\documentstyle[a4,proof,iman,extra,times]{llncs}

\newif\ifCADE

\CADEtrue

\title{A Fixedpoint Approach to Implementing\\ 

  (Co)Inductive Definitions\thanks{J. Grundy and S. Thompson made detailed

    comments; the referees were also helpful.  Research funded by

    SERC grants GR/G53279, GR/H40570 and by the ESPRIT Project 6453

    `Types'.}}

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

\institute{Computer Laboratory, University of Cambridge, England}

\date{\today} 

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

\newcommand\sbs{\subseteq}

\let\To=\Rightarrow

\newcommand\pow{{\cal P}}

%%%\let\pow=\wp

\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}

%CADE%\pagestyle{empty}

%CADE%\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 other conveniences.  It also

  handles coinductive definitions: simply replace the least fixedpoint by a

  greatest fixedpoint.  This represents the first automated support for

  coinductive definitions.

  The method has been implemented in Isabelle's formalization of ZF set

  theory.  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.  \ifCADE\else Recursive datatypes are examined in detail, as

  well as one example of a {\bf codatatype}: lazy lists.  The appendices

  are simple user's manuals for this Isabelle/ZF package.\fi

\end{abstract}

%

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

%CADE%\thispagestyle{empty} 

%CADE%\end{titlepage}

%CADE%\tableofcontents\cleardoublepage\pagestyle{headings}

\section{Introduction}

Several theorem provers provide commands for formalizing recursive data

structures, like lists and trees.  Examples include Boyer and Moore's shell

principle~\cite{bm79} and Melham's recursive type package for the HOL

system~\cite{melham89}.  Such data structures are called {\bf datatypes}

below, by analogy with {\tt datatype} definitions in Standard~ML\@.

A datatype is but one example of an {\bf inductive definition}.  This

specifies the least set closed under given rules~\cite{aczel77}.  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{paulin92} and again the HOL system~\cite{camilleri92}.

The dual notion is that of a {\bf coinductive definition}.  This specifies

the greatest set closed under given rules.  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 {\bf codatatypes} below.

Not all inductive definitions are meaningful.  {\bf 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.  The package has several advantages:

\begin{itemize}

\item It allows reference to any operators that have been proved monotone.

  Thus it accepts all provably monotone inductive definitions, including

  iterated definitions.

\item It accepts a wide class of datatype definitions, though at present

  restricted to finite branching.

\item It handles coinductive and codatatype definitions.  Most of

  the discussion below applies equally to inductive and coinductive

  definitions, and most of the code is shared.  To my knowledge, this is

  the only package supporting coinductive definitions.

\item Definitions may be mutually recursive.

\end{itemize}

The package is implemented in Isabelle~\cite{isabelle-intro}, using ZF set

theory \cite{paulson-set-I,paulson-set-II}.  However, the fixedpoint

approach is independent of Isabelle.  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.  The package consists of several Standard ML

functors~\cite{paulson91}; it accepts its argument and returns its result

as ML structures.\footnote{This use of ML modules is not essential; the

  package could also be implemented as a function on records.}

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.  To define recursive

functions, the Isabelle/ZF theory provides well-founded recursion and other

logical tools~\cite{paulson-set-II}.

{\bf 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.  \ifCADE\else The appendices are simple

user's manuals for this Isabelle/ZF 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 {\bf bounded by}~$D$ if $h(D)\sbs D$, and

{\bf 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 `formulae.'  Isabelle/ZF provides a suitable bounding set

for finitely branching (co)datatype definitions; see~\S\ref{univ-sec}

below.  Bounding sets are also called {\bf 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{paulin92}.  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,

specified as strings.  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 `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 {\bf side-conditions}.  These are

premises consisting of arbitrary formulae 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

\begin{eqnarray*}

  \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}

\end{eqnarray*} 

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 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, as the theorem {\tt unfold}, a fixedpoint equation such

as

\begin{eqnarray*}

  \Fin(A) & = &

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

     \{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}

\end{eqnarray*}

It also contains, as the theorem {\tt dom\_subset}, an inclusion such as 

$\Fin(A)\sbs\pow(A)$.

\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/ZF defines the

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

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

\begin{eqnarray*}

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

\end{eqnarray*}   

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

\begin{eqnarray*}

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

\end{eqnarray*} 

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.  The user

supplies the package with type-checking rules to apply.  Usually these are

general purpose rules from the 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.  There is one

case for each introduction rule.  The elimination rule

for $\Fin(A)$ is

\[ \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.

The rule states that if $x\in\Fin(A)$ then either $x=\emptyset$ or else

$x=\{a\}\un b$ for some $a\in A$ and $b\in\Fin(A)$; it is a simple consequence

of {\tt unfold}.

The package also returns a function for generating simplified instances of

the case analysis rule.  It works for datatypes and for inductive

definitions involving datatypes, such as an inductively defined relation

between lists.  It instantiates {\tt elim} with a user-supplied term then

simplifies the cases using freeness of the underlying datatype.  The

simplified rules perform `rule inversion' on the inductive definition.

Section~\S\ref{mkcases} presents an example.

\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 and inductively defined relations.  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 following form.

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 Isabelle/ZF theory defines the {\bf rank} of a

set~\cite[\S3.4]{paulson-set-II}, which supports well-founded induction and

recursion over datatypes.  The package proves a rule for mutual induction

and inductive relations.

\subsection{Mutual induction}

The mutual induction rule is called {\tt

mutual\_induct}.  It differs from the basic rule in several 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,~$n$.

\item If the domain of some $R_i$ is the Cartesian product

$A_1\times\cdots\times A_m$, then the corresponding predicate $P_i$ takes $m$

arguments and the corresponding conjunct of the conclusion is

\[ (\forall z_1\ldots z_m.\pair{z_1,\ldots,z_m}\in R_i\imp P_i(z_1,\ldots,z_m))

\]

\end{itemize}

The last point above simplifies reasoning about inductively defined

relations.  It eliminates the need to express properties of $z_1$,

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

\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 fixedpoint satisfying 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 $\quniv(A)$, a set closed under variant forms of

sum and product for representing infinite data structures

(see~\S\ref{univ-sec}).  Coinductive definitions use these variant sums and

products.

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*{z=\LNil\disj \bigl(\exists a\,l.\,

            z=\LCons(a,l) \conj a\in A \conj l\in X\un\llist(A) \bigr)}

           {[z\in X]_z}}

%     \begin{array}[t]{@{}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}.

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

This section presents several examples: 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 ML invocation (note that $\cons(a,b)$ abbreviates

$\{a\}\un b$ in Isabelle/ZF):

\begin{ttbox}

structure Fin = Inductive_Fun

 (val thy        = Arith.thy addconsts [(["Fin"],"i=>i")]

  val rec_doms   = [("Fin","Pow(A)")]

  val sintrs     = ["0 : Fin(A)",

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

  val monos      = []

  val con_defs   = []

  val type_intrs = [empty_subsetI, cons_subsetI, PowI]

  val type_elims = [make_elim PowD]);

\end{ttbox}

We apply the functor {\tt Inductive\_Fun} to a structure describing the

desired inductive definition.  The parent theory~{\tt thy} is obtained from

{\tt Arith.thy} by adding the unary function symbol~$\Fin$.  Its domain is

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

introduction rules.  For type-checking, the structure supplies 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 the two

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

involves mostly introduction rules.  

ML is Isabelle's top level, so such functor invocations can take place at

any time.  The result structure is declared with the name~{\tt Fin}; we can

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

rule as {\tt Fin.induct} and so forth.  There are plans to integrate the

package better into Isabelle so that users can place inductive definitions

in Isabelle theory files instead of applying functors.

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

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

Paulin-Mohring~\cite{paulin92}, 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.

\begin{figure}

\begin{ttbox}

structure ListN = Inductive_Fun

 (val thy        = ListFn.thy addconsts [(["listn"],"i=>i")]

  val rec_doms   = [("listn", "nat*list(A)")]

  val sintrs     = 

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

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

  val monos      = []

  val con_defs   = []

  val type_intrs = nat_typechecks @ List.intrs @ [SigmaI]

  val type_elims = [SigmaE2]);

\end{ttbox}

\hrule

\caption{Defining lists of $n$ elements} \label{listn-fig}

\end{figure} 

The Isabelle/ZF version of this example suggests a general treatment of

varying parameters.  Here, we use the existing datatype definition of

$\lst(A)$, with constructors $\Nil$ and~$\Cons$.  Then incorporate the

parameter~$n$ into the inductive set itself, defining $\listn(A)$ as a

relation.  It consists 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/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)}

\]

Figure~\ref{listn-fig} presents the ML invocation.  A theory of lists,

extended with a declaration of $\listn$, is the parent theory.  The domain

is specified as $\nat\times\lst(A)$.  The type-checking rules include those

for 0, $\succ$, $\Nil$ and $\Cons$.  Because $\listn(A)$ is a set of pairs,

type-checking also requires introduction and elimination rules to express

both directions of the equivalence $\pair{a,b}\in A\times B \bimp a\in A

\conj b\in B$. 

The package returns introduction, elimination and induction rules for

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

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

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

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

\]

This rule requires the induction formula to be a 

unary property of pairs,~$P(\pair{n,l})$.  The alternative rule, {\tt

ListN.mutual\_induct}, uses a binary property instead:

\[ \infer{\forall n\,l. \pair{n,l}\in\listn(A) \imp P(n,l)}

         {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}}}

\]

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.  

Unlike in Coq, the definition does not declare a new datatype.  A `list of

$n$ elements' really is a list and 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 $+$ here denotes addition on the natural numbers and @ denotes append.

\subsection{A demonstration of rule inversion}\label{mkcases}

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 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.  For

example, 

\begin{ttbox}

ListN.mk_cases List.con_defs "<i,Cons(a,l)> : listn(A)"

\end{ttbox}

yields a rule with only two premises:

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

          \infer*{Q}

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

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

               \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 similarly 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 the rule

\[ \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, the Isabelle/ZF theory includes a short proof

of the diamond property for parallel contraction on combinators.

\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 {\bf 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 {\bf 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, we invoke \verb|CoInductive_Fun|:

\begin{ttbox}

structure LList_Eq = CoInductive_Fun

 (val thy = LList.thy addconsts [(["lleq"],"i=>i")]

  val rec_doms   = [("lleq", "llist(A) * llist(A)")]

  val sintrs     = 

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

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

  val monos      = []

  val con_defs   = []

  val type_intrs = LList.intrs @ [SigmaI]

  val type_elims = [SigmaE2]);

\end{ttbox}

Again, {\tt addconsts} declares a constant for $\lleq$ in the parent theory. 

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

rules include the introduction rules for lazy lists as well as rules

for both directions of the equivalence

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

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'.\,

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

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

%     \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 {\bf accessible} or {\bf 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{paulin92}, but it causes

difficulties for other systems.  Its premise does not conform to 

the structure of introduction rules for HOL's inductive definition

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

(\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 ML invocation 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~$r$.  The package is

supplied the theorem {\tt Pow\_mono}, which asserts that $\pow$ is monotonic.

\begin{ttbox}

structure Acc = Inductive_Fun

 (val thy        = WF.thy addconsts [(["acc"],"i=>i")]

  val rec_doms   = [("acc", "field(r)")]

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

  val monos      = [Pow_mono]

  val con_defs   = []

  val type_intrs = []

  val type_elims = []);

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

                   a\in\field(r)]_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 {\bf primitive recursive} if it

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

\begin{itemize}

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

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

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

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

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

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

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

such that

\begin{eqnarray*}

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

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

\end{eqnarray*} 

\item All {\bf 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}

structure Primrec = Inductive_Fun

 (val thy        = Primrec0.thy

  val rec_doms   = [("primrec", "list(nat)->nat")]

  val sintrs     = 

        ["SC : primrec",

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

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

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

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

  val monos      = [list_mono]

  val con_defs   = [SC_def,CONST_def,PROJ_def,COMP_def,PREC_def]

  val type_intrs = pr0_typechecks

  val type_elims = []);

\end{ttbox}

\hrule

\caption{Inductive definition of the primitive recursive functions} 

\label{primrec-fig}

\end{figure}

\def\fs{{\it fs}} 

Szasz was using ALF, but Coq and 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

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 ML invocation.  Theory {\tt

  Primrec0.thy} defines the constants $\SC$, $\CONST$, etc.  These are not

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

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

Isabelle/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}

Conceptually, our two forms of definition are distinct.  A (co)inductive

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

creates a new set.  But the package reduces the latter to the former.  A

set having strong closure properties must serve as the domain of the

(co)inductive definition.  Constructing this set requires some theoretical

effort, which must be done anyway to show that (co)datatypes exist.  It is

not obvious that standard set theory is suitable for defining codatatypes.

Isabelle/ZF defines the standard notion of Cartesian product $A\times B$,

containing ordered pairs $\pair{a,b}$.  Now the $m$-tuple

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

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

Isabelle/ZF also defines the disjoint sum $A+B$, containing injections

$\Inl(a)\equiv\pair{0,a}$ and $\Inr(b)\equiv\pair{1,b}$.

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/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/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 $\quniv(A_1\un\cdots\un A_k)$.  This approach using

standard ZF set theory~\cite{paulson-final} is an alternative to adopting

Aczel's Anti-Foundation Axiom~\cite{aczel88}.

\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:

\begin{eqnarray*}

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

    \qquad i = 1, \ldots, k

\end{eqnarray*}

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.  Here are two examples:

\begin{itemize}

\item $\split(\lambda x.\split(f(x)))$ performs case analysis for

$A\times (B\times C)$, as is easily verified:

\begin{eqnarray*}

  \split(\lambda x.\split(f(x)), \pair{a,b,c}) 

    & = & (\lambda x.\split(f(x))(a,\pair{b,c}) \\

    & = & \split(f(a), \pair{b,c}) \\

    & = & f(a,b,c)

\end{eqnarray*}

\item $\case(f,\case(g,h))$ performs case analysis for $A+(B+C)$; let us

verify one of the three equations:

\begin{eqnarray*}

  \case(f,\case(g,h), \Inr(\Inl(b))) 

    & = & \case(g,h,\Inl(b)) \\

    & = & g(b)

\end{eqnarray*}

\end{itemize}

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

\ifCADE The package has processed all the datatypes discussed in

my earlier paper~\cite{paulson-set-II} and the codatatype of lazy lists.

Space limitations preclude discussing these examples here, but they are

distributed with Isabelle.  \typeout{****Omitting datatype examples from

  CADE version!} \else

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

examine some examples.  These include lists and trees/forests, which I have

discussed extensively in another paper~\cite{paulson-set-II}.