doc-src/Inductive/ind-defs.tex

Changed function cabs to also allow abstraction over Vars.

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−%% $Id$
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−\documentclass[12pt,a4paper]{article}
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−\usepackage{latexsym,../iman,../extra,../ttbox,../proof,../pdfsetup}
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−
+
−\newif\ifshort%''Short'' means a published version, not the documentation
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−\shortfalse%%%%%\shorttrue
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−
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−\title{A Fixedpoint Approach to\\ 
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−  (Co)Inductive and (Co)Datatype Definitions%
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−  \thanks{J. Grundy and S. Thompson made detailed comments.  Mads Tofte and
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−    the referees were also helpful.  The research was funded by the SERC
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−    grants GR/G53279, GR/H40570 and by the ESPRIT Project 6453 ``Types''.}}
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−
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−\author{Lawrence C. Paulson\\{\tt [e-mail]}\\
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−        Computer Laboratory, University of Cambridge, England}
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−\date{\today} 
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−\setcounter{secnumdepth}{2} \setcounter{tocdepth}{2}
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−
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−\newcommand\sbs{\subseteq}
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−\let\To=\Rightarrow
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−
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−\newcommand\defn[1]{{\bf#1}}
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−
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−\newcommand\pow{{\cal P}}
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−\newcommand\RepFun{\hbox{\tt RepFun}}
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−\newcommand\cons{\hbox{\tt cons}}
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−\def\succ{\hbox{\tt succ}}
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−\newcommand\split{\hbox{\tt split}}
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−\newcommand\fst{\hbox{\tt fst}}
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−\newcommand\snd{\hbox{\tt snd}}
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−\newcommand\converse{\hbox{\tt converse}}
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−\newcommand\domain{\hbox{\tt domain}}
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−\newcommand\range{\hbox{\tt range}}
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−\newcommand\field{\hbox{\tt field}}
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−\newcommand\lfp{\hbox{\tt lfp}}
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−\newcommand\gfp{\hbox{\tt gfp}}
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−\newcommand\id{\hbox{\tt id}}
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−\newcommand\trans{\hbox{\tt trans}}
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−\newcommand\wf{\hbox{\tt wf}}
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−\newcommand\nat{\hbox{\tt nat}}
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−\newcommand\rank{\hbox{\tt rank}}
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−\newcommand\univ{\hbox{\tt univ}}
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−\newcommand\Vrec{\hbox{\tt Vrec}}
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−\newcommand\Inl{\hbox{\tt Inl}}
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−\newcommand\Inr{\hbox{\tt Inr}}
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−\newcommand\case{\hbox{\tt case}}
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−\newcommand\lst{\hbox{\tt list}}
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−\newcommand\Nil{\hbox{\tt Nil}}
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−\newcommand\Cons{\hbox{\tt Cons}}
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−\newcommand\lstcase{\hbox{\tt list\_case}}
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−\newcommand\lstrec{\hbox{\tt list\_rec}}
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−\newcommand\length{\hbox{\tt length}}
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−\newcommand\listn{\hbox{\tt listn}}
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−\newcommand\acc{\hbox{\tt acc}}
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−\newcommand\primrec{\hbox{\tt primrec}}
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−\newcommand\SC{\hbox{\tt SC}}
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−\newcommand\CONST{\hbox{\tt CONST}}
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−\newcommand\PROJ{\hbox{\tt PROJ}}
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−\newcommand\COMP{\hbox{\tt COMP}}
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−\newcommand\PREC{\hbox{\tt PREC}}
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−
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−\newcommand\quniv{\hbox{\tt quniv}}
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−\newcommand\llist{\hbox{\tt llist}}
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−\newcommand\LNil{\hbox{\tt LNil}}
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−\newcommand\LCons{\hbox{\tt LCons}}
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−\newcommand\lconst{\hbox{\tt lconst}}
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−\newcommand\lleq{\hbox{\tt lleq}}
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−\newcommand\map{\hbox{\tt map}}
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−\newcommand\term{\hbox{\tt term}}
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−\newcommand\Apply{\hbox{\tt Apply}}
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−\newcommand\termcase{\hbox{\tt term\_case}}
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−\newcommand\rev{\hbox{\tt rev}}
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−\newcommand\reflect{\hbox{\tt reflect}}
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−\newcommand\tree{\hbox{\tt tree}}
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−\newcommand\forest{\hbox{\tt forest}}
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−\newcommand\Part{\hbox{\tt Part}}
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−\newcommand\TF{\hbox{\tt tree\_forest}}
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−\newcommand\Tcons{\hbox{\tt Tcons}}
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−\newcommand\Fcons{\hbox{\tt Fcons}}
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−\newcommand\Fnil{\hbox{\tt Fnil}}
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−\newcommand\TFcase{\hbox{\tt TF\_case}}
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−\newcommand\Fin{\hbox{\tt Fin}}
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−\newcommand\QInl{\hbox{\tt QInl}}
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−\newcommand\QInr{\hbox{\tt QInr}}
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−\newcommand\qsplit{\hbox{\tt qsplit}}
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−\newcommand\qcase{\hbox{\tt qcase}}
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−\newcommand\Con{\hbox{\tt Con}}
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−\newcommand\data{\hbox{\tt data}}
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−
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−\binperiod     %%%treat . like a binary operator
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−
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−\begin{document}
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−\pagestyle{empty}
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−\begin{titlepage}
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−\maketitle 
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−\begin{abstract}
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−  This paper presents a fixedpoint approach to inductive definitions.
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−  Instead of using a syntactic test such as ``strictly positive,'' the
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−  approach lets definitions involve any operators that have been proved
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−  monotone.  It is conceptually simple, which has allowed the easy
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−  implementation of mutual recursion and iterated definitions.  It also
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−  handles coinductive definitions: simply replace the least fixedpoint by a
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−  greatest fixedpoint.  
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−
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−  The method has been implemented in two of Isabelle's logics, \textsc{zf} set
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−  theory and higher-order logic.  It should be applicable to any logic in
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−  which the Knaster-Tarski theorem can be proved.  Examples include lists of
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−  $n$ elements, the accessible part of a relation and the set of primitive
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−  recursive functions.  One example of a coinductive definition is
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−  bisimulations for lazy lists.  Recursive datatypes are examined in detail,
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−  as well as one example of a \defn{codatatype}: lazy lists.
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−
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−  The Isabelle package has been applied in several large case studies,
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−  including two proofs of the Church-Rosser theorem and a coinductive proof of
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−  semantic consistency.  The package can be trusted because it proves theorems
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−  from definitions, instead of asserting desired properties as axioms.
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−\end{abstract}
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−%
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−\bigskip
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−\centerline{Copyright \copyright{} \number\year{} by Lawrence C. Paulson}
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−\thispagestyle{empty} 
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−\end{titlepage}
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−\tableofcontents\cleardoublepage\pagestyle{plain}
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−
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−\setcounter{page}{1}
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−
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−\section{Introduction}
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−Several theorem provers provide commands for formalizing recursive data
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−structures, like lists and trees.  Robin Milner implemented one of the first
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−of these, for Edinburgh \textsc{lcf}~\cite{milner-ind}.  Given a description
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−of the desired data structure, Milner's package formulated appropriate
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−definitions and proved the characteristic theorems.  Similar is Melham's
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−recursive type package for the Cambridge \textsc{hol} system~\cite{melham89}.
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−Such data structures are called \defn{datatypes}
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−below, by analogy with datatype declarations in Standard~\textsc{ml}\@.
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−Some logics take datatypes as primitive; consider Boyer and Moore's shell
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−principle~\cite{bm79} and the Coq type theory~\cite{paulin-tlca}.
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−
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−A datatype is but one example of an \defn{inductive definition}.  Such a
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−definition~\cite{aczel77} specifies the least set~$R$ \defn{closed under}
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−given rules: applying a rule to elements of~$R$ yields a result within~$R$.
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−Inductive definitions have many applications.  The collection of theorems in a
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−logic is inductively defined.  A structural operational
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−semantics~\cite{hennessy90} is an inductive definition of a reduction or
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−evaluation relation on programs.  A few theorem provers provide commands for
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−formalizing inductive definitions; these include Coq~\cite{paulin-tlca} and
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−again the \textsc{hol} system~\cite{camilleri92}.
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−
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−The dual notion is that of a \defn{coinductive definition}.  Such a definition
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−specifies the greatest set~$R$ \defn{consistent with} given rules: every
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−element of~$R$ can be seen as arising by applying a rule to elements of~$R$.
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−Important examples include using bisimulation relations to formalize
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−equivalence of processes~\cite{milner89} or lazy functional
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−programs~\cite{abramsky90}.  Other examples include lazy lists and other
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−infinite data structures; these are called \defn{codatatypes} below.
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−
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−Not all inductive definitions are meaningful.  \defn{Monotone} inductive
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−definitions are a large, well-behaved class.  Monotonicity can be enforced
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−by syntactic conditions such as ``strictly positive,'' but this could lead to
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−monotone definitions being rejected on the grounds of their syntactic form.
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−More flexible is to formalize monotonicity within the logic and allow users
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−to prove it.
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−
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−This paper describes a package based on a fixedpoint approach.  Least
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−fixedpoints yield inductive definitions; greatest fixedpoints yield
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−coinductive definitions.  Most of the discussion below applies equally to
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−inductive and coinductive definitions, and most of the code is shared.  
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−
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−The package supports mutual recursion and infinitely-branching datatypes and
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−codatatypes.  It allows use of any operators that have been proved monotone,
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−thus accepting all provably monotone inductive definitions, including
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−iterated definitions.
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−
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−The package has been implemented in
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−Isabelle~\cite{paulson-markt,paulson-isa-book} using 
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−\textsc{zf} set theory \cite{paulson-set-I,paulson-set-II}; part of it has
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−since been ported to Isabelle/\textsc{hol} (higher-order logic).  The
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−recursion equations are specified as introduction rules for the mutually
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−recursive sets.  The package transforms these rules into a mapping over sets,
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−and attempts to prove that the mapping is monotonic and well-typed.  If
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−successful, the package makes fixedpoint definitions and proves the
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−introduction, elimination and (co)induction rules.  Users invoke the package
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−by making simple declarations in Isabelle theory files.
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−
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−Most datatype packages equip the new datatype with some means of expressing
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−recursive functions.  This is the main omission from my package.  Its
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−fixedpoint operators define only recursive sets.  The Isabelle/\textsc{zf}
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−theory provides well-founded recursion~\cite{paulson-set-II}, which is harder
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−to use than structural recursion but considerably more general.
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−Slind~\cite{slind-tfl} has written a package to automate the definition of
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−well-founded recursive functions in Isabelle/\textsc{hol}.
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−
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−\paragraph*{Outline.} Section~2 introduces the least and greatest fixedpoint
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−operators.  Section~3 discusses the form of introduction rules, mutual
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−recursion and other points common to inductive and coinductive definitions.
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−Section~4 discusses induction and coinduction rules separately.  Section~5
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−presents several examples, including a coinductive definition.  Section~6
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−describes datatype definitions.  Section~7 presents related work.
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−Section~8 draws brief conclusions.  \ifshort\else The appendices are simple
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−user's manuals for this Isabelle package.\fi
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−
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−Most of the definitions and theorems shown below have been generated by the
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−package.  I have renamed some variables to improve readability.
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− 
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−\section{Fixedpoint operators}
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−In set theory, the least and greatest fixedpoint operators are defined as
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−follows:
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−\begin{eqnarray*}
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−   \lfp(D,h)  & \equiv & \inter\{X\sbs D. h(X)\sbs X\} \\
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−   \gfp(D,h)  & \equiv & \union\{X\sbs D. X\sbs h(X)\}
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−\end{eqnarray*}   
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−Let $D$ be a set.  Say that $h$ is \defn{bounded by}~$D$ if $h(D)\sbs D$, and
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−\defn{monotone below~$D$} if
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−$h(A)\sbs h(B)$ for all $A$ and $B$ such that $A\sbs B\sbs D$.  If $h$ is
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−bounded by~$D$ and monotone then both operators yield fixedpoints:
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−\begin{eqnarray*}
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−   \lfp(D,h)  & = & h(\lfp(D,h)) \\
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−   \gfp(D,h)  & = & h(\gfp(D,h)) 
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−\end{eqnarray*}   
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−These equations are instances of the Knaster-Tarski theorem, which states
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−that every monotonic function over a complete lattice has a
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−fixedpoint~\cite{davey-priestley}.  It is obvious from their definitions
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−that $\lfp$ must be the least fixedpoint, and $\gfp$ the greatest.
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−
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−This fixedpoint theory is simple.  The Knaster-Tarski theorem is easy to
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−prove.  Showing monotonicity of~$h$ is trivial, in typical cases.  We must
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−also exhibit a bounding set~$D$ for~$h$.  Frequently this is trivial, as when
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−a set of theorems is (co)inductively defined over some previously existing set
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−of formul{\ae}.  Isabelle/\textsc{zf} provides suitable bounding sets for
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−infinitely-branching (co)datatype definitions; see~\S\ref{univ-sec}.  Bounding
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−sets are also called \defn{domains}.
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−
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−The powerset operator is monotone, but by Cantor's theorem there is no
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−set~$A$ such that $A=\pow(A)$.  We cannot put $A=\lfp(D,\pow)$ because
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−there is no suitable domain~$D$.  But \S\ref{acc-sec} demonstrates
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−that~$\pow$ is still useful in inductive definitions.
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−
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−\section{Elements of an inductive or coinductive definition}\label{basic-sec}
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−Consider a (co)inductive definition of the sets $R_1$, \ldots,~$R_n$, in
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−mutual recursion.  They will be constructed from domains $D_1$,
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−\ldots,~$D_n$, respectively.  The construction yields not $R_i\sbs D_i$ but
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−$R_i\sbs D_1+\cdots+D_n$, where $R_i$ is contained in the image of~$D_i$
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−under an injection.  Reasons for this are discussed
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−elsewhere~\cite[\S4.5]{paulson-set-II}.
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−
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−The definition may involve arbitrary parameters $\vec{p}=p_1$,
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−\ldots,~$p_k$.  Each recursive set then has the form $R_i(\vec{p})$.  The
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−parameters must be identical every time they occur within a definition.  This
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−would appear to be a serious restriction compared with other systems such as
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−Coq~\cite{paulin-tlca}.  For instance, we cannot define the lists of
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−$n$ elements as the set $\listn(A,n)$ using rules where the parameter~$n$
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−varies.  Section~\ref{listn-sec} describes how to express this set using the
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−inductive definition package.
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−
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−To avoid clutter below, the recursive sets are shown as simply $R_i$
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−instead of~$R_i(\vec{p})$.
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−
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−\subsection{The form of the introduction rules}\label{intro-sec}
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−The body of the definition consists of the desired introduction rules.  The
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−conclusion of each rule must have the form $t\in R_i$, where $t$ is any term.
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−Premises typically have the same form, but they can have the more general form
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−$t\in M(R_i)$ or express arbitrary side-conditions.
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−
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−The premise $t\in M(R_i)$ is permitted if $M$ is a monotonic operator on
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−sets, satisfying the rule 
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−\[ \infer{M(A)\sbs M(B)}{A\sbs B} \]
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−The user must supply the package with monotonicity rules for all such premises.
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−
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−The ability to introduce new monotone operators makes the approach
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−flexible.  A suitable choice of~$M$ and~$t$ can express a lot.  The
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−powerset operator $\pow$ is monotone, and the premise $t\in\pow(R)$
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−expresses $t\sbs R$; see \S\ref{acc-sec} for an example.  The \emph{list of}
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−operator is monotone, as is easily proved by induction.  The premise
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−$t\in\lst(R)$ avoids having to encode the effect of~$\lst(R)$ using mutual
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−recursion; see \S\ref{primrec-sec} and also my earlier
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−paper~\cite[\S4.4]{paulson-set-II}.
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−
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−Introduction rules may also contain \defn{side-conditions}.  These are
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−premises consisting of arbitrary formul{\ae} not mentioning the recursive
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−sets. Side-conditions typically involve type-checking.  One example is the
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−premise $a\in A$ in the following rule from the definition of lists:
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−\[ \infer{\Cons(a,l)\in\lst(A)}{a\in A & l\in\lst(A)} \]
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−
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−\subsection{The fixedpoint definitions}
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−The package translates the list of desired introduction rules into a fixedpoint
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−definition.  Consider, as a running example, the finite powerset operator
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−$\Fin(A)$: the set of all finite subsets of~$A$.  It can be
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−defined as the least set closed under the rules
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−\[  \emptyset\in\Fin(A)  \qquad 
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−    \infer{\{a\}\un b\in\Fin(A)}{a\in A & b\in\Fin(A)} 
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−\]
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−
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−The domain in a (co)inductive definition must be some existing set closed
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−under the rules.  A suitable domain for $\Fin(A)$ is $\pow(A)$, the set of all
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−subsets of~$A$.  The package generates the definition
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−\[  \Fin(A) \equiv \lfp(\pow(A), \,
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−  \begin{array}[t]{r@{\,}l}
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−      \lambda X. \{z\in\pow(A). & z=\emptyset \disj{} \\
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−                  &(\exists a\,b. z=\{a\}\un b\conj a\in A\conj b\in X)\})
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−  \end{array}
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−\]
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−The contribution of each rule to the definition of $\Fin(A)$ should be
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−obvious.  A coinductive definition is similar but uses $\gfp$ instead
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−of~$\lfp$.
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−
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−The package must prove that the fixedpoint operator is applied to a
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−monotonic function.  If the introduction rules have the form described
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−above, and if the package is supplied a monotonicity theorem for every
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−$t\in M(R_i)$ premise, then this proof is trivial.\footnote{Due to the
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−  presence of logical connectives in the fixedpoint's body, the
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−  monotonicity proof requires some unusual rules.  These state that the
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−  connectives $\conj$, $\disj$ and $\exists$ preserve monotonicity with respect
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−  to the partial ordering on unary predicates given by $P\sqsubseteq Q$ if and
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−  only if $\forall x.P(x)\imp Q(x)$.}
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−
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−The package returns its result as an \textsc{ml} structure, which consists of named
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−components; we may regard it as a record.  The result structure contains
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−the definitions of the recursive sets as a theorem list called {\tt defs}.
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−It also contains some theorems; {\tt dom\_subset} is an inclusion such as 
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−$\Fin(A)\sbs\pow(A)$, while {\tt bnd\_mono} asserts that the fixedpoint
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−definition is monotonic.
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−
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−Internally the package uses the theorem {\tt unfold}, a fixedpoint equation
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−such as
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−\[
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−  \begin{array}[t]{r@{\,}l}
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−     \Fin(A) = \{z\in\pow(A). & z=\emptyset \disj{} \\
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−             &(\exists a\,b. z=\{a\}\un b\conj a\in A\conj b\in \Fin(A))\}
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−  \end{array}
+
−\]
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−In order to save space, this theorem is not exported.  
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−
+
−
+
−\subsection{Mutual recursion} \label{mutual-sec}
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−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
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−binary disjoint sum, typically $\Inl$ and~$\Inr$, to express injections
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−$h_{1n}$, \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~$h(z)$:
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−\[ \Part(A,h)  \equiv \{x\in A. \exists z. x=h(z)\}. \]   
+
−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
+
−\[ R_i \equiv \Part(R,h_{in}), \qquad i=1,\ldots, n.  \] 
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−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
+
−\[  \emptyset\in\pow(A)  \qquad 
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−    \infer{\{a\}\un b\in\pow(A)}{a\in A & b\in\pow(A)} 
+
−\]
+
−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}