%% $Id$

 \documentclass[12pt,a4paper]{article}

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

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

 \shortfalse%%%%%\shorttrue

 \title{A Fixedpoint Approach to\\ 

   (Co)Inductive and (Co)Datatype Definitions%

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

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

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

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

         Computer Laboratory, University of Cambridge, England}

 \date{\today} 

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

 \newcommand\sbs{\subseteq}

 \let\To=\Rightarrow

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

 \newcommand\pow{{\cal P}}

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

 \binperiod     %%%treat . like a binary operator

 \begin{document}

 \pagestyle{empty}

 \begin{titlepage}

 \maketitle 

 \begin{abstract}

   This paper presents a fixedpoint approach to inductive definitions.

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

   approach lets definitions involve any operators that have been proved

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

   implementation of mutual recursion and iterated definitions.  It also

   handles coinductive definitions: simply replace the least fixedpoint by a

   greatest fixedpoint.  

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

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

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

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

   recursive functions.  One example of a coinductive definition is

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

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

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

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

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

   from definitions, instead of asserting desired properties as axioms.

 \end{abstract}

 %

 \bigskip

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

 \thispagestyle{empty} 

 \end{titlepage}

 \tableofcontents\cleardoublepage\pagestyle{plain}

 \setcounter{page}{1}

 \section{Introduction}

 Several theorem provers provide commands for formalizing recursive data

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

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

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

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

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

 Such data structures are called \defn{datatypes}

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

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

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

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

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

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

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

 logic is inductively defined.  A structural operational

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

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

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

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

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

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

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

 Important examples include using bisimulation relations to formalize

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

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

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

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

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

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

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

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

 to prove it.

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

 fixedpoints yield inductive definitions; greatest fixedpoints yield

 coinductive definitions.  Most of the discussion below applies equally to

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

 The package supports mutual recursion and infinitely-branching datatypes and

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

 thus accepting all provably monotone inductive definitions, including

 iterated definitions.

 The package has been implemented in

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

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

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

 recursion equations are specified as introduction rules for the mutually

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

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

 successful, the package makes fixedpoint definitions and proves the

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

 by making simple declarations in Isabelle theory files.

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

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

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

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

 to use than structural recursion but considerably more general.

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

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

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

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

 recursion and other points common to inductive and coinductive definitions.

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

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

 describes datatype definitions.  Section~7 presents related work.

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

 user's manuals for this Isabelle package.\fi

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

 package.  I have renamed some variables to improve readability.

 \section{Fixedpoint operators}

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

 follows:

 \begin{eqnarray*}

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

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

 \end{eqnarray*}   

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

 \defn{monotone below~$D$} if

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

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

 \begin{eqnarray*}

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

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

 \end{eqnarray*}   

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

 that every monotonic function over a complete lattice has a

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

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

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

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

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

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

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

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

 sets are also called \defn{domains}.

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

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

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

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

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

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

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

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

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

 under an injection.  Reasons for this are discussed

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

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

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

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

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

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

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

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

 inductive definition package.

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

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

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

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

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

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

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

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

 sets, satisfying the rule 

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

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

 The ability to introduce new monotone operators makes the approach

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

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

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

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

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

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

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

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

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

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

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

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

 \subsection{The fixedpoint definitions}

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

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

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

 defined as the least set closed under the rules

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

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

 \]

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

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

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

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

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

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

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

   \end{array}

 \]

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

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

 of~$\lfp$.

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

 monotonic function.  If the introduction rules have the form described

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

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

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

   monotonicity proof requires some unusual rules.  These state that the

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

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

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

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

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

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

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

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

 definition is monotonic.

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

 such as

 \[

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

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

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

   \end{array}

 \]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

 \subsection{Proving the introduction rules}

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

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

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

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

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

 closed under all the introduction rules.  This essentially involves replacing

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

 attempting to prove the result.

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

 in the rules, the package must prove

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

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

 \]

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

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

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

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

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

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

 definition through!

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

 intrs}.

 \subsection{The case analysis rule}

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

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

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

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

 is written

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

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

 \]

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

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

 \section{Induction and coinduction rules}

 Here we must consider inductive and coinductive definitions separately.  For

 an inductive definition, the package returns an induction rule derived

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

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

 basic coinduction rule.

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

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

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

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

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

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

 product; inductively defined relations are treated slightly differently.

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

 introduction rule:

 \begin{itemize}

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

 is~$P(t)$.

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

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

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

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

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

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

 then the minor premise discharges the single assumption

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

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

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

 exploited by appealing to properties of~$M$.

 \end{itemize}

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

 but includes an induction hypothesis:

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

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

 \] 

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

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

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

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

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

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

 \subsection{Modified induction rules}

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

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

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

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

 inductively defined relations, eliminating the need to express properties of

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

 Occasionally it may require you to split up the induction variable

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

   split} appears in the rule.

 The mutual induction rule is called {\tt

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

 \begin{itemize}

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

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

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

 refers to all the recursive sets:

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

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

 \]

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

 \ldots,~$n$.

 \end{itemize}

 %

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

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

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

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

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

 A coinductive definition yields a primitive coinduction rule, with no

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

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

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

 greatest set consistent with the rules

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

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

 \]

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

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

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

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

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

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

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

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

     \infer*{

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

        z=\LNil\disj 

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

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

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

 \]

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

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

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

 is the set of natural numbers.)

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

 represents a slight strengthening of the greatest fixedpoint property.  I

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

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

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

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

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

 This section presents several examples from the literature: the finite

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

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

 \subsection{The finite powerset operator}

 This operator has been discussed extensively above.  Here is the

 corresponding invocation in an Isabelle theory file.  Note that

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

 \begin{ttbox}

 Finite = Arith + 

 consts      Fin :: i=>i

 inductive

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

   intrs

     emptyI  "0 : Fin(A)"

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

   type_intrs  empty_subsetI, cons_subsetI, PowI

   type_elims "[make_elim PowD]"

 end

 \end{ttbox}

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

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

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

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

 rules:

 \[ \emptyset\sbs A              \qquad

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

 \]

 A further introduction rule and an elimination rule express both

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

 involves mostly introduction rules.  

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

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

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

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

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

 rule is {\tt Fin.induct}.

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

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

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

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

 But her introduction rules

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

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

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

 \]

 are not acceptable to the inductive definition package:

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

 The Isabelle version of this example suggests a general treatment of

 varying parameters.  It uses the existing datatype definition of

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

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

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

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

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

 rules are

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

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

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

 \]

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

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

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

 \begin{ttbox}

 ListN = List +

 consts  listn :: i=>i

 inductive

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

   intrs

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

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

   type_intrs "nat_typechecks @ list.intrs"

 end

 \end{ttbox}

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

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

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

 package always includes the rules for ordered pairs.

 The package returns introduction, elimination and induction rules for

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

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

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

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

 \]

 This rule lets the induction formula to be a 

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

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

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

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

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

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

 $n$-element list.  

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

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

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

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

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

 \]

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

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

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

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

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

           \infer*{Q}

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

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

                a\in A \\

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

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

 \]

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

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

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

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

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

 Here is a sample session: 

 \begin{ttbox}

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

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

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

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

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

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

 \end{ttbox}

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

 second rule is

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

           \infer*{Q}

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

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

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

 \]

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

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

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

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

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

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

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

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

 \]

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

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

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

 diamond property for parallel contraction on combinators.  Ole Rasmussen used

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

 residuals~\cite{rasmussen95}.

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

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

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

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

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

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

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

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

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

 relation on lazy lists is the greatest fixedpoint of some monotonic

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

 bisimulation for lazy lists, define equivalence to be the greatest

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

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

 coinduction principle for equalities.

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

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

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

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

 \]

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

 bisimulation.  Equivalence can be coinductively defined as the greatest

 fixedpoint for the introduction rules

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

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

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

 \]

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

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

 \bgroup\leftmargini=\parindent

 \begin{ttbox}

 consts    lleq :: i=>i

 coinductive

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

   intrs

     LNil  "<LNil,LNil> : lleq(A)"

     LCons "[| a:A; <l,l'>:lleq(A) |] ==> <LCons(a,l),LCons(a,l')>: lleq(A)"

   type_intrs  "llist.intrs"

 \end{ttbox}

 \egroup

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

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

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

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

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

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

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

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

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

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

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

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

      \end{array}  

     }{[z\in X]_z}

 \]

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

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

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

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

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

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

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

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

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

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

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

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

 introduction rule of the form 

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

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

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

 inductive definition package of the Cambridge \textsc{hol}

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

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

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

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

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

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

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

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

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

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

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

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

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

 \begin{ttbox}

 consts    acc :: i=>i

 inductive

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

   intrs

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

   monos      Pow_mono

 \end{ttbox}

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

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

 $\acc(\prec)$.  

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

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

 induction rule, {\tt acc.induct}:

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

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

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

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

                                 a &\, \in\field(r)

                    \end{array}

                    \right]_a}}

 \]

 The strange induction hypothesis is equivalent to

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

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

 of~$\prec$.

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

 introduction rules with arbitrary premises of the form $\forall

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

 equivalently as 

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

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

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

 where $M=\lst$. 

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

 The primitive recursive functions are traditionally defined inductively, as

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

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

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

 the notion of composition, is less easily circumvented.

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

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

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

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

 \begin{itemize}

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

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

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

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

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

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

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

 such that

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

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

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

   recursive, such that

 \begin{eqnarray*}

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

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

 \end{eqnarray*} 

 \end{itemize}

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

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

 primitive recursive, Nora Szasz was unable to formalize this definition

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

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

 each operation on primitive recursive functions combined just two functions.

 \begin{figure}

 \begin{ttbox}

 Primrec_defs = Main +

 consts SC :: i

  \(\vdots\)

 defs

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

  \(\vdots\)

 end

 Primrec = Primrec_defs +

 consts prim_rec :: i

 inductive

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

  intrs

    SC     "SC : primrec"

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

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

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

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

  monos       list_mono

  con_defs    SC_def, CONST_def, PROJ_def, COMP_def, PREC_def

  type_intrs "nat_typechecks @ list.intrs @                      

              [lam_type, list_case_type, drop_type, map_type,    

               apply_type, rec_type]"

 end

 \end{ttbox}

 \hrule

 \caption{Inductive definition of the primitive recursive functions} 

 \label{primrec-fig}

 \end{figure}

 \def\fs{{\it fs}} 

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

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

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

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

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

 $\COMP$:

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

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

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

 an unusual induction hypothesis:

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

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

 \]

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

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

 requires structural induction on lists, yielding two subcases: either

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

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

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

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

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

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

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

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

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

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

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

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

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

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

 the case operator inverts the constructors and can prove freeness theorems

 involving any pair of constructors.

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

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

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

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

 as domains for (co)inductive definitions.

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

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

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

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

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

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

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

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

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

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

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

 constructors for~$R_i$ are pairwise distinct.  

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

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

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

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

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

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

 The standard pairs and injections can only yield well-founded

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

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

 contain non-well-founded objects.

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

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

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

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

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

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

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

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

 elsewhere~\cite{paulson-mscs}.

 \subsection{The case analysis operator}

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

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

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

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

 for products and sums.

 Datatype definitions employ standard products and sums, whose operators are

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

 \begin{eqnarray*}

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

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

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

 \end{eqnarray*}

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

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

 constructor:

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

     \qquad i = 1, \ldots, k

 \]

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

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

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

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

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

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

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

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

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

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

 $\qsplit$ and~$\qcase$.

 \medskip

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

 examine some examples.  Further details are available

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

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

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

 file:

 \begin{ttbox} 

 consts  list :: i=>i

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

 \end{ttbox}

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

 \begin{ttbox}

 consts  llist :: i=>i

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

 \end{ttbox}

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

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

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

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

 can be overridden.

 \ifshort

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

 \else

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

   list.induct}:

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

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

 \] 

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

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

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

 consequence, which justifies structural recursion on lists

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

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

 \par

 \fi

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

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

 injections are monotonic and need not have greater rank than their

 components, fixedpoint operators can create cyclic constructions.  For

 example, the definition

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

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

 \ifshort

 \typeout{****SHORT VERSION}

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

 \else

 \medskip

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

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

 The list constructors are defined as follows:

 \begin{eqnarray*}

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

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

 \end{eqnarray*}

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

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

 Let us verify the two equations:

 \begin{eqnarray*}

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

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

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

      & = & c\\[1ex]

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

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

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

      & = & h(x,y)

 \end{eqnarray*} 

 \fi

 \ifshort

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

 \else

 \subsection{Example: mutual recursion}

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

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

 $\Fnil$ and~$\Fcons$:

 \begin{ttbox}

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

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

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

 \end{ttbox}

 The three introduction rules define the mutual recursion.  The

 distinguishing feature of this example is its two induction rules.

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

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

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

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

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

                \end{array}

          \right]_{a,f}}

      & P(\Fnil)

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

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

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

                 \end{array}

          \right]_{t,f}} }

 \] 

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

 recursive sets.  Although such reasoning can be useful

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

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

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

 the induction hypotheses:

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

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

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

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

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

                \end{array}

          \right]_{a,f}}

      & Q(\Fnil)

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

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

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

                 \end{array}

          \right]_{t,f}} }

 \] 

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

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

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

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

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

 \begin{eqnarray*}

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

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

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

 \end{eqnarray*} 

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

 forests:

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

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

 \]

 \fi

 \subsection{Example: a four-constructor datatype}

 A bigger datatype will illustrate some efficiency 

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

 corresponding arities.

 \begin{ttbox}

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

 datatype  "data(A,B)" = Con0

                       | Con1 ("a: A")

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

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

 \end{ttbox}

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

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

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

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

 The constructors are defined by the equations

 \begin{eqnarray*}

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

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

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

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

 \end{eqnarray*} 

 The case analysis operator is

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

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

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

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

    \end{array} 

 \]

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

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

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

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

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

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

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

 advantage of binary notation over unary.

 The result structure contains the case operator and constructor definitions as

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

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

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

 \subsection{Proving freeness theorems}

 There are two kinds of freeness theorems:

 \begin{itemize}

 \item \defn{injectiveness} theorems, such as

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

 \item \defn{distinctness} theorems, such as

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

 \end{itemize}

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

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

 proving desired theorems --- either manually or during

 simplification or classical reasoning.

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

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

 equations.  The theorem list contains logical equivalences such as

 \begin{eqnarray*}

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

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

                 & \vdots &                             \\

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

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

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

 \end{eqnarray*}

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

 The theorem list \verb|free_SEs| enables the classical

 reasoner to perform similar replacements.  It consists of elimination rules

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

 assumptions.

 Such incremental unfolding combines freeness reasoning with other proof

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

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

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

 restores the defined constants.

 \section{Related work}\label{related}

 The use of least fixedpoints to express inductive definitions seems

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

 Most automated logics can only express inductive definitions by asserting

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

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

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

 wellordering types) express datatype definitions, but the version underlying

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

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

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

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

 circumvent these problems before finally extending the Calculus with rule

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

 the Calculus of Constructions with inductive and coinductive

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

 Higher-order logic can express inductive definitions through quantification

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

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

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

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

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

 Melham~\cite{melham89} describes the development.  The natural numbers are

 defined as shown above, but lists are defined as functions over the natural

 numbers.  Unlabelled trees are defined using G\"odel numbering; a labelled

 tree consists of an unlabelled tree paired with a list of labels.  Melham's

 datatype package expresses the user's datatypes in terms of labelled trees.

 It has been highly successful, but a fixedpoint approach might have yielded

 greater power with less effort.

 Elsa Gunter~\cite{gunter-trees} reports an ongoing project to generalize the

 Cambridge \textsc{hol} system with mutual recursion and infinitely-branching

 trees.  She retains many features of Melham's approach.

 Melham's inductive definition package~\cite{camilleri92} also uses

 quantification over predicates.  But instead of formalizing the notion of

 monotone function, it requires definitions to consist of finitary rules, a

 syntactic form that excludes many monotone inductive definitions.

 \textsc{pvs}~\cite{pvs-language} is another proof assistant based on

 higher-order logic.  It supports both inductive definitions and datatypes,

 apparently by asserting axioms.  Datatypes may not be iterated in general, but

 may use recursion over the built-in $\lst$ type.

 The earliest use of least fixedpoints is probably Robin Milner's.  Brian

 Monahan extended this package considerably~\cite{monahan84}, as did I in

 unpublished work.\footnote{The datatype package described in my \textsc{lcf}

   book~\cite{paulson87} does {\it not\/} make definitions, but merely asserts

   axioms.} \textsc{lcf} is a first-order logic of domain theory; the relevant

 fixedpoint theorem is not Knaster-Tarski but concerns fixedpoints of

 continuous functions over domains.  \textsc{lcf} is too weak to express

 recursive predicates.  The Isabelle package might be the first to be based on

 the Knaster-Tarski theorem.

 \section{Conclusions and future work}

 Higher-order logic and set theory are both powerful enough to express

 inductive definitions.  A growing number of theorem provers implement one

 of these~\cite{IMPS,saaltink-fme}.  The easiest sort of inductive

 definition package to write is one that asserts new axioms, not one that

 makes definitions and proves theorems about them.  But asserting axioms

 could introduce unsoundness.

 The fixedpoint approach makes it fairly easy to implement a package for

 (co)in\-duc\-tive definitions that does not assert axioms.  It is efficient:

 it processes most definitions in seconds and even a 60-constructor datatype

 requires only a few minutes.  It is also simple: The first working version took

 under a week to code, consisting of under 1100 lines (35K bytes) of Standard

 \textsc{ml}.

 In set theory, care is needed to ensure that the inductive definition yields

 a set (rather than a proper class).  This problem is inherent to set theory,

 whether or not the Knaster-Tarski theorem is employed.  We must exhibit a

 bounding set (called a domain above).  For inductive definitions, this is

 often trivial.  For datatype definitions, I have had to formalize much set

 theory.  To justify infinitely-branching datatype definitions, I have had to

 develop a theory of cardinal arithmetic~\cite{paulson-gr}, such as the theorem

 that if $\kappa$ is an infinite cardinal and $|X(\alpha)| \le \kappa$ for all

 $\alpha<\kappa$ then $|\union\sb{\alpha<\kappa} X(\alpha)| \le \kappa$.  

 The need for such efforts is not a drawback of the fixedpoint approach, for

 the alternative is to take such definitions on faith.

 Care is also needed to ensure that the greatest fixedpoint really yields a

 coinductive definition.  In set theory, standard pairs admit only well-founded

 constructions.  Aczel's anti-foundation axiom~\cite{aczel88} could be used to

 get non-well-founded objects, but it does not seem easy to mechanize.

 Isabelle/\textsc{zf} instead uses a variant notion of ordered pairing, which

 can be generalized to a variant notion of function.  Elsewhere I have

 proved that this simple approach works (yielding final coalgebras) for a broad

 class of definitions~\cite{paulson-mscs}.

 Several large studies make heavy use of inductive definitions.  L\"otzbeyer

 and Sandner have formalized two chapters of a semantics book~\cite{winskel93},

 proving the equivalence between the operational and denotational semantics of

 a simple imperative language.  A single theory file contains three datatype

 definitions (of arithmetic expressions, boolean expressions and commands) and

 three inductive definitions (the corresponding operational rules).  Using

 different techniques, Nipkow~\cite{nipkow-CR} and Rasmussen~\cite{rasmussen95}

 have both proved the Church-Rosser theorem; inductive definitions specify

 several reduction relations on $\lambda$-terms.  Recently, I have applied

 inductive definitions to the analysis of cryptographic

 protocols~\cite{paulson-markt}. 

 To demonstrate coinductive definitions, Frost~\cite{frost95} has proved the

 consistency of the dynamic and static semantics for a small functional

 language.  The example is due to Milner and Tofte~\cite{milner-coind}.  It

 concerns an extended correspondence relation, which is defined coinductively.

 A codatatype definition specifies values and value environments in mutual

 recursion.  Non-well-founded values represent recursive functions.  Value

 environments are variant functions from variables into values.  This one key

 definition uses most of the package's novel features.

 The approach is not restricted to set theory.  It should be suitable for any

 logic that has some notion of set and the Knaster-Tarski theorem.  I have

 ported the (co)inductive definition package from Isabelle/\textsc{zf} to

 Isabelle/\textsc{hol} (higher-order logic).  

 \begin{footnotesize}

 \bibliographystyle{plain}

 \bibliography{../manual}

 \end{footnotesize}

 %%%%%\doendnotes

 \end{document}