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doc-src/Inductive/ind-defs.tex

author | wenzelm |

Mon Aug 28 13:52:38 2000 +0200 (2000-08-28) | |

changeset 9695 | ec7d7f877712 |

parent 7829 | c2672c537894 |

child 42637 | 381fdcab0f36 |

permissions | -rw-r--r-- |

proper setup of iman.sty/extra.sty/ttbox.sty;

1 %% $Id$

2 \documentclass[12pt,a4paper]{article}

3 \usepackage{latexsym,../iman,../extra,../ttbox,../proof,../pdfsetup}

5 \newif\ifshort%''Short'' means a published version, not the documentation

6 \shortfalse%%%%%\shorttrue

8 \title{A Fixedpoint Approach to\\

9 (Co)Inductive and (Co)Datatype Definitions%

10 \thanks{J. Grundy and S. Thompson made detailed comments. Mads Tofte and

11 the referees were also helpful. The research was funded by the SERC

12 grants GR/G53279, GR/H40570 and by the ESPRIT Project 6453 ``Types''.}}

14 \author{Lawrence C. Paulson\\{\tt lcp@cl.cam.ac.uk}\\

15 Computer Laboratory, University of Cambridge, England}

16 \date{\today}

17 \setcounter{secnumdepth}{2} \setcounter{tocdepth}{2}

19 \newcommand\sbs{\subseteq}

20 \let\To=\Rightarrow

22 \newcommand\defn[1]{{\bf#1}}

24 \newcommand\pow{{\cal P}}

25 \newcommand\RepFun{\hbox{\tt RepFun}}

26 \newcommand\cons{\hbox{\tt cons}}

27 \def\succ{\hbox{\tt succ}}

28 \newcommand\split{\hbox{\tt split}}

29 \newcommand\fst{\hbox{\tt fst}}

30 \newcommand\snd{\hbox{\tt snd}}

31 \newcommand\converse{\hbox{\tt converse}}

32 \newcommand\domain{\hbox{\tt domain}}

33 \newcommand\range{\hbox{\tt range}}

34 \newcommand\field{\hbox{\tt field}}

35 \newcommand\lfp{\hbox{\tt lfp}}

36 \newcommand\gfp{\hbox{\tt gfp}}

37 \newcommand\id{\hbox{\tt id}}

38 \newcommand\trans{\hbox{\tt trans}}

39 \newcommand\wf{\hbox{\tt wf}}

40 \newcommand\nat{\hbox{\tt nat}}

41 \newcommand\rank{\hbox{\tt rank}}

42 \newcommand\univ{\hbox{\tt univ}}

43 \newcommand\Vrec{\hbox{\tt Vrec}}

44 \newcommand\Inl{\hbox{\tt Inl}}

45 \newcommand\Inr{\hbox{\tt Inr}}

46 \newcommand\case{\hbox{\tt case}}

47 \newcommand\lst{\hbox{\tt list}}

48 \newcommand\Nil{\hbox{\tt Nil}}

49 \newcommand\Cons{\hbox{\tt Cons}}

50 \newcommand\lstcase{\hbox{\tt list\_case}}

51 \newcommand\lstrec{\hbox{\tt list\_rec}}

52 \newcommand\length{\hbox{\tt length}}

53 \newcommand\listn{\hbox{\tt listn}}

54 \newcommand\acc{\hbox{\tt acc}}

55 \newcommand\primrec{\hbox{\tt primrec}}

56 \newcommand\SC{\hbox{\tt SC}}

57 \newcommand\CONST{\hbox{\tt CONST}}

58 \newcommand\PROJ{\hbox{\tt PROJ}}

59 \newcommand\COMP{\hbox{\tt COMP}}

60 \newcommand\PREC{\hbox{\tt PREC}}

62 \newcommand\quniv{\hbox{\tt quniv}}

63 \newcommand\llist{\hbox{\tt llist}}

64 \newcommand\LNil{\hbox{\tt LNil}}

65 \newcommand\LCons{\hbox{\tt LCons}}

66 \newcommand\lconst{\hbox{\tt lconst}}

67 \newcommand\lleq{\hbox{\tt lleq}}

68 \newcommand\map{\hbox{\tt map}}

69 \newcommand\term{\hbox{\tt term}}

70 \newcommand\Apply{\hbox{\tt Apply}}

71 \newcommand\termcase{\hbox{\tt term\_case}}

72 \newcommand\rev{\hbox{\tt rev}}

73 \newcommand\reflect{\hbox{\tt reflect}}

74 \newcommand\tree{\hbox{\tt tree}}

75 \newcommand\forest{\hbox{\tt forest}}

76 \newcommand\Part{\hbox{\tt Part}}

77 \newcommand\TF{\hbox{\tt tree\_forest}}

78 \newcommand\Tcons{\hbox{\tt Tcons}}

79 \newcommand\Fcons{\hbox{\tt Fcons}}

80 \newcommand\Fnil{\hbox{\tt Fnil}}

81 \newcommand\TFcase{\hbox{\tt TF\_case}}

82 \newcommand\Fin{\hbox{\tt Fin}}

83 \newcommand\QInl{\hbox{\tt QInl}}

84 \newcommand\QInr{\hbox{\tt QInr}}

85 \newcommand\qsplit{\hbox{\tt qsplit}}

86 \newcommand\qcase{\hbox{\tt qcase}}

87 \newcommand\Con{\hbox{\tt Con}}

88 \newcommand\data{\hbox{\tt data}}

90 \binperiod %%%treat . like a binary operator

92 \begin{document}

93 \pagestyle{empty}

94 \begin{titlepage}

95 \maketitle

96 \begin{abstract}

97 This paper presents a fixedpoint approach to inductive definitions.

98 Instead of using a syntactic test such as ``strictly positive,'' the

99 approach lets definitions involve any operators that have been proved

100 monotone. It is conceptually simple, which has allowed the easy

101 implementation of mutual recursion and iterated definitions. It also

102 handles coinductive definitions: simply replace the least fixedpoint by a

103 greatest fixedpoint.

105 The method has been implemented in two of Isabelle's logics, \textsc{zf} set

106 theory and higher-order logic. It should be applicable to any logic in

107 which the Knaster-Tarski theorem can be proved. Examples include lists of

108 $n$ elements, the accessible part of a relation and the set of primitive

109 recursive functions. One example of a coinductive definition is

110 bisimulations for lazy lists. Recursive datatypes are examined in detail,

111 as well as one example of a \defn{codatatype}: lazy lists.

113 The Isabelle package has been applied in several large case studies,

114 including two proofs of the Church-Rosser theorem and a coinductive proof of

115 semantic consistency. The package can be trusted because it proves theorems

116 from definitions, instead of asserting desired properties as axioms.

117 \end{abstract}

118 %

119 \bigskip

120 \centerline{Copyright \copyright{} \number\year{} by Lawrence C. Paulson}

121 \thispagestyle{empty}

122 \end{titlepage}

123 \tableofcontents\cleardoublepage\pagestyle{plain}

125 \setcounter{page}{1}

127 \section{Introduction}

128 Several theorem provers provide commands for formalizing recursive data

129 structures, like lists and trees. Robin Milner implemented one of the first

130 of these, for Edinburgh \textsc{lcf}~\cite{milner-ind}. Given a description

131 of the desired data structure, Milner's package formulated appropriate

132 definitions and proved the characteristic theorems. Similar is Melham's

133 recursive type package for the Cambridge \textsc{hol} system~\cite{melham89}.

134 Such data structures are called \defn{datatypes}

135 below, by analogy with datatype declarations in Standard~\textsc{ml}\@.

136 Some logics take datatypes as primitive; consider Boyer and Moore's shell

137 principle~\cite{bm79} and the Coq type theory~\cite{paulin-tlca}.

139 A datatype is but one example of an \defn{inductive definition}. Such a

140 definition~\cite{aczel77} specifies the least set~$R$ \defn{closed under}

141 given rules: applying a rule to elements of~$R$ yields a result within~$R$.

142 Inductive definitions have many applications. The collection of theorems in a

143 logic is inductively defined. A structural operational

144 semantics~\cite{hennessy90} is an inductive definition of a reduction or

145 evaluation relation on programs. A few theorem provers provide commands for

146 formalizing inductive definitions; these include Coq~\cite{paulin-tlca} and

147 again the \textsc{hol} system~\cite{camilleri92}.

149 The dual notion is that of a \defn{coinductive definition}. Such a definition

150 specifies the greatest set~$R$ \defn{consistent with} given rules: every

151 element of~$R$ can be seen as arising by applying a rule to elements of~$R$.

152 Important examples include using bisimulation relations to formalize

153 equivalence of processes~\cite{milner89} or lazy functional

154 programs~\cite{abramsky90}. Other examples include lazy lists and other

155 infinite data structures; these are called \defn{codatatypes} below.

157 Not all inductive definitions are meaningful. \defn{Monotone} inductive

158 definitions are a large, well-behaved class. Monotonicity can be enforced

159 by syntactic conditions such as ``strictly positive,'' but this could lead to

160 monotone definitions being rejected on the grounds of their syntactic form.

161 More flexible is to formalize monotonicity within the logic and allow users

162 to prove it.

164 This paper describes a package based on a fixedpoint approach. Least

165 fixedpoints yield inductive definitions; greatest fixedpoints yield

166 coinductive definitions. Most of the discussion below applies equally to

167 inductive and coinductive definitions, and most of the code is shared.

169 The package supports mutual recursion and infinitely-branching datatypes and

170 codatatypes. It allows use of any operators that have been proved monotone,

171 thus accepting all provably monotone inductive definitions, including

172 iterated definitions.

174 The package has been implemented in

175 Isabelle~\cite{paulson-markt,paulson-isa-book} using

176 \textsc{zf} set theory \cite{paulson-set-I,paulson-set-II}; part of it has

177 since been ported to Isabelle/\textsc{hol} (higher-order logic). The

178 recursion equations are specified as introduction rules for the mutually

179 recursive sets. The package transforms these rules into a mapping over sets,

180 and attempts to prove that the mapping is monotonic and well-typed. If

181 successful, the package makes fixedpoint definitions and proves the

182 introduction, elimination and (co)induction rules. Users invoke the package

183 by making simple declarations in Isabelle theory files.

185 Most datatype packages equip the new datatype with some means of expressing

186 recursive functions. This is the main omission from my package. Its

187 fixedpoint operators define only recursive sets. The Isabelle/\textsc{zf}

188 theory provides well-founded recursion~\cite{paulson-set-II}, which is harder

189 to use than structural recursion but considerably more general.

190 Slind~\cite{slind-tfl} has written a package to automate the definition of

191 well-founded recursive functions in Isabelle/\textsc{hol}.

193 \paragraph*{Outline.} Section~2 introduces the least and greatest fixedpoint

194 operators. Section~3 discusses the form of introduction rules, mutual

195 recursion and other points common to inductive and coinductive definitions.

196 Section~4 discusses induction and coinduction rules separately. Section~5

197 presents several examples, including a coinductive definition. Section~6

198 describes datatype definitions. Section~7 presents related work.

199 Section~8 draws brief conclusions. \ifshort\else The appendices are simple

200 user's manuals for this Isabelle package.\fi

202 Most of the definitions and theorems shown below have been generated by the

203 package. I have renamed some variables to improve readability.

205 \section{Fixedpoint operators}

206 In set theory, the least and greatest fixedpoint operators are defined as

207 follows:

208 \begin{eqnarray*}

209 \lfp(D,h) & \equiv & \inter\{X\sbs D. h(X)\sbs X\} \\

210 \gfp(D,h) & \equiv & \union\{X\sbs D. X\sbs h(X)\}

211 \end{eqnarray*}

212 Let $D$ be a set. Say that $h$ is \defn{bounded by}~$D$ if $h(D)\sbs D$, and

213 \defn{monotone below~$D$} if

214 $h(A)\sbs h(B)$ for all $A$ and $B$ such that $A\sbs B\sbs D$. If $h$ is

215 bounded by~$D$ and monotone then both operators yield fixedpoints:

216 \begin{eqnarray*}

217 \lfp(D,h) & = & h(\lfp(D,h)) \\

218 \gfp(D,h) & = & h(\gfp(D,h))

219 \end{eqnarray*}

220 These equations are instances of the Knaster-Tarski theorem, which states

221 that every monotonic function over a complete lattice has a

222 fixedpoint~\cite{davey-priestley}. It is obvious from their definitions

223 that $\lfp$ must be the least fixedpoint, and $\gfp$ the greatest.

225 This fixedpoint theory is simple. The Knaster-Tarski theorem is easy to

226 prove. Showing monotonicity of~$h$ is trivial, in typical cases. We must

227 also exhibit a bounding set~$D$ for~$h$. Frequently this is trivial, as when

228 a set of theorems is (co)inductively defined over some previously existing set

229 of formul{\ae}. Isabelle/\textsc{zf} provides suitable bounding sets for

230 infinitely-branching (co)datatype definitions; see~\S\ref{univ-sec}. Bounding

231 sets are also called \defn{domains}.

233 The powerset operator is monotone, but by Cantor's theorem there is no

234 set~$A$ such that $A=\pow(A)$. We cannot put $A=\lfp(D,\pow)$ because

235 there is no suitable domain~$D$. But \S\ref{acc-sec} demonstrates

236 that~$\pow$ is still useful in inductive definitions.

238 \section{Elements of an inductive or coinductive definition}\label{basic-sec}

239 Consider a (co)inductive definition of the sets $R_1$, \ldots,~$R_n$, in

240 mutual recursion. They will be constructed from domains $D_1$,

241 \ldots,~$D_n$, respectively. The construction yields not $R_i\sbs D_i$ but

242 $R_i\sbs D_1+\cdots+D_n$, where $R_i$ is contained in the image of~$D_i$

243 under an injection. Reasons for this are discussed

244 elsewhere~\cite[\S4.5]{paulson-set-II}.

246 The definition may involve arbitrary parameters $\vec{p}=p_1$,

247 \ldots,~$p_k$. Each recursive set then has the form $R_i(\vec{p})$. The

248 parameters must be identical every time they occur within a definition. This

249 would appear to be a serious restriction compared with other systems such as

250 Coq~\cite{paulin-tlca}. For instance, we cannot define the lists of

251 $n$ elements as the set $\listn(A,n)$ using rules where the parameter~$n$

252 varies. Section~\ref{listn-sec} describes how to express this set using the

253 inductive definition package.

255 To avoid clutter below, the recursive sets are shown as simply $R_i$

256 instead of~$R_i(\vec{p})$.

258 \subsection{The form of the introduction rules}\label{intro-sec}

259 The body of the definition consists of the desired introduction rules. The

260 conclusion of each rule must have the form $t\in R_i$, where $t$ is any term.

261 Premises typically have the same form, but they can have the more general form

262 $t\in M(R_i)$ or express arbitrary side-conditions.

264 The premise $t\in M(R_i)$ is permitted if $M$ is a monotonic operator on

265 sets, satisfying the rule

266 \[ \infer{M(A)\sbs M(B)}{A\sbs B} \]

267 The user must supply the package with monotonicity rules for all such premises.

269 The ability to introduce new monotone operators makes the approach

270 flexible. A suitable choice of~$M$ and~$t$ can express a lot. The

271 powerset operator $\pow$ is monotone, and the premise $t\in\pow(R)$

272 expresses $t\sbs R$; see \S\ref{acc-sec} for an example. The \emph{list of}

273 operator is monotone, as is easily proved by induction. The premise

274 $t\in\lst(R)$ avoids having to encode the effect of~$\lst(R)$ using mutual

275 recursion; see \S\ref{primrec-sec} and also my earlier

276 paper~\cite[\S4.4]{paulson-set-II}.

278 Introduction rules may also contain \defn{side-conditions}. These are

279 premises consisting of arbitrary formul{\ae} not mentioning the recursive

280 sets. Side-conditions typically involve type-checking. One example is the

281 premise $a\in A$ in the following rule from the definition of lists:

282 \[ \infer{\Cons(a,l)\in\lst(A)}{a\in A & l\in\lst(A)} \]

284 \subsection{The fixedpoint definitions}

285 The package translates the list of desired introduction rules into a fixedpoint

286 definition. Consider, as a running example, the finite powerset operator

287 $\Fin(A)$: the set of all finite subsets of~$A$. It can be

288 defined as the least set closed under the rules

289 \[ \emptyset\in\Fin(A) \qquad

290 \infer{\{a\}\un b\in\Fin(A)}{a\in A & b\in\Fin(A)}

291 \]

293 The domain in a (co)inductive definition must be some existing set closed

294 under the rules. A suitable domain for $\Fin(A)$ is $\pow(A)$, the set of all

295 subsets of~$A$. The package generates the definition

296 \[ \Fin(A) \equiv \lfp(\pow(A), \,

297 \begin{array}[t]{r@{\,}l}

298 \lambda X. \{z\in\pow(A). & z=\emptyset \disj{} \\

299 &(\exists a\,b. z=\{a\}\un b\conj a\in A\conj b\in X)\})

300 \end{array}

301 \]

302 The contribution of each rule to the definition of $\Fin(A)$ should be

303 obvious. A coinductive definition is similar but uses $\gfp$ instead

304 of~$\lfp$.

306 The package must prove that the fixedpoint operator is applied to a

307 monotonic function. If the introduction rules have the form described

308 above, and if the package is supplied a monotonicity theorem for every

309 $t\in M(R_i)$ premise, then this proof is trivial.\footnote{Due to the

310 presence of logical connectives in the fixedpoint's body, the

311 monotonicity proof requires some unusual rules. These state that the

312 connectives $\conj$, $\disj$ and $\exists$ preserve monotonicity with respect

313 to the partial ordering on unary predicates given by $P\sqsubseteq Q$ if and

314 only if $\forall x.P(x)\imp Q(x)$.}

316 The package returns its result as an \textsc{ml} structure, which consists of named

317 components; we may regard it as a record. The result structure contains

318 the definitions of the recursive sets as a theorem list called {\tt defs}.

319 It also contains some theorems; {\tt dom\_subset} is an inclusion such as

320 $\Fin(A)\sbs\pow(A)$, while {\tt bnd\_mono} asserts that the fixedpoint

321 definition is monotonic.

323 Internally the package uses the theorem {\tt unfold}, a fixedpoint equation

324 such as

325 \[

326 \begin{array}[t]{r@{\,}l}

327 \Fin(A) = \{z\in\pow(A). & z=\emptyset \disj{} \\

328 &(\exists a\,b. z=\{a\}\un b\conj a\in A\conj b\in \Fin(A))\}

329 \end{array}

330 \]

331 In order to save space, this theorem is not exported.

334 \subsection{Mutual recursion} \label{mutual-sec}

335 In a mutually recursive definition, the domain of the fixedpoint construction

336 is the disjoint sum of the domain~$D_i$ of each~$R_i$, for $i=1$,

337 \ldots,~$n$. The package uses the injections of the

338 binary disjoint sum, typically $\Inl$ and~$\Inr$, to express injections

339 $h_{1n}$, \ldots, $h_{nn}$ for the $n$-ary disjoint sum $D_1+\cdots+D_n$.

341 As discussed elsewhere \cite[\S4.5]{paulson-set-II}, Isabelle/\textsc{zf} defines the

342 operator $\Part$ to support mutual recursion. The set $\Part(A,h)$

343 contains those elements of~$A$ having the form~$h(z)$:

344 \[ \Part(A,h) \equiv \{x\in A. \exists z. x=h(z)\}. \]

345 For mutually recursive sets $R_1$, \ldots,~$R_n$ with

346 $n>1$, the package makes $n+1$ definitions. The first defines a set $R$ using

347 a fixedpoint operator. The remaining $n$ definitions have the form

348 \[ R_i \equiv \Part(R,h_{in}), \qquad i=1,\ldots, n. \]

349 It follows that $R=R_1\un\cdots\un R_n$, where the $R_i$ are pairwise disjoint.

352 \subsection{Proving the introduction rules}

353 The user supplies the package with the desired form of the introduction

354 rules. Once it has derived the theorem {\tt unfold}, it attempts

355 to prove those rules. From the user's point of view, this is the

356 trickiest stage; the proofs often fail. The task is to show that the domain

357 $D_1+\cdots+D_n$ of the combined set $R_1\un\cdots\un R_n$ is

358 closed under all the introduction rules. This essentially involves replacing

359 each~$R_i$ by $D_1+\cdots+D_n$ in each of the introduction rules and

360 attempting to prove the result.

362 Consider the $\Fin(A)$ example. After substituting $\pow(A)$ for $\Fin(A)$

363 in the rules, the package must prove

364 \[ \emptyset\in\pow(A) \qquad

365 \infer{\{a\}\un b\in\pow(A)}{a\in A & b\in\pow(A)}

366 \]

367 Such proofs can be regarded as type-checking the definition.\footnote{The

368 Isabelle/\textsc{hol} version does not require these proofs, as \textsc{hol}

369 has implicit type-checking.} The user supplies the package with

370 type-checking rules to apply. Usually these are general purpose rules from

371 the \textsc{zf} theory. They could however be rules specifically proved for a

372 particular inductive definition; sometimes this is the easiest way to get the

373 definition through!

375 The result structure contains the introduction rules as the theorem list {\tt

376 intrs}.

378 \subsection{The case analysis rule}

379 The elimination rule, called {\tt elim}, performs case analysis. It is a

380 simple consequence of {\tt unfold}. There is one case for each introduction

381 rule. If $x\in\Fin(A)$ then either $x=\emptyset$ or else $x=\{a\}\un b$ for

382 some $a\in A$ and $b\in\Fin(A)$. Formally, the elimination rule for $\Fin(A)$

383 is written

384 \[ \infer{Q}{x\in\Fin(A) & \infer*{Q}{[x=\emptyset]}

385 & \infer*{Q}{[x=\{a\}\un b & a\in A &b\in\Fin(A)]_{a,b}} }

386 \]

387 The subscripted variables $a$ and~$b$ above the third premise are

388 eigenvariables, subject to the usual ``not free in \ldots'' proviso.

391 \section{Induction and coinduction rules}

392 Here we must consider inductive and coinductive definitions separately. For

393 an inductive definition, the package returns an induction rule derived

394 directly from the properties of least fixedpoints, as well as a modified rule

395 for mutual recursion. For a coinductive definition, the package returns a

396 basic coinduction rule.

398 \subsection{The basic induction rule}\label{basic-ind-sec}

399 The basic rule, called {\tt induct}, is appropriate in most situations.

400 For inductive definitions, it is strong rule induction~\cite{camilleri92}; for

401 datatype definitions (see below), it is just structural induction.

403 The induction rule for an inductively defined set~$R$ has the form described

404 below. For the time being, assume that $R$'s domain is not a Cartesian

405 product; inductively defined relations are treated slightly differently.

407 The major premise is $x\in R$. There is a minor premise for each

408 introduction rule:

409 \begin{itemize}

410 \item If the introduction rule concludes $t\in R_i$, then the minor premise

411 is~$P(t)$.

413 \item The minor premise's eigenvariables are precisely the introduction

414 rule's free variables that are not parameters of~$R$. For instance, the

415 eigenvariables in the $\Fin(A)$ rule below are $a$ and $b$, but not~$A$.

417 \item If the introduction rule has a premise $t\in R_i$, then the minor

418 premise discharges the assumption $t\in R_i$ and the induction

419 hypothesis~$P(t)$. If the introduction rule has a premise $t\in M(R_i)$

420 then the minor premise discharges the single assumption

421 \[ t\in M(\{z\in R_i. P(z)\}). \]

422 Because $M$ is monotonic, this assumption implies $t\in M(R_i)$. The

423 occurrence of $P$ gives the effect of an induction hypothesis, which may be

424 exploited by appealing to properties of~$M$.

425 \end{itemize}

426 The induction rule for $\Fin(A)$ resembles the elimination rule shown above,

427 but includes an induction hypothesis:

428 \[ \infer{P(x)}{x\in\Fin(A) & P(\emptyset)

429 & \infer*{P(\{a\}\un b)}{[a\in A & b\in\Fin(A) & P(b)]_{a,b}} }

430 \]

431 Stronger induction rules often suggest themselves. We can derive a rule for

432 $\Fin(A)$ whose third premise discharges the extra assumption $a\not\in b$.

433 The package provides rules for mutual induction and inductive relations. The

434 Isabelle/\textsc{zf} theory also supports well-founded induction and recursion

435 over datatypes, by reasoning about the \defn{rank} of a

436 set~\cite[\S3.4]{paulson-set-II}.

439 \subsection{Modified induction rules}

441 If the domain of $R$ is a Cartesian product $A_1\times\cdots\times A_m$

442 (however nested), then the corresponding predicate $P_i$ takes $m$ arguments.

443 The major premise becomes $\pair{z_1,\ldots,z_m}\in R$ instead of $x\in R$;

444 the conclusion becomes $P(z_1,\ldots,z_m)$. This simplifies reasoning about

445 inductively defined relations, eliminating the need to express properties of

446 $z_1$, \ldots,~$z_m$ as properties of the tuple $\pair{z_1,\ldots,z_m}$.

447 Occasionally it may require you to split up the induction variable

448 using {\tt SigmaE} and {\tt dom\_subset}, especially if the constant {\tt

449 split} appears in the rule.

451 The mutual induction rule is called {\tt

452 mutual\_induct}. It differs from the basic rule in two respects:

453 \begin{itemize}

454 \item Instead of a single predicate~$P$, it uses $n$ predicates $P_1$,

455 \ldots,~$P_n$: one for each recursive set.

457 \item There is no major premise such as $x\in R_i$. Instead, the conclusion

458 refers to all the recursive sets:

459 \[ (\forall z.z\in R_1\imp P_1(z))\conj\cdots\conj

460 (\forall z.z\in R_n\imp P_n(z))

461 \]

462 Proving the premises establishes $P_i(z)$ for $z\in R_i$ and $i=1$,

463 \ldots,~$n$.

464 \end{itemize}

465 %

466 If the domain of some $R_i$ is a Cartesian product, then the mutual induction

467 rule is modified accordingly. The predicates are made to take $m$ separate

468 arguments instead of a tuple, and the quantification in the conclusion is over

469 the separate variables $z_1$, \ldots, $z_m$.

471 \subsection{Coinduction}\label{coind-sec}

472 A coinductive definition yields a primitive coinduction rule, with no

473 refinements such as those for the induction rules. (Experience may suggest

474 refinements later.) Consider the codatatype of lazy lists as an example. For

475 suitable definitions of $\LNil$ and $\LCons$, lazy lists may be defined as the

476 greatest set consistent with the rules

477 \[ \LNil\in\llist(A) \qquad

478 \infer[(-)]{\LCons(a,l)\in\llist(A)}{a\in A & l\in\llist(A)}

479 \]

480 The $(-)$ tag stresses that this is a coinductive definition. A suitable

481 domain for $\llist(A)$ is $\quniv(A)$; this set is closed under the variant

482 forms of sum and product that are used to represent non-well-founded data

483 structures (see~\S\ref{univ-sec}).

485 The package derives an {\tt unfold} theorem similar to that for $\Fin(A)$.

486 Then it proves the theorem {\tt coinduct}, which expresses that $\llist(A)$

487 is the greatest solution to this equation contained in $\quniv(A)$:

488 \[ \infer{x\in\llist(A)}{x\in X & X\sbs \quniv(A) &

489 \infer*{

490 \begin{array}[b]{r@{}l}

491 z=\LNil\disj

492 \bigl(\exists a\,l.\, & z=\LCons(a,l) \conj a\in A \conj{}\\

493 & l\in X\un\llist(A) \bigr)

494 \end{array} }{[z\in X]_z}}

495 \]

496 This rule complements the introduction rules; it provides a means of showing

497 $x\in\llist(A)$ when $x$ is infinite. For instance, if $x=\LCons(0,x)$ then

498 applying the rule with $X=\{x\}$ proves $x\in\llist(\nat)$. (Here $\nat$

499 is the set of natural numbers.)

501 Having $X\un\llist(A)$ instead of simply $X$ in the third premise above

502 represents a slight strengthening of the greatest fixedpoint property. I

503 discuss several forms of coinduction rules elsewhere~\cite{paulson-coind}.

505 The clumsy form of the third premise makes the rule hard to use, especially in

506 large definitions. Probably a constant should be declared to abbreviate the

507 large disjunction, and rules derived to allow proving the separate disjuncts.

510 \section{Examples of inductive and coinductive definitions}\label{ind-eg-sec}

511 This section presents several examples from the literature: the finite

512 powerset operator, lists of $n$ elements, bisimulations on lazy lists, the

513 well-founded part of a relation, and the primitive recursive functions.

515 \subsection{The finite powerset operator}

516 This operator has been discussed extensively above. Here is the

517 corresponding invocation in an Isabelle theory file. Note that

518 $\cons(a,b)$ abbreviates $\{a\}\un b$ in Isabelle/\textsc{zf}.

519 \begin{ttbox}

520 Finite = Arith +

521 consts Fin :: i=>i

522 inductive

523 domains "Fin(A)" <= "Pow(A)"

524 intrs

525 emptyI "0 : Fin(A)"

526 consI "[| a: A; b: Fin(A) |] ==> cons(a,b) : Fin(A)"

527 type_intrs empty_subsetI, cons_subsetI, PowI

528 type_elims "[make_elim PowD]"

529 end

530 \end{ttbox}

531 Theory {\tt Finite} extends the parent theory {\tt Arith} by declaring the

532 unary function symbol~$\Fin$, which is defined inductively. Its domain is

533 specified as $\pow(A)$, where $A$ is the parameter appearing in the

534 introduction rules. For type-checking, we supply two introduction

535 rules:

536 \[ \emptyset\sbs A \qquad

537 \infer{\{a\}\un B\sbs C}{a\in C & B\sbs C}

538 \]

539 A further introduction rule and an elimination rule express both

540 directions of the equivalence $A\in\pow(B)\bimp A\sbs B$. Type-checking

541 involves mostly introduction rules.

543 Like all Isabelle theory files, this one yields a structure containing the

544 new theory as an \textsc{ml} value. Structure {\tt Finite} also has a

545 substructure, called~{\tt Fin}. After declaring \hbox{\tt open Finite;} we

546 can refer to the $\Fin(A)$ introduction rules as the list {\tt Fin.intrs}

547 or individually as {\tt Fin.emptyI} and {\tt Fin.consI}. The induction

548 rule is {\tt Fin.induct}.

551 \subsection{Lists of $n$ elements}\label{listn-sec}

552 This has become a standard example of an inductive definition. Following

553 Paulin-Mohring~\cite{paulin-tlca}, we could attempt to define a new datatype

554 $\listn(A,n)$, for lists of length~$n$, as an $n$-indexed family of sets.

555 But her introduction rules

556 \[ \hbox{\tt Niln}\in\listn(A,0) \qquad

557 \infer{\hbox{\tt Consn}(n,a,l)\in\listn(A,\succ(n))}

558 {n\in\nat & a\in A & l\in\listn(A,n)}

559 \]

560 are not acceptable to the inductive definition package:

561 $\listn$ occurs with three different parameter lists in the definition.

563 The Isabelle version of this example suggests a general treatment of

564 varying parameters. It uses the existing datatype definition of

565 $\lst(A)$, with constructors $\Nil$ and~$\Cons$, and incorporates the

566 parameter~$n$ into the inductive set itself. It defines $\listn(A)$ as a

567 relation consisting of pairs $\pair{n,l}$ such that $n\in\nat$

568 and~$l\in\lst(A)$ and $l$ has length~$n$. In fact, $\listn(A)$ is the

569 converse of the length function on~$\lst(A)$. The Isabelle/\textsc{zf} introduction

570 rules are

571 \[ \pair{0,\Nil}\in\listn(A) \qquad

572 \infer{\pair{\succ(n),\Cons(a,l)}\in\listn(A)}

573 {a\in A & \pair{n,l}\in\listn(A)}

574 \]

575 The Isabelle theory file takes, as parent, the theory~{\tt List} of lists.

576 We declare the constant~$\listn$ and supply an inductive definition,

577 specifying the domain as $\nat\times\lst(A)$:

578 \begin{ttbox}

579 ListN = List +

580 consts listn :: i=>i

581 inductive

582 domains "listn(A)" <= "nat*list(A)"

583 intrs

584 NilI "<0,Nil>: listn(A)"

585 ConsI "[| a:A; <n,l>:listn(A) |] ==> <succ(n), Cons(a,l)>: listn(A)"

586 type_intrs "nat_typechecks @ list.intrs"

587 end

588 \end{ttbox}

589 The type-checking rules include those for 0, $\succ$, $\Nil$ and $\Cons$.

590 Because $\listn(A)$ is a set of pairs, type-checking requires the

591 equivalence $\pair{a,b}\in A\times B \bimp a\in A \conj b\in B$. The

592 package always includes the rules for ordered pairs.

594 The package returns introduction, elimination and induction rules for

595 $\listn$. The basic induction rule, {\tt listn.induct}, is

596 \[ \infer{P(z_1,z_2)}{\pair{z_1,z_2}\in\listn(A) & P(0,\Nil) &

597 \infer*{P(\succ(n),\Cons(a,l))}

598 {[a\in A & \pair{n,l}\in\listn(A) & P(n,l)]_{a,l,n}}}

599 \]

600 This rule lets the induction formula to be a

601 binary property of pairs, $P(n,l)$.

602 It is now a simple matter to prove theorems about $\listn(A)$, such as

603 \[ \forall l\in\lst(A). \pair{\length(l),\, l}\in\listn(A) \]

604 \[ \listn(A)``\{n\} = \{l\in\lst(A). \length(l)=n\} \]

605 This latter result --- here $r``X$ denotes the image of $X$ under $r$

606 --- asserts that the inductive definition agrees with the obvious notion of

607 $n$-element list.

609 A ``list of $n$ elements'' really is a list, namely an element of ~$\lst(A)$.

610 It is subject to list operators such as append (concatenation). For example,

611 a trivial induction on $\pair{m,l}\in\listn(A)$ yields

612 \[ \infer{\pair{m\mathbin{+} m',\, l@l'}\in\listn(A)}

613 {\pair{m,l}\in\listn(A) & \pair{m',l'}\in\listn(A)}

614 \]

615 where $+$ denotes addition on the natural numbers and @ denotes append.

617 \subsection{Rule inversion: the function \texttt{mk\_cases}}

618 The elimination rule, {\tt listn.elim}, is cumbersome:

619 \[ \infer{Q}{x\in\listn(A) &

620 \infer*{Q}{[x = \pair{0,\Nil}]} &

621 \infer*{Q}

622 {\left[\begin{array}{l}

623 x = \pair{\succ(n),\Cons(a,l)} \\

624 a\in A \\

625 \pair{n,l}\in\listn(A)

626 \end{array} \right]_{a,l,n}}}

627 \]

628 The \textsc{ml} function {\tt listn.mk\_cases} generates simplified instances of

629 this rule. It works by freeness reasoning on the list constructors:

630 $\Cons(a,l)$ is injective in its two arguments and differs from~$\Nil$. If

631 $x$ is $\pair{i,\Nil}$ or $\pair{i,\Cons(a,l)}$ then {\tt listn.mk\_cases}

632 deduces the corresponding form of~$i$; this is called rule inversion.

633 Here is a sample session:

634 \begin{ttbox}

635 listn.mk_cases "<i,Nil> : listn(A)";

636 {\out "[| <?i, []> : listn(?A); ?i = 0 ==> ?Q |] ==> ?Q" : thm}

637 listn.mk_cases "<i,Cons(a,l)> : listn(A)";

638 {\out "[| <?i, Cons(?a, ?l)> : listn(?A);}

639 {\out !!n. [| ?a : ?A; <n, ?l> : listn(?A); ?i = succ(n) |] ==> ?Q }

640 {\out |] ==> ?Q" : thm}

641 \end{ttbox}

642 Each of these rules has only two premises. In conventional notation, the

643 second rule is

644 \[ \infer{Q}{\pair{i, \Cons(a,l)}\in\listn(A) &

645 \infer*{Q}

646 {\left[\begin{array}{l}

647 a\in A \\ \pair{n,l}\in\listn(A) \\ i = \succ(n)

648 \end{array} \right]_{n}}}

649 \]

650 The package also has built-in rules for freeness reasoning about $0$

651 and~$\succ$. So if $x$ is $\pair{0,l}$ or $\pair{\succ(i),l}$, then {\tt

652 listn.mk\_cases} can deduce the corresponding form of~$l$.

654 The function {\tt mk\_cases} is also useful with datatype definitions. The

655 instance from the definition of lists, namely {\tt list.mk\_cases}, can

656 prove that $\Cons(a,l)\in\lst(A)$ implies $a\in A $ and $l\in\lst(A)$:

657 \[ \infer{Q}{\Cons(a,l)\in\lst(A) &

658 & \infer*{Q}{[a\in A &l\in\lst(A)]} }

659 \]

660 A typical use of {\tt mk\_cases} concerns inductive definitions of evaluation

661 relations. Then rule inversion yields case analysis on possible evaluations.

662 For example, Isabelle/\textsc{zf} includes a short proof of the

663 diamond property for parallel contraction on combinators. Ole Rasmussen used

664 {\tt mk\_cases} extensively in his development of the theory of

665 residuals~\cite{rasmussen95}.

668 \subsection{A coinductive definition: bisimulations on lazy lists}

669 This example anticipates the definition of the codatatype $\llist(A)$, which

670 consists of finite and infinite lists over~$A$. Its constructors are $\LNil$

671 and~$\LCons$, satisfying the introduction rules shown in~\S\ref{coind-sec}.

672 Because $\llist(A)$ is defined as a greatest fixedpoint and uses the variant

673 pairing and injection operators, it contains non-well-founded elements such as

674 solutions to $\LCons(a,l)=l$.

676 The next step in the development of lazy lists is to define a coinduction

677 principle for proving equalities. This is done by showing that the equality

678 relation on lazy lists is the greatest fixedpoint of some monotonic

679 operation. The usual approach~\cite{pitts94} is to define some notion of

680 bisimulation for lazy lists, define equivalence to be the greatest

681 bisimulation, and finally to prove that two lazy lists are equivalent if and

682 only if they are equal. The coinduction rule for equivalence then yields a

683 coinduction principle for equalities.

685 A binary relation $R$ on lazy lists is a \defn{bisimulation} provided $R\sbs

686 R^+$, where $R^+$ is the relation

687 \[ \{\pair{\LNil,\LNil}\} \un

688 \{\pair{\LCons(a,l),\LCons(a,l')} . a\in A \conj \pair{l,l'}\in R\}.

689 \]

690 A pair of lazy lists are \defn{equivalent} if they belong to some

691 bisimulation. Equivalence can be coinductively defined as the greatest

692 fixedpoint for the introduction rules

693 \[ \pair{\LNil,\LNil} \in\lleq(A) \qquad

694 \infer[(-)]{\pair{\LCons(a,l),\LCons(a,l')} \in\lleq(A)}

695 {a\in A & \pair{l,l'}\in \lleq(A)}

696 \]

697 To make this coinductive definition, the theory file includes (after the

698 declaration of $\llist(A)$) the following lines:

699 \bgroup\leftmargini=\parindent

700 \begin{ttbox}

701 consts lleq :: i=>i

702 coinductive

703 domains "lleq(A)" <= "llist(A) * llist(A)"

704 intrs

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"

708 \end{ttbox}

709 \egroup

710 The domain of $\lleq(A)$ is $\llist(A)\times\llist(A)$. The type-checking

711 rules include the introduction rules for $\llist(A)$, whose

712 declaration is discussed below (\S\ref{lists-sec}).

714 The package returns the introduction rules and the elimination rule, as

715 usual. But instead of induction rules, it returns a coinduction rule.

716 The rule is too big to display in the usual notation; its conclusion is

717 $x\in\lleq(A)$ and its premises are $x\in X$,

718 ${X\sbs\llist(A)\times\llist(A)}$ and

719 \[ \infer*{z=\pair{\LNil,\LNil}\disj \bigl(\exists a\,l\,l'.\,

720 \begin{array}[t]{@{}l}

721 z=\pair{\LCons(a,l),\LCons(a,l')} \conj a\in A \conj{}\\

722 \pair{l,l'}\in X\un\lleq(A) \bigr)

723 \end{array}

724 }{[z\in X]_z}

725 \]

726 Thus if $x\in X$, where $X$ is a bisimulation contained in the

727 domain of $\lleq(A)$, then $x\in\lleq(A)$. It is easy to show that

728 $\lleq(A)$ is reflexive: the equality relation is a bisimulation. And

729 $\lleq(A)$ is symmetric: its converse is a bisimulation. But showing that

730 $\lleq(A)$ coincides with the equality relation takes some work.

732 \subsection{The accessible part of a relation}\label{acc-sec}

733 Let $\prec$ be a binary relation on~$D$; in short, $(\prec)\sbs D\times D$.

734 The \defn{accessible} or \defn{well-founded} part of~$\prec$, written

735 $\acc(\prec)$, is essentially that subset of~$D$ for which $\prec$ admits

736 no infinite decreasing chains~\cite{aczel77}. Formally, $\acc(\prec)$ is

737 inductively defined to be the least set that contains $a$ if it contains

738 all $\prec$-predecessors of~$a$, for $a\in D$. Thus we need an

739 introduction rule of the form

740 \[ \infer{a\in\acc(\prec)}{\forall y.y\prec a\imp y\in\acc(\prec)} \]

741 Paulin-Mohring treats this example in Coq~\cite{paulin-tlca}, but it causes

742 difficulties for other systems. Its premise is not acceptable to the

743 inductive definition package of the Cambridge \textsc{hol}

744 system~\cite{camilleri92}. It is also unacceptable to the Isabelle package

745 (recall \S\ref{intro-sec}), but fortunately can be transformed into the

746 acceptable form $t\in M(R)$.

748 The powerset operator is monotonic, and $t\in\pow(R)$ is equivalent to

749 $t\sbs R$. This in turn is equivalent to $\forall y\in t. y\in R$. To

750 express $\forall y.y\prec a\imp y\in\acc(\prec)$ we need only find a

751 term~$t$ such that $y\in t$ if and only if $y\prec a$. A suitable $t$ is

752 the inverse image of~$\{a\}$ under~$\prec$.

754 The definition below follows this approach. Here $r$ is~$\prec$ and

755 $\field(r)$ refers to~$D$, the domain of $\acc(r)$. (The field of a

756 relation is the union of its domain and range.) Finally $r^{-}``\{a\}$

757 denotes the inverse image of~$\{a\}$ under~$r$. We supply the theorem {\tt

758 Pow\_mono}, which asserts that $\pow$ is monotonic.

759 \begin{ttbox}

760 consts acc :: i=>i

761 inductive

762 domains "acc(r)" <= "field(r)"

763 intrs

764 vimage "[| r-``\{a\}: Pow(acc(r)); a: field(r) |] ==> a: acc(r)"

765 monos Pow_mono

766 \end{ttbox}

767 The Isabelle theory proceeds to prove facts about $\acc(\prec)$. For

768 instance, $\prec$ is well-founded if and only if its field is contained in

769 $\acc(\prec)$.

771 As mentioned in~\S\ref{basic-ind-sec}, a premise of the form $t\in M(R)$

772 gives rise to an unusual induction hypothesis. Let us examine the

773 induction rule, {\tt acc.induct}:

774 \[ \infer{P(x)}{x\in\acc(r) &

775 \infer*{P(a)}{\left[

776 \begin{array}{r@{}l}

777 r^{-}``\{a\} &\, \in\pow(\{z\in\acc(r).P(z)\}) \\

778 a &\, \in\field(r)

779 \end{array}

780 \right]_a}}

781 \]

782 The strange induction hypothesis is equivalent to

783 $\forall y. \pair{y,a}\in r\imp y\in\acc(r)\conj P(y)$.

784 Therefore the rule expresses well-founded induction on the accessible part

785 of~$\prec$.

787 The use of inverse image is not essential. The Isabelle package can accept

788 introduction rules with arbitrary premises of the form $\forall

789 \vec{y}.P(\vec{y})\imp f(\vec{y})\in R$. The premise can be expressed

790 equivalently as

791 \[ \{z\in D. P(\vec{y}) \conj z=f(\vec{y})\} \in \pow(R) \]

792 provided $f(\vec{y})\in D$ for all $\vec{y}$ such that~$P(\vec{y})$. The

793 following section demonstrates another use of the premise $t\in M(R)$,

794 where $M=\lst$.

796 \subsection{The primitive recursive functions}\label{primrec-sec}

797 The primitive recursive functions are traditionally defined inductively, as

798 a subset of the functions over the natural numbers. One difficulty is that

799 functions of all arities are taken together, but this is easily

800 circumvented by regarding them as functions on lists. Another difficulty,

801 the notion of composition, is less easily circumvented.

803 Here is a more precise definition. Letting $\vec{x}$ abbreviate

804 $x_0,\ldots,x_{n-1}$, we can write lists such as $[\vec{x}]$,

805 $[y+1,\vec{x}]$, etc. A function is \defn{primitive recursive} if it

806 belongs to the least set of functions in $\lst(\nat)\to\nat$ containing

807 \begin{itemize}

808 \item The \defn{successor} function $\SC$, such that $\SC[y,\vec{x}]=y+1$.

809 \item All \defn{constant} functions $\CONST(k)$, such that

810 $\CONST(k)[\vec{x}]=k$.

811 \item All \defn{projection} functions $\PROJ(i)$, such that

812 $\PROJ(i)[\vec{x}]=x_i$ if $0\leq i<n$.

813 \item All \defn{compositions} $\COMP(g,[f_0,\ldots,f_{m-1}])$,

814 where $g$ and $f_0$, \ldots, $f_{m-1}$ are primitive recursive,

815 such that

816 \[ \COMP(g,[f_0,\ldots,f_{m-1}])[\vec{x}] =

817 g[f_0[\vec{x}],\ldots,f_{m-1}[\vec{x}]]. \]

819 \item All \defn{recursions} $\PREC(f,g)$, where $f$ and $g$ are primitive

820 recursive, such that

821 \begin{eqnarray*}

822 \PREC(f,g)[0,\vec{x}] & = & f[\vec{x}] \\

823 \PREC(f,g)[y+1,\vec{x}] & = & g[\PREC(f,g)[y,\vec{x}],\, y,\, \vec{x}].

824 \end{eqnarray*}

825 \end{itemize}

826 Composition is awkward because it combines not two functions, as is usual,

827 but $m+1$ functions. In her proof that Ackermann's function is not

828 primitive recursive, Nora Szasz was unable to formalize this definition

829 directly~\cite{szasz93}. So she generalized primitive recursion to

830 tuple-valued functions. This modified the inductive definition such that

831 each operation on primitive recursive functions combined just two functions.

833 \begin{figure}

834 \begin{ttbox}

835 Primrec_defs = Main +

836 consts SC :: i

837 \(\vdots\)

838 defs

839 SC_def "SC == lam l:list(nat).list_case(0, \%x xs.succ(x), l)"

840 \(\vdots\)

841 end

843 Primrec = Primrec_defs +

844 consts prim_rec :: i

845 inductive

846 domains "primrec" <= "list(nat)->nat"

847 intrs

848 SC "SC : primrec"

849 CONST "k: nat ==> CONST(k) : primrec"

850 PROJ "i: nat ==> PROJ(i) : primrec"

851 COMP "[| g: primrec; fs: list(primrec) |] ==> COMP(g,fs): primrec"

852 PREC "[| f: primrec; g: primrec |] ==> PREC(f,g): primrec"

853 monos list_mono

854 con_defs SC_def, CONST_def, PROJ_def, COMP_def, PREC_def

855 type_intrs "nat_typechecks @ list.intrs @

856 [lam_type, list_case_type, drop_type, map_type,

857 apply_type, rec_type]"

858 end

859 \end{ttbox}

860 \hrule

861 \caption{Inductive definition of the primitive recursive functions}

862 \label{primrec-fig}

863 \end{figure}

864 \def\fs{{\it fs}}

866 Szasz was using \textsc{alf}, but Coq and \textsc{hol} would also have

867 problems accepting this definition. Isabelle's package accepts it easily

868 since $[f_0,\ldots,f_{m-1}]$ is a list of primitive recursive functions and

869 $\lst$ is monotonic. There are five introduction rules, one for each of the

870 five forms of primitive recursive function. Let us examine the one for

871 $\COMP$:

872 \[ \infer{\COMP(g,\fs)\in\primrec}{g\in\primrec & \fs\in\lst(\primrec)} \]

873 The induction rule for $\primrec$ has one case for each introduction rule.

874 Due to the use of $\lst$ as a monotone operator, the composition case has

875 an unusual induction hypothesis:

876 \[ \infer*{P(\COMP(g,\fs))}

877 {[g\in\primrec & \fs\in\lst(\{z\in\primrec.P(z)\})]_{\fs,g}}

878 \]

879 The hypothesis states that $\fs$ is a list of primitive recursive functions,

880 each satisfying the induction formula. Proving the $\COMP$ case typically

881 requires structural induction on lists, yielding two subcases: either

882 $\fs=\Nil$ or else $\fs=\Cons(f,\fs')$, where $f\in\primrec$, $P(f)$, and

883 $\fs'$ is another list of primitive recursive functions satisfying~$P$.

885 Figure~\ref{primrec-fig} presents the theory file. Theory {\tt Primrec}

886 defines the constants $\SC$, $\CONST$, etc. These are not constructors of

887 a new datatype, but functions over lists of numbers. Their definitions,

888 most of which are omitted, consist of routine list programming. In

889 Isabelle/\textsc{zf}, the primitive recursive functions are defined as a subset of

890 the function set $\lst(\nat)\to\nat$.

892 The Isabelle theory goes on to formalize Ackermann's function and prove

893 that it is not primitive recursive, using the induction rule {\tt

894 primrec.induct}. The proof follows Szasz's excellent account.

897 \section{Datatypes and codatatypes}\label{data-sec}

898 A (co)datatype definition is a (co)inductive definition with automatically

899 defined constructors and a case analysis operator. The package proves that

900 the case operator inverts the constructors and can prove freeness theorems

901 involving any pair of constructors.

904 \subsection{Constructors and their domain}\label{univ-sec}

905 A (co)inductive definition selects a subset of an existing set; a (co)datatype

906 definition creates a new set. The package reduces the latter to the former.

907 Isabelle/\textsc{zf} supplies sets having strong closure properties to serve

908 as domains for (co)inductive definitions.

910 Isabelle/\textsc{zf} defines the Cartesian product $A\times

911 B$, containing ordered pairs $\pair{a,b}$; it also defines the

912 disjoint sum $A+B$, containing injections $\Inl(a)\equiv\pair{0,a}$ and

913 $\Inr(b)\equiv\pair{1,b}$. For use below, define the $m$-tuple

914 $\pair{x_1,\ldots,x_m}$ to be the empty set~$\emptyset$ if $m=0$, simply $x_1$

915 if $m=1$ and $\pair{x_1,\pair{x_2,\ldots,x_m}}$ if $m\geq2$.

917 A datatype constructor $\Con(x_1,\ldots,x_m)$ is defined to be

918 $h(\pair{x_1,\ldots,x_m})$, where $h$ is composed of $\Inl$ and~$\Inr$.

919 In a mutually recursive definition, all constructors for the set~$R_i$ have

920 the outer form~$h_{in}$, where $h_{in}$ is the injection described

921 in~\S\ref{mutual-sec}. Further nested injections ensure that the

922 constructors for~$R_i$ are pairwise distinct.

924 Isabelle/\textsc{zf} defines the set $\univ(A)$, which contains~$A$ and

925 furthermore contains $\pair{a,b}$, $\Inl(a)$ and $\Inr(b)$ for $a$,

926 $b\in\univ(A)$. In a typical datatype definition with set parameters

927 $A_1$, \ldots, $A_k$, a suitable domain for all the recursive sets is

928 $\univ(A_1\un\cdots\un A_k)$. This solves the problem for

929 datatypes~\cite[\S4.2]{paulson-set-II}.

931 The standard pairs and injections can only yield well-founded

932 constructions. This eases the (manual!) definition of recursive functions

933 over datatypes. But they are unsuitable for codatatypes, which typically

934 contain non-well-founded objects.

936 To support codatatypes, Isabelle/\textsc{zf} defines a variant notion of

937 ordered pair, written~$\pair{a;b}$. It also defines the corresponding variant

938 notion of Cartesian product $A\otimes B$, variant injections $\QInl(a)$

939 and~$\QInr(b)$ and variant disjoint sum $A\oplus B$. Finally it defines the

940 set $\quniv(A)$, which contains~$A$ and furthermore contains $\pair{a;b}$,

941 $\QInl(a)$ and $\QInr(b)$ for $a$, $b\in\quniv(A)$. In a typical codatatype

942 definition with set parameters $A_1$, \ldots, $A_k$, a suitable domain is

943 $\quniv(A_1\un\cdots\un A_k)$. Details are published

944 elsewhere~\cite{paulson-mscs}.

946 \subsection{The case analysis operator}

947 The (co)datatype package automatically defines a case analysis operator,

948 called {\tt$R$\_case}. A mutually recursive definition still has only one

949 operator, whose name combines those of the recursive sets: it is called

950 {\tt$R_1$\_\ldots\_$R_n$\_case}. The case operator is analogous to those

951 for products and sums.

953 Datatype definitions employ standard products and sums, whose operators are

954 $\split$ and $\case$ and satisfy the equations

955 \begin{eqnarray*}

956 \split(f,\pair{x,y}) & = & f(x,y) \\

957 \case(f,g,\Inl(x)) & = & f(x) \\

958 \case(f,g,\Inr(y)) & = & g(y)

959 \end{eqnarray*}

960 Suppose the datatype has $k$ constructors $\Con_1$, \ldots,~$\Con_k$. Then

961 its case operator takes $k+1$ arguments and satisfies an equation for each

962 constructor:

963 \[ R\hbox{\_case}(f_1,\ldots,f_k, {\tt Con}_i(\vec{x})) = f_i(\vec{x}),

964 \qquad i = 1, \ldots, k

965 \]

966 The case operator's definition takes advantage of Isabelle's representation of

967 syntax in the typed $\lambda$-calculus; it could readily be adapted to a

968 theorem prover for higher-order logic. If $f$ and~$g$ have meta-type $i\To i$

969 then so do $\split(f)$ and $\case(f,g)$. This works because $\split$ and

970 $\case$ operate on their last argument. They are easily combined to make

971 complex case analysis operators. For example, $\case(f,\case(g,h))$ performs

972 case analysis for $A+(B+C)$; let us verify one of the three equations:

973 \[ \case(f,\case(g,h), \Inr(\Inl(b))) = \case(g,h,\Inl(b)) = g(b) \]

974 Codatatype definitions are treated in precisely the same way. They express

975 case operators using those for the variant products and sums, namely

976 $\qsplit$ and~$\qcase$.

978 \medskip

980 To see how constructors and the case analysis operator are defined, let us

981 examine some examples. Further details are available

982 elsewhere~\cite{paulson-set-II}.

985 \subsection{Example: lists and lazy lists}\label{lists-sec}

986 Here is a declaration of the datatype of lists, as it might appear in a theory

987 file:

988 \begin{ttbox}

989 consts list :: i=>i

990 datatype "list(A)" = Nil | Cons ("a:A", "l: list(A)")

991 \end{ttbox}

992 And here is a declaration of the codatatype of lazy lists:

993 \begin{ttbox}

994 consts llist :: i=>i

995 codatatype "llist(A)" = LNil | LCons ("a: A", "l: llist(A)")

996 \end{ttbox}

998 Each form of list has two constructors, one for the empty list and one for

999 adding an element to a list. Each takes a parameter, defining the set of

1000 lists over a given set~$A$. Each is automatically given the appropriate

1001 domain: $\univ(A)$ for $\lst(A)$ and $\quniv(A)$ for $\llist(A)$. The default

1002 can be overridden.

1004 \ifshort

1005 Now $\lst(A)$ is a datatype and enjoys the usual induction rule.

1006 \else

1007 Since $\lst(A)$ is a datatype, it has a structural induction rule, {\tt

1008 list.induct}:

1009 \[ \infer{P(x)}{x\in\lst(A) & P(\Nil)

1010 & \infer*{P(\Cons(a,l))}{[a\in A & l\in\lst(A) & P(l)]_{a,l}} }

1011 \]

1012 Induction and freeness yield the law $l\not=\Cons(a,l)$. To strengthen this,

1013 Isabelle/\textsc{zf} defines the rank of a set and proves that the standard

1014 pairs and injections have greater rank than their components. An immediate

1015 consequence, which justifies structural recursion on lists

1016 \cite[\S4.3]{paulson-set-II}, is

1017 \[ \rank(l) < \rank(\Cons(a,l)). \]

1018 \par

1019 \fi

1020 But $\llist(A)$ is a codatatype and has no induction rule. Instead it has

1021 the coinduction rule shown in \S\ref{coind-sec}. Since variant pairs and

1022 injections are monotonic and need not have greater rank than their

1023 components, fixedpoint operators can create cyclic constructions. For

1024 example, the definition

1025 \[ \lconst(a) \equiv \lfp(\univ(a), \lambda l. \LCons(a,l)) \]

1026 yields $\lconst(a) = \LCons(a,\lconst(a))$.

1028 \ifshort

1029 \typeout{****SHORT VERSION}

1030 \typeout{****Omitting discussion of constructors!}

1031 \else

1032 \medskip

1033 It may be instructive to examine the definitions of the constructors and

1034 case operator for $\lst(A)$. The definitions for $\llist(A)$ are similar.

1035 The list constructors are defined as follows:

1036 \begin{eqnarray*}

1037 \Nil & \equiv & \Inl(\emptyset) \\

1038 \Cons(a,l) & \equiv & \Inr(\pair{a,l})

1039 \end{eqnarray*}

1040 The operator $\lstcase$ performs case analysis on these two alternatives:

1041 \[ \lstcase(c,h) \equiv \case(\lambda u.c, \split(h)) \]

1042 Let us verify the two equations:

1043 \begin{eqnarray*}

1044 \lstcase(c, h, \Nil) & = &

1045 \case(\lambda u.c, \split(h), \Inl(\emptyset)) \\

1046 & = & (\lambda u.c)(\emptyset) \\

1047 & = & c\\[1ex]

1048 \lstcase(c, h, \Cons(x,y)) & = &

1049 \case(\lambda u.c, \split(h), \Inr(\pair{x,y})) \\

1050 & = & \split(h, \pair{x,y}) \\

1051 & = & h(x,y)

1052 \end{eqnarray*}

1053 \fi

1056 \ifshort

1057 \typeout{****Omitting mutual recursion example!}

1058 \else

1059 \subsection{Example: mutual recursion}

1060 In mutually recursive trees and forests~\cite[\S4.5]{paulson-set-II}, trees

1061 have the one constructor $\Tcons$, while forests have the two constructors

1062 $\Fnil$ and~$\Fcons$:

1063 \begin{ttbox}

1064 consts tree, forest, tree_forest :: i=>i

1065 datatype "tree(A)" = Tcons ("a: A", "f: forest(A)")

1066 and "forest(A)" = Fnil | Fcons ("t: tree(A)", "f: forest(A)")

1067 \end{ttbox}

1068 The three introduction rules define the mutual recursion. The

1069 distinguishing feature of this example is its two induction rules.

1071 The basic induction rule is called {\tt tree\_forest.induct}:

1072 \[ \infer{P(x)}{x\in\TF(A) &

1073 \infer*{P(\Tcons(a,f))}

1074 {\left[\begin{array}{l} a\in A \\

1075 f\in\forest(A) \\ P(f)

1076 \end{array}

1077 \right]_{a,f}}

1078 & P(\Fnil)

1079 & \infer*{P(\Fcons(t,f))}

1080 {\left[\begin{array}{l} t\in\tree(A) \\ P(t) \\

1081 f\in\forest(A) \\ P(f)

1082 \end{array}

1083 \right]_{t,f}} }

1084 \]

1085 This rule establishes a single predicate for $\TF(A)$, the union of the

1086 recursive sets. Although such reasoning can be useful

1087 \cite[\S4.5]{paulson-set-II}, a proper mutual induction rule should establish

1088 separate predicates for $\tree(A)$ and $\forest(A)$. The package calls this

1089 rule {\tt tree\_forest.mutual\_induct}. Observe the usage of $P$ and $Q$ in

1090 the induction hypotheses:

1091 \[ \infer{(\forall z. z\in\tree(A)\imp P(z)) \conj

1092 (\forall z. z\in\forest(A)\imp Q(z))}

1093 {\infer*{P(\Tcons(a,f))}

1094 {\left[\begin{array}{l} a\in A \\

1095 f\in\forest(A) \\ Q(f)

1096 \end{array}

1097 \right]_{a,f}}

1098 & Q(\Fnil)

1099 & \infer*{Q(\Fcons(t,f))}

1100 {\left[\begin{array}{l} t\in\tree(A) \\ P(t) \\

1101 f\in\forest(A) \\ Q(f)

1102 \end{array}

1103 \right]_{t,f}} }

1104 \]

1105 Elsewhere I describe how to define mutually recursive functions over trees and

1106 forests \cite[\S4.5]{paulson-set-II}.

1108 Both forest constructors have the form $\Inr(\cdots)$,

1109 while the tree constructor has the form $\Inl(\cdots)$. This pattern would

1110 hold regardless of how many tree or forest constructors there were.

1111 \begin{eqnarray*}

1112 \Tcons(a,l) & \equiv & \Inl(\pair{a,l}) \\

1113 \Fnil & \equiv & \Inr(\Inl(\emptyset)) \\

1114 \Fcons(a,l) & \equiv & \Inr(\Inr(\pair{a,l}))

1115 \end{eqnarray*}

1116 There is only one case operator; it works on the union of the trees and

1117 forests:

1118 \[ {\tt tree\_forest\_case}(f,c,g) \equiv

1119 \case(\split(f),\, \case(\lambda u.c, \split(g)))

1120 \]

1121 \fi

1124 \subsection{Example: a four-constructor datatype}

1125 A bigger datatype will illustrate some efficiency

1126 refinements. It has four constructors $\Con_0$, \ldots, $\Con_3$, with the

1127 corresponding arities.

1128 \begin{ttbox}

1129 consts data :: [i,i] => i

1130 datatype "data(A,B)" = Con0

1131 | Con1 ("a: A")

1132 | Con2 ("a: A", "b: B")

1133 | Con3 ("a: A", "b: B", "d: data(A,B)")

1134 \end{ttbox}

1135 Because this datatype has two set parameters, $A$ and~$B$, the package

1136 automatically supplies $\univ(A\un B)$ as its domain. The structural

1137 induction rule has four minor premises, one per constructor, and only the last

1138 has an induction hypothesis. (Details are left to the reader.)

1140 The constructors are defined by the equations

1141 \begin{eqnarray*}

1142 \Con_0 & \equiv & \Inl(\Inl(\emptyset)) \\

1143 \Con_1(a) & \equiv & \Inl(\Inr(a)) \\

1144 \Con_2(a,b) & \equiv & \Inr(\Inl(\pair{a,b})) \\

1145 \Con_3(a,b,c) & \equiv & \Inr(\Inr(\pair{a,b,c})).

1146 \end{eqnarray*}

1147 The case analysis operator is

1148 \[ {\tt data\_case}(f_0,f_1,f_2,f_3) \equiv

1149 \case(\begin{array}[t]{@{}l}

1150 \case(\lambda u.f_0,\; f_1),\, \\

1151 \case(\split(f_2),\; \split(\lambda v.\split(f_3(v)))) )

1152 \end{array}

1153 \]

1154 This may look cryptic, but the case equations are trivial to verify.

1156 In the constructor definitions, the injections are balanced. A more naive

1157 approach is to define $\Con_3(a,b,c)$ as $\Inr(\Inr(\Inr(\pair{a,b,c})))$;

1158 instead, each constructor has two injections. The difference here is small.

1159 But the \textsc{zf} examples include a 60-element enumeration type, where each

1160 constructor has 5 or~6 injections. The naive approach would require 1 to~59

1161 injections; the definitions would be quadratic in size. It is like the

1162 advantage of binary notation over unary.

1164 The result structure contains the case operator and constructor definitions as

1165 the theorem list \verb|con_defs|. It contains the case equations, such as

1166 \[ {\tt data\_case}(f_0,f_1,f_2,f_3,\Con_3(a,b,c)) = f_3(a,b,c), \]

1167 as the theorem list \verb|case_eqns|. There is one equation per constructor.

1169 \subsection{Proving freeness theorems}

1170 There are two kinds of freeness theorems:

1171 \begin{itemize}

1172 \item \defn{injectiveness} theorems, such as

1173 \[ \Con_2(a,b) = \Con_2(a',b') \bimp a=a' \conj b=b' \]

1175 \item \defn{distinctness} theorems, such as

1176 \[ \Con_1(a) \not= \Con_2(a',b') \]

1177 \end{itemize}

1178 Since the number of such theorems is quadratic in the number of constructors,

1179 the package does not attempt to prove them all. Instead it returns tools for

1180 proving desired theorems --- either manually or during

1181 simplification or classical reasoning.

1183 The theorem list \verb|free_iffs| enables the simplifier to perform freeness

1184 reasoning. This works by incremental unfolding of constructors that appear in

1185 equations. The theorem list contains logical equivalences such as

1186 \begin{eqnarray*}

1187 \Con_0=c & \bimp & c=\Inl(\Inl(\emptyset)) \\

1188 \Con_1(a)=c & \bimp & c=\Inl(\Inr(a)) \\

1189 & \vdots & \\

1190 \Inl(a)=\Inl(b) & \bimp & a=b \\

1191 \Inl(a)=\Inr(b) & \bimp & {\tt False} \\

1192 \pair{a,b} = \pair{a',b'} & \bimp & a=a' \conj b=b'

1193 \end{eqnarray*}

1194 For example, these rewrite $\Con_1(a)=\Con_1(b)$ to $a=b$ in four steps.

1196 The theorem list \verb|free_SEs| enables the classical

1197 reasoner to perform similar replacements. It consists of elimination rules

1198 to replace $\Con_0=c$ by $c=\Inl(\Inl(\emptyset))$ and so forth, in the

1199 assumptions.

1201 Such incremental unfolding combines freeness reasoning with other proof

1202 steps. It has the unfortunate side-effect of unfolding definitions of

1203 constructors in contexts such as $\exists x.\Con_1(a)=x$, where they should

1204 be left alone. Calling the Isabelle tactic {\tt fold\_tac con\_defs}

1205 restores the defined constants.

1208 \section{Related work}\label{related}

1209 The use of least fixedpoints to express inductive definitions seems

1210 obvious. Why, then, has this technique so seldom been implemented?

1212 Most automated logics can only express inductive definitions by asserting

1213 axioms. Little would be left of Boyer and Moore's logic~\cite{bm79} if their

1214 shell principle were removed. With \textsc{alf} the situation is more

1215 complex; earlier versions of Martin-L\"of's type theory could (using

1216 wellordering types) express datatype definitions, but the version underlying

1217 \textsc{alf} requires new rules for each definition~\cite{dybjer91}. With Coq

1218 the situation is subtler still; its underlying Calculus of Constructions can

1219 express inductive definitions~\cite{huet88}, but cannot quite handle datatype

1220 definitions~\cite{paulin-tlca}. It seems that researchers tried hard to

1221 circumvent these problems before finally extending the Calculus with rule

1222 schemes for strictly positive operators. Recently Gim{\'e}nez has extended

1223 the Calculus of Constructions with inductive and coinductive

1224 types~\cite{gimenez-codifying}, with mechanized support in Coq.

1226 Higher-order logic can express inductive definitions through quantification

1227 over unary predicates. The following formula expresses that~$i$ belongs to the

1228 least set containing~0 and closed under~$\succ$:

1229 \[ \forall P. P(0)\conj (\forall x.P(x)\imp P(\succ(x))) \imp P(i) \]

1230 This technique can be used to prove the Knaster-Tarski theorem, which (in its

1231 general form) is little used in the Cambridge \textsc{hol} system.

1232 Melham~\cite{melham89} describes the development. The natural numbers are

1233 defined as shown above, but lists are defined as functions over the natural

1234 numbers. Unlabelled trees are defined using G\"odel numbering; a labelled

1235 tree consists of an unlabelled tree paired with a list of labels. Melham's

1236 datatype package expresses the user's datatypes in terms of labelled trees.

1237 It has been highly successful, but a fixedpoint approach might have yielded

1238 greater power with less effort.

1240 Elsa Gunter~\cite{gunter-trees} reports an ongoing project to generalize the

1241 Cambridge \textsc{hol} system with mutual recursion and infinitely-branching

1242 trees. She retains many features of Melham's approach.

1244 Melham's inductive definition package~\cite{camilleri92} also uses

1245 quantification over predicates. But instead of formalizing the notion of

1246 monotone function, it requires definitions to consist of finitary rules, a

1247 syntactic form that excludes many monotone inductive definitions.

1249 \textsc{pvs}~\cite{pvs-language} is another proof assistant based on

1250 higher-order logic. It supports both inductive definitions and datatypes,

1251 apparently by asserting axioms. Datatypes may not be iterated in general, but

1252 may use recursion over the built-in $\lst$ type.

1254 The earliest use of least fixedpoints is probably Robin Milner's. Brian

1255 Monahan extended this package considerably~\cite{monahan84}, as did I in

1256 unpublished work.\footnote{The datatype package described in my \textsc{lcf}

1257 book~\cite{paulson87} does {\it not\/} make definitions, but merely asserts

1258 axioms.} \textsc{lcf} is a first-order logic of domain theory; the relevant

1259 fixedpoint theorem is not Knaster-Tarski but concerns fixedpoints of

1260 continuous functions over domains. \textsc{lcf} is too weak to express

1261 recursive predicates. The Isabelle package might be the first to be based on

1262 the Knaster-Tarski theorem.

1265 \section{Conclusions and future work}

1266 Higher-order logic and set theory are both powerful enough to express

1267 inductive definitions. A growing number of theorem provers implement one

1268 of these~\cite{IMPS,saaltink-fme}. The easiest sort of inductive

1269 definition package to write is one that asserts new axioms, not one that

1270 makes definitions and proves theorems about them. But asserting axioms

1271 could introduce unsoundness.

1273 The fixedpoint approach makes it fairly easy to implement a package for

1274 (co)in\-duc\-tive definitions that does not assert axioms. It is efficient:

1275 it processes most definitions in seconds and even a 60-constructor datatype

1276 requires only a few minutes. It is also simple: The first working version took

1277 under a week to code, consisting of under 1100 lines (35K bytes) of Standard

1278 \textsc{ml}.

1280 In set theory, care is needed to ensure that the inductive definition yields

1281 a set (rather than a proper class). This problem is inherent to set theory,

1282 whether or not the Knaster-Tarski theorem is employed. We must exhibit a

1283 bounding set (called a domain above). For inductive definitions, this is

1284 often trivial. For datatype definitions, I have had to formalize much set

1285 theory. To justify infinitely-branching datatype definitions, I have had to

1286 develop a theory of cardinal arithmetic~\cite{paulson-gr}, such as the theorem

1287 that if $\kappa$ is an infinite cardinal and $|X(\alpha)| \le \kappa$ for all

1288 $\alpha<\kappa$ then $|\union\sb{\alpha<\kappa} X(\alpha)| \le \kappa$.

1289 The need for such efforts is not a drawback of the fixedpoint approach, for

1290 the alternative is to take such definitions on faith.

1292 Care is also needed to ensure that the greatest fixedpoint really yields a

1293 coinductive definition. In set theory, standard pairs admit only well-founded

1294 constructions. Aczel's anti-foundation axiom~\cite{aczel88} could be used to

1295 get non-well-founded objects, but it does not seem easy to mechanize.

1296 Isabelle/\textsc{zf} instead uses a variant notion of ordered pairing, which

1297 can be generalized to a variant notion of function. Elsewhere I have

1298 proved that this simple approach works (yielding final coalgebras) for a broad

1299 class of definitions~\cite{paulson-mscs}.

1301 Several large studies make heavy use of inductive definitions. L\"otzbeyer

1302 and Sandner have formalized two chapters of a semantics book~\cite{winskel93},

1303 proving the equivalence between the operational and denotational semantics of

1304 a simple imperative language. A single theory file contains three datatype

1305 definitions (of arithmetic expressions, boolean expressions and commands) and

1306 three inductive definitions (the corresponding operational rules). Using

1307 different techniques, Nipkow~\cite{nipkow-CR} and Rasmussen~\cite{rasmussen95}

1308 have both proved the Church-Rosser theorem; inductive definitions specify

1309 several reduction relations on $\lambda$-terms. Recently, I have applied

1310 inductive definitions to the analysis of cryptographic

1311 protocols~\cite{paulson-markt}.

1313 To demonstrate coinductive definitions, Frost~\cite{frost95} has proved the

1314 consistency of the dynamic and static semantics for a small functional

1315 language. The example is due to Milner and Tofte~\cite{milner-coind}. It

1316 concerns an extended correspondence relation, which is defined coinductively.

1317 A codatatype definition specifies values and value environments in mutual

1318 recursion. Non-well-founded values represent recursive functions. Value

1319 environments are variant functions from variables into values. This one key

1320 definition uses most of the package's novel features.

1322 The approach is not restricted to set theory. It should be suitable for any

1323 logic that has some notion of set and the Knaster-Tarski theorem. I have

1324 ported the (co)inductive definition package from Isabelle/\textsc{zf} to

1325 Isabelle/\textsc{hol} (higher-order logic).

1328 \begin{footnotesize}

1329 \bibliographystyle{plain}

1330 \bibliography{../manual}

1331 \end{footnotesize}

1332 %%%%%\doendnotes

1334 \end{document}