying both sides of an equation.
444 The following simple tactics are occasionally useful:
445 \begin{ttdescription}
446 \item[\ttindexbold{strip_tac} $i$] applies \texttt{allI} and \texttt{impI}
447 repeatedly to remove all outermost universal quantifiers and implications
449 \item[\ttindexbold{case_tac} {\tt"}$P${\tt"} $i$] performs case distinction on
450 $P$ for subgoal $i$: the latter is replaced by two identical subgoals with
451 the added assumptions $P$ and $\lnot P$, respectively.
452 \item[\ttindexbold{smp_tac} $j$ $i$] applies $j$ times \texttt{spec} and then
453 \texttt{mp} in subgoal $i$, which is typically useful when forward-chaining
454 from an induction hypothesis. As a generalization of \texttt{mp_tac},
455 if there are assumptions $\forall \vec{x}. P \vec{x} \imp Q \vec{x}$ and
456 $P \vec{a}$, ($\vec{x}$ being a vector of $j$ variables)
457 then it replaces the universally quantified implication by $Q \vec{a}$.
458 It may instantiate unknowns. It fails if it can do nothing.
465 \it name &\it meta-type & \it description \\
466 \index{{}@\verb'{}' symbol}
467 \verb|{}| & $\alpha\,set$ & the empty set \\
468 \cdx{insert} & $[\alpha,\alpha\,set]\To \alpha\,set$
469 & insertion of element \\
470 \cdx{Collect} & $(\alpha\To bool)\To\alpha\,set$
472 \cdx{INTER} & $[\alpha\,set,\alpha\To\beta\,set]\To\beta\,set$
473 & intersection over a set\\
474 \cdx{UNION} & $[\alpha\,set,\alpha\To\beta\,set]\To\beta\,set$
476 \cdx{Inter} & $(\alpha\,set)set\To\alpha\,set$
477 &set of sets intersection \\
478 \cdx{Union} & $(\alpha\,set)set\To\alpha\,set$
479 &set of sets union \\
480 \cdx{Pow} & $\alpha\,set \To (\alpha\,set)set$
482 \cdx{range} & $(\alpha\To\beta )\To\beta\,set$
483 & range of a function \\[1ex]
484 \cdx{Ball}~~\cdx{Bex} & $[\alpha\,set,\alpha\To bool]\To bool$
485 & bounded quantifiers
488 \subcaption{Constants}
491 \begin{tabular}{llrrr}
492 \it symbol &\it name &\it meta-type & \it priority & \it description \\
493 \sdx{INT} & \cdx{INTER1} & $(\alpha\To\beta\,set)\To\beta\,set$ & 10 &
495 \sdx{UN} & \cdx{UNION1} & $(\alpha\To\beta\,set)\To\beta\,set$ & 10 &
505 \begin{tabular}{rrrr}
506 \it symbol & \it meta-type & \it priority & \it description \\
507 \tt `` & $[\alpha\To\beta ,\alpha\,set]\To \beta\,set$
509 \sdx{Int} & $[\alpha\,set,\alpha\,set]\To\alpha\,set$
510 & Left 70 & intersection ($\int$) \\
511 \sdx{Un} & $[\alpha\,set,\alpha\,set]\To\alpha\,set$
512 & Left 65 & union ($\un$) \\
513 \tt: & $[\alpha ,\alpha\,set]\To bool$
514 & Left 50 & membership ($\in$) \\
515 \tt <= & $[\alpha\,set,\alpha\,set]\To bool$
516 & Left 50 & subset ($\subseteq$)
520 \caption{Syntax of the theory \texttt{Set}} \label{hol-set-syntax}
525 \begin{center} \tt\frenchspacing
528 \it external & \it internal & \it description \\
529 $a$ \ttilde: $b$ & \ttilde($a$ : $b$) & \rm not in\\
530 {\ttlbrace}$a@1$, $\ldots${\ttrbrace} & insert $a@1$ $\ldots$ {\ttlbrace}{\ttrbrace} & \rm finite set \\
531 {\ttlbrace}$x$. $P[x]${\ttrbrace} & Collect($\lambda x. P[x]$) &
533 \sdx{INT} $x$:$A$. $B[x]$ & INTER $A$ $\lambda x. B[x]$ &
535 \sdx{UN}{\tt\ } $x$:$A$. $B[x]$ & UNION $A$ $\lambda x. B[x]$ &
537 \sdx{ALL} $x$:$A$.\ $P[x]$ or \texttt{!} $x$:$A$.\ $P[x]$ &
538 Ball $A$ $\lambda x.\ P[x]$ &
539 \rm bounded $\forall$ \\
540 \sdx{EX}{\tt\ } $x$:$A$.\ $P[x]$ or \texttt{?} $x$:$A$.\ $P[x]$ &
541 Bex $A$ $\lambda x.\ P[x]$ & \rm bounded $\exists$
544 \subcaption{Translations}
547 \[\begin{array}{rclcl}
548 term & = & \hbox{other terms\ldots} \\
549 & | & "{\ttlbrace}{\ttrbrace}" \\
550 & | & "{\ttlbrace} " term\; ("," term)^* " {\ttrbrace}" \\
551 & | & "{\ttlbrace} " id " . " formula " {\ttrbrace}" \\
552 & | & term " `` " term \\
553 & | & term " Int " term \\
554 & | & term " Un " term \\
555 & | & "INT~~" id ":" term " . " term \\
556 & | & "UN~~~" id ":" term " . " term \\
557 & | & "INT~~" id~id^* " . " term \\
558 & | & "UN~~~" id~id^* " . " term \\[2ex]
559 formula & = & \hbox{other formulae\ldots} \\
560 & | & term " : " term \\
561 & | & term " \ttilde: " term \\
562 & | & term " <= " term \\
563 & | & "ALL " id ":" term " . " formula
564 & | & "!~" id ":" term " . " formula \\
565 & | & "EX~~" id ":" term " . " formula
566 & | & "?~" id ":" term " . " formula \\
569 \subcaption{Full Grammar}
570 \caption{Syntax of the theory \texttt{Set} (continued)} \label{hol-set-syntax2}
574 \section{A formulation of set theory}
575 Historically, higher-order logic gives a foundation for Russell and
576 Whitehead's theory of classes. Let us use modern terminology and call them
577 {\bf sets}, but note that these sets are distinct from those of ZF set theory,
578 and behave more like ZF classes.
581 Sets are given by predicates over some type~$\sigma$. Types serve to
582 define universes for sets, but type-checking is still significant.
584 There is a universal set (for each type). Thus, sets have complements, and
585 may be defined by absolute comprehension.
587 Although sets may contain other sets as elements, the containing set must
588 have a more complex type.
590 Finite unions and intersections have the same behaviour in HOL as they do
591 in~ZF. In HOL the intersection of the empty set is well-defined, denoting the
592 universal set for the given type.
594 \subsection{Syntax of set theory}\index{*set type}
595 HOL's set theory is called \thydx{Set}. The type $\alpha\,set$ is essentially
596 the same as $\alpha\To bool$. The new type is defined for clarity and to
597 avoid complications involving function types in unification. The isomorphisms
598 between the two types are declared explicitly. They are very natural:
599 \texttt{Collect} maps $\alpha\To bool$ to $\alpha\,set$, while \hbox{\tt op :}
600 maps in the other direction (ignoring argument order).
602 Figure~\ref{hol-set-syntax} lists the constants, infixes, and syntax
603 translations. Figure~\ref{hol-set-syntax2} presents the grammar of the new
604 constructs. Infix operators include union and intersection ($A\un B$
605 and $A\int B$), the subset and membership relations, and the image
606 operator~{\tt``}\@. Note that $a$\verb|~:|$b$ is translated to
609 The $\{a@1,\ldots\}$ notation abbreviates finite sets constructed in
610 the obvious manner using~\texttt{insert} and~$\{\}$:
612 \{a, b, c\} & \equiv &
613 \texttt{insert} \, a \, ({\tt insert} \, b \, ({\tt insert} \, c \, \{\}))
616 The set \hbox{\tt{\ttlbrace}$x$.\ $P[x]${\ttrbrace}} consists of all $x$ (of
617 suitable type) that satisfy~$P[x]$, where $P[x]$ is a formula that may contain
618 free occurrences of~$x$. This syntax expands to \cdx{Collect}$(\lambda x.
619 P[x])$. It defines sets by absolute comprehension, which is impossible in~ZF;
620 the type of~$x$ implicitly restricts the comprehension.
622 The set theory defines two {\bf bounded quantifiers}:
624 \forall x\in A. P[x] &\hbox{abbreviates}& \forall x. x\in A\imp P[x] \\
625 \exists x\in A. P[x] &\hbox{abbreviates}& \exists x. x\in A\conj P[x]
627 The constants~\cdx{Ball} and~\cdx{Bex} are defined
628 accordingly. Instead of \texttt{Ball $A$ $P$} and \texttt{Bex $A$ $P$} we may
629 write\index{*"! symbol}\index{*"? symbol}
630 \index{*ALL symbol}\index{*EX symbol}
632 \hbox{\tt ALL~$x$:$A$.\ $P[x]$} and \hbox{\tt EX~$x$:$A$.\ $P[x]$}. The
633 original notation of Gordon's {\sc hol} system is supported as well:
634 \texttt{!}\ and \texttt{?}.
636 Unions and intersections over sets, namely $\bigcup@{x\in A}B[x]$ and
637 $\bigcap@{x\in A}B[x]$, are written
638 \sdx{UN}~\hbox{\tt$x$:$A$.\ $B[x]$} and
639 \sdx{INT}~\hbox{\tt$x$:$A$.\ $B[x]$}.
641 Unions and intersections over types, namely $\bigcup@x B[x]$ and $\bigcap@x
642 B[x]$, are written \sdx{UN}~\hbox{\tt$x$.\ $B[x]$} and
643 \sdx{INT}~\hbox{\tt$x$.\ $B[x]$}. They are equivalent to the previous
644 union and intersection operators when $A$ is the universal set.
646 The operators $\bigcup A$ and $\bigcap A$ act upon sets of sets. They are
647 not binders, but are equal to $\bigcup@{x\in A}x$ and $\bigcap@{x\in A}x$,
652 \begin{figure} \underscoreon
654 \tdx{mem_Collect_eq} (a : {\ttlbrace}x. P x{\ttrbrace}) = P a
655 \tdx{Collect_mem_eq} {\ttlbrace}x. x:A{\ttrbrace} = A
657 \tdx{empty_def} {\ttlbrace}{\ttrbrace} == {\ttlbrace}x. False{\ttrbrace}
658 \tdx{insert_def} insert a B == {\ttlbrace}x. x=a{\ttrbrace} Un B
659 \tdx{Ball_def} Ball A P == ! x. x:A --> P x
660 \tdx{Bex_def} Bex A P == ? x. x:A & P x
661 \tdx{subset_def} A <= B == ! x:A. x:B
662 \tdx{Un_def} A Un B == {\ttlbrace}x. x:A | x:B{\ttrbrace}
663 \tdx{Int_def} A Int B == {\ttlbrace}x. x:A & x:B{\ttrbrace}
664 \tdx{set_diff_def} A - B == {\ttlbrace}x. x:A & x~:B{\ttrbrace}
665 \tdx{Compl_def} -A == {\ttlbrace}x. ~ x:A{\ttrbrace}
666 \tdx{INTER_def} INTER A B == {\ttlbrace}y. ! x:A. y: B x{\ttrbrace}
667 \tdx{UNION_def} UNION A B == {\ttlbrace}y. ? x:A. y: B x{\ttrbrace}
668 \tdx{INTER1_def} INTER1 B == INTER {\ttlbrace}x. True{\ttrbrace} B
669 \tdx{UNION1_def} UNION1 B == UNION {\ttlbrace}x. True{\ttrbrace} B
670 \tdx{Inter_def} Inter S == (INT x:S. x)
671 \tdx{Union_def} Union S == (UN x:S. x)
672 \tdx{Pow_def} Pow A == {\ttlbrace}B. B <= A{\ttrbrace}
673 \tdx{image_def} f``A == {\ttlbrace}y. ? x:A. y=f x{\ttrbrace}
674 \tdx{range_def} range f == {\ttlbrace}y. ? x. y=f x{\ttrbrace}
676 \caption{Rules of the theory \texttt{Set}} \label{hol-set-rules}
680 \begin{figure} \underscoreon
682 \tdx{CollectI} [| P a |] ==> a : {\ttlbrace}x. P x{\ttrbrace}
683 \tdx{CollectD} [| a : {\ttlbrace}x. P x{\ttrbrace} |] ==> P a
684 \tdx{CollectE} [| a : {\ttlbrace}x. P x{\ttrbrace}; P a ==> W |] ==> W
686 \tdx{ballI} [| !!x. x:A ==> P x |] ==> ! x:A. P x
687 \tdx{bspec} [| ! x:A. P x; x:A |] ==> P x
688 \tdx{ballE} [| ! x:A. P x; P x ==> Q; ~ x:A ==> Q |] ==> Q
690 \tdx{bexI} [| P x; x:A |] ==> ? x:A. P x
691 \tdx{bexCI} [| ! x:A. ~ P x ==> P a; a:A |] ==> ? x:A. P x
692 \tdx{bexE} [| ? x:A. P x; !!x. [| x:A; P x |] ==> Q |] ==> Q
693 \subcaption{Comprehension and Bounded quantifiers}
695 \tdx{subsetI} (!!x. x:A ==> x:B) ==> A <= B
696 \tdx{subsetD} [| A <= B; c:A |] ==> c:B
697 \tdx{subsetCE} [| A <= B; ~ (c:A) ==> P; c:B ==> P |] ==> P
699 \tdx{subset_refl} A <= A
700 \tdx{subset_trans} [| A<=B; B<=C |] ==> A<=C
702 \tdx{equalityI} [| A <= B; B <= A |] ==> A = B
703 \tdx{equalityD1} A = B ==> A<=B
704 \tdx{equalityD2} A = B ==> B<=A
705 \tdx{equalityE} [| A = B; [| A<=B; B<=A |] ==> P |] ==> P
707 \tdx{equalityCE} [| A = B; [| c:A; c:B |] ==> P;
708 [| ~ c:A; ~ c:B |] ==> P
710 \subcaption{The subset and equality relations}
712 \caption{Derived rules for set theory} \label{hol-set1}
716 \begin{figure} \underscoreon
718 \tdx{emptyE} a : {\ttlbrace}{\ttrbrace} ==> P
720 \tdx{insertI1} a : insert a B
721 \tdx{insertI2} a : B ==> a : insert b B
722 \tdx{insertE} [| a : insert b A; a=b ==> P; a:A ==> P |] ==> P
724 \tdx{ComplI} [| c:A ==> False |] ==> c : -A
725 \tdx{ComplD} [| c : -A |] ==> ~ c:A
727 \tdx{UnI1} c:A ==> c : A Un B
728 \tdx{UnI2} c:B ==> c : A Un B
729 \tdx{UnCI} (~c:B ==> c:A) ==> c : A Un B
730 \tdx{UnE} [| c : A Un B; c:A ==> P; c:B ==> P |] ==> P
732 \tdx{IntI} [| c:A; c:B |] ==> c : A Int B
733 \tdx{IntD1} c : A Int B ==> c:A
734 \tdx{IntD2} c : A Int B ==> c:B
735 \tdx{IntE} [| c : A Int B; [| c:A; c:B |] ==> P |] ==> P
737 \tdx{UN_I} [| a:A; b: B a |] ==> b: (UN x:A. B x)
738 \tdx{UN_E} [| b: (UN x:A. B x); !!x.[| x:A; b:B x |] ==> R |] ==> R
740 \tdx{INT_I} (!!x. x:A ==> b: B x) ==> b : (INT x:A. B x)
741 \tdx{INT_D} [| b: (INT x:A. B x); a:A |] ==> b: B a
742 \tdx{INT_E} [| b: (INT x:A. B x); b: B a ==> R; ~ a:A ==> R |] ==> R
744 \tdx{UnionI} [| X:C; A:X |] ==> A : Union C
745 \tdx{UnionE} [| A : Union C; !!X.[| A:X; X:C |] ==> R |] ==> R
747 \tdx{InterI} [| !!X. X:C ==> A:X |] ==> A : Inter C
748 \tdx{InterD} [| A : Inter C; X:C |] ==> A:X
749 \tdx{InterE} [| A : Inter C; A:X ==> R; ~ X:C ==> R |] ==> R
751 \tdx{PowI} A<=B ==> A: Pow B
752 \tdx{PowD} A: Pow B ==> A<=B
754 \tdx{imageI} [| x:A |] ==> f x : f``A
755 \tdx{imageE} [| b : f``A; !!x.[| b=f x; x:A |] ==> P |] ==> P
757 \tdx{rangeI} f x : range f
758 \tdx{rangeE} [| b : range f; !!x.[| b=f x |] ==> P |] ==> P
760 \caption{Further derived rules for set theory} \label{hol-set2}
764 \subsection{Axioms and rules of set theory}
765 Figure~\ref{hol-set-rules} presents the rules of theory \thydx{Set}. The
766 axioms \tdx{mem_Collect_eq} and \tdx{Collect_mem_eq} assert
767 that the functions \texttt{Collect} and \hbox{\tt op :} are isomorphisms. Of
768 course, \hbox{\tt op :} also serves as the membership relation.
770 All the other axioms are definitions. They include the empty set, bounded
771 quantifiers, unions, intersections, complements and the subset relation.
772 They also include straightforward constructions on functions: image~({\tt``})
775 %The predicate \cdx{inj_on} is used for simulating type definitions.
776 %The statement ${\tt inj_on}~f~A$ asserts that $f$ is injective on the
777 %set~$A$, which specifies a subset of its domain type. In a type
778 %definition, $f$ is the abstraction function and $A$ is the set of valid
779 %representations; we should not expect $f$ to be injective outside of~$A$.
781 %\begin{figure} \underscoreon
783 %\tdx{Inv_f_f} inj f ==> Inv f (f x) = x
784 %\tdx{f_Inv_f} y : range f ==> f(Inv f y) = y
787 % [| Inv f x=Inv f y; x: range f; y: range f |] ==> x=y
790 %\tdx{monoI} [| !!A B. A <= B ==> f A <= f B |] ==> mono f
791 %\tdx{monoD} [| mono f; A <= B |] ==> f A <= f B
793 %\tdx{injI} [| !! x y. f x = f y ==> x=y |] ==> inj f
794 %\tdx{inj_inverseI} (!!x. g(f x) = x) ==> inj f
795 %\tdx{injD} [| inj f; f x = f y |] ==> x=y
797 %\tdx{inj_onI} (!!x y. [| f x=f y; x:A; y:A |] ==> x=y) ==> inj_on f A
798 %\tdx{inj_onD} [| inj_on f A; f x=f y; x:A; y:A |] ==> x=y
800 %\tdx{inj_on_inverseI}
801 % (!!x. x:A ==> g(f x) = x) ==> inj_on f A
802 %\tdx{inj_on_contraD}
803 % [| inj_on f A; x~=y; x:A; y:A |] ==> ~ f x=f y
805 %\caption{Derived rules involving functions} \label{hol-fun}
809 \begin{figure} \underscoreon
811 \tdx{Union_upper} B:A ==> B <= Union A
812 \tdx{Union_least} [| !!X. X:A ==> X<=C |] ==> Union A <= C
814 \tdx{Inter_lower} B:A ==> Inter A <= B
815 \tdx{Inter_greatest} [| !!X. X:A ==> C<=X |] ==> C <= Inter A
817 \tdx{Un_upper1} A <= A Un B
818 \tdx{Un_upper2} B <= A Un B
819 \tdx{Un_least} [| A<=C; B<=C |] ==> A Un B <= C
821 \tdx{Int_lower1} A Int B <= A
822 \tdx{Int_lower2} A Int B <= B
823 \tdx{Int_greatest} [| C<=A; C<=B |] ==> C <= A Int B
825 \caption{Derived rules involving subsets} \label{hol-subset}
829 \begin{figure} \underscoreon \hfuzz=4pt%suppress "Overfull \hbox" message
831 \tdx{Int_absorb} A Int A = A
832 \tdx{Int_commute} A Int B = B Int A
833 \tdx{Int_assoc} (A Int B) Int C = A Int (B Int C)
834 \tdx{Int_Un_distrib} (A Un B) Int C = (A Int C) Un (B Int C)
836 \tdx{Un_absorb} A Un A = A
837 \tdx{Un_commute} A Un B = B Un A
838 \tdx{Un_assoc} (A Un B) Un C = A Un (B Un C)
839 \tdx{Un_Int_distrib} (A Int B) Un C = (A Un C) Int (B Un C)
841 \tdx{Compl_disjoint} A Int (-A) = {\ttlbrace}x. False{\ttrbrace}
842 \tdx{Compl_partition} A Un (-A) = {\ttlbrace}x. True{\ttrbrace}
843 \tdx{double_complement} -(-A) = A
844 \tdx{Compl_Un} -(A Un B) = (-A) Int (-B)
845 \tdx{Compl_Int} -(A Int B) = (-A) Un (-B)
847 \tdx{Union_Un_distrib} Union(A Un B) = (Union A) Un (Union B)
848 \tdx{Int_Union} A Int (Union B) = (UN C:B. A Int C)
850 \tdx{Inter_Un_distrib} Inter(A Un B) = (Inter A) Int (Inter B)
851 \tdx{Un_Inter} A Un (Inter B) = (INT C:B. A Un C)
854 \caption{Set equalities} \label{hol-equalities}
856 %\tdx{Un_Union_image} (UN x:C.(A x) Un (B x)) = Union(A``C) Un Union(B``C)
857 %\tdx{Int_Inter_image} (INT x:C.(A x) Int (B x)) = Inter(A``C) Int Inter(B``C)
859 Figures~\ref{hol-set1} and~\ref{hol-set2} present derived rules. Most are
860 obvious and resemble rules of Isabelle's ZF set theory. Certain rules, such
861 as \tdx{subsetCE}, \tdx{bexCI} and \tdx{UnCI}, are designed for classical
862 reasoning; the rules \tdx{subsetD}, \tdx{bexI}, \tdx{Un1} and~\tdx{Un2} are
863 not strictly necessary but yield more natural proofs. Similarly,
864 \tdx{equalityCE} supports classical reasoning about extensionality, after the
865 fashion of \tdx{iffCE}. See the file \texttt{HOL/Set.ML} for proofs
866 pertaining to set theory.
868 Figure~\ref{hol-subset} presents lattice properties of the subset relation.
869 Unions form least upper bounds; non-empty intersections form greatest lower
870 bounds. Reasoning directly about subsets often yields clearer proofs than
871 reasoning about the membership relation. See the file \texttt{HOL/subset.ML}.
873 Figure~\ref{hol-equalities} presents many common set equalities. They
874 include commutative, associative and distributive laws involving unions,
875 intersections and complements. For a complete listing see the file {\tt
879 \texttt{Blast_tac} proves many set-theoretic theorems automatically.
880 Hence you seldom need to refer to the theorems above.
886 \it name &\it meta-type & \it description \\
887 \cdx{inj}~~\cdx{surj}& $(\alpha\To\beta )\To bool$
888 & injective/surjective \\
889 \cdx{inj_on} & $[\alpha\To\beta ,\alpha\,set]\To bool$
890 & injective over subset\\
891 \cdx{inv} & $(\alpha\To\beta)\To(\beta\To\alpha)$ & inverse function
897 \tdx{inj_def} inj f == ! x y. f x=f y --> x=y
898 \tdx{surj_def} surj f == ! y. ? x. y=f x
899 \tdx{inj_on_def} inj_on f A == !x:A. !y:A. f x=f y --> x=y
900 \tdx{inv_def} inv f == (\%y. @x. f(x)=y)
902 \caption{Theory \thydx{Fun}} \label{fig:HOL:Fun}
905 \subsection{Properties of functions}\nopagebreak
906 Figure~\ref{fig:HOL:Fun} presents a theory of simple properties of functions.
907 Note that ${\tt inv}~f$ uses Hilbert's $\varepsilon$ to yield an inverse
908 of~$f$. See the file \texttt{HOL/Fun.ML} for a complete listing of the derived
909 rules. Reasoning about function composition (the operator~\sdx{o}) and the
910 predicate~\cdx{surj} is done simply by expanding the definitions.
912 There is also a large collection of monotonicity theorems for constructions
913 on sets in the file \texttt{HOL/mono.ML}.
916 \section{Generic packages}
917 \label{sec:HOL:generic-packages}
919 HOL instantiates most of Isabelle's generic packages, making available the
920 simplifier and the classical reasoner.
922 \subsection{Simplification and substitution}
924 Simplification tactics tactics such as \texttt{Asm_simp_tac} and \texttt{Full_simp_tac} use the default simpset
925 (\texttt{simpset()}), which works for most purposes. A quite minimal
926 simplification set for higher-order logic is~\ttindexbold{HOL_ss};
927 even more frugal is \ttindexbold{HOL_basic_ss}. Equality~($=$), which
928 also expresses logical equivalence, may be used for rewriting. See
929 the file \texttt{HOL/simpdata.ML} for a complete listing of the basic
930 simplification rules.
932 See \iflabelundefined{chap:classical}{the {\em Reference Manual\/}}%
933 {Chaps.\ts\ref{substitution} and~\ref{simp-chap}} for details of substitution
936 \begin{warn}\index{simplification!of conjunctions}%
937 Reducing $a=b\conj P(a)$ to $a=b\conj P(b)$ is sometimes advantageous. The
938 left part of a conjunction helps in simplifying the right part. This effect
939 is not available by default: it can be slow. It can be obtained by
940 including \ttindex{conj_cong} in a simpset, \verb$addcongs [conj_cong]$.
943 \begin{warn}\index{simplification!of \texttt{if}}\label{if-simp}%
944 By default only the condition of an \ttindex{if} is simplified but not the
945 \texttt{then} and \texttt{else} parts. Of course the latter are simplified
946 once the condition simplifies to \texttt{True} or \texttt{False}. To ensure
947 full simplification of all parts of a conditional you must remove
948 \ttindex{if_weak_cong} from the simpset, \verb$delcongs [if_weak_cong]$.
951 If the simplifier cannot use a certain rewrite rule --- either because
952 of nontermination or because its left-hand side is too flexible ---
953 then you might try \texttt{stac}:
954 \begin{ttdescription}
955 \item[\ttindexbold{stac} $thm$ $i,$] where $thm$ is of the form $lhs = rhs$,
956 replaces in subgoal $i$ instances of $lhs$ by corresponding instances of
957 $rhs$. In case of multiple instances of $lhs$ in subgoal $i$, backtracking
958 may be necessary to select the desired ones.
960 If $thm$ is a conditional equality, the instantiated condition becomes an
961 additional (first) subgoal.
964 HOL provides the tactic \ttindex{hyp_subst_tac}, which substitutes for an
965 equality throughout a subgoal and its hypotheses. This tactic uses HOL's
966 general substitution rule.
968 \subsubsection{Case splitting}
969 \label{subsec:HOL:case:splitting}
971 HOL also provides convenient means for case splitting during rewriting. Goals
972 containing a subterm of the form \texttt{if}~$b$~{\tt then\dots else\dots}
973 often require a case distinction on $b$. This is expressed by the theorem
976 \Var{P}(\mbox{\tt if}~\Var{b}~{\tt then}~\Var{x}~\mbox{\tt else}~\Var{y})~=~
977 ((\Var{b} \to \Var{P}(\Var{x})) \land (\lnot \Var{b} \to \Var{P}(\Var{y})))
980 For example, a simple instance of $(*)$ is
982 x \in (\mbox{\tt if}~x \in A~{\tt then}~A~\mbox{\tt else}~\{x\})~=~
983 ((x \in A \to x \in A) \land (x \notin A \to x \in \{x\}))
985 Because $(*)$ is too general as a rewrite rule for the simplifier (the
986 left-hand side is not a higher-order pattern in the sense of
987 \iflabelundefined{chap:simplification}{the {\em Reference Manual\/}}%
988 {Chap.\ts\ref{chap:simplification}}), there is a special infix function
989 \ttindexbold{addsplits} of type \texttt{simpset * thm list -> simpset}
990 (analogous to \texttt{addsimps}) that adds rules such as $(*)$ to a
993 by(simp_tac (simpset() addsplits [split_if]) 1);
995 The effect is that after each round of simplification, one occurrence of
996 \texttt{if} is split acording to \texttt{split_if}, until all occurences of
997 \texttt{if} have been eliminated.
999 It turns out that using \texttt{split_if} is almost always the right thing to
1000 do. Hence \texttt{split_if} is already included in the default simpset. If
1001 you want to delete it from a simpset, use \ttindexbold{delsplits}, which is
1002 the inverse of \texttt{addsplits}:
1004 by(simp_tac (simpset() delsplits [split_if]) 1);
1007 In general, \texttt{addsplits} accepts rules of the form
1009 \Var{P}(c~\Var{x@1}~\dots~\Var{x@n})~=~ rhs
1011 where $c$ is a constant and $rhs$ is arbitrary. Note that $(*)$ is of the
1012 right form because internally the left-hand side is
1013 $\Var{P}(\mathtt{If}~\Var{b}~\Var{x}~~\Var{y})$. Important further examples
1014 are splitting rules for \texttt{case} expressions (see~{\S}\ref{subsec:list}
1015 and~{\S}\ref{subsec:datatype:basics}).
1017 Analogous to \texttt{Addsimps} and \texttt{Delsimps}, there are also
1018 imperative versions of \texttt{addsplits} and \texttt{delsplits}
1020 \ttindexbold{Addsplits}: thm list -> unit
1021 \ttindexbold{Delsplits}: thm list -> unit
1023 for adding splitting rules to, and deleting them from the current simpset.
1025 \subsection{Classical reasoning}
1027 HOL derives classical introduction rules for $\disj$ and~$\exists$, as well as
1028 classical elimination rules for~$\imp$ and~$\bimp$, and the swap rule; recall
1029 Fig.\ts\ref{hol-lemmas2} above.
1031 The classical reasoner is installed. Tactics such as \texttt{Blast_tac} and
1032 {\tt Best_tac} refer to the default claset (\texttt{claset()}), which works
1033 for most purposes. Named clasets include \ttindexbold{prop_cs}, which
1034 includes the propositional rules, and \ttindexbold{HOL_cs}, which also
1035 includes quantifier rules. See the file \texttt{HOL/cladata.ML} for lists of
1036 the classical rules,
1037 and \iflabelundefined{chap:classical}{the {\em Reference Manual\/}}%
1038 {Chap.\ts\ref{chap:classical}} for more discussion of classical proof methods.
1041 %FIXME outdated, both from the Isabelle and SVC perspective
1042 % \section{Calling the decision procedure SVC}\label{sec:HOL:SVC}
1044 % \index{SVC decision procedure|(}
1046 % The Stanford Validity Checker (SVC) is a tool that can check the validity of
1047 % certain types of formulae. If it is installed on your machine, then
1048 % Isabelle/HOL can be configured to call it through the tactic
1049 % \ttindex{svc_tac}. It is ideal for large tautologies and complex problems in
1050 % linear arithmetic. Subexpressions that SVC cannot handle are automatically
1051 % replaced by variables, so you can call the tactic on any subgoal. See the
1052 % file \texttt{HOL/ex/svc_test.ML} for examples.
1054 % svc_tac : int -> tactic
1055 % Svc.trace : bool ref \hfill{\bf initially false}
1058 % \begin{ttdescription}
1059 % \item[\ttindexbold{svc_tac} $i$] attempts to prove subgoal~$i$ by translating
1060 % it into a formula recognized by~SVC\@. If it succeeds then the subgoal is
1061 % removed. It fails if SVC is unable to prove the subgoal. It crashes with
1062 % an error message if SVC appears not to be installed. Numeric variables may
1063 % have types \texttt{nat}, \texttt{int} or \texttt{real}.
1065 % \item[\ttindexbold{Svc.trace}] is a flag that, if set, causes \texttt{svc_tac}
1066 % to trace its operations: abstraction of the subgoal, translation to SVC
1067 % syntax, SVC's response.
1068 % \end{ttdescription}
1070 % Here is an example, with tracing turned on:
1073 % {\out val it : bool = true}
1074 % Goal "(#3::nat)*a <= #2 + #4*b + #6*c & #11 <= #2*a + b + #2*c & \ttback
1075 % \ttback a + #3*b <= #5 + #2*c --> #2 + #3*b <= #2*a + #6*c";
1078 % {\out Subgoal abstracted to}
1079 % {\out #3 * a <= #2 + #4 * b + #6 * c &}
1080 % {\out #11 <= #2 * a + b + #2 * c & a + #3 * b <= #5 + #2 * c -->}
1081 % {\out #2 + #3 * b <= #2 * a + #6 * c}
1082 % {\out Calling SVC:}
1083 % {\out (=> (<= 0 (F_c) ) (=> (<= 0 (F_b) ) (=> (<= 0 (F_a) )}
1084 % {\out (=> (AND (<= {* 3 (F_a) } {+ {+ 2 {* 4 (F_b) } } }
1085 % {\out {* 6 (F_c) } } ) (AND (<= 11 {+ {+ {* 2 (F_a) } (F_b) }}
1086 % {\out {* 2 (F_c) } } ) (<= {+ (F_a) {* 3 (F_b) } } {+ 5 }
1087 % {\out {* 2 (F_c) } } ) ) ) (< {+ 2 {* 3 (F_b) } } {+ 1 {+ }
1088 % {\out {* 2 (F_a) } {* 6 (F_c) } } } ) ) ) ) ) }
1089 % {\out SVC Returns:}
1092 % {\out #3 * a <= #2 + #4 * b + #6 * c &}
1093 % {\out #11 <= #2 * a + b + #2 * c & a + #3 * b <= #5 + #2 * c -->}
1094 % {\out #2 + #3 * b <= #2 * a + #6 * c}
1095 % {\out No subgoals!}
1100 % Calling \ttindex{svc_tac} entails an above-average risk of
1101 % unsoundness. Isabelle does not check SVC's result independently. Moreover,
1102 % the tactic translates the submitted formula using code that lies outside
1103 % Isabelle's inference core. Theorems that depend upon results proved using SVC
1104 % (and other oracles) are displayed with the annotation \texttt{[!]} attached.
1105 % You can also use \texttt{\#der (rep_thm $th$)} to examine the proof object of
1106 % theorem~$th$, as described in the \emph{Reference Manual}.
1109 % To start, first download SVC from the Internet at URL
1111 % http://agamemnon.stanford.edu/~levitt/vc/index.html
1113 % and install it using the instructions supplied. SVC requires two environment
1115 % \begin{ttdescription}
1116 % \item[\ttindexbold{SVC_HOME}] is an absolute pathname to the SVC
1117 % distribution directory.
1119 % \item[\ttindexbold{SVC_MACHINE}] identifies the type of computer and
1121 % \end{ttdescription}
1122 % You can set these environment variables either using the Unix shell or through
1123 % an Isabelle \texttt{settings} file. Isabelle assumes SVC to be installed if
1124 % \texttt{SVC_HOME} is defined.
1126 % \paragraph*{Acknowledgement.} This interface uses code supplied by S{\o}ren
1130 % \index{SVC decision procedure|)}
1135 \section{Types}\label{sec:HOL:Types}
1136 This section describes HOL's basic predefined types ($\alpha \times \beta$,
1137 $\alpha + \beta$, $nat$ and $\alpha \; list$) and ways for introducing new
1138 types in general. The most important type construction, the
1139 \texttt{datatype}, is treated separately in {\S}\ref{sec:HOL:datatype}.
1142 \subsection{Product and sum types}\index{*"* type}\index{*"+ type}
1143 \label{subsec:prod-sum}
1145 \begin{figure}[htbp]
1147 \it symbol & \it meta-type & & \it description \\
1148 \cdx{Pair} & $[\alpha,\beta]\To \alpha\times\beta$
1149 & & ordered pairs $(a,b)$ \\
1150 \cdx{fst} & $\alpha\times\beta \To \alpha$ & & first projection\\
1151 \cdx{snd} & $\alpha\times\beta \To \beta$ & & second projection\\
1152 \cdx{split} & $[[\alpha,\beta]\To\gamma, \alpha\times\beta] \To \gamma$
1153 & & generalized projection\\
1155 $[\alpha\,set, \alpha\To\beta\,set]\To(\alpha\times\beta)set$ &
1156 & general sum of sets
1158 %\tdx{fst_def} fst p == @a. ? b. p = (a,b)
1159 %\tdx{snd_def} snd p == @b. ? a. p = (a,b)
1160 %\tdx{split_def} split c p == c (fst p) (snd p)
1161 \begin{ttbox}\makeatletter
1162 \tdx{Sigma_def} Sigma A B == UN x:A. UN y:B x. {\ttlbrace}(x,y){\ttrbrace}
1164 \tdx{Pair_eq} ((a,b) = (a',b')) = (a=a' & b=b')
1165 \tdx{Pair_inject} [| (a, b) = (a',b'); [| a=a'; b=b' |] ==> R |] ==> R
1166 \tdx{PairE} [| !!x y. p = (x,y) ==> Q |] ==> Q
1168 \tdx{fst_conv} fst (a,b) = a
1169 \tdx{snd_conv} snd (a,b) = b
1170 \tdx{surjective_pairing} p = (fst p,snd p)
1172 \tdx{split} split c (a,b) = c a b
1173 \tdx{split_split} R(split c p) = (! x y. p = (x,y) --> R(c x y))
1175 \tdx{SigmaI} [| a:A; b:B a |] ==> (a,b) : Sigma A B
1177 \tdx{SigmaE} [| c:Sigma A B; !!x y.[| x:A; y:B x; c=(x,y) |] ==> P
1180 \caption{Type $\alpha\times\beta$}\label{hol-prod}
1183 Theory \thydx{Prod} (Fig.\ts\ref{hol-prod}) defines the product type
1184 $\alpha\times\beta$, with the ordered pair syntax $(a, b)$. General
1185 tuples are simulated by pairs nested to the right:
1187 \begin{tabular}{c|c}
1188 external & internal \\
1190 $\tau@1 \times \dots \times \tau@n$ & $\tau@1 \times (\dots (\tau@{n-1} \times \tau@n)\dots)$ \\
1192 $(t@1,\dots,t@n)$ & $(t@1,(\dots,(t@{n-1},t@n)\dots)$ \\
1195 In addition, it is possible to use tuples
1196 as patterns in abstractions:
1198 {\tt\%($x$,$y$). $t$} \quad stands for\quad \texttt{split(\%$x$\thinspace$y$.\ $t$)}
1200 Nested patterns are also supported. They are translated stepwise:
1202 \hbox{\tt\%($x$,$y$,$z$).\ $t$}
1203 & \leadsto & \hbox{\tt\%($x$,($y$,$z$)).\ $t$} \\
1204 & \leadsto & \hbox{\tt split(\%$x$.\%($y$,$z$).\ $t$)}\\
1205 & \leadsto & \hbox{\tt split(\%$x$.\ split(\%$y$ $z$.\ $t$))}
1207 The reverse translation is performed upon printing.
1209 The translation between patterns and \texttt{split} is performed automatically
1210 by the parser and printer. Thus the internal and external form of a term
1211 may differ, which can affects proofs. For example the term {\tt
1212 (\%(x,y).(y,x))(a,b)} requires the theorem \texttt{split} (which is in the
1213 default simpset) to rewrite to {\tt(b,a)}.
1215 In addition to explicit $\lambda$-abstractions, patterns can be used in any
1216 variable binding construct which is internally described by a
1217 $\lambda$-abstraction. Some important examples are
1219 \item[Let:] \texttt{let {\it pattern} = $t$ in $u$}
1220 \item[Quantifiers:] \texttt{ALL~{\it pattern}:$A$.~$P$}
1221 \item[Choice:] {\underscoreon \tt SOME~{\it pattern}.~$P$}
1222 \item[Set operations:] \texttt{UN~{\it pattern}:$A$.~$B$}
1223 \item[Sets:] \texttt{{\ttlbrace}{\it pattern}.~$P${\ttrbrace}}
1226 There is a simple tactic which supports reasoning about patterns:
1227 \begin{ttdescription}
1228 \item[\ttindexbold{split_all_tac} $i$] replaces in subgoal $i$ all
1229 {\tt!!}-quantified variables of product type by individual variables for
1230 each component. A simple example:
1232 {\out 1. !!p. (\%(x,y,z). (x, y, z)) p = p}
1233 by(split_all_tac 1);
1234 {\out 1. !!x xa ya. (\%(x,y,z). (x, y, z)) (x, xa, ya) = (x, xa, ya)}
1238 Theory \texttt{Prod} also introduces the degenerate product type \texttt{unit}
1239 which contains only a single element named {\tt()} with the property
1241 \tdx{unit_eq} u = ()
1245 Theory \thydx{Sum} (Fig.~\ref{hol-sum}) defines the sum type $\alpha+\beta$
1246 which associates to the right and has a lower priority than $*$: $\tau@1 +
1247 \tau@2 + \tau@3*\tau@4$ means $\tau@1 + (\tau@2 + (\tau@3*\tau@4))$.
1249 The definition of products and sums in terms of existing types is not
1250 shown. The constructions are fairly standard and can be found in the
1251 respective theory files. Although the sum and product types are
1252 constructed manually for foundational reasons, they are represented as
1253 actual datatypes later (see {\S}\ref{subsec:datatype:rep_datatype}).
1254 Therefore, the theory \texttt{Datatype} should be used instead of
1255 \texttt{Sum} or \texttt{Prod}.
1259 \it symbol & \it meta-type & & \it description \\
1260 \cdx{Inl} & $\alpha \To \alpha+\beta$ & & first injection\\
1261 \cdx{Inr} & $\beta \To \alpha+\beta$ & & second injection\\
1262 \cdx{sum_case} & $[\alpha\To\gamma, \beta\To\gamma, \alpha+\beta] \To\gamma$
1265 \begin{ttbox}\makeatletter
1266 \tdx{Inl_not_Inr} Inl a ~= Inr b
1268 \tdx{inj_Inl} inj Inl
1269 \tdx{inj_Inr} inj Inr
1271 \tdx{sumE} [| !!x. P(Inl x); !!y. P(Inr y) |] ==> P s
1273 \tdx{sum_case_Inl} sum_case f g (Inl x) = f x
1274 \tdx{sum_case_Inr} sum_case f g (Inr x) = g x
1276 \tdx{surjective_sum} sum_case (\%x. f(Inl x)) (\%y. f(Inr y)) s = f s
1277 \tdx{sum.split_case} R(sum_case f g s) = ((! x. s = Inl(x) --> R(f(x))) &
1278 (! y. s = Inr(y) --> R(g(y))))
1280 \caption{Type $\alpha+\beta$}\label{hol-sum}
1292 \it symbol & \it meta-type & \it priority & \it description \\
1293 \cdx{0} & $\alpha$ & & zero \\
1294 \cdx{Suc} & $nat \To nat$ & & successor function\\
1295 \tt * & $[\alpha,\alpha]\To \alpha$ & Left 70 & multiplication \\
1296 \tt div & $[\alpha,\alpha]\To \alpha$ & Left 70 & division\\
1297 \tt mod & $[\alpha,\alpha]\To \alpha$ & Left 70 & modulus\\
1298 \tt dvd & $[\alpha,\alpha]\To bool$ & Left 70 & ``divides'' relation\\
1299 \tt + & $[\alpha,\alpha]\To \alpha$ & Left 65 & addition\\
1300 \tt - & $[\alpha,\alpha]\To \alpha$ & Left 65 & subtraction
1302 \subcaption{Constants and infixes}
1304 \begin{ttbox}\makeatother
1305 \tdx{nat_induct} [| P 0; !!n. P n ==> P(Suc n) |] ==> P n
1307 \tdx{Suc_not_Zero} Suc m ~= 0
1308 \tdx{inj_Suc} inj Suc
1309 \tdx{n_not_Suc_n} n~=Suc n
1310 \subcaption{Basic properties}
1312 \caption{The type of natural numbers, \tydx{nat}} \label{hol-nat1}
1317 \begin{ttbox}\makeatother
1319 (Suc m)+n = Suc(m+n)
1328 \tdx{mod_less} m<n ==> m mod n = m
1329 \tdx{mod_geq} [| 0<n; ~m<n |] ==> m mod n = (m-n) mod n
1331 \tdx{div_less} m<n ==> m div n = 0
1332 \tdx{div_geq} [| 0<n; ~m<n |] ==> m div n = Suc((m-n) div n)
1334 \caption{Recursion equations for the arithmetic operators} \label{hol-nat2}
1337 \subsection{The type of natural numbers, \textit{nat}}
1338 \index{nat@{\textit{nat}} type|(}
1340 The theory \thydx{Nat} defines the natural numbers in a roundabout but
1341 traditional way. The axiom of infinity postulates a type~\tydx{ind} of
1342 individuals, which is non-empty and closed under an injective operation. The
1343 natural numbers are inductively generated by choosing an arbitrary individual
1344 for~0 and using the injective operation to take successors. This is a least
1345 fixedpoint construction.
1347 Type~\tydx{nat} is an instance of class~\cldx{ord}, which makes the overloaded
1348 functions of this class (especially \cdx{<} and \cdx{<=}, but also \cdx{min},
1349 \cdx{max} and \cdx{LEAST}) available on \tydx{nat}. Theory \thydx{Nat}
1350 also shows that {\tt<=} is a linear order, so \tydx{nat} is
1351 also an instance of class \cldx{linorder}.
1353 Theory \thydx{NatArith} develops arithmetic on the natural numbers. It defines
1354 addition, multiplication and subtraction. Theory \thydx{Divides} defines
1355 division, remainder and the ``divides'' relation. The numerous theorems
1356 proved include commutative, associative, distributive, identity and
1357 cancellation laws. See Figs.\ts\ref{hol-nat1} and~\ref{hol-nat2}. The
1358 recursion equations for the operators \texttt{+}, \texttt{-} and \texttt{*} on
1359 \texttt{nat} are part of the default simpset.
1361 Functions on \tydx{nat} can be defined by primitive or well-founded recursion;
1362 see {\S}\ref{sec:HOL:recursive}. A simple example is addition.
1363 Here, \texttt{op +} is the name of the infix operator~\texttt{+}, following
1364 the standard convention.
1368 "Suc m + n = Suc (m + n)"
1370 There is also a \sdx{case}-construct
1373 case \(e\) of 0 => \(a\) | Suc \(m\) => \(b\)
1375 Note that Isabelle insists on precisely this format; you may not even change
1376 the order of the two cases.
1377 Both \texttt{primrec} and \texttt{case} are realized by a recursion operator
1378 \cdx{nat_rec}, which is available because \textit{nat} is represented as
1379 a datatype (see {\S}\ref{subsec:datatype:rep_datatype}).
1381 %The predecessor relation, \cdx{pred_nat}, is shown to be well-founded.
1382 %Recursion along this relation resembles primitive recursion, but is
1383 %stronger because we are in higher-order logic; using primitive recursion to
1384 %define a higher-order function, we can easily Ackermann's function, which
1385 %is not primitive recursive \cite[page~104]{thompson91}.
1386 %The transitive closure of \cdx{pred_nat} is~$<$. Many functions on the
1387 %natural numbers are most easily expressed using recursion along~$<$.
1389 Tactic {\tt\ttindex{induct_tac} "$n$" $i$} performs induction on variable~$n$
1390 in subgoal~$i$ using theorem \texttt{nat_induct}. There is also the derived
1391 theorem \tdx{less_induct}:
1393 [| !!n. [| ! m. m<n --> P m |] ==> P n |] ==> P n
1397 \subsection{Numerical types and numerical reasoning}
1399 The integers (type \tdx{int}) are also available in HOL, and the reals (type
1400 \tdx{real}) are available in the logic image \texttt{HOL-Complex}. They support
1401 the expected operations of addition (\texttt{+}), subtraction (\texttt{-}) and
1402 multiplication (\texttt{*}), and much else. Type \tdx{int} provides the
1403 \texttt{div} and \texttt{mod} operators, while type \tdx{real} provides real
1404 division and other operations. Both types belong to class \cldx{linorder}, so
1405 they inherit the relational operators and all the usual properties of linear
1406 orderings. For full details, please survey the theories in subdirectories
1407 \texttt{Integ}, \texttt{Real}, and \texttt{Complex}.
1409 All three numeric types admit numerals of the form \texttt{$sd\ldots d$},
1410 where $s$ is an optional minus sign and $d\ldots d$ is a string of digits.
1411 Numerals are represented internally by a datatype for binary notation, which
1412 allows numerical calculations to be performed by rewriting. For example, the
1413 integer division of \texttt{54342339} by \texttt{3452} takes about five
1414 seconds. By default, the simplifier cancels like terms on the opposite sites
1415 of relational operators (reducing \texttt{z+x<x+y} to \texttt{z<y}, for
1416 instance. The simplifier also collects like terms, replacing \texttt{x+y+x*3}
1419 Sometimes numerals are not wanted, because for example \texttt{n+3} does not
1420 match a pattern of the form \texttt{Suc $k$}. You can re-arrange the form of
1421 an arithmetic expression by proving (via \texttt{subgoal_tac}) a lemma such as
1422 \texttt{n+3 = Suc (Suc (Suc n))}. As an alternative, you can disable the
1423 fancier simplifications by using a basic simpset such as \texttt{HOL_ss}
1424 rather than the default one, \texttt{simpset()}.
1426 Reasoning about arithmetic inequalities can be tedious. Fortunately, HOL
1427 provides a decision procedure for \emph{linear arithmetic}: formulae involving
1428 addition and subtraction. The simplifier invokes a weak version of this
1429 decision procedure automatically. If this is not sufficent, you can invoke the
1430 full procedure \ttindex{Lin_Arith.tac} explicitly. It copes with arbitrary
1431 formulae involving {\tt=}, {\tt<}, {\tt<=}, {\tt+}, {\tt-}, {\tt Suc}, {\tt
1432 min}, {\tt max} and numerical constants. Other subterms are treated as
1433 atomic, while subformulae not involving numerical types are ignored. Quantified
1434 subformulae are ignored unless they are positive universal or negative
1435 existential. The running time is exponential in the number of
1436 occurrences of {\tt min}, {\tt max}, and {\tt-} because they require case
1438 If {\tt k} is a numeral, then {\tt div k}, {\tt mod k} and
1439 {\tt k dvd} are also supported. The former two are eliminated
1440 by case distinctions, again blowing up the running time.
1441 If the formula involves explicit quantifiers, \texttt{Lin_Arith.tac} may take
1442 super-exponential time and space.
1444 If \texttt{Lin_Arith.tac} fails, try to find relevant arithmetic results in
1445 the library. The theories \texttt{Nat} and \texttt{NatArith} contain
1446 theorems about {\tt<}, {\tt<=}, \texttt{+}, \texttt{-} and \texttt{*}.
1447 Theory \texttt{Divides} contains theorems about \texttt{div} and
1448 \texttt{mod}. Use Proof General's \emph{find} button (or other search
1449 facilities) to locate them.
1451 \index{nat@{\textit{nat}} type|)}
1455 \index{#@{\tt[]} symbol}
1456 \index{#@{\tt\#} symbol}
1457 \index{"@@{\tt\at} symbol}
1460 \it symbol & \it meta-type & \it priority & \it description \\
1461 \tt[] & $\alpha\,list$ & & empty list\\
1462 \tt \# & $[\alpha,\alpha\,list]\To \alpha\,list$ & Right 65 &
1464 \cdx{null} & $\alpha\,list \To bool$ & & emptiness test\\
1465 \cdx{hd} & $\alpha\,list \To \alpha$ & & head \\
1466 \cdx{tl} & $\alpha\,list \To \alpha\,list$ & & tail \\
1467 \cdx{last} & $\alpha\,list \To \alpha$ & & last element \\
1468 \cdx{butlast} & $\alpha\,list \To \alpha\,list$ & & drop last element \\
1469 \tt\at & $[\alpha\,list,\alpha\,list]\To \alpha\,list$ & Left 65 & append \\
1470 \cdx{map} & $(\alpha\To\beta) \To (\alpha\,list \To \beta\,list)$
1472 \cdx{filter} & $(\alpha \To bool) \To (\alpha\,list \To \alpha\,list)$
1473 & & filter functional\\
1474 \cdx{set}& $\alpha\,list \To \alpha\,set$ & & elements\\
1475 \sdx{mem} & $\alpha \To \alpha\,list \To bool$ & Left 55 & membership\\
1476 \cdx{foldl} & $(\beta\To\alpha\To\beta) \To \beta \To \alpha\,list \To \beta$ &
1478 \cdx{concat} & $(\alpha\,list)list\To \alpha\,list$ & & concatenation \\
1479 \cdx{rev} & $\alpha\,list \To \alpha\,list$ & & reverse \\
1480 \cdx{length} & $\alpha\,list \To nat$ & & length \\
1481 \tt! & $\alpha\,list \To nat \To \alpha$ & Left 100 & indexing \\
1482 \cdx{take}, \cdx{drop} & $nat \To \alpha\,list \To \alpha\,list$ &&
1483 take/drop a prefix \\
1486 $(\alpha \To bool) \To \alpha\,list \To \alpha\,list$ &&
1489 \subcaption{Constants and infixes}
1491 \begin{center} \tt\frenchspacing
1492 \begin{tabular}{rrr}
1493 \it external & \it internal & \it description \\{}
1494 [$x@1$, $\dots$, $x@n$] & $x@1$ \# $\cdots$ \# $x@n$ \# [] &
1495 \rm finite list \\{}
1496 [$x$:$l$. $P$] & filter ($\lambda x{.}P$) $l$ &
1497 \rm list comprehension
1500 \subcaption{Translations}
1501 \caption{The theory \thydx{List}} \label{hol-list}
1506 \begin{ttbox}\makeatother
1516 (x#xs) @ ys = x # xs @ ys
1518 set [] = \ttlbrace\ttrbrace
1519 set (x#xs) = insert x (set xs)
1522 x mem (y#ys) = (if y=x then True else x mem ys)
1525 concat(x#xs) = x @ concat(xs)
1528 rev(x#xs) = rev(xs) @ [x]
1531 length(x#xs) = Suc(length(xs))
1534 xs!(Suc n) = (tl xs)!n
1536 \caption{Simple list processing functions}
1537 \label{fig:HOL:list-simps}
1541 \begin{ttbox}\makeatother
1543 map f (x#xs) = f x # map f xs
1546 filter P (x#xs) = (if P x then x#filter P xs else filter P xs)
1549 foldl f a (x#xs) = foldl f (f a x) xs
1552 take n (x#xs) = (case n of 0 => [] | Suc(m) => x # take m xs)
1555 drop n (x#xs) = (case n of 0 => x#xs | Suc(m) => drop m xs)
1558 takeWhile P (x#xs) = (if P x then x#takeWhile P xs else [])
1561 dropWhile P (x#xs) = (if P x then dropWhile P xs else xs)
1563 \caption{Further list processing functions}
1564 \label{fig:HOL:list-simps2}
1568 \subsection{The type constructor for lists, \textit{list}}
1570 \index{list@{\textit{list}} type|(}
1572 Figure~\ref{hol-list} presents the theory \thydx{List}: the basic list
1573 operations with their types and syntax. Type $\alpha \; list$ is
1574 defined as a \texttt{datatype} with the constructors {\tt[]} and {\tt\#}.
1575 As a result the generic structural induction and case analysis tactics
1576 \texttt{induct\_tac} and \texttt{cases\_tac} also become available for
1577 lists. A \sdx{case} construct of the form
1579 case $e$ of [] => $a$ | \(x\)\#\(xs\) => b
1581 is defined by translation. For details see~{\S}\ref{sec:HOL:datatype}. There
1582 is also a case splitting rule \tdx{split_list_case}
1585 P(\mathtt{case}~e~\mathtt{of}~\texttt{[] =>}~a ~\texttt{|}~
1586 x\texttt{\#}xs~\texttt{=>}~f~x~xs) ~= \\
1587 ((e = \texttt{[]} \to P(a)) \land
1588 (\forall x~ xs. e = x\texttt{\#}xs \to P(f~x~xs)))
1591 which can be fed to \ttindex{addsplits} just like
1592 \texttt{split_if} (see~{\S}\ref{subsec:HOL:case:splitting}).
1594 \texttt{List} provides a basic library of list processing functions defined by
1595 primitive recursion (see~{\S}\ref{sec:HOL:primrec}). The recursion equations
1596 are shown in Figs.\ts\ref{fig:HOL:list-simps} and~\ref{fig:HOL:list-simps2}.
1598 \index{list@{\textit{list}} type|)}
1601 \subsection{Introducing new types} \label{sec:typedef}
1603 The HOL-methodology dictates that all extensions to a theory should be
1604 \textbf{definitional}. The type definition mechanism that meets this
1605 criterion is \ttindex{typedef}. Note that \emph{type synonyms}, which are
1606 inherited from Pure and described elsewhere, are just syntactic abbreviations
1607 that have no logical meaning.
1610 Types in HOL must be non-empty; otherwise the quantifier rules would be
1611 unsound, because $\exists x. x=x$ is a theorem \cite[{\S}7]{paulson-COLOG}.
1613 A \bfindex{type definition} identifies the new type with a subset of
1614 an existing type. More precisely, the new type is defined by
1615 exhibiting an existing type~$\tau$, a set~$A::\tau\,set$, and a
1616 theorem of the form $x:A$. Thus~$A$ is a non-empty subset of~$\tau$,
1617 and the new type denotes this subset. New functions are defined that
1618 establish an isomorphism between the new type and the subset. If
1619 type~$\tau$ involves type variables $\alpha@1$, \ldots, $\alpha@n$,
1620 then the type definition creates a type constructor
1621 $(\alpha@1,\ldots,\alpha@n)ty$ rather than a particular type.
1623 \begin{figure}[htbp]
1625 typedef : 'typedef' ( () | '(' name ')') type '=' set witness;
1627 type : typevarlist name ( () | '(' infix ')' );
1629 witness : () | '(' id ')';
1631 \caption{Syntax of type definitions}
1632 \label{fig:HOL:typedef}
1635 The syntax for type definitions is shown in Fig.~\ref{fig:HOL:typedef}. For
1636 the definition of `typevarlist' and `infix' see
1637 \iflabelundefined{chap:classical}
1638 {the appendix of the {\em Reference Manual\/}}%
1639 {Appendix~\ref{app:TheorySyntax}}. The remaining nonterminals have the
1642 \item[\it type:] the new type constructor $(\alpha@1,\dots,\alpha@n)ty$ with
1643 optional infix annotation.
1644 \item[\it name:] an alphanumeric name $T$ for the type constructor
1645 $ty$, in case $ty$ is a symbolic name. Defaults to $ty$.
1646 \item[\it set:] the representing subset $A$.
1647 \item[\it witness:] name of a theorem of the form $a:A$ proving
1648 non-emptiness. It can be omitted in case Isabelle manages to prove
1649 non-emptiness automatically.
1651 If all context conditions are met (no duplicate type variables in
1652 `typevarlist', no extra type variables in `set', and no free term variables
1653 in `set'), the following components are added to the theory:
1655 \item a type $ty :: (term,\dots,term)term$
1659 Rep_T &::& (\alpha@1,\dots,\alpha@n)ty \To \tau \\
1660 Abs_T &::& \tau \To (\alpha@1,\dots,\alpha@n)ty
1662 \item a definition and three axioms
1665 T{\tt_def} & T \equiv A \\
1666 {\tt Rep_}T & Rep_T\,x \in T \\
1667 {\tt Rep_}T{\tt_inverse} & Abs_T\,(Rep_T\,x) = x \\
1668 {\tt Abs_}T{\tt_inverse} & y \in T \Imp Rep_T\,(Abs_T\,y) = y
1671 stating that $(\alpha@1,\dots,\alpha@n)ty$ is isomorphic to $A$ by $Rep_T$
1672 and its inverse $Abs_T$.
1674 Below are two simple examples of HOL type definitions. Non-emptiness is
1675 proved automatically here.
1677 typedef unit = "{\ttlbrace}True{\ttrbrace}"
1680 ('a, 'b) "*" (infixr 20)
1681 = "{\ttlbrace}f . EX (a::'a) (b::'b). f = (\%x y. x = a & y = b){\ttrbrace}"
1684 Type definitions permit the introduction of abstract data types in a safe
1685 way, namely by providing models based on already existing types. Given some
1686 abstract axiomatic description $P$ of a type, this involves two steps:
1688 \item Find an appropriate type $\tau$ and subset $A$ which has the desired
1689 properties $P$, and make a type definition based on this representation.
1690 \item Prove that $P$ holds for $ty$ by lifting $P$ from the representation.
1692 You can now forget about the representation and work solely in terms of the
1693 abstract properties $P$.
1696 If you introduce a new type (constructor) $ty$ axiomatically, i.e.\ by
1697 declaring the type and its operations and by stating the desired axioms, you
1698 should make sure the type has a non-empty model. You must also have a clause
1701 arities \(ty\) :: (term,\thinspace\(\dots\),{\thinspace}term){\thinspace}term
1703 in your theory file to tell Isabelle that $ty$ is in class \texttt{term}, the
1704 class of all HOL types.
1708 \section{Datatype definitions}
1709 \label{sec:HOL:datatype}
1712 Inductive datatypes, similar to those of \ML, frequently appear in
1713 applications of Isabelle/HOL. In principle, such types could be defined by
1714 hand via \texttt{typedef} (see {\S}\ref{sec:typedef}), but this would be far too
1715 tedious. The \ttindex{datatype} definition package of Isabelle/HOL (cf.\
1716 \cite{Berghofer-Wenzel:1999:TPHOL}) automates such chores. It generates an
1717 appropriate \texttt{typedef} based on a least fixed-point construction, and
1718 proves freeness theorems and induction rules, as well as theorems for
1719 recursion and case combinators. The user just has to give a simple
1720 specification of new inductive types using a notation similar to {\ML} or
1723 The current datatype package can handle both mutual and indirect recursion.
1724 It also offers to represent existing types as datatypes giving the advantage
1725 of a more uniform view on standard theories.
1729 \label{subsec:datatype:basics}
1731 A general \texttt{datatype} definition is of the following form:
1734 \mathtt{datatype} & (\vec{\alpha})t@1 & = &
1735 C^1@1~\tau^1@{1,1}~\ldots~\tau^1@{1,m^1@1} ~\mid~ \ldots ~\mid~
1736 C^1@{k@1}~\tau^1@{k@1,1}~\ldots~\tau^1@{k@1,m^1@{k@1}} \\
1738 \mathtt{and} & (\vec{\alpha})t@n & = &
1739 C^n@1~\tau^n@{1,1}~\ldots~\tau^n@{1,m^n@1} ~\mid~ \ldots ~\mid~
1740 C^n@{k@n}~\tau^n@{k@n,1}~\ldots~\tau^n@{k@n,m^n@{k@n}}
1743 where $\vec{\alpha} = (\alpha@1,\ldots,\alpha@h)$ is a list of type variables,
1744 $C^j@i$ are distinct constructor names and $\tau^j@{i,i'}$ are {\em
1745 admissible} types containing at most the type variables $\alpha@1, \ldots,
1746 \alpha@h$. A type $\tau$ occurring in a \texttt{datatype} definition is {\em
1747 admissible} if and only if
1749 \item $\tau$ is non-recursive, i.e.\ $\tau$ does not contain any of the
1750 newly defined type constructors $t@1,\ldots,t@n$, or
1751 \item $\tau = (\vec{\alpha})t@{j'}$ where $1 \leq j' \leq n$, or
1752 \item $\tau = (\tau'@1,\ldots,\tau'@{h'})t'$, where $t'$ is
1753 the type constructor of an already existing datatype and $\tau'@1,\ldots,\tau'@{h'}$
1754 are admissible types.
1755 \item $\tau = \sigma \to \tau'$, where $\tau'$ is an admissible
1756 type and $\sigma$ is non-recursive (i.e. the occurrences of the newly defined
1757 types are {\em strictly positive})
1759 If some $(\vec{\alpha})t@{j'}$ occurs in a type $\tau^j@{i,i'}$
1762 (\ldots,\ldots ~ (\vec{\alpha})t@{j'} ~ \ldots,\ldots)t'
1764 this is called a {\em nested} (or \emph{indirect}) occurrence. A very simple
1765 example of a datatype is the type \texttt{list}, which can be defined by
1767 datatype 'a list = Nil
1770 Arithmetic expressions \texttt{aexp} and boolean expressions \texttt{bexp} can be modelled
1771 by the mutually recursive datatype definition
1773 datatype 'a aexp = If_then_else ('a bexp) ('a aexp) ('a aexp)
1774 | Sum ('a aexp) ('a aexp)
1775 | Diff ('a aexp) ('a aexp)
1778 and 'a bexp = Less ('a aexp) ('a aexp)
1779 | And ('a bexp) ('a bexp)
1780 | Or ('a bexp) ('a bexp)
1782 The datatype \texttt{term}, which is defined by
1784 datatype ('a, 'b) term = Var 'a
1785 | App 'b ((('a, 'b) term) list)
1787 is an example for a datatype with nested recursion. Using nested recursion
1788 involving function spaces, we may also define infinitely branching datatypes, e.g.
1790 datatype 'a tree = Atom 'a | Branch "nat => 'a tree"
1795 Types in HOL must be non-empty. Each of the new datatypes
1796 $(\vec{\alpha})t@j$ with $1 \leq j \leq n$ is non-empty if and only if it has a
1797 constructor $C^j@i$ with the following property: for all argument types
1798 $\tau^j@{i,i'}$ of the form $(\vec{\alpha})t@{j'}$ the datatype
1799 $(\vec{\alpha})t@{j'}$ is non-empty.
1801 If there are no nested occurrences of the newly defined datatypes, obviously
1802 at least one of the newly defined datatypes $(\vec{\alpha})t@j$
1803 must have a constructor $C^j@i$ without recursive arguments, a \emph{base
1804 case}, to ensure that the new types are non-empty. If there are nested
1805 occurrences, a datatype can even be non-empty without having a base case
1806 itself. Since \texttt{list} is a non-empty datatype, \texttt{datatype t = C (t
1807 list)} is non-empty as well.
1810 \subsubsection{Freeness of the constructors}
1812 The datatype constructors are automatically defined as functions of their
1814 \[ C^j@i :: [\tau^j@{i,1},\dots,\tau^j@{i,m^j@i}] \To (\alpha@1,\dots,\alpha@h)t@j \]
1815 These functions have certain {\em freeness} properties. They construct
1818 C^j@i~x@1~\dots~x@{m^j@i} \neq C^j@{i'}~y@1~\dots~y@{m^j@{i'}} \qquad
1819 \mbox{for all}~ i \neq i'.
1821 The constructor functions are injective:
1823 (C^j@i~x@1~\dots~x@{m^j@i} = C^j@i~y@1~\dots~y@{m^j@i}) =
1824 (x@1 = y@1 \land \dots \land x@{m^j@i} = y@{m^j@i})
1826 Since the number of distinctness inequalities is quadratic in the number of
1827 constructors, the datatype package avoids proving them separately if there are
1828 too many constructors. Instead, specific inequalities are proved by a suitable
1829 simplification procedure on demand.\footnote{This procedure, which is already part
1830 of the default simpset, may be referred to by the ML identifier
1831 \texttt{DatatypePackage.distinct_simproc}.}
1833 \subsubsection{Structural induction}
1835 The datatype package also provides structural induction rules. For
1836 datatypes without nested recursion, this is of the following form:
1838 \infer{P@1~x@1 \land \dots \land P@n~x@n}
1840 \Forall x@1 \dots x@{m^1@1}.
1841 \List{P@{s^1@{1,1}}~x@{r^1@{1,1}}; \dots;
1842 P@{s^1@{1,l^1@1}}~x@{r^1@{1,l^1@1}}} & \Imp &
1843 P@1~\left(C^1@1~x@1~\dots~x@{m^1@1}\right) \\
1845 \Forall x@1 \dots x@{m^1@{k@1}}.
1846 \List{P@{s^1@{k@1,1}}~x@{r^1@{k@1,1}}; \dots;
1847 P@{s^1@{k@1,l^1@{k@1}}}~x@{r^1@{k@1,l^1@{k@1}}}} & \Imp &
1848 P@1~\left(C^1@{k@1}~x@1~\ldots~x@{m^1@{k@1}}\right) \\
1850 \Forall x@1 \dots x@{m^n@1}.
1851 \List{P@{s^n@{1,1}}~x@{r^n@{1,1}}; \dots;
1852 P@{s^n@{1,l^n@1}}~x@{r^n@{1,l^n@1}}} & \Imp &
1853 P@n~\left(C^n@1~x@1~\ldots~x@{m^n@1}\right) \\
1855 \Forall x@1 \dots x@{m^n@{k@n}}.
1856 \List{P@{s^n@{k@n,1}}~x@{r^n@{k@n,1}}; \dots
1857 P@{s^n@{k@n,l^n@{k@n}}}~x@{r^n@{k@n,l^n@{k@n}}}} & \Imp &
1858 P@n~\left(C^n@{k@n}~x@1~\ldots~x@{m^n@{k@n}}\right)
1865 \left\{\left(r^j@{i,1},s^j@{i,1}\right),\ldots,
1866 \left(r^j@{i,l^j@i},s^j@{i,l^j@i}\right)\right\} = \\[2ex]
1867 && \left\{(i',i'')~\left|~
1868 1\leq i' \leq m^j@i \land 1 \leq i'' \leq n \land
1869 \tau^j@{i,i'} = (\alpha@1,\ldots,\alpha@h)t@{i''}\right.\right\}
1872 i.e.\ the properties $P@j$ can be assumed for all recursive arguments.
1874 For datatypes with nested recursion, such as the \texttt{term} example from
1875 above, things are a bit more complicated. Conceptually, Isabelle/HOL unfolds
1878 datatype ('a,'b) term = Var 'a
1879 | App 'b ((('a, 'b) term) list)
1881 to an equivalent definition without nesting:
1883 datatype ('a,'b) term = Var
1884 | App 'b (('a, 'b) term_list)
1885 and ('a,'b) term_list = Nil'
1886 | Cons' (('a,'b) term) (('a,'b) term_list)
1888 Note however, that the type \texttt{('a,'b) term_list} and the constructors {\tt
1889 Nil'} and \texttt{Cons'} are not really introduced. One can directly work with
1890 the original (isomorphic) type \texttt{(('a, 'b) term) list} and its existing
1891 constructors \texttt{Nil} and \texttt{Cons}. Thus, the structural induction rule for
1892 \texttt{term} gets the form
1894 \infer{P@1~x@1 \land P@2~x@2}
1896 \Forall x.~P@1~(\mathtt{Var}~x) \\
1897 \Forall x@1~x@2.~P@2~x@2 \Imp P@1~(\mathtt{App}~x@1~x@2) \\
1899 \Forall x@1~x@2. \List{P@1~x@1; P@2~x@2} \Imp P@2~(\mathtt{Cons}~x@1~x@2)
1902 Note that there are two predicates $P@1$ and $P@2$, one for the type \texttt{('a,'b) term}
1903 and one for the type \texttt{(('a, 'b) term) list}.
1905 For a datatype with function types such as \texttt{'a tree}, the induction rule
1909 {\Forall a.~P~(\mathtt{Atom}~a) &
1910 \Forall ts.~(\forall x.~P~(ts~x)) \Imp P~(\mathtt{Branch}~ts)}
1913 \medskip In principle, inductive types are already fully determined by
1914 freeness and structural induction. For convenience in applications,
1915 the following derived constructions are automatically provided for any
1918 \subsubsection{The \sdx{case} construct}
1920 The type comes with an \ML-like \texttt{case}-construct:
1923 \mbox{\tt case}~e~\mbox{\tt of} & C^j@1~x@{1,1}~\dots~x@{1,m^j@1} & \To & e@1 \\
1925 \mid & C^j@{k@j}~x@{k@j,1}~\dots~x@{k@j,m^j@{k@j}} & \To & e@{k@j}
1928 where the $x@{i,j}$ are either identifiers or nested tuple patterns as in
1929 {\S}\ref{subsec:prod-sum}.
1931 All constructors must be present, their order is fixed, and nested patterns
1932 are not supported (with the exception of tuples). Violating this
1933 restriction results in strange error messages.
1936 To perform case distinction on a goal containing a \texttt{case}-construct,
1937 the theorem $t@j.$\texttt{split} is provided:
1939 \begin{array}{@{}rcl@{}}
1940 P(t@j_\mathtt{case}~f@1~\dots~f@{k@j}~e) &\!\!\!=&
1941 \!\!\! ((\forall x@1 \dots x@{m^j@1}. e = C^j@1~x@1\dots x@{m^j@1} \to
1942 P(f@1~x@1\dots x@{m^j@1})) \\
1943 &&\!\!\! ~\land~ \dots ~\land \\
1944 &&~\!\!\! (\forall x@1 \dots x@{m^j@{k@j}}. e = C^j@{k@j}~x@1\dots x@{m^j@{k@j}} \to
1945 P(f@{k@j}~x@1\dots x@{m^j@{k@j}})))
1948 where $t@j$\texttt{_case} is the internal name of the \texttt{case}-construct.
1949 This theorem can be added to a simpset via \ttindex{addsplits}
1950 (see~{\S}\ref{subsec:HOL:case:splitting}).
1952 Case splitting on assumption works as well, by using the rule
1953 $t@j.$\texttt{split_asm} in the same manner. Both rules are available under
1954 $t@j.$\texttt{splits} (this name is \emph{not} bound in ML, though).
1956 \begin{warn}\index{simplification!of \texttt{case}}%
1957 By default only the selector expression ($e$ above) in a
1958 \texttt{case}-construct is simplified, in analogy with \texttt{if} (see
1959 page~\pageref{if-simp}). Only if that reduces to a constructor is one of
1960 the arms of the \texttt{case}-construct exposed and simplified. To ensure
1961 full simplification of all parts of a \texttt{case}-construct for datatype
1962 $t$, remove $t$\texttt{.}\ttindexbold{case_weak_cong} from the simpset, for
1963 example by \texttt{delcongs [thm "$t$.weak_case_cong"]}.
1966 \subsubsection{The function \cdx{size}}\label{sec:HOL:size}
1968 Theory \texttt{NatArith} declares a generic function \texttt{size} of type
1969 $\alpha\To nat$. Each datatype defines a particular instance of \texttt{size}
1970 by overloading according to the following scheme:
1971 %%% FIXME: This formula is too big and is completely unreadable
1973 size(C^j@i~x@1~\dots~x@{m^j@i}) = \!
1976 0 & \!\mbox{if $Rec^j@i = \emptyset$} \\
1977 1+\sum\limits@{h=1}^{l^j@i}size~x@{r^j@{i,h}} &
1978 \!\mbox{if $Rec^j@i = \left\{\left(r^j@{i,1},s^j@{i,1}\right),\ldots,
1979 \left(r^j@{i,l^j@i},s^j@{i,l^j@i}\right)\right\}$}
1983 where $Rec^j@i$ is defined above. Viewing datatypes as generalised trees, the
1984 size of a leaf is 0 and the size of a node is the sum of the sizes of its
1987 \subsection{Defining datatypes}
1989 The theory syntax for datatype definitions is shown in
1990 Fig.~\ref{datatype-grammar}. In order to be well-formed, a datatype
1991 definition has to obey the rules stated in the previous section. As a result
1992 the theory is extended with the new types, the constructors, and the theorems
1993 listed in the previous section.
1997 datatype : 'datatype' typedecls;
1999 typedecls: ( newtype '=' (cons + '|') ) + 'and'
2001 newtype : typevarlist id ( () | '(' infix ')' )
2003 cons : name (argtype *) ( () | ( '(' mixfix ')' ) )
2005 argtype : id | tid | ('(' typevarlist id ')')
2008 \caption{Syntax of datatype declarations}
2009 \label{datatype-grammar}
2012 Most of the theorems about datatypes become part of the default simpset and
2013 you never need to see them again because the simplifier applies them
2014 automatically. Only induction or case distinction are usually invoked by hand.
2015 \begin{ttdescription}
2016 \item[\ttindexbold{induct_tac} {\tt"}$x${\tt"} $i$]
2017 applies structural induction on variable $x$ to subgoal $i$, provided the
2018 type of $x$ is a datatype.
2019 \item[\texttt{induct_tac}
2020 {\tt"}$x@1$ $\ldots$ $x@n${\tt"} $i$] applies simultaneous
2021 structural induction on the variables $x@1,\ldots,x@n$ to subgoal $i$. This
2022 is the canonical way to prove properties of mutually recursive datatypes
2023 such as \texttt{aexp} and \texttt{bexp}, or datatypes with nested recursion such as
2026 In some cases, induction is overkill and a case distinction over all
2027 constructors of the datatype suffices.
2028 \begin{ttdescription}
2029 \item[\ttindexbold{case_tac} {\tt"}$u${\tt"} $i$]
2030 performs a case analysis for the term $u$ whose type must be a datatype.
2031 If the datatype has $k@j$ constructors $C^j@1$, \dots $C^j@{k@j}$, subgoal
2032 $i$ is replaced by $k@j$ new subgoals which contain the additional
2033 assumption $u = C^j@{i'}~x@1~\dots~x@{m^j@{i'}}$ for $i'=1$, $\dots$,~$k@j$.
2036 Note that induction is only allowed on free variables that should not occur
2037 among the premises of the subgoal. Case distinction applies to arbitrary terms.
2042 For the technically minded, we exhibit some more details. Processing the
2043 theory file produces an \ML\ structure which, in addition to the usual
2044 components, contains a structure named $t$ for each datatype $t$ defined in
2045 the file. Each structure $t$ contains the following elements:
2047 val distinct : thm list
2048 val inject : thm list
2051 val cases : thm list
2056 val simps : thm list
2058 \texttt{distinct}, \texttt{inject}, \texttt{induct}, \texttt{size}
2059 and \texttt{split} contain the theorems
2060 described above. For user convenience, \texttt{distinct} contains
2061 inequalities in both directions. The reduction rules of the {\tt
2062 case}-construct are in \texttt{cases}. All theorems from {\tt
2063 distinct}, \texttt{inject} and \texttt{cases} are combined in \texttt{simps}.
2064 In case of mutually recursive datatypes, \texttt{recs}, \texttt{size}, \texttt{induct}
2065 and \texttt{simps} are contained in a separate structure named $t@1_\ldots_t@n$.
2068 \subsection{Representing existing types as datatypes}\label{subsec:datatype:rep_datatype}
2070 For foundational reasons, some basic types such as \texttt{nat}, \texttt{*}, {\tt
2071 +}, \texttt{bool} and \texttt{unit} are not defined in a \texttt{datatype} section,
2072 but by more primitive means using \texttt{typedef}. To be able to use the
2073 tactics \texttt{induct_tac} and \texttt{case_tac} and to define functions by
2074 primitive recursion on these types, such types may be represented as actual
2075 datatypes. This is done by specifying the constructors of the desired type,
2076 plus a proof of the induction rule, as well as theorems
2077 stating the distinctness and injectivity of constructors in a {\tt
2078 rep_datatype} section. For the sum type this works as follows:
2080 rep_datatype (sum) Inl Inr
2084 assume x: "!!x::'a. P (Inl x)" and y: "!!y::'b. P (Inr y)"
2085 then show "P s" by (auto intro: sumE [of s])
2088 The datatype package automatically derives additional theorems for recursion
2089 and case combinators from these rules. Any of the basic HOL types mentioned
2090 above are represented as datatypes. Try an induction on \texttt{bool}
2094 \subsection{Examples}
2096 \subsubsection{The datatype $\alpha~mylist$}
2098 We want to define a type $\alpha~mylist$. To do this we have to build a new
2099 theory that contains the type definition. We start from the theory
2100 \texttt{Datatype} instead of \texttt{Main} in order to avoid clashes with the
2101 \texttt{List} theory of Isabelle/HOL.
2104 datatype 'a mylist = Nil | Cons 'a ('a mylist)
2107 After loading the theory, we can prove $Cons~x~xs\neq xs$, for example. To
2108 ease the induction applied below, we state the goal with $x$ quantified at the
2109 object-level. This will be stripped later using \ttindex{qed_spec_mp}.
2111 Goal "!x. Cons x xs ~= xs";
2113 {\out ! x. Cons x xs ~= xs}
2114 {\out 1. ! x. Cons x xs ~= xs}
2116 This can be proved by the structural induction tactic:
2118 by (induct_tac "xs" 1);
2120 {\out ! x. Cons x xs ~= xs}
2121 {\out 1. ! x. Cons x Nil ~= Nil}
2122 {\out 2. !!a mylist.}
2123 {\out ! x. Cons x mylist ~= mylist ==>}
2124 {\out ! x. Cons x (Cons a mylist) ~= Cons a mylist}
2126 The first subgoal can be proved using the simplifier. Isabelle/HOL has
2127 already added the freeness properties of lists to the default simplification
2132 {\out ! x. Cons x xs ~= xs}
2133 {\out 1. !!a mylist.}
2134 {\out ! x. Cons x mylist ~= mylist ==>}
2135 {\out ! x. Cons x (Cons a mylist) ~= Cons a mylist}
2137 Similarly, we prove the remaining goal.
2139 by (Asm_simp_tac 1);
2141 {\out ! x. Cons x xs ~= xs}
2144 qed_spec_mp "not_Cons_self";
2145 {\out val not_Cons_self = "Cons x xs ~= xs" : thm}
2147 Because both subgoals could have been proved by \texttt{Asm_simp_tac}
2148 we could have done that in one step:
2150 by (ALLGOALS Asm_simp_tac);
2154 \subsubsection{The datatype $\alpha~mylist$ with mixfix syntax}
2156 In this example we define the type $\alpha~mylist$ again but this time
2157 we want to write \texttt{[]} for \texttt{Nil} and we want to use infix
2158 notation \verb|#| for \texttt{Cons}. To do this we simply add mixfix
2159 annotations after the constructor declarations as follows:
2162 datatype 'a mylist =
2164 Cons 'a ('a mylist) (infixr "#" 70)
2167 Now the theorem in the previous example can be written \verb|x#xs ~= xs|.
2170 \subsubsection{A datatype for weekdays}
2172 This example shows a datatype that consists of 7 constructors:
2175 datatype days = Mon | Tue | Wed | Thu | Fri | Sat | Sun
2178 Because there are more than 6 constructors, inequality is expressed via a function
2179 \verb|days_ord|. The theorem \verb|Mon ~= Tue| is not directly
2180 contained among the distinctness theorems, but the simplifier can
2181 prove it thanks to rewrite rules inherited from theory \texttt{NatArith}:
2186 You need not derive such inequalities explicitly: the simplifier will dispose
2187 of them automatically.
2191 \section{Recursive function definitions}\label{sec:HOL:recursive}
2192 \index{recursive functions|see{recursion}}
2194 Isabelle/HOL provides two main mechanisms of defining recursive functions.
2196 \item \textbf{Primitive recursion} is available only for datatypes, and it is
2197 somewhat restrictive. Recursive calls are only allowed on the argument's
2198 immediate constituents. On the other hand, it is the form of recursion most
2199 often wanted, and it is easy to use.
2201 \item \textbf{Well-founded recursion} requires that you supply a well-founded
2202 relation that governs the recursion. Recursive calls are only allowed if
2203 they make the argument decrease under the relation. Complicated recursion
2204 forms, such as nested recursion, can be dealt with. Termination can even be
2205 proved at a later time, though having unsolved termination conditions around
2206 can make work difficult.%
2207 \footnote{This facility is based on Konrad Slind's TFL
2208 package~\cite{slind-tfl}. Thanks are due to Konrad for implementing TFL
2209 and assisting with its installation.}
2212 Following good HOL tradition, these declarations do not assert arbitrary
2213 axioms. Instead, they define the function using a recursion operator. Both
2214 HOL and ZF derive the theory of well-founded recursion from first
2215 principles~\cite{paulson-set-II}. Primitive recursion over some datatype
2216 relies on the recursion operator provided by the datatype package. With
2217 either form of function definition, Isabelle proves the desired recursion
2218 equations as theorems.
2221 \subsection{Primitive recursive functions}
2222 \label{sec:HOL:primrec}
2223 \index{recursion!primitive|(}
2226 Datatypes come with a uniform way of defining functions, {\bf primitive
2227 recursion}. In principle, one could introduce primitive recursive functions
2228 by asserting their reduction rules as new axioms, but this is not recommended:
2229 \begin{ttbox}\slshape
2231 consts app :: ['a list, 'a list] => 'a list
2233 app_Nil "app [] ys = ys"
2234 app_Cons "app (x#xs) ys = x#app xs ys"
2237 Asserting axioms brings the danger of accidentally asserting nonsense, as
2238 in \verb$app [] ys = us$.
2240 The \ttindex{primrec} declaration is a safe means of defining primitive
2241 recursive functions on datatypes:
2244 consts app :: ['a list, 'a list] => 'a list
2247 "app (x#xs) ys = x#app xs ys"
2250 Isabelle will now check that the two rules do indeed form a primitive
2251 recursive definition. For example
2256 is rejected with an error message ``\texttt{Extra variables on rhs}''.
2260 The general form of a primitive recursive definition is
2263 {\it reduction rules}
2265 where \textit{reduction rules} specify one or more equations of the form
2266 \[ f \, x@1 \, \dots \, x@m \, (C \, y@1 \, \dots \, y@k) \, z@1 \,
2267 \dots \, z@n = r \] such that $C$ is a constructor of the datatype, $r$
2268 contains only the free variables on the left-hand side, and all recursive
2269 calls in $r$ are of the form $f \, \dots \, y@i \, \dots$ for some $i$. There
2270 must be at most one reduction rule for each constructor. The order is
2271 immaterial. For missing constructors, the function is defined to return a
2274 If you would like to refer to some rule by name, then you must prefix
2275 the rule with an identifier. These identifiers, like those in the
2276 \texttt{rules} section of a theory, will be visible at the \ML\ level.
2278 The primitive recursive function can have infix or mixfix syntax:
2279 \begin{ttbox}\underscoreon
2280 consts "@" :: ['a list, 'a list] => 'a list (infixr 60)
2283 "(x#xs) @ ys = x#(xs @ ys)"
2286 The reduction rules become part of the default simpset, which
2287 leads to short proof scripts:
2288 \begin{ttbox}\underscoreon
2289 Goal "(xs @ ys) @ zs = xs @ (ys @ zs)";
2290 by (induct\_tac "xs" 1);
2291 by (ALLGOALS Asm\_simp\_tac);
2294 \subsubsection{Example: Evaluation of expressions}
2295 Using mutual primitive recursion, we can define evaluation functions \texttt{evala}
2296 and \texttt{eval_bexp} for the datatypes of arithmetic and boolean expressions mentioned in
2297 {\S}\ref{subsec:datatype:basics}:
2300 evala :: "['a => nat, 'a aexp] => nat"
2301 evalb :: "['a => nat, 'a bexp] => bool"
2304 "evala env (If_then_else b a1 a2) =
2305 (if evalb env b then evala env a1 else evala env a2)"
2306 "evala env (Sum a1 a2) = evala env a1 + evala env a2"
2307 "evala env (Diff a1 a2) = evala env a1 - evala env a2"
2308 "evala env (Var v) = env v"
2309 "evala env (Num n) = n"
2311 "evalb env (Less a1 a2) = (evala env a1 < evala env a2)"
2312 "evalb env (And b1 b2) = (evalb env b1 & evalb env b2)"
2313 "evalb env (Or b1 b2) = (evalb env b1 & evalb env b2)"
2315 Since the value of an expression depends on the value of its variables,
2316 the functions \texttt{evala} and \texttt{evalb} take an additional
2317 parameter, an {\em environment} of type \texttt{'a => nat}, which maps
2318 variables to their values.
2320 Similarly, we may define substitution functions \texttt{substa}
2321 and \texttt{substb} for expressions: The mapping \texttt{f} of type
2322 \texttt{'a => 'a aexp} given as a parameter is lifted canonically
2323 on the types \texttt{'a aexp} and \texttt{'a bexp}:
2326 substa :: "['a => 'b aexp, 'a aexp] => 'b aexp"
2327 substb :: "['a => 'b aexp, 'a bexp] => 'b bexp"
2330 "substa f (If_then_else b a1 a2) =
2331 If_then_else (substb f b) (substa f a1) (substa f a2)"
2332 "substa f (Sum a1 a2) = Sum (substa f a1) (substa f a2)"
2333 "substa f (Diff a1 a2) = Diff (substa f a1) (substa f a2)"
2334 "substa f (Var v) = f v"
2335 "substa f (Num n) = Num n"
2337 "substb f (Less a1 a2) = Less (substa f a1) (substa f a2)"
2338 "substb f (And b1 b2) = And (substb f b1) (substb f b2)"
2339 "substb f (Or b1 b2) = Or (substb f b1) (substb f b2)"
2341 In textbooks about semantics one often finds {\em substitution theorems},
2342 which express the relationship between substitution and evaluation. For
2343 \texttt{'a aexp} and \texttt{'a bexp}, we can prove such a theorem by mutual
2344 induction, followed by simplification:
2347 "evala env (substa (Var(v := a')) a) =
2348 evala (env(v := evala env a')) a &
2349 evalb env (substb (Var(v := a')) b) =
2350 evalb (env(v := evala env a')) b";
2351 by (induct_tac "a b" 1);
2352 by (ALLGOALS Asm_full_simp_tac);
2355 \subsubsection{Example: A substitution function for terms}
2356 Functions on datatypes with nested recursion, such as the type
2357 \texttt{term} mentioned in {\S}\ref{subsec:datatype:basics}, are
2358 also defined by mutual primitive recursion. A substitution
2359 function \texttt{subst_term} on type \texttt{term}, similar to the functions
2360 \texttt{substa} and \texttt{substb} described above, can
2361 be defined as follows:
2364 subst_term :: "['a => ('a,'b) term, ('a,'b) term] => ('a,'b) term"
2366 "['a => ('a,'b) term, ('a,'b) term list] => ('a,'b) term list"
2369 "subst_term f (Var a) = f a"
2370 "subst_term f (App b ts) = App b (subst_term_list f ts)"
2372 "subst_term_list f [] = []"
2373 "subst_term_list f (t # ts) =
2374 subst_term f t # subst_term_list f ts"
2376 The recursion scheme follows the structure of the unfolded definition of type
2377 \texttt{term} shown in {\S}\ref{subsec:datatype:basics}. To prove properties of
2378 this substitution function, mutual induction is needed:
2381 "(subst_term ((subst_term f1) o f2) t) =
2382 (subst_term f1 (subst_term f2 t)) &
2383 (subst_term_list ((subst_term f1) o f2) ts) =
2384 (subst_term_list f1 (subst_term_list f2 ts))";
2385 by (induct_tac "t ts" 1);
2386 by (ALLGOALS Asm_full_simp_tac);
2389 \subsubsection{Example: A map function for infinitely branching trees}
2390 Defining functions on infinitely branching datatypes by primitive
2391 recursion is just as easy. For example, we can define a function
2392 \texttt{map_tree} on \texttt{'a tree} as follows:
2395 map_tree :: "('a => 'b) => 'a tree => 'b tree"
2398 "map_tree f (Atom a) = Atom (f a)"
2399 "map_tree f (Branch ts) = Branch (\%x. map_tree f (ts x))"
2401 Note that all occurrences of functions such as \texttt{ts} in the
2402 \texttt{primrec} clauses must be applied to an argument. In particular,
2403 \texttt{map_tree f o ts} is not allowed.
2405 \index{recursion!primitive|)}
2409 \subsection{General recursive functions}
2410 \label{sec:HOL:recdef}
2411 \index{recursion!general|(}
2414 Using \texttt{recdef}, you can declare functions involving nested recursion
2415 and pattern-matching. Recursion need not involve datatypes and there are few
2416 syntactic restrictions. Termination is proved by showing that each recursive
2417 call makes the argument smaller in a suitable sense, which you specify by
2418 supplying a well-founded relation.
2420 Here is a simple example, the Fibonacci function. The first line declares
2421 \texttt{fib} to be a constant. The well-founded relation is simply~$<$ (on
2422 the natural numbers). Pattern-matching is used here: \texttt{1} is a
2423 macro for \texttt{Suc~0}.
2425 consts fib :: "nat => nat"
2426 recdef fib "less_than"
2429 "fib (Suc(Suc x)) = (fib x + fib (Suc x))"
2432 With \texttt{recdef}, function definitions may be incomplete, and patterns may
2433 overlap, as in functional programming. The \texttt{recdef} package
2434 disambiguates overlapping patterns by taking the order of rules into account.
2435 For missing patterns, the function is defined to return a default value.
2437 %For example, here is a declaration of the list function \cdx{hd}:
2439 %consts hd :: 'a list => 'a
2443 %Because this function is not recursive, we may supply the empty well-founded
2446 The well-founded relation defines a notion of ``smaller'' for the function's
2447 argument type. The relation $\prec$ is \textbf{well-founded} provided it
2448 admits no infinitely decreasing chains
2449 \[ \cdots\prec x@n\prec\cdots\prec x@1. \]
2450 If the function's argument has type~$\tau$, then $\prec$ has to be a relation
2451 over~$\tau$: it must have type $(\tau\times\tau)set$.
2453 Proving well-foundedness can be tricky, so Isabelle/HOL provides a collection
2454 of operators for building well-founded relations. The package recognises
2455 these operators and automatically proves that the constructed relation is
2456 well-founded. Here are those operators, in order of importance:
2458 \item \texttt{less_than} is ``less than'' on the natural numbers.
2459 (It has type $(nat\times nat)set$, while $<$ has type $[nat,nat]\To bool$.
2461 \item $\mathop{\mathtt{measure}} f$, where $f$ has type $\tau\To nat$, is the
2462 relation~$\prec$ on type~$\tau$ such that $x\prec y$ if and only if
2464 Typically, $f$ takes the recursive function's arguments (as a tuple) and
2465 returns a result expressed in terms of the function \texttt{size}. It is
2466 called a \textbf{measure function}. Recall that \texttt{size} is overloaded
2467 and is defined on all datatypes (see {\S}\ref{sec:HOL:size}).
2469 \item $\mathop{\mathtt{inv_image}} R\;f$ is a generalisation of
2470 \texttt{measure}. It specifies a relation such that $x\prec y$ if and only
2472 is less than $f(y)$ according to~$R$, which must itself be a well-founded
2475 \item $R@1\texttt{<*lex*>}R@2$ is the lexicographic product of two relations.
2477 is a relation on pairs and satisfies $(x@1,x@2)\prec(y@1,y@2)$ if and only
2479 is less than $y@1$ according to~$R@1$ or $x@1=y@1$ and $x@2$
2480 is less than $y@2$ according to~$R@2$.
2482 \item \texttt{finite_psubset} is the proper subset relation on finite sets.
2485 We can use \texttt{measure} to declare Euclid's algorithm for the greatest
2486 common divisor. The measure function, $\lambda(m,n). n$, specifies that the
2487 recursion terminates because argument~$n$ decreases.
2489 recdef gcd "measure ((\%(m,n). n) ::nat*nat=>nat)"
2490 "gcd (m, n) = (if n=0 then m else gcd(n, m mod n))"
2493 The general form of a well-founded recursive definition is
2495 recdef {\it function} {\it rel}
2496 congs {\it congruence rules} {\bf(optional)}
2497 simpset {\it simplification set} {\bf(optional)}
2498 {\it reduction rules}
2502 \item \textit{function} is the name of the function, either as an \textit{id}
2503 or a \textit{string}.
2505 \item \textit{rel} is a HOL expression for the well-founded termination
2508 \item \textit{congruence rules} are required only in highly exceptional
2511 \item The \textit{simplification set} is used to prove that the supplied
2512 relation is well-founded. It is also used to prove the \textbf{termination
2513 conditions}: assertions that arguments of recursive calls decrease under
2514 \textit{rel}. By default, simplification uses \texttt{simpset()}, which
2515 is sufficient to prove well-foundedness for the built-in relations listed
2518 \item \textit{reduction rules} specify one or more recursion equations. Each
2519 left-hand side must have the form $f\,t$, where $f$ is the function and $t$
2520 is a tuple of distinct variables. If more than one equation is present then
2521 $f$ is defined by pattern-matching on components of its argument whose type
2522 is a \texttt{datatype}.
2524 The \ML\ identifier $f$\texttt{.simps} contains the reduction rules as
2528 With the definition of \texttt{gcd} shown above, Isabelle/HOL is unable to
2529 prove one termination condition. It remains as a precondition of the
2533 {\out ["! m n. n ~= 0 --> m mod n < n}
2534 {\out ==> gcd (?m,?n) = (if ?n=0 then ?m else gcd (?n, ?m mod ?n))"] }
2537 The theory \texttt{HOL/ex/Primes} illustrates how to prove termination
2538 conditions afterwards. The function \texttt{Tfl.tgoalw} is like the standard
2539 function \texttt{goalw}, which sets up a goal to prove, but its argument
2540 should be the identifier $f$\texttt{.simps} and its effect is to set up a
2541 proof of the termination conditions:
2543 Tfl.tgoalw thy [] gcd.simps;
2545 {\out ! m n. n ~= 0 --> m mod n < n}
2546 {\out 1. ! m n. n ~= 0 --> m mod n < n}
2548 This subgoal has a one-step proof using \texttt{simp_tac}. Once the theorem
2549 is proved, it can be used to eliminate the termination conditions from
2550 elements of \texttt{gcd.simps}. Theory \texttt{HOL/Subst/Unify} is a much
2551 more complicated example of this process, where the termination conditions can
2552 only be proved by complicated reasoning involving the recursive function
2555 Isabelle/HOL can prove the \texttt{gcd} function's termination condition
2556 automatically if supplied with the right simpset.
2558 recdef gcd "measure ((\%(m,n). n) ::nat*nat=>nat)"
2559 simpset "simpset() addsimps [mod_less_divisor, zero_less_eq]"
2560 "gcd (m, n) = (if n=0 then m else gcd(n, m mod n))"
2563 If all termination conditions were proved automatically, $f$\texttt{.simps}
2564 is added to the simpset automatically, just as in \texttt{primrec}.
2565 The simplification rules corresponding to clause $i$ (where counting starts
2566 at 0) are called $f$\texttt{.}$i$ and can be accessed as \texttt{thms
2568 which returns a list of theorems. Thus you can, for example, remove specific
2569 clauses from the simpset. Note that a single clause may give rise to a set of
2570 simplification rules in order to capture the fact that if clauses overlap,
2571 their order disambiguates them.
2573 A \texttt{recdef} definition also returns an induction rule specialised for
2574 the recursive function. For the \texttt{gcd} function above, the induction
2578 {\out "(!!m n. n ~= 0 --> ?P n (m mod n) ==> ?P m n) ==> ?P ?u ?v" : thm}
2580 This rule should be used to reason inductively about the \texttt{gcd}
2581 function. It usually makes the induction hypothesis available at all
2582 recursive calls, leading to very direct proofs. If any termination conditions
2583 remain unproved, they will become additional premises of this rule.
2585 \index{recursion!general|)}
2589 \section{Inductive and coinductive definitions}
2590 \index{*inductive|(}
2591 \index{*coinductive|(}
2593 An {\bf inductive definition} specifies the least set~$R$ closed under given
2594 rules. (Applying a rule to elements of~$R$ yields a result within~$R$.) For
2595 example, a structural operational semantics is an inductive definition of an
2596 evaluation relation. Dually, a {\bf coinductive definition} specifies the
2597 greatest set~$R$ consistent with given rules. (Every element of~$R$ can be
2598 seen as arising by applying a rule to elements of~$R$.) An important example
2599 is using bisimulation relations to formalise equivalence of processes and
2600 infinite data structures.
2602 A theory file may contain any number of inductive and coinductive
2603 definitions. They may be intermixed with other declarations; in
2604 particular, the (co)inductive sets {\bf must} be declared separately as
2605 constants, and may have mixfix syntax or be subject to syntax translations.
2607 Each (co)inductive definition adds definitions to the theory and also
2608 proves some theorems. Each definition creates an \ML\ structure, which is a
2609 substructure of the main theory structure.
2611 This package is related to the ZF one, described in a separate
2613 \footnote{It appeared in CADE~\cite{paulson-CADE}; a longer version is
2614 distributed with Isabelle.} %
2615 which you should refer to in case of difficulties. The package is simpler
2616 than ZF's thanks to HOL's extra-logical automatic type-checking. The types of
2617 the (co)inductive sets determine the domain of the fixedpoint definition, and
2618 the package does not have to use inference rules for type-checking.
2621 \subsection{The result structure}
2622 Many of the result structure's components have been discussed in the paper;
2623 others are self-explanatory.
2625 \item[\tt defs] is the list of definitions of the recursive sets.
2627 \item[\tt mono] is a monotonicity theorem for the fixedpoint operator.
2629 \item[\tt unfold] is a fixedpoint equation for the recursive set (the union of
2630 the recursive sets, in the case of mutual recursion).
2632 \item[\tt intrs] is the list of introduction rules, now proved as theorems, for
2633 the recursive sets. The rules are also available individually, using the
2634 names given them in the theory file.
2636 \item[\tt elims] is the list of elimination rule. This is for compatibility
2637 with ML scripts; within the theory the name is \texttt{cases}.
2639 \item[\tt elim] is the head of the list \texttt{elims}. This is for
2642 \item[\tt mk_cases] is a function to create simplified instances of {\tt
2643 elim} using freeness reasoning on underlying datatypes.
2646 For an inductive definition, the result structure contains the
2647 rule \texttt{induct}. For a
2648 coinductive definition, it contains the rule \verb|coinduct|.
2650 Figure~\ref{def-result-fig} summarises the two result signatures,
2651 specifying the types of all these components.
2659 val intrs : thm list
2660 val elims : thm list
2662 val mk_cases : string -> thm
2663 {\it(Inductive definitions only)}
2665 {\it(coinductive definitions only)}
2670 \caption{The {\ML} result of a (co)inductive definition} \label{def-result-fig}
2673 \subsection{The syntax of a (co)inductive definition}
2674 An inductive definition has the form
2676 inductive {\it inductive sets}
2677 intrs {\it introduction rules}
2678 monos {\it monotonicity theorems}
2680 A coinductive definition is identical, except that it starts with the keyword
2681 \texttt{coinductive}.
2683 The \texttt{monos} section is optional; if present it is specified by a list
2687 \item The \textit{inductive sets} are specified by one or more strings.
2689 \item The \textit{introduction rules} specify one or more introduction rules in
2690 the form \textit{ident\/}~\textit{string}, where the identifier gives the name of
2691 the rule in the result structure.
2693 \item The \textit{monotonicity theorems} are required for each operator
2694 applied to a recursive set in the introduction rules. There {\bf must}
2695 be a theorem of the form $A\subseteq B\Imp M(A)\subseteq M(B)$, for each
2696 premise $t\in M(R@i)$ in an introduction rule!
2698 \item The \textit{constructor definitions} contain definitions of constants
2699 appearing in the introduction rules. In most cases it can be omitted.
2703 \subsection{*Monotonicity theorems}
2705 Each theory contains a default set of theorems that are used in monotonicity
2706 proofs. New rules can be added to this set via the \texttt{mono} attribute.
2707 Theory \texttt{Inductive} shows how this is done. In general, the following
2708 monotonicity theorems may be added:
2710 \item Theorems of the form $A\subseteq B\Imp M(A)\subseteq M(B)$, for proving
2711 monotonicity of inductive definitions whose introduction rules have premises
2712 involving terms such as $t\in M(R@i)$.
2713 \item Monotonicity theorems for logical operators, which are of the general form
2714 $\List{\cdots \to \cdots;~\ldots;~\cdots \to \cdots} \Imp
2716 For example, in the case of the operator $\lor$, the corresponding theorem is
2718 \infer{P@1 \lor P@2 \to Q@1 \lor Q@2}
2719 {P@1 \to Q@1 & P@2 \to Q@2}
2721 \item De Morgan style equations for reasoning about the ``polarity'' of expressions, e.g.
2723 (\lnot \lnot P) ~=~ P \qquad\qquad
2724 (\lnot (P \land Q)) ~=~ (\lnot P \lor \lnot Q)
2726 \item Equations for reducing complex operators to more primitive ones whose
2727 monotonicity can easily be proved, e.g.
2729 (P \to Q) ~=~ (\lnot P \lor Q) \qquad\qquad
2730 \mathtt{Ball}~A~P ~\equiv~ \forall x.~x \in A \to P~x
2734 \subsection{Example of an inductive definition}
2735 Two declarations, included in a theory file, define the finite powerset
2736 operator. First we declare the constant~\texttt{Fin}. Then we declare it
2737 inductively, with two introduction rules:
2739 consts Fin :: 'a set => 'a set set
2742 emptyI "{\ttlbrace}{\ttrbrace} : Fin A"
2743 insertI "[| a: A; b: Fin A |] ==> insert a b : Fin A"
2745 The resulting theory structure contains a substructure, called~\texttt{Fin}.
2746 It contains the \texttt{Fin}$~A$ introduction rules as the list \texttt{Fin.intrs},
2747 and also individually as \texttt{Fin.emptyI} and \texttt{Fin.consI}. The induction
2748 rule is \texttt{Fin.induct}.
2750 For another example, here is a theory file defining the accessible part of a
2751 relation. The paper \cite{paulson-CADE} discusses a ZF version of this
2752 example in more detail.
2754 Acc = WF + Inductive +
2756 consts acc :: "('a * 'a)set => 'a set" (* accessible part *)
2760 accI "ALL y. (y, x) : r --> y : acc r ==> x : acc r"
2764 The Isabelle distribution contains many other inductive definitions. Simple
2765 examples are collected on subdirectory \texttt{HOL/Induct}. The theory
2766 \texttt{HOL/Induct/LList} contains coinductive definitions. Larger examples
2767 may be found on other subdirectories of \texttt{HOL}, such as \texttt{IMP},
2768 \texttt{Lambda} and \texttt{Auth}.
2770 \index{*coinductive|)} \index{*inductive|)}
2773 \section{Executable specifications}
2774 \index{code generator}
2776 For validation purposes, it is often useful to {\em execute} specifications.
2777 In principle, specifications could be ``executed'' using Isabelle's
2778 inference kernel, i.e. by a combination of resolution and simplification.
2779 Unfortunately, this approach is rather inefficient. A more efficient way
2780 of executing specifications is to translate them into a functional
2781 programming language such as ML. Isabelle's built-in code generator
2784 \railalias{verblbrace}{\texttt{\ttlbrace*}}
2785 \railalias{verbrbrace}{\texttt{*\ttrbrace}}
2786 \railterm{verblbrace}
2787 \railterm{verbrbrace}
2791 codegen : ( 'code_module' | 'code_library' ) modespec ? name ? \\
2792 ( 'file' name ) ? ( 'imports' ( name + ) ) ? \\
2793 'contains' ( ( name '=' term ) + | term + );
2795 modespec : '(' ( name * ) ')';
2797 \caption{Code generator invocation syntax}
2798 \label{fig:HOL:codegen-invocation}
2803 constscode : 'consts_code' (codespec +);
2805 codespec : const template attachment ?;
2807 typescode : 'types_code' (tycodespec +);
2809 tycodespec : name template attachment ?;
2813 template: '(' string ')';
2815 attachment: 'attach' modespec ? verblbrace text verbrbrace;
2817 \caption{Code generator configuration syntax}
2818 \label{fig:HOL:codegen-configuration}
2821 \subsection{Invoking the code generator}
2823 The code generator is invoked via the \ttindex{code_module} and
2824 \ttindex{code_library} commands (see Fig.~\ref{fig:HOL:codegen-invocation}),
2825 which correspond to {\em incremental} and {\em modular} code generation,
2828 \item[Modular] For each theory, an ML structure is generated, containing the
2829 code generated from the constants defined in this theory.
2830 \item[Incremental] All the generated code is emitted into the same structure.
2831 This structure may import code from previously generated structures, which
2832 can be specified via \texttt{imports}.
2833 Moreover, the generated structure may also be referred to in later invocations
2834 of the code generator.
2836 After the \texttt{code_module} and \texttt{code_library} keywords, the user
2837 may specify an optional list of ``modes'' in parentheses. These can be used
2838 to instruct the code generator to emit additional code for special purposes,
2839 e.g.\ functions for converting elements of generated datatypes to Isabelle terms,
2840 or test data generators. The list of modes is followed by a module name.
2841 The module name is optional for modular code generation, but must be specified
2842 for incremental code generation.
2843 The code can either be written to a file,
2844 in which case a file name has to be specified after the \texttt{file} keyword, or be
2845 loaded directly into Isabelle's ML environment. In the latter case,
2846 the \texttt{ML} theory command can be used to inspect the results
2848 The terms from which to generate code can be specified after the
2849 \texttt{contains} keyword, either as a list of bindings, or just as
2850 a list of terms. In the latter case, the code generator just produces
2851 code for all constants and types occuring in the term, but does not
2852 bind the compiled terms to ML identifiers.
2857 test = "foldl op + (0::int) [1,2,3,4,5]"
2859 binds the result of compiling the term
2860 \texttt{foldl op + (0::int) [1,2,3,4,5]}
2861 (i.e.~\texttt{15}) to the ML identifier \texttt{Test.test}.
2863 \subsection{Configuring the code generator}
2865 When generating code for a complex term, the code generator recursively
2866 calls itself for all subterms.
2867 When it arrives at a constant, the default strategy of the code
2868 generator is to look up its definition and try to generate code for it.
2869 Constants which have no definitions that
2870 are immediately executable, may be associated with a piece of ML
2871 code manually using the \ttindex{consts_code} command
2872 (see Fig.~\ref{fig:HOL:codegen-configuration}).
2873 It takes a list whose elements consist of a constant (given in usual term syntax
2874 -- an explicit type constraint accounts for overloading), and a
2875 mixfix template describing the ML code. The latter is very much the
2876 same as the mixfix templates used when declaring new constants.
2877 The most notable difference is that terms may be included in the ML
2878 template using antiquotation brackets \verb|{*|~$\ldots$~\verb|*}|.
2879 A similar mechanism is available for
2880 types: \ttindex{types_code} associates type constructors with
2881 specific ML code. For example, the declaration
2889 in theory \texttt{Product_Type} describes how the product type of Isabelle/HOL
2890 should be compiled to ML. Sometimes, the code associated with a
2891 constant or type may need to refer to auxiliary functions, which
2892 have to be emitted when the constant is used. Code for such auxiliary
2893 functions can be declared using \texttt{attach}. For example, the
2894 \texttt{wfrec} function from theory \texttt{Wellfounded_Recursion}
2895 is implemented as follows:
2898 "wfrec" ("\bs<module>wfrec?")
2900 fun wfrec f x = f (wfrec f) x;
2903 If the code containing a call to \texttt{wfrec} resides in an ML structure
2904 different from the one containing the function definition attached to
2905 \texttt{wfrec}, the name of the ML structure (followed by a ``\texttt{.}'')
2906 is inserted in place of ``\texttt{\bs<module>}'' in the above template.
2907 The ``\texttt{?}'' means that the code generator should ignore the first
2908 argument of \texttt{wfrec}, i.e.\ the termination relation, which is
2909 usually not executable.
2911 Another possibility of configuring the code generator is to register
2912 theorems to be used for code generation. Theorems can be registered
2913 via the \ttindex{code} attribute. It takes an optional name as
2914 an argument, which indicates the format of the theorem. Currently
2915 supported formats are equations (this is the default when no name
2916 is specified) and horn clauses (this is indicated by the name
2917 \texttt{ind}). The left-hand sides of equations may only contain
2918 constructors and distinct variables, whereas horn clauses must have
2919 the same format as introduction rules of inductive definitions.
2920 For example, the declaration
2922 lemma Suc_less_eq [iff, code]: "(Suc m < Suc n) = (m < n)" \(\langle\ldots\rangle\)
2923 lemma [code]: "((n::nat) < 0) = False" by simp
2924 lemma [code]: "(0 < Suc n) = True" by simp
2926 in theory \texttt{Nat} specifies three equations from which to generate
2927 code for \texttt{<} on natural numbers.
2929 \subsection{Specific HOL code generators}
2931 The basic code generator framework offered by Isabelle/Pure has
2932 already been extended with additional code generators for specific
2933 HOL constructs. These include datatypes, recursive functions and
2934 inductive relations. The code generator for inductive relations
2935 can handle expressions of the form $(t@1,\ldots,t@n) \in r$, where
2936 $r$ is an inductively defined relation. If at least one of the
2937 $t@i$ is a dummy pattern ``$_$'', the above expression evaluates to a
2938 sequence of possible answers. If all of the $t@i$ are proper
2939 terms, the expression evaluates to a boolean value.
2942 theory Test = Lambda:
2946 test1 = "Abs (Var 0) \(\circ\) Var 0 -> Var 0"
2947 test2 = "Abs (Abs (Var 0 \(\circ\) Var 0) \(\circ\) (Abs (Var 0) \(\circ\) Var 0)) -> _"
2950 In the above example, \texttt{Test.test1} evaluates to the boolean
2951 value \texttt{true}, whereas \texttt{Test.test2} is a sequence whose
2952 elements can be inspected using \texttt{Seq.pull} or similar functions.
2954 ML \{* Seq.pull Test.test2 *\} -- \{* This is the first answer *\}
2955 ML \{* Seq.pull (snd (the it)) *\} -- \{* This is the second answer *\}
2958 underlying the HOL code generator is described more detailed in
2959 \cite{Berghofer-Nipkow:2002}. More examples that illustrate the usage
2960 of the code generator can be found e.g.~in theories
2961 \texttt{MicroJava/J/JListExample} and \texttt{MicroJava/JVM/JVMListExample}.
2963 \section{The examples directories}
2965 Directory \texttt{HOL/Auth} contains theories for proving the correctness of
2966 cryptographic protocols~\cite{paulson-jcs}. The approach is based upon
2967 operational semantics rather than the more usual belief logics. On the same
2968 directory are proofs for some standard examples, such as the Needham-Schroeder
2969 public-key authentication protocol and the Otway-Rees
2972 Directory \texttt{HOL/IMP} contains a formalization of various denotational,
2973 operational and axiomatic semantics of a simple while-language, the necessary
2974 equivalence proofs, soundness and completeness of the Hoare rules with
2975 respect to the denotational semantics, and soundness and completeness of a
2976 verification condition generator. Much of development is taken from
2977 Winskel~\cite{winskel93}. For details see~\cite{nipkow-IMP}.
2979 Directory \texttt{HOL/Hoare} contains a user friendly surface syntax for Hoare
2980 logic, including a tactic for generating verification-conditions.
2982 Directory \texttt{HOL/MiniML} contains a formalization of the type system of
2983 the core functional language Mini-ML and a correctness proof for its type
2984 inference algorithm $\cal W$~\cite{milner78,nipkow-W}.
2986 Directory \texttt{HOL/Lambda} contains a formalization of untyped
2987 $\lambda$-calculus in de~Bruijn notation and Church-Rosser proofs for $\beta$
2988 and $\eta$ reduction~\cite{Nipkow-CR}.
2990 Directory \texttt{HOL/Subst} contains Martin Coen's mechanization of a theory
2991 of substitutions and unifiers. It is based on Paulson's previous
2992 mechanisation in LCF~\cite{paulson85} of Manna and Waldinger's
2993 theory~\cite{mw81}. It demonstrates a complicated use of \texttt{recdef},
2994 with nested recursion.
2996 Directory \texttt{HOL/Induct} presents simple examples of (co)inductive
2997 definitions and datatypes.
2999 \item Theory \texttt{PropLog} proves the soundness and completeness of
3000 classical propositional logic, given a truth table semantics. The only
3001 connective is $\imp$. A Hilbert-style axiom system is specified, and its
3002 set of theorems defined inductively. A similar proof in ZF is described
3003 elsewhere~\cite{paulson-set-II}.
3005 \item Theory \texttt{Term} defines the datatype \texttt{term}.
3007 \item Theory \texttt{ABexp} defines arithmetic and boolean expressions
3008 as mutually recursive datatypes.
3010 \item The definition of lazy lists demonstrates methods for handling
3011 infinite data structures and coinduction in higher-order
3012 logic~\cite{paulson-coind}.%
3013 \footnote{To be precise, these lists are \emph{potentially infinite} rather
3014 than lazy. Lazy implies a particular operational semantics.}
3015 Theory \thydx{LList} defines an operator for
3016 corecursion on lazy lists, which is used to define a few simple functions
3017 such as map and append. A coinduction principle is defined
3018 for proving equations on lazy lists.
3020 \item Theory \thydx{LFilter} defines the filter functional for lazy lists.
3021 This functional is notoriously difficult to define because finding the next
3022 element meeting the predicate requires possibly unlimited search. It is not
3023 computable, but can be expressed using a combination of induction and
3026 \item Theory \thydx{Exp} illustrates the use of iterated inductive definitions
3027 to express a programming language semantics that appears to require mutual
3028 induction. Iterated induction allows greater modularity.
3031 Directory \texttt{HOL/ex} contains other examples and experimental proofs in
3034 \item Theory \texttt{Recdef} presents many examples of using \texttt{recdef}
3035 to define recursive functions. Another example is \texttt{Fib}, which
3036 defines the Fibonacci function.
3038 \item Theory \texttt{Primes} defines the Greatest Common Divisor of two
3039 natural numbers and proves a key lemma of the Fundamental Theorem of
3040 Arithmetic: if $p$ is prime and $p$ divides $m\times n$ then $p$ divides~$m$
3043 \item Theory \texttt{Primrec} develops some computation theory. It
3044 inductively defines the set of primitive recursive functions and presents a
3045 proof that Ackermann's function is not primitive recursive.
3047 \item File \texttt{cla.ML} demonstrates the classical reasoner on over sixty
3048 predicate calculus theorems, ranging from simple tautologies to
3049 moderately difficult problems involving equality and quantifiers.
3051 \item File \texttt{meson.ML} contains an experimental implementation of the {\sc
3052 meson} proof procedure, inspired by Plaisted~\cite{plaisted90}. It is
3053 much more powerful than Isabelle's classical reasoner. But it is less
3054 useful in practice because it works only for pure logic; it does not
3055 accept derived rules for the set theory primitives, for example.
3057 \item File \texttt{mesontest.ML} contains test data for the {\sc meson} proof
3058 procedure. These are mostly taken from Pelletier \cite{pelletier86}.
3060 \item File \texttt{set.ML} proves Cantor's Theorem, which is presented in
3061 {\S}\ref{sec:hol-cantor} below, and the Schr{\"o}der-Bernstein Theorem.
3063 \item Theory \texttt{MT} contains Jacob Frost's formalization~\cite{frost93} of
3064 Milner and Tofte's coinduction example~\cite{milner-coind}. This
3065 substantial proof concerns the soundness of a type system for a simple
3066 functional language. The semantics of recursion is given by a cyclic
3067 environment, which makes a coinductive argument appropriate.
3072 \section{Example: Cantor's Theorem}\label{sec:hol-cantor}
3073 Cantor's Theorem states that every set has more subsets than it has
3074 elements. It has become a favourite example in higher-order logic since
3075 it is so easily expressed:
3076 \[ \forall f::\alpha \To \alpha \To bool. \exists S::\alpha\To bool.
3077 \forall x::\alpha. f~x \not= S
3080 Viewing types as sets, $\alpha\To bool$ represents the powerset
3081 of~$\alpha$. This version states that for every function from $\alpha$ to
3082 its powerset, some subset is outside its range.
3084 The Isabelle proof uses HOL's set theory, with the type $\alpha\,set$ and
3085 the operator \cdx{range}.
3089 The set~$S$ is given as an unknown instead of a
3090 quantified variable so that we may inspect the subset found by the proof.
3092 Goal "?S ~: range\thinspace(f :: 'a=>'a set)";
3094 {\out ?S ~: range f}
3095 {\out 1. ?S ~: range f}
3097 The first two steps are routine. The rule \tdx{rangeE} replaces
3098 $\Var{S}\in \texttt{range} \, f$ by $\Var{S}=f~x$ for some~$x$.
3100 by (resolve_tac [notI] 1);
3102 {\out ?S ~: range f}
3103 {\out 1. ?S : range f ==> False}
3105 by (eresolve_tac [rangeE] 1);
3107 {\out ?S ~: range f}
3108 {\out 1. !!x. ?S = f x ==> False}
3110 Next, we apply \tdx{equalityCE}, reasoning that since $\Var{S}=f~x$,
3111 we have $\Var{c}\in \Var{S}$ if and only if $\Var{c}\in f~x$ for
3114 by (eresolve_tac [equalityCE] 1);
3116 {\out ?S ~: range f}
3117 {\out 1. !!x. [| ?c3 x : ?S; ?c3 x : f x |] ==> False}
3118 {\out 2. !!x. [| ?c3 x ~: ?S; ?c3 x ~: f x |] ==> False}
3120 Now we use a bit of creativity. Suppose that~$\Var{S}$ has the form of a
3121 comprehension. Then $\Var{c}\in\{x.\Var{P}~x\}$ implies
3122 $\Var{P}~\Var{c}$. Destruct-resolution using \tdx{CollectD}
3123 instantiates~$\Var{S}$ and creates the new assumption.
3125 by (dresolve_tac [CollectD] 1);
3127 {\out {\ttlbrace}x. ?P7 x{\ttrbrace} ~: range f}
3128 {\out 1. !!x. [| ?c3 x : f x; ?P7(?c3 x) |] ==> False}
3129 {\out 2. !!x. [| ?c3 x ~: {\ttlbrace}x. ?P7 x{\ttrbrace}; ?c3 x ~: f x |] ==> False}
3131 Forcing a contradiction between the two assumptions of subgoal~1
3132 completes the instantiation of~$S$. It is now the set $\{x. x\not\in
3133 f~x\}$, which is the standard diagonal construction.
3137 {\out {\ttlbrace}x. x ~: f x{\ttrbrace} ~: range f}
3138 {\out 1. !!x. [| x ~: {\ttlbrace}x. x ~: f x{\ttrbrace}; x ~: f x |] ==> False}
3140 The rest should be easy. To apply \tdx{CollectI} to the negated
3141 assumption, we employ \ttindex{swap_res_tac}:
3143 by (swap_res_tac [CollectI] 1);
3145 {\out {\ttlbrace}x. x ~: f x{\ttrbrace} ~: range f}
3146 {\out 1. !!x. [| x ~: f x; ~ False |] ==> x ~: f x}
3150 {\out {\ttlbrace}x. x ~: f x{\ttrbrace} ~: range f}
3153 How much creativity is required? As it happens, Isabelle can prove this
3154 theorem automatically. The default classical set \texttt{claset()} contains
3155 rules for most of the constructs of HOL's set theory. We must augment it with
3156 \tdx{equalityCE} to break up set equalities, and then apply best-first search.
3157 Depth-first search would diverge, but best-first search successfully navigates
3158 through the large search space. \index{search!best-first}
3162 {\out ?S ~: range f}
3163 {\out 1. ?S ~: range f}
3165 by (best_tac (claset() addSEs [equalityCE]) 1);
3167 {\out {\ttlbrace}x. x ~: f x{\ttrbrace} ~: range f}
3170 If you run this example interactively, make sure your current theory contains
3171 theory \texttt{Set}, for example by executing \ttindex{context}~{\tt Set.thy}.
3172 Otherwise the default claset may not contain the rules for set theory.
3173 \index{higher-order logic|)}
3175 %%% Local Variables:
3177 %%% TeX-master: "logics-HOL"