1 \documentstyle[a4,alltt,iman,extra,proof209,12pt]{article}
2 \newif\ifshort%''Short'' means a published version, not the documentation
3 \shortfalse%%%%%\shorttrue
5 \title{A Fixedpoint Approach to\\
6 (Co)Inductive and (Co)Datatype Definitions%
7 \thanks{J. Grundy and S. Thompson made detailed comments. Mads Tofte and
8 the referees were also helpful. The research was funded by the SERC
9 grants GR/G53279, GR/H40570 and by the ESPRIT Project 6453 ``Types''.}}
11 \author{Lawrence C. Paulson\\{\tt lcp@cl.cam.ac.uk}\\
12 Computer Laboratory, University of Cambridge, England}
14 \setcounter{secnumdepth}{2} \setcounter{tocdepth}{2}
16 \newcommand\sbs{\subseteq}
19 \newcommand\emph[1]{{\em#1\/}}
20 \newcommand\defn[1]{{\bf#1}}
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22 \newcommand\texttt[1]{{\tt#1}}
24 \newcommand\pow{{\cal P}}
26 \newcommand\RepFun{\hbox{\tt RepFun}}
27 \newcommand\cons{\hbox{\tt cons}}
28 \def\succ{\hbox{\tt succ}}
29 \newcommand\split{\hbox{\tt split}}
30 \newcommand\fst{\hbox{\tt fst}}
31 \newcommand\snd{\hbox{\tt snd}}
32 \newcommand\converse{\hbox{\tt converse}}
33 \newcommand\domain{\hbox{\tt domain}}
34 \newcommand\range{\hbox{\tt range}}
35 \newcommand\field{\hbox{\tt field}}
36 \newcommand\lfp{\hbox{\tt lfp}}
37 \newcommand\gfp{\hbox{\tt gfp}}
38 \newcommand\id{\hbox{\tt id}}
39 \newcommand\trans{\hbox{\tt trans}}
40 \newcommand\wf{\hbox{\tt wf}}
41 \newcommand\nat{\hbox{\tt nat}}
42 \newcommand\rank{\hbox{\tt rank}}
43 \newcommand\univ{\hbox{\tt univ}}
44 \newcommand\Vrec{\hbox{\tt Vrec}}
45 \newcommand\Inl{\hbox{\tt Inl}}
46 \newcommand\Inr{\hbox{\tt Inr}}
47 \newcommand\case{\hbox{\tt case}}
48 \newcommand\lst{\hbox{\tt list}}
49 \newcommand\Nil{\hbox{\tt Nil}}
50 \newcommand\Cons{\hbox{\tt Cons}}
51 \newcommand\lstcase{\hbox{\tt list\_case}}
52 \newcommand\lstrec{\hbox{\tt list\_rec}}
53 \newcommand\length{\hbox{\tt length}}
54 \newcommand\listn{\hbox{\tt listn}}
55 \newcommand\acc{\hbox{\tt acc}}
56 \newcommand\primrec{\hbox{\tt primrec}}
57 \newcommand\SC{\hbox{\tt SC}}
58 \newcommand\CONST{\hbox{\tt CONST}}
59 \newcommand\PROJ{\hbox{\tt PROJ}}
60 \newcommand\COMP{\hbox{\tt COMP}}
61 \newcommand\PREC{\hbox{\tt PREC}}
63 \newcommand\quniv{\hbox{\tt quniv}}
64 \newcommand\llist{\hbox{\tt llist}}
65 \newcommand\LNil{\hbox{\tt LNil}}
66 \newcommand\LCons{\hbox{\tt LCons}}
67 \newcommand\lconst{\hbox{\tt lconst}}
68 \newcommand\lleq{\hbox{\tt lleq}}
69 \newcommand\map{\hbox{\tt map}}
70 \newcommand\term{\hbox{\tt term}}
71 \newcommand\Apply{\hbox{\tt Apply}}
72 \newcommand\termcase{\hbox{\tt term\_case}}
73 \newcommand\rev{\hbox{\tt rev}}
74 \newcommand\reflect{\hbox{\tt reflect}}
75 \newcommand\tree{\hbox{\tt tree}}
76 \newcommand\forest{\hbox{\tt forest}}
77 \newcommand\Part{\hbox{\tt Part}}
78 \newcommand\TF{\hbox{\tt tree\_forest}}
79 \newcommand\Tcons{\hbox{\tt Tcons}}
80 \newcommand\Fcons{\hbox{\tt Fcons}}
81 \newcommand\Fnil{\hbox{\tt Fnil}}
82 \newcommand\TFcase{\hbox{\tt TF\_case}}
83 \newcommand\Fin{\hbox{\tt Fin}}
84 \newcommand\QInl{\hbox{\tt QInl}}
85 \newcommand\QInr{\hbox{\tt QInr}}
86 \newcommand\qsplit{\hbox{\tt qsplit}}
87 \newcommand\qcase{\hbox{\tt qcase}}
88 \newcommand\Con{\hbox{\tt Con}}
89 \newcommand\data{\hbox{\tt data}}
91 \binperiod %%%treat . like a binary operator
98 This paper presents a fixedpoint approach to inductive definitions.
99 Instead of using a syntactic test such as ``strictly positive,'' the
100 approach lets definitions involve any operators that have been proved
101 monotone. It is conceptually simple, which has allowed the easy
102 implementation of mutual recursion and iterated definitions. It also
103 handles coinductive definitions: simply replace the least fixedpoint by a
106 The method has been implemented in two of Isabelle's logics, \textsc{zf} set
107 theory and higher-order logic. It should be applicable to any logic in
108 which the Knaster-Tarski theorem can be proved. Examples include lists of
109 $n$ elements, the accessible part of a relation and the set of primitive
110 recursive functions. One example of a coinductive definition is
111 bisimulations for lazy lists. Recursive datatypes are examined in detail,
112 as well as one example of a \defn{codatatype}: lazy lists.
114 The Isabelle package has been applied in several large case studies,
115 including two proofs of the Church-Rosser theorem and a coinductive proof of
116 semantic consistency. The package can be trusted because it proves theorems
117 from definitions, instead of asserting desired properties as axioms.
121 \centerline{Copyright \copyright{} \number\year{} by Lawrence C. Paulson}
122 \thispagestyle{empty}
124 \tableofcontents\cleardoublepage\pagestyle{plain}
128 \section{Introduction}
129 Several theorem provers provide commands for formalizing recursive data
130 structures, like lists and trees. Robin Milner implemented one of the first
131 of these, for Edinburgh \textsc{lcf}~\cite{milner-ind}. Given a description
132 of the desired data structure, Milner's package formulated appropriate
133 definitions and proved the characteristic theorems. Similar is Melham's
134 recursive type package for the Cambridge \textsc{hol} system~\cite{melham89}.
135 Such data structures are called \defn{datatypes}
136 below, by analogy with datatype declarations in Standard~\textsc{ml}\@.
137 Some logics take datatypes as primitive; consider Boyer and Moore's shell
138 principle~\cite{bm79} and the Coq type theory~\cite{paulin-tlca}.
140 A datatype is but one example of an \defn{inductive definition}. Such a
141 definition~\cite{aczel77} specifies the least set~$R$ \defn{closed under}
142 given rules: applying a rule to elements of~$R$ yields a result within~$R$.
143 Inductive definitions have many applications. The collection of theorems in a
144 logic is inductively defined. A structural operational
145 semantics~\cite{hennessy90} is an inductive definition of a reduction or
146 evaluation relation on programs. A few theorem provers provide commands for
147 formalizing inductive definitions; these include Coq~\cite{paulin-tlca} and
148 again the \textsc{hol} system~\cite{camilleri92}.
150 The dual notion is that of a \defn{coinductive definition}. Such a definition
151 specifies the greatest set~$R$ \defn{consistent with} given rules: every
152 element of~$R$ can be seen as arising by applying a rule to elements of~$R$.
153 Important examples include using bisimulation relations to formalize
154 equivalence of processes~\cite{milner89} or lazy functional
155 programs~\cite{abramsky90}. Other examples include lazy lists and other
156 infinite data structures; these are called \defn{codatatypes} below.
158 Not all inductive definitions are meaningful. \defn{Monotone} inductive
159 definitions are a large, well-behaved class. Monotonicity can be enforced
160 by syntactic conditions such as ``strictly positive,'' but this could lead to
161 monotone definitions being rejected on the grounds of their syntactic form.
162 More flexible is to formalize monotonicity within the logic and allow users
165 This paper describes a package based on a fixedpoint approach. Least
166 fixedpoints yield inductive definitions; greatest fixedpoints yield
167 coinductive definitions. Most of the discussion below applies equally to
168 inductive and coinductive definitions, and most of the code is shared.
170 The package supports mutual recursion and infinitely-branching datatypes and
171 codatatypes. It allows use of any operators that have been proved monotone,
172 thus accepting all provably monotone inductive definitions, including
173 iterated definitions.
175 The package has been implemented in
176 Isabelle~\cite{paulson-markt,paulson-isa-book} using
177 \textsc{zf} set theory \cite{paulson-set-I,paulson-set-II}; part of it has
178 since been ported to Isabelle/\textsc{hol} (higher-order logic). The
179 recursion equations are specified as introduction rules for the mutually
180 recursive sets. The package transforms these rules into a mapping over sets,
181 and attempts to prove that the mapping is monotonic and well-typed. If
182 successful, the package makes fixedpoint definitions and proves the
183 introduction, elimination and (co)induction rules. Users invoke the package
184 by making simple declarations in Isabelle theory files.
186 Most datatype packages equip the new datatype with some means of expressing
187 recursive functions. This is the main omission from my package. Its
188 fixedpoint operators define only recursive sets. The Isabelle/\textsc{zf}
189 theory provides well-founded recursion~\cite{paulson-set-II}, which is harder
190 to use than structural recursion but considerably more general.
191 Slind~\cite{slind-tfl} has written a package to automate the definition of
192 well-founded recursive functions in Isabelle/\textsc{hol}.
194 \paragraph*{Outline.} Section~2 introduces the least and greatest fixedpoint
195 operators. Section~3 discusses the form of introduction rules, mutual
196 recursion and other points common to inductive and coinductive definitions.
197 Section~4 discusses induction and coinduction rules separately. Section~5
198 presents several examples, including a coinductive definition. Section~6
199 describes datatype definitions. Section~7 presents related work.
200 Section~8 draws brief conclusions. \ifshort\else The appendices are simple
201 user's manuals for this Isabelle package.\fi
203 Most of the definitions and theorems shown below have been generated by the
204 package. I have renamed some variables to improve readability.
206 \section{Fixedpoint operators}
207 In set theory, the least and greatest fixedpoint operators are defined as
210 \lfp(D,h) & \equiv & \inter\{X\sbs D. h(X)\sbs X\} \\
211 \gfp(D,h) & \equiv & \union\{X\sbs D. X\sbs h(X)\}
213 Let $D$ be a set. Say that $h$ is \defn{bounded by}~$D$ if $h(D)\sbs D$, and
214 \defn{monotone below~$D$} if
215 $h(A)\sbs h(B)$ for all $A$ and $B$ such that $A\sbs B\sbs D$. If $h$ is
216 bounded by~$D$ and monotone then both operators yield fixedpoints:
218 \lfp(D,h) & = & h(\lfp(D,h)) \\
219 \gfp(D,h) & = & h(\gfp(D,h))
221 These equations are instances of the Knaster-Tarski theorem, which states
222 that every monotonic function over a complete lattice has a
223 fixedpoint~\cite{davey&priestley}. It is obvious from their definitions
224 that $\lfp$ must be the least fixedpoint, and $\gfp$ the greatest.
226 This fixedpoint theory is simple. The Knaster-Tarski theorem is easy to
227 prove. Showing monotonicity of~$h$ is trivial, in typical cases. We must
228 also exhibit a bounding set~$D$ for~$h$. Frequently this is trivial, as when
229 a set of theorems is (co)inductively defined over some previously existing set
230 of formul{\ae}. Isabelle/\textsc{zf} provides suitable bounding sets for
231 infinitely-branching (co)datatype definitions; see~\S\ref{univ-sec}. Bounding
232 sets are also called \defn{domains}.
234 The powerset operator is monotone, but by Cantor's theorem there is no
235 set~$A$ such that $A=\pow(A)$. We cannot put $A=\lfp(D,\pow)$ because
236 there is no suitable domain~$D$. But \S\ref{acc-sec} demonstrates
237 that~$\pow$ is still useful in inductive definitions.
239 \section{Elements of an inductive or coinductive definition}\label{basic-sec}
240 Consider a (co)inductive definition of the sets $R_1$, \ldots,~$R_n$, in
241 mutual recursion. They will be constructed from domains $D_1$,
242 \ldots,~$D_n$, respectively. The construction yields not $R_i\sbs D_i$ but
243 $R_i\sbs D_1+\cdots+D_n$, where $R_i$ is contained in the image of~$D_i$
244 under an injection. Reasons for this are discussed
245 elsewhere~\cite[\S4.5]{paulson-set-II}.
247 The definition may involve arbitrary parameters $\vec{p}=p_1$,
248 \ldots,~$p_k$. Each recursive set then has the form $R_i(\vec{p})$. The
249 parameters must be identical every time they occur within a definition. This
250 would appear to be a serious restriction compared with other systems such as
251 Coq~\cite{paulin-tlca}. For instance, we cannot define the lists of
252 $n$ elements as the set $\listn(A,n)$ using rules where the parameter~$n$
253 varies. Section~\ref{listn-sec} describes how to express this set using the
254 inductive definition package.
256 To avoid clutter below, the recursive sets are shown as simply $R_i$
257 instead of~$R_i(\vec{p})$.
259 \subsection{The form of the introduction rules}\label{intro-sec}
260 The body of the definition consists of the desired introduction rules. The
261 conclusion of each rule must have the form $t\in R_i$, where $t$ is any term.
262 Premises typically have the same form, but they can have the more general form
263 $t\in M(R_i)$ or express arbitrary side-conditions.
265 The premise $t\in M(R_i)$ is permitted if $M$ is a monotonic operator on
266 sets, satisfying the rule
267 \[ \infer{M(A)\sbs M(B)}{A\sbs B} \]
268 The user must supply the package with monotonicity rules for all such premises.
270 The ability to introduce new monotone operators makes the approach
271 flexible. A suitable choice of~$M$ and~$t$ can express a lot. The
272 powerset operator $\pow$ is monotone, and the premise $t\in\pow(R)$
273 expresses $t\sbs R$; see \S\ref{acc-sec} for an example. The \emph{list of}
274 operator is monotone, as is easily proved by induction. The premise
275 $t\in\lst(R)$ avoids having to encode the effect of~$\lst(R)$ using mutual
276 recursion; see \S\ref{primrec-sec} and also my earlier
277 paper~\cite[\S4.4]{paulson-set-II}.
279 Introduction rules may also contain \defn{side-conditions}. These are
280 premises consisting of arbitrary formul{\ae} not mentioning the recursive
281 sets. Side-conditions typically involve type-checking. One example is the
282 premise $a\in A$ in the following rule from the definition of lists:
283 \[ \infer{\Cons(a,l)\in\lst(A)}{a\in A & l\in\lst(A)} \]
285 \subsection{The fixedpoint definitions}
286 The package translates the list of desired introduction rules into a fixedpoint
287 definition. Consider, as a running example, the finite powerset operator
288 $\Fin(A)$: the set of all finite subsets of~$A$. It can be
289 defined as the least set closed under the rules
290 \[ \emptyset\in\Fin(A) \qquad
291 \infer{\{a\}\un b\in\Fin(A)}{a\in A & b\in\Fin(A)}
294 The domain in a (co)inductive definition must be some existing set closed
295 under the rules. A suitable domain for $\Fin(A)$ is $\pow(A)$, the set of all
296 subsets of~$A$. The package generates the definition
297 \[ \Fin(A) \equiv \lfp(\pow(A), \,
298 \begin{array}[t]{r@{\,}l}
299 \lambda X. \{z\in\pow(A). & z=\emptyset \disj{} \\
300 &(\exists a\,b. z=\{a\}\un b\conj a\in A\conj b\in X)\})
303 The contribution of each rule to the definition of $\Fin(A)$ should be
304 obvious. A coinductive definition is similar but uses $\gfp$ instead
307 The package must prove that the fixedpoint operator is applied to a
308 monotonic function. If the introduction rules have the form described
309 above, and if the package is supplied a monotonicity theorem for every
310 $t\in M(R_i)$ premise, then this proof is trivial.\footnote{Due to the
311 presence of logical connectives in the fixedpoint's body, the
312 monotonicity proof requires some unusual rules. These state that the
313 connectives $\conj$, $\disj$ and $\exists$ preserve monotonicity with respect
314 to the partial ordering on unary predicates given by $P\sqsubseteq Q$ if and
315 only if $\forall x.P(x)\imp Q(x)$.}
317 The package returns its result as an \textsc{ml} structure, which consists of named
318 components; we may regard it as a record. The result structure contains
319 the definitions of the recursive sets as a theorem list called {\tt defs}.
320 It also contains some theorems; {\tt dom\_subset} is an inclusion such as
321 $\Fin(A)\sbs\pow(A)$, while {\tt bnd\_mono} asserts that the fixedpoint
322 definition is monotonic.
324 Internally the package uses the theorem {\tt unfold}, a fixedpoint equation
327 \begin{array}[t]{r@{\,}l}
328 \Fin(A) = \{z\in\pow(A). & z=\emptyset \disj{} \\
329 &(\exists a\,b. z=\{a\}\un b\conj a\in A\conj b\in \Fin(A))\}
332 In order to save space, this theorem is not exported.
335 \subsection{Mutual recursion} \label{mutual-sec}
336 In a mutually recursive definition, the domain of the fixedpoint construction
337 is the disjoint sum of the domain~$D_i$ of each~$R_i$, for $i=1$,
338 \ldots,~$n$. The package uses the injections of the
339 binary disjoint sum, typically $\Inl$ and~$\Inr$, to express injections
340 $h_{1n}$, \ldots, $h_{nn}$ for the $n$-ary disjoint sum $D_1+\cdots+D_n$.
342 As discussed elsewhere \cite[\S4.5]{paulson-set-II}, Isabelle/\textsc{zf} defines the
343 operator $\Part$ to support mutual recursion. The set $\Part(A,h)$
344 contains those elements of~$A$ having the form~$h(z)$:
345 \[ \Part(A,h) \equiv \{x\in A. \exists z. x=h(z)\}. \]
346 For mutually recursive sets $R_1$, \ldots,~$R_n$ with
347 $n>1$, the package makes $n+1$ definitions. The first defines a set $R$ using
348 a fixedpoint operator. The remaining $n$ definitions have the form
349 \[ R_i \equiv \Part(R,h_{in}), \qquad i=1,\ldots, n. \]
350 It follows that $R=R_1\un\cdots\un R_n$, where the $R_i$ are pairwise disjoint.
353 \subsection{Proving the introduction rules}
354 The user supplies the package with the desired form of the introduction
355 rules. Once it has derived the theorem {\tt unfold}, it attempts
356 to prove those rules. From the user's point of view, this is the
357 trickiest stage; the proofs often fail. The task is to show that the domain
358 $D_1+\cdots+D_n$ of the combined set $R_1\un\cdots\un R_n$ is
359 closed under all the introduction rules. This essentially involves replacing
360 each~$R_i$ by $D_1+\cdots+D_n$ in each of the introduction rules and
361 attempting to prove the result.
363 Consider the $\Fin(A)$ example. After substituting $\pow(A)$ for $\Fin(A)$
364 in the rules, the package must prove
365 \[ \emptyset\in\pow(A) \qquad
366 \infer{\{a\}\un b\in\pow(A)}{a\in A & b\in\pow(A)}
368 Such proofs can be regarded as type-checking the definition.\footnote{The
369 Isabelle/\textsc{hol} version does not require these proofs, as \textsc{hol}
370 has implicit type-checking.} The user supplies the package with
371 type-checking rules to apply. Usually these are general purpose rules from
372 the \textsc{zf} theory. They could however be rules specifically proved for a
373 particular inductive definition; sometimes this is the easiest way to get the
376 The result structure contains the introduction rules as the theorem list {\tt
379 \subsection{The case analysis rule}
380 The elimination rule, called {\tt elim}, performs case analysis. It is a
381 simple consequence of {\tt unfold}. There is one case for each introduction
382 rule. If $x\in\Fin(A)$ then either $x=\emptyset$ or else $x=\{a\}\un b$ for
383 some $a\in A$ and $b\in\Fin(A)$. Formally, the elimination rule for $\Fin(A)$
385 \[ \infer{Q}{x\in\Fin(A) & \infer*{Q}{[x=\emptyset]}
386 & \infer*{Q}{[x=\{a\}\un b & a\in A &b\in\Fin(A)]_{a,b}} }
388 The subscripted variables $a$ and~$b$ above the third premise are
389 eigenvariables, subject to the usual ``not free in \ldots'' proviso.
392 \section{Induction and coinduction rules}
393 Here we must consider inductive and coinductive definitions separately. For
394 an inductive definition, the package returns an induction rule derived
395 directly from the properties of least fixedpoints, as well as a modified rule
396 for mutual recursion. For a coinductive definition, the package returns a
397 basic coinduction rule.
399 \subsection{The basic induction rule}\label{basic-ind-sec}
400 The basic rule, called {\tt induct}, is appropriate in most situations.
401 For inductive definitions, it is strong rule induction~\cite{camilleri92}; for
402 datatype definitions (see below), it is just structural induction.
404 The induction rule for an inductively defined set~$R$ has the form described
405 below. For the time being, assume that $R$'s domain is not a Cartesian
406 product; inductively defined relations are treated slightly differently.
408 The major premise is $x\in R$. There is a minor premise for each
411 \item If the introduction rule concludes $t\in R_i$, then the minor premise
414 \item The minor premise's eigenvariables are precisely the introduction
415 rule's free variables that are not parameters of~$R$. For instance, the
416 eigenvariables in the $\Fin(A)$ rule below are $a$ and $b$, but not~$A$.
418 \item If the introduction rule has a premise $t\in R_i$, then the minor
419 premise discharges the assumption $t\in R_i$ and the induction
420 hypothesis~$P(t)$. If the introduction rule has a premise $t\in M(R_i)$
421 then the minor premise discharges the single assumption
422 \[ t\in M(\{z\in R_i. P(z)\}). \]
423 Because $M$ is monotonic, this assumption implies $t\in M(R_i)$. The
424 occurrence of $P$ gives the effect of an induction hypothesis, which may be
425 exploited by appealing to properties of~$M$.
427 The induction rule for $\Fin(A)$ resembles the elimination rule shown above,
428 but includes an induction hypothesis:
429 \[ \infer{P(x)}{x\in\Fin(A) & P(\emptyset)
430 & \infer*{P(\{a\}\un b)}{[a\in A & b\in\Fin(A) & P(b)]_{a,b}} }
432 Stronger induction rules often suggest themselves. We can derive a rule for
433 $\Fin(A)$ whose third premise discharges the extra assumption $a\not\in b$.
434 The package provides rules for mutual induction and inductive relations. The
435 Isabelle/\textsc{zf} theory also supports well-founded induction and recursion
436 over datatypes, by reasoning about the \defn{rank} of a
437 set~\cite[\S3.4]{paulson-set-II}.
440 \subsection{Modified induction rules}
442 If the domain of $R$ is a Cartesian product $A_1\times\cdots\times A_m$
443 (however nested), then the corresponding predicate $P_i$ takes $m$ arguments.
444 The major premise becomes $\pair{z_1,\ldots,z_m}\in R$ instead of $x\in R$;
445 the conclusion becomes $P(z_1,\ldots,z_m)$. This simplifies reasoning about
446 inductively defined relations, eliminating the need to express properties of
447 $z_1$, \ldots,~$z_m$ as properties of the tuple $\pair{z_1,\ldots,z_m}$.
448 Occasionally it may require you to split up the induction variable
449 using {\tt SigmaE} and {\tt dom\_subset}, especially if the constant {\tt
450 split} appears in the rule.
452 The mutual induction rule is called {\tt
453 mutual\_induct}. It differs from the basic rule in two respects:
455 \item Instead of a single predicate~$P$, it uses $n$ predicates $P_1$,
456 \ldots,~$P_n$: one for each recursive set.
458 \item There is no major premise such as $x\in R_i$. Instead, the conclusion
459 refers to all the recursive sets:
460 \[ (\forall z.z\in R_1\imp P_1(z))\conj\cdots\conj
461 (\forall z.z\in R_n\imp P_n(z))
463 Proving the premises establishes $P_i(z)$ for $z\in R_i$ and $i=1$,
467 If the domain of some $R_i$ is a Cartesian product, then the mutual induction
468 rule is modified accordingly. The predicates are made to take $m$ separate
469 arguments instead of a tuple, and the quantification in the conclusion is over
470 the separate variables $z_1$, \ldots, $z_m$.
472 \subsection{Coinduction}\label{coind-sec}
473 A coinductive definition yields a primitive coinduction rule, with no
474 refinements such as those for the induction rules. (Experience may suggest
475 refinements later.) Consider the codatatype of lazy lists as an example. For
476 suitable definitions of $\LNil$ and $\LCons$, lazy lists may be defined as the
477 greatest set consistent with the rules
478 \[ \LNil\in\llist(A) \qquad
479 \infer[(-)]{\LCons(a,l)\in\llist(A)}{a\in A & l\in\llist(A)}
481 The $(-)$ tag stresses that this is a coinductive definition. A suitable
482 domain for $\llist(A)$ is $\quniv(A)$; this set is closed under the variant
483 forms of sum and product that are used to represent non-well-founded data
484 structures (see~\S\ref{univ-sec}).
486 The package derives an {\tt unfold} theorem similar to that for $\Fin(A)$.
487 Then it proves the theorem {\tt coinduct}, which expresses that $\llist(A)$
488 is the greatest solution to this equation contained in $\quniv(A)$:
489 \[ \infer{x\in\llist(A)}{x\in X & X\sbs \quniv(A) &
491 \begin{array}[b]{r@{}l}
493 \bigl(\exists a\,l.\, & z=\LCons(a,l) \conj a\in A \conj{}\\
494 & l\in X\un\llist(A) \bigr)
495 \end{array} }{[z\in X]_z}}
497 This rule complements the introduction rules; it provides a means of showing
498 $x\in\llist(A)$ when $x$ is infinite. For instance, if $x=\LCons(0,x)$ then
499 applying the rule with $X=\{x\}$ proves $x\in\llist(\nat)$. (Here $\nat$
500 is the set of natural numbers.)
502 Having $X\un\llist(A)$ instead of simply $X$ in the third premise above
503 represents a slight strengthening of the greatest fixedpoint property. I
504 discuss several forms of coinduction rules elsewhere~\cite{paulson-coind}.
506 The clumsy form of the third premise makes the rule hard to use, especially in
507 large definitions. Probably a constant should be declared to abbreviate the
508 large disjunction, and rules derived to allow proving the separate disjuncts.
511 \section{Examples of inductive and coinductive definitions}\label{ind-eg-sec}
512 This section presents several examples from the literature: the finite
513 powerset operator, lists of $n$ elements, bisimulations on lazy lists, the
514 well-founded part of a relation, and the primitive recursive functions.
516 \subsection{The finite powerset operator}
517 This operator has been discussed extensively above. Here is the
518 corresponding invocation in an Isabelle theory file. Note that
519 $\cons(a,b)$ abbreviates $\{a\}\un b$ in Isabelle/\textsc{zf}.
524 domains "Fin(A)" <= "Pow(A)"
527 consI "[| a: A; b: Fin(A) |] ==> cons(a,b) : Fin(A)"
528 type_intrs "[empty_subsetI, cons_subsetI, PowI]"
529 type_elims "[make_elim PowD]"
532 Theory {\tt Finite} extends the parent theory {\tt Arith} by declaring the
533 unary function symbol~$\Fin$, which is defined inductively. Its domain is
534 specified as $\pow(A)$, where $A$ is the parameter appearing in the
535 introduction rules. For type-checking, we supply two introduction
537 \[ \emptyset\sbs A \qquad
538 \infer{\{a\}\un B\sbs C}{a\in C & B\sbs C}
540 A further introduction rule and an elimination rule express both
541 directions of the equivalence $A\in\pow(B)\bimp A\sbs B$. Type-checking
542 involves mostly introduction rules.
544 Like all Isabelle theory files, this one yields a structure containing the
545 new theory as an \textsc{ml} value. Structure {\tt Finite} also has a
546 substructure, called~{\tt Fin}. After declaring \hbox{\tt open Finite;} we
547 can refer to the $\Fin(A)$ introduction rules as the list {\tt Fin.intrs}
548 or individually as {\tt Fin.emptyI} and {\tt Fin.consI}. The induction
549 rule is {\tt Fin.induct}.
552 \subsection{Lists of $n$ elements}\label{listn-sec}
553 This has become a standard example of an inductive definition. Following
554 Paulin-Mohring~\cite{paulin-tlca}, we could attempt to define a new datatype
555 $\listn(A,n)$, for lists of length~$n$, as an $n$-indexed family of sets.
556 But her introduction rules
557 \[ \hbox{\tt Niln}\in\listn(A,0) \qquad
558 \infer{\hbox{\tt Consn}(n,a,l)\in\listn(A,\succ(n))}
559 {n\in\nat & a\in A & l\in\listn(A,n)}
561 are not acceptable to the inductive definition package:
562 $\listn$ occurs with three different parameter lists in the definition.
564 The Isabelle version of this example suggests a general treatment of
565 varying parameters. It uses the existing datatype definition of
566 $\lst(A)$, with constructors $\Nil$ and~$\Cons$, and incorporates the
567 parameter~$n$ into the inductive set itself. It defines $\listn(A)$ as a
568 relation consisting of pairs $\pair{n,l}$ such that $n\in\nat$
569 and~$l\in\lst(A)$ and $l$ has length~$n$. In fact, $\listn(A)$ is the
570 converse of the length function on~$\lst(A)$. The Isabelle/\textsc{zf} introduction
572 \[ \pair{0,\Nil}\in\listn(A) \qquad
573 \infer{\pair{\succ(n),\Cons(a,l)}\in\listn(A)}
574 {a\in A & \pair{n,l}\in\listn(A)}
576 The Isabelle theory file takes, as parent, the theory~{\tt List} of lists.
577 We declare the constant~$\listn$ and supply an inductive definition,
578 specifying the domain as $\nat\times\lst(A)$:
583 domains "listn(A)" <= "nat*list(A)"
585 NilI "<0,Nil>: listn(A)"
586 ConsI "[| a: A; <n,l>: listn(A) |] ==> <succ(n), Cons(a,l)>: listn(A)"
587 type_intrs "nat_typechecks @ list.intrs"
590 The type-checking rules include those for 0, $\succ$, $\Nil$ and $\Cons$.
591 Because $\listn(A)$ is a set of pairs, type-checking requires the
592 equivalence $\pair{a,b}\in A\times B \bimp a\in A \conj b\in B$. The
593 package always includes the rules for ordered pairs.
595 The package returns introduction, elimination and induction rules for
596 $\listn$. The basic induction rule, {\tt listn.induct}, is
597 \[ \infer{P(z_1,z_2)}{\pair{z_1,z_2}\in\listn(A) & P(0,\Nil) &
598 \infer*{P(\succ(n),\Cons(a,l))}
599 {[a\in A & \pair{n,l}\in\listn(A) & P(n,l)]_{a,l,n}}}
601 This rule lets the induction formula to be a
602 binary property of pairs, $P(n,l)$.
603 It is now a simple matter to prove theorems about $\listn(A)$, such as
604 \[ \forall l\in\lst(A). \pair{\length(l),\, l}\in\listn(A) \]
605 \[ \listn(A)``\{n\} = \{l\in\lst(A). \length(l)=n\} \]
606 This latter result --- here $r``X$ denotes the image of $X$ under $r$
607 --- asserts that the inductive definition agrees with the obvious notion of
610 A ``list of $n$ elements'' really is a list, namely an element of ~$\lst(A)$.
611 It is subject to list operators such as append (concatenation). For example,
612 a trivial induction on $\pair{m,l}\in\listn(A)$ yields
613 \[ \infer{\pair{m\mathbin{+} m',\, l@l'}\in\listn(A)}
614 {\pair{m,l}\in\listn(A) & \pair{m',l'}\in\listn(A)}
616 where $+$ denotes addition on the natural numbers and @ denotes append.
618 \subsection{Rule inversion: the function {\tt mk\_cases}}
619 The elimination rule, {\tt listn.elim}, is cumbersome:
620 \[ \infer{Q}{x\in\listn(A) &
621 \infer*{Q}{[x = \pair{0,\Nil}]} &
623 {\left[\begin{array}{l}
624 x = \pair{\succ(n),\Cons(a,l)} \\
626 \pair{n,l}\in\listn(A)
627 \end{array} \right]_{a,l,n}}}
629 The \textsc{ml} function {\tt listn.mk\_cases} generates simplified instances of
630 this rule. It works by freeness reasoning on the list constructors:
631 $\Cons(a,l)$ is injective in its two arguments and differs from~$\Nil$. If
632 $x$ is $\pair{i,\Nil}$ or $\pair{i,\Cons(a,l)}$ then {\tt listn.mk\_cases}
633 deduces the corresponding form of~$i$; this is called rule inversion.
634 Here is a sample session:
636 listn.mk_cases list.con_defs "<i,Nil> : listn(A)";
637 {\out "[| <?i, []> : listn(?A); ?i = 0 ==> ?Q |] ==> ?Q" : thm}
638 listn.mk_cases list.con_defs "<i,Cons(a,l)> : listn(A)";
639 {\out "[| <?i, Cons(?a, ?l)> : listn(?A);}
640 {\out !!n. [| ?a : ?A; <n, ?l> : listn(?A); ?i = succ(n) |] ==> ?Q }
641 {\out |] ==> ?Q" : thm}
643 Each of these rules has only two premises. In conventional notation, the
645 \[ \infer{Q}{\pair{i, \Cons(a,l)}\in\listn(A) &
647 {\left[\begin{array}{l}
648 a\in A \\ \pair{n,l}\in\listn(A) \\ i = \succ(n)
649 \end{array} \right]_{n}}}
651 The package also has built-in rules for freeness reasoning about $0$
652 and~$\succ$. So if $x$ is $\pair{0,l}$ or $\pair{\succ(i),l}$, then {\tt
653 listn.mk\_cases} can deduce the corresponding form of~$l$.
655 The function {\tt mk\_cases} is also useful with datatype definitions. The
656 instance from the definition of lists, namely {\tt list.mk\_cases}, can
657 prove that $\Cons(a,l)\in\lst(A)$ implies $a\in A $ and $l\in\lst(A)$:
658 \[ \infer{Q}{\Cons(a,l)\in\lst(A) &
659 & \infer*{Q}{[a\in A &l\in\lst(A)]} }
661 A typical use of {\tt mk\_cases} concerns inductive definitions of evaluation
662 relations. Then rule inversion yields case analysis on possible evaluations.
663 For example, Isabelle/\textsc{zf} includes a short proof of the
664 diamond property for parallel contraction on combinators. Ole Rasmussen used
665 {\tt mk\_cases} extensively in his development of the theory of
666 residuals~\cite{rasmussen95}.
669 \subsection{A coinductive definition: bisimulations on lazy lists}
670 This example anticipates the definition of the codatatype $\llist(A)$, which
671 consists of finite and infinite lists over~$A$. Its constructors are $\LNil$
672 and~$\LCons$, satisfying the introduction rules shown in~\S\ref{coind-sec}.
673 Because $\llist(A)$ is defined as a greatest fixedpoint and uses the variant
674 pairing and injection operators, it contains non-well-founded elements such as
675 solutions to $\LCons(a,l)=l$.
677 The next step in the development of lazy lists is to define a coinduction
678 principle for proving equalities. This is done by showing that the equality
679 relation on lazy lists is the greatest fixedpoint of some monotonic
680 operation. The usual approach~\cite{pitts94} is to define some notion of
681 bisimulation for lazy lists, define equivalence to be the greatest
682 bisimulation, and finally to prove that two lazy lists are equivalent if and
683 only if they are equal. The coinduction rule for equivalence then yields a
684 coinduction principle for equalities.
686 A binary relation $R$ on lazy lists is a \defn{bisimulation} provided $R\sbs
687 R^+$, where $R^+$ is the relation
688 \[ \{\pair{\LNil,\LNil}\} \un
689 \{\pair{\LCons(a,l),\LCons(a,l')} . a\in A \conj \pair{l,l'}\in R\}.
691 A pair of lazy lists are \defn{equivalent} if they belong to some
692 bisimulation. Equivalence can be coinductively defined as the greatest
693 fixedpoint for the introduction rules
694 \[ \pair{\LNil,\LNil} \in\lleq(A) \qquad
695 \infer[(-)]{\pair{\LCons(a,l),\LCons(a,l')} \in\lleq(A)}
696 {a\in A & \pair{l,l'}\in \lleq(A)}
698 To make this coinductive definition, the theory file includes (after the
699 declaration of $\llist(A)$) the following lines:
703 domains "lleq(A)" <= "llist(A) * llist(A)"
705 LNil "<LNil,LNil> : lleq(A)"
706 LCons "[| a:A; <l,l'>: lleq(A) |] ==> <LCons(a,l),LCons(a,l')>: lleq(A)"
707 type_intrs "llist.intrs"
709 The domain of $\lleq(A)$ is $\llist(A)\times\llist(A)$. The type-checking
710 rules include the introduction rules for $\llist(A)$, whose
711 declaration is discussed below (\S\ref{lists-sec}).
713 The package returns the introduction rules and the elimination rule, as
714 usual. But instead of induction rules, it returns a coinduction rule.
715 The rule is too big to display in the usual notation; its conclusion is
716 $x\in\lleq(A)$ and its premises are $x\in X$,
717 ${X\sbs\llist(A)\times\llist(A)}$ and
718 \[ \infer*{z=\pair{\LNil,\LNil}\disj \bigl(\exists a\,l\,l'.\,
719 \begin{array}[t]{@{}l}
720 z=\pair{\LCons(a,l),\LCons(a,l')} \conj a\in A \conj{}\\
721 \pair{l,l'}\in X\un\lleq(A) \bigr)
725 Thus if $x\in X$, where $X$ is a bisimulation contained in the
726 domain of $\lleq(A)$, then $x\in\lleq(A)$. It is easy to show that
727 $\lleq(A)$ is reflexive: the equality relation is a bisimulation. And
728 $\lleq(A)$ is symmetric: its converse is a bisimulation. But showing that
729 $\lleq(A)$ coincides with the equality relation takes some work.
731 \subsection{The accessible part of a relation}\label{acc-sec}
732 Let $\prec$ be a binary relation on~$D$; in short, $(\prec)\sbs D\times D$.
733 The \defn{accessible} or \defn{well-founded} part of~$\prec$, written
734 $\acc(\prec)$, is essentially that subset of~$D$ for which $\prec$ admits
735 no infinite decreasing chains~\cite{aczel77}. Formally, $\acc(\prec)$ is
736 inductively defined to be the least set that contains $a$ if it contains
737 all $\prec$-predecessors of~$a$, for $a\in D$. Thus we need an
738 introduction rule of the form
739 \[ \infer{a\in\acc(\prec)}{\forall y.y\prec a\imp y\in\acc(\prec)} \]
740 Paulin-Mohring treats this example in Coq~\cite{paulin-tlca}, but it causes
741 difficulties for other systems. Its premise is not acceptable to the
742 inductive definition package of the Cambridge \textsc{hol}
743 system~\cite{camilleri92}. It is also unacceptable to the Isabelle package
744 (recall \S\ref{intro-sec}), but fortunately can be transformed into the
745 acceptable form $t\in M(R)$.
747 The powerset operator is monotonic, and $t\in\pow(R)$ is equivalent to
748 $t\sbs R$. This in turn is equivalent to $\forall y\in t. y\in R$. To
749 express $\forall y.y\prec a\imp y\in\acc(\prec)$ we need only find a
750 term~$t$ such that $y\in t$ if and only if $y\prec a$. A suitable $t$ is
751 the inverse image of~$\{a\}$ under~$\prec$.
753 The definition below follows this approach. Here $r$ is~$\prec$ and
754 $\field(r)$ refers to~$D$, the domain of $\acc(r)$. (The field of a
755 relation is the union of its domain and range.) Finally $r^{-}``\{a\}$
756 denotes the inverse image of~$\{a\}$ under~$r$. We supply the theorem {\tt
757 Pow\_mono}, which asserts that $\pow$ is monotonic.
761 domains "acc(r)" <= "field(r)"
763 vimage "[| r-``\{a\}: Pow(acc(r)); a: field(r) |] ==> a: acc(r)"
766 The Isabelle theory proceeds to prove facts about $\acc(\prec)$. For
767 instance, $\prec$ is well-founded if and only if its field is contained in
770 As mentioned in~\S\ref{basic-ind-sec}, a premise of the form $t\in M(R)$
771 gives rise to an unusual induction hypothesis. Let us examine the
772 induction rule, {\tt acc.induct}:
773 \[ \infer{P(x)}{x\in\acc(r) &
776 r^{-}``\{a\} &\, \in\pow(\{z\in\acc(r).P(z)\}) \\
781 The strange induction hypothesis is equivalent to
782 $\forall y. \pair{y,a}\in r\imp y\in\acc(r)\conj P(y)$.
783 Therefore the rule expresses well-founded induction on the accessible part
786 The use of inverse image is not essential. The Isabelle package can accept
787 introduction rules with arbitrary premises of the form $\forall
788 \vec{y}.P(\vec{y})\imp f(\vec{y})\in R$. The premise can be expressed
790 \[ \{z\in D. P(\vec{y}) \conj z=f(\vec{y})\} \in \pow(R) \]
791 provided $f(\vec{y})\in D$ for all $\vec{y}$ such that~$P(\vec{y})$. The
792 following section demonstrates another use of the premise $t\in M(R)$,
795 \subsection{The primitive recursive functions}\label{primrec-sec}
796 The primitive recursive functions are traditionally defined inductively, as
797 a subset of the functions over the natural numbers. One difficulty is that
798 functions of all arities are taken together, but this is easily
799 circumvented by regarding them as functions on lists. Another difficulty,
800 the notion of composition, is less easily circumvented.
802 Here is a more precise definition. Letting $\vec{x}$ abbreviate
803 $x_0,\ldots,x_{n-1}$, we can write lists such as $[\vec{x}]$,
804 $[y+1,\vec{x}]$, etc. A function is \defn{primitive recursive} if it
805 belongs to the least set of functions in $\lst(\nat)\to\nat$ containing
807 \item The \defn{successor} function $\SC$, such that $\SC[y,\vec{x}]=y+1$.
808 \item All \defn{constant} functions $\CONST(k)$, such that
809 $\CONST(k)[\vec{x}]=k$.
810 \item All \defn{projection} functions $\PROJ(i)$, such that
811 $\PROJ(i)[\vec{x}]=x_i$ if $0\leq i<n$.
812 \item All \defn{compositions} $\COMP(g,[f_0,\ldots,f_{m-1}])$,
813 where $g$ and $f_0$, \ldots, $f_{m-1}$ are primitive recursive,
815 \[ \COMP(g,[f_0,\ldots,f_{m-1}])[\vec{x}] =
816 g[f_0[\vec{x}],\ldots,f_{m-1}[\vec{x}]]. \]
818 \item All \defn{recursions} $\PREC(f,g)$, where $f$ and $g$ are primitive
821 \PREC(f,g)[0,\vec{x}] & = & f[\vec{x}] \\
822 \PREC(f,g)[y+1,\vec{x}] & = & g[\PREC(f,g)[y,\vec{x}],\, y,\, \vec{x}].
825 Composition is awkward because it combines not two functions, as is usual,
826 but $m+1$ functions. In her proof that Ackermann's function is not
827 primitive recursive, Nora Szasz was unable to formalize this definition
828 directly~\cite{szasz93}. So she generalized primitive recursion to
829 tuple-valued functions. This modified the inductive definition such that
830 each operation on primitive recursive functions combined just two functions.
840 SC_def "SC == lam l:list(nat).list_case(0, \%x xs.succ(x), l)"
843 domains "primrec" <= "list(nat)->nat"
846 CONST "k: nat ==> CONST(k) : primrec"
847 PROJ "i: nat ==> PROJ(i) : primrec"
848 COMP "[| g: primrec; fs: list(primrec) |] ==> COMP(g,fs): primrec"
849 PREC "[| f: primrec; g: primrec |] ==> PREC(f,g): primrec"
851 con_defs "[SC_def,CONST_def,PROJ_def,COMP_def,PREC_def]"
852 type_intrs "nat_typechecks @ list.intrs @
853 [lam_type, list_case_type, drop_type, map_type,
854 apply_type, rec_type]"
858 \caption{Inductive definition of the primitive recursive functions}
863 Szasz was using \textsc{alf}, but Coq and \textsc{hol} would also have
864 problems accepting this definition. Isabelle's package accepts it easily
865 since $[f_0,\ldots,f_{m-1}]$ is a list of primitive recursive functions and
866 $\lst$ is monotonic. There are five introduction rules, one for each of the
867 five forms of primitive recursive function. Let us examine the one for
869 \[ \infer{\COMP(g,\fs)\in\primrec}{g\in\primrec & \fs\in\lst(\primrec)} \]
870 The induction rule for $\primrec$ has one case for each introduction rule.
871 Due to the use of $\lst$ as a monotone operator, the composition case has
872 an unusual induction hypothesis:
873 \[ \infer*{P(\COMP(g,\fs))}
874 {[g\in\primrec & \fs\in\lst(\{z\in\primrec.P(z)\})]_{\fs,g}}
876 The hypothesis states that $\fs$ is a list of primitive recursive functions,
877 each satisfying the induction formula. Proving the $\COMP$ case typically
878 requires structural induction on lists, yielding two subcases: either
879 $\fs=\Nil$ or else $\fs=\Cons(f,\fs')$, where $f\in\primrec$, $P(f)$, and
880 $\fs'$ is another list of primitive recursive functions satisfying~$P$.
882 Figure~\ref{primrec-fig} presents the theory file. Theory {\tt Primrec}
883 defines the constants $\SC$, $\CONST$, etc. These are not constructors of
884 a new datatype, but functions over lists of numbers. Their definitions,
885 most of which are omitted, consist of routine list programming. In
886 Isabelle/\textsc{zf}, the primitive recursive functions are defined as a subset of
887 the function set $\lst(\nat)\to\nat$.
889 The Isabelle theory goes on to formalize Ackermann's function and prove
890 that it is not primitive recursive, using the induction rule {\tt
891 primrec.induct}. The proof follows Szasz's excellent account.
894 \section{Datatypes and codatatypes}\label{data-sec}
895 A (co)datatype definition is a (co)inductive definition with automatically
896 defined constructors and a case analysis operator. The package proves that
897 the case operator inverts the constructors and can prove freeness theorems
898 involving any pair of constructors.
901 \subsection{Constructors and their domain}\label{univ-sec}
902 A (co)inductive definition selects a subset of an existing set; a (co)datatype
903 definition creates a new set. The package reduces the latter to the former.
904 Isabelle/\textsc{zf} supplies sets having strong closure properties to serve
905 as domains for (co)inductive definitions.
907 Isabelle/\textsc{zf} defines the Cartesian product $A\times
908 B$, containing ordered pairs $\pair{a,b}$; it also defines the
909 disjoint sum $A+B$, containing injections $\Inl(a)\equiv\pair{0,a}$ and
910 $\Inr(b)\equiv\pair{1,b}$. For use below, define the $m$-tuple
911 $\pair{x_1,\ldots,x_m}$ to be the empty set~$\emptyset$ if $m=0$, simply $x_1$
912 if $m=1$ and $\pair{x_1,\pair{x_2,\ldots,x_m}}$ if $m\geq2$.
914 A datatype constructor $\Con(x_1,\ldots,x_m)$ is defined to be
915 $h(\pair{x_1,\ldots,x_m})$, where $h$ is composed of $\Inl$ and~$\Inr$.
916 In a mutually recursive definition, all constructors for the set~$R_i$ have
917 the outer form~$h_{in}$, where $h_{in}$ is the injection described
918 in~\S\ref{mutual-sec}. Further nested injections ensure that the
919 constructors for~$R_i$ are pairwise distinct.
921 Isabelle/\textsc{zf} defines the set $\univ(A)$, which contains~$A$ and
922 furthermore contains $\pair{a,b}$, $\Inl(a)$ and $\Inr(b)$ for $a$,
923 $b\in\univ(A)$. In a typical datatype definition with set parameters
924 $A_1$, \ldots, $A_k$, a suitable domain for all the recursive sets is
925 $\univ(A_1\un\cdots\un A_k)$. This solves the problem for
926 datatypes~\cite[\S4.2]{paulson-set-II}.
928 The standard pairs and injections can only yield well-founded
929 constructions. This eases the (manual!) definition of recursive functions
930 over datatypes. But they are unsuitable for codatatypes, which typically
931 contain non-well-founded objects.
933 To support codatatypes, Isabelle/\textsc{zf} defines a variant notion of
934 ordered pair, written~$\pair{a;b}$. It also defines the corresponding variant
935 notion of Cartesian product $A\otimes B$, variant injections $\QInl(a)$
936 and~$\QInr(b)$ and variant disjoint sum $A\oplus B$. Finally it defines the
937 set $\quniv(A)$, which contains~$A$ and furthermore contains $\pair{a;b}$,
938 $\QInl(a)$ and $\QInr(b)$ for $a$, $b\in\quniv(A)$. In a typical codatatype
939 definition with set parameters $A_1$, \ldots, $A_k$, a suitable domain is
940 $\quniv(A_1\un\cdots\un A_k)$. Details are published
941 elsewhere~\cite{paulson-final}.
943 \subsection{The case analysis operator}
944 The (co)datatype package automatically defines a case analysis operator,
945 called {\tt$R$\_case}. A mutually recursive definition still has only one
946 operator, whose name combines those of the recursive sets: it is called
947 {\tt$R_1$\_\ldots\_$R_n$\_case}. The case operator is analogous to those
948 for products and sums.
950 Datatype definitions employ standard products and sums, whose operators are
951 $\split$ and $\case$ and satisfy the equations
953 \split(f,\pair{x,y}) & = & f(x,y) \\
954 \case(f,g,\Inl(x)) & = & f(x) \\
955 \case(f,g,\Inr(y)) & = & g(y)
957 Suppose the datatype has $k$ constructors $\Con_1$, \ldots,~$\Con_k$. Then
958 its case operator takes $k+1$ arguments and satisfies an equation for each
960 \[ R\hbox{\_case}(f_1,\ldots,f_k, {\tt Con}_i(\vec{x})) = f_i(\vec{x}),
961 \qquad i = 1, \ldots, k
963 The case operator's definition takes advantage of Isabelle's representation of
964 syntax in the typed $\lambda$-calculus; it could readily be adapted to a
965 theorem prover for higher-order logic. If $f$ and~$g$ have meta-type $i\To i$
966 then so do $\split(f)$ and $\case(f,g)$. This works because $\split$ and
967 $\case$ operate on their last argument. They are easily combined to make
968 complex case analysis operators. For example, $\case(f,\case(g,h))$ performs
969 case analysis for $A+(B+C)$; let us verify one of the three equations:
970 \[ \case(f,\case(g,h), \Inr(\Inl(b))) = \case(g,h,\Inl(b)) = g(b) \]
971 Codatatype definitions are treated in precisely the same way. They express
972 case operators using those for the variant products and sums, namely
973 $\qsplit$ and~$\qcase$.
977 To see how constructors and the case analysis operator are defined, let us
978 examine some examples. Further details are available
979 elsewhere~\cite{paulson-set-II}.
982 \subsection{Example: lists and lazy lists}\label{lists-sec}
983 Here is a declaration of the datatype of lists, as it might appear in a theory
987 datatype "list(A)" = Nil | Cons ("a:A", "l: list(A)")
989 And here is a declaration of the codatatype of lazy lists:
992 codatatype "llist(A)" = LNil | LCons ("a: A", "l: llist(A)")
995 Each form of list has two constructors, one for the empty list and one for
996 adding an element to a list. Each takes a parameter, defining the set of
997 lists over a given set~$A$. Each is automatically given the appropriate
998 domain: $\univ(A)$ for $\lst(A)$ and $\quniv(A)$ for $\llist(A)$. The default
1002 Now $\lst(A)$ is a datatype and enjoys the usual induction rule.
1004 Since $\lst(A)$ is a datatype, it enjoys a structural induction rule, {\tt
1006 \[ \infer{P(x)}{x\in\lst(A) & P(\Nil)
1007 & \infer*{P(\Cons(a,l))}{[a\in A & l\in\lst(A) & P(l)]_{a,l}} }
1009 Induction and freeness yield the law $l\not=\Cons(a,l)$. To strengthen this,
1010 Isabelle/\textsc{zf} defines the rank of a set and proves that the standard
1011 pairs and injections have greater rank than their components. An immediate
1012 consequence, which justifies structural recursion on lists
1013 \cite[\S4.3]{paulson-set-II}, is
1014 \[ \rank(l) < \rank(\Cons(a,l)). \]
1017 But $\llist(A)$ is a codatatype and has no induction rule. Instead it has
1018 the coinduction rule shown in \S\ref{coind-sec}. Since variant pairs and
1019 injections are monotonic and need not have greater rank than their
1020 components, fixedpoint operators can create cyclic constructions. For
1021 example, the definition
1022 \[ \lconst(a) \equiv \lfp(\univ(a), \lambda l. \LCons(a,l)) \]
1023 yields $\lconst(a) = \LCons(a,\lconst(a))$.
1026 \typeout{****SHORT VERSION}
1027 \typeout{****Omitting discussion of constructors!}
1030 It may be instructive to examine the definitions of the constructors and
1031 case operator for $\lst(A)$. The definitions for $\llist(A)$ are similar.
1032 The list constructors are defined as follows:
1034 \Nil & \equiv & \Inl(\emptyset) \\
1035 \Cons(a,l) & \equiv & \Inr(\pair{a,l})
1037 The operator $\lstcase$ performs case analysis on these two alternatives:
1038 \[ \lstcase(c,h) \equiv \case(\lambda u.c, \split(h)) \]
1039 Let us verify the two equations:
1041 \lstcase(c, h, \Nil) & = &
1042 \case(\lambda u.c, \split(h), \Inl(\emptyset)) \\
1043 & = & (\lambda u.c)(\emptyset) \\
1045 \lstcase(c, h, \Cons(x,y)) & = &
1046 \case(\lambda u.c, \split(h), \Inr(\pair{x,y})) \\
1047 & = & \split(h, \pair{x,y}) \\
1054 \typeout{****Omitting mutual recursion example!}
1056 \subsection{Example: mutual recursion}
1057 In mutually recursive trees and forests~\cite[\S4.5]{paulson-set-II}, trees
1058 have the one constructor $\Tcons$, while forests have the two constructors
1059 $\Fnil$ and~$\Fcons$:
1061 consts tree, forest, tree_forest :: i=>i
1062 datatype "tree(A)" = Tcons ("a: A", "f: forest(A)")
1063 and "forest(A)" = Fnil | Fcons ("t: tree(A)", "f: forest(A)")
1065 The three introduction rules define the mutual recursion. The
1066 distinguishing feature of this example is its two induction rules.
1068 The basic induction rule is called {\tt tree\_forest.induct}:
1069 \[ \infer{P(x)}{x\in\TF(A) &
1070 \infer*{P(\Tcons(a,f))}
1071 {\left[\begin{array}{l} a\in A \\
1072 f\in\forest(A) \\ P(f)
1076 & \infer*{P(\Fcons(t,f))}
1077 {\left[\begin{array}{l} t\in\tree(A) \\ P(t) \\
1078 f\in\forest(A) \\ P(f)
1082 This rule establishes a single predicate for $\TF(A)$, the union of the
1083 recursive sets. Although such reasoning is sometimes useful
1084 \cite[\S4.5]{paulson-set-II}, a proper mutual induction rule should establish
1085 separate predicates for $\tree(A)$ and $\forest(A)$. The package calls this
1086 rule {\tt tree\_forest.mutual\_induct}. Observe the usage of $P$ and $Q$ in
1087 the induction hypotheses:
1088 \[ \infer{(\forall z. z\in\tree(A)\imp P(z)) \conj
1089 (\forall z. z\in\forest(A)\imp Q(z))}
1090 {\infer*{P(\Tcons(a,f))}
1091 {\left[\begin{array}{l} a\in A \\
1092 f\in\forest(A) \\ Q(f)
1096 & \infer*{Q(\Fcons(t,f))}
1097 {\left[\begin{array}{l} t\in\tree(A) \\ P(t) \\
1098 f\in\forest(A) \\ Q(f)
1102 Elsewhere I describe how to define mutually recursive functions over trees and
1103 forests \cite[\S4.5]{paulson-set-II}.
1105 Both forest constructors have the form $\Inr(\cdots)$,
1106 while the tree constructor has the form $\Inl(\cdots)$. This pattern would
1107 hold regardless of how many tree or forest constructors there were.
1109 \Tcons(a,l) & \equiv & \Inl(\pair{a,l}) \\
1110 \Fnil & \equiv & \Inr(\Inl(\emptyset)) \\
1111 \Fcons(a,l) & \equiv & \Inr(\Inr(\pair{a,l}))
1113 There is only one case operator; it works on the union of the trees and
1115 \[ {\tt tree\_forest\_case}(f,c,g) \equiv
1116 \case(\split(f),\, \case(\lambda u.c, \split(g)))
1121 \subsection{Example: a four-constructor datatype}
1122 A bigger datatype will illustrate some efficiency
1123 refinements. It has four constructors $\Con_0$, \ldots, $\Con_3$, with the
1124 corresponding arities.
1126 consts data :: [i,i] => i
1127 datatype "data(A,B)" = Con0
1129 | Con2 ("a: A", "b: B")
1130 | Con3 ("a: A", "b: B", "d: data(A,B)")
1132 Because this datatype has two set parameters, $A$ and~$B$, the package
1133 automatically supplies $\univ(A\un B)$ as its domain. The structural
1134 induction rule has four minor premises, one per constructor, and only the last
1135 has an induction hypothesis. (Details are left to the reader.)
1137 The constructors are defined by the equations
1139 \Con_0 & \equiv & \Inl(\Inl(\emptyset)) \\
1140 \Con_1(a) & \equiv & \Inl(\Inr(a)) \\
1141 \Con_2(a,b) & \equiv & \Inr(\Inl(\pair{a,b})) \\
1142 \Con_3(a,b,c) & \equiv & \Inr(\Inr(\pair{a,b,c})).
1144 The case analysis operator is
1145 \[ {\tt data\_case}(f_0,f_1,f_2,f_3) \equiv
1146 \case(\begin{array}[t]{@{}l}
1147 \case(\lambda u.f_0,\; f_1),\, \\
1148 \case(\split(f_2),\; \split(\lambda v.\split(f_3(v)))) )
1151 This may look cryptic, but the case equations are trivial to verify.
1153 In the constructor definitions, the injections are balanced. A more naive
1154 approach is to define $\Con_3(a,b,c)$ as $\Inr(\Inr(\Inr(\pair{a,b,c})))$;
1155 instead, each constructor has two injections. The difference here is small.
1156 But the \textsc{zf} examples include a 60-element enumeration type, where each
1157 constructor has 5 or~6 injections. The naive approach would require 1 to~59
1158 injections; the definitions would be quadratic in size. It is like the
1159 advantage of binary notation over unary.
1161 The result structure contains the case operator and constructor definitions as
1162 the theorem list \verb|con_defs|. It contains the case equations, such as
1163 \[ {\tt data\_case}(f_0,f_1,f_2,f_3,\Con_3(a,b,c)) = f_3(a,b,c), \]
1164 as the theorem list \verb|case_eqns|. There is one equation per constructor.
1166 \subsection{Proving freeness theorems}
1167 There are two kinds of freeness theorems:
1169 \item \defn{injectiveness} theorems, such as
1170 \[ \Con_2(a,b) = \Con_2(a',b') \bimp a=a' \conj b=b' \]
1172 \item \defn{distinctness} theorems, such as
1173 \[ \Con_1(a) \not= \Con_2(a',b') \]
1175 Since the number of such theorems is quadratic in the number of constructors,
1176 the package does not attempt to prove them all. Instead it returns tools for
1177 proving desired theorems --- either manually or during
1178 simplification or classical reasoning.
1180 The theorem list \verb|free_iffs| enables the simplifier to perform freeness
1181 reasoning. This works by incremental unfolding of constructors that appear in
1182 equations. The theorem list contains logical equivalences such as
1184 \Con_0=c & \bimp & c=\Inl(\Inl(\emptyset)) \\
1185 \Con_1(a)=c & \bimp & c=\Inl(\Inr(a)) \\
1187 \Inl(a)=\Inl(b) & \bimp & a=b \\
1188 \Inl(a)=\Inr(b) & \bimp & {\tt False} \\
1189 \pair{a,b} = \pair{a',b'} & \bimp & a=a' \conj b=b'
1191 For example, these rewrite $\Con_1(a)=\Con_1(b)$ to $a=b$ in four steps.
1193 The theorem list \verb|free_SEs| enables the classical
1194 reasoner to perform similar replacements. It consists of elimination rules
1195 to replace $\Con_0=c$ by $c=\Inl(\Inl(\emptyset))$ and so forth, in the
1198 Such incremental unfolding combines freeness reasoning with other proof
1199 steps. It has the unfortunate side-effect of unfolding definitions of
1200 constructors in contexts such as $\exists x.\Con_1(a)=x$, where they should
1201 be left alone. Calling the Isabelle tactic {\tt fold\_tac con\_defs}
1202 restores the defined constants.
1205 \section{Related work}\label{related}
1206 The use of least fixedpoints to express inductive definitions seems
1207 obvious. Why, then, has this technique so seldom been implemented?
1209 Most automated logics can only express inductive definitions by asserting
1210 axioms. Little would be left of Boyer and Moore's logic~\cite{bm79} if their
1211 shell principle were removed. With \textsc{alf} the situation is more
1212 complex; earlier versions of Martin-L\"of's type theory could (using
1213 wellordering types) express datatype definitions, but the version underlying
1214 \textsc{alf} requires new rules for each definition~\cite{dybjer91}. With Coq
1215 the situation is subtler still; its underlying Calculus of Constructions can
1216 express inductive definitions~\cite{huet88}, but cannot quite handle datatype
1217 definitions~\cite{paulin-tlca}. It seems that researchers tried hard to
1218 circumvent these problems before finally extending the Calculus with rule
1219 schemes for strictly positive operators. Recently Gim{\'e}nez has extended
1220 the Calculus of Constructions with inductive and coinductive
1221 types~\cite{gimenez-codifying}, with mechanized support in Coq.
1223 Higher-order logic can express inductive definitions through quantification
1224 over unary predicates. The following formula expresses that~$i$ belongs to the
1225 least set containing~0 and closed under~$\succ$:
1226 \[ \forall P. P(0)\conj (\forall x.P(x)\imp P(\succ(x))) \imp P(i) \]
1227 This technique can be used to prove the Knaster-Tarski theorem, which (in its
1228 general form) is little used in the Cambridge \textsc{hol} system.
1229 Melham~\cite{melham89} describes the development. The natural numbers are
1230 defined as shown above, but lists are defined as functions over the natural
1231 numbers. Unlabelled trees are defined using G\"odel numbering; a labelled
1232 tree consists of an unlabelled tree paired with a list of labels. Melham's
1233 datatype package expresses the user's datatypes in terms of labelled trees.
1234 It has been highly successful, but a fixedpoint approach might have yielded
1235 greater power with less effort.
1237 Elsa Gunter~\cite{gunter-trees} reports an ongoing project to generalize the
1238 Cambridge \textsc{hol} system with mutual recursion and infinitely-branching
1239 trees. She retains many features of Melham's approach.
1241 Melham's inductive definition package~\cite{camilleri92} also uses
1242 quantification over predicates. But instead of formalizing the notion of
1243 monotone function, it requires definitions to consist of finitary rules, a
1244 syntactic form that excludes many monotone inductive definitions.
1246 \textsc{pvs}~\cite{pvs-language} is another proof assistant based on
1247 higher-order logic. It supports both inductive definitions and datatypes,
1248 apparently by asserting axioms. Datatypes may not be iterated in general, but
1249 may use recursion over the built-in $\lst$ type.
1251 The earliest use of least fixedpoints is probably Robin Milner's. Brian
1252 Monahan extended this package considerably~\cite{monahan84}, as did I in
1253 unpublished work.\footnote{The datatype package described in my \textsc{lcf}
1254 book~\cite{paulson87} does {\it not\/} make definitions, but merely asserts
1255 axioms.} \textsc{lcf} is a first-order logic of domain theory; the relevant
1256 fixedpoint theorem is not Knaster-Tarski but concerns fixedpoints of
1257 continuous functions over domains. \textsc{lcf} is too weak to express
1258 recursive predicates. The Isabelle package might be the first to be based on
1259 the Knaster-Tarski theorem.
1262 \section{Conclusions and future work}
1263 Higher-order logic and set theory are both powerful enough to express
1264 inductive definitions. A growing number of theorem provers implement one
1265 of these~\cite{IMPS,saaltink-fme}. The easiest sort of inductive
1266 definition package to write is one that asserts new axioms, not one that
1267 makes definitions and proves theorems about them. But asserting axioms
1268 could introduce unsoundness.
1270 The fixedpoint approach makes it fairly easy to implement a package for
1271 (co)in\-duc\-tive definitions that does not assert axioms. It is efficient:
1272 it processes most definitions in seconds and even a 60-constructor datatype
1273 requires only a few minutes. It is also simple: The first working version took
1274 under a week to code, consisting of under 1100 lines (35K bytes) of Standard
1277 In set theory, care is needed to ensure that the inductive definition yields
1278 a set (rather than a proper class). This problem is inherent to set theory,
1279 whether or not the Knaster-Tarski theorem is employed. We must exhibit a
1280 bounding set (called a domain above). For inductive definitions, this is
1281 often trivial. For datatype definitions, I have had to formalize much set
1282 theory. To justify infinitely-branching datatype definitions, I have had to
1283 develop a theory of cardinal arithmetic~\cite{paulson-gr}, such as the theorem
1284 that if $\kappa$ is an infinite cardinal and $|X(\alpha)| \le \kappa$ for all
1285 $\alpha<\kappa$ then $|\union\sb{\alpha<\kappa} X(\alpha)| \le \kappa$.
1286 The need for such efforts is not a drawback of the fixedpoint approach, for
1287 the alternative is to take such definitions on faith.
1289 Care is also needed to ensure that the greatest fixedpoint really yields a
1290 coinductive definition. In set theory, standard pairs admit only well-founded
1291 constructions. Aczel's anti-foundation axiom~\cite{aczel88} could be used to
1292 get non-well-founded objects, but it does not seem easy to mechanize.
1293 Isabelle/\textsc{zf} instead uses a variant notion of ordered pairing, which
1294 can be generalized to a variant notion of function. Elsewhere I have
1295 proved that this simple approach works (yielding final coalgebras) for a broad
1296 class of definitions~\cite{paulson-final}.
1298 Several large studies make heavy use of inductive definitions. L\"otzbeyer
1299 and Sandner have formalized two chapters of a semantics book~\cite{winskel93},
1300 proving the equivalence between the operational and denotational semantics of
1301 a simple imperative language. A single theory file contains three datatype
1302 definitions (of arithmetic expressions, boolean expressions and commands) and
1303 three inductive definitions (the corresponding operational rules). Using
1304 different techniques, Nipkow~\cite{nipkow-CR} and Rasmussen~\cite{rasmussen95}
1305 have both proved the Church-Rosser theorem; inductive definitions specify
1306 several reduction relations on $\lambda$-terms. Recently, I have applied
1307 inductive definitions to the analysis of cryptographic
1308 protocols~\cite{paulson-markt}.
1310 To demonstrate coinductive definitions, Frost~\cite{frost95} has proved the
1311 consistency of the dynamic and static semantics for a small functional
1312 language. The example is due to Milner and Tofte~\cite{milner-coind}. It
1313 concerns an extended correspondence relation, which is defined coinductively.
1314 A codatatype definition specifies values and value environments in mutual
1315 recursion. Non-well-founded values represent recursive functions. Value
1316 environments are variant functions from variables into values. This one key
1317 definition uses most of the package's novel features.
1319 The approach is not restricted to set theory. It should be suitable for any
1320 logic that has some notion of set and the Knaster-Tarski theorem. I have
1321 ported the (co)inductive definition package from Isabelle/\textsc{zf} to
1322 Isabelle/\textsc{hol} (higher-order logic). V\"olker~\cite{voelker95}
1323 is investigating how to port the (co)datatype package. \textsc{hol}
1324 represents sets by unary predicates; defining the corresponding types may
1325 cause complications.
1328 \begin{footnotesize}
1329 \bibliographystyle{springer}
1330 \bibliography{string-abbrv,atp,theory,funprog,isabelle,crossref}
1334 \ifshort\typeout{****Omitting appendices}
1338 \section{Inductive and coinductive definitions: users guide}
1339 A theory file may contain any number of inductive and coinductive
1340 definitions. They may be intermixed with other declarations; in
1341 particular, the (co)inductive sets \defn{must} be declared separately as
1342 constants, and may have mixfix syntax or be subject to syntax translations.
1344 The syntax is rather complicated. Please consult the examples above and the
1345 theory files on the \textsc{zf} source directory.
1347 Each (co)inductive definition adds definitions to the theory and also proves
1348 some theorems. Each definition creates an \textsc{ml} structure, which is a
1349 substructure of the main theory structure.
1351 Inductive and datatype definitions can take up considerable storage. The
1352 introduction rules are replicated in slightly different forms as fixedpoint
1353 definitions, elimination rules and induction rules. L\"otzbeyer and Sandner's
1354 six definitions occupy over 600K in total. Defining the 60-constructor
1355 datatype requires nearly 560K\@.
1357 \subsection{The result structure}
1358 Many of the result structure's components have been discussed
1359 in~\S\ref{basic-sec}; others are self-explanatory.
1361 \item[\tt thy] is the new theory containing the recursive sets.
1363 \item[\tt defs] is the list of definitions of the recursive sets.
1365 \item[\tt bnd\_mono] is a monotonicity theorem for the fixedpoint operator.
1367 \item[\tt dom\_subset] is a theorem stating inclusion in the domain.
1369 \item[\tt intrs] is the list of introduction rules, now proved as theorems, for
1370 the recursive sets. The rules are also available individually, using the
1371 names given them in the theory file.
1373 \item[\tt elim] is the elimination rule.
1375 \item[\tt mk\_cases] is a function to create simplified instances of {\tt
1376 elim}, using freeness reasoning on some underlying datatype.
1379 For an inductive definition, the result structure contains two induction
1380 rules, {\tt induct} and \verb|mutual_induct|. (To save storage, the latter
1381 rule is just {\tt True} unless more than one set is being defined.) For a
1382 coinductive definition, it contains the rule \verb|coinduct|.
1384 Figure~\ref{def-result-fig} summarizes the two result signatures,
1385 specifying the types of all these components.
1393 val dom_subset : thm
1394 val intrs : thm list
1396 val mk_cases : thm list -> string -> thm
1397 {\it(Inductive definitions only)}
1399 val mutual_induct: thm
1400 {\it(Coinductive definitions only)}
1405 \caption{The result of a (co)inductive definition} \label{def-result-fig}
1408 \subsection{The syntax of a (co)inductive definition}
1409 An inductive definition has the form
1412 domains {\it domain declarations}
1413 intrs {\it introduction rules}
1414 monos {\it monotonicity theorems}
1415 con_defs {\it constructor definitions}
1416 type_intrs {\it introduction rules for type-checking}
1417 type_elims {\it elimination rules for type-checking}
1419 A coinductive definition is identical, but starts with the keyword
1422 The {\tt monos}, {\tt con\_defs}, {\tt type\_intrs} and {\tt type\_elims}
1423 sections are optional. If present, each is specified as a string, which
1424 must be a valid \textsc{ml} expression of type {\tt thm list}. It is simply
1425 inserted into the {\tt .thy.ML} file; if it is ill-formed, it will trigger
1426 \textsc{ml} error messages. You can then inspect the file on your directory.
1429 \item[\it domain declarations] consist of one or more items of the form
1430 {\it string\/}~{\tt <=}~{\it string}, associating each recursive set with
1433 \item[\it introduction rules] specify one or more introduction rules in
1434 the form {\it ident\/}~{\it string}, where the identifier gives the name of
1435 the rule in the result structure.
1437 \item[\it monotonicity theorems] are required for each operator applied to
1438 a recursive set in the introduction rules. There \defn{must} be a theorem
1439 of the form $A\sbs B\Imp M(A)\sbs M(B)$, for each premise $t\in M(R_i)$
1440 in an introduction rule!
1442 \item[\it constructor definitions] contain definitions of constants
1443 appearing in the introduction rules. The (co)datatype package supplies
1444 the constructors' definitions here. Most (co)inductive definitions omit
1445 this section; one exception is the primitive recursive functions example
1446 (\S\ref{primrec-sec}).
1448 \item[\it type\_intrs] consists of introduction rules for type-checking the
1449 definition, as discussed in~\S\ref{basic-sec}. They are applied using
1450 depth-first search; you can trace the proof by setting
1452 \verb|trace_DEPTH_FIRST := true|.
1454 \item[\it type\_elims] consists of elimination rules for type-checking the
1455 definition. They are presumed to be safe and are applied as much as
1456 possible, prior to the {\tt type\_intrs} search.
1459 The package has a few notable restrictions:
1461 \item The theory must separately declare the recursive sets as
1464 \item The names of the recursive sets must be identifiers, not infix
1467 \item Side-conditions must not be conjunctions. However, an introduction rule
1468 may contain any number of side-conditions.
1470 \item Side-conditions of the form $x=t$, where the variable~$x$ does not
1471 occur in~$t$, will be substituted through the rule \verb|mutual_induct|.
1474 Isabelle/\textsc{hol} uses a simplified syntax for inductive definitions,
1475 thanks to type-checking. There are no \texttt{domains}, \texttt{type\_intrs}
1476 or \texttt{type\_elims} parts.
1479 \section{Datatype and codatatype definitions: users guide}
1480 This section explains how to include (co)datatype declarations in a theory
1481 file. Please include {\tt Datatype} as a parent theory; this makes available
1482 the definitions of $\univ$ and $\quniv$.
1485 \subsection{The result structure}
1486 The result structure extends that of (co)inductive definitions
1487 (Figure~\ref{def-result-fig}) with several additional items:
1489 val con_defs : thm list
1490 val case_eqns : thm list
1491 val free_iffs : thm list
1492 val free_SEs : thm list
1493 val mk_free : string -> thm
1495 Most of these have been discussed in~\S\ref{data-sec}. Here is a summary:
1497 \item[\tt con\_defs] is a list of definitions: the case operator followed by
1498 the constructors. This theorem list can be supplied to \verb|mk_cases|, for
1501 \item[\tt case\_eqns] is a list of equations, stating that the case operator
1502 inverts each constructor.
1504 \item[\tt free\_iffs] is a list of logical equivalences to perform freeness
1505 reasoning by rewriting. A typical application has the form
1507 by (asm_simp_tac (ZF_ss addsimps free_iffs) 1);
1510 \item[\tt free\_SEs] is a list of safe elimination rules to perform freeness
1511 reasoning. It can be supplied to \verb|eresolve_tac| or to the classical
1514 by (fast_tac (ZF_cs addSEs free_SEs) 1);
1517 \item[\tt mk\_free] is a function to prove freeness properties, specified as
1518 strings. The theorems can be expressed in various forms, such as logical
1519 equivalences or elimination rules.
1522 The result structure also inherits everything from the underlying
1523 (co)inductive definition, such as the introduction rules, elimination rule,
1524 and (co)induction rule.
1527 \subsection{The syntax of a (co)datatype definition}
1528 A datatype definition has the form
1530 datatype <={\it domain}
1531 {\it datatype declaration} and {\it datatype declaration} and \ldots
1532 monos {\it monotonicity theorems}
1533 type_intrs {\it introduction rules for type-checking}
1534 type_elims {\it elimination rules for type-checking}
1536 A codatatype definition is identical save that it starts with the keyword {\tt
1539 The {\tt monos}, {\tt type\_intrs} and {\tt type\_elims} sections are
1540 optional. They are treated like their counterparts in a (co)inductive
1541 definition, as described above. The package supplements your type-checking
1542 rules (if any) with additional ones that should cope with any
1543 finitely-branching (co)datatype definition.
1546 \item[\it domain] specifies a single domain to use for all the mutually
1547 recursive (co)datatypes. If it (and the preceeding~{\tt <=}) are
1548 omitted, the package supplies a domain automatically. Suppose the
1549 definition involves the set parameters $A_1$, \ldots, $A_k$. Then
1550 $\univ(A_1\un\cdots\un A_k)$ is used for a datatype definition and
1551 $\quniv(A_1\un\cdots\un A_k)$ is used for a codatatype definition.
1553 These choices should work for all finitely-branching (co)datatype
1554 definitions. For examples of infinitely-branching datatypes,
1555 see file {\tt ZF/ex/Brouwer.thy}.
1557 \item[\it datatype declaration] has the form
1559 {\it string\/} {\tt =} {\it constructor} {\tt|} {\it constructor} {\tt|}
1562 The {\it string\/} is the datatype, say {\tt"list(A)"}. Each
1563 {\it constructor\/} has the form
1565 {\it name\/} {\tt(} {\it premise} {\tt,} {\it premise} {\tt,} \ldots {\tt)}
1568 The {\it name\/} specifies a new constructor while the {\it premises\/} its
1569 typing conditions. The optional {\it mixfix\/} phrase may give
1570 the constructor infix, for example.
1572 Mutually recursive {\it datatype declarations\/} are separated by the
1576 Isabelle/\textsc{hol}'s datatype definition package is (as of this writing)
1577 entirely different from Isabelle/\textsc{zf}'s. The syntax is different, and
1578 instead of making an inductive definition it asserts axioms.
1581 In the definitions of the constructors, the right-hand sides may overlap.
1582 For instance, the datatype of combinators has constructors defined by
1584 {\tt K} & \equiv & \Inl(\emptyset) \\
1585 {\tt S} & \equiv & \Inr(\Inl(\emptyset)) \\
1586 p{\tt\#}q & \equiv & \Inr(\Inl(\pair{p,q}))
1588 Unlike in previous versions of Isabelle, \verb|fold_tac| now ensures that the
1589 longest right-hand sides are folded first.